
// Make newform 912.6.a.y in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_912_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_912_6_a_y();".
// 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_912_6_a_y();".
// 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 := [-24661341, -16081740, 1610577, 92430, -4725, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [-121791601863, -23681399193, 820304928, 17990538, -288577, -4393], [62101268487, -30899430423, -1434178872, 103907598, -142247, -23423], [58788666567, -17649022743, -1434178872, 103907598, -142247, -23423], [-53106861087, 11916019503, 639322632, -28829198, -64113, 5543], [97030518849, -9893470161, -539262904, 28357426, 3471, -6041]];
    Rf_basisdens := [1, 828150480, 3312601920, 3312601920, 552100320, 184033440];
    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_912_a();"
function MakeCharacter_912_a()
    N := 912;
    order := 1;
    char_gens := [799, 229, 305, 97];
    v := [1, 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_912_a_Hecke(Kf)
    return MakeCharacter_912_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 := 6;
    raw_aps := [[0, 0, 0, 0, 0, 0], [9, 0, 0, 0, 0, 0], [-11, 0, -1, 0, 0, 0], [25, 0, 0, 0, 1, 0], [34, 0, 0, 0, 0, 1], [-49, 1, 2, 1, 3, 1], [223, -4, 3, -3, 6, 3], [-361, 0, 0, 0, 0, 0], [-210, -5, -31, 4, 3, 1], [1221, 11, 3, 27, 0, 5], [-282, 21, -31, 2, -9, 3], [2362, 12, -25, -30, -3, -22], [2460, 10, 20, -54, -42, 4], [781, -2, 19, 35, 18, -11], [1790, 21, -106, 83, 32, -37], [7896, -86, -108, 104, 2, 0], [10251, -4, -147, 11, 0, -84], [-13608, 40, -305, -189, 57, 32], [7671, -75, 115, -135, -100, 45], [1683, -9, -417, -81, -116, 125], [-9166, 126, -162, 182, 92, -3], [24253, -20, 118, 121, -99, 6], [26756, 15, -40, -230, -266, 79], [-16346, 37, 46, 248, -288, 179], [-109, -83, 323, 325, -74, 99], [-17010, -51, 873, -82, -127, -149], [38258, 117, 115, -546, 41, 83], [72312, 65, -918, 120, -332, 213], [-5559, 83, 1968, -135, 75, -201], [-3875, 227, 1099, -367, -716, -269], [133869, -114, 650, -85, 721, 380], [98352, 0, 482, 240, -140, 257], [-12605, -170, -375, 1451, -112, 163], [196663, -23, 2579, -11, 410, -30], [-57615, 946, -53, 74, 592, -246], [192659, 208, 1548, -641, 665, 450], [-46610, -546, 4120, 342, 1252, 340], [99678, -3, 2248, -930, 1404, -129], [-31383, 999, -55, 647, -960, -43], [95068, 70, -2698, -1776, -252, -216], [134813, -1149, 2901, 801, 1332, -1097], [-116394, -1652, 423, -908, 2469, 818], [-137154, -1159, -3270, -2155, 824, 987], [44925, 1397, -1223, -237, -110, -31], [-70402, 163, -5385, -190, 2399, -687], [50995, -590, -4242, -218, -1483, -1102], [78257, 2239, -553, 2555, -1358, -1493], [168540, 1985, -2239, 2602, -1043, 831], [131689, -173, 2961, 1077, 1884, 2115], [117222, -1768, 5231, -1345, 2399, 2068], [249491, -42, 7121, -1931, -180, 831], [50604, 81, -8320, 705, 1068, 697], [-148368, -200, -916, 226, -4158, -322], [275728, -3086, 1660, -2714, 3238, 453], [355748, 1515, -1212, 5162, -1468, -3053], [16298, -401, -7522, 319, -952, -2423], [249478, 888, -3404, 1394, 3816, 1098], [685878, 111, -7664, -3568, -68, -2465], [-498390, 4102, 821, -6075, -5119, 988], [270484, 208, -7476, -9578, -1178, 832], [407823, -171, -815, 4339, -1452, -484], [933124, -5398, -2084, 3598, -270, 404], [710683, -2181, 