
// Make newform 1280.2.o.u in Magma, downloaded from the LMFDB on 23 February 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1280_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1280_o_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1280_2_o_u();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1280_2_o_u();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [1, 0, 18, 0, 69, 0, 90, 0, 50, 0, 12, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -23, 0, -29, 0, -10, 0, -1, 0, 0, 0], [2, 2, 23, 0, 52, 0, 39, 0, 11, 0, 1, 0], [2, -2, 23, 0, 52, 0, 39, 0, 11, 0, 1, 0], [6, 13, 48, 40, 81, 60, 49, 40, 12, 11, 1, 1], [-6, 13, -48, 40, -81, 60, -49, 40, -12, 11, -1, 1], [7, 5, 38, 61, 60, 89, 40, 50, 11, 12, 1, 1], [7, -5, 38, -61, 60, -89, 40, -50, 11, -12, 1, -1], [-3, 23, -51, 98, -110, 120, -79, 60, -22, 13, -2, 1], [-3, -23, -51, -98, -110, -120, -79, -60, -22, -13, -2, -1], [0, -13, 0, -86, 0, -141, 0, -89, 0, -23, 0, -2], [0, 13, 0, 109, 0, 170, 0, 99, 0, 24, 0, 2]];
    Rf_basisdens := [1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1280_o();"
function MakeCharacter_1280_o()
    N := 1280;
    order := 4;
    char_gens := [511, 261, 257];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1280_o_Hecke();"
function MakeCharacter_1280_o_Hecke( : Kf := false)
    N := 1280;
    order := 4;
    char_gens := [511, 261, 257];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, -1, 0, 1, 0, 1, -1, 0], [0, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 1, 0, 1, 0, 0, -1], [1, 0, 2, 0, 0, 0, -1, -1, -1, 1, -1, 0], [0, 0, 1, -1, 0, 0, 0, 0, -1, 1, 0, -1], [1, 1, -1, 0, 1, 0, 0, 1, 0, -1, -1, 1], [2, 0, 0, 0, -2, 2, -1, -1, -1, -1, 0, 0], [0, 0, -1, 1, 1, 1, -1, 1, 1, -1, 2, 0], [-2, 0, 0, 0, 0, 0, 0, 1, 0, 1, -2, 0], [0, -1, -1, -1, -1, 1, 0, 0, 0, 0, 0, 0], [-2, 0, 0, -1, 0, 2, 2, 0, -2, 0, -2, 0], [-1, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, -1], [0, 1, -4, 0, 0, 0, 0, 1, 0, -1, 0, 1], [0, 0, -1, 1, -2, -2, -2, 2, -1, 1, 4, 1], [0, 0, 2, -2, -2, -2, -1, 1, 0, 0, 0, 0], [-2, 0, 1, 0, -2, 0, -2, 0, -2, 0, 2, 0], [0, 0, 1, -1, -1, -1, -1, 1, 1, -1, 2, 0], [-1, 1, 0, 0, 0, -2, 1, -1, -1, -1, -1, -1], [-4, 