
// Make newform 8001.2.a.s in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8001_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_8001_2_a_s();".
// 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_8001_2_a_s();".
// 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 := [-20, -200, 1614, -1882, -3163, 5385, 2075, -5422, -298, 2648, -235, -668, 112, 83, -18, -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], [3, -5, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, -1, -7, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1833530, -32996882, 249231930, -240231533, -399632645, 508824053, 189480617, -351255852, -7403894, 106160280, -15371453, -13969268, 3569234, 545740, -233047, 16172], [-3035278, -7448006, 77060622, -20078726, -187228846, 74881691, 193857030, -75361440, -100836497, 35796817, 26857942, -8968081, -3471014, 1127781, 171907, -55506], [-1805692, -52297458, 219774482, -57970538, -476307526, 250701988, 404529319, -224913536, -170332316, 90669787, 37509563, -18731714, -4128253, 1937788, 179031, -79649], [482636, 932298, 1013322, 32187666, -71773574, -53417264, 131065711, 22495878, -88425697, 3013336, 27243628, -3798508, -3856011, 777233, 202993, -49011], [3426510, 50732374, -325487958, 170356469, 709532970, -461426392, -605454280, 378583411, 256052832, -146738516, -56580819, 29665237, 6241792, -3028354, -271059, 123393], [4758460, 46767048, -158630492, -76420466, 444594657, 12589986, -493686395, 52335901, 264664955, -45326027, -71647986, 15315132, 9361224, -2288559, -467479, 125049], [725548, 26121758, -166221504, 51084111, 461790873, -196229912, -484210085, 193199050, 244582021, -89947843, -62975173, 21971299, 7911672, -2688667, -383386, 129018], [-4842758, 2540988, 88655714, 31220698, -332083099, -30567173, 434181743, -24053391, -256493020, 36812718, 73837244, -14807929, -10047088, 2401726, 515922, -137239], [5228678, 60680346, -321397232, 44791941, 754990315, -255070845, -697758738, 252475352, 320134980, -113889853, -76732984, 26680262, 9144531, -3134036, -426142, 145145], [7379728, 51868034, -399924308, 177265028, 901431563, -500952602, -797772424, 425502097, 348630065, -170414762, -79277367, 35638598, 8967247, -3767459, -398038, 158942], [7468534, 124105592, -540456742, -43714386, 1389436646, -279881591, -1405569986, 358046232, 698572039, -187902897, -178585472, 49725923, 22367158, -6442221, -1082571, 322032]];
    Rf_basisdens := [1, 1, 1, 1, 1, 739814, 739814, 739814, 369907, 739814, 739814, 739814, 739814, 739814, 739814, 739814];
    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_8001_a();"
function MakeCharacter_8001_a()
    N := 8001;
