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\theoremstyle{plain}
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\begin{center}
\vskip 1cm{\LARGE\bf Explicit Congruences for Modular Forms\\
\vskip .1in
of Higher Weights
}
\vskip 1cm
\large
Qing Lu\\
School of Mathematical Sciences\\
Beijing Normal University\\
Beijing 100875\\
China\\
\href{mailto:qlu@bnu.edu.cn}{\texttt{qlu@bnu.edu.cn}}
\end{center}

\vskip .2 in
\begin{abstract}
We give explicit formulas for congruences between Eisenstein series and certain cusp forms of even weight at least twelve. These formulas generalize several classical congruences of Ramanujan type and express the relevant Fourier coefficients through additive convolutions involving the Ramanujan tau function and divisor functions. As a consequence, we prove that every irregular prime occurs as the modulus of at least one Ramanujan-type congruence.
\end{abstract}

\section{Introduction}\label{secIntroduction}

Ramanujan \cite{Ramanujan,RamanujanCollected} observed the following famous congruence:
\begin{equation}\label{eq691}
\sigma_{11}(n) \equiv \tau(n) \pmod{691},
\end{equation}
for all $n\ge 1$, where $\sigma_k$ is the divisor function
\[
\sigma_k(n)=\sum_{d\mid n, d>0}d^k,
\]
for integers $k\ge 0$ and $n\ge 1$. Throughout, let $z$ lie in the complex upper half-plane and put $q=e^{2\pi iz}$, so that $|q|<1$. The Ramanujan tau function $\tau(n)$ is defined by the expansion
\[
\sum_{n=1}^\infty \tau(n)q^n
=
q\prod_{n=1}^\infty (1-q^n)^{24}
=q-24q^2+252q^3-1472q^4+4830q^5+\cdots.
\]
 From a modern perspective, this congruence is the manifestation of a congruence between the Fourier coefficients of two modular forms of weight $12$: the unique normalized cusp form
\[
\Delta(z)=q\prod_{n=1}^\infty (1-q^n)^{24}, \quad q=e^{2\pi iz},
\]
and the Eisenstein series
\[
G_{12}(z)=\frac{691}{65520}+\sum_{n=1}^\infty \sigma_{11}(n)q^n,
\quad q=e^{2\pi iz}.
\]
This congruence has fueled numerous profound investigations. Important contributions include those of Serre \cite{Serre317,Serre1974}, Deligne \cite{DS}, Swinnerton-Dyer \cite{SD,SD2}, Katz \cite{Katz}, Ribet \cite{Ribet}, Manin \cite{Manin}, and many others. Diamond \cite{DiamondCongruence} and Ciolan, Languasco, and Moree \cite{CLM} give further results.

In this paper, we prove explicit congruences between Eisenstein series $G_k$ and certain cusp forms for all even weights $k\ge 12$. Datskovsky and Guerzhoy \cite{DG} noticed the existence of congruences of this kind, but to the best of our knowledge, the explicit formulas we obtain for all $k\ge 28$ are new.
We also show that for every irregular prime $p$, there is a Ramanujan-type congruence modulo $p$ in at least one weight.

\section{Main theorem}\label{secMain}

To state our theorem, we first introduce the following notation.

For a nonzero integer $n$, let
\[
\mathbb{Z}_{(n)}=\bigl\{a/b\in\mathbb{Q} : a,b\in\mathbb{Z},\ b\ne0,\ \gcd(b,n)=1\bigr\}.
\]
We extend congruence modulo $n$ to rational numbers by writing
\[
a\equiv b\pmod n\quad \text{ if } a-b\in n\mathbb{Z}_{(n)}.
\]
For two modular forms $f,g$ with rational Fourier coefficients, we write $f\equiv g$ (mod $n$) if their corresponding Fourier coefficients are congruent modulo $n$.

