
// Make newform 4017.2.a.f in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4017_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_4017_2_a_f();".
// 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_4017_2_a_f();".
// 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 := [4, 11, -378, -352, 2261, 1430, -5579, -2223, 7285, 1491, -5460, -278, 2388, -154, -594, 88, 77, -16, -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, 0, 0], [0, 1, 0, 0, 0, 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, 0, 0, 0], [-30277713, -65076335, 670807958, 67933382, -1910156217, 70345533, 1628375777, -125435006, -136140833, -44364492, -445048447, 115056639, 230717285, -53503587, -46409280, 9779883, 3886444, -626428, -94088], [8487263, -92926590, -385111717, 685986403, 1726428678, -2166951074, -3299395776, 3263812942, 3112759541, -2649906620, -1520636534, 1213875850, 377207518, -310348080, -39656466, 40884782, -4314, -2142369, 193859], [-18473720, 74082323, 900642533, -38526093, -3694969428, -87045064, 6113602981, -464378927, -5111196996, 1009320407, 2280606051, -720016087, -523591869, 235624707, 49187691, -36097151, 830096, 2086417, -301002], [-7102352, -50846708, -298043445, -61613093, 1414007301, 1068552050, -2757965584, -2137689600, 2842754969, 1786828862, -1668257465, -761598784, 571733266, 173915817, -112671033, -20247569, 11786836, 943097, -505639], [-73754683, -49334292, 1412281769, 883967110, -5081746752, -2506162403, 7923527727, 2902932841, -6682423192, -1680536573, 3292783051, 519460555, -963497294, -84744672, 162869580, 6574721, -14520768, -173763, 522452], [8797055, 90613854, 919589882, 508159444, -5271494415, -2586060788, 10924134189, 3367642315, -11061827209, -1718391077, 6037093933, 288782608, -1828942592, 37272603, 300102768, -16582591, -24023172, 1334850, 683216], [9751737, -46280392, -137691452, 1416269452, 273029682, -4845433314, 352120375, 6779610320, -1281877039, -4869475809, 1244174596, 1949589303, -579561356, -438788274, 142470306, 51743466, -17742577, -2477135, 878636], [-11793478, 361699234, -1196579108, -2646783813, 5904177015, 6490474048, -10915609351, -7250315182, 10277505300, 4181996197, -5429734446, -1301382665, 1654523163, 214341540, -284915496, -16785145, 25408444, 445742, -898500], [-3170015, 