
// Make newform 8037.2.a.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8037_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_8037_2_a_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_8037_2_a_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 := [-25, 62, 179, -456, -487, 1250, 618, -1639, -320, 1115, 4, -390, 47, 65, -13, -4, 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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-82, 81, 456, -584, -602, 1404, -64, -1290, 560, 459, -332, -42, 71, -7, -5, 1], [125, -198, -673, 1063, 1064, -2000, -355, 1643, -389, -569, 301, 59, -69, 6, 5, -1], [-89, -88, 816, 350, -2407, -399, 3181, -3, -2110, 245, 715, -140, -116, 29, 7, -2], [112, 222, -1217, -1371, 4250, 2735, -6552, -1989, 5006, 321, -1891, 166, 331, -60, -21, 5], [400, -214, -3127, 931, 8648, -1520, -11035, 1746, 7228, -1357, -2444, 558, 397, -105, -24, 7], [-418, 214, 3148, -931, -8651, 1520, 11035, -1746, -7228, 1357, 2444, -558, -397, 105, 24, -7], [493, -336, -3791, 1596, 10475, -2653, -13188, 2709, 8474, -1821, -2815, 685, 451, -123, -27, 8], [268, 204, -2736, -1358, 9274, 2835, -13687, -1700, 9997, -283, -3649, 501, 624, -131, -39, 10], [433, 37, -3843, -383, 11809, 933, -16288, 25, 11313, -1074, -3987, 689, 666, -153, -41, 11], [308, 220, -3170, -1443, 10745, 2933, -15933, -1618, 11702, -505, -4288, 630, 735, -159, -46, 12], [864, -568, -6687, 2683, 18658, -4548, -23761, 4820, 15424, -3336, -5164, 1273, 832, -230, -50, 15], [-1055, 731, 7802, -3545, -20617, 6334, 24914, -6438, -15365, 3941, 4924, -1344, -767, 225, 45, -14], [-1299, 466, 10328, -1615, -29676, 2038, 39289, -3339, -26570, 3766, 9205, -1841, -1521, 375, 93, -26]];
    Rf_basisdens := [1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3];
    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_8037_a();"
function MakeCharacter_8037_a()
    N := 8037;
    order := 1;
    char_gens := [4466, 1693, 7525];
    v := [1, 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;

function MakeCharacter_8037_a_Hecke(Kf)
    return MakeCharacter_8037_a();
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 := 2;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-2, 1, -1, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, 0, 0, 0], [1, -1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, -2, 0, 1, -1, 1, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, -1, 0, 0, -2, -1, 0, 1, 0, 1, 1, 0, 0, -1], [0, 0, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, -2, 1, 1, 