
// Make newform 784.2.x.p in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_784_x();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_784_2_x_p();".
// 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_784_x();"
function MakeCharacter_784_x()
    N := 784;
    order := 12;
    char_gens := [687, 197, 689];
    v := [12, 3, 8];
    // 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_784_x_Hecke(Kf)
    return MakeCharacter_784_x();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_784_2_x_p();".
// 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_784_2_x_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_784_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4294967296, 0, 0, 0, -32311038967808, 0, 0, 0, 240532834278703104, 0, 0, 0, -19096564361003532288, 0, 0, 0, 1373137317150878334976, 0, 0, 0, -10114973896195002335232, 0, 0, 0, 59502764111697222303744, 0, 0, 0, -78460929427749666291712, 0, 0, 0, 72004952424752735125504, 0, 0, 0, -31949050074833315954688, 0, 0, 0, 9831047739401653125120, 0, 0, 0, -1823413996320809680896, 0, 0, 0, 241458515147210006528, 0, 0, 0, -22758390154016522240, 0, 0, 0, 1627396910840127488, 0, 0, 0, -88448475147563008, 0, 0, 0, 3747155880329984, 0, 0, 0, -122067528105984, 0, 0, 0, 3083985715200, 0, 0, 0, -58586222592, 0, 0, 0, 849400992, 0, 0, 0, -8966400, 0, 0, 0, 69072, 0, 0, 0, -336, 0, 0, 0, 1]>,<5,R![60336661037197280935936, 0, 0, 0, -751198847848347384938496, 0, 0, 0, 6009447160366829444005888, 0, 0, 0, -27836113651082923379523584, 0, 0, 0, 92560209871836212382138368, 0, 0, 0, -214307052860740945829691392, 0, 0, 0, 370730570978095952534962176, 0, 0, 0, -451216726118344511929188352, 0, 0, 0, 402066795035085285202395136, 0, 0, 0, -230617296993716713008136192, 0, 0, 0, 92511098480941416888598528, 0, 0, 0, -19349397154483816932835328, 0, 0, 0, 2798689268917314753372160, 0, 0, 0, -177299015397965436387328, 0, 0, 0, 7643664485211123138560, 0, 0, 0, -216645259463118340096, 0, 0, 0, 4515884835488973568, 0, 0, 0, -66735808233768960, 0, 0, 0, 736377886682112, 0, 0, 0, -5731077684224, 0, 0, 0, 33435938208, 0, 0, 0, -139298048, 0, 0, 0, 426416, 0, 0, 0, -832, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;