
// Make newform 8037.2.a.m in Magma, downloaded from the LMFDB on 24 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_m();".
// 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_m();".
// 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 := [-17, 83, -49, -229, 225, 196, -213, -76, 84, 14, -15, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, -14, -14, 23, 7, -9, -1, 1, 0, 0, 0, 0], [4, 2, -11, -33, 8, 73, -5, -48, 1, 12, 0, -1], [-13, 53, 16, -139, 0, 95, -1, -24, 0, 2, 0, 0], [-4, 28, -24, -99, 76, 116, -53, -59, 13, 13, -1, -1], [-10, 29, 43, -106, -70, 125, 43, -61, -11, 13, 1, -1], [21, -65, -53, 182, 31, -170, -5, 72, 0, -14, 0, 1], [18, -59, -20, 175, -42, -165, 43, 71, -12, -14, 1, 1], [21, -63, -21, 176, -42, -165, 43, 71, -12, -14, 1, 1], [32, -89, -70, 249, 35, -217, -5, 84, 0, -15, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1, 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_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], [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1], [-1, 0, 0, 1, 0, 0, -1, 1, 0, -1, 0, 1], [-1, 1, -1, -1, 0, 0, 0, -1, 1, 0, 1, 0], [1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, -1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, -1, 0, -1, -1, -1, 1, 0, 1, -1], [-2, 0, -2, 0, 0, -2, -1, 2, 0, 1, -1, 0], [-3, 0, 1, 2, -1, 0, -2, 1, -2, 0, 0, -1], [2, -1, -1, -1, -1, -2, 1, -1, 2, 2, -1, -2], [1, -1, 2, 0, -2, -1, 2, 0, -3, 2, -2, 1], [0, -2, 2, -1, 0, 3, 2, -3, -2, 0, -2, 3], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, -1, 0, -2, 2, 0, -1, -1, 1, -2, -1], [-2, -2, -2, 1, 1, 0, -2, -1, 0, 1, 1, -1], [0, 1, 0, 2, 0, -1, 0, -1, 2, -1, 0, -3], [-4, -2, 1, -1, 0, -1, -1, 2, 0, -1, 3, -1], [0, 0, 3, 2, 1, 1, -2, -1, -3, 0, -1, 2], [-3, 0, -4, -1, 4, 2, 1, -3, 2, 1, -1, -2], [-1, 0, 1, -2, 0, 1, 1, 3, 1, -2, -1, 5], [4, 2, -3, 0, 2, 0, -2, -2, 2, 2, 2, -3], [0, 1, 2, -3, 2, 0, 1, 1, 0, -2, 2, 1], [-4, 2, -1, 2, 1, -2, -1, 1, 1, -2, 2, -6], [-1, 1, 2, 0, 3, 2, 0, -2, 1, -2, 2, -1], [-2, -2, -3, -1, -1, 1, 1, 0, 0, -1, 1, 1], [5, -1, 3, 0, 1, 5, 2, -5, 0, -2, 0, 4], [0, 0, -2, 1, 2, 0, -1, -1, 3, 1, 0, -2], [2, 3, -1, -4, -1, -1, -1, -3, 5, -5, 2, -1], [-4, 0, 0, 1, 4, 2, 1, -3, -4, 3, -1, 0], [-4, -2, 