0], [-1, -1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 1, 0, 1, -1, 0], [0, 0, 0, 1, 1, -1, -2, 1, -1, 2, 1, 0, 0, -1, 1, 0], [1, -1, -1, 0, -1, 1, 1, 1, -1, 0, 0, 0, 1, -1, -2, 1], [-3, 0, 0, 0, 1, 1, -1, -1, -1, -1, 1, 0, 0, 1, 0, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, -2, -1, 2, 1, -1, 2, 1, 0, -1, 0, 1, 0], [2, 1, 0, 0, 0, 1, 0, 0, -1, -1, 0, 1, 0, 0, -1, 0], [1, -2, 1, -1, -1, -1, -1, -1, -1, 3, 0, 1, -2, -1, 0, 0], [-5, 0, -1, 0, 0, -1, -1, -1, 0, -1, -3, 1, 0, 1, 1, 0], [1, 0, 1, 2, 0, -1, 3, 0, 1, 0, -1, -1, -1, 1, 0, 2], [1, -3, 1, -1, 0, 1, 0, 1, 0, 2, 0, 0, 1, -1, -1, 1], [-4, 0, 1, -1, 0, 0, 2, -3, 4, -2, -2, 1, -2, 5, 1, 1], [-1, -1, 0, -1, -2, 1, 0, -1, 1, -1, -1, 1, 0, 2, 1, 1], [1, -2, 1, -2, 0, -1, -5, 1, -2, 5, 4, 2, -1, -2, 0, -1], [2, -4, 1, 3, -2, -1, 3, 2, -2, 3, 1, -1, 0, -1, -1, 1], [1, 0, -1, 1, 2, -1, 1, 0, 0, -1, -3, 0, -1, 1, 2, 0], [-7, 0, 0, 0, 2, 1, 0, 0, 0, -2, -2, 0, 2, 1, 0, 0], [-1, 1, 1, 0, 1, 1, 0, -1, 0, -2, 1, 0, 0, 2, -1, -1], [2, -3, 3, -2, -1, 1, -2, -2, -1, 1, -1, 2, -1, 1, -1, -1], [-2, 4, -1, 0, 1, -2, 4, 0, 2, -2, -2, -2, 0, 2, 1, 2], [-7, -4, -1, -2, 1, 2, -4, -1, 3, 0, 1, 3, 0, 1, 3, -3], [7, -3, 1, 0, -1, -3, -1, 1, -1, 1, 0, 1, 0, 0, 1, 0], [-3, -2, 1, -2, -1, 1, 0, -1, 1, 0, -1, -1, 0, 1, 0, -1], [-1, -1, 2, 0, -1, -1, -3, -1, -4, 0, 0, -1, 2, -2, -1, -3], [-6, 2, -3, -2, -2, 5, 1, -1, -2, -3, 3, 1, 0, -2, -2, -2], [1, 5, -1, 1, 0, 2, 4, 3, -1, 1, 0, -3, 1, -1, -1, 2], [-10, 4, -3, 0, 2, 2, -3, -2, 2, -6, -1, -2, 3, 1, -1, -1], [-4, 1, -2, 0, 0, -1, -1, 2, 0, 1, 1, 0, -2, 0, 1, -3], [0, 2, -1, 1, 0, -3, 2, 0, 1, 1, -1, 0, 0, 1, 2, 0], [7, -1, 4, 2, -1, -2, 0, 1, -3, 0, 1, 0, 0, 0, -1, 0], [2, 3, -1, 3, 3, 2, 4, 0, 2, -3, -1, -3, 1, 1, -2, 3], [-2, 0, -1, -4, -2, 2, 0, 2, 0, 2, 3, 1, -2, 1, 0, 0], [-6, 3, -3, 1, 4, 1, -2, 0, -1, -3, 1, -5, 5, -2, -1, 2], [-7, 4, -3, 0, 3, 1, -2, -1, 2, -3, -2, -3, 3, -1, 1, -1], [1, -2, 1, 0, 0, 2, 2, 1, -1, 0, 0, 2, 0, -2, -1, 1], [3, -3, 1, 0, -3, -1, 4, 3, -1, 3, 1, 0, -3, 0, 1, -1], [-6, 1, -2, -2, 1, 2, 5, -2, 5, -2, -4, 2, -2, 5, 2, 2], [-7, 2, -4, 2, 2, 1, 1, 1, 3, -3, 1, -1, 0, 0, -1, 1], [-1, 3, -4, 2, 4, -2, 2, -1, -1, -2, -2, -3, 2, -2, -1, 0], [-1, 1, 2, 0, -4, 4, 6, 1, 2, -3, 0, 0, 2, 2, -3, 3], [-5, -2, 4, -2, -2, 3, 4, -3, 4, -1, -5, 2, 0, 3, -1, 0], [-1, -2, 1, -2, 3, 0, 2, -2, 4, 2, -2, 2, 0, 3, 1, -2], [3, 0, 2, 2, 4, -3, -5, 2, -2, 0, -2, 0, 4, -2, -1, 0], [2, -4, 3, -3, 0, 3, -2, -1, -3, 2, 0, 3, 1, -1, 0, -1], [2, 0, 0, -3, -1, -4, -1, 2, -3, 2, 