// Make newform 1849.4.a.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1849_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_1849_4_a_e();". // 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_1849_4_a_e();". // 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 := [-35688, 0, 39376, 0, -11252, 0, 1289, 0, -62, 0, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [6504, 0, -19240, 0, -22388, 0, 2915, 0, -71, 0], [0, -633336, 0, 407096, 0, -54692, 0, 2447, 0, -35], [0, 441304, 0, -264984, 0, 43924, 0, -2603, 0, 47], [0, 752376, 0, 336328, 0, -110332, 0, 8017, 0, -157], [0, -866264, 0, 286232, 0, -43924, 0, 2603, 0, -47], [317640, 0, -565640, 0, 121052, 0, -7913, 0, 149, 0], [420056, 0, -264984, 0, 43924, 0, -2603, 0, 47, 0], [-731976, 0, 597512, 0, -121052, 0, 7913, 0, -149, 0]]; Rf_basisdens := [1, 1, 63744, 31872, 21248, 63744, 21248, 31872, 10624, 31872]; 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_1849_a();" function MakeCharacter_1849_a() N := 1849; order := 1; char_gens := [3]; v := [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_1849_a_Hecke(Kf) return MakeCharacter_1849_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, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, -4, 0, 0, 0, 0, -1, -4, 2], [16, 0, -4, 0, 0, 0, 0, -1, -3, -3], [33, 0, 5, 0, 0, 0, 0, -11, -2, 0], [0, -8, 0, 1, -2, 3, -2, 0, 0, 0], [46, 0, -9, 0, 0, 0, 0, 14, 7, 7], [0, -14, 0, 6, -7, -8, -2, 0, 0, 0], [8, 0, -7, 0, 0, 0, 0, -25, -5, -1], [0, 3, 0, -3, 21, -6, -4, 0, 0, 0], [82, 0, -19, 0, 0, 0, 0, 22, 27, 5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-129, 0, -11, 0, 0, 0, 0, 2, 25, -17], [129, 0, 12, 0, 0, 0, 0, 9, -44, -16], [71, 0, 6, 0, 0, 0, 0, 74, -11, 1], [0, -83, 0, 9, 12, 16, 2, 0, 0, 0], [321, 0, 58, 0, 0, 0, 0, 51, 44, 6], [0, -76, 0, 2, 106, 6, -4, 0, 0, 0], [0, -69, 0, 9, 46, 10, 2, 0, 0, 0], [-143, 0, 45, 0, 0, 0, 0, -67, 29, -25], [608, 0, 14, 0, 0, 0, 0, -27, 47, 33], [0, 111, 0, -19, 68, -18, -22, 0, 0, 0], [-13, 0, -125, 0, 0, 0, 0, 96, 32, -10], [-409, 0, 76, 0, 0, 0, 0, -9, 18, 94], [-251, 0, 105, 0, 0, 0, 0, 72, 74, -10], [75, 0, 144, 0, 0, 0, 0, -96, -71, -67], [90, 0, -172, 0, 0, 0, 0, -53, -85, 95], [0, 10, 0, 18, -20, -48, 0, 0, 0, 0], [329, 0, 115, 0, 0, 0, 0, -133, -176, -80], [0, 257, 0, 45, 25, -28, 0, 0, 0, 0], [0, -412, 0, 16, 150, 4, 8, 0, 0, 0], [522, 0, -136, 0, 0, 0, 0, 95, 9, -61], [0, 277, 0, 35, -105, 82, 0, 0, 0, 0], [0, 21, 0, -39, -52, -68, 48, 0, 0, 0], [0, -10, 0, 29, 110, 9, 22, 0, 0, 0], [0, 417, 0, -12, 44, -83, -26, 0, 0, 0], [70, 0, -162, 0, 0, 0, 0, 83, -87, -49], [476, 0, -120, 0, 0, 0, 0, -112, 226, 230], [0, -305, 0, -14, -272, 121, -22, 0, 0, 0], [1211, 0, 88, 0, 0, 0, 0, 96, -77, 79], [0, -586, 0, 