
// Make newform 1050.3.h.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1050_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1050_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_1050_3_h_a();".
// 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_1050_3_h_a();".
// 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, 0, 0, -1, 0, 0, 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, 1, 0, -1, 0, -1, 0, 0], [0, 0, 2, 0, 0, 0, -1, 0], [0, 1, 0, 1, 0, 1, 0, -2], [-2, 0, 0, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0], [0, 1, 0, 1, 0, -1, 0, 0], [0, 1, 0, -1, 0, 1, 0, 2]];
    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_1050_h();"
function MakeCharacter_1050_h()
    N := 1050;
    order := 2;
    char_gens := [701, 127, 451];
    v := [2, 1, 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_1050_h_Hecke();"
function MakeCharacter_1050_h_Hecke(Kf)
    N := 1050;
    order := 2;
    char_gens := [701, 127, 451];
    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, -1, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 3, -1, -2, 0, 1, 0, 0], [-6, 0, 0, 0, 0, 0, -3, 0], [0, 0, 8, -2, 0, 0, 0, 0], [0, 0, 2, 11, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, -8], [0, -9, 0, 0, 0, -3, 0, 0], [-30, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, 0, 0, -12], [0, -36, 0, 0, 0, -10, 0, 0], [0, 0, 0, 0, 7, 0, 0, -7], [0, -30, 0, 0, 0, 16, 0, 0], [0, 0, -28, -4, 0, 0, 0, 0], [0, -12, 0, 0, 0, 27, 0, 0], [0, 0, 0, 0, 14, 0, 0, -20], [0, 0, 0, 0, 2, 0, 0, -4], [0, 12, 0, 0, 0, 22, 0, 0], [-30, 0, 0, 0, 0, 0, 57, 0], [0, 0, 4, 26, 0, 0, 0, 0], [-32, 0, 0, 0, 0, 0, -72, 0], [0, 0, 32, 20, 0, 0, 0, 0], [0, 0, 0, 0, 27, 0, 0, 21], [0, 0, -44, -10, 0, 0, 0, 0], [0, 0, 0, 0, 21, 0, 0, 27], [0, 0, 28, -52, 0, 0, 0, 0], [0, -3, 0, 0, 0, -21, 0, 0], [2, 0, 0, 0, 0, 0, -24, 0], [0, -12, 0, 0, 0, 45, 0, 0], [0, 24, 0, 0, 0, 28, 0, 0], [0, 0, 0, 0, 12, 0, 0, 12], [0, 96, 0, 0, 0, 15, 0, 0], [0, 0, 0, 0, 58, 0, 0, -8], [-126, 0, 0, 0, 0, 0, 60, 0], [4, 0, 0, 0, 0, 0, 54, 0], [0, 0, -84, 36, 0, 0, 0, 0], [0, -36, 0, 0, 0, 62, 0, 0], [0, 0, 68, 32, 0, 0, 0, 0], [0, 0, 18, 3, 0, 0, 0, 0], [102, 0, 0, 0, 0, 0, 63, 0], [0, 0, 0, 0, -8, 0, 0, -38], [6, 0, 0, 0, 0, 0, 75, 0], [0, -36, 0, 0, 0, -14, 0, 0], [0, -24, 0, 0, 0, -75, 0, 0], [0, 0, 0, 0, -1, 0, 0, 68], [64, 0, 0, 0, 0, 0, 30, 0], [0, 0, -18, 36, 0, 0, 0, 0], [0, 0, 132, 24, 0, 0, 0, 0], [0, 0, 0, 0, 36, 0, 0, 6], [0, 108, 0, 0, 0, -105, 0, 0], [-150, 0, 0, 0, 0, 0, 93, 0], [0, 0, 0, 0, 50, 0, 0, 2], [0, 0, 0, 0, 8, 0, 0, -32], [0, 0, 138, -21, 0, 0, 0, 0], [0, -111, 0, 0, 0, 123, 0, 0], [0, 0, 0, 0, -97, 0, 0, -59], [0, 0, 0, 0, -26, 0, 0, 52], [0, 