Properties

Label 4.37.b.b
Level 4
Weight 37
Character orbit 4.b
Analytic conductor 32.837
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4 = 2^{2} \)
Weight: \( k \) = \( 37 \)
Character orbit: \([\chi]\) = 4.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.8365034637\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{240}\cdot 3^{24}\cdot 5^{6}\cdot 7^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 11077 + \beta_{1} ) q^{2} \) \( + ( 50 + 198 \beta_{1} + \beta_{2} ) q^{3} \) \( + ( -350666438 + 11150 \beta_{1} + 12 \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 363505039903 - 2225516 \beta_{1} + 146 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{5} \) \( + ( -13601811751576 + 2206189 \beta_{1} - 34093 \beta_{2} + 217 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{6} \) \( + ( -82015045 - 328199576 \beta_{1} + 73757 \beta_{2} + 4063 \beta_{3} - 6 \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{7} \) \( + ( 743597395761271 - 470929655 \beta_{1} - 1526123 \beta_{2} + 11977 \beta_{3} - 355 \beta_{4} - 351 \beta_{5} + 9 \beta_{7} - \beta_{8} ) q^{8} \) \( + ( -58448994965113227 + 135361742291 \beta_{1} - 8938479 \beta_{2} - 31791 \beta_{3} + 6644 \beta_{4} + 2378 \beta_{5} - 89 \beta_{7} - \beta_{8} + \beta_{9} ) q^{9} \) \(+O(q^{10})\) \( q\) \(+(11077 + \beta_{1}) q^{2}\) \(+(50 + 198 \beta_{1} + \beta_{2}) q^{3}\) \(+(-350666438 + 11150 \beta_{1} + 12 \beta_{2} + \beta_{3}) q^{4}\) \(+(363505039903 - 2225516 \beta_{1} + 146 \beta_{2} + 2 \beta_{3} + \beta_{4}) q^{5}\) \(+(-13601811751576 + 2206189 \beta_{1} - 34093 \beta_{2} + 217 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{7}) q^{6}\) \(+(-82015045 - 328199576 \beta_{1} + 73757 \beta_{2} + 4063 \beta_{3} - 6 \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7}) q^{7}\) \(+(743597395761271 - 470929655 \beta_{1} - 1526123 \beta_{2} + 11977 \beta_{3} - 355 \beta_{4} - 351 \beta_{5} + 9 \beta_{7} - \beta_{8}) q^{8}\) \(+(-58448994965113227 + 135361742291 \beta_{1} - 8938479 \beta_{2} - 31791 \beta_{3} + 6644 \beta_{4} + 2378 \beta_{5} - 89 \beta_{7} - \beta_{8} + \beta_{9}) q^{9}\) \(+(-148633835474836756 + 388194666285 \beta_{1} - 134731899 \beta_{2} - 2166565 \beta_{3} + 53570 \beta_{4} + 3866 \beta_{5} - 9 \beta_{6} - 111 \beta_{7} - 5 \beta_{8} - 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14}) q^{10}\) \(+(984018105830 + 3936048020861 \beta_{1} + 7577063 \beta_{2} - 4681635 \beta_{3} + 60846 \beta_{4} - 59172 \beta_{5} + 354 \beta_{6} + 3049 \beta_{7} + 28 \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{12} + \beta_{14} + \beta_{15}) q^{11}\) \(+(-2680179909634429287 - 13193612758197 \beta_{1} + 897456977 \beta_{2} + 1383235 \beta_{3} - 340249 \beta_{4} + 81194 \beta_{5} - 585 \beta_{6} - 35714 \beta_{7} - 265 \beta_{8} + 39 \beta_{9} - 7 \beta_{10} - 31 \beta_{11} - 70 \beta_{12} + 15 \beta_{13} - 6 \beta_{14} - 2 \beta_{15}) q^{12}\) \(+(12163234415351318317 - 65888152378161 \beta_{1} + 4294540820 \beta_{2} + 221759774 \beta_{3} - 7589710 \beta_{4} - 1307983 \beta_{5} + 1103 \beta_{6} - 90135 \beta_{7} + 283 \beta_{8} + 396 \beta_{9} - 53 \beta_{10} + 39 \beta_{11} - 261 \beta_{12} + 38 \beta_{13}) q^{13}\) \(+(22512054645520516676 - 3652171977162 \beta_{1} + 59063919167 \beta_{2} - 359748173 \beta_{3} + 2395301 \beta_{4} - 4075328 \beta_{5} + 8320 \beta_{6} + 73877 \beta_{7} - 3126 \beta_{8} - 1268 \beta_{9} - 14 \beta_{10} - 213 \beta_{11} - 326 \beta_{12} + 160 \beta_{13} - 280 \beta_{14} + 56 \beta_{15}) q^{14}\) \(+(-82350124979353 - 329053539003520 \beta_{1} - 172408458577 \beta_{2} + 1076184160 \beta_{3} - 4813975 \beta_{4} + 4230723 \beta_{5} - 642 \beta_{6} - 91848 \beta_{7} + 37100 \beta_{8} + 81 \beta_{9} + 1518 \beta_{10} - 836 \beta_{11} - 1343 \beta_{12} - 320 \beta_{13} + 1998 \beta_{14} - 50 \beta_{15}) q^{15}\) \(+(-\)\(14\!\cdots\!99\)\( + 975463181933136 \beta_{1} + 394502811483 \beta_{2} - 753004726 \beta_{3} + 237562202 \beta_{4} + 80732528 \beta_{5} + 457563 \beta_{6} - 2289979 \beta_{7} - 60212 \beta_{8} + 10649 \beta_{9} + 1604 \beta_{10} + 452 \beta_{11} + 5179 \beta_{12} + 436 \beta_{13} - 2856 \beta_{14} - 120 \beta_{15}) q^{16}\) \(+(\)\(26\!\cdots\!54\)\( - 8638735924673347 \beta_{1} + 557668406453 \beta_{2} + 47798182413 \beta_{3} - 799376346 \beta_{4} - 172602700 \beta_{5} + 379066 \beta_{6} - 68391 \beta_{7} - 192423 \beta_{8} + 24905 \beta_{9} + 4862 \beta_{10} + 24582 \beta_{11} + 18530 \beta_{12} - 4964 \beta_{13}) q^{17}\) \(+(\)\(86\!\cdots\!49\)\( - 59950331268154209 \beta_{1} + 6968306904646 \beta_{2} + 132828202830 \beta_{3} - 1969355532 \beta_{4} - 118949884 \beta_{5} - 8030958 \beta_{6} + 7562182 \beta_{7} + 222350 \beta_{8} - 142458 \beta_{9} + 23002 \beta_{10} + 29076 \beta_{11} + 25410 \beta_{12} + 2624 \beta_{13} - 1974 \beta_{14} - 3008 \beta_{15}) q^{18}\) \(+(-17934009248853034 - 71755650620106401 \beta_{1} + 10021836327165 \beta_{2} + 216104627729 \beta_{3} - 1156837188 \beta_{4} + 1023555678 \beta_{5} + 2513484 \beta_{6} + 111591905 \beta_{7} + 3602172 \beta_{8} + 73697 \beta_{9} + 42031 \beta_{10} + 25576 \beta_{11} - 56848 \beta_{12} + 8320 \beta_{13} - 52369 \beta_{14} + 879 \beta_{15}) q^{19}\) \(+(\)\(65\!\cdots\!72\)\( - 149964131903267964 \beta_{1} + 30970007288044 \beta_{2} + 372790957378 \beta_{3} - 6035723536 \beta_{4} - 2147568160 \beta_{5} + 103702260 \beta_{6} - 168683420 \beta_{7} + 3076280 \beta_{8} + 666700 \beta_{9} + 144760 \beta_{10} + 320760 \beta_{11} - 91220 \beta_{12} + 53160 \beta_{13} + 116080 \beta_{14} + 10960 \beta_{15}) q^{20}\) \(+(-\)\(10\!\cdots\!90\)\( - 174451247780161601 \beta_{1} + 11661032178090 \beta_{2} - 473936420972 \beta_{3} + 319465557 \beta_{4} - 1768268555 \beta_{5} - 1830573 \beta_{6} - 2452723555 \beta_{7} - 22139969 \beta_{8} - 880708 \beta_{9} + 511063 \beta_{10} - 1384189 \beta_{11} - 1674097 \beta_{12} + 129406 \beta_{13}) q^{21}\) \(+(-\)\(27\!\cdots\!48\)\( + 43529701053049227 \beta_{1} + 244067103740323 \beta_{2} + 3787600726573 \beta_{3} - 52469455468 \beta_{4} + 3121197213 \beta_{5} - 490144832 \beta_{6} + 141035537 \beta_{7} + 6856348 \beta_{8} + 5710024 \beta_{9} + 153612 \beta_{10} + 1981586 \beta_{11} + 714748 \beta_{12} + 430528 \beta_{13} + 409840 \beta_{14} + 73680 \beta_{15}) q^{22}\) \(+(208950824162332071 + 838298625394866584 \beta_{1} - 1246878999782277 \beta_{2} + 771659158794 \beta_{3} + 19334811999 \beta_{4} - 22183434919 \beta_{5} - 76594324 \beta_{6} + 17012749710 \beta_{7} - 14635324 \beta_{8} + 108691 \beta_{9} + 329266 \beta_{10} - 3922284 \beta_{11} - 4768709 \beta_{12} - 1041856 \beta_{13} - 41262 \beta_{14} + 1746 \beta_{15}) q^{23}\) \(+(\)\(31\!\cdots\!72\)\( - 2737001136089550448 \beta_{1} + 1296090158505756 \beta_{2} - 13777532301256 \beta_{3} + 566287003832 \beta_{4} + 3769990736 \beta_{5} + 1839199308 \beta_{6} + 466013188 \beta_{7} - 27071904 \beta_{8} - 26410876 \beta_{9} - 3406384 \beta_{10} + 8916176 \beta_{11} + 16562060 \beta_{12} + 2174736 \beta_{13} - 1802016 \beta_{14} - 345440 \beta_{15}) q^{24}\) \(+(\)\(38\!\cdots\!15\)\( + 9458936004151106220 \beta_{1} - 619173445295070 \beta_{2} - 20008468289590 \beta_{3} - 367332848270 \beta_{4} + 166992112350 \beta_{5} - 533794350 \beta_{6} - 82749628100 \beta_{7} + 453421700 \beta_{8} + 19791350 \beta_{9} - 7207050 \beta_{10} - 1276450 \beta_{11} - 3779750 \beta_{12} - 11046900 \beta_{13}) q^{25}\) \(+(-\)\(43\!\cdots\!20\)\( + 12891685997407136497 \beta_{1} + 4752286744688777 \beta_{2} - 69880471982465 \beta_{3} - 1932631164838 \beta_{4} + 200280685714 \beta_{5} - 444696765 \beta_{6} + 9997742005 \beta_{7} - 277364961 \beta_{8} - 72474431 \beta_{9} - 20590259 \beta_{10} + 37856502 \beta_{11} + 75932779 \beta_{12} + 9616512 \beta_{13} - 11058131 \beta_{14} - 1060736 \beta_{15}) q^{26}\) \(+(9709873670097739016 + 38904799500928124875 \beta_{1} - 32787848360392222 \beta_{2} - 136034954570889 \beta_{3} + 795017851516 \beta_{4} - 767372551462 \beta_{5} - 2138638980 \beta_{6} + 343114955191 \beta_{7} - 1375775452 \beta_{8} - 15372553 \beta_{9} - 725167 \beta_{10} - 79804680 \beta_{11} - 96107592 \beta_{12} - 26417792 \beta_{13} + 21026577 \beta_{14} - 407791 \beta_{15}) q^{27}\) \(+(-\)\(18\!\cdots\!06\)\( + 25612923814436302922 \beta_{1} + 9569761898614910 \beta_{2} - 5133915171238 \beta_{3} + 19097788390802 \beta_{4} + 482929235436 \beta_{5} - 21262278286 \beta_{6} + 22806623044 \beta_{7} - 399406478 \beta_{8} + 362030994 \beta_{9} - 36514898 \beta_{10} + 160178398 \beta_{11} + 76526924 \beta_{12} + 32304450 \beta_{13} + 9200716 \beta_{14} + 6411972 \beta_{15}) q^{28}\) \(+(-\)\(59\!\cdots\!41\)\( + 31047114101017300220 \beta_{1} - 2069158546527246 \beta_{2} + 57433681361514 \beta_{3} + 2773194676785 \beta_{4} + 787774931600 \beta_{5} - 558283528 \beta_{6} - 641190818112 \beta_{7} - 3331395192 \beta_{8} - 192924320 \beta_{9} + 63283024 \beta_{10} - 254803728 \beta_{11} - 476503400 \beta_{12} + 28508576 \beta_{13}) q^{29}\) \(+(\)\(22\!\cdots\!56\)\( - 3645991214842424102 \beta_{1} + 1809945772914677 \beta_{2} - 353218800223391 \beta_{3} - 78136021159473 \beta_{4} - 3692527344440 \beta_{5} + 127440000000 \beta_{6} - 87467173465 \beta_{7} + 2293968190 \beta_{8} + 275881700 \beta_{9} + 105715510 \beta_{10} + 547064465 \beta_{11} - 338730610 \beta_{12} + 28065760 \beta_{13} + 155521080 \beta_{14} + 9347880 \beta_{15}) q^{30}\) \(+(-92623238487411116422 - \)\(37\!