60, 200, -86, -20, 0, 0, 0], [-229, 0, -285, 0, 0, 0, 0, -152, -224, -230], [1031, 0, 168, 0, 0, 0, 0, 136, 179, -129], [0, 98, 0, -40, 230, -202, 40, 0, 0, 0], [0, -313, 0, -41, -83, -84, 52, 0, 0, 0], [0, 77, 0, -95, 150, 112, 80, 0, 0, 0], [0, -170, 0, 93, 526, -119, 6, 0, 0, 0], [-444, 0, -22, 0, 0, 0, 0, 533, 151, 231], [0, -240, 0, 72, -30, -88, 36, 0, 0, 0], [2923, 0, -389, 0, 0, 0, 0, 91, 7, -27], [0, 214, 0, 76, 358, 110, -12, 0, 0, 0], [-573, 0, -252, 0, 0, 0, 0, -627, -2, 156], [0, -231, 0, -1, 122, -46, 62, 0, 0, 0], [0, -994, 0, -94, -92, -40, -12, 0, 0, 0], [926, 0, -62, 0, 0, 0, 0, 469, -233, 91], [-807, 0, -525, 0, 0, 0, 0, 404, -116, 172], [0, -297, 0, -52, 396, 39, -44, 0, 0, 0], [-3610, 0, -39, 0, 0, 0, 0, 737, 287, -3], [1654, 0, -262, 0, 0, 0, 0, -125, -233, 161], [1935, 0, 358, 0, 0, 0, 0, -22, -277, -369], [1950, 0, 404, 0, 0, 0, 0, -111, -181, -427], [3121, 0, 38, 0, 0, 0, 0, -259, -542, -276], [0, -1667, 0, -91, 636, 88, 26, 0, 0, 0], [998, 0, 84, 0, 0, 0, 0, 223, -303, 353], [0, 1509, 0, -127, -486, 392, -36, 0, 0, 0], [-2098, 0, -21, 0, 0, 0, 0, 475, -259, 423], [0, -161, 0, 64, -208, 119, -18, 0, 0, 0], [0, -1346, 0, -10, -75, 544, -90, 0, 0, 0], [-792, 0, 262, 0, 0, 0, 0, -289, -685, 139], [-1622, 0, 201, 0, 0, 0, 0, -553, -527, -343], [-1381, 0, 895, 0, 0, 0, 0, 742, 129, 175], [0, 1148, 0, -71, -62, 251, 58, 0, 0, 0], [1091, 0, -420, 0, 0, 0, 0, 441, -256, 430], [0, -824, 0, -108, 302, 208, -44, 0, 0, 0], [0, -1182, 0, -168, -294, -330, 16, 0, 0, 0], [6529, 0, 436, 0, 0, 0, 0, -142, 85, 17], [132, 0, 745, 0, 0, 0, 0, 243, 763, 597], [0, 547, 0, -175, 230, -98, 30, 0, 0, 0], [0, 385, 0, -83, 466, 152, 216, 0, 0, 0], [0, 650, 0, 373, -306, -167, 34, 0, 0, 0], [5161, 0, 279, 0, 0, 0, 0, -283, -1346, -474], [0, -91, 0, 217, 296, -472, 166, 0, 0, 0], [-2914, 0, 1019, 0, 0, 0, 0, -751, -213, -697], [915, 0, -542, 0, 0, 0, 0, 38, -827, -819], [0, -435, 0, 395, -30, -170, 30, 0, 0, 0], [0, 1351, 0, -133, -66, -272, 186, 0, 0, 0], [5421, 0, -798, 0, 0, 0, 0, 1026, 929, 709], [0, 1037, 0, -251, -884, 296, 52, 0, 0, 0], [0, -1971, 0, 145, -15, 524, -140, 0, 0, 0], [-1728, 0, -917, 0, 0, 0, 0, -1040, 443, -553], [9122, 0, -847, 0, 0, 0, 0, -261, 640, -214], [0, -1041, 0, -224, -1052, -405, -22, 0, 0, 0], [0, -912, 0, -320, 815, 608, -60, 0, 0, 0], [0, 260, 0, -390, -94, -286, 56, 0, 0, 0], [1319, 0, 464, 0, 0, 0, 0, -583, 596, -616], [0, -1959, 0, -29, 358, -358, 298, 0, 0, 0], [0, -1558, 0, 494, -613, -592, -48, 0, 0, 0], [-5795, 0, -454, 0, 0, 0, 0, 1535, -90, -358], [-166, 0, 228, 0, 0, 0, 0, 2205, 557, 851], [-4554, 0, 40, 0, 0, 0, 0, -457, 1001, 65], [5531, 0, 192, 0, 0, 0, 0, 1235, 1478, -376], [-6207, 0, 134, 0, 0, 0, 0, -53, -1016, -1208], [0, -2552, 0, 