
// Make newform 1344.4.b.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1344_b();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1344_4_b_j();".
// 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_1344_b();"
function MakeCharacter_1344_b()
    N := 1344;
    order := 2;
    char_gens := [127, 1093, 449, 577];
    v := [1, 2, 2, 1];
    // 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_1344_b_Hecke(Kf)
    return MakeCharacter_1344_b();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1344_4_b_j();".
// 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_1344_4_b_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1344_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![21744258850152591130624, 0, 11900654047146757062656, 0, 2280546313077341814784, 0, 189479901741741375488, 0, 7915503942246199296, 0, 185088997044646912, 0, 2624120172789824, 0, 23705103544640, 0, 139508402672, 0, 532572832, 0, 1270076, 0, 1716, 0, 1]>,<19,R![80820427220073054208, 39053762646145105920, 5975264429705576448, 289176941968752640, 826282403204096, -231932005560832, -3296023911616, 38813878976, 692579632, -1990400, -46324, 28, 1]>],Snew);
    return Vf;
end function;