normalized Hecke eigenform

正规化 Hecke 特征形式

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Other names / variants: normalized eigenform

Translation evidence: Literature usage

Meaning in context

A nonzero modular form on $\Gamma_1(N)$ that is a simultaneous eigenvector for its Hecke algebra and has Fourier coefficient $a_1=1$ at infinity.

$\Gamma_1(N)$ 上对其 Hecke 代数为共同特征向量、且在无穷尖点的傅里叶系数满足 $a_1=1$ 的非零模形式。

Example

$\Delta=q-24q^2+252q^3+\cdots$, with $q=e^{2\pi i\tau}$, is normalized and satisfies $T_2\Delta=-24\Delta$.

Usage and distinctions

Normalization fixes scalar ambiguity but does not assert that the form is new at its level; a general eigenform need not be a cusp form.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 李文威 — 模形式初步，第六章，作者源文件Definitions def:eigenform, def:normalized-eigenform and Corollary prop:Hecke-Fourier-eigenvalue · Checked 2026-10-15
2. 李文威 — 模形式初步，第五章 Hecke 算子通论，作者源文件Section sec:Hecke-full-level, Example eg:full-level-eigenform-1: Delta is normalized · Checked 2026-10-15

Translation sources

1. 李文威 — 模形式初步，第六章，作者源文件定义 def:normalized-eigenform 正规化 Hecke 特征形式 · Checked 2026-10-15

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