Properties

Label 4.35.b.a
Level 4
Weight 35
Character orbit 4.b
Analytic conductor 29.290
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 35 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.2902616171\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{240}\cdot 3^{26}\cdot 5^{6}\cdot 7^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1711 + \beta_{1} ) q^{2} + ( 19 - 76 \beta_{1} - \beta_{2} ) q^{3} + ( -1252054245 - 1834 \beta_{1} + 7 \beta_{2} + \beta_{4} ) q^{4} + ( -1335724915 - 164285 \beta_{1} + 12 \beta_{2} - \beta_{3} - 7 \beta_{4} ) q^{5} + ( 1302295996831 + 128246 \beta_{1} - 7775 \beta_{2} - 3 \beta_{3} - 67 \beta_{4} - \beta_{5} ) q^{6} + ( -71490302 + 285961666 \beta_{1} + 136602 \beta_{2} - \beta_{3} - 230 \beta_{4} + \beta_{6} + \beta_{9} ) q^{7} + ( 142772685191850 - 1237915894 \beta_{1} + 726942 \beta_{2} + 330 \beta_{3} - 1654 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} + \beta_{8} - \beta_{9} ) q^{8} + ( -4239281223234021 - 11518610453 \beta_{1} + 691919 \beta_{2} + 2103 \beta_{3} + 101670 \beta_{4} + 23 \beta_{5} + 50 \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q +(-1711 + \beta_{1}) q^{2} +(19 - 76 \beta_{1} - \beta_{2}) q^{3} +(-1252054245 - 1834 \beta_{1} + 7 \beta_{2} + \beta_{4}) q^{4} +(-1335724915 - 164285 \beta_{1} + 12 \beta_{2} - \beta_{3} - 7 \beta_{4}) q^{5} +(1302295996831 + 128246 \beta_{1} - 7775 \beta_{2} - 3 \beta_{3} - 67 \beta_{4} - \beta_{5}) q^{6} +(-71490302 + 285961666 \beta_{1} + 136602 \beta_{2} - \beta_{3} - 230 \beta_{4} + \beta_{6} + \beta_{9}) q^{7} +(142772685191850 - 1237915894 \beta_{1} + 726942 \beta_{2} + 330 \beta_{3} - 1654 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} + \beta_{8} - \beta_{9}) q^{8} +(-4239281223234021 - 11518610453 \beta_{1} + 691919 \beta_{2} + 2103 \beta_{3} + 101670 \beta_{4} + 23 \beta_{5} + 50 \beta_{6} + \beta_{7}) q^{9} +(-2819611628195211 - 1600581235 \beta_{1} + 89035409 \beta_{2} + 245 \beta_{3} - 224142 \beta_{4} - 92 \beta_{5} + 429 \beta_{6} - 7 \beta_{8} + 2 \beta_{9} + \beta_{10}) q^{10} +(65176391521 - 260702130631 \beta_{1} + 238838199 \beta_{2} + 6206 \beta_{3} - 1719290 \beta_{4} - 2157 \beta_{5} - 956 \beta_{6} + \beta_{7} + 21 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{14}) q^{11} +(149325040526146211 + 1317872231977 \beta_{1} + 2629214802 \beta_{2} - 112517 \beta_{3} - 574789 \beta_{4} - 7667 \beta_{5} - 3868 \beta_{6} - 39 \beta_{7} - 81 \beta_{8} + 424 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} + \beta_{12} - 3 \beta_{14}) q^{12} +(-327092299927016875 - 7559261622320 \beta_{1} + 491848336 \beta_{2} - 484812 \beta_{3} - 66369014 \beta_{4} - 29895 \beta_{5} + 29281 \beta_{6} - 10 \beta_{7} - 890 \beta_{8} + 68 \beta_{9} + 46 \beta_{10} + 7 \beta_{11} - 4 \beta_{12} - 4 \beta_{14} - \beta_{15}) q^{13} +(-4911518264679731836 - 487758156733 \beta_{1} + 7018115890 \beta_{2} + 508451 \beta_{3} + 263788095 \beta_{4} + 126089 \beta_{5} - 8621 \beta_{6} + 1006 \beta_{7} - 419 \beta_{8} - 22452 \beta_{9} - 16 \beta_{10} - 29 \beta_{11} + 12 \beta_{12} + \beta_{13} + 28 \beta_{14} - 4 \beta_{15}) q^{14} +(26929731970483 - 107718503108025 \beta_{1} - 65466535755 \beta_{2} + 1456573 \beta_{3} - 213253257 \beta_{4} + 710127 \beta_{5} - 395278 \beta_{6} + 270 \beta_{7} + 28057 \beta_{8} - 47294 \beta_{9} - 614 \beta_{10} + 45 \beta_{11} - 32 \beta_{12} - 4 \beta_{13} + 42 \beta_{14} - 24 \beta_{15}) q^{15} +(30948845541364495589 + 145882774988110 \beta_{1} + 22940476336 \beta_{2} - 31822649 \beta_{3} - 1123677915 \beta_{4} + 943712 \beta_{5} - 402371 \beta_{6} - 10580 \beta_{7} + 2176 \beta_{8} + 351776 \beta_{9} - 208 \beta_{10} - 120 \beta_{11} + 4 \beta_{12} + 36 \beta_{13} + 244 \beta_{14} - 80 \beta_{15}) q^{16} +(-37729472786402715952 + 164664879931753 \beta_{1} - 10521354933 \beta_{2} - 206661679 \beta_{3} + 560834532 \beta_{4} - 10845363 \beta_{5} - 337992 \beta_{6} - 487 \beta_{7} - 219748 \beta_{8} + 32120 \beta_{9} - 1892 \beta_{10} - 290 \beta_{11} + 120 \beta_{12} - 128 \beta_{13} + 120 \beta_{14} - 226 \beta_{15}) q^{17} +(-\)\(19\!\cdots\!41\)\( - 4259133941166313 \beta_{1} - 755274954158 \beta_{2} - 200820434 \beta_{3} - 12061948960 \beta_{4} + 3686568 \beta_{5} - 7239994 \beta_{6} + 16032 \beta_{7} + 49346 \beta_{8} - 6010604 \beta_{9} - 1742 \beta_{10} + 32 \beta_{11} - 512 \beta_{12} + 592 \beta_{13} - 1536 \beta_{14} - 576 \beta_{15}) q^{18} +(-112734679695383 + 450902263744329 \beta_{1} + 8346772045783 \beta_{2} - 19639462 \beta_{3} + 18184633040 \beta_{4} + 110684923 \beta_{5} - 4056290 \beta_{6} - 18665 \beta_{7} + 820205 \beta_{8} - 1436199 \beta_{9} + 27673 \beta_{10} - 1098 \beta_{11} + 1600 \beta_{12} - 1848 \beta_{13} - 1031 \beta_{14} - 848 \beta_{15}) q^{19} +(-\)\(20\!\cdots\!54\)\( - 3025234656919596 \beta_{1} + 3506244106678 \beta_{2} + 4385498356 \beta_{3} - 12720442666 \beta_{4} + 46456928 \beta_{5} + 46728252 \beta_{6} + 358912 \beta_{7} + 43232 \beta_{8} + 33933232 \beta_{9} - 4096 \beta_{10} + 192 \beta_{11} - 1632 \beta_{12} + 5808 \beta_{13} - 8416 \beta_{14} - 960 \beta_{15}) q^{20} +(\)\(32\!\cdots\!56\)\( + 8880461186883161 \beta_{1} - 531695820884 \beta_{2} - 5474009607 \beta_{3} - 87932374563 \beta_{4} - 1160842579 \beta_{5} - 810683 \beta_{6} - 29434 \beta_{7} + 1756366 \beta_{8} + 1914900 \beta_{9} + 7350 \beta_{10} + 33899 \beta_{11} - 372 \beta_{12} - 15616 \beta_{13} - 372 \beta_{14} + 1443 \beta_{15}) q^{21} +(\)\(44\!\cdots\!69\)\( + 439110350151692 \beta_{1} - 37378820389737 \beta_{2} - 20030747415 \beta_{3} - 222465713103 \beta_{4} + 412469171 \beta_{5} - 127113254 \beta_{6} - 1349660 \beta_{7} - 2105242 \beta_{8} - 100059288 \beta_{9} + 33312 \beta_{10} + 4378 \beta_{11} + 9064 \beta_{12} + 36894 \beta_{13} + 40264 \beta_{14} + 10120 \beta_{15}) q^{22} +(-17792957731820875 + 71169295033760553 \beta_{1} - 67593338047237 \beta_{2} - 2988222855 \beta_{3} + 1267127429409 \beta_{4} + 4392299413 \beta_{5} + 158775364 \beta_{6} + 181954 \beta_{7} - 22618429 \beta_{8} + 60908364 \beta_{9} - 628426 \beta_{10} + 349703 \beta_{11} - 36960 \beta_{12} - 82444 \beta_{13} + 22758 \beta_{14} + 25528 \beta_{15}) q^{23} +(\)\(13\!\cdots\!84\)\( + 150673701066631320 \beta_{1} - 126483058892016 \beta_{2} + 217963572964 \beta_{3} + 1434060173068 \beta_{4} + 1768591144 \beta_{5} - 82537068 \beta_{6} - 3372208 \beta_{7} - 10407144 \beta_{8} - 105567320 \beta_{9} + 179008 \beta_{10} + 275680 \beta_{11} + 50800 \beta_{12} + 149808 \beta_{13} + 170160 \beta_{14} + 58944 \beta_{15}) q^{24} +(\)\(76\!\cdots\!19\)\( + 81090319649121224 \beta_{1} - 3161506400090 \beta_{2} - 66654495412 \beta_{3} - 6701311275430 \beta_{4} - 18311358040 \beta_{5} + 56766634 \beta_{6} - 459138 \beta_{7} + 76891244 \beta_{8} + 24425688 \beta_{9} + 639660 \beta_{10} + 1736422 \beta_{11} - 41064 \beta_{12} - 253056 \beta_{13} - 41064 \beta_{14} + 73446 \beta_{15}) q^{25} +(-\)\(12\!\cdots\!27\)\( - 339930121906161287 \beta_{1} + 544990146270773 \beta_{2} - 821729628175 \beta_{3} - 8387962915822 \beta_{4} + 2786810356 \beta_{5} + 3480473369 \beta_{6} + 29870656 \beta_{7} - 5525123 \beta_{8} + 2220677546 \beta_{9} + 26981 \beta_{10} + 3230016 \beta_{11} - 70656 \beta_{12} + 312096 \beta_{13} - 670720 \beta_{14} + 31104 \beta_{15}) q^{26} +(5361209784044796 - 21511471118067789 \beta_{1} + 2598804127567362 \beta_{2} - 102056523370 \beta_{3} + 33360728891896 \beta_{4} + 13127650453 \beta_{5} - 216531478 \beta_{6} - 5106415 \beta_{7} + 100718083 \beta_{8} - 264538105 \beta_{9} + 9097343 \beta_{10} + 7300258 \beta_{11} + 516800 \beta_{12} - 195496 \beta_{13} - 452577 \beta_{14} - 134640 \beta_{15}) q^{27} +(\)\(55\!\cdots\!50\)\( - 4907016407961630410 \beta_{1} - 1138890659296532 \beta_{2} + 2826067920514 \beta_{3} + 29557927746 \beta_{4} - 2778097202 \beta_{5} - 10526690120 \beta_{6} - 4410474 \beta_{7} + 73830554 \beta_{8} - 7401355920 \beta_{9} - 1760744 \beta_{10} + 20394228 \beta_{11} - 835194 \beta_{12} - 342144 \beta_{13} - 2286738 \beta_{14} - 689664 \beta_{15}) q^{28} +(-\)\(25\!\cdots\!55\)\( - 1169832476694191821 \beta_{1} + 124875228877220 \beta_{2} - 591582083129 \beta_{3} - 180270454053223 \beta_{4} + 157206268680 \beta_{5} - 6735185800 \beta_{6} + 4511272 \beta_{7} - 1386007616 \beta_{8} - 301860320 \beta_{9} - 14940640 \beta_{10} + 36150512 \beta_{11} + 1026624 \beta_{12} + 2018048 \beta_{13} + 1026624 \beta_{14} - 1310576 \beta_{15}) q^{29} +(\)\(18\!\cdots\!72\)\( + 186121074655639473 \beta_{1} + 10410727180145150 \beta_{2} - 11319315519111 \beta_{3} - 105072456087507 \beta_{4} - 6543637813 \beta_{5} - 26962883567 \beta_{6} - 309163846 \beta_{7} + 129240895 \beta_{8} - 2439210684 \beta_{9} - 6618288 \beta_{10} + 90026369 \beta_{11} - 177276 \beta_{12} - 4895413 \beta_{13} + 7958580 \beta_{14} - 2049324 \beta_{15}) q^{30} +(6065883016596761943 - 24264610252174229189 \beta_{1} - 12595575800252555 \beta_{2} - 1786236864016 \beta_{3} + 540248860085627 \beta_{4} - 468984250987 \beta_{5} - 49127322083 \beta_{6} + 60841810 \beta_{7} + 1984137411 \beta_{8} - 5677493291 \beta_{9} - 87543002 \beta_{10} + 167733719 \beta_{11} - 4780640 \beta_{12} + 10097076 \beta_{13} + 7158742 \beta_{14} - 1934792 \beta_{15}) q^{31} +(\)\(12\!\cdots\!56\)\( + 31077773392714521112 \beta_{1} - 6060054933878208 \beta_{2} + 2834202471116 \beta_{3} + 155693436404644 \beta_{4} - 7065112640 \beta_{5} + 66076706244 \beta_{6} + 463917936 \beta_{7} - 471372864 \beta_{8} + 65982096832 \beta_{9} - 4855872 \beta_{10} + 322303648 \beta_{11} + 8842320 \beta_{12} - 16243760 \beta_{13} + 21597968 \beta_{14} + 1918400 \beta_{15}) q^{32} +(\)\(46\!