
// Make newform 605.2.g.q in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_605_g();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_605_2_g_q();".
// 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_605_g();"
function MakeCharacter_605_g()
    N := 605;
    order := 5;
    char_gens := [122, 486];
    v := [5, 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_605_g_Hecke(Kf)
    return MakeCharacter_605_g();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_605_2_g_q();".
// 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_605_2_g_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_605_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![81, 0, 405, 0, 1782, 0, 7722, 0, 33399, 0, 21978, 0, 12267, 0, 6534, 0, 3195, 0, 462, 0, 66, 0, 9, 0, 1]>,<3,R![2401, 1029, 1470, 728, 795, 108, 283, 54, 129, 38, 12, 3, 1]>],Snew);
    return Vf;
end function;