
// Make newform 10000.2.a.bi in Magma, downloaded from the LMFDB on 01 June 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_bi();".
// 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_bi();".
// 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]];
    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, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 0], [1, 0, 1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, -1], [0, 1, 0, 0, 0, 3, -2, 1], [2, 0, 0, 2, -1, 0, 0, 0], [0, -1, 0, 0, 0, -1, 3, -2], [-4, 0, -2, 0, 1, 0, 0, 0], [2, 0, -3, 1, 4, 0, 0, 0], [0, -1, 0, 0, 0, 1, 3, 2], [-5, 0, -1, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 3, -3, 3], [0, 0, 0, 0, 0, -1, 5, 1], [0, 1, 0, 0, 0, -2, 4, -4], [1, 0, 0, -1, -3, 0, 0, 0], [-5, 0, 1, 0, -2, 0, 0, 0], [0, -1, 0, 0, 0, 5, -2, 5], [-2, 0, -2, -3, 1, 0, 0, 0], [0, -3, 0, 0, 0, -3, 1, 6], [-2, 0, -4, 2, -3, 0, 0, 0], [0, 1, 0, 0, 0, -2, 0, -6], [-11, 0, 3, -3, -6, 0, 0, 0], [0, 4, 0, 0, 0, 1, -3, -7], [-10, 0, 2, -1, -2, 0, 0, 0], [0, -4, 0, 0, 0, -4, 1, 6], [0, 6, 0, 0, 0, 1, -2, -5], [-3, 0, -5, -1, 6, 0, 0, 0], [0, -4, 0, 0, 0, 3, -2, 4], [0, 7, 0, 0, 0, -4, 1, -7], [6, 0, 4, -1, 1, 0, 0, 0], [0, -3, 0, 0, 0, 1, 5, -2], [-1, 0, 1, 2, -10, 0, 0, 0], [-10, 0, 0, -3, 0, 0, 0, 0], [0, 0, 5, -4, -1, 0, 0, 0], [0, -2, 0, 0, 0, 3, 0, 3], [0, -3, 0, 0, 0, -1, -3, 2], [0, 6, 0, 0, 0, 0, 1, 2], [0, -5, 0, 0, 0, 1, -5, 2], [2, 0, 1, -3, 0, 0, 0, 0], [-4, 0, 3, 0, -2, 0, 0, 0], [-1, 0, -1, 3, 4, 0, 0, 0], [0, 0, 0, 0, 0, -6, 0, -6], [0, -9, 0, 0, 0, -1, -2, 1], [-4, 0, 5, -1, -1, 0, 0, 0], [1, 0, 1, -6, 8, 0, 0, 0], [0, 7, 0, 0, 0, -6, 5, -7], [0, -5, 0, 0, 0, 4, 1, 13], [4, 0, 1, 6, 6, 0, 0, 0], [0, 5, 0, 0, 0, 1, 4, -1], [3, 0, 5, -4, 2, 0, 0, 0], [-9, 0, 5, -6, -10, 0, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0], [0, 3, 0, 0, 0, -3, -1, 3], [0, 2, 0, 0, 0, 6, -10, -5], [-4, 0, 3, -4, -4, 0, 0, 0], [-10, 0, -5, 4, -4, 0, 0, 0], [0, -5, 0, 0, 0, -1, -1, 2], [-4, 0, 2, -2, 11, 0, 0, 0], [0, -2, 0, 0, 0, -5, 1, 1], [0, -6, 0, 0, 0, -2, -2, 2], [0, -6, 0, 0, 0, -10, 0, 4], [-11, 0, 0, -2, -11, 0, 0, 0], [0, 7, 0, 0, 0, -4, -3, -9], [0, -3, 0, 0, 0, 0, -4, 6], [-18, 0, -1, -4, 2, 0, 0, 0], [0, 12, 0, 0, 0, 2, 0, -7], [0, -2, 0, 0, 0, 