
// Make newform 1600.3.h.n in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_h_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_1600_3_h_n();".
// 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_1600_3_h_n();".
// 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, 0, 0, 2, 0, 0], [0, 0, 0, 0, 4, 0, 4, 0], [0, 4, 0, -2, 0, 2, 0, -2], [2, 0, 0, 0, 4, 0, -4, 0], [0, 0, 0, 4, 0, -2, 0, 4], [0, 0, 0, 4, 0, 0, 0, -4], [-8, 0, 16, 0, -8, 0, 8, 0]];
    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_1600_h();"
function MakeCharacter_1600_h()
    N := 1600;
    order := 2;
    char_gens := [1151, 901, 577];
    v := [1, 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_1600_h_Hecke();"
function MakeCharacter_1600_h_Hecke(Kf)
    N := 1600;
    order := 2;
    char_gens := [1151, 901, 577];
    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, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 1, 0], [0, 0, -1, 0, 0, 0, 0, 1], [0, 2, 0, 0, 0, -1, 0, 0], [0, 3, 0, 0, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -3, 0, 0, -1, 0], [-2, 0, 0, 0, 2, 0, 0, 0], [0, 0, -5, 0, 0, 0, 0, -1], [0, 2, 0, 0, 0, 5, 0, 0], [-28, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, 3, 0, 0, -2, 0], [0, 0, 0, 13, 0, 0, -3, 0], [0, 22, 0, 0, 0, -5, 0, 0], [0, 0, -6, 0, 0, 0, 0, -3], [-32, 0, 0, 0, 13, 0, 0, 0], [0, 0, 0, 11, 0, 0, 10, 0], [0, 0, -1, 0, 0, 0, 0, 5], [0, 33, 0, 0, 0, 16, 0, 0], [0, 0, -10, 0, 0, 0, 0, -4], [0, 0, 0, -13, 0, 0, 6, 0], [22, 0, 0, 0, -20, 0, 0, 0], [0, 33, 0, 0, 0, -6, 0, 0], [-82, 0, 0, 0, -4, 0, 0, 0], [0, 0, 0, -23, 0, 0, 11, 0], [0, 0, 0, -17, 0, 0, -12, 0], [-32, 0, 0, 0, -37, 0, 0, 0], [0, 63, 0, 0, 0, -10, 0, 0], [0, 0, 0, -23, 0, 0, 19, 0], [0, 0, 21, 0, 0, 0, 0, 7], [0, 13, 0, 0, 0, 6, 0, 0], [0, 0, -2, 0, 0, 0, 0, 15], [48, 0, 0, 0, 19, 0, 0, 0], [0, 0, 31, 0, 0, 0, 0, -9], [0, 42, 0, 0, 0, -27, 0, 0], [0, 0, 0, -45, 0, 0, -28, 0], [0, 0, 0, -15, 0, 0, -9, 0], [0, 102, 0, 0, 0, -5, 0, 0], [0, 0, 38, 0, 0, 0, 0, -3], [-122, 0, 0, 0, -8, 0, 0, 0], [0, 0, 29, 0, 0, 0, 0, 11], [0, 73, 0, 0, 0, 8, 0, 0], [0, -8, 0, 0, 0, 35, 0, 0], [0, 0, 32, 0, 0, 0, 0, -8], [0, 0, 21, 0, 0, 0, 0, -7], [0, 0, 0, 13, 0, 0, 35, 0], [0, 0, 0, -1, 0, 0, -8, 0], [118, 0, 0, 0, -10, 0, 0, 0], [0, 3, 0, 0, 0, -70, 0, 0], [0, 0, -18, 0, 0, 0, 0, 10], [-68, 0, 0, 0, -15, 0, 0, 0], [0, 0, -13, 0, 0, 0, 0, -13], [0, 133, 0, 0, 0, -2, 0, 0], [0, 0, 0, -79, 0, 0, 29, 0], [-192, 0, 0, 0, 35, 0, 0, 0], [0, 0, -43, 0, 0, 0, 0, 3], [0, -128, 0, 0, 0, -45, 0, 0], [-28, 0, 0, 0, 5, 0, 0, 0], [0, 0, 0, 15, 0, 