
// Make newform 6026.2.a.f in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6026_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_6026_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_6026_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 := [-18, -78, 1159, 576, -9137, 2042, 25315, -14682, -28084, 21886, 14837, -14776, -3723, 5359, 248, -1083, 78, 115, -17, -5, 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], [0, 1, 0, 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, 0], [-5029558767, -6210157485, 80734359230, 266688463738, -154167082409, -1142544646218, -50250467098, 1839263430391, 330166097514, -1492751196985, -311582977642, 686826922760, 136061253901, -187493827325, -31119918252, 30194840133, 3599346675, -2663674000, -165884442, 99716990], [-1903874094, 19091308350, -239014121539, -514352194880, 1484823148105, 1927564243203, -3564239814652, -2433986096297, 4014750146469, 1387964422985, -2433681484336, -353315039944, 845435323468, 12131055271, -168614176551, 12991637169, 17942676336, -2581287385, -788021802, 152897921], [10210385484, 12667611897, -655658120918, -868959764302, 3828596607116, 3181938636582, -8681537826470, -3483690211072, 9252549939678, 1465883080024, -5302318876388, -79154083532, 1740985774982, -134696224735, -328492457904, 46406228355, 33136669317, -6233624999, -1383955827, 309851458], [11277017919, -13818490434, -444981502009, -479591138117, 2669559331153, 2022451755138, -6497668918675, -2230153018763, 7345735584573, 822808635896, -4429190235145, 94977561293, 1522754372488, -160904820680, -299751293436, 48925034667, 31439546223, -6450050938, -1360457148, 322669970], [7853033133, -6970520862, -443269509616, -481303130510, 2669559331153, 2022451755138, -6497668918675, -2230153018763, 7345735584573, 822808635896, -4429190235145, 94977561293, 1522754372488, -160904820680, -299751293436, 48925034667, 31439546223, -6450050938, -1360457148, 322669970], [3805064997, 13425493694, -141470821499, -263557449692, 880564478145, 917906878673, -2228055002861, -1049940919284, 2647408333697, 474253792532, -1685927227003, -27928299023, 611952043660, -50992377522, -126771603270, 19013317515, 13927919237, -2750855046, -627942667, 146343330], [-21496095405, -123020445429, 862543526110, 2213572376246, -5106166101469, -7746619103055, 12597173558332, 9136296553166, -14170725778413, -4771815292994, 8410243049992, 1050361806526, -2832716706445, 14470325336, 545581112568, -50354083845, -56024761614, 8657638597, 2377631685, -477486794], [18210231207, 129553986273, -827450554963, -1822112773976, 5021707486708, 5657252453124, -12398618288461, -5708901648527, 13944905779887, 2096214081833, -8265873305065, 96329231267, 