4, -2, 0, -2, 1, -2], [-6, -1, 1, -2, -2, 2, -5, -2, -1, 4, 5, 1, 1, -2, -1, -1], [-1, -2, -1, -1, 2, 1, -2, -1, 2, -3, 1, 1, -2, 4, 3, -3], [-5, 2, -3, -2, 0, 2, 0, 0, 3, 1, -2, 4, -2, 2, 1, 3], [-5, 9, 0, 2, -1, 1, -3, 0, 2, -1, -1, 0, 1, 0, -2, 2], [-2, 4, 2, -3, 3, 1, -1, 0, -3, -2, 2, -1, 1, 0, 0, -3], [-1, -2, 4, 4, 2, -4, -3, 1, -2, 3, 3, -4, 4, -1, 1, 2], [-5, 0, -2, -4, -1, 4, -3, -3, 6, -3, -2, 5, -4, 3, 1, -4], [-13, 4, -1, 0, -3, 2, 0, 2, -1, 0, 1, 0, 1, -1, -1, 2], [-11, -4, 1, -3, 2, 4, -4, -3, 5, -4, 2, 3, -1, 4, -1, -1], [-4, -2, 0, -1, 0, 1, 0, 0, 1, 1, 1, 1, 0, -4, 0, -4], [-1, 5, -2, 5, -1, -7, -1, -1, 3, 2, -4, 0, 0, 0, 1, 3], [-4, 3, 0, 1, -1, 0, 4, 0, -3, 4, 1, -3, 1, -1, -1, 2], [-4, 2, 1, -5, -1, 2, -6, -2, -3, 0, 5, 1, -2, -3, 1, -2], [2, 4, -2, 0, 1, 0, -2, 3, 0, -1, 3, 0, -1, -1, 2, -3], [5, 0, 0, 0, -2, -2, -2, -1, -3, -3, 3, -2, 1, 3, 3, 0], [-5, 2, 2, -4, 6, 1, -3, -1, 2, -4, -1, 0, 8, 1, -1, 0], [-7, 2, -2, 0, 2, -1, -5, -3, 5, -1, 1, 0, -1, 4, 0, -2], [0, 1, 0, 4, -3, -2, 4, 4, -5, 1, -1, -5, 5, -2, -6, 5], [5, -7, 2, -1, -2, -2, -3, 1, -6, 1, 4, 0, 0, -3, -2, 0], [3, -2, 2, 0, 0, -5, -3, 1, -2, 7, 3, 0, 0, -5, 2, -2], [6, -6, -1, -3, -1, 0, -2, -1, 2, 5, -5, 4, -2, 0, 2, 1], [-7, -1, 0, 2, -2, 5, 0, -3, -2, -5, -4, 3, 1, 0, -4, 2], [1, -6, 1, -3, 1, 3, -3, -2, -5, -2, 1, 3, 2, 0, 3, 0], [6, -1, 2, 5, -3, -4, 4, -3, 2, 0, -4, -3, -2, 4, -2, 6], [2, -1, 6, 3, 0, -1, -2, 0, -2, 1, -1, -2, 4, -3, -7, 4], [6, 0, 3, -3, -1, 0, -4, 2, -6, 3, 1, -1, -1, -2, 0, -4], [6, -1, 4, -1, -3, -1, 2, 2, 3, 4, -2, 1, -1, 2, 3, 2], [1, 4, -1, -1, -2, -1, 3, 5, 2, 7, 3, -2, -5, -2, 5, 2], [-4, 1, -1, 5, -1, -1, 5, 2, 5, 0, -3, -3, 2, 0, -6, 3], [1, -4, 3, 3, 0, -4, -3, -1, -5, -1, 2, -1, 3, -1, 2, -1], [4, 4, -1, 5, 2, -7, 1, -2, -1, -2, -1, -4, 0, 0, 3, -4], [-2, -2, 7, -6, -3, 2, -13, -6, -7, 3, 2, 7, 0, -1, 0, -5], [0, 7, 0, -3, -3, 0, 4, -1, 0, -1, 6, 0, -3, 2, 5, -1], [2, 1, 3, -2, -1, -1, -1, 2, -2, 1, 3, -3, -3, -4, 1, 0], [2, -2, 2, -2, -1, -3, 0, -2, 1, 7, -3, 5, -3, -1, 0, -2], [6, -4, 3, 0, 1, -2, -8, 2, -4, 5, 2, 1, 1, -4, 4, -3], [-1, -1, -8, 3, 1, 1, -2, 5, 0, 1, 8, -4, -1, 0, 2, 1], [-5, 3, 3, 0, -1, 1, -6, -3, -5, 0, 0, 2, 5, -4, -2, 1], [4, 2, 0, 1, -3, -4, -2, 6, 1, 7, 3, -2, 0, -4, 1, 2], [-8, 5, 0, 1, 5, -1, -4, 0, 3, -4, 0, -1, -1, 2, 3, -4], [-1, 5, -3, 6, 4, 0, 2, 3, 2, -5, -3, 1, 6, -1, -6, 1], [2, 6, 2, 2, 1, 0, 2, 0, -3, 4, 0, -3, 7, -3, -7, 4], [5, -3, 0, 1, 4, -3, 1, -1, 4, -5, -9, 4, -2, 4, 3, 0], [0, -4, 5, -3, 4, -5, -6, 0, -6, 8, 