// Make newform 507.4.a.q in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_507_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_507_4_a_q();". // 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_507_4_a_q();". // 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 := [5144, 9428, -966, -4745, -150, 772, 29, -48, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0], [513836, 1425628, -58625, -334818, -27838, 27181, 1734, -719, -3], [-2502316, -1425628, 257473, 334818, 27838, -27181, -1734, 719, 3], [3265492, -623892, -2016507, 170474, 248606, -16209, -10310, 483, 127], [-168892, -606900, -558855, 389090, 139382, -35013, -8974, 831, 171], [2264092, 3151596, -82469, -860826, -141702, 61649, 14878, -1291, -383], [2379924, 1774556, -2400467, -772630, 493806, 79991, -33670, -2293, 743], [-1800516, -2318484, 1205827, 771566, -300654, -67775, 24206, 1757, -591]]; Rf_basisdens := [1, 1, 198848, 198848, 198848, 198848, 198848, 198848, 99424]; 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_507_a();" function MakeCharacter_507_a() N := 507; order := 1; char_gens := [170, 340]; 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_507_a_Hecke(Kf) return MakeCharacter_507_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 := [[1, -1, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0], [5, -3, 0, 0, 0, 1, 1, -2, 1], [1, -1, 0, -2, 1, -1, -2, 1, 2], [2, 0, -2, 4, -1, -3, 1, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [-10, -5, -1, 0, 2, -2, -1, -10, 5], [-8, 1, 7, 8, 2, -6, 3, -8, -9], [31, -11, 1, 5, 9, -3, 0, 2, -14], [-52, -18, -6, -11, 12, 11, 2, 9, -4], [-52, -2, 4, 5, 4, 5, -22, 9, -6], [82, 7, -21, -15, 11, 2, 12, 1, -17], [41, 58, -8, 5, -9, -19, 11, -14, 19], [10, -40, -6, -5, -17, -30, -3, 15, 16], [148, 59, -9, -32, -28, -22, -21, 58, 3], [157, -28, 29, 26, 30, 23, -4, -14, -6], [214, 18, 11, -29, 16, -10, -38, -25, -20], [280, -81, -3, -42, 6, 34, -25, -22, -1], [35, 69, -45, -20, 22, 63, 67, -25, 2], [241, -12, 34, -48, -4, -15, 34, 49, -47], [131, -125, -22, 8, -22, 67, 17, -62, 9], [-7, 111, -38, -53, 20, -55, 55, -36, 44], [150, 43, -32, 58, 29, 58, 44, -54, -35], [307, 55, 35, -40, 66, 9, -105, 59, -40], [194, 80, 29, -16, -4, -8, 12, 9, 55], [-73, -1, 30, 94, -45, 93, 30, -119, 72], [-310, 144, 73, 41, -45, -184, -49, -40, -35], [-69, 208, 45, -40, -13, -55, 57, 77, -43], [185, -126, 214, 81, -21, -133, -137, -16, -37], [-366, 57, 177, 17, 3, 2, -148, -11, -53], [300, 196, 47, 42, 60, -44, -78, 195, -181], [397, 410, -47, -171, 33, -9, -159, 213, -8], [378, 128, 98, 113, 143, -44, -151, 91, -280], [-391, 56, -126, 13, -53, 129, -15, 48, 163], [436, 33, -21, -57, -58, 165, 133, 1, -103], [-1154, -127, 109, -66, -47, 57, 64, 106, -125], [-130, -130, 252, 263, -55, 96, 127, -133, -66], [-503, -31, 29, -24, -142, -151, 115, 83, -62], [-78, 278, -254, -7, -203, 20, 271, -45, 86], [344, 444, -8, -171, 154, -255, -22, 125, -82], [-247, 271, -440, -102, -62, -73, -233, 142, 197], [259, 229, -165, 8, 98, -91, 79, -113, 114], [-1395, -111, 219, 140, -110, -263, -15, 215, 86], [1562, -202, 54, 71, 117, 186, 339, 163, -314], [2050, 83, 208, -106, -69, -88, -238, 40, -75], [1306, -145, 