Recall that the Bernoulli numbers $B_k$ are defined by the standard generating function
\[
\frac{x}{e^x-1}=\sum_{k=0}^{\infty}B_k\frac{x^k}{k!}.
\]
Thus $B_0=1$, $B_1=-\frac12$, and $B_k=0$ for every odd $k>1$. For each even integer $k\ge4$, let
\[
G_k(z)=-\frac{B_k}{2k}
+\sum_{n=1}^{\infty}\sigma_{k-1}(n)q^n,
\qquad q=e^{2\pi iz}.
\]
Then $G_k(z)$ is the Eisenstein series of weight $k$. We define $G_0(z)=1$. We extend the definition of $\sigma_{k-1}(n)$ by setting
\[
\sigma_{k-1}(0)=-\frac{B_k}{2k}
\qquad (\text{$k\ge4$ even}),
\]
and
\[
\sigma_{-1}(n)=
\begin{cases}
1, & \text{if $n=0$;}\\
0, & \text{if $n\ge 1$.}
\end{cases}
\]
Thus, for every even $k\ge4$,
\[
G_k(z)=\sum_{n=0}^{\infty}\sigma_{k-1}(n)q^n,
\]
with $G_0=1$. For even $k\ge4$, let
\[
E_k(z)=1+\sum_{n=1}^{\infty}
\frac{\sigma_{k-1}(n)}{\sigma_{k-1}(0)}q^n,
\]
and set $E_0=1$. Then $E_k$ is the normalized Eisenstein series of weight $k$.


For even $k\ge4$, let $\sigma_{k-1}(0)=N_k/D_k$ where $N_k, D_k\in\mathbb{Z}$, $N_k>0$, and $\gcd(N_k, D_k)=1$. We also set $N_0=N_2=1$.
For $j\ge 1$, the value $N_{2j}$ is the absolute value of the $j$-th term of \seqnum{A001067}, which lists the numerators of $B_{2j}/(2j)$.

For an even integer $k\ge 4$, let $\mathcal{S}_k$ denote the complex vector space of cusp forms of $\SL_2(\mathbb{Z})$ of weight $k$, and let $m$ be its 
dimension.
By the dimension formula of Diamond and Shurman \cite[Sec.\ 3.5]{GTM228},
\begin{equation}\label{eqDimFormula}
m=
\begin{cases}
\lfloor k/12 \rfloor -1,
 & \text{if $k\equiv 2$ (mod $12$);} \\
\lfloor k/12 \rfloor, & \text{otherwise.}
\end{cases}
\end{equation}

For two functions $f,g:\mathbb{Z}_{\ge 0}\to\mathbb{Q}$, define their \emph{additive convolution} by
\[
\bigl(f\ast g\bigr)(n)=\sum_{i=0}^n f(i)g(n-i).
\]
Here $n\ge 0$, and $\ast$ is not to be confused with Dirichlet convolution. 
For formal power series, the coefficients of a product are exactly the additive convolution of the coefficient sequences, i.e.,
\begin{equation}\label{eqConvolution}
\Biggl(\sum_{n=0}^\infty f(n)q^n\Biggr) \Biggl(\sum_{n=0}^\infty g(n)q^n\Biggr)
= \sum_{n=0}^\infty \bigl(f*g\bigr)(n)q^n.
\end{equation}

We extend the definition of Ramanujan's tau function by $\tau(0)=0$, and write $\tau^{\ast 1}=\tau$ and $\tau^{\ast(r+1)}=\tau\ast(\tau^{\ast r})$ for integers $r\ge 1$.
The sequence $(\tau^{\ast 2}(n))_{n\ge 2}$ is listed in \seqnum{A010839}, and $(\tau^{\ast 3}(n))_{n\ge 3}$ is listed in \seqnum{A035118}.
\begin{theorem}\label{thmMain}
Let $k$ be an even integer, $k\ge 12$, $k\ne 14$, and let $m$ be defined as in \eqref{eqDimFormula}. Let $C$ be the $m\times m$ matrix $(c_{ij})$, where
\[
c_{ij}=\frac{1}{\sigma_{k-12i-1}(0)}\bigl(\tau^{\ast i}\ast\sigma_{k-12i-1}\bigr)(j),
\qquad 1\le i,j\le m.
\]
Let
\[
\begin{pmatrix}\gamma_1 & \gamma_2 & \cdots & \gamma_m\end{pmatrix} = \begin{pmatrix}\sigma_{k-1}(1) &  \sigma_{k-1}(2) & \cdots   & \sigma_{k-1}(m)\end{pmatrix}C^{-1}.
\]
Then
\begin{equation}\label{eqMain}
G_k\equiv \sum_{i=1}^m \gamma_i\Delta^i  E_{k-12i}\pmod{N_k'},
\end{equation}
where
\[
N_k'=\max \Biggl(
d : d\mid N_k,\; \gcd(d, N_{k-12i})=1
\textnormal{ for every integer }i\textnormal{ with }
1\le i\le \lfloor k/12 \rfloor
\Biggr).
\]