2775395086510, -296476727960, -531636343350, 89385080193, 54163977555, -11409071752, -2275865718, 550477613], [12725619606, 7395908349, -741450867083, -1130822268817, 4609864588088, 4417753791675, -11746626266969, -4598705287915, 13445110060605, 1516003317697, -8055235042100, 284298537844, 2725240933223, -334836535351, -525192157785, 94239617940, 53784538836, -11766008126, -2270260821, 562516600], [-15043298778, -98272405995, 1077516056294, 2217972391426, -6931059756389, -7487287629810, 17620952974694, 7662576443257, -20128396990494, -2722915606537, 12065066105960, -252098034391, -4091073927500, 460823919703, 791153125932, -135968571153, -81372015369, 17340713522, 3451389795, -840747436], [-6583539255, -25266898338, 334887564545, 583750175422, -2191104403268, -1963791412764, 5660449894751, 1860096940774, -6556627302303, -438853391215, 3983590226324, -271585338031, -1368544326728, 200567866756, 267919145877, -52656595317, -27863463012, 6440353895, 1193388237, -306455260], [3636815862, 28438633600, -316169845004, -494972035686, 2119661708299, 1522471869796, -5441991275711, -1212682461482, 6281885236696, -5754095250, -3812158841505, 436085177527, 1309458273684, -234380734790, -256415042700, 56303390241, 26675284060, -6603875713, -1142759006, 306841820], [21903959223, 110179494222, -1351112921560, -2585383086992, 8535466217248, 9087254168172, -21643300706941, -9712589542436, 24888060042309, 3787701103535, -15082153681294, 90802281113, 5180055874786, -521211788708, -1015296823950, 165046457061, 105828203667, -21817310038, -4546314924, 1084667243], [26342084538, 167597701761, -1588589428406, -3344500909942, 9847996099748, 11437902014430, -24392018763977, -12573673222303, 27580460910462, 5529256958347, -16492312061762, -493304121437, 5600687059850, -412087129414, -1087065350931, 154602262989, 112365319416, -21455653031, -4793379879, 1089425851], [133089489, 740541705, -6391055815, -13420672217, 39427171861, 44859980994, -98477264995, -47391522458, 111562121574, 18907936892, -66542626897, -294234073, 22488696304, -2109921236, -4338879945, 685127538, 445558719, -90366943, -18879387, 4456799], [44758625463, 204713127345, -2180302356475, -4131055705850, 13409940203227, 14054638663002, -33358718318455, -14792189063120, 37854401980263, 5678248146920, -22682866537195, 160415769779, 7710757987687, -779896124435, -1496966561592, 243686192967, 154671482871, -31890712168, -6592015260, 1571933387], [16459377834, 81393303662, -859091649656, -1558745920940, 5245158178278, 5219898632999, -12843632042951, -5460601016910, 14459887772984, 2074169438390, -8637298145164, 81393002008, 2934376722508, -298853019066, -570148648821, 92842904574, 59005753619, -12156366042, -2519966401, 600429471]];