0, 0, 0, 2, -2, 2, 2, 0, 0, 0, 0], [-6, -1, 0, -1, 0, -2, 1, 3, -1, 3, -6, 1], [0, 0, -2, 2, 2, 2, 0, 0, 2, -2, 0, -2], [-1, -1, 0, 0, 2, 0, 3, 3, 3, -3, 1, -1], [0, 0, 0, 0, 2, 2, 0, 0, 2, -2, 6, 0], [5, -1, -1, 0, 1, 0, 0, 1, 0, -1, -5, -1], [0, -2, 3, 0, 0, 0, 0, 0, 0, 0, 0, -2], [-4, 2, -4, -4, 0, 0, 1, 1, 0, 0, 0, 0], [-1, -1, 0, 2, 0, 2, 2, -2, -2, -2, -1, 1], [-5, -2, 0, 1, 0, 3, 2, 3, -2, 3, -5, 2], [-4, 1, -4, -4, 2, -2, 3, 3, 2, 2, 0, 0], [1, -2, 4, 0, -4, 0, 0, -2, 0, 2, -1, -2], [0, 0, 1, -1, -2, -2, 2, -2, -1, 1, 4, -1], [-4, -2, -2, -2, 2, -2, 1, 1, 0, 0, 0, 0], [0, 0, -3, 3, -3, -3, -1, 1, -1, 1, 6, 0], [-2, -2, 0, 4, 0, 0, 1, 0, -1, 0, -2, 2], [-4, 0, 0, 1, 0, 0, 0, -4, 0, -4, -4, 0], [-1, -1, 0, -3, 0, 1, 1, -2, -1, -2, -1, 1], [6, 0, 4, 0, -4, 0, -3, 0, -3, 0, -6, 0], [0, 0, 1, -1, 2, 2, -2, 2, -1, 1, 4, 3], [0, 0, 0, 0, 2, 2, 0, 0, 2, -2, -6, -2], [0, 0, 3, -3, -1, -1, 3, -3, -1, 1, -2, -4], [1, 0, 0, -2, 0, 4, -1, 3, 1, 3, 1, 0], [-6, 1, 0, 0, 0, 4, 1, -4, -1, -4, -6, -1], [-4, 2, 6, 6, -2, 2, -4, -4, -4, -4, 0, 0], [8, -1, 2, 2, -4, 4, -5, -5, 0, 0, 0, 0], [5, 2, -1, 0, -1, 0, -5, 0, -5, 0, -5, 2], [0, 1, -1, 0, -4, 0, 1, 1, 1, -1, 0, 1], [2, 2, -4, -4, 2, -2, 3, 3, 1, 1, 0, 0], [5, -2, 0, 6, 0, 4, -1, -3, 1, -3, 5, 2], [-4, -2, 6, 6, 0, 0, -4, -4, 0, 0, 0, 0], [4, -3, 5, 5, 1, -1, -4, -4, 0, 0, 0, 0], [4, -3, -2, -2, -2, 2, 1, 1, 2, 2, 0, 0], [3, 3, 2, 0, 2, 0, -2, 2, -2, -2, -3, 3], [5, 2, -7, 0, 3, 0, 1, -2, 1, 2, -5, 2], [0, 0, 2, 2, -2, 2, -3, -3, 0, 0, 0, 0], [0, 0, 1, -1, 3, 3, 3, -3, -1, 1, 10, -2], [8, 2, 0, -4, 0, 0, 3, -2, -3, -2, 8, -2], [-4, -3, 1, 1, 1, -1, 4, 4, 2, 2, 0, 0], [-8, -3, 0, -3, 0, 4, 1, 1, -1, 1, -8, 3], [-6, 0, 4, 0, 0, 0, 2, 3, 2, -3, 6, 0], [-6, 2, -3, 0, 2, 0, -2, -4, -2, 4, 6, 2], [0, 0, -5, 5, 3, 3, 3, -3, 1, -1, -2, 2], [-5, 1, 0, -6, 0, -2, -2, 0, 2, 0, -5, -1], [-2, 1, 0, 4, 0, 0, -4, 1, 4, 1, -2, -1], [0, 1, -2, -2, 4, -4, -3, -3, 0, 0, 0, 0], [5, -2, 0, 0, -8, 0, -2, -2, -2, 2, -5, -2], [-2, -3, -3, 0, 2, 0, 7, 1, 7, -1, 2, -3], [2, 2, 0, 0, 2, -2, -5, -5, -3, -3, 0, 0], [-3, 2, 0, -4, 0, -4, -2, 2, 2, 2, -3, -2], [-16, -4, 4, 4, 2, -2, 4, 4, 4, 4, 0, 0], [-5, -2, 