
// Make newform 403.2.l.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_403_l();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_403_2_l_c();".
// 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_l();"
function MakeCharacter_403_l()
    N := 403;
    order := 6;
    char_gens := [249, 313];
    v := [3, 2];
    // 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_l_Hecke(Kf)
    return MakeCharacter_403_l();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_403_2_l_c();".
// 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_l_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_403_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 0, 4854, 0, 268819, 0, 4672934, 0, 25704838, 0, 71173928, 0, 118141291, 0, 128354894, 0, 96106914, 0, 51210844, 0, 19794820, 0, 5599630, 0, 1158124, 0, 172998, 0, 18156, 0, 1269, 0, 53, 0, 1]>,<5,R![1143094552336, 0, -63474740739652, 0, 2441345174872037, 0, -43565609993361387, 0, 534712939134233900, 0, -4439490088149773915, 0, 27290322564228270281, 0, -117718365920208544411, 0, 368345264161814676415, 0, -735290235013811159202, 0, 1052024460675605468800, 0, -1120766713636376149290, 0, 929772716510261795424, 0, -614055154770373206327, 0, 329834194252115455520, 0, -146082291947731340918, 0, 54064861489593130339, 0, -16893764694651895470, 0, 4497926567790278941, 0, -1027559072266115088, 0, 202572538358819730, 0, -34597862606814692, 0, 5133336968003540, 0, -662435486603809, 0, 74343814088984, 0, -7243084454734, 0, 610507733168, 0, -44258416065, 0, 2735930332, 0, -142332208, 0, 6114046, 0, -210316, 0, 5518, 0, -100, 0, 1]>],Snew);
    return Vf;
end function;