\end{theorem}
\begin{remark}
When $k=14$, the modulus $N_k'$ defined above equals $1$. In this case, the corresponding congruence trivially holds.
\end{remark}

\begin{corollary}\label{corMain} With the notation above, we have
\begin{equation}\label{eqCor}
\sigma_{k-1}(n)\equiv\sum_{i=1}^m  \frac{\gamma_i}{\sigma_{k-12i-1}(0)} \bigl(\tau^{\ast i}\ast\sigma_{k-12i-1}\bigr)(n) \pmod{N_k'},
\end{equation}
for all $n\ge 1$.
\end{corollary}
\begin{proof}[Proof of Theorem \ref{thmMain} and Corollary \ref{corMain}]
Choose $r,s\in\mathbb{Z}_{\ge 0}$ such that
\[
4r+6s=k,
\] and let $h=E_4^r E_6^s$. It is clear that
\[
G_k-\sigma_{k-1}(0)h\in \mathcal{S}_k.
\]
Since the Eisenstein series
\[
E_4(z)=1+240\sum_{n=1}^\infty \sigma_3(n)q^n
\]
and
\[
E_6(z)=1-504\sum_{n=1}^\infty \sigma_5(n)q^n
\quad (q=e^{2\pi iz})
\]
 have integral Fourier coefficients (listed in \seqnum{A004009} and \seqnum{A013973}, respectively),
we have
\begin{equation}\label{eqGkCong}
G_k\equiv G_k-\sigma_{k-1}(0)h\pmod{N_k}.
\end{equation}
Datskovsky and Guerzhoy \cite{DG} already obtained this congruence.