    Rf_basisdens := [1, 1, 1, 1711992393, 1711992393, 1711992393, 1711992393, 1711992393, 570664131, 1711992393, 1711992393, 1711992393, 1711992393, 570664131, 570664131, 1711992393, 1711992393, 6661449, 1711992393, 570664131];
    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_6026_a();"
function MakeCharacter_6026_a()
    N := 6026;
    order := 1;
    char_gens := [787, 4325];
    v := [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_6026_a_Hecke(Kf)
    return MakeCharacter_6026_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 := [[1, 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, 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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 1, -1, -1, -1, -1, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 0], [-1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, -1, 0], [-2, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, -1, 1, -1, 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, -1, 1, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 1, -1, -1], [0, 0, -1, -1, 0, 0, 0, 0, 1, -1, 1, 1, 0, 1, 2, 0, 0, 0, 0, -2], [1, -2, 1, 0, 1, 1, 1, -2, 0, -1, 1, 1, 2, 0, 1, -1, 0, 0, 0, -1], [-3, 0, 1, -1, -1, 2, -1, 2, 0, 1, 0, -1, -1, 0, -2, 0, -2, -1, 1, 1], [-1, 1, 1, 0, -2, 1, 0, 1, -2, 1, 0, -1, 0, 1, -1, 1, 0, 1, 0, 1], [-2, 0, 2, 2, 1, 0, 1, 1, 0, 0, 1, -1, 1, -1, -1, 0, 0, 1, 1, 0], [-2, 2, -2, 0, 1, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 1, 1, 0, 1, -1], [-2, 2, 2, 0, -1, 0, 1, 1, -1, 1, -1, -1, 1, 0, -2, 1, 1, 1, -1, 2], [1, 2, -2, -3, 1, -3, 1, -2, -1, -1, -2, 0, -1, 1, 2, -2, 2, -2, 1, 0], [-1, 0, 0, 0, 0, 2, -2, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1], [-3, -1, 0, -3, 4, 0, 0, -6, 2, -2, -2, 0, -2, -1, 0, -2, 0, -3, 2, 0], [-5, -2, 3, -1, 0, 2, 1, 1, 1, 0, 1, -2, 0, -3, -2, 0, -1, 1, -1, -1], [-1, -3, 0, 0, 2, 0, -1, -3, 3, -1, -1, 3, 0, 0, 1, -1, -1, -2, 1, 1], [-3, -3, 2, 0, 3, 2, -1, -1, 1, 0, 0, -1, -1, -2, -2, -1, -1, 0, 1, -1], [-1, 0, 1, -1, 0, -2, 0, -1, 0, 0, 0, 0, 1, -1, 2, -2, 1, 0, -1, 0], [-2, -2, 2, 1, 1, -1, 1, 0, 2, -1, 0, 0, 1, -2, 0, 0, 1, 3, -2, 0], [-1, 2, -3, -2, -2, -1, -3, 1, -1, 1, -1, 0, -4, 1, -2, 0, -1, -3, 3, 0], [2, 1, -1, 1, 3, -2, 1, -4, -1, 0, -3, 2, 0, 2, 2, 0, 2, 0, 2, 0], [-1, 0, 0, -1, -1, -2, -2, -1, 2, 1, 0, 1, -2, 0, 0, -1, 0, -1, 1, 1], [-11, -2, 3, 2, 0, 2, -1, 5, 0, 4, 0, -6, -2, -3, -7, 2, -3, 2, 2, 1], [-3, 1, 1, 4, -3, 1, 1, 7, -2, 2, 3, -3, 0, -1, -4, 4, -3, 1, -1, -1], [-2, -5, 2, 5, 2, 3, 3, -2, 1, 1, 4, 0, 2, 0, 1, -1, -2, 3, 0, -2], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 5, 0, -2, 2, 1, 2, -1, -1, 1, -3, -1, -1, -3, 1, -2, 2, -1, 3], [0, -1, 