
// Make newform 441.2.bb.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_bb();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_441_2_bb_f();".
// 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_441_bb();"
function MakeCharacter_441_bb()
    N := 441;
    order := 21;
    char_gens := [344, 199];
    v := [21, 16];
    // 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_441_bb_Hecke(Kf)
    return MakeCharacter_441_bb();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_441_2_bb_f();".
// 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_441_2_bb_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_bb();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![594248265625, 0, 5018126443750, 0, 16967630697500, 0, -35350252414750, 0, -33424120413600, 0, 202322557442950, 0, -281156261177659, 0, -67743796120875, 0, 784370827914856, 0, -1306065579092596, 0, 1288546276169768, 0, -872558084536691, 0, 387796515392777, 0, -76532644739701, 0, -27724060883347, 0, 27559936596337, 0, -8528835553988, 0, -563583917275, 0, 996291679162, 0, 234902561447, 0, -296107331055, 0, 121657807853, 0, 30916744585, 0, -28187914277, 0, 5495639214, 0, 326604426, 0, -300919572, 0, 55526661, 0, -2951363, 0, -902466, 0, 222939, 0, -18148, 0, 670, 0, -177, 0, 79, 0, -13, 0, 1]>],Snew);
    return Vf;
end function;