// Make newform 567.4.a.g in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_567_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_567_4_a_g();". // 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_567_4_a_g();". // 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 := [-4616, 8592, -2506, -1941, 708, 138, -49, -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, 1, 0, 0, 0, 0, 0, 0], [-13, 0, 1, 0, 0, 0, 0, 0], [14, -20, -1, 1, 0, 0, 0, 0], [24200, -11632, -8352, 1851, 657, -81, -14, 1], [61640, -52264, -9066, 10857, 171, -681, 4, 13], [19504, -17633, -6738, 4950, 699, -369, -19, 8], [-25436, 30460, 3687, -6354, -228, 381, 5, -7]]; Rf_basisdens := [1, 1, 1, 1, 744, 744, 186, 186]; 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_567_a();" function MakeCharacter_567_a() N := 567; order := 1; char_gens := [407, 325]; 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_567_a_Hecke(Kf) return MakeCharacter_567_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 := 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 := 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 := 4; raw_aps := [[0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, 0, -1, 0, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0], [-2, -1, -1, 0, 1, 1, 0, 0], [7, 7, -2, 1, 0, 0, 0, -1], [-21, 4, -2, 1, 2, -2, -1, 1], [17, 11, 1, 1, 2, 1, 1, 3], [-25, -4, -5, -2, -1, -1, 1, -4], [-80, 4, 5, 0, 5, -1, -1, 4], [3, 10, 5, -3, -3, 1, -4, 3], [31, -2, -13, -5, -1, 3, 2, -3], [-62, 3, 15, 0, -5, 3, 0, 6], [-7, -2, 7, -7, -9, -7, 4, 1], [-80, 33, -2, 2, 8, 0, 1, -4], [-89, 23, 25, 9, -9, 1, 7, 5], [16, 10, 17, 10, 13, -5, -5, -4], [78, -5, 4, -6, 7, -2, -9, -4], [-93, 4, -28, 19, 34, 0, 1, 1], [-122, 19, 3, -1, 1, -9, 17, -11], [110, 13, 0, -6, -8, 10, 21, -16], [38, -60, -60, 7, 36, 2, -11, 5], [-361, -20, 2, 1, -30, -4, -15, -15], [-538, 9, -31, 2, 35, 25, 2, 28], [44, -164, 45, 18, 1, -5, -21, 14], [-293, 53, 75, -25, 32, -5, 1, -7], [555, -145, 23, 3, 17, 5, 9, 19], [-327, 33, 64, -23, -20, 0, -6, -27], [204, -203, 2, 8, 60, 26, 19, -2], [-648, -39, -104, 27, 28, 6, -16, -17], [28, -330, 53, 21, -55, 9, -10, -11], [-841, 32, -40, -33, -69, -4, -31, 25], [-939, -22, 59, -11, -41, -25, 36, -9], [-731, -225, -61, 11, -82, -29, 25, -31], [52, -39, 9, -10, 107, -17, 22, -4], [88, -183, 113, 25, -71, -7, 23, -5], [461, 79, 65, 3, -64, -47, -47, 13], [-115, -78, -10, -27, -106, -30, -37, -29], [-645, -13, -4, 35, 62, 12, 36, 21], [-601, -120, -33, -25, 21, 25, 34, -13], [422, -189, 144, 32, 54, 4, -47, 38], [234, -282, 120, -28, -23, -32, -6, 12], [963, -101, 61, 42, -21, -79, -84, 42], [-238, -415, -85, -19, -75, -21, -45, 49], [-368, 302, -77, 28, 91, -43, -53, 20], [-803, 477, -139, -53, -31, -1, 9, 51], [739, -91, 76, 25, -42, -8, 