0, 1, -1, 0, 1, 0, -2, 0, -1, 2, 0, 0, 1, 0, -1, 1, -1, 2], [-3, -6, 4, 3, 4, 1, 1, -2, 2, -1, 1, 1, 1, -2, 1, -1, -1, 3, 1, -2], [-4, 2, 0, 0, -3, 2, -1, 1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 2, -2, 0], [2, -5, -1, 0, 0, 1, 2, 0, 1, 1, 1, 1, 1, -1, 2, -3, 0, -2, 0, -3], [-4, 0, 3, 2, -1, 1, -2, 5, 0, 2, -1, -1, 1, -2, -2, -1, 1, 2, -1, 1], [2, -1, -5, 3, 0, -1, 2, -2, -4, -1, 0, 0, 0, -1, 2, -1, 2, -1, -3, -2], [-1, -2, -1, -1, 5, 1, 1, -6, -1, -1, -4, 3, 3, -1, 2, -3, 4, 2, 1, -2], [2, -2, 1, 0, 4, 2, 1, -7, 2, -2, 1, 4, 3, 0, 2, 0, 4, 3, -1, -1], [4, 1, -1, -1, 1, -2, -3, -3, -2, -2, -2, 2, -2, 1, 1, -1, 2, -2, 4, 1], [-1, -1, 2, 3, -1, 3, -4, 3, 0, 1, 2, 1, 3, -2, -2, 1, -2, 1, 2, 1], [-5, 5, -5, -3, -4, -2, -4, 5, 1, 1, -1, -3, -4, 3, -2, 3, -3, -6, 0, 1], [-3, 0, -4, 0, 1, -1, 2, 2, -1, 0, 1, 0, 0, 0, 2, -3, 1, 0, -1, -1], [2, 0, -2, 2, 1, 0, 1, 2, -4, -1, 0, -1, 0, 0, 0, 0, 2, 0, -1, 0], [-1, 3, 3, 3, -1, 0, 2, 1, -2, 1, 0, -1, 2, -1, -2, 4, 3, 3, -1, 0], [6, 4, -7, -4, -1, -5, -3, -3, 1, -2, -3, 7, 1, 3, 7, -2, 4, -2, -4, -2], [-1, -3, 4, 0, 1, 2, 1, -1, 2, 3, 1, -1, 2, -2, -3, -1, 0, 0, 0, -1], [4, -3, -3, 0, 2, -3, 1, -7, 0, 1, -4, 3, 1, 0, 4, -5, 0, -3, 0, 1], [8, -2, -5, -5, 2, -4, 3, -9, -4, -4, -5, 7, 2, 4, 7, -4, 3, -3, -1, -1], [-1, 1, -1, 1, -2, 0, 4, 3, -3, 0, 2, -2, 2, 2, -1, 2, -1, 1, 0, 0], [-3, -3, 0, -1, 3, 2, -1, -2, 1, 0, 2, 0, 0, 0, 1, -2, 0, 1, 0, -2], [-2, -4, 4, 2, 4, 3, 0, 0, 1, -1, 2, 2, 3, -2, -2, -1, -1, 3, 0, 0], [1, 3, -3, 2, -1, -3, -2, 4, 0, 1, 0, -2, -3, 2, -1, 3, -4, -7, 3, 2], [-3, -1, -2, -3, -3, -1, -2, -1, 1, 1, -2, -2, -2, -3, 0, -1, -2, -3, -2, 2], [-5, 4, 0, -2, 4, -3, -3, -1, 0, -2, -5, -1, -1, -1, -1, 1, 3, 1, 3, 2], [-1, 3, 0, 2, -6, 0, -1, 5, -2, 1, 2, -4, 1, 0, -3, 3, -1, -1, -1, 1], [-4, -3, 0, 1, -3, 2, -1, 3, 2, 0, 2, -2, -5, -1, -3, 3, -6, -3, 2, 1], [0, 0, -2, 4, -1, -1, 0, 7, -1, 3, 2, -4, 0, 1, -3, 3, -1, -3, 3, -2], [9, 2, -8, -1, -4, -1, 1, 2, -4, 1, 0, 0, 0, 6, 5, 2, 3, 1, -3, -2], [-10, -3, 2, -1, 0, 3, 1, 2, 2, 0, 1, 0, -1, -3, -4, 2, -4, 0, -2, 1], [4, -5, -2, -3, -3, 0, -1, 1, 0, 1, -1, 0, -3, 2, -1, -2, -3, -5, 4, -2], [4, -3, 0, -1, 9, 0, 6, -10, 0, -4, -1, 1, 3, 0, 5, -3, 5, 4, 0, -3], [6, -2, 3, 0, 5, 2, 0, -4, 0, 1, -2, 3, 3, 2, 0, -1, 6, 4, 1, -2], [-1, -4, 7, -1, 6, 2, 7, -8, 2, -5, 3, 1, 6, -2, 3, -1, 2, 4, -5, 1], [-3, 0, -3, 0, -2, 1, -1, 4, 0, 6, 1, -4, -2, 3, -3, -1, -2, -5, 3, -1], [-1, 2, -1, -2, -3, -1, -3, 0, -2, 1, -3, 4, -1, 3, -3, 1, 0, -1, 0, 6], [-3, -1, 4, 0, 2, 2, -4, -2, 0, 0, -4, 2, 2, -2, 0, 0, 3, 7, 1, -1], [-9, -5, 6, -1, -3, 6, -3, 3, 3, 4, 3, -3, -1, -4, -5, 1, -6, 0, 1, -1], [-3, 4, -1, -3, -9, 1, -4, 10, -2, 5, 1, -4, -3, 2, -5, 