
// Make newform 1280.3.g.e in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1280_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1280_g_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_3_g_e();".
// 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_3_g_e();".
// 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, -1, 0, 1, 0, -1, 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, 2, 0, -1, 0, 2], [0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 4, 0, 4, 0], [0, 4, 0, -2, 0, 2, 0, -2], [-4, 0, 8, 0, -6, 0, 2, 0], [2, 0, 0, 0, 4, 0, -4, 0], [0, 0, 0, 8, 0, 0, 0, -8]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 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_g();"
function MakeCharacter_1280_g()
    N := 1280;
    order := 2;
    char_gens := [511, 261, 257];
    v := [1, 1, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1280_g_Hecke();"
function MakeCharacter_1280_g_Hecke(Kf)
    N := 1280;
    order := 2;
    char_gens := [511, 261, 257];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+1)]];
    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 := 3;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, -2, 0, 0, 1], [0, -2, -2, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -2, 0, -1, 0, 0], [0, -4, -1, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, -2, 0, 0], [0, 10, -2, 0, 0, 0, 0, 0], [28, 0, 0, 0, 0, 0, 3, 0], [0, 0, 0, 0, -3, 0, 0, 1], [0, 0, 0, 5, 0, -3, 0, 0], [0, 10, 22, 0, 0, 0, 0, 0], [0, 0, 0, 0, 12, 0, 0, 3], [0, 26, 16, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, -5], [0, 0, 0, -4, 0, -10, 0, 0], [-66, 0, 0, 0, 0, 0, -16, 0], [0, 0, 0, 14, 0, 8, 0, 0], [0, 0, 0, 0, -13, 0, 0, 3], [22, 0, 0, 0, 0, 0, -20, 0], [-66, 0, 0, 0, 0, 0, 6, 0], [0, 8, -41, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 11, 0, 0], [0, 0, 0, 0, -17, 0, 0, -6], [0, 74, -16, 0, 0, 0, 0, 0], [126, 0, 0, 0, 0, 0, -10, 0], [0, 0, 0, -2, 0, 19, 0, 0], [0, 0, 0, 0, -42, 0, 0, -7], [26, 0, 0, 0, 0, 0, 6, 0], [0, 0, 0, 0, 4, 0, 0, -15], [0, 38, -24, 0, 0, 0, 0, 0], [0, 0, 0, -22, 0, 18, 0, 0], [0, 54, 42, 0, 0, 0, 0, 0], [0, 0, 0, 0, -45, 0, 0, -14], [0, 0, 0, 12, 0, 9, 0, 0], [0, -10, -102, 0, 0, 0, 0, 0], [0, 0, 0, 0, 76, 0, 0, -3], [0, 16, -61, 0, 0, 0, 0, 0], [0, 0, 0, 40, 0, 22, 0, 0], [146, 0, 0, 0, 0, 0, 8, 0], [0, 70, 8, 0, 0, 0, 0, 0], [0, 0, 0, 24, 0, -16, 0, 0], [0, 0, 0, 0, -42, 0, 0, 7], [0, 0, 0, -24, 0, -35, 0, 0], [0, 0, 0, 0, 1, 0, 0, 4], [0, -20, -59, 0, 0, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 70, 0], [0, 0, 0, 8, 0, -20, 0, 0], [-68, 0, 0, 0, 0, 0, -15, 0], [0, 0, 0, 0, -26, 0, 0, -13], [-266, 0, 0, 0, 0, 0, 2, 0], [0, 0, 0, -25, 