/* 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![3, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [3,a], [3,a-1], [2,2], [5,a+1], [5,a-2], [11,a+5], [23,a+4], [23,a-5], [31,a+9], [31,a-10], [37,a+13], [37,a-14], [47,a+20], [47,a-21], [7,7], [53,a+12], [53,a-13], [59,a+7], [59,a-8], [67,a+24], [67,a-25], [71,a+14], [71,a-15], [89,a+36], [89,a-37], [97,a+29], [97,a-30], [103,a+17], [103,a-18], [113,a+10], [113,a-11], [137,a+58], [137,a-59], [157,a+48], [157,a-49], [163,a+28], [163,a-29], [13,13], [179,a+69], [179,a-70], [181,a+64], [181,a-65], [191,a+53], [191,a-54], [199,a+31], [199,a-32], [223,a+49], [223,a-50], [229,a+101], [229,a-102], [251,a+47], [251,a-48], [257,a+97], [257,a-98], [269,a+63], [269,a-64], [17,17], [311,a+30], [311,a-31], [313,a+68], [313,a-69], [317,a+92], [317,a-93], [331,a+104], [331,a-105], [353,a+32], [353,a-33], [19,19], [367,a+128], [367,a-129], [379,a+43], [379,a-44], [383,a+19], [383,a-20], [389,a+177], [389,a-178], [397,a+165], [397,a-166], [401,a+148], [401,a-149], [419,a+124], [419,a-125], [421,a+35], [421,a-36], [433,a+46], [433,a-47], [443,a+81], [443,a-82], [449,a+183], [449,a-184], [463,a+207], [463,a-208], [467,a+179], [467,a-180], [487,a+169], [487,a-170], [499,a+86], [499,a-87], [509,a+22], [509,a-23], [521,a+39], [521,a-40], [577,a+184], [577,a-185], [587,a+281], [587,a-282], [599,a+181], [599,a-182], [617,a+74], [617,a-75], [619,a+82], [619,a-83], [631,a+168], [631,a-169], [641,a+271], [641,a-272], [643,a+283], [643,a-284], [647,a+98], [647,a-99], [653,a+25], [653,a-26], [661,a+44], [661,a-45], [683,a+145], [683,a-146], [691,a+45], [691,a-46], [709,a+331], [709,a-332], [719,a+282], [719,a-283], [727,a+233], [727,a-234], [751,a+199], [751,a-200], [757,a+61], [757,a-62], [773,a+186], [773,a-187], [797,a+234], [797,a-235], [823,a+368], [823,a-369], [829,a+111], [829,a-112], [839,a+150], [839,a-151], [29,29], [859,a+113], [859,a-114], [863,a+309], [863,a-310], [881,a+170], [881,a-171], [883,a+142], [883,a-143], [907,a+352], [907,a-353], [911,a+440], [911,a-441], [929,a+405], [929,a-406], [947,a+266], [947,a-267], [971,a+300], [971,a-301], [977,a+401], [977,a-402], [983,a+444], [983,a-445], [991,a+54], [991,a-55], [1013,a+213], [1013,a-214], [1021,a+427], [1021,a-428], [1039,a+483], [1039,a-484], [1049,a+496], [1049,a-497], [1061,a+218], [1061,a-219], [1087,a+423], [1087,a-424], [1093,a+482], [1093,a-483], [1103,a+57], [1103,a-58], [1109,a+166], [1109,a-167], [1123,a+192], [1123,a-193], [1153,a+131], [1153,a-132], [1171,a+76], [1171,a-77], [1181,a+59], [1181,a-60], [1193,a+34], [1193,a-35], [1213,a+115], [1213,a-116], [1237,a+105], [1237,a-106], [1259,a+510], [1259,a-511], [1277,a+292], [1277,a-293], [1279,a+138], [1279,a-139], [1291,a+357], [1291,a-358], [1301,a+460], [1301,a-461], [1303,a+62], [1303,a-63], [1307,a+635], [1307,a-636], [1321,a+350], [1321,a-351], [1367,a+559], [1367,a-560], [1373,a+143], [1373,a-144], [1409,a+37], [1409,a-38], [1423,a+339], [1423,a-340], [1433,a+449], [1433,a-450], [1439,a+310], [1439,a-311], [1453,a+277], [1453,a-278], [1483,a+333], [1483,a-334], [1489,a+200], [1489,a-201], [1499,a+235], [1499,a-236], [1511,a+394], [1511,a-395], [1523,a+418], [1523,a-419], [1543,a+263], [1543,a-264], [1549,a+204], [1549,a-205], [1567,a+88], [1567,a-89], [1571,a+153], [1571,a-154], [1607,a+565], [1607,a-566], [1609,a+469], [1609,a-470], [1621,a+672], [1621,a-673], [1637,a+783], [1637,a-784], [41,41], [1697,a+631], [1697,a-632], [1699,a+276], [1699,a-277], [1709,a+675], [1709,a-676], [1721,a+619], [1721,a-620], [1741,a+466], [1741,a-467], [1747,a+568], [1747,a-569], [1753,a+72], [1753,a-73], [1783,a+202], [1783,a-203], [1787,a+313], [1787,a-314], [1831,a+624], [1831,a-625], [43,43], [1871,a+722], [1871,a-723], [1873,a+389], [1873,a-390], [1879,a+874], [1879,a-875], [1901,a+75], [1901,a-76], [1907,a+434], [1907,a-435], [1951,a+132], [1951,a-133], [1973,a+900], [1973,a-901], [2003,a+77], [2003,a-78], [2011,a+372], [2011,a-373], [2017,a+597], [2017,a-598], [2027,a+862], [2027,a-863], [2029,a+906], [2029,a-907], [2039,a+607], [2039,a-608], [2069,a+528], [2069,a-529], [2083,a+990], [2083,a-991], [2099,a+588], [2099,a-589], [2113,a+967], [2113,a-968], [2137,a+1028], [2137,a-1029], [2143,a+103], [2143,a-104], [2161,a+80], [2161,a-81], [2179,a+1019], [2179,a-1020], [2203,a+494], [2203,a-495], [2237,a+878], [2237,a-879], [2267,a+395], [2267,a-396], [2269,a+106], [2269,a-107], [2281,a+744], [2281,a-745], [2293,a+504], [2293,a-505], [2297,a+321], [2297,a-322], [2311,a+598], [2311,a-599], [2333,a+241], [2333,a-242], [2341,a+621], [2341,a-622], [2347,a+955], [2347,a-956], [2357,a+1084], [2357,a-1085], [2377,a+808], [2377,a-809], [2381,a+84], [2381,a-85], [2399,a+254], [2399,a-255], [2423,a+584], [2423,a-585], [2447,a+852], [2447,a-853], [2467,a+914], [2467,a-915], [2473,a+590], [2473,a-591], [2531,a+1112], [2531,a-1113], [2539,a+1100], [2539,a-1101], [2557,a+758], [2557,a-759], [2579,a+778], [2579,a-779], [2621,a+1177], [2621,a-1178], [2633,a+1106], [2633,a-1107], [2663,a+1202], [2663,a-1203], [2671,a+89], [2671,a-90], [2677,a+909], [2677,a-910], [2687,a+155], [2687,a-156], [2689,a+355], [2689,a-356], [2693,a+556], [2693,a-557], [2699,a+348], [2699,a-349], [2707,a+201], [2707,a-202], [2711,a+432], [2711,a-433], [2729,a+514], [2729,a-515], [2731,a+90], [2731,a-91], [2753,a+532], [2753,a-533