79078185, 333663343, 499205258, -1578545622, -2203868323, 2953197273, 3357982253, -2982909800, -2664298648, 1835184089, 1199138972, -701472610, -304111437, 159886860, 40045288, -19576350, -2109846, 977686], [-20840648, 94367681, 671237324, -1002901935, -2999440299, 3726959093, 5188235767, -6214385801, -4231425600, 5380497584, 1636153110, -2570195272, -210763494, 676127358, -38552868, -90912362, 13408784, 4839121, -1006596], [-1711033, 7664027, -480758306, -140104997, 2302393775, 120589701, -4127234417, 562746675, 3628535304, -1066464572, -1656310546, 725834826, 371781427, -233045118, -28351011, 35421487, -2297555, -2041117, 340719], [33234527, 163481722, -227428052, -1293543924, 306252117, 4020429766, -112710592, -6164640280, 286658199, 5073257425, -555378540, -2332250702, 398312019, 595242228, -131370000, -78194119, 20237422, 4091147, -1173795], [21257525, 355144607, -808126617, -2833548478, 4354535292, 7316823022, -8918685770, -9045153015, 9166163902, 6102042916, -5303138830, -2357316008, 1803133061, 518525517, -356551428, -60186352, 37818140, 2850571, -1656344], [6865267, 159048874, -1069751136, -218917406, 6362522892, -2090427664, -13149766900, 6739037490, 12419678693, -7991010202, -5703282992, 4592467277, 1150153099, -1357359756, -25058403, 196981321, -22037027, -11038000, 2105177], [-10034449, 64549695, 345567051, -668587229, -1492732591, 2447706825, 2467406610, -4115548654, -1830878338, 3618039992, 521784388, -1750271618, 54323182, 464512414, -64697412, -62837734, 13123579, 3359640, -855796], [-25074710, -24618681, 963132781, -328728832, -4978722603, 3352119782, 9654354819, -8469958429, -8597176955, 9280859786, 3503256218, -5118043786, -434695399, 1474350357, -106884618, -210313979, 34882692, 11645805, -2617292]];
    Rf_basisdens := [1, 1, 1, 8327001, 8327001, 8327001, 8327001, 8327001, 8327001, 8327001, 8327001, 8327001, 8327001, 2775667, 8327001, 8327001, 8327001, 2775667, 8327001];
    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_4017_a();"
function MakeCharacter_4017_a()
    N := 4017;
    order := 1;
    char_gens := [1340, 1237, 1756];
    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_4017_a_Hecke(Kf)
