Properties

Label 8.19.d.b
Level 8
Weight 19
Character orbit 8.d
Analytic conductor 16.431
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 19 \)
Character orbit: \([\chi]\) \(=\) 8.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.4308910168\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} + 56971 x^{14} - 14457953 x^{13} + 5222713963 x^{12} - 1887111404561 x^{11} + 551417417640415 x^{10} - 120118761233202509 x^{9} + 30966377422568512537 x^{8} - 9791029903061915769567 x^{7} + 2760409144896820517353041 x^{6} - 573126276959276881630152915 x^{5} + 113556201106829558152115445753 x^{4} - 24602953293990202118839424410059 x^{3} + 5109186069446462658360003036354309 x^{2} - 959560421721305781394985245626434751 x + 326887106263049715278881269448539579330\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{120}\cdot 3^{13}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + ( 27 + \beta_{1} ) q^{2} + ( 203 - 4 \beta_{1} - \beta_{2} ) q^{3} + ( -27759 + 29 \beta_{1} - \beta_{2} + \beta_{3} ) q^{4} + ( 141 + 376 \beta_{1} - \beta_{3} - \beta_{4} ) q^{5} + ( -959222 + 336 \beta_{1} + 105 \beta_{2} - 4 \beta_{3} + \beta_{6} ) q^{6} + ( -2343 - 6255 \beta_{1} + 6 \beta_{2} + 13 \beta_{3} - \beta_{5} + \beta_{7} ) q^{7} + ( 19045904 - 29705 \beta_{1} + 286 \beta_{2} + 38 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} + \beta_{6} + \beta_{8} ) q^{8} + ( 144517361 - 264799 \beta_{1} - 1079 \beta_{2} + 327 \beta_{3} - 8 \beta_{4} + 40 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} ) q^{9} +O(q^{10})\) \( q + ( 27 + \beta_{1} ) q^{2} + ( 203 - 4 \beta_{1} - \beta_{2} ) q^{3} + ( -27759 + 29 \beta_{1} - \beta_{2} + \beta_{3} ) q^{4} + ( 141 + 376 \beta_{1} - \beta_{3} - \beta_{4} ) q^{5} + ( -959222 + 336 \beta_{1} + 105 \beta_{2} - 4 \beta_{3} + \beta_{6} ) q^{6} + ( -2343 - 6255 \beta_{1} + 6 \beta_{2} + 13 \beta_{3} - \beta_{5} + \beta_{7} ) q^{7} + ( 19045904 - 29705 \beta_{1} + 286 \beta_{2} + 38 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} + \beta_{6} + \beta_{8} ) q^{8} + ( 144517361 - 264799 \beta_{1} - 1079 \beta_{2} + 327 \beta_{3} - 8 \beta_{4} + 40 \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} ) q^{9} + ( -98112569 + 11167 \beta_{1} + 4077 \beta_{2} + 365 \beta_{3} + 43 \beta_{4} - 32 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{12} + \beta_{14} ) q^{10} + ( -154461923 + 30051 \beta_{1} - 17 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - 13 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{11} + ( 297404604 - 995846 \beta_{1} - 10720 \beta_{2} + 315 \beta_{3} + 337 \beta_{4} + 77 \beta_{5} - 95 \beta_{6} - 16 \beta_{7} - 3 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - 3 \beta_{14} + \beta_{15} ) q^{12} + ( -373642 - 1001016 \beta_{1} + 258 \beta_{2} + 4474 \beta_{3} - 370 \beta_{4} - 78 \beta_{5} + 45 \beta_{6} + 12 \beta_{7} + 11 \beta_{8} - \beta_{9} + 2 \beta_{10} - 6 \beta_{12} - 2 \beta_{15} ) q^{13} + ( 1635169561 - 156403 \beta_{1} + 29445 \beta_{2} - 5669 \beta_{3} + 186 \beta_{4} + 345 \beta_{5} - 6 \beta_{6} + 49 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} + 33 \beta_{10} + 15 \beta_{11} + 11 \beta_{12} - 3 \beta_{13} + 20 \beta_{14} + 2 \beta_{15} ) q^{14} + ( 3252978 + 8693940 \beta_{1} - 3467 \beta_{2} - 28194 \beta_{3} - 4204 \beta_{4} + 656 \beta_{5} + 804 \beta_{6} + 53 \beta_{7} + 129 \beta_{8} - 10 \beta_{9} + 13 \beta_{10} + 23 \beta_{11} + 44 \beta_{12} + 8 \beta_{13} - 4 \beta_{15} ) q^{15} + ( -12012334234 + 20299243 \beta_{1} - 138336 \beta_{2} - 33870 \beta_{3} + 766 \beta_{4} - 5414 \beta_{5} - 446 \beta_{6} + 338 \beta_{7} + 125 \beta_{8} + 14 \beta_{9} - 37 \beta_{10} + 21 \beta_{11} - 20 \beta_{12} - 26 \beta_{13} - 14 \beta_{14} + 2 \beta_{15} ) q^{16} + ( -11019686314 + 20309203 \beta_{1} - 361199 \beta_{2} - 73117 \beta_{3} + 1496 \beta_{4} - 3842 \beta_{5} + 357 \beta_{6} - 187 \beta_{8} + 51 \beta_{10} - 136 \beta_{14} ) q^{17} + ( -65211677717 + 151933931 \beta_{1} + 667798 \beta_{2} - 287902 \beta_{3} - 9740 \beta_{4} - 13722 \beta_{5} + 216 \beta_{6} - 1662 \beta_{7} + 484 \beta_{8} - 34 \beta_{9} - 160 \beta_{10} + 96 \beta_{11} + 54 \beta_{12} + 62 \beta_{13} + 36 \beta_{14} + 4 \beta_{15} ) q^{18} + ( -52038659583 + 122904315 \beta_{1} - 1323461 \beta_{2} - 170212 \beta_{3} - 214 \beta_{4} - 21170 \beta_{5} - 8509 \beta_{6} + 15 \beta_{7} - 1098 \beta_{8} - 49 \beta_{9} + 295 \beta_{10} + 247 \beta_{11} + 15 \beta_{12} - 15 \beta_{13} + 287 \beta_{14} ) q^{19} + ( 78491756816 - 107414078 \beta_{1} - 3284868 \beta_{2} + 17254 \beta_{3} + 6914 \beta_{4} + 19074 \beta_{5} - 1670 \beta_{6} - 5492 \beta_{7} - 264 \beta_{8} - 108 \beta_{9} + 1002 \beta_{10} + 246 \beta_{11} - 46 \beta_{12} - 60 \beta_{13} - 106 \beta_{14} - 34 \beta_{15} ) q^{20} + ( 19787025 + 53250264 \beta_{1} + 39110 \beta_{2} - 413933 \beta_{3} + 4551 \beta_{4} - 2938 \beta_{5} - 31229 \beta_{6} - 844 \beta_{7} + 4205 \beta_{8} + 97 \beta_{9} + 774 \beta_{10} + 184 \beta_{11} - 378 \beta_{12} + 64 \beta_{13} + 66 \beta_{15} ) q^{21} + ( 3686037654 - 154385864 \beta_{1} + 3018725 \beta_{2} + 36872 \beta_{3} + 97352 \beta_{4} - 2540 \beta_{5} - 1575 \beta_{6} + 16252 \beta_{7} + 288 \beta_{8} - 124 \beta_{9} + 1720 \beta_{10} + 328 \beta_{11} - 300 \beta_{12} + 388 \beta_{13} - 264 \beta_{14} - 72 \beta_{15} ) q^{22} + ( -14345788 - 36265650 \beta_{1} + 593485 \beta_{2} - 1311672 \beta_{3} - 53340 \beta_{4} - 9754 \beta_{5} + 64756 \beta_{6} - 1349 \beta_{7} + 