
// Make newform 637.2.x.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_637_x();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_637_2_x_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_637_x();"
function MakeCharacter_637_x()
    N := 637;
    order := 12;
    char_gens := [248, 197];
    v := [10, 5];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_637_x_Hecke(Kf)
    return MakeCharacter_637_x();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_637_2_x_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_637_2_x_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_637_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, -90, 405, -1182, 2047, -556, -832, 1246, -27448, 14276, 78776, 17454, 27225, -27104, -21543, -846, -3423, 4342, 4036, 910, 864, -228, -138, -46, -31, 6, 5, 2, 1]>],Snew);
    return Vf;
end function;