
// Make newform 10000.2.a.bo in Magma, downloaded from the LMFDB on 23 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_bo();".
// 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_bo();".
// 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 := [45, 0, -285, 15, 556, 4, -440, -29, 144, 11, -20, -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], [648531, 455907, -3087181, -410492, 3473001, 159642, -1274457, -27635, 180827, -678, -8733, 307], [568359, 190755, -2266309, -590618, 2541583, 642787, -939928, -214775, 135204, 24756, -6738, -802], [577755, 347808, -3551140, -1154914, 4411093, 1296397, -1723439, -433991, 261955, 48406, -13969, -1351], [536010, 62361, -1604243, -692225, 1275695, 746510, -323536, -281581, 21083, 42209, 529, -2231], [348303, 282024, -1635878, -8724, 1905609, -513662, -848573, 293370, 160053, -56193, -10598, 3607], [888045, 553611, -3041720, -2248279, 2723982, 2409179, -798674, -861089, 78063, 118948, -1703, -5601], [-140112, 601323, 592910, -2310683, -910206, 2324195, 587287, -854757, -144518, 129472, 11191, -7021], [-269892, -1041702, 948330, 3065911, -365489, -3070705, -263751, 1136338, 123435, -171681, -11720, 9252], [-1338864, -1041531, 5256201, 4101418, -5542082, -4587698, 1757068, 1678758, -183751, -234641, 4724, 11122], [608583, 1236204, -2909367, -5681070, 2888145, 6888818, -451469, -2649292, -88219, 393757, 14554, -20291], [-914526, -1951401, 2716718, 8855582, -2213534, -10129172, 15784, 3801976, 196840, -557759, -22984, 28451]];
    Rf_basisdens := [1, 63003, 63003, 63003, 63003, 63003, 63003, 63003, 63003, 63003, 63003, 63003];
    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, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 1, -1, -1, 0, 0, 1, 0, 1, 0, 1, 0], [-2, 0, 2, 0, 0, -1, 1, 1, 1, 1, 0, 0], [0, 0, 1, 0, 0, 0, 0, 2, -1, 0, -1, 0], [-2, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0], [1, -1, -1, 0, 0, 0, -1, -1, -2, 0, -1, -1], [1, -1, 0, 1, 1, 2, -1, 1, -1, 0, 0, -1], [-4, 0, 1, 0, 0, -2, 0, -1, 0, -1, 1, 1], [-2, -1, 0, -2, 1, 0, 1, -1, 1, 1, 1, 0], [3, 0, 0, 1, -1, 0, 0, 1, -2, 0, 0, -1], [-5, 0, 2, 0, 1, 0, 0, 2, 1, 0, 0, 1], [1, -1, 1, 1, 1, -2, -3, 1, 0, -1, 0, 0], [-1, -1, -1, 0, -1, 1, 2, 0, 0, -1, 0, 2], [-2, 0, -1, -2, -2, -1, 1, -2, 0, -1, -1, 0], [-1, 1, 1, 1, 0, 2, -1, 1, 1, 1, 1, -1], [-4, -2, 1, 3, 3, 2, -3, -1, -1, 0, 0, 1], [0, 0, -3, 1, -1, -3, -1, 0, -1, -1, 0, 0], [-2, -3, 4, 2, 2, 1, -2, -1, -1, 0, 0, 1], [-8, 1, 4, -1, -1, -4, 3, -1, 0, 0, -1, 0], [-4, -3, 3, 0, 2, 0, -1, -1, -1, 1, 0, 0], [7, 0, -6, 1, 0, 0, -1, -2, -3, 1, -1, -2], [-3, 0, -1, -1, -2, -3, 3, -3, 2, -1, 0, 2], [-3, -3, 7, 2, 1, 0, -3, 5, -1, 0, 0, 0], [-1, 0, 0, -1, -3, -3, 1, -3, 0, 0, 0, 2], [1, 