\cdots\!06\)\( - 44878519509915268463 \beta_{1} + 3617867391290801 \beta_{2} + 6954430915773 \beta_{3} - 2809362341403618 \beta_{4} + 1662721536597 \beta_{5} + 34430773058 \beta_{6} - 10206185 \beta_{7} + 6693514696 \beta_{8} - 4911043632 \beta_{9} + 176124168 \beta_{10} + 585585572 \beta_{11} - 13402224 \beta_{12} + 20278016 \beta_{13} - 13402224 \beta_{14} + 7648740 \beta_{15}) q^{33} +(\)\(28\!\cdots\!56\)\( - 37411041087967708664 \beta_{1} + 199774471512341978 \beta_{2} + 19297265563862 \beta_{3} + 72715689423984 \beta_{4} - 130696142392 \beta_{5} - 79108665810 \beta_{6} + 1500800672 \beta_{7} - 2460868726 \beta_{8} - 154672647388 \beta_{9} + 92383290 \beta_{10} + 989407264 \beta_{11} + 10780160 \beta_{12} - 19114416 \beta_{13} - 71337472 \beta_{14} + 20344256 \beta_{15}) q^{34} +(-12111512268722487902 + 48433888918877405792 \beta_{1} + 159725302347340690 \beta_{2} - 19867559583092 \beta_{3} + 6089846396547514 \beta_{4} - 32637233424 \beta_{5} + 321376129274 \beta_{6} - 218290994 \beta_{7} - 36749536752 \beta_{8} + 58446307380 \beta_{9} + 538367330 \beta_{10} + 1533348246 \beta_{11} + 29497920 \beta_{12} - 4613304 \beta_{13} - 85914814 \beta_{14} + 31910832 \beta_{15}) q^{35} +(\)\(35\!\cdots\!11\)\( - \)\(18\!\cdots\!30\)\( \beta_{1} - 594310633478294401 \beta_{2} - 181523461342808 \beta_{3} - 3820510073935231 \beta_{4} + 453180994240 \beta_{5} - 10793854536 \beta_{6} - 5747620864 \beta_{7} - 3052471360 \beta_{8} + 45665941600 \beta_{9} + 293380096 \beta_{10} + 2444027264 \beta_{11} - 63717056 \beta_{12} + 51455584 \beta_{13} - 145189824 \beta_{14} + 25289856 \beta_{15}) q^{36} +(\)\(59\!\cdots\!65\)\( + \)\(22\!\cdots\!64\)\( \beta_{1} - 10209560520023392 \beta_{2} - 93990643745184 \beta_{3} - 13512860689260298 \beta_{4} - 9335072350707 \beta_{5} - 1063275945163 \beta_{6} + 860132878 \beta_{7} + 6450765950 \beta_{8} + 8593306516 \beta_{9} - 1355247034 \beta_{10} + 3228349139 \beta_{11} + 114447532 \beta_{12} - 144844800 \beta_{13} + 114447532 \beta_{14} + 9061675 \beta_{15}) q^{37} +(-\)\(77\!\cdots\!85\)\( - 415265635406941004 \beta_{1} + 1928125120020685445 \beta_{2} + 436545620342251 \beta_{3} - 618679131748605 \beta_{4} + 5195740732201 \beta_{5} + 958655077726 \beta_{6} + 258716108 \beta_{7} + 7807626978 \beta_{8} + 876286879096 \beta_{9} - 711073952 \beta_{10} + 4117952926 \beta_{11} - 135234696 \beta_{12} + 280812490 \beta_{13} + 498106392 \beta_{14} - 76060456 \beta_{15}) q^{38} +(\)\(15\!\cdots\!38\)\( - \)\(63\!\cdots\!18\)\( \beta_{1} + 459433613013117490 \beta_{2} - 53115664532183 \beta_{3} + 20541352415704250 \beta_{4} + 28672461261764 \beta_{5} - 2725445094221 \beta_{6} - 518482880 \beta_{7} + 193274709020 \beta_{8} - 298753209893 \beta_{9} - 1647017504 \beta_{10} + 5200449956 \beta_{11} - 108836480 \beta_{12} - 411838160 \beta_{13} + 783171360 \beta_{14} - 181147872 \beta_{15}) q^{39} +(-\)\(20\!\cdots\!76\)\( - \)\(20\!\cdots\!32\)\( \beta_{1} - 6753918718308032244 \beta_{2} - 625520089505596 \beta_{3} - 374612382894140 \beta_{4} + 10026917222454 \beta_{5} - 3273782608758 \beta_{6} + 36202708224 \beta_{7} + 2990404762 \beta_{8} - 3069215700122 \beta_{9} - 3725339648 \beta_{10} + 2614603264 \beta_{11} + 304973568 \beta_{12} + 539846400 \beta_{13} + 666053376 \beta_{14} - 266382336 \beta_{15}) q^{40} +(\)\(29\!\cdots\!86\)\( - \)\(46\!\cdots\!04\)\( \beta_{1} + 34067308677759334 \beta_{2} - 431170552281696 \beta_{3} - 18350427452225730 \beta_{4} - 39709874646156 \beta_{5} + 2496523862206 \beta_{6} - 16267189610 \beta_{7} - 339809764364 \beta_{8} + 79998894376 \beta_{9} + 7289171060 \beta_{10} + 2464537322 \beta_{11} - 669526872 \beta_{12} - 438206848 \beta_{13} - 669526872 \beta_{14} - 372575702 \beta_{15}) q^{41} +(\)\(14\!\cdots\!80\)\( + \)\(32\!\cdots\!92\)\( \beta_{1} + 20041645866325254572 \beta_{2} - 190398606440668 \beta_{3} - 766224930625232 \beta_{4} - 8143313873552 \beta_{5} + 596071883092 \beta_{6} - 53486824896 \beta_{7} - 136174272820 \beta_{8} + 1257280829624 \beta_{9} + 3739114220 \beta_{10} - 8364953280 \beta_{11} + 1015442432 \beta_{12} + 87263008 \beta_{13} - 2734144512 \beta_{14} - 124507776 \beta_{15}) q^{42} +(\)\(61\!\cdots\!29\)\( - \)\(24\!\cdots\!02\)\( \beta_{1} - 949753620786767287 \beta_{2} + 268217706537380 \beta_{3} - 86086946008320360 \beta_{4} - 63622336551786 \beta_{5} - 8839075908716 \beta_{6} + 6221048246 \beta_{7} - 137536589254 \beta_{8} + 95697581730 \beta_{9} - 3488643862 \beta_{10} - 20415162316 \beta_{11} + 80048000 \beta_{12} + 776429040 \beta_{13} - 5468670166 \beta_{14} + 229557152 \beta_{15}) q^{43} +(\)\(45\!\cdots\!57\)\( + \)\(45\!\cdots\!63\)\( \beta_{1} - 39940166992880560178 \beta_{2} + 3548336378479221 \beta_{3} + 22818503703242165 \beta_{4} - 44424908175037 \beta_{5} + 39215491930268 \beta_{6} - 104952134601 \beta_{7} - 104193334591 \beta_{8} + 11627691826648 \beta_{9} + 29929941564 \beta_{10} - 37254306078 \beta_{11} - 785033073 \beta_{12} - 2329206528 \beta_{13} - 1673644461 \beta_{14} + 881937408 \beta_{15}) q^{44} +(-\)\(15\!\cdots\!63\)\( + \)\(97\!\cdots\!42\)\( \beta_{1} - 727875365906270992 \beta_{2} + 4113887152408350 \beta_{3} + 422731338852353172 \beta_{4} + 328731728813315 \beta_{5} - 35310466162933 \beta_{6} + 153422026186 \beta_{7} + 1890339032562 \beta_{8} - 460341676116 \beta_{9} - 26561097270 \beta_{10} - 83570889579 \beta_{11} + 2549458548 \beta_{12} + 4182995712 \beta_{13} + 2549458548 \beta_{14} + 2016202653 \beta_{15}) q^{45} +(-\)\(12\!\cdots\!16\)\( - \)\(10\!\cdots\!91\)\( \beta_{1} + 74788080887111787542 \beta_{2} - 8491260235020987 \beta_{3} + 28540604678284457 \beta_{4} - 63592786854481 \beta_{5} + 57515706452077 \beta_{6} + 375857298194 \beta_{7} + 1268622812451 \beta_{8} - 26657673351500 \beta_{9} - 14980055024 \beta_{10} - 102949431139 \beta_{11} - 5066735116 \beta_{12} - 6666635649 \beta_{13} + 11582593508 \beta_{14} + 2254091780 \beta_{15}) q^{46} +(-\)\(15\!\cdots\!91\)\( + \)\(62\!\cdots\!81\)\( \beta_{1} - 25970190847802600221 \beta_{2} + 1922291044896286 \beta_{3} - 649753963498918123 \beta_{4} - 468842689263113 \beta_{5} + 6020817687129 \beta_{6} - 29411073186 \beta_{7} - 3665863380655 \beta_{8} + 8329934414697 \beta_{9} + 65227317258 \beta_{10} - 188749252603 \beta_{11} + 1612036320 \beta_{12} + 7958460188 \beta_{13} + 29207630362 \beta_{14} + 2695498408 \beta_{15}) q^{47} +(\)\(66\!\cdots\!20\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} - \)\(19\!\cdots\!36\)\( \beta_{2} + 18256351258617592 \beta_{3} + 252496443204232040 \beta_{4} - 149069419196928 \beta_{5} + 87801817953576 \beta_{6} - 126083540896 \beta_{7} + 1675228772608 \beta_{8} + 24722162667520 \beta_{9} - 179751049856 \beta_{10} - 230619447744 \beta_{11} - 792878816 \beta_{12} - 7809796576 \beta_{13} - 1685429088 \beta_{14} + 1269823872 \beta_{15}) q^{48} +(-\)\(35\!\cdots\!35\)\( + \)\(61\!\cdots\!60\)\( \beta_{1} - 4201235298332769844 \beta_{2} + 1545647874810508 \beta_{3} + 1217705397465469816 \beta_{4} + 140738278746748 \beta_{5} - 213175301217208 \beta_{6} - 1024172025708 \beta_{7} - 2100698979296 \beta_{8} + 554712743808 \beta_{9} + 47415571680 \beta_{10} - 292637210704 \beta_{11} - 4362079296 \beta_{12} + 3774638592 \beta_{13} - 4362079296 \beta_{14} - 2464689936 \beta_{15}) q^{49} +(\)\(12\!\cdots\!75\)\( + \)\(76\!\cdots\!91\)\( \beta_{1} + \)\(31\!\cdots\!80\)\( \beta_{2} - 11044478952437572 \beta_{3} - 76842502583004464 \beta_{4} - 143166696081136 \beta_{5} + 347467052275724 \beta_{6} - 1231292377152 \beta_{7} - 6322203364780 \beta_{8} + 65487342543496 \beta_{9} + 47154230580 \beta_{10} - 360718761792 \beta_{11} + 15782310912 \beta_{12} + 6378520032 \beta_{13} - 34923684864 \beta_{14} - 6624562560 \beta_{15}) q^{50} +(\)\(59\!\cdots\!82\)\( - \)\(23\!\cdots\!71\)\( \beta_{1} + 13342896199961181684 \beta_{2} + 5184335456958750 \beta_{3} - 1112152753634591114 \beta_{4} + 1097451164311891 \beta_{5} - 917529380422796 \beta_{6} + 170167969553 \beta_{7} + 20051484599381 \beta_{8} - 55415497790115 \beta_{9} - 329280516753 \beta_{10} - 242233953776 \beta_{11} - 9650342400 \beta_{12} - 21301530944 \beta_{13} - 116267607441 \beta_{14} - 14828201856 \beta_{15}) q^{51} +(-\)\(18\!\cdots\!38\)\( - \)\(13\!\cdots\!32\)\( \beta_{1} - \)\(28\!\cdots\!42\)\( \beta_{2} - 30503412495961020 \beta_{3} - 273756196140167210 \beta_{4} + 696008280630752 \beta_{5} + 1055626226133932 \beta_{6} + 2240211812864 \beta_{7} - 7074489057184 \beta_{8} - 248422828160912 \beta_{9} + 850648313856 \beta_{10} - 209622725696 \beta_{11} + 16638532640 \beta_{12} + 45615706992 \beta_{13} + 35443435424 \beta_{14} - 19336755904 \beta_{15}) q^{52} +(-\)\(17\!\cdots\!47\)\( + \)\(28\!\cdots\!86\)\( \beta_{1} - 17769283517116235168 \beta_{2} - 28896650269166070 \beta_{3} - 638246406741856640 \beta_{4} - 5162731429314777 \beta_{5} - 1009651929422609 \beta_{6} + 5584398149154 \beta_{7} - 19113859091958 \beta_{8} + 9714864851004 \beta_{9} + 128434185474 \beta_{10} + 302408608881 \beta_{11} - 14252645532 \beta_{12} - 67793733888 \beta_{13} - 14252645532 \beta_{14} - 21298780455 \beta_{15}) q^{53} +(\)\(37\!\cdots\!26\)\( + 85724985419317036518 \beta_{1} + \)\(26\!\cdots\!42\)\( \beta_{2} + 155871322858297288 \beta_{3} - 170847727432813264 \beta_{4} + 1469956663124456 \beta_{5} + 2179142289061618 \beta_{6} + 383474099284 \beta_{7} + 33665261683406 \beta_{8} + 200745399954376 \beta_{9} - 88534767200 \beta_{10} + 709844746418 \beta_{11} - 16281918008 \beta_{12} + 89980507558 \beta_{13} + 50295044136 \beta_{14} - 11951861400 \beta_{15}) q^{54} +(\)\(19\!\cdots\!22\)\( - \)\(78\!\cdots\!74\)\( \beta_{1} + \)\(13\!