// Make newform 1575.4.a.bo in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1575_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_1575_4_a_bo();". // 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_1575_4_a_bo();". // 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 := [-10, 30, 7, -18, -1, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0], [10, -20, 27, 3, -2], [70, -320, -84, 24, 4], [155, 140, -138, -12, 8], [-785, 100, 582, 18, -32]]; Rf_basisdens := [1, 15, 15, 15, 15]; 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_1575_a();" function MakeCharacter_1575_a() N := 1575; order := 1; char_gens := [1226, 127, 451]; v := [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_1575_a_Hecke(Kf) return MakeCharacter_1575_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], [7, 0, 0, 0, 0], [-15, -8, 0, 0, -1], [-1, -6, -1, 0, 1], [-18, 6, -1, 8, 2], [33, -2, 1, -6, 1], [27, 12, -3, 2, -1], [-72, 8, 2, 6, 0], [65, -26, -1, 8, -1], [-74, 24, 2, 10, 8], [-171, -32, -3, -20, -7], [-122, -16, 2, 2, -8], [-138, 36, 6, 10, -8], [-15, 14, -11, -14, 1], [-144, 28, 12, -28, 8], [118, -36, 16, 18, -10], [52, 64, -12, -2, -6], [-291, 16, 8, -2, 17], [-87, 102, 13, 24, -1], [-182, -96, 14, 36, -2], [-312, -48, 4, 20, -4], [-81, -72, 15, 8, -25], [83, 162, -1, -20, -7], [-507, -24, 1, -40, -35], [-338, 148, -2, -60, 10], [-261, -24, -7, 66, 39], [794, -88, 40, -52, -28], [27, -394, 27, -102, 7], [-494, 336, -2, 36, 38], [-294, 44, -56, -12, 46], [1365, -186, 19, -34, 1], [1103, 118, 41, 106, 7], [630, -200, 48, -54, -6], [92, -168, -50, 0, 56], [357, 66, -61, -96, -5], [-784, -376, -40, 30, 50], [-738, -324, -22, -206, -28], [-978, -150, -19, -2, 20], [-1791, 128, 32, -92, -29], [-960, 396, -82, 86, 12], [-393, -792, 12, 62, 23], [-1930, -516, 2, -68, -70], [-969, -342, -53, -186, 15], [-241, -114, 27, 64, -11], [-382, 188, 20, -68, -6], [-108, 392, -16, 4, 120], [-2610, -336, -54, -76, 54], [500, -412, 14, -14, -24], [-3153, -254, -11, 62, 23], [-1875, 584, -60, 322, -91], [16, -160, 138, -124, -6], [-1560, -220, 76, 380, 8], [-912, 130, 129, -16, -72], [-1527, 748, 27, -74, -3], [99, 944, 39, -180, 83], [727, -30, -11, -336, -7], [-3842, -608, -2, -378, 4], [1860, 440, 82, 112, -58], [1294, 436, -22, 308, -10], [-1650, 166, -89, -426, -20], [4058, 732, -30, 48, -90], [-1380, 1492, -102, -172, 40], [-1365, 222, 29, -124, 33], [-303, -74, 81, 254, -107], [3078, 2188, -96, 124, -66], [-3204, 16, -84, 244, 192], [1587, 720, -191, 342, -13], [-14, 1476, -4, 278, -186], [-2778, -38, -95, -48, -94], [-2061, -1104, 148, -350, -89], [3420, 104, -140, -192, 60], [-4638, 16, -66, 90, -64], [3212, -8, -178, 360, -132], [-1164, -1028, 216, -222, 94], [-888, 720, -246, 414, 92], [-3937, -2002, -35, -56, -215], [-3120, 680, -142, -140, 186], [5312, 416, -122, -468, 150], [-2718, 376, -14, 88, -142], [-426, 700, 32, 664, -8], [-2307, -936, -44, -986, -191], [-245, 2070, -143, 392, -131], [5103, 414, 55, -156, -103], [3807, -536, -79, -226, -33], [-1566, 1272, 