1, -6, 0, -3, 0, 1, -5, 0, 0, 0, 0], [3, 2, -1, -3, 0, 1, 3, -1, 1, 2, 2, 1], [-3, 0, 3, 0, 0, -4, 0, -4, 0, -2, -1, 3], [4, 1, -5, 0, -2, 0, -1, -5, -1, -2, -2, -2], [0, 0, 0, 3, 0, 1, 1, 2, -2, 2, -3, -3], [0, 2, -2, -2, -1, 1, 3, -2, 1, -2, -2, 1], [-4, -2, 4, 0, 1, 3, 1, 4, 2, 1, 1, -1], [1, -2, -2, -1, 1, 1, -3, 4, -2, -1, 0, 1], [0, 2, -3, -1, -1, 5, 5, 0, 4, 0, -1, 1], [-3, 2, -3, -2, -3, 2, 2, -1, -1, 0, 0, 0], [2, 1, 2, -5, 1, 3, 2, 2, 4, -1, 3, 2], [-1, 0, -1, 0, -1, -2, 4, 0, 4, 0, -1, 0], [1, 1, 5, 0, 0, 2, -3, 4, 1, 0, 0, -3], [-7, -2, 2, 1, 1, -2, 0, -5, -1, -1, 0, 0], [8, 1, 1, 2, -1, -2, 1, -2, 0, 1, -1, 0], [1, 1, -3, 1, -1, 2, -1, -3, 0, -3, 0, 0], [-4, -4, 9, 3, 0, -5, 0, 0, 0, 0, -4, 1], [0, 2, -4, 0, 3, 6, -2, -1, 0, 2, 0, -1], [-5, -2, 5, 1, 0, -4, -2, 1, -1, -1, -1, 1], [0, 0, -8, -1, 0, -2, 1, -5, 0, -3, 1, 0], [3, 2, 4, 2, 0, -4, 2, -2, -1, 0, -3, -1], [-8, 0, 0, 1, 0, 6, 2, -1, -1, 1, 1, -1], [9, 2, -5, -6, -4, 3, 1, 0, 2, 0, 2, -2], [-9, 1, 3, 1, 2, -4, -3, 0, -1, 0, 0, -2], [0, 0, -7, -5, 1, 3, 5, -7, 2, -1, 3, 4], [7, 1, 0, 1, 2, 2, 0, 3, 3, 1, 4, 3], [4, 3, -11, -1, -1, 4, 2, 0, 2, 1, 4, 1], [-4, -1, -3, -1, -4, -5, 2, -7, 0, -3, -2, 0], [-4, -5, 8, 1, 1, 4, 1, -1, -1, -3, -2, 3], [4, -1, -1, -2, 4, -2, 1, 0, 2, 0, -1, 2], [-8, -1, 2, 1, 3, -1, -2, 3, 2, 3, 0, 0], [5, 0, -4, 1, -2, -1, -1, -5, 0, -5, -2, 0], [7, -1, -1, 4, 1, 4, -5, -1, -3, 2, -3, -6], [-4, 5, 0, 2, 1, 3, -2, 0, 0, 1, 1, -3], [0, 0, -9, -2, -1, 1, -3, -7, 0, -1, 2, 3], [-5, -2, -8, 3, 2, 2, 0, -5, -2, 3, -2, -3], [1, 0, -2, 3, 2, 2, -4, 6, -1, -2, 0, 0], [-2, 1, 6, 1, -2, -2, -5, 3, 1, 3, 1, -1], [5, -1, 2, 6, -2, -1, -4, 10, -4, -2, -5, -4], [-7, -1, -1, 2, 0, -3, -2, -3, -1, -1, 0, 1], [-3, 0, 2, 0, -4, -6, 4, -4, 3, -1, -2, 4], [-4, 5, -9, -2, 0, -2, 4, 2, 3, 0, 3, 3], [-3, 1, 6, 0, -2, -5, 2, 0, 4, -2, 1, 4], [1, 3, -3, -1, -1, 7, -2, 0, 0, -3, 2, -1], [0, 1, -4, 1, -3, 8, -1, 0, 1, 2, 2, -2], [2, 0, -4, -3, -2, 0, 6, 3, 1, -2, 0, 5], [4, -4, -3, 2, -1, 3, -7, -7, -7, 2, -1, -2], [-9, -1, 2, -1, -1, 3, 1, 0, -3, -2, -1, 0], [-1, 1, 6, -1, -4, -7, 0, -1, -2, -1, -2, -2], [0, -2, -4, -5, 4, -1, -1, -2, -2, 1, 2, 1], [-1, -4, -2, -3, -2, -9, -1, -2, 0, -4, -2, 1], [7, 2, -6, -3, 0, 7, 2, 0, 1, -3, 1, 4], [-2, 0, 9, 5, 3, 0, -4, 2, 2, 2, 0, -1], [1, -4, 3, -3, 0, -3, -1, 3, -2, -2, 0, 1], [6, 3, -13, 4, 2, 5, -4, -3, -4, 0, 0, -5], [3, 0, 3, 4, 0, 0, -4, -1, -3, -1, -4, -4], [-1, -2, -2, 4, -4, -2, -3, -9, -4, -2, -4, -1], [0, 4, 1, -3, -4, -7, 1, 6, 4, -3, -3, 1], [-3, 0, -6, -1, 2, 0, 0, 3, 0, -3, 1, -1], [5, 4, -7, -5, 0, -1, 6, 1, 4, -1, 3, 2], [-1, 1, 1, 3, 4, 11, -1, 4, 0, 4, 1, -1], [-17, 0, 13, 1, 2, -5, -5, 5, 4, -1, -1, -2], [-1, -2, 7, 6, 4, 6, 2, 0, 3, 6, 