2, -4, 3, -5, 0, -2], [6, 10, 0, 4, 2, -3, 8, 0, 6, -6, -2, -5, -5, 3, 3, -1], [-8, 4, -1, 1, 3, 0, 13, 2, 9, -4, -3, -3, 4, 5, 1, 2], [5, 4, -4, 3, 2, -5, 2, 1, -1, 2, -5, -1, 4, -3, -4, 2], [10, -3, 0, -1, 2, -3, 1, -2, 5, -3, -3, 2, -1, 4, 1, 2], [-1, 4, 1, 3, -1, -3, 7, 1, -6, 0, 4, -5, 3, -2, -4, 6], [-8, 2, 0, 0, -1, 1, -2, 3, -3, 1, 1, 1, -1, -3, 3, 0], [3, 4, 0, -4, -1, 5, -8, -3, -3, -3, 2, 3, 1, -2, -6, -2], [14, -4, -1, 2, -3, -5, -1, 6, -8, 11, 6, -4, 0, -5, 2, 2], [-8, -7, 2, 1, -2, -3, -7, -2, -3, 0, 2, -1, 0, 0, 3, -3], [-2, -3, -2, -2, -1, 3, -7, 4, -9, 1, 4, 0, 4, -3, 0, -4], [5, -6, 2, 6, -3, -8, 5, 3, 0, 4, -6, 4, -3, -4, 0, 1], [-12, 7, -5, -4, -3, 3, 3, -4, -1, 1, 5, -3, -4, -2, 3, -2], [8, -6, 3, 2, -5, 3, -2, 2, -6, 5, 2, -1, -1, -5, -9, 1], [-14, 12, -4, -1, 2, 1, 2, -3, 11, -5, 1, 1, -4, 6, 7, 0], [-8, 4, 2, -3, -1, 0, 0, -2, 2, -2, -6, 4, 1, 3, 2, 0], [6, -10, 2, 2, -3, 0, -4, 4, -2, 6, 4, 4, -7, -3, 0, -5], [-7, -5, 2, 3, -2, 2, -3, -1, -2, 0, -1, 0, 4, 1, -7, 2], [9, -3, 3, 3, 2, -4, 11, 3, -2, 0, -4, -7, 2, 3, -1, 6], [4, -6, 2, 0, 4, 4, -13, 0, -4, 0, 6, 3, 5, -2, -1, -1], [-11, 6, -2, -1, 3, 2, -2, -2, -1, -5, 0, 0, 2, -2, 2, -4], [7, -5, 5, 4, -5, -4, 8, 6, -2, 5, -1, -3, -5, 2, 2, 0], [1, 5, 1, 3, 1, -6, 10, -2, 7, -5, -9, -4, -3, 6, 0, 0], [0, -5, 2, -6, 1, 3, -5, -4, 4, 3, -3, 11, -2, 5, -3, -5], [4, 1, -2, -2, -6, 3, 13, 2, -3, -4, -2, 2, 1, 0, -1, 3], [-4, 4, -4, 1, -1, 9, 6, 2, 4, -5, 1, -2, -1, -2, 0, 0], [-12, -3, 0, -3, 3, 6, -11, -6, -2, -1, 0, 5, 4, 3, -1, 0], [-1, 1, 1, 4, 5, 2, 2, 7, 4, 2, -1, 2, 0, -4, -2, -1], [13, 0, -4, 3, -3, -3, -9, 7, -5, 7, 6, -3, -3, -4, 2, -1], [1, 4, 4, 4, -5, 4, 8, 5, -2, 1, -4, -2, 7, 1, -5, 7], [-13, 6, -5, -1, 7, 3, -3, -7, 7, -2, -2, 3, -4, 4, 4, 0], [21, -6, 7, 1, -1, -9, -10, 0, -11, 12, 2, 4, -1, -5, 1, 1], [-12, 9, -5, -3, -2, 4, 5, -4, 7, 1, 5, 2, -4, 0, 2, 1], [-1, 3, 4, -1, -2, 4, 2, -5, 4, -5, -7, 10, -4, 4, -4, 0], [-1, 6, -2, 2, -1, -1, 10, -3, 7, -9, -7, 0, 0, 2, 0, 1], [-1, 4, -6, 5, 5, 7, 8, -5, 7, -14, -8, -2, 4, 6, -4, 6], [-1, -3, -2, 1, -3, 0, 11, -1, 15, -5, -6, 4, -8, 10, 9, -1], [-9, 4, -3, 1, 1, 7, -2, -4, 11, -12, -5, 5, 2, 5, -2, 0], [9, -5, 5, -2, -7, 3, 6, -2, -7, -3, 2, 1, 2, 2, -9, 2], [-16, 0, -4, 0, 0, -3, 4, 0, 3, 0, 1, 3, 2, 0, -1, -1], [-13, 7, 0, -2, 4, 1, -1, -3, 9, -1, 1, 0, 1, 5, 4, -1], [5, -1, 5, 6, -3, -5, 3, 11, -1, 9, 7, -7, -2, -5, 0, 4], [0, -4, -3, 4, -1, 5, 0, 1, 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9, -1, -1, 7, -11, 0, -3, 0], [-16, -10, 