6, -3, -8], [-2, 0, 0, 4, -8, 0, 0, 0], [0, 0, 0, 0, 0, -3, -1, 9], [-7, 0, 4, 0, -3, 0, 0, 0], [0, -2, 0, 0, 0, -1, -1, 9], [0, 2, 0, 0, 0, -7, 16, 0], [-6, 0, 1, 3, -4, 0, 0, 0], [0, 5, 0, 0, 0, 16, -8, 4], [1, 0, 1, 6, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, 9, -12], [-6, 0, 2, 1, 6, 0, 0, 0], [1, 0, -3, 2, -8, 0, 0, 0], [6, 0, 9, -3, 10, 0, 0, 0], [8, 0, -3, 10, 8, 0, 0, 0], [-2, 0, 5, 3, 0, 0, 0, 0], [0, -12, 0, 0, 0, -6, -1, 2], [-1, 0, -6, 4, 13, 0, 0, 0], [0, 2, 0, 0, 0, 5, 10, -5], [-2, 0, -3, 6, -13, 0, 0, 0], [0, -8, 0, 0, 0, 8, -12, 0], [-9, 0, -1, -4, 4, 0, 0, 0], [0, 13, 0, 0, 0, 5, 1, -12], [0, -6, 0, 0, 0, 3, -13, -3], [-10, 0, 6, 4, -5, 0, 0, 0], [0, -5, 0, 0, 0, 1, 11, 2], [3, 0, -3, -4, -6, 0, 0, 0], [-8, 0, -3, -1, -9, 0, 0, 0], [0, -7, 0, 0, 0, 3, 10, 11], [-15, 0, -10, 2, -9, 0, 0, 0], [0, 0, 7, 1, 10, 0, 0, 0], [0, 0, 0, 0, 0, -1, -10, 16], [7, 0, -3, 1, 14, 0, 0, 0], [0, 17, 0, 0, 0, 0, -6, -8], [0, 2, 0, 0, 0, 4, -10, 16], [0, 8, 0, 0, 0, -2, -12, -1], [-16, 0, 3, -1, -20, 0, 0, 0], [2, 0, 5, 5, 10, 0, 0, 0], [0, 15, 0, 0, 0, 2, 1, 5], [0, 3, 0, 0, 0, -15, -1, -10], [0, -3, 0, 0, 0, 6, -5, 8], [4, 0, 2, -1, 18, 0, 0, 0], [-14, 0, -5, -3, 5, 0, 0, 0], [0, 3, 0, 0, 0, 4, -1, 16], [0, 6, 0, 0, 0, 10, -10, -13], [0, 19, 0, 0, 0, 4, -2, -2], [-26, 0, -3, -1, -22, 0, 0, 0], [-18, 0, -1, 3, -2, 0, 0, 0], [3, 0, 8, -12, -5, 0, 0, 0], [0, 4, 0, 0, 0, -8, -2, -18], [0, 13, 0, 0, 0, -7, 0, -19], [0, -4, 0, 0, 0, 5, -1, 13], [29, 0, -9, 7, 30, 0, 0, 0], [23, 0, 0, 1, -1, 0, 0, 0], [0, 7, 0, 0, 0, -10, 5, -9], [0, -6, 0, 0, 0, -20, 14, -7], [0, 3, 0, 0, 0, -6, -10, -10], [7, 0, 0, 1, 19, 0, 0, 0], [-6, 0, -16, -4, 20, 0, 0, 0], [1, 0, 7, -6, 10, 0, 0, 0], [18, 0, -3, 5, 22, 0, 0, 0], [0, -2, 0, 0, 0, 5, -4, -4], [0, 12, 0, 0, 0, 6, 3, -10], [-13, 0, 2, -5, -7, 0, 0, 0], [0, -2, 0, 0, 0, 12, -12, 10], [-24, 0, 2, -5, -2, 0, 0, 0], [0, 0, 0, 0, 0, -23, 18, -11], [-29, 0, 6, 3, -3, 0, 0, 0], [20, 0, -11, 8, 26, 0, 0, 0], [0, 10, 0, 0, 0, 15, -4, -6], [0, 15, 0, 0, 0, 16, -3, 1], [0, -7, 0, 0, 0, -6, 6, 0], [-19, 0, -8, 6, -3, 0, 0, 0], [21, 0, 1, 4, 8, 0, 0, 0], [20, 0, -2, 1, 23, 0, 0, 0], [0, -11, 0, 0, 0, -14, 7, -11], [0, -2, 0, 0, 0, -19, 16, 4], [16, 0, -6, 3, 13, 0, 0, 0], [7, 0, 2, 3, 