\cdots\!24\)\( \beta_{1} + 95890696080038288 \beta_{2} - 3394974081010903 \beta_{3} - 4813590036271 \beta_{4} + 6919054429957 \beta_{5} + 33722708651 \beta_{6} - 927396167761 \beta_{7} + 10489523620 \beta_{8} + 515310843 \beta_{9} + 15924466 \beta_{10} + 1335779252 \beta_{11} + 117383443 \beta_{12} + 166401088 \beta_{13} - 430711150 \beta_{14} + 10143378 \beta_{15}) q^{31}\) \(+(\)\(29\!\cdots\!56\)\( - \)\(14\!\cdots\!72\)\( \beta_{1} - 221571233653601916 \beta_{2} + 1175756085168824 \beta_{3} + 147794152545400 \beta_{4} - 544682155648 \beta_{5} - 130831648700 \beta_{6} + 326901762172 \beta_{7} - 2050684848 \beta_{8} - 1602391764 \beta_{9} + 681613872 \beta_{10} + 1260490800 \beta_{11} - 1088332860 \beta_{12} - 273138320 \beta_{13} + 122654752 \beta_{14} - 80909216 \beta_{15}) q^{32}\) \(+(-\)\(54\!\cdots\!88\)\( - \)\(73\!\cdots\!99\)\( \beta_{1} + 43994233757114103 \beta_{2} + 16263609365101495 \beta_{3} - 23087061678296 \beta_{4} - 19211038176670 \beta_{5} + 138188383452 \beta_{6} + 10302122691109 \beta_{7} - 34599920851 \beta_{8} + 2257247359 \beta_{9} + 298046692 \beta_{10} + 7801236116 \beta_{11} + 4122049868 \beta_{12} + 61619848 \beta_{13}) q^{33}\) \(+(-\)\(56\!\cdots\!66\)\( + \)\(27\!\cdots\!76\)\( \beta_{1} - 263687744558676162 \beta_{2} - 8681405261525770 \beta_{3} - 253696020986652 \beta_{4} + 43443670667828 \beta_{5} - 468652446150 \beta_{6} + 1194992913566 \beta_{7} - 64436844234 \beta_{8} + 5842400574 \beta_{9} + 953595858 \beta_{10} + 2126611940 \beta_{11} - 3490275542 \beta_{12} - 1073411008 \beta_{13} - 1331208766 \beta_{14} - 40516032 \beta_{15}) q^{34}\) \(+(-71070034206720870692 - \)\(28\!\cdots\!30\)\( \beta_{1} - 28395050863661908 \beta_{2} - 71313511430740780 \beta_{3} + 15857660666830 \beta_{4} + 28616278598402 \beta_{5} + 267870026442 \beta_{6} - 25862650552232 \beta_{7} - 60908372000 \beta_{8} + 1295779744 \beta_{9} + 1139518702 \beta_{10} + 16038297576 \beta_{11} + 2999267698 \beta_{12} + 777920640 \beta_{13} + 4732763182 \beta_{14} - 153547730 \beta_{15}) q^{35}\) \(+(-\)\(57\!\cdots\!30\)\( + \)\(86\!\cdots\!74\)\( \beta_{1} - 5866504417088488556 \beta_{2} - 57048771963916047 \beta_{3} + 481813066952032 \beta_{4} - 15974927918400 \beta_{5} + 3018213565320 \beta_{6} + 2283551187432 \beta_{7} - 174829190352 \beta_{8} + 3786611960 \beta_{9} - 2835774800 \beta_{10} + 7425162672 \beta_{11} - 19074474312 \beta_{12} - 1127735152 \beta_{13} - 2898915744 \beta_{14} + 735835168 \beta_{15}) q^{36}\) \(+(-\)\(19\!\cdots\!67\)\( + \)\(73\!\cdots\!63\)\( \beta_{1} - 508996226196663188 \beta_{2} + 90472130783478190 \beta_{3} + 420785261527246 \beta_{4} + 96368496817693 \beta_{5} + 557393131363 \beta_{6} + 38651976974997 \beta_{7} + 118015770143 \beta_{8} + 11145421884 \beta_{9} - 9529694737 \beta_{10} + 34538635803 \beta_{11} - 15757580385 \beta_{12} + 5353668910 \beta_{13}) q^{37}\) \(+(\)\(49\!\cdots\!40\)\( - \)\(79\!\cdots\!23\)\( \beta_{1} + 7093694308137395569 \beta_{2} - 75239984444329377 \beta_{3} + 1146658856670348 \beta_{4} - 164824645699233 \beta_{5} - 3845761158080 \beta_{6} + 16612863111819 \beta_{7} - 198023257708 \beta_{8} - 29858198056 \beta_{9} - 10832711452 \beta_{10} + 46045787862 \beta_{11} - 43312869196 \beta_{12} + 27742528 \beta_{13} + 6656979408 \beta_{14} - 140133264 \beta_{15}) q^{38}\) \(+(\)\(41\!\cdots\!21\)\( + \)\(16\!\cdots\!12\)\( \beta_{1} - 12057445762384316661 \beta_{2} - 537028054206831747 \beta_{3} + 491082205155170 \beta_{4} - 195286579413944 \beta_{5} + 1636913474973 \beta_{6} + 77619319351859 \beta_{7} - 1356858479024 \beta_{8} + 18964809628 \beta_{9} - 27508546976 \beta_{10} + 91528668336 \beta_{11} - 15549781500 \beta_{12} + 1383327488 \beta_{13} - 32449853472 \beta_{14} + 1683351520 \beta_{15}) q^{39}\) \(+(-\)\(86\!\cdots\!54\)\( + \)\(79\!\cdots\!30\)\( \beta_{1} - 52109275187243595206 \beta_{2} - 125818103808815550 \beta_{3} - 7550818318538070 \beta_{4} + 576506037342514 \beta_{5} - 2827711857056 \beta_{6} + 19868931974946 \beta_{7} - 377096426610 \beta_{8} + 150520037408 \beta_{9} + 1305944704 \beta_{10} + 197667695232 \beta_{11} + 41651082336 \beta_{12} - 2788686720 \beta_{13} + 29983723264 \beta_{14} - 4904129280 \beta_{15}) q^{40}\) \(+(\)\(61\!\cdots\!46\)\( + \)\(38\!\cdots\!48\)\( \beta_{1} - 2664088113160447838 \beta_{2} + 448008132351261962 \beta_{3} + 1541341184773006 \beta_{4} + 669181184719194 \beta_{5} + 2250236469518 \beta_{6} - 326610270670416 \beta_{7} - 2082425189160 \beta_{8} - 83828833106 \beta_{9} + 55066315834 \beta_{10} + 99998563794 \beta_{11} - 41442895226 \beta_{12} - 41928386476 \beta_{13}) q^{41}\) \(+(-\)\(12\!\cdots\!56\)\( - \)\(90\!\cdots\!76\)\( \beta_{1} + \)\(16\!\cdots\!32\)\( \beta_{2} - 257332688341416668 \beta_{3} + 23347375904195096 \beta_{4} + 854237186739832 \beta_{5} + 49199464317324 \beta_{6} + 42606641933924 \beta_{7} - 979575904604 \beta_{8} + 191990055300 \beta_{9} + 45626179788 \beta_{10} + 512523753080 \beta_{11} + 308348293292 \beta_{12} + 17269801088 \beta_{13} - 9551093076 \beta_{14} + 3906401408 \beta_{15}) q^{42}\) \(+(-\)\(23\!\cdots\!62\)\( - \)\(93\!\cdots\!04\)\( \beta_{1} - 94445479529937151857 \beta_{2} - 1100718761392410078 \beta_{3} - 539413298842528 \beta_{4} + 1054829175442740 \beta_{5} - 39071408592 \beta_{6} + 914487295858482 \beta_{7} + 4831973534392 \beta_{8} + 144490859186 \beta_{9} + 195122443862 \beta_{10} + 35291149104 \beta_{11} - 459647438088 \beta_{12} - 71963382016 \beta_{13} + 131429748694 \beta_{14} - 14220841002 \beta_{15}) q^{43}\) \(+(\)\(18\!\cdots\!95\)\( - \)\(27\!\cdots\!19\)\( \beta_{1} - \)\(35\!\cdots\!21\)\( \beta_{2} + 224435934355493477 \beta_{3} - 35687653792311647 \beta_{4} - 1935260846428666 \beta_{5} - 51110430377391 \beta_{6} + 143870974382610 \beta_{7} - 1764359111663 \beta_{8} - 781160210847 \beta_{9} + 4601617791 \beta_{10} + 613565847639 \beta_{11} + 133046362614 \beta_{12} + 160275416121 \beta_{13} - 200349891402 \beta_{14} + 23329505682 \beta_{15}) q^{44}\) \(+(\)\(94\!\cdots\!81\)\( - \)\(34\!\cdots\!47\)\( \beta_{1} + 22391752556653720152 \beta_{2} + 1435143727218179454 \beta_{3} - 34448130107440828 \beta_{4} - 6505584987752845 \beta_{5} + 7222944862485 \beta_{6} - 1091600140103765 \beta_{7} - 3537067137175 \beta_{8} + 1227226570660 \beta_{9} - 54740372895 \beta_{10} + 261292138965 \beta_{11} - 987451119495 \beta_{12} + 37588394610 \beta_{13}) q^{45}\) \(+(-\)\(57\!\cdots\!96\)\( + \)\(89\!\cdots\!46\)\( \beta_{1} + \)\(11\!\cdots\!97\)\( \beta_{2} + 201077840978317797 \beta_{3} - 3208714191682245 \beta_{4} + 3599979464944600 \beta_{5} - 11590283090688 \beta_{6} - 650937346667421 \beta_{7} - 3156839746986 \beta_{8} + 708761784116 \beta_{9} - 106333282130 \beta_{10} - 664891947067 \beta_{11} - 1091341218842 \beta_{12} + 255530451296 \beta_{13} - 127805272680 \beta_{14} - 34772779704 \beta_{15}) q^{46}\) \(+(\)\(15\!\cdots\!44\)\( + \)\(63\!\cdots\!84\)\( \beta_{1} + \)\(21\!\cdots\!30\)\( \beta_{2} + 2777620668081599841 \beta_{3} + 7539926564967495 \beta_{4} - 8990395224430009 \beta_{5} + 24860408520431 \beta_{6} - 1632226529805289 \beta_{7} - 6838773511412 \beta_{8} - 277757112855 \beta_{9} - 297405041922 \beta_{10} - 281250537636 \beta_{11} + 796930070553 \beta_{12} + 132555911360 \beta_{13} - 142305842722 \beta_{14} + 95092892126 \beta_{15}) q^{47}\) \(+(-\)\(94\!\cdots\!88\)\( + \)\(33\!\cdots\!76\)\( \beta_{1} - \)\(31\!\cdots\!52\)\( \beta_{2} - 1008650204975304064 \beta_{3} + 165242392417817216 \beta_{4} + 11762633552282880 \beta_{5} + 228984455677248 \beta_{6} - 736984530600000 \beta_{7} + 7473488873472 \beta_{8} + 1720121599424 \beta_{9} + 211229731584 \beta_{10} - 3940468950272 \beta_{11} - 918390931136 \beta_{12} - 619378321664 \beta_{13} + 944825000448 \beta_{14} - 69350480384 \beta_{15}) q^{48}\) \(+(-\)\(70\!\cdots\!63\)\( + \)\(10\!\cdots\!16\)\( \beta_{1} - 65752197820801175484 \beta_{2} - 15000769681066856876 \beta_{3} + 207966210325582400 \beta_{4} + 23628603298219512 \beta_{5} - 77773190315616 \beta_{6} + 9636152720292524 \beta_{7} + 46255387349788 \beta_{8} - 5443860854108 \beta_{9} - 301087271184 \beta_{10} - 3142666533392 \beta_{11} + 6355504314272 \beta_{12} + 216957373152 \beta_{13}) q^{49}\) \(+(\)\(69\!\cdots\!95\)\( + \)\(37\!\cdots\!75\)\( \beta_{1} + \)\(56\!\cdots\!80\)\( \beta_{2} + 6379393313141836300 \beta_{3} - 469609645343812600 \beta_{4} - 23258384926530520 \beta_{5} - 977322355037820 \beta_{6} + 4016915670787020 \beta_{7} + 47985129090700 \beta_{8} - 9758664425940 \beta_{9} - 55396537020 \beta_{10} - 7187680307160 \beta_{11} + 2886110184420 \beta_{12} - 2403107318400 \beta_{13} + 1121427946980 \beta_{14} + 192807408000 \beta_{15}) q^{50}\) \(+(-\)\(64\!\cdots\!44\)\( - \)\(26\!\cdots\!13\)\( \beta_{1} + \)\(22\!\cdots\!22\)\( \beta_{2} + 45251854238333215573 \beta_{3} - 30533684534427378 \beta_{4} + 7567698181038764 \beta_{5} - 375022583223678 \beta_{6} - 20579703026740127 \beta_{7} - 42476851113668 \beta_{8} - 3871496571463 \beta_{9} - 2934406350199 \beta_{10} - 4466710417472 \beta_{11} + 9888425844670 \beta_{12} + 1201768573952 \beta_{13} - 1823709808503 \beta_{14} - 507008239479 \beta_{15}) q^{51}\) \(+(\)\(10\!\cdots\!44\)\( - \)\(45\!\cdots\!76\)\( \beta_{1} - \)\(12\!\cdots\!64\)\( \beta_{2} + 19725059209648411938 \beta_{3} + 531340821083870256 \beta_{4} + 5998165300381792 \beta_{5} + 304135506871012 \beta_{6} - 1895001216982124 \beta_{7} + 71025202089048 \beta_{8} + 5677259798364 \beta_{9} - 2937653190120 \beta_{10} - 11483867645032 \beta_{11} + 944802458428 \beta_{12} - 1099233618808 \beta_{13} - 3306650342224 \beta_{14} + 35447336592 \beta_{15}) q^{52}\) \(+(\)\(13\!\cdots\!13\)\( - \)\(11\!