7277 \beta_{8} - 290 \beta_{9} + 1257 \beta_{10} + 555 \beta_{11} - 388 \beta_{12} - 408 \beta_{13} + 140 \beta_{15} ) q^{23} + ( -122935259524 + 308596732 \beta_{1} - 9759428 \beta_{2} - 955048 \beta_{3} - 519642 \beta_{4} - 80694 \beta_{5} + 830 \beta_{6} + 37252 \beta_{7} + 460 \beta_{8} - 1220 \beta_{9} + 94 \beta_{10} + 1282 \beta_{11} + 784 \beta_{12} + 172 \beta_{13} - 132 \beta_{14} - 68 \beta_{15} ) q^{24} + ( -957003866621 + 1466227094 \beta_{1} + 42906376 \beta_{2} - 1315900 \beta_{3} + 112032 \beta_{4} - 164622 \beta_{5} + 192434 \beta_{6} - 896 \beta_{7} - 4718 \beta_{8} + 640 \beta_{9} + 3094 \beta_{10} + 960 \beta_{11} - 896 \beta_{12} + 896 \beta_{13} + 2984 \beta_{14} ) q^{25} + ( 261517796185 - 22506447 \beta_{1} + 11766659 \beta_{2} - 762477 \beta_{3} - 820563 \beta_{4} + 141464 \beta_{5} - 7454 \beta_{6} - 66527 \beta_{7} - 991 \beta_{8} + 1752 \beta_{9} + 4840 \beta_{10} + 664 \beta_{11} - 2351 \beta_{12} + 152 \beta_{13} - 841 \beta_{14} - 144 \beta_{15} ) q^{26} + ( 722749586676 - 441159405 \beta_{1} - 119202918 \beta_{2} + 6189660 \beta_{3} - 301806 \beta_{4} - 22314 \beta_{5} - 464817 \beta_{6} + 1995 \beta_{7} + 1902 \beta_{8} - 1653 \beta_{9} - 477 \beta_{10} - 1677 \beta_{11} + 1995 \beta_{12} - 1995 \beta_{13} - 6309 \beta_{14} ) q^{27} + ( -768185672448 + 1699413300 \beta_{1} - 57223240 \beta_{2} - 1244292 \beta_{3} + 2336916 \beta_{4} - 202956 \beta_{5} + 30852 \beta_{6} - 90664 \beta_{7} - 848 \beta_{8} - 1944 \beta_{9} - 13980 \beta_{10} + 4956 \beta_{11} + 5748 \beta_{12} - 1976 \beta_{13} + 3516 \beta_{14} + 524 \beta_{15} ) q^{28} + ( 936681463 + 2515080240 \beta_{1} + 996104 \beta_{2} - 15654851 \beta_{3} - 181931 \beta_{4} + 7576 \beta_{5} - 679816 \beta_{6} + 8864 \beta_{7} - 3616 \beta_{8} + 2008 \beta_{9} + 712 \beta_{10} - 4856 \beta_{11} + 5328 \beta_{12} + 4544 \beta_{13} - 976 \beta_{15} ) q^{29} + ( -2271027220243 + 291713297 \beta_{1} + 210531017 \beta_{2} + 7722839 \beta_{3} + 6905890 \beta_{4} - 550003 \beta_{5} + 3586 \beta_{6} + 57829 \beta_{7} + 663 \beta_{8} + 7721 \beta_{9} - 29099 \beta_{10} - 2149 \beta_{11} - 5817 \beta_{12} + 641 \beta_{13} - 636 \beta_{14} + 1194 \beta_{15} ) q^{30} + ( -570310113 - 1473143531 \beta_{1} + 14639649 \beta_{2} - 30176825 \beta_{3} - 884596 \beta_{4} - 325929 \beta_{5} + 1268220 \beta_{6} + 24356 \beta_{7} - 72705 \beta_{8} - 4694 \beta_{9} - 9773 \beta_{10} - 12087 \beta_{11} + 5780 \beta_{12} - 4360 \beta_{13} - 2236 \beta_{15} ) q^{31} + ( 8837733011788 - 12930762194 \beta_{1} + 78508080 \beta_{2} + 25953668 \beta_{3} - 9237068 \beta_{4} + 2099132 \beta_{5} + 74172 \beta_{6} - 63612 \beta_{7} - 96462 \beta_{8} + 3388 \beta_{9} - 2346 \beta_{10} + 7434 \beta_{11} + 4488 \beta_{12} - 12564 \beta_{13} + 9140 \beta_{14} + 1044 \beta_{15} ) q^{32} + ( -58437702314 - 5571354099 \beta_{1} + 194542801 \beta_{2} + 73233447 \beta_{3} - 569224 \beta_{4} + 1867780 \beta_{5} + 1531587 \beta_{6} - 6784 \beta_{7} + 175427 \beta_{8} + 6016 \beta_{9} - 15379 \beta_{10} - 47296 \beta_{11} - 6784 \beta_{12} + 6784 \beta_{13} - 26832 \beta_{14} ) q^{33} + ( 5036562013036 - 11605727790 \beta_{1} - 116896862 \beta_{2} + 24697158 \beta_{3} - 14605924 \beta_{4} + 2027794 \beta_{5} + 254728 \beta_{6} + 373286 \beta_{7} - 115668 \beta_{8} + 1530 \beta_{9} - 52768 \beta_{10} - 2720 \beta_{11} + 1666 \beta_{12} + 15130 \beta_{13} + 4012 \beta_{14} + 2380 \beta_{15} ) q^{34} + ( 1280901307448 + 808587170 \beta_{1} - 139334796 \beta_{2} + 158450980 \beta_{3} - 2911732 \beta_{4} + 1296272 \beta_{5} - 718014 \beta_{6} + 1738 \beta_{7} + 203756 \beta_{8} - 886 \beta_{9} + 11170 \beta_{10} - 38414 \beta_{11} + 1738 \beta_{12} - 1738 \beta_{13} + 58714 \beta_{14} ) q^{35} + ( 17299127683547 - 67549621617 \beta_{1} - 942213579 \beta_{2} + 150133571 \beta_{3} + 29692200 \beta_{4} + 3987272 \beta_{5} + 251176 \beta_{6} + 982464 \beta_{7} - 255736 \beta_{8} - 1600 \beta_{9} - 58928 \beta_{10} - 4560 \beta_{11} - 41976 \beta_{12} - 4672 \beta_{13} - 37944 \beta_{14} - 4760 \beta_{15} ) q^{36} + ( -970540462 - 2219204904 \beta_{1} + 91571406 \beta_{2} - 243687522 \beta_{3} + 5198922 \beta_{4} - 2796578 \beta_{5} - 954645 \beta_{6} - 63596 \beta_{7} - 118787 \beta_{8} + 17225 \beta_{9} + 28110 \beta_{10} - 110176 \beta_{11} - 90442 \beta_{12} - 21760 \beta_{13} + 8338 \beta_{15} ) q^{37} + ( 30819548555718 - 54812590920 \beta_{1} + 1950321853 \beta_{2} + 136468328 \beta_{3} + 31818184 \beta_{4} + 10310740 \beta_{5} + 1642417 \beta_{6} - 1573828 \beta_{7} - 377440 \beta_{8} - 28988 \beta_{9} + 20408 \beta_{10} + 94280 \beta_{11} + 53012 \beta_{12} + 15556 \beta_{13} + 35960 \beta_{14} - 11976 \beta_{15} ) q^{38} + ( 8111159187 + 22411051531 \beta_{1} + 164414250 \beta_{2} - 563389841 \beta_{3} - 679168 \beta_{4} - 4300123 \beta_{5} - 5896064 \beta_{6} - 154589 \beta_{7} + 414568 \beta_{8} + 26704 \beta_{9} + 125928 \beta_{10} - 103432 \beta_{11} + 35616 \beta_{12} + 18496 \beta_{13} + 21280 \beta_{15} ) q^{39} + ( -32466695210056 + 83536108948 \beta_{1} + 4672723056 \beta_{2} - 94409752 \beta_{3} - 31257968 \beta_{4} - 785456 \beta_{5} + 1122656 \beta_{6} - 2258488 \beta_{7} - 24900 \beta_{8} - 17480 \beta_{9} - 378436 \beta_{10} + 89348 \beta_{11} - 30912 \beta_{12} + 66712 \beta_{13} - 131752 \beta_{14} - 9384 \beta_{15} ) q^{40} + ( -37150700752560 + 81411661314 \beta_{1} - 3586638568 \beta_{2} + 515386956 \beta_{3} - 13329824 \beta_{4} - 10510506 \beta_{5} - 12482282 \beta_{6} + 83712 \beta_{7} + 75894 \beta_{8} - 9472 \beta_{9} + 309762 \beta_{10} - 11136 \beta_{11} + 83712 \beta_{12} - 83712 \beta_{13} + 118392 \beta_{14} ) q^{41} + ( -13894056899740 - 1018471340 \beta_{1} - 7916103236 \beta_{2} + 24291628 \beta_{3} - 31220964 \beta_{4} - 19517048 \beta_{5} + 2024040 \beta_{6} + 2441860 \beta_{7} - 344188 \beta_{8} - 34648 \beta_{9} - 315240 \beta_{10} + 401896 \beta_{11} + 77524 \beta_{12} - 113944 \beta_{13} + 33828 \beta_{14} - 23664 \beta_{15} ) q^{42} + ( -158640141082833 + 319881880850 \beta_{1} + 1185895167 \beta_{2} + 445475288 \beta_{3} + 1590948 \beta_{4} - 37978164 \beta_{5} + 15804446 \beta_{6} - 69098 \beta_{7} + 324892 \beta_{8} + 57238 \beta_{9} + 301830 \beta_{10} - 230618 \beta_{11} - 69098 \beta_{12} + 69098 \beta_{13} - 260170 \beta_{14} ) q^{43} + ( 4616043889340 + 463202026 \beta_{1} - 9216003840 \beta_{2} - 210873037 \beta_{3} + 12229977 \beta_{4} + 16689333 \beta_{5} + 1090857 \beta_{6} + 451056 \beta_{7} - 268315 \beta_{8} + 110976 \beta_{9} + 345746 \beta_{10} + 610670 \beta_{11} + 225637 \beta_{12} + 93568 \beta_{13} + 183013 \beta_{14} + 27401 \beta_{15} ) q^{44} + ( 109301391516 + 294163786960 \beta_{1} + 372069974 \beta_{2} - 2309154052 \beta_{3} - 96419584 \beta_{4} + 11142262 \beta_{5} + 34462787 \beta_{6} - 204108 \beta_{7} + 3777981 \beta_{8} - 250303 \beta_{9} + 658710 \beta_{10} + 116712 \beta_{11} + 650310 \beta_{12} + 704 \beta_{13} - 43262 \beta_{15} ) q^{45} + ( 9610976563331 + 4169767391 \beta_{1} + 16568875495 \beta_{2} + 76167801 \beta_{3} - 38798338 \beta_{4} - 37247709 \beta_{5} - 8691490 \beta_{6} + 2478091 \beta_{7} - 1347591 \beta_{8} - 89145 \beta_{9} - 272613 \beta_{10} + 548853 \beta_{11} - 225911 \beta_{12} - 229393 \beta_{13} - 326020 \beta_{14} + 79798 \beta_{15} ) q^{46} + ( -61086862793 - 159779447815 \beta_{1} + 970275095 \beta_{2} - 1891189097 \beta_{3} - 38465756 \beta_{4} - 45430677 \beta_{5} - 35344396 \beta_{6} + 532958 \beta_{7} + 4342581 \beta_{8} + 45678 \beta_{9} + 1007537 \beta_{10} - 389437 \beta_{11} - 674788 \beta_{12} + 129448 \beta_{13} - 131028 \beta_{15} ) q^{47} + ( 56007601224804 - 117304464078 \beta_{1} + 38835353504 \beta_{2} + 466057388 \beta_{3} + 248839460 \beta_{4} + 47234412 \beta_{5} - 1298244 \beta_{6} + 8788780 \beta_{7} - 1848034 \beta_{8} + 248084 \beta_{9} + 1218530 \beta_{10} + 1003454 \beta_{11} - 190680 \beta_{12} + 60452 \beta_{13} + 984588 \beta_{14} + 52716 \beta_{15} ) q^{48} + ( -199145865761759 - 3088553512 \beta_{1} - 11386118280 \beta_{2} + 1536195784 \beta_{3} - 44656960 \beta_{4} + 2208736 \beta_{5} - 44283800 \beta_{6} - 272768 \beta_{7} - 600920 \beta_{8} - 551808 \beta_{9} + 363096 \beta_{10} - 612672 \beta_{11} - 272768 \beta_{12} + 272768 \beta_{13} - 214784 \beta_{14} ) q^{49} + ( 353906131210607 - 1013499551303 \beta_{1} - 54177511772 \beta_{2} + 1260576300 \beta_{3} + 404547576 \beta_{4} + 53820804 \beta_{5} - 11461488 \beta_{6} - 18709908 \beta_{7} - 1677864 \beta_{8} - 237100 \beta_{9} + 944512 \beta_{10} + 236800 \beta_{11} - 370588 \beta_{12} + 43028 \beta_{13} - 518248 \beta_{14} + 154840 \beta_{15} ) q^{50} + ( 182350450580796 + 256536763841 \beta_{1} - 267180058 \beta_{2} + 2567574600 \beta_{3} - 36354330 \beta_{4} - 9341194 \beta_{5} + 11752253 \beta_{6} - 70159 \beta_{7} - 86462 \beta_{8} + 64753 \beta_{9} + 2473041 \beta_{10} - 541807 \beta_{11} - 70159 \beta_{12} + 70159 \beta_{13} + 235569 \beta_{14} ) q^{51} + ( -287011036376464 + 269359566158 \beta_{1} - 72420499868 \beta_{2} - 437632822 \beta_{3} - 506714610 \beta_{4} + 47385998 \beta_{5} + 11006294 \beta_{6} - 24621356 \beta_{7} - 1224760 \beta_{8} - 262132 \beta_{9} - 3450522 \beta_{10} - 440134 \beta_{11} - 1563202 \beta_{12} + 180572 \beta_{13} - 94438 \beta_{14} - 95982 \beta_{15} ) q^{52} + ( 325800404124 + 868088755760 \beta_{1} - 1321298022 \beta_{2} - 784658516 \beta_{3} + 151407496 \beta_{4} + 84270842 \beta_{5} - 48954987 \beta_{6} + 8056812 \beta_{7} - 5500117 \beta_{8} + 451207 \beta_{9} - 412198 \beta_{10} - 1393768 \beta_{11} - 1557398 \beta_{12} - 200384 \beta_{13} + 119438 \beta_{15} ) q^{53} + ( -85855506788352 + 774138419112 \beta_{1} + 129418313238 \beta_{2} + 441139860 \beta_{3} - 907629336 \beta_{4} - 177278556 \beta_{5} + 51765522 \beta_{6} + 30665004 \beta_{7} + 5695008 \beta_{8} + 261588 \beta_{9} - 2954088 \beta_{10} + 275304 \beta_{11} + 1265508 \beta_{12} + 349140 \beta_{13} + 1459800 \beta_{14} - 360936 \beta_{15} ) q^{54} + ( -666278553983 - 1772491433103 \beta_{1} + 3224600274 \beta_{2} - 128858939 \beta_{3} + 49708560 \beta_{4} - 185947025 \beta_{5} + 85912272 \beta_{6} - 2196867 \beta_{7} - 12061636 \beta_{8} - 7544 \beta_{9} - 2212308 \beta_{10} - 341372 \beta_{11} + 2601616 \beta_{12} - 324896 \beta_{13} + 517328 \beta_{15} ) q^{55} + ( 511152183880560 - 785654872536 \beta_{1} + 123938719968 \beta_{2} + 2238222160 \beta_{3} + 834151840 \beta_{4} + 404219680 \beta_{5} + 30387520 \beta_{6} + 23358096 \beta_{7} - 1136904 \beta_{8} - 536464 \beta_{9} - 3048584 \beta_{10} - 2555384 \beta_{11} + 1605632 \beta_{12} - 644560 \beta_{13} - 4352720 \beta_{14} - 173264 \beta_{15} ) q^{56} + ( 594806489376398 - 3872151223579 \beta_{1} + 41857503297 \beta_{2} - 2257203129 \beta_{3} + 119567096 \beta_{4} + 590488412 \beta_{5} + 247058827 \beta_{6} + 1392384 \beta_{7} + 14097739 \beta_{8} + 3060480 \beta_{9} - 1275067 \beta_{10} + 151680 \beta_{11} + 1392384 \beta_{12} - 1392384 \beta_{13} + 70800 \beta_{14} ) q^{57} + ( -656518212284747 + 15648929469 \beta_{1} - 163517776281 \beta_{2} + 1480415847 \beta_{3} + 967535265 \beta_{4} - 557414432 \beta_{5} + 67666362 \beta_{6} + 6495837 \beta_{7} - 8283811 \beta_{8} + 1183040 \beta_{9} - 3320384 \beta_{10} - 1847744 \beta_{11} + 1849565 \beta_{12} + 1099584 \beta_{13} + 2801379 \beta_{14} - 674688 \beta_{15} ) q^{58} + ( 36567868545583 + 4704100749028 \beta_{1} - 8912283637 \beta_{2} - 3584638940 \beta_{3} - 7026416 \beta_{4} - 745137404 \beta_{5} - 243091968 \beta_{6} + 1098240 \beta_{7} + 15907960 \beta_{8} - 887616 \beta_{9} - 17898504 \beta_{10} + 1424904 \beta_{11} + 1098240 \beta_{12} - 1098240 \beta_{13} + 3077200 \beta_{14} ) q^{59} + ( 1854063795378944 - 2571761293820 \beta_{1} - 342059617128 \beta_{2} - 779621012 \beta_{3} - 1194562844 \beta_{4} + 1043613188 \beta_{5} - 76642668 \beta_{6} + 41223992 \beta_{7} + 636272 \beta_{8} - 100344 \beta_{9} + 5540084 \beta_{10} - 2988340 \beta_{11} + 6644804 \beta_{12} - 2455704 \beta_{13} - 3909972 \beta_{14} + 136252 \beta_{15} ) q^{60} + ( 2704224345690 + 7170750562160 \beta_{1} - 19253968854 \beta_{2} + 16819234902 \beta_{3} + 631074626 \beta_{4} + 985348970 \beta_{5} - 387999435 \beta_{6} - 71170964 \beta_{7} - 27262117 \beta_{8} + 2034695 \beta_{9} - 5523222 \beta_{10} + 546440 \beta_{11} - 2424342 \beta_{12} + 1295808 \beta_{13} + 15502 \beta_{15} ) q^{61} + ( 388129512477508 + 66165092084 \beta_{1} + 341479262836 \beta_{2} + 583048524 \beta_{3} + 411109768 \beta_{4} - 783513884 \beta_{5} - 145472280 \beta_{6} - 99636828 \beta_{7} - 29772500 \beta_{8} + 1242132 \beta_{9} + 12041636 \beta_{10} - 9722596 \beta_{11} - 4534772 \beta_{12} + 2550004 \beta_{13} - 3096720 \beta_{14} + 1046792 \beta_{15} ) q^{62} + ( -5073499763358 - 13548728993912 \beta_{1} + 11717425223 \beta_{2} + 33943810454 \beta_{3} + 338526332 \beta_{4} - 1093539644 \beta_{5} + 508011308 \beta_{6} + 8602731 \beta_{7} - 26652573 \beta_{8} - 2098078 \beta_{9} - 7629753 \beta_{10} + 5610789 \beta_{11} - 1222524 \beta_{12} - 1823464 \beta_{13} - 1085708 \beta_{15} ) q^{63} + ( -3523720260151352 + 9326434895796 \beta_{1} + 314968836448 \beta_{2} - 13220316136 \beta_{3} - 1210572840 \beta_{4} + 656264648 \beta_{5} - 228964344 \beta_{6} - 150600040 \beta_{7} + 19189964 \beta_{8} - 2245144 \beta_{9} - 4483708 \beta_{10} - 4656708 \beta_{11} - 6603280 \beta_{12} - 806904 \beta_{13} + 10935672 \beta_{14} + 168120 \beta_{15} ) q^{64} + ( -1327613616065186 - 14585655118182 \beta_{1} + 112014754400 \beta_{2} - 36014086620 \beta_{3} + 632724000 \beta_{4} + 1838100870 \beta_{5} + 336298430 \beta_{6} - 8070656 \beta_{7} - 43877474 \beta_{8} - 5156352 \beta_{9} - 10676582 \beta_{10} + 3058432 \beta_{11} - 8070656 \beta_{12} + 8070656 \beta_{13} - 2639304 \beta_{14} ) q^{65} + ( -1469552303122602 - 11463542572 \beta_{1} - 350531179042 \beta_{2} - 6139286086 \beta_{3} - 2781696028 \beta_{4} - 2889078418 \beta_{5} - 186800648 \beta_{6} + 171992666 \beta_{7} + 113705620 \beta_{8} + 2691974 \beta_{9} + 5167456 \beta_{10} - 10575008 \beta_{11} - 8584962 \beta_{12} - 1361818 \beta_{13} - 6822060 \beta_{14} + 1786292 \beta_{15} ) q^{66} + ( 10152965312380941 + 3098341329467 \beta_{1} - 47807795393 \beta_{2} - 17321025888 \beta_{3} + 174569866 \beta_{4} - 695524030 \beta_{5} - 259561893 \beta_{6} + 1165063 \beta_{7} - 110874530 \beta_{8} - 1238137 \beta_{9} + 47521111 \beta_{10} + 12312247 \beta_{11} + 1165063 \beta_{12} - 1165063 \beta_{13} - 13885209 \beta_{14} ) q^{67} + ( -4417685182833678 + 5404254311514 \beta_{1} - 326783054642 \beta_{2} - 17039002726 \beta_{3} + 4645759864 \beta_{4} + 1780160984 \beta_{5} + 302481272 \beta_{6} + 191284544 \beta_{7} + 28608280 \beta_{8} - 1195712 \beta_{9} + 29534576 \beta_{10} - 1298800 \beta_{11} - 12313576 \beta_{12} + 1572160 \beta_{13} + 23961432 \beta_{14} + 455736 \beta_{15} ) q^{68} + ( 12620878611879 + 33523905548216 \beta_{1} - 72316684870 \beta_{2} + 36846097333 \beta_{3} - 1744213743 \beta_{4} + 4076816026 \beta_{5} + 608838277 \beta_{6} + 374510636 \beta_{7} + 73091451 \beta_{8} - 6049449 \beta_{9} + 1414138 \beta_{10} + 28376024 \beta_{11} + 20841290 \beta_{12} + 3219264 \beta_{13} - 1477586 \beta_{15} ) q^{69} + ( 265876663128040 + 1337854136688 \beta_{1} + 265006856792 \beta_{2} + 3213713224 \beta_{3} + 5028892976 \beta_{4} - 4869454920 \beta_{5} + 31358144 \beta_{6} - 151167832 \beta_{7} + 209887040 \beta_{8} - 390824 \beta_{9} + 22209360 \beta_{10} + 1098928 \beta_{11} + 84536 \beta_{12} - 7036584 \beta_{13} - 823984 \beta_{14} - 1350192 \beta_{15} ) q^{70} + ( -17723235669512 - 47271400841570 \beta_{1} + 55467468323 \beta_{2} + 77351014212 \beta_{3} - 1790644932 \beta_{4} - 4745752754 \beta_{5} - 858233236 \beta_{6} - 7444807 \beta_{7} + 85599131 \beta_{8} + 1815058 \beta_{9} + 8843839 \beta_{10} + 20095309 \beta_{11} - 14463900 \beta_{12} + 3004376 \beta_{13} - 587308 \beta_{15} ) q^{71} + ( -2293528174955888 + 17658003106237 \beta_{1} + 111275930282 \beta_{2} - 69633994782 \beta_{3} - 11495935069 \beta_{4} + 6574541789 \beta_{5} + 846018235 \beta_{6} - 34701792 \beta_{7} + 84992299 \beta_{8} + 6293472 \beta_{9} + 70186736 \beta_{10} + 5952528 \beta_{11} + 31214976 \beta_{12} + 3490656 \beta_{13} - 9658656 \beta_{14} + 1248480 \beta_{15} ) q^{72} + ( -496517408737064 - 48556493170279 \beta_{1} - 408961585695 \beta_{2} - 75998034321 \beta_{3} + 196613560 \beta_{4} + 5645518472 \beta_{5} - 1678301625 \beta_{6} + 16153216 \beta_{7} - 124537721 \beta_{8} + 1468544 \beta_{9} + 