29, 0, 0, 0], [0, 5, 0, 0, 0, -5, 6, -15], [0, 7, 0, 0, 0, 11, -5, 17], [-3, 0, 1, -8, -24, 0, 0, 0], [0, -12, 0, 0, 0, -13, -2, -4], [0, 9, 0, 0, 0, -8, -1, -9], [12, 0, -5, 14, 6, 0, 0, 0], [0, -3, 0, 0, 0, -7, 14, -3], [0, -1, 0, 0, 0, -4, -3, 5], [0, -6, 0, 0, 0, 17, -10, 7], [-19, 0, 19, -4, -22, 0, 0, 0], [-26, 0, 2, -2, -3, 0, 0, 0], [-20, 0, -2, 1, -11, 0, 0, 0], [0, 0, 0, 0, 0, 17, -6, 20], [-13, 0, -3, 4, -10, 0, 0, 0], [0, -15, 0, 0, 0, 3, 16, 7], [0, 2, 0, 0, 0, -15, -3, -7], [0, -19, 0, 0, 0, 5, 0, 13], [42, 0, -4, 6, 7, 0, 0, 0], [0, 11, 0, 0, 0, 9, 11, -4], [0, 16, 0, 0, 0, -16, 5, -18], [-17, 0, 1, -10, -10, 0, 0, 0], [0, 1, 0, 0, 0, 6, -14, 4], [-3, 0, 9, -10, -8, 0, 0, 0], [0, 0, 0, 0, 0, 12, 8, 25], [-2, 0, -9, -1, -4, 0, 0, 0], [-20, 0, 2, 4, -13, 0, 0, 0], [6, 0, 2, 4, -9, 0, 0, 0], [0, 11, 0, 0, 0, 10, -20, 4], [-21, 0, -5, -4, -20, 0, 0, 0], [2, 0, 0, 15, 20, 0, 0, 0], [36, 0, -8, 6, 36, 0, 0, 0], [-17, 0, 11, -4, 4, 0, 0, 0], [0, 3, 0, 0, 0, -9, 15, -6], [-8, 0, -12, 11, -9, 0, 0, 0], [0, -17, 0, 0, 0, -11, 11, 2], [-1, 0, 7, 5, -32, 0, 0, 0], [0, 1, 0, 0, 0, 9, 17, -5], [0, 17, 0, 0, 0, -1, -8, -7], [0, -9, 0, 0, 0, -10, -6, 14], [-15, 0, 9, -16, 4, 0, 0, 0], [0, 17, 0, 0, 0, 0, -4, -6], [0, -2, 0, 0, 0, 20, -10, 9], [24, 0, -3, -6, 14, 0, 0, 0], [8, 0, -5, 12, -7, 0, 0, 0], [0, 14, 0, 0, 0, -15, -7, -31], [0, -5, 0, 0, 0, 14, -23, 13], [22, 0, -12, 3, 27, 0, 0, 0], [-36, 0, 7, -2, -26, 0, 0, 0], [0, -18, 0, 0, 0, 18, -22, 5], [0, 9, 0, 0, 0, -6, -1, -8], [22, 0, 22, 0, -10, 0, 0, 0], [0, 2, 0, 0, 0, -7, -4, 6], [0, -11, 0, 0, 0, 3, -18, 5], [0, -23, 0, 0, 0, -11, 17, -16], [16, 0, 3, -4, 40, 0, 0, 0], [-2, 0, 1, -15, -8, 0, 0, 0], [0, 11, 0, 0, 0, 24, -7, 17], [10, 0, 5, 0, 27, 0, 0, 0], [-15, 0, -1, -5, -2, 0, 0, 0], [0, -10, 0, 0, 0, 11, -8, -8], [26, 0, -4, 3, 45, 0, 0, 0], [0, -12, 0, 0, 0, 1, -9, -9], [-3, 0, 1, 16, 12, 0, 0, 0], [47, 0, -3, 1, 14, 0, 0, 0], [0, 8, 0, 0, 0, 0, -13, -8], [-22, 0, -2, 1, -18, 0, 0, 0], [0, 15, 0, 0, 0, 4, -14, 4], [0, -15, 0, 0, 0, 4, 1, 12], [6, 0, 10, -3, 9, 0, 0, 0], [12, 0, -5, -5, 14, 0, 0, 0], [0, -26, 0, 0, 0, -2, -2, 5], [-13, 0, -22, 6, -7, 0, 0, 0], [0, -23, 0, 0, 0, -3, 4, -7], [0, -1, 0, 0, 0, 5, -29, -2], [-12, 0, -1, 1, -34, 0, 0, 0], [24, 