
// Make newform 10000.2.a.bj in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_10000_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_10000_2_a_bj();".
// 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_10000_2_a_bj();".
// 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, -31, 0, 36, 0, -11, 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, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0], [1, 0, -5, 0, 1, 0, 0, 0], [-7, 0, 16, 0, -8, 0, 1, 0], [0, -7, 0, 16, 0, -8, 0, 1], [0, 27, 0, -40, 0, 12, 0, -1], [0, -43, 0, 44, 0, -12, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1, 4, 4, 4, 4];
    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_10000_a();"
function MakeCharacter_10000_a()
    N := 10000;
    order := 1;
    char_gens := [8751, 2501, 9377];
    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_10000_a_Hecke(Kf)
    return MakeCharacter_10000_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, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 1], [-2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 1, 0, 0, 0, 0, 1, -2], [-2, 0, 1, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 1, 1, 0], [4, 0, -2, 1, 3, 0, 0, 0], [-1, 0, 2, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 1, 0, 4], [2, 0, 1, -2, -1, 0, 0, 0], [0, -1, 0, 0, 0, -1, 2, 3], [0, 3, 0, 0, 0, -2, 1, -1], [0, -3, 0, 0, 0, -2, 2, -3], [-4, 0, -1, -2, 2, 0, 0, 0], [3, 0, 1, 3, 1, 0, 0, 0], [0, 4, 0, 0, 0, 0, 2, -2], [-7, 0, -3, 1, -2, 0, 0, 0], [0, -1, 0, 0, 0, 2, -2, 3], [2, 0, -3, 4, 4, 0, 0, 0], [0, 4, 0, 0, 0, -1, 5, -4], [0, 0, 1, -1, -7, 0, 0, 0], [0, 4, 0, 0, 0, -4, -4, -5], [7, 0, 1, -2, 9, 0, 0, 0], [0, -3, 0, 0, 0, -5, 2, 1], [0, 0, 0, 0, 0, -3, 3, -6], [-6, 0, 3, -2, -9, 0, 0, 0], [0, 0, 0, 0, 0, -1, 7, -1], [0, 0, 0, 0, 0, 2, 2, 4], [-5, 0, -3, 1, 2, 0, 0, 0], [0, 7, 0, 0, 0, 2, -2, -5], [-5, 0, 2, -5, -6, 0, 0, 0], [1, 0, -3, 1, -5, 0, 0, 0], [2, 0, -5, 0, 8, 0, 0, 0], [0, -1, 0, 0, 0, 1, 0, 2], [0, -4, 0, 0, 0, -7, 9, -2], [0, -3, 0, 0, 0, -9, 8, -5], [0, 1, 0, 0, 0, 1, -6, 0], [-5, 0, -2, -3, 4, 0, 0, 0], [0, 0, 5, -5, 1, 0, 0, 0], [-12, 0, 2, -4, -2, 0, 0, 0], [0, -1, 0, 0, 0, 7, 1, 3], [0, -3, 0, 0, 0, -3, 1, 5], [-11, 0, -3, 1, -4, 0, 0, 0], [1, 0, 4, -3, 2, 0, 0, 0], [0, -4, 0, 0, 0, -12, 0, -2], [0, 6, 0, 0, 0, -6, -4, -8], [-2, 0, -3, -2, -3, 0, 0, 0], [0, -1, 0, 0, 0, 7, 1, 11], [-3, 