    return MakeCharacter_4017_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], [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, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 1, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1], [-2, 1, -1, 2, 0, 0, 1, -1, 1, 0, 1, 0, 1, 0, 0, -1, 1, -1, -1], [-1, 0, 1, 0, -1, -1, 0, -1, 0, 1, -1, -1, 0, -1, 0, 0, 2, 1, 0], [0, -1, 1, -1, 0, 0, 0, 1, 0, 1, -2, 0, -1, 1, 1, 1, 0, 0, 1], [-2, 0, 1, 0, 0, -1, 0, -1, 1, 1, -1, 0, 1, 0, -1, 1, 1, 2, -1], [-1, 0, 1, 0, -1, -1, 2, 0, 1, 1, -1, 0, 0, -1, -1, 1, 0, 1, -1], [-1, 2, -1, 1, 0, 0, 0, 0, 0, -1, 1, 1, 1, -2, 0, 0, 0, -1, 0], [-1, -1, 0, -1, 0, 1, 0, 1, 0, -1, -1, 1, -1, 1, 1, 0, 1, 0, -1], [-1, 0, 1, 1, -1, 1, 0, 0, 0, 1, 0, -1, 1, 2, 1, -1, 1, 0, 0], [-1, -2, -1, 0, 0, 0, 0, 0, -1, 2, 0, -2, -1, 0, 2, -3, 2, 0, 1], [0, 1, -2, -1, 1, -2, -1, 2, 0, -2, 1, 0, 0, -1, -1, 0, 0, 0, 1], [-3, -2, 1, 0, 1, 4, -3, -2, -3, 1, 1, 2, -1, -1, 0, -2, -2, 0, 0], [-3, 2, -2, -2, 2, -2, 0, 1, 2, -3, 0, 0, 0, 0, -1, 0, 0, 0, 2], [-2, 2, -2, 2, 1, 2, 1, 0, 1, -2, 1, 2, 0, 2, -1, 0, -2, -2, -1], [-3, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, -2, 0, -1, -1, 0, 0, 0], [-2, 2, 0, 0, 0, 0, 1, 1, 2, -2, 0, 0, 0, 1, 1, 3, -1, 0, 1], [2, 1, 1, 0, -1, -1, 3, 1, 2, -1, -1, 0, 2, 2, 1, 1, 1, 1, -2], [-2, 4, -2, -1, 1, 1, -1, 1, 0, -4, 1, 2, 0, 0, 1, 1, -1, 0, 2], [-1, 0, 0, 1, -2, -1, 1, -1, -2, 0, 0, 2, -1, -1, -1, -1, 0, -2, -1], [-1, 1, -1, 0, 0, 3, 0, 0, 2, 1, 1, 1, 0, 1, -1, 1, -2, 1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, -3, 1, -2, 2, 1, -2, 0, 1, 0, -1, -2, -2, 1, -1, 0, 2, 5, 1], [-1, -1, 2, -1, -3, -1, 0, 1, -1, 2, 0, -3, 3, -2, 1, 2, -2, -2, 2], [1, -2, -1, -1, 2, -1, -1, -1, -2, 0, 0, 1, -2, -2, -4, -1, 0, 2, 1], [1, -2, 3, 0, 0, -4, 0, 0, 1, 3, -1, -4, 1, 0, 1, -1, 4, 2, -1], [-1, 3, 1, 4, -3, -3, 4, -3, 0, 2, 1, -1, 3, -3, -2, 0, 1, -4, -2], [1, -5, 5, 2, -1, 2, -1, -1, -2, 6, 0, -3, 3, 0, 2, -2, 4, 1, -3], [-3, -2, -1, 0, 0, 0, -5, -1, -3, 2, 2, -1, -2, -2, -1, -2, -1, 2, 3], [4, -3, 2, 1, -3, -1, 2, 0, 0, 1, -1, 1, 0, -1, -2, 1, -1, 1, -4], [-4, 1, 1, -3, 2, 0, 1, 1, 1, -1, 0, 2, 1, -1, -2, 3, -2, 1, 1], [-4, -4, -2, -1, 5, 4, -5, 1, 0, -1, 0, 0, -1, 2, 2, -3, 3, 4, 3], [-5, 1, 0, 0, -1, 3, 0, -3, -2, -1, 1, 2, 1, 1, 0, 0, -2, -1, 0], [-1, 0, 1, 0, -2, -3, 3, -1, -3, 1, -1, 0, 0, -1, 1, 1, -1, -1, -1], [1, 1, -2, 0, -1, -4, 