432, -1269, 180, -104, 0, 0, 0], [0, 291, 0, 195, 430, 496, -10, 0, 0, 0], [0, -230, 0, 118, 2432, 200, -48, 0, 0, 0], [0, 2419, 0, 583, -1576, -100, 274, 0, 0, 0], [-4199, 0, -2003, 0, 0, 0, 0, -199, -894, -270], [0, -3100, 0, -230, 1620, -90, -260, 0, 0, 0], [0, 1062, 0, -210, -502, 692, -380, 0, 0, 0], [-6177, 0, 2018, 0, 0, 0, 0, -928, -1549, -369], [-2447, 0, -1211, 0, 0, 0, 0, 1598, -1158, -12], [-9988, 0, -932, 0, 0, 0, 0, -700, -832, -1108], [0, -746, 0, -256, -1388, -2, 232, 0, 0, 0], [0, -1577, 0, 195, 1770, -412, -270, 0, 0, 0], [4099, 0, 58, 0, 0, 0, 0, 2920, 2157, 1001], [0, 336, 0, 288, 1062, 136, -8, 0, 0, 0], [0, -4096, 0, 227, -350, -683, 70, 0, 0, 0], [4217, 0, 66, 0, 0, 0, 0, 2347, 1536, 1038], [4703, 0, 62, 0, 0, 0, 0, 1555, 2474, 554], [0, -2005, 0, -529, -952, -272, -102, 0, 0, 0], [0, 472, 0, -275, 50, -329, -190, 0, 0, 0], [11151, 0, -608, 0, 0, 0, 0, 1455, 1068, -206], [0, 3899, 0, -217, -486, 92, 204, 0, 0, 0], [-4193, 0, -1142, 0, 0, 0, 0, 1318, 139, -1369], [-10542, 0, -1270, 0, 0, 0, 0, 1719, 391, 991], [2994, 0, -221, 0, 0, 0, 0, 644, -74, 548], [0, -3948, 0, -254, -2392, -266, 188, 0, 0, 0], [0, 2063, 0, -95, -725, 98, -620, 0, 0, 0], [0, -1381, 0, 91, 513, 1340, -352, 0, 0, 0], [0, 1336, 0, 672, 530, -952, -104, 0, 0, 0], [0, -2735, 0, -247, 1844, -772, -56, 0, 0, 0], [0, 3455, 0, 373, 2161, -434, 216, 0, 0, 0], [0, 150, 0, 412, -624, -398, 756, 0, 0, 0], [18263, 0, 1293, 0, 0, 0, 0, -542, 259, -1363], [0, 2474, 0, 486, -723, -924, -378, 0, 0, 0], [895, 0, -2446, 0, 0, 0, 0, -1946, 689, 433], [-6205, 0, -80, 0, 0, 0, 0, 790, -2045, -245], [13623, 0, 1547, 0, 0, 0, 0, -74, 1583, 845], [0, 2145, 0, 476, -1932, 801, -102, 0, 0, 0], [-8392, 0, 312, 0, 0, 0, 0, -1199, -1869, -729], [-9381, 0, -870, 0, 0, 0, 0, -3003, -2076, 70], [16596, 0, 1504, 0, 0, 0, 0, 1562, 1640, -912], [0, -255, 0, -1097, 219, 418, 124, 0, 0, 0], [0, -4977, 0, -221, 2272, -148, 212, 0, 0, 0], [18365, 0, 778, 0, 0, 0, 0, -3267, -372, 1276], [14558, 0, 477, 0, 0, 0, 0, -499, -50, 1744], [0, -306, 0, 51, -4022, 615, 318, 0, 0, 0], [0, 3986, 0, -280, -374, 530, -64, 0, 0, 0], [7579, 0, 74, 0, 0, 0, 0, 1111, -2572, -632], [-1220, 0, 1119, 0, 0, 0, 0, -3793, -1757, -1499], [10839, 0, 1086, 0, 0, 0, 0, -3017, -2960, -2538], [0, -2862, 0, -628, -2974, -10, 96, 0, 0, 0], [-4690, 0, -1258, 0, 0, 0, 0, -3413, -4363, 319], [0, -2351, 0, -691, 1062, 848, 372, 0, 0, 0], [15905, 0, 667, 0, 0, 0, 0, 237, -4610, -190], [0, -60, 0, -642, 2394, -182, 24, 0, 0, 0], [0, 6853, 0, -537, -2494, 714, -134, 0, 0, 0], [7696, 0, -2096, 0, 0, 0, 0, -717, -2593, -1097], [12124, 0, 1208, 0, 0, 0, 0, 2245, -1833, 1601], [0, 1187, 0, 215, 164, -1472, 234, 0, 0, 0], [-14855, 0, -477, 0, 0, 0, 0, -2082, -522, -54], [-2103, 