
// Make newform 630.2.t.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_630_t();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_630_2_t_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_630_t();"
function MakeCharacter_630_t()
    N := 630;
    order := 6;
    char_gens := [281, 127, 451];
    v := [5, 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_630_t_Hecke(Kf)
    return MakeCharacter_630_t();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_630_2_t_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_630_2_t_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_630_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![14197824, 0, 534385824, 0, 5843448337, 0, 27687320028, 0, 62930119704, 0, 70453440704, 0, 38039493624, 0, 10580239590, 0, 1701189678, 0, 169671660, 0, 10843488, 0, 444572, 0, 11301, 0, 162, 0, 1]>],Snew);
    return Vf;
end function;