18713721 \beta_{10} + 45384896 \beta_{11} + 16153216 \beta_{12} - 16153216 \beta_{13} + 22850304 \beta_{14} ) q^{73} + ( 600895717270119 - 88975407601 \beta_{1} - 117468382403 \beta_{2} + 3629071533 \beta_{3} - 12806395309 \beta_{4} - 8518579288 \beta_{5} - 369911906 \beta_{6} - 153634913 \beta_{7} - 257236065 \beta_{8} - 16354712 \beta_{9} + 68364760 \beta_{10} + 1660072 \beta_{11} + 9561967 \beta_{12} - 1062232 \beta_{13} - 1066807 \beta_{14} - 1374960 \beta_{15} ) q^{74} + ( -23981617611039173 + 115043877481702 \beta_{1} + 867549304923 \beta_{2} - 84038511900 \beta_{3} + 3317820796 \beta_{4} - 16255233536 \beta_{5} + 2589450122 \beta_{6} - 10412238 \beta_{7} + 79087676 \beta_{8} + 8009970 \beta_{9} - 139874678 \beta_{10} + 15722970 \beta_{11} - 10412238 \beta_{12} + 10412238 \beta_{13} + 9311682 \beta_{14} ) q^{75} + ( -9433830988216996 + 31989668447194 \beta_{1} + 318847394816 \beta_{2} - 64670918149 \beta_{3} + 10832833201 \beta_{4} + 9053724333 \beta_{5} - 1876892159 \beta_{6} - 421954448 \beta_{7} + 154849885 \beta_{8} - 542336 \beta_{9} - 59971006 \beta_{10} - 25973314 \beta_{11} + 9030621 \beta_{12} + 19428736 \beta_{13} - 68489891 \beta_{14} - 2968351 \beta_{15} ) q^{76} + ( 39874159890477 + 106126939009704 \beta_{1} - 179813390658 \beta_{2} - 20248342457 \beta_{3} + 2593850491 \beta_{4} + 12899183582 \beta_{5} + 2402900727 \beta_{6} - 1421742876 \beta_{7} - 80052663 \beta_{8} - 2854659 \beta_{9} - 9264258 \beta_{10} - 6903288 \beta_{11} - 42835026 \beta_{12} - 22779456 \beta_{13} + 5185146 \beta_{15} ) q^{77} + ( -5813213670893333 + 206059590727 \beta_{1} - 1241708572113 \beta_{2} + 20394417777 \beta_{3} + 9542553454 \beta_{4} - 19210557269 \beta_{5} + 81781294 \beta_{6} + 806871347 \beta_{7} - 420781103 \beta_{8} - 17117137 \beta_{9} - 70885629 \beta_{10} + 16121741 \beta_{11} + 36051713 \beta_{12} - 5706985 \beta_{13} + 20953020 \beta_{14} - 3122362 \beta_{15} ) q^{78} + ( -50335823714506 - 133915088732626 \beta_{1} + 232619729880 \beta_{2} - 2804549058 \beta_{3} + 13504967024 \beta_{4} - 15337274910 \beta_{5} - 3688447056 \beta_{6} - 74042302 \beta_{7} - 54980844 \beta_{8} + 19025560 \beta_{9} - 3987804 \beta_{10} - 15954068 \beta_{11} + 37590704 \beta_{12} + 15891872 \beta_{13} + 11836016 \beta_{15} ) q^{79} + ( 1667966663075040 - 33467889672880 \beta_{1} - 2226193019456 \beta_{2} + 67304231072 \beta_{3} + 435284416 \beta_{4} + 18949915840 \beta_{5} - 3758020352 \beta_{6} + 1037832480 \beta_{7} - 164003344 \beta_{8} + 8652000 \beta_{9} - 28621712 \beta_{10} - 53858672 \beta_{11} - 99318784 \beta_{12} + 12280928 \beta_{13} - 23669152 \beta_{14} - 6218144 \beta_{15} ) q^{80} + ( 6156533540834751 - 168523808130483 \beta_{1} - 1050653351271 \beta_{2} + 82173023631 \beta_{3} - 3628094664 \beta_{4} + 23696781900 \beta_{5} - 2765217309 \beta_{6} + 19655424 \beta_{7} + 280437411 \beta_{8} - 1642752 \beta_{9} - 41992275 \beta_{10} - 61142400 \beta_{11} + 19655424 \beta_{12} - 19655424 \beta_{13} - 60162480 \beta_{14} ) q^{81} + ( 20595696230926666 - 38301994694670 \beta_{1} + 3544776465900 \beta_{2} + 115778594948 \beta_{3} - 3605853208 \beta_{4} - 13456455028 \beta_{5} + 2262615472 \beta_{6} - 1059087036 \beta_{7} + 534499720 \beta_{8} - 4976772 \beta_{9} - 96946048 \beta_{10} + 37772032 \beta_{11} + 59685676 \beta_{12} - 7642564 \beta_{13} + 52359880 \beta_{14} - 9657720 \beta_{15} ) q^{82} + ( 54183119369348623 + 95932967258780 \beta_{1} + 678798194475 \beta_{2} + 385451220844 \beta_{3} - 4837721760 \beta_{4} - 9031937668 \beta_{5} + 2271396200 \beta_{6} - 11302392 \beta_{7} + 95775352 \beta_{8} + 13083208 \beta_{9} + 288987120 \beta_{10} - 70492128 \beta_{11} - 11302392 \beta_{12} + 11302392 \beta_{13} + 97451512 \beta_{14} ) q^{83} + ( 28232102314683968 - 15363077129816 \beta_{1} + 4717457553328 \beta_{2} + 20008889720 \beta_{3} - 7224486232 \beta_{4} + 36648239528 \beta_{5} + 7219573768 \beta_{6} - 1130925328 \beta_{7} - 409344416 \beta_{8} + 47130512 \beta_{9} + 9226696 \beta_{10} + 26582968 \beta_{11} - 24853272 \beta_{12} - 34881712 \beta_{13} + 75455544 \beta_{14} + 6137816 \beta_{15} ) q^{84} + ( 95467142730505 + 254452958204416 \beta_{1} - 340664035062 \beta_{2} - 306289235365 \beta_{3} + 1278693199 \beta_{4} + 23810281498 \beta_{5} - 877990551 \beta_{6} + 4188449052 \beta_{7} - 379891809 \beta_{8} + 11108531 \beta_{9} - 25024374 \beta_{10} - 121420528 \beta_{11} - 3835982 \beta_{12} - 820352 \beta_{13} - 3646874 \beta_{15} ) q^{85} + ( 79264089077073234 - 168618433057056 \beta_{1} - 4172696227191 \beta_{2} + 351235817428 \beta_{3} - 26468352560 \beta_{4} - 13776070328 \beta_{5} + 301231097 \beta_{6} + 555852888 \beta_{7} + 489637952 \beta_{8} + 2541992 \beta_{9} - 54140624 \beta_{10} - 36659504 \beta_{11} - 47840056 \beta_{12} + 29824424 \beta_{13} - 43020112 \beta_{14} + 20144176 \beta_{15} ) q^{86} + ( -123131327343954 - 327252108942748 \beta_{1} + 693702916263 \beta_{2} - 263499232718 \beta_{3} - 46271834116 \beta_{4} - 40797269592 \beta_{5} + 4969151788 \beta_{6} + 276119167 \beta_{7} - 425981253 \beta_{8} - 23216110 \beta_{9} - 27320929 \beta_{10} - 148584467 \beta_{11} - 32356508 \beta_{12} - 15423784 \beta_{13} - 34938924 \beta_{15} ) q^{87} + ( -47398096876966564 + 7637487356860 \beta_{1} - 5786328425508 \beta_{2} - 77112445736 \beta_{3} + 54561974006 \beta_{4} + 37717810042 \beta_{5} + 13249131470 \beta_{6} - 104128348 \beta_{7} - 253303988 \beta_{8} - 10755556 \beta_{9} - 391794738 \beta_{10} + 104835218 \beta_{11} + 