25947, 3633, -3564, -2737], [75072, 7169, 5126, 9595, -5154, -4407], [367856, 3927, -2144, -2736, 4228, -3721], [-96057, 7683, -3887, 6251, -7052, -1455], [294549, 5269, -1591, -209, 7624, -455], [202863, -1069, -12675, -3687, 5300, -155], [-60890, -1526, -6270, 1298, 5650, 5367], [-792614, -1214, -10215, -9941, 2509, -1618], [1902362, -8440, 13642, 7834, 350, -10], [-673826, -3255, 20774, -7263, -7924, 2035], [918038, 6447, -304, -4724, -5228, 6839], [643688, 978, 24713, 940, -19815, -1808], [1301360, -5092, 5902, 1852, -8694, -5974], [-1109220, -6378, 26218, -5578, 5522, 7870], [-26893, -3806, -2151, 16078, 4188, 2158], [-11390, -1002, -26753, 1433, -13145, 4288], [2027335, 9171, 7797, -6797, -15716, -2087], [687616, 5880, 14228, 2810, 9374, 8808], [-362620, 8699, 32776, 260, -7618, -2447], [-758940, -8114, -19721, 10270, 8563, -2944], [-772347, -1841, 50429, -9645, 568, 9201], [418357, 881, -85807, -5809, -3698, -1585], [168973, -11936, -23662, 13319, 141, -3838], [-467064, 5972, -32352, 19208, -8642, -6651], [1597886, -1717, 35994, 820, 2004, 1583], [752534, -6993, -13062, 5270, 7198, 3618], [-1058571, 5384, 16569, 25956, -17798, -4238], [-241985, -9702, -30850, -19654, 19421, 4132], [-380894, 10220, -21196, 14614, -7488, -4891], [-386478, -9433, 62247, -1846, 11401, -10917], [1396951, 9558, 8144, -20721, -22435, -4560], [-464068, -551, 2844, -3570, 13198, 16029], [4297275, 942, -67959, 15269, 21330, 13715], [-1694146, -5931, 46729, 8162, -44321, 5197], [-824355, 151, -71839, 20103, 5586, -2639], [-4895901, -256, -18923, -1065, -8522, 10100], [2445478, 10150, 28176, -5514, -22154, -3600], [-4766210, -4988, 21569, 4719, 20561, 3482], [4206295, -6381, 44031, -47, -11960, -5911], [-4403037, 1460, -2679, 380, 6100, 10020], [-1329086, -25593, -46536, -31210, 6884, -13267], [-4634381, 3260, -19315, 12879, -53550, -6010], [685354, -6364, -10760, -10740, 40226, 24894], [-1071310, -13101, -84714, -18130, 19014, 1394], [2749380, -16318, 51398, -12918, -2032, 33], [-2115232, -19434, -61774, 30886, 2508, -10182], [-2236089, 20293, 49441, 21031, -9470, -13005], [-2347967, -2393, 100891, -28353, -33440, 12863], [-2098311, -312, 2516, 11595, -32601, 1942], [-5121138, 658, 104837, -28169, -779, 20918], [-5774915, -704, 56855, -18701, -31556, -7321], [305096, 20456, -109500, 20460, -18648, 6464], [-1098287, -1292, 166590, -300, -20557, 3088], [-4292657, -954, 49617, -6637, -2666, -24528], [2901923, 7991, -16285, -20047, 48698, -6944], [-7194480, -12049, -1564, 7455, -41416, 5621], [-2380989, -19158, -53573, -20182, 58286, 27040], [472484, 2620, -12076, 19280, 7440, 28774], [-138394, -6960, -10729, -6918, 15905, 18734], [230031, 1883, -62825, 72721, 25350, -26957], [-1324047, 19093, -43765, -14301, -6392, 38787], [196665, 1291, 102129, 61045, -31988, -12445], [343317, -36690, -64083, 22253, 39100, -16595], [-4426184, -3647, 72829, 14982, 40187, -2059], [-5079668, -20780, -43944, 5504, -26526, -20810], [-1338138, -32627, 18890, 18431, 96856, 3933], [-537385, -11364, 153036, -65470, 30787, 5206], [7451578, -7120, 56918, 57224, -27682, -8444], [5412531, 7435, 166205, -38689, 65592, 19438], [-10600703, 369, 76199, -32819, -11414, 26739], [-5348996, 3759, 154171, 16090, 5239, -44831], [10773578, 37304, -14191, 39379, -72739, -8158], [3749725, 