The main step in numerical evaluation of classical Sl₂(Z) modular forms is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, we show that every generalized pentagonal number c ≥ 5 can be written as c = 2a + b where a, b are smaller generalized pentagonal numbers. We also give a baby-step giant-step algorithm that uses O(N / (log N)^r) multiplications for every r ≥ 0, beating the lower bound of N multiplications required when computing the terms explicitly. These results lead to speed-ups in practice.