.. _sec-creation:

.. _creation:

Creation of Admissible Representations
======================================

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

.. magma:function:: LocalComponent(M, p)
   :input_types: ModSym, RngIntElt
   :output_types: RepLoc
   :label: LocalComponent_ModSym_RngIntElt

   This returns the admissible representation of
   :math:`{\operatorname{GL}}({\mathbb{Q}}_p)` associated to the cuspidal eigenform
   specified by :math:`M`. Here :math:`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``).

.. magma:example:: Example: Creation Example
   :label: creation-example

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

   .. code-block:: magma

      > 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)