34196, -2275, 83103, -18290, -30353], [-6680364, -9696, -94256, -22240, -35536, 9093], [-107259, -14669, -73629, 2581, 60564, -25651], [2276698, -8854, 5814, 49118, -72504, 14546], [3642811, 15333, 26109, 37823, -110112, -6337], [-10473461, 12258, -22323, -63569, 13432, -21351], [951379, -14405, 246431, -59419, -104004, 1693], [-2177185, 19892, -190343, 43830, 2784, -14982], [1495785, 14788, -144228, 10054, 33621, 30750], [374845, -53231, 34277, 13017, -6772, 7179], [10077804, 39082, 40935, 33920, -76201, -52632], [10624810, -22610, 52126, 6972, 34648, -28814], [1896152, -5114, -162816, 4156, -110958, 34688], [-2260947, -34058, -131641, 37737, -85362, 23842], [-7513591, 56528, -53203, -37207, -67416, -535], [-19767198, -6014, -244228, 5354, 66174, -2698], [3014080, 53946, 241883, -36486, 95051, -4052], [-6234925, 99922, -235745, 91243, -7638, -36193], [5040723, -53066, -34169, 3711, 24696, 38999], [-125553, -3887, 234825, 153395, -19934, -25561], [-6404214, 20476, -104200, 60394, 33838, -6094], [-14035954, 19308, -206058, -69268, 39928, 12204], [617036, 3551, -150472, -24450, 24438, -11065], [-11321486, 26140, 108482, 72712, 115542, -24684], [1864248, -69870, -240980, 7692, 2184, 45945], [12432150, -78950, 267552, 5490, -105336, -31808], [-2656174, 76201, 103678, -126480, 76126, -15027], [-10265199, -31819, 527423, -80027, -28860, 27771], [1331688, -27753, -281082, 24658, 62484, -45325], [-3078000, 3966, 439484, 84700, 2788, 9674], [6329120, 14820, -197552, 101294, -1194, 43788], [-8724036, -101034, -232040, 85464, 6448, -38174], [-20620660, 31145, 3881, -116190, -13387, 12427], [14913924, -5942, 14987, 51149, -142559, 6038], [13497379, -48817, -127965, 33633, 31102, -12665], [-9902739, 95924, -405227, -8396, -215338, -94314], [11976038, -23126, 525618, 91666, -105840, 1062], [12182802, -66096, 32519, 21620, -31415, -24690], [-7808370, 23851, 429995, -29488, -116727, -21707], [15896324, 76303, 189964, -9708, -53996, 25783], [-3979967, 48702, -151920, -46261, -45425, -63924], [-15965871, -38454, 128173, -19907, -4342, 72209], [21767134, -66967, -302454, 107368, 6850, 9697], [6731797, -96524, -63259, 20188, 45726, 46030], [-4067320, 46911, 191949, 250914, 21993, 37053], [-17071300, 16380, -111509, 32483, -136519, 91060], [-19324509, -38036, -150558, 162934, -129575, -38498], [16535631, -45802, -20915, 195043, 39164, 12684], [22560378, -51476, 155543, -227812, 12337, 32230], [-1478353, 23035, 180153, 71675, -137800, -10371], [-35965226, 29465, 453142, -126393, -12500, -64963], [-14189417, 9276, -420383, -3332, 51800, -5884], [14182742, 14784, -316561, -76312, -151507, -71910], [506565, 116737, 191231, -164841, -29230, -38511], [18224625, 22517, -182867, -63649, 188590, 27591], [15778658, -141139, -8939, 260800, -3065, -32197], [6715801, 59135, 469479, 147511, -136832, -169499], [4682974, -78264, -376074, -17904, -95770, 59713], [19746850, 9986, 78612, -215574, -167650, -43468], [-10376014, -126992, 312948, -243436, 14600, 18286], [11909434, 98817, 482928, -85456, -188340, 33529], [4767907, -54339, 942801, -26105, 61756, 48933], [-13085820, -96816, 307954, -14574, 262066, 79013], [-9100188, 153112, -270290, 207892, -111660, -138090], [1520040, -77676, -267238, -169366, 124336, -26872], [21002270, -11533, 267990, -102087, 250764, 82071], [-9979504, 24476, 