Computing bases of modular forms using the graded algebra structure
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We develop a new algorithm to compute a basis for \(M_k(\Gamma _0(N))\), the space of weight k holomorphic modular forms on \(\Gamma _0(N)\), in the case when the graded algebra of modular forms over \(\Gamma _0(N)\) is generated at weight two. Our tests show that this algorithm significantly outperforms a commonly used algorithm which relies more heavily on modular symbols.
KeywordsHolomorphic modular forms on \(\Gamma _0(N)\) Bases of modular forms Eta-quotients
Mathematics Subject ClassificationPrimary 11F11 11F30 11Y16 Secondary 11F20
The second and third authors received funding for this project through the James Madison University Tickle Scholarship Fund, while the fourth author received support from a James Madison University College of Science and Mathematics Faculty Summer Grant.
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