0, 24, 0], [0, -38, 0, 0, 0, 23, 0, 0], [0, 0, 0, 11, 0, 0, -34, 0], [0, 0, 5, 0, 0, 0, 0, -17], [0, -87, 0, 0, 0, 88, 0, 0], [0, 132, 0, 0, 0, 23, 0, 0], [0, 0, -35, 0, 0, 0, 0, 11], [0, 63, 0, 0, 0, -32, 0, 0], [0, 0, 0, 91, 0, 0, -38, 0], [-242, 0, 0, 0, -54, 0, 0, 0], [0, -7, 0, 0, 0, 72, 0, 0], [0, 0, -8, 0, 0, 0, 0, 4], [0, 0, 0, 157, 0, 0, -17, 0], [0, -128, 0, 0, 0, 77, 0, 0], [0, 0, 16, 0, 0, 0, 0, -15], [0, 0, 0, -43, 0, 0, -57, 0], [48, 0, 0, 0, -77, 0, 0, 0], [0, 62, 0, 0, 0, -47, 0, 0], [282, 0, 0, 0, 36, 0, 0, 0], [272, 0, 0, 0, -75, 0, 0, 0], [0, 0, 14, 0, 0, 0, 0, 35], [328, 0, 0, 0, 57, 0, 0, 0], [0, 0, -63, 0, 0, 0, 0, 11], [0, -117, 0, 0, 0, -30, 0, 0], [0, 0, 24, 0, 0, 0, 0, 32], [0, 0, 0, 19, 0, 0, 44, 0], [-528, 0, 0, 0, -51, 0, 0, 0], [0, -277, 0, 0, 0, 60, 0, 0], [278, 0, 0, 0, -40, 0, 0, 0], [0, 0, 0, -167, 0, 0, 1, 0], [0, 0, 0, -13, 0, 0, 116, 0], [0, 0, 86, 0, 0, 0, 0, -8], [0, 0, 0, -67, 0, 0, -79, 0], [0, 0, 19, 0, 0, 0, 0, -13], [0, 0, -12, 0, 0, 0, 0, -57], [0, 0, 0, 73, 0, 0, 107, 0], [-602, 0, 0, 0, 70, 0, 0, 0], [202, 0, 0, 0, -180, 0, 0, 0], [0, 0, 0, -161, 0, 0, 52, 0], [438, 0, 0, 0, 40, 0, 0, 0], [0, 0, 0, -1, 0, 0, -20, 0], [0, 352, 0, 0, 0, 43, 0, 0], [0, 0, 0, -89, 0, 0, -92, 0], [112, 0, 0, 0, -27, 0, 0, 0], [0, 0, 61, 0, 0, 0, 0, 5], [0, 93, 0, 0, 0, 118, 0, 0], [0, 0, 0, -153, 0, 0, 34, 0], [0, -47, 0, 0, 0, -48, 0, 0], [0, 0, 6, 0, 0, 0, 0, -54], [172, 0, 0, 0, 41, 0, 0, 0], [0, 0, 0, -251, 0, 0, 69, 0], [0, -328, 0, 0, 0, -35, 0, 0], [0, -27, 0, 0, 0, 110, 0, 0], [0, 0, 100, 0, 0, 0, 0, 1], [0, 0, 99, 0, 0, 0, 0, 13], [212, 0, 0, 0, 141, 0, 0, 0], [0, 0, 0, 271, 0, 0, 26, 0], [0, 0, 0, 117, 0, 0, -19, 0], [0, 232, 0, 0, 0, 21, 0, 0], [0, 0, -46, 0, 0, 0, 0, 7], [48, 0, 0, 0, -171, 0, 0, 0], [0, -57, 0, 0, 0, -130, 0, 0], [0, -68, 0, 0, 0, -153, 0, 0], [0, 0, 0, -61, 0, 0, -22, 0], [0, 0, 43, 0, 0, 0, 0, 1], [568, 0, 0, 0, -31, 0, 0, 0], [518, 0, 0, 0, 158, 0, 0, 0], [0, 0, 38, 0, 0, 0, 0, 22], [0, 0, 0, -55, 0, 0, 77, 0], [0, -118, 0, 0, 0, 223, 0, 0], [0, 0, 18, 0, 0, 0, 0, 5], [0, 0, 0, 213, 0, 0, -67, 0], [0, 0, -141, 0, 0, 0, 0, 29], [0, -288, 0, 0, 0, 67, 0, 0], [122, 0, 0, 0, 172, 0, 0, 0], [102, 0, 0, 0, 164, 0, 0, 0], [0, 52, 0, 0, 0, -71, 0, 0], [0, 0, 0, -169, 0, 0, 58, 0], [0, 372, 0, 0, 0, -105, 0, 0], [-218, 0, 0, 0, -100, 0, 0, 0], [0, 0, -61, 0, 0, 0, 0, 11], [-232, 0, 0, 0, 201, 0, 0, 0], [0, 0, 0, -115, 0, 0, 67, 0], [0, 0, 0, 267, 0, 0, 110, 0], [-352, 0, 0, 0, 35, 0, 0, 0], [0, 0, -110, 0, 0, 0, 0, -52], [0, 132, 0, 0, 0, -295, 0, 0], [0, 213, 0, 0, 0, -130, 0, 