    order := 1;
    char_gens := [3557, 1144, 7750];
    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_8001_a_Hecke(Kf)
    return MakeCharacter_8001_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, -1, 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, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1], [0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 0, 0, 0], [2, 0, -1, 0, -1, 1, 1, -1, -1, -1, 0, -1, 1, 1, 1, 0], [2, 1, 0, -1, 0, 0, 0, 1, 0, 1, -1, 0, -1, -1, -1, 1], [0, -1, 1, 1, 0, 0, -1, -1, -1, 0, 0, -1, 1, 0, 0, -1], [3, -2, 0, 0, -1, -1, 1, -1, -1, -2, 1, 0, 0, 1, 1, 0], [0, 0, 1, -1, 0, 0, 0, 1, 0, 1, 0, 0, -1, -1, 0, 1], [-1, -1, 0, 1, -1, 0, 0, -1, -1, 0, -1, -2, 0, 1, -1, 0], [1, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 1, 1, 0, 0], [3, 0, -1, -1, 0, 0, 0, 0, 1, -1, 0, 1, 0, 0, 0, 0], [2, 1, 1, 0, 0, 1, 0, -1, 2, 1, 1, 0, -1, -2, -1, 0], [0, 1, -1, -1, 0, 0, 0, 1, -1, 0, -2, -1, 0, 1, 0, 1], [2, -2, 0, -1, 1, 0, 0, 0, 1, 1, 0, 0, 0, -1, 1, -1], [4, -2, 2, 0, 0, 0, 0, -1, 1, 1, 0, 1, 0, -2, -1, 0], [-2, 3, 0, 1, 0, 0, 0, 1, 1, 1, 0, -1, -1, 0, -1, 1], [-1, -1, 1, 0, -1, 1, 1, 0, 0, -1, 1, 1, 0, 1, 1, 0], [0, -1, 0, 0, -1, 0, 0, 2, 0, 0, 0, 1, 1, 1, 1, 1], [0, -1, 1, -2, -1, -1, 0, 1, 0, -2, 1, 3, 0, 0, 1, 0], [2, -5, 0, -2, 1, -3, 0, 0, 0, -2, 1, 2, -1, 1, 2, -1], [-2, -1, 0, 0, 0, 0, 0, 0, -1, 2, 1, 0, 1, 0, 0, -1], [2, 1, 0, 0, -1, -1, 0, 1, 1, 0, 1, 2, -1, -1, 0, 1], [-3, -1, -1, 0, -1, 1, 0, 0, -2, -1, 0, -1, 2, 3, 2, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, -3, 1, 1, -1, 1, -1, -2, 0, -1, 0, 0, -1, 1, 0, 0], [-2, -1, 2, -2, 1, 0, 0, 2, 2, 2, 0, 2, 0, -2, 0, 0], [0, -2, 3, 1, 1, 2, 1, -1, 2, 1, 1, 1, 0, 0, -1, -1], [-1, 6, 0, 1, 1, 3, 0, 0, 3, 3, -1, -1, -1, -3, -2, 0], [5, 1, -2, 0, 0, 0, 0, -4, -3, -1, -2, -1, 1, 2, 2, -1], [2, -1, -3, 1, -1, 0, 0, 0, 0, -2, 1, 0, 0, 2, 2, 0], [-1, -1, -1, 3, 1, 2, 0, -4, -2, 2, 0, -3, 3, 1, -1, -2], [-1, -1, 0, 1, 0, -1, 1, 1, -1, 0, 0, -1, 0, 2, 1, 0], [-3, 3, 1, 0, 1, 0, 0, 2, 2, 1, 1, 2, -1, -1, -1, 1], [7, -5, 3, -2, 0, -2, 0, 0, 0, 0, 3, 1, 0, -1, 1, -3], [4, -4, 0, 2, 0, -1, -2, -2, -3, -3, 3, 0, 4, 2, 3, -3], [-1, 4, -3, 2, 0, 3, 1, -3, -1, 4, -2, -5, 3, 2, -1, -1], [2, 3, 0, -2, 1, 0, 0, 1, 0, 0, -1, 1, -1, 1, 2, 0], [-3, -1, 0, -1, 1, -1, 0, 0, -1, 1, 2, -1, 0, 1, 2, -3], [2, -8, 6, -2, 1, -3, 0, 1, 6, -1, 6, 5, -4, -3, 0, -3], [2, 5, -1, 2, 0, 2, 0, -2, -3, 3, -4, -4, 2, 1, -3, 1], [4, -1, 4, -1, 0, -1, -1, 0, 0, 2, 1, 3, -2, -2, 0, 0], [3, 7, -2, -1, -2, 0, 1, 0, 0, 0, -2, 0, 0, 0, -1, 3], [1, 2, 3, 0, 0, 1, 0, 5, 3, 1, 0, 2, 0, -2, -2, 1], [0, -1, -4, 0, 2, 0, -1, -1, -3, 0, -1, -1, 1, 