
// Make newform 33.6.e.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_33_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_33_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_33_6_e_a();".
// 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_33_6_e_a();".
// 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 := [25735126080400, -161017247606600, 398041877081860, -118816044595930, 63231264058241, -6173773919508, 2527225278875, -53496436514, 107078602608, -7877307064, 4158720649, 14028509, 129076450, 1969563, 1399353, -6599, 14927, -389, 142, -3, 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], [-333710633705359606546591908472570406149490666995526302565789676612492291617551399904106492971189166459501362600, 728161420633720298120988380448229541613514624682676964719824787066599642904556422088023020109086148200816394740, 130265317832184286444495889702232474400765281888764518678092873615768641432787482093641932838160597276294700750, 24022927568707175917810406923488295015046919398575954487999582462567147821145123835176313939649100591035831949, 3957946096069169515460884142646121870578043294013261345931836221413948378539635534188024837265845577617107336, 1502390940010275147208050559902074776535321081105975515706684136550652543385136296722790178532724672294618535, 243249252244404667343539206981055794232828807934385123979151089223698512627187862539652418622322668343398778, 40235497843061540965384995039165484144225136232003967966582872475167872180237530185406549670231691906039084, 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    Rf_basisdens := [1, 787939043862135316828470647675136878795595953256394127669757643741400467612878698954941248194723672886177786580, 8667329482483488485113177124426505666751555485820335404367334081155405143741665688504353730141960401747955652380, 2166832370620872121278294281106626416687888871455083851091833520288851285935416422126088432535490100436988913095, 17334658964966976970226354248853011333503110971640670808734668162310810287483331377008707460283920803495911304760, 4333664741241744242556588562213252833375777742910167702183667040577702571870832844252176865070980200873977826190, 8667329482483488485113177124426505666751555485820335404367334081155405143741665688504353730141960401747955652380, 399719901073173521956449502624301588339248235895062228178612713154724974419078138222443785326686929807812218776460840, 36338172824833956541495409329481962576295294172278384379873883014065906765370739838403980484244266346164747161496440, 199859950536586760978224751312150794169624117947531114089306356577362487209539069111221892663343464903906109388230420, 36338172824833956541495409329481962576295294172278384379873883014065906765370739838403980484244266346164747161496440, 199859950536586760978224751312150794169624117947531114089306356577362487209539069111221892663343464903906109388230420, 199859950536586760978224751312150794169624117947531114089306356577362487209539069111221892663343464903906109388230420, 399719901073173521956449502624301588339248235895062228178612713154724974419078138222443785326686929807812218776460840, 99929975268293380489112375656075397084812058973765557044653178288681243604769534555610946331671732451953054694115210, 399719901073173521956449502624301588339248235895062228178612713154724974419078138222443785326686929807812218776460840, 399719901073173521956449502624301588339248235895062228178612713154724974419078138222443785326686929807812218776460840, 199859950536586760978224751312150794169624117947531114089306356577362487209539069111221892663343464903906109388230420, 199859950536586760978224751312150794169624117947531114089306356577362487209539069111221892663343464903906109388230420, 399719901073173521956449502624301588339248235895062228178612713154724974419078138222443785326686929807812218776460840];