103, -55, 52, 11, 51, -229, -17], [-201, 280, -62, -226, -118, 187, -46, 213, -181], [-419, 448, 200, 57, 215, -169, -351, 0, -141], [398, 134, 119, 76, -328, -314, -298, 57, 79], [765, 24, 38, -178, 52, -85, 106, 23, 483], [-1028, -513, -71, 54, 100, -206, 105, -294, 123], [1372, -123, 157, 233, 173, 110, 318, -151, -95], [-1627, 671, -220, -79, 49, 183, -64, 311, -215], [1606, 81, 20, -299, 128, -176, -31, 63, -41], [1511, -189, 15, -16, -364, -77, -31, 103, 172], [-479, -75, -283, -50, -116, 167, 43, 413, 98], [-574, -447, 266, -227, 69, -208, -136, 324, -182], [-1540, -481, -64, 126, 106, 102, 319, -445, -28], [-1319, 324, -22, 541, 287, 505, -7, 80, -71], [2295, -325, -219, 199, 23, 303, -62, 2, 490], [916, -61, 461, 12, 202, -124, -667, -114, 183], [4228, -67, 181, -32, 79, -459, -136, 78, -65], [-709, 226, 544, 574, -86, -559, 168, -45, -25], [-135, -843, 1103, 838, 126, 373, -35, -457, -788], [1408, 139, 324, 349, 325, 94, 286, 0, -340], [3437, 15, -34, 405, 398, -307, -381, 306, -116], [-963, 904, -748, -833, 35, -61, -55, 372, 1], [-1648, 774, -388, -75, -562, 445, 246, -73, 318], [-1150, -15, -140, -108, -251, 762, 290, 272, -277], [-680, -584, 630, 92, 464, -156, -912, 54, -298], [4073, 131, -641, -268, -104, 21, 113, -395, 94], [963, -761, 553, 112, -32, 467, 217, -133, -668], [-569, 875, 470, 589, -350, 139, 353, -728, -82], [1341, 300, -194, 837, -335, -21, -325, 120, -271], [340, -853, 293, 151, -275, -200, -46, 469, 59], [1769, 517, -685, 130, 154, 909, 269, 363, -468], [75, -1436, 70, 50, -185, 474, -79, 73, -103], [3346, -163, -829, 248, 200, 924, 211, -434, 635], [823, -102, 1052, 876, 348, 605, -334, -391, -655], [911, 919, -244, -14, 453, 591, 660, 307, -308], [-1469, -364, 426, 18, -249, -1416, -861, -21, 411], [1400, 1901, -1181, -1378, -10, -502, -103, -2, 605], [1343, 537, -105, -470, 542, 141, 85, -25, -172], [-1691, 905, 299, 167, -114, 148, 281, 140, -154], [-367, -48, -615, 741, 36, -389, 320, -430, -313], [-1141, -225, 167, -559, 102, 312, 777, 446, -974], [3638, -1639, 115, 696, 132, 56, -505, -508, 401], [2968, -655, 899, 718, 243, 1233, 388, -564, -91], [4688, 262, 663, -252, 671, -188, 167, -874, -664], [1608, 904, -492, 1176, -192, 524, 368, -228, 56], [2206, 360, -889, -454, 282, 262, 862, -171, 131], [2464, -157, 645, -1003, -353, -1634, 114, -295, 187], [100, 1744, -119, -70, -776, -1436, 282, -551, 433], [-3338, -676, 630, 606, 855, 1059, 57, -396, -724], [2044, -1634, 350, 1298, -18, -32, -44, 630, -38], [2013, 841, -349, 795, -1059, -163, 872, -400, -734], [1704, 392, -824, -944, -248, -446, 864, 238, -230], [756, -976, -110, -1393, -791, 644, 415, -735, 1556], [-3125, -1143, 427, -767, -125, 47, -706, 592, -410], [4673, -358, 106, 683, 739, 825, -553, 178, -273], [1642, -2531, 2723, 1503, 361, -270, -528, -273, -1781], [-135, 470, -610, -1384, -230, -1081, -68, 1827, 515], [5980, 485, -1268, 696, -264, 486, 261, -1233, -318], [99, -3972, 302, -1027, 505, 601, -585, 308, -213], [2165, 605, -1413, -543, -183, -869, 52, -162, 1468], [1122, -3927, 1594, 1254, 337, -58, 1080, -2432, 621], [-7703, -2066, 2362, 2443, 1112, 948, -746, -1444, -389], [-336, -284, 780, -1190, -1182, -414, 