For $1\le i\le m$, extend the notation $c_{ij}$ to all integers $j\ge 1$ by the same formula. By \eqref{eqConvolution},
\[
c_{ij}=\frac{1}{\sigma_{k-12i-1}(0)}\bigl(\tau^{\ast i}\ast\sigma_{k-12i-1}\bigr)(j)
\]
 is defined so that it is the $j$-th Fourier coefficient of $\Delta^i E_{k-12i}$, i.e.,
\[
\Delta^i E_{k-12i} = \sum_{j=1}^\infty c_{ij} q^j.
\]
Each $\Delta^iE_{k-12i}$ is a cusp form of weight $k$ with leading term $q^i$. Thus $c_{ij}=0$ for $i>j$ and $c_{ii}=1$, so $C$ is upper triangular with every diagonal entry equal to $1$. The distinct leading powers imply linear independence. Since there are $m=\dim\mathcal{S}_k$ such forms, they form a basis of $\mathcal{S}_k$. Hence there exist $\tilde{\gamma}_1,\tilde{\gamma}_2,\ldots,\tilde{\gamma}_m\in\mathbb{C}$ such that
\begin{equation}\label{eqGkLinear}
G_k-\sigma_{k-1}(0) h=\sum_{i=1}^m \tilde{\gamma}_i \Delta^i E_{k-12i}.
\end{equation}
Comparing the Fourier expansion of both sides of \eqref{eqGkLinear} up to the $q^m$-term gives a linear system with rational coefficients. For each $i=1,2,\ldots, m$, the diagonal entry $c_{ii}=1$ as it is the leading Fourier coefficient of $\Delta^i E_{k-12i}$. Therefore, the upper triangular matrix $C$ is invertible over $\mathbb{Q}$. The first $m$ Fourier coefficients of $G_k-\sigma_{k-1}(0)h$ are rational, and therefore
\[
\tilde{\gamma}_1,\tilde{\gamma}_2,\ldots,\tilde{\gamma}_m\in\mathbb{Q}.
\]
Combining \eqref{eqGkLinear} with the congruence
\[
G_k\equiv G_k-\sigma_{k-1}(0)h\pmod{N_k}
\]
now yields
\[
G_k\equiv\sum_{i=1}^m \tilde{\gamma}_i \Delta^i E_{k-12i}\pmod{N_k}.
\]
Comparing the first $m$ Fourier coefficients in this congruence, we obtain
\[
\begin{pmatrix}\tilde{\gamma}_1 & \tilde{\gamma}_2 & \cdots & \tilde{\gamma}_m\end{pmatrix} C
\equiv \begin{pmatrix}\sigma_{k-1}(1) & \sigma_{k-1}(2) &\cdots & \sigma_{k-1}(m) \end{pmatrix} \pmod{N_k}.
\]
We may write $C=I+N$, where $I$ is the identity matrix and $N$ a nilpotent matrix satisfying $N^m=0$.
Hence
\[
C^{-1} = (I+N)^{-1} = I - N + N^2 - N^3 + \cdots + (-1)^{m-1}N^{m-1}.
\]
For each $i$, the Fourier coefficients of $E_{k-12i}$ have denominators whose prime factors divide $N_{k-12i}$. Hence the same is true for the coefficients $c_{ij}$ of $\Delta^iE_{k-12i}$. By the definition of $N_k'$, all these coefficients lie in the localization $\mathbb{Z}_{(N_k')}$. The displayed finite expansion of $C^{-1}$ therefore shows that every entry of $C^{-1}$ also lies in $\mathbb{Z}_{(N_k')}$.

Reducing the preceding vector congruence modulo $N_k'$ and multiplying by $C^{-1}$ inside $\mathbb{Z}_{(N_k')}$ gives
\[
\begin{pmatrix}
\tilde{\gamma}_1 & \tilde{\gamma}_2 & \cdots & \tilde{\gamma}_m
\end{pmatrix} 
\equiv
\begin{pmatrix}
\sigma_{k-1}(1) & \sigma_{k-1}(2) & \dots & \sigma_{k-1}(m)
\end{pmatrix}
C^{-1} \pmod{N_k'}.
\]
Since all Fourier coefficients of the basis elements $\Delta^iE_{k-12i}$ lie in $\mathbb{Z}_{(N_k')}$, substitution back into~\eqref{eqGkCong} yields
\[
G_k\equiv \sum_{i=1}^m \gamma_i\Delta^i E_{k-12i} \pmod{N_k'}.
\]
Corollary~\ref{corMain} follows by taking Fourier coefficients of both sides.

\end{proof}
\begin{remark}
If we use the Miller basis described by Stein \cite[Sec.\ 2.3]{Stein} instead of $\Delta^i E_{k-12i}$, we obtain congruences modulo $N_k$. However, the present basis $\Delta^i E_{k-12i}$ yields the particularly simple formula \eqref{eqCor} in Corollary \ref{corMain}, which involves only the additive convolution of $\tau$ and $\sigma$.

\end{remark}

\section{Explicit formulas in the cases of low dimensions}\label{secExplicit}

In this section, we apply the main theorem to produce congruences of Ramanujan type in the cases $m=\dim\mathcal{S}_k\le 3$. For $k\le 26$, we recover Ramanujan's congruence \eqref{eq691} and other congruences in the literature. For $k\ge 28$, our results are new to the best of our knowledge.

