Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fourier expansions at cusps   In this article we study the number fields generated by the Fourier
coefficients of modular forms at arbitrary cusps. We give upper bounds for
these number fields which are cyclotomic extensions of the field generated by
the Fourier coefficients at $\infty$, and determine them explicitly for
newforms with trivial Nebentypus. The main tool is an extension of a result of
Shimura on the compatibility between the actions of $\mathrm{SL}_2(\mathbb{Z})$
and $\mathrm{Aut}(\mathbb{C})$ on the space of modular forms. We give two new
proofs of this result: one based on products of Eisenstein series, and the
other using the theory of algebraic modular forms.

25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org