0, -3, 0, -5, 2, 3, -2, 3, -5, 2], [8, 2, 4, 0, 0, 0, -1, 2, -1, -2, -8, 2], [0, 0, -1, 1, 4, 4, 4, -4, -1, 1, 0, 1], [9, -2, 3, 0, 3, 0, 3, -4, 3, 4, -9, -2], [8, 0, 2, 2, -2, 2, -3, -3, -4, -4, 0, 0], [8, -1, 0, 4, 0, 4, -1, -2, 1, -2, 8, 1], [-6, 4, 2, 2, 0, 0, -2, -2, -2, -2, 0, 0], [0, 0, -3, 3, -3, -3, -2, 2, 2, -2, -2, 5], [0, 0, 7, -7, -4, -4, 0, 0, -1, 1, 0, -3], [0, 0, -2, 2, -2, -2, -5, 5, 0, 0, 8, -2], [0, 0, 1, -1, -5, -5, -1, 1, 3, -3, -6, -2], [-5, -1, 0, 0, 0, 2, -1, -1, 1, -1, -5, 1], [0, -4, -4, -4, 2, -2, 4, 4, 0, 0, 0, 0], [-4, 1, 0, 5, 0, 4, 1, -3, -1, -3, -4, -1], [0, 0, 1, -1, 3, 3, 4, -4, 6, -6, 2, -5], [3, -1, 10, 0, -2, 0, -6, 0, -6, 0, -3, -1], [0, 0, -4, 4, 2, 2, 0, 0, -2, 2, -6, 0], [1, -3, 1, 0, 1, 0, 2, 3, 2, -3, -1, -3], [4, 1, 3, 0, 4, 0, -1, 3, -1, -3, -4, 1], [-8, 2, -6, -6, 2, -2, 2, 2, 4, 4, 0, 0], [-1, 2, 0, -1, 0, 11, 0, -3, 0, -3, -1, -2], [0, 5, 2, 2, -4, 4, -3, -3, -4, -4, 0, 0], [0, 0, 3, -3, -4, -4, 4, -4, -5, 5, 8, 1], [5, -1, -1, 0, -3, 0, 4, 5, 4, -5, -5, -1], [10, 6, -4, -4, 2, -2, 3, 3, -3, -3, 0, 0], [2, 6, 2, 2, 6, -6, -4, -4, 0, 0, 0, 0], [0, 5, 0, -3, 0, -4, 3, -5, -3, -5, 0, -5], [0, 0, 0, 0, -2, -2, 4, -4, -6, 6, 2, 0], [-8, -3, -5, 0, 4, 0, 1, 1, 1, -1, 8, -3], [0, -2, 0, 0, 0, 0, 2, 7, -2, 7, 0, 2], [-4, -2, 0, 1, 0, 8, -2, 2, 2, 2, -4, 2], [0, 0, -5, 5, -1, -1, -2, 2, -2, 2, 10, 3], [8, 3, -8, -8, 4, -4, 1, 1, -4, -4, 0, 0], [-5, 1, -6, 0, 2, 0, 0, 0, 0, 0, 5, 1], [10, -1, 5, 0, -10, 0, -5, 1, -5, -1, -10, -1], [5, -4, 0, -8, 0, -4, 0, 8, 0, 8, 5, 4], [-4, -2, 2, 2, 2, -2, -4, -4, 4, 4, 0, 0], [4, 1, 9, 9, -3, 3, -8, -8, 2, 2, 0, 0], [-1, 1, 0, 7, 0, -3, -1, 0, 1, 0, -1, -1], [-10, 1, 0, 0, 0, 0, 1, 4, 1, -4, 10, 1], [5, 2, 2, 0, 8, 0, 1, -1, 1, 1, -5, 2], [0, 0, 3, -3, 2, 2, -2, 2, -5, 5, -12, -3], [0, 0, 3, -3, -5, -5, -1, 1, 1, -1, -2, -10], [-4, -7, 5, 5, -3, 3, 0, 0, 4, 4, 0, 0], [-2, 1, 0, -1, 0, -10, 3, 1, -3, 1, -2, -1], [11, -2, 0, -1, 0, 3, -4, 5, 4, 5, 11, 2], [6, 1, -4, 0, 4, 0, 4, -3, 4, 3, -6, 1], [0, 0, -5, 5, 4, 4, 4, -4, 3, -3, 16, -3], [0, 0, 0, 0, 4, 4, 1, -1, -8, 8, -16, 4], [3, -3, 0, 4, 0, 2, 1, -3, -1, -3, 3, 3], [-14, -1, 0, 4, 0, 8, -1, 2, 1, 2, -14, 1], [-6, 