5, -2, -3, 0, 4], [-2, 1, 0, 2, -3, 0, -3, 3, 1, 2, -2, -2, -1, -4, -2, 1, 0, 0, -3, 5], [1, 2, -3, 3, 6, -1, -2, -7, 1, -3, -3, 3, 1, 2, 1, -2, 2, 4, 1, -1], [4, 2, -2, -6, 2, -3, 5, -5, -4, -3, -6, -2, 1, 2, 3, -3, 2, -4, -2, 6], [-1, -2, 3, -4, 8, 1, 1, -13, 4, -4, -5, 5, 6, 0, 4, -5, 3, 1, 0, 1], [3, -1, 4, 3, -6, -1, 2, 2, 2, 1, 3, 1, 0, -1, 1, 0, -1, 5, -2, 2], [4, 4, 1, -3, 0, -5, 1, -9, 3, -4, -4, 6, 3, 3, 9, -2, 4, 1, -5, 5], [0, 2, -1, 2, 1, -4, 0, 5, -3, -1, -3, -3, -2, 1, -4, 3, 3, 1, 2, -1], [0, -7, -1, 1, 1, 3, -1, -1, 3, 1, 2, 1, -1, 0, -2, -3, -3, -5, 2, -1], [-8, 4, -1, 2, -6, -2, -5, 9, 2, 6, 3, -2, -5, -1, -6, 3, -3, -2, 1, 4], [5, 4, -5, 2, 1, -6, -1, -2, -4, -1, -6, 2, 0, 1, -1, 0, 3, 1, 2, 1], [4, 2, -9, -3, -4, -4, 4, 0, -4, -2, 1, 3, 1, 3, 7, 0, 4, 5, -7, 0], [-4, 2, -3, -1, -4, -1, 2, 2, -4, 1, 2, -2, 0, -2, 3, -2, -1, -1, -2, 3], [7, 3, -1, 2, -1, -2, 3, 2, -2, -3, 3, 0, 1, 2, 2, 5, 2, 2, -1, -3], [-4, -4, 9, 6, 4, 5, 4, -3, 1, -1, 3, -1, 8, -3, -3, 0, 3, 10, -1, -1], [2, -1, -4, -7, 4, -4, 1, -7, 3, -3, -5, 0, -4, 2, 6, -4, -2, -12, 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-5, -7, -14, 10, -4, 5, -3, -7, 5, 0], [7, -10, -1, 2, 7, 4, 12, -24, 0, -7, -8, 14, 12, 0, 20, -9, 11, 15, -17, -2], [-6, -10, 20, -2, -3, 11, 0, 13, 3, 9, 10, -15, -4, -4, -17, 5, -12, 6, 3, 4], [2, -11, -3, -8, 6, 1, -7, -5, 4, 5, -9, 3, 1, 3, 3, -9, -1, -14, 4, 1], [-3, -2, -4, -1, 9, 10, 1, -7, -14, 3, -10, 0, 4, 2, -4, -5, 11, 5, 4, 6], [-4, 5, -14, 1, -2, -2, -6, 0, -8, 5, -6, 10, 1, 2, -1, -1, 7, 9, -2, 0], [4, 8, -5, -20, 13, -14, -4, -23, 5, -14, -13, 10, -4, 9, 12, -12, 10, -12, 19, -5], [-5, 4, 7, 4, 0, -1, -4, -6, 3, -2, 4, 3, 8, -9, -6, 4, 6, 5, -1, 10], [3, -5, 5, -2, 3, 4, -10, 4, 4, 6, 2, 6, -5, 7, -7, -6, -3, -11, 14, -2], [-6, 2, 5, 12, -8, 4, -1, 9, -6, 8, 1, -6, -2, -3, -10, 3, -1, 13, -2, 5], [14, -2, 2, -1, 7, -3, 5, -5, -2, -12, -5, 2, 1, -8, 9, -5, 8, 3, -4, -1], [-20, -7, 4, -9, -3, -1, 4, 10, 0, -3, 7, -3, -8, 1, -2, 1, -4, -4, 1, 0], [6, -14, 8, 14, -13, 9, 1, 20, 7, 7, 19, -8, 2, -3, -3, 9, -11, 13, -7, -3], [-9, -8, 1, -10, -3, 1, -2, 1, 9, 1, -4, -11, -13, -4, 2, 3, -12, -12, -3, 0], [3, 1, -9, -1, -6, -2, -2, 4, 8, 1, 4, 2, 1, 3, 8, 4, 1, 3, -2, -11], [0, 13, -3, 6, -8, 2, -2, 29, -4, -1, 5, -6, -6, 8, -5, 10, -2, 1, 1, -2], [-16, -5, 3, 0, -2, -6, 4, 10, 1, 3, -3, -8, -5, 4, -4, 3, -4, 1, 3, 5], [-7, -11, -1, -4, 20, -11, -1, -24, 9, -4, -18, 12, -4, 5, 6, -7, 5, -2, 7, 5], [-13, -8, 12, 11, -5, 9, -2, 21, -3, 10, 17, -14, -6, 4, -16, 11, -13, 8, 4, -3], [19, 2, -16, -6, -13, -12, 8, 14, -6, 4, -4, -2, -8, 10, 7, 6, 1, -9, -4, -1], [-5, -9, 8, 19, 11, 7, 8, 9, -1, 7, 11, 0, 11, -4, -6, -3, -8, 12, 0, -2], [-33, -10, 