120, -140, -184], [-7690, -364, 302, 312, 194], [735, 1352, 279, 224, -201], [1482, -1152, 254, 216, -14], [-1290, -672, -206, 208, -54], [-3408, -196, -110, -380, 244], [-3300, 384, -156, -896, 40], [-4767, 1072, -232, 436, 203], [2632, 920, 94, -16, -16], [780, 2844, -176, -250, 10], [-495, -712, 237, 596, -63], [-687, -2248, 65, 28, 185], [1156, -360, -84, 84, 228], [-2790, 2260, 328, 96, 40], [4360, -1272, 24, 158, 254], [81, 1686, 229, 862, 181], [1374, 2936, 106, 0, 126], [2076, -64, -50, -1324, 182], [-1644, 796, -300, -668, -168], [-1637, -3234, 269, 876, -47], [-4446, 752, 122, -184, 442], [9354, 230, -37, 608, -202], [711, 2568, 324, 518, 303], [-10154, -152, 300, -256, -224], [-672, -1528, 40, 148, 196], [-544, 1740, -344, 322, 594], [-153, -1054, -419, 578, 547], [4039, 742, 469, -130, 3], [-350, -1088, 442, 644, -90], [2760, -928, 342, 236, -410], [-2674, 3588, -258, 340, -170], [-7026, -2028, 66, -810, -256], [-3117, 318, 173, 1454, -125], [2067, 2536, 52, 352, 5], [-5874, -52, -4, -518, -498], [-4488, -784, -480, -452, -280], [3474, -390, -447, 670, -256], [2619, -3024, -51, 42, -361], [7993, -302, -77, -962, -679], [13596, -1320, -318, 1414, 12], [-4594, -1576, -20, 16, 508], [-7110, -4676, 454, 960, -242], [-6854, -1460, -522, -796, -50], [10115, -550, -277, -916, -479], [-5744, -3780, -188, -92, -12], [11187, -2764, -375, -170, 787], [-9690, -5188, 332, 112, 390], [13234, 360, -50, 346, -432], [27645, 1040, -55, -8, -163], [-2406, 128, -4, -1700, -432], [-6174, -466, 587, 734, 376], [3760, 5296, -544, 640, 52], [-17298, -1574, -371, 318, 20], [6942, -2368, -424, 1324, 360], [14203, 86, 105, 58, 515], [6810, 6256, -24, 674, 362], [802, 4152, -346, 920, -294], [-4101, 960, 37, -778, -485], [-1908, -4484, -658, -14, 100], [-690, -1904, 756, -356, -330], [711, 4894, -35, 1712, 369], [-20964, -3550, -231, -264, -44], [-9641, 3042, 827, 786, -725], [7095, -5260, -271, 558, 335], [268, 1132, 556, -186, 322], [4533, -4112, 365, -216, 545], [-8250, 1208, 74, 2730, 788], [-11238, -2236, 878, 342, 96], [-6394, 4104, 266, 730, -460], [4743, 6120, -92, -914, 15], [-11738, 6420, 528, -80, -818], [14157, -3400, -507, -16, 397], [5005, 5090, -813, 16, 227], [-3111, 1040, -271, -732, -31], [18573, -408, -461, -626, 633], [-5973, 5022, 155, -890, -181], [-16294, -1832, -114, 500, -638], [9840, -9224, 18, 1520, -460], [-573, 4778, 349, -386, -533], [-252, -4140, -804, 54, -114], [4030, 1532, -260, -952, 86], [-10001, -5154, -211, 240, 409], [-19142, -5444, 648, 508, -216], [-24054, 2806, 323, -1846, -364], [-4755, -1160, 516, -2072, -1053], [-20046, -2868, 180, 1810, 118], [-39351, 312, -376, 10, -359], [-6074, 3548, -838, -1844, -102], [-4439, 5274, 497, 1780, -289], [13179, 648, -757, 84, 427], [20762, -912, -450, 512, 446], [-4080, -2944, 134, -1562, 804], [8270, 2988, 378, -1692, -1614], [-2078, 8812, -148, -474, -870], [-15502, -2864, -314, -4280, -970], [744, -3880, -578, 816, 492], [12692, -1300, 956, -2246, 666], [-11322, 498, -131, -604, 1054], [-14667, 