132, 0, 0, 0, 52, 0, 0], [-66, 0, 0, 0, 0, 0, -60, 0], [0, 0, -198, 24, 0, 0, 0, 0], [0, 0, 74, 167, 0, 0, 0, 0], [0, 0, -54, 24, 0, 0, 0, 0], [0, 0, 0, 0, -28, 0, 0, 76], [0, 0, -136, -116, 0, 0, 0, 0], [0, -396, 0, 0, 0, -9, 0, 0], [256, 0, 0, 0, 0, 0, -54, 0], [0, -72, 0, 0, 0, 95, 0, 0], [0, -273, 0, 0, 0, 33, 0, 0], [0, 0, 0, 0, 82, 0, 0, -20], [0, 0, -182, 139, 0, 0, 0, 0], [102, 0, 0, 0, 0, 0, 3, 0], [0, 0, 10, -124, 0, 0, 0, 0], [0, -216, 0, 0, 0, 59, 0, 0], [128, 0, 0, 0, 0, 0, -162, 0], [0, 0, 116, -172, 0, 0, 0, 0], [78, 0, 0, 0, 0, 0, 216, 0], [0, 0, -348, 48, 0, 0, 0, 0], [222, 0, 0, 0, 0, 0, -336, 0], [0, 0, 0, 0, -14, 0, 0, 22], [0, 0, 0, 0, -14, 0, 0, 140], [32, 0, 0, 0, 0, 0, -228, 0], [318, 0, 0, 0, 0, 0, -189, 0], [0, 0, 96, 228, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, -156], [0, 207, 0, 0, 0, 261, 0, 0], [210, 0, 0, 0, 0, 0, 276, 0], [0, -72, 0, 0, 0, 191, 0, 0], [0, 0, 0, 0, 19, 0, 0, -139], [0, -216, 0, 0, 0, -76, 0, 0], [0, 0, 168, -144, 0, 0, 0, 0], [0, 0, 0, 0, 56, 0, 0, 148], [0, 114, 0, 0, 0, -82, 0, 0], [198, 0, 0, 0, 0, 0, 315, 0], [-136, 0, 0, 0, 0, 0, -438, 0], [0, 0, 164, 92, 0, 0, 0, 0], [0, 0, 0, 0, -121, 0, 0, -143], [0, 0, 0, 0, -35, 0, 0, -169], [0, 0, -100, -272, 0, 0, 0, 0], [-376, 0, 0, 0, 0, 0, 468, 0], [0, 216, 0, 0, 0, -218, 0, 0], [0, -60, 0, 0, 0, 375, 0, 0], [0, 0, 360, -84, 0, 0, 0, 0], [-690, 0, 0, 0, 0, 0, 12, 0], [-316, 0, 0, 0, 0, 0, 36, 0], [0, 0, 248, 16, 0, 0, 0, 0], [0, 0, 168, -168, 0, 0, 0, 0], [0, 0, 110, -307, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, -669, 0], [0, 0, 0, 0, 24, 0, 0, -264], [0, 0, 202, -268, 0, 0, 0, 0], [0, 168, 0, 0, 0, -19, 0, 0], [0, -288, 0, 0, 0, 27, 0, 0], [0, 0, 0, 0, -46, 0, 0, -88], [-160, 0, 0, 0, 0, 0, -684, 0], [66, 0, 0, 0, 0, 0, -672, 0], [0, 0, -30, 144, 0, 0, 0, 0], [0, 0, 0, -384, 0, 0, 0, 0], [0, -420, 0, 0, 0, -69, 0, 0], [-654, 0, 0, 0, 0, 0, 177, 0], [0, 0, 0, 0, 118, 0, 0, 304], [0, -636, 0, 0, 0, -146, 0, 0], [0, 0, -678, 15, 0, 0, 0, 0], [0, 147, 0, 0, 0, 513, 0, 0], [0, 0, 0, 0, -42, 0, 0, -24], [-234, 0, 0, 0, 0, 0, -60, 0], [14, 0, 0, 0, 0, 0, 144, 0], [0, 0, 0, 0, 176, 0, 0, 244], [0, 0, 156, -60, 0, 0, 0, 0], [0, 0, 480, -186, 0, 0, 0, 0], [-1028, 0, 0, 0, 0, 0, 216, 0], [0, 723, 0, 0, 0, -123, 0, 0], [244, 0, 0, 0, 0, 0, 138, 0], [0, -216, 0, 0, 0, -487, 0, 0], [0, 0, 0, 0, -317, 0, 0, 89], [0, 0, 0, 0, -152, 0, 0, -260], [0, 0, -362, 37, 0, 0, 0, 0], [0, 0, 76, -88, 0, 0, 0, 0], [0, 0, 186, 279, 0, 0, 0, 0], [198, 0, 0, 0, 0, 0, 60, 0], [0, 0, 0, 0, -114, 0, 0, -216], [882, 0, 0, 0, 0, 0, 276, 0], [0, 468, 0, 0, 0, 140, 0, 0], [0, 111, 0, 0, 0, 405, 0, 0], [0, 0, 0, 0, 140, 0, 0, -286], [0, 0, 0, 0, -290, 0, 0, 