38, -67], [947, -455, -128, -45, 2, -8, 10, 49], [-915, -108, -28, 83, -57, 4, -7, 1], [-992, 193, -121, 16, 112, -69, -14, -2], [19, -75, -117, 52, -151, -11, 54, -78], [865, 778, -96, 20, 82, -30, 50, 52], [-575, 549, 25, -99, -113, 69, 13, -13], [-654, 290, 229, -100, -22, -71, 21, -132], [-978, 34, -272, 72, -8, 48, -70, 16], [-74, 598, 22, -93, 110, -60, -7, 37], [-1449, 200, -204, 9, 247, -32, -67, 25], [352, 169, 37, -8, -285, 33, -60, -112], [1209, 92, -172, 69, -20, -82, 17, -109], [77, -325, -144, 122, -92, 44, -3, -12], [-1068, 966, -142, -58, -143, 76, -18, 0], [-2225, 374, -18, 25, -73, 94, -15, -15], [978, -530, -205, -10, -152, -145, -19, -150], [-544, 447, -29, 84, -23, 79, 112, -4], [-972, -58, 215, -162, -205, -139, 3, -26], [-412, 525, -381, -92, -65, 89, 14, 94], [2775, 536, 80, 33, 230, 44, 3, 199], [594, -1202, 79, -32, -95, -111, -17, -188], [-3252, 527, -118, -50, 66, 94, 61, -64], [-1570, -108, 238, 36, 314, 30, -46, 64], [-3535, -167, -107, 41, 265, 217, 129, 3], [327, 21, 116, -184, -334, 66, -105, 78], [-3227, 958, -291, 129, 401, 1, -56, 79], [1581, 204, -449, -41, 149, -61, -120, -15], [640, -226, -33, 130, 431, -215, -77, 192], [259, -664, 671, 143, -39, 51, -66, 81], [-5424, -615, -62, 126, -24, -22, 93, 24], [-1743, -828, 85, 41, -9, 7, -34, 1], [-410, 1246, 193, 33, 31, -31, -66, 3], [1794, 1224, 252, -34, -268, -80, -26, -70], [-4708, -369, 400, -84, -107, -104, 103, -40], [440, -883, 452, -140, 282, 52, 125, -46], [-774, 1092, 279, 28, -39, -3, -53, -298], [1311, 1065, 90, 309, 484, 156, 106, 143], [3524, 62, -581, 256, -311, 205, 63, 142], [-3186, -730, -302, -90, -200, -42, 158, -110], [1597, -993, -614, 162, 218, -78, 11, -80], [180, -68, 126, -165, -146, -128, 57, -331], [-4058, -536, 1065, -174, -102, 95, -5, 26], [611, -434, 19, -122, 305, -165, -205, 24], [-2957, 1040, 197, 1, -525, -15, -34, -353], [-2908, 555, -236, -108, -462, 54, -85, 94], [470, -1135, 65, -259, -655, 117, -145, -193], [4827, -486, 477, 113, 443, 119, -6, -39], [-2270, 1556, -282, 42, -126, 14, 112, 42], [-7718, -765, 439, 18, 281, 159, 62, 82], [1981, -1203, -559, -395, 61, 121, 229, 167], [-7043, -923, 392, -433, -308, -158, 74, -155], [3095, 92, 500, -321, -91, -112, -147, -71], [-1583, 1119, -172, -249, 352, 102, 118, 23], [2649, 963, -197, 55, -59, -43, 17, 169], [-6796, -2, 105, 94, -275, 311, 95, 34], [-6811, -19, 232, 77, -453, 18, 4, 173], [-1659, 1939, -346, 245, 456, -128, 2, -371], [-906, 2070, -395, -260, -13, 155, 207, -138], [-253, 671, 297, 113, -1087, -39, -193, 155], [-5419, 1783, 120, 35, -161, -10, -120, -285], [1100, -623, 294, -6, 834, 316, 107, 280], [4740, -892, -130, -76, 622, 362, 166, 164], [3316, -13, -185, -182, -607, 127, -326, 48], [-8119, -309, 716, 247, 42, 140, 340, 281], [-4609, 4029, -364, 259, 620, 338, 