2620, -1489, 1414, 1001], [-6321, 5112, 351, 2376, -873], [28856, 4620, 60, 978, -6], [-6104, 1616, -160, 204, 1304], [-2222, -4196, -820, -860, 244], [-44586, -3852, 786, 144, 250], [-12485, -1662, 151, 3108, 473], [-9525, 2312, 921, 1714, -209], [9548, -3976, -1310, -48, 108], [12705, 456, -687, 5136, 457], [-15315, -4208, -1021, -3210, 513], [-35970, 990, 1111, -816, 530], [-18606, -7028, -780, 300, -1144], [-19792, -52, -200, 1798, 130], [-15492, 2354, 117, -72, 220], [1134, -8212, 54, -2986, 192], [-14370, 3536, 808, -3562, 170], [34039, -274, 107, 2360, -275], [26033, 7390, 557, 1680, 15], [-13700, -4008, -1450, -2752, -294], [12144, -2968, 814, -56, 92], [-4470, 9338, -1123, 450, 1076], [21457, -966, 29, 2804, 735], [23955, -1392, 1169, -2922, -989], [32778, -1784, -792, -1944, 1672], [7459, -3846, -53, -2398, 775], [17378, -6004, -410, 2548, 322], [30777, -4016, -187, -2156, -391], [-40868, -3880, -428, -904, -508], [-24048, 4128, -392, -1420, 292], [-1446, 12224, -448, 524, 930], [-12214, -3360, 552, -2076, -744], [18168, -416, -272, 5528, 300], [2463, 472, -1973, -1176, 219], [-6207, -1876, 623, 2126, -1299], [-7365, 118, -1123, 1006, -1029], [-4402, 1192, -1204, -1736, 1156], [-17351, 4406, -777, -1884, -29], [-13098, 120, 1028, -3300, 312], [-32400, 3456, 1248, 1156, 100], [41964, 9704, -684, 1444, 404], [35626, -2892, 1164, -2140, -1120], [-25932, 2314, -1175, 608, 1420], [15327, -1488, 780, 150, 791], [-37914, -6244, 174, 400, -394], [-25491, -10256, 1212, 1156, -105], [18462, -8024, 1466, -1018, -160], [-16299, -13410, 1137, -310, -763], [-1316, -11152, 262, -1384, -1312], [-15150, 768, 296, -2384, -2224], [-50506, 4548, 1106, -2000, -646], [7092, -3820, 108, 4154, 322], [220, 1312, 222, -1804, 874], [-2049, 8650, -1209, -722, 1959], [-18099, 12928, 132, 1156, -993], [13446, -6828, -1322, 80, 930], [-11643, 2224, 971, -474, 1161], [43147, -5410, -3, 2782, 23], [17040, -16168, 348, 1400, 312], [-13114, 8716, 2644, -2120, -364], [21528, -1190, -1655, 820, 884], [22206, -14600, -404, 684, 430], [-23806, -11260, -246, 1660, -1258], [12594, 13568, -902, 5912, 482], [23266, -14192, 1242, -1520, 174], [8751, 6004, 1653, 1214, -121], [-12404, -5428, -2984, -3794, 850], [41127, 6656, 255, 3088, 971], [-14667, 2692, -621, -6086, -371], [-4352, -12864, -422, -3224, 114], [-14394, -10186, -1729, -442, -692], [-60231, -12344, 624, 1012, 851], [-46178, -13604, 40, -5000, 248], [-8782, -6476, -126, -2044, 442], [28122, 4846, -81, -4266, -352], [39099, -8370, -27, 5224, -539], [38130, 6072, -490, 1168, 578], [7491, 15000, -547, 2310, 491], [10288, -4548, -1282, 4606, 800], [-25259, -8406, -149, 2684, 1255], [55098, -9018, 1431, -4908, -1398], [43897, 434, -741, 2854, 2009], [-25902, 2552, 244, 4950, -42], [-1611, 2720, 3129, 5748, -563], [53160, 14792, 504, 1170, 82], [-30993, -7506, -299, -4654, -885], [9006, -7236, 1200, -1038, -246], [52990, 12456, 3058, 3406, -300], [-23633, -3790, -169, -1472, -3567], [-15778, -13804, -1632, -784, -762], [36987, -5894, -1501, -2376, 