2, 1], [-7, 1, 12, 4, -1, -3, 0, 5, -4, 1, -4, -3], [-1, 2, 2, 1, 0, -7, -1, -6, -1, -3, -1, 2], [-4, 0, -2, 2, 0, -5, -1, -4, 0, -1, 1, 0], [-6, -5, 11, -1, 1, -3, 4, 6, 1, 2, -1, 0], [6, 3, -1, 6, -1, -8, -3, 0, 2, -2, -2, 2], [10, -3, -3, -1, -4, -1, 2, 7, -4, -2, -1, 2], [-9, -5, -3, -6, 0, 1, 0, -3, 1, 2, 3, 0], [1, -4, -2, 0, 3, 4, -3, 6, -6, 1, -1, -4], [-4, 2, 2, -2, -1, -3, 5, 4, -3, 2, 0, 3], [5, 6, -5, -4, -2, 7, 7, 6, 7, 5, 3, 1], [-1, 2, 4, -1, -4, 2, -4, 4, -3, -3, -2, -4], [-7, 6, 4, -5, -4, -2, 8, -5, 4, 0, -3, 0], [-20, -5, 5, 3, 1, -3, -4, 5, -2, 1, 0, -2], [-10, -5, 6, -3, 2, -6, -1, 0, 1, 0, 6, 3], [-1, 2, 6, -3, -1, -2, 4, -4, -1, -5, 1, 2], [4, 1, -14, -2, 4, 10, -1, 0, 3, 2, 3, 3], [-13, 1, 2, -1, -2, 3, -2, 4, -1, -2, 1, -2], [-16, 4, 2, -3, -1, -4, 7, -1, 11, 1, 2, 3], [-3, 0, 9, -3, -6, -4, 4, 10, 5, -2, -3, 2], [3, 1, -2, -5, -5, -5, 5, -2, -2, 2, -1, -1], [-5, 1, 0, 4, -5, -2, -1, 2, 1, -1, -1, 1], [-5, 2, 14, 4, 5, 5, -1, 13, 0, 2, 1, 1], [7, 6, -10, -3, 0, 10, -4, 0, -1, -3, 5, -2], [2, -1, -19, -4, 0, 2, -3, -5, -3, -3, -1, -2], [-8, -5, 7, 8, 1, -1, -4, 6, -6, 4, -4, -4], [-9, 1, 9, 2, 0, 0, 5, 8, 5, 4, 1, -3], [-11, -4, 11, -4, 0, -10, 6, 5, 2, 2, -1, 1], [-4, -4, -1, 5, -1, -3, -7, -2, -7, 3, 0, -4], [-8, -2, 0, 2, 7, 2, -4, -2, 2, 3, 4, -2], [4, 5, 3, -1, 7, 2, -6, 4, 2, 3, 4, -3], [18, -3, -10, 1, 1, 3, -2, 2, 1, -1, -1, 6], [-17, 2, 12, -1, 0, -5, 5, -2, 5, -2, 0, 0], [-21, -2, 11, -1, 5, 3, 1, -3, 4, 4, 7, 2], [0, 4, -19, -6, -4, 0, 4, -5, 5, 1, -1, -1], [-20, 1, 24, 1, 2, -7, -6, 1, 1, -4, 1, 3], [13, -1, -15, 3, 0, 15, -10, -3, -5, 0, 0, -9], [1, 3, 4, 1, -2, -1, -2, 0, -1, -5, -3, -2], [-8, 3, 14, -4, 6, -7, 4, 7, 8, 6, 9, 1], [-5, -2, 6, 0, 4, -10, 2, 0, 1, 4, -1, -1], [15, 3, -2, 5, 3, 12, -7, -7, -3, 0, 0, 1], [9, -4, -10, -3, 2, -1, -7, 0, -3, 2, 2, 2], [0, 3, -4, -2, 0, 6, 5, -2, 4, 8, -2, 0], [-15, 4, 8, -6, -3, 6, 10, 3, 3, 2, 5, 2], [14, -2, -11, -6, -4, 1, 7, -5, -4, -8, -2, 3], [-2, 4, 1, -3, -2, -4, 4, 5, 5, 1, -3, 0], [14, -2, 2, -1, 0, 6, -5, 3, -7, 2, -4, -3], [-4, 2, -1, 4, 3, -7, -3, 2, 0, 0, 0, 0], [2, 7, -6, -1, -3, 14, 3, 9, 8, 4, 3, -4], [6, 3, -8, -7, -7, -11, 4, 5, 3, -2, -1, 1], [2, 3, -6, 1, 2, 11, 2, -6, 0, -3, -3, -3], [-12, -1, 1, 3, 3, 0, 1, -14, 2, 5, 3, 1], [-6, 2, 2, -6, 2, -5, 1, 6, 9, 0, 8, 5], [13, 0, -2, -6, -5, -5, 5, -2, 4, -5, 3, 4], [2, -5, -1, -3, -3, 12, -1, -1, -1, -2, -2, 0], [9, 0, 2, 9, 7, 9, -15, -5, -7, 1, -1, -5], [2, -1, -8, 6, -1, -9, -4, -14, 0, 0, -3, 0], [13, -2, 4, 8, 5, 0, -4, 5, 2, 3, 2, -3], [-12, -3, 5, 0, 8, -7, -3, -3, 1, 2, 8, 2], [4, 2, -15, -1, 3, 9, -9, -4, -7, 6, -1, -11], [-15, -3, 8, 3, 2, -3, -4, -13, -6, -3, 2, 0], [-1, -7, -5, 4, 2, 9, -6, 4, -7, -1, -5, -5], [-3, -10, 