0, 4, 1, -4, 0, 0, 0], [9, 0, 5, 0, 9, 0, 0, 0], [1, 0, 2, 1, 4, 0, 0, 0], [0, 10, 0, 0, 0, -2, -1, -10], [0, 1, 0, 0, 0, 8, -1, 13], [0, 0, -3, 4, -5, 0, 0, 0], [-6, 0, 2, -4, 10, 0, 0, 0], [0, -9, 0, 0, 0, -5, -2, 6], [2, 0, -8, 6, 3, 0, 0, 0], [0, 1, 0, 0, 0, 8, -7, 7], [0, 5, 0, 0, 0, 4, -7, 2], [0, 12, 0, 0, 0, 6, -4, 0], [-9, 0, 1, 3, 10, 0, 0, 0], [0, -2, 0, 0, 0, 10, 0, 10], [0, -2, 0, 0, 0, -1, 9, -6], [2, 0, -4, -2, 2, 0, 0, 0], [0, -11, 0, 0, 0, -4, -3, 13], [0, -5, 0, 0, 0, 7, -8, 1], [-2, 0, 1, 3, -5, 0, 0, 0], [0, 5, 0, 0, 0, 4, -3, -3], [-11, 0, 3, 5, 4, 0, 0, 0], [0, 3, 0, 0, 0, -2, -7, -15], [0, -1, 0, 0, 0, -13, 2, -9], [9, 0, 2, -1, -12, 0, 0, 0], [0, -10, 0, 0, 0, 1, -3, 0], [7, 0, 1, -1, 7, 0, 0, 0], [0, 1, 0, 0, 0, 5, -2, 9], [-3, 0, -7, 9, -3, 0, 0, 0], [7, 0, -3, 0, 3, 0, 0, 0], [-9, 0, -4, 5, -16, 0, 0, 0], [-2, 0, 1, 3, 11, 0, 0, 0], [-15, 0, 2, 1, -8, 0, 0, 0], [0, -4, 0, 0, 0, 6, -9, -4], [7, 0, -11, 1, 6, 0, 0, 0], [0, 2, 0, 0, 0, 5, 1, 6], [-4, 0, -4, -7, 11, 0, 0, 0], [0, 0, 0, 0, 0, 8, -8, -4], [-1, 0, 11, -2, -7, 0, 0, 0], [0, -8, 0, 0, 0, 15, -1, 18], [0, 5, 0, 0, 0, 12, 3, -5], [-20, 0, 1, -4, 6, 0, 0, 0], [0, 6, 0, 0, 0, -1, -7, 2], [1, 0, -2, 9, -2, 0, 0, 0], [9, 0, 5, -5, -12, 0, 0, 0], [0, 4, 0, 0, 0, 4, 6, -6], [-21, 0, 0, 4, -1, 0, 0, 0], [-1, 0, 5, -5, -11, 0, 0, 0], [0, 5, 0, 0, 0, 1, -12, -11], [-6, 0, -7, 9, 1, 0, 0, 0], [0, 8, 0, 0, 0, 3, -11, 10], [0, -15, 0, 0, 0, -2, -1, 10], [0, 3, 0, 0, 0, 2, 5, -15], [9, 0, -11, 10, 11, 0, 0, 0], [23, 0, 0, 5, 0, 0, 0, 0], [0, -6, 0, 0, 0, -4, 2, -6], [0, -6, 0, 0, 0, 5, -3, 20], [0, -8, 0, 0, 0, 8, 4, 9], [-17, 0, 8, -11, -12, 0, 0, 0], [-7, 0, -6, -3, 23, 0, 0, 0], [0, -1, 0, 0, 0, 10, 7, 7], [0, -18, 0, 0, 0, -2, 8, 1], [0, 9, 0, 0, 0, 6, 0, 1], [5, 0, 2, 3, 6, 0, 0, 0], [13, 0, 6, -7, 14, 0, 0, 0], [-21, 0, 11, -7, -12, 0, 0, 0], [0, 16, 0, 0, 0, -2, -6, -8], [0, 10, 0, 0, 0, 2, -4, -6], [0, 0, 0, 0, 0, -6, -10, -7], [-18, 0, -8, 2, -14, 0, 0, 0], [-2, 0, -5, 0, 12, 0, 0, 0], [0, -11, 0, 0, 0, 2, -5, -8], [0, 5, 0, 0, 0, -2, -1, -7], [0, 8, 0, 0, 0, -3, -3, 0], [16, 0, 9, 2, 2, 0, 0, 0], [-10, 0, 11, -2, -15, 0, 0, 0], [-13, 0, 5, -1, -15, 0, 0, 0], [-13, 0, -10, -3, 2, 0, 0, 0], [0, 9, 0, 0, 0, -5, 14, -7], [0, 3, 0, 0, 0, 15, -10, -1], [6, 0, -13, 10, 6, 0, 0, 0], [0, -12, 0, 0, 0, -8, 14, -8], [-1, 0, -10, 5, 2, 0, 0, 0], [0, 3, 0, 0, 0, 14, 8, 