68922768 \beta_{12} - 18811124 \beta_{13} + 59223644 \beta_{14} + 9586844 \beta_{15} ) q^{88} + ( -55335680999493680 - 287296726698639 \beta_{1} + 270072825361 \beta_{2} + 301852089551 \beta_{3} - 3744920136 \beta_{4} + 45229444704 \beta_{5} + 7403008079 \beta_{6} - 102600832 \beta_{7} + 606852751 \beta_{8} - 31295104 \beta_{9} - 20627119 \beta_{10} - 258349504 \beta_{11} - 102600832 \beta_{12} + 102600832 \beta_{13} - 45484576 \beta_{14} ) q^{89} + ( -76730372454634539 + 10660786193837 \beta_{1} + 9503307925303 \beta_{2} + 318910052647 \beta_{3} + 110222065225 \beta_{4} - 65531680600 \beta_{5} - 4013000102 \beta_{6} + 782725245 \beta_{7} - 1589653635 \beta_{8} + 100083880 \beta_{9} - 455055720 \beta_{10} - 2798616 \beta_{11} - 177328947 \beta_{12} - 18291992 \beta_{13} - 120043989 \beta_{14} + 42316688 \beta_{15} ) q^{90} + ( -23559698732469928 + 474829355529230 \beta_{1} - 3321443643044 \beta_{2} + 569407455340 \beta_{3} - 11875685740 \beta_{4} - 66103010720 \beta_{5} - 13878288626 \beta_{6} + 63964102 \beta_{7} + 806205076 \beta_{8} - 44397306 \beta_{9} - 282195122 \beta_{10} - 187626946 \beta_{11} + 63964102 \beta_{12} - 63964102 \beta_{13} - 285198218 \beta_{14} ) q^{91} + ( 88318262250488064 + 2338126307388 \beta_{1} + 5466213493992 \beta_{2} + 62498310996 \beta_{3} - 80247127716 \beta_{4} + 61080863804 \beta_{5} - 19448696660 \beta_{6} + 2362362952 \beta_{7} - 799965808 \beta_{8} - 69474184 \beta_{9} + 17557580 \beta_{10} + 223609844 \beta_{11} + 35014140 \beta_{12} - 32604392 \beta_{13} + 148679188 \beta_{14} + 2215300 \beta_{15} ) q^{92} + ( 259141629946488 + 691288354715504 \beta_{1} - 773733543520 \beta_{2} - 1218995049528 \beta_{3} - 27726445832 \beta_{4} + 66947942400 \beta_{5} - 12992076984 \beta_{6} - 9844147232 \beta_{7} + 1246087624 \beta_{8} + 1395000 \beta_{9} + 242210592 \beta_{10} - 48455824 \beta_{11} + 289308624 \beta_{12} + 133341312 \beta_{13} - 28040848 \beta_{15} ) q^{93} + ( 41965674584692846 - 6580289686234 \beta_{1} - 7729392046314 \beta_{2} - 142455570966 \beta_{3} - 65194881684 \beta_{4} - 70035537714 \beta_{5} + 766238988 \beta_{6} - 3650508738 \beta_{7} - 1951218486 \beta_{8} + 129800950 \beta_{9} + 157324574 \beta_{10} + 273240194 \beta_{11} - 80446070 \beta_{12} - 37594618 \beta_{13} + 15543864 \beta_{14} - 43833092 \beta_{15} ) q^{94} + ( -228634841692431 - 607454474956655 \beta_{1} + 1251978327018 \beta_{2} - 542422520619 \beta_{3} + 107055559440 \beta_{4} - 69016290097 \beta_{5} + 15035505744 \beta_{6} - 239071563 \beta_{7} + 1274946996 \beta_{8} - 60253704 \beta_{9} + 285784356 \beta_{10} - 59597268 \beta_{11} - 60705552 \beta_{12} - 95033952 \beta_{13} + 26251056 \beta_{15} ) q^{95} + ( -30168495249686744 + 54199764548676 \beta_{1} - 16390683039968 \beta_{2} - 260276422600 \beta_{3} + 67277404184 \beta_{4} + 72166347272 \beta_{5} - 31636646264 \beta_{6} - 5013997768 \beta_{7} - 1193604 \beta_{8} - 66991288 \beta_{9} + 81832756 \beta_{10} + 477385484 \beta_{11} + 454232816 \beta_{12} - 77814040 \beta_{13} + 48555864 \beta_{14} + 17137304 \beta_{15} ) q^{96} + ( -163510572989439410 - 207786813920717 \beta_{1} + 3261772743737 \beta_{2} - 218736870933 \beta_{3} + 9673268120 \beta_{4} + 32262536326 \beta_{5} + 18096327525 \beta_{6} + 95684480 \beta_{7} - 653308795 \beta_{8} + 225587584 \beta_{9} + 804623123 \beta_{10} + 354499136 \beta_{11} + 95684480 \beta_{12} - 95684480 \beta_{13} + 605494680 \beta_{14} ) q^{97} + ( -5181723666711461 - 195283752709583 \beta_{1} + 12576923101200 \beta_{2} + 74704320304 \beta_{3} + 34744138464 \beta_{4} - 45097321584 \beta_{5} + 5836088384 \beta_{6} + 5651135024 \beta_{7} + 1885958496 \beta_{8} - 75413808 \beta_{9} + 610482432 \beta_{10} + 272924928 \beta_{11} + 16094992 \beta_{12} + 178237392 \beta_{13} + 15847264 \beta_{14} - 66261920 \beta_{15} ) q^{98} + ( -50789338878567019 + 708775916138180 \beta_{1} - 4040823225671 \beta_{2} + 226551011052 \beta_{3} - 7722142208 \beta_{4} - 104162429156 \beta_{5} - 20359305064 \beta_{6} + 71090424 \beta_{7} - 908484904 \beta_{8} - 88857288 \beta_{9} + 210901600 \beta_{10} + 145928592 \beta_{11} + 71090424 \beta_{12} - 71090424 \beta_{13} - 60568344 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 426q^{2} + 3264q^{3} - 444332q^{4} - 15348708q^{6} + 304914744q^{8} + 2313856176q^{9} + O(q^{10}) \) \( 16q + 426q^{2} + 3264q^{3} - 444332q^{4} - 15348708q^{6} + 304914744q^{8} + 2313856176q^{9} - 1569837600q^{10} - 2471571264q^{11} + 4764364056q^{12} + 26163923904q^{14} - 192320075504q^{16} - 176439301344q^{17} - 1044291511122q^{18} - 833365634368q^{19} + 1256486405760q^{20} + 59927319356q^{22} - 1968891910512q^{24} - 15320509140080q^{25} + 4184514840864q^{26} + 11565649473408q^{27} - 12301604294400q^{28} - 36336510039360q^{30} + 141481742931936q^{32} - 900457491648q^{33} + 80653465357268q^{34} + 20487495736320q^{35} + 277183098847068q^{36} + 493456694265564q^{38} - 519930573603840q^{40} - 594931562445024q^{41} - 222362598288000q^{42} - 2540155017577792q^{43} + 73781630442072q^{44} + 153882272211264q^{46} + 897135071530464q^{48} - 3186415704132848q^{49} + 5668141028706330q^{50} + 2916050119466880q^{51} - 4594372123628160q^{52} - 1377307163285640q^{54} + 8184134137073664q^{56} + 9540488340840768q^{57} - 10505700099162720q^{58} + 556807886314176q^{59} + 29677718651516160q^{60} + 6212402633091840q^{62} - 56432885385732032q^{64} - 21153177930524160q^{65} - 23515570511959320q^{66} + 162428574934613696q^{67} - 70717864614203736q^{68} + 