0, 680, 0, 0, 0, 0, 1363, 5138, 2292], [-7199, 0, 3489, 0, 0, 0, 0, 2242, -55, -1877], [0, 4439, 0, 665, -820, 1742, 220, 0, 0, 0], [0, 4947, 0, 501, 3198, -702, 188, 0, 0, 0], [0, 860, 0, 168, -4811, 492, -546, 0, 0, 0], [0, 3712, 0, 628, -2766, 2280, 124, 0, 0, 0], [14626, 0, -2736, 0, 0, 0, 0, 1342, -3800, 1248], [0, -2625, 0, -189, -2367, -1064, 128, 0, 0, 0], [0, 1830, 0, 327, -2714, -1613, -794, 0, 0, 0], [0, 1095, 0, -679, -1692, 1826, -752, 0, 0, 0], [-3413, 0, 2327, 0, 0, 0, 0, 3338, -66, -2496], [0, -370, 0, 430, -610, 1200, -180, 0, 0, 0], [4087, 0, 1619, 0, 0, 0, 0, -403, 953, 1283], [0, 4421, 0, 173, -2601, -1376, -176, 0, 0, 0], [0, 5533, 0, 158, -4536, 1591, 24, 0, 0, 0], [18505, 0, -2784, 0, 0, 0, 0, -4754, -269, 807], [0, -3830, 0, -1001, -818, -201, 82, 0, 0, 0], [0, -6667, 0, 217, 908, -2640, 276, 0, 0, 0], [23393, 0, -4260, 0, 0, 0, 0, -1478, 587, -1397], [0, 8885, 0, -1183, 3416, -1028, 126, 0, 0, 0], [0, -873, 0, -961, 5390, 368, -170, 0, 0, 0], [0, -1075, 0, 115, 2940, 1038, -40, 0, 0, 0], [0, -4520, 0, -178, -4314, -498, 596, 0, 0, 0], [0, -2665, 0, -2, 1692, -167, 1144, 0, 0, 0], [0, 5188, 0, 584, -1569, -2172, -424, 0, 0, 0], [10333, 0, -2311, 0, 0, 0, 0, 518, -495, 2919], [0, 4293, 0, -29, -1042, 1374, -1172, 0, 0, 0], [6860, 0, 351, 0, 0, 0, 0, 4611, 1791, 3537], [0, -4029, 0, 542, 7220, 293, 666, 0, 0, 0], [17837, 0, 680, 0, 0, 0, 0, 1918, 2743, -1153], [0, -898, 0, -1771, -2698, -279, 562, 0, 0, 0], [0, -4317, 0, -1365, 78, 368, 540, 0, 0, 0], [0, -3834, 0, 972, 5370, 218, 256, 0, 0, 0], [-14237, 0, -1045, 0, 0, 0, 0, 3950, 2754, 840], [21524, 0, -44, 0, 0, 0, 0, 5213, -1985, -33], [-4209, 0, -4567, 0, 0, 0, 0, -9429, 9, -2009], [0, -229, 0, -571, 316, -970, 352, 0, 0, 0], [19607, 0, -6396, 0, 0, 0, 0, 3029, -1462, -318], [0, 6512, 0, 802, -4134, -1926, -924, 0, 0, 0], [0, -3562, 0, 652, -1742, -2222, 388, 0, 0, 0], [0, -1894, 0, -1034, 1838, -2268, -552, 0, 0, 0], [0, 4844, 0, -768, -1809, -884, 736, 0, 0, 0], [0, -6293, 0, -1471, 267, 550, -748, 0, 0, 0], [0, -7768, 0, -1650, -300, 1782, 40, 0, 0, 0], [1246, 0, 4288, 0, 0, 0, 0, -309, -6751, -1309], [0, 198, 0, -556, -28, 1022, 1112, 0, 0, 0], [-29150, 0, -416, 0, 0, 0, 0, 365, -1335, 683], [0, 3412, 0, 1202, 7606, 914, -404, 0, 0, 0], [8719, 0, 1146, 0, 0, 0, 0, 3097, 3778, -3078], [11708, 0, 3966, 0, 0, 0, 0, -6485, -2981, 1077], [-19301, 0, -5382, 0, 0, 0, 0, 85, 142, -4404], [0, -1727, 0, -467, -526, -924, -836, 0, 0, 0], [2039, 0, 2071, 0, 0, 0, 0, 3323, -1774, -4596], [0, -4039, 0, 9, 4152, 444, 92, 0, 0, 0], [0, -1826, 0, -1110, -6830, 844, -20, 0, 0, 0], [0, -10045, 0, 1781, -6232, -614, 188, 0, 0, 0], [4868, 0, -4238, 0, 0, 0, 0, -1829, -2215, -3391], [0, 11337, 0, -736, 3292, 181, 52, 0, 