3], [12, 0, 8, 4, -3, 0, 0, 0], [-20, 0, 2, -6, -20, 0, 0, 0], [0, -8, 0, 0, 0, 8, -19, 4], [0, 2, 0, 0, 0, 6, 16, 8], [0, 1, 0, 0, 0, 10, -16, 1], [-13, 0, 4, -14, 1, 0, 0, 0], [15, 0, 4, -3, 30, 0, 0, 0], [-1, 0, -13, 1, 12, 0, 0, 0], [0, 4, 0, 0, 0, 4, -12, 4], [0, 1, 0, 0, 0, 9, -10, -5], [21, 0, -2, 1, 7, 0, 0, 0], [-26, 0, -9, 2, -2, 0, 0, 0], [0, -1, 0, 0, 0, -3, 13, -1], [0, 3, 0, 0, 0, -1, -18, -9], [-9, 0, 14, -7, -8, 0, 0, 0], [0, 13, 0, 0, 0, -13, 4, -17], [0, 3, 0, 0, 0, 19, -7, 11], [-14, 0, -2, 4, -20, 0, 0, 0], [0, -14, 0, 0, 0, -2, 14, -6], [0, 0, 0, 0, 0, -8, -10, -14], [0, -2, 0, 0, 0, 23, -23, 12], [7, 0, 6, -7, 2, 0, 0, 0], [-18, 0, 11, 0, -14, 0, 0, 0], [15, 0, 6, 0, 7, 0, 0, 0], [0, 8, 0, 0, 0, 9, -15, 9], [-5, 0, -3, 6, -11, 0, 0, 0], [0, -18, 0, 0, 0, -14, 14, 0], [0, -8, 0, 0, 0, 18, -10, 3], [0, -8, 0, 0, 0, 6, -14, 6], [2, 0, -15, 0, 18, 0, 0, 0], [0, 21, 0, 0, 0, 0, -4, -23], [0, 3, 0, 0, 0, 7, -16, 9], [-25, 0, 10, -5, -26, 0, 0, 0], [0, -8, 0, 0, 0, -13, 23, -16], [7, 0, 5, -1, 15, 0, 0, 0], [0, 14, 0, 0, 0, 7, 0, 10], [13, 0, 6, 5, -12, 0, 0, 0], [-4, 0, -8, -4, -9, 0, 0, 0], [-16, 0, -3, -4, 10, 0, 0, 0], [0, -13, 0, 0, 0, 13, 8, 30], [-7, 0, 0, -11, 22, 0, 0, 0], [-5, 0, 11, -2, -11, 0, 0, 0], [2, 0, -8, -4, -4, 0, 0, 0], [-17, 0, -9, -2, -19, 0, 0, 0], [0, -20, 0, 0, 0, -9, -11, 18], [21, 0, 2, 4, -3, 0, 0, 0], [0, -22, 0, 0, 0, 13, -19, 8], [-6, 0, -4, -12, 16, 0, 0, 0], [0, 8, 0, 0, 0, 15, -1, 9], [0, -5, 0, 0, 0, -22, 1, -20], [0, -10, 0, 0, 0, 1, -1, -20], [-3, 0, 4, 11, -24, 0, 0, 0], [0, 9, 0, 0, 0, -19, 6, -22], [0, -17, 0, 0, 0, -8, 27, -13], [2, 0, -1, -1, -17, 0, 0, 0], [-26, 0, 9, -8, -12, 0, 0, 0], [0, 7, 0, 0, 0, -12, -1, -9], [0, 2, 0, 0, 0, 10, -18, -16], [-17, 0, -3, 11, -12, 0, 0, 0], [34, 0, -3, 6, 17, 0, 0, 0], [0, 23, 0, 0, 0, 8, -11, -1], [0, -14, 0, 0, 0, -19, 16, -22], [6, 0, -6, 12, 4, 0, 0, 0], [0, 8, 0, 0, 0, 0, -17, -4], [0, -21, 0, 0, 0, 1, 1, 23], [0, -12, 0, 0, 0, 7, -1, 26], [0, 0, -17, -4, 25, 0, 0, 0], [-21, 0, 4, -3, -42, 0, 0, 0], [0, -4, 0, 0, 0, -10, 2, -12], [-2, 0, 0, 10, -29, 0, 0, 0], [26, 0, 16, 4, -10, 0, 0, 0], [0, 0, 0, 0, 0, 3, -9, -23], [-9, 0, 1, 1, -2, 0, 0, 0], [0, -13, 0, 0, 0, 0, -7, 27], [-19, 0, 1, -11, -5, 0, 0, 0], [-6, 0, 2, 6, 0, 0, 0, 0], [0, 4, 0, 0, 0, -4, -11, 16], [-7, 0, -14, 3, 16, 0, 0, 0], [0, -8, 0, 0, 0, -5, 7, 16], [0, -15, 0, 0, 0, 