509554, 128358, 121628, 109510], [-16984664, 14163, -35777, 24506, -120089, -122031], [3679337, 122015, 81718, -269615, -54283, 4883], [-22138369, -15569, 83513, -91795, -277272, -193183], [32865698, 95055, 58732, 164984, -125036, -149765], [-32221439, -36972, 603973, -225346, -272622, 46696], [29089385, 132756, 173048, -142930, -222531, 3340], [31670522, 111138, -214002, 78534, -64656, -150638], [2736893, 115318, 243395, 212963, 30564, 18321], [19129038, -95550, 112010, -376878, -151052, 55512], [-8642648, -50672, -367188, 17668, -249650, -23614], [9571586, 50849, -725843, -188410, -495, -71401], [13348733, 159746, -260488, -83692, 108761, -16578], [-10959514, 60207, -1315632, -29282, -16648, -41167], [5806009, 141821, -298411, -21877, 323448, -9689], [41301464, -84354, 56592, 129522, 197346, -96396], [-18052998, -28149, 755312, -299126, -209440, 53281], [-28822340, 31250, -1003020, -197550, 236878, -51492], [9582728, -153526, 335740, 66262, 155684, -252930], [-26058147, 118096, 85075, -49500, -59292, -34568], [-4640125, 30977, -412702, 166091, -18453, 14965], [48573336, -100983, 150853, -84534, 285565, 21103], [1590989, -71128, -18285, 240149, -66326, 56778], [7768875, -271720, -3392, -299374, 11907, 21638], [2834607, -96686, 164817, -85501, -346916, -49118], [-61826860, 61482, 887511, 137431, 166961, -111174], [19160187, 88344, -484795, -170613, -285930, -244632], [-13453475, 232907, 75295, 90277, -126302, 162249], [11243691, 203368, 86708, 562607, -219967, -44926], [48337308, -26480, -135818, -77774, -349008, -76187], [23251636, -25320, -255667, -287309, 123447, -124632], [-22161852, 138794, -767992, 149668, 12824, -73352], [-17939124, -93915, 504893, 193980, -548291, 3743], [96626563, -126678, -176199, -199773, -87634, 53486], [-39438723, -74319, -173085, -300899, -65864, 115718], [-54077080, -76861, -275444, 186283, -334960, 76921], [17143796, -19538, 336504, 266832, -109856, 8525], [12379897, -227358, 71301, -522778, -19390, 8200], [33346806, 120167, -458522, -135500, 203374, -243817], [31114375, 27215, 714411, 175987, -164580, -261515], [-5706852, 116587, -514118, -114240, -627380, 208739], [4655275, -112022, -723553, -190069, 479540, 146165], [-30584626, 58837, 156610, -420236, 133894, -127383], [-33137519, -237129, 227372, -321735, 156223, 152479], [29913225, 6980, 73483, 102569, 175790, -165966], [-16926386, -95291, 1672230, -295801, 139960, -35021], [-70921835, 268104, -136720, -341958, -118175, 167228], [-20730715, 125192, -50109, 306249, 575368, -92600], [2815308, -271682, 314444, 56340, 582480, 83754], [646776, -154755, -194576, 581227, 124404, -68143], [-20918650, 113678, 636932, 252134, -95352, -21016], [32812997, -327858, 428047, 89315, -338252, -135185], [-71326166, 11171, 353341, -279378, -58893, -231157], [-11910557, -27959, 851865, 207753, -127924, 96285], [37550213, -61798, 746953, 405340, 43522, 89822], [7981580, 6587, -110568, 537420, 191110, 173161], [-22642860, 364940, 670939, 169155, 37741, -150742], [18713089, 403191, -785655, 509195, -17004, -64743], [-28836544, 32798, -451654, -279482, -119872, 66120], [-20772780, 99121, 977692, 108648, -73624, -305339], [-26918822, -366043, 76731, -439104, 623613, -52425], [6447791, 174154, -1084815, -175245, 99740, -139598], [-47134754, 87618, 210697, 