453], [42494, 5036, -1906, -2040, 1118], [-30909, -14368, -1227, -2878, 1383], [-17578, -572, 2512, 204, -860], [-5038, 19908, 992, 1804, -1948], [-2364, 14972, 1656, 100, -1780], [-5472, 15524, 1496, 518, 1194], [39350, 488, 2206, 3892, -998], [63060, -11092, 428, -2654, -726], [12706, 3236, -2732, 4446, -366], [88468, -5528, 1632, -3732, -820], [-34905, -5296, 1048, -4798, -1789], [-7490, -764, 814, 3696, -1094], [15297, -5586, -1219, 2098, 3217], [13829, -3962, 783, -3424, -821], [10353, 10120, 953, 5976, 2785], [-33006, -3200, 500, -5722, -754], [11772, 5296, 648, 984, 860], [24099, -86, 3725, 1302, -829], [-47154, -2096, 538, -408, 726], [-70282, 1616, 834, 4774, -2608], [21639, 8896, 2111, -3008, -2553], [26345, 12058, -279, 3156, 551], [-6576, 12450, 2461, 2618, -466], [-68388, 6204, -292, -5596, -356], [-28904, 1960, 288, -5072, 1116], [44053, 9634, -973, -3808, 1907], [69519, 2550, -283, -2410, -2601], [-16248, 148, -3024, -676, 880], [68001, -16232, 299, 3230, -1283], [61144, -3180, -372, -2364, -1956], [-63344, 17120, -1232, 4552, -8], [11361, 6408, 2163, -1914, 2733], [2412, -2444, 798, 7830, -676], [-38487, 15320, 212, -458, -2979], [-11056, -5300, -4432, 2314, 1894], [32856, -5772, -4184, 166, -558], [-27408, 12944, 18, -7674, -2192], [-13428, -480, -670, -5980, 1278], [31990, 5896, -94, 2418, 564], [-3729, -1476, -359, -930, 1919], [25400, 10624, -998, -4008, 886], [-4182, 10188, -2688, 4076, 2014], [-40305, 7224, -1408, 2642, 3587], [-30157, -14910, -265, 3364, 2297], [-55092, -2360, -1362, -1152, 602], [-27793, -11094, -685, -1798, -1125], [-32688, -4656, 2648, 9372, 64], [24501, 5968, -2411, -136, -1499], [18844, 16312, -1788, 6608, -1220], [-60831, 4218, -1075, 2138, -3015], [-4012, 31104, -1598, 7524, 614], [12128, 7204, 1684, -884, 1820], [40298, 1316, -318, -3608, -682], [993, -12572, -469, 3770, 401], [27753, 3522, 283, 4854, 1485], [-35438, -16960, -1452, -10672, -12], [17499, 4198, 2249, 862, -1973], [-16581, 6946, 1525, 4876, 185], [28362, -2312, -1810, -4172, 1806], [38962, 21936, 30, -1520, 2974], [-258, 4936, 2994, 8060, -590], [-27170, 24296, 1816, 2636, -3532], [-14688, 14974, -1665, 568, 2288], [38944, -9920, -3854, 2616, -482], [29290, 14372, -566, 9432, -102], [4246, -4152, 1602, -1198, 1772], [57015, 2922, -1631, 3610, -525], [71415, 7504, 463, -2212, -945], [-27588, 9104, 1170, 4728, 568], [61815, -4642, -907, 8994, 1187], [-37125, 464, 3496, 11196, -319], [-880, -14268, -4322, -3950, 1020], [-19950, -2968, -4442, 2198, 1028], [17802, 13004, -326, 1888, -2218], [7926, 262, -1397, -4130, 1940], [-16979, 9062, 1253, -1730, 369], [37132, 29352, 744, 2440, -464], [-4350, 1944, -1644, -4510, 2426], [40800, -32184, -628, 64, 584], [-8422, -20880, 668, 1060, -1768], [21882, -2942, 977, -6280, 270], [48618, 11748, 2510, 928, 1158], [69534, -468, -1000, -5252, -3318], [47751, 33038, -901, 2738, 2219], [68319, 42652, -1131, 5146, 51], [-10930, 18712, 50, -8158, -1132], [61683, -10472, -793, 9340, 615], [42987, 4692, -1131, 12874, 