1, 1, -1, -3, 0, 2, -5, 0, -2, 1, 0, -2, -1], [-1, 2, 1, 1, -1, 0, -1, -1, 0, 2, -1, 0, 3, 1, 3, 0, 1, -1, 0], [0, 0, 4, 2, -2, 1, -1, 0, 0, 1, -2, -1, 0, 0, 0, 1, 0, -1, -4], [1, 1, 2, -1, 1, 3, 3, -1, -4, -5, 0, 5, -1, 0, 0, 2, -4, -4, -2], [1, -1, 3, 1, -2, 0, 4, -1, 0, 1, -4, 1, 1, 2, 3, 1, 2, 0, -3], [1, 0, -1, -1, -5, -3, 4, 6, 3, -1, -3, -4, 0, 4, 3, 4, -2, -2, 1], [0, 1, -3, 0, -1, -5, 2, 0, 4, 0, -1, -3, 0, 1, -1, 0, 1, 2, 2], [3, -1, 5, 4, -3, 2, 2, -4, 2, 2, 0, 3, 2, 3, -3, 1, -5, 0, -4], [-7, -2, -2, 0, 2, -1, -2, -1, -4, 2, 2, -2, -3, -3, -1, -2, 0, 0, 2], [2, -3, 1, 0, 2, 1, -1, 2, -1, -1, 3, 0, 2, 0, 4, 0, -1, -5, 1], [-3, -1, 2, 0, 1, 2, 0, -2, 1, -1, -1, 3, -1, 1, -3, 1, -4, 0, 0], [5, -2, 4, 1, 1, 5, -2, -2, -3, 2, 2, 1, 4, -3, 0, -2, -1, 1, -4], [-2, -1, -1, -6, 2, -2, -1, 4, 0, -3, -2, -3, -2, 2, 1, 1, -2, 1, 6], [-1, 1, -2, -1, 1, 0, 3, 4, 1, -4, -1, -2, 1, 1, 4, -1, 0, -4, 1], [-5, 2, -5, 0, 4, 1, 0, 2, 4, -2, 1, -2, -2, 1, 1, 0, 2, 0, 1], [2, 2, 0, -3, 2, 4, 2, 2, 5, -2, -1, 0, 3, 2, 1, 2, -1, 2, -1], [-1, -7, 2, 1, 0, 0, -3, -4, -6, 6, 1, -2, 0, -3, 1, -5, 3, 0, 1], [1, 0, 2, -3, 3, 2, 2, -1, -1, -2, 0, 0, 0, 2, 3, 2, -2, -1, 3], [0, -5, 3, 0, -2, 0, -1, 1, -2, 6, -3, -5, 2, -1, 4, -4, 4, 3, 2], [-3, -2, -1, -3, 3, 3, -2, -4, -3, -2, 2, 3, 1, -3, -3, 0, -4, 2, -1], [4, 3, -1, 0, 0, -4, 2, 0, 2, -1, 3, -1, 1, 0, -2, 0, 0, -2, 3], [-3, 0, -3, -3, 4, -2, -1, 2, 0, 0, 0, -2, -2, 0, 0, 3, -1, 1, 4], [1, -1, -3, -2, 0, -2, -6, -1, -2, -1, -1, 1, -4, -1, -3, -4, -1, 3, 4], [-5, 0, -3, -1, -2, 0, 1, 4, 4, 1, -2, -5, -3, 5, 2, 1, 2, 0, 5], [1, -4, 2, -1, 2, 5, -1, -3, -4, -2, 1, 4, -1, -2, -2, -2, -2, 3, -3], [0, -3, 4, -6, -1, 2, -2, -2, -4, 1, -1, 4, -2, -1, -1, 4, -5, 2, 3], [5, -4, 7, 2, -3, 0, 0, -4, -1, -1, -1, 3, 0, 0, -2, 0, -2, 0, -3], [-7, 0, 0, 1, -6, 1, -2, 3, 3, 3, 1, -4, 2, 1, 2, 3, -2, -2, 0], [3, 7, 1, 2, -7, -3, 4, 1, 4, 1, 1, -4, 4, 0, 3, 3, 0, -4, -1], [2, 3, -7, -1, 1, -4, 1, 7, 3, -4, -1, -2, -1, 4, 1, 1, 0, 0, 2], [-2, 1, -3, 1, -1, -3, 1, 0, -2, -1, 1, 3, 2, -2, -1, -1, 1, -4, 0], [-2, 2, -5, -7, 7, 6, -2, 2, -1, -9, -1, 9, -2, -1, -2, 4, -5, 3, 1], [-1, 6, 3, 8, 2, -1, 1, -5, 4, 4, 2, 1, 5, 0, 3, -3, 4, 0, -4], [-2, -5, -2, 2, 0, -4, 0, 2, 0, 1, 4, 0, -3, -1, -3, -1, -2, -3, 1], [5, -1, -1, -2, 