/* 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![1, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2,i+1], [5,i+2], [5,i-2], [3,3], [13,i+5], [13,i-5], [17,i+4], [17,i-4], [29,i+12], [29,i-12], [37,i+6], [37,i-6], [41,i+9], [41,i-9], [7,7], [53,i+23], [53,i-23], [61,i+11], [61,i-11], [73,i+27], [73,i-27], [89,i+34], [89,i-34], [97,i+22], [97,i-22], [101,i+10], [101,i-10], [109,i+33], [109,i-33], [113,i+15], [113,i-15], [11,11], [137,i+37], [137,i-37], [149,i+44], [149,i-44], [157,i+28], [157,i-28], [173,i+80], [173,i-80], [181,i+19], [181,i-19], [193,i+81], [193,i-81], [197,i+14], [197,i-14], [229,i+107], [229,i-107], [233,i+89], [233,i-89], [241,i+64], [241,i-64], [257,i+16], [257,i-16], [269,i+82], [269,i-82], [277,i+60], [277,i-60], [281,i+53], [281,i-53], [293,i+138], [293,i-138], [313,i+25], [313,i-25], [317,i+114], [317,i-114], [337,i+148], [337,i-148], [349,i+136], [349,i-136], [353,i+42], [353,i-42], [19,19], [373,i+104], [373,i-104], [389,i+115], [389,i-115], [397,i+63], [397,i-63], [401,i+20], [401,i-20], [409,i+143], [409,i-143], [421,i+29], [421,i-29], [433,i+179], [433,i-179], [449,i+67], [449,i-67], [457,i+109], [457,i-109], [461,i+48], [461,i-48], [509,i+208], [509,i-208], [521,i+235], [521,i-235], [23,23], [541,i+52], [541,i-52], [557,i+118], [557,i-118], [569,i+86], [569,i-86], [577,i+24], [577,i-24], [593,i+77], [593,i-77], [601,i+125], [601,i-125], [613,i+35], [613,i-35], [617,i+194], [617,i-194], [641,i+154], [641,i-154], [653,i+149], [653,i-149], [661,i+106], [661,i-106], [673,i+58], [673,i-58], [677,i+26], [677,i-26], [701,i+135], [701,i-135], [709,i+96], [709,i-96], [733,i+353], [733,i-353], [757,i+87], [757,i-87], [761,i+39], [761,i-39], [769,i+62], [769,i-62], [773,i+317], [773,i-317], [797,i+215], [797,i-215], [809,i+318], [809,i-318], [821,i+295], [821,i-295], [829,i+246], [829,i-246], [853,i+333], [853,i-333], [857,i+207], [857,i-207], [877,i+151], [877,i-151], [881,i+387], [881,i-387], [929,i+324], [929,i-324], [937,i+196], [937,i-196], [941,i+97], [941,i-97], [953,i+442], [953,i-442], [31,31], [977,i+252], [977,i-252], [997,i+161], [997,i-161], [1009,i+469], [1009,i-469], [1013,i+45], [1013,i-45], [1021,i+374], [1021,i-374], [1033,i+355], [1033,i-355], [1049,i+426], [1049,i-426], [1061,i+103], [1061,i-103], [1069,i+249], [1069,i-249], [1093,i+530], [1093,i-530], [1097,i+341], [1097,i-341], [1109,i+354], [1109,i-354], [1117,i+214], [1117,i-214], [1129,i+168], [1129,i-168], [1153,i+140], [1153,i-140], [1181,i+243], [1181,i-243], [1193,i+186], [1193,i-186], [1201,i+49], [1201,i-49], [1213,i+495], [1213,i-495], [1217,i+78], [1217,i-78], [1229,i+597], [1229,i-597], [1237,i+546], [1237,i-546], [1249,i+585], [