2, 1, 0, 12, 6, -2, -3, -5, -2, 3, 8, 5, -7, 8], [-13, 2, -3, 1, 8, 4, -9, -3, -4, 3, 7, 4, 1, -2, -4, -8], [-5, 0, -7, -3, 10, -1, -10, -5, 6, -5, 4, 0, -5, 4, 10, -2], [-1, -14, 2, -2, -12, 2, -5, 1, -5, -5, 3, 3, -2, 8, 8, -3], [-6, 5, -6, -3, -2, 9, 13, -4, 12, -6, 0, 4, -6, 5, 6, 8], [-12, 17, -8, 8, 6, 3, 3, -2, 16, -2, -7, -3, -1, 4, -6, 6], [4, -5, 1, 9, -2, 4, 7, -5, 2, -8, -8, -2, 13, 9, -5, 5], [-2, 8, 4, 1, -1, -4, -5, 2, -8, -4, 1, -7, -2, -4, 5, -1], [-11, -3, 11, -3, 5, -8, -22, -13, 4, 2, 1, 1, 1, 6, 4, -1], [-22, -8, -4, -5, -8, 0, 12, -10, 16, -3, -10, 10, -12, 4, 9, -7], [-5, -5, 1, -10, -6, 4, -11, -1, -4, 2, 6, 13, -4, -2, -5, -5], [4, -5, 6, 6, 7, -9, -3, -6, 1, 5, -10, -4, 0, 5, 0, 6], [-1, 8, -8, -3, -13, 3, 17, 1, 2, -4, -1, 0, -5, 1, 5, 4], [-8, -7, -10, 4, -4, 10, 1, 7, -5, -1, -3, -1, 8, -8, -5, -4], [26, 9, 4, 6, 8, -11, -7, 7, -9, -1, -3, -8, 5, -8, 0, -5], [-18, 9, -18, 9, 3, 6, 2, -2, 7, -10, -6, 5, -1, 1, 7, 0], [38, -16, 12, 0, -4, -9, -12, 0, -17, 17, 6, -4, -1, -6, -1, 2], [-15, 4, -9, 2, 4, 7, -14, -4, 11, -8, -2, -1, 13, 4, -3, 0], [-9, 9, -2, -6, 9, 9, 1, -6, 7, -15, -1, -1, -5, 4, 0, -5], [0, 8, -1, -3, 3, 5, -24, 4, -11, 1, 11, -5, 4, -10, 0, -6], [8, 12, 0, 7, -6, -9, 16, 4, 1, 0, -7, -5, -4, 5, 1, 0], [17, -15, 4, 7, -1, -2, -7, 5, -6, 8, 0, 2, -3, -14, -8, -1], [-20, 8, -3, -10, 5, 4, 1, -4, 0, 4, 4, 0, 10, -7, -10, 0], [12, -5, 5, -2, -3, -8, 3, 6, -5, -1, 3, 0, 1, 4, 7, -7], [-18, 3, -17, 4, 0, 11, -6, 2, 10, -16, 4, 6, 1, 6, 0, -2], [-8, -5, 10, -6, -2, -2, 3, 0, 4, -1, -3, -2, -8, 4, -1, 7], [5, -22, 1, 9, 9, 7, -1, 1, 0, -6, 2, -4, 10, -5, -12, 3], [-16, -6, -1, -6, 4, -6, 0, -8, 6, 2, -3, -1, -2, 7, 5, 2], [-6, 15, 0, -3, 6, -3, -6, 5, 3, -9, 9, -6, 3, -5, 5, -4], [-5, -9, 3, -5, -1, -3, -7, -7, 9, -1, -2, 0, -5, 3, 7, 2], [10, 2, 10, 8, 3, -1, -5, -3, -3, 1, 0, -8, 10, -4, -10, 5], [15, -16, 8, 7, 1, -12, -3, 4, -14, 17, 0, 2, 5, -5, -3, 2], [0, 6, -4, 5, 1, -7, 22, 7, -8, -4, 4, -14, -1, 3, 8, 4], [2, 13, 1, -4, 7, 5, 14, -4, 8, -13, -1, -13, 3, 5, -4, 6], [2, 4, -1, 0, -1, 4, -22, 2, -7, 9, 13, -2, 9, -7, -3, 2], [-5, -3, -7, 5, -3, -3, 10, 9, 3, 4, 1, -5, 8, 1, -1, 1], [12, 0, 1, 1, 2, -2, 3, 10, -10, 2, -1, -11, 16, -8, 3, -3], [-2, -2, 2, 5, 0, -9, -5, -2, -10, 2, 10, -14, 4, -9, -1, 1], [-6, -3, 10, 2, 4, -4, -23, 0, -9, 7, 7, -6, 7, 2, 0, 1], [-9, 4, 2, 3, -10, 2, 23, -1, 10, -15, -8, -2, -4, 7, 5, 5], [0, 3, -1, 0, 11, 0, -9, -4, -2, 7, 3, 0, 2, -6, -3, 1], [-9, 18, 3, -6, 1, -6, 5, -3, -7, -7, 1, -8, 13, 2, 0, 