150, 1072, 2], [-3745, 707, -1399, 390, -1368, -301, 2141, 239, 1070], [-2139, 1708, 494, -12, 945, 374, 53, -1399, -515], [395, -1284, -963, -561, 1286, 2129, 938, -754, -1875], [5888, -2489, 119, -479, 483, 746, -1244, -31, 887], [1132, -680, -1218, -916, -342, 456, 342, 464, 2204], [-13070, 669, -1647, -1288, 654, -316, -133, 724, 699], [-9666, -1221, -846, -310, -283, -616, 666, 758, 1403], [4389, -1233, 1097, -955, 989, -393, -2492, 1008, -1454], [-1890, 659, 947, 270, -430, 494, 637, -190, -1295], [-6949, 1602, 316, -577, 549, -1707, -723, -184, 1553], [6957, 1507, 866, -23, -934, -1943, 1391, -150, 408], [4467, -84, 1665, -270, -1359, -1563, 615, 567, -235], [-893, 1983, -569, -1472, -2040, 553, -77, -225, 1480], [-1271, 2900, -539, -980, -1251, 639, 403, 1487, -1709], [-5458, 1269, 842, 100, -307, -1404, -2066, 902, -245], [-8755, 4447, 838, -674, 34, -105, 1259, -240, 23], [-8979, -1028, -536, -915, -661, 1145, 777, -634, 497], [-387, 2178, 204, 1349, 609, 209, -199, -224, -643], [957, -864, 1456, 128, -1102, -3355, -1320, 621, 471], [2926, 364, -1152, 1914, -1482, -164, 622, -254, -36], [5230, 829, -184, 1368, -929, -908, 602, -532, 1473], [1303, 1679, 357, -630, 586, -649, 149, 277, 1012], [-13100, 2461, -1109, -333, 59, 1994, 1086, 353, 17], [3254, -4527, -1399, -364, 1078, -802, 511, -540, 1143], [-1758, -445, 1083, -158, 427, 2431, 1472, -298, 357], [3203, 6152, 364, -721, -851, -403, -609, 1856, -689], [-4099, 2289, 3259, 520, -1190, -1865, 537, -209, 6], [-4318, 1822, -2358, -246, -494, -1116, 912, 1378, 986], [8369, -1569, 1048, -192, 863, 719, -2574, 849, -330], [5699, 900, -736, 67, -677, 1127, 2225, -124, -727], [-855, 6792, -2844, -2577, 750, 774, -1590, 1116, 2205], [6326, 1132, -3342, -833, 791, 1406, 969, 421, 444], [-13330, 1781, 539, 404, -234, -580, 235, 982, -1127], [7459, -1098, 1173, 2334, 1070, -279, -590, -1814, -232], [-9061, -813, -2754, 1584, -547, -3, -722, 49, -16], [6534, -3284, -1212, -135, 586, 1165, 1464, -2235, 1676], [-1241, 23, -1059, 1141, -2267, 1813, 1580, 1346, -258], [-8018, -2242, 280, 561, -1261, -1088, -307, 2003, 1950], [12165, 973, -1539, 973, 283, 1283, 1464, -856, -3434], [12974, 1856, -3242, -3660, -902, -206, 2538, 846, 1838], [-2141, -2423, -723, -2031, 2131, 187, 1238, -2238, 1528], [3018, 3127, -1257, -2500, -3642, 538, -299, 168, 3517], [15400, 162, 2322, -253, 187, -1164, 1677, 1595, -1770], [-7106, 121, -11, -1700, -954, -418, 1097, 1170, 83], [8832, -2871, -627, 276, 1484, 1240, -915, -1398, 2245], [3269, 2098, 690, -1372, 1302, -313, -1906, -1829, 2497], [3825, -5212, 4244, 3331, 1853, 601, 811, -1320, -2883], [-11137, -1634, -452, -936, -1262, 1805, 1738, -3449, 1749], [8681, -2330, 2583, -638, 558, -3027, -1134, -2026, 536], [-12236, -3064, 3274, 219, 315, -1698, -771, -2175, 1448], [-8904, 1149, -369, -147, 698, -387, -3929, -2261, 3089], [11108, 1572, -482, 482, -1577, -1471, -1379, 1662, 3108], [19569, -3280, -1777, -1042, -162, -1731, 1836, 1538, -184], [8911, -3373, 811, 3616, 686, -2761, -393, -1511, -1550], [8903, -3777, -721, -654, -1367, 1480, 214, -417, -2374], [5367, 88, 682, 354, -431, -1682, -3351, 2931, -169], [-4830, 885, -2613, 