2, 5, 3, 6, -10, -1, -10, 0, -6, -4], [1, -4, 4, 2, -1, -12, -1, -3, 2, -4, -3, 1], [5, -3, -4, 9, 5, 2, -9, -14, -2, 2, -1, 3], [-6, 3, -1, 3, -6, -18, -1, -12, -1, -5, -4, 2], [3, -12, 16, 5, 4, -10, -10, -4, -4, -5, -3, 3], [-10, -8, -1, 6, 4, 3, -5, -1, -11, 3, -7, -8], [-13, 2, 1, -3, -3, -14, 0, -5, 4, -2, 1, 0], [4, -1, 23, 1, 0, -5, -9, 9, -7, -2, -3, -1], [-21, 2, 10, -7, -4, -10, 10, -3, 6, -4, 5, 6], [2, 2, -19, 2, -6, 7, 1, -16, -9, -1, -6, -7], [4, 4, -9, 7, -4, -1, 4, -1, -7, -2, -3, -1], [5, -5, 4, -3, 2, 1, 0, 11, 4, -1, 2, 2], [-8, -9, 3, 3, 1, -3, 2, 0, -7, 4, 1, 1], [4, -4, -4, 8, -4, -1, 0, 2, -3, -2, -10, -5], [-4, 5, -8, 1, 7, 12, -7, 5, -4, 4, 1, -6], [7, -3, 13, 4, 10, -5, 0, 5, 1, 2, -1, 2], [6, -4, -12, -4, 2, 11, -5, 2, -7, 2, 2, 1], [-21, -3, 11, -2, -1, -4, 0, 9, -6, 4, 0, -3], [3, -5, -8, -2, 1, 4, 3, -10, 4, 0, 6, 8], [23, -4, -7, 7, -2, 2, -2, 4, -4, 0, -8, -4], [-6, 3, -5, 3, 1, -9, 6, 1, -1, 0, 0, -1], [9, -6, 8, 0, 3, 7, 1, -6, -7, -2, -4, 3], [25, 3, -21, 2, -4, 5, -1, -14, 3, -1, 5, 3], [-15, 4, 1, -5, 5, -12, 10, 13, 5, 2, 3, -1], [-18, -2, 3, 4, 6, 2, 0, 3, 2, 4, 3, 3], [7, 4, -4, -5, -5, 2, 4, 12, -1, 2, 4, 4], [0, -6, -1, -4, 0, -7, -3, -10, 1, -4, 1, 7], [-15, -5, 15, -2, -1, -11, -2, -9, -2, -4, -6, -1], [6, 2, 5, 1, 4, 13, -1, 11, -2, 4, -3, -2], [17, 4, -28, -3, -1, 7, -1, -4, -3, 6, 1, -4], [3, 0, -1, 6, -1, 2, -3, 2, -6, 3, 0, -2], [10, 0, -19, -7, 1, -2, 4, 1, 8, 2, 10, 2], [-4, -2, 11, 4, -3, -2, -2, -5, -1, 0, -4, 3], [6, 2, 3, 9, -1, 2, -6, 23, -5, -1, -5, -7], [9, 2, -12, 0, -3, 16, 4, 7, 0, 5, 0, -5], [-5, 8, -8, 3, -3, -8, 4, -13, 4, -1, 1, -2], [-19, -5, 16, 0, 3, 1, 0, 15, -2, -1, -3, -3], [9, -4, -6, 9, -4, 0, -9, -10, -2, 0, -2, -1], [1, 6, 4, 5, -9, -13, 9, -10, -5, -8, -7, -2], [2, 5, 6, -1, 0, -9, -1, 0, -6, -3, -2, -3], [4, 5, -5, 3, 2, -1, 8, -3, 2, 1, -3, 0], [-4, -1, 13, 4, 4, 7, -11, 5, 4, 4, 4, -3], [-8, 3, -6, 0, 5, 4, 3, -14, 8, -5, 6, 4], [-3, 0, -2, 4, 0, 7, -2, 4, 2, -1, -4, -2], [-16, -4, 2, 3, 6, 5, -3, -2, 3, -2, 2, 6], [-10, -3, 4, 0, 6, -12, 3, -9, -3, -1, 0, 5], [-13, 2, 10, -3, -4, -8, 4, -7, 5, -7, 0, 7], [2, -6, -2, -2, 3, 9, 2, -4, -2, 0, -3, -2], [18, -7, 4, 5, 12, 13, -2, 5, -2, 6, 4, -2], [-10, 1, 3, -6, -1, -12, 1, 7, -7, -1, -4, -2], [4, 9, -10, 7, 1, 12, 1, 15, -2, 0, -1, -6], [16, 7, 9, 4, 4, -3, 0, 9, -1, -1, -1, -2], [0, -3, 4, -5, 1, -1, 0, -16, -7, -4, 1, 5], [-1, -1, 5, 2, -5, -6, -3, 9, -8, -1, 1, -1], [-4, -6, 11, 8, 0, 4, -6, 1, -7, -2, 0, 4], [-8, -1, 1, 0, 6, -1, -2, 8, 2, 1, 0, 4], [-12, -1, -15, 9, 1, 15, -2, -7, -2, -1, -2, -4], [-10, -3, -5, 7, 5, 1, -12, 1, -4, -4, 2, 2], [-15, -6, 9, 1, -6, -13, -3, -14, -1, -3, -1, 0], [11, 3, -13, 1, 0, -3, -12, 