2, 2, -1], [-3, 6, -1, 2, -2, 3, 0, -2, -2, 0, -2, -2, 5, 0, 1, -3], [5, 2, 0, -1, -1, -1, 1, 4, 2, 0, -1, 1, -4, -2, -2, 3], [6, 0, 0, -2, 0, -2, -1, -1, -1, 0, 0, 0, 1, -3, 0, 0], [1, -2, 0, 0, -1, -1, 1, -1, 0, -2, 0, 2, -3, 1, 0, 1], [-2, -2, 3, 2, 1, 1, 1, 0, 1, 3, 1, -1, -1, 1, -1, 0], [1, 0, -1, 3, 0, 2, -1, -2, -4, 1, 1, -5, 3, 1, -2, -2], [1, 4, -3, 2, 1, 1, -1, 0, 0, 0, -3, -3, 0, 0, -1, 0], [2, 1, 1, 2, -1, 3, 0, -1, 2, 1, 2, 0, 2, 0, -1, -3], [-2, -3, 2, -2, 0, -4, 0, 0, 1, -4, 5, 4, -1, -1, 2, -1], [-1, -2, -1, -1, 2, -1, 0, 1, -1, 2, 0, -1, -4, -2, -1, 1], [-4, -1, 3, 0, 2, -1, 1, 2, 4, 3, 1, 2, -4, -2, -1, 0], [4, 4, 0, 1, -2, 1, 1, 1, 1, 0, -4, 0, -2, -1, -2, 5], [0, 4, 1, 1, 2, 2, 2, 2, 5, 2, 1, -2, -2, -2, -2, 0], [-2, 0, -3, -1, 1, -2, -1, 5, -2, 3, -2, -2, 1, 1, -1, 1], [-2, 3, -2, -3, 0, 1, -1, 2, 1, 3, 1, 3, -1, -1, -1, 0], [3, -2, 0, 0, 0, -3, -2, -2, 0, 0, 0, -3, -1, 2, -1, -1], [5, -2, 0, 0, -2, -3, 0, 1, -4, -3, -2, 1, 1, -1, 0, 1], [-1, 1, 0, -1, 2, 0, -1, 3, -1, 2, -3, -2, -4, 1, -1, 1], [3, 0, 1, 2, -1, 1, -1, -1, 0, -4, -1, 0, 3, 3, 2, 1], [-3, 10, -2, 0, 1, 2, -1, 2, 4, 3, -2, 1, 0, -4, -2, 1], [2, 1, -8, -1, 1, 1, 1, 3, -5, -2, -1, -1, 0, 3, 2, 3], [4, -1, -1, 0, -1, 0, -1, 2, -2, 2, -1, -1, 0, -1, -1, -1], [6, -2, -4, -3, -1, 1, 3, -2, -1, -5, 1, 1, 3, 3, 3, 0], [4, -1, -3, -2, -1, -1, 0, -5, -2, -3, -2, 0, 2, 0, 3, -1], [4, 0, -5, 2, -1, 1, 0, 0, -1, 0, -2, -2, 0, 0, -1, 1], [-2, -3, 0, 0, 1, 1, 0, 0, 3, -1, 4, 2, 1, 2, 3, -3], [5, -1, -2, 1, -3, 0, 3, -1, -5, -6, -1, 0, -1, 4, 1, 2], [-2, 1, -7, 1, -2, 2, -1, 0, -7, -2, -2, -2, 4, 3, 2, -1], [4, 1, 1, 0, -1, 4, -1, -2, 1, 5, -2, -3, 3, 0, -1, -4], [8, 3, 5, -1, -1, 0, 1, 4, 4, 2, 2, 3, -4, -5, -2, 0], [2, -3, 2, 0, 1, -3, -3, 0, -1, -4, 5, 5, 3, 0, 4, -5], [-1, 5, -1, -1, -5, 2, 1, 3, -3, -1, -1, -2, 2, 1, 2, 2], [-2, -5, 0, 2, -2, -3, 1, 1, -1, -2, 1, 0, -1, -1, 0, 2], [-1, 1, 0, 3, 0, 3, 3, -4, 0, 3, 0, -6, 4, 4, 0, -2], [6, -1, -2, -1, -1, 0, 2, 3, 1, -1, 2, 3, 0, 0, -1, 4], [-1, -2, 3, 2, 4, 1, 1, -2, 4, 4, 1, -2, -2, -1, -1, -3], [1, -8, 3, -4, 1, -6, 1, 2, 2, -3, 5, 3, -6, -2, 0, 1], [1, -3, 2, -5, 0, -2, 1, -1, 0, -1, 1, 3, -2, -2, -1, -2], [4, 10, -1, 1, 1, 2, 1, -1, 2, 6, -1, -1, -1, -5, -4, 3], [4, -3, -5, -2, -1, 0, 2, 1, -1, -1, 0, 1, 0, 0, -1, 0], [1, -1, 0, 0, -2, -2, 0, 3, 0, -3, 5, 5, 1, -1, 2, -3], [2, 0, -1, 0, 0, -2, 2, -2, -2, -3, 3, -2, 1, 3, 3, 0], [-6, 5, -3, 2, 1, 1, -3, 1, -2, 2, -3, -1, 2, 0, -2, 3], [-4, 1, -2, -1, -2, 3, 1, -4, 1, -2, -2, 0, -2, 1, 0, 4], [2, -5, 3, -2, 0, -6, 1, -1, 0, -7, 2, 6, 1, 2, 3, -1], [0, -7, -4, -1, 4, -1, -1, -2, 0, -1, -3, -3, 3, 4, 1, -4], [8, 1, -2, 0, 