27, 6, 13, 18, 8, 0, 8, 0, 4, -4, 11, -15, -18, -3, -1, 16, 0, 3], [8, 5, -11, -12, 0, -8, 11, -15, 1, -9, -4, 2, -2, 6, 16, -4, 1, -2, -7, 9], [-3, 14, 10, 0, -6, 6, 2, 10, -1, 7, -1, -15, 4, -2, -17, 11, -7, -7, 0, 9], [-7, -3, 16, 8, 11, 3, 2, -7, 3, -1, 3, -4, 4, -11, -11, 1, -8, 3, 6, 10], [-11, 7, -2, -6, -19, 14, -2, 13, -8, 11, 5, -4, 9, -2, -7, 8, -6, 11, -13, 0], [7, -28, 11, 2, 10, 12, 0, -21, 10, -2, 9, 14, 5, -6, 7, -6, -4, -4, 8, -17], [16, 7, -11, 17, 6, -9, 6, -3, -17, 1, -7, 10, 14, 10, 12, -2, 16, 5, -2, -5], [20, -8, -2, 13, 5, -6, 16, -11, -5, -5, 4, 5, 13, 14, 15, -5, 7, 8, -11, -5], [-16, 6, 4, 8, -16, 4, -8, 36, 4, 13, 9, -14, -5, -1, -12, 5, -13, -5, -1, -2], [-21, -5, 5, 14, 2, 4, -3, 8, -4, 5, 4, -3, 4, -12, 1, -4, 8, 9, -1, -13], [19, 8, -6, -17, 7, -8, -3, -14, 9, -3, -3, 5, -2, 10, 12, -7, 0, -15, 2, 3], [-4, -19, -1, -11, 16, -2, 4, -15, 4, 0, -8, 9, 4, -6, 7, -16, 3, -4, 6, -14], [15, 8, -13, -6, -9, -9, -2, 5, -3, 7, -1, -1, -6, 15, 0, -4, -1, -11, 3, -2], [12, 17, -12, 6, -7, -5, -2, 8, -7, -3, -3, 8, 1, 2, 6, 1, 0, -8, 0, 5], [-2, -5, 12, -8, 4, 14, -12, -14, 8, -5, -8, 14, 10, -6, 5, -9, -3, -5, -1, 6], [-2, 6, -4, -12, -5, -5, -6, -3, 4, -3, 1, 2, -12, 6, -1, 7, 5, -7, 1, -1], [-14, 5, 2, 4, -2, -2, 3, 11, -4, -3, 10, -2, 2, -5, 6, -2, -1, 6, -1, -13], [-16, 14, -8, -14, -1, -6, -1, 4, 0, -1, -2, -14, -13, 3, -7, -1, -2, -13, 1, 3], [-9, 0, 4, 5, -3, 2, 9, -5, 0, 0, -1, 2, 1, -7, 0, 6, -6, 4, -8, 3], [-8, 3, -9, -1, 0, -6, -4, 5, -3, -1, 3, -3, -5, 13, 1, -2, -3, -8, 5, 0], [-4, -6, -6, -14, 18, -3, -2, -28, 13, -6, -10, 4, -10, -1, 9, -13, -3, -12, 3, 0], [-7, 14, 8, 8, -23, 10, -4, 24, 0, 8, 17, -18, -5, -6, -15, 19, -19, -1, -5, 11], [-12, -1, 12, 7, 18, -3, 9, -15, 3, 3, -1, -1, 10, -9, 1, -12, 4, 13, 1, 1], [-12, -2, 17, 11, 0, 6, 6, -2, 3, 4, -1, -2, 5, -6, -11, 1, -7, 17, -1, 12], [0, 1, -10, -4, -12, -4, -1, 3, 0, 0, -4, 5, -8, 7, 3, 0, -3, -4, -1, 7], [17, 4, -6, -4, -13, -4, -9, 16, -6, 7, -6, 1, -1, 10, 2, -1, 3, -6, -4, 2], [4, 0, -1, -3, -5, -5, -2, -5, 2, -4, 2, -4, -7, 1, 8, -3, 1, -12, 8, -8], [-9, 2, 6, -12, 13, -2, 3, -15, 3, -9, -11, 1, 2, 4, 9, -7, 6, 4, 2, 6], [-4, -11, -1, -18, -1, 7, -3, 0, 3, -6, 1, 2, 2, -1, 3, -4, -3, -4, -3, 4], [-14, -1, 15, 2, 1, -3, -8, 5, 11, -2, 2, -2, -7, -9, -8, -3, -2, 0, 3, -3], [1, 17, -7, -10, -3, -9, 3, -9, 4, -1, -9, 4, 2, 13, 14, 1, 5, -10, -3, -1], [0, -2, 9, -14, 27, -3, 14, -36, 10, -12, -8, 9, 12, -1, 16, -12, 10, 0, 0, -2], [-37, 12, 8, 7, -3, 9, -6, 9, -2, 4, -1, -7, -1, -6, -21, 12, -5, 1, 3, 13], [10, -20, -13, -8, 3, -1, 2, -5, -4, -5, -10, 6, -6, 5, 6, -11, -6, -12, 4, -4], [-8, 8, -19, 1, -9, -10, -13, 13, 5, 14, -1, -4, -13, 4, 