    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_33_e();"
function MakeCharacter_33_e()
    N := 33;
    order := 5;
    char_gens := [23, 13];
    v := [5, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_33_e_Hecke();"
function MakeCharacter_33_e_Hecke(Kf)
    N := 33;
    order := 5;
    char_gens := [23, 13];
    char_values := [[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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 6;
    raw_aps := [[0, -1, 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, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 1, 0, -2, 0, 1, 2, -1, 1, 3, 1, -1, 1, -1, 0, 0, 0, 0, 0, 0], [-1, -3, 0, -2, 0, -2, -1, 8, -50, -7, 2, 1, 0, -1, -1, -2, 0, 0, 0, 2], [-2, 0, 6, 5, 1, 22, 7, 64, -71, -50, 1, 8, 1, 5, 7, 0, -4, -7, -3, 2], [21, 32, -14, 10, -7, -5, -9, 236, -25, -3, -1, -3, 0, -7, -6, -3, -2, 3, 2, -7], [-379, -21, -2, -37, 0, 29, -15, -255, 255, 379, 6, 0, -3, -27, -12, -19, -8, 10, -16, -10], [151, -29, -28, 22, -22, 53, 0, 144, -19, -631, -5, -29, -19, -9, -32, 16, -4, 22, -5, -19], [554, -26, 61, 206, 32, -142, -143, -32, -843, -863, -21, -20, 32, 71, 76, -6, -1, -32, 26, 0], [-16, 179, -17, -32, 55, 195, -27, -872, -665, 888, 32, 16, 55, -16, -33, -59, -50, 0, 35, 17], [694, -142, -61, 193, -37, 28, -209, 284, -785, 54, 16, 54, 0, -45, -53, -36, -65, 4, 26, -37], [-19, -702, -41, -38, 49, -683, 24, 446, 101, -427, 38, 19, 49, -19, -60, 13, -32, 0, -52, -46], [-109, 471, -176, -447, -42, 981, 0, -85, 11, -1631, 4, -18, 11, -187, -98, -91, 77, 42, 4, 11], [1430, -143, 41, 770, -87, 436, 569, -122, 3316, 3152, -31, -164, 122, 36, 148, 21, 133, -122, 143, 0], [-2974, -958, 122, 799, 36, -514, 0, -3133, 156, 512, 149, -123, 156, -34, -170, -203, -39, -36, 149, 156], [-1921, -1448, -17, -545, -202, -428, 699, -843, 2160, -441, -154, -441, 0, 149, -48, 459, 226, 120, -13, -202], [-59, -2463, -48, -118, -14, -2404, -2560, -10108, 596, 10167, 118, 59, -14, -59, -107, -128, -93, 0, -152, -127], [-2762, 50, -109, 1507, 0, -1335, -231, -9410, 9410, 2762, -281, -125, -183, 717, 637, 545, 297, -188, 594, 188], [-5109, 85, 248, 1304, 412, -2248, -2625, 131, 13652, 13691, -338, 39, -131, 28, -395, 46, -377, 131, -85, 0], [25748, 1910, 54, -513, 0, 290, 2187, 21869, -21869, -25748, 277, -211, 114, -564, -142, -341, -12, -42, -24, 42], [-264, 1797, 751, -528, 643, 2061, -1139, -3130, -19039, 3394, 528, 264, 643, -264, 487, -225, -296, 0, -241, -9], 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-30188], [622855, -43985, 114613, -227628, 63558, -401206, -432302, -68749, 2096904, 2077683, -50317, -19221, 68749, -23093, -29425, -24764, -31096, -68749, 43985, 0], [1568493, 24407, -22631, 8308, 19030, -72774, -196474, -8253, 529767, 586834, -66633, 57067, 8253, 170815, 79775, -32660, -123700, -8253, -24407, 0], [4901878, -337737, -285554, -535841, -57618, -153657, 516709, 4850736, -4853848, -105648, 19132, -105648, 0, -234706, -76750, -55147, 96039, 191921, 48009, -57618], [-1943149, -1071466, 66811, 600413, 0, -496441, -1108627, -982898, 982898, 1943149, -37161, 152923, 12697, 9811, -72467, -94161, -48951, -17860, -97902, 17860], [-440224, 25420, 21789, -166515, -71805, 101738, 33228, -3064, -4299593, -4245689, -14606, 53904, 3064, 62241, 22215, -28484, -68510, -3064, -25420, 0], [3076394, 6392, 144260, -266868, 157832, 423946, 141142, 2810138, -3291606, 373044, 125726, 373044, 0, 8030, 32106, -311792, -266114, -172494, -50902, 157832], [1092538, 237537, 81508, -193253, 172965, -731979, 0, 1136822, 203522, 3041695, 74610, 217249, 203522, -122014, 137993, -282778, 74841, -172965, 74610, 203522], [4524592, -378744, -51739, 746158, 0, -950256, -226385, 331824, -331824, -4524592, 