228, 87], [2366, -1757, -779, -500, -561, 159, -114, -510], [206, 261, 984, -74, -295, 92, 147, 138], [8392, 1094, -205, 191, 121, -153, 202, -321], [5801, 620, 555, -390, 335, 129, 159, -30], [3004, 1720, -865, 130, 151, -159, -245, 196], [3240, -361, 376, -168, -84, 38, 147, -342], [3191, -2104, 118, -323, -616, -520, -437, -301], [8304, -1997, -60, -342, -346, 416, 379, 356], [-6841, -1880, -166, -407, -427, 90, -73, -255], [-7950, 2234, 825, -215, -815, 13, 210, -413], [2770, -2098, -297, -666, -867, 29, -149, -10], [4558, -1786, 200, -36, -528, -276, -26, 128], [-1623, -2335, -1424, 239, 227, 210, 86, 11], [5468, -3544, 64, -198, -892, -520, -426, 42], [4012, 1164, -810, -100, 254, -434, 58, -544], [6164, 3362, -981, 144, -307, 21, 123, -174], [-3439, -3792, 289, -305, -381, -187, 40, 221], [-3256, -581, -351, 742, -266, 57, -44, 424], [8940, 3004, -813, 308, 549, 89, -5, 154], [7937, 1206, 480, -257, -1280, -280, -301, -35], [6132, -2344, 130, -29, -8, 508, 255, 169], [4865, -4350, 952, -199, -282, 106, 23, 451], [-3648, -782, -1714, 394, 242, -150, 218, -38], [-5153, -3015, 249, -655, -857, 205, 105, -201], [-5740, 1561, -437, 324, -76, 167, -90, 474], [6343, -4890, 1693, 245, 99, -287, -232, -149], [1822, 2974, -243, 574, 974, 523, 105, 306], [1619, -1368, 448, -169, -254, -98, 111, 273], [3219, -415, 717, -25, -1291, 87, 101, 155], [2010, -381, -458, -42, -1150, 166, -149, -100], [7248, 1864, 136, 112, 144, 252, 144, -364], [10402, 2266, -1673, -42, 643, 173, -261, 156], [-3450, -1719, 1420, 520, -310, -322, 303, -198], [2279, 2033, -554, -649, -492, -570, -716, -103], [4923, 891, 1293, 119, 446, 25, -239, 305], [-7449, -1229, -765, 37, -87, -297, 169, -169], [4311, -3757, -126, -65, 956, -170, -518, -147], [2485, -5475, 560, -272, -76, -286, 75, -18], [-7465, -2308, 1218, 89, 722, 468, 35, 431], [-10201, 3082, 831, -547, -1119, -479, 70, -499], [-12483, -98, -995, 19, 959, 219, 92, 723], [3323, -2289, -1381, 413, 369, 185, -113, 557], [3621, -1525, 2492, -177, -670, 500, 522, 247], [11094, -641, -143, 701, -1231, 109, -111, 51], [17318, -158, 103, -609, -651, 379, -82, -813], [-11162, 2032, 551, -306, -801, -327, -479, 770], [-3581, -1824, 474, -19, 494, -566, -701, -919], [-11412, -2220, 1268, 508, 23, -504, 44, -176], [6939, -1379, 1069, 255, -791, -345, -295, 889], [3530, 1628, 32, 552, 2080, 156, -512, -16], [10595, -3519, -1162, -674, 596, -16, 699, -252], [7334, -4570, -1277, -32, 410, 359, 163, 188], [4283, 3779, 2460, -265, 254, -66, -154, 1027], [2325, -1671, 95, -403, -39, -143, 557, -935], [-19898, 3779, 58, 32, 562, -786, -437, 774], [-3379, -918, 825, 1129, -10, -265, 240, -279], [-3329, -1520, -1874, -373, -840, 140, -323, 361], [-5591, 2401, 457, -25, -630, 469, 219, -51], [15932, -1126, -373, 306, -457, 997, 735, 