
// Make newform 605.2.j.k in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_605_j();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_605_2_j_k();".
// 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_j();"
function MakeCharacter_605_j()
    N := 605;
    order := 10;
    char_gens := [122, 486];
    v := [5, 6];
    // 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_j_Hecke(Kf)
    return MakeCharacter_605_j();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_605_2_j_k();".
// 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_j_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_605_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![81, 0, -783, 0, 7272, 0, -67452, 0, 625633, 0, -242476, 0, 72431, 0, -19632, 0, 5021, 0, -696, 0, 92, 0, -11, 0, 1]>,<19,R![228509902503936, 0, 78285799931904, 0, 22588470300672, 0, 6347678690304, 0, 1776497930496, 0, 105930021888, 0, 5025364416, 0, 216903744, 0, 8492688, 0, 185328, 0, 3852, 0, 72, 0, 1]>],Snew);
    return Vf;
end function;