2], [0, -9, -8, -2, 2, 4, 4, 2, 7, 1, -3, -3, 2, -3, 6, 0], [-47, 15, -8, -8, 5, 20, 13, -7, 21, -24, -9, 6, -8, 19, 0, 0], [39, -1, 7, 6, 2, -3, -15, 9, -32, 14, 18, -10, 17, -13, -10, 6], [15, 4, -3, -6, 7, -2, 1, 3, -6, 4, 5, -10, 8, -10, -2, -2], [28, -17, 5, -3, 1, 1, -6, 0, 2, 6, -1, 13, -3, 3, -6, -3], [-19, -8, 2, 1, 2, 1, -4, -2, 11, -2, 1, 10, 1, 7, 4, 1], [18, -4, -1, -3, 12, -2, -11, 4, 3, 7, 7, -3, 1, -2, -4, -5], [11, -15, 7, -5, -4, 6, 11, 9, 1, 4, 7, 2, -6, 4, -5, 3], [3, 13, 6, -3, 4, -7, -10, -9, -2, -6, -6, 2, -4, 11, 4, 2], [-18, -15, 0, 4, -8, -1, 5, 2, 4, 4, 9, 4, -8, 0, 3, 1], [-3, 11, 6, 5, 4, 4, 11, 1, 3, -10, 5, -5, 2, -3, -4, 7], [-10, -15, 11, -6, 4, -2, -11, -4, 0, 4, 0, -3, 8, 7, 1, 3], [6, 17, 5, 8, 1, -4, -7, 0, -15, 2, -3, -6, 5, -6, -9, -1], [27, -15, 11, 1, -9, -9, -15, 10, -22, 23, 13, -4, -6, -21, -2, -6], [10, 4, -7, 10, 1, -6, 18, 6, -1, 9, -7, -1, 4, -8, 0, 1], [16, 0, 8, 9, -3, 1, 3, 7, -11, 0, -1, -1, 9, -4, -6, 1], [10, -9, 5, 2, 11, -16, -11, -2, -6, -1, -4, -6, 8, 3, 3, -1], [14, 3, -1, -3, -5, -2, 15, 6, -3, 17, 7, 1, -6, -10, -1, 5], [16, 5, -4, 8, -4, -10, -3, 10, -15, 15, 17, -11, -2, -14, -2, 2], [-33, 16, -5, -2, 14, 5, -2, -7, 15, -8, -3, -5, -9, 0, 1, -4], [-3, -15, 6, 1, 8, 4, 6, 0, 13, -13, -1, 8, 1, 10, 7, -2], [-3, 0, 8, -1, -1, -11, -3, -5, 0, 11, -3, 0, -7, 9, 5, 2], [-1, 6, 4, 0, -12, 0, 16, 5, -13, 3, 2, 2, 5, -10, -7, 4], [-23, 3, -12, 14, 3, 0, 15, 1, 26, -8, -10, 3, -16, 10, 4, 3], [34, -10, 10, 11, 4, -9, -5, 11, -11, 7, 9, -4, 5, -9, -1, -3], [20, -11, 3, -3, -11, -1, -1, 4, -12, 13, 20, 7, -9, -13, 2, 3], [0, 2, -5, 6, -3, 1, 13, -4, 15, -9, -3, -13, 2, 7, -5, 3], [-36, 11, -10, -7, 0, 4, -8, -12, 9, -13, -11, 9, -6, 15, 4, 3], [16, -20, 7, -3, 1, -2, -5, -2, -3, 19, 7, 3, 4, -1, 0, -4], [0, 5, -1, -7, -2, -8, 2, 6, 7, 6, 0, -3, -13, 4, 10, 2], [13, -10, 20, 4, -5, -13, 5, 1, -16, 12, 1, -7, 0, 1, -1, -2], [7, -6, -11, 3, 4, 2, -14, 7, -14, 10, 20, -1, -2, -11, 3, -5], [-24, -8, 4, 2, 0, -8, -8, 0, -6, 7, 4, -6, -8, -3, 2, 6], [-3, 6, -9, 10, 3, 6, -5, 12, -1, 8, 16, -9, 0, -4, 2, 1], [20, -11, 0, 19, 0, -12, -6, 18, -3, 8, 2, -13, 0, -7, 1, 1], [-6, -5, -4, 6, 1, -10, -6, 6, 2, 8, 10, -13, -2, -2, 12, 1], [6, -5, -5, 1, -1, 0, 1, 4, 9, -3, 2, 8, -11, -10, 4, -2], [2, 6, -1, 2, 0, 5, 10, 2, 17, 4, -8, 13, -3, 5, -1, -4], [-18, 3, 0, -17, -2, 13, -15, -16, 1, -15, 11, 2, -1, 1, -1, -4], [10, 1, -2, 6, 6, 0, -6, 5, -8, 6, 10, -9, 7, -9, -11, 0], [-39, 14, -16, -5, 0, 15, -6, -7, 11, -10, 7, -1, -6, 3, 4, -4], [2, -14, 