
// Make newform 403.2.f.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_403_f();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_403_2_f_b();".
// 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_403_f();"
function MakeCharacter_403_f()
    N := 403;
    order := 3;
    char_gens := [249, 313];
    v := [1, 3];
    // 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_403_f_Hecke(Kf)
    return MakeCharacter_403_f();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_403_2_f_b();".
// 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_403_2_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_403_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -6, 428, 5250, 142094, 606680, 3238389, -868860, 11677925, -6294788, 26202710, -17343049, 37655990, -23737304, 37108454, -22040906, 26353935, -13952538, 13545805, -6453428, 5235581, -2199020, 1515928, -561455, 335322, -107638, 55661, -15232, 6908, -1558, 605, -104, 35, -4, 1]>],Snew);
    return Vf;
end function;