], [2777,a+353], [2777,a-354], [2797,a+1169], [2797,a-1170], [2803,a+458], [2803,a-459], [2819,a+1068], [2819,a-1069], [2843,a+206], [2843,a-207], [2861,a+993], [2861,a-994], [2887,a+944], [2887,a-945], [2897,a+814], [2897,a-815], [2909,a+929], [2909,a-930], [2927,a+1436], [2927,a-1437], [2953,a+121], [2953,a-122], [2957,a+698], [2957,a-699], [2963,a+816], [2963,a-817], [2971,a+795], [2971,a-796], [3001,a+636], [3001,a-637], [3019,a+285], [3019,a-286], [3023,a+779], [3023,a-780], [3037,a+920], [3037,a-921], [3041,a+95], [3041,a-96], [3061,a+1132], [3061,a-1133], [3067,a+371], [3067,a-372], [3083,a+55], [3083,a-56], [3089,a+1118], [3089,a-1119], [3169,a+1475], [3169,a-1476], [3191,a+324], [3191,a-325], [3217,a+1053], [3217,a-1054], [3221,a+491], [3221,a-492], [3257,a+1322], [3257,a-1323], [3259,a+759], [3259,a-760], [3271,a+221], [3271,a-222], [3301,a+99], [3301,a-100], [3323,a+684], [3323,a-685], [3331,a+1129], [3331,a-1130], [3347,a+729], [3347,a-730], [3359,a+1076], [3359,a-1077], [3371,a+1059], [3371,a-1060], [3389,a+774], [3389,a-775], [3391,a+1321], [3391,a-1322], [3413,a+615], [3413,a-616], [3433,a+999], [3433,a-1000], [3457,a+1362], [3457,a-1363], [3463,a+567], [3463,a-568], [3469,a+658], [3469,a-659], [3491,a+686], [3491,a-687], [3499,a+817], [3499,a-818], [3529,a+398], [3529,a-399], [3547,a+1505], [3547,a-1506], [3557,a+516], [3557,a-517], [3613,a+1304], [3613,a-1305], [3617,a+312], [3617,a-313], [3623,a+1252], [3623,a-1253], [3631,a+1332], [3631,a-1333], [3677,a+1180], [3677,a-1181], [3697,a+1119], [3697,a-1120], [3701,a+182], [3701,a-183], [3719,a+642], [3719,a-643], [61,61], [3727,a+136], [3727,a-137], [3733,a+725], [3733,a-726], [3767,a+1323], [3767,a-1324], [3793,a+1736], [3793,a-1737], [3821,a+1032], [3821,a-1033], [3833,a+1608], [3833,a-1609], [3851,a+782], [3851,a-783], [3853,a+107], [3853,a-108], [3877,a+1610], [3877,a-1611], [3881,a+1709], [3881,a-1710], [3917,a+1687], [3917,a-1688], [3919,a+1699], [3919,a-1700], [3931,a+1289], [3931,a-1290], [3943,a+667], [3943,a-668], [3947,a+1600], [3947,a-1601], [4007,a+316], [4007,a-317], [4013,a+987], [4013,a-988], [4019,a+1493], [4019,a-1494], [4027,a+1115], [4027,a-1116], [4049,a+915], [4049,a-916], [4051,a+246], [4051,a-247], [4057,a+1373], [4057,a-1374], [4073,a+1503], [4073,a-1504], [4079,a+1560], [4079,a-1561], [4093,a+796], [4093,a-797], [4129,a+924], [4129,a-925], [4139,a+1482], [4139,a-1483], [4159,a+558], [4159,a-559], [4211,a+1645], [4211,a-1646], [4217,a+251], [4217,a-252], [4229,a+1704], [4229,a-1705], [4261,a+629], [4261,a-630], [4271,a+1039], [4271,a-1040], [4273,a+438], [4273,a-439], [4283,a+689], [4283,a-690], [4327,a+654], [4327,a-655], [4337,a+