0, 0, -9, 0, -6, 6, 8, -6, 8, -6, 0], [-8, -5, 2, 2, 4, -4, 7, 7, 4, 4, 0, 0], [0, 0, -4, 4, 8, 8, -3, 3, 8, -8, 8, 2], [10, -4, 0, 0, -6, 6, -5, -5, 3, 3, 0, 0], [-4, 2, -2, -2, 0, 0, 4, 4, 4, 4, 0, 0], [-1, -4, 0, 3, 0, 3, 2, 7, -2, 7, -1, 4], [2, -2, 4, 0, -4, 0, -5, -6, -5, 6, -2, -2], [0, 0, 1, -1, -10, -10, -2, 2, -1, 1, -12, -1], [5, 0, 5, 0, -5, 0, -3, -2, -3, 2, -5, 0], [0, 0, -1, 1, 3, 3, -5, 5, -5, 5, 6, 0], [16, 4, 0, 4, 0, -8, -3, 0, 3, 0, 16, -4], [2, 0, -2, -2, 8, -8, 2, 2, -2, -2, 0, 0], [0, 0, 0, 0, 2, 2, 0, 0, -10, 10, -8, 2], [-2, -1, 12, 0, 4, 0, -9, -6, -9, 6, 2, -1], [6, -2, 9, 0, 6, 0, -4, -2, -4, 2, -6, -2], [2, -1, 0, 12, 0, 4, 2, -9, -2, -9, 2, 1], [0, 0, -4, 4, -4, -4, -6, 6, -2, 2, -12, 8], [-4, -7, 2, 2, -2, 2, 1, 1, -2, -2, 0, 0], [0, 0, -4, 4, 0, 0, -5, 5, 0, 0, -16, 6], [-7, 5, 3, 0, 1, 0, -6, -1, -6, 1, 7, 5], [6, 5, 1, 0, 10, 0, -1, 1, -1, -1, -6, 5], [0, -4, -2, -2, 10, -10, 5, 5, 12, 12, 0, 0], [8, 6, -2, -2, -4, 4, -2, -2, -8, -8, 0, 0], [6, -1, -12, 0, 0, 0, 7, -2, 7, 2, -6, -1], [1, 4, 8, 0, 0, 0, -6, 0, -6, 0, -1, 4], [0, 0, -5, 5, 10, 10, -2, 2, 3, -3, 12, 9], [1, -1, -3, 0, 5, 0, -2, 3, -2, -3, -1, -1], [18, 0, 0, 16, 0, 4, -9, -6, 9, -6, 18, 0], [8, 7, -1, -1, -5, 5, 0, 0, -2, -2, 0, 0], [16, 2, 0, -7, 0, 0, 2, -6, -2, -6, 16, -2], [11, 6, 0, -9, 0, -1, 0, -7, 0, -7, 11, -6], [-4, -4, 3, 0, 0, 0, 8, -4, 8, 4, 4, -4], [0, 0, -13, 13, 5, 5, 3, -3, 5, -5, -6, 8], [12, -6, 2, 2, 2, -2, 8, 8, -4, -4, 0, 0], [0, 0, -9, 9, -7, -7, 0, 0, -2, 2, -2, 1], [-3, 0, -6, 0, 12, 0, -1, 5, -1, -5, 3, 0], [0, 0, 4, -4, 6, 6, 0, 0, -2, 2, 6, 2], [22, 2, -3, 0, -6, 0, -6, -4, -6, 4, -22, 2], [-21, -1, 0, 2, 0, -6, 4, 6, -4, 6, -21, 1], [-13, 5, 0, -11, 0, -3, 5, 2, -5, 2, -13, -5], [-4, 3, -4, -4, -10, 10, 1, 1, -6, -6, 0, 0], [7, -3, -4, 0, 2, 0, 5, 9, 5, -9, -7, -3], [5, -4, 9, 0, -1, 0, 3, -8, 3, 8, -5, -4], [0, 0, 5, -5, 9, 9, 11, -11, 5, -5, 6, -10], [-12, -1, 0, 8, 0, 4, -4, 1, 4, 1, -12, 1], [0, 0, 4, -4, -2, -2, -4, 4, 2, -2, 8, -6], [10, 1, 4, 0, 12, 0, 1, 0, 1, 0, -10, 1], [0, 0, 3, -3, -8, -8, -4, 4, -1, 1, -16, 1], [0, 0, 4, -4, -6, -6, -4, 4, 2, -2, 14, -6], [0, 0, 9, -9, -3, -3, -1, 1, -13, 13, -34, 