0, 0], [-15391, 0, 4564, 0, 0, 0, 0, -174, -1385, 3847], [0, 3255, 0, 1643, 1744, -2572, 814, 0, 0, 0], [-8962, 0, -2697, 0, 0, 0, 0, 3852, 2401, 4101], [0, 8050, 0, 747, 2034, 1627, 686, 0, 0, 0], [-21385, 0, 3606, 0, 0, 0, 0, -5291, 5784, -2268], [15698, 0, -2618, 0, 0, 0, 0, -3749, -2875, 3829], [0, 11737, 0, -737, 7674, -150, -96, 0, 0, 0], [0, -5726, 0, -792, 5936, -4318, -496, 0, 0, 0], [0, -6140, 0, 1441, -7522, 2671, 178, 0, 0, 0], [0, -3376, 0, -556, -5763, 3420, -978, 0, 0, 0], [45626, 0, 2530, 0, 0, 0, 0, -8237, -2157, 2129], [4120, 0, -1293, 0, 0, 0, 0, -2990, 2821, 37], [0, 334, 0, 810, -3570, -1936, 60, 0, 0, 0], [16806, 0, -1380, 0, 0, 0, 0, 1343, 4801, 95], [8069, 0, 7347, 0, 0, 0, 0, -2386, -4609, -4371], [0, 2912, 0, 2014, -848, 326, -728, 0, 0, 0], [-9507, 0, -6724, 0, 0, 0, 0, 5715, 5054, 4762], [0, 6709, 0, 632, -2004, -2459, 546, 0, 0, 0], [13752, 0, -2427, 0, 0, 0, 0, 2058, 6043, -1569], [0, 8917, 0, 209, -357, 1720, -992, 0, 0, 0], [0, 6140, 0, 823, -7326, 1053, 1474, 0, 0, 0], [-838, 0, -1967, 0, 0, 0, 0, 3999, -900, -3494], [-16253, 0, 1350, 0, 0, 0, 0, -5522, -5621, 2175], [0, 1478, 0, -1916, 1932, 302, 52, 0, 0, 0], [34395, 0, -391, 0, 0, 0, 0, -459, 411, -1267], [0, 12280, 0, 340, 2500, -284, -180, 0, 0, 0], [890, 0, -5148, 0, 0, 0, 0, 3125, 1651, 2957], [39955, 0, -2744, 0, 0, 0, 0, 1015, -642, 36], [-6151, 0, -3541, 0, 0, 0, 0, 1146, 4198, 4398], [25293, 0, -7396, 0, 0, 0, 0, 5458, 5837, 953], [43356, 0, 926, 0, 0, 0, 0, -7157, -2809, -4573], [-37580, 0, -150, 0, 0, 0, 0, -5255, 2765, -1245], [0, 6506, 0, 748, 3404, -1386, 984, 0, 0, 0], [0, 752, 0, 2960, 5059, 1312, -1830, 0, 0, 0], [0, 11019, 0, -513, 981, 616, 916, 0, 0, 0], [0, 1792, 0, -2005, -750, -1383, -470, 0, 0, 0], [-4733, 0, -2526, 0, 0, 0, 0, 11205, 6456, 5798], [0, -3501, 0, 1179, 3204, -1944, -86, 0, 0, 0], [5782, 0, -739, 0, 0, 0, 0, -6698, 5063, 397], [0, -6736, 0, 1270, -1320, 2094, 1140, 0, 0, 0], [0, 225, 0, -22, -7556, 4383, -566, 0, 0, 0], [56402, 0, 1458, 0, 0, 0, 0, 4208, -5188, 2920], [6237, 0, 3190, 0, 0, 0, 0, 2176, -683, 1105], [0, 8021, 0, 15, -4552, 618, -1262, 0, 0, 0], [0, -13503, 0, -1547, -151, -2580, -56, 0, 0, 0], [0, -12749, 0, -391, 5127, 1530, 212, 0, 0, 0], [7564, 0, 9529, 0, 0, 0, 0, -2164, 2487, -2239], [0, 2808, 0, -420, -3820, 4940, -1040, 0, 0, 0], [28142, 0, 5494, 0, 0, 0, 0, -435, 6081, -287], [-30759, 0, 3962, 0, 0, 0, 0, 5725, -1546, 246], [0, 13385, 0, -2458, 1524, -61, 834, 0, 0, 0], [0, -11931, 0, -567, 9004, 304, -346, 0, 0, 0], [0, 215, 0, 973, -5036, 1478, 984, 0, 0, 0], [23629, 0, 177, 0, 0, 0, 0, 5954, 15221, 6979], [0, -1809, 0, 1651, 11924, 272, 868, 0, 0, 0], [-44289, 0, -3774, 0, 0, 0, 0, -471, 9052, -2238], [0, -5589, 0, 372, 1416, -1277, 736, 