360], [401, 181, 400, -473, 1219, 18, 454, -323], [5753, 1052, 594, 438, 1466, 120, 336, 366], [-7791, 3356, -544, -664, -1072, -84, -472, -208], [-9087, 3743, -2142, 399, 1898, -502, -558, 145], [-4742, -6943, -1553, 696, 1139, 327, 170, -482], [-15545, 2810, -227, 163, 2013, 835, 176, 251], [-5757, -3794, -419, -449, 915, -79, -368, 29], [-7419, 2125, 1939, -3, -219, 229, -463, 597], [16004, -2138, 68, -302, 1163, 228, 440, -618], [6081, -1128, -1377, 926, -697, 475, 49, 452], [10868, -4041, 1566, 288, -878, 512, -155, 974], [4080, 2294, 1236, -488, -718, -176, -206, -516], [-10914, 515, 910, -870, -1026, 392, -255, 360], [3537, -5358, 895, 188, -2649, -481, 389, -566], [-9106, 2599, 444, 496, 722, 330, 781, 782], [11713, -530, 677, -587, -963, 321, 282, 155], [-5994, 1212, -1363, -778, -86, 985, 693, -1074], [-22836, 6020, 2129, 311, -2087, -3, 190, 201], [4516, 8139, -1022, 440, 2142, 484, 105, 366], [2878, -324, 927, 0, 321, 165, 195, 516], [7463, 1502, 43, 193, 2921, 89, 1084, 113], [-17912, 1478, -2181, 40, -1577, -827, -639, 454], [-5013, -3827, 14, -21, 677, 204, -188, -1075], [8023, -4486, 1047, 287, -881, -279, -1090, 799], [-43603, 4177, 363, -71, -1467, 343, -129, 351], [74, 2278, 2618, 349, -2694, -552, 297, 167], [8226, -1683, 140, -430, -444, 928, 95, 520], [8643, 2312, -172, 87, -386, -736, 359, -137], [-1889, -1717, 1928, -805, -750, 516, -36, 405], [-1551, 1511, -3954, 155, 642, -504, -280, -1097], [4793, -3513, 1760, 95, -1830, -1040, -340, -383], [14349, -5180, -1786, 663, 453, -266, 1025, -993], [1148, 8381, 1585, 552, -1345, -199, 804, 634], [27041, 5637, -2172, 927, 765, 190, -166, 283], [-7113, -993, 142, 379, 539, 40, 506, -587], [-17570, -3484, 1424, -150, -730, 164, 90, 838], [-1045, 6945, 2153, -211, 2215, 51, -417, 495], [-6894, 1932, 866, 344, -114, -1250, -358, 568], [9501, -1507, -128, -17, 1309, 1042, -522, 631], [-9954, -90, 969, -121, -3093, 25, -724, -47], [-1968, 8695, -1822, -134, -2800, 1200, 257, 520], [10600, 2482, -1058, 676, -762, -858, -60, -292], [15879, -195, 452, -215, 1777, -526, 1020, -1215], [15015, -4698, -2456, 181, 2780, -218, -1147, -431], [-5736, -8698, 673, -196, 499, 15, -365, -312], [-4449, 1074, 853, 1024, 671, 89, -393, 1602], [-38529, -3836, -2813, 61, 1353, 221, -428, 725], [-10237, 5278, -578, -723, -182, 1654, 547, 59], [2816, -162, 2555, 702, 2251, 561, 157, 890], [7184, 6138, -2901, 1262, 1503, 73, 557, -986], [-4692, -1974, 390, 308, -2240, 298, 192, 1200], [-24820, -9125, -232, 459, -1726, -230, 894, -225], [-5353, 4526, 2417, -273, -2733, -397, 1168, -1113], [38358, -13302, 1156, 268, -1186, -176, 38, -368], [-25103, -3114, -2424, 109, -1340, -202, 635, 883], [-14760, -4051, 3621, 214, -1663, 545, -1114, -608], [31063, 2394, -3553, 1124, 2395, -473, -67, -206], [4054, 2145, -1137, 