1269, -119, 3776, -470, -557, 1669], [8303, 4737, -3925, -1876, -994, 2911, 1133, 1597, 736], [4311, 1930, -2971, -352, 1212, 683, -276, 3464, -168], [-3587, 2525, -169, 1008, 746, -869, -2017, 453, 1376], [-2601, -4846, 2256, -1059, 2286, 2200, 1540, -1284, -59], [-8498, 803, 2031, 1199, -2211, -976, 1316, -779, -2969], [-2470, -655, -1575, -2136, -8, 1408, -2007, 1548, 555], [7003, -3358, -928, 474, 3500, 1929, -1728, -2759, -1257], [-9341, 1618, -836, -2501, -811, 1341, -3047, 3716, -2495], [14474, -2523, -4275, -245, 2203, -852, 1390, -829, 2201], [6512, 2005, -3549, 259, -2467, 200, -1540, 565, 2597], [-506, -2528, 1776, 2605, 3239, 3870, -1031, -4201, -176], [4827, -2445, 3383, 1019, 2487, -4443, -2708, 320, -3170], [16591, 1058, -91, 1783, -982, -3019, -2988, 28, 939], [-5262, -709, 3557, -1971, 2727, 62, -1032, 1141, -4711], [7133, -797, -3451, -3793, -1225, -1739, -958, 4120, 3294], [376, -748, 1039, 696, -2126, -2680, -1288, -1423, 3887], [6561, 12, -1576, -1097, -1336, 620, 2112, -176, 2581], [2505, 4934, -4504, -473, 191, 2429, 2835, 692, -1831], [-2137, -4863, 4571, -2331, 3141, -947, -1826, 178, -1798], [-10236, 5567, -1297, -2554, 904, 4862, -425, -1866, -467], [-8577, -1001, 878, 1937, 1622, -3625, 1083, -3088, -2350], [10638, -2407, 295, -1264, -212, -3234, -637, 1204, 4773], [10483, 4049, -4939, 704, -3736, -121, 893, -2245, 2466], [7188, -6031, 6438, 3937, 2366, 514, -5927, 2341, -5393], [16740, 87, 1091, 1145, 173, 1758, 2008, 357, -2601], [-886, -4928, 4026, 1447, 692, -553, 2606, 823, -5410], [14938, 8344, -1321, -490, 971, -464, -1969, 676, -582], [1862, -1085, -1905, 816, -1532, 2670, 4911, -740, 2801], [-972, 3467, -1030, -816, -3738, -2732, -647, 5641, -352], [-10968, 3220, -1193, -2037, -425, 1894, -1929, 4620, -4397], [13845, 264, 2304, 1432, -1806, 1009, 2806, -2611, 691], [19586, -1765, 449, 2674, -817, 5571, 3386, -3266, -875], [16850, 1480, -4206, -1140, -568, 902, -1344, 2644, 2344], [11872, 8622, 762, -3011, -815, 434, -1307, 1841, -3584], [-2209, 291, 1815, -2315, -2261, -3071, -1884, 2692, 1866], [4784, 2609, 189, 2941, 667, -3886, -1004, -3211, 939], [10631, 4655, -85, 3490, -3543, 32, 1682, -1157, -354], [8073, -2473, -1373, -237, 495, 6789, 5294, -2626, -1440], [5671, -2989, -592, -1002, 2780, 879, 27, 1126, -1905], [11324, 1307, 1400, 4122, 2029, 628, 684, -4500, -3705], [-2422, 8275, 5517, -1239, 1548, -321, 1485, 1257, -3203], [-48165, -4485, 1512, 426, -436, 1461, 2893, 70, -2401], [-23091, 1674, -2251, -4396, -3211, 1299, 421, 915, 6025], [11065, -3144, -3076, 2043, -4667, 1435, 1827, -4860, 2621], [6816, 2604, 4616, 1941, -3293, -3426, -2659, 1993, -540], [-7546, -1390, 1495, -2238, -5086, -664, -2468, 883, -747], [40338, 8989, -2427, -118, -3459, -917, 250, 498, -511], [-12725, 2230, -1718, -91, -933, -4755, -2125, 174, -1253], [-662, -6775, -4129, 177, 169, 2438, -2378, 1159, 2939], [-3509, 543, -4085, 2091, 2089, 4609, 690, -138, -850], [-4374, -5672, 2575, 3671, -312, 2922, 704, -4911, 3244], [-13249, 2617, 2073, -15, -2624, -7154, 353, 1836, -1154], [-21290, -5072, -5400, 1036, -2392, -378, 6400, -6382, 2562], [9165, -4135, 7051, 6298, 4302, 5933, 