152], [0, 0, -316, -152, 0, 0, 0, 0], [0, 0, 170, 95, 0, 0, 0, 0], [0, 0, 0, 0, -7, 0, 0, -88], [0, 303, 0, 0, 0, 33, 0, 0], [0, 144, 0, 0, 0, -329, 0, 0], [0, 0, 0, 0, 69, 0, 0, 39], [0, -198, 0, 0, 0, 604, 0, 0], [0, 0, -508, 32, 0, 0, 0, 0], [0, 90, 0, 0, 0, -340, 0, 0], [-150, 0, 0, 0, 0, 0, -207, 0], [-484, 0, 0, 0, 0, 0, 426, 0], [0, 0, 0, 0, 133, 0, 0, -445], [0, 0, -280, 112, 0, 0, 0, 0], [0, 0, 0, 0, 31, 0, 0, 29], [0, -117, 0, 0, 0, 129, 0, 0], [0, 288, 0, 0, 0, 477, 0, 0], [0, -276, 0, 0, 0, 436, 0, 0], [0, 0, 0, 0, -50, 0, 0, 512], [0, -1116, 0, 0, 0, 111, 0, 0], [0, 0, 92, -472, 0, 0, 0, 0], [916, 0, 0, 0, 0, 0, -90, 0], [0, 0, 324, 564, 0, 0, 0, 0], [664, 0, 0, 0, 0, 0, -828, 0], [0, 0, 202, -197, 0, 0, 0, 0], [1506, 0, 0, 0, 0, 0, 153, 0], [0, 0, 0, 0, -54, 0, 0, 48], [-450, 0, 0, 0, 0, 0, 63, 0], [0, -252, 0, 0, 0, -376, 0, 0], [0, 0, 0, 0, -85, 0, 0, -100], [0, 0, 0, 0, -281, 0, 0, 53], [716, 0, 0, 0, 0, 0, 0, 0], [-78, 0, 0, 0, 0, 0, -132, 0], [0, 0, 284, 76, 0, 0, 0, 0], [0, 0, 0, 0, -390, 0, 0, -12], [0, -180, 0, 0, 0, -704, 0, 0], [0, 0, 0, 0, 172, 0, 0, -352], [0, 36, 0, 0, 0, -580, 0, 0], [0, 0, -278, -449, 0, 0, 0, 0], [0, 111, 0, 0, 0, -399, 0, 0], [0, 0, 0, 0, 35, 0, 0, 85], [0, -624, 0, 0, 0, 143, 0, 0], [0, 0, -236, 464, 0, 0, 0, 0], [-8, 0, 0, 0, 0, 0, 300, 0], [0, 0, 0, 0, 164, 0, 0, 316], [0, 0, -1060, -74, 0, 0, 0, 0], [0, -3, 0, 0, 0, -105, 0, 0], [-416, 0, 0, 0, 0, 0, -306, 0], [0, 0, 0, 0, -175, 0, 0, -185], [0, 981, 0, 0, 0, 147, 0, 0], [0, 0, 298, 439, 0, 0, 0, 0], [100, 0, 0, 0, 0, 0, 156, 0], [0, -684, 0, 0, 0, 236, 0, 0], [0, 0, -150, -561, 0, 0, 0, 0], [0, 0, -160, 44, 0, 0, 0, 0], [1206, 0, 0, 0, 0, 0, -360, 0], [0, 0, 0, 0, -127, 0, 0, 620], [0, 0, 104, 418, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, -274], [0, 0, 0, 0, -260, 0, 0, 200], [0, 0, -146, 13, 0, 0, 0, 0], [0, 0, 0, 0, -257, 0, 0, 268], [0, -741, 0, 0, 0, -375, 0, 0], [1506, 0, 0, 0, 0, 0, 120, 0], [0, 0, 0, 0, 479, 0, 0, -328], [0, 0, 0, 0, 0, 559, 0, 0], [0, 0, 0, 0, -159, 0, 0, -477], [0, -774, 0, 0, 0, -254, 0, 0], [0, 0, -44, 616, 0, 0, 0, 0], [0, 0, 0, 0, 272, 0, 0, -212], [0, 0, 0, 0, -340, 0, 0, -64], [0, -42, 0, 0, 0, -562, 0, 0], [0, 0, 0, 0, 279, 0, 0, -207], [0, 291, 0, 0, 0, 273, 0, 0], [0, -336, 0, 0, 0, -1065, 0, 0], [514, 0, 0, 0, 0, 0, 648, 0], [0, 0, 0, 0, -210, 0, 0, 420], [810, 0, 0, 0, 0, 0, 36, 0], [0, 654, 0, 0, 0, -350, 0, 0], [0, 0, 40, 4, 0, 0, 0, 0], [-508, 0, 0, 0, 0, 0, -108, 0], [0, 0, -902, 247, 0, 0, 0, 0], [-438, 0, 0, 0, 0, 0, 9, 0], [0, 0, -1372, -116, 0, 0, 0, 0], [1446, 0, 0, 0, 0, 0, 555, 0], [0, 408, 0, 0, 0, -113, 0, 0], [0, 0, 0, 0, 333, 0, 0, 384], [820, 0, 0, 0, 0, 0, -390, 0], [-654, 0, 0, 0, 