0, 0, 0], [-15096, 0, -6589, 0, 0, 0, 0, -10689, 1359, 4985], [0, 10037, 0, -1362, -4696, -4365, -46, 0, 0, 0], [-7892, 0, -3365, 0, 0, 0, 0, -3831, -947, 265], [-41074, 0, 8969, 0, 0, 0, 0, 3152, 1926, 3180], [-19121, 0, -4787, 0, 0, 0, 0, 7615, 1942, 4246], [0, 13076, 0, -1292, -4081, -4436, 1314, 0, 0, 0], [0, 1690, 0, 622, 6309, 1704, 644, 0, 0, 0], [0, 15443, 0, 99, -14878, -608, 1292, 0, 0, 0], [54988, 0, 4513, 0, 0, 0, 0, 9422, -4499, 2151], [0, -1166, 0, -902, 1127, 5512, -206, 0, 0, 0], [0, 6417, 0, 769, -2148, 2516, 1252, 0, 0, 0], [-32420, 0, 5253, 0, 0, 0, 0, 5216, 6931, 5121], [-37779, 0, 4668, 0, 0, 0, 0, -2769, -5540, -6252], [-1242, 0, 3236, 0, 0, 0, 0, 14335, 8429, 2287], [29225, 0, 5129, 0, 0, 0, 0, 11779, 11479, -27], [0, 3123, 0, -221, -5543, -1608, -188, 0, 0, 0], [31775, 0, -14648, 0, 0, 0, 0, 7427, 1382, 2534], [-9238, 0, 3126, 0, 0, 0, 0, -5355, 4609, -3863], [-18209, 0, -7959, 0, 0, 0, 0, 10970, 2894, 2742], [7780, 0, 10824, 0, 0, 0, 0, 1489, -4331, 3153], [-24769, 0, 2610, 0, 0, 0, 0, -3320, -5475, -7555], [-26796, 0, 1036, 0, 0, 0, 0, 10388, 2920, -2668], [-27427, 0, 11725, 0, 0, 0, 0, 309, 2138, -4260], [0, -3567, 0, 2788, 2736, 2861, -916, 0, 0, 0], [22922, 0, 10698, 0, 0, 0, 0, -4553, 10741, 2671], [47848, 0, -15970, 0, 0, 0, 0, -8861, 1323, 2665], [0, -16048, 0, -264, 6133, -3772, 288, 0, 0, 0], [0, 15860, 0, -2788, -1272, 3712, 1596, 0, 0, 0], [-29381, 0, -782, 0, 0, 0, 0, 495, 82, -2304], [0, 6738, 0, 911, -4862, 1939, 1478, 0, 0, 0], [0, -14535, 0, 2307, 4206, -3358, 2256, 0, 0, 0], [0, 1188, 0, -3843, 9282, -725, -474, 0, 0, 0], [0, 18866, 0, -804, 1364, -6218, 88, 0, 0, 0], [0, -5019, 0, 2113, -3836, 1064, 1234, 0, 0, 0], [45505, 0, -877, 0, 0, 0, 0, -12388, -1618, 3316], [0, -578, 0, 1481, 9122, 3593, -282, 0, 0, 0], [32168, 0, 203, 0, 0, 0, 0, 77, -3439, 1601], [20722, 0, 3227, 0, 0, 0, 0, 9044, -5313, 6665], [-32769, 0, -6634, 0, 0, 0, 0, 5634, 4085, 11553], [71323, 0, 8500, 0, 0, 0, 0, -6978, 3337, 4493], [-7752, 0, -9351, 0, 0, 0, 0, 11873, 217, -2637], [0, -13242, 0, -1580, -4050, -562, 280, 0, 0, 0], [-41859, 0, 4082, 0, 0, 0, 0, -14569, -4350, -2856], [0, 8942, 0, 1396, -1502, 3514, 1008, 0, 0, 0], [0, 2745, 0, -5369, 10303, -1134, -612, 0, 0, 0], [-45247, 0, 720, 0, 0, 0, 0, -8826, -8257, -5865], [0, -8228, 0, -422, -14786, 5190, -1236, 0, 0, 0], [-3201, 0, -16959, 0, 0, 0, 0, -2007, -3624, 1494], [0, -12723, 0, 1189, 42, 3776, -988, 0, 0, 0], [-22917, 0, -8996, 0, 0, 0, 0, -1827, -4718, 11748], [3506, 0, -5293, 0, 0, 0, 0, 8265, -10772, -2006], [0, -15564, 0, -2548, -139, -200, 846, 0, 0, 0], [0, 3307, 0, 4319, -2488, 956, 862, 0, 0, 0], [57610, 0, -2704, 0, 0, 0, 0, 221, 10991, 1667], [0, -711, 0, 651, 10038, -2410, -1152, 0, 0, 0], [0, 