2, -7, -11], [1, 0, -1, 9, 16, 0, 0, 0], [-11, 0, 11, -7, 3, 0, 0, 0], [0, 13, 0, 0, 0, 18, 3, 1], [-3, 0, -4, -7, -11, 0, 0, 0], [0, 8, 0, 0, 0, -12, -14, -4], [0, 21, 0, 0, 0, 1, -2, 8], [7, 0, -1, -8, 11, 0, 0, 0], [14, 0, -4, 8, 22, 0, 0, 0], [23, 0, 0, 9, 32, 0, 0, 0], [0, 11, 0, 0, 0, 13, -14, -5], [0, 2, 0, 0, 0, -14, -12, -14], [2, 0, 2, 13, 5, 0, 0, 0], [0, -11, 0, 0, 0, -28, 16, 1], [17, 0, 10, -11, 10, 0, 0, 0], [0, -21, 0, 0, 0, -10, -3, 21], [43, 0, -13, 11, 18, 0, 0, 0], [0, -6, 0, 0, 0, 7, 5, 13], [4, 0, 12, 12, 0, 0, 0, 0], [-28, 0, 12, -4, -14, 0, 0, 0], [-16, 0, 3, -1, -39, 0, 0, 0], [0, -16, 0, 0, 0, 4, -16, -6], [0, 4, 0, 0, 0, 0, -14, -12], [32, 0, 2, -3, 11, 0, 0, 0], [0, 0, 0, 0, 0, -6, 3, -8], [7, 0, -27, 9, 16, 0, 0, 0], [-53, 0, 18, -11, -40, 0, 0, 0], [0, 19, 0, 0, 0, -1, 10, -7], [8, 0, 6, 8, 24, 0, 0, 0], [0, -5, 0, 0, 0, 11, 2, 13], [13, 0, -6, -3, 17, 0, 0, 0], [0, 0, 0, 0, 0, -1, 10, -12], [-33, 0, -15, 11, -2, 0, 0, 0], [0, -2, 0, 0, 0, 23, -21, 22], [-13, 0, 12, -9, -24, 0, 0, 0], [1, 0, -26, 5, 20, 0, 0, 0], [0, 6, 0, 0, 0, -2, -14, -2], [0, 17, 0, 0, 0, -3, -2, -16], [25, 0, 7, 6, -21, 0, 0, 0], [0, 13, 0, 0, 0, -7, -2, -11], [-28, 0, -6, -4, -19, 0, 0, 0], [0, -32, 0, 0, 0, 0, -8, 24], [5, 0, -6, -11, 24, 0, 0, 0], [1, 0, 6, 8, -3, 0, 0, 0], [0, -28, 0, 0, 0, -4, -2, 6], [0, -19, 0, 0, 0, 0, 13, 0], [0, 17, 0, 0, 0, -23, 14, -20], [0, -4, 0, 0, 0, 8, -34, 12], [0, 3, 0, 0, 0, -17, 20, 19], [1, 0, 10, -7, -11, 0, 0, 0], [0, -5, 0, 0, 0, -6, -4, 3], [0, 7, 0, 0, 0, 12, -5, 18], [-6, 0, 10, -10, -22, 0, 0, 0], [27, 0, -2, 3, 17, 0, 0, 0], [-14, 0, -9, 3, -13, 0, 0, 0], [0, -12, 0, 0, 0, -23, 33, -16], [0, -19, 0, 0, 0, 6, 10, 29], [31, 0, 1, 3, -13, 0, 0, 0], [0, 11, 0, 0, 0, 4, 13, -27], [0, 6, 0, 0, 0, 16, 1, 30], [-11, 0, -4, -1, 16, 0, 0, 0], [0, -16, 0, 0, 0, -24, 12, -1], [0, 8, 0, 0, 0, 7, 15, 6], [0, 6, 0, 0, 0, -1, 31, -2], [-8, 0, 17, -3, -9, 0, 0, 0], [5, 0, -9, 8, 5, 0, 0, 0], [-22, 0, -7, 4, 22, 0, 0, 0], [0, -7, 0, 0, 0, -8, 5, 23], [-15, 0, 7, 1, -18, 0, 0, 0], [0, 21, 0, 0, 0, -6, 13, -17], [-19, 0, 9, -3, -21, 0, 0, 0], [0, 11, 0, 0, 0, -16, -21, -13], [-2, 0, -12, 24, 2, 0, 0, 0], [0, 1, 0, 0, 0, -14, 38, -7], [0, -1, 0, 0, 0, 1, -28, 14], [39, 0, 20, -7, 20, 0, 0, 0], [44, 0, 8, 0, 29, 0, 0, 0], [-12, 0, 7, 1, -25, 0, 0, 0], [0, 18, 0, 0, 0, -2, -10, -2], [0, 11, 0, 0, 0, 1, -4, -18], [-11, 