\cdots\!63\)\( \beta_{1} + \)\(77\!\cdots\!48\)\( \beta_{2} - 32721613838457567762 \beta_{3} - 392120314703783376 \beta_{4} - 191134628087158513 \beta_{5} - 167559213860775 \beta_{6} + 8682532130575047 \beta_{7} + 73937217433629 \beta_{8} - 6596701560684 \beta_{9} - 1147544376555 \beta_{10} - 9699121157607 \beta_{11} + 1450441223181 \beta_{12} + 1500259765146 \beta_{13}) q^{53}\) \(+(-\)\(26\!\cdots\!08\)\( + \)\(42\!\cdots\!50\)\( \beta_{1} + \)\(23\!\cdots\!68\)\( \beta_{2} + 25975146110885716680 \beta_{3} - 703798483886559886 \beta_{4} + 214030980581805730 \beta_{5} + 1310909001051072 \beta_{6} - 20229461568039052 \beta_{7} + 67456686080620 \beta_{8} + 25284159441448 \beta_{9} + 1245166289692 \beta_{10} - 26843012557014 \beta_{11} - 12042097732788 \beta_{12} + 5688991608512 \beta_{13} - 4343966163408 \beta_{14} - 695767174256 \beta_{15}) q^{54}\) \(+(\)\(35\!\cdots\!15\)\( + \)\(14\!\cdots\!40\)\( \beta_{1} + \)\(68\!\cdots\!85\)\( \beta_{2} + \)\(12\!\cdots\!15\)\( \beta_{3} + 173315779721672110 \beta_{4} - 254124148855980840 \beta_{5} + 998954220174315 \beta_{6} + 35461424485038165 \beta_{7} + 658978645852400 \beta_{8} + 2655549885460 \beta_{9} + 16271596062160 \beta_{10} - 31932552154800 \beta_{11} - 19567467570660 \beta_{12} - 3191950657280 \beta_{13} + 12585103011920 \beta_{14} + 2130036183120 \beta_{15}) q^{55}\) \(+(-\)\(11\!\cdots\!64\)\( - \)\(18\!\cdots\!24\)\( \beta_{1} - \)\(28\!\cdots\!80\)\( \beta_{2} + 41115298398650182288 \beta_{3} - 630389545736160880 \beta_{4} + 195878650314761568 \beta_{5} - 5745889421808920 \beta_{6} - 1584278633089416 \beta_{7} + 26203276756800 \beta_{8} - 105129097044616 \beta_{9} + 21233207952992 \beta_{10} - 29616747866016 \beta_{11} + 4819400666216 \beta_{12} + 9454396505056 \beta_{13} + 9736821713472 \beta_{14} + 838945377984 \beta_{15}) q^{56}\) \(+(-\)\(10\!\cdots\!04\)\( - \)\(20\!\cdots\!85\)\( \beta_{1} + \)\(13\!\cdots\!97\)\( \beta_{2} - 26412614472683425491 \beta_{3} - 189474002491695264 \beta_{4} - 274186344924405378 \beta_{5} - 352483180805268 \beta_{6} - 152573262391035153 \beta_{7} - 482335853132745 \beta_{8} + 122299730642853 \beta_{9} + 11684990968452 \beta_{10} - 31707152966412 \beta_{11} - 41787309978468 \beta_{12} - 8079176015928 \beta_{13}) q^{57}\) \(+(\)\(20\!\cdots\!48\)\( - \)\(63\!\cdots\!59\)\( \beta_{1} + \)\(41\!\cdots\!05\)\( \beta_{2} + 10007082298001802187 \beta_{3} + 4004486040883966946 \beta_{4} + 320643684844533306 \beta_{5} + 10020826581102087 \beta_{6} + 16880225747457601 \beta_{7} - 390695331517205 \beta_{8} - 11437424629587 \beta_{9} + 3530265429073 \beta_{10} + 81313915945886 \beta_{11} + 87957460165519 \beta_{12} + 4064491692544 \beta_{13} + 5653321839569 \beta_{14} + 1291678069248 \beta_{15}) q^{58}\) \(+(\)\(11\!\cdots\!22\)\( + \)\(47\!\cdots\!82\)\( \beta_{1} - \)\(14\!\cdots\!49\)\( \beta_{2} - \)\(18\!\cdots\!50\)\( \beta_{3} + 860688245756302654 \beta_{4} - 789815029067761346 \beta_{5} - 6307586327527798 \beta_{6} + 224047690060499654 \beta_{7} - 982904773159256 \beta_{8} + 5293208601294 \beta_{9} - 21308160985848 \beta_{10} - 1487135714952 \beta_{11} - 44619258361302 \beta_{12} - 9829941940864 \beta_{13} - 35345418792760 \beta_{14} - 6733146341176 \beta_{15}) q^{59}\) \(+(-\)\(43\!\cdots\!42\)\( + \)\(23\!\cdots\!90\)\( \beta_{1} - \)\(28\!\cdots\!98\)\( \beta_{2} + 7388911664701848150 \beta_{3} - 14009919299406916770 \beta_{4} + 1847850059295341492 \beta_{5} + 3527241553353342 \beta_{6} + 34081351078995868 \beta_{7} + 88870591339390 \beta_{8} + 429022008108894 \beta_{9} - 84153417280158 \beta_{10} + 288995701109426 \beta_{11} + 133286240501588 \beta_{12} - 4696744311250 \beta_{13} - 32680278492588 \beta_{14} - 4638587806180 \beta_{15}) q^{60}\) \(+(\)\(13\!\cdots\!93\)\( - \)\(11\!\cdots\!57\)\( \beta_{1} + \)\(73\!\cdots\!64\)\( \beta_{2} + \)\(13\!\cdots\!06\)\( \beta_{3} + 5886317405495041050 \beta_{4} - 1555909162851635447 \beta_{5} + 146471779006495 \beta_{6} - 22766387391711823 \beta_{7} + 943225797827163 \beta_{8} - 696728562615668 \beta_{9} - 2589759145933 \beta_{10} + 41443344084367 \beta_{11} + 9476886163019 \beta_{12} - 17191879308170 \beta_{13}) q^{61}\) \(+(\)\(25\!\cdots\!04\)\( - \)\(40\!\cdots\!52\)\( \beta_{1} - \)\(86\!\cdots\!00\)\( \beta_{2} - \)\(32\!\cdots\!80\)\( \beta_{3} + 10962151642223759672 \beta_{4} + 3722648692855717976 \beta_{5} - 29729871894802048 \beta_{6} + 116555437522609808 \beta_{7} + 2405238833510352 \beta_{8} - 222283800526368 \beta_{9} - 50048338276080 \beta_{10} + 222985550853528 \beta_{11} - 63687095790384 \beta_{12} - 25783931749120 \beta_{13} + 27897499161920 \beta_{14} + 1777715887040 \beta_{15}) q^{62}\) \(+(\)\(10\!\cdots\!77\)\( + \)\(43\!\cdots\!16\)\( \beta_{1} - \)\(19\!\cdots\!23\)\( \beta_{2} - \)\(24\!\cdots\!60\)\( \beta_{3} + 7287177815420971895 \beta_{4} - 5906562492866836787 \beta_{5} + 30510730774827786 \beta_{6} - 309655670115095920 \beta_{7} - 5462585780480108 \beta_{8} + 35778512526799 \beta_{9} - 74513749591358 \beta_{10} + 436630848057348 \beta_{11} + 54848603321967 \beta_{12} + 20301434060096 \beta_{13} + 198284394210 \beta_{14} + 13641600812770 \beta_{15}) q^{63}\) \(+(-\)\(16\!\cdots\!32\)\( + \)\(29\!\cdots\!32\)\( \beta_{1} + \)\(12\!\cdots\!92\)\( \beta_{2} - \)\(15\!\cdots\!88\)\( \beta_{3} + 2252418836451201056 \beta_{4} + 7321599615903575552 \beta_{5} + 67712292711557104 \beta_{6} - 96314026717705968 \beta_{7} - 4513787717062976 \beta_{8} - 616043164348848 \beta_{9} + 153124639716160 \beta_{10} - 218758631253184 \beta_{11} - 602666375853584 \beta_{12} - 35545736341952 \beta_{13} + 137069914285952 \beta_{14} + 10912536053376 \beta_{15}) q^{64}\) \(+(-\)\(12\!\cdots\!20\)\( - \)\(51\!\cdots\!20\)\( \beta_{1} + \)\(33\!\cdots\!50\)\( \beta_{2} + \)\(22\!\cdots\!70\)\( \beta_{3} + 11979635884233489130 \beta_{4} - 8575943888734549570 \beta_{5} + 18666902036556330 \beta_{6} + 1276631259208706280 \beta_{7} - 2046397690927680 \beta_{8} + 2669812748332050 \beta_{9} - 171229348255410 \beta_{10} + 1088418124698390 \beta_{11} - 59103752696430 \beta_{12} + 127327113277500 \beta_{13}) q^{65}\) \(+(-\)\(51\!\cdots\!40\)\( - \)\(46\!\cdots\!26\)\( \beta_{1} - \)\(76\!\cdots\!18\)\( \beta_{2} - \)\(34\!\cdots\!14\)\( \beta_{3} - 40856745708650921620 \beta_{4} + 13748415208892038364 \beta_{5} - 46077731322255234 \beta_{6} - 129207170233273430 \beta_{7} - 21845149915208062 \beta_{8} + 1081312066552746 \beta_{9} + 65544840575318 \beta_{10} - 140818542356596 \beta_{11} - 1139255341584946 \beta_{12} + 27556355797440 \beta_{13} - 163327948149786 \beta_{14} - 21110637795392 \beta_{15}) q^{66}\) \(+(\)\(26\!\cdots\!14\)\( + \)\(10\!\cdots\!71\)\( \beta_{1} + \)\(19\!\cdots\!53\)\( \beta_{2} - \)\(27\!\cdots\!33\)\( \beta_{3} + 15528766608085997502 \beta_{4} - 13734677704957396216 \beta_{5} - 68396246768482814 \beta_{6} - 2152977464572420033 \beta_{7} + 11876990331473988 \beta_{8} + 255789125199063 \beta_{9} + 330736321448883 \beta_{10} + 1034507016420048 \beta_{11} + 181686861290358 \beta_{12} + 97685585064192 \beta_{13} + 359661539516979 \beta_{14} - 1874368593357 \beta_{15}) q^{67}\) \(+(\)\(21\!\cdots\!16\)\( - \)\(60\!\cdots\!24\)\( \beta_{1} + \)\(10\!\cdots\!84\)\( \beta_{2} + \)\(21\!\cdots\!42\)\( \beta_{3} + 22291992454469918176 \beta_{4} + 21775390095912565696 \beta_{5} - 73335555359624728 \beta_{6} + 823614768952650952 \beta_{7} + 709290222264688 \beta_{8} - 376843445824360 \beta_{9} - 38130240202000 \beta_{10} + 744629942295024 \beta_{11} - 423225474307240 \beta_{12} - 34874363408560 \beta_{13} - 501321037731104 \beta_{14} + 4856587990432 \beta_{15}) q^{68}\) \(+(\)\(24\!\cdots\!58\)\( - \)\(35\!\cdots\!51\)\( \beta_{1} + \)\(23\!\cdots\!90\)\( \beta_{2} + \)\(13\!\cdots\!60\)\( \beta_{3} + 1450485998339189879 \beta_{4} - 56588331509226940169 \beta_{5} + 21622746192010065 \beta_{6} + 1884534994653723503 \beta_{7} - 13288297984472843 \beta_{8} - 5610109304449612 \beta_{9} + 468687640411277 \beta_{10} + 691106640929137 \beta_{11} + 1023827399237989 \beta_{12} - 29963623397110 \beta_{13}) q^{69}\) \(+(\)\(19\!\cdots\!64\)\( - \)\(25\!\cdots\!48\)\( \beta_{1} - \)\(20\!\cdots\!52\)\( \beta_{2} + \)\(77\!\cdots\!16\)\( \beta_{3} - \)\(19\!\cdots\!52\)\( \beta_{4} + 63098427075199464780 \beta_{5} + 350751336143378560 \beta_{6} - 56364326644858480 \beta_{7} + 64706466428317240 \beta_{8} - 107924350682480 \beta_{9} + 587933202698520 \beta_{10} + 768999117190820 \beta_{11} + 1995912139534840 \beta_{12} - 345707230006400 \beta_{13} + 285719697004000 \beta_{14} + 67585705491360 \beta_{15}) q^{70}\) \(+(\)\(10\!\cdots\!13\)\( + \)\(40\!\cdots\!84\)\( \beta_{1} + \)\(37\!\cdots\!61\)\( \beta_{2} + \)\(75\!\cdots\!90\)\( \beta_{3} + 57259353408211776993 \beta_{4} - 62501104021591218737 \beta_{5} + 330484909341257944 \beta_{6} + 2114149043459410382 \beta_{7} + 9234794090312412 \beta_{8} - 111770584548251 \beta_{9} - 396501539764850 \beta_{10} - 1686092304135732 \beta_{11} + 117570552183565 \beta_{12} + 10992295470016 \beta_{13} - 1174557324120082 \beta_{14} - 113045024438290 \beta_{15}) q^{71}\) \(+(-\)\(11\!\cdots\!61\)\( - \)\(39\!\cdots\!07\)\( \beta_{1} + \)\(26\!\cdots\!01\)\( \beta_{2} + \)\(73\!\cdots\!73\)\( \beta_{3} + 61515538918589351341 \beta_{4} + \)\(10\!\cdots\!77\)\( \beta_{5} - 502495161610158144 \beta_{6} - 3359155759362603367 \beta_{7} + 23228182145009455 \beta_{8} + 4499789166047552 \beta_{9} + 121343149218048 \beta_{10} - 770805157107456 \beta_{11} + 3047580398328768 \beta_{12} + 14156701494528 \beta_{13} + 1098111361586688 \beta_{14} - 123997050453504 \beta_{15}) q^{72}\) \(+(-\)\(84\!