2, 4, 2, -1, -1, 1, 0, -1, -1, -2], [3, -1, -4, -4, -3, 2, 2, -2, 0, 4, 0, -2], [-5, -2, -4, -1, 0, 1, -5, -2, 3, 0, 2, -4], [-1, -3, 0, 1, 0, -2, -1, -1, 2, -3, -1, 0], [1, -3, 4, 4, 0, 0, 2, 3, -2, 1, -4, 0], [2, 0, 1, -1, -1, -2, -1, -2, 2, 3, -5, -1], [-4, -3, 1, 1, -4, 0, 0, 1, -2, 4, -4, 1], [2, -4, 1, 0, 0, 2, 2, 0, -1, -2, 2, 0], [-6, -1, -1, 2, 2, 0, 1, -2, 2, -1, 1, -3], [-7, 2, -3, -2, -3, -7, -3, 7, 3, 0, 4, 1], [8, -3, 0, -1, 1, 4, 4, -7, 2, 0, 1, 0], [2, -1, 5, 2, 3, 3, 0, 1, -3, 3, -4, 5], [4, -5, 4, 2, 0, 0, 4, -3, -4, 6, -3, 1], [0, 3, 1, -6, 0, 0, -1, 0, 2, -1, 5, 2], [-8, 0, 0, 4, 4, -1, -3, -1, -2, -3, -2, -5], [-2, 2, 2, 3, 0, -5, 2, 5, -1, 1, -4, 2], [-4, 4, 2, 0, 4, -1, -3, -1, 4, -5, 1, -1], [2, 0, 1, 3, -2, -2, -2, 3, -1, 0, 2, -1], [-1, -1, -3, -3, 1, -3, 1, -3, 2, -2, 5, -4], [-2, -1, 3, 2, 1, -4, 0, 1, 0, -3, -1, 2], [-9, 3, -6, 0, -7, -7, -6, 5, 0, 2, 3, -5], [0, 9, -2, 1, 5, 2, 1, -1, 0, 2, -3, -2], [16, -4, 4, -1, 2, 1, 10, -4, -1, 6, -6, 4], [4, 1, -2, 3, 6, 5, 1, -5, 3, 1, -1, -3], [6, -3, 6, 3, -1, 2, 4, -2, -11, 8, -2, 8], [0, 4, 3, -3, -3, 5, 1, -1, 0, -1, 4, 3], [-4, 1, 2, 3, -2, -1, -7, 6, 2, -6, 0, 1], [5, -5, 5, 0, -4, 3, 6, -5, -8, 7, -1, 1], [-3, -2, 1, 2, -1, -4, 0, 4, 0, -3, -1, -1], [-9, -3, -3, 3, -2, -1, 2, 1, -1, -2, -1, -3], [5, 0, 5, -4, 0, 5, 7, -2, 0, 1, 0, 7], [-9, 4, -2, 0, 0, -2, -4, 5, 5, -3, 3, -1], [2, 3, 7, 8, -1, 1, -4, 3, -4, 0, 0, 2], [0, 7, -2, -3, -3, 1, 4, -1, 6, -6, 3, -3], [-1, -3, -1, -1, 2, 0, 0, 3, 1, -2, 3, 5], [6, -1, 1, 1, -1, 4, 0, -1, -7, 4, -6, 6], [-8, 2, 6, 0, -3, -1, -3, 8, -3, -2, 2, 11], [14, -9, -3, -2, -6, 0, 4, -3, -5, 14, -6, -5], [-3, 0, 6, 1, -1, -1, -2, 1, 1, 0, 3, -4], [-1, 1, 1, 0, -3, 0, 1, 5, -6, 2, -1, 0], [-2, -3, -1, -2, -4, -3, 2, 4, 0, 5, -1, 6], [-7, 2, 1, -3, -2, -1, -2, 3, 0, 4, 2, -5], [5, 4, 5, -7, -4, 0, 8, 1, 1, -3, 5, 5], [6, -6, 1, -2, -1, 7, 5, -6, -3, 1, -6, 9], [-6, 4, 12, 3, 4, 2, 1, 3, -6, -5, 0, 10], [2, 4, -8, 4, 6, 0, -4, 0, 7, 0, -1, -10], [2, 2, -3, -1, 4, -4, 2, 4, -1, 2, -1, 4], [3, -5, 5, -6, 0, 3, 9, -6, 2, 2, -3, -4], [0, 5, 5, 1, -1, 0, 2, 6, -6, 2, -3, 13], [-5, 12, -2, 4, 6, -2, -2, 5, 3, -7, 6, -7], [5, -3, 1, 0, -4, 2, 1, -1, -4, 5, -5, 6], [-4, -4, -2, -5, -5, -6, 