0, 0, -660, 0], [0, 0, -682, -344, 0, 0, 0, 0], [0, 0, -276, 840, 0, 0, 0, 0], [0, 0, 0, 0, 94, 0, 0, -170], [0, -720, 0, 0, 0, 465, 0, 0], [-1158, 0, 0, 0, 0, 0, -327, 0], [0, 0, 0, 0, 58, 0, 0, 692], [0, -189, 0, 0, 0, 1017, 0, 0], [0, 0, 0, 0, 46, 0, 0, -656], [0, 0, -1226, -76, 0, 0, 0, 0], [596, 0, 0, 0, 0, 0, -1020, 0], [0, 0, 690, -393, 0, 0, 0, 0], [0, 0, 0, 0, -146, 0, 0, -736], [0, 0, 748, -124, 0, 0, 0, 0], [0, 0, 0, 0, -16, 0, 0, -728], [-548, 0, 0, 0, 0, 0, 348, 0], [0, 153, 0, 0, 0, 795, 0, 0], [0, 1344, 0, 0, 0, 239, 0, 0], [0, 0, 0, 0, -291, 0, 0, -561], [0, 729, 0, 0, 0, 483, 0, 0], [0, 0, 0, 0, 394, 0, 0, -470], [0, 0, -278, -173, 0, 0, 0, 0], [-906, 0, 0, 0, 0, 0, 423, 0], [-304, 0, 0, 0, 0, 0, 1188, 0], [0, 0, -234, 336, 0, 0, 0, 0], [0, 0, -174, 579, 0, 0, 0, 0], [0, 0, -168, 24, 0, 0, 0, 0], [0, 672, 0, 0, 0, 632, 0, 0], [0, -291, 0, 0, 0, 819, 0, 0], [0, 0, 0, 0, 104, 0, 0, 1034], [0, 0, -4, 232, 0, 0, 0, 0], [0, 0, 342, -291, 0, 0, 0, 0], [0, 0, 0, 0, 402, 0, 0, 96], [-6, 0, 0, 0, 0, 0, 120, 0], [0, 0, 0, 0, 537, 0, 0, -321], [0, -1968, 0, 0, 0, -94, 0, 0], [0, -1020, 0, 0, 0, -225, 0, 0], [0, 0, 0, 0, -142, 0, 0, -16], [0, -750, 0, 0, 0, 332, 0, 0], [0, 0, 492, -390, 0, 0, 0, 0], [-1468, 0, 0, 0, 0, 0, 126, 0], [0, 0, -564, 582, 0, 0, 0, 0], [0, 0, 1228, 236, 0, 0, 0, 0], [0, -1263, 0, 0, 0, -357, 0, 0], [1292, 0, 0, 0, 0, 0, 1092, 0], [628, 0, 0, 0, 0, 0, -1092, 0], [0, 0, 0, 0, 428, 0, 0, -224], [0, 0, 424, 208, 0, 0, 0, 0], [-2440, 0, 0, 0, 0, 0, -420, 0], [0, 0, -200, -944, 0, 0, 0, 0], [0, 0, 0, 0, 442, 0, 0, 556], [0, 0, -364, 808, 0, 0, 0, 0], [42, 0, 0, 0, 0, 0, -543, 0], [0, 1656, 0, 0, 0, -431, 0, 0], [0, -480, 0, 0, 0, -735, 0, 0], [0, 0, 0, 0, 610, 0, 0, -668], [0, 0, 0, 0, 289, 0, 0, -337], [2610, 0, 0, 0, 0, 0, -384, 0], [0, 0, 1456, 148, 0, 0, 0, 0], [0, 408, 0, 0, 0, 957, 0, 0], [0, 0, 0, 0, -682, 0, 0, 460], [0, -456, 0, 0, 0, 1019, 0, 0], [0, 0, 0, 0, 67, 0, 0, 41], [0, 0, 0, 0, 181, 0, 0, -356], [0, 0, 854, -343, 0, 0, 0, 0], [0, 0, 0, 0, 322, 0, 0, -592], [0, 0, -1772, -40, 0, 0, 0, 0], [0, 0, -88, 460, 0, 0, 0, 0], [0, -360, 0, 0, 0, -1281, 0, 0], [2536, 0, 0, 0, 0, 0, 396, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1050_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1050_3_h_a();".
// 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_1050_3_h_a(:prec:=8)
    chi := MakeCharacter_1050_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_1050_3_h_a();".
// 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_1050_3_h_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1050_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<11,R![18, 12, 1]>],Snew);
    return Vf;
end function;