1983], [35811, 13192, 884, -2848, 1973], [77060, -19640, -784, 7176, 1896], [-57032, -15400, -2400, -3640, -612], [-22140, -16890, 1303, 6214, -594], [34222, -23840, -398, -6952, -1254], [-88246, 25444, -2132, 6164, -420], [2505, -384, 3884, -5520, -1721], [46547, -9318, -425, -12742, 55], [73803, -7220, 4473, 338, -3617], [59886, -3728, 1264, 3958, 430], [587, 38602, -1983, -5832, 1349], [-5498, -10736, -1018, 910, -3380], [-12174, -7212, 2148, 1012, 3894], [-1893, 22448, -2084, 12714, 2143], [-1397, -2802, 1109, -10128, -35], [68913, -8816, 1197, 2860, -1167], [-63239, 7206, 237, 12644, -997], [35793, -6112, 2229, 3320, -239], [-11685, -14410, -73, 3618, -1225], [131464, -5032, 504, 1192, -2284], [43305, -5670, 2745, -2706, 965], [17171, 46, -731, 3866, -1029], [54816, -17324, -1372, 4530, -1046], [-75622, 17708, -1928, -1000, -502], [51707, -15814, -101, -1380, -1431], [-70476, 20536, 2404, 1030, -1774], [6762, -6108, -1814, -10554, -3208], [-41078, 24300, 744, -1636, 3716], [-65004, 12258, 2525, 3822, -1314], [18903, -104, -3872, 800, -1143], [-84338, -43628, 194, -8032, -1006], [41673, 4360, 1296, 3662, -1259], [-22396, -1848, -2228, 7696, -1452], [-64363, 1706, 329, 9664, 4191], [8469, 1400, -2967, -5896, -279], [-11756, -14644, 4624, 2940, -2520], [-46644, -14272, 3210, -198, -3320], [-73902, -1404, -2436, -5602, 1634], [-8361, 2374, -4125, -3418, 4551], [-21618, -28152, -2918, -7176, 3578], [1596, -14834, 3163, -4320, -1244], [7425, -12960, 2657, -12620, -3055], [-28303, -21446, -451, -4592, -6125], [-54302, -15272, -4054, -15546, 288], [-22146, 5120, 3856, -2844, -2700], [15200, 4800, 1040, 11268, 1744], [-138480, -12444, 2608, 4796, 24], [-47847, 562, 3427, 2776, -625], [20877, -3782, 379, -4574, 221], [-64041, -20752, -5659, -2122, 2015], [-238, -17196, 2500, 6348, 3458], [-21170, 3084, 4762, -2604, -3838], [3591, -8944, -2329, -8940, 2127], [121737, -9320, -1092, -10086, 1849], [92694, -19884, 2846, -9736, -4914], [-18786, -11798, 5021, -1440, 1074], [-33054, -2428, 3298, 4414, 2376], [47016, -17944, -1538, -850, 5420], [17779, -3314, 3441, -5932, 4081], [48567, 34308, -2319, -2450, 1639], [47760, 19584, -986, 4168, 420], [-78357, 12590, -715, 4100, 1129], [-30018, -10530, 715, 4774, -2148], [102141, 3928, 1611, 194, -395], [11508, -1904, -1030, -6760, -1966], [25785, 30242, -3989, -1382, 2197], [204042, 4720, -1276, -1076, -1358]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1575_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1575_4_a_bo();". // 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_1575_4_a_bo(:prec:=5) chi := MakeCharacter_1575_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_1575_4_a_bo();". // 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_1575_4_a_bo( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1575_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![-80, 200, -17, -33, 1, 1]>,<11,R![55852416, 1472448, -140456, -2100, 66, 1]>,<13,R![-41380960, 4336080, 15504, -4152, 2, 1]>],Snew); return Vf; end function;