9, 0, -2, -4, 0], [-1, -1, -1, 10, 1, 3, 8, -2, -2, 5, -4, 2], [9, 0, -4, -5, -3, 1, 1, -6, 1, 0, -1, 0], [-8, 8, -2, 0, -4, 0, -5, 6, -2, 5, 1, -14], [7, 10, 7, 3, 3, 3, -11, 12, 2, 1, 1, -5], [11, -1, -1, -7, 0, -6, -7, 4, 5, -5, 1, 3], [-8, 2, 7, 2, 5, -11, 3, -6, 1, 3, 5, -1], [-12, 5, -5, 9, -6, -10, -5, -6, -8, -3, -6, -1], [17, -2, -1, 6, 4, 15, -9, 2, -2, 4, 2, -2], [15, -8, -3, -4, 2, -2, 6, 8, -8, -5, -3, -2], [14, -1, -14, 7, 2, 4, -5, 5, -1, 8, 2, -1], [6, 4, -14, -9, 3, 11, -1, 9, 5, 0, 5, 3], [5, 6, -7, -3, -2, -2, -6, 9, 5, -4, 6, 6], [9, 0, 8, 0, -6, 0, 0, 9, -3, 3, 1, -4], [-7, -2, 4, 3, 4, 21, -3, 7, -5, 1, 3, -1], [10, -1, -18, -7, 4, 4, 5, -13, 3, -4, 4, 11], [-12, -1, 5, 7, 10, 13, -8, 18, -4, 8, -1, -7], [8, 0, -20, -1, -3, 8, 0, -14, -10, -1, -5, -7], [20, -1, 9, -2, 1, 3, 6, -8, -6, -4, 2, 6], [27, 2, -3, 5, -2, -3, -8, -5, -8, 2, -6, -8], [10, 9, -23, -4, -1, -8, -7, -7, 2, 0, 3, 1], [-1, 7, -26, -1, -9, -11, 12, -16, 7, -1, -3, 3], [9, 1, 2, 0, -9, 11, 14, -5, 3, -3, -6, 6], [-8, 3, -4, 0, -1, 11, -6, 5, -5, -1, -4, -6], [-1, 4, -10, -5, -6, -6, 4, 7, 2, -1, 0, -2], [-3, 3, 5, -8, -1, 4, 9, -5, 2, 5, 7, -5], [-3, -8, 9, 12, 7, 17, -5, 3, -6, 7, -4, -9], [25, 1, -6, -1, 1, 14, 1, 11, -2, 1, -9, -2], [-17, -5, 12, 4, 17, 4, -4, 0, 4, 8, 2, 0], [4, 4, 3, -9, -3, -11, -1, 3, 4, 1, 3, 0], [11, -6, -26, 0, 6, 6, -8, -5, -4, -7, -3, 0], [-20, 5, 12, -7, 5, 8, 11, 13, 13, 10, 9, -2], [-13, 11, 15, -4, -4, 2, 13, -5, 12, 6, 5, 3], [-5, 7, 12, -5, 5, 4, 15, -2, 13, 7, 4, 4], [-2, -3, 3, 6, -1, -9, -16, 3, -6, 2, -8, -5], [-2, 6, 25, 0, -2, 3, 5, 14, 3, -2, 1, -1], [-1, -6, -1, 3, 1, 8, 0, 6, 1, -5, -7, 0], [-10, 2, 14, 4, 5, -6, -6, 21, 5, 4, 0, 0], [-1, -4, -6, -6, 2, 17, 3, 12, -4, 9, 4, -5], [-11, -9, 21, 3, 0, -12, -7, 5, 2, 0, 4, 0], [-9, 11, 10, 3, 0, -4, 5, 3, 11, 1, 4, 3], [1, -3, 0, -3, -4, -2, 13, -6, 9, -1, -6, 5], [13, -2, 22, -7, 7, 8, -2, 3, -3, 0, 0, 3], [-4, 1, 15, -6, 0, -13, 1, -6, 6, -2, 7, 4], [-3, 9, -37, 0, 1, 7, -4, -11, 3, 1, 7, -2], [-13, 1, 10, -10, -3, -5, 15, 4, 9, 4, 0, 3], [-8, -4, -13, 3, -8, -6, -4, -13, -4, -7, 2, 0], [-13, -8, 20, 5, 5, -10, -17, 4, -11, 4, 3, -4], [5, 5, -20, -5, -2, 4, -1, -2, 5, 2, 8, 10], [12, 1, 19, 1, -8, -5, -5, 18, 0, -3, -5, 1], [20, -4, -9, 3, -3, 5, -3, -8, -8, 0, -3, 3], [-20, -3, 9, -4, 3, -8, 4, -9, -3, -3, 0, 6], [10, -6, -5, 2, 2, -7, 1, -10, -2, -3, 6, 4], [-1, -1, 12, 3, 0, -11, 4, -5, 10, 5, 4, 6], [-17, -11, 6, -2, 2, -8, 3, -6, -4, -6, -4, 0], [6, 2, -12, 5, 0, -2, -10, -4, -4, 7, 1, -5], [26, 3, -25, -2, -5, 3, -2, 0, -9, -1, -3, -5], [-1, -3, 6, 8, -7, 4, -9, -3, -3, -6, -4, 2], [17, 1, -8, -2, -4, -7, -7, 8, 5, -3, 4, 5], [-5, 4, -5, -14, -2, -12, 2, -5, 13, -3, 14, 