3, 0, -5, -1, -3, -2, 1, 1, 1, -2, -2, -5, 0, 4, 0], [-7, 3, -5, 1, 4, 4, -6, 1, -5, 0, 8, 0, 4, -3, 4, -4, 1, -3, 3], [7, -6, 1, -2, -3, -6, 4, 3, 1, 6, -5, -2, -1, 2, 0, 1, 0, 1, 2], [8, 3, 4, 2, -8, -3, 6, -1, 3, 2, -1, -3, 3, 0, -2, 4, 2, -1, -3], [-3, 5, 0, 0, -2, 0, 6, 3, 1, -6, -3, 2, 1, 2, -1, 3, -4, -2, -3], [8, 1, 1, 5, -2, -3, 4, -1, 0, 6, 2, -1, 5, -2, 1, -2, 3, 0, -2], [-2, -4, -1, 2, 1, 4, 0, 0, 6, 1, 0, -1, 4, 2, 1, -2, 6, 1, -5], [7, 6, 3, -4, -7, -4, 5, 4, 3, -4, -1, 1, 4, -1, -1, 12, -8, -4, 3], [-8, 4, 0, 8, 3, 2, -1, -4, -1, 2, 7, 0, 3, -2, 1, -5, 0, 0, 0], [-5, -3, -4, -1, 0, 3, -5, -2, -4, -3, -1, 3, -4, 1, -5, 0, -7, -1, 1], [-5, 1, -7, 4, 0, 1, -1, 3, 1, 0, 5, 0, -1, -1, -1, -3, 1, -2, -1], [-2, -5, -3, -4, 6, -1, -4, 1, -4, -3, 0, 0, -3, 2, 3, -2, -2, 0, 6], [-5, -3, 0, -4, -2, 0, -2, 1, -4, -4, 2, 2, -4, -1, -2, 6, -6, -3, 2], [0, 1, -5, -1, -4, -2, 3, 3, -1, -5, 1, -2, -3, 0, 4, -1, -1, -7, 3], [-2, -5, -2, 1, 0, -5, 0, -1, 2, 0, -3, -2, -2, 3, -2, -2, 3, 1, -1], [-2, -5, -7, -3, 7, 1, -4, 4, 5, 0, 0, -7, 0, 2, 4, -3, 7, 7, 3], [-10, 7, 0, -4, 5, 0, 4, -2, 2, -7, -1, 6, -3, 3, -2, 5, -1, 4, -1], [-4, 0, -5, -1, 1, 1, -2, 1, -4, 1, 2, 3, -1, -1, -2, -1, -3, -4, 3], [-3, -7, -3, -3, -2, 0, -7, 6, 0, 2, -1, -4, -4, 0, 3, -2, 1, 2, 3], [-7, -9, 0, 0, -4, 3, -5, -2, -8, 4, 1, 1, 0, -4, -3, -3, -3, -3, -1], [0, -4, 3, 1, 8, 4, -6, -2, 0, 3, -1, -1, 0, 3, 5, -5, 7, 5, -2], [-3, 4, -8, 4, -2, -3, 4, 4, 0, -5, 1, -1, -2, -1, 5, 0, -1, -11, 3], [-1, -3, 2, -3, 0, 0, 3, 2, -2, 1, -7, 2, 1, 3, 1, 2, -1, -1, 0], [-3, -2, -5, 2, 4, -2, -3, -1, -4, -2, -1, 2, -6, -4, -3, -5, 2, 3, 0], [6, -4, 10, -2, -1, -2, -3, -5, -2, 0, -3, 4, 0, -3, -4, 3, -3, 3, -5], [5, -6, 2, 0, 5, 0, 1, 2, 1, -2, 0, 2, 0, 1, 0, -2, 0, 1, -1], [-4, 5, 2, 2, 0, -4, -3, -3, -9, 0, -1, 5, -4, -5, -6, 1, -8, -1, 0], [5, 4, 1, 5, -5, 0, 4, 2, 2, 5, 3, -5, 9, -2, 1, -1, 2, -6, -2], [-1, -6, -7, -5, 4, 0, -3, 1, -5, -3, -2, 1, -2, -1, 2, -4, 3, 1, 3], [1, 2, 5, -1, -4, 1, 6, -2, 3, 4, -2, -1, 7, -1, -4, 5, -2, 7, -7], [9, 1, 1, 5, -1, -5, -1, -4, 4, -1, 2, 2, -1, -4, -7, 1, -4, -3, 0], [0, 8, -5, 3, -6, 3, -3, 4, 2, 1, 1, -1, 2, 4, 4, -2, 0, 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4, -9, 2], [18, -2, 15, 19, -11, 2, 13, -3, 13, 6, -2, -5, 21, 