1084, 319, -73, -1342, 868], [-16617, 1964, -1139, 645, -1139, 1371, 1028, -467], [-19405, -2458, -1407, 619, 1991, 2005, 136, -39], [-7713, 11276, 1243, 619, -8, 865, 774, 1483], [17238, 1301, -164, 2041, 1388, -98, 94, -1011], [9352, -2267, -1688, 710, 692, -108, 271, 1196], [1061, -2586, 711, -1223, -3062, 1193, 50, -995], [4003, 2444, -833, -1761, -723, -1025, -374, -1627], [25970, 920, 3874, 562, 2966, 310, -380, 294], [-34909, -179, 2919, -1297, 347, -1329, -715, -173], [22967, 4926, -511, -615, -1727, -1497, -1130, -1591], [19836, 5483, -2598, 944, 1704, 672, 1541, 1594], [-886, -8791, -2497, 1116, 1595, 1333, -186, 1752], [9366, 1994, -3121, 920, 1095, 729, 303, 982], [-3692, 5109, 1041, -620, -1297, 509, 528, 1596], [4630, 10826, 679, -604, 3969, 393, 111, 748], [-2095, -2591, 1459, -705, 825, -429, -571, 229], [17617, -5326, 2808, -337, -1100, 354, 23, 1735], [13504, -2042, 3267, -664, -4317, 1217, 427, 286], [-3749, -6891, -416, -999, 997, -766, -564, -771], [45956, 2609, -810, -1660, 632, -162, -907, -1150], [6047, 1038, 844, 858, 1168, 1114, 1192, -94], [-4398, -4851, 3665, 130, -205, 491, 306, 1968], [-1500, 338, -1420, -940, 1228, 2012, 850, 152], [32129, 3706, 2877, -68, 895, 809, 227, 1546], [16295, 738, -1950, -1021, -1126, 442, 1171, -1321], [-8280, 10423, -1132, 138, 78, -398, 255, -84], [14021, -9190, -3658, 1177, -836, 330, -1081, 787], [4136, 7341, 990, 1450, 3870, 242, 277, -196], [22711, 8745, 2043, -897, 170, -381, -337, -1487], [31594, -2692, 456, -480, -3615, -460, 52, -232], [-21259, -3581, 350, 299, 1902, 1928, 22, -767], [-3843, 4933, -1729, 1567, 3353, -51, 1163, 1965], [3387, -4036, 1518, -1809, -2280, -234, 291, -2067], [-55318, 14401, -733, -1978, -897, -1189, 148, -1388], [-2976, 7239, -4797, 1352, 3926, 943, -1064, 1226], [-22794, -8790, -1890, 84, 4532, 614, 952, 860], [-25810, 5284, 3554, 836, 372, -106, 1594, -296], [34500, 9281, -740, -978, 2680, 1644, 623, 964], [-18279, 5474, -7237, -1821, 257, 837, 616, -757], [6828, -2943, -691, -908, 1183, -1057, 728, -272], [2007, -2983, -6079, 1435, -1309, 645, 743, -1337], [17265, -1586, -1294, -541, 6982, -658, -1, 1275], [-13210, -1693, 338, -238, 2760, 114, -1299, 1204], [23535, 15703, 86, 1881, 4030, 280, -960, 1115], [14169, 11274, 6186, -329, -2446, -1006, -627, 1977], [-35464, -1592, -2279, -46, 425, -169, -837, -862], [20029, 6495, -4594, 379, -266, -1522, -790, -1235], [-40857, -6205, -263, 717, -2377, -929, -1019, -1753], [24358, -1265, 4718, 142, -1876, 462, -1417, -12], [1544, -5048, -2444, -1630, 584, 1868, 1770, 374], [-31816, 11270, 2050, 1834, -353, 1108, 650, 1084], [-27149, 8095, 500, 921, 2424, -996, 802, 1223], [-65808, 5359, 3957, 596, 1167, -1945, 382, 472], [22203, -15221, 2637, 1505, 3577, -1925, -515, -1195], [-21279, 128, 8632, 