0, 18, -1, -6, 0, 0, 0], [0, -40, 0, 0, 0, -11, 10, 4], [27, 0, -1, 7, 35, 0, 0, 0], [22, 0, 1, 8, 22, 0, 0, 0], [0, -28, 0, 0, 0, -17, 7, 1], [28, 0, -15, 8, 28, 0, 0, 0], [0, 24, 0, 0, 0, -15, 17, -10], [0, -4, 0, 0, 0, -39, 16, -12], [0, -11, 0, 0, 0, 1, 18, 24], [3, 0, 4, 17, 12, 0, 0, 0], [0, 14, 0, 0, 0, -11, 35, -18], [-2, 0, 11, -12, -16, 0, 0, 0], [0, -3, 0, 0, 0, 12, -16, -11], [0, -1, 0, 0, 0, 15, -38, 19], [11, 0, 2, 8, 23, 0, 0, 0], [-7, 0, 4, 11, 28, 0, 0, 0], [0, 21, 0, 0, 0, 1, 2, -3], [0, 2, 0, 0, 0, 10, -28, -16], [-4, 0, 4, -10, 20, 0, 0, 0], [-22, 0, 13, -6, -31, 0, 0, 0], [0, -6, 0, 0, 0, -6, -2, 2], [0, -21, 0, 0, 0, -10, 23, -3], [-3, 0, -1, 8, 9, 0, 0, 0], [-1, 0, -6, -3, -26, 0, 0, 0], [-1, 0, -23, -9, 20, 0, 0, 0], [0, -7, 0, 0, 0, -25, 8, 6], [41, 0, 9, -11, 21, 0, 0, 0], [43, 0, -11, -3, -4, 0, 0, 0], [0, 20, 0, 0, 0, 3, 10, -16], [-3, 0, -1, -3, -39, 0, 0, 0], [0, 12, 0, 0, 0, 5, -3, -42], [0, -9, 0, 0, 0, -29, 30, -22], [0, 0, -13, -9, -11, 0, 0, 0], [-4, 0, -1, 0, -50, 0, 0, 0], [0, 24, 0, 0, 0, 39, -19, 14], [0, -3, 0, 0, 0, 1, -4, 15], [0, -23, 0, 0, 0, -17, 17, -11], [26, 0, -2, -6, 25, 0, 0, 0], [0, 24, 0, 0, 0, -19, -8, -32], [7, 0, 10, -21, 10, 0, 0, 0], [0, -24, 0, 0, 0, -12, 0, 4], [55, 0, 21, -7, 8, 0, 0, 0], [0, 1, 0, 0, 0, 9, 20, 9], [35, 0, 13, 2, 9, 0, 0, 0], [0, 18, 0, 0, 0, -10, 17, -14], [-9, 0, 11, 3, -13, 0, 0, 0], [0, 2, 0, 0, 0, 16, -26, -16], [0, -21, 0, 0, 0, 27, -38, 10], [0, 4, 0, 0, 0, 21, 13, 4], [4, 0, -21, -4, 35, 0, 0, 0], [4, 0, 5, 10, -18, 0, 0, 0], [0, 10, 0, 0, 0, 4, -34, 1], [28, 0, -14, -8, 6, 0, 0, 0], [-22, 0, 21, -17, -29, 0, 0, 0], [0, -22, 0, 0, 0, 3, -17, 18], [-40, 0, -6, -8, 10, 0, 0, 0], [0, -8, 0, 0, 0, 22, -27, 8], [22, 0, -9, 8, 2, 0, 0, 0], [0, 21, 0, 0, 0, 10, 12, -11], [13, 0, -9, -2, 31, 0, 0, 0], [0, 0, 0, 0, 0, 36, -28, 10], [-18, 0, 1, 4, -48, 0, 0, 0], [0, 22, 0, 0, 0, 3, -6, -6], [23, 0, -7, -1, 28, 0, 0, 0], [29, 0, -7, 9, -16, 0, 0, 0], [0, 13, 0, 0, 0, -1, 24, 14], [0, 1, 0, 0, 0, 8, -7, 29], [0, 12, 0, 0, 0, 12, -4, -39], [-12, 0, 11, 13, 31, 0, 0, 0], [0, -3, 0, 0, 0, 11, -10, -3], [18, 0, 6, -16, 44, 0, 0, 0], [0, -19, 0, 0, 0, -11, -2, 35], [0, 44, 0, 0, 0, -3, -3, -19], [0, -16, 0, 0, 0, 1, -18, 4], [0, -23, 0, 0, 0, -35, 18, 3], [-25, 0, -26, 2, -23, 0, 0, 0], [24, 0, 16, -2, -14, 0, 0, 0], [13, 0, 5, 9, -28, 0, 0, 0], [0, -18, 0, 0, 0, 