1249,i-585], [1277,i+113], [1277,i-113], [1289,i+479], [1289,i-479], [1297,i+36], [1297,i-36], [1301,i+51], [1301,i-51], [1321,i+257], [1321,i-257], [1361,i+614], [1361,i-614], [1373,i+668], [1373,i-668], [1381,i+366], [1381,i-366], [1409,i+452], [1409,i-452], [1429,i+620], [1429,i-620], [1433,i+542], [1433,i-542], [1453,i+497], [1453,i-497], [1481,i+465], [1481,i-465], [1489,i+225], [1489,i-225], [1493,i+432], [1493,i-432], [1549,i+88], [1549,i-88], [1553,i+339], [1553,i-339], [1597,i+610], [1597,i-610], [1601,i+40], [1601,i-40], [1609,i+523], [1609,i-523], [1613,i+127], [1613,i-127], [1621,i+166], [1621,i-166], [1637,i+316], [1637,i-316], [1657,i+783], [1657,i-783], [1669,i+220], [1669,i-220], [1693,i+92], [1693,i-92], [1697,i+414], [1697,i-414], [1709,i+390], [1709,i-390], [1721,i+473], [1721,i-473], [1733,i+410], [1733,i-410], [1741,i+59], [1741,i-59], [1753,i+713], [1753,i-713], [1777,i+775], [1777,i-775], [1789,i+724], [1789,i-724], [1801,i+824], [1801,i-824], [43,43], [1861,i+61], [1861,i-61], [1873,i+737], [1873,i-737], [1877,i+137], [1877,i-137], [1889,i+331], [1889,i-331], [1901,i+218], [1901,i-218], [1913,i+712], [1913,i-712], [1933,i+598], [1933,i-598], [1949,i+589], [1949,i-589], [1973,i+259], [1973,i-259], [1993,i+834], [1993,i-834], [1997,i+412], [1997,i-412], [2017,i+229], [2017,i-229], [2029,i+992], [2029,i-992], [2053,i+244], [2053,i-244], [2069,i+164], [2069,i-164], [2081,i+102], [2081,i-102], [2089,i+789], [2089,i-789], [2113,i+65], [2113,i-65], [2129,i+372], [2129,i-372], [2137,i+296], [2137,i-296], [2141,i+419], [2141,i-419], [2153,i+232], [2153,i-232], [2161,i+147], [2161,i-147], [47,47], [2213,i+1083], [2213,i-1083], [2221,i+790], [2221,i-790], [2237,i+1021], [2237,i-1021], [2269,i+982], [2269,i-982], [2273,i+290], [2273,i-290], [2281,i+710], [2281,i-710], [2293,i+600], [2293,i-600], [2297,i+365], [2297,i-365], [2309,i+688], [2309,i-688], [2333,i+108], [2333,i-108], [2341,i+153], [2341,i-153], [2357,i+633], [2357,i-633], [2377,i+1134], [2377,i-1134], [2381,i+69], [2381,i-69], [2389,i+285], [2389,i-285], [2393,i+971], [2393,i-971], [2417,i+592], [2417,i-592], [2437,i+398], [2437,i-398], [2441,i+672], [2441,i-672], [2473,i+567], [2473,i-567], [2477,i+915], [2477,i-915], [2521,i+71], [2521,i-71], [2549,i+357], [2549,i-357], [2557,i+611], [2557,i-611], [2593,i+918], [2593,i-918], [2609,i+389], [2609,i-389], [2617,i+667], [2617,i-667], [2621,i+472], [2621,i-472], [2633,i+1224], [2633,i-1224], [2657,i+163], [2657,i-163], [2677,i+550], [2677,i-550], [2689,i+1142], [2689,i-1142], [2693,i+859], [2693,i-859], [2713,i+887], [2713,i-887], [2729,i+1102], [2729,i-1102], [2741,i+656], [2741,i-656], [2749,i+640], [2749,i-640], [2753,i+794