0], [-5, -1, 0, 12, 0, -6, 3, 5, -3, 5, -5, 1], [16, -10, 10, 10, -6, 6, -6, -6, 4, 4, 0, 0], [0, 0, -4, 4, 2, 2, 2, -2, -4, 4, -12, -2], [12, -3, 14, 14, 2, -2, -3, -3, 2, 2, 0, 0], [0, 0, 8, -8, -6, -6, -8, 8, -2, 2, 10, -2], [-15, -3, 3, 0, 1, 0, 0, 1, 0, -1, 15, -3], [8, 0, -4, -4, 0, 0, 1, 1, 0, 0, 0, 0], [-1, 2, 0, -3, 0, -17, 6, 3, -6, 3, -1, -2], [-16, -1, -2, -2, 0, 0, -5, -5, 4, 4, 0, 0], [-12, -1, 8, 0, -12, 0, 2, -9, 2, 9, 12, -1], [-9, -3, 4, 0, -2, 0, 11, 1, 11, -1, 9, -3], [-11, 3, -7, 0, 1, 0, 0, -3, 0, 3, 11, 3], [2, 0, 4, 4, 2, -2, -9, -9, 3, 3, 0, 0], [-8, -2, 0, 16, 0, 12, 2, -5, -2, -5, -8, 2], [10, 1, 0, -13, 0, -2, -1, 1, 1, 1, 10, -1], [0, 0, -5, 5, -1, -1, -2, 2, 2, -2, 2, -5], [0, 0, -1, 1, 3, 3, -5, 5, -1, 1, -10, 12], [-5, -3, 0, 2, 0, 2, 8, 4, -8, 4, -5, 3], [-8, -1, 0, -11, 0, -16, 1, 9, -1, 9, -8, 1], [4, -3, -8, -8, -6, 6, -1, -1, 6, 6, 0, 0], [0, 0, 4, -4, 0, 0, -1, 1, -8, 8, -8, 4], [10, -2, -15, 0, 6, 0, 6, 4, 6, -4, -10, -2], [1, 0, 0, -14, 0, -20, -3, 11, 3, 11, 1, 0], [16, -1, -1, -1, 7, -7, 4, 4, 6, 6, 0, 0], [2, 7, 0, 0, 8, 0, 2, -1, 2, 1, -2, 7], [-3, -4, 12, 0, 4, 0, 4, -4, 4, 4, 3, -4], [1, -3, 3, 0, -19, 0, 2, -1, 2, 1, -1, -3], [-22, -4, -4, -4, -2, 2, 3, 3, -1, -1, 0, 0], [0, 0, -3, 3, 1, 1, -5, 5, -11, 11, -18, 6], [8, -1, 0, -12, 0, -8, 0, 1, 0, 1, 8, 1], [0, 0, -10, 10, 8, 8, 6, -6, 2, -2, 20, 4], [0, 0, -11, 11, 0, 0, 4, -4, 3, -3, 16, 3], [-2, -8, 0, -8, 0, -4, 7, 14, -7, 14, -2, 8], [28, 8, 0, 0, 10, -10, -6, -6, 4, 4, 0, 0], [2, -3, 0, 11, 0, 2, -3, -9, 3, -9, 2, 3], [0, 0, 12, -12, 6, 6, 2, -2, -4, 4, -4, -2], [-8, 5, -6, -6, 12, -12, 1, 1, 4, 4, 0, 0], [5, 6, -2, 0, 12, 0, 5, 1, 5, -1, -5, 6], [0, 0, 6, -6, -6, -6, 3, -3, 0, 0, -24, -14], [-15, -4, 1, 0, 3, 0, -3, -6, -3, 6, 15, -4], [22, 3, -3, 0, 14, 0, 1, -5, 1, 5, -22, 3], [12, 2, -2, -2, -6, 6, -12, -12, 0, 0, 0, 0], [16, -1, 11, 11, -9, 9, -4, -4, -8, -8, 0, 0], [-9, 4, 0, 1, 0, 7, -4, -7, 4, -7, -9, -4], [-2, 2, 0, 0, -2, 2, -2, -2, -6, -6, 0, 0], [27, -3, 0, -3, 0, 9, 1, 6, -1, 6, 27, 3], [-24, 5, -12, 0, 0, 0, 3, -2, 3, 2, 24, 5], [0, 0, 4, -4, 12, 12, 1, -1, 0, 0, 16, 0], [-16, 2, 2, 2, 4, -4, 2, 2, 0, 0, 0, 0], [0, 0, 