3, 1, 3, -1, 1, -3], [-19, 3, -3, 3, 1, -6, -9, 7, 5, -5, 7, -1], [5, 2, -2, -2, 5, 4, -3, -8, 2, 3, -4, -3], [-9, 7, -10, -2, 8, -3, -1, -1, 9, -11, 2, -7], [-5, 3, 2, -2, -4, -1, -2, 8, 4, -2, 5, 1], [-4, 0, -4, -2, 2, 3, 1, -3, 2, -4, 7, -5], [0, -3, -3, 1, 3, 5, -8, -3, -1, 1, -4, -1], [0, -8, 6, 4, -4, -3, 7, 5, -5, 7, -4, 4], [7, -2, 0, -4, 3, 5, 7, -6, -3, 1, -8, 12], [-14, 3, -8, 3, 7, -3, -4, -2, 4, -1, 6, -13], [-1, -1, 5, -2, -2, 7, -3, 4, 2, -6, -3, 9], [7, -1, -3, 5, 2, 6, 4, -9, 0, 2, 1, -6], [-5, -5, 3, 2, 6, 0, 1, -3, 0, -1, 5, -2], [-7, 3, 7, -4, -2, 4, 0, -1, -3, -7, -1, 1], [16, 1, -1, 3, 11, 2, 7, -10, 8, -2, -1, -7], [-18, 7, -1, 0, 1, -3, -10, 8, 1, -7, 5, -6], [-19, -5, -5, 12, 1, 3, -1, -2, -6, 5, -6, -6], [1, 3, -2, 5, 5, 5, 0, -7, 8, -5, -1, -2], [-12, 4, -2, 2, -3, -9, -7, 4, -5, 9, 2, -5], [1, -7, 8, 0, -5, 4, -3, 3, -7, 2, -5, 16], [-10, 0, -1, 11, 2, -4, 1, 6, 3, -5, -8, -2], [3, -2, 2, 6, 1, 3, -2, -3, -3, -5, 1, -6], [-7, 5, 3, 7, -2, -11, -11, 8, 0, -7, 6, 0], [-20, 12, -5, -5, 0, 3, -5, 3, 11, -11, 6, 4], [6, 2, 0, 5, -2, 1, -4, 4, -4, 4, -8, 7], [5, 6, 1, -5, -4, -8, 2, 2, 3, 6, 1, -5], [-2, 0, 6, 6, -5, -4, -5, 9, -4, 5, -2, 7], [2, 7, -3, -2, 4, -1, -4, 5, 5, -6, 5, 0], [4, -6, -4, -3, -4, -9, 0, 1, 3, 8, 2, -10], [6, -7, -5, -11, 2, 7, 11, -13, 6, 1, -4, 6], [1, 4, 7, 6, 5, 3, 2, 0, -1, -6, 1, -1], [-10, 1, -4, 3, 3, -4, 3, 0, 1, 1, 1, 3], [18, -11, 7, 1, 2, 14, 10, -9, -6, 5, -9, 9], [12, 5, -2, -13, -4, 2, 14, -10, 5, 4, 8, 1], [-16, -2, 5, 6, -3, 0, -5, 3, -4, 5, -6, 9], [4, 3, 5, -8, -2, 0, 13, -3, -6, 1, 0, 13], [-10, 6, 2, 1, -2, -10, -1, 10, -6, -1, 2, 6], [4, 2, 1, 0, 0, -1, 3, -2, -3, 1, 1, 1], [-4, 11, -12, 0, -2, 0, 0, 3, 5, -7, -2, -8], [6, 4, -1, -10, -6, -1, -4, -2, 1, -2, 4, -4], [-7, 0, 1, -3, 1, 1, -1, 3, -3, 5, -6, 7], [-22, 9, -2, 4, 0, -6, -11, 12, 6, -10, 10, 1], [-13, 4, 1, -3, -6, -7, -3, 7, -12, 8, 4, 8], [0, 0, 1, 3, 6, 1, 1, -6, 4, 3, -5, -6], [8, 4, -6, -13, 5, 3, 10, -6, 6, -2, 2, 4], [1, 0, 2, 0, 0, 0, 7, 0, -1, 3, 3, 5], [-5, 9, 1, -2, -2, 3, 0, 5, -1, 4, -1, -1], [-4, -4, 8, -1, -8, -8, 1, 4, -4, -7, 11, 1], [1, 6, 2, -7, -4, 2, 1, 6, 2, -8, 6, 3], [-7, -2, 1, -3, -2, 1, 1, 1, 3, -1, -8, 4], [2, -4, 1, 1, -6, 11, -3, 