0, 29, 0, 0], [0, -70, -96, 0, 0, 0, 0, 0], [0, 0, 0, -40, 0, 6, 0, 0], [0, -90, 128, 0, 0, 0, 0, 0], [28, 0, 0, 0, 0, 0, -5, 0], [0, 0, 0, 0, -15, 0, 0, -12], [0, -46, -38, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, 17], [0, 0, 0, 12, 0, 34, 0, 0], [174, 0, 0, 0, 0, 0, -88, 0], [0, -46, 132, 0, 0, 0, 0, 0], [0, 0, 0, 0, -70, 0, 0, 11], [-126, 0, 0, 0, 0, 0, 32, 0], [0, 0, 0, 0, 91, 0, 0, -19], [0, 108, -121, 0, 0, 0, 0, 0], [-14, 0, 0, 0, 0, 0, 72, 0], [0, 0, 0, -4, 0, 8, 0, 0], [0, 0, 0, 70, 0, -17, 0, 0], [0, -154, -128, 0, 0, 0, 0, 0], [0, 0, 0, 0, -32, 0, 0, 15], [0, 0, 0, 50, 0, 57, 0, 0], [0, -154, -24, 0, 0, 0, 0, 0], [0, 94, 62, 0, 0, 0, 0, 0], [282, 0, 0, 0, 0, 0, 36, 0], [272, 0, 0, 0, 0, 0, -75, 0], [0, 0, 0, 0, 28, 0, 0, 35], [0, -114, 164, 0, 0, 0, 0, 0], [0, 0, 0, -52, 0, 22, 0, 0], [-234, 0, 0, 0, 0, 0, -30, 0], [0, 0, 0, 56, 0, 64, 0, 0], [0, 0, 0, 0, -19, 0, 0, -22], [528, 0, 0, 0, 0, 0, 51, 0], [-554, 0, 0, 0, 0, 0, 60, 0], [0, -80, -139, 0, 0, 0, 0, 0], [0, 0, 0, 83, 0, -1, 0, 0], [0, 0, 0, 0, 13, 0, 0, -58], [0, 0, 0, -78, 0, 16, 0, 0], [0, 0, 0, 73, 0, 79, 0, 0], [0, 0, 0, 0, 38, 0, 0, -13], [0, 0, 0, 0, -24, 0, 0, -57], [0, 0, 0, 90, 0, 107, 0, 0], [0, -140, -301, 0, 0, 0, 0, 0], [-202, 0, 0, 0, 0, 0, 180, 0], [0, 0, 0, 0, 161, 0, 0, -26], [0, 80, -219, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 10], [0, -86, 352, 0, 0, 0, 0, 0], [0, 0, 0, 0, -89, 0, 0, -46], [112, 0, 0, 0, 0, 0, -27, 0], [0, 0, 0, 0, 122, 0, 0, 5], [-186, 0, 0, 0, 0, 0, -118, 0], [0, 0, 0, 0, -153, 0, 0, 17], [-94, 0, 0, 0, 0, 0, -48, 0], [0, 0, 0, -48, 0, -108, 0, 0], [-172, 0, 0, 0, 0, 0, -41, 0], [0, 0, 0, -91, 0, 69, 0, 0], [0, 70, -328, 0, 0, 0, 0, 0], [-54, 0, 0, 0, 0, 0, 110, 0], [0, 0, 0, 0, -200, 0, 0, -1], [0, 0, 0, -112, 0, -26, 0, 0], [212, 0, 0, 0, 0, 0, 141, 0], [0, 0, 0, 0, 271, 0, 0, 13], [0, 0, 0, -49, 0, 19, 0, 0], [0, 42, -232, 0, 0, 0, 0, 0], [0, 0, 0, 0, -92, 0, 0, 7], [0, 342, 24, 0, 0, 0, 0, 0], [-114, 0, 0, 0, 0, 0, -130, 0], [0, -306, 68, 0, 0, 0, 0, 0], [0, 0, 0, 0, 61, 0, 0, 11], [0, 0, 0, 0, -86, 0, 0, -1], [0, -62, -284, 0, 0, 0, 0, 0], [0, 316, -259, 0, 0, 0, 0, 0], [0, 0, 0, -60, 0, -44, 0, 0], [0, 0, 0, -11, 0, -77, 0, 0], [0, 446, 118, 0, 0, 0, 0, 0], [0, 0, 0, 0, 36, 0, 0, 5], [0, 0, 0, 73, 0, -67, 0, 0], [0, 0, 0, -112, 0, 58, 0, 0], [0, 134, 288, 0, 0, 0, 0, 0], [-122, 0, 0, 0, 0, 0, -172, 0], [-102, 0, 0, 0, 0, 0, -164, 0], [0, 142, 52, 0, 0, 0, 0, 0], [0, 0, 0, 0, 169, 0, 0, -29], [0, 210, 372, 0, 0, 0, 0, 0], [-218, 0, 0, 0, 0, 0, -100, 0], [0, 0, 0, 0, -122, 0, 0, 11], [0, -402, -116, 0, 0, 0, 0, 0], [0, 0, 0, -24, 0, 67, 0, 0], [0, 0, 0, 0, 267, 0, 0, 55], [0, -70, -176, 0, 0, 0, 0, 