\cdots\!30\)\( \beta_{2} - 19691066583884273 \beta_{3} + 7953308167647491190 \beta_{4} + 6121038667196940 \beta_{5} - 2735251571646091 \beta_{6} - 707679848912 \beta_{7} - 21627881593964 \beta_{8} + 233473927164861 \beta_{9} + 730682597296 \beta_{10} + 1696326644188 \beta_{11} + 20598802048 \beta_{12} - 93840799536 \beta_{13} + 314010229360 \beta_{14} + 12647564512 \beta_{15}) q^{55} +(\)\(59\!\cdots\!56\)\( + \)\(61\!\cdots\!52\)\( \beta_{1} + \)\(24\!\cdots\!08\)\( \beta_{2} - 280499822072329896 \beta_{3} - 4793477118083040696 \beta_{4} - 238957488529680 \beta_{5} + 3732538517374328 \beta_{6} - 7351374298144 \beta_{7} + 19707146751632 \beta_{8} + 225822956308976 \beta_{9} - 3141311634560 \beta_{10} + 2877153852736 \beta_{11} - 67739162976 \beta_{12} + 63659267872 \beta_{13} - 148310961120 \beta_{14} + 46335583616 \beta_{15}) q^{56} +(\)\(17\!\cdots\!78\)\( + \)\(23\!\cdots\!59\)\( \beta_{1} - \)\(14\!\cdots\!33\)\( \beta_{2} - 39952553121652977 \beta_{3} - 18496832966931414942 \beta_{4} + 1892983684283991 \beta_{5} - 10199457609963378 \beta_{6} - 25014063949851 \beta_{7} + 92492429440776 \beta_{8} - 19207242015984 \beta_{9} - 1310145828984 \beta_{10} + 4012053570084 \beta_{11} + 126164948112 \beta_{12} - 1559249664 \beta_{13} + 126164948112 \beta_{14} + 101916774180 \beta_{15}) q^{57} +(-\)\(19\!\cdots\!75\)\( - \)\(25\!\cdots\!03\)\( \beta_{1} - \)\(24\!\cdots\!99\)\( \beta_{2} + 130108895379610893 \beta_{3} - 31287359097561470 \beta_{4} - 807445466928124 \beta_{5} + 12296237386663749 \beta_{6} + 14186835001600 \beta_{7} - 173433808555695 \beta_{8} - 1020124442740302 \beta_{9} - 172868168423 \beta_{10} + 6188629826816 \beta_{11} - 113049718784 \beta_{12} - 139113850752 \beta_{13} + 161280593920 \beta_{14} + 121609254400 \beta_{15}) q^{58} +(\)\(12\!\cdots\!27\)\( - \)\(50\!\cdots\!86\)\( \beta_{1} - \)\(28\!\cdots\!85\)\( \beta_{2} - 69071839506817368 \beta_{3} + 27415438362038410562 \beta_{4} - 17902187617138406 \beta_{5} - 16180990656579962 \beta_{6} + 535133267256 \beta_{7} - 163542336247018 \beta_{8} - 898643004645486 \beta_{9} + 726558533976 \beta_{10} + 6851934193346 \beta_{11} + 41384529600 \beta_{12} + 302392435160 \beta_{13} - 344193692936 \beta_{14} + 142223082256 \beta_{15}) q^{59} +(\)\(68\!\cdots\!42\)\( + \)\(18\!\cdots\!46\)\( \beta_{1} + \)\(75\!\cdots\!80\)\( \beta_{2} + 24692358029591390 \beta_{3} - 3424380979824264802 \beta_{4} + 3383560971725522 \beta_{5} + 28864190025149512 \beta_{6} + 991222356618 \beta_{7} - 62013302600122 \beta_{8} + 2278262987284880 \beta_{9} + 8310833810024 \beta_{10} + 8515315421388 \beta_{11} + 82920797594 \beta_{12} - 529045622144 \beta_{13} + 208241868594 \beta_{14} + 92033170944 \beta_{15}) q^{60} +(-\)\(46\!\cdots\!59\)\( + \)\(58\!\cdots\!48\)\( \beta_{1} - \)\(35\!\cdots\!36\)\( \beta_{2} + 430703228162650684 \beta_{3} - 28363719455925793638 \beta_{4} + 56122498461767545 \beta_{5} - 26263781531507759 \beta_{6} + 87068535603598 \beta_{7} - 75637264889130 \beta_{8} - 115323167511740 \beta_{9} + 4491295485470 \beta_{10} + 7809830008543 \beta_{11} - 384397707044 \beta_{12} + 727022712576 \beta_{13} - 384397707044 \beta_{14} - 60738499529 \beta_{15}) q^{61} +(\)\(41\!\cdots\!08\)\( + \)\(40\!\cdots\!56\)\( \beta_{1} - \)\(77\!\cdots\!00\)\( \beta_{2} - 1427049471823258304 \beta_{3} - 18547982959721365856 \beta_{4} - 5126025034484032 \beta_{5} + 35227964399567000 \beta_{6} - 54669936498960 \beta_{7} + 682989096598824 \beta_{8} - 1266516403448224 \beta_{9} + 2163354690432 \beta_{10} + 5308549136600 \beta_{11} + 666852935520 \beta_{12} - 824782252984 \beta_{13} - 1380912841760 \beta_{14} - 180037370144 \beta_{15}) q^{62} +(\)\(49\!\cdots\!29\)\( - \)\(19\!\cdots\!23\)\( \beta_{1} - \)\(10\!\cdots\!69\)\( \beta_{2} + 87632104626280075 \beta_{3} - 1314908651909813191 \beta_{4} - 39605391951757855 \beta_{5} - 67466313360655994 \beta_{6} + 7078716457282 \beta_{7} + 590645635961879 \beta_{8} + 4115422695955270 \beta_{9} - 10059225080042 \beta_{10} + 872345873075 \beta_{11} - 396587994080 \beta_{12} + 797581971076 \beta_{13} - 1517672574330 \beta_{14} - 503190309096 \beta_{15}) q^{63} +(-\)\(29\!\cdots\!24\)\( + \)\(12\!\cdots\!20\)\( \beta_{1} + \)\(18\!\cdots\!20\)\( \beta_{2} + 2219146254151748656 \beta_{3} + 20391304056597647760 \beta_{4} - 33635478450023680 \beta_{5} + 76736484361033232 \beta_{6} + 69816860578240 \beta_{7} + 94347424291584 \beta_{8} - 5701948073174272 \beta_{9} - 10936263233792 \beta_{10} - 7616484035968 \beta_{11} + 425569393984 \beta_{12} - 376309011648 \beta_{13} + 826205239360 \beta_{14} - 681609632000 \beta_{15}) q^{64} +(\)\(46\!\cdots\!88\)\( + \)\(31\!\cdots\!48\)\( \beta_{1} - \)\(20\!\cdots\!78\)\( \beta_{2} - 976501655190828040 \beta_{3} + 77063941290794482938 \beta_{4} - 33471880571587020 \beta_{5} - 125544296729584742 \beta_{6} - 220888879336326 \beta_{7} - 737637956937732 \beta_{8} + 140490057879096 \beta_{9} - 4212038418180 \beta_{10} - 19402934311026 \beta_{11} + 221459673912 \beta_{12} - 245096572032 \beta_{13} + 221459673912 \beta_{14} - 777261165618 \beta_{15}) q^{65} +(-\)\(77\!\cdots\!98\)\( + \)\(45\!\cdots\!46\)\( \beta_{1} - \)\(25\!\cdots\!94\)\( \beta_{2} - 3355384749279214190 \beta_{3} - 34813080395310475456 \beta_{4} + 1774183908461272 \beta_{5} + 185663547657919034 \beta_{6} + 34226887786080 \beta_{7} - 2238381745644066 \beta_{8} + 13463158991621548 \beta_{9} - 7717381830290 \beta_{10} - 33916201080608 \beta_{11} - 1556841532928 \beta_{12} + 1440957232176 \beta_{13} + 4649440925184 \beta_{14} - 786906800064 \beta_{15}) q^{66} +(\)\(14\!\cdots\!63\)\( - \)\(56\!\cdots\!85\)\( \beta_{1} - \)\(59\!\cdots\!67\)\( \beta_{2} + 771800516648612410 \beta_{3} - \)\(13\!\cdots\!62\)\( \beta_{4} + 162495861735704781 \beta_{5} - 207001940097580912 \beta_{6} - 24403314220197 \beta_{7} - 33650556535413 \beta_{8} - 20390420238510325 \beta_{9} + 26788886948613 \beta_{10} - 53883007345044 \beta_{11} + 911806262400 \beta_{12} - 2780519400048 \beta_{13} + 9724212796101 \beta_{14} - 129822416544 \beta_{15}) q^{67} +(\)\(20\!\cdots\!74\)\( + \)\(29\!\cdots\!88\)\( \beta_{1} + \)\(14\!\cdots\!22\)\( \beta_{2} + 2529719688014261576 \beta_{3} - 44871946370508767046 \beta_{4} + 159369903757448640 \beta_{5} + 238543656456191896 \beta_{6} - 215564130184192 \beta_{7} + 238793960284864 \beta_{8} - 10932374232070688 \beta_{9} - 22673023557632 \beta_{10} - 76874395931776 \beta_{11} - 2304630591936 \beta_{12} + 4269227256800 \beta_{13} - 5088916116160 \beta_{14} + 635399931520 \beta_{15}) q^{68} +(-\)\(13\!\cdots\!84\)\( + \)\(90\!\cdots\!39\)\( \beta_{1} - \)\(58\!\cdots\!40\)\( \beta_{2} + 2325754031878090899 \beta_{3} + \)\(59\!\cdots\!03\)\( \beta_{4} - 415027393946328897 \beta_{5} - 330821016965229881 \beta_{6} + 357074221472018 \beta_{7} + 2172352083223130 \beta_{8} + 912953203567260 \beta_{9} - 28305328848750 \beta_{10} - 97442419339463 \beta_{11} + 2700675870084 \beta_{12} - 5704965944576 \beta_{13} + 2700675870084 \beta_{14} + 2051390796129 \beta_{15}) q^{69} +(-\)\(83\!\cdots\!44\)\( - \)\(81\!\cdots\!76\)\( \beta_{1} + \)\(56\!\cdots\!60\)\( \beta_{2} + 12084303284754497016 \beta_{3} + 41219816918176442248 \beta_{4} + 54937034551055432 \beta_{5} + 399183134759379988 \beta_{6} + 346158201939592 \beta_{7} + 6368248589065196 \beta_{8} - 12598391848908720 \beta_{9} + 7755147885120 \beta_{10} - 121597835815148 \beta_{11} + 37488274000 \beta_{12} + 5892377883612 \beta_{13} - 6392393631728 \beta_{14} + 3171772948624 \beta_{15}) q^{70} +(\)\(28\!\cdots\!03\)\( - \)\(11\!\cdots\!29\)\( \beta_{1} + \)\(18\!\cdots\!81\)\( \beta_{2} + 1859130029743447219 \beta_{3} - \)\(39\!\cdots\!85\)\( \beta_{4} + 160885428103586627 \beta_{5} - 409646336265289064 \beta_{6} + 5945315945566 \beta_{7} - 3791945820878939 \beta_{8} + 79356209668035552 \beta_{9} - 36554054806 \beta_{10} - 118436498224079 \beta_{11} + 967081069920 \beta_{12} - 5603820482132 \beta_{13} - 26823767609030 \beta_{14} + 4206336612872 \beta_{15}) q^{71} +(-\)\(48\!\cdots\!58\)\( + \)\(35\!\cdots\!66\)\( \beta_{1} - \)\(38\!\cdots\!10\)\( \beta_{2} - 16231409999845217382 \beta_{3} - \)\(16\!\cdots\!54\)\( \beta_{4} - 620190823038072449 \beta_{5} + 770513199941280769 \beta_{6} + 26742122342912 \beta_{7} - 1498539950773047 \beta_{8} + 51233624457777975 \beta_{9} + 173287880878080 \beta_{10} - 123432007566336 \beta_{11} + 3582521826816 \beta_{12} + 2463532752384 \beta_{13} + 9384939283968 \beta_{14} + 3986122106880 \beta_{15}) q^{72} +(-\)\(13\!\cdots\!72\)\( + \)\(17\!\cdots\!67\)\( \beta_{1} - \)\(10\!\cdots\!29\)\( \beta_{2} - 707799054712247501 \beta_{3} + \)\(29\!\cdots\!50\)\( \beta_{4} + 88166367819376867 \beta_{5} - 683653705985251366 \beta_{6} - 69359045087835 \beta_{7} + 2014207881429856 \beta_{8} - 141101655980928 \beta_{9} + 130414402678176 \beta_{10} - 62376784815472 \beta_{11} - 10567426881216 \beta_{12} + 2033073839616 \beta_{13} - 10567426881216 \beta_{14} + 1802640051792 \beta_{15}) q^{73} +(\)\(37\!\cdots\!65\)\( + \)\(59\!\cdots\!85\)\( \beta_{1} + \)\(17\!\cdots\!21\)\( \beta_{2} + 16485778606660287577 \beta_{3} + \)\(14\!\cdots\!82\)\( \beta_{4} - 244495476872929132 \beta_{5} + 841669029490138753 \beta_{6} - 1002335860391104 \beta_{7} - 14519173920770971 \beta_{8} - 58139666534962310 \beta_{9} + 40157948397709 \beta_{10} - 27132609118656 \beta_{11} + 11071933750272 \beta_{12} - 9139040544864 \beta_{13} - 15738702801920 \beta_{14} + 214115729280 \beta_{15}) q^{74} +(\)\(98\!\cdots\!27\)\( - \)\(39\!\cdots\!92\)\( \beta_{1} - \)\(85\!\cdots\!25\)\( \beta_{2} + 595267777643626252 \beta_{3} + \)\(35\!