Newspaces with Nebentypus: An Explicit Dimension Formula and Classification of Trivial Newspaces

2024/07/11 by Erick Ross, Ross, Erick · 3 citations
Engineering · Mathematics · #11F06 (Secondary) #11F11 (Primary) #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2407.08881

openalex publication_date 2024/07/11 · openalex created_date 2024/07/16 · openalex updated_date 2026/07/28

Abstract

Consider N ≥ 1, k ≥ 2, and χ a Dirichlet character modulo N such that χ(-1) = (-1)^k. For any bound B, one can show that dim S_k(Γ₀(N),χ) ≤ B for only finitely many triples (N,k,χ). It turns out that this property does not extend to the newspace; there exists an infinite family of triples (N,k,χ) for which dim S_k^new(Γ₀(N),χ) = 0. However, we classify this case entirely. We also show that excluding the infinite family for which dim S_k^new(Γ₀(N),χ) = 0, dim S_k^new(Γ₀(N),χ) ≤ B for only finitely many triples (N,k,χ). In order to show these results, we derive an explicit dimension formula for the newspace S_k^new(Γ₀(N),χ). We also use this explicit dimension formula to prove a character equidistribution property and disprove a conjecture from Greg Martin that dim S₂^new(Γ₀(N)) takes on all possible non-negative integers.

Cited by

• Dimensions of spaces of modular forms
• Hecke Eigenvalue Equidistribution over the Newspaces with Nebentypus
• Dimension sequences of modular forms

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