152359, -191882, 236565, -67280, 73726, 136818, -12216, 63955, -24432, -63955], [37805, 510923, -119528, 75610, 14722, 473118, 307480, 3710921, 2900093, -3748726, -75610, -37805, 14722, 37805, -81723, 70970, 80699, 0, -60727, -55638], [-6677635, -114606, 39379, -51206, -104551, -721800, -512658, -50299, 948984, 770071, 30229, -178913, 50299, -206714, -61879, 64307, 209142, -50299, 114606, 0], [-82532, 1555742, 62394, -165064, 63986, 1638274, 696384, 2925504, -5625150, -2842972, 165064, 82532, 63986, -82532, -20138, -256854, -19714, 0, -48536, 96814], [-6332012, -18372, 61576, -403340, -119208, -138975, 547507, -5394075, 6299432, -86628, -144167, -86628, 0, 167105, 24959, 175592, 19767, -149359, 52347, -119208], [2557837, -168069, -12773, 316136, 0, -353600, -143378, 3919167, -3919167, -2557837, 24691, -114053, 27570, 133708, 148973, 171172, 76589, -63816, 153178, 63816], [5060459, -270815, 369950, 327182, -6062, -549129, 0, 5116826, -5661, -6415067, -37361, 50305, -5661, 375611, 243964, 199320, 56768, 6062, -37361, -5661], [-2386405, -1009894, -205988, -269702, -100919, -324385, 338561, 555252, 2659391, -373905, -68859, -373905, 0, -192831, -32060, 150457, 223466, 186667, -49520, -100919], [5462467, 962520, 90083, 306251, -80597, -164968, -294201, -2874874, -5361984, -181080, -12050, -181080, 0, 49921, -68547, 323598, 84371, 140868, -16112, -80597], [-882150, 191463, 168327, -580543, 163757, 945966, 806277, 28619, 897411, 1251718, 214618, 354307, -28619, -93347, -70192, -162844, -139689, 28619, -191463, 0], [563519, 297292, 139679, -267210, 150148, 267400, 0, 593601, 106154, 7207629, 343, 180230, 106154, 33525, 262244, -24140, -13912, -150148, 343, 106154], [5178658, -211006, 180732, 536618, 0, -411016, -155876, -2319414, 2319414, -5178658, 55130, 142764, 9482, 19036, 49870, -106566, -17162, -163570, -34324, 163570], [-477135, 815928, -112867, -244867, 0, 238429, 709499, -8384294, 8384294, 477135, -106429, 107089, -347711, 223253, 25181, 229691, -113527, 226394, -227054, -226394], [-6220711, 1181467, -151297, -773954, 0, 664985, 1139139, 403636, -403636, 6220711, -42328, -190613, -30612, -138501, 14512, -29532, 81644, 69653, 163288, -69653], [-7675178, 551115, 515419, -675006, 112761, 229445, 0, -7799069, 220626, 10981832, 12216, -11130, 220626, 294793, 111793, -97703, -16026, -112761, 12216, 220626], [-27799, -586096, 173809, -55598, 1885, -558297, -83669, 1674275, -538912, -1646476, 55598, 27799, 1885, -27799, 146010, 65915, 37601, 0, -288373, -195174], [10305071, -516337, -157760, 1043641, -54005, -280489, -972607, 2806323, -10133305, -225771, -71034, -225771, 0, 230004, 17029, 105700, 226484, 138421, 54718, -54005], [10751456, -466279, 102482, -577183, 89486, 275709, 473299, -1411841, -10897222, 235252, 103884, 235252, 0, -42642, -14398, -158829, -186223, -67941, -40457, 89486], [48328, -1345139, 48515, 96656, 129867, -1393467, -1002293, -7642775, 3202084, 7594447, -96656, -48328, 129867, 48328, 96843, -63113, 63138, 0, -51717, -85235], [-2297952, -648446, -227845, 586481, -10056, 489604, 0, -2359917, -52688, -3032235, -5536, -72021, -52688, -175157, -204820, -80111, -104597, 10056, -5536, -52688], [3813728, 1297274, -145654, 769281, 0, -822951, 1205290, 4708292, -4708292, -3813728, -91984, -213179, 114088, 17021, 370171, 70691, 159509, -13855, 319018, 13855], [8930444, -132684, 332149, 1125885, -3542, -739104, -693291, -189889, -7136491, -7211970, -29666, -75479, 189889, 285636, 388654, -57205, 45813, -189889, 132684, 0], [4111844, 53245, -37063, 922766, 82527, -601976, -604225, 9460, 1791298, 1888328, 94781, 97030, -9460, 324918, 366454, -43785, -2249, 9460, -53245, 0], [-7881048, -769334, 15266, -466338, 0, 407000, -694730, -8993704, 8993704, 7881048, 74604, 75744, 