\cdots\!46\)\( \beta_{4} - 818951135073173596 \beta_{5} - 1322961669094881314 \beta_{6} + 114535968318694 \beta_{7} + 6920640113382332 \beta_{8} - 218284862302259480 \beta_{9} - 187915663812950 \beta_{10} + 89747256404834 \beta_{11} - 10750932096320 \beta_{12} + 18224335078744 \beta_{13} + 29422022801994 \beta_{14} - 6534988392432 \beta_{15}) q^{75} +(-\)\(65\!\cdots\!25\)\( - \)\(84\!\cdots\!67\)\( \beta_{1} - \)\(27\!\cdots\!70\)\( \beta_{2} - 25310903611789505057 \beta_{3} + \)\(11\!\cdots\!51\)\( \beta_{4} + 2065520389475254089 \beta_{5} + 969418068052548052 \beta_{6} + 1469025667338789 \beta_{7} + 3716126175559363 \beta_{8} + 28782756578712968 \beta_{9} - 484153382927180 \beta_{10} + 249801352810918 \beta_{11} + 8023717116717 \beta_{12} - 26319980451072 \beta_{13} + 13536873926265 \beta_{14} - 10275096148992 \beta_{15}) q^{76} +(\)\(19\!\cdots\!16\)\( + \)\(31\!\cdots\!75\)\( \beta_{1} - \)\(18\!\cdots\!00\)\( \beta_{2} + 14260688212604845847 \beta_{3} - \)\(18\!\cdots\!21\)\( \beta_{4} + 2667783815602926867 \beta_{5} - 1399513357600261189 \beta_{6} - 1894364973086806 \beta_{7} - 15947663201598766 \beta_{8} - 4986423476236180 \beta_{9} - 171346280953430 \beta_{10} + 370074099276805 \beta_{11} + 11807377154100 \beta_{12} + 35327658936064 \beta_{13} + 11807377154100 \beta_{14} - 14828877552115 \beta_{15}) q^{77} +(\)\(10\!\cdots\!84\)\( + \)\(11\!\cdots\!97\)\( \beta_{1} + \)\(49\!\cdots\!18\)\( \beta_{2} - 25574553673813144507 \beta_{3} - \)\(84\!\cdots\!03\)\( \beta_{4} + 261445139892485807 \beta_{5} + 1157157751190826485 \beta_{6} - 212720519579806 \beta_{7} + 23553434114329051 \beta_{8} + 58885478033240276 \beta_{9} - 171364892334448 \beta_{10} + 761043330035173 \beta_{11} - 32278559685868 \beta_{12} - 36586446117481 \beta_{13} + 106200333671556 \beta_{14} - 21678468470364 \beta_{15}) q^{78} +(\)\(67\!\cdots\!36\)\( - \)\(26\!\cdots\!92\)\( \beta_{1} + \)\(14\!\cdots\!52\)\( \beta_{2} - 10220459004015426210 \beta_{3} + \)\(33\!\cdots\!64\)\( \beta_{4} - 1417067347472753632 \beta_{5} - 808835372402925366 \beta_{6} - 208659558683544 \beta_{7} + 10714574139777248 \beta_{8} + 463910969784205634 \beta_{9} + 436943003368152 \beta_{10} + 835468334995624 \beta_{11} + 20751788824320 \beta_{12} + 35632322337504 \beta_{13} + 72406315115352 \beta_{14} - 14873649704640 \beta_{15}) q^{79} +(\)\(22\!\cdots\!26\)\( - \)\(20\!\cdots\!52\)\( \beta_{1} - \)\(58\!\cdots\!08\)\( \beta_{2} + 26577561865782935734 \beta_{3} - \)\(17\!\cdots\!18\)\( \beta_{4} - 6748068031148382784 \beta_{5} + 876966367537690546 \beta_{6} - 3346203148544136 \beta_{7} - 2288608498864896 \beta_{8} - 300186683503956160 \beta_{9} + 654347147596512 \beta_{10} + 1274411510853584 \beta_{11} - 47878448456088 \beta_{12} - 20717665119832 \beta_{13} - 109694185865528 \beta_{14} - 10702437113888 \beta_{15}) q^{80} +(-\)\(16\!\cdots\!65\)\( + \)\(13\!\cdots\!01\)\( \beta_{1} - \)\(67\!\cdots\!63\)\( \beta_{2} - 62331247973280923127 \beta_{3} - \)\(52\!\cdots\!14\)\( \beta_{4} + 536323692030812289 \beta_{5} - 745888256480821662 \beta_{6} + 7777109053223523 \beta_{7} + 4369968631936440 \beta_{8} - 4077352593849744 \beta_{9} - 426237173947080 \beta_{10} + 1286759444517180 \beta_{11} + 36617601829104 \beta_{12} - 3191021561088 \beta_{13} + 36617601829104 \beta_{14} + 6587024001084 \beta_{15}) q^{81} +(-\)\(79\!\cdots\!94\)\( + \)\(28\!\cdots\!46\)\( \beta_{1} + \)\(71\!\cdots\!64\)\( \beta_{2} - \)\(14\!\cdots\!48\)\( \beta_{3} - \)\(85\!\cdots\!96\)\( \beta_{4} + 181672104237432976 \beta_{5} + 912105741792603660 \beta_{6} + 5850153500370112 \beta_{7} - 9666585232299020 \beta_{8} + 394534979151288136 \beta_{9} + 190776068446932 \beta_{10} + 1366949272091072 \beta_{11} + 23914884195328 \beta_{12} + 36554454064224 \beta_{13} - 231142360994816 \beta_{14} + 29916700356736 \beta_{15}) q^{82} +(\)\(14\!\cdots\!61\)\( - \)\(58\!\cdots\!10\)\( \beta_{1} + \)\(12\!\cdots\!65\)\( \beta_{2} - 16173219035344422416 \beta_{3} + \)\(56\!\cdots\!58\)\( \beta_{4} + 3576725542115056558 \beta_{5} - 226275101973592794 \beta_{6} - 47189266527164 \beta_{7} - 49638726669963550 \beta_{8} - 922956523270403010 \beta_{9} + 225660790738892 \beta_{10} + 1415089506303218 \beta_{11} + 27718844510400 \beta_{12} - 87812950477288 \beta_{13} - 405625079531412 \beta_{14} + 52313912286352 \beta_{15}) q^{83} +(-\)\(30\!\cdots\!92\)\( + \)\(14\!\cdots\!92\)\( \beta_{1} - \)\(58\!\cdots\!92\)\( \beta_{2} + \)\(27\!\cdots\!12\)\( \beta_{3} + \)\(31\!\cdots\!44\)\( \beta_{4} + 19267990545074526336 \beta_{5} - 1585811853916527792 \beta_{6} - 1838152068048896 \beta_{7} - 12128623046447488 \beta_{8} - 48831092132297152 \beta_{9} + 267506060017664 \beta_{10} + 782104274678016 \beta_{11} + 66697806487424 \beta_{12} + 128928313907776 \beta_{13} + 194504775414144 \beta_{14} + 66302917921536 \beta_{15}) q^{84} +(\)\(12\!\cdots\!18\)\( - \)\(70\!\cdots\!47\)\( \beta_{1} + \)\(45\!\cdots\!68\)\( \beta_{2} + \)\(14\!\cdots\!77\)\( \beta_{3} - \)\(48\!\cdots\!83\)\( \beta_{4} - 14296651882738952065 \beta_{5} + 3261252821015901783 \beta_{6} - 18160317417628886 \beta_{7} + 74333513261322538 \beta_{8} + 18252964537356316 \beta_{9} + 2075978835216770 \beta_{10} + 929040588186129 \beta_{11} - 161513909941148 \beta_{12} - 181851037850112 \beta_{13} - 161513909941148 \beta_{14} + 68905957961497 \beta_{15}) q^{85} +(\)\(42\!\cdots\!21\)\( + \)\(40\!\cdots\!46\)\( \beta_{1} - \)\(11\!\cdots\!17\)\( \beta_{2} - 93028872577274956449 \beta_{3} - \)\(17\!\cdots\!45\)\( \beta_{4} + 876597649000941397 \beta_{5} - 5398991349753133684 \beta_{6} - 6916997156525384 \beta_{7} - 97747404408722188 \beta_{8} - 478712439791503824 \beta_{9} + 439827313004480 \beta_{10} - 985053453586420 \beta_{11} + 95686067741232 \beta_{12} + 203581451998340 \beta_{13} - 7475285230992 \beta_{14} + 74493735902704 \beta_{15}) q^{86} +(-\)\(73\!\cdots\!21\)\( + \)\(29\!\cdots\!99\)\( \beta_{1} - \)\(38\!\cdots\!71\)\( \beta_{2} + 11413961643442262293 \beta_{3} - \)\(69\!\cdots\!81\)\( \beta_{4} + 13058220761187076863 \beta_{5} + 9792031914806409986 \beta_{6} + 762159128088294 \beta_{7} - 3474174281758199 \beta_{8} + 1735804322798525818 \beta_{9} - 2973826603875326 \beta_{10} - 1387962842574987 \beta_{11} - 202136205888800 \beta_{12} - 207321901072804 \beta_{13} + 844559128654098 \beta_{14} + 11101089678888 \beta_{15}) q^{87} +(-\)\(11\!\cdots\!32\)\( + \)\(44\!\cdots\!00\)\( \beta_{1} + \)\(18\!\cdots\!52\)\( \beta_{2} - \)\(13\!\cdots\!24\)\( \beta_{3} + \)\(30\!\cdots\!04\)\( \beta_{4} - 40423592414080555176 \beta_{5} - 10484639128599699028 \beta_{6} + 19000390612042160 \beta_{7} + 17996357259745896 \beta_{8} + 1436404901653746392 \beta_{9} - 3535779652282176 \beta_{10} - 4193016247080672 \beta_{11} + 133598318176400 \beta_{12} + 164906741445072 \beta_{13} + 223821838637648 \beta_{14} + 550237527488 \beta_{15}) q^{88} +(-\)\(15\!\cdots\!52\)\( - \)\(37\!\cdots\!09\)\( \beta_{1} + \)\(22\!\cdots\!71\)\( \beta_{2} - \)\(58\!\cdots\!09\)\( \beta_{3} + \)\(25\!\cdots\!18\)\( \beta_{4} - 5811760667707598061 \beta_{5} + 15893429575724013106 \beta_{6} + 24158352018582877 \beta_{7} - 70659802598982608 \beta_{8} + 31671817930072480 \beta_{9} - 2387828711852560 \beta_{10} - 6134585136282376 \beta_{11} + 186304080830688 \beta_{12} - 69926262184960 \beta_{13} + 186304080830688 \beta_{14} - 76089870942152 \beta_{15}) q^{89} +(\)\(16\!\cdots\!21\)\( - \)\(15\!\cdots\!47\)\( \beta_{1} - \)\(61\!\cdots\!59\)\( \beta_{2} + \)\(67\!\cdots\!89\)\( \beta_{3} + \)\(13\!\cdots\!70\)\( \beta_{4} + 2960500980080148564 \beta_{5} - 25036650936131019967 \beta_{6} - 20841767953682496 \beta_{7} + 361011207360756917 \beta_{8} - 2172636042874599974 \beta_{9} - 1678404760087331 \beta_{10} - 7447388505192256 \beta_{11} - 300108554046464 \beta_{12} - 65989495199264 \beta_{13} + 1398341594569728 \beta_{14} - 212660055653760 \beta_{15}) q^{90} +(-\)\(17\!\cdots\!02\)\( + \)\(69\!\cdots\!08\)\( \beta_{1} + \)\(41\!\cdots\!06\)\( \beta_{2} + \)\(10\!\cdots\!88\)\( \beta_{3} - \)\(43\!\cdots\!42\)\( \beta_{4} - 17558528952362397140 \beta_{5} + 23397341399887995930 \beta_{6} - 1524592449874446 \beta_{7} + 201768352792092148 \beta_{8} - 2106152708303483528 \beta_{9} + 4023279630313662 \beta_{10} - 10985448635545690 \beta_{11} + 256434351067200 \beta_{12} + 294370512871944 \beta_{13} - 451010035053666 \beta_{14} - 198981502454736 \beta_{15}) q^{91} +(-\)\(35\!\cdots\!58\)\( - \)\(12\!\cdots\!78\)\( \beta_{1} + \)\(83\!\cdots\!88\)\( \beta_{2} - \)\(80\!\cdots\!46\)\( \beta_{3} - \)\(59\!\cdots\!90\)\( \beta_{4} + 79024008639830573530 \beta_{5} - 29078919023694398616 \beta_{6} - 17518098545514126 \beta_{7} + 3666146523377822 \beta_{8} + 82468253154587088 \beta_{9} + 9619770631945992 \beta_{10} - 10116556833030532 \beta_{11} - 652237817567550 \beta_{12} - 492600890001280 \beta_{13} - 1647553772803142 \beta_{14} - 317534562318848 \beta_{15}) q^{92} +(-\)\(29\!\cdots\!40\)\( - \)\(94\!\cdots\!12\)\( \beta_{1} + \)\(56\!\cdots\!28\)\( \beta_{2} + \)\(18\!\cdots\!48\)\( \beta_{3} + \)\(68\!\cdots\!76\)\( \beta_{4} + 55044480192599983156 \beta_{5} + 39122553389102078036 \beta_{6} - 289431466230584 \beta_{7} - 289869428537504680 \beta_{8} - 46129181637702192 \beta_{9} - 4698091861350216 \beta_{10} - 14326943798984228 \beta_{11} + 344068525268976 \beta_{12} + 781828611274240 \beta_{13} + 344068525268976 \beta_{14} - 284634779061252 \beta_{15}) q^{93} +(-\)\(10\!\cdots\!00\)\( - \)\(26\!\cdots\!10\)\( \beta_{1} - \)\(84\!\cdots\!20\)\( \beta_{2} + \)\(10\!\cdots\!30\)\( \beta_{3} + \)\(48\!\cdots\!06\)\( \beta_{4} - 24423115332254620446 \beta_{5} - 38688058382824567842 \beta_{6} + 57305184596611084 \beta_{7} - 734706805554913598 \beta_{8} + 3636231563061651832 \beta_{9} + 1610061208636768 \beta_{10} - 11851678539785282 \beta_{11} + 274097312277496 \beta_{12} - 960438295872342 \beta_{13} - 3481858783232360 \beta_{14} - 105558530753192 \beta_{15}) q^{94} +(-\)\(40\!