4248100663257600q^{70} - 36801123950767128q^{72} - 7655712852492256q^{73} + 9613779604149984q^{74} - 384388788469704000q^{75} - 151130166278324584q^{76} - 93022771634070720q^{78} + 26870186366192640q^{80} + 99506910981790800q^{81} + 329788507250077556q^{82} + 866357336138514624q^{83} + 451843591950574080q^{84} + 1269201398006865468q^{86} - 758460699051663856q^{88} - 883646550464284128q^{89} - 1227675520905095520q^{90} - 379834560933460992q^{91} + 1413121475102841600q^{92} + 671428514272869504q^{94} - 483149691255826368q^{96} - 2614894858526780128q^{97} - 81635840192738166q^{98} - 816916348833567168q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 5 x^{15} + 56971 x^{14} - 14457953 x^{13} + 5222713963 x^{12} - 1887111404561 x^{11} + 551417417640415 x^{10} - 120118761233202509 x^{9} + 30966377422568512537 x^{8} - 9791029903061915769567 x^{7} + 2760409144896820517353041 x^{6} - 573126276959276881630152915 x^{5} + 113556201106829558152115445753 x^{4} - 24602953293990202118839424410059 x^{3} + 5109186069446462658360003036354309 x^{2} - 959560421721305781394985245626434751 x + 326887106263049715278881269448539579330\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\(24291957863492033 \nu^{15} - 6610998248766402450 \nu^{14} + \)\(17\!\cdots\!41\)\( \nu^{13} - \)\(62\!\cdots\!86\)\( \nu^{12} + \)\(16\!\cdots\!53\)\( \nu^{11} - \)\(63\!\cdots\!50\)\( \nu^{10} + \)\(10\!\cdots\!01\)\( \nu^{9} - \)\(36\!\cdots\!26\)\( \nu^{8} + \)\(91\!\cdots\!43\)\( \nu^{7} - \)\(11\!\cdots\!50\)\( \nu^{6} + \)\(53\!\cdots\!07\)\( \nu^{5} - \)\(98\!\cdots\!46\)\( \nu^{4} + \)\(13\!\cdots\!35\)\( \nu^{3} - \)\(25\!\cdots\!82\)\( \nu^{2} + \)\(51\!\cdots\!67\)\( \nu + \)\(25\!\cdots\!30\)\(\)\()/ \)\(17\!\cdots\!20\)\( \)
\(\beta_{3}\)\(=\)\((\)\(24291957863492033 \nu^{15} - 6610998248766402450 \nu^{14} + \)\(17\!\cdots\!41\)\( \nu^{13} - \)\(62\!\cdots\!86\)\( \nu^{12} + \)\(16\!\cdots\!53\)\( \nu^{11} - \)\(63\!\cdots\!50\)\( \nu^{10} + \)\(10\!\cdots\!01\)\( \nu^{9} - \)\(36\!\cdots\!26\)\( \nu^{8} + \)\(91\!\cdots\!43\)\( \nu^{7} - \)\(11\!\cdots\!50\)\( \nu^{6} + \)\(53\!\cdots\!07\)\( \nu^{5} - \)\(98\!\cdots\!46\)\( \nu^{4} + \)\(13\!\cdots\!35\)\( \nu^{3} + \)\(68\!\cdots\!98\)\( \nu^{2} + \)\(13\!\cdots\!87\)\( \nu + \)\(52\!\cdots\!10\)\(\)\()/ \)\(17\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(98539827758330624921 \nu^{15} + \)\(38\!\cdots\!34\)\( \nu^{14} + \)\(49\!\cdots\!13\)\( \nu^{13} - \)\(29\!\cdots\!66\)\( \nu^{12} + \)\(16\!\cdots\!45\)\( \nu^{11} - \)\(14\!\cdots\!30\)\( \nu^{10} + \)\(41\!\cdots\!01\)\( \nu^{9} - \)\(15\!\cdots\!86\)\( \nu^{8} + \)\(25\!\cdots\!47\)\( \nu^{7} - \)\(62\!\cdots\!86\)\( \nu^{6} + \)\(12\!\cdots\!19\)\( \nu^{5} - \)\(25\!\cdots\!10\)\( \nu^{4} + \)\(44\!\cdots\!83\)\( \nu^{3} - \)\(17\!\cdots\!06\)\( \nu^{2} - \)\(47\!\cdots\!69\)\( \nu + \)\(12\!\cdots\!10\)\(\)\()/ \)\(93\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(1002149113385542979 \nu^{15} - 36559913062610411190 \nu^{14} + \)\(58\!\cdots\!23\)\( \nu^{13} - \)\(16\!\cdots\!98\)\( \nu^{12} + \)\(55\!\cdots\!19\)\( \nu^{11} - \)\(20\!\cdots\!10\)\( \nu^{10} + \)\(56\!\cdots\!83\)\( \nu^{9} - \)\(12\!\cdots\!38\)\( \nu^{8} + \)\(33\!\cdots\!69\)\( \nu^{7} - \)\(98\!\cdots\!30\)\( \nu^{6} + \)\(28\!\cdots\!41\)\( \nu^{5} - \)\(59\!\cdots\!98\)\( \nu^{4} + \)\(11\!\cdots\!25\)\( \nu^{3} - \)\(21\!\cdots\!86\)\( \nu^{2} + \)\(47\!\cdots\!01\)\( \nu - \)\(86\!\cdots\!10\)\(\)\()/ \)\(29\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(9893998270234396379 \nu^{15} + \)\(20\!\cdots\!54\)\( \nu^{14} + \)\(33\!\cdots\!67\)\( \nu^{13} - \)\(11\!\cdots\!26\)\( \nu^{12} + \)\(14\!\cdots\!43\)\( \nu^{11} + \)\(14\!\cdots\!38\)\( \nu^{10} + \)\(15\!\cdots\!31\)\( \nu^{9} + \)\(13\!\cdots\!58\)\( \nu^{8} - \)\(36\!\cdots\!83\)\( \nu^{7} + \)\(41\!\cdots\!42\)\( \nu^{6} - \)\(14\!\cdots\!87\)\( \nu^{5} + \)\(40\!\cdots\!70\)\( \nu^{4} - \)\(85\!\cdots\!75\)\( \nu^{3} + \)\(17\!\cdots\!74\)\( \nu^{2} - \)\(58\!\cdots\!75\)\( \nu + \)\(17\!\cdots\!90\)\(\)\()/ \)\(17\!\cdots\!20\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(56\!\cdots\!15\)\( \nu^{15} - \)\(67\!\cdots\!18\)\( \nu^{14} + \)\(70\!\cdots\!27\)\( \nu^{13} + \)\(24\!\cdots\!26\)\( \nu^{12} + \)\(48\!\cdots\!07\)\( \nu^{11} - \)\(37\!\cdots\!46\)\( \nu^{10} + \)\(31\!\cdots\!99\)\( \nu^{9} - \)\(12\!\cdots\!82\)\( \nu^{8} - \)\(19\!\cdots\!91\)\( \nu^{7} - \)\(26\!\cdots\!54\)\( \nu^{6} + \)\(26\!\cdots\!21\)\( \nu^{5} - \)\(49\!\cdots\!06\)\( \nu^{4} + \)\(34\!\cdots\!65\)\( \nu^{3} + \)\(34\!\cdots\!30\)\( \nu^{2} - \)\(58\!\cdots\!83\)\( \nu - \)\(92\!\cdots\!50\)\(\)\()/ \)\(93\!\cdots\!60\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(12\!\cdots\!85\)\( \nu^{15} + \)\(10\!\cdots\!58\)\( \nu^{14} - \)\(71\!\cdots\!05\)\( \nu^{13} + \)\(12\!\cdots\!78\)\( \nu^{12} - \)\(74\!\cdots\!37\)\( \nu^{11} + \)\(18\!\cdots\!18\)\( \nu^{10} - \)\(57\!\cdots\!33\)\( \nu^{9} + \)\(11\!\cdots\!42\)\( \nu^{8} - \)\(26\!\cdots\!31\)\( \nu^{7} + \)\(86\!\cdots\!26\)\( \nu^{6} - \)\(31\!\cdots\!63\)\( \nu^{5} + \)\(55\!\cdots\!90\)\( \nu^{4} + \)\(27\!\cdots\!33\)\( \nu^{3} + \)\(22\!\cdots\!38\)\( \nu^{2} - \)\(14\!\cdots\!39\)\( \nu - \)\(44\!\cdots\!50\)\(\)\()/ \)\(46\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(12\!\cdots\!73\)\( \nu^{15} - \)\(24\!\cdots\!50\)\( \nu^{14} + \)\(51\!\cdots\!23\)\( \nu^{13} - \)\(26\!\cdots\!66\)\( \nu^{12} + \)\(45\!\cdots\!27\)\( \nu^{11} - \)\(16\!