989, -4485, -6524], [28846, 328, -1641, -5053, 1095, -1866, -2805, 2696, 5171], [492, -3536, 3512, 1256, 4980, 662, -1392, -778, -750], [-33051, -2332, -911, 221, -1427, 6495, 5441, -3029, -2772], [-6711, 1489, -1549, 2614, 6642, -345, 3297, -1553, -4384], [-4114, -10422, 2776, 1454, -2381, 1551, -3933, -2514, 1678], [11921, 1655, 419, 6720, -2612, -5, -1095, -1863, -2510], [-7707, -7841, 1397, -2044, 5040, -1785, -6157, 3595, -4636], [-5012, -2527, -1591, -6913, 1193, 3678, 1906, 2475, 3955], [6425, -291, -5629, -552, 900, -5381, 2635, 235, 3030], [-11349, -8181, -1602, 2305, 635, 2863, 722, -1599, -4561], [16140, 1118, 6725, 5338, -4306, -2184, 1372, -1657, 2171], [-15909, -12165, 1599, -5173, 3497, -2571, -5008, 3556, -1466], [-4907, -10819, 351, 2155, -5389, -89, -1226, -164, 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-136, -4285], [-19830, -1334, -624, 6681, -425, -2306, 2433, -3063, 4732], [49044, 14293, -5087, -8216, 372, 286, -757, 1286, 1343], [6630, 5374, 2254, 6107, -3777, -1982, -1277, 6399, -1256], [7286, -5173, 5369, 3044, -632, -486, -2437, -3836, -1799], [41770, -3651, -4786, -2860, -4129, 466, -2126, 7898, 4241], [9676, -3506, -5983, -547, -1542, 6654, 1176, 215, 1862], [3048, 5594, -2439, -1662, -7757, -1014, -2107, 4202, 2842], [-58896, 9351, -1415, -3351, -2955, -54, 5526, 1849, -89], [41416, -7583, 1647, -1218, -446, 2656, 971, 870, -5525], [-32620, 3208, 4253, 142, -883, 4772, 2019, 3448, 298], [2917, 931, -2875, -3580, -11681, -4612, 1400, 3673, 4852], [22811, 5229, 2213, -2660, -717, -10764, -1594, 53, -756], [672, -2786, 2109, 3976, -332, 1052, -2282, -5743, -1873], [-13395, 15524, 1350, -2064, 746, -5273, -4756, 6733, -7221], [24345, 4529, 4021, -3166, 3288, -3755, -1927, -1721, -1628], [8622, 3979, -3989, -9540, 1046, -4764, -7721, 4744, 3859], [41653, 2172, -8152, -6545, -1159, -1233, 27, 4506, -1537], [-9981, 8208, 2335, -6587, 6043, -2277, -8173, 8415, -1886], [12330, 3891, -3161, -1577, 2457, -4616, -5648, 5537, 3259], [38424, 13385, -6223, -8176, -850, -3646, 3407, 2582, -843], [12541, -2502, -5983, -958, 2482, 6039, -1390, -4304, 1122], [-22877, 5998, -2934, 4501, -4605, 3753, 5799, -1858, 8045], [552, -592, -5152, -5085, 7165, 2572, -6609, -2603, 5278], [25372, 3674, -1927, 4324, 2436, 3246, -3458, -219, -499], [-50527, 7866, -7119, -2620, -1464, 4141, 2712, 3254, 2410], [18483, -4854, -1377, -3480, 2785, -1707, -8287, -1807, 2323], [-12595, -14816, 2006, 3127, 6316, 3454, -4532, -4180, -971], [25707, 4049, -7844, -7721, -3849, 3563, 3038, -5025, 9357], [16297, 4042, 502, 5753, -2085, -3011, -3677, -1498, 2421], [16615, 837, -1066, 2053, -3086, -1675, 3587, -6058, 1382], [-180, -5683, 260, -3022, -2873, 6392, 1084, 2556, 6505], [12419, -646, -3048, 894, 284, -4269, -4280, -2189, 4767], [14149, 8086, 4773, -4769, 4927, 371, -4479, 3359, -11024], [9704, 3515, -2675, -3459, 3365, 5076, 7264, 2017, -937], [42159, 4448, -4804, -2253, 1749, -8585, -2729, -1150, -1599], [25268, -1242, 3646, -822, -8982, -6266, -5152, -528, 8172], [20240, 7166, -2626, 6566, 4042, -4056, -748, -3926, 2358], [-16748, 6537, 1848, 3034, -1504, 800, -1613, -1753, -7832], [31798, 4219, -850, 3048, -3354, 