\cdots\!26\)\( - \)\(52\!\cdots\!89\)\( \beta_{1} + \)\(35\!\cdots\!09\)\( \beta_{2} - \)\(22\!\cdots\!67\)\( \beta_{3} + \)\(24\!\cdots\!52\)\( \beta_{4} - 64192930194279832062 \beta_{5} - 168603236704028064 \beta_{6} - 12977164213755576541 \beta_{7} + 18322882616341003 \beta_{8} + 6358065124221877 \beta_{9} + 351859015320528 \beta_{10} - 9435712674946352 \beta_{11} - 2257186689757600 \beta_{12} - 530483484594528 \beta_{13}) q^{73}\) \(+(\)\(48\!\cdots\!56\)\( - \)\(20\!\cdots\!79\)\( \beta_{1} - \)\(29\!\cdots\!83\)\( \beta_{2} + \)\(89\!\cdots\!91\)\( \beta_{3} - \)\(26\!\cdots\!90\)\( \beta_{4} + 76053613812526134290 \beta_{5} - 11907813906887133 \beta_{6} - 79021154943648619 \beta_{7} - 117107146205360289 \beta_{8} - 7362904052636575 \beta_{9} - 1936622771059315 \beta_{10} - 4129856745882570 \beta_{11} + 1971775419415115 \beta_{12} + 1315396947851904 \beta_{13} + 482528449655021 \beta_{14} - 73737980674432 \beta_{15}) q^{74}\) \(+(\)\(44\!\cdots\!10\)\( + \)\(17\!\cdots\!00\)\( \beta_{1} + \)\(20\!\cdots\!15\)\( \beta_{2} + \)\(40\!\cdots\!00\)\( \beta_{3} + \)\(25\!\cdots\!50\)\( \beta_{4} - \)\(28\!\cdots\!10\)\( \beta_{5} - 1593295934637050010 \beta_{6} + 20229815018545202660 \beta_{7} - 57953307472850000 \beta_{8} - 2946811562972420 \beta_{9} - 745013843310410 \beta_{10} - 12900577046918280 \beta_{11} - 2974876469703090 \beta_{12} - 1290961618268800 \beta_{13} + 521408738358390 \beta_{14} + 494741428626550 \beta_{15}) q^{75}\) \(+(-\)\(98\!\cdots\!31\)\( + \)\(50\!\cdots\!87\)\( \beta_{1} + \)\(32\!\cdots\!57\)\( \beta_{2} - \)\(11\!\cdots\!41\)\( \beta_{3} - \)\(40\!\cdots\!01\)\( \beta_{4} + \)\(23\!\cdots\!26\)\( \beta_{5} + 816054600500849019 \beta_{6} + 10460393854209601222 \beta_{7} + 2306921761055931 \beta_{8} - 9028095638821333 \beta_{9} - 1308434556971915 \beta_{10} - 11416436345087043 \beta_{11} + 4076444328865170 \beta_{12} + 1599506818096755 \beta_{13} + 99672842212306 \beta_{14} + 386889346380102 \beta_{15}) q^{76}\) \(+(\)\(14\!\cdots\!10\)\( - \)\(24\!\cdots\!03\)\( \beta_{1} + \)\(16\!\cdots\!38\)\( \beta_{2} - \)\(34\!\cdots\!96\)\( \beta_{3} + 36232816958719180899 \beta_{4} - \)\(37\!\cdots\!33\)\( \beta_{5} - 281796309359973179 \beta_{6} - 13403001814287011205 \beta_{7} + 204363467827152633 \beta_{8} - 4987613453280124 \beta_{9} - 4182269389594639 \beta_{10} - 9872861491647195 \beta_{11} - 6386062213892887 \beta_{12} - 211705042874222 \beta_{13}) q^{77}\) \(+(-\)\(11\!\cdots\!60\)\( + \)\(18\!\cdots\!54\)\( \beta_{1} + \)\(30\!\cdots\!17\)\( \beta_{2} + \)\(15\!\cdots\!09\)\( \beta_{3} + \)\(10\!\cdots\!95\)\( \beta_{4} + \)\(62\!\cdots\!52\)\( \beta_{5} - 2533266017207680128 \beta_{6} - 5156627610186652701 \beta_{7} + 351811722375248102 \beta_{8} + 4051648572898964 \beta_{9} - 1591313121355906 \beta_{10} - 13189764159272451 \beta_{11} + 4913597248570998 \beta_{12} + 808002430478944 \beta_{13} - 3294330448718376 \beta_{14} - 231972723187192 \beta_{15}) q^{78}\) \(+(\)\(75\!\cdots\!54\)\( + \)\(30\!\cdots\!04\)\( \beta_{1} + \)\(29\!\cdots\!62\)\( \beta_{2} + \)\(20\!\cdots\!62\)\( \beta_{3} + \)\(42\!\cdots\!28\)\( \beta_{4} - \)\(44\!\cdots\!20\)\( \beta_{5} + 4238052257670451030 \beta_{6} - 25575645150124213086 \beta_{7} + 119905216541427968 \beta_{8} - 3501144792924448 \beta_{9} + 3935661681276744 \beta_{10} - 7513617137702944 \beta_{11} + 3664093154021848 \beta_{12} - 365453201508864 \beta_{13} + 7194819290826312 \beta_{14} - 1119110925826488 \beta_{15}) q^{79}\) \(+(\)\(42\!\cdots\!58\)\( - \)\(93\!\cdots\!16\)\( \beta_{1} - \)\(13\!\cdots\!34\)\( \beta_{2} + \)\(14\!\cdots\!52\)\( \beta_{3} - \)\(29\!\cdots\!64\)\( \beta_{4} + \)\(53\!\cdots\!40\)\( \beta_{5} + 2323520424165632390 \beta_{6} - 45421125465189277510 \beta_{7} + 64742451777529560 \beta_{8} - 13835823416230110 \beta_{9} - 3999525679206840 \beta_{10} + 3894671769070920 \beta_{11} - 824589596066490 \beta_{12} - 1668799649507800 \beta_{13} - 9899673598517200 \beta_{14} - 244372949200240 \beta_{15}) q^{80}\) \(+(-\)\(25\!\cdots\!55\)\( - \)\(51\!\cdots\!71\)\( \beta_{1} + \)\(33\!\cdots\!71\)\( \beta_{2} + \)\(38\!\cdots\!15\)\( \beta_{3} + \)\(54\!\cdots\!96\)\( \beta_{4} - \)\(81\!\cdots\!70\)\( \beta_{5} + 608527194743671956 \beta_{6} + 74366931733897951425 \beta_{7} - 276402107260889607 \beta_{8} - 12005385626536917 \beta_{9} + 7090268914887804 \beta_{10} + 30057340048525836 \beta_{11} + 42622112776424292 \beta_{12} - 109176462668232 \beta_{13}) q^{81}\) \(+(\)\(27\!\cdots\!10\)\( + \)\(57\!\cdots\!34\)\( \beta_{1} + \)\(19\!\cdots\!80\)\( \beta_{2} + \)\(28\!\cdots\!76\)\( \beta_{3} - 11250672396886998840 \beta_{4} + \)\(52\!\cdots\!00\)\( \beta_{5} + 740073052987819908 \beta_{6} + 18059232690389343244 \beta_{7} - 731615520710864340 \beta_{8} + 31214077693385772 \beta_{9} + 14568289513191588 \beta_{10} + 11401704070477480 \beta_{11} - 7344026994877468 \beta_{12} - 10511156517666944 \beta_{13} + 4263246219560644 \beta_{14} + 1106795332914048 \beta_{15}) q^{82}\) \(+(\)\(11\!\cdots\!18\)\( + \)\(45\!\cdots\!94\)\( \beta_{1} + \)\(14\!\cdots\!29\)\( \beta_{2} - \)\(10\!\cdots\!94\)\( \beta_{3} + \)\(65\!\cdots\!30\)\( \beta_{4} - \)\(58\!\cdots\!50\)\( \beta_{5} - 11291069312817749182 \beta_{6} - 98792620466381183886 \beta_{7} + 330864045952913528 \beta_{8} + 18452937402740410 \beta_{9} - 5058425786573132 \beta_{10} + 70784158375939704 \beta_{11} + 29374322366723218 \beta_{12} + 11759735407656320 \beta_{13} - 20523690344365452 \beta_{14} + 822523112665716 \beta_{15}) q^{83}\) \(+(-\)\(23\!\cdots\!36\)\( - \)\(11\!\cdots\!00\)\( \beta_{1} - \)\(66\!\cdots\!88\)\( \beta_{2} + \)\(26\!\cdots\!04\)\( \beta_{3} + 27727355596131270976 \beta_{4} + \)\(11\!\cdots\!00\)\( \beta_{5} - 5968941822699239952 \beta_{6} + \)\(14\!\cdots\!52\)\( \beta_{7} - 98209278126741600 \beta_{8} + 26855751348390160 \beta_{9} + 36075133156726944 \beta_{10} - 4649988484857184 \beta_{11} - 59410956651250288 \beta_{12} - 12086886691804960 \beta_{13} + 34452461211991872 \beta_{14} - 2108411958506560 \beta_{15}) q^{84}\) \(+(-\)\(12\!\cdots\!80\)\( - \)\(69\!\cdots\!95\)\( \beta_{1} + \)\(45\!\cdots\!10\)\( \beta_{2} + \)\(17\!\cdots\!80\)\( \beta_{3} + \)\(42\!\cdots\!05\)\( \beta_{4} - \)\(99\!\cdots\!65\)\( \beta_{5} + 1662896239921810945 \beta_{6} + 83431307746596764695 \beta_{7} - 991781600259430475 \beta_{8} + 68978832481118420 \beta_{9} + 9055034080424885 \beta_{10} + 46927804411847705 \beta_{11} - 32157603685454315 \beta_{12} + 16144206731044570 \beta_{13}) q^{85}\) \(+(\)\(63\!\cdots\!64\)\( - \)\(10\!\cdots\!17\)\( \beta_{1} + \)\(56\!\cdots\!29\)\( \beta_{2} - \)\(12\!\cdots\!53\)\( \beta_{3} - \)\(68\!\cdots\!22\)\( \beta_{4} + \)\(12\!\cdots\!73\)\( \beta_{5} + 13162088246040709760 \beta_{6} - 34522948765685992965 \beta_{7} + 965780277478581320 \beta_{8} - 28467701819274896 \beta_{9} - 6937198563225752 \beta_{10} + 47316275711094300 \beta_{11} - 134594861953759352 \beta_{12} + 10399749628720256 \beta_{13} + 11460577032893472 \beta_{14} - 1591415558978976 \beta_{15}) q^{86}\) \(+(\)\(31\!\cdots\!55\)\( + \)\(12\!\cdots\!28\)\( \beta_{1} - \)\(12\!\cdots\!65\)\( \beta_{2} - \)\(75\!\cdots\!08\)\( \beta_{3} + \)\(21\!\cdots\!09\)\( \beta_{4} - \)\(17\!\cdots\!77\)\( \beta_{5} + 34306092649630082670 \beta_{6} + 36708371144456045664 \beta_{7} - 1153895331244090036 \beta_{8} + 18863146355620089 \beta_{9} - 8996984933662026 \beta_{10} + 90301727799837244 \beta_{11} - 33537865218136751 \beta_{12} - 2893255259695424 \beta_{13} + 584505949274262 \beta_{14} + 3860693896444054 \beta_{15}) q^{87}\) \(+(\)\(24\!\cdots\!52\)\( + \)\(18\!\cdots\!88\)\( \beta_{1} - \)\(67\!\cdots\!72\)\( \beta_{2} - \)\(23\!\cdots\!40\)\( \beta_{3} + \)\(80\!\cdots\!40\)\( \beta_{4} + \)\(16\!\cdots\!76\)\( \beta_{5} - 8069175003866879916 \beta_{6} - \)\(39\!\cdots\!36\)\( \beta_{7} - 970260512673645920 \beta_{8} + 208082453120368732 \beta_{9} - 63920792009371728 \beta_{10} + 170507326604677296 \beta_{11} + 5673526194559764 \beta_{12} + 20177654756224112 \beta_{13} - 50149054970425312 \beta_{14} + 7454514458572896 \beta_{15}) q^{88}\) \(+(\)\(28\!\cdots\!58\)\( - \)\(68\!\cdots\!05\)\( \beta_{1} + \)\(44\!\cdots\!01\)\( \beta_{2} + \)\(22\!\cdots\!33\)\( \beta_{3} - \)\(77\!\cdots\!88\)\( \beta_{4} - \)\(13\!\cdots\!74\)\( \beta_{5} - 157939529512935304 \beta_{6} - \)\(25\!\cdots\!25\)\( \beta_{7} + 1880329560691515003 \beta_{8} - 186704282449335299 \beta_{9} - 53953827673717640 \beta_{10} + 67735854437497944 \beta_{11} - 169408042953743144 \beta_{12} - 12920753909828752 \beta_{13}) q^{89}\) \(+(-\)\(22\!\cdots\!92\)\( + \)\(98\!\cdots\!25\)\( \beta_{1} + \)\(63\!\cdots\!17\)\( \beta_{2} - \)\(38\!\cdots\!05\)\( \beta_{3} - \)\(69\!\cdots\!70\)\( \beta_{4} + \)\(15\!\cdots\!62\)\( \beta_{5} + 49104391568233467 \beta_{6} + 74724118105746105373 \beta_{7} - 2154187949094225625 \beta_{8} - 142700937993644631 \beta_{9} - 51728228085993563 \beta_{10} + 194129600038964646 \beta_{11} + 265973670927565923 \beta_{12} + 9816445825312640 \beta_{13} - 44407457680989243 \beta_{14} - 1363625128922240 \beta_{15}) q^{90}\) \(+(\)\(27\!\cdots\!28\)\( + \)\(11\!\cdots\!90\)\( \beta_{1} - \)\(69\!\cdots\!12\)\( \beta_{2} - \)\(14\!\cdots\!44\)\( \beta_{3} + \)\(57\!\cdots\!70\)\( \beta_{4} - \)\(55\!\cdots\!54\)\( \beta_{5} - 75154967060829308782 \beta_{6} + \)\(64\!