# Eisenstein Series

The intrinsics below require that the base ring of $M$ has characteristic $0$. To compute mod $p$ eigenforms, use the `Reduction` intrinsic (see Section [Reductions and Embeddings](reductions-embeddings.md#sec-reductions-and-embeddings)).

## `EisensteinSeries(M): ModFrm -> List`

List of the Eisenstein series associated to the modular forms space $M$. By “associated to” we mean that the Eisenstein series lies in $M{\otimes}{\mathbb{C}}$.

## `IsEisensteinSeries(f): ModFrmElt -> BoolElt`

Returns `true` if the modular form $f$ was created using `EisensteinSeries`.

## `EisensteinData(f): ModFrmElt -> Tup`

The data $<\chi, \psi, t, \chi', \psi'>$ that defines the Eisenstein series (modular form) $f$. Here $\chi$ is a primitive character of conductor $S$, $\psi$ is primitive of conductor $M$, and $MSt$ divides $N$, where $N$ is the level of $f$. (The additional characters $\chi'$ and $\psi'$ are equal to $\chi$ and $\psi$ respectively, except they take values in the big field ${\mathbb{Q}}(\zeta_{\phi(N)})^*$ instead of ${\mathbb{Q}}(\zeta_{n})^*$, where $n$ is the order of $\chi$ or $\psi$.) The Eisenstein series associated to $(\chi,\psi,t)$ has $q$-expansion

$$
c_0 + \sum_{m\geq 1} \left(\sum_{n|m}\psi(n)n^{k-1}\chi(m/n)\right)q^{mt},
$$

where $c_0=0$ if $S>1$ and $c_0=L(1-k,\psi)/2$ if $S=1$.

## `Example: Eisenstein Series (ex-bee467)`

We illustrate the above intrinsics by computing the Eisenstein series in $M_3(\Gamma_1(12))$.

```magma
> M := ModularForms(Gamma1(12),3); M;
Space of modular forms on Gamma_1(12) of weight 3 and dimension 13
over Integer Ring.
> E := EisensteinSubspace(M); E;
Space of modular forms on Gamma_1(12) of weight 3 and dimension 10
over Integer Ring.
> s := EisensteinSeries(E); s;
[*
-1/9 + q - 3*q^2 + q^3 + 13*q^4 - 24*q^5 - 3*q^6 + 50*q^7 + O(q^8),
-1/9 + q^2 - 3*q^4 + q^6 + O(q^8),
-1/9 + q^4 + O(q^8),
-1/4 + q + q^2 - 8*q^3 + q^4 + 26*q^5 - 8*q^6 - 48*q^7 + O(q^8),
-1/4 + q^3 + q^6 + O(q^8),
q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + 27*q^6 + 50*q^7 + O(q^8),
q^2 + 3*q^4 + 9*q^6 + O(q^8),
q^4 + O(q^8),
q + 4*q^2 + 8*q^3 + 16*q^4 + 26*q^5 + 32*q^6 + 48*q^7 + O(q^8),
q^3 + 4*q^6 + O(q^8)
*]
> a := EisensteinData(s[1]); a;
<1, $.1, 1, 1, $.2>
> Parent(a[2]);
Group of Dirichlet characters of modulus 3 over Rational Field
> Order(a[2]);
2
> Parent(a[5]);
Group of Dirichlet characters of modulus 12 over Cyclotomic Field of
order 4 and degree 2
> Parent(s[1]);
Space of modular forms on Gamma_1(12) of weight 3 and dimension 10
over Rational Field.
> IsEisensteinSeries(s[1]);
true

```