
// Make newform 210.3.k.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_210_k();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_210_3_k_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_210_k();"
function MakeCharacter_210_k()
    N := 210;
    order := 4;
    char_gens := [71, 127, 31];
    v := [2, 3, 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_210_k_Hecke(Kf)
    return MakeCharacter_210_k();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_210_3_k_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_210_3_k_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_210_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<23,R![46925513731096576, 9771061808840704, 1017289330270208, -445641433815040, 306021443987712, 42466444868096, 4324471602688, -1702030780544, 255512355136, -7605725568, 54721952, 5640688, 1114532, -11600, 72, 12, 1]>],Snew);
    return Vf;
end function;