110035, -60963, -114644], [-111943518, -20108, 819075, 531048, 339093, -32810], [5544263, -295804, 878977, 205927, 879292, -8925], [-24520778, -269614, 1053802, -103242, -150356, -103884], [118613226, 97382, -10334, -259338, -123294, -47300], [-3196341, 249898, -804185, -212765, -318986, 33233], [-66309881, 61151, -836915, -690723, 294998, -56741], [-24707271, 520922, -338063, 393294, -130738, -230268], [-60912052, -439602, -509643, -266882, 185509, 213724], [-57490116, 29002, 408220, 97380, 107864, 41110], [-2356336, 197382, 805754, -453870, -464598, -57877], [29947057, -70816, -617662, 416230, 188367, 275470], [-33756727, 129533, -1365217, 75163, 245800, -25971], [-10591128, -81383, -553608, 239738, -509450, -126499], [-76945552, -99836, 897798, 669512, 31756, 68483], [-103690692, -340762, -894163, -4698, 102637, -218860], [27831755, -57935, -1575183, 628461, 420044, -115315], [69673814, 295606, 163608, -450872, 107088, 117011], [-93582001, 179663, -510117, -445757, -413220, -147075], [46052972, -93499, -491130, 338254, -2076, -119859], [28599194, -3753, -300418, -99045, -89472, -339255], [53535197, -85581, 168536, -1032183, -138625, 188939], [-91274277, -388275, 414437, -374005, 42212, 51152], [-93404752, 104685, 461718, 105941, 284786, 183085], [-58541060, -160745, -259594, 639050, 388732, -198856], [131997031, 278189, 1156411, -218231, -692334, -19645], [109299355, 158792, -472422, -207242, -327099, -92856], [-21768669, -532075, 1251513, -871469, 579556, 24873], [37666738, 280447, -401677, -58406, -750401, 185637], [36526012, 147518, 1074586, 188656, -1212464, 90597], [56382401, 206879, -608197, 107949, 408936, 37135], [76837811, 555313, 968751, -344301, -14812, -226601], [-118655843, 99207, 1047380, -203451, 376731, 71423], [43952210, -184403, 2721465, 681854, -395563, -59333], [-8547572, -127619, 546023, -176506, 377287, 27331], [16902123, -663070, 435885, -171228, 399512, 364000], [103908606, -240525, 1636184, 23866, -252556, -44901], [9225539, -402027, -1450875, 550199, 847816, 193358], [-138367388, 335213, -1481830, -725514, 160776, 232656], [28274194, -378726, 419554, 226770, -617278, 79452], [21111559, 19112, -967308, 443062, -249937, -445276], [-40943002, -79739, -674084, 906174, -223468, -523395], [-42380663, -537856, -1949293, 84559, -417650, 181041], [-41080840, 136080, 536020, -786766, -56058, 4700], [-69100226, -212198, 1339010, 1147450, 427208, 240090], [-62141963, -21153, -1918402, -517113, -9163, -243081], [26345548, 533041, -274890, 163519, -348578, 419979], [-40789252, 154702, 377504, -553552, -720722, -74096], [114474548, 475481, 134816, 171243, -797300, -435089], [87485408, -421237, 1191751, -467946, 591445, 105787], [-76588720, 16215, 1667604, -930730, -908568, 11927], [51575050, -107282, 183686, 187658, -770600, 205010], [90693796, 91223, -144811, -565378, -747735, 67961], [118604214, -82548, 999968, 341600, 107700, -338312], [-26932562, -113898, 196400, -500338, -630652, -226173], [91012058, 27648, 681048, -403788, -291446, -270640], [-187065180, -92969, 467858, 274978, 312440, 165912], [-21602983, 382410, -2072059, 475189, 555718, -207213], [67088826, -166194, 2887944, 36186, -155386, -62018], [92071370, -384539, -670542, -1036602, 327546, 272310], [119184688, -31392, -2727072, 118260, 418554, 215110], [35193170, -357121, 475165, 