0, 6, 10, -26, 0, 0, 0], [5, 0, 5, 6, -12, 0, 0, 0], [0, -6, 0, 0, 0, -21, 8, -20], [0, -3, 0, 0, 0, 20, 11, 17], [-20, 0, -12, 6, -1, 0, 0, 0], [0, -1, 0, 0, 0, -2, -4, -16], [7, 0, 25, 0, -6, 0, 0, 0], [0, -2, 0, 0, 0, 19, -21, 13], [-8, 0, -7, -7, -1, 0, 0, 0], [0, 0, 0, 0, 0, 6, 17, 14], [1, 0, -15, 9, 4, 0, 0, 0], [54, 0, 1, 2, 14, 0, 0, 0], [-32, 0, 5, -8, -4, 0, 0, 0], [0, -5, 0, 0, 0, 17, -4, 21], [0, 21, 0, 0, 0, -8, -5, -3], [13, 0, -4, 1, 13, 0, 0, 0], [0, -29, 0, 0, 0, -5, 3, 13], [24, 0, 5, 1, 7, 0, 0, 0], [-3, 0, -15, -8, -8, 0, 0, 0], [0, 6, 0, 0, 0, -29, 12, -16], [-2, 0, -5, -14, -18, 0, 0, 0], [0, -3, 0, 0, 0, -1, -17, 23], [26, 0, -11, 5, 15, 0, 0, 0], [0, -8, 0, 0, 0, -1, -17, -5], [13, 0, -16, 6, 21, 0, 0, 0], [0, 5, 0, 0, 0, 0, 30, 4], [-6, 0, -15, 1, -2, 0, 0, 0], [2, 0, -3, 7, -32, 0, 0, 0], [0, -3, 0, 0, 0, -23, 22, -19], [0, 1, 0, 0, 0, 2, 18, 0], [-14, 0, 6, 13, 10, 0, 0, 0], [0, 5, 0, 0, 0, 9, 7, -19], [9, 0, 6, -11, 3, 0, 0, 0], [0, -3, 0, 0, 0, -30, 19, -7], [-10, 0, -1, -9, -24, 0, 0, 0], [28, 0, -18, -1, 7, 0, 0, 0], [0, 23, 0, 0, 0, 0, 5, 1], [0, 11, 0, 0, 0, 0, -15, -11], [0, -8, 0, 0, 0, 19, 10, 11], [0, -1, 0, 0, 0, -4, 29, -1], [0, 20, 0, 0, 0, 2, 3, -20], [-20, 0, 8, -9, -13, 0, 0, 0], [0, 0, 0, 0, 0, -15, -6, 1], [0, -25, 0, 0, 0, 13, -22, 15], [24, 0, 0, -10, 16, 0, 0, 0], [21, 0, -6, 10, 23, 0, 0, 0], [-44, 0, 3, -2, 6, 0, 0, 0], [0, 11, 0, 0, 0, 19, -25, 14], [0, 3, 0, 0, 0, 24, -4, 14], [-1, 0, 3, 4, -24, 0, 0, 0], [0, 0, 0, 0, 0, -13, 25, -7], [0, -18, 0, 0, 0, 8, -13, 0], [1, 0, 19, -6, -6, 0, 0, 0], [0, -28, 0, 0, 0, -4, 4, 17], [0, -15, 0, 0, 0, -8, 0, 16], [0, 1, 0, 0, 0, -5, 1, 4], [25, 0, 5, -3, 28, 0, 0, 0], [38, 0, -12, 17, 4, 0, 0, 0], [17, 0, 2, 5, -33, 0, 0, 0], [0, -2, 0, 0, 0, 4, -6, -7], [-2, 0, 10, 13, -5, 0, 0, 0], [0, -26, 0, 0, 0, 25, -29, 15], [-1, 0, -5, -14, 8, 0, 0, 0], [0, 20, 0, 0, 0, 9, -7, -29], [-30, 0, 5, 4, 14, 0, 0, 0], [0, 5, 0, 0, 0, -1, 1, -20], [0, 3, 0, 0, 0, 21, -5, 16], [-38, 0, -9, -7, -2, 0, 0, 0], [9, 0, -4, -11, 25, 0, 0, 0], [-6, 0, 6, 14, -10, 0, 0, 0], [0, 8, 0, 0, 0, 18, 22, 0], [0, -17, 0, 0, 0, 13, -13, 8], [10, 0, 11, -17, 10, 0, 0, 0], [0, -24, 0, 0, 0, 1, 1, 9], [22, 0, -18, 3, 14, 0, 0, 0], [4, 0, 5, -4, -37, 0, 0, 