0], [0, 0, 0, -162, 0, -104, 0, 0], [0, 590, 132, 0, 0, 0, 0, 0], [426, 0, 0, 0, 0, 0, -130, 0], [0, 0, 0, 0, 52, 0, 0, 3], [0, 0, 0, 113, 0, 79, 0, 0], [0, 142, 142, 0, 0, 0, 0, 0], [-348, 0, 0, 0, 0, 0, -131, 0], [0, 0, 0, 0, 103, 0, 0, 16], [0, 0, 0, 95, 0, -9, 0, 0], [0, 0, 0, 0, 27, 0, 0, 26], [0, 0, 0, -194, 0, 54, 0, 0], [0, 0, 0, -80, 0, 44, 0, 0], [968, 0, 0, 0, 0, 0, 127, 0], [-294, 0, 0, 0, 0, 0, -10, 0], [0, -160, 541, 0, 0, 0, 0, 0], [0, 0, 0, 0, -259, 0, 0, 35], [-866, 0, 0, 0, 0, 0, 96, 0], [0, 0, 0, -108, 0, -91, 0, 0], [0, 0, 0, 0, -246, 0, 0, -99], [-1226, 0, 0, 0, 0, 0, 34, 0], [0, 0, 0, 63, 0, 61, 0, 0], [0, 0, 0, -54, 0, -122, 0, 0], [0, 346, 158, 0, 0, 0, 0, 0], [-502, 0, 0, 0, 0, 0, -252, 0], [0, -218, 712, 0, 0, 0, 0, 0], [0, 0, 0, 0, -96, 0, 0, 11], [0, -440, -119, 0, 0, 0, 0, 0], [0, 0, 0, 214, 0, 270, 0, 0], [-426, 0, 0, 0, 0, 0, -164, 0], [0, 0, 0, 182, 0, 204, 0, 0], [-578, 0, 0, 0, 0, 0, 264, 0], [0, 0, 0, 0, -278, 0, 0, 31], [0, -200, 79, 0, 0, 0, 0, 0], [0, 0, 0, 14, 0, 103, 0, 0], [0, 106, 384, 0, 0, 0, 0, 0], [0, 0, 0, -148, 0, 11, 0, 0], [0, 0, 0, 0, -66, 0, 0, -101], [0, 754, -78, 0, 0, 0, 0, 0], [1166, 0, 0, 0, 0, 0, 24, 0], [0, 0, 0, -23, 0, 55, 0, 0], [0, -12, 321, 0, 0, 0, 0, 0], [0, 458, -288, 0, 0, 0, 0, 0], [0, 0, 0, 0, -433, 0, 0, -15], [72, 0, 0, 0, 0, 0, -47, 0], [0, 0, 0, 224, 0, 234, 0, 0], [326, 0, 0, 0, 0, 0, -118, 0], [0, 0, 0, 0, 37, 0, 0, -130], [0, 0, 0, 0, -62, 0, 0, -35], [0, 410, -264, 0, 0, 0, 0, 0], [0, 0, 0, 0, 329, 0, 0, -9], [-1086, 0, 0, 0, 0, 0, 42, 0], [-668, 0, 0, 0, 0, 0, 205, 0], [0, -142, 388, 0, 0, 0, 0, 0], [-1006, 0, 0, 0, 0, 0, 120, 0], [0, 0, 0, -240, 0, -49, 0, 0], [0, 180, -541, 0, 0, 0, 0, 0], [0, 0, 0, -162, 0, -78, 0, 0], [0, 154, -2, 0, 0, 0, 0, 0], [-502, 0, 0, 0, 0, 0, -276, 0], [0, 0, 0, 0, 396, 0, 0, 45], [0, -890, 162, 0, 0, 0, 0, 0], [0, 0, 0, 6, 0, -60, 0, 0], [0, 0, 0, 0, 455, 0, 0, -13], [-418, 0, 0, 0, 0, 0, -248, 0], [0, 0, 0, 0, 34, 0, 0, -85], [674, 0, 0, 0, 0, 0, 240, 0], [0, -50, -316, 0, 0, 0, 0, 0], [0, 0, 0, -57, 0, -115, 0, 0], [0, 0, 0, 0, -265, 0, 0, 65], [0, 0, 0, 196, 0, 60, 0, 0], [-412, 0, 0, 0, 0, 0, -185, 0], [0, 0, 0, -76, 0, -265, 0, 0], [1042, 0, 0, 0, 0, 0, 272, 0], [0, 0, 0, 232, 0, 209, 0, 0], [0, 150, 738, 0, 0, 0, 0, 0], [0, -554, 184, 0, 0, 0, 0, 0], [0, 0, 0, -126, 0, -256, 0, 0], [498, 0, 0, 0, 0, 0, 224, 0], [0, 0, 0, -188, 0, 81, 0, 0], [0, 0, 0, 0, 273, 0, 0, 35], [0, -186, 336, 0, 0, 0, 0, 0], [-546, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 64, 0, -24, 0, 0], [0, 0, 0, -84, 0, 69, 0, 0], [0, 0, 0, 0, -374, 0, 0, 43], [0, 222, -642, 0, 0, 0, 0, 0], [0, 0, 0, 0, 688, 0, 0, -43], [0, 0, 0, -222, 0, -206, 