], [2753,i-794], [2777,i+190], [2777,i-190], [2789,i+167], [2789,i-167], [2797,i+603], [2797,i-603], [2801,i+1258], [2801,i-1258], [2833,i+1357], [2833,i-1357], [2837,i+416], [2837,i-416], [2857,i+896], [2857,i-896], [2861,i+1202], [2861,i-1202], [2897,i+1120], [2897,i-1120], [2909,i+878], [2909,i-878], [2917,i+54], [2917,i-54], [2953,i+1226], [2953,i-1226], [2957,i+1222], [2957,i-1222], [2969,i+964], [2969,i-964], [3001,i+1353], [3001,i-1353], [3037,i+281], [3037,i-281], [3041,i+774], [3041,i-774], [3049,i+475], [3049,i-475], [3061,i+501], [3061,i-501], [3089,i+393], [3089,i-393], [3109,i+727], [3109,i-727], [3121,i+79], [3121,i-79], [3137,i+56], [3137,i-56], [3169,i+1325], [3169,i-1325], [3181,i+282], [3181,i-282], [3209,i+484], [3209,i-484], [3217,i+1436], [3217,i-1436], [3221,i+234], [3221,i-234], [3229,i+839], [3229,i-839], [3253,i+1598], [3253,i-1598], [3257,i+291], [3257,i-291], [3301,i+1212], [3301,i-1212], [3313,i+407], [3313,i-407], [3329,i+1600], [3329,i-1600], [3361,i+900], [3361,i-900], [3373,i+1105], [3373,i-1105], [3389,i+1344], [3389,i-1344], [3413,i+1471], [3413,i-1471], [3433,i+1651], [3433,i-1651], [3449,i+1122], [3449,i-1122], [3457,i+708], [3457,i-708], [3461,i+1453], [3461,i-1453], [3469,i+1003], [3469,i-1003], [59,59], [3517,i+596], [3517,i-596], [3529,i+808], [3529,i-808], [3533,i+548], [3533,i-548], [3541,i+852], [3541,i-852], [3557,i+943], [3557,i-943], [3581,i+364], [3581,i-364], [3593,i+1153], [3593,i-1153], [3613,i+85], [3613,i-85], [3617,i+1234], [3617,i-1234], [3637,i+1027], [3637,i-1027], [3673,i+994], [3673,i-994], [3677,i+1309], [3677,i-1309], [3697,i+1131], [3697,i-1131], [3701,i+1279], [3701,i-1279], [3709,i+1609], [3709,i-1609], [3733,i+851], [3733,i-851], [3761,i+604], [3761,i-604], [3769,i+1445], [3769,i-1445], [3793,i+803], [3793,i-803], [3797,i+742], [3797,i-742], [3821,i+376], [3821,i-376], [3833,i+361], [3833,i-361], [3853,i+1305], [3853,i-1305], [3877,i+502], [3877,i-502], [3881,i+197], [3881,i-197], [3889,i+454], [3889,i-454], [3917,i+835], [3917,i-835], [3929,i+226], [3929,i-226], [3989,i+481], [3989,i-481], [4001,i+899], [4001,i-899], [4013,i+1230], [4013,i-1230], [4021,i+723], [4021,i-723], [4049,i+884], [4049,i-884], [4057,i+1857], [4057,i-1857], [4073,i+549], [4073,i-549], [4093,i+1059], [4093,i-1059], [4129,i+895], [4129,i-895], [4133,i+733], [4133,i-733], [4153,i+1643], [4153,i-1643], [4157,i+1761], [4157,i-1761], [4177,i+457], [4177,i-457], [4201,i+1154], [4201,i-1154], [4217,i+1911], [4217,i-1911], [4229,i+2082], [4229,i-2082], [4241,i+1044], [4241,i-1044], [4253,i+561], [4253,i-561], [4261,i+721], [4261,i-721], [4273,i+1200], [4273,i-1200], [4289,i+528], [4289,i-528], [4297,i+1972], [4297,i-1