8], [18, -5, -12, 6, -4, -3, -8, 6, 2, -5, -1, 6], [-1, 5, -8, -8, -4, 7, 0, 6, 6, 3, 7, -1], [-7, 11, -4, -10, -4, -1, 16, -2, 10, 5, 12, 7], [-10, 10, -14, -6, 0, 3, -3, -2, 5, -2, 10, 1], [-7, -4, 16, 1, 4, 4, 0, 12, -6, 7, -1, -7], [-2, -5, 19, 11, -5, -5, 0, 10, -1, -3, -2, -2], [-7, -4, -5, -8, -3, 0, 13, 8, 5, 3, 6, 3], [3, 1, 2, 7, 8, 9, -10, -5, -4, 1, 0, -4], [8, -4, -2, 3, 3, -17, 1, -7, 4, 4, 3, 0], [15, 7, -7, -2, -9, 14, 5, -2, 2, 7, 2, 1], [6, 6, -17, 2, 6, 3, 1, -3, 10, 0, 2, 11], [-42, -8, 16, -4, 5, -17, -1, 4, 3, -9, 0, 7], [9, 13, -7, -5, -7, -10, 11, -8, 11, -5, 5, 4], [13, 5, -2, 1, -1, 19, 6, 7, 10, 6, 2, -4], [-1, 5, 2, 0, 0, 14, 10, -7, 7, 8, 10, 2], [-6, -7, -9, 1, 12, 7, 0, -6, -5, 2, 4, -4], [3, -1, 1, -11, 2, 16, 9, -3, 11, -3, 6, 5], [0, -2, 16, 9, -5, 0, -6, 6, -8, -5, -11, -9], [-10, -6, -2, 0, 2, -15, -5, -2, -7, 0, -4, -5], [-12, -7, -1, -4, 1, -9, 0, 2, -12, 2, -7, -4], [-11, -1, -3, 0, -6, -14, 1, -2, 1, -2, 0, 7], [-19, -13, 14, 8, 3, -12, -11, 7, -10, 2, -3, 1], [6, 7, -28, -16, 3, 2, 12, -16, 5, 0, 3, 3], [-2, 4, -6, 4, -2, 7, -1, -3, -6, 1, -3, -2], [13, 9, -16, -12, -12, 14, 11, -14, 2, -1, 2, 0], [17, -10, -12, -5, 4, 5, -9, 4, -10, -8, -4, 3], [-5, -2, 3, 16, 4, 20, -20, 1, -7, 1, -6, -12], [7, 0, -17, -2, -3, -14, 12, -6, 10, -8, 1, 10], [11, 7, 2, 1, -7, -8, 11, 1, 1, -8, -10, -3], [-9, -10, 13, -1, 7, 6, 1, -4, 4, -2, 2, 13], [40, 8, 5, -3, -1, 3, 7, 13, 4, 8, 1, -3], [-17, 8, 14, -5, -4, -5, 3, -9, 3, 8, 5, -1], [11, -4, -14, 5, -9, -6, 4, -4, -1, -5, -6, -2], [-3, 9, 1, 3, 1, -5, -10, 6, 1, 7, 3, -7], [-7, 7, -16, -8, 2, -5, 1, -16, 8, -2, 7, 6], [-20, 4, 15, -1, -4, -16, 6, -5, -1, -5, 3, 3], [-3, 9, 4, -3, -5, 14, 7, 14, 8, -2, 0, 1], [14, -3, 4, 3, -1, -2, 9, 19, 4, -2, -1, -2], [0, 7, -15, -8, -3, 3, -6, -3, -2, -5, 2, -11], [-5, 7, 13, -15, 4, 4, 7, 9, 6, 0, 14, 2], [-4, 6, -10, 0, 0, 22, -2, 7, 5, 2, 1, -2], [-2, 10, -10, -2, 5, 9, 3, 19, -1, 10, 4, -12], [-4, 5, -26, -10, -3, 10, 20, -14, 9, 2, 7, 7], [11, 2, 6, -11, 10, 6, 2, 12, 7, 2, 12, -2], [5, 2, 1, -2, -8, -11, -3, -5, -10, -1, -14, -9], [-16, -2, 21, -12, 5, -7, 1, -9, 11, 6, 10, 10], [-6, 11, 15, -7, -4, 8, 8, -4, 5, 0, 0, 2], [7, 3, -28, 4, 0, -3, 0, -4, -1, 3, 4, 5], [-1, -6, -5, 10, -4, -16, -1, 4, -9, -5, -11, -3], [7, 3, 22, -2, 5, -16, -1, 20, 9, 0, 1, 2], [6, -4, -7, 9, -1, 14, 0, 3, -6, -10, -5, 6], [37, 2, -13, -2, 2, 0, -10, 4, -7, 5, -1, -8], [-8, -3, 22, 6, 9, 9, 2, 2, -7, -5, 0, 3], [6, -7, 22, -9, -2, -11, -10, -8, 1, -4, 1, 8], [25, 8, -22, -17, 0, 4, 18, 18, 16, 4, 8, 7], [15, -11, 13, 5, 3, -8, -5, 0, -3, 2, -2, -4], [-24, -7, 16, 3, 5, -11, -12, 6, 0, -11, -4, 5], [-5, -1, 14, 8, 2, 2, -5, -16, 0, 6, -1, -4], [26, 7, -17, 7, -2, 9, 6, -10, 