
// Make newform 432.2.u.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_432_u();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_432_2_u_e();".
// 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_432_u();"
function MakeCharacter_432_u()
    N := 432;
    order := 9;
    char_gens := [271, 325, 353];
    v := [9, 9, 7];
    // 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_432_u_Hecke(Kf)
    return MakeCharacter_432_u();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_432_2_u_e();".
// 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_432_2_u_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_432_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 201, 10800, -83125, 273360, -644256, 1385830, -1826226, 1168164, 389045, 260358, 763902, 454615, -6159, -23877, -4858, -486, -1152, 1702, 309, 45, -19, 12, 0, 1]>],Snew);
    return Vf;
end function;