972], [4337,i+886], [4337,i-886], [4349,i+608], [4349,i-608], [4357,i+66], [4357,i-66], [4373,i+1904], [4373,i-1904], [4397,i+505], [4397,i-505], [4409,i+332], [4409,i-332], [4421,i+952], [4421,i-952], [4441,i+2146], [4441,i-2146], [4457,i+1880], [4457,i-1880], [4481,i+276], [4481,i-276], [67,67], [4493,i+2213], [4493,i-2213], [4513,i+95], [4513,i-95], [4517,i+1474], [4517,i-1474], [4549,i+1260], [4549,i-1260], [4561,i+2205], [4561,i-2205], [4597,i+2129], [4597,i-2129], [4621,i+152], [4621,i-152], [4637,i+2044], [4637,i-2044], [4649,i+1846], [4649,i-1846], [4657,i+1912], [4657,i-1912], [4673,i+1993], [4673,i-1993], [4721,i+1697], [4721,i-1697], [4729,i+1365], [4729,i-1365], [4733,i+897], [4733,i-897], [4789,i+1481], [4789,i-1481], [4793,i+1480], [4793,i-1480], [4801,i+1403], [4801,i-1403], [4813,i+1868], [4813,i-1868], [4817,i+1291], [4817,i-1291], [4861,i+493], [4861,i-493], [4877,i+719], [4877,i-719], [4889,i+730], [4889,i-730], [4909,i+1613], [4909,i-1613], [4933,i+1194], [4933,i-1194], [4937,i+849], [4937,i-849], [4957,i+359], [4957,i-359], [4969,i+1076], [4969,i-1076], [4973,i+223], [4973,i-223], [4993,i+158], [4993,i-158], [5009,i+539], [5009,i-539], [5021,i+1363], [5021,i-1363], [71,71], [5077,i+858], [5077,i-858], [5081,i+2412], [5081,i-2412], [5101,i+101], [5101,i-101], [5113,i+2025], [5113,i-2025], [5153,i+227], [5153,i-227], [5189,i+2446], [5189,i-2446], [5197,i+1969], [5197,i-1969], [5209,i+2098], [5209,i-2098], [5233,i+2253], [5233,i-2253], [5237,i+369], [5237,i-369], [5261,i+827], [5261,i-827], [5273,i+944], [5273,i-944], [5281,i+1673], [5281,i-1673], [5297,i+2313], [5297,i-2313], [5309,i+1804], [5309,i-1804], [5333,i+2630], [5333,i-2630], [5381,i+1739], [5381,i-1739], [5393,i+665], [5393,i-665], [5413,i+429], [5413,i-429], [5417,i+368], [5417,i-368], [5437,i+630], [5437,i-630], [5441,i+2452], [5441,i-2452], [5449,i+635], [5449,i-635], [5477,i+74], [5477,i-74], [5501,i+1115], [5501,i-1115], [5521,i+765], [5521,i-765], [5557,i+2478], [5557,i-2478], [5569,i+973], [5569,i-973], [5573,i+2017], [5573,i-2017], [5581,i+1437], [5581,i-1437], [5641,i+1429], [5641,i-1429], [5653,i+310], [5653,i-310], [5657,i+1670], [5657,i-1670], [5669,i+1046], [5669,i-1046], [5689,i+2124], [5689,i-2124], [5693,i+1193], [5693,i-1193], [5701,i+385], [5701,i-385], [5717,i+2416], [5717,i-2416], [5737,i+1126], [5737,i-1126], [5741,i+2378], [5741,i-2378], [5749,i+806], [5749,i-806], [5801,i+1145], [5801,i-1145], [5813,i+796], [5813,i-796], [5821,i+1242], [5821,i-1242], [5849,i+2839], [5849,i-2839], [5857,i+1310], [5857,i-1310], [5861,i+754], [5861,i-754], [5869,i+1042], [5869,i-1042], [5881,i+1098], [5881,i-1098], [5897,i+543], [5897,i-543], [5953,i+2403], [5953,