2, -2, -2, -2, 10, -10, 0, 0, 4, -6], [-3, 2, 23, 0, -5, 0, -9, -4, -9, 4, 3, 2], [14, 0, 9, 0, -14, 0, -2, -8, -2, 8, -14, 0], [8, -4, 12, 12, -8, 8, -7, -7, -12, -12, 0, 0], [11, 7, 0, 0, 0, -6, 3, -3, -3, -3, 11, -7], [-4, -2, -6, -6, 4, -4, 12, 12, 4, 4, 0, 0], [7, -3, 0, 17, 0, -3, -9, -4, 9, -4, 7, 3], [-8, -3, 6, 6, 4, -4, -7, -7, 0, 0, 0, 0], [-4, -7, 0, 0, 12, 0, 7, -2, 7, 2, 4, -7], [0, 0, 13, -13, 2, 2, -2, 2, 3, -3, 4, -5], [0, 0, 1, -1, 5, 5, -1, 1, 9, -9, 30, -2], [12, 1, 9, 9, -7, 7, -4, -4, 2, 2, 0, 0], [22, 6, 0, 3, 0, 2, -2, -4, 2, -4, 22, -6], [23, 0, 0, -7, 0, 3, 2, -5, -2, -5, 23, 0], [0, 0, -3, 3, -5, -5, 0, 0, 6, -6, -22, -1], [-4, 7, 0, 0, 0, 0, 2, -5, 2, 5, 4, 7], [0, 0, 15, -15, -10, -10, -2, 2, -13, 13, -12, 5], [0, 0, -3, 3, 1, 1, -1, 1, 3, -3, -26, -8], [2, -3, 0, -5, 0, 2, -5, 5, 5, 5, 2, 3], [4, -11, -2, -2, -2, 2, 1, 1, 6, 6, 0, 0], [-3, -4, 9, 0, 19, 0, 11, 0, 11, 0, 3, -4], [2, 14, -8, -8, 6, -6, -5, -5, -3, -3, 0, 0], [9, 2, 0, -4, 0, -4, -2, 6, 2, 6, 9, -2], [-24, 0, 8, 8, -6, 6, 4, 4, 8, 8, 0, 0], [-5, 2, 0, -11, 0, -1, 6, -1, -6, -1, -5, -2], [-20, -1, -6, -6, 2, -2, 7, 7, 6, 6, 0, 0], [0, 0, -1, 1, 2, 2, -10, 10, -1, 1, -20, 9], [-7, -3, 9, 0, -3, 0, 4, 9, 4, -9, 7, -3], [0, 3, 0, 0, 0, 4, -9, -14, 9, -14, 0, -3], [6, 4, 2, 2, -8, 8, -2, -2, -6, -6, 0, 0], [11, 2, 0, 11, 0, -5, 2, -1, -2, -1, 11, -2], [0, 0, -5, 5, 1, 1, -4, 4, -4, 4, 14, -3], [10, -2, 4, 0, 0, 0, 3, 6, 3, -6, -10, -2], [0, 0, -7, 7, 0, 0, 4, -4, -9, 9, 0, -1], [0, 0, -12, 12, 10, 10, 4, -4, 10, -10, -2, 0], [14, -1, -7, 0, 6, 0, 1, 11, 1, -11, -14, -1], [-16, -6, 0, 0, 0, -8, -9, 6, 9, 6, -16, 6], [-13, -1, 14, 0, -14, 0, 2, -4, 2, 4, 13, -1], [-19, 3, -7, 0, 5, 0, 4, -7, 4, 7, 19, 3], [16, -7, 15, 0, -8, 0, -5, -5, -5, 5, -16, -7], [2, 14, -4, -4, 10, -10, -1, -1, -3, -3, 0, 0], [4, -4, 16, 0, 16, 0, 0, 1, 0, -1, -4, -4], [-15, -6, -10, 0, 12, 0, 13, -3, 13, 3, 15, -6], [0, 0, 7, -7, 12, 12, 0, 0, -1, 1, 8, 5], [-38, 2, -4, -4, 2, -2, 7, 7, -3, -3, 0, 0], [6, -6, 6, 6, -6, 6, 4, 4, -4, -4, 0, 0], [-8, -5, 0, -11, 0, -12, 9, 17, -9, 17, -8, 5], [2, 2, -12, 0, -8, 0, 0, 1, 0, -1, -2, 2], [0, 0, -12, 12, 2, 2, 4, -4, -2, 2, 14, -6], [-8, 