0, -1, -2, 1, -4, -6, -4, 0, 0, 3, 2, 2], [2, 2, -1, 0, 1, 3, 1, 0, -1, 2, 2, -2, 3, 1, 1, -1], [6, 5, 2, -2, 6, -3, -1, -2, 6, 4, 2, 3, -2, -6, 0, -2], [0, -3, -6, -2, 0, 1, 0, -6, -2, -7, 0, 2, 0, 2, 3, 0], [-7, -6, 1, -1, -1, -1, 1, 1, 2, -7, 4, 3, -1, -1, 4, -4], [10, -7, -3, -1, 1, 0, 1, -2, -5, -4, 4, 0, 3, 4, 4, -4], [8, 0, 3, -3, 1, -3, 2, 3, 7, 1, 2, 5, -7, -3, -1, 0], [-4, 2, 3, -3, 1, -1, 1, 2, 2, 0, 4, 5, 0, -2, 2, 0], [9, 1, 2, -2, 4, 0, 3, -1, 4, 5, 1, -2, -1, -1, 0, -1], [4, 9, -2, 1, 1, 1, 0, -8, 1, -1, -1, -4, 2, -1, -1, -3], [0, 0, -4, 0, -2, -1, -2, 2, 1, -3, -1, 4, -4, -2, -2, 6], [0, -6, 4, 2, 0, -1, -2, 4, 2, 1, 4, 6, -3, -3, 1, -3], [-7, 6, -2, -1, 3, 2, -2, 1, 0, 3, -5, -3, -1, 0, -2, 1], [3, -3, 5, -2, -3, 1, -2, 2, 1, -2, 3, 5, 2, -4, 3, -1], [5, 10, -4, 2, -1, 3, 2, -7, -1, -1, -3, -1, 6, 1, 0, 0], [4, -2, 4, 2, -4, 1, -2, -1, -1, -2, 1, 0, 3, -4, 0, -1], [1, 6, 2, -2, -1, -2, 1, -2, 2, -4, 0, 5, -5, -3, -3, 7], [9, -1, -5, 1, 1, -1, -3, -1, 2, -2, 2, 2, -1, -1, -1, -1], [-1, 0, 5, 2, 3, 0, -1, -4, 4, 0, 0, 1, -1, -2, -3, -3], [-3, -5, 4, -2, 0, 0, 3, -1, 2, -1, 5, 5, 0, 0, 2, -3], [4, 1, 0, 0, 0, 2, 0, -3, -1, 0, -2, -1, 0, -1, 2, -3], [6, -3, 8, 0, 1, -2, 0, 2, 8, -5, 6, 7, -7, -2, -1, 2], [1, 5, 0, 1, -1, -1, 0, 0, -5, -2, -2, 0, 1, 3, 1, 5], [1, 5, -8, 0, 0, 0, -4, -4, 0, -3, -3, -1, 5, 0, 0, -3], [-5, 7, -9, 1, -3, -1, 1, 0, -7, -3, -2, -5, 3, 3, 0, 5], [1, 5, -2, 2, 1, 2, -3, 1, 1, 7, -3, -6, 2, -2, -4, -2], [-4, 13, -7, 2, 0, 5, -3, 1, -2, 5, -10, -4, 2, 0, -7, 6], [-1, 4, -1, -9, 2, -1, 1, 4, 7, 2, -1, 6, -6, -7, -1, 1], [6, -6, 1, -6, -2, -2, 2, -1, 3, 1, -2, 1, -4, 2, 1, 0], [-4, 1, 5, 0, 2, -4, -4, 3, 3, 0, 0, 3, -2, -1, -4, 3], [7, 3, -2, 1, 2, 5, -1, 0, 1, 6, -3, -1, 4, -1, 3, -2], [0, -3, 2, -1, 2, -3, 1, 0, -2, -2, 1, -1, -4, 0, 0, 2], [5, 5, 5, -1, 3, 1, 4, 0, 7, 7, -1, -2, -4, -4, -6, -1], [6, 0, -1, -5, 1, -4, 0, 3, -1, -3, 1, 1, -5, 6, 1, 1], [-1, -11, 11, -1, 4, -2, -1, 2, 5, 3, 5, 3, -6, -3, -4, -1], [-1, 6, -2, -2, 0, 2, 0, 0, -1, 0, 2, 4, 3, 0, -3, 2], [7, -1, -3, -2, -2, 1, 1, 6, -2, -3, 1, 2, 4, -2, 4, 0], [-1, -3, 1, 0, 3, 2, -1, -3, -5, 4, 0, 1, 0, -1, -1, 0], [-6, 1, 1, -4, 1, -1, 1, 4, 0, -2, 4, 4, 4, 3, 2, -3], [-6, 4, -2, -2, 0, -1, 3, 5, 1, 8, 0, -2, -2, -6, -4, 1], [2, -8, 2, -2, 1, -5, 2, -2, -3, -1, 3, 0, -1, 2, 4, -5], [-1, 5, 6, 1, 0, -2, 0, 5, 5, 2, 2, 4, 1, -4, 1, -1], [-3, 3, -5, -1, -3, 6, 1, -1, -3, -1, -3, 0, 2, 3, 2, 2], [-6, 4, -8, -2, -3, 2, -3, 4, -4, -2, -1, -1, 1, 5, 2, 3], [0, 6, 4, 1, 2, 1, -5, -4, 3, 7, -2, -3, 4, -2, -3, -3], [3, 5, -1, -1, 0, -2, -2, 1, 0, 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2, -3, 