1, -5, -2, -7, 7], [-3, 6, 7, 3, 11, 4, 3, -4, -7, 4, -9, 8], [-9, -1, -3, 4, -1, -3, -1, -5, 4, 6, -9, 0], [2, -11, -4, 7, -2, -5, 0, 1, -1, 5, -1, -9], [-11, 0, -9, -7, 0, 2, -4, -3, 10, -11, 8, -13], [5, 4, -9, -1, 0, -6, 1, 3, 8, 5, -2, 0], [16, -15, 2, 0, -1, 0, 1, -8, -12, 8, -1, -4], [0, 3, 0, 2, 12, 3, -3, -8, -3, -1, -5, -6], [-9, -2, 3, 4, -11, -11, -1, 11, -4, 5, -1, 2], [2, -3, 7, 1, -7, 7, -5, 5, -13, 6, -3, 9], [-9, -4, 1, -1, 0, 3, 0, 1, -3, 5, -2, 17], [8, 4, -5, 4, 8, 3, 3, -1, -2, 9, -12, -5], [-3, -4, -3, 5, 1, -3, -3, 3, -7, 2, -6, -1], [-8, 7, -3, -2, -6, -5, -5, 6, 10, -8, 11, -10], [13, -6, 1, -1, 2, 3, 2, -4, 4, -8, 2, -3], [-18, -5, 6, 8, 1, -4, -5, 4, -5, 4, -5, 2], [-14, 7, -14, 4, 14, 0, 1, 3, 12, -4, 0, -8], [-25, 3, 4, 4, -2, -3, -11, 15, -11, 1, -9, 4], [12, 12, 1, 5, 10, 0, 1, 6, 2, -12, 7, -1], [-13, 4, 0, 12, 4, -2, -3, 6, 5, -8, 0, -9], [-11, 0, 0, 1, -5, -2, -4, 7, -3, -3, 5, 4], [-5, 11, -9, 11, 12, -4, -1, 2, 5, -3, 5, -17], [-1, 4, 4, -6, 10, 13, 6, -4, -2, -2, -1, 7], [3, -1, -4, -1, 6, 0, -8, -4, 10, -2, -8, 1], [-15, 5, -2, 0, -2, -8, -8, 12, 14, -13, 9, -5], [-12, 4, 2, 4, 6, -5, -1, -1, -6, 4, 2, -1], [-9, 1, 1, 11, -4, 1, -7, -3, -3, 8, -2, -15], [-3, 9, -7, -7, 8, 5, 5, -7, -4, 8, -5, 4], [-9, 2, -4, 3, -2, -9, -5, 17, -3, 10, -11, -7], [0, -2, -1, 0, -3, -3, 4, -5, 0, -2, 9, -9], [4, -3, 3, -6, -12, -2, 6, -3, -10, 6, 1, -1], [2, -7, 18, 9, 4, 0, -6, 6, -10, 5, -4, 0], [-2, 9, 4, -5, -3, 7, 1, 0, 1, 1, 1, 15], [6, -4, 5, 8, -2, 2, 2, -2, -14, 15, -8, 9], [-7, -9, 4, -2, -6, 4, -6, 8, -13, 0, 9, 3], [-17, 3, -12, -2, 7, 3, 3, -6, 0, 7, -2, -5], [7, 2, 0, 2, 9, 4, 0, -8, 2, -2, -5, -17], [-8, -5, 2, -4, -2, 5, 3, 1, -3, -6, -2, 8], [-6, 7, -12, -7, 2, -3, -4, -5, 7, -6, 14, -1], [3, -5, -1, 11, 2, -6, 4, -3, 6, -5, -13, -7], [-7, 13, 7, 6, 0, 4, -4, -2, 7, -7, 2, -7], [-11, 11, -3, -6, 0, -6, -11, 3, 21, -19, 13, -7], [-11, 4, -6, 3, 7, 1, -1, -3, 5, 0, 10, -6], [10, -8, 2, -2, 0, 7, 6, 2, -13, 6, 3, 9], [-7, 2, 10, 8, -4, -11, -10, 10, -3, 6, 3, 7], [20, -4, 7, 1, -10, 8, 1, -1, -2, 0, -7, 9], [-5, -15, 9, 5, -1, 6, -3, 8, -7, -4, -12, 3], [-12, -1, 7, 10, -8, -4, -16, 2, -14, 3, 1, 1], [-9, 19, 2, 2, 7, -7, -2, 9, -2, -2, 0, -3], [-10, 6, -11, -4, 7, 3, 