# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{37}^{\mathrm{new}}(4, [\chi])$$.