The **$q$-expansion** of a modular form $f(z)$ is its Fourier expansion at the cusp $z=i\infty$, expressed as a power series $\sum_{n=0}^{\infty} a_n q^n$ in the variable $q=e^{2\pi iz}$.

For cusp forms, the constant coefficient $a_0$ of the $q$-expansion is zero.

For newforms, we have $a_1=1$ and the coefficients $a_n$ are algebraic integers in a number field $K \subseteq \C$.

Accordingly, we define the **$q$-expansion** of a newform orbit $[f]$ to be the $q$-expansion of any newform $f$ in the orbit, but with coefficients $a_n \in K$ (without an embedding into $\C$). Each embedding $K \hookrightarrow \C$ then gives rise to an embedded newform whose $q$-expansion has $a_n \in \C$, as above.

**Knowl status:**

- Review status: reviewed
- Last edited by John Voight on 2020-02-15 13:58:33

**Referred to by:**

- cmf.analytic_rank
- cmf.dualform
- cmf.eisenstein
- cmf.embedding
- cmf.hecke_operator
- cmf.hecke_ring_generators
- cmf.lfunction
- cmf.selfdual
- cmf.trace_bound
- rcs.cande.cmf
- lmfdb/classical_modular_forms/templates/cmf_browse.html (line 44)
- lmfdb/classical_modular_forms/templates/cmf_newform_common.html (line 187)
- lmfdb/classical_modular_forms/templates/cmf_newform_common.html (line 231)
- lmfdb/classical_modular_forms/templates/cmf_newform_list.html (line 29)

**History:**(expand/hide all)

- 2020-02-15 13:58:33 by John Voight (Reviewed)
- 2020-02-15 13:58:12 by John Voight
- 2020-01-03 12:33:03 by John Voight
- 2020-01-03 06:08:09 by Andrew Sutherland
- 2020-01-03 04:36:09 by Andrew Sutherland
- 2019-04-29 09:28:57 by David Farmer (Reviewed)
- 2018-12-07 20:29:35 by Andrew Sutherland (Reviewed)

**Differences**(show/hide)