0], [0, -8, 0, 0, 0, -13, 14, -22], [-6, 0, -16, 12, 9, 0, 0, 0], [0, -9, 0, 0, 0, 11, -29, 4], [0, 21, 0, 0, 0, -8, 19, -10], [0, -3, 0, 0, 0, -22, 10, -8], [0, 0, -10, -5, 34, 0, 0, 0], [0, -11, 0, 0, 0, 10, -8, -2], [37, 0, 10, -7, -5, 0, 0, 0], [0, -11, 0, 0, 0, -24, 8, -10], [0, -6, 0, 0, 0, 19, 0, 22], [-20, 0, 14, -11, -31, 0, 0, 0], [-21, 0, -7, 12, 10, 0, 0, 0], [0, 24, 0, 0, 0, 6, -11, 2], [0, 17, 0, 0, 0, -12, -17, -13], [24, 0, -13, 0, -8, 0, 0, 0], [8, 0, -5, -2, -22, 0, 0, 0], [0, 9, 0, 0, 0, 1, -6, 17], [0, -8, 0, 0, 0, 24, 8, 47], [-11, 0, 5, 4, -44, 0, 0, 0], [16, 0, 6, 3, 14, 0, 0, 0], [-43, 0, -6, 2, -21, 0, 0, 0], [0, -1, 0, 0, 0, 7, 15, 14], [4, 0, -11, -1, 6, 0, 0, 0], [-22, 0, -14, -1, 15, 0, 0, 0], [0, 8, 0, 0, 0, -16, -4, 3], [19, 0, -1, 6, -20, 0, 0, 0], [0, 16, 0, 0, 0, 23, 4, -11], [0, -7, 0, 0, 0, 26, -12, 36], [43, 0, -5, 7, 12, 0, 0, 0], [6, 0, -4, 4, 21, 0, 0, 0], [0, -1, 0, 0, 0, -22, 38, -14], [0, -27, 0, 0, 0, 7, 11, 15], [0, 1, 0, 0, 0, 6, -5, 21], [16, 0, 2, -10, 11, 0, 0, 0], [0, -4, 0, 0, 0, 10, -28, 5], [11, 0, -3, 2, 14, 0, 0, 0], [0, -36, 0, 0, 0, -14, 22, -12], [-72, 0, 5, -11, -5, 0, 0, 0], [0, -6, 0, 0, 0, -8, -19, -10], [24, 0, 13, -11, -10, 0, 0, 0], [0, 20, 0, 0, 0, 5, -6, -32], [16, 0, 31, -7, -6, 0, 0, 0], [0, 7, 0, 0, 0, -14, 25, -1], [0, 22, 0, 0, 0, -3, -8, 5], [0, -15, 0, 0, 0, 8, 2, 30], [-19, 0, 19, 5, -22, 0, 0, 0], [17, 0, 6, -9, -25, 0, 0, 0], [0, 30, 0, 0, 0, 17, 15, -31], [-21, 0, -9, -5, 44, 0, 0, 0], [27, 0, -1, 1, 16, 0, 0, 0], [0, 1, 0, 0, 0, -24, 2, -28], [0, 0, -4, 8, 12, 0, 0, 0], [0, 25, 0, 0, 0, -3, 25, -25], [30, 0, 14, 10, -7, 0, 0, 0], [0, -15, 0, 0, 0, -19, 13, -28], [20, 0, 3, -9, -14, 0, 0, 0], [0, 11, 0, 0, 0, -11, -20, 5], [5, 0, -8, -1, 13, 0, 0, 0], [0, -27, 0, 0, 0, 4, 7, 38], [-12, 0, -15, 1, 19, 0, 0, 0], [1, 0, 6, 16, -1, 0, 0, 0], [0, -3, 0, 0, 0, 22, -8, 12], [0, 0, 0, 0, 0, 4, 0, -7], [0, 8, 0, 0, 0, 12, -12, 3], [-25, 0, -3, -13, 10, 0, 0, 0], [0, 30, 0, 0, 0, -14, -11, -32], [9, 0, -9, -9, 2, 0, 0, 0], [0, -6, 0, 0, 0, -2, 17, 20], [0, -36, 0, 0, 0, 9, -14, 16], [0, 12, 0, 0, 0, -16, 16, 9], [0, 0, 0, 0, 0, -24, 5, -26], [-4, 0, 10, -15, 27, 0, 0, 0], [-2, 0, -13, 14, 2, 0, 0, 0], [7, 0, 36, -2, -21, 0, 0, 0], [0, 14, 0, 0, 0, 11, -2, 