947], [4337,a-948], [4339,a+1223], [4339,a-1224], [4349,a+649], [4349,a-650], [4357,a+1615], [4357,a-1616], [4409,a+771], [4409,a-772], [4423,a+1134], [4423,a-1135], [4447,a+1504], [4447,a-1505], [4481,a+1232], [4481,a-1233], [4493,a+1219], [4493,a-1220], [4513,a+1998], [4513,a-1999], [4519,a+863], [4519,a-864], [4547,a+1795], [4547,a-1796], [4591,a+151], [4591,a-152], [4603,a+117], [4603,a-118], [4621,a+390], [4621,a-391], [4643,a+2038], [4643,a-2039], [4651,a+1244], [4651,a-1245], [4657,a+860], [4657,a-861], [4673,a+2040], [4673,a-2041], [4679,a+2168], [4679,a-2169], [4691,a+660], [4691,a-661], [4723,a+798], [4723,a-799], [4733,a+461], [4733,a-462], [4783,a+980], [4783,a-981], [4789,a+2260], [4789,a-2261], [4799,a+1341], [4799,a-1342], [4801,a+2035], [4801,a-2036], [4871,a+896], [4871,a-897], [4877,a+209], [4877,a-210], [4889,a+1339], [4889,a-1340], [4909,a+2286], [4909,a-2287], [4931,a+2014], [4931,a-2015], [4933,a+2330], [4933,a-2331], [4937,a+608], [4937,a-609], [4943,a+2179], [4943,a-2180], [4951,a+2239], [4951,a-2240], [4973,a+70], [4973,a-71], [4987,a+273], [4987,a-274], [4999,a+821], [4999,a-822], [5003,a+122], [5003,a-123], [5009,a+430], [5009,a-431], [5021,a+1153], [5021,a-1154], [5039,a+2001], [5039,a-2002], [5087,a+1774], [5087,a-1775], [5107,a+2067], [5107,a-2068], [5113,a+918], [5113,a-919], [5119,a+490], [5119,a-491], [5153,a+2437], [5153,a-2438], [5171,a+278], [5171,a-279], [5179,a+2242], [5179,a-2243], [5197,a+964], [5197,a-965], [5237,a+1363], [5237,a-1364], [5261,a+602], [5261,a-603], [5273,a+1029], [5273,a-1030], [5281,a+1784], [5281,a-1785], [5303,a+488], [5303,a-489], [73,73], [5333,a+848], [5333,a-849], [5347,a+163], [5347,a-164], [5351,a+633], [5351,a-634], [5393,a+2233], [5393,a-2234], [5399,a+872], [5399,a-873], [5413,a+1953], [5413,a-1954], [5417,a+1860], [5417,a-1861], [5437,a+2259], [5437,a-2260], [5443,a+2373], [5443,a-2374], [5449,a+2547], [5449,a-2548], [5471,a+1079], [5471,a-1080], [5479,a+1209], [5479,a-1210], [5483,a+1772], [5483,a-1773], [5501,a+222], [5501,a-223], [5503,a+2099], [5503,a-2100], [5527,a+2236], [5527,a-2237], [5531,a+949], [5531,a-950], [5569,a+247], [5569,a-248], [5581,a+2356], [5581,a-2357], [5591,a+129], [5591,a-130], [5641,a+2005], [5641,a-2006], [5647,a+2739], [5647,a-2740], [5657,a+457], [5657,a-458], [5659,a+880], [5659,a-881], [5669,a+1398], [5669,a-1399], [5701,a+506], [5701,a-507], [5743,a+293], [5743,a-294], [5779,a+2183], [5779,a-2184], [5791,a+1612], [5791,a-1613], [5801,a+734], [5801,a-735], [5813,a+424], [5813,a-425], [5839,a+1360], [5839,a-1361], [5857,a+1421], [5857,a-1422], [5861,a+296], [5861,a-297], [5867,a+2713], [5867,a-2714], [5879,a+2824], [5879,a-