
// Make newform 570.2.x.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_570_x();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_570_2_x_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_570_x();"
function MakeCharacter_570_x()
    N := 570;
    order := 12;
    char_gens := [191, 457, 211];
    v := [12, 9, 2];
    // 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_570_x_Hecke(Kf)
    return MakeCharacter_570_x();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_570_2_x_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_570_2_x_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![21025, 223880, 1191968, 3787460, 7880205, 10831448, 9722664, 5240040, 1717934, 786896, 876920, 387176, 89914, 8472, 7800, 3400, 761, 24, 8, 4, 1]>],Snew);
    return Vf;
end function;