0, 3, 7, 1], [2, -14, -3, 17, 13, 15, -23, -3, -14, 9, -13, -12], [17, -7, -11, 4, 8, 15, -8, -13, -9, 4, 1, 4], [25, 8, -40, -4, -3, 18, 18, -15, 7, -4, 1, 6], [-7, 2, -4, -8, 1, 5, -3, 19, -8, -4, 3, -7], [14, 6, -27, -6, 4, 15, -4, 3, 0, -7, -5, 3], [-18, 0, 4, 5, -5, -5, 9, 8, -1, -4, -5, -4], [0, -5, -14, 9, 7, 11, -12, -21, -4, 7, 3, -6], [1, 8, -7, -15, -13, 3, 21, 6, 5, 4, 2, -6], [-8, -2, 1, -6, 0, 7, 7, 6, 7, 0, 4, 15], [17, 9, 10, -2, 4, -10, 5, -2, 15, -3, 9, 12], [30, -1, -9, 8, 9, 6, 1, 17, -5, 7, -2, -6], [50, 8, -16, -3, 1, 6, 4, 5, 2, -6, -5, 2], [-21, -1, 13, -4, -9, -15, 10, -10, -2, 0, -6, -4], [0, -4, 1, 8, 2, 19, -3, 6, 0, 9, 4, 4], [11, -1, -8, -5, 8, 1, -8, -2, -12, 0, 0, -11], [-14, 3, -10, -7, 0, 12, 1, 13, 1, 8, 10, 1], [-12, -1, 23, 2, -5, 5, -6, 0, -4, -2, 0, -6], [6, -9, -9, -5, 8, -5, 6, -5, 5, 7, 8, -3], [10, -4, 3, 3, -3, -15, -3, 0, -1, -6, -3, 13], [2, 4, -19, -1, 1, 13, 13, -17, -1, 10, 0, -3], [-1, -17, 24, 6, 5, -17, -16, -11, -8, -10, -11, 3], [-13, -9, 10, 3, 2, 1, -19, 10, -13, -4, -2, -4], [3, -2, 0, 10, 5, 0, 0, -6, 3, -2, 2, 3], [13, -6, -6, -3, 5, 26, -4, -11, 1, 4, 2, 7], [23, 0, 12, 6, 3, 21, -17, 18, -3, 6, 0, -7], [-3, 9, 13, -10, -10, -6, 15, -13, 5, 1, 8, 2], [26, -2, -47, -2, -13, 15, -4, -13, -8, -3, -2, -9], [-24, 0, 19, -4, -1, 6, 2, 10, -7, 4, -2, -7], [1, 1, 4, 0, 0, 16, -7, 18, 8, 4, 4, 3], [-5, 0, 21, 2, -11, 0, 16, 7, -2, -3, -4, 1], [28, -2, -11, 5, -4, -7, -23, -5, -15, -9, -10, -7], [31, -5, 11, 4, -4, 5, 8, 0, 1, 7, -3, 3], [8, 1, -3, -4, 4, -11, -19, 4, -13, 1, -2, -7], [-27, -17, 52, 7, 13, -14, -9, 20, -3, 1, -6, 2], [28, 1, -33, 8, -3, 10, -3, 10, -16, -2, -14, -12], [7, -8, -2, 5, -5, 20, -13, -21, -15, 0, -2, -1], [-14, 5, 14, 6, -5, -10, -1, -2, 1, -5, -3, -2], [-10, 8, -12, -6, 0, -3, 7, 9, -1, -7, 2, -4], [19, 7, 0, 7, 7, 3, -14, -12, -3, 2, 5, 0], [-18, -5, -10, 0, 0, -4, 11, -17, 13, -5, 3, 7], [0, 2, 7, -10, 5, -8, -4, 4, 7, -4, 1, -1], [-25, -15, 15, -6, 3, -6, 3, 7, 5, 1, 5, 7], [-5, -7, -17, -10, -3, -8, 3, -11, 5, 2, 0, -2], [-19, 4, -14, 15, 3, 5, 1, 0, -3, -2, -1, -2], [-6, 9, 11, 2, 9, 20, -5, 9, 3, 7, -4, -8], [-7, -13, 8, 7, 14, 6, -7, -3, -1, 3, 6, 9], [11, -13, -10, 0, -4, -13, -22, -10, -18, -1, -4, -1], [1, -8, -5, 10, 7, 3, -7, 11, 0, -2, -8, 1], [-9, -16, 23, 14, 7, -5, -13, 6, -20, -2, -10, -12], [21, -2, 17, 14, -1, 0, -10, 18, -17, 0, -18, -10], [15, -4, -21, 0, 0, -6, 4, -12, -3, 6, -5, 3], [4, 1, -1, -9, -4, 3, 2, -4, 10, -3, 5, -3], [-23, -3, -10, -8, -6, -17, 9, -18, 4, -11, -6, 8], [28, 10, -12, -4, -8, 10, 0, 7, 9, 0, 5, -4], [-15, 7, 8, 9, 2, -10, 1, -4, 7, 1, -1, 1], [7, -7, -9, 4, 6, -4, -7, -9, -15, -10, -5, -4], [3, -12, 6, -2, 3, 1, -11, 0, -11, 6, -2, 