2825], [5897,a+823], [5897,a-824], [5923,a+666], [5923,a-667], [5927,a+639], [5927,a-640], [5987,a+1721], [5987,a-1722], [6007,a+1858], [6007,a-1859], [6011,a+1826], [6011,a-1827], [6029,a+1765], [6029,a-1766], [6037,a+1567], [6037,a-1568], [6043,a+1435], [6043,a-1436], [6053,a+1237], [6053,a-1238], [6073,a+1768], [6073,a-1769], [6121,a+135], [6121,a-136], [6131,a+391], [6131,a-392], [6143,a+1201], [6143,a-1202], [6163,a+2213], [6163,a-2214], [6197,a+2191], [6197,a-2192], [6229,a+683], [6229,a-684], [79,79], [6257,a+1863], [6257,a-1864], [6263,a+306], [6263,a-307], [6271,a+1533], [6271,a-1534], [6301,a+765], [6301,a-766], [6317,a+2435], [6317,a-2436], [6323,a+79], [6323,a-80], [6329,a+1982], [6329,a-1983], [6337,a+3036], [6337,a-3037], [6359,a+1915], [6359,a-1916], [6361,a+662], [6361,a-663], [6367,a+1764], [6367,a-1765], [6373,a+178], [6373,a-179], [6389,a+2571], [6389,a-2572], [6427,a+240], [6427,a-241], [6449,a+2209], [6449,a-2210], [6451,a+1690], [6451,a-1691], [6469,a+311], [6469,a-312], [6473,a+2457], [6473,a-2458], [6491,a+1208], [6491,a-1209], [6521,a+2113], [6521,a-2114], [6571,a+2411], [6571,a-2412], [6581,a+140], [6581,a-141], [6637,a+546], [6637,a-547], [6653,a+2426], [6653,a-2427], [6659,a+3125], [6659,a-3126], [6689,a+805], [6689,a-806], [6691,a+595], [6691,a-596], [6703,a+1498], [6703,a-1499], [6719,a+1899], [6719,a-1900], [6733,a+864], [6733,a-865], [6737,a+2536], [6737,a-2537], [6763,a+318], [6763,a-319], [6779,a+2992], [6779,a-2993], [6781,a+1813], [6781,a-1814], [6791,a+1162], [6791,a-1163], [6803,a+2635], [6803,a-2636], [6823,a+1123], [6823,a-1124], [6829,a+1061], [6829,a-1062], [6857,a+2984], [6857,a-2985], [6869,a+1423], [6869,a-1424], [83,83], [6911,a+2902], [6911,a-2903], [6917,a+249], [6917,a-250], [6961,a+144], [6961,a-145], [6967,a+830], [6967,a-831], [6977,a+323], [6977,a-324], [6983,a+1425], [6983,a-1426], [6997,a+2304], [6997,a-2305], [7001,a+1576], [7001,a-1577], [7019,a+3228], [7019,a-3229], [7027,a+1492], [7027,a-1493], [7043,a+2982], [7043,a-2983], [7109,a+3481], [7109,a-3482], [7121,a+2566], [7121,a-2567], [7129,a+1266], [7129,a-1267], [7151,a+815], [7151,a-816], [7159,a+2807], [7159,a-2808], [7177,a+2841], [7177,a-2842], [7187,a+2332], [7187,a-2333], [7219,a+1722], [7219,a-1723], [7243,a+2272], [7243,a-2273], [7247,a+2023], [7247,a-2024], [7253,a+147], [7253,a-148], [7283,a+3513], [7283,a-3514], [7297,a+1063], [7297,a-1064], [7307,a+1909], [7307,a-1910], [7309,a+573], [7309,a-574], [7331,a+741], [7331,a-742], [7349,a+3508], [7349,a-3509], [7351,a+2276], [7351,a-2277], [7393,a+724], [7393,a-725], [7417,a+447], [7417,a-448], [7451,a+149], [7451,a-150], [7459,a+2778], [7459,a-2779], [7481,a