\cdots\!18\)\( + \)\(16\!\cdots\!14\)\( \beta_{1} + \)\(18\!\cdots\!10\)\( \beta_{2} + 74495818312779035117 \beta_{3} - \)\(55\!\cdots\!66\)\( \beta_{4} - 68637666891268834724 \beta_{5} + 56901757741461860687 \beta_{6} - 1719108408884008 \beta_{7} - 41860534955956540 \beta_{8} - 994593751944943953 \beta_{9} + 9623550256032712 \beta_{10} - 12649866573367948 \beta_{11} + 638678698133376 \beta_{12} + 1059515211444464 \beta_{13} - 2548648149560824 \beta_{14} + 18651570529696 \beta_{15}) q^{95} +(-\)\(84\!\cdots\!72\)\( + \)\(65\!\cdots\!44\)\( \beta_{1} + \)\(42\!\cdots\!88\)\( \beta_{2} - \)\(13\!\cdots\!40\)\( \beta_{3} + \)\(11\!\cdots\!80\)\( \beta_{4} - \)\(17\!\cdots\!04\)\( \beta_{5} - 79741173627464544224 \beta_{6} - 45506700919123072 \beta_{7} + 98851455868308992 \beta_{8} - 5922691178876600832 \beta_{9} - 16381512434508288 \beta_{10} - 6241922115762944 \beta_{11} + 642352747022976 \beta_{12} - 1051329046329728 \beta_{13} + 2287131684831360 \beta_{14} + 161217528860160 \beta_{15}) q^{96} +(\)\(44\!\cdots\!88\)\( - \)\(20\!\cdots\!59\)\( \beta_{1} + \)\(12\!\cdots\!67\)\( \beta_{2} - \)\(35\!\cdots\!39\)\( \beta_{3} - \)\(67\!\cdots\!40\)\( \beta_{4} + 51160027829584091881 \beta_{5} + 76753671818290994408 \beta_{6} - 93619933206040635 \beta_{7} + 494291124166575148 \beta_{8} - 134080550149629864 \beta_{9} + 18379787618849388 \beta_{10} + 2798038450870214 \beta_{11} - 1449267028684008 \beta_{12} + 757430226246528 \beta_{13} - 1449267028684008 \beta_{14} + 494291637320646 \beta_{15}) q^{97} +(\)\(10\!\cdots\!77\)\( - \)\(24\!\cdots\!47\)\( \beta_{1} - \)\(31\!\cdots\!72\)\( \beta_{2} - \)\(19\!\cdots\!04\)\( \beta_{3} + \)\(68\!\cdots\!28\)\( \beta_{4} + 9931956653341350304 \beta_{5} - 75411892397339867624 \beta_{6} + 40524566239378560 \beta_{7} + 1190492742465128008 \beta_{8} + 6706073482150914512 \beta_{9} + 1215844681283592 \beta_{10} + 9905767813266048 \beta_{11} + 198231362131968 \beta_{12} - 314633665971648 \beta_{13} + 1218433774848000 \beta_{14} + 734481620460288 \beta_{15}) q^{98} +(-\)\(66\!\cdots\!69\)\( + \)\(26\!\cdots\!46\)\( \beta_{1} - \)\(68\!\cdots\!93\)\( \beta_{2} - \)\(36\!\cdots\!48\)\( \beta_{3} + \)\(74\!\cdots\!30\)\( \beta_{4} + 44902095848128236802 \beta_{5} + 96807772785414161750 \beta_{6} + 15498205152466304 \beta_{7} - 610492060831682066 \beta_{8} + 11296493028161642762 \beta_{9} - 37657657325468272 \beta_{10} + 26874252168329170 \beta_{11} - 2496318625979200 \beta_{12} - 461334693178216 \beta_{13} + 8542311720969264 \beta_{14} + 615042423583632 \beta_{15}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 27372q^{2} - 20032875248q^{4} - 21372255840q^{5} + 20836736461728q^{6} + 2284358011394112q^{8} - 67828545645364080q^{9} + O(q^{10}) \) \( 16q - 27372q^{2} - 20032875248q^{4} - 21372255840q^{5} + 20836736461728q^{6} + 2284358011394112q^{8} - 67828545645364080q^{9} - 45113792455239160q^{10} + 2389205919902175360q^{12} - 5233507036411659616q^{13} - 78584294183795706432q^{14} + \)\(49\!\cdots\!16\)\(q^{16} - \)\(60\!\cdots\!44\)\(q^{17} - \)\(30\!\cdots\!08\)\(q^{18} - \)\(33\!\cdots\!00\)\(q^{20} + \)\(51\!\cdots\!96\)\(q^{21} + \)\(71\!\cdots\!80\)\(q^{22} + \)\(21\!\cdots\!68\)\(q^{24} + \)\(12\!\cdots\!60\)\(q^{25} - \)\(20\!\cdots\!96\)\(q^{26} + \)\(89\!\cdots\!60\)\(q^{28} - \)\(40\!\cdots\!56\)\(q^{29} + \)\(29\!\cdots\!80\)\(q^{30} + \)\(20\!\cdots\!28\)\(q^{32} + \)\(73\!\cdots\!60\)\(q^{33} + \)\(46\!\cdots\!76\)\(q^{34} + \)\(56\!\cdots\!84\)\(q^{36} + \)\(95\!\cdots\!56\)\(q^{37} - \)\(12\!\cdots\!60\)\(q^{38} - \)\(32\!\cdots\!00\)\(q^{40} + \)\(46\!\cdots\!92\)\(q^{41} + \)\(23\!\cdots\!60\)\(q^{42} + \)\(73\!\cdots\!20\)\(q^{44} - \)\(24\!\cdots\!80\)\(q^{45} - \)\(19\!\cdots\!72\)\(q^{46} + \)\(10\!\cdots\!20\)\(q^{48} - \)\(56\!\cdots\!60\)\(q^{49} + \)\(20\!\cdots\!20\)\(q^{50} - \)\(29\!\cdots\!04\)\(q^{52} - \)\(28\!\cdots\!36\)\(q^{53} + \)\(60\!\cdots\!36\)\(q^{54} + \)\(95\!\cdots\!28\)\(q^{56} + \)\(27\!\cdots\!80\)\(q^{57} - \)\(31\!\cdots\!36\)\(q^{58} + \)\(11\!\cdots\!20\)\(q^{60} - \)\(73\!\cdots\!28\)\(q^{61} + \)\(66\!\cdots\!40\)\(q^{62} - \)\(46\!\cdots\!88\)\(q^{64} + \)\(73\!\cdots\!80\)\(q^{65} - \)\(12\!\cdots\!20\)\(q^{66} + \)\(32\!\cdots\!84\)\(q^{68} - \)\(20\!\cdots\!04\)\(q^{69} - \)\(13\!\cdots\!00\)\(q^{70} - \)\(77\!\cdots\!92\)\(q^{72} - \)\(21\!\cdots\!96\)\(q^{73} + \)\(60\!\cdots\!76\)\(q^{74} - \)\(10\!\cdots\!00\)\(q^{76} + \)\(31\!\cdots\!60\)\(q^{77} + \)\(17\!\cdots\!40\)\(q^{78} + \)\(36\!\cdots\!20\)\(q^{80} - \)\(26\!\cdots\!48\)\(q^{81} - \)\(12\!\cdots\!64\)\(q^{82} - \)\(48\!\cdots\!44\)\(q^{84} + \)\(20\!\cdots\!60\)\(q^{85} + \)\(68\!\cdots\!28\)\(q^{86} - \)\(18\!\cdots\!20\)\(q^{88} - \)\(24\!\cdots\!76\)\(q^{89} + \)\(27\!\cdots\!20\)\(q^{90} - \)\(57\!\cdots\!40\)\(q^{92} - \)\(46\!\cdots\!20\)\(q^{93} - \)\(17\!\cdots\!92\)\(q^{94} - \)\(13\!\cdots\!32\)\(q^{96} + \)\(71\!\cdots\!56\)\(q^{97} + \)\(16\!\cdots\!32\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 72626617369828 x^{14} - 508386321588656 x^{13} + \)\(20\!\cdots\!38\)\( x^{12} - \)\(12\!\cdots\!64\)\( x^{11} + \)\(29\!\cdots\!24\)\( x^{10} - \)\(14\!\cdots\!48\)\( x^{9} + \)\(20\!\cdots\!41\)\( x^{8} - \)\(83\!\cdots\!08\)\( x^{7} + \)\(69\!\cdots\!92\)\( x^{6} - \)\(20\!\cdots\!68\)\( x^{5} + \)\(87\!\cdots\!92\)\( x^{4} - \)\(17\!\cdots\!44\)\( x^{3} + \)\(58\!\cdots\!32\)\( x^{2} - \)\(58\!\cdots\!52\)\( x + \)\(10\!\cdots\!04\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(26\!\cdots\!75\)\( \nu^{15} - \)\(30\!\cdots\!17\)\( \nu^{14} - \)\(18\!\cdots\!31\)\( \nu^{13} - \)\(22\!\cdots\!33\)\( \nu^{12} - \)\(53\!\cdots\!49\)\( \nu^{11} - \)\(63\!\cdots\!51\)\( \nu^{10} - \)\(75\!\cdots\!93\)\( \nu^{9} - \)\(89\!\cdots\!63\)\( \nu^{8} - \)\(53\!\cdots\!16\)\( \nu^{7} - \)\(63\!\cdots\!84\)\( \nu^{6} - \)\(17\!\cdots\!16\)\( \nu^{5} - \)\(21\!\cdots\!08\)\( \nu^{4} - \)\(22\!\cdots\!92\)\( \nu^{3} - \)\(26\!\cdots\!80\)\( \nu^{2} - \)\(74\!\cdots\!92\)\( \nu - \)\(89\!\cdots\!28\)\(\)\()/ \)\(51\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(49\!\cdots\!25\)\( \nu^{15} + \)\(58\!\cdots\!23\)\( \nu^{14} + \)\(35\!\cdots\!89\)\( \nu^{13} + \)\(42\!\cdots\!27\)\( \nu^{12} + \)\(10\!\cdots\!31\)\( \nu^{11} + \)\(12\!\cdots\!69\)\( \nu^{10} + \)\(14\!\cdots\!67\)\( \nu^{9} + \)\(16\!\cdots\!97\)\( \nu^{8} + \)\(10\!\cdots\!04\)\( \nu^{7} + \)\(12\!\cdots\!96\)\( \nu^{6} + \)\(34\!\cdots\!04\)\( \nu^{5} + \)\(40\!\cdots\!52\)\( \nu^{4} + \)\(42\!\cdots\!48\)\( \nu^{3} + \)\(49\!\cdots\!20\)\( \nu^{2} + \)\(14\!\cdots\!48\)\( \nu + \)\(17\!\cdots\!32\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(14\!\cdots\!45\)\( \nu^{15} - \)\(97\!\cdots\!67\)\( \nu^{14} - \)\(10\!\cdots\!25\)\( \nu^{13} - \)\(70\!\cdots\!55\)\( \nu^{12} - \)\(30\!\cdots\!51\)\( \nu^{11} - \)\(20\!\cdots\!13\)\( \nu^{10} - \)\(44\!\cdots\!47\)\( \nu^{9} - \)\(28\!\cdots\!09\)\( \nu^{8} - \)\(33\!\cdots\!28\)\( \nu^{7} - \)\(20\!\cdots\!36\)\( \nu^{6} - \)\(12\!\cdots\!32\)\( \nu^{5} - \)\(67\!\cdots\!16\)\( \nu^{4} - \)\(20\!\cdots\!48\)\( \nu^{3} - \)\(82\!\cdots\!56\)\( \nu^{2} - \)\(88\!\cdots\!68\)\( \nu - \)\(28\!\cdots\!92\)\(\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(23\!\cdots\!45\)\( \nu^{15} + \)\(91\!\cdots\!03\)\( \nu^{14} + \)\(17\!\cdots\!53\)\( \nu^{13} + \)\(66\!\cdots\!91\)\( \nu^{12} + \)\(49\!\cdots\!91\)\( \nu^{11} + \)\(18\!\cdots\!45\)\( \nu^{10} + \)\(69\!\cdots\!11\)\( \nu^{9} + \)\(26\!\cdots\!65\)\( \nu^{8} + \)\(49\!\cdots\!16\)\( \nu^{7} + \)\(19\!\cdots\!24\)\( \nu^{6} + \)\(16\!\cdots\!72\)\( \nu^{5} + \)\(63\!\cdots\!40\)\( \nu^{4} + \)\(20\!\cdots\!88\)\( \nu^{3} + \)\(77\!\cdots\!68\)\( \nu^{2} + \)\(84\!\cdots\!04\)\( \nu + \)\(25\!\cdots\!44\)\(\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(14\!\cdots\!41\)\( \nu^{15} + \)\(93\!\cdots\!29\)\( \nu^{14} - \)\(10\!\cdots\!77\)\( \nu^{13} + \)\(67\!\cdots\!05\)\( \nu^{12} - \)\(29\!\cdots\!67\)\( \nu^{11} + \)\(19\!\cdots\!67\)\( \nu^{10} - \)\(42\!\cdots\!55\)\( \nu^{9} + \)\(26\!\cdots\!35\)\( \nu^{8} - \)\(30\!\cdots\!48\)\( \nu^{7} + \)\(18\!\cdots\!12\)\( \nu^{6} - \)\(10\!\cdots\!88\)\( \nu^{5} + \)\(62\!\cdots\!04\)\( \nu^{4} - \)\(12\!\cdots\!52\)\( \nu^{3} + \)\(76\!\cdots\!84\)\( \nu^{2} - \)\(43\!\cdots\!88\)\( \nu + \)\(26\!\cdots\!48\)\(\)\()/ \)\(36\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(49\!\cdots\!15\)\( \nu^{15} + \)\(67\!\cdots\!87\)\( \nu^{14} - \)\(35\!\cdots\!07\)\( \nu^{13} + \)\(48\!\cdots\!59\)\( \nu^{12} - \)\(10\!\cdots\!65\)\( \nu^{11} + \)\(13\!\cdots\!97\)\( \nu^{10} - \)\(14\!\cdots\!45\)\( \nu^{9} + \)\(19\!\cdots\!81\)\( \nu^{8} - \)\(10\!\cdots\!28\)\( \nu^{7} + \)\(13\!\cdots\!00\)\( \nu^{6} - \)\(34\!\cdots\!76\)\( \nu^{5} + \)\(46\!\cdots\!52\)\( \nu^{4} - \)\(41\!\cdots\!00\)\( \nu^{3} + \)\(57\!\cdots\!08\)\( \nu^{2} - \)\(14\!\cdots\!80\)\( \nu + \)\(19\!\cdots\!00\)\(\)\()/ \)\(39\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(88\!\cdots\!21\)\( \nu^{15} - \)\(78\!\cdots\!87\)\( \nu^{14} - \)\(63\!\cdots\!73\)\( \nu^{13} - \)\(57\!\cdots\!63\)\( \nu^{12} - \)\(18\!\cdots\!83\)\( \nu^{11} - \)\(16\!\cdots\!25\)\( \nu^{10} - \)\(25\!\cdots\!27\)\( \nu^{9} - \)\(22\!\cdots\!81\)\( \nu^{8} - \)\(18\!\cdots\!20\)\( \nu^{7} - \)\(16\!\cdots\!04\)\( \nu^{6} - \)\(61\!\cdots\!48\)\( \nu^{5} - \)\(55\!\cdots\!96\)\( \nu^{4} - \)\(75\!\cdots\!00\)\( \nu^{3} - \)\(68\!\cdots\!08\)\( \nu^{2} - \)\(34\!\cdots\!76\)\( \nu + \)\(21\!\cdots\!16\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(59\!