\begin{corollary}
For $k=12, 16, 18, 20, 22, 26$,  we have
\begin{equation}\label{eqDim1Gk}
G_k \equiv \Delta E_{k-12} \pmod{N_k'}
\end{equation}
and
\begin{equation}\label{eqDim1Sigma}
\sigma_{k-1}(n) \equiv \frac{1}{\sigma_{k-13}(0)} (\tau\ast\sigma_{k-13})(n) \pmod{N_k'}.
\end{equation}
\end{corollary}
\begin{proof}
For $k=12, 16, 18, 20, 22, 26$, we have $\dim \mathcal{S}_k=1$.
Therefore the matrix $C=C^{-1}=(1)$ is the $1\times 1$ identity matrix.
Hence $\gamma_1= \sigma_{k-1}(1)=1$ and
Theorem \ref{thmMain} directly implies \eqref{eqDim1Gk} and \eqref{eqDim1Sigma}.
\end{proof}

Substituting in $k=12,16,18,20,22,26$, we obtain congruences of Ramanujan type:
{\allowdisplaybreaks
\begin{align*}
\sigma_{11}(n)
  &\equiv \tau(n)
  \pmod{691},\\
\sigma_{15}(n)
  &\equiv \tau(n)
  +240\sum_{m=1}^{n-1}\sigma_3(m)\tau(n-m)
  \pmod{3617},\\
\sigma_{17}(n)
  &\equiv \tau(n)
  -504\sum_{m=1}^{n-1}\sigma_5(m)\tau(n-m)
  \pmod{43867},\\
\sigma_{19}(n)
  &\equiv \tau(n)
  +480\sum_{m=1}^{n-1}\sigma_7(m)\tau(n-m)
  \pmod{283\cdot617},\\
\sigma_{21}(n)
  &\equiv \tau(n)
  -264\sum_{m=1}^{n-1}\sigma_9(m)\tau(n-m)
  \pmod{131\cdot593},\\
\sigma_{25}(n)
  &\equiv \tau(n)
  -24\sum_{m=1}^{n-1}\sigma_{13}(m)\tau(n-m)
  \pmod{657931}.
\end{align*}
}

Congruence \eqref{eqDim1Gk} can be deduced from formulae given by Ramanujan \cite[Table 1]{Ramanujan}. The congruence modulo $691$ appears in Ramanujan's manuscript \cite{R-unpublished}, in which he also studied the Fourier coefficients of $\Delta E_{k-12}$ for the above $k$. In fact, he used the notation $\tau_t$ for our $\tau\ast\sigma_{2t-1}$. 

Wilton \cite{Wilton} published proofs of the congruences modulo $691$ and $3617$. Swinnerton-Dyer \cite{SD} and Manin \cite{Manin} gave modern proofs of all six congruences in this corollary. Rankin \cite{Rankin} and Berndt and Ono \cite{R-unpublished} discuss the history of these congruences.
\begin{corollary}
For $k=24, 28, 30, 32, 34, 38$,
we have
\begin{equation}\label{eqDim2Gk}
G_k\equiv \Delta E_{k-12}+\biggl(2^{k-1}+25-\frac{1}{\sigma_{k-13}(0)}\biggr)\Delta^2E_{k-24}\pmod{N_k'}
\end{equation}
and
\begin{equation}\label{eqDim2Sigma}
\begin{aligned}
\sigma_{k-1}(n)
  &\equiv
  \frac{1}{\sigma_{k-13}(0)}
  (\tau \ast \sigma_{k-13})(n)\\
  &\quad +\frac{1}{\sigma_{k-25}(0)}
  \biggl(2^{k-1}+25-\frac{1}{\sigma_{k-13}(0)}\biggr)
  (\tau \ast \tau \ast \sigma_{k-25})(n)
  \pmod{N_k'}.
\end{aligned}
\end{equation}
\end{corollary}
\begin{proof}
For $k=24, 28, 30, 32, 34, 38$, we have $\dim \mathcal{S}_k=2$.
The space $\mathcal{S}_k$ is spanned by $\Delta E_{k-12}$ and $\Delta^2 E_{k-24}$.