3, 7, 0, 0, 0, -3, 5, -3, -5, 8, 3], [17, 2, 0, 6, 0, -16, -9, -3, 9, -3, 17, -2], [4, -6, 6, 6, -8, 8, 0, 0, -4, -4, 0, 0], [3, -11, 14, 0, 6, 0, 12, -8, 12, 8, -3, -11], [-15, 4, -11, 0, 3, 0, 11, 4, 11, -4, 15, 4], [8, -4, -17, 0, 4, 0, 16, 0, 16, 0, -8, -4], [-6, 4, 8, 8, -2, 2, -1, -1, -5, -5, 0, 0], [-4, -7, 1, 1, -7, 7, 12, 12, -2, -2, 0, 0], [-20, -1, -2, -2, -6, 6, -1, -1, -10, -10, 0, 0], [1, -3, -31, 0, 13, 0, 14, 7, 14, -7, -1, -3], [0, 0, 3, -3, -11, -11, -1, 1, -5, 5, -6, 10], [7, -1, 0, -7, 0, -23, -3, 10, 3, 10, 7, 1], [0, 0, -4, 4, 2, 2, 12, -12, 14, -14, 2, -10], [-26, -3, -27, 0, 10, 0, 17, 19, 17, -19, 26, -3], [0, 0, -3, 3, -3, -3, -13, 13, -3, 3, -2, -6], [33, 0, 0, 2, 0, 12, -1, -13, 1, -13, 33, 0], [10, 3, 0, -16, 0, 16, -5, -6, 5, -6, 10, -3], [36, 4, 12, 12, 0, 0, -6, -6, 4, 4, 0, 0], [0, 0, 1, -1, -7, -7, -10, 10, -4, 4, -10, 1], [0, 0, -2, 2, -6, -6, -3, 3, 0, 0, 8, 2], [22, -8, 1, 0, -2, 0, 4, 6, 4, -6, -22, -8], [-13, -9, 0, 0, 0, 14, 3, 9, -3, 9, -13, 9], [-24, -7, 6, 6, -4, 4, -3, -3, 0, 0, 0, 0], [-10, 2, -24, 0, 0, 0, 3, 6, 3, -6, 10, 2], [4, -10, 2, 2, 2, -2, 9, 9, 8, 8, 0, 0], [0, 0, 9, -9, -3, -3, -1, 1, 1, -1, 6, 6], [-6, -8, 12, 0, 12, 0, 14, 3, 14, -3, 6, -8], [0, 0, -7, 7, -4, -4, -8, 8, -9, 9, -16, 3], [-12, 3, -1, 0, 16, 0, -1, 3, -1, -3, 12, 3], [-7, -10, 0, -8, 0, 0, 6, 4, -6, 4, -7, 10], [2, 2, 0, 4, 0, 8, 5, -8, -5, -8, 2, -2], [-8, -2, 2, 2, -6, 6, -2, -2, 8, 8, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1280_o_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1280_2_o_u();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1280_2_o_u(:prec:=12)
    chi := MakeCharacter_1280_o();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1280_2_o_u();".
// 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_1280_2_o_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1280_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![64, -128, 128, 384, 592, -32, 32, 80, 88, 8, 0, 0, 1]>,<7,R![64, -640, 3200, -5632, 3472, 6944, 4768, 1232, 184, 40, 32, 8, 1]>,<13,R![937024, -1874048, 1874048, -875072, 213808, -13376, 7744, -4000, 1036, -8, 8, -4, 1]>],Snew);
    return Vf;
end function;