1558, -2699, -4553, 8122], [-7239, 15738, -5028, -3890, 1263, -5100, -8713, 12945, 961], [-56129, 15997, -5529, -1902, -3782, 2727, 14741, -2233, 884], [-15433, -3028, -4749, 1172, -1945, -3589, 7605, -2035, 3351], [15870, 1860, 9408, 5721, -2948, 3575, -1034, -1955, -4760], [-6561, -6462, 1252, -5662, -5386, 75, 3088, -2439, 279], [14752, -4758, -675, 3773, -4635, -9266, -2489, -162, 7861], [6856, -11391, 6147, 7971, -669, 2532, -222, -4731, -2577], [24590, -1049, 190, 2727, -6046, -9026, -9157, 1961, 5323], [10231, 18195, -2860, -7758, -1388, 1509, 233, 6766, -393], [-7319, 2608, 3921, 7736, 7817, 2869, 383, 3307, -5923], [-37787, 9461, -3155, -3645, -3360, -1206, -5343, 4134, -9942], [10095, -4242, 5152, -7392, -1540, 1537, 2442, -5135, 6131], [-32768, 10629, -8041, -11740, -3747, 4765, 90, 1150, 9033], [8711, -15638, -3067, -1255, -8709, 5759, 2261, -65, 8368], [-13649, 4400, 3669, -5346, -1055, -1257, -10745, 663, 4863], [-48937, -8015, 3290, 3361, -3675, 3763, 10132, -7813, -4521], [-14053, 18882, 298, 3083, -5407, -5037, -7759, 5354, -2839], [58345, 3164, -4379, -2246, -7118, -1105, -704, 5060, 5670], [-44256, 7984, 1826, 3601, 4193, -2666, -1231, -1773, -5604], [2119, 376, 3540, -3302, 1002, -3655, -1496, -27, 5901], [-4209, -5798, 1176, -5284, 7148, -2923, -12036, 9341, -6909], [-5768, 13516, 2108, 1510, 3246, -2750, 12122, 360, -13350], [36354, -17294, 3112, -1603, -467, 3678, 2351, -5717, 7118], [-31825, 12806, -4885, 3969, -5162, 6673, 8064, 2694, -7033], [-14490, 16385, 3339, 8546, 1716, -11874, -165, 8462, -7023], [-7736, 8871, -10151, -10502, 2818, 6270, 11953, -720, 6483], [-6757, 7072, -7336, 1183, -5271, -11897, 319, 1364, 1651], [16219, 13030, 5044, 2471, 12, -3656, 5746, 6518, -5187], [-13001, -10617, 8826, 7150, -2286, 6813, -1461, -866, 2577], [-5582, -19684, -7172, 2937, 6175, 7304, -845, -3167, -7350], [7611, -2632, -1440, 416, 5272, 16601, 1050, -9833, -2355], [48527, 12930, -10882, -35, 1247, 2055, 14095, -3478, 435], [-12239, 16240, 354, 7311, 963, 1857, -359, 6932, -6971], [10766, -7270, 1828, 1027, 5786, 19133, 1662, -569, -3008], [-18458, 14157, -3498, -3277, -4836, -4322, -4661, 117, -3595], [9551, -14075, -1111, 6055, -12164, 2012, 11901, -578, -534], [-53417, 526, 6908, 5716, -6612, -9967, 494, 8001, -3507], [259, 5128, 3358, 12599, -2590, 2790, 5850, -2810, -16617], [-43204, 1162, 8548, 2256, 1822, -8550, -2976, 3950, -5232], [3768, 5355, -10331, -1986, -3224, -8148, -4117, 12612, 3023], [-18577, -19984, -2664, -4478, -11172, -1875, 1806, -2525, 7889], [-41558, 5181, -4537, 3222, -8074, 15414, 10021, -4196, -765], [40706, 8850, -5098, -8468, -2250, -3254, 3376, 2500, 8438], [16270, -5378, -6425, 5515, -1889, -1214, 4581, -3266, 4427], [36278, 1797, -173, 6460, 7898, 12348, 807, -1858, 517], [-2127, -163, -4082, 4092, -5040, 4999, 11047, 3520, -11457], [34581, -7562, -3126, 3261, 3987, -4311, -9293, -11878, 1579], [-48354, 2900, 8349, 2529, -6155, 1596, 13481, -7318, 7883], [-32404, -4269, 323, -1240, -1736, 9136, 7589, -5208, -5837], [-49551, 10172, -14650, -7887, 3131, -3979, 1749, 6522, 6613], [-68981, 15163, 4021, -4095, 8162, 7000, 151, -6564, -10818], [-27235, -22258, -924, -7131, 