741, -119, 788, -607, 2441], [-21530, -3750, -953, -2322, -2540, 159, 1309, -2770], [-5368, 21574, 2632, 845, 2058, 30, 529, 95], [17270, -2018, 6124, -885, -4426, 658, 877, -1235], [-13782, 3436, -43, 1126, 83, -1005, -595, 2626], [-30234, 1772, -4255, 1026, -1315, 635, -2127, 1696], [24827, 11548, -6140, -1685, 1434, -1692, -1845, -693], [47511, -11599, 4191, -1181, -2095, -1213, 353, -3605], [-26387, -8514, -707, -195, 2658, 739, 2200, -571], [-56746, 2898, -380, -1525, 268, -1478, -423, -2151], [47296, 4423, -3122, 716, -1994, 104, -367, -1958], [23561, -3039, -4533, 1793, -2641, 643, 55, -343], [-486, 8510, 1286, -24, -991, 1578, 1564, 1220], [25024, -4646, 865, -972, -807, -1685, 53, -798], [-18663, 14810, 1433, 535, 1080, -701, -218, 2555], [13495, -3326, -4287, -2175, 1887, -583, -1160, -1909], [-12967, -3399, -106, 1181, -4356, -1070, 50, -185], [-29441, -67, 1028, -993, 1370, 410, -1206, 1057], [-26653, -16471, 3569, 1335, 95, -107, 375, 375], [9618, -2313, 897, 681, -113, -399, -2735, 575], [-10456, -19623, 3096, -560, 1550, 240, 1411, 794], [41673, -10079, 1848, -817, -804, -2128, -2504, 1219], [10102, -6823, -5221, -2447, 59, -173, 1087, -2079], [32273, -5589, 1353, -1381, -4421, -157, 591, 2253], [-5657, -5018, 2032, -873, -1171, -2276, -1883, -1865], [-9031, 9611, 4588, 1981, 1056, -744, -360, 681], [26994, -15159, -133, -678, -3089, 9, -344, -940], [-71191, -438, -357, -27, 2161, -1621, -2102, 2507], [-4965, -9896, 3476, 2375, -2378, 794, -809, 1853], [-60213, -8474, 5183, -801, 167, -645, 1606, 197], [-33459, -13086, 1512, -1459, -1488, -56, -1267, -2141], [-30795, 11745, -2510, -1768, 1334, 1852, 1653, -870], [-13718, 688, -275, 1160, -545, 191, -883, 1116], [5635, 31898, -4259, -615, 36, 1591, 1708, 953], [-1007, 3977, 1955, -1484, 5421, 363, 422, 556], [-2061, -5289, -2292, 1697, 1750, 2158, 2874, 159], [29254, 19848, 416, -860, 1544, 476, 1056, -2340], [18239, -14089, -3035, -1817, 2109, 667, 2699, 371], [11336, -2622, -3180, 395, 1142, -1898, -1163, -667], [-254, -9512, -3627, 2322, 3283, -1977, -921, 2694], [-41867, 5684, 5492, 154, 1600, -254, -906, -722], [-42585, 24070, -835, 1130, 6425, -21, -237, 654], [5131, 906, -5940, 687, 2472, -324, 1353, -549], [-6089, -4083, -559, 3523, 681, -1729, -1979, 3341], [30691, 9562, 4244, 657, 964, 706, 2385, 1553], [-72564, 2458, -1183, 1587, -179, -2359, -1492, 379], [-4197, -6983, 1099, -147, -1195, 247, -317, -295], [-57116, -6476, 942, -100, -516, -1234, -710, -2180], [-3857, -18885, -2630, -1221, -5422, -384, -2236, 1423], [-35769, -11881, -4782, -2809, 3668, 2010, -298, -1219], [-34371, -6905, 129, 627, -586, 1901, -1289, 1133], [18149, -17188, -1431, -49, 5495, -1317, -2482, 35], [-18636, 30557, 3876, -1220, -3865, -2272, -1327, 880], [1155, -2900, 3011, 501, -4195, -1265, 