3, 5, -9, 6, -11, 14], [4, 0, 2, -1, -2, -1, 13, -2, -1, 0, 1, -6], [-4, 7, 0, 3, 11, 7, 4, -3, -3, -4, -4, -4], [-5, -6, -1, -8, -4, 0, -9, 12, 5, -8, 4, 0], [11, 3, -8, -5, -6, -7, 0, 5, 11, -5, 4, 3], [-7, 1, -2, -4, -1, -4, 1, 4, -3, -12, 8, 17], [-23, 6, -8, -3, 9, 7, -2, -2, 7, -12, 11, 2], [2, -8, -9, 0, 13, 7, 2, -16, 11, -14, -7, -4], [-7, -7, -7, -2, 0, -3, 2, 11, -3, 8, -3, -4], [7, -16, 8, -2, -9, 5, 5, -2, -23, 14, -10, 11], [-8, -9, -8, -3, 1, -3, 0, -8, 6, 1, 2, -4], [-2, -3, 1, -5, -1, 9, 13, -6, -14, 8, -1, -3], [7, -8, -1, 4, -12, 2, 5, -2, 7, -5, -3, 4], [12, -4, 11, 4, -9, -5, -3, 0, -10, 1, 12, 10], [25, -13, -18, -4, -5, 2, 3, -5, 0, 18, -4, -2], [-3, -1, 15, -7, -10, 4, 4, -3, -13, 1, -2, 8], [-7, 6, -6, -7, -10, -12, -2, 13, -5, 8, 6, 7], [6, -3, -5, 0, 13, 7, -1, -16, 0, -7, 9, -8], [-6, -4, 1, 8, -3, 0, -8, 9, -14, 9, -4, 5], [-19, 9, -17, -1, 7, -6, 1, -2, 16, -1, -4, -21], [13, -5, -9, -3, -6, -4, 10, 6, -5, 7, -9, 5], [-10, -10, 0, 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10, -4, -6, -1, 3, 6, -5, 3], [22, 8, 3, -20, -15, -9, 10, -7, 16, -8, 17, -8], [9, 4, -22, -6, -3, -10, -1, -9, 25, 12, -5, -20], [-11, -10, 15, 19, -9, -8, 0, 2, -25, 20, -4, 2], [-8, 2, -11, 1, 2, -2, 11, 6, 0, -2, -10, 4], [17, 5, 6, 10, 4, 2, -4, -2, 3, -7, 8, -13], [-55, 17, -7, 4, 9, -9, -4, -5, 8, -15, 15, -10], [-30, 0, -18, 1, -4, -5, 8, 4, -6, 7, 4, -2], [-30, 12, -6, -5, -8, -10, -19, 22, 6, -22, 22, 2], [-3, 8, 2, 10, 12, -7, -6, 1, -9, 6, 0, 3], [19, 0, -5, 1, 12, 6, -9, 11, 2, -11, -1, 11], [-36, 9, -13, 5, 2, -23, -24, 21, 5, -10, 26, -25], [5, 5, 18, -3, -5, 5, 7, 8, -4, 10, -7, 9], [33, 1, 11, -4, -5, -2, 3, 13, -3, 12, -15, 29], [24, -12, 24, -7, 1, 7, 9, -5, -11, 9, 2, 6], [4, -7, 13, -6, -4, 11, 11, -4, -9, -10, 17, 24], [-21, -16, -17, 10, 9, -5, -19, 1, -4, 1, 8, -13], [-16, 2, -11, 11, -6, 9, 3, -3, -13, 28, -9, 9], [10, 5, -1, -10, -7, 13, 10, -8, -11, 5, -14, 19], [8, 3, -24, -4, 7, 4, 1, -19, 7, 1, 6, -8], [-9, 15, 0, 6, 3, 6, -16, 0, -4, -7, 0, 10], [34, 1, -9, 5, 9, -2, 8, -19, 22, 2, -6, -23], [-7, -4, 12, 18, -5, 1, -8, 11, -12, -10, 5, 18], [-33, 7, -1, 9, -1, 0, -7, 5, -6, 0, 1, 4], [2, 12, -15, -8, -2, -7, -5, 2, 19, 13, 9, -9], [22, -16, 25, 7, -4, 