0], [0, 0, -26, 0, 0, 0, 0, -3], [0, 0, 0, -147, 0, 0, -79, 0], [0, 142, 0, 0, 0, -71, 0, 0], [-348, 0, 0, 0, -131, 0, 0, 0], [0, 0, 0, 103, 0, 0, 32, 0], [0, 0, 0, -199, 0, 0, 9, 0], [0, 0, 0, 27, 0, 0, 52, 0], [0, 0, -221, 0, 0, 0, 0, 27], [0, 0, -102, 0, 0, 0, 0, 22], [-968, 0, 0, 0, -127, 0, 0, 0], [0, -147, 0, 0, 0, -10, 0, 0], [-1082, 0, 0, 0, -80, 0, 0, 0], [0, 0, 0, 259, 0, 0, -70, 0], [0, 433, 0, 0, 0, -96, 0, 0], [0, 0, 0, 125, 0, 0, 91, 0], [0, 0, -123, 0, 0, 0, 0, -99], [0, 613, 0, 0, 0, -34, 0, 0], [0, 0, 0, 65, 0, 0, 61, 0], [0, 0, 7, 0, 0, 0, 0, -61], [0, -158, 0, 0, 0, 173, 0, 0], [502, 0, 0, 0, 252, 0, 0, 0], [0, 712, 0, 0, 0, 109, 0, 0], [0, 0, 48, 0, 0, 0, 0, -11], [238, 0, 0, 0, -220, 0, 0, 0], [0, 0, -79, 0, 0, 0, 0, -135], [0, 213, 0, 0, 0, 164, 0, 0], [0, 0, -80, 0, 0, 0, 0, -102], [-578, 0, 0, 0, 264, 0, 0, 0], [0, 0, -139, 0, 0, 0, 0, 31], [158, 0, 0, 0, 100, 0, 0, 0], [0, 0, 0, -75, 0, 0, 103, 0], [768, 0, 0, 0, -53, 0, 0, 0], [0, 0, 0, -307, 0, 0, 11, 0], [0, 0, 33, 0, 0, 0, 0, 101], [0, -78, 0, 0, 0, -377, 0, 0], [0, 583, 0, 0, 0, 24, 0, 0], [0, 0, 0, 101, 0, 0, -55, 0], [-642, 0, 0, 0, -6, 0, 0, 0], [0, -288, 0, 0, 0, -229, 0, 0], [0, 0, 0, -433, 0, 0, -30, 0], [72, 0, 0, 0, -47, 0, 0, 0], [0, 0, 107, 0, 0, 0, 0, 117], [0, 163, 0, 0, 0, -118, 0, 0], [0, 0, 0, -37, 0, 0, 260, 0], [0, 0, 31, 0, 0, 0, 0, 35], [528, 0, 0, 0, 205, 0, 0, 0], [0, 0, 0, -329, 0, 0, 18, 0], [0, 543, 0, 0, 0, -42, 0, 0], [-668, 0, 0, 0, 205, 0, 0, 0], [0, -388, 0, 0, 0, -71, 0, 0], [0, 503, 0, 0, 0, -120, 0, 0], [0, 0, 0, -431, 0, 0, -49, 0], [-1082, 0, 0, 0, -90, 0, 0, 0], [0, 0, -123, 0, 0, 0, 0, -39], [0, 2, 0, 0, 0, 77, 0, 0], [502, 0, 0, 0, 276, 0, 0, 0], [0, 0, -198, 0, 0, 0, 0, -45], [0, 162, 0, 0, 0, 445, 0, 0], [0, 0, -36, 0, 0, 0, 0, 30], [0, 0, 0, 455, 0, 0, -26, 0], [-418, 0, 0, 0, -248, 0, 0, 0], [0, 0, 17, 0, 0, 0, 0, -85], [0, -337, 0, 0, 0, -240, 0, 0], [-632, 0, 0, 0, 25, 0, 0, 0], [0, 0, 0, 1, 0, 0, -115, 0], [0, 0, 0, -265, 0, 0, 130, 0], [0, 0, 166, 0, 0, 0, 0, 30], [412, 0, 0, 0, 185, 0, 0, 0], [0, 0, 0, 113, 0, 0, -265, 0], [1042, 0, 0, 0, 272, 0, 0, 0], [0, 0, 0, -255, 0, 0, -209, 0], [0, -738, 0, 0, 0, 75, 0, 0], [368, 0, 0, 0, 277, 0, 0, 0], [0, 0, 2, 0, 0, 0, 0, -128], [-498, 0, 0, 0, -224, 0, 0, 0], [0, 0, 0, 457, 0, 0, -81, 0], [0, 0, 0, -273, 0, 0, -70, 0], [-672, 0, 0, 0, -93, 0, 0, 0], [0, 273, 0, 0, 0, 0, 0, 0], [0, 0, -76, 0, 0, 0, 0, 12], [0, 0, 0, 237, 0, 0, -69, 0], [0, 0, -187, 0, 0, 0, 0, 43], [0, 642, 0, 0, 0, 111, 0, 0], [0, 0, 344, 0, 0, 0, 0, -43], [0, 0, -119, 0, 0, 0, 0, -103], [-108, 