\cdots\!36\)\( \beta_{7} - 2515999891344930032 \beta_{8} - 45766850881791116 \beta_{9} + 34010443202664642 \beta_{10} - 147794932799885336 \beta_{11} - 201018894108954166 \beta_{12} - 63496745501400960 \beta_{13} + 113812757401569282 \beta_{14} - 16639472375638014 \beta_{15}) q^{91}\) \(+(-\)\(10\!\cdots\!50\)\( - \)\(57\!\cdots\!62\)\( \beta_{1} - \)\(16\!\cdots\!58\)\( \beta_{2} + \)\(22\!\cdots\!66\)\( \beta_{3} + \)\(13\!\cdots\!74\)\( \beta_{4} - \)\(18\!\cdots\!12\)\( \beta_{5} + 29180587455871316102 \beta_{6} + \)\(11\!\cdots\!52\)\( \beta_{7} - 405634436885423738 \beta_{8} - 479325257944509978 \beta_{9} - 51635576170544166 \beta_{10} + 154296858403007466 \beta_{11} + 260581334058704388 \beta_{12} + 33972690103314806 \beta_{13} - 25406390906062588 \beta_{14} - 6233011955917396 \beta_{15}) q^{92}\) \(+(-\)\(14\!\cdots\!52\)\( + \)\(26\!\cdots\!92\)\( \beta_{1} - \)\(17\!\cdots\!44\)\( \beta_{2} - \)\(27\!\cdots\!12\)\( \beta_{3} + \)\(18\!\cdots\!08\)\( \beta_{4} + \)\(49\!\cdots\!48\)\( \beta_{5} - 3777639461218803060 \beta_{6} - \)\(57\!\cdots\!04\)\( \beta_{7} + 736811052847313660 \beta_{8} + 709491772149878224 \beta_{9} + 37702517208353708 \beta_{10} - 8294117934659108 \beta_{11} + 306617361041318908 \beta_{12} - 129798749555340904 \beta_{13}) q^{93}\) \(+(-\)\(43\!\cdots\!64\)\( + \)\(70\!\cdots\!28\)\( \beta_{1} - \)\(90\!\cdots\!50\)\( \beta_{2} + \)\(68\!\cdots\!50\)\( \beta_{3} + \)\(88\!\cdots\!50\)\( \beta_{4} - \)\(32\!\cdots\!24\)\( \beta_{5} - 56202621165503188096 \beta_{6} + 77986525506235367386 \beta_{7} - 2114025690582882172 \beta_{8} + 464347474945190584 \beta_{9} + 28446853364046484 \beta_{10} + 41623415629025886 \beta_{11} + 317505630319159140 \beta_{12} + 8969458691199040 \beta_{13} + 7881914229347216 \beta_{14} + 8044744059098544 \beta_{15}) q^{94}\) \(+(-\)\(75\!\cdots\!75\)\( - \)\(30\!\cdots\!40\)\( \beta_{1} + \)\(32\!\cdots\!35\)\( \beta_{2} + \)\(14\!\cdots\!25\)\( \beta_{3} - \)\(51\!\cdots\!70\)\( \beta_{4} + \)\(42\!\cdots\!40\)\( \beta_{5} + \)\(12\!\cdots\!85\)\( \beta_{6} + 96572822096916099295 \beta_{7} + 5036468657382980400 \beta_{8} + 56838024788784740 \beta_{9} - 27514087895890600 \beta_{10} - 208396644139652880 \beta_{11} + 42296654111557700 \beta_{12} + 37993258028522240 \beta_{13} - 214030282234397480 \beta_{14} + 30732417557454040 \beta_{15}) q^{95}\) \(+(\)\(65\!\cdots\!16\)\( - \)\(10\!\cdots\!60\)\( \beta_{1} + \)\(49\!\cdots\!00\)\( \beta_{2} + \)\(70\!\cdots\!12\)\( \beta_{3} - \)\(31\!\cdots\!08\)\( \beta_{4} - \)\(53\!\cdots\!08\)\( \beta_{5} + 47988728363209462272 \beta_{6} - \)\(30\!\cdots\!68\)\( \beta_{7} + 4806064097781962752 \beta_{8} - 517050082427771392 \beta_{9} + 293625798052026368 \beta_{10} - 582301433033517056 \beta_{11} + 1558941901132288 \beta_{12} - 59637631003445248 \beta_{13} + 258649620211372032 \beta_{14} - 26845783379595264 \beta_{15}) q^{96}\) \(+(\)\(28\!\cdots\!86\)\( + \)\(36\!\cdots\!81\)\( \beta_{1} - \)\(23\!\cdots\!59\)\( \beta_{2} - \)\(14\!\cdots\!91\)\( \beta_{3} - \)\(61\!\cdots\!82\)\( \beta_{4} + \)\(36\!\cdots\!92\)\( \beta_{5} - 4059922566933605118 \beta_{6} + \)\(52\!\cdots\!13\)\( \beta_{7} - 6990555213939498539 \beta_{8} - 1633111880935693931 \beta_{9} + 175523170578808326 \beta_{10} - 944179300683893714 \beta_{11} - 448549378436054710 \beta_{12} + 230172687374540844 \beta_{13}) q^{97}\) \(+(\)\(64\!\cdots\!21\)\( - \)\(72\!\cdots\!99\)\( \beta_{1} - \)\(56\!\cdots\!56\)\( \beta_{2} + \)\(13\!\cdots\!20\)\( \beta_{3} + \)\(41\!\cdots\!48\)\( \beta_{4} - \)\(16\!\cdots\!00\)\( \beta_{5} - 17906471627199420024 \beta_{6} - \)\(64\!\cdots\!48\)\( \beta_{7} + 23053916477859294008 \beta_{8} + 311898745854711768 \beta_{9} + 173245080646954344 \beta_{10} - 1271457329249384496 \beta_{11} - 1456979565361302456 \beta_{12} + 55257672153076992 \beta_{13} + 223793358804565032 \beta_{14} - 6483304499110656 \beta_{15}) q^{98}\) \(+(-\)\(43\!\cdots\!22\)\( - \)\(17\!\cdots\!54\)\( \beta_{1} + \)\(24\!\cdots\!27\)\( \beta_{2} + \)\(65\!\cdots\!46\)\( \beta_{3} - \)\(29\!\cdots\!10\)\( \beta_{4} + \)\(26\!\cdots\!10\)\( \beta_{5} - \)\(15\!\cdots\!70\)\( \beta_{6} - \)\(25\!\cdots\!02\)\( \beta_{7} + 14070780902241836168 \beta_{8} - 110987647112549482 \beta_{9} - 1372660076028208 \beta_{10} - 262504351932924552 \beta_{11} + 931244311472338842 \beta_{12} + 190591136429845888 \beta_{13} - 160568425722029424 \beta_{14} - 13494817197667696 \beta_{15}) q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut +\mathstrut 177228q^{2} \) \(\mathstrut -\mathstrut 5610707696q^{4} \) \(\mathstrut +\mathstrut 5816089539360q^{5} \) \(\mathstrut -\mathstrut 217628996575488q^{6} \) \(\mathstrut +\mathstrut 11897560228206528q^{8} \) \(\mathstrut -\mathstrut 935184460817545968q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut +\mathstrut 177228q^{2} \) \(\mathstrut -\mathstrut 5610707696q^{4} \) \(\mathstrut +\mathstrut 5816089539360q^{5} \) \(\mathstrut -\mathstrut 217628996575488q^{6} \) \(\mathstrut +\mathstrut 11897560228206528q^{8} \) \(\mathstrut -\mathstrut 935184460817545968q^{9} \) \(\mathstrut -\mathstrut 2378142919315561000q^{10} \) \(\mathstrut -\mathstrut 42882825786868930560q^{12} \) \(\mathstrut +\mathstrut \)\(19\!\cdots\!76\)\(q^{13} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!28\)\(q^{14} \) \(\mathstrut -\mathstrut \)\(22\!\cdots\!84\)\(q^{16} \) \(\mathstrut +\mathstrut \)\(41\!\cdots\!76\)\(q^{17} \) \(\mathstrut +\mathstrut \)\(13\!\cdots\!68\)\(q^{18} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!40\)\(q^{20} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!96\)\(q^{21} \) \(\mathstrut -\mathstrut \)\(43\!\cdots\!40\)\(q^{22} \) \(\mathstrut +\mathstrut \)\(51\!\cdots\!68\)\(q^{24} \) \(\mathstrut +\mathstrut \)\(61\!\cdots\!00\)\(q^{25} \) \(\mathstrut -\mathstrut \)\(70\!\cdots\!40\)\(q^{26} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!00\)\(q^{28} \) \(\mathstrut -\mathstrut \)\(95\!\cdots\!52\)\(q^{29} \) \(\mathstrut +\mathstrut \)\(36\!\cdots\!20\)\(q^{30} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!48\)\(q^{32} \) \(\mathstrut -\mathstrut \)\(87\!\cdots\!60\)\(q^{33} \) \(\mathstrut -\mathstrut \)\(90\!\cdots\!20\)\(q^{34} \) \(\mathstrut -\mathstrut \)\(92\!\cdots\!92\)\(q^{36} \) \(\mathstrut -\mathstrut \)\(30\!\cdots\!84\)\(q^{37} \) \(\mathstrut +\mathstrut \)\(78\!\cdots\!80\)\(q^{38} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!00\)\(q^{40} \) \(\mathstrut +\mathstrut \)\(99\!\cdots\!12\)\(q^{41} \) \(\mathstrut -\mathstrut \)\(19\!\cdots\!00\)\(q^{42} \) \(\mathstrut +\mathstrut \)\(30\!\cdots\!60\)\(q^{44} \) \(\mathstrut +\mathstrut \)\(15\!\cdots\!20\)\(q^{45} \) \(\mathstrut -\mathstrut \)\(91\!\cdots\!28\)\(q^{46} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!20\)\(q^{48} \) \(\mathstrut -\mathstrut \)\(11\!\cdots\!48\)\(q^{49} \) \(\mathstrut +\mathstrut \)\(11\!\cdots\!00\)\(q^{50} \) \(\mathstrut +\mathstrut \)\(16\!\cdots\!56\)\(q^{52} \) \(\mathstrut +\mathstrut \)\(20\!\cdots\!56\)\(q^{53} \) \(\mathstrut -\mathstrut \)\(42\!\cdots\!84\)\(q^{54} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!68\)\(q^{56} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!80\)\(q^{57} \) \(\mathstrut +\mathstrut \)\(33\!\cdots\!56\)\(q^{58} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!00\)\(q^{60} \) \(\mathstrut +\mathstrut \)\(21\!\cdots\!32\)\(q^{61} \) \(\mathstrut +\mathstrut \)\(40\!\cdots\!80\)\(q^{62} \) \(\mathstrut -\mathstrut \)\(26\!\cdots\!96\)\(q^{64} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{65} \) \(\mathstrut -\mathstrut \)\(81\!\cdots\!60\)\(q^{66} \) \(\mathstrut +\mathstrut \)\(35\!\cdots\!56\)\(q^{68} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!56\)\(q^{69} \) \(\mathstrut +\mathstrut \)\(31\!\cdots\!80\)\(q^{70} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!32\)\(q^{72} \) \(\mathstrut -\mathstrut \)\(13\!\cdots\!64\)\(q^{73} \) \(\mathstrut +\mathstrut \)\(76\!\cdots\!40\)\(q^{74} \) \(\mathstrut -\mathstrut \)\(15\!\cdots\!60\)\(q^{76} \) \(\mathstrut +\mathstrut \)\(22\!\cdots\!00\)\(q^{77} \) \(\mathstrut -\mathstrut \)\(18\!\cdots\!60\)\(q^{78} \) \(\mathstrut +\mathstrut \)\(67\!\cdots\!60\)\(q^{80} \) \(\mathstrut -\mathstrut \)\(40\!\cdots\!00\)\(q^{81} \) \(\mathstrut +\mathstrut \)\(43\!\cdots\!36\)\(q^{82} \) \(\mathstrut -\mathstrut \)\(37\!\cdots\!64\)\(q^{84} \) \(\mathstrut -\mathstrut \)\(20\!\cdots\!00\)\(q^{85} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!32\)\(q^{86} \) \(\mathstrut +\mathstrut \)\(39\!\cdots\!80\)\(q^{88} \) \(\mathstrut +\mathstrut \)\(46\!\cdots\!88\)\(q^{89} \) \(\mathstrut -\mathstrut \)\(36\!\cdots\!00\)\(q^{90} \) \(\mathstrut -\mathstrut \)\(17\!\cdots\!80\)\(q^{92} \) \(\mathstrut -\mathstrut \)\(23\!\cdots\!80\)\(q^{93} \) \(\mathstrut -\mathstrut \)\(69\!\cdots\!32\)\(q^{94} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!72\)\(q^{96} \) \(\mathstrut +\mathstrut \)\(45\!\cdots\!96\)\(q^{97} \) \(\mathstrut +\mathstrut \)\(10\!\cdots\!28\)\(q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut +\mathstrut \) \(6516989503065492\) \(x^{14}\mathstrut +\mathstrut \) \(17\!\cdots\!98\) \(x^{12}\mathstrut +\mathstrut \) \(23\!\cdots\!44\) \(x^{10}\mathstrut +\mathstrut \) \(17\!\cdots\!45\) \(x^{8}\mathstrut +\mathstrut \) \(72\!\cdots\!20\) \(x^{6}\mathstrut +\mathstrut \) \(15\!\cdots\!00\) \(x^{4}\mathstrut +\mathstrut \) \(15\!\cdots\!00\) \(x^{2}\mathstrut +\mathstrut \) \(51\!\cdots\!00\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(11\!\cdots\!75\) \(\nu^{15}\mathstrut +\mathstrut \) \(59\!\cdots\!12\) \(\nu^{14}\mathstrut -\mathstrut \) \(66\!\cdots\!00\) \(\nu^{13}\mathstrut +\mathstrut \) \(34\!\cdots\!04\) \(\nu^{12}\mathstrut -\mathstrut \) \(14\!\cdots\!50\) \(\nu^{11}\mathstrut +\mathstrut \) \(77\!\cdots\!76\) \(\nu^{10}\mathstrut -\mathstrut \) \(15\!\cdots\!00\) \(\nu^{9}\mathstrut +\mathstrut \) \(85\!\cdots\!08\) \(\nu^{8}\mathstrut -\mathstrut \) \(88\!\cdots\!75\) \(\nu^{7}\mathstrut +\mathstrut \) \(49\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(24\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(14\!\cdots\!20\) \(\nu^{4}\mathstrut -\mathstrut \) \(31\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(14\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(71\!\cdots\!00\)\()/\)\(46\!\cdots\!00\)