1, -1, -5, -4, -5, 1, 1, -2, -3, 2, 1], [-8, 5, -13, 2, -3, 7, 0, -7, -15, -7, -4, -7, 15, 11, 11, -5], [-17, 5, -3, 8, 3, 3, -2, -5, -2, -2, -4, -3, 4, 2, 1, -1], [-2, -3, 4, 0, 2, -5, 0, -6, 0, 0, 5, -7, 3, 5, 2, -8], [9, -2, -8, -1, 2, -1, 4, 0, -1, -10, 0, 4, 1, 2, 5, 1], [3, -9, 10, -4, 2, 0, -8, 6, 8, 2, 5, 7, -6, -7, -3, -3], [-20, 7, -8, -4, -1, 2, 3, 10, 1, 5, -3, -3, -4, 4, -1, 4], [16, -13, 3, -5, -4, 1, 0, 0, -3, 0, 0, 0, 0, 1, 1, -3], [5, -2, 3, -1, 1, -1, -6, -1, 5, -10, -2, 10, -5, -1, -1, 4], [11, -12, 0, 0, -1, 2, -4, -1, 1, -4, 0, 1, 4, 7, 4, -8], [-6, -10, -2, -3, -5, -3, -4, 11, -2, -15, 7, 10, -1, 4, 11, -2], [5, 7, 1, -3, 0, 3, 5, -5, -2, -8, 0, 4, 1, 3, 7, -2], [4, 0, 4, 6, 0, 5, 2, -6, 7, 3, 0, -5, 3, 3, -1, -3], [8, -6, 13, -1, -2, 2, -1, 1, 8, 2, 4, 9, -8, -7, 2, -4], [13, 0, 14, 1, -4, 0, -1, -8, 7, 5, 0, 5, -3, -6, -4, 0], [-2, -5, 1, 0, -3, -2, 0, 3, -5, -7, 4, 1, 6, 2, 5, -6], [19, 4, 4, -8, -3, -4, -1, 9, 7, -3, 7, 12, -5, -9, 4, -3], [1, -4, -3, 4, -1, 7, -2, -6, -7, 2, 0, 0, 10, 1, 3, -6], [2, -10, -3, -6, 2, -1, 4, 1, -4, 0, 3, 1, -1, 10, 6, -3], [-15, 15, -5, -4, 3, 7, 4, 3, 1, 12, -11, -4, 5, 0, -5, 5], [-4, -5, 7, 0, 2, 1, 0, -10, 7, 12, 0, -4, -2, -3, -7, -7], [9, -20, 5, -4, -2, -2, 2, -9, -2, -8, 4, 8, 4, 2, 11, -4], [1, -6, -4, 1, -1, -2, -2, 1, -5, -8, 4, 6, 5, 0, 1, -1], [-18, 2, 5, 2, -2, -3, -1, 9, -2, 5, -1, -2, 5, 0, -3, 1], [2, -3, 8, 5, 2, -1, -2, 9, 3, 6, 3, -1, -4, -4, -8, -6], [-13, 7, -6, -2, -6, -2, -1, 0, -9, -12, 0, 1, 1, 7, 7, -3], [-2, 11, 1, 3, -9, 7, -1, -3, 2, -4, -2, 5, 1, -8, -4, 3], [-15, 4, 8, 6, -2, 4, 2, -1, -4, 7, -2, -2, 1, 4, 0, 4], [14, -20, 13, -1, 4, -13, 0, 5, 5, -5, 12, 9, -5, -5, 4, 0], [-13, 6, -2, -3, -4, -2, 3, 5, -2, -2, 0, 2, -5, 0, -3, 14], [0, 2, -9, 10, -3, 2, -1, -11, -13, -7, -7, -10, 11, 7, 3, -1], [-3, -3, -1, -8, 3, -9, -2, 6, 1, -12, -4, 10, -5, -1, 7, 3], [-13, 0, -3, 5, -1, 5, -1, 9, 7, 3, -4, 1, -3, -5, 0, 0], [-7, -1, 6, -4, 1, -2, 3, 5, 9, 6, 2, 8, -5, -10, -9, 6], [-12, 4, -10, 3, -5, 8, 2, 1, -8, -9, 0, -3, 17, 5, 7, -3], [6, -11, 5, 1, 2, -2, -1, 3, -3, 3, 0, 2, 5, -2, 2, -6], [3, -8, 1, -3, 1, -9, 0, -3, 9, -10, 8, 10, -8, -1, 2, 1], [1, -6, 2, 8, -7, -2, 1, -1, -7, -13, 6, 0, 3, 5, 4, -5], [7, -14, -1, -3, -4, -11, -3, 0, -8, -21, 6, 12, -1, 0, 10, 3], [-4, 3, 2, 5, 3, -3, -5, -2, 1, -2, -3, -3, 5, -2, 2, -3], [6, -9, 8, -3, 3, -4, -1, 5, 2, -5, -3, 15, -13, -6, -1, 6], [-11, 6, -12, 0, -5, 6, 0, -7, -7, -5, -9, -10, 8, 5, 6, -1], [-4, -5, -4, 14, -3, 13, 0, -17, -13, 4, -3, -14, 15, 3, -2, -5], [-18, 8, 4, 1, 1, 4, 0, 7, 3, 3, 3, 1, 1, -2, -3, 1], [-12, 1, -6, 1, -7, 4, -3, -2, -4, -3, -8, 0, 4, 7, 3, 