i-2403], [5981,i+1317], [5981,i-1317], [6029,i+1801], [6029,i-1801], [6037,i+2652], [6037,i-2652], [6053,i+2832], [6053,i-2832], [6073,i+2524], [6073,i-2524], [6089,i+455], [6089,i-455], [6101,i+247], [6101,i-247], [6113,i+1089], [6113,i-1089], [6121,i+2583], [6121,i-2583], [6133,i+865], [6133,i-865], [6173,i+2447], [6173,i-2447], [6197,i+2007], [6197,i-2007], [6217,i+2372], [6217,i-2372], [6221,i+1121], [6221,i-1121], [6229,i+1451], [6229,i-1451], [79,79], [6257,i+1584], [6257,i-1584], [6269,i+1523], [6269,i-1523], [6277,i+1033], [6277,i-1033], [6301,i+2184], [6301,i-2184], [6317,i+1963], [6317,i-1963], [6329,i+2219], [6329,i-2219], [6337,i+178], [6337,i-178], [6353,i+1392], [6353,i-1392], [6361,i+1751], [6361,i-1751], [6373,i+1879], [6373,i-1879], [6389,i+2092], [6389,i-2092], [6397,i+1302], [6397,i-1302], [6421,i+825], [6421,i-825], [6449,i+1854], [6449,i-1854], [6469,i+2977], [6469,i-2977], [6473,i+1808], [6473,i-1808], [6481,i+729], [6481,i-729], [6521,i+2364], [6521,i-2364], [6529,i+2311], [6529,i-2311], [6553,i+3186], [6553,i-3186], [6569,i+3038], [6569,i-3038], [6577,i+1624], [6577,i-1624], [6581,i+2727], [6581,i-2727], [6637,i+2828], [6637,i-2828], [6653,i+752], [6653,i-752], [6661,i+658], [6661,i-658], [6673,i+2437], [6673,i-2437], [6689,i+2759], [6689,i-2759], [6701,i+1721], [6701,i-1721], [6709,i+2150], [6709,i-2150], [6733,i+2217], [6733,i-2217], [6737,i+2393], [6737,i-2393], [6761,i+1775], [6761,i-1775], [6781,i+995], [6781,i-995], [6793,i+709], [6793,i-709], [6829,i+1596], [6829,i-1596], [6833,i+1307], [6833,i-1307], [6841,i+1625], [6841,i-1625], [6857,i+1348], [6857,i-1348], [6869,i+998], [6869,i-998], [83,83], [6917,i+263], [6917,i-263], [6949,i+932], [6949,i-932], [6961,i+344], [6961,i-344], [6977,i+2063], [6977,i-2063], [6997,i+1796], [6997,i-1796], [7001,i+1198], [7001,i-1198], [7013,i+2480], [7013,i-2480], [7057,i+84], [7057,i-84], [7069,i+188], [7069,i-188], [7109,i+304], [7109,i-304], [7121,i+778], [7121,i-778], [7129,i+267], [7129,i-267], [7177,i+1965], [7177,i-1965], [7193,i+967], [7193,i-967], [7213,i+1999], [7213,i-1999], [7229,i+3572], [7229,i-3572], [7237,i+2502], [7237,i-2502], [7253,i+2211], [7253,i-2211], [7297,i+3553], [7297,i-3553], [7309,i+2717], [7309,i-2717], [7321,i+121], [7321,i-121], [7333,i+2909], [7333,i-2909], [7349,i+2061], [7349,i-2061], [7369,i+607], [7369,i-607], [7393,i+2361], [7393,i-2361], [7417,i+2737], [7417,i-2737], [7433,i+983], [7433,i-983], [7457,i+1275], [7457,i-1275], [7477,i+1652], [7477,i-1652], [7481,i+1408], [7481,i-1408], [7489,i+1591], [7489,i-1591], [7517,i+3409], [7517,i-3409], [7529,i+2445], [7529,i-2445], [7537,i+1049], [7537,i-1049], [7541,i+2867], [7541,i-2867], [7549,i+2931], [7549,i-2931], [7561,i+2923], [7561,i-2923], [7573,i+3743], [7573,i-3743], [7577,i+1540], [7577,i-1540], [7589,i+3270], [7589,i-3270], [7621,i+2038], [7621,i-2038], [7649,i+2363], [7649,i-2363], [7669,i+2292], [7669,i-2292], [7673,i+277], [7673,i-277], [7681,i+3383], [7681,i-3383], [7717,i+2953], [7717,i-2953], [7741,i+3199], [7741,i-3199], [7753,i+2555], [7753,i-2555], [7757,i+812], [7757,i-812], [7789,i+3378], [7789,i-3378], [7793,i+2214], [7793,i-2214], [7817,i+2564], [7817,i-2564], [7829,i+2037], [7829,i-2037], [7841,i+198], [7841,i-198], [7853,i+1759], [7853,i-1759], [7873,i+3590], [7873,i-3590]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesList := [* 0, -2, -2, 1, -2, -2, 2, 2, 6, 6, 6, 6, -6, -6, -14, -2, -2, -2, -2, 10, 10, -6, -6, 2, 2, -18, -18, -2, -2, 18, 18, -6, -6, -6, 14, 14, -2, -2, 6, 6, 6, 6, 2, 2, -18, -18, 22, 22, 10, 10, 18, 18, 2, 2, -10, -10, -26, -26, 26, 26, -18, -18, -6, -6, 6, 6, 18, 18, 30, 30, 2, 2, -22, -10, -10, -2, -2, 14, 14, -30, -30, -6, -6, -10, -10, -14, -14, -14, -14, -22, -22, -26, -26, 6, 6, 26, 26, 18, -18, -18, -26, -26, 10, 10, 2, 2, -14, -14, -38, -38, 38, 38, 42, 42, -14, -14, 6, 6, -10, -10, 34, 34, -2, -2, 6, 6, -10, -10, 14, 14, 38, 38, -22, -22, 2, 2, -18, -18, 22, 22, 26, 26, 30, 30, -50, -50, -10, -10, 42, 42, -18, -18, 50, 50, 50, 50, 42, 42, 6, 6, -54, -54, 2, -30, -30, -26, -26, -46, -46, -50, -50, 30, 30, 10, 10, 26, 26, 30, 30, 14, 14, 54, 54, -54, -54, -18, -18, -50, -50, 10, 10, -62, -62, 22, 22, -22, -22, -14, -14, 46, 46, 2, 2, 38, 38, 38, 38, 34, 34, 6, 6, -22, -22, 18, 18, -18, -18, -22, -22, -62, -62, 54, 54, -10, -10, 18, 18, -10, -10, -6, -6, -66, -66, 10, 10, 50, 50, -50, -50, 46, 46, 18, 18, 78, 78, -30, -30, 10, 10, 54, 54, 38, 38, 78, 78, 58, 58, 22, 22, -66, -66, -14, -14, 38, 38, 26, 26, 62, 62, -50, -50, -38, -38, -14, -14, 62, 62, 10, 10, -70, -58, -58, 50, 50, -18, -18, 2, 2, -26, -26, -38, -38, -18, -18, -74, -74, -66, -66, 74, 74, -58, -58, 2, 2, -34, -34, -58, -58, 30, 30, -46, -46, -22, -22, 2, 2, -78, -78, -38, -38, -58, -58, 58, 58, 18, 18, -94, 30, 30, -2, -2, -42, -42, -2, -2, 66, 66, -22, -22, 22, 22, -54, -54, 78, 78, -26, -26, -42, -42, -34, -34, -22, -22, 22, 22, -10, -10, 26, 26, 34, 34, 38, 38, 42, 42, 42, 42, -42, -42, -38, -38, -2, -2, -82, -82, 34, 34, 66, 66, 26, 26, -42, -42, -70, -70, -30, -30, -58, -58, -62, -62, -98, -98, 58, 58, -6, -6, 46, 46, 62, 62, 18, 18, -22, -22, 46, 46, -2, -2, 66, 66, -46, -46, 46, 46, 42, 42, 22, 22, -30, -30, -10, -10, -58, -58, 10, 10, -10, -10, -6, -6, -38, -38, 62, 62, -14, -14, -22, -22, -10, -10, 50, 50, 70, 70, -14, -14, -30, -30, 34, 34, -66, -66, -6, -6, 82, 82, -18, -18, 14, 14, 38, 38, 90, 90, 70, 70, -14, -14, 