0, 0, 0, 333, 0, 0, 0], [0, 0, 0, 319, 0, 0, -202, 0], [0, 0, 0, -183, 0, 0, 275, 0], [1432, 0, 0, 0, -175, 0, 0, 0], [0, 622, 0, 0, 0, -13, 0, 0], [0, 0, -412, 0, 0, 0, 0, 33], [0, 0, 133, 0, 0, 0, 0, 161], [0, 0, 0, -629, 0, 0, -6, 0], [0, 0, 11, 0, 0, 0, 0, 67], [0, 0, 0, -67, 0, 0, 273, 0], [838, 0, 0, 0, -106, 0, 0, 0], [0, -367, 0, 0, 0, 0, 0, 0], [0, 0, 84, 0, 0, 0, 0, -64], [0, 0, 0, 353, 0, 0, 339, 0], [0, 0, 111, 0, 0, 0, 0, 85], [0, 0, -174, 0, 0, 0, 0, -55], [0, 0, 0, 313, 0, 0, 229, 0], [0, -1098, 0, 0, 0, 41, 0, 0], [762, 0, 0, 0, -236, 0, 0, 0], [0, 0, 0, -3, 0, 0, 217, 0], [-1008, 0, 0, 0, 77, 0, 0, 0], [0, 242, 0, 0, 0, 87, 0, 0], [0, 0, -356, 0, 0, 0, 0, -15], [1078, 0, 0, 0, -48, 0, 0, 0], [0, 0, 0, 547, 0, 0, -10, 0], [0, 872, 0, 0, 0, -5, 0, 0], [0, 63, 0, 0, 0, -48, 0, 0], [0, 0, 0, -403, 0, 0, -135, 0], [0, 0, 0, -169, 0, 0, -180, 0], [198, 0, 0, 0, 190, 0, 0, 0], [0, -1428, 0, 0, 0, 97, 0, 0], [0, -177, 0, 0, 0, 304, 0, 0], [0, 0, 220, 0, 0, 0, 0, -25], [318, 0, 0, 0, 658, 0, 0, 0], [922, 0, 0, 0, -252, 0, 0, 0], [0, 0, 0, -61, 0, 0, 12, 0], [0, -238, 0, 0, 0, -385, 0, 0], [-2002, 0, 0, 0, 228, 0, 0, 0], [0, 0, 0, 483, 0, 0, 244, 0], [0, -657, 0, 0, 0, -330, 0, 0], [0, 0, -432, 0, 0, 0, 0, 0], [0, -687, 0, 0, 0, -150, 0, 0], [0, 0, 0, 117, 0, 0, -67, 0], [0, 0, 0, -37, 0, 0, -220, 0], [1558, 0, 0, 0, 30, 0, 0, 0], [1372, 0, 0, 0, 129, 0, 0, 0], [0, 0, -31, 0, 0, 0, 0, -79], [0, 0, 0, -283, 0, 0, -211, 0], [0, 0, 97, 0, 0, 0, 0, 121], [0, 0, 0, 589, 0, 0, 3, 0], [2838, 0, 0, 0, -32, 0, 0, 0], [0, 0, 0, 287, 0, 0, -354, 0], [0, 0, -85, 0, 0, 0, 0, 31], [0, -507, 0, 0, 0, 356, 0, 0], [0, 892, 0, 0, 0, 295, 0, 0], [0, 0, -22, 0, 0, 0, 0, 204], [1032, 0, 0, 0, 337, 0, 0, 0], [-312, 0, 0, 0, 481, 0, 0, 0], [0, 0, 0, 95, 0, 0, 510, 0], [0, 93, 0, 0, 0, -260, 0, 0], [0, 0, -113, 0, 0, 0, 0, 23], [0, 562, 0, 0, 0, 295, 0, 0], [-912, 0, 0, 0, -421, 0, 0, 0], [0, 0, -85, 0, 0, 0, 0, 95], [0, 882, 0, 0, 0, -65, 0, 0], [0, 0, 404, 0, 0, 0, 0, -117], [0, 0, 0, -389, 0, 0, 148, 0], [0, -157, 0, 0, 0, 618, 0, 0], [0, -338, 0, 0, 0, 441, 0, 0], [0, 0, 170, 0, 0, 0, 0, 40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_3_h_n();".
// 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_1600_3_h_n(:prec:=8)
    chi := MakeCharacter_1600_h();
    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_1600_3_h_n();".
// 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_1600_3_h_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![80, 0, -20, 0, 1]>,<7,R![2000, 0, -100, 0, 1]>],Snew);
    return Vf;
end function;