0], [9, -7, -10, 5, 2, 3, -2, -5, -12, 2, -6, -13, 7, 12, 7, -8], [17, -6, -7, 3, 0, -2, -3, -9, -2, -8, -2, -3, 1, 5, 3, -2], [18, -1, -11, 9, -5, 2, -7, 2, -10, -8, -1, -7, 10, 7, 2, -2], [-18, 9, -3, -3, 2, -2, -1, 5, 6, 10, -1, 1, -3, -7, 0, -3], [-9, 22, -3, 11, -2, 4, 3, 1, -2, 4, -5, -9, 5, 1, -5, 4], [-22, -3, 7, -2, 11, 0, -4, 1, 1, 5, 2, 1, 0, 0, 1, -6], [22, 0, 4, 3, -4, 3, -2, -3, 5, -8, 2, 5, 1, -4, 2, 1], [8, 8, -6, -1, 0, -3, -7, 5, -3, -1, 0, 1, 0, -1, -1, 9], [12, 4, -6, -4, 3, 3, 0, 5, 9, 8, -1, 6, -6, -9, -4, 4], [2, 0, 5, 4, 1, -1, -4, -10, 2, -5, 6, -5, 4, 7, 5, -8], [-16, 11, -14, -3, 1, 1, 2, 8, 0, -5, -3, 5, 1, -1, 5, 1], [8, -7, -8, -2, 1, 1, 4, 6, -1, 4, 4, -3, -4, -2, 4, -1], [-27, -8, 8, 0, 0, 1, 5, -5, -3, 3, 0, -4, -3, 6, 2, -5], [-11, -7, 2, 6, 1, 5, -1, 5, -2, 1, -1, -3, 0, 5, -4, 7], [0, 16, 0, 1, -3, -1, -3, 12, 4, -1, -7, 8, -4, -8, -3, 8], [-9, -1, -6, 1, 1, 2, 3, -13, -4, -3, -3, -1, 10, 8, 2, -6], [-6, 5, -4, -1, 0, -4, -7, 5, 3, -6, -4, 8, 1, -4, -4, -1], [14, 2, -7, -8, 2, 2, 6, -6, -1, 5, -2, -4, 1, -1, 1, -4], [9, -5, 4, -4, -7, -1, -3, -1, -1, -1, 1, -1, 4, -1, -2, 1], [-2, 2, -4, 5, -1, 1, -7, 0, 0, 8, 2, -6, 1, 3, -5, 3], [14, 5, 2, -4, -4, -1, 3, -3, 3, 4, -1, 4, -1, 0, 2, -2], [-9, 4, -4, 5, -4, 5, 8, -8, -6, -2, -2, -8, -1, 9, -3, 5], [11, 6, 3, 8, -3, 0, 5, -4, 4, 7, -5, -18, -2, 3, -8, 5], [-23, -1, -2, -4, 5, -6, -7, 0, 7, -1, 2, -1, 1, 1, -2, -2], [25, -2, -3, -5, 6, 0, -2, -1, 2, 4, 0, 2, -10, -6, -5, 2], [18, 5, -5, 3, -1, 1, -1, -12, -5, 2, -2, -8, 8, 7, 0, -5], [-9, 9, -5, -4, -2, 7, 3, 11, 5, 4, -1, 9, 6, -6, -3, 4], [4, 9, 3, 1, -7, 1, -1, 2, -4, -6, 2, 7, 1, -4, 4, 5], [6, -15, 5, 11, -3, 1, 3, -17, -7, -10, 8, -4, 6, 7, 10, -9], [-6, -6, -3, -2, 0, -9, -1, 9, 5, 1, 7, 5, -4, -3, -3, 3], [1, -7, 7, 5, -1, -3, 1, 2, 2, -1, 2, -8, -7, 8, 0, -5], [15, -4, 5, -5, -4, 0, -3, 3, 6, -6, -1, 7, 0, -7, 4, -1], [7, 18, -14, -2, -2, 2, -1, -8, -3, -1, 2, -7, 7, 4, 0, -4], [-10, 14, -2, 8, -1, 3, -5, -4, -4, 1, -8, -4, 5, 6, 1, -4], [10, -5, 11, -3, 0, -1, -1, 7, 4, 1, 4, 1, -5, -5, -2, -5], [-15, 0, 6, -7, -2, -1, 10, -3, 4, -7, 2, 5, -3, -5, 2, -1], [-10, 6, -3, 2, 3, 0, -2, -6, -2, 1, 0, 6, 4, -5, 5, -7], [20, 9, -5, -1, 2, 1, -1, -15, 1, 1, -9, -7, 0, 4, -3, -6], [-8, 10, 3, 2, -3, -1, 2, 12, 5, -7, 2, 6, -1, 2, 3, 4], [-4, -10, -2, 5, -8, -3, -2, -9, -14, -12, 1, -2, 9, 9, 12, -3], [17, -6, 11, -10, 10, -10, 2, 0, 15, 2, 11, 11, -13, -13, -1, -4], [5, -3, 4, 2, 4, 0, -6, 1, -3, 5, 0, -4, -4, 0, 0, 1], [7, 7, -6, 3, -2, 4, 1, -4, -11, 2, -2, -3, 5, 3, 1, 5], [5, -10, -11, 6, -3, 2, 0, -9, -1, -4, -4, -6, 3, 8, 3, 1], [5, 5, -5, -7, 1, -1, 2, 20, -2, 