
// Make newform 315.2.be.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_315_be();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_315_2_be_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_315_be();"
function MakeCharacter_315_be()
    N := 315;
    order := 6;
    char_gens := [281, 127, 136];
    v := [1, 6, 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_315_be_Hecke(Kf)
    return MakeCharacter_315_be();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_315_2_be_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_315_2_be_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_315_be();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![27, 1620, 29079, -199260, 389349, 65448, -1020689, 451506, 1761639, -1546254, -1634448, 2062989, 1085831, -1848624, -425133, 1124622, 93651, -506289, 16041, 163524, -16806, -39501, 6648, 6711, -1417, -828, 213, 63, -18, -3, 1]>],Snew);
    return Vf;
end function;