4, -7, -4, -1, 8, 8, 0, 8, 1, 6, -10, 2, 1, -5], [28, -5, 1, 9, 0, -13, 10, 6, -8, -4, -5, -7, 4, -4, -3, 2], [25, -12, 7, -6, -3, -4, -9, -5, -26, 0, 3, 3, 5, -4, 4, -1], [-3, -7, 10, 0, -3, 1, -1, 5, 9, 0, 6, 5, -1, -2, 0, -1], [-18, 9, -1, -4, -2, -1, 12, 0, 11, -7, -9, -5, -5, 6, 7, -7], [-15, 1, -5, 14, 9, -9, 1, -1, 2, 2, -8, -9, -1, -5, -9, -2], [-27, 7, -7, -1, 5, -4, 4, -3, 6, 0, -6, 0, -10, 6, 10, -12], [0, -9, -5, -3, 3, 5, -7, 0, 5, 0, 2, 5, -4, 2, 2, -3], [23, -6, 8, 1, 2, -9, -5, 8, 4, 20, -7, -3, 5, 2, -5, 2], [-27, 3, -5, -3, 2, 13, 3, 11, 4, 10, -1, 4, -1, 0, 4, -3], [4, 18, -8, 11, 7, -3, 5, 0, 16, -15, -19, 2, -7, 10, 4, 4], [6, -4, -2, 7, -1, 0, 8, -2, 3, -2, 0, -9, 3, 1, -5, 3], [-15, -3, -15, 5, 0, 6, 15, 7, 4, 0, -1, -3, -2, 0, -1, 12], [11, -6, 15, 5, -3, -7, -10, 1, -16, 8, 6, -4, 0, -3, -4, -7], [-11, -10, -7, 2, 5, 8, 1, 1, 2, 3, 5, 0, 2, -6, 2, -8], [-9, 5, 0, 1, -7, -19, -6, -1, -21, 7, -4, -9, 15, -7, -4, 4], [7, -20, 3, -9, -6, 4, -10, -5, -18, 15, 15, 8, 3, -8, -4, 6], [1, 2, 8, 0, 10, -5, -10, 3, 2, 8, -1, -9, 3, 0, 1, -1], [11, -4, 7, -5, -5, -1, -11, -2, -13, 8, 5, 8, -11, -2, 9, -7], [-2, -9, -1, -14, 10, 6, -36, -6, -20, 6, 12, 8, 7, -11, -5, -8], [-24, -1, 0, -2, -8, 4, -12, -8, -11, -3, 13, 4, -1, 2, -3, -5], [-8, 14, -14, -8, -10, 19, 11, -6, 9, -6, 0, 2, -7, 8, -5, 4], [-8, -1, -15, 6, 3, 6, 18, -1, 18, -19, -1, -5, -5, 5, 3, 0], [33, -28, 9, 13, -10, -15, -2, 5, -6, 26, 4, -1, -8, -4, 2, 1], [-10, 3, 1, -6, -10, 19, 19, -8, 20, -6, 5, 7, -12, 14, 0, 3], [2, -18, 17, 7, 0, -11, -28, -7, 2, 4, -8, 6, 7, 3, -1, 2], [-22, 5, -8, -3, 6, 4, -4, 4, -10, 2, 13, -4, -8, -13, -5, -6], [-9, 7, 3, 1, -4, 8, 23, -9, 18, -15, -14, 4, -15, 26, 1, 3], [1, -16, -6, 4, 11, -1, -18, 3, 5, -3, -7, 4, 8, 4, 6, -1], [-6, -6, -7, 0, -5, 1, 11, 5, 5, 4, 2, -1, -16, 7, 6, 3], [5, -4, -1, 0, -2, 10, 18, -3, 16, -6, -14, 13, 0, 5, -4, 10], [9, -16, 2, 0, -1, 4, -20, 7, 5, 15, 4, -2, -4, -8, 3, -7], [-14, 1, -3, 4, 16, -3, -14, -1, 8, 2, -6, -3, 10, 2, 4, 5], [-10, -4, -3, -9, -4, 3, 8, -10, 12, -4, 7, 6, -13, 3, 13, -4], [-9, -8, 2, 13, 5, -1, -4, 5, -5, -12, 1, -12, 12, -5, -3, 0], [26, -10, 7, 11, 4, -5, 12, 10, -1, -5, -4, 2, 6, 0, -2, 2], [-37, 7, -5, -5, 14, 6, -13, -6, 3, -7, 9, -7, 7, -2, -6, -10], [2, -5, 1, -8, 8, 5, -5, -10, 5, -3, -6, 11, -9, 11, 1, -9], [22, -10, 6, 9, 0, 7, -6, 7, -9, 14, 11, -12, 9, -4, -1, 3], [4, 0, 6, -4, -1, -3, -7, -6, 0, -4, 0, -2, 3, -4, -3, -5], [-15, 10, -9, 9, -3, 7, 9, 9, 2, 0, -5, -2, 6, -8, -10, 9], [-1, -2, 