\(\beta_{2}\)\(=\)\((\)\(11\!\cdots\!75\) \(\nu^{15}\mathstrut -\mathstrut \) \(59\!\cdots\!12\) \(\nu^{14}\mathstrut +\mathstrut \) \(66\!\cdots\!00\) \(\nu^{13}\mathstrut -\mathstrut \) \(34\!\cdots\!04\) \(\nu^{12}\mathstrut +\mathstrut \) \(14\!\cdots\!50\) \(\nu^{11}\mathstrut -\mathstrut \) \(77\!\cdots\!76\) \(\nu^{10}\mathstrut +\mathstrut \) \(15\!\cdots\!00\) \(\nu^{9}\mathstrut -\mathstrut \) \(85\!\cdots\!08\) \(\nu^{8}\mathstrut +\mathstrut \) \(88\!\cdots\!75\) \(\nu^{7}\mathstrut -\mathstrut \) \(49\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(24\!\cdots\!00\) \(\nu^{5}\mathstrut -\mathstrut \) \(14\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(31\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(18\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(18\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(71\!\cdots\!00\)\()/\)\(23\!\cdots\!00\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(41\!\cdots\!39\) \(\nu^{15}\mathstrut +\mathstrut \) \(38\!\cdots\!52\) \(\nu^{14}\mathstrut -\mathstrut \) \(23\!\cdots\!88\) \(\nu^{13}\mathstrut +\mathstrut \) \(21\!\cdots\!84\) \(\nu^{12}\mathstrut -\mathstrut \) \(48\!\cdots\!42\) \(\nu^{11}\mathstrut +\mathstrut \) \(44\!\cdots\!96\) \(\nu^{10}\mathstrut -\mathstrut \) \(47\!\cdots\!36\) \(\nu^{9}\mathstrut +\mathstrut \) \(43\!\cdots\!68\) \(\nu^{8}\mathstrut -\mathstrut \) \(22\!\cdots\!55\) \(\nu^{7}\mathstrut +\mathstrut \) \(20\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(45\!\cdots\!40\) \(\nu^{5}\mathstrut +\mathstrut \) \(42\!\cdots\!20\) \(\nu^{4}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(20\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(14\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(51\!\cdots\!00\)\()/\)\(77\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(53\!\cdots\!33\) \(\nu^{15}\mathstrut -\mathstrut \) \(65\!\cdots\!04\) \(\nu^{14}\mathstrut -\mathstrut \) \(30\!\cdots\!36\) \(\nu^{13}\mathstrut -\mathstrut \) \(36\!\cdots\!88\) \(\nu^{12}\mathstrut -\mathstrut \) \(68\!\cdots\!74\) \(\nu^{11}\mathstrut -\mathstrut \) \(78\!\cdots\!52\) \(\nu^{10}\mathstrut -\mathstrut \) \(74\!\cdots\!92\) \(\nu^{9}\mathstrut -\mathstrut \) \(81\!\cdots\!56\) \(\nu^{8}\mathstrut -\mathstrut \) \(42\!\cdots\!85\) \(\nu^{7}\mathstrut -\mathstrut \) \(43\!\cdots\!60\) \(\nu^{6}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(11\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(14\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(81\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(59\!\cdots\!00\)\()/\)\(11\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(39\!\cdots\!05\) \(\nu^{15}\mathstrut +\mathstrut \) \(54\!\cdots\!12\) \(\nu^{14}\mathstrut +\mathstrut \) \(22\!\cdots\!60\) \(\nu^{13}\mathstrut +\mathstrut \) \(34\!\cdots\!64\) \(\nu^{12}\mathstrut +\mathstrut \) \(50\!\cdots\!90\) \(\nu^{11}\mathstrut +\mathstrut \) \(85\!\cdots\!56\) \(\nu^{10}\mathstrut +\mathstrut \) \(53\!\cdots\!20\) \(\nu^{9}\mathstrut +\mathstrut \) \(10\!\cdots\!68\) \(\nu^{8}\mathstrut +\mathstrut \) \(29\!\cdots\!25\) \(\nu^{7}\mathstrut +\mathstrut \) \(69\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(82\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(22\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(31\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(47\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(12\!\cdots\!00\)\()/\)\(23\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(90\!\cdots\!09\) \(\nu^{15}\mathstrut +\mathstrut \) \(23\!\cdots\!20\) \(\nu^{14}\mathstrut +\mathstrut \) \(19\!\cdots\!28\) \(\nu^{13}\mathstrut +\mathstrut \) \(13\!\cdots\!80\) \(\nu^{12}\mathstrut +\mathstrut \) \(91\!\cdots\!42\) \(\nu^{11}\mathstrut +\mathstrut \) \(28\!\cdots\!80\) \(\nu^{10}\mathstrut +\mathstrut \) \(17\!\cdots\!36\) \(\nu^{9}\mathstrut +\mathstrut \) \(29\!\cdots\!20\) \(\nu^{8}\mathstrut +\mathstrut \) \(15\!\cdots\!45\) \(\nu^{7}\mathstrut +\mathstrut \) \(14\!\cdots\!00\) \(\nu^{6}\mathstrut +\mathstrut \) \(69\!\cdots\!40\) \(\nu^{5}\mathstrut +\mathstrut \) \(35\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(12\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(32\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(64\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(90\!\cdots\!00\)\()/\)\(46\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(81\!\cdots\!23\) \(\nu^{15}\mathstrut +\mathstrut \) \(48\!\cdots\!72\) \(\nu^{14}\mathstrut +\mathstrut \) \(46\!\cdots\!16\) \(\nu^{13}\mathstrut +\mathstrut \) \(27\!\cdots\!84\) \(\nu^{12}\mathstrut +\mathstrut \) \(10\!\cdots\!74\) \(\nu^{11}\mathstrut +\mathstrut \) \(59\!\cdots\!36\) \(\nu^{10}\mathstrut +\mathstrut \) \(10\!\cdots\!92\) \(\nu^{9}\mathstrut +\mathstrut \) \(61\!\cdots\!08\) \(\nu^{8}\mathstrut +\mathstrut \) \(59\!\cdots\!15\) \(\nu^{7}\mathstrut +\mathstrut \) \(32\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(16\!\cdots\!80\) \(\nu^{5}\mathstrut +\mathstrut \) \(83\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(19\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(92\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(78\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(34\!\cdots\!00\)\()/\)\(15\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(19\!\cdots\!71\) \(\nu^{15}\mathstrut -\mathstrut \) \(72\!\cdots\!00\) \(\nu^{14}\mathstrut -\mathstrut \) \(10\!\cdots\!32\) \(\nu^{13}\mathstrut -\mathstrut \) \(28\!\cdots\!80\) \(\nu^{12}\mathstrut -\mathstrut \) \(22\!\cdots\!58\) \(\nu^{11}\mathstrut -\mathstrut \) \(14\!\cdots\!00\) \(\nu^{10}\mathstrut -\mathstrut \) \(24\!\cdots\!64\) \(\nu^{9}\mathstrut -\mathstrut \) \(26\!\cdots\!60\) \(\nu^{8}\mathstrut -\mathstrut \) \(14\!\cdots\!15\) \(\nu^{7}\mathstrut -\mathstrut \) \(23\!\cdots\!60\) \(\nu^{6}\mathstrut -\mathstrut \) \(46\!\cdots\!60\) \(\nu^{5}\mathstrut -\mathstrut \) \(96\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(86\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(15\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(80\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(77\!\cdots\!00\)\()/\)\(14\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(40\!\cdots\!11\) \(\nu^{15}\mathstrut +\mathstrut \) \(10\!\cdots\!76\) \(\nu^{14}\mathstrut -\mathstrut \) \(21\!\cdots\!12\) \(\nu^{13}\mathstrut +\mathstrut \) \(55\!\cdots\!52\) \(\nu^{12}\mathstrut -\mathstrut \) \(39\!\cdots\!98\) \(\nu^{11}\mathstrut +\mathstrut \) \(11\!\cdots\!68\) \(\nu^{10}\mathstrut -\mathstrut \) \(32\!\cdots\!84\) \(\nu^{9}\mathstrut +\mathstrut \) \(10\!\cdots\!64\) \(\nu^{8}\mathstrut -\mathstrut \) \(10\!\cdots\!35\) \(\nu^{7}\mathstrut +\mathstrut \) \(49\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(59\!\cdots\!60\) \(\nu^{5}\mathstrut +\mathstrut \) \(10\!\cdots\!60\) \(\nu^{4}\mathstrut +\mathstrut \) \(44\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(97\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(50\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(82\!\cdots\!00\)\()/\)\(23\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(12\!\cdots\!51\) \(\nu^{15}\mathstrut -\mathstrut \) \(10\!\cdots\!44\) \(\nu^{14}\mathstrut +\mathstrut \) \(67\!\cdots\!92\) \(\nu^{13}\mathstrut -\mathstrut \) \(57\!\cdots\!08\) \(\nu^{12}\mathstrut +\mathstrut \) \(13\!\cdots\!58\) \(\nu^{11}\mathstrut -\mathstrut \) \(11\!\cdots\!72\) \(\nu^{10}\mathstrut +\mathstrut \) \(12\!\cdots\!64\) \(\nu^{9}\mathstrut -\mathstrut \) \(11\!\cdots\!96\) \(\nu^{8}\mathstrut +\mathstrut \) \(51\!\cdots\!75\) \(\nu^{7}\mathstrut -\mathstrut \) \(51\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(38\!\cdots\!60\) \(\nu^{5}\mathstrut -\mathstrut \) \(97\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(18\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(48\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(27\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(65\!\cdots\!00\)\()/\)\(11\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(56\!\cdots\!63\) \(\nu^{15}\mathstrut +\mathstrut \) \(66\!\cdots\!24\) \(\nu^{14}\mathstrut +\mathstrut \) \(30\!\cdots\!96\) \(\nu^{13}\mathstrut +\mathstrut \) \(36\!\cdots\!48\) \(\nu^{12}\mathstrut +\mathstrut \) \(62\!\cdots\!74\) \(\nu^{11}\mathstrut +\mathstrut \) \(73\!\cdots\!12\) \(\nu^{10}\mathstrut +\mathstrut \) \(59\!\cdots\!92\) \(\nu^{9}\mathstrut +\mathstrut \) \(69\!\cdots\!76\) \(\nu^{8}\mathstrut +\mathstrut \) \(25\!\cdots\!95\) \(\nu^{7}\mathstrut +\mathstrut \) \(29\!\cdots\!00\) \(\nu^{6}\mathstrut +\mathstrut \) \(39\!\cdots\!80\) \(\nu^{5}\mathstrut +\mathstrut \) \(45\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(63\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(25\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(42\!