300, 0, -3020, -2725, -9392, -1200, 0, 0, 0], [64027, 0, -9340, 0, 0, 0, 0, 7534, -8339, -1807], [4632, 0, 3852, 0, 0, 0, 0, -8077, 5049, -4381], [0, -17514, 0, -1328, -9094, 398, 1216, 0, 0, 0], [0, -12140, 0, 1434, 15226, -26, 572, 0, 0, 0], [0, -16768, 0, -2552, 794, 7880, -496, 0, 0, 0], [0, -7295, 0, 281, 6148, -6324, 308, 0, 0, 0], [29613, 0, 15642, 0, 0, 0, 0, 1431, 6060, 1384], [0, -9291, 0, 3039, 7632, 2034, -2498, 0, 0, 0], [0, 5578, 0, -4936, 2322, 1062, -1328, 0, 0, 0], [0, -13225, 0, 1993, 8144, 898, -196, 0, 0, 0], [-33608, 0, 1786, 0, 0, 0, 0, 6627, 8903, 7427], [-4020, 0, -8280, 0, 0, 0, 0, -6825, 11183, -2039], [0, 2618, 0, -3782, -13321, -1904, 2474, 0, 0, 0], [-23372, 0, 2540, 0, 0, 0, 0, -17266, 5328, 1160], [0, 23096, 0, 3380, -8540, -404, -160, 0, 0, 0], [-86571, 0, 4492, 0, 0, 0, 0, -1411, 10546, 2754], [-28881, 0, 5000, 0, 0, 0, 0, -19552, -1855, -375], [0, -9631, 0, 1457, 18356, -2344, -2074, 0, 0, 0], [1259, 0, 4658, 0, 0, 0, 0, -22016, -10685, -8889], [0, 23561, 0, 2051, 4398, 4814, -2272, 0, 0, 0], [-56137, 0, -5158, 0, 0, 0, 0, 12690, 6007, 1783], [0, -21475, 0, -251, 3102, -736, 722, 0, 0, 0], [0, 5968, 0, 1940, -7770, -3612, 2120, 0, 0, 0], [0, -12191, 0, -851, -923, -592, -528, 0, 0, 0], [-37975, 0, -1705, 0, 0, 0, 0, 19635, 4480, 14050], [-3286, 0, -8431, 0, 0, 0, 0, -2779, -3757, 2263], [0, -9413, 0, -3991, 19599, 586, -648, 0, 0, 0], [0, 23659, 0, 1737, 9866, -2034, -114, 0, 0, 0], [0, -20479, 0, 751, 12396, -2326, -2724, 0, 0, 0], [0, 22762, 0, -1941, 2542, -669, -3578, 0, 0, 0], [8193, 0, 12186, 0, 0, 0, 0, -5235, 6954, -8988], [0, -2880, 0, -1644, -17106, -7260, -1756, 0, 0, 0], [0, -18830, 0, 1495, 4870, -3581, -870, 0, 0, 0], [-24777, 0, 3681, 0, 0, 0, 0, -13310, 10789, -3553], [0, -6085, 0, -771, 4462, -9506, 1262, 0, 0, 0], [0, -33554, 0, -653, 13706, -1377, -1314, 0, 0, 0], [0, -13272, 0, 3592, -4899, -4020, -1064, 0, 0, 0], [-6977, 0, -3633, 0, 0, 0, 0, 6711, 4055, -2631], [0, -15993, 0, 120, -2580, 7647, -1660, 0, 0, 0], [-119710, 0, -4909, 0, 0, 0, 0, -17423, 7987, -4353], [0, -34754, 0, -520, 16122, -406, 872, 0, 0, 0], [0, -26870, 0, 2607, -12554, 5387, 86, 0, 0, 0], [65659, 0, -272, 0, 0, 0, 0, 98, -14379, -7811], [-368, 0, -13289, 0, 0, 0, 0, -513, 5243, -9583], [0, 4885, 0, -1481, -5998, 4714, 2062, 0, 0, 0], [0, 19229, 0, 5905, 1918, 7616, 478, 0, 0, 0], [32619, 0, -6252, 0, 0, 0, 0, -14471, -4554, -4134], [-112404, 0, -15807, 0, 0, 0, 0, -9934, -17991, -2289], [-36533, 0, 4807, 0, 0, 0, 0, 5753, 19044, 4704], [0, 2883, 0, 3287, -15474, -1196, -354, 0, 0, 0], [-83355, 0, 1896, 0, 0, 0, 0, 3197, -7996, -10622], [150559, 0, -2995, 0, 0, 0, 0, 14387, -2606, 5850], [0, -19273, 0, -3647, 8274, 7830, 364, 0, 0, 0], [-89962, 0, -12966, 