6, -7, 9, -9, 8, -5, 1, -6, -1, -1, 1, 12, 4, 5], [-6, -4, -8, -6, -9, 9, 21, -2, 9, -2, 7, 7, -15, 3, 1, -10], [-4, -16, -6, -3, -8, 13, -3, 0, -8, -1, 2, 9, 3, -3, -5, -6], [-2, 2, 12, -6, -14, -5, 1, -4, -8, 13, -12, 5, 5, -3, -1, 1], [7, 1, -10, 3, 7, -13, -15, 5, -17, 2, 7, -7, 7, -12, 9, -6], [12, -22, 14, 1, -9, -2, -16, 2, -14, 11, -5, 0, 3, -4, -4, -6], [-13, 7, -1, -8, 12, 3, -1, -8, 7, -4, 0, -6, 6, 6, 2, 2], [19, 11, 2, 11, 1, -6, 14, 5, 6, -4, 5, -11, -8, 4, 7, 6], [8, -7, 8, -8, -8, -12, 7, 6, -14, 13, 13, 0, -16, -10, 2, 1], [-9, 2, -20, 3, -4, 6, -3, 5, -8, -5, 6, -2, -4, -8, 3, 1], [-5, -1, -9, 7, 6, 13, 0, -1, -2, -12, -6, 3, 10, 9, 1, 6], [-16, 5, -8, 2, 1, 14, -2, 3, 9, -13, -3, 10, 8, 6, -11, 8], [2, 3, 2, 7, -5, 1, 33, 0, 23, -18, -17, -5, -7, 13, -2, 10], [10, 6, 4, -5, -6, 5, -1, 0, -6, 11, 6, 9, -9, 3, 2, -6], [-2, -9, -5, -10, -13, 16, -5, 7, -8, 4, 19, 16, -6, -4, 2, -9], [21, 1, 3, 4, -13, -9, 9, 3, -12, 22, 5, 4, -16, -5, 4, 1], [-6, -8, 13, 8, -9, -6, -10, 4, -13, -1, 1, -2, 9, -8, -1, 4], [3, 27, 0, 0, 4, -2, 9, 7, -5, -5, 6, -2, 9, -9, -4, 5], [-7, 1, -1, 0, 2, -5, 6, -7, 0, -11, 1, -4, -1, -5, 3, -2], [22, -9, 11, -6, -6, 0, 8, 2, -9, 17, 5, 6, -10, 3, 7, -4], [15, -1, 13, 4, 0, -11, -17, -10, -6, 0, -2, 5, 4, 0, 0, -3], [1, -11, 6, 4, -4, -8, -2, -11, -3, 3, -4, 1, 6, 0, -5, 12], [-30, 1, 2, -5, -6, -4, 14, 2, -6, 1, -5, -1, 5, -1, 3, -1], [-3, -5, -4, -5, 0, 9, 1, -1, -5, -11, -8, -3, 2, 1, -6, -1], [16, -6, -3, 6, -2, -1, 26, 0, -1, -5, -5, 8, 2, 5, -2, 9], [12, -2, -5, 7, 10, -1, -18, 16, -18, 19, 10, -5, 14, -25, -8, -11], [7, 6, 2, 6, -2, -1, 17, 11, -4, 0, -6, -8, -2, 0, 7, 4], [-3, -4, -3, 0, -2, 0, -15, -1, 3, 17, -2, 5, -7, -12, -5, 6], [0, -13, -1, 17, -3, -4, 20, 14, 11, 11, -4, -5, 0, 5, -5, 7], [25, -2, 2, 8, 7, -4, 2, 11, -12, 11, -4, -2, 13, -8, -8, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8037_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8037_2_a_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_8037_2_a_n(:prec:=16)
    chi := MakeCharacter_8037_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_8037_2_a_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_8037_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8037_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-25, 62, 179, -456, -487, 1250, 618, -1639, -320, 1115, 4, -390, 47, 65, -13, -4, 1]>,<5,R![-176, 824, 11080, 31308, 27093, -16418, -34504, -1285, 16349, 2275, -4109, -481, 559, 38, -38, -1, 1]>],Snew);
    return Vf;
end function;