50, 50, -30, -30, -66, -66, 22, 22, 14, 14, -54, -54, -102, -102, 34, 34, 14, 14, -98, -98, -102, -98, -98, 74, 74, 118, 118, -74, -74, -50, -50, -26, -26, -6, -6, 30, 30, 114, 114, 38, 38, -38, -38, 102, 102, -14, -14, 30, 30, -2, -2, 70, 70, -110, -110, 58, 58, -14, -14, 94, 94, 22, 22, -70, -70, 94, 94, 102, 102, -6, -6, 114, 114, -106, -106, 74, 74, 30, 30, 2, 2, 54, 54, -26, -26, -46, -46, 90, 90, 26, 26, -50, -50, 34, 34, -66, -66, -70, -70, 54, 54, 18, 18, 42, 42, -6, -6, -114, -114, 34, 34, -10, -10, 102, 102, -14, -14, -30, -30, -54, -54, -30, -30, -90, -90, -90, -90, 62, 62, -90, -90, 90, 90, 94, 94, 90, 90, 42, 42, 66, 66, -118, 54, 54, -30, -30, 30, 30, 70, 70, 18, 18, 118, 118, 46, 46, 38, 38, -38, -38, 50, 50, -78, -78, -78, -78, -102, -102, 38, 38, -74, -74, 10, 10, 66, 66, -50, -50, -78, -78, 78, 78, -42, -42, 74, 74, -98, -98, 86, 86, 10, 10, -82, -82, -86, -86, -74, -74, 2, 2, 66, 66, -42, -42, -78, -74, -74, -6, -6, -34, -34, -6, -6, 114, 114, -82, -82, 14, 14, -38, -38, -14, -14, 30, 30, -42, -42, 10, 10, 98, 98, -78, -78, -90, -90, 14, 14, -34, -34, 114, 114, 118, 118, -102, -102, -82, -82, -110, -110, 74, 74, 126, 126, 6, 6, -46, -46, 38, 38, -62, -62, -114, -114, 110, 110, -22, -22, -10, -10, -54, -54, 30, 30, -6, -6, -10, -10, -10, -10, -82, -82, -86, -86, -10, -10, -90, -90, -102, -102, 14, 14, 14, 14, -86, -86, 66, 66, -114, -114, 126, 126, -6, -6, 90, 90, 98, 98, -10, -10, 102, 102, 86, 86, -114, -114, -6, -6, -22, -22, -18, -18, -14, -14, 106, 106, -74, -74, -10, -10, 30, 30, 106, 106, 38, 38, 118, 118, -94, 130, 130, -138, -138, 70, 70, -114, -114, -74, -74, 42, 42, -62, -62, 18, 18, 26, 26, -154, -154, -82, -82, 30, 30, 70, 70, -78, -78, 54, 54, -70, -70, 82, 82, -70, -70, -62, -62, 154, 154, -86, -86, 82, 82, -34, -34, 14, 14, 54, 54, 86, 86, 50, 50, -30, -30, 38, 38, 86, 86, 30, 30, 2, 2, 42, 42, 14, 14, -118, -118, 46, 46, -126, -126, -70, -70, -22, -22, -18, -18, -150, -18, -18, 54, 54, -110, -110, 50, 50, -122, -122, -86, -86, 126, 126, -110, -110, 46, 46, 30, 30, 82, 82, -38, -38, 10, 10, 106, 106, -50, -50, -106, -106, -138, -138, -2, -2, -158, -158, -82, -82, 154, 154, -74, -74, -34, -34, -54, -54, -94, -94, 122, 122, -6, -6, 18, 18, 70, 70, 122, 122, 2, 2, 6, 6, 10, 10, 50, 50, 46, 46, 94, 94, -22, -22, 70, 70, -102, -102, 30, 30, 22, 22, 18, 18, 70, 70, 138, 138, 2, 2, -74, -74, -82, -82, 10, 10, -90, -90, -114, -114, -46, -46, 138, 138, 14, 14, 66, 66, 6, 6, 34, 34 *]; heckeEigenvalues := AssociativeArray(); for i in [1..#heckeEigenvaluesList] do heckeEigenvalues[primes[i]] := heckeEigenvaluesList[i]; end for; ALEigenvalues := AssociativeArray(); 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;