/* This code can be loaded, or copied and pasted, into Magma. It will load the data associated to the BMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. At the *bottom* of the file, there is code to recreate the Bianchi modular form in Magma, by creating the BMF space and cutting out the corresponding Hecke irreducible subspace. From there, you can ask for more eigenvalues or modify as desired. It is commented out, as this computation may be lengthy. */ P := PolynomialRing(Rationals()); g := P![2, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2,a], [3,a+1], [3,a-1], [11,a+3], [11,a-3], [17,a+7], [17,a-7], [19,a+6], [19,a-6], [5,5], [41,a+11], [41,a-11], [43,a+16], [43,a-16], [7,7], [59,a+23], [59,a-23], [67,a+20], [67,a-20], [73,a+12], [73,a-12], [83,a+9], [83,a-9], [89,a+40], [89,a-40], [97,a+17], [97,a-17], [107,a+31], [107,a-31], [113,a+26], [113,a-26], [131,a+28], [131,a-28], [137,a+51], [137,a-51], [139,a+50], [139,a-50], [163,a+18], [163,a-18], [13,13], [179,a+78], [179,a-78], [193,a+34], [193,a-34], [211,a+93], [211,a-93], [227,a+15], [227,a-15], [233,a+109], [233,a-109], [241,a+38], [241,a-38], [251,a+91], [251,a-91], [257,a+68], [257,a-68], [281,a+29], [281,a-29], [283,a+127], [283,a-127], [307,a+108], [307,a-108], [313,a+130], [313,a-130], [331,a+75], [331,a-75], [337,a+141], [337,a-141], [347,a+107], [347,a-107], [353,a+46], [353,a-46], [379,a+120], [379,a-120], [401,a+143], [401,a-143], [409,a+35], [409,a-35], [419,a+96], [419,a-96], [433,a+69], [433,a-69], [443,a+21], [443,a-21], [449,a+214], [449,a-214], [457,a+37], [457,a-37], [467,a+126], [467,a-126], [491,a+94], [491,a-94], [499,a+67], [499,a-67], [521,a+163], [521,a-163], [523,a+56], [523,a-56], [23,23], [547,a+190], [547,a-190], [563,a+261], [563,a-261], [569,a+216], [569,a-216], [571,a+153], [571,a-153], [577,a+239], [577,a-239], [587,a+207], [587,a-207], [593,a+260], [593,a-260], [601,a+104], [601,a-104], [617,a+43], [617,a-43], [619,a+288], [619,a-288], [641,a+62], [641,a-62], [643,a+206], [643,a-206], [659,a+77], [659,a-77], [673,a+262], [673,a-262], [683,a+64], [683,a-64], [691,a+151], [691,a-151], [739,a+346], [739,a-346], [761,a+197], [761,a-197], [769,a+213], [769,a-213], [787,a+347], [787,a-347], [809,a+283], [809,a-283], [811,a+215], [811,a-215], [827,a+122], [827,a-122], [29,29], [857,a+318], [857,a-318], [859,a+296], [859,a-296], [881,a+396], [881,a-396], [883,a+42], [883,a-42], [907,a+173], [907,a-173], [929,a+276], [929,a-276], [937,a+53], [937,a-53], [947,a+250], [947,a-250], [953,a+180], [953,a-180], [31,31], [971,a+179], [971,a-179], [977,a+311], [977,a-311], [1009,a+55], [1009,a-55], [1019,a+241], [1019,a-241], [1033,a+167], [1033,a-167], [1049,a+238], [1049,a-238], [1051,a+401], [1051,a-401], [1091,a+33], [1091,a-33], [1097,a+532], [1097,a-532], [1123,a+253], [1123,a-253], [1129,a+468], [1129,a-468], [1153,a+48], [1153,a-48], [1163,a+552], [1163,a-552], [1171,a+278], [1171,a-278], [1187,a+504], [1187,a-504], [1193,a+162], [1193,a-162], [1201,a+350], [1201,a-350], [1217,a+148], [1217,a-148], [1249,a+523], [1249,a-523], [1259,a+403], [1259,a-403], [1283,a+168], [1283,a-168], [1289,a+390], [1289,a-390], [1291,a+88], [1291,a-88], [1297,a+222], [1297,a-222], [1307,a+386], [1307,a-386], [1321,a+606], [1321,a-606], [1361,a+471], [1361,a-471], [37,37], [1409,a+65], [1409,a-65], [1427,a+217], [1427,a-217], [1433,a+165], [1433,a-165], [1451,a+536], [1451,a-536], [1459,a+54], [1459,a-54], [1481,a+315], [1481,a-315], [1483,a+604], [1483,a-604], [1489,a+412], [1489,a-412], [1499,a+716], [1499,a-716], [1523,a+39], [1523,a-39], [1531,a+166], [1531,a-166], [1553,a+398], [1553,a-398], [1571,a+322], [1571,a-322], [1579,a+292], [1579,a-292], [1601,a+98], [1601,a-98], [1609,a+624], [1609,a-624], [1619,a+457], [1619,a-457], [1627,a+121], [1627,a-121], [1657,a+135], [1657,a-135], [1667,a+100], [1667,a-100], [1697,a+693], [1697,a-693], [1699,a+580], [1699,a-580], [1721,a+176], [1721,a-176], [1723,a+655], [1723,a-655], [1747,a+517], [1747,a-517], [1753,a+299], [1753,a-299], [1777,a+73], [1777,a-73], [1787,a+465], [1787,a-465], [1801,a+60], [1801,a-60], [1811,a+383], [1811,a-383], [1867,a+608], [1867,a-608], [1873,a+106], [1873,a-106], [1889,a+285], [1889,a-285], [1907,a+131], [1907,a-131], [1913,a+686], [1913,a-686], [1931,a+623], [1931,a-623], [1979,a+951], [1979,a-951], [1987,a+516], [1987,a-516], [1993,a+414], [1993,a-414], [2003,a+452], [2003,a-452], [2011,a+889], [2011,a-889], [2017,a+110], [2017,a-110], [2027,a+45], [2027,a-45], [2081,a+79], [2081,a-79], [2083,a+1002], [2083,a-1002], [2089,a+488], [2089,a-488], [2099,a+862], [2099,a-862], [2113,a+615], [2113,a-615], [2129,a+466], [2129,a-466], [2131,a+571], [2131,a-571], [2137,a+894], [2137,a-894], [2153,a+813], [2153,a-813], [2161,a+793], [2161,a-793], [2179,a+66], [2179,a-66], [2203,a+868], [2203,a-868], [47,47], [2243,a+116], [2243,a-116], [2251,a+427], [2251,a-427], [2267,a+202], [2267,a-202], [2273,a+1065], [2273,a-1065], [2281,a+388], [2281,a-388], [2297,a+83], [2297,a-83], [2339,a+282], [2339,a-282], [2347,a+356], [2347,a-356], [2371,a+1059], [2371,a-1059], [2377,a+1088], [2377,a-1088], [2393,a+1056], [2393,a-1056], [2411,a+1163], [2411,a-1163], [2417,a+860], [2417,a-860], [2441,a+1034], [2441,a-1034], [2459,a+843], [2459,a-843], [2467,a+149], [2467,a-149], [2473,a+404], [2473,a-404], [2521,a+1157], [2521,a-1157], [2531,a+289], [2531,a-289], [2539,a+538], [2539,a-538], [2579,a+1123], [2579,a-1123], [2593,a+72], [2593,a-72], [2609,a+1279], [2609,a-1279], [2617,a+509], [2617,a-509], [2633,a+671], [2633,a-671], [2657,a+1045], [2657,a-1045], [2659,a+1244], [2659,a-1244], [2683,a+995], [2683,a-995], [2689,a+220], [2689,a-220], [2699,a+1164], [2699,a-1164], [2707,a+821], [2707,a-821], [2713,a+980], [2713,a-980], [2729,a+1017], [2729,a-1017], [2731,a+128], [2731,a-128], [2753,a+1052], [2753,a-1052], [2777,a+1129], [2777,a-1129], [2801,a+275], [2801,a-275], [2803,a+339], [2803,a-339], [53,53], [2819,a+305], [2819,a-305], [2833,a+828