---
tag: A50Y012RBP
kind: concept
name: Cusp form
msc: [11F11]
specializes: [GWKN5VG78A]
sources:
  - url: https://en.wikipedia.org/wiki/Cusp_form
same_as:
  wikidata: Q2400580
  wikipedia: Cusp form
---

## Summary

A cusp form is a [[modular form|GWKN5VG78A]] that vanishes at every cusp.
For the full [[modular group|VEKV1EM77B]], this means that the constant term of its $q$-expansion is $0$.
The cusp forms of weight $k$ form a subspace $S_k$ of $M_k$, on which the [[Petersson inner product|EMZTGD63V3]] is defined.
The coefficients of cusp forms grow more slowly than those of Eisenstein series, and they carry deep arithmetic information.

## Explanations

### L1 human

A modular form has an expansion $a_0 + a_1 q + a_2 q^2 + \cdots$ with $q = e^{2 \pi i \tau}$.
As $\tau$ goes up to infinity, $q$ goes to $0$, and the value tends to $a_0$.
A cusp form is one with $a_0 = 0$, so it dies off at infinity.
The first example is the discriminant $\Delta = q \prod_{n \ge 1} (1 - q^n)^{24}$ of weight $12$.
Its coefficients are the values of the [[Ramanujan tau function|6DNZAGCSQS]].