
// Make newform 756.2.n.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_756_n();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_n_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_n();"
function MakeCharacter_756_n()
    N := 756;
    order := 6;
    char_gens := [379, 29, 325];
    v := [3, 4, 5];
    // 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_n_Hecke(Kf)
    return MakeCharacter_756_n();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_n_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_n_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_756_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2187, 0, 1161621, 0, 132747768, 0, 5363577000, 0, 75910939122, 0, 455117788606, 0, 1431477867050, 0, 2651265558960, 0, 3103925065489, 0, 2421374784095, 0, 1313492100018, 0, 511974945462, 0, 146814741693, 0, 31455323817, 0, 5076052073, 0, 617515849, 0, 56237622, 0, 3769914, 0, 180273, 0, 5811, 0, 113, 0, 1]>],Snew);
    return Vf;
end function;