15, -5, -6], [-5, 0, 3, 9, -28, 0, 0, 0], [15, 0, 5, 5, -46, 0, 0, 0], [0, -22, 0, 0, 0, 14, 0, 42], [16, 0, 28, -8, 16, 0, 0, 0], [-40, 0, 10, 0, -26, 0, 0, 0], [0, -15, 0, 0, 0, -1, 20, 36], [9, 0, 21, 2, -37, 0, 0, 0], [0, -39, 0, 0, 0, -10, -5, 26], [-34, 0, -2, -16, 0, 0, 0, 0], [0, 8, 0, 0, 0, 30, -18, 33], [0, -10, 0, 0, 0, -18, 14, 10], [0, 10, 0, 0, 0, -12, -15, -18], [38, 0, -10, 6, 10, 0, 0, 0], [0, -7, 0, 0, 0, -23, 20, -5], [16, 0, -21, -8, 2, 0, 0, 0], [0, 27, 0, 0, 0, 7, -26, -12], [0, 5, 0, 0, 0, -14, 5, -19], [0, 10, 0, 0, 0, 22, 4, 14], [39, 0, -16, 3, 13, 0, 0, 0], [0, -29, 0, 0, 0, 31, -22, 27], [43, 0, 1, -7, 20, 0, 0, 0], [0, -21, 0, 0, 0, -10, -9, 25], [-11, 0, 8, 9, -36, 0, 0, 0], [0, 11, 0, 0, 0, 6, -4, 27], [-40, 0, -8, 8, -14, 0, 0, 0], [-12, 0, 15, -13, 9, 0, 0, 0], [-21, 0, -20, -5, -8, 0, 0, 0], [-20, 0, 5, -10, -24, 0, 0, 0], [51, 0, 1, -2, 1, 0, 0, 0], [0, 1, 0, 0, 0, -19, 39, 1], [0, -27, 0, 0, 0, 18, -5, 19], [0, -10, 0, 0, 0, 3, -6, 38], [-30, 0, -17, 0, 16, 0, 0, 0], [-35, 0, -4, -17, 28, 0, 0, 0], [0, -6, 0, 0, 0, -4, -31, 6], [22, 0, -9, -2, -11, 0, 0, 0], [0, -6, 0, 0, 0, 34, -36, 20], [15, 0, 20, -3, -2, 0, 0, 0], [0, 4, 0, 0, 0, -13, 11, -4], [0, -17, 0, 0, 0, 34, -23, 23], [0, 19, 0, 0, 0, 4, -13, -7], [2, 0, 16, 8, 12, 0, 0, 0], [0, -28, 0, 0, 0, -8, 15, -8], [-24, 0, -20, 10, -32, 0, 0, 0], [-10, 0, -9, 26, -4, 0, 0, 0], [0, -5, 0, 0, 0, 7, -8, -5], [0, 2, 0, 0, 0, -43, 23, -26], [0, 12, 0, 0, 0, -15, -9, -26], [21, 0, 14, -19, 39, 0, 0, 0], [0, 6, 0, 0, 0, 5, -17, -3], [0, -1, 0, 0, 0, 5, -2, -19], [-41, 0, 9, -19, 10, 0, 0, 0], [0, -25, 0, 0, 0, 14, -19, 20], [0, -1, 0, 0, 0, -14, -35, -13], [0, -9, 0, 0, 0, 8, -7, 1], [-13, 0, -17, 7, -7, 0, 0, 0], [30, 0, -6, 2, 60, 0, 0, 0], [-28, 0, 22, 6, 2, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_10000_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_10000_2_a_bj();".
// 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_10000_2_a_bj(:prec:=8)
    chi := MakeCharacter_10000_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_10000_2_a_bj();".
// 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_10000_2_a_bj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_10000_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 0, -84, 0, 61, 0, -14, 0, 1]>,<7,R![16, 0, -116, 0, 121, 0, -21, 0, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;