 From
\[
\Delta(z)=q\prod_{n\ge 1}(1-q^n)^{24}=\sum_{n\ge 1} \tau(n)q^n=q-24q^2+
\text{(higher order terms)},
\]
we calculate the first few terms of the Fourier expansion of $\Delta E_{k-12}$ and $\Delta^2 E_{k-24}$:
\begin{align*}
\Delta E_{k-12}(z)
  &= q+\biggl(-24+\frac{1}{\sigma_{k-13}(0)}\biggr)q^2+\cdots,\\
\Delta^2E_{k-24}(z)
  &= q^2+\cdots.
\end{align*}
Therefore the matrix $C$ and its inverse in Theorem \ref{thmMain} are
\[
C=
\begin{pmatrix} 1 & -24+\frac{1}{\sigma_{k-13}(0)}\\
0 & 1\end{pmatrix},
\quad
C^{-1}=
\begin{pmatrix}1 &  24-\frac{1}{\sigma_{k-13}(0)}\\
0 & 1 \end{pmatrix},
\]
and thus
\[
\begin{pmatrix}\gamma_1 & \gamma_2\end{pmatrix}
=\begin{pmatrix}1 & \sigma_{k-1}(2)\end{pmatrix}C^{-1}
=\begin{pmatrix}1 & \sigma_{k-1}(2)+24-\frac{1}{\sigma_{k-13}(0)}\end{pmatrix}.
\]
Theorem \ref{thmMain} gives congruence \eqref{eqDim2Gk}, namely,
\[
G_k\equiv \Delta E_{k-12}+\biggl(2^{k-1}+25-\displaystyle\frac{1}{\sigma_{k-13}(0)}\biggr)\Delta^2E_{k-24}\pmod{N_k'}
\]
and \eqref{eqDim2Sigma} follows.
\end{proof}
For example, for $k=24$ we have
$B_{24}=-(103\times 2294797)/2730$.
Therefore $N_{24}'=103\times 2294797$.
As $\sigma_{11}(0)=-B_{12}/24=691/65520$, we have
\[
G_{24}\equiv \Delta E_{12}+\biggl(2^{23}+25-\frac{65520}{691}\biggr)\Delta^2\pmod{103\times 2294797},
\]
and
\begin{align*}
\sigma_{23}(n)
  &\equiv
  \frac{65520}{691}
  \sum_{i=0}^{n}\tau(i)\sigma_{11}(n-i)\\
  &\quad +\biggl(2^{23}+25-\frac{65520}{691}\biggr)
  \sum_{i=0}^{n}\tau(i)\tau(n-i)
  \pmod{103\cdot2294797}.
\end{align*}
Datskovsky and Guerzhoy \cite{DG} obtained the same explicit formula for $k=24$. Their proof involved congruence of Hecke eigenforms modulo prime ideals.