5, 8, 4, 1, -12, -20], [-4, -11, -14, -3, 13, 14, -15, 5, -10, 6, 3, -3, -17, 3, 9, -19, 4, 6, 6], [-25, -3, 17, -6, -7, 3, -2, -4, -7, 16, -1, 0, 5, -11, 2, 10, 2, 11, -5], [2, -17, 8, -5, 3, 8, -3, -8, -1, 2, -6, -1, -5, 12, 4, 1, -6, 3, -6], [-11, -11, -2, -4, 13, 0, -20, 2, -12, 13, 7, -16, -5, 1, 14, -8, 10, 6, 23], [18, 5, 14, 2, -6, -9, 16, 0, 15, -2, -3, -7, 14, 1, -7, 7, 8, -2, -12], [-2, -12, 7, 0, 9, 2, -18, -9, -6, 4, 0, 11, -10, -4, -10, -8, 13, 4, -6], [3, -1, -3, 2, 6, 9, 2, -2, -1, -7, 7, -3, 12, -3, 0, -4, -7, -7, 5], [20, 6, 6, 2, -2, -3, 11, -9, 12, -4, -6, 4, 3, 0, -13, 6, -2, 4, -10], [-8, 19, -11, -6, 7, -15, -7, 5, -3, -7, 2, -13, -2, 0, 1, 1, 5, 1, 24], [0, -11, 6, -5, 10, -3, -16, 6, 7, 7, -4, -6, -7, 3, 4, -5, 14, 14, -2], [9, 10, 2, 7, -11, 9, 2, 0, -7, -7, 9, 5, 10, 0, 2, -6, -2, -12, -2], [-10, 6, -5, 2, 6, 0, -9, 1, 17, -7, 6, -9, 3, -3, -3, -3, 2, 7, 6], [10, -5, -13, 6, -10, -9, 4, 6, -7, -1, 1, -1, -2, 2, 11, -2, 10, -9, 2], [1, 4, -8, -14, 16, 1, -2, 1, -6, -18, -5, 9, -14, 6, -5, 2, -14, 13, 8], [-15, -1, 8, 7, -11, 11, -1, -6, 2, 8, 11, 5, 11, -2, 4, 3, -14, -9, -6], [2, 4, 4, 11, 4, 3, -2, -4, 2, -9, 12, 12, 1, -1, -12, 1, -8, -14, -5], [-16, -8, -13, 9, 1, 17, -4, 3, 7, 3, 8, 0, -1, 6, 5, 0, -3, -3, -7], [-1, -2, 5, 8, -6, 1, 4, -3, 10, 8, 5, -8, 3, 3, 9, -10, -1, -5, 5], [12, 5, 4, -2, 1, 1, 4, 5, 17, -11, -8, -7, 7, 17, 10, 6, 11, 10, -9], [2, 5, 8, -2, -5, 5, 0, -3, 1, -3, 5, 0, 11, -13, 2, 3, -3, -4, -3], [8, 9, -8, -7, 6, 9, 6, 0, 13, -15, 6, 12, -1, 4, -12, 6, -15, 12, -4], [3, -14, -5, 8, -9, 1, 7, -3, 8, 16, -7, -2, 2, 3, 7, -1, 7, -1, -10], [-15, 7, -10, 3, 0, -9, 12, 1, 4, -1, 1, -1, 4, -1, 0, -9, 9, -10, -1], [-26, -15, 4, -23, 6, 0, -11, 3, -7, -1, -10, 3, -10, -3, 4, 1, -7, 10, 5], [5, 5, -23, -13, 9, 11, -3, 13, 1, -16, 3, 5, -4, 1, 2, -7, -5, 0, 6], [-1, -24, 9, 4, -8, 6, -2, -5, -9, 20, -12, -11, -3, 1, 1, -2, 7, 4, -4], [0, 1, -6, 1, 12, 5, -2, 1, 12, -1, -4, 8, -2, 6, -7, -6, 10, 18, -6], [-3, 25, -4, -3, -3, 3, 7, 12, 6, -12, 6, 3, -1, 10, 7, 11, -2, -12, -6], [2, 4, 3, 8, 0, -3, 13, -5, 18, -3, -3, 10, 9, 4, -13, 8, -5, 12, -15], [10, -4, 2, 0, -4, -3, 8, 10, 3, -8, -10, 6, -3, 9, -4, 10, -4, -4, -3], [15, 18, -10, 5, -4, -5, 3, 2, 0, -15, 11, 9, 2, -8, -8, 1, -17, -14, 8], [6, 