118, -773], [-27964, -6659, -1150, -690, -900, -544, 1295, -2852], [51701, -12074, -1734, 505, -3168, 2562, -2175, 2213], [59166, -8485, -9092, -2102, -1832, -890, -1659, -3008], [3197, -6377, 1320, -4729, -6016, -2050, -1038, 1415], [-33869, -8083, 3431, -3732, -3193, -331, -1654, -1574], [-13875, 5108, -2331, -635, -7661, 679, 2466, -435], [49870, -5307, -2273, -921, 7881, -1137, 387, 2013], [-20469, 392, -2027, -2335, 2205, 1721, 908, 523], [59535, -1702, 10444, -377, 68, 208, 899, 1873], [1545, -8207, -5946, 1953, -3077, 268, 2190, 743], [-19884, -8288, -4790, 840, -1538, 1462, 1842, 2108], [23399, -461, -3532, -1475, -186, 2348, -966, -2169], [24400, 16283, -3350, -886, 8432, 1428, -1595, 2648], [49467, 16491, 2771, -313, -6002, -1241, -2055, -267], [16126, -7141, 6670, -2445, -3076, 272, 132, -317], [20765, -31065, 4593, 1299, -5527, 1053, 513, -1269], [8395, -1055, -1595, 2115, 2501, -2153, -1895, -893], [55583, -10959, -4739, -2969, -985, -871, 159, -2973], [-29848, -10596, 413, 2084, -573, -1933, -1295, -1132], [-18911, 8771, 5775, -1365, -2793, 497, 2269, -877], [-21731, -5377, 2588, -735, -915, -2310, 262, 1601], [-47624, 4115, -4322, -1505, 1376, 2412, 1764, -537], [14610, 22825, 4132, -80, -1888, -1352, -2749, 1102], [2046, 19276, 2934, -1802, -6426, -3114, -1056, 1334], [-10302, -696, 12197, -2216, -4967, -313, 2073, -72], [51153, -14554, -1825, -2683, 481, 1397, 892, -4327], [53212, 15447, -4186, 5494, 4426, 1054, 721, 476], [-25676, 628, 2372, -1058, -1598, 144, 1546, -1826], [9160, 5903, 5234, 2527, 1422, -2204, -2044, 71], [73517, 10807, 12469, 49, 293, -991, -1181, 4835], [2317, 10317, 283, 1487, -5271, -3035, -1837, -2303], [-55661, -638, 741, -688, -6931, 227, -519, -784], [20731, 6340, -10222, 1039, 3484, -2342, 2575, -367], [78220, -7830, 3019, 1542, 4319, 3429, -7, 3796], [19874, -14261, 1639, 2149, 9825, 1431, 1635, 951], [73656, 5020, -8124, -2334, -2326, -340, -650, -670], [25164, -28343, -1394, -1404, -2368, 1120, 391, 4282], [-39201, 640, 7137, 1163, 1249, -1159, -2854, -2283], [-21558, -8304, -4602, 234, -185, 2272, -2248, 2620], [87518, 15586, 51, 2305, 2335, 35, 3284, -4367], [-8910, -9385, -1376, 3588, 560, -886, -51, 454], [40617, 3338, -6321, 2579, 103, -467, -428, 503], [-11442, 4830, -6052, -2240, 1998, 860, 754, -1384], [-24447, 17964, -3206, -841, 5129, -1070, 1617, -1929], [9437, 8981, 2203, 2856, 2519, -679, -1386, 4386], [1517, 12552, 6569, -4196, -913, 845, 441, -2970], [-10332, 823, 4779, -315, -7493, 427, -271, -1235], [10272, -9173, 1987, -3070, -780, -583, 1462, -1736], [-50396, 18812, 6440, -376, 656, 2864, -1520, 3068], [-18154, -324, -5289, 4208, -519, -623, 1499, -470], [-8339, -4602, -4771, 3451, 2712, 2831, -698, 3587], [4283, -7328, -1382, -1349, 1902, -4278, -2625, -2011], [-8910, -13987, -3101, 