3, -3, 2, -8, 3, 2, 11], [-39, 1, -8, 9, 1, 5, 0, 8, -12, 3, -13, -8, 8, 10, -2, 9], [8, 6, -7, -3, 0, 1, -2, 1, -2, 0, -5, -7, 3, -6, -2, -1], [-5, -5, 0, -14, 1, -4, 9, 3, 3, 1, 1, 4, -4, 1, 2, 0], [14, -2, 8, -4, 1, 5, -4, -3, 3, 21, -6, 1, -4, -10, -10, -1], [14, 4, -3, 8, 0, 1, -4, -3, -5, -5, -9, -4, -2, 0, -3, 11], [-4, 3, 12, 6, 5, 3, -1, -1, -3, 13, 0, -5, -3, -5, -7, 1], [-2, -4, -12, -3, 3, -9, 6, 4, -5, -5, 4, -1, 4, 6, 11, -4], [4, -4, -8, -6, -5, -5, 0, -4, -14, -10, 3, -5, 11, 6, 10, -2], [-3, 15, 4, -4, 3, 7, -1, -2, 6, 5, 1, 13, 2, -3, -2, 1], [-5, -14, 1, -6, -2, -6, 3, 6, -5, -12, 9, 3, 4, 11, 9, -4], [-3, 12, 11, 5, 2, 4, 0, -1, 9, -4, 10, 7, 0, -4, -1, -8], [8, 6, -8, -2, -8, 6, 7, -11, -6, -5, -2, -9, 3, 10, 5, 4], [4, -20, 2, 4, -5, -8, -3, 10, -1, -16, 2, 7, -2, 1, 0, 1], [-5, 11, -5, -3, 1, 0, 2, 3, -1, -1, -3, -4, -7, -3, -4, 6], [0, -4, 2, 5, -1, 2, 0, 10, -6, 0, 1, -3, 5, 1, -2, 3], [-10, 9, -11, 3, -3, -2, -1, 9, -8, -8, 0, -3, -4, 9, 2, 6], [8, 14, 0, -4, -2, -1, 5, 0, -2, 2, 5, -1, -3, 1, 0, 3], [-13, 1, 1, 0, -5, 4, 5, -4, 3, 3, 4, -1, -1, -2, -8, -4], [12, -8, 4, 4, -3, 2, 0, -3, 3, 3, -2, -12, -1, 0, -7, 3], [-2, 8, 6, 5, 4, 11, -5, -4, -1, 12, -4, -4, 3, -5, -8, -1], [-18, -8, 12, 2, 0, 2, 1, -8, 6, 1, 4, -1, 3, 2, 4, -5], [-10, -10, 9, -2, -2, 0, 3, -6, 3, 1, 4, 6, 2, 3, 3, -6], [28, -7, 0, 0, -3, -5, 0, 6, 8, -8, 1, 11, -6, -4, 0, 7], [18, -14, 4, -2, 1, -6, 4, -1, -5, -11, 12, 4, 1, 8, 13, -3], [9, -12, -1, 4, 5, 3, -1, -9, -9, 3, 5, -2, 4, 0, 2, -8], [-1, -22, 14, -6, 3, -7, 2, -13, 1, -11, 7, 13, 0, 6, 4, -2], [-10, -16, -7, -4, 2, -12, 1, 2, -9, -18, 6, 9, -4, 9, 15, -7], [35, -4, -1, -1, 3, -1, -1, -15, -3, -9, 2, 5, 11, -2, 6, -4], [-4, -11, 6, 3, 4, -4, -5, 13, 3, -3, 4, 11, -6, -1, -3, -1], [6, -15, -7, 0, -2, -4, 4, -3, -2, -10, -4, -1, 2, 5, 8, -7], [5, 14, 3, 0, 3, -4, -5, 8, -2, 3, -4, 0, -6, -5, -6, 7], [-2, -6, -9, 6, 3, 8, -7, -5, 0, 8, -7, -9, 3, 1, -6, -1], [10, 4, -2, 1, 3, 5, -3, -7, 2, -3, -7, 6, -2, -8, -5, 6], [-1, 13, -5, -2, -4, 1, 2, 4, -2, 8, -15, -8, -8, 3, -9, 12], [-13, 2, -9, 4, 1, 4, 1, -3, -7, 6, -6, -15, -1, 7, 0, 4], [4, 1, -7, 0, 0, 4, 6, 6, 1, 7, -2, 0, 0, -1, -5, -1], [-6, 14, -8, 5, 3, 6, 2, -2, 1, 5, -9, -8, -2, -1, -13, 3], [-10, 0, 3, 1, -8, -2, 2, -6, 2, -11, 3, -2, -5, -3, -3, 6], [-4, 11, 4, 0, -10, 4, -2, -1, -5, -16, -2, 7, 6, 7, 6, -3], [-7, 11, 3, 5, 5, -1, 3, -6, -3, 7, 2, -6, 3, 0, -6, 1], [4, 8, -13, 1, -7, 0, 2, 0, -4, -4, 1, 0, 11, 6, 2, 1], [-14, 6, 2, 9, -7, 4, 3, 10, -5, -1, -3, 2, 3, -1, 1, 13], [8, -6, 3, 2, -3, 8, 4, -2, 2, -7, 4, 8, 4, 7, 3, 3], [3, -11, 3, 5, 1, -1, 0, 3, -5, 8, -3, -14, -1, 