1, -13, 14, -5, -15, -7, 0, 6, 5, 12, -9, -1, -9, 5, -12, 5, -15, 3], [8, 17, -15, -3, 13, -7, 7, 3, 8, -4, -3, 4, 3, 9, 2, 0, 5, 12, 7], [9, -8, 13, 13, -8, -11, 7, -2, -4, 22, -1, -15, 2, -10, 9, -8, 21, -6, -8], [-10, 0, 2, -7, 5, 6, -16, -8, -12, -3, 5, 8, -5, -6, -10, -9, -1, 11, -2], [3, -10, 11, 5, -2, -11, 9, -6, -1, 2, -7, 6, 2, -10, -7, 8, -6, -3, -9], [14, -21, 2, -18, 5, -14, 3, 7, -13, 6, -8, -6, -14, -5, 0, -6, 12, 11, 10], [-3, 9, -10, 1, 1, -2, -7, -1, 5, 2, -3, -1, 2, -4, -2, -8, 7, 13, -1], [-19, -18, -3, -6, 4, 6, -15, -14, -6, 14, 0, 2, -12, -16, -11, -17, 6, 18, 3], [7, -28, 6, -1, 9, 3, -16, 9, 0, 10, 8, -7, 0, 0, 7, -8, 1, 1, 7], [1, -4, 13, 0, -1, 14, -4, -9, -5, 12, -15, 9, 6, 1, 8, 2, -2, 8, -6], [-5, 0, 8, 4, 3, 9, -10, -7, -13, 7, 1, 3, -1, -2, 2, -5, 5, -1, -5], [4, 2, 5, -11, 2, 3, -3, -2, 0, -5, 2, 5, 3, -7, -7, 10, 2, 9, 3], [-14, 1, -3, 3, -2, 18, -4, -8, -3, 4, 2, 2, 0, 2, 0, -4, -12, 5, 5], [21, 18, 8, 5, -13, -28, 20, -1, 11, 2, -15, -5, 9, -1, 2, 16, 8, 2, -10], [18, 4, 2, 2, -4, -9, -2, 4, 4, 2, -2, 7, 3, -6, -7, 7, -7, 3, -11], [-27, -1, -12, 13, 5, 1, -3, 2, 9, 7, 6, -1, -10, 3, -1, -10, 13, 7, -7], [30, 11, 2, -1, -9, -16, 20, 0, 1, -3, -8, 7, 8, -9, -4, 14, 3, -3, -10], [-13, -4, -3, -10, 25, -7, 1, 8, 5, -2, -4, -4, -3, 9, 4, 1, 8, 14, 9], [6, -6, -15, -8, 9, 8, -4, 7, -6, -5, -3, -1, -19, 2, 7, -8, 7, 1, 16], [6, 18, -12, 1, 1, 7, 0, -1, 1, -5, 5, 7, 3, 2, -2, -4, -8, -2, 1], [26, 6, 11, 7, -17, 3, 10, -6, 5, 2, -1, 8, 8, 9, -9, 4, -9, -5, -13], [14, -11, 16, 3, -3, -16, 5, -4, 8, 20, -17, -17, 4, 5, -1, 5, 17, 21, -6], [-4, -7, 12, 1, -6, 7, -7, -4, -26, 17, -1, -5, -3, 1, 9, -6, 12, -3, -4], [-8, 11, 6, -3, -18, -6, 18, 5, 14, -5, -11, -3, 8, 9, -3, 22, -10, 1, -7], [4, 1, 6, -3, -4, 4, -10, -6, -4, -4, 6, -7, 0, -5, -5, 1, 3, 2, 3], [3, 17, 8, 11, -9, 1, 1, -9, 9, 1, 0, 1, 14, -3, -10, 9, 5, 2, -19], [-7, 11, -8, 0, -3, -12, 6, 8, 6, 11, 7, -20, 10, 1, 11, 5, 4, -16, 11], [7, -3, 17, 9, -12, 16, 5, -8, 5, 8, -6, 7, 7, 6, 4, 2, 2, 3, -14], [28, 7, 0, 4, -1, -3, 8, -5, 3, -1, 8, 8, 10, -8, -8, 4, -9, -8, -10], [-25, -4, 0, 10, -4, -5, -1, 0, 0, 12, 1, -17, 3, 3, 8, -9, 8, -2, 7], [11, 12, 13, -2, -11, -11, 23, 3, -2, -3, -7, 0, 14, 1, -2, 16, -9, -9, -8], [15, -6, 9, -1, -1, -9, 4, 3, -3, -2, -11, 