], [2833,a-828], [2843,a+780], [2843,a-780], [2851,a+757], [2851,a-757], [2857,a+791], [2857,a-791], [2897,a+991], [2897,a-991], [2939,a+230], [2939,a-230], [2953,a+1395], [2953,a-1395], [2963,a+400], [2963,a-400], [2969,a+1328], [2969,a-1328], [2971,a+336], [2971,a-336], [3001,a+627], [3001,a-627], [3011,a+1458], [3011,a-1458], [3019,a+731], [3019,a-731], [3041,a+448], [3041,a-448], [3049,a+1183], [3049,a-1183], [3067,a+629], [3067,a-629], [3083,a+136], [3083,a-136], [3089,a+553], [3089,a-553], [3121,a+953], [3121,a-953], [3137,a+97], [3137,a-97], [3163,a+568], [3163,a-568], [3169,a+1372], [3169,a-1372], [3187,a+348], [3187,a-348], [3203,a+330], [3203,a-330], [3209,a+1043], [3209,a-1043], [3217,a+1161], [3217,a-1161], [3251,a+57], [3251,a-57], [3257,a+1600], [3257,a-1600], [3259,a+431], [3259,a-431], [3299,a+1331], [3299,a-1331], [3307,a+614], [3307,a-614], [3313,a+1385], [3313,a-1385], [3323,a+522], [3323,a-522], [3329,a+789], [3329,a-789], [3331,a+1367], [3331,a-1367], [3347,a+470], [3347,a-470], [3361,a+142], [3361,a-142], [3371,a+1151], [3371,a-1151], [3433,a+476], [3433,a-476], [3449,a+1029], [3449,a-1029], [3457,a+195], [3457,a-195], [3467,a+1030], [3467,a-1030], [3491,a+1592], [3491,a-1592], [3499,a+1186], [3499,a-1186], [3529,a+84], [3529,a-84], [3539,a+1718], [3539,a-1718], [3547,a+861], [3547,a-861], [3571,a+1556], [3571,a-1556], [3593,a+640], [3593,a-640], [3617,a+667], [3617,a-667], [3643,a+803], [3643,a-803], [3659,a+1434], [3659,a-1434], [3673,a+201], [3673,a-201], [3691,a+349], [3691,a-349], [3697,a+1144], [3697,a-1144], [61,61], [3739,a+1226], [3739,a-1226], [3761,a+1649], [3761,a-1649], [3769,a+319], [3769,a-319], [3779,a+1670], [3779,a-1670], [3793,a+622], [3793,a-622], [3803,a+185], [3803,a-185], [3833,a+788], [3833,a-788], [3851,a+152], [3851,a-152], [3881,a+1323], [3881,a-1323], [3889,a+1019], [3889,a-1019], [3907,a+1491], [3907,a-1491], [3923,a+1825], [3923,a-1825], [3929,a+294], [3929,a-294], [3931,a+266], [3931,a-266], [3947,a+1402], [3947,a-1402], [4001,a+1016], [4001,a-1016], [4003,a+514], [4003,a-514], [4019,a+1595], [4019,a-1595], [4027,a+1219], [4027,a-1219], [4049,a+1420], [4049,a-1420], [4051,a+90], [4051,a-90], [4057,a+1062], [4057,a-1062], [4073,a+635], [4073,a-635], [4091,a+300], [4091,a-300], [4099,a+1184], [4099,a-1184], [4129,a+1609], [4129,a-1609], [4139,a+193], [4139,a-193], [4153,a+495], [4153,a-495], [4177,a+1216], [4177,a-1216], [4201,a+1542], [4201,a-1542], [4211,a+1537], [4211,a-1537], [4217,a+961], [4217,a-961], [4219,a+656], [4219,a-656], [4241,a+1383], [4241,a-1383], [4243,a+1436], [4243,a-1436], [4259,a+2073], [4259,a-2073], [4273,a+832], [4273,a-832], [4283,a+589], [4283,a-589], [4289,a+1822], [4289,a-1822], [4297,a+727], [4297,a-727], [4337,a+384], [4337,a-384], [4339,a+771], [4339,a-771], [4363,a+1036], [4363,a-1036], [4409,a+115], [4409,a-115], [4441,a+618], [4441,a-618], [4451,a+542], [4451,a-542], [4457,a+291], [4457,a-291], [4481,a+284], [4481,a-284], [4483,a+164], [4483,a-164], [4507,a+1480], [4507,a-1480], [4513,a+1875], [4513,a-1875], [4523,a+2041], [4523,a-2041], [4547,a+1876], [4547,a-1876], [4561,a+749], [4561,a-749], [4603,a+2189], [4603,a-2189], [4643,a+2173], [4643,a-2173], [4649,a+728], [4649,a-728], [4651,a+1041], [4651,a-1041], [4657,a+679], [4657,a-679], [4673,a+1117], [4673,a-1117], [4691,a+744], [4691,a-744], [4721,a+119], [4721,a-119], [4723,a+2302], [4723,a-2302], [4729,a+1281], [4729,a-1281], [4787,a+1448], [4787,a-1448], [4793,a+1181], [4793,a-1181], [4801,a+1735], [4801,a-1735], [4817,a+170], [4817,a-170], [4889,a+1842], [4889,a-1842], [4931,a+172], [4931,a-172], [4937,a+1568], [4937,a-1568], [4969,a+518], [4969,a-518], [4987,a+754], [4987,a-754], [4993,a+1527], [4993,a-1527], [5003,a+1813], [5003,a-1813], [5009,a+1703], [5009,a-1703], [5011,a+1781], [5011,a-1781], [71,71], [5051,a+1586], [5051,a-1586], [5059,a+1710], [5059,a-1710], [5081,a+1118], [5081,a-1118], [5099,a+1182], [5099,a-1182], [5107,a+2435], [5107,a-2435], [5113,a+864], [5113,a-864], [5147,a+766], [5147,a-766], [5153,a+373], [5153,a-373], [5171,a+999], [5171,a-999], [5179,a+765], [5179,a-765], [5209,a+125], [5209,a-125], [5227,a+1025], [5227,a-1025], [5233,a+2508], [5233,a-2508], [5273,a+993], [5273,a-993], [5281,a+178], [5281,a-178], [5297,a+1488], [5297,a-1488], [5323,a+1461], [5323,a-1461], [5347,a+1758], [5347,a-1758], [5387,a+1315], [5387,a-1315], [5393,a+1104], [5393,a-1104], [5417,a+1771], [5417,a-1771], [5419,a+2048], [5419,a-2048], [5441,a+2068], [5441,a-2068], [5443,a+557], [5443,a-557], [5449,a+567], [5449,a-567], [5483,a+2305], [5483,a-2305], [5507,a+1171], [5507,a-1171], [5521,a+182], [5521,a-182], [5531,a+2257], [5531,a-2257], [5563,a+873], [5563,a-873], [5569,a+1973], [5569,a-1973], [5641,a+1178], [5641,a-1178], [5651,a+1167], [5651,a-1167], [5657,a+1433], [5657,a-1433], [5659,a+2399], [5659,a-2399], [5683,a+1050], [5683,a-1050], [5689,a+320], [5689,a-320], [5737,a+2321], [5737,a-2321], [5779,a+2692], [5779,a-2692], [5801,a+1594], [5801,a-1594], [5827,a+229], [5827,a-229], [5843,a+772], [5843,a-772], [5849,a+562], [5849,a-562], [5851,a+2099], [5851,a-2099], [5857,a+1193], [5857,a-1193], [5867,a+2660], [5867,a-2660], [5881,a+2505], [5881,a-2505], [5897,a+133], [5897,a-133], [5923,a+2253], [5923,a-2253], [5939,a+759], [5939,a-759], [5953,a+551], [5953,a-551], [5987,a+2102], [5987,a-2102], [6011,a+959], [6011,a-959], [6043,a+830], [6043,a-830], [6067,a+2646], [6067,a-2646], [6073,a+1854], [6073,a-1854], [6089,a+366], [6089,a-366], [6091,a+1345], [6091,a-1345], [6113,a+708], [6113,a-708], [6121,a+2154], [6121,a-2154], [6131,a+669], [6131,a-669], [6163,a+2780], [6163,a-2780], [6203,a+1820], [6203,a-1820], [6211,a+595], [6211,a-595], [6217,a+2584], [6217,a-2584], [79,79], [6257,a+137], [6257,a-137], [6299,a+2345], [6299,a-2345], [6323,a+646], [6323,a-646], [6329,a+2936], [6329,a-2936], [6337,a+2868], [6337,a-2868], [6353,a+2836], [6353,a-2836], [6361,a+825], [6361,a-825], [6379,a+2982], [6379,a-2982], [6427,a+2017