+1592], [7481,a-1593], [7489,a+193], [7489,a-194], [7507,a+2473], [7507,a-2474], [7517,a+433], [7517,a-434], [7529,a+643], [7529,a-644], [7547,a+1225], [7547,a-1226], [7549,a+336], [7549,a-337], [7561,a+2048], [7561,a-2049], [7573,a+1484], [7573,a-1485], [7577,a+712], [7577,a-713], [7583,a+2801], [7583,a-2802], [7591,a+500], [7591,a-501], [7621,a+3182], [7621,a-3183], [7639,a+338], [7639,a-339], [7643,a+2771], [7643,a-2772], [7649,a+2484], [7649,a-2485], [7681,a+2589], [7681,a-2590], [7687,a+1628], [7687,a-1629], [7703,a+1316], [7703,a-1317], [7723,a+196], [7723,a-197], [7727,a+2478], [7727,a-2479], [7753,a+152], [7753,a-153], [7759,a+3626], [7759,a-3627], [7789,a+2814], [7789,a-2815], [7793,a+794]]; primes := [ideal : I in primesArray]; heckePol := x; K := Rationals(); e := 1; heckeEigenvaluesList := [* 0, -1, 1, -2, 2, 0, 4, -4, 0, 0, -2, -2, -8, 8, -6, -6, 6, 0, 0, 12, 12, -8, 8, -14, 14, 2, 2, 16, 16, 6, -6, 22, -22, -2, -2, 4, 4, 26, -4, 4, -10, -10, 0, 0, 0, 0, -16, -16, -10, -10, 12, -12, 22, -22, 10, -10, -14, -12, 12, -14, -14, -18, 18, -20, -20, -14, 14, 18, 8, 8, -20, -20, -36, 36, 10, -10, 22, 22, 30, -30, -4, 4, -10, -10, -6, -6, 24, -24, 6, -6, 24, 24, -8, 8, -8, -8, 20, 20, -30, 30, 30, -30, 2, 2, 8, -8, 36, -36, 2, -2, 44, 44, -32, -32, -30, 30, 36, 36, -12, 12, 46, -46, 22, 22, 44, -44, 12, 12, 6, 6, -24, 24, 8, 8, -32, -32, -42, -42, -14, 14, 42, -42, 16, 16, 14, 14, 20, -20, -38, 4, 4, 4, -4, -18, 18, -44, -44, -12, -12, 0, 0, -30, 30, -52, 52, 0, 0, 42, -42, 36, -36, -40, -40, -6, 6, -10, -10, 56, 56, -10, 10, 42, -42, -48, -48, -26, -26, 24, -24, -6, 6, -4, -4, 34, 34, 12, 12, -18, 18, 14, -14, 14, 14, 22, 22, 36, -36, 38, -38, 40, 40, -20, -20, -10, 10, 16, 16, -48, 48, -62, -62, 32, -32, -54, 54, 50, -50, -56, -56, -26, 26, 20, -20, 14, 14, 36, 36, 34, 34, 60, -60, -20, 20, 16, -16, 24, 24, 14, 14, 8, 8, -12, 12, -48, 48, 10, 10, 70, 70, 18, -18, -62, -2, 2, 20, 20, -14, 14, -18, 18, -42, -42, 68, 68, -54, -54, 56, 56, 28, -28, -40, -40, 66, 60, -60, 66, 66, -64, -64, -22, 22, 48, -48, -48, -48, -26, 26, 16, -16, 60, 60, -62, -62, -52, 52, 30, 30, -4, 4, -30, 30, -36, -36, 24, -24, 34, 34, 58, 58, -16, -16, 50, 50, -4, -4, 44, 44, 62, -62, -72, 72, 70, 70, -70, -70, 94, 94, 82, -82, -40, -40, -34, 34, 38, 38, 28, 28, -62, 62, -22, -22, 10, -10, 40, -40, -24, 24, 12, -12, 92, 92, 26, 26, 88, -88, -36, -36, -18, -18, -84, 84, 78, -78, -6, 6, -24, 24, -8, -8, -58, -58, -48, 48, 34, 34, 14, -14, 60, -60, 68, 68, 52, -52, -30, 30, 52, 52, 34, -34, 58, -58, -2, -2, 4, 4, 100, -100, -36, 36, -90, 90, -88, -88, -62, 62, 6, -6, -32, 32, -6, -6, -58, 58, -4, 4, 68, 68, 10, 10, -20, -20, 96, -96, 62, 62, 