8286, -2, -3320, 3054, 9883], [41259, -5333, 2569, 5075, -4075, 7969, -1876, 11226, -8428], [-24487, 32878, -9500, -9323, 1367, -9577, -4977, 12124, 3013], [-60758, -5112, 2430, -7374, -3071, -2927, 315, 14928, 7366], [9625, 22055, 1911, -847, -4670, -5062, 4029, 6058, -7710], [-8208, 6349, -17973, -10763, 942, 10743, 8625, -5453, 6289], [34623, 2426, 3543, 5534, 1985, -191, -6355, -9845, 4655], [-54537, 6293, 5979, -1848, -10816, 7967, -5147, 5871, 1610], [-19704, 7552, 1912, 4082, -3020, -3258, 2644, -16618, 4460], [-14761, 7930, 8808, 7896, -5528, 2915, 8620, -14081, -423], [80248, 13079, -5758, -2818, -7136, -8732, 5703, -1213, 2012], [-31061, 2283, -6631, -11358, -6222, -8363, 3101, -3935, 12724], [-21843, 11331, -11788, -10476, -553, -1567, 650, -125, 2174], [-7488, 18114, -9020, -1412, -4237, 8363, -993, 196, 2974], [49985, 20066, -8130, -4071, 5367, -5125, -5361, 10304, -1839], [-18850, 486, 9450, 9050, 3918, 10716, 8296, -16734, -11662], [15781, -6329, 2643, 11766, -1860, 8017, -707, 2329, 13912], [-32797, -14619, -6673, -2313, -12403, 5513, 1956, -1908, 15442], [2405, -17755, 6416, 4369, -9708, -7017, -8481, -3248, -1224], [-91772, 2174, 2312, -6220, -1990, -2262, 4360, -1866, -9084], [-45, 33898, -8906, -7426, -3708, -7919, 3916, -7185, 13359], [99660, -10512, -1156, 10037, 45, -2928, -9095, 4877, 9868], [37559, 3480, -3306, 8487, 9851, 2921, -7575, -270, -2109], [78631, -14932, -3366, 4257, -5249, 2435, 3637, -12332, 5425], [-11543, -25597, -6490, -10568, 4873, 9117, 11966, -8007, 6922], [-8855, 2733, 1355, -10408, -4920, 5265, -6323, 4019, 9682], [12800, 6741, 5272, 1755, -981, -7766, -3178, -2844, -300], [7726, -25022, 15649, 13006, 1091, -5646, 2359, -11170, 822], [-76146, 11168, 1349, 19116, 2998, 478, 9648, 3205, -1817], [8045, 5388, 4514, 4932, 10755, 4868, -8213, -7863, -2799], [36330, -7651, -14672, -2243, 329, 5826, 13858, -1174, 10538], [22333, -19148, 2880, 1427, 8809, 24055, 2751, -11394, -3741], [11855, 4450, 1732, 12902, 10792, 2197, -2724, 4901, -18791], [-32807, 18662, -12010, -3225, -17148, -300, 18270, -960, 8787], [-79865, 18044, 9757, 12155, 1939, -5823, -2701, -5345, 3138], [-47839, -981, -9293, -11357, 3157, 2271, 15148, -13422, 4284], [33690, 6456, 12467, 1981, 6069, 232, -851, 12080, -13915], [-22107, -10014, -6480, 199, -2535, 7999, 6625, -9730, 15931], [-21011, -7071, 8050, 9981, -2722, 5615, 335, 1008, -13040], [47478, 7782, 17250, 13451, 4811, 11142, -3977, -4593, -16180], [17689, -13180, 2235, -7959, 1162, 9593, 1268, -1208, 2743], [-52523, -7720, 13578, -3554, -7167, -2, 1663, -8269, -2973], [-109910, 10011, -7421, 498, -328, 9744, 11053, -1610, -1293], [17240, -18469, 11715, -1605, 15744, 3903, -4813, -16985, 4265], [-55461, 936, -10818, 7389, 4511, -2771, 9941, 5918, -6123], [4795, -5607, -1185, 2264, 12256, 1021, -7327, 8371, -11770], [33009, -13316, 4150, -2197, -11467, -6499, -3897, -1924, -433], [11806, 2186, 3279, -8805, -3360, 1004, 2046, 2873, 5634], [103641, 3915, -6959, -7650, 1436, -5301, -5909, -1921, 8598], [-48995, 14185, 8081, 7404, -1720, -26581, -8535, 9927, -1662], [-31341, -3471, 9725, 25431, 6849, -10203, -4116, -7062, -12800], [29722, -9956, -414, 1866, 