], [6427,a-2017], [6449,a+1130], [6449,a-1130], [6451,a+1578], [6451,a-1578], [6473,a+2794], [6473,a-2794], [6481,a+267], [6481,a-267], [6491,a+2478], [6491,a-2478], [6521,a+1173], [6521,a-1173], [6529,a+2727], [6529,a-2727], [6547,a+919], [6547,a-919], [6553,a+2027], [6553,a-2027], [6563,a+81], [6563,a-81], [6569,a+3244], [6569,a-3244], [6571,a+1418], [6571,a-1418], [6577,a+2562], [6577,a-2562], [6619,a+3019], [6619,a-3019], [6659,a+1891], [6659,a-1891], [6673,a+1607], [6673,a-1607], [6689,a+826], [6689,a-826], [6691,a+1779], [6691,a-1779], [6737,a+2207], [6737,a-2207], [6761,a+668], [6761,a-668], [6763,a+2773], [6763,a-2773], [6779,a+247], [6779,a-247], [6793,a+880], [6793,a-880], [6803,a+1848], [6803,a-1848], [6827,a+1671], [6827,a-1671], [6833,a+1338], [6833,a-1338], [6841,a+2506], [6841,a-2506], [6857,a+543], [6857,a-543], [6883,a+674], [6883,a-674], [6899,a+2129], [6899,a-2129], [6907,a+2330], [6907,a-2330], [6947,a+486], [6947,a-486], [6961,a+1174], [6961,a-1174], [6971,a+2363], [6971,a-2363], [6977,a+2418], [6977,a-2418], [7001,a+3375], [7001,a-3375], [7019,a+1791], [7019,a-1791], [7027,a+616], [7027,a-616], [7043,a+1552], [7043,a-1552], [7057,a+2221], [7057,a-2221], [7121,a+2750], [7121,a-2750], [7129,a+1661], [7129,a-1661], [7177,a+605], [7177,a-605], [7187,a+487], [7187,a-487], [7193,a+768], [7193,a-768], [7211,a+208], [7211,a-208], [7219,a+2846], [7219,a-2846], [7243,a+2386], [7243,a-2386], [7283,a+1529], [7283,a-1529], [7297,a+1202], [7297,a-1202], [7307,a+3021], [7307,a-3021], [7321,a+1342], [7321,a-1342], [7331,a+2326], [7331,a-2326], [7369,a+2825], [7369,a-2825], [7393,a+2153], [7393,a-2153], [7411,a+1890], [7411,a-1890], [7417,a+2937], [7417,a-2937], [7433,a+2145], [7433,a-2145], [7451,a+2443], [7451,a-2443], [7457,a+1929], [7457,a-1929], [7459,a+1440], [7459,a-1440], [7481,a+3414], [7481,a-3414], [7489,a+874], [7489,a-874], [7499,a+3042], [7499,a-3042], [7507,a+3187], [7507,a-3187], [7523,a+3330], [7523,a-3330], [7529,a+1023], [7529,a-1023], [7537,a+2089], [7537,a-2089], [7547,a+2919], [7547,a-2919], [7561,a+2394], [7561,a-2394], [7577,a+3745], [7577,a-3745], [7603,a+3726], [7603,a-3726], [7643,a+2885], [7643,a-2885], [7649,a+2216], [7649,a-2216], [7673,a+717], [7673,a-717], [7681,a+712], [7681,a-712], [7691,a+3129], [7691,a-3129], [7699,a+3704], [7699,a-3704], [7723,a+1762], [7723,a-1762], [7753,a+2029], [7753,a-2029], [7793,a+3200], [7793,a-3200], [7817,a+944], [7817,a-944], [7841,a+1246], [7841,a-1246], [7867,a+2761], [7867,a-2761], [7873,a+3057], [7873,a-3057], [7883,a+3748], [7883,a-3748], [7907,a+1818], [7907,a-1818], [7937,a+1165]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesList := [* -1, 1, 1, -6, -6, 3, 3, 1, 1, -10, 0, 0, 8, 8, -13, 9, 9, 5, 5, -7, -7, -6, -6, -12, -12, -10, -10, -9, -9, 6, 6, 0, 0, -9, -9, -4, -4, 20, 20, -1, 0, 0, 14, 14, 5, 5, -15, -15, -6, -6, 8, 8, 6, 6, 12, 12, 0, 0, -22, -22, 20, 20, -19, -19, -1, -1, -4, -4, 18, 18, -15, -15, -7, -7, 0, 0, 32, 32, -12, -12, 2, 