90, -90, -50, -50, -68, -68, -64, 64, -10, 10, 26, 26, 68, -68, 18, 18, -90, 90, -78, 78, 36, 36, -88, -88, 70, 70, -36, 36, -20, -20, -32, 32, -84, 84, 32, -32, -6, 6, 40, 40, -34, 34, 26, 26, -78, -78, 56, 56, 30, 30, -28, 28, 36, 36, 50, 50, 52, 52, 42, -42, -66, -66, -42, 42, -16, 16, 8, 8, -78, 78, 58, 58, -18, 18, -80, 80, 42, -72, -72, -74, -74, 52, -52, -94, -94, -30, 30, 94, -94, 12, -12, -34, -34, -98, -98, -2, 2, 78, -78, -96, -96, 60, 60, 56, 56, -48, 48, 108, -108, -46, 46, -56, 56, -52, -52, 10, -10, -60, -60, 98, 98, -94, 94, 100, -100, -66, -66, 10, 10, 20, -20, -56, -56, 120, -120, -102, 102, 86, -86, -90, -90, -100, 100, -14, -14, 44, -44, 32, 32, -42, 42, -20, -20, -126, 126, -122, -122, -126, 126, -16, -16, -88, -88, -22, 22, 114, -114, -14, -14, 80, 80, -68, 68, -8, -8, -4, -4, -50, -50, -44, 44, -92, -92, 2, 2, -86, 86, 0, 0, 100, -100, -36, -36, -54, 54, -64, -64, 110, 110, -36, 36, 130, 130, -60, 60, 2, -2, -130, 130, -34, -34, 40, -40, 54, 54, -102, 102, 36, -36, 120, 120, 66, -66, -52, -52, -120, -120, -4, 4, 74, -74, 50, -50, -24, 24, -48, 48, -28, -28, 74, 74, -40, -40, 66, -66, -120, 120, 44, 44, 38, 38, 78, -78, 58, -58, 54, -54, -110, -110, 4, -4, 66, 34, -34, 28, 28, 80, -80, -66, 66, 40, -40, -74, -74, 62, -62, 78, 78, -4, -4, -14, -14, -88, 88, 40, 40, 4, -4, -82, 82, 56, 56, 128, 128, 32, -32, 114, 114, 38, 38, 40, -40, -38, -38, -128, -128, -42, 42, 116, 116, 50, -50, 118, 118, 16, 16, 44, 44, 40, 40, -70, 70, 34, -34, 16, 16, 2, 2, 62, -62, 72, -72, -36, 36, -98, 98, 84, 84, -108, 108, -52, 52, 128, 128, 0, 0, 90, -90, -2, -2, -4, -4, 94, -94, -46, -46, -38, -38, -120, 120, 24, -24, -156, -156, 138, -138, 150, 150, -22, 42, -42, -96, 96, -72, -72, 30, 30, 18, -18, -96, 96, -90, 90, 2, 2, -4, 4, 10, 10, 112, 112, 134, 134, -14, 14, -28, -28, 26, -26, -20, -20, -90, -90, 106, -106, 60, -60, -70, 70, 108, 108, 110, -110, -82, -82, -14, 14, 120, -120, -34, 34, 68, 68, -64, -64, -40, 40, 46, 46, -42, 42, -76, -76, -60, 60, -82, -82, -148, 148, 44, -44, -16, -16, 46, 46, 118, -118, 50, -50, -86, -128, 128, 58, -58, 130, 130, -48, -48, -162, 162, 44, -44, -58, -58, -18, 18, -56, 56, -52, -52, -24, 24, 30, -30, 42, -42, 106, 106, -132, 132, -120, -120, 82, 82, -88, 88, 100, 100, 116, 116, 48, -48, 114, -114, -84, 84, 82, 82, 112, -112, -50, -50, -12, 12, 150, -150, -160, -160, -166, -166, 42, 42, 72, -72, -100, -100, 30, -30, 130, 130, 108, 108, 22, -22, 94, -94, 8, -8, -106, -106, -70, -70, -74, -74, -18, 18, -24, 24, 32, 32, 70, 70, -16, -16, -104, 104, 46, -46, -70, -70, 112, 112, -136, 136, 44, 44, 32, -32, 26, 26, -64, -64, -50, -50, 66 *]; 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;