For $k=24,28,30,32,34,38$, we list $N_k'$ and the residues of $\gamma_1,\gamma_2$ modulo $N_k'$ in Table \ref{tabDimTwo}:
\begin{table}[H]
\begin{center}
\begin{tabular}{cccc}
\hline
$k$ & $N_k'$ & $\gamma_1$ & $\gamma_2$ (mod $N_k'$)\\
\hline
$24$ & $103\times2294797$ & $1$ & $107586203$\\
$28$ & $9349\times362903$ & $1$ & $2008360611$\\
$30$ & $1721\times1001259881$ & $1$ & $419043597883$\\
$32$ & $37\times683\times305065927$ & $1$ & $2426412849149$\\
$34$ & $151628697551$ & $1$ & $144283399404$\\
$38$ & $154210205991661$ & $1$ & $80609153788200$\\
\hline
\end{tabular}
\end{center}
\caption{Values of $N_k'$ and residues of $\gamma_1$ and $\gamma_2$ modulo $N_k'$.}
\label{tabDimTwo}
\end{table}
\begin{corollary}
When $k=36, 40, 42, 44, 46, 50$, take
\begin{align*}
\gamma_2 &= 2^{k-1}+25-\displaystyle\frac{1}{\sigma_{k-13}(0)},\\
\gamma_3 &= 3^{k-1}+2^{k-1}\cdot 48 +949
-\displaystyle\frac{2^{k-13}+25}{\sigma_{k-13}(0)}-\displaystyle\frac{2^{k-1}+25}{\sigma_{k-25}(0)}+\displaystyle\frac{1}{\sigma_{k-13}(0)\sigma_{k-25}(0)}.
\end{align*}
Then we have
\[
G_k \equiv \Delta E_{k-12}
+ \gamma_2 \Delta^2 E_{k-24}
+ \gamma_3 \Delta^3 E_{k-36} \pmod{N_k'},
\]
and
\begin{equation*}
\begin{aligned}
\sigma_{k-1}(n)
  &\equiv
  \frac{1}{\sigma_{k-13}(0)}
  \bigl(\tau\ast\sigma_{k-13}\bigr)(n)\\
  &\quad+
  \frac{\gamma_2}{\sigma_{k-25}(0)}
  \bigl(\tau^{\ast 2}\ast\sigma_{k-25}\bigr)(n)\\
  &\quad+
  \frac{\gamma_3}{\sigma_{k-37}(0)}
  \bigl(\tau^{\ast 3}\ast\sigma_{k-37}\bigr)(n)
  \pmod{N_k'}.
\end{aligned}
\end{equation*}
\end{corollary}
\begin{proof}
When $k=36, 40, 42, 44, 46, 50$, we have $\dim \mathcal{S}_k = 3$, and the space $\mathcal{S}_k$ is spanned by $\Delta E_{k-12}$, $\Delta^2 E_{k-24}$, and $\Delta^3 E_{k-36}$.
 From the calculation of the Fourier coefficients of the three modular forms, we see
\[
C=
\begin{pmatrix}
1 & -24+\frac{1}{\sigma_{k-13}(0)} & 252+\frac{2^{k-13}-23}{\sigma_{k-13}(0)}\\
0 & 1 & -48+\frac{1}{\sigma_{k-25}(0)}\\
0 & 0 & 1
\end{pmatrix},\quad
C^{-1} =
\begin{pmatrix}
1 & -c_{12} & c_{12}c_{23}-c_{13}\\
0 & 1 & -c_{23}\\
0 & 0 & 1
\end{pmatrix}
\]
and
\[
\begin{pmatrix}\gamma_1 & \gamma_2 & \gamma_3\end{pmatrix}
=
\begin{pmatrix}\sigma_{k-1}(1) & \sigma_{k-1}(2) & \sigma_{k-1}(3) \end{pmatrix}C^{-1}.
\]
So
\begin{align*}
\gamma_1
  &=1,\\
\gamma_2
  &=2^{k-1}+25-\frac{1}{\sigma_{k-13}(0)},\\
\gamma_3
  &=3^{k-1}+2^{k-1}\cdot48+949
  -\frac{2^{k-13}+25}{\sigma_{k-13}(0)}
  -\frac{2^{k-1}+25}{\sigma_{k-25}(0)}
  +\frac{1}{\sigma_{k-13}(0)\sigma_{k-25}(0)}.
\end{align*}
Theorem \ref{thmMain} gives the claimed congruences.
\end{proof}

Table \ref{tabDimThree} lists $N_k'$ and the residues of $\gamma_1$, $\gamma_2$, and $\gamma_3$ modulo $N_k'$ for each $k$.
\begin{table}[H]
\begin{center}
\resizebox{\textwidth}{!}{
\begin{tabular}{ccccc}
\hline
$k$ & $N_k'$ & $\gamma_1$
& $\gamma_2$ (mod $N_k'$)
& $\gamma_3$ (mod $N_k'$)\\
\hline
$36$
& $26315271553053477373$
& $1$
& $8390735099767322508$
& $23753298026670817518$\\
$40$
& $137616929\times1897170067619$
& $1$
& $143325441834485554993$
& $249170693679242863157$\\
$42$
& $1520097643918070802691$
& $1$
& $1268541799120853738102$
& $213658362938250302834$\\
$44$
& $59\times8089\times2947939\times1798482437$
& $1$
& $2310744223299992745919$
& $1741035903638575027145$\\
$46$
& $383799511\times67568238839737$
& $1$
& $25393878538540355611918$
& $6784674507998026661076$\\
$50$
& $417202699\times47464429777438199$
& $1$
& $13556968179673327086390836$
& $15511858160808671948447160$\\
\hline
\end{tabular}
}
\end{center}
\caption{Values of $N_k'$ and residues of $\gamma_1$, $\gamma_2$, and $\gamma_3$ modulo $N_k'$.}
\label{tabDimThree}
\end{table}
In all the above cases, $N_k$ does not share common prime factors with any of the $N_{k-12i}$ ($1\le i\le \frac{k}{12}$), and therefore $N_k'$ coincides with $N_k$. In fact, the smallest $k$ such that $N_k'\ne N_k$ is $k=68$.

