
// Make newform 1369.2.a.o in Magma, downloaded from the LMFDB on 04 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1369_a();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1369_2_a_o();".
// 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_1369_a();"
function MakeCharacter_1369_a()
    N := 1369;
    order := 1;
    char_gens := [2];
    v := [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_1369_a_Hecke( : Kf := Kf)
    return MakeCharacter_1369_a();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1369_2_a_o();".
// 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_1369_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1369_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-19, -228, 817, 11305, -21679, -117914, 275386, 266361, -967879, -53315, 1553695, -507157, -1336470, 768031, 642907, -539821, -157361, 217788, 5844, -52486, 7212, 7238, -1956, -470, 215, 1, -9, 1]>,<3,R![-1369, -8214, 54924, 269690, -594156, -2633955, 1574888, 7935702, -2245979, -11972770, 2184224, 10729719, -1535581, -6186095, 763567, 2391826, -260347, -632577, 59542, 114675, -8910, -14001, 833, 1099, -44, -50, 1, 1]>],Snew);
    return Vf;
end function;