Creation of Admissible Representations

One starts with a classical cuspidal eigenform, given as a space of modular symbols.

LocalComponent(M, p): ModSym, RngIntElt → RepLoc

This returns the admissible representation of \({\operatorname{GL}}({\mathbb{Q}}_p)\) associated to the cuspidal eigenform specified by \(M\). Here \(M\) must be a space of modular symbols that is cuspidal and contains only a single Galois conjugacy class of newforms. (Such spaces are created using NewformDecomposition).

Example: Creation Example

We create the local component at \(11\) of the representation associated to the newform of level \(11\) and weight \(2\). We specify the newform as a space of modular symbols of level \(11\), weight \(2\) and sign \(+1.\)

> S11 := CuspidalSubspace(ModularSymbols(11, 2, 1));
> newform_spaces := NewformDecomposition(S11);
> newform_spaces;
[
Modular symbols space for Gamma_0(11) of weight 2 and dimension 1
   over Rational Field
]
> Eigenform(newform_spaces[1]);
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
> LocalComponent(newform_spaces[1], 11);
Steinberg Representation of GL(2,Q_11)