newform in the classical cuspidal sense

新形式

Number Theory · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: primitive cusp form

Translation evidence: Literature usage

Meaning in context

A normalized Hecke eigenform in $S_k(\Gamma_1(N))$ lying in the Petersson orthogonal complement of the old subspace generated by degeneracy images from proper divisor levels.

$S_k(\Gamma_1(N))$ 中的正规化 Hecke 特征形式，且属于由真因子级的退化映射像生成的旧空间的 Petersson 正交补。

Example

$\Delta=q\prod_{n\ge1}(1-q^n)^{24}$ is a weight-$12$, level-$1$ newform, with $q=e^{2\pi i\tau}$.

Usage and distinctions

Being normalized or being in the new subspace alone is insufficient: the definition also requires simultaneous Hecke eigenvector status.

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:oldform, def:newform and paragraph stating that the old space at N=1 is zero · 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:newform 新形式，又称本原形式 · Checked 2026-10-15

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