1], [-2, 0, 5, -1, 7, 0, 0, 0], [-24, 0, -17, -3, 9, 0, 0, 0], [0, 34, 0, 0, 0, -2, -6, 2], [30, 0, 9, -2, -16, 0, 0, 0], [-29, 0, -5, -5, -44, 0, 0, 0], [0, -24, 0, 0, 0, -7, -14, 29], [1, 0, -11, -8, 36, 0, 0, 0], [0, 13, 0, 0, 0, 9, 6, -21], [20, 0, 3, 10, -38, 0, 0, 0], [0, 2, 0, 0, 0, 7, -25, -37], [0, -9, 0, 0, 0, 11, 10, 31], [0, 11, 0, 0, 0, 19, 3, -7], [13, 0, -15, 19, 24, 0, 0, 0], [0, 30, 0, 0, 0, -9, 28, -12], [38, 0, -26, 16, 37, 0, 0, 0], [0, -7, 0, 0, 0, 11, -3, -14], [0, 26, 0, 0, 0, -5, 13, -35], [0, -11, 0, 0, 0, 0, -25, 13], [25, 0, -4, 2, 5, 0, 0, 0], [0, 7, 0, 0, 0, -13, 19, 13], [0, 0, 5, -3, 31, 0, 0, 0], [0, -7, 0, 0, 0, -4, 5, 0], [-3, 0, 15, -10, 26, 0, 0, 0], [0, -3, 0, 0, 0, 15, -11, 30], [-30, 0, 13, -14, -14, 0, 0, 0], [4, 0, -21, -4, 32, 0, 0, 0], [66, 0, -5, 5, 6, 0, 0, 0], [-21, 0, 6, -5, -27, 0, 0, 0], [-18, 0, 2, -13, 26, 0, 0, 0], [0, 19, 0, 0, 0, -17, 16, 1], [0, 20, 0, 0, 0, 23, -11, 5], [0, 6, 0, 0, 0, 14, -10, 31], [-3, 0, -6, -13, 35, 0, 0, 0], [-1, 0, -11, 0, -14, 0, 0, 0], [0, 20, 0, 0, 0, -10, 3, -6], [-10, 0, 8, -4, -56, 0, 0, 0], [0, 21, 0, 0, 0, -5, 22, -21], [42, 0, -5, 5, 44, 0, 0, 0], [0, -7, 0, 0, 0, 22, -14, 20], [0, -26, 0, 0, 0, 0, -18, 27], [0, 15, 0, 0, 0, 12, -9, -8], [32, 0, -24, 2, 48, 0, 0, 0], [0, 13, 0, 0, 0, -13, -19, -35], [7, 0, 27, -17, -8, 0, 0, 0], [-22, 0, -10, -6, -21, 0, 0, 0], [0, -16, 0, 0, 0, -35, 34, -4], [0, -9, 0, 0, 0, -8, 16, -26], [0, -29, 0, 0, 0, 8, -6, 40], [-27, 0, -16, -8, 17, 0, 0, 0], [0, 20, 0, 0, 0, -12, -7, -24], [0, 34, 0, 0, 0, 7, 8, -14], [7, 0, -38, 8, 25, 0, 0, 0], [0, 3, 0, 0, 0, 17, 0, -21], [0, -13, 0, 0, 0, -14, 21, 16], [0, -6, 0, 0, 0, -10, 6, 27], [-56, 0, -29, 17, -20, 0, 0, 0], [-10, 0, 29, -16, -22, 0, 0, 0], [-48, 0, 12, -4, -36, 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_bi();".
// 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_bi(: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_bi();".
// 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_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_10000_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 0, -31, 0, 36, 0, -11, 0, 1]>,<7,R![16, 0, -56, 0, 61, 0, -21, 0, 1]>,<11,R![-20, 40, -5, -5, 1]>],Snew);
    return Vf;
end function;