
// Make newform 756.2.bi.c in Magma, downloaded from the LMFDB on 04 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_756_bi();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_bi_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_bi();"
function MakeCharacter_756_bi()
    N := 756;
    order := 6;
    char_gens := [379, 29, 325];
    v := [3, 2, 3];
    // 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_bi_Hecke( : Kf := Kf)
    return MakeCharacter_756_bi();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_756_2_bi_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_bi_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_756_bi();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3370896, 0, -465712416, 0, 61591708476, 0, -371823512760, 0, 1683338096629, 0, -2526088619506, 0, 2423038916979, 0, -1618039070798, 0, 818404822472, 0, -319950684900, 0, 99265144659, 0, -24482028372, 0, 4848544452, 0, -766064542, 0, 96876664, 0, -9662658, 0, 757017, 0, -45000, 0, 1986, 0, -58, 0, 1]>],Snew);
    return Vf;
end function;