1, -1, -8, -12, 1, 1], [18, 1, -1, 2, 4, -9, 5, -21, -1, -3, -9, 7, 7, 4, 14, -7, 16, 5, 0, 1], [-17, -5, 5, 6, 20, 7, -3, -4, 6, -1, 0, -5, -8, -12, -20, 0, -4, -3, 15, -6], [-24, 2, 5, 2, 9, -3, 3, -3, -6, -2, -5, -3, 4, -1, -1, -7, 0, 7, 5, -8], [-1, -17, 2, 8, 11, 1, 14, 2, -1, 5, 0, -1, 1, 1, -4, 1, 2, 17, -6, -13], [9, 3, -8, 2, -8, -6, 7, -5, -3, 0, 7, 1, 7, -2, 14, 4, 4, 9, -15, -3], [-4, 8, 0, -10, 2, 4, -4, -6, 3, 2, -7, -3, -6, -10, -4, -8, -5, -7, 5, 6], [4, 19, -21, -6, -14, -6, -8, 0, -2, -6, 3, 7, -9, 5, 12, 2, 1, -15, -4, -1], [0, 1, 0, -4, 7, 0, 5, -21, -4, -3, -5, 7, 11, -9, 10, -14, 8, -5, -5, 2], [-9, -14, 12, 1, 4, 14, 3, -8, 3, 0, 12, 0, 13, -12, 0, -8, -7, 15, -7, 3], [33, 16, -18, -6, -4, -10, 4, -11, -12, -4, -10, 8, -3, 14, 10, 0, 9, -10, -2, 12], [-14, -22, 29, 22, 29, 17, 3, -19, 15, -3, 4, 9, 16, -16, -2, -8, -2, 20, 7, -5], [7, -11, -5, -1, 11, 7, -7, -18, 5, -6, -5, 8, 2, 0, 14, -6, 7, 5, -4, 2], [-2, -7, -1, -5, -10, -6, -6, 2, 5, -4, 3, 12, -5, 6, 11, 3, -8, -9, -6, 8], [15, 1, -2, -15, 3, -11, -9, -12, 9, -14, -4, 14, -8, 9, 18, -5, 7, -15, 2, 1], [8, -18, 1, -13, 25, -6, 6, -27, 8, -13, -7, 4, -3, 8, 5, -9, 2, -12, 8, -3], [19, 8, -6, 4, -3, -10, -3, -3, -6, -5, 4, 14, 5, 16, 21, -6, 13, 10, -4, 4], [-15, -4, 24, 23, -3, 16, -7, 9, -1, 4, 13, 2, 15, -12, -11, 8, 2, 24, -7, 5], [16, 11, -12, -3, -1, -12, 9, -1, -5, -10, 1, -1, 0, 3, 4, 8, 12, -9, 3, 0], [-4, 22, -4, -17, 10, -13, -10, -11, 15, -13, -9, 9, -2, 2, 14, 1, 11, -7, 4, -3], [-8, 5, -1, 17, 7, -9, 6, -2, -2, 6, -7, 10, 10, -4, 6, -10, 11, 15, 1, 4], [5, 9, 2, -3, 3, 6, -8, 11, 3, -2, 3, -2, 1, -6, -10, 9, -12, -4, 6, 3], [6, -6, 5, -8, -2, 3, -2, 4, -5, 2, -3, -6, -9, 1, -11, -2, -8, -3, 8, 6], [6, -6, 18, 0, 25, -4, 5, -27, 3, -7, -10, 6, 7, 2, -1, -6, 9, 5, 10, 8], [37, 5, -23, -7, 15, -6, 3, -28, -8, -10, -11, 20, 9, 16, 20, -10, 20, -3, -2, -10], [-4, -11, 6, 2, -2, -1, 12, -12, -4, 0, 11, 2, 8, -4, 9, -12, 0, -2, 2, 0], [-23, -14, 17, 0, 11, 19, -6, -7, 10, 4, -1, -2, -5, -20, -23, -8, -10, -2, 14, -4], [-15, -4, 3, -13, 10, 2, 9, -20, 3, -7, -7, 2, -3, -2, 5, -6, 12, 1, -6, 6], [11, 27, -30, -21, -22, -29, -10, 20, -3, 0, -11, -3, -20, 18, 6, 10, -1, -29, -1, 9], [-1, 11, -11, -13, -9, -4, 2, 14, -12, 1, -8, -18, -15, 1, -15, -2, -7, -19, 6, 4], [0, 12, -13, 10, -20, -2, -2, 24, -1, 1, 18, 5, -9, 3, -4, 14, -10, -6, -6, -5], [-21, 21, -16, -7, -20, -3, -15, 15, -2, 7, 0, 1, -8, 7, -5, 2, -11, -19, 5, 7], [-17, -9, 17, 3, -2, 13, -4, 15, 0, 3, 5, -4, 2, -8, -10, 2, -3, 17, -9, -5], [-15, 4, 13, 17, 4, 7, 11, -6, -9, 7, 0, -13, 10, -2, -13, -1, 13, 13, 2, 6], [-10, 5, -4, 21, -17, 7, -4, 16, -15, 9, 2, 7, 9, -1, -7, 13, -4, 14, -9, 5], [-16, 29, -4, 