
// Make newform 756.2.bj.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_756_bj();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_bj_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_756_bj();"
function MakeCharacter_756_bj()
    N := 756;
    order := 6;
    char_gens := [379, 29, 325];
    v := [3, 4, 1];
    // 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_756_bj_Hecke(Kf)
    return MakeCharacter_756_bj();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_bj_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_756_2_bj_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_756_bj();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2187, -61965, 297027, 8165610, -9406746, -492887916, 595928664, 19753255134, 65278104627, 17899675743, -203705898953, -157369611474, 473863254980, 571531286526, -460210584489, -792767579133, 319582795966, 731469773715, -59033992819, -395961525822, -20190644910, 151867168200, 20890995180, -42407918664, -8011025565, 9125487573, 2058578001, -1530151344, -382546807, 203865297, 54740905, -21439284, -6029808, 1777332, 518868, -112518, -33705, 5289, 1632, -165, -52, 3, 1]>],Snew);
    return Vf;
end function;