0, 0], [108, 0, 0, 0, 0, 0, -333, 0], [0, 0, 0, 0, -319, 0, 0, 101], [0, 0, 0, 46, 0, 275, 0, 0], [-1432, 0, 0, 0, 0, 0, 175, 0], [0, 26, 622, 0, 0, 0, 0, 0], [0, 0, 0, 0, 824, 0, 0, -33], [0, 0, 0, -294, 0, -322, 0, 0], [0, 0, 0, 0, -629, 0, 0, -3], [0, 0, 0, 0, 22, 0, 0, 67], [0, 0, 0, 103, 0, 273, 0, 0], [0, 212, 419, 0, 0, 0, 0, 0], [-734, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 20, 0, -128, 0, 0], [0, 0, 0, 346, 0, 339, 0, 0], [0, 0, 0, 0, -222, 0, 0, -85], [0, 0, 0, 0, 348, 0, 0, 55], [0, 0, 0, -271, 0, -229, 0, 0], [0, -82, -1098, 0, 0, 0, 0, 0], [762, 0, 0, 0, 0, 0, -236, 0], [0, 0, 0, -107, 0, -217, 0, 0], [-1008, 0, 0, 0, 0, 0, 77, 0], [0, 174, -242, 0, 0, 0, 0, 0], [0, 0, 0, 0, -712, 0, 0, -15], [0, 96, 539, 0, 0, 0, 0, 0], [0, 0, 0, 0, 547, 0, 0, -5], [0, -10, -872, 0, 0, 0, 0, 0], [126, 0, 0, 0, 0, 0, -48, 0], [0, 0, 0, 269, 0, 135, 0, 0], [0, 0, 0, 0, 169, 0, 0, 90], [0, 380, -99, 0, 0, 0, 0, 0], [0, 194, 1428, 0, 0, 0, 0, 0], [354, 0, 0, 0, 0, 0, -304, 0], [0, 0, 0, 0, 440, 0, 0, -25], [0, -1316, 159, 0, 0, 0, 0, 0], [-922, 0, 0, 0, 0, 0, 252, 0], [0, 0, 0, 0, 61, 0, 0, -6], [0, 770, -238, 0, 0, 0, 0, 0], [0, 456, 1001, 0, 0, 0, 0, 0], [0, 0, 0, 0, -483, 0, 0, -122], [1314, 0, 0, 0, 0, 0, 330, 0], [0, 0, 0, 432, 0, 0, 0, 0], [1374, 0, 0, 0, 0, 0, 150, 0], [0, 0, 0, 25, 0, -67, 0, 0], [0, 0, 0, 0, -37, 0, 0, -110], [0, -60, 779, 0, 0, 0, 0, 0], [-1372, 0, 0, 0, 0, 0, -129, 0], [0, 0, 0, 0, 62, 0, 0, 79], [0, 0, 0, 247, 0, 211, 0, 0], [0, 0, 0, -218, 0, -242, 0, 0], [0, 0, 0, -296, 0, -3, 0, 0], [0, 64, 1419, 0, 0, 0, 0, 0], [0, 0, 0, 0, 287, 0, 0, -177], [0, 0, 0, -54, 0, 62, 0, 0], [-1014, 0, 0, 0, 0, 0, 356, 0], [0, 590, -892, 0, 0, 0, 0, 0], [0, 0, 0, 182, 0, 408, 0, 0], [-1032, 0, 0, 0, 0, 0, -337, 0], [0, 962, 156, 0, 0, 0, 0, 0], [0, 0, 0, 0, -95, 0, 0, -255], [-186, 0, 0, 0, 0, 0, 260, 0], [0, 0, 0, 0, -226, 0, 0, 23], [0, 590, -562, 0, 0, 0, 0, 0], [0, 842, -456, 0, 0, 0, 0, 0], [0, 0, 0, 10, 0, 190, 0, 0], [0, 130, 882, 0, 0, 0, 0, 0], [0, 0, 0, 0, -808, 0, 0, 117], [0, 0, 0, 0, 389, 0, 0, -74], [314, 0, 0, 0, 0, 0, -618, 0], [0, -882, -338, 0, 0, 0, 0, 0], [0, 0, 0, -210, 0, -80, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1280_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1280_3_g_e();".
// 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_3_g_e(:prec:=8)
    chi := MakeCharacter_1280_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_3_g_e();".
// 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_3_g_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1280_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![80, 0, -20, 0, 1]>],Snew);
    return Vf;
end function;