By William Stein

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Lan95]). The space Mk (Γ0 (N )) decomposes as a direct sum Mk (Γ0 (N )) = Mk (Γ0 (N ))old ⊕ Mk (Γ0 (N ))new , where Mk (Γ0 (N ))old is the subspace generated by all images αd (Mk (Γ0 (N )) where N runs through proper divisors of N and d runs through all divisors of N/N . The new subspace Mk (Γ0 (N ))new can be defined as either the intersection of the kernels of all maps βd to lower level, or the largest Hecke-stable complement of Mk (Γ0 (N ))old . Atkin and Lehner [AL70] proved that the space Sk (Γ0 (N )) is built out of new subspaces, in the following sense.

I learned about rational reconstruction in the context of computing echelon forms from Allan Steel, who is one of the developers of Magma. , and takes advantage of asymptotically fast matrix multiplication algorithms. The matrix multiplies are done using a modular CRT technique. This is probably better in many cases, especially for dense matrices. 62 CHAPTER 5. LINEAR ALGEBRA 2. 11-8. 3 is much more efficient (both in time and memory usage) than MAGMA. , when intersecting subspaces), MAGMA is often an order of magnitude faster.

We say f is holomorphic at i∞ if its q-expansion an q n has no nonzero coefficient an for n < 0. To make sense of holomorphicity of f at α ∈ Q, let γ ∈ SL2 (Z) be such that γ(∞) = α. Then we say f is holomorphic at α if f |[γ]k is holomorphic at infinity. Note that formally f |[γ]k (∞) = (cz + d)−k f (α), where (c, d) is the bottom row of γ and the factor (cz + d)−k does not affect holomorphicity at α. Another subtlety hidden in this definition is that f |[γ]k is a modular form for the conjugate group G = γ −1 Γ1 (N )γ, which need not equal Γ1 (N ).

### Algorithms for Computing with Modular Forms by William Stein

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