\cdots\!77\)\( \nu^{15} + \)\(11\!\cdots\!89\)\( \nu^{14} - \)\(42\!\cdots\!21\)\( \nu^{13} + \)\(82\!\cdots\!53\)\( \nu^{12} - \)\(12\!\cdots\!11\)\( \nu^{11} + \)\(23\!\cdots\!15\)\( \nu^{10} - \)\(17\!\cdots\!71\)\( \nu^{9} + \)\(32\!\cdots\!83\)\( \nu^{8} - \)\(12\!\cdots\!28\)\( \nu^{7} + \)\(23\!\cdots\!28\)\( \nu^{6} - \)\(40\!\cdots\!44\)\( \nu^{5} + \)\(78\!\cdots\!72\)\( \nu^{4} - \)\(50\!\cdots\!00\)\( \nu^{3} + \)\(96\!\cdots\!52\)\( \nu^{2} - \)\(20\!\cdots\!80\)\( \nu + \)\(32\!\cdots\!76\)\(\)\()/ \)\(59\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(10\!\cdots\!75\)\( \nu^{15} - \)\(14\!\cdots\!03\)\( \nu^{14} + \)\(74\!\cdots\!51\)\( \nu^{13} - \)\(10\!\cdots\!67\)\( \nu^{12} + \)\(21\!\cdots\!09\)\( \nu^{11} - \)\(29\!\cdots\!89\)\( \nu^{10} + \)\(29\!\cdots\!93\)\( \nu^{9} - \)\(41\!\cdots\!57\)\( \nu^{8} + \)\(21\!\cdots\!96\)\( \nu^{7} - \)\(29\!\cdots\!96\)\( \nu^{6} + \)\(71\!\cdots\!16\)\( \nu^{5} - \)\(99\!\cdots\!12\)\( \nu^{4} + \)\(87\!\cdots\!72\)\( \nu^{3} - \)\(12\!\cdots\!60\)\( \nu^{2} + \)\(29\!\cdots\!92\)\( \nu - \)\(45\!\cdots\!12\)\(\)\()/ \)\(60\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(18\!\cdots\!93\)\( \nu^{15} - \)\(13\!\cdots\!25\)\( \nu^{14} + \)\(13\!\cdots\!65\)\( \nu^{13} - \)\(99\!\cdots\!89\)\( \nu^{12} + \)\(37\!\cdots\!95\)\( \nu^{11} - \)\(28\!\cdots\!03\)\( \nu^{10} + \)\(52\!\cdots\!91\)\( \nu^{9} - \)\(39\!\cdots\!91\)\( \nu^{8} + \)\(37\!\cdots\!64\)\( \nu^{7} - \)\(28\!\cdots\!36\)\( \nu^{6} + \)\(12\!\cdots\!16\)\( \nu^{5} - \)\(95\!\cdots\!28\)\( \nu^{4} + \)\(15\!\cdots\!08\)\( \nu^{3} - \)\(11\!\cdots\!76\)\( \nu^{2} + \)\(89\!\cdots\!88\)\( \nu - \)\(45\!\cdots\!32\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(43\!\cdots\!87\)\( \nu^{15} - \)\(27\!\cdots\!23\)\( \nu^{14} + \)\(31\!\cdots\!87\)\( \nu^{13} - \)\(20\!\cdots\!03\)\( \nu^{12} + \)\(90\!\cdots\!65\)\( \nu^{11} - \)\(57\!\cdots\!49\)\( \nu^{10} + \)\(12\!\cdots\!13\)\( \nu^{9} - \)\(80\!\cdots\!85\)\( \nu^{8} + \)\(90\!\cdots\!20\)\( \nu^{7} - \)\(57\!\cdots\!96\)\( \nu^{6} + \)\(30\!\cdots\!16\)\( \nu^{5} - \)\(19\!\cdots\!12\)\( \nu^{4} + \)\(37\!\cdots\!52\)\( \nu^{3} - \)\(23\!\cdots\!08\)\( \nu^{2} + \)\(16\!\cdots\!08\)\( \nu - \)\(77\!\cdots\!76\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(21\!\cdots\!39\)\( \nu^{15} + \)\(49\!\cdots\!67\)\( \nu^{14} - \)\(15\!\cdots\!71\)\( \nu^{13} + \)\(36\!\cdots\!19\)\( \nu^{12} - \)\(44\!\cdots\!77\)\( \nu^{11} + \)\(10\!\cdots\!69\)\( \nu^{10} - \)\(62\!\cdots\!17\)\( \nu^{9} + \)\(14\!\cdots\!33\)\( \nu^{8} - \)\(44\!\cdots\!12\)\( \nu^{7} + \)\(10\!\cdots\!16\)\( \nu^{6} - \)\(15\!\cdots\!20\)\( \nu^{5} + \)\(34\!\cdots\!68\)\( \nu^{4} - \)\(18\!\cdots\!80\)\( \nu^{3} + \)\(43\!\cdots\!40\)\( \nu^{2} - \)\(12\!\cdots\!00\)\( \nu + \)\(16\!\cdots\!72\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(84\!\cdots\!53\)\( \nu^{15} + \)\(62\!\cdots\!47\)\( \nu^{14} + \)\(61\!\cdots\!61\)\( \nu^{13} + \)\(45\!\cdots\!27\)\( \nu^{12} + \)\(17\!\cdots\!71\)\( \nu^{11} + \)\(12\!\cdots\!69\)\( \nu^{10} + \)\(24\!\cdots\!59\)\( \nu^{9} + \)\(17\!\cdots\!65\)\( \nu^{8} + \)\(17\!\cdots\!80\)\( \nu^{7} + \)\(12\!\cdots\!12\)\( \nu^{6} + \)\(58\!\cdots\!00\)\( \nu^{5} + \)\(42\!\cdots\!40\)\( \nu^{4} + \)\(72\!\cdots\!56\)\( \nu^{3} + \)\(51\!\cdots\!32\)\( \nu^{2} + \)\(27\!\cdots\!40\)\( \nu + \)\(17\!\cdots\!28\)\(\)\()/ \)\(36\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(16\!\cdots\!15\)\( \nu^{15} + \)\(20\!\cdots\!93\)\( \nu^{14} + \)\(12\!\cdots\!71\)\( \nu^{13} + \)\(14\!\cdots\!93\)\( \nu^{12} + \)\(34\!\cdots\!57\)\( \nu^{11} + \)\(42\!\cdots\!35\)\( \nu^{10} + \)\(49\!\cdots\!09\)\( \nu^{9} + \)\(60\!\cdots\!55\)\( \nu^{8} + \)\(34\!\cdots\!60\)\( \nu^{7} + \)\(42\!\cdots\!72\)\( \nu^{6} + \)\(11\!\cdots\!72\)\( \nu^{5} + \)\(14\!\cdots\!56\)\( \nu^{4} + \)\(14\!\cdots\!56\)\( \nu^{3} + \)\(17\!\cdots\!88\)\( \nu^{2} + \)\(36\!\cdots\!36\)\( \nu + \)\(64\!\cdots\!84\)\(\)\()/ \)\(59\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(11\!\cdots\!65\)\( \nu^{15} + \)\(24\!\cdots\!55\)\( \nu^{14} + \)\(86\!\cdots\!01\)\( \nu^{13} + \)\(17\!\cdots\!27\)\( \nu^{12} + \)\(24\!\cdots\!59\)\( \nu^{11} + \)\(50\!\cdots\!61\)\( \nu^{10} + \)\(34\!\cdots\!71\)\( \nu^{9} + \)\(71\!\cdots\!93\)\( \nu^{8} + \)\(24\!\cdots\!88\)\( \nu^{7} + \)\(50\!\cdots\!20\)\( \nu^{6} + \)\(82\!\cdots\!28\)\( \nu^{5} + \)\(16\!\cdots\!12\)\( \nu^{4} + \)\(10\!\cdots\!32\)\( \nu^{3} + \)\(20\!\cdots\!08\)\( \nu^{2} + \)\(41\!\cdots\!44\)\( \nu + \)\(70\!\cdots\!12\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 76 \beta_{1} + 5\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + 50 \beta_{6} + 23 \beta_{5} + 101670 \beta_{4} + 2103 \beta_{3} + 691967 \beta_{2} - 11518606805 \beta_{1} - 20916462922900926\)\()/2304\)
\(\nu^{3}\)\(=\)\((\)\(134640 \beta_{15} + 452577 \beta_{14} + 195496 \beta_{13} - 516800 \beta_{12} - 7300258 \beta_{11} - 9097343 \beta_{10} + 264538105 \beta_{9} - 100718083 \beta_{8} + 5106487 \beta_{7} + 216535078 \beta_{6} - 13127648797 \beta_{5} - 33360721571656 \beta_{4} + 102056674786 \beta_{3} - 35953167477080604 \beta_{2} - 2513420976571071987 \beta_{1} - 877613635645576662\)\()/110592\)
\(\nu^{4}\)\(=\)\((\)\(548919743877 \beta_{15} + 3051470439708 \beta_{14} - 265916899456 \beta_{13} + 3051462684692 \beta_{12} + 107229895307701 \beta_{11} - 35519837274334 \beta_{10} - 339777266515972 \beta_{9} + 364163246916706 \beta_{8} - 3521202962963074 \beta_{7} - 270622124220301027 \beta_{6} - 51200258791712081 \beta_{5} - 859633241682207170375 \beta_{4} - 13962298126586982955 \beta_{3} - 3445846660595459834328 \beta_{2} + 58968637429443028877075283 \beta_{1} + 62660693321128185779942629285926\)\()/442368\)
\(\nu^{5}\)\(=\)\((\)\(-46890716744372370480 \beta_{15} - 478631722396561530630 \beta_{14} - 137028309650429876300 \beta_{13} + 279694901247341532160 \beta_{12} + 3189226227001352873135 \beta_{11} + 4741788665855006693530 \beta_{10} - 32622936324964294462223 \beta_{9} + 33054377014425428356895 \beta_{8} - 2592420404111471913250 \beta_{7} + 157584216197060232968421 \beta_{6} + 7762660992681586033780945 \beta_{5} + 14960986714214796833413085195 \beta_{4} - 45606723352278845515968176 \beta_{3} + 13097283199589815493998514692373 \beta_{2} + 987143031855832351199437406343823 \beta_{1} + 588682878469177680790346320707135\)\()/2359296\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(13\!\cdots\!19\)\( \beta_{15} - \)\(33\!\cdots\!84\)\( \beta_{14} + \)\(82\!\cdots\!64\)\( \beta_{13} - \)\(33\!\cdots\!84\)\( \beta_{12} - \)\(20\!\cdots\!59\)\( \beta_{11} + \)\(38\!\cdots\!70\)\( \beta_{10} + \)\(61\!\cdots\!08\)\( \beta_{9} - \)\(10\!\cdots\!50\)\( \beta_{8} + \)\(40\!\cdots\!76\)\( \beta_{7} + \)\(37\!\cdots\!11\)\( \beta_{6} + \)\(71\!\cdots\!65\)\( \beta_{5} + \)\(13\!\cdots\!55\)\( \beta_{4} + \)\(21\!\cdots\!01\)\( \beta_{3} + \)\(46\!\cdots\!18\)\( \beta_{2} - \)\(80\!\cdots\!59\)\( \beta_{1} - \)\(68\!\cdots\!76\)\(\)\()/28311552\)
\(\nu^{7}\)\(=\)\((\)\(\)\(39\!\cdots\!04\)\( \beta_{15} + \)\(85\!\cdots\!76\)\( \beta_{14} + \)\(19\!\cdots\!68\)\( \beta_{13} - \)\(38\!\cdots\!08\)\( \beta_{12} - \)\(35\!\cdots\!09\)\( \beta_{11} - \)\(63\!\cdots\!12\)\( \beta_{10} - \)\(82\!\cdots\!33\)\( \beta_{9} - \)\(23\!\cdots\!99\)\( \beta_{8} + \)\(33\!\cdots\!96\)\( \beta_{7} - \)\(66\!\cdots\!95\)\( \beta_{6} - \)\(10\!\cdots\!97\)\( \beta_{5} - \)\(17\!\cdots\!05\)\( \beta_{4} + \)\(52\!\cdots\!32\)\( \beta_{3} - \)\(14\!\cdots\!39\)\( \beta_{2} - \)\(12\!\cdots\!71\)\( \beta_{1} - \)\(96\!\cdots\!41\)\(\)\()/ 150994944 \)
\(\nu^{8}\)\(=\)\((\)\(\)\(24\!\cdots\!84\)\( \beta_{15} + \)\(36\!\cdots\!64\)\( \beta_{14} - \)\(21\!\cdots\!44\)\( \beta_{13} + \)\(36\!\cdots\!12\)\( \beta_{12} + \)\(39\!\cdots\!22\)\( \beta_{11} - \)\(39\!\cdots\!24\)\( \beta_{10} - \)\(10\!\cdots\!10\)\( \beta_{9} + \)\(19\!\cdots\!30\)\( \beta_{8} - \)\(58\!\cdots\!12\)\( \beta_{7} - \)\(58\!\cdots\!59\)\( \beta_{6} - \)\(14\!\cdots\!62\)\( \beta_{5} - \)\(24\!\cdots\!75\)\( \beta_{4} - \)\(37\!\cdots\!59\)\( \beta_{3} - \)\(69\!\cdots\!42\)\( \beta_{2} + \)\(12\!\cdots\!68\)\( \beta_{1} + \)\(96\!\cdots\!73\)\(\)\()/ 226492416 \)
\(\nu^{9}\)\(=\)\((\)\(-\)\(63\!\cdots\!08\)\( \beta_{15} - \)\(29\!\cdots\!12\)\( \beta_{14} - \)\(57\!\cdots\!76\)\( \beta_{13} + \)\(11\!\cdots\!68\)\( \beta_{12} + \)\(89\!\cdots\!07\)\( \beta_{11} + \)\(19\!\cdots\!88\)\( \beta_{10} + \)\(54\!\cdots\!03\)\( \beta_{9} + \)\(18\!\cdots\!21\)\( \beta_{8} - \)\(10\!\cdots\!92\)\( \beta_{7} + \)\(33\!\cdots\!07\)\( \beta_{6} + \)\(27\!\cdots\!67\)\( \beta_{5} + \)\(44\!\cdots\!25\)\( \beta_{4} - \)\(13\!\cdots\!14\)\( \beta_{3} + \)\(40\!\cdots\!41\)\( \beta_{2} + \)\(38\!\cdots\!29\)\( \beta_{1} + \)\(33\!\cdots\!25\)\(\)\()/ 226492416 \)
\(\nu^{10}\)\(=\)\((\)\(-\)\(23\!\cdots\!48\)\( \beta_{15} - \)\(21\!\cdots\!68\)\( \beta_{14} + \)\(28\!\cdots\!92\)\( \beta_{13} - \)\(21\!\cdots\!68\)\( \beta_{12} - \)\(42\!\cdots\!80\)\( \beta_{11} + \)\(20\!\cdots\!20\)\( \beta_{10} + \)\(10\!\cdots\!04\)\( \beta_{9} - \)\(18\!\cdots\!24\)\( \beta_{8} + \)\(52\!\cdots\!84\)\( \beta_{7} + \)\(54\!\cdots\!31\)\( \beta_{6} + \)\(19\!\cdots\!12\)\( \beta_{5} + \)\(26\!\cdots\!39\)\( \beta_{4} + \)\(40\!\cdots\!81\)\( \beta_{3} + \)\(62\!\cdots\!16\)\( \beta_{2} - \)\(11\!\cdots\!70\)\( \beta_{1} - \)\(87\!\cdots\!21\)\(\)\()/ 113246208 \)