0, 0, 0, 0, -5347, -823, 1753], [-19127, 0, 21570, 0, 0, 0, 0, -13636, -11197, -10125], [88961, 0, 10534, 0, 0, 0, 0, 16627, 12080, 1018], [8474, 0, -14481, 0, 0, 0, 0, 13719, 2861, -1217], [0, 6008, 0, -1228, 11597, -10440, 2146, 0, 0, 0], [0, -10402, 0, -3582, -6236, 5676, 504, 0, 0, 0], [-138928, 0, 10230, 0, 0, 0, 0, -16029, 13247, -5255], [0, 17989, 0, -1489, -1902, 5470, 148, 0, 0, 0], [-51265, 0, 11179, 0, 0, 0, 0, -4331, 12549, -7817], [18197, 0, 1723, 0, 0, 0, 0, 307, -13464, 4996], [0, 17623, 0, 3031, -1002, -6316, -642, 0, 0, 0], [0, 2527, 0, -2500, -11460, -5113, -10, 0, 0, 0], [0, 34561, 0, -2446, -30588, 4975, 2912, 0, 0, 0], [-38757, 0, -12188, 0, 0, 0, 0, -12589, -932, 3280], [23725, 0, -4571, 0, 0, 0, 0, 4839, -6581, -8827], [68563, 0, -18521, 0, 0, 0, 0, -2242, 6172, 7148], [70783, 0, 5803, 0, 0, 0, 0, -1543, -21919, -10569], [0, -12933, 0, -2839, -20157, 1978, -1772, 0, 0, 0], [0, 15905, 0, -4541, 2652, -7226, 772, 0, 0, 0], [0, -26447, 0, -481, 6277, 2882, 2792, 0, 0, 0], [-71179, 0, 6837, 0, 0, 0, 0, -3742, -15703, -73], [0, -22188, 0, 6339, -19302, 2685, 518, 0, 0, 0], [-90450, 0, -4408, 0, 0, 0, 0, 2335, 4145, 16719], [-30255, 0, -6037, 0, 0, 0, 0, 11673, -12437, -5679], [0, 1364, 0, 1170, -10342, 54, 2220, 0, 0, 0], [0, -13183, 0, -3983, -3099, -4964, -1664, 0, 0, 0], [50756, 0, 16484, 0, 0, 0, 0, 9027, -6491, 1231], [0, -15707, 0, 5375, 2450, -9646, 1190, 0, 0, 0], [-10379, 0, 1704, 0, 0, 0, 0, -9413, 10846, 13150], [-94361, 0, -12070, 0, 0, 0, 0, 25313, 18292, 5906], [103483, 0, 18785, 0, 0, 0, 0, 6829, 9743, 13005], [39577, 0, 23464, 0, 0, 0, 0, -1770, 14801, -1927], [0, 36555, 0, 1559, 6036, 2180, -324, 0, 0, 0], [0, 13797, 0, -744, -17772, -87, -4152, 0, 0, 0], [27879, 0, -112, 0, 0, 0, 0, 15500, 14527, 6111], [0, -14017, 0, -6033, 3814, 7560, -2724, 0, 0, 0], [63105, 0, 14724, 0, 0, 0, 0, -16223, -8028, 11578], [0, 32654, 0, -754, 6678, 4108, -3812, 0, 0, 0], [0, 9624, 0, -1823, -14954, -3065, -1474, 0, 0, 0], [0, 1941, 0, -2015, -1479, 6680, -764, 0, 0, 0], [0, 26741, 0, -2455, -15230, -2304, 1450, 0, 0, 0], [-140228, 0, -10966, 0, 0, 0, 0, -8181, -10895, -2117], [0, -31, 0, 4427, 7876, -3894, -3064, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1849_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1849_4_a_e();". // 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_1849_4_a_e(:prec:=10) chi := MakeCharacter_1849_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_1849_4_a_e();". // 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_1849_4_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1849_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![-35688, 0, 39376, 0, -11252, 0, 1289, 0, -62, 0, 1]>],Snew); return Vf; end function;