2308, 979, -1233, -1170, 456], [25757, -12463, -3846, -1459, -4996, 1128, 260, 1037], [51933, -11602, 592, -2573, -3731, 1692, 2037, -3381], [5595, 14998, -7436, -381, -3904, 2104, -677, 2513], [14469, 428, -1010, 603, 6161, 3914, 555, 25], [27635, 19776, -3147, -1429, -5049, -1109, -1082, 791], [29239, -5650, -14952, 5872, 7116, -680, -1630, 3164], [9867, -13502, -1175, 1343, 10933, 641, 2630, -1419], [-16030, 14853, -10408, 6274, 5296, 4602, 257, 4144], [-53972, 17124, 1327, 2576, 4793, -1079, 439, -1522], [53814, -5772, 1240, -3586, 1794, -3412, -2806, -38], [-14806, -12798, -7746, 3758, -3057, -1646, -2074, 2194], [-36689, -9925, 2173, -904, -2461, 1503, 466, 3146], [88814, -16140, -461, -2936, -3907, 4085, 3207, -1398], [-1512, 15714, -5681, 178, 753, -355, -1097, 1466], [-18472, -18780, -5910, 1228, -1661, -2914, -1834, 424], [-116546, -9622, 9261, -622, 45, 1071, 291, 3556], [23426, -4301, 729, -2968, -1553, -1585, -2722, -6258], [77529, -12636, -3295, 3540, 10649, 15, -149, -148], [-32552, -9506, 3819, 3880, -3386, -2085, 295, 1036], [28166, 21076, -1469, -72, 1287, -1371, 521, 658], [-71658, 31110, 8772, -746, 9320, 692, -528, 366], [13466, -7769, -15153, 1640, 2069, 1233, 128, -1082], [-54674, 21552, -6969, 1058, -1925, 393, -1867, 1876], [13060, -33331, 5960, -1172, 3780, -1616, 1055, -2386], [36917, 21797, -930, -468, 3286, 746, 539, 1730], [119369, -12284, 2963, 2741, 672, 3413, 448, 5129], [25289, 542, -2705, 721, 425, -5455, -2480, -3151], [-63181, 10264, -4548, 2693, -1026, -5026, -5573, -647], [33936, -9081, 9257, 2208, 3585, -2591, -1362, -3846], [-8868, -11888, 3559, -2966, -6157, -3155, 683, -6116], [40612, 15847, 1147, -4462, 3265, 3737, -154, -1100], [39091, 18080, -3918, 2539, 315, 3634, 1217, 2039], [9527, 9346, -6249, -62, -1819, -1425, -2093, -4840], [4880, 9210, -1408, 3874, 6447, 842, -1680, -744], [-53777, 21094, 7287, 1665, -707, 997, -2106, 3881], [4500, -788, -2110, -534, -5590, -1278, -128, 974], [7752, -20994, -7208, 538, 1426, 96, -2160, 654], [101656, -25378, -3731, 1891, 9477, 865, 800, 421], [-6924, -29222, 2933, 3434, -2532, -1627, -2429, 190], [-52938, 6474, -5450, -2908, 7818, -2254, 1388, -4292]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_567_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_567_4_a_g();". // 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_567_4_a_g(:prec:=8) chi := MakeCharacter_567_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); 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_567_4_a_g();". // 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_567_4_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_567_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![-4616, -8592, -2506, 1941, 708, -138, -49, 3, 1]>],Snew); return Vf; end function;