\(\nu^{11}\)\(=\)\((\)\(\)\(20\!\cdots\!24\)\( \beta_{15} + \)\(42\!\cdots\!36\)\( \beta_{14} + \)\(69\!\cdots\!60\)\( \beta_{13} - \)\(15\!\cdots\!00\)\( \beta_{12} - \)\(10\!\cdots\!27\)\( \beta_{11} - \)\(25\!\cdots\!76\)\( \beta_{10} - \)\(10\!\cdots\!79\)\( \beta_{9} + \)\(38\!\cdots\!79\)\( \beta_{8} + \)\(13\!\cdots\!80\)\( \beta_{7} - \)\(61\!\cdots\!41\)\( \beta_{6} - \)\(30\!\cdots\!91\)\( \beta_{5} - \)\(52\!\cdots\!63\)\( \beta_{4} + \)\(16\!\cdots\!88\)\( \beta_{3} - \)\(48\!\cdots\!69\)\( \beta_{2} - \)\(52\!\cdots\!05\)\( \beta_{1} - \)\(50\!\cdots\!59\)\(\)\()/ 150994944 \)
\(\nu^{12}\)\(=\)\((\)\(\)\(27\!\cdots\!20\)\( \beta_{15} + \)\(10\!\cdots\!72\)\( \beta_{14} - \)\(45\!\cdots\!12\)\( \beta_{13} + \)\(10\!\cdots\!72\)\( \beta_{12} + \)\(58\!\cdots\!98\)\( \beta_{11} - \)\(72\!\cdots\!04\)\( \beta_{10} - \)\(13\!\cdots\!82\)\( \beta_{9} + \)\(21\!\cdots\!70\)\( \beta_{8} - \)\(64\!\cdots\!28\)\( \beta_{7} - \)\(65\!\cdots\!59\)\( \beta_{6} - \)\(31\!\cdots\!50\)\( \beta_{5} - \)\(35\!\cdots\!39\)\( \beta_{4} - \)\(56\!\cdots\!43\)\( \beta_{3} - \)\(71\!\cdots\!54\)\( \beta_{2} + \)\(13\!\cdots\!84\)\( \beta_{1} + \)\(10\!\cdots\!77\)\(\)\()/75497472\)
\(\nu^{13}\)\(=\)\((\)\(\)\(50\!\cdots\!56\)\( \beta_{15} - \)\(64\!\cdots\!44\)\( \beta_{14} - \)\(92\!\cdots\!96\)\( \beta_{13} + \)\(23\!\cdots\!64\)\( \beta_{12} + \)\(13\!\cdots\!64\)\( \beta_{11} + \)\(38\!\cdots\!56\)\( \beta_{10} + \)\(18\!\cdots\!84\)\( \beta_{9} - \)\(14\!\cdots\!20\)\( \beta_{8} - \)\(18\!\cdots\!36\)\( \beta_{7} + \)\(11\!\cdots\!05\)\( \beta_{6} + \)\(37\!\cdots\!80\)\( \beta_{5} + \)\(69\!\cdots\!05\)\( \beta_{4} - \)\(22\!\cdots\!37\)\( \beta_{3} + \)\(66\!\cdots\!92\)\( \beta_{2} + \)\(80\!\cdots\!94\)\( \beta_{1} + \)\(83\!\cdots\!25\)\(\)\()/ 113246208 \)
\(\nu^{14}\)\(=\)\((\)\(-\)\(33\!\cdots\!86\)\( \beta_{15} + \)\(47\!\cdots\!96\)\( \beta_{14} + \)\(80\!\cdots\!52\)\( \beta_{13} + \)\(47\!\cdots\!56\)\( \beta_{12} - \)\(88\!\cdots\!48\)\( \beta_{11} - \)\(12\!\cdots\!24\)\( \beta_{10} + \)\(17\!\cdots\!86\)\( \beta_{9} - \)\(25\!\cdots\!22\)\( \beta_{8} + \)\(89\!\cdots\!88\)\( \beta_{7} + \)\(87\!\cdots\!59\)\( \beta_{6} + \)\(56\!\cdots\!16\)\( \beta_{5} + \)\(52\!\cdots\!39\)\( \beta_{4} + \)\(86\!\cdots\!35\)\( \beta_{3} + \)\(91\!\cdots\!34\)\( \beta_{2} - \)\(16\!\cdots\!22\)\( \beta_{1} - \)\(14\!\cdots\!59\)\(\)\()/56623104\)
\(\nu^{15}\)\(=\)\((\)\(-\)\(94\!\cdots\!60\)\( \beta_{15} + \)\(51\!\cdots\!76\)\( \beta_{14} + \)\(64\!\cdots\!96\)\( \beta_{13} - \)\(18\!\cdots\!08\)\( \beta_{12} - \)\(89\!\cdots\!71\)\( \beta_{11} - \)\(30\!\cdots\!80\)\( \beta_{10} - \)\(17\!\cdots\!35\)\( \beta_{9} + \)\(16\!\cdots\!63\)\( \beta_{8} + \)\(14\!\cdots\!88\)\( \beta_{7} - \)\(10\!\cdots\!65\)\( \beta_{6} - \)\(23\!\cdots\!39\)\( \beta_{5} - \)\(49\!\cdots\!95\)\( \beta_{4} + \)\(16\!\cdots\!60\)\( \beta_{3} - \)\(49\!\cdots\!85\)\( \beta_{2} - \)\(65\!\cdots\!93\)\( \beta_{1} - \)\(71\!\cdots\!91\)\(\)\()/ 452984832 \)

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
0.500000 + 3.36022e6i
0.500000 3.36022e6i
0.500000 + 182748.i
0.500000 182748.i
0.500000 2.05624e6i
0.500000 + 2.05624e6i
0.500000 + 4.30163e6i
0.500000 4.30163e6i
0.500000 4.39438e6i
0.500000 + 4.39438e6i
0.500000 190511.i
0.500000 + 190511.i
0.500000 + 1.72842e6i
0.500000 1.72842e6i
0.500000 4.02935e6i
0.500000 + 4.02935e6i
−130415. 13105.5i 1.61291e8i 1.68364e10 + 3.41831e9i −2.13902e10 −2.11380e12 + 2.10348e13i 7.96031e13i −2.15092e15 6.66449e14i −9.33752e15 2.78961e15 + 2.80330e14i
3.2 −130415. + 13105.5i 1.61291e8i 1.68364e10 3.41831e9i −2.13902e10 −2.11380e12 2.10348e13i 7.96031e13i −2.15092e15 + 6.66449e14i −9.33752e15 2.78961e15 2.80330e14i
3.3 −88896.6 96318.6i 8.77192e6i −1.37468e9 + 1.71248e10i 8.59192e11 −8.44899e11 + 7.79793e11i 3.22173e14i 1.77164e15 1.38993e15i 1.66002e16 −7.63792e16 8.27562e16i
3.4 −88896.6 + 96318.6i 8.77192e6i −1.37468e9 1.71248e10i 8.59192e11 −8.44899e11 7.79793e11i 3.22173e14i 1.77164e15 + 1.38993e15i 1.66002e16 −7.63792e16 + 8.27562e16i
3.5 −83785.2 100796.i 9.86995e7i −3.13996e9 + 1.68905e10i −1.17209e12 9.94856e12 8.26956e12i 2.80726e14i 1.96558e15 1.09867e15i 6.93558e15 9.82041e16 + 1.18143e17i
3.6 −83785.2 + 100796.i 9.86995e7i −3.13996e9 1.68905e10i −1.17209e12 9.94856e12 + 8.26956e12i 2.80726e14i 1.96558e15 + 1.09867e15i 6.93558e15 9.82041e16 1.18143e17i
3.7 −13913.7 130331.i 2.06478e8i −1.67927e10 + 3.62680e9i −9.83896e10 −2.69106e13 + 2.87289e12i 2.44416e14i 7.06335e14 + 2.13815e15i −2.59561e16 1.36897e15 + 1.28233e16i
3.8 −13913.7 + 130331.i 2.06478e8i −1.67927e10 3.62680e9i −9.83896e10 −2.69106e13 2.87289e12i 2.44416e14i 7.06335e14 2.13815e15i −2.59561e16 1.36897e15 1.28233e16i
3.9 13540.0 130371.i 2.10930e8i −1.68132e10 3.53043e9i 1.26172e12 2.74991e13 + 2.85599e12i 2.66990e14i −6.87916e14 + 2.14415e15i −2.78144e16 1.70837e16 1.64492e17i
3.10 13540.0 + 130371.i 2.10930e8i −1.68132e10 + 3.53043e9i 1.26172e12 2.74991e13 2.85599e12i 2.66990e14i −6.87916e14 2.14415e15i −2.78144e16 1.70837e16 + 1.64492e17i
3.11 49362.6 121422.i 9.14451e6i −1.23065e10 1.19874e10i −6.20904e11 1.11034e12 + 4.51397e11i 3.03953e14i −2.06301e15 + 9.02552e14i 1.65936e16 −3.06494e16 + 7.53911e16i
3.12 49362.6 + 121422.i 9.14451e6i −1.23065e10 + 1.19874e10i −6.20904e11 1.11034e12 4.51397e11i 3.03953e14i −2.06301e15 9.02552e14i 1.65936e16 −3.06494e16 7.53911e16i
3.13 114483. 63824.3i 8.29643e7i 9.03280e9 1.46136e10i 6.47366e11 −5.29513e12 9.49799e12i 1.29684e14i 1.01400e14 2.24952e15i 9.79411e15 7.41124e16 4.13177e16i
3.14 114483. + 63824.3i 8.29643e7i 9.03280e9 + 1.46136e10i 6.47366e11 −5.29513e12 + 9.49799e12i 1.29684e14i 1.01400e14 + 2.24952e15i 9.79411e15 7.41124e16 + 4.13177e16i
3.15 125939. 36320.7i 1.93409e8i 1.45415e10 9.14841e9i −8.66188e11 7.02475e12 + 2.43577e13i 8.12413e13i 1.49906e15 1.68030e15i −2.07298e16 −1.09087e17 + 3.14606e16i
3.16 125939. + 36320.7i 1.93409e8i 1.45415e10 + 9.14841e9i −8.66188e11 7.02475e12 2.43577e13i 8.12413e13i 1.49906e15 + 1.68030e15i −2.07298e16 −1.09087e17 3.14606e16i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.35.b.a 16
4.b odd 2 1 inner 4.35.b.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.35.b.a 16 1.a even 1 1 trivial
4.35.b.a 16 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{35}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 27372 T + 10391050816 T^{2} - 483864769056768 T^{3} - 90697820359441252352 T^{4} - \)\(13\!\cdots\!56\)\( T^{5} - \)\(35\!\cdots\!16\)\( T^{6} + \)\(83\!\cdots\!04\)\( T^{7} + \)\(59\!\cdots\!92\)\( T^{8} + \)\(14\!\cdots\!36\)\( T^{9} - \)\(10\!\cdots\!96\)\( T^{10} - \)\(70\!\cdots\!24\)\( T^{11} - \)\(79\!\cdots\!72\)\( T^{12} - \)\(72\!\cdots\!32\)\( T^{13} + \)\(26\!\cdots\!56\)\( T^{14} + \)\(12\!\cdots\!68\)\( T^{15} + \)\(75\!\cdots\!96\)\( T^{16} \)
$3$ \( 1 - 99503180774650512 T^{2} + \)\(52\!\cdots\!64\)\( T^{4} - \)\(20\!\cdots\!16\)\( T^{6} + \)\(64\!\cdots\!24\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{10} + \)\(39\!\cdots\!92\)\( T^{12} - \)\(78\!\cdots\!52\)\( T^{14} + \)\(13\!\cdots\!18\)\( T^{16} - \)\(21\!\cdots\!72\)\( T^{18} + \)\(30\!\cdots\!32\)\( T^{20} - \)\(36\!\cdots\!40\)\( T^{22} + \)\(38\!\cdots\!84\)\( T^{24} - \)\(34\!\cdots\!16\)\( T^{26} + \)\(24\!\cdots\!04\)\( T^{28} - \)\(12\!\cdots\!52\)\( T^{30} + \)\(35\!\cdots\!81\)\( T^{32} \)
$5$ \( ( 1 + 10686127920 T + \)\(20\!\cdots\!60\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!00\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!50\)\( T^{8} - \)\(12\!\cdots\!00\)\( T^{9} + \)\(63\!\cdots\!00\)\( T^{10} - \)\(43\!\cdots\!00\)\( T^{11} + \)\(26\!\cdots\!00\)\( T^{12} - \)\(81\!\cdots\!00\)\( T^{13} + \)\(78\!\cdots\!00\)\( T^{14} + \)\(24\!\cdots\!00\)\( T^{15} + \)\(13\!\cdots\!25\)\( T^{16} )^{2} \)
$7$ \( 1 - \)\(43\!\cdots\!12\)\( T^{2} + \)\(98\!\cdots\!24\)\( T^{4} - \)\(15\!\cdots\!36\)\( T^{6} + \)\(19\!\cdots\!64\)\( T^{8} - \)\(19\!\cdots\!40\)\( T^{10} + \)\(15\!\cdots\!72\)\( T^{12} - \)\(10\!\cdots\!12\)\( T^{14} + \)\(63\!\cdots\!78\)\( T^{16} - \)\(31\!\cdots\!12\)\( T^{18} + \)\(13\!\cdots\!72\)\( T^{20} - \)\(47\!\cdots\!40\)\( T^{22} + \)\(14\!\cdots\!64\)\( T^{24} - \)\(34\!\cdots\!36\)\( T^{26} + \)\(62\!\cdots\!24\)\( T^{28} - \)\(79\!\cdots\!12\)\( T^{30} + \)\(54\!\cdots\!01\)\( T^{32} \)
$11$ \( 1 - \)\(18\!\cdots\!56\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(70\!\cdots\!00\)\( T^{8} - \)\(30\!\cdots\!68\)\( T^{10} + \)\(11\!\cdots\!28\)\( T^{12} - \)\(35\!\cdots\!00\)\( T^{14} + \)\(98\!\cdots\!90\)\( T^{16} - \)\(23\!\cdots\!00\)\( T^{18} + \)\(48\!\cdots\!08\)\( T^{20} - \)\(86\!\cdots\!88\)\( T^{22} + \)\(12\!\cdots\!00\)\( T^{24} - \)\(15\!\cdots\!20\)\( T^{26} + \)\(14\!\cdots\!00\)\( T^{28} - \)\(95\!\cdots\!16\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \)
$13$ \( ( 1 + 2616753518205829808 T + \)\(29\!\cdots\!16\)\( T^{2} + \)\(19\!\cdots\!48\)\( T^{3} + \)\(45\!\cdots\!08\)\( T^{4} + \)\(37\!\cdots\!76\)\( T^{5} + \)\(55\!\cdots\!84\)\( T^{6} + \)\(38\!\cdots\!56\)\( T^{7} + \)\(50\!\cdots\!42\)\( T^{8} + \)\(28\!\cdots\!84\)\( T^{9} + \)\(31\!\cdots\!64\)\( T^{10} + \)\(15\!\cdots\!44\)\( T^{11} + \)\(14\!\cdots\!28\)\( T^{12} + \)\(45\!\cdots\!52\)\( T^{13} + \)\(51\!\cdots\!76\)\( T^{14} + \)\(34\!\cdots\!32\)\( T^{15} + \)\(98\!\cdots\!81\)\( T^{16} )^{2} \)
$17$ \( ( 1 + \)\(30\!\cdots\!72\)\( T + \)\(28\!\cdots\!76\)\( T^{2} + \)\(47\!\cdots\!92\)\( T^{3} + \)\(44\!\cdots\!88\)\( T^{4} + \)\(62\!\cdots\!04\)\( T^{5} + \)\(48\!\cdots\!04\)\( T^{6} + \)\(54\!\cdots\!24\)\( T^{7} + \)\(37\!\cdots\!82\)\( T^{8} + \)\(37\!\cdots\!96\)\( T^{9} + \)\(22\!\cdots\!64\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{11} + \)\(97\!\cdots\!28\)\( T^{12} + \)\(72\!\cdots\!08\)\( T^{13} + \)\(29\!\cdots\!96\)\( T^{14} + \)\(21\!\cdots\!48\)\( T^{15} + \)\(48\!\cdots\!61\)\( T^{16} )^{2} \)