3219, 9511, -423, -20042, 7854], [-10018, -14331, -258, -3535, 7221, -3804, -14664, 4124, -2026], [27531, 5590, 4869, -6579, 12596, -15153, -8204, 6760, -13661], [89826, -39170, 10116, 7404, 3978, -10118, 6276, -11534, -1668], [-18500, 14738, -632, -3164, 4384, 6936, -8050, 210, -4988], [11959, -1714, -9984, -14643, 5248, 11950, -10794, 4160, 5837], [-61868, 14291, 6174, -567, -5752, 2532, -4929, -3127, 8419], [-58077, -1744, 4266, 5208, -8606, 7857, -12478, 13847, -8147], [42058, 4019, -11325, 4861, -12733, 378, 10592, -2377, 11099], [-16513, -13722, -10038, 5815, -4072, 4758, 13810, -6646, -9687], [12080, 27717, -12831, -1349, -243, -7344, 7880, -4419, 14863], [10685, 9084, 11452, -5112, -3930, -12983, -13790, 3697, 13183], [-10321, 2162, -1333, 6747, -3289, 9173, -10943, 19115, -2456], [-38935, 631, -17285, 9985, -7053, 14983, 15358, -14152, 6670], [-2701, 2070, -2316, 12440, -4771, 9766, 20489, -17129, 3931], [-84073, -1045, -14496, 12318, -3256, 6761, 4717, 7290, -11845], [74644, 17911, -11431, 4389, -2703, 8892, 17530, -10217, 1113], [-65227, 8130, -15816, -12073, 7807, 8597, 13687, -4362, 2765], [-7290, 41248, -11148, -5996, -9438, -24984, -11334, 14378, 15014], [41747, -4582, -14618, -8494, 5890, 735, -14122, 1807, 11731], [-18808, -12793, -11238, -3074, -7449, -1348, 12092, 7034, 10657], [-17659, 6903, 10211, 6886, 1419, -28930, -8962, 8793, 2244], [16977, -15244, 18008, 11287, 12332, 2112, 9808, -2404, -11935], [54761, -13500, 7126, 2395, -4439, -1417, 10091, -9992, 2809], [-37497, -13166, -7353, -13839, 11099, 10567, 9055, 1481, 8546], [12845, -7998, -456, -9605, -10145, -1691, -9, -7214, 8245], [39417, -5530, -4770, 1152, -4586, -2943, 4718, 6209, 8151], [-29527, -17152, -5090, 14158, -7331, -14322, 1371, -7535, -1729], [-64080, 8206, -4594, 2609, -285, -17600, -13073, 4989, 3652], [-5479, -22091, 6769, -1332, -7756, -12821, -1357, 7205, -4624], [13343, -8516, 25509, 18483, -3280, -6179, 12008, -2320, -8383], [64227, -7027, 3165, -946, -15858, 13881, 5391, -3719, 2086], [60915, -27353, -2377, -1449, -165, 7475, 16760, 1130, -20058], [-81325, 13082, 10414, 6830, 15606, 5455, -22514, -3285, -8793], [-102642, -5079, 19799, -7629, 5848, 7739, -1503, 2327, -7925], [42273, -7101, -1873, -2687, 9041, 10235, 7332, -19658, 3564], [3295, -24478, -14313, -3580, -976, 17427, 13348, -7032, 12550], [1917, 9261, -935, -17313, 7143, -9429, -17460, 3936, 1842], [-29021, -31207, -6695, 1638, 6826, 6785, 7451, 9479, -13936], [61906, 10321, -689, 12008, 10600, 9608, 12949, 6066, -18271]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_507_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_507_4_a_q();". // 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_507_4_a_q(:prec:=9) chi := MakeCharacter_507_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_507_4_a_q();". // 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_507_4_a_q( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_507_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![-9464, -3640, 8946, 1303, -2521, 8, 251, -20, -8, 1]>,<5,R![15899689, -65644332, 23660364, 19041401, -2089839, -130261, 18187, -28, -41, 1]>],Snew); return Vf; end function;