107784, -115812, -229518, -56474, -135082, 119816, -270164, -119816], [-809850, 732567, 416072, -673756, 289219, 774219, 0, -751039, 287420, -2985667, 100119, 348030, 287420, 128652, 613061, -22389, 57012, -289219, 100119, 287420], [1210791, 71806, 446133, 1509377, -150409, 150662, 178134, 73750, -6512180, -6442318, 97334, 69862, -73750, 126013, 151541, 1944, 27472, 73750, -71806, 0], [9131696, 138582, -591168, -153720, -200632, 872966, 0, 9116558, -54042, -10410842, 6530, -215770, -54042, -537126, -270384, -572, 131452, 200632, 6530, -54042], [7313, 1921827, 79544, 14626, 201798, 1914514, 321286, -11927477, 9481319, 11920164, -14626, -7313, 201798, 7313, 86857, -75994, -109919, 0, 54987, -69558], [419343, 174507, 183294, 105254, 0, 310624, -58077, 15957, -15957, -419343, -232584, 361449, 35385, 281084, 210971, -134794, 54429, -237723, 108858, 237723], [1943938, 294560, -399497, -924537, -279700, -353152, 949711, 5418505, -1868945, -354693, -25174, -354693, 0, -165303, -254526, 208181, 73452, 302804, -1541, -279700], [-3277977, -158889, -189277, -23428, 0, -50691, -274047, 7201961, -7201961, 3277977, -115158, -26057, -70714, -204253, -197132, -130134, -48062, 237339, -96124, -237339], [621210, -2135288, 45947, 459085, 0, -361362, -2187064, -6187289, 6187289, -621210, -51776, -87891, 8757, -162064, 91874, -259787, 185614, -231561, 371228, 231561], [2095587, 888033, 26982, 770517, -769, 38986, -590590, 3084283, -2196655, 100299, -179927, 100299, 0, 270511, 179158, -191162, -39755, -398840, 61313, -769], [2639689, -1277987, -95400, 435915, 0, -490223, -1319079, 757497, -757497, -2639689, -41092, 27430, -187919, 110417, -139270, 164725, -81738, 177138, -163476, -177138], [-2858891, 35023, -117612, -1755650, 253693, 2058266, 1790232, -130236, -6107228, -5906946, -67752, 200282, 130236, 46319, -56456, -165259, -268034, -130236, -35023, 0], [49738, 1232340, -386802, 99476, -359894, 1182602, 2517946, -3593936, 4532158, 3544198, -99476, -49738, -359894, 49738, -337064, 12150, 111866, 0, -66596, -54206], [-5134508, -183576, -234169, 1472100, 45188, -1154760, -929679, -24436, 9020346, 8677630, -117635, -342716, 24436, -219803, -153862, 159140, 225081, -24436, 183576, 0], [8824704, 612258, 322848, -493644, 145130, -404622, 0, 8943318, -11600, -15998978, 7614, 263744, -11600, 334448, 273270, 21126, -38116, -145130, 7614, -11600], [-12402055, 2263968, -41619, -1513579, 181661, 42310, 1387219, -7592808, 12603748, -20032, 126360, -20032, 0, -234403, 55301, -139230, 139351, 210410, -62342, 181661], [-124844, -582424, 401792, -249688, 170180, -457580, 1889218, -1780330, 4563748, 1905174, 249688, 124844, 170180, -124844, 276948, -632388, 12234, 0, 221706, 483628], [-14178972, -236892, -40261, 643798, -76740, 2222952, 2467737, -273558, 2836800, 2636574, 44559, -200226, 273558, -4593, 276858, -36666, 244785, -273558, 236892, 0], [-195491, -1035305, -96490, -390982, 213588, -839814, 1667087, -5678927, -1605307, 5874418, 390982, 195491, 213588, -195491, -291981, -140752, -305178, 0, -34827, 50977]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_33_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_33_6_e_a();".
// 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_33_6_e_a(:prec:=20)
    chi := MakeCharacter_33_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_33_6_e_a();".
// 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_33_6_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_33_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![327810433847296, 620772490756096, 426573912739840, -103914240984576, 71841567812096, -9190650459648, 3685236198080, -115987806144, 157022432704, 7530196616, 5575959496, 352751112, 141550233, 3779199, 1440147, 24816, 15553, 592, 144, 2, 1]>],Snew);
    return Vf;
end function;