$19$ \( 1 - \)\(21\!\cdots\!16\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!20\)\( T^{8} - \)\(84\!\cdots\!48\)\( T^{10} + \)\(37\!\cdots\!28\)\( T^{12} - \)\(14\!\cdots\!40\)\( T^{14} + \)\(46\!\cdots\!50\)\( T^{16} - \)\(12\!\cdots\!40\)\( T^{18} + \)\(30\!\cdots\!68\)\( T^{20} - \)\(61\!\cdots\!08\)\( T^{22} + \)\(10\!\cdots\!20\)\( T^{24} - \)\(13\!\cdots\!00\)\( T^{26} + \)\(14\!\cdots\!20\)\( T^{28} - \)\(10\!\cdots\!96\)\( T^{30} + \)\(43\!\cdots\!21\)\( T^{32} \)
$23$ \( 1 - \)\(72\!\cdots\!32\)\( T^{2} + \)\(34\!\cdots\!84\)\( T^{4} - \)\(12\!\cdots\!76\)\( T^{6} + \)\(36\!\cdots\!24\)\( T^{8} - \)\(99\!\cdots\!40\)\( T^{10} + \)\(24\!\cdots\!12\)\( T^{12} - \)\(55\!\cdots\!12\)\( T^{14} + \)\(11\!\cdots\!98\)\( T^{16} - \)\(22\!\cdots\!72\)\( T^{18} + \)\(38\!\cdots\!32\)\( T^{20} - \)\(61\!\cdots\!40\)\( T^{22} + \)\(90\!\cdots\!04\)\( T^{24} - \)\(11\!\cdots\!76\)\( T^{26} + \)\(13\!\cdots\!04\)\( T^{28} - \)\(11\!\cdots\!52\)\( T^{30} + \)\(60\!\cdots\!41\)\( T^{32} \)
$29$ \( ( 1 + \)\(20\!\cdots\!28\)\( T + \)\(21\!\cdots\!84\)\( T^{2} + \)\(36\!\cdots\!44\)\( T^{3} + \)\(22\!\cdots\!64\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!52\)\( T^{6} - \)\(10\!\cdots\!72\)\( T^{7} + \)\(11\!\cdots\!38\)\( T^{8} - \)\(56\!\cdots\!32\)\( T^{9} + \)\(50\!\cdots\!72\)\( T^{10} - \)\(24\!\cdots\!60\)\( T^{11} + \)\(17\!\cdots\!44\)\( T^{12} + \)\(14\!\cdots\!44\)\( T^{13} + \)\(46\!\cdots\!04\)\( T^{14} + \)\(22\!\cdots\!08\)\( T^{15} + \)\(59\!\cdots\!41\)\( T^{16} )^{2} \)
$31$ \( 1 - \)\(32\!\cdots\!16\)\( T^{2} + \)\(54\!\cdots\!20\)\( T^{4} - \)\(65\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!20\)\( T^{8} - \)\(51\!\cdots\!48\)\( T^{10} + \)\(35\!\cdots\!28\)\( T^{12} - \)\(21\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!50\)\( T^{16} - \)\(56\!\cdots\!40\)\( T^{18} + \)\(24\!\cdots\!68\)\( T^{20} - \)\(88\!\cdots\!08\)\( T^{22} + \)\(28\!\cdots\!20\)\( T^{24} - \)\(75\!\cdots\!00\)\( T^{26} + \)\(16\!\cdots\!20\)\( T^{28} - \)\(24\!\cdots\!96\)\( T^{30} + \)\(19\!\cdots\!21\)\( T^{32} \)
$37$ \( ( 1 - \)\(47\!\cdots\!28\)\( T + \)\(89\!\cdots\!96\)\( T^{2} - \)\(42\!\cdots\!48\)\( T^{3} + \)\(40\!\cdots\!48\)\( T^{4} - \)\(19\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!84\)\( T^{6} - \)\(59\!\cdots\!56\)\( T^{7} + \)\(30\!\cdots\!02\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{9} + \)\(55\!\cdots\!64\)\( T^{10} - \)\(17\!\cdots\!84\)\( T^{11} + \)\(76\!\cdots\!68\)\( T^{12} - \)\(16\!\cdots\!52\)\( T^{13} + \)\(73\!\cdots\!56\)\( T^{14} - \)\(81\!\cdots\!12\)\( T^{15} + \)\(35\!\cdots\!81\)\( T^{16} )^{2} \)
$41$ \( ( 1 - \)\(23\!\cdots\!96\)\( T + \)\(37\!\cdots\!40\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(67\!\cdots\!40\)\( T^{4} - \)\(19\!\cdots\!68\)\( T^{5} + \)\(77\!\cdots\!48\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(62\!\cdots\!50\)\( T^{8} - \)\(83\!\cdots\!80\)\( T^{9} + \)\(36\!\cdots\!08\)\( T^{10} - \)\(61\!\cdots\!08\)\( T^{11} + \)\(14\!\cdots\!40\)\( T^{12} - \)\(43\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!40\)\( T^{14} - \)\(16\!\cdots\!16\)\( T^{15} + \)\(47\!\cdots\!81\)\( T^{16} )^{2} \)
$43$ \( 1 - \)\(42\!\cdots\!72\)\( T^{2} + \)\(86\!\cdots\!24\)\( T^{4} - \)\(11\!\cdots\!16\)\( T^{6} + \)\(10\!\cdots\!64\)\( T^{8} - \)\(76\!\cdots\!00\)\( T^{10} + \)\(43\!\cdots\!52\)\( T^{12} - \)\(20\!\cdots\!12\)\( T^{14} + \)\(76\!\cdots\!18\)\( T^{16} - \)\(23\!\cdots\!12\)\( T^{18} + \)\(61\!\cdots\!52\)\( T^{20} - \)\(12\!\cdots\!00\)\( T^{22} + \)\(21\!\cdots\!64\)\( T^{24} - \)\(27\!\cdots\!16\)\( T^{26} + \)\(24\!\cdots\!24\)\( T^{28} - \)\(14\!\cdots\!72\)\( T^{30} + \)\(40\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 - \)\(48\!\cdots\!32\)\( T^{2} + \)\(13\!\cdots\!44\)\( T^{4} - \)\(25\!\cdots\!36\)\( T^{6} + \)\(37\!\cdots\!04\)\( T^{8} - \)\(46\!\cdots\!60\)\( T^{10} + \)\(48\!\cdots\!52\)\( T^{12} - \)\(42\!\cdots\!92\)\( T^{14} + \)\(32\!\cdots\!78\)\( T^{16} - \)\(21\!\cdots\!12\)\( T^{18} + \)\(12\!\cdots\!92\)\( T^{20} - \)\(59\!\cdots\!60\)\( T^{22} + \)\(24\!\cdots\!64\)\( T^{24} - \)\(82\!\cdots\!36\)\( T^{26} + \)\(21\!\cdots\!84\)\( T^{28} - \)\(40\!\cdots\!72\)\( T^{30} + \)\(41\!\cdots\!81\)\( T^{32} \)
$53$ \( ( 1 + \)\(14\!\cdots\!68\)\( T + \)\(20\!\cdots\!36\)\( T^{2} + \)\(37\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!88\)\( T^{4} + \)\(45\!\cdots\!16\)\( T^{5} + \)\(12\!\cdots\!04\)\( T^{6} + \)\(30\!\cdots\!76\)\( T^{7} + \)\(59\!\cdots\!22\)\( T^{8} + \)\(13\!\cdots\!44\)\( T^{9} + \)\(22\!\cdots\!44\)\( T^{10} + \)\(33\!\cdots\!44\)\( T^{11} + \)\(63\!\cdots\!48\)\( T^{12} + \)\(50\!\cdots\!72\)\( T^{13} + \)\(11\!\cdots\!16\)\( T^{14} + \)\(34\!\cdots\!52\)\( T^{15} + \)\(10\!\cdots\!41\)\( T^{16} )^{2} \)
$59$ \( 1 - \)\(11\!\cdots\!16\)\( T^{2} + \)\(65\!\cdots\!60\)\( T^{4} - \)\(27\!\cdots\!60\)\( T^{6} + \)\(87\!\cdots\!60\)\( T^{8} - \)\(23\!\cdots\!08\)\( T^{10} + \)\(53\!\cdots\!68\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{14} + \)\(18\!\cdots\!70\)\( T^{16} - \)\(27\!\cdots\!20\)\( T^{18} + \)\(36\!\cdots\!88\)\( T^{20} - \)\(42\!\cdots\!88\)\( T^{22} + \)\(41\!\cdots\!60\)\( T^{24} - \)\(33\!\cdots\!60\)\( T^{26} + \)\(21\!\cdots\!60\)\( T^{28} - \)\(94\!\cdots\!56\)\( T^{30} + \)\(22\!\cdots\!61\)\( T^{32} \)
$61$ \( ( 1 + \)\(36\!\cdots\!64\)\( T + \)\(19\!\cdots\!40\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!40\)\( T^{4} + \)\(40\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!68\)\( T^{6} + \)\(25\!\cdots\!20\)\( T^{7} + \)\(72\!\cdots\!50\)\( T^{8} + \)\(12\!\cdots\!20\)\( T^{9} + \)\(32\!\cdots\!08\)\( T^{10} + \)\(51\!\cdots\!72\)\( T^{11} + \)\(11\!\cdots\!40\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{13} + \)\(31\!\cdots\!40\)\( T^{14} + \)\(29\!\cdots\!84\)\( T^{15} + \)\(40\!\cdots\!21\)\( T^{16} )^{2} \)
$67$ \( 1 - \)\(37\!\cdots\!12\)\( T^{2} + \)\(91\!\cdots\!84\)\( T^{4} - \)\(19\!\cdots\!96\)\( T^{6} + \)\(35\!\cdots\!64\)\( T^{8} - \)\(56\!\cdots\!60\)\( T^{10} + \)\(82\!\cdots\!32\)\( T^{12} - \)\(11\!\cdots\!12\)\( T^{14} + \)\(14\!\cdots\!58\)\( T^{16} - \)\(16\!\cdots\!92\)\( T^{18} + \)\(18\!\cdots\!92\)\( T^{20} - \)\(18\!\cdots\!60\)\( T^{22} + \)\(17\!\cdots\!04\)\( T^{24} - \)\(14\!\cdots\!96\)\( T^{26} + \)\(10\!\cdots\!44\)\( T^{28} - \)\(60\!\cdots\!72\)\( T^{30} + \)\(24\!\cdots\!21\)\( T^{32} \)
$71$ \( 1 - \)\(42\!\cdots\!16\)\( T^{2} + \)\(95\!\cdots\!80\)\( T^{4} - \)\(16\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!80\)\( T^{8} - \)\(31\!\cdots\!88\)\( T^{10} + \)\(35\!\cdots\!88\)\( T^{12} - \)\(36\!\cdots\!60\)\( T^{14} + \)\(33\!\cdots\!30\)\( T^{16} - \)\(27\!\cdots\!60\)\( T^{18} + \)\(21\!\cdots\!48\)\( T^{20} - \)\(14\!\cdots\!28\)\( T^{22} + \)\(83\!\cdots\!80\)\( T^{24} - \)\(43\!\cdots\!40\)\( T^{26} + \)\(19\!\cdots\!80\)\( T^{28} - \)\(66\!\cdots\!36\)\( T^{30} + \)\(12\!\cdots\!81\)\( T^{32} \)
$73$ \( ( 1 + \)\(10\!\cdots\!48\)\( T + \)\(11\!\cdots\!56\)\( T^{2} + \)\(40\!\cdots\!48\)\( T^{3} + \)\(61\!\cdots\!08\)\( T^{4} - \)\(59\!\cdots\!24\)\( T^{5} + \)\(20\!\cdots\!84\)\( T^{6} - \)\(32\!\cdots\!84\)\( T^{7} + \)\(50\!\cdots\!42\)\( T^{8} - \)\(73\!\cdots\!56\)\( T^{9} + \)\(10\!\cdots\!04\)\( T^{10} - \)\(67\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!88\)\( T^{12} + \)\(23\!\cdots\!52\)\( T^{13} + \)\(15\!\cdots\!96\)\( T^{14} + \)\(31\!\cdots\!12\)\( T^{15} + \)\(66\!\cdots\!21\)\( T^{16} )^{2} \)
$79$ \( 1 - \)\(29\!\cdots\!76\)\( T^{2} + \)\(43\!\cdots\!20\)\( T^{4} - \)\(43\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!20\)\( T^{8} - \)\(18\!\cdots\!48\)\( T^{10} + \)\(93\!\cdots\!08\)\( T^{12} - \)\(38\!\cdots\!40\)\( T^{14} + \)\(13\!\cdots\!50\)\( T^{16} - \)\(42\!\cdots\!40\)\( T^{18} + \)\(11\!\cdots\!68\)\( T^{20} - \)\(24\!\cdots\!88\)\( T^{22} + \)\(45\!\cdots\!20\)\( T^{24} - \)\(67\!\cdots\!00\)\( T^{26} + \)\(74\!\cdots\!20\)\( T^{28} - \)\(55\!\cdots\!96\)\( T^{30} + \)\(20\!\cdots\!81\)\( T^{32} \)
$83$ \( 1 - \)\(17\!\cdots\!92\)\( T^{2} + \)\(14\!\cdots\!24\)\( T^{4} - \)\(80\!\cdots\!16\)\( T^{6} + \)\(32\!\cdots\!04\)\( T^{8} - \)\(10\!\cdots\!40\)\( T^{10} + \)\(27\!\cdots\!72\)\( T^{12} - \)\(60\!\cdots\!32\)\( T^{14} + \)\(11\!\cdots\!18\)\( T^{16} - \)\(19\!\cdots\!12\)\( T^{18} + \)\(27\!\cdots\!32\)\( T^{20} - \)\(32\!\cdots\!40\)\( T^{22} + \)\(31\!\cdots\!44\)\( T^{24} - \)\(24\!\cdots\!16\)\( T^{26} + \)\(13\!\cdots\!84\)\( T^{28} - \)\(51\!\cdots\!52\)\( T^{30} + \)\(95\!\cdots\!21\)\( T^{32} \)
$89$ \( ( 1 + \)\(12\!\cdots\!88\)\( T + \)\(96\!\cdots\!84\)\( T^{2} + \)\(72\!\cdots\!24\)\( T^{3} + \)\(41\!\cdots\!84\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!92\)\( T^{6} + \)\(56\!\cdots\!08\)\( T^{7} + \)\(27\!\cdots\!98\)\( T^{8} + \)\(10\!\cdots\!28\)\( T^{9} + \)\(44\!\cdots\!52\)\( T^{10} + \)\(14\!\cdots\!00\)\( T^{11} + \)\(54\!\cdots\!24\)\( T^{12} + \)\(18\!\cdots\!24\)\( T^{13} + \)\(45\!\cdots\!44\)\( T^{14} + \)\(11\!\cdots\!28\)\( T^{15} + \)\(17\!\cdots\!21\)\( T^{16} )^{2} \)
$97$ \( ( 1 - \)\(35\!\cdots\!28\)\( T + \)\(13\!\cdots\!76\)\( T^{2} + \)\(16\!\cdots\!72\)\( T^{3} + \)\(67\!\cdots\!48\)\( T^{4} + \)\(33\!\cdots\!84\)\( T^{5} + \)\(38\!\cdots\!84\)\( T^{6} + \)\(11\!\cdots\!84\)\( T^{7} + \)\(19\!\cdots\!62\)\( T^{8} + \)\(39\!\cdots\!96\)\( T^{9} + \)\(49\!\cdots\!24\)\( T^{10} + \)\(15\!\cdots\!56\)\( T^{11} + \)\(10\!\cdots\!08\)\( T^{12} + \)\(93\!\cdots\!28\)\( T^{13} + \)\(26\!\cdots\!56\)\( T^{14} - \)\(25\!\cdots\!92\)\( T^{15} + \)\(25\!\cdots\!41\)\( T^{16} )^{2} \)
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