
// Make newform 315.2.t.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_315_t();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_315_2_t_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_315_t();"
function MakeCharacter_315_t()
    N := 315;
    order := 6;
    char_gens := [281, 127, 136];
    v := [1, 6, 1];
    // 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_t_Hecke(Kf)
    return MakeCharacter_315_t();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_315_2_t_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_315_2_t_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_315_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![81, 0, 9153, 0, 167499, 0, 1330254, 0, 5674327, 0, 14222139, 0, 22161051, 0, 22563914, 0, 15612084, 0, 7544406, 0, 2587491, 0, 632922, 0, 109596, 0, 13118, 0, 1032, 0, 48, 0, 1]>],Snew);
    return Vf;
end function;