11, 25, 4, -23, 5, -23, 25], [-16, -7, -10, 0, 8, -8, -6, 10, -4, -1, -7, -4], [-2, -6, 0, -12, -9, 10, -1, 2, -8, -5, 8, 21], [-19, 10, 4, -2, 3, -6, -6, 14, 5, 12, -3, 9], [1, -10, -6, 2, -7, -1, 0, 5, 9, -4, -2, -5], [37, -16, -1, 3, 13, 0, 13, -6, -11, 15, -14, 5], [1, 5, 22, 0, 16, 8, 13, 2, 0, -31, 9, 17], [34, 9, 1, -3, -4, -1, 2, -12, 21, -4, 5, 2], [-16, 9, -14, 13, 5, -11, -1, 13, -10, 20, -1, -8], [-7, -3, 1, 7, -16, -6, 2, 23, -5, -19, -4, 6], [-3, -3, -16, -17, -13, 7, -5, 2, 25, -4, 2, -18], [4, -6, 14, 12, -6, 6, -7, 9, 5, -12, 6, 9], [32, -2, 4, -20, 2, 9, 8, -5, -4, 3, -8, 24], [11, -10, -1, -8, -7, 10, 23, -22, -10, 11, -5, 9], [23, 5, -10, -35, 1, 4, 10, -12, 14, 5, -4, 4], [-43, 2, -5, -1, -14, -7, -17, 5, 13, -18, 8, 4], [8, 3, -3, 10, 6, -18, 2, -4, 14, -3, -9, -11], [-9, -20, 5, 2, -22, -12, 1, 16, -17, 20, -9, 16], [4, 0, 5, 11, 8, 10, -1, -13, 4, 1, 16, -1], [-33, -20, -19, 15, 8, -15, -33, -2, 14, -8, 5, -26], [-15, 16, 4, 6, 7, 4, -16, -4, 4, -15, 10, -11], [-9, 10, 9, 0, 14, 4, -2, 1, -12, -6, -2, 23], [31, 17, 9, 11, 2, -4, 7, -16, 14, -16, -1, -10], [-4, -4, -12, -9, -3, 7, -3, 0, -18, 21, -15, 23], [-17, -5, 16, -3, -1, -3, 6, 8, -13, -11, 0, 2], [-20, -21, -2, 6, -11, 7, -21, 13, -10, -12, 13, 2], [9, 4, -5, -23, -14, 8, 5, -9, -17, 10, 20, 7], [-17, -3, -16, 2, 22, 8, -4, -2, -6, -1, -16, 6], [30, -17, 7, 0, -4, -6, 6, -17, 11, 9, -2, -13], [0, 4, 16, -6, -15, -4, 9, -3, -2, -6, -4, 10], [-10, 3, -26, -12, 15, 14, 6, -20, 13, -12, 25, -18], [19, -9, 15, 14, -7, 3, 0, -9, -10, 19, -8, -13], [-25, 15, -20, 0, 6, -5, -8, 12, 11, 9, 2, 2], [-1, -12, 19, 12, -6, 5, 4, 0, -33, 10, -13, 6], [13, -25, -3, 4, 18, 25, 9, -27, -13, 9, -39, -2]];
    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_m();".
// 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_m(:prec:=12)
    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_m();".
// 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_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8037_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-17, 83, -49, -229, 225, 196, -213, -76, 84, 14, -15, -1, 1]>,<5,R![51, -172, -78, 729, -511, -565, 771, -83, -245, 112, -2, -7, 1]>],Snew);
    return Vf;
end function;