\cdots\!00\)\()/\)\(46\!\cdots\!00\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(12\!\cdots\!89\) \(\nu^{15}\mathstrut -\mathstrut \) \(24\!\cdots\!44\) \(\nu^{14}\mathstrut -\mathstrut \) \(70\!\cdots\!88\) \(\nu^{13}\mathstrut -\mathstrut \) \(13\!\cdots\!88\) \(\nu^{12}\mathstrut -\mathstrut \) \(15\!\cdots\!22\) \(\nu^{11}\mathstrut -\mathstrut \) \(27\!\cdots\!52\) \(\nu^{10}\mathstrut -\mathstrut \) \(15\!\cdots\!76\) \(\nu^{9}\mathstrut -\mathstrut \) \(26\!\cdots\!96\) \(\nu^{8}\mathstrut -\mathstrut \) \(79\!\cdots\!85\) \(\nu^{7}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(18\!\cdots\!40\) \(\nu^{5}\mathstrut -\mathstrut \) \(22\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(17\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(73\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(47\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(64\!\cdots\!00\)\()/\)\(97\!\cdots\!00\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(15\!\cdots\!95\) \(\nu^{15}\mathstrut +\mathstrut \) \(25\!\cdots\!44\) \(\nu^{14}\mathstrut -\mathstrut \) \(89\!\cdots\!40\) \(\nu^{13}\mathstrut +\mathstrut \) \(14\!\cdots\!88\) \(\nu^{12}\mathstrut -\mathstrut \) \(19\!\cdots\!90\) \(\nu^{11}\mathstrut +\mathstrut \) \(30\!\cdots\!92\) \(\nu^{10}\mathstrut -\mathstrut \) \(21\!\cdots\!20\) \(\nu^{9}\mathstrut +\mathstrut \) \(31\!\cdots\!16\) \(\nu^{8}\mathstrut -\mathstrut \) \(12\!\cdots\!55\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!20\) \(\nu^{6}\mathstrut -\mathstrut \) \(34\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(42\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(43\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(46\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(20\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(17\!\cdots\!00\)\()/\)\(46\!\cdots\!00\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(34\!\cdots\!39\) \(\nu^{15}\mathstrut +\mathstrut \) \(71\!\cdots\!44\) \(\nu^{14}\mathstrut -\mathstrut \) \(19\!\cdots\!28\) \(\nu^{13}\mathstrut +\mathstrut \) \(39\!\cdots\!68\) \(\nu^{12}\mathstrut -\mathstrut \) \(41\!\cdots\!82\) \(\nu^{11}\mathstrut +\mathstrut \) \(84\!\cdots\!92\) \(\nu^{10}\mathstrut -\mathstrut \) \(43\!\cdots\!36\) \(\nu^{9}\mathstrut +\mathstrut \) \(86\!\cdots\!76\) \(\nu^{8}\mathstrut -\mathstrut \) \(22\!\cdots\!75\) \(\nu^{7}\mathstrut +\mathstrut \) \(43\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(57\!\cdots\!40\) \(\nu^{5}\mathstrut +\mathstrut \) \(10\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(59\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(10\!\cdots\!00\) \(\nu^{2}\mathstrut -\mathstrut \) \(14\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(32\!\cdots\!00\)\()/\)\(51\!\cdots\!00\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(32\!\cdots\!63\) \(\nu^{15}\mathstrut -\mathstrut \) \(90\!\cdots\!20\) \(\nu^{14}\mathstrut -\mathstrut \) \(15\!\cdots\!16\) \(\nu^{13}\mathstrut -\mathstrut \) \(51\!\cdots\!20\) \(\nu^{12}\mathstrut -\mathstrut \) \(26\!\cdots\!74\) \(\nu^{11}\mathstrut -\mathstrut \) \(11\!\cdots\!00\) \(\nu^{10}\mathstrut -\mathstrut \) \(13\!\cdots\!32\) \(\nu^{9}\mathstrut -\mathstrut \) \(11\!\cdots\!60\) \(\nu^{8}\mathstrut +\mathstrut \) \(70\!\cdots\!65\) \(\nu^{7}\mathstrut -\mathstrut \) \(59\!\cdots\!00\) \(\nu^{6}\mathstrut +\mathstrut \) \(89\!\cdots\!20\) \(\nu^{5}\mathstrut -\mathstrut \) \(14\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(25\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(15\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(17\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(56\!\cdots\!00\)\()/\)\(46\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(198\) \(\beta_{1}\mathstrut +\mathstrut \) \(50\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(89\) \(\beta_{7}\mathstrut +\mathstrut \) \(2378\) \(\beta_{5}\mathstrut +\mathstrut \) \(6644\) \(\beta_{4}\mathstrut -\mathstrut \) \(31791\) \(\beta_{3}\mathstrut -\mathstrut \) \(8938479\) \(\beta_{2}\mathstrut +\mathstrut \) \(135361742291\) \(\beta_{1}\mathstrut -\mathstrut \) \(208543630262112348\)\()/256\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(407791\) \(\beta_{15}\mathstrut +\mathstrut \) \(21026577\) \(\beta_{14}\mathstrut -\mathstrut \) \(26417792\) \(\beta_{13}\mathstrut -\mathstrut \) \(96107592\) \(\beta_{12}\mathstrut -\mathstrut \) \(79804680\) \(\beta_{11}\mathstrut -\mathstrut \) \(725167\) \(\beta_{10}\mathstrut -\mathstrut \) \(15372553\) \(\beta_{9}\mathstrut -\mathstrut \) \(1375775452\) \(\beta_{8}\mathstrut +\mathstrut \) \(343114955191\) \(\beta_{7}\mathstrut -\mathstrut \) \(2138638980\) \(\beta_{6}\mathstrut -\mathstrut \) \(767372551462\) \(\beta_{5}\mathstrut +\mathstrut \) \(795017851516\) \(\beta_{4}\mathstrut -\mathstrut \) \(136034954570889\) \(\beta_{3}\mathstrut -\mathstrut \) \(332977118954390464\) \(\beta_{2}\mathstrut -\mathstrut \) \(20532676076683527041\) \(\beta_{1}\mathstrut -\mathstrut \) \(5299589859602173084\)\()/4096\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(27294115667058\) \(\beta_{13}\mathstrut +\mathstrut \) \(10655528194106073\) \(\beta_{12}\mathstrut +\mathstrut \) \(7514335012131459\) \(\beta_{11}\mathstrut +\mathstrut \) \(1772567228721951\) \(\beta_{10}\mathstrut -\mathstrut \) \(115572322879383570\) \(\beta_{9}\mathstrut +\mathstrut \) \(43470449657526939\) \(\beta_{8}\mathstrut +\mathstrut \) \(28610549839549179183\) \(\beta_{7}\mathstrut +\mathstrut \) \(152131798685917989\) \(\beta_{6}\mathstrut -\mathstrut \) \(471937617230945288871\) \(\beta_{5}\mathstrut -\mathstrut \) \(612412939169372380719\) \(\beta_{4}\mathstrut +\mathstrut \) \(13081537698891540569712\) \(\beta_{3}\mathstrut +\mathstrut \) \(1854792558808224422089212\) \(\beta_{2}\mathstrut -\mathstrut \) \(28121886913447525417647029301\) \(\beta_{1}\mathstrut +\mathstrut \) \(17207655784145280320163541081258932\)\()/16384\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(181676081806636225310130\) \(\beta_{15}\mathstrut -\mathstrut \) \(2463839071735942811309490\) \(\beta_{14}\mathstrut +\mathstrut \) \(4060381078373312640682560\) \(\beta_{13}\mathstrut +\mathstrut \) \(16599908231020969254737955\) \(\beta_{12}\mathstrut +\mathstrut \) \(10866820643702738258410260\) \(\beta_{11}\mathstrut -\mathstrut \) \(247492807611303476684370\) \(\beta_{10}\mathstrut +\mathstrut \) \(1004175342803314722650415\) \(\beta_{9}\mathstrut +\mathstrut \) \(186757975616842236031476180\) \(\beta_{8}\mathstrut -\mathstrut \) \(57291004667482368513180490221\) \(\beta_{7}\mathstrut +\mathstrut \) \(97648606839513969564714171\) \(\beta_{6}\mathstrut +\mathstrut \) \(159953927684267272319706817869\) \(\beta_{5}\mathstrut -\mathstrut \) \(167919180922909170008481267087\) \(\beta_{4}\mathstrut +\mathstrut \) \(29183793236336188166299714754541\) \(\beta_{3}\mathstrut +\mathstrut \) \(33216020335375583242764028171438248\) \(\beta_{2}\mathstrut -\mathstrut \) \(3213727326400799500139171090854996188\) \(\beta_{1}\mathstrut -\mathstrut \) \(786838347652366325542887728338203894\)\()/262144\)
\(\nu^{6}\)\(=\)\((\)\(37\!\cdots\!34\) \(\beta_{13}\mathstrut -\mathstrut \) \(23\!\cdots\!65\) \(\beta_{12}\mathstrut -\mathstrut \) \(18\!\cdots\!05\) \(\beta_{11}\mathstrut -\mathstrut \) \(31\!\cdots\!01\) \(\beta_{10}\mathstrut +\mathstrut \) \(13\!\cdots\!30\) \(\beta_{9}\mathstrut -\mathstrut \) \(13\!\cdots\!07\) \(\beta_{8}\mathstrut -\mathstrut \) \(50\!\cdots\!09\) \(\beta_{7}\mathstrut -\mathstrut \) \(33\!\cdots\!37\) \(\beta_{6}\mathstrut +\mathstrut \) \(65\!\cdots\!85\) \(\beta_{5}\mathstrut +\mathstrut \) \(53\!\cdots\!89\) \(\beta_{4}\mathstrut -\mathstrut \) \(29\!\cdots\!78\) \(\beta_{3}\mathstrut -\mathstrut \) \(25\!\cdots\!24\) \(\beta_{2}\mathstrut +\mathstrut \) \(39\!\cdots\!49\) \(\beta_{1}\mathstrut -\mathstrut \) \(16\!\cdots\!14\)\()/1048576\)
\(\nu^{7}\)\(=\)\((\)\(45\!\cdots\!64\) \(\beta_{15}\mathstrut +\mathstrut \) \(27\!\cdots\!92\) \(\beta_{14}\mathstrut -\mathstrut \) \(52\!\cdots\!48\) \(\beta_{13}\mathstrut -\mathstrut \) \(22\!\cdots\!29\) \(\beta_{12}\mathstrut -\mathstrut \) \(12\!\cdots\!08\) \(\beta_{11}\mathstrut +\mathstrut \) \(29\!\cdots\!96\) \(\beta_{10}\mathstrut -\mathstrut \) \(39\!\cdots\!67\) \(\beta_{9}\mathstrut -\mathstrut \) \(24\!\cdots\!64\) \(\beta_{8}\mathstrut +\mathstrut \) \(77\!\cdots\!57\) \(\beta_{7}\mathstrut +\mathstrut \) \(22\!\cdots\!03\) \(\beta_{6}\mathstrut -\mathstrut \) \(24\!\cdots\!47\) \(\beta_{5}\mathstrut +\mathstrut \) \(26\!\cdots\!21\) \(\beta_{4}\mathstrut -\mathstrut \) \(50\!\cdots\!69\) \(\beta_{3}\mathstrut -\mathstrut \) \(36\!\cdots\!24\) \(\beta_{2}\mathstrut +\mathstrut \) \(82\!\cdots\!90\) \(\beta_{1}\mathstrut +\mathstrut \) \(20\!\cdots\!82\)\()/16777216\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(11\!\cdots\!24\) \(\beta_{13}\mathstrut +\mathstrut \) \(47\!\cdots\!51\) \(\beta_{12}\mathstrut +\mathstrut \) \(38\!\cdots\!84\) \(\beta_{11}\mathstrut +\mathstrut \) \(52\!\cdots\!68\) \(\beta_{10}\mathstrut -\mathstrut \) \(19\!\cdots\!39\) \(\beta_{9}\mathstrut -\mathstrut \) \(18\!\cdots\!16\) \(\beta_{8}\mathstrut +\mathstrut \) \(96\!\cdots\!97\) \(\beta_{7}\mathstrut +\mathstrut \) \(66\!\cdots\!67\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\!\cdots\!54\) \(\beta_{5}\mathstrut -\mathstrut \) \(55\!\cdots\!00\) \(\beta_{4}\mathstrut +\mathstrut \) \(58\!\cdots\!32\) \(\beta_{3}\mathstrut +\mathstrut \) \(41\!\cdots\!93\) \(\beta_{2}\mathstrut -\mathstrut \) \(62\!\cdots\!72\) \(\beta_{1}\mathstrut +\mathstrut \) \(23\!\cdots\!83\)\()/8388608\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(59\!\cdots\!00\) \(\beta_{15}\mathstrut -\mathstrut \) \(23\!\cdots\!00\) \(\beta_{14}\mathstrut +\mathstrut \) \(51\!\cdots\!08\) \(\beta_{13}\mathstrut +\mathstrut \) \(23\!\cdots\!36\) \(\beta_{12}\mathstrut +\mathstrut \) \(10\!\cdots\!64\) \(\beta_{11}\mathstrut -\mathstrut \) \(75\!\cdots\!64\) \(\beta_{10}\mathstrut -\mathstrut \) \(13\!\cdots\!48\) \(\beta_{9}\mathstrut +\mathstrut \) \(25\!\cdots\!36\) \(\beta_{8}\mathstrut -\mathstrut \) \(77\!\cdots\!72\) \(\beta_{7}\mathstrut -\mathstrut \) \(57\!\cdots\!80\) \(\beta_{6}\mathstrut +\mathstrut \) \(27\!\cdots\!63\) \(\beta_{5}\mathstrut -\mathstrut \) \(29\!\cdots\!35\) \(\beta_{4}\mathstrut +\mathstrut \) \(60\!\cdots\!75\) \(\beta_{3}\mathstrut +\mathstrut \) \(33\!\cdots\!07\) \(\beta_{2}\mathstrut -\mathstrut \) \(10\!\cdots\!26\) \(\beta_{1}\mathstrut -\mathstrut \) \(26\!\cdots\!85\)\()/8388608\)