\section{Remarks on irregular primes}\label{secIrregular}
Recall that an odd prime $p$ is irregular if and only if $p$ divides the numerator of $B_k$ for some even integer $k$ with $2\le k\le p-3$ (see entry \seqnum{A000928} in \cite{oeis}).  
By the Clausen--von Staudt and Kummer congruences, $N_k$ is a product of irregular primes.
Conversely, every irregular prime $p$ appears as a prime factor of $N_k$ for some $k$. Consequently, it is also a factor of $N_k'$ for some $k$.  (To see this, it suffices to choose the least even $k>0$ such that $p\mid N_k$.)

Corollary \ref{corMain} thereby provides at least one Ramanujan-type congruence for each irregular prime.
Jensen \cite{Jensen} proved that there are infinitely many irregular primes. Carlitz \cite{Carlitz}, Ireland and Rosen \cite[Chap.\ 15]{IR}, and Luca, Pizarro-Madariaga, and Pomerance \cite{irreg} give further discussion.

The smallest irregular prime is $37$. Since it is a prime factor of $N_{32}'$, we have $\gamma_1=1$ and $\gamma_2\equiv 25$ (mod $37$),
\[
G_{32} \equiv
\Delta E_{20} + 25 \Delta^2 E_{8} \pmod{37}
\]
and
\[
\sigma_{31}(n)
 \equiv 22(\tau\ast\sigma_{19})(n) +12(\tau\ast\tau\ast\sigma_7)(n)
 \pmod{37}.
\]
The second smallest irregular prime is $59$, which is a prime factor of $N_{44}'$. We have $\gamma_1=1$, $\gamma_2\equiv 16$ (mod $59$), $\gamma_3\equiv 4$ (mod $59$).
Thus we obtain
\[
 G_{44} \equiv \Delta E_{32} + 16\Delta^2 E_{20} + 4\Delta^3 E_8
 \pmod{59}
\]
and
\[
\sigma_{43}(n) \equiv 27(\tau\ast\sigma_{31})(n)
+19(\tau^{\ast 2}\ast \sigma_{19})(n)
+32(\tau^{\ast 3}\ast\sigma_{7})(n)\pmod{59}.
\]

\section{Acknowledgment}\label{secAcknowledgment}
This work was supported by the National Natural Science Foundation of China (grant number 12271037). The author thanks Zheng for suggesting this problem and for providing comments on drafts of this paper. The author also thanks the referees for the careful reading and helpful suggestions.

The author used the mathematics software system \emph{SageMath} for numerical calculations with exact integer and rational arithmetic; the table entries involve no floating-point approximations.


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\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2020 {\it Mathematics Subject Classification}:
Primary 11F33; Secondary 11F11, 11B68.

\noindent \emph{Keywords: } modular form, Eisenstein series, Ramanujan congruence, irregular prime, Bernoulli number.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000928},
\seqnum{A001067},
\seqnum{A004009},
\seqnum{A010839},
\seqnum{A013973}, and
\seqnum{A035118}.)


\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received February 1 2026;
revised versions received February 2 2026; September 3 2026; September 17 2026.
Published in {\it Journal of Integer Sequences}, September 17 2026.

\bigskip
\hrule
\bigskip

\noindent
Return to \href{https://cs.uwaterloo.ca/journals/JIS/}{Journal of Integer Sequences home page}.
\vskip .1in

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