12, -19, -4, -6, 12, -5, 9, -2, -6, 6, -5, -11, 6, 1, 4, -9, 15], [-1, 8, -5, -22, 11, -11, 10, -16, 4, -6, -12, 0, -5, 8, 7, -2, -4, -18, 7, -2], [0, -1, 5, 8, -7, 5, 7, -1, -7, -5, 2, -4, 6, -3, 2, 9, 0, 14, -11, 3], [10, 19, -19, -1, 6, -11, 4, -12, 1, -10, -4, 9, 3, 14, 14, 0, 0, -15, 6, -1], [0, 4, -8, -1, -2, 0, -9, 2, 7, 4, 3, 3, 2, 7, 5, 2, 2, -12, -3, -4], [22, 18, 6, -2, -3, -7, 4, 0, -1, -1, -4, -5, 8, 9, 9, 10, 3, -1, -6, 4], [-17, -13, 8, 5, 17, 16, -14, -11, 10, -1, -7, 0, 6, -18, -7, -6, -7, -3, 8, -3], [32, -9, -16, 7, 5, 1, 8, -5, -6, -5, 3, 10, 4, 10, 17, 2, 2, 3, 3, -13], [11, -2, -4, -2, -2, 4, 4, -6, 9, -7, 2, 14, 8, 7, 13, 5, -4, -11, -8, 4], [6, 19, -17, -9, -5, -8, 10, -3, -8, -9, -7, -3, -1, 4, 19, 2, 17, 11, -9, -1], [14, -4, 1, -8, 16, 2, 5, -24, 10, -8, -3, 18, 10, 2, 21, -2, 10, 5, -4, -9], [-22, 4, 10, 12, -6, 9, 8, 23, -4, 6, 7, -2, 2, 1, -12, 8, -8, 21, -10, 3], [-4, 8, 9, -15, 5, -9, -8, -7, -4, -1, -22, 1, -5, 2, -9, -4, 9, -2, 8, 14], [-4, 11, -10, -7, -18, -13, -12, 29, -5, 9, -6, -14, -19, 0, -20, 12, -5, -15, 7, 2], [-3, -7, 0, 6, 16, -3, -5, -16, 8, 0, -4, 2, -2, -6, -4, -1, -14, -5, 4, -5], [2, 11, 8, 8, -16, -5, 3, 20, -15, 10, -2, -17, -2, 4, -17, 13, 6, 0, 2, 14], [18, 15, -20, -15, 3, -23, 8, -21, -6, -10, -13, 8, -3, 13, 20, -7, 8, -15, 0, -2], [-5, -13, -7, -5, 2, 10, -5, -5, 0, 5, -1, -2, -3, -7, -12, -2, -6, -11, 12, -5], [-6, -29, 7, -1, 22, 8, -1, -33, 12, -3, -12, 11, 2, -17, 1, -13, 2, 12, 0, -3], [25, -1, -4, 11, 21, 4, 14, -22, 3, -10, 8, 17, 19, 11, 25, -7, 7, 11, -4, -13], [-22, 5, 0, -9, 14, -1, -4, -17, 16, -6, -3, 4, -5, -7, 3, -1, -1, -4, -1, -4], [9, 8, 9, -4, -23, 4, 8, 10, -2, 4, 13, -1, -1, 4, 3, 3, -9, 8, -18, 6], [-13, -7, 4, -24, -2, -10, -1, -17, 1, -8, -13, 5, -4, -3, 4, -10, 1, -4, -2, 14], [1, -22, 21, -7, 5, 20, -2, -1, 19, -3, 5, 2, 7, -8, -4, -1, -3, 11, 1, -5], [-13, -9, 4, -14, -14, 5, -2, 4, -2, -1, -5, -4, -14, -2, -3, -3, -4, -19, -3, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6026_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6026_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_6026_2_a_f(:prec:=20)
    chi := MakeCharacter_6026_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_6026_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_6026_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6026_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-18, 78, 1159, -576, -9137, -2042, 25315, 14682, -28084, -21886, 14837, 14776, -3723, -5359, 248, 1083, 78, -115, -17, 5, 1]>,<5,R![-27, -279, 1366, 4899, -12300, -30034, 30717, 69769, -33296, -76849, 17176, 44069, -3579, -13685, -193, 2285, 190, -189, -25, 6, 1]>],Snew);
    return Vf;
end function;