
// Make newform 756.2.be.c in Magma, downloaded from the LMFDB on 04 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_756_be();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_be_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_756_be();"
function MakeCharacter_756_be()
    N := 756;
    order := 6;
    char_gens := [379, 29, 325];
    v := [3, 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_756_be_Hecke( : Kf := Kf)
    return MakeCharacter_756_be();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_be_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_756_2_be_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_756_be();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16384, 0, -212992, 0, 2023424, 0, -8007680, 0, 22607872, 0, -28779008, 0, 25635072, 0, -13085056, 0, 4735808, 0, -969088, 0, 142048, 0, -13544, 0, 936, 0, -38, 0, 1]>,<19,R![6083328, -21667584, 30663616, -17589696, 1509472, 2282496, -315516, -258900, 69483, 6303, -3292, -105, 152, -21, 1]>],Snew);
    return Vf;
end function;