
// Make newform 315.2.r.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_315_r();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_315_2_r_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_r();"
function MakeCharacter_315_r()
    N := 315;
    order := 6;
    char_gens := [281, 127, 136];
    v := [2, 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_315_r_Hecke(Kf)
    return MakeCharacter_315_r();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_315_2_r_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_r_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_315_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2401, 0, 285442, 0, 8588765, 0, 108864579, 0, 688476409, 0, 2559973616, 0, 6189307170, 0, 10365948952, 0, 12544393598, 0, 11294088604, 0, 7721909198, 0, 4066263964, 0, 1663907457, 0, 531336191, 0, 132360046, 0, 25581175, 0, 3791037, 0, 422073, 0, 34121, 0, 1889, 0, 64, 0, 1]>],Snew);
    return Vf;
end function;