2, -18, -18, -18, -18, 17, 17, 18, 18, -36, -36, -4, -4, -36, -36, 11, 11, -37, 44, 44, -12, -12, -24, -24, -4, -4, 11, 11, -12, -12, -30, -30, -28, -28, -6, -6, -10, -10, 6, 6, -22, -22, -45, -45, 44, 44, 36, 36, -10, -10, -16, -16, -21, -21, 5, 5, -31, -31, 9, 9, 11, 11, 33, 33, 23, -12, -12, 14, 14, -18, -18, -34, -34, -37, -37, 33, 33, -7, -7, -48, -48, 30, 30, -46, 12, 12, -12, -12, 20, 20, 39, 39, -1, -1, 30, 30, 8, 8, -33, -33, 48, 48, 8, 8, -58, -58, -34, -34, -36, -36, 56, 56, -12, -12, 12, 12, 41, 41, 3, 3, -28, -28, -6, -6, -15, -15, -3, -3, -67, -67, 29, 29, 27, 27, -46, -46, 30, 30, -70, 18, 18, 27, 27, 66, 66, 66, 66, 29, 29, -30, -30, 26, 26, -58, -58, 18, 18, -12, -12, 74, 74, 36, 36, -60, -60, -4, -4, -69, -69, 62, 62, -30, -30, -28, -28, 38, 38, 60, 60, 66, 66, 17, 17, 3, 3, -13, -13, 47, 47, -55, -55, 14, 14, 6, 6, 20, 20, 0, 0, -4, -4, -79, -79, -18, -18, 36, 36, 66, 66, 60, 60, 3, 3, -46, -46, 14, 14, 69, 69, -4, -4, -34, -34, 81, 81, 90, 90, -4, -4, -16, -16, -54, -54, -37, -37, 78, 78, -37, -37, 23, 23, -9, -9, 2, 2, -7, -7, -73, -73, -94, 30, 30, 74, 74, -48, -48, 24, 24, -49, -49, -57, -57, 84, 84, -37, -37, -40, -40, 2, 2, 72, 72, 48, 48, -33, -33, -90, -90, -15, -15, 74, 74, 44, 44, 38, 38, 90, 90, -61, -61, -45, -45, 38, 38, -3, -3, 44, 44, -30, -30, -66, -66, 5, 5, -34, -34, -82, -82, -54, -54, -10, -10, 14, 14, 48, 48, 68, 68, 66, 66, -72, -72, 84, 84, 23, 23, -97, -24, -24, -64, -64, -27, -27, 62, 62, 77, 77, 51, 51, -69, -69, 2, 2, 21, 21, 3, 3, -88, -88, 50, 50, -42, -42, -94, -94, -63, -63, -46, -46, 44, 44, 54, 54, 21, 21, 35, 35, -18, -18, -4, -4, -16, -16, 29, 29, 0, 0, -54, -54, -58, -58, -75, -75, -108, -108, -79, -79, 51, 51, 68, 68, -43, -43, 24, 24, -21, -21, -70, -70, -33, -33, 23, 23, 60, 60, -76, -76, 0, 0, 56, 56, -90, -90, -15, -15, 116, 116, 20, 20, 96, 96, 92, 92, -55, -55, 24, 24, -63, -63, 53, 53, -30, -30, -79, -79, -34, -34, -55, -55, -22, -49, -49, -72, -72, -10, -10, 12, 12, 32, 32, -99, -99, 30, 30, -84, -84, 6, 6, -52, -52, -43, -43, 24, 24, 102, 102, 98, 98, 81, 81, -93, -93, 44, 44, -87, -87, -31, -31, 66, 66, -106, -106, 26, 26, 69, 69, 84, 84, -76, -76, 35, 35, 84, 84, 95, 95, 110, 110, -46, -46, -63, -63, -102, -102, 8, 8, 18, 18, 86, 86, -57, -57, 53, 53, 108, 108, 18, 18, -40, -40, -33, -33, 92, 92, -64, -64, 117, 117, 86, 86, -36, -36, 66, 66, 75, 75, 29, 29, 98, 98, -76, -76, -12, -12, -18, -18, -55, -55, 104, 104, -36, -36, 48, 48, 125, 125, 50, 50, -132, -132, -12, -12, 75, 75, -64, -64, -67, -67, 36, 36, 9, 9, 68, 68, 18, 18, -54, -54, -21, -21, -45, -45, -16, -16, -52, -52, 8, 8, 96, 96, -48, -48, -64, -64, -106, 60, 60, -70, -70, -12, -12, -12, -12, 131, 131, -16, -16, 30, 30, 57, 57, 108, 108, -58, -58, 98, 98, -73, -73, -4, -4, -30, -30, -100, -100, -96, -96, -121, -121, -61, -61, 39, 39, -111, -111, -132, -132, 20, 20, -129, -129, -82, -82, 50, 50, -96, -96, 102, 102, -22, -22, -27, -27, -124, -124, 134, 134, -10, -10, 45, 45, -18, -18, 116, 116, -49, -49, 38, 38, 140, 