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).

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 of Weight 3 and Level 12 (ex-bee467)

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

> 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

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