
// Make newform 429.2.s.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_429_s();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_429_2_s_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_429_s();"
function MakeCharacter_429_s()
    N := 429;
    order := 6;
    char_gens := [287, 79, 67];
    v := [6, 6, 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_429_s_Hecke(Kf)
    return MakeCharacter_429_s();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_429_2_s_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_429_2_s_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_429_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4096, -6144, -42304, 68064, 384665, -11934, -1061417, -165498, 2000562, 266826, -2216245, -226008, 1757192, 89220, -908886, -22782, 345596, 2496, -94487, 18, 19570, -54, -2938, 6, 325, 0, -23, 0, 1]>],Snew);
    return Vf;
end function;