
// Make newform 216.2.q.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_216_q();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_216_2_q_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_216_q();"
function MakeCharacter_216_q()
    N := 216;
    order := 9;
    char_gens := [55, 109, 137];
    v := [9, 9, 5];
    // 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_216_q_Hecke(Kf)
    return MakeCharacter_216_q();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_216_2_q_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_216_2_q_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_216_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![298321984, -94305120, 1841144544, 698422760, 1448244684, -1305811008, 2086299117, -3464549073, 1542258090, -238445301, 939935934, -680231610, 303760740, -257296176, 130158990, 24619671, -31621464, 1642140, 3428967, -761355, 100485, 53718, -19116, -1908, 2772, 261, 27, -19, -12, 0, 1]>],Snew);
    return Vf;
end function;