\(\nu^{10}\)\(=\)\((\)\(32\!\cdots\!00\) \(\beta_{13}\mathstrut -\mathstrut \) \(11\!\cdots\!93\) \(\beta_{12}\mathstrut -\mathstrut \) \(89\!\cdots\!52\) \(\beta_{11}\mathstrut -\mathstrut \) \(99\!\cdots\!16\) \(\beta_{10}\mathstrut +\mathstrut \) \(36\!\cdots\!97\) \(\beta_{9}\mathstrut +\mathstrut \) \(21\!\cdots\!04\) \(\beta_{8}\mathstrut -\mathstrut \) \(21\!\cdots\!19\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\!\cdots\!05\) \(\beta_{6}\mathstrut +\mathstrut \) \(20\!\cdots\!12\) \(\beta_{5}\mathstrut +\mathstrut \) \(67\!\cdots\!18\) \(\beta_{4}\mathstrut -\mathstrut \) \(13\!\cdots\!42\) \(\beta_{3}\mathstrut -\mathstrut \) \(80\!\cdots\!97\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\!\cdots\!84\) \(\beta_{1}\mathstrut -\mathstrut \) \(41\!\cdots\!67\)\()/8388608\)
\(\nu^{11}\)\(=\)\((\)\(27\!\cdots\!96\) \(\beta_{15}\mathstrut +\mathstrut \) \(80\!\cdots\!88\) \(\beta_{14}\mathstrut -\mathstrut \) \(20\!\cdots\!20\) \(\beta_{13}\mathstrut -\mathstrut \) \(90\!\cdots\!97\) \(\beta_{12}\mathstrut -\mathstrut \) \(35\!\cdots\!96\) \(\beta_{11}\mathstrut -\mathstrut \) \(78\!\cdots\!72\) \(\beta_{10}\mathstrut +\mathstrut \) \(18\!\cdots\!25\) \(\beta_{9}\mathstrut -\mathstrut \) \(10\!\cdots\!12\) \(\beta_{8}\mathstrut +\mathstrut \) \(30\!\cdots\!85\) \(\beta_{7}\mathstrut +\mathstrut \) \(34\!\cdots\!67\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\!\cdots\!21\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\!\cdots\!79\) \(\beta_{4}\mathstrut -\mathstrut \) \(27\!\cdots\!91\) \(\beta_{3}\mathstrut -\mathstrut \) \(12\!\cdots\!18\) \(\beta_{2}\mathstrut +\mathstrut \) \(49\!\cdots\!26\) \(\beta_{1}\mathstrut +\mathstrut \) \(12\!\cdots\!48\)\()/16777216\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(38\!\cdots\!28\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\!\cdots\!24\) \(\beta_{12}\mathstrut +\mathstrut \) \(97\!\cdots\!96\) \(\beta_{11}\mathstrut +\mathstrut \) \(91\!\cdots\!00\) \(\beta_{10}\mathstrut -\mathstrut \) \(35\!\cdots\!96\) \(\beta_{9}\mathstrut +\mathstrut \) \(93\!\cdots\!32\) \(\beta_{8}\mathstrut +\mathstrut \) \(22\!\cdots\!60\) \(\beta_{7}\mathstrut +\mathstrut \) \(15\!\cdots\!44\) \(\beta_{6}\mathstrut -\mathstrut \) \(19\!\cdots\!21\) \(\beta_{5}\mathstrut -\mathstrut \) \(33\!\cdots\!07\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\!\cdots\!07\) \(\beta_{3}\mathstrut +\mathstrut \) \(78\!\cdots\!99\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\!\cdots\!50\) \(\beta_{1}\mathstrut +\mathstrut \) \(38\!\cdots\!59\)\()/4194304\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(14\!\cdots\!12\) \(\beta_{15}\mathstrut -\mathstrut \) \(34\!\cdots\!36\) \(\beta_{14}\mathstrut +\mathstrut \) \(97\!\cdots\!56\) \(\beta_{13}\mathstrut +\mathstrut \) \(44\!\cdots\!31\) \(\beta_{12}\mathstrut +\mathstrut \) \(14\!\cdots\!40\) \(\beta_{11}\mathstrut +\mathstrut \) \(90\!\cdots\!56\) \(\beta_{10}\mathstrut -\mathstrut \) \(13\!\cdots\!71\) \(\beta_{9}\mathstrut +\mathstrut \) \(55\!\cdots\!36\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\!\cdots\!99\) \(\beta_{7}\mathstrut -\mathstrut \) \(22\!\cdots\!49\) \(\beta_{6}\mathstrut +\mathstrut \) \(60\!\cdots\!28\) \(\beta_{5}\mathstrut -\mathstrut \) \(66\!\cdots\!78\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\!\cdots\!62\) \(\beta_{3}\mathstrut +\mathstrut \) \(59\!\cdots\!15\) \(\beta_{2}\mathstrut -\mathstrut \) \(27\!\cdots\!04\) \(\beta_{1}\mathstrut -\mathstrut \) \(67\!\cdots\!91\)\()/4194304\)
\(\nu^{14}\)\(=\)\((\)\(87\!\cdots\!24\) \(\beta_{13}\mathstrut -\mathstrut \) \(26\!\cdots\!23\) \(\beta_{12}\mathstrut -\mathstrut \) \(20\!\cdots\!52\) \(\beta_{11}\mathstrut -\mathstrut \) \(16\!\cdots\!72\) \(\beta_{10}\mathstrut +\mathstrut \) \(68\!\cdots\!67\) \(\beta_{9}\mathstrut -\mathstrut \) \(82\!\cdots\!88\) \(\beta_{8}\mathstrut -\mathstrut \) \(47\!\cdots\!53\) \(\beta_{7}\mathstrut -\mathstrut \) \(31\!\cdots\!87\) \(\beta_{6}\mathstrut +\mathstrut \) \(38\!\cdots\!72\) \(\beta_{5}\mathstrut +\mathstrut \) \(14\!\cdots\!62\) \(\beta_{4}\mathstrut -\mathstrut \) \(28\!\cdots\!70\) \(\beta_{3}\mathstrut -\mathstrut \) \(15\!\cdots\!91\) \(\beta_{2}\mathstrut +\mathstrut \) \(23\!\cdots\!28\) \(\beta_{1}\mathstrut -\mathstrut \) \(73\!\cdots\!61\)\()/4194304\)
\(\nu^{15}\)\(=\)\((\)\(12\!\cdots\!48\) \(\beta_{15}\mathstrut +\mathstrut \) \(24\!\cdots\!44\) \(\beta_{14}\mathstrut -\mathstrut \) \(75\!\cdots\!72\) \(\beta_{13}\mathstrut -\mathstrut \) \(35\!\cdots\!65\) \(\beta_{12}\mathstrut -\mathstrut \) \(10\!\cdots\!44\) \(\beta_{11}\mathstrut -\mathstrut \) \(10\!\cdots\!40\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\!\cdots\!97\) \(\beta_{9}\mathstrut -\mathstrut \) \(45\!\cdots\!60\) \(\beta_{8}\mathstrut +\mathstrut \) \(11\!\cdots\!33\) \(\beta_{7}\mathstrut +\mathstrut \) \(20\!\cdots\!71\) \(\beta_{6}\mathstrut -\mathstrut \) \(49\!\cdots\!35\) \(\beta_{5}\mathstrut +\mathstrut \) \(54\!\cdots\!57\) \(\beta_{4}\mathstrut -\mathstrut \) \(12\!\cdots\!53\) \(\beta_{3}\mathstrut -\mathstrut \) \(45\!\cdots\!92\) \(\beta_{2}\mathstrut +\mathstrut \) \(23\!\cdots\!62\) \(\beta_{1}\mathstrut +\mathstrut \) \(57\!\cdots\!94\)\()/16777216\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.10776e7i
1.10776e7i
4.46418e7i
4.46418e7i
3.83971e7i
3.83971e7i
8.47589e6i
8.47589e6i
2.99676e7i
2.99676e7i
2.95956e7i
2.95956e7i
2.71780e7i
2.71780e7i
1.85096e7i
1.85096e7i
−227674. 129939.i 1.77242e8i 3.49511e10 + 5.91675e10i 5.55508e12 −2.30307e13 + 4.03533e13i 2.49971e15i −2.69262e14 1.80124e16i 1.18680e17 −1.26475e18 7.21823e17i
3.2 −227674. + 129939.i 1.77242e8i 3.49511e10 5.91675e10i 5.55508e12 −2.30307e13 4.03533e13i 2.49971e15i −2.69262e14 + 1.80124e16i 1.18680e17 −1.26475e18 + 7.21823e17i
3.3 −210329. 156464.i 7.14268e8i 1.97575e10 + 6.58180e10i 8.84656e11 1.11757e14 1.50232e14i 1.08208e15i 6.14256e15 1.69348e16i −3.60085e17 −1.86069e17 1.38417e17i
3.4 −210329. + 156464.i 7.14268e8i 1.97575e10 6.58180e10i 8.84656e11 1.11757e14 + 1.50232e14i 1.08208e15i 6.14256e15 + 1.69348e16i −3.60085e17 −1.86069e17 + 1.38417e17i
3.5 −167207. 201894.i 6.14354e8i −1.28029e10 + 6.75163e10i −3.33360e12 −1.24034e14 + 1.02724e14i 2.74304e15i 1.57719e16 8.70438e15i −2.27336e17 5.57402e17 + 6.73034e17i
3.6 −167207. + 201894.i 6.14354e8i −1.28029e10 6.75163e10i −3.33360e12 −1.24034e14 1.02724e14i 2.74304e15i 1.57719e16 + 8.70438e15i −2.27336e17 5.57402e17 6.73034e17i
3.7 −39525.7 259147.i 1.35614e8i −6.55949e10 + 2.04859e10i −1.24080e12 3.51440e13 5.36025e12i 4.20496e14i 7.90156e15 + 1.61890e16i 1.31703e17 4.90435e16 + 3.21549e17i
3.8 −39525.7 + 259147.i 1.35614e8i −6.55949e10 2.04859e10i −1.24080e12 3.51440e13 + 5.36025e12i 4.20496e14i 7.90156e15 1.61890e16i 1.31703e17 4.90435e16 3.21549e17i
3.9 98385.8 242981.i 4.79482e8i −4.93600e10 4.78117e10i 6.60636e12 −1.16505e14 4.71742e13i 2.27014e14i −1.64737e16 + 7.28954e15i −7.98083e16 6.49972e17 1.60522e18i
3.10 98385.8 + 242981.i 4.79482e8i −4.93600e10 + 4.78117e10i 6.60636e12 −1.16505e14 + 4.71742e13i 2.27014e14i −1.64737e16 7.28954e15i −7.98083e16 6.49972e17 + 1.60522e18i
3.11 178949. 191564.i 4.73530e8i −4.67423e9 6.85603e10i 4.97998e11 9.07113e13 + 8.47375e13i 9.39063e14i −1.39702e16 1.13734e16i −7.41357e16 8.91161e16 9.53986e16i
3.12 178949. + 191564.i 4.73530e8i −4.67423e9 + 6.85603e10i 4.97998e11 9.07113e13 8.47375e13i 9.39063e14i −1.39702e16 + 1.13734e16i −7.41357e16 8.91161e16 + 9.53986e16i
3.13 194816. 175403.i 4.34848e8i 7.18697e9 6.83426e10i −7.52461e12 −7.62737e13 8.47153e13i 2.26538e15i −1.05874e16 1.45748e16i −3.89984e16 −1.46591e18 + 1.31984e18i
3.14 194816. + 175403.i 4.34848e8i 7.18697e9 + 6.83426e10i −7.52461e12 −7.62737e13 + 8.47153e13i 2.26538e15i −1.05874e16 + 1.45748e16i −3.89984e16 −1.46591e18 1.31984e18i
3.15 261200. 22229.7i 2.96154e8i 6.77312e10 1.16128e10i 1.46296e12 −6.58343e12 7.73554e13i 2.38729e15i 1.74332e16 4.53891e15i 6.23874e16 3.82125e17 3.25212e16i
3.16 261200. + 22229.7i 2.96154e8i 6.77312e10 + 1.16128e10i 1.46296e12 −6.58343e12 + 7.73554e13i 2.38729e15i 1.74332e16 + 4.53891e15i 6.23874e16 3.82125e17 + 3.25212e16i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.16
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{16} + \cdots\) acting on \(S_{37}^{\mathrm{new}}(4, [\chi])\).