398594, -166749, 153805], [-78997774, -66208, 2119344, -454226, -41520, -366754], [-90789685, -629231, -1075105, 17389, -201706, 342233], [-55405089, -152001, -302401, -221671, -268672, -407087], [-34315990, 665704, -546090, 1050158, -77028, -47960], [88696770, 48774, -95906, -1051242, 277748, -170162], [96594845, -52300, -1023515, -244700, -540130, 369474], [-11261704, 956756, -1924583, 1573731, -136857, -489692], [97835204, 217188, -2001208, 753732, 369502, -13534], [121664698, -125317, -597044, 467650, 1650172, 613857], [83585604, 212640, 1523806, -1665154, 316336, 375501], [-98528895, -364357, -3193165, -316187, 524724, -93654], [150257692, -491828, 2891312, 296234, 641202, -179053], [-60109970, 535516, -26659, -1488240, -752593, -287718], [-150011061, -919743, -9483, -143829, 925208, 461489], [56156552, 115754, -3152654, -762590, -981194, -625451], [143002290, -17855, 2090270, -735624, 1115198, 447177], [46463793, -174613, -2141154, -1143339, 64677, -172797], [-32413415, -177854, -3275547, 813691, 226062, -30791], [94029983, -678655, 182545, 741073, 441100, -309901], [171973871, 230714, -3177750, 4091, 236571, 260708], [71160459, -931883, -1000909, 307963, -896310, 102831], [151436751, 615897, -63741, 438043, -1052496, -259395], [-88250690, -281246, 444610, -385660, -869982, 59594], [-2629949, 198505, -3994977, 283561, 150648, -316557], [125210057, -251247, -1091483, -585893, 1025528, 83119], [156585825, -31872, -511099, -107830, 1124362, 131516], [-129390041, -365735, 2426667, -136831, 1451784, -286621], [108535006, -1154106, -262878, -1348544, -110190, 441992], [6467589, 418916, -2407777, -152210, 312168, 425326], [-101305746, -314623, 2446137, 1006986, 1311967, 258207], [-41773365, -406267, 3759384, -391869, -2875, -1539], [37643931, 552804, -3596847, 232067, 921654, 229822], [-27437496, 379, -3128923, 997622, -1049507, 183657], [194075736, 205965, -1726180, 377550, 1247546, -353247], [-93853865, 78452, -2870823, -714685, 568360, 328553], [147419275, 311661, -960907, 106741, -1234738, 330599], [140432662, -142136, -1095911, 164351, 1796581, 443228], [-2378514, -444438, 4024478, -716802, -800990, 336952], [-82222422, -362570, 3059216, -692462, -1830134, -467502], [-239228003, -153823, -1187727, -707491, -68928, 298639], [-166001477, 780688, -1769147, -760267, 1069122, -212183], [59863551, -1251951, -1626287, -389927, 178020, 269743], [-74844450, -599063, -718911, -541654, 496393, -589301], [61534026, -655365, 1396801, 155542, -1505449, -249895], [132924672, -23280, -2183404, 1099930, 382310, 256360], [75245458, 906997, 1543422, 431780, -117538, 860285], [-115371557, -551337, -2650331, 228945, -1286826, -629131], [71481200, 325547, -1745448, -972953, 602256, -168637], [-139949880, 305818, -4164642, -153622, -667900, 71724], [-104805699, -406552, -67978, 1149886, -1343393, -173622], [148650984, 314630, -1852509, 493055, 1557373, -128924], [-45222673, -1220337, -1536021, -256481, 558636, -392053], [264709252, -159763, 2682271, -1362430, 1874531, 616571], [167867018, 657264, 691254, 479238, -377148, -75799], [-271096397, -9410, 3921749, 468939, -132000, 302997], [64620736, 175776, -476276, -371882, 696898, -286890], [253044398, 29338, -1293986, -637258, 670322, 502896], [64579383, -1259536, -1665405, -799371, -642102, 147325], [125457113, -948338, -178668, 