140, -124, -124, -123, -123, -4, -4, -36, -36, -69, -69, 20, 20, -118, -118, 87, 87, -148, -148, -51, -51, -61, -61, -72, -72, -49, -49, 69, 69, 90, 90, 14, 14, 20, 20, -10, -10, 93, 93, 62, 62, 78, 78, -88, -88, 135, 135, 80, 80, -54, -54, 2, 2, 125, 125, -58, -63, -63, -111, -111, 12, 12, -156, -156, -34, -34, 135, 135, 80, 80, -55, -55, -106, -106, -48, -48, -145, -145, 126, 126, 110, 110, -105, -105, -87, -87, -58, -58, -22, -22, 101, 101, 93, 93, -42, -42, 92, 92, -64, -64, -58, -58, -48, -48, 23, 23, -33, -33, -67, -67, -3, -3, -39, -39, -76, -76, -9, -9, 80, 80, 54, 54, -42, -42, 156, 156, 38, 38, -69, -69, 44, 44, -36, -36, 68, 68, -108, -108, -121, -121, -126, -126, -27, -27, 105, 105, -45, -45, -82, -82, -15, -15, 56, 56, -18, -18, 65, 65, 14, 14, 108, 108, 102, 102, 9, 9, -52, -52, -28, -28, 72, 72, 26, 26, -54, -54, 86, 86, 0, 0, -55, -55, 32, 32, 122, 122, -97, -97, 33, 33, 129, 129, -78, -78, 26, 26, -66, -66, -70, -70, -63, -63, 56, 56, -9, -9, -90, -90, 50, 50, 48, 48, 20, 20, -12, -12, 89, 89, -174, -174, -117, -117, 126, 126, 50, 50, -159, -159, 86, 86, 56, 56, -103, -103, 66, 66, 78, 78, 36, 36, -106, -106, 62, 62, -42, -42, 63, 63, 84 *]; heckeEigenvalues := AssociativeArray(); for i in [1..#heckeEigenvaluesList] do heckeEigenvalues[primes[i]] := heckeEigenvaluesList[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; ALEigenvalues[ideal] := -1; ALEigenvalues[ideal] := -1; // EXAMPLE: // pp := Factorization(2*ZF)[1][1]; // heckeEigenvalues[pp]; print "To reconstruct the Bianchi newform f, type f, iso := Explode(make_newform());"; function make_newform(); M := BianchiCuspForms(F, NN); S := NewSubspace(M); // SetVerbose("Bianchi", 1); NFD := NewformDecomposition(S); newforms := [* Eigenform(U) : U in NFD *]; if #newforms eq 0 then; print "No Bianchi newforms at this level"; return 0; end if; print "Testing ", #newforms, " possible newforms"; newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *]; print #newforms, " newforms have the correct Hecke field"; if #newforms eq 0 then; print "No Bianchi newform found with the correct Hecke field"; return 0; end if; autos := Automorphisms(K); xnewforms := [* *]; for f in newforms do; if K eq RationalField() then; Append(~xnewforms, [* f, autos[1] *]); else; flag, iso := IsIsomorphic(K,BaseField(f)); for a in autos do; Append(~xnewforms, [* f, a*iso *]); end for; end if; end for; newforms := xnewforms; for P in primes do; if Valuation(NN,P) eq 0 then; xnewforms := [* *]; for f_iso in newforms do; f, iso := Explode(f_iso); if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then; Append(~xnewforms, f_iso); end if; end for; newforms := xnewforms; if #newforms eq 0 then; print "No Bianchi newform found which matches the Hecke eigenvalues"; return 0; else if #newforms eq 1 then; print "success: unique match"; return newforms[1]; end if; end if; end if; end for; print #newforms, "Bianchi newforms found which match the Hecke eigenvalues"; return newforms[1]; end function;