3, -6, -4], [2, 10, -11, 1, -2, 3, -4, -3, -3, 14, -8, -6, 3, 2, -6, -5], [-5, -15, 14, -7, 5, -1, 4, -4, 0, 6, 4, 2, -1, -7, 3, -9], [-13, 3, -1, 10, -13, 3, 3, 0, -7, -3, -5, -9, 7, 1, 2, 3], [20, 8, -10, 9, -1, 12, -1, -8, -12, 3, -5, -6, 7, 3, 1, 1], [-11, 7, -2, 3, 1, -2, 2, -3, -3, -6, -5, 2, 2, -7, -3, 2], [-8, 5, 0, 4, 7, 0, -7, 9, -3, 14, -10, -7, 2, -1, -10, 4], [-20, 15, 2, 0, -3, -3, -2, 3, 4, 0, 5, 9, 4, -10, -4, 3], [-8, 5, -16, 8, -10, 3, -2, -5, -6, -6, -4, -8, 10, 7, 0, 6], [18, -2, 5, 8, 0, 5, 5, -20, 2, 6, 0, -14, 4, -3, -7, -6], [-6, -10, 5, 1, 2, -2, -11, 4, -4, 2, -3, 8, -7, 2, 2, 0], [21, -19, 7, -5, 2, -8, 8, -3, 3, 1, 7, -6, -4, 4, 2, 1], [6, -17, -4, -12, 1, -4, 6, -5, -8, -2, 2, -2, 2, 0, 8, -4], [30, -18, 7, -5, 2, -12, -3, 8, 4, 1, 6, 7, -9, -10, 2, -5], [-5, 5, -8, -2, 4, 2, -3, 6, 1, 2, 5, 6, -4, 0, -1, -1], [1, 8, 4, 1, 3, 7, 3, -13, -5, -6, 5, 5, 6, -3, 3, -6], [-19, 9, 1, 4, 4, -4, -2, 10, 6, 2, -3, -3, 2, -3, -9, -1], [-3, 5, 0, 5, 6, 5, 3, -4, 7, 8, 2, -4, 3, -3, 1, -8], [-18, -6, 6, 5, -2, 4, -1, -6, -3, -5, 2, 0, 0, -3, -5, 4], [18, 0, -2, 0, -1, 10, -3, 6, 1, 8, 1, 5, -3, -3, -4, -1], [20, 8, -7, -10, 4, 4, 6, 0, 5, 9, 3, 0, -3, 4, 5, -2], [-28, 13, -2, -10, 9, 2, -4, 18, 9, 21, -6, 8, -11, -9, -9, 8], [-5, 13, 6, 4, 1, 2, -7, 7, 11, 11, 2, -2, -2, -3, -2, -10], [22, 5, -1, 3, -5, 8, -2, 2, -2, 4, -4, 0, 12, 0, 4, -4], [-4, 4, -2, -4, -5, 3, 0, 1, 6, -1, 0, 7, 1, -11, -1, -2], [25, -12, -5, -1, -4, -3, 1, 8, -5, -11, 11, 5, 6, 1, 8, -3], [2, -12, 10, 2, -6, 0, 6, -2, -2, -12, 10, 8, 0, 2, 12, -12], [-6, 9, -4, 8, -6, 2, 7, -8, -4, 4, -6, -17, 7, 8, 2, 5], [0, 7, -8, -5, 4, 2, -1, 2, -3, -3, -2, 10, -3, 1, 3, -3], [16, -2, 12, -6, -3, -1, -3, 4, 11, -6, 4, 9, -10, -5, -4, -2], [-15, 16, -19, 3, 1, 8, 4, 4, -1, 1, -6, -3, 3, 0, -2, -1], [4, 9, 0, -6, 2, 5, -2, -8, 4, 4, 1, -1, 9, -1, 0, -2], [-6, -14, 3, -9, 5, -11, 3, -9, 5, -2, 10, 5, -8, -10, 2, -4], [21, -2, -5, -9, 3, -7, 1, 0, -4, -5, -6, 0, -6, 0, 6, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8001_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8001_2_a_s();".
// 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_8001_2_a_s(:prec:=16)
    chi := MakeCharacter_8001_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_8001_2_a_s();".
// 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_8001_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-20, 200, 1614, 1882, -3163, -5385, 2075, 5422, -298, -2648, -235, 668, 112, -83, -18, 4, 1]>,<5,R![46352, 247912, 396800, 42868, -397916, -224398, 131760, 120392, -14196, -28542, -1293, 3467, 435, -210, -36, 5, 1]>],Snew);
    return Vf;
end function;