1774312, 411237, 639204], [68893871, 699028, -242843, 724349, 120464, -408495], [-135399343, -1147239, 1116643, -979953, 1250614, -681581], [148135547, 1075189, 1720039, -734523, -740564, -318801], [47910498, 1284937, 1044990, -903152, -972706, -636551], [206720630, -751028, -2762590, -1681034, -503154, 685478], [146783859, -169118, 748127, -274145, 340988, -71979], [-281352093, -677029, 1231799, 109025, -566760, 562909], [-27426471, 70507, -3721157, 250355, 338376, 948451], [-56460442, 264762, 1884494, 265366, -1968934, 384212], [-332965628, 526511, -431560, 454234, -253106, -266349], [-280044165, 272147, 225921, 134063, -201104, 1147796], [23491583, -1231731, -2352301, -904829, -68056, 67507], [192258906, 558760, -3141594, 172860, 411442, 186132], [123100075, 738128, 5939590, -1030293, -710321, 333962], [-184444599, 270655, 824167, 2640893, -1073724, -563865], [-105460645, 661853, -1145865, 464457, -1835200, 370623], [295605644, 220421, -3030563, 352142, -139305, 316449], [-30898337, 1230977, 857090, -390653, -1975873, -638007], [-83775809, 554656, 4455639, 767517, 1215158, -700843], [-114568967, 289140, 2702270, -869165, 1450517, -455570], [12858352, -322833, 1257188, -1468290, 1188132, 863665], [42029485, 1158035, 1635863, -1733111, -1649752, 773105], [197510613, 478844, 257020, -561430, -2743635, 341448], [116888456, -1009436, 5493377, 554065, 1410339, 290008], [-63366700, -1159575, -816528, -544870, 2116524, 20349], [12974511, -958659, -1245691, 492709, 1449982, -13641], [134836716, -664710, -2575920, -1413042, -1875202, -345547], [-84703468, 867392, -4550786, 1406894, 256808, -157524], [-27517947, -1072502, 1673473, -1330946, 669378, -303768], [-66722456, -757795, 1937518, 353260, -1496848, -483619], [-370504551, -719667, -3404575, 718819, -787826, -504584], [279764432, 389836, 3992844, -710948, 1789710, 512370], [-35326948, -899943, 732765, 2648882, 1654391, -34755], [-21064841, 1187361, -371837, 260105, -2558332, -812913], [66112731, -750924, 3110888, -969965, 1150357, -42678], [-132920126, 195992, 861950, -1123420, -2195838, -193776], [-362036942, 193672, 327438, -970284, -1157964, 88060], [-110674963, -578967, 4047381, -272923, -999522, -362257], [94403373, 853363, -2162312, -679789, -144549, -1105009], [23197830, -342423, 748990, 2082179, 829232, -17509], [-198637201, 1032767, -4102533, 643159, -2679828, -1369153], [344298352, 545028, 2977248, 623506, -288898, -641298], [26621653, -514911, 5246113, -1421961, -2114410, 723015], [-80075583, -324814, 470751, 86357, -997272, 353028], [-237710179, -1219456, -457741, -1298897, 1289444, -942147], [238066515, 529539, 3743019, 2108909, -1003190, -589488], [218914684, 187883, 225218, 455641, -1035558, 363853]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_912_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_912_6_a_y();".
// 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_912_6_a_y(:prec:=6)
    chi := MakeCharacter_912_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_912_6_a_y();".
// 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_912_6_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_912_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<5,R![-26557430352, 602356212, 29529208, -445577, -9557, 65, 1]>],Snew);
    return Vf;
end function;