9, -9, -1, -7, 6, -4, 5, -4], [-18, -7, -9, -14, -4, -11, -2, 0, -11, -11, -3, 8, -8, -5, -10, 6, -17, -8, 10], [-2, 2, -5, -1, 3, -8, 13, 1, 0, -5, -5, 6, -3, 3, 8, 4, 5, -5, -2], [5, 7, -4, 3, 9, 3, -5, -2, 15, 7, 10, -9, 12, -2, -1, -15, 23, 11, 1], [14, -11, -11, -1, -10, -13, 2, 0, -11, 2, 0, 1, 5, -7, -1, 4, -12, -4, 12], [-2, 7, -9, -6, 8, 8, -5, -1, 12, 1, 3, -3, 8, -4, 2, -9, 13, 10, 4], [-15, 12, -17, 2, 4, 9, 2, -5, -5, -10, 12, 7, 3, -6, -6, -2, -5, 0, -3], [-2, -9, 6, -9, 10, 8, -7, -3, -10, -3, 2, 20, -8, -8, -10, 1, -10, -2, -7], [11, -4, 6, 14, -3, 15, -8, -19, -14, 16, 11, 2, 12, -15, 0, -11, 12, -1, -11], [-15, -9, 7, -4, -2, 11, 3, -4, -2, 7, 1, -2, 2, 1, 15, 2, -2, 2, -1], [3, 6, -11, -1, 3, 2, -4, -3, 5, 8, 9, -1, -5, -2, -1, -3, 1, 2, 1], [0, 11, -6, 4, 6, -3, -8, -1, 0, 15, 6, -16, 8, 3, 10, -5, 13, 5, 5], [-33, -18, -1, -7, -7, 4, -10, 8, -6, 1, -1, -10, -4, 1, 7, 3, 13, 0, 3], [-33, 2, -11, -11, 12, -1, -3, 0, -2, -7, -6, 5, -12, -5, -4, -1, 3, 2, 6], [-10, -7, -2, -9, 1, -4, -6, 10, -7, 2, -11, -8, -9, 9, 12, 3, 1, -3, 10], [0, -12, 7, -19, 11, 1, -5, -1, -1, -2, -14, 8, -14, 7, 0, 11, 4, 27, 2], [1, -9, -2, -4, -4, -3, 11, 12, -1, -1, -7, -4, 1, 10, 4, 11, -11, -2, 1], [25, 12, 9, 9, -11, -2, -3, -12, 0, 22, 6, -9, 10, -2, 3, -11, 5, 3, 5], [1, 10, 14, 9, -20, 3, 14, -12, 0, 3, 0, 5, 5, -3, -15, 3, -18, -9, -19], [-1, 13, -3, -23, 8, -2, -3, -3, -2, -15, -6, 22, -16, -2, -17, 19, -26, 7, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4017_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4017_2_a_f();".
// 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_4017_2_a_f(:prec:=19)
    chi := MakeCharacter_4017_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_4017_2_a_f();".
// 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_4017_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4017_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-4, 11, 378, -352, -2261, 1430, 5579, -2223, -7285, 1491, 5460, -278, -2388, -154, 594, 88, -77, -16, 4, 1]>,<23,R![-8242928272, 12979811948, 12769069416, -20472639619, -5827532004, 11334759526, 941458935, -2845281766, -104993838, 376748205, 18626126, -27230040, -2302565, 1002200, 125118, -15622, -2877, 26, 23, 1]>],Snew);
    return Vf;
end function;