5], [7, -3, -12, 1, 4, 24, -1, 6, -5, 2, 4, -2], [29, -6, 15, 3, -5, 5, 1, -16, -5, -10, -4, 10], [16, 10, -6, -8, -1, 1, 13, 3, 13, 8, -1, 6], [-3, 4, -6, 6, 7, 1, -13, -5, 6, 6, 7, 3], [25, 8, -6, 4, -2, 0, 7, -12, 5, -4, -7, 7], [-15, 5, 13, 2, 2, -13, 6, -13, 12, -3, 8, 11], [-19, -5, 0, 16, 9, 9, -10, -10, -14, 0, -2, -5], [14, 6, -32, -3, -2, 4, 13, -1, 4, 6, 0, 0], [-26, -7, 31, 0, 10, 1, -2, 8, -9, -2, -3, 3], [-11, 2, 5, 10, -1, -6, 3, 7, -9, -4, 0, 4], [-24, -4, 14, -1, -4, -2, -4, -4, -4, 6, -7, -3], [13, 26, -33, -9, -9, 13, 3, -3, 5, -2, 2, -9], [1, 5, -9, 11, 1, 9, -9, 12, -6, 2, -4, -19], [-2, 3, 10, 7, -1, -13, -2, 24, -10, 2, -10, -15], [13, 14, -28, -15, 1, 2, 20, -7, 12, 1, 11, 6], [-32, 3, -7, 7, -2, -9, 1, -13, 7, 2, -3, 1], [35, 8, -24, -3, -4, 15, -1, 12, 4, 6, 4, -3], [-11, -5, 24, 2, 6, 23, 8, 14, -6, 13, -2, -7], [-6, -6, 12, 4, 0, -37, 1, -7, 5, 1, -6, 0], [-27, 3, 21, -5, 2, -14, 5, -5, 13, 9, 5, 5], [7, 2, -17, 7, 3, 1, -5, 0, -15, -10, -6, -6], [12, 8, -3, -5, 5, 3, -3, 4, -2, -2, 0, -1], [2, -6, -18, -5, 3, -11, -11, -16, 0, -10, 2, 5], [-1, -10, -6, 13, 14, 20, -16, 7, 1, 8, 1, -4], [-11, -18, -5, 2, 7, 10, -12, -4, -4, 3, -1, -2], [-25, 7, 3, -13, -6, -16, 1, -15, 4, -4, 12, 12], [-15, -2, 21, 0, -5, -16, 0, -5, 3, -5, 4, 16], [-5, -6, 6, 3, 6, 7, -11, -2, -4, 4, -7, -11], [30, 7, -5, 0, -4, -4, 1, 7, 13, -5, 4, 3], [-38, -10, 36, 12, 1, -25, 1, 2, -1, 1, -10, 5], [-15, -2, 21, 0, 0, -18, -2, 5, 2, 4, -7, -1], [-12, -9, 22, 3, -2, -4, -13, 12, 1, -3, 2, 0], [-6, 9, -4, 11, 12, 12, -4, 6, -1, 6, 1, 0], [4, -2, 2, -6, -1, -1, 5, 10, -13, -9, -8, 1], [3, 3, -2, 6, 1, 14, -5, 3, 6, -11, 8, 11], [10, -11, -1, -4, 2, -1, 0, -15, -11, 1, -3, 2], [0, -4, 11, -2, 6, 6, -8, 32, -6, -2, -1, 2], [16, 2, 4, 7, -4, -3, 6, -9, -6, -8, -4, 6], [41, -5, -23, 9, 1, 8, -17, 4, -10, -15, -14, -2], [-29, -1, 24, 5, 3, -5, -4, 4, 7, 14, 11, -6], [2, 3, -5, -9, 6, -7, -2, -2, 15, -2, 11, 8], [15, 4, -20, 1, 7, 26, -14, 6, -9, 0, -1, -3], [-29, -3, 6, -4, -1, 4, -2, -4, -9, 4, 0, -3], [24, -2, -37, 4, -3, 10, 0, -7, 4, -8, -3, 9]];
    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_bo();".
// 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_bo(:prec:=12)
    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_bo();".
// 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_bo( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_10000_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-5, 110, 1165, 2760, 726, -2624, -708, 773, 195, -93, -23, 4, 1]>,<7,R![16, -848, 12224, -30798, 13407, 9353, -5896, -919, 824, 29, -48, 0, 1]>,<11,R![-8609, 29479, 4252, -78417, 44673, 17573, -14022, -1322, 1533, 48, -68, -1, 1]>],Snew);
    return Vf;
end function;