\cdots\!26\)\( \nu^{10} + \)\(69\!\cdots\!91\)\( \nu^{9} - \)\(10\!\cdots\!74\)\( \nu^{8} + \)\(39\!\cdots\!73\)\( \nu^{7} - \)\(12\!\cdots\!26\)\( \nu^{6} + \)\(17\!\cdots\!49\)\( \nu^{5} - \)\(39\!\cdots\!46\)\( \nu^{4} + \)\(12\!\cdots\!61\)\( \nu^{3} - \)\(20\!\cdots\!02\)\( \nu^{2} + \)\(32\!\cdots\!49\)\( \nu - \)\(71\!\cdots\!10\)\(\)\()/ \)\(93\!\cdots\!60\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(14\!\cdots\!79\)\( \nu^{15} - \)\(32\!\cdots\!90\)\( \nu^{14} + \)\(14\!\cdots\!77\)\( \nu^{13} - \)\(17\!\cdots\!22\)\( \nu^{12} + \)\(59\!\cdots\!81\)\( \nu^{11} - \)\(11\!\cdots\!50\)\( \nu^{10} + \)\(63\!\cdots\!77\)\( \nu^{9} - \)\(33\!\cdots\!22\)\( \nu^{8} + \)\(92\!\cdots\!71\)\( \nu^{7} - \)\(16\!\cdots\!50\)\( \nu^{6} + \)\(27\!\cdots\!99\)\( \nu^{5} - \)\(12\!\cdots\!02\)\( \nu^{4} + \)\(33\!\cdots\!75\)\( \nu^{3} - \)\(70\!\cdots\!94\)\( \nu^{2} + \)\(10\!\cdots\!39\)\( \nu - \)\(15\!\cdots\!50\)\(\)\()/ \)\(93\!\cdots\!60\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(58\!\cdots\!57\)\( \nu^{15} + \)\(56\!\cdots\!30\)\( \nu^{14} + \)\(90\!\cdots\!77\)\( \nu^{13} + \)\(16\!\cdots\!42\)\( \nu^{12} - \)\(18\!\cdots\!23\)\( \nu^{11} + \)\(72\!\cdots\!38\)\( \nu^{10} - \)\(33\!\cdots\!07\)\( \nu^{9} + \)\(12\!\cdots\!46\)\( \nu^{8} - \)\(11\!\cdots\!61\)\( \nu^{7} + \)\(18\!\cdots\!98\)\( \nu^{6} - \)\(17\!\cdots\!05\)\( \nu^{5} + \)\(64\!\cdots\!02\)\( \nu^{4} - \)\(47\!\cdots\!53\)\( \nu^{3} + \)\(10\!\cdots\!54\)\( \nu^{2} - \)\(26\!\cdots\!85\)\( \nu + \)\(51\!\cdots\!70\)\(\)\()/ \)\(31\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(26\!\cdots\!55\)\( \nu^{15} + \)\(61\!\cdots\!62\)\( \nu^{14} - \)\(37\!\cdots\!71\)\( \nu^{13} - \)\(29\!\cdots\!70\)\( \nu^{12} - \)\(31\!\cdots\!03\)\( \nu^{11} + \)\(29\!\cdots\!66\)\( \nu^{10} - \)\(22\!\cdots\!15\)\( \nu^{9} + \)\(28\!\cdots\!98\)\( \nu^{8} - \)\(99\!\cdots\!73\)\( \nu^{7} + \)\(65\!\cdots\!58\)\( \nu^{6} - \)\(50\!\cdots\!61\)\( \nu^{5} + \)\(45\!\cdots\!78\)\( \nu^{4} - \)\(22\!\cdots\!17\)\( \nu^{3} + \)\(28\!\cdots\!98\)\( \nu^{2} + \)\(30\!\cdots\!55\)\( \nu - \)\(12\!\cdots\!90\)\(\)\()/ \)\(93\!\cdots\!60\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(28\!\cdots\!85\)\( \nu^{15} + \)\(14\!\cdots\!42\)\( \nu^{14} + \)\(20\!\cdots\!51\)\( \nu^{13} + \)\(74\!\cdots\!94\)\( \nu^{12} + \)\(13\!\cdots\!27\)\( \nu^{11} - \)\(16\!\cdots\!82\)\( \nu^{10} + \)\(17\!\cdots\!31\)\( \nu^{9} + \)\(40\!\cdots\!78\)\( \nu^{8} - \)\(22\!\cdots\!95\)\( \nu^{7} - \)\(17\!\cdots\!30\)\( \nu^{6} - \)\(87\!\cdots\!03\)\( \nu^{5} - \)\(25\!\cdots\!38\)\( \nu^{4} - \)\(17\!\cdots\!19\)\( \nu^{3} + \)\(58\!\cdots\!86\)\( \nu^{2} - \)\(14\!\cdots\!27\)\( \nu - \)\(50\!\cdots\!90\)\(\)\()/ \)\(93\!\cdots\!60\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(37\!\cdots\!81\)\( \nu^{15} + \)\(24\!\cdots\!66\)\( \nu^{14} - \)\(82\!\cdots\!37\)\( \nu^{13} + \)\(65\!\cdots\!18\)\( \nu^{12} - \)\(38\!\cdots\!65\)\( \nu^{11} + \)\(62\!\cdots\!86\)\( \nu^{10} - \)\(14\!\cdots\!93\)\( \nu^{9} + \)\(46\!\cdots\!26\)\( \nu^{8} - \)\(18\!\cdots\!03\)\( \nu^{7} + \)\(57\!\cdots\!22\)\( \nu^{6} - \)\(10\!\cdots\!15\)\( \nu^{5} + \)\(17\!\cdots\!42\)\( \nu^{4} - \)\(33\!\cdots\!39\)\( \nu^{3} + \)\(11\!\cdots\!30\)\( \nu^{2} - \)\(38\!\cdots\!87\)\( \nu + \)\(43\!\cdots\!50\)\(\)\()/ \)\(93\!\cdots\!60\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(48\!\cdots\!73\)\( \nu^{15} + \)\(48\!\cdots\!58\)\( \nu^{14} + \)\(62\!\cdots\!35\)\( \nu^{13} + \)\(10\!\cdots\!66\)\( \nu^{12} + \)\(32\!\cdots\!15\)\( \nu^{11} - \)\(51\!\cdots\!06\)\( \nu^{10} - \)\(10\!\cdots\!41\)\( \nu^{9} - \)\(89\!\cdots\!62\)\( \nu^{8} + \)\(38\!\cdots\!65\)\( \nu^{7} - \)\(53\!\cdots\!18\)\( \nu^{6} + \)\(99\!\cdots\!89\)\( \nu^{5} - \)\(19\!\cdots\!02\)\( \nu^{4} + \)\(41\!\cdots\!97\)\( \nu^{3} - \)\(13\!\cdots\!46\)\( \nu^{2} + \)\(37\!\cdots\!13\)\( \nu - \)\(75\!\cdots\!50\)\(\)\()/ \)\(31\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - \beta_{2} - 23 \beta_{1} - 28487\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{8} + \beta_{6} + 7 \beta_{5} - 7 \beta_{4} - 40 \beta_{3} + 364 \beta_{2} - 29939 \beta_{1} + 21248286\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{15} - 14 \beta_{14} - 26 \beta_{13} - 20 \beta_{12} + 21 \beta_{11} - 37 \beta_{10} + 14 \beta_{9} + 21 \beta_{8} + 338 \beta_{7} - 550 \beta_{6} - 6142 \beta_{5} + 1494 \beta_{4} - 33766 \beta_{3} - 172136 \beta_{2} + 23435883 \beta_{1} - 14107139986\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(49 \beta_{15} + 685 \beta_{14} - 574 \beta_{13} + 443 \beta_{12} + 294 \beta_{11} + 154 \beta_{10} + 98 \beta_{9} - 6622 \beta_{8} - 6722 \beta_{7} + 8682 \beta_{6} + 178142 \beta_{5} - 586498 \beta_{4} + 1902368 \beta_{3} + 6162555 \beta_{2} - 985830109 \beta_{1} + 658313469092\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(1596 \beta_{15} + 585492 \beta_{14} + 55590 \beta_{13} - 469138 \beta_{12} - 350217 \beta_{11} - 280807 \beta_{10} - 164482 \beta_{9} + 2197126 \beta_{8} - 8578078 \beta_{7} - 15338071 \beta_{6} + 16965091 \beta_{5} + 15039853 \beta_{4} - 1101189579 \beta_{3} + 18825716223 \beta_{2} + 722502344999 \beta_{1} - 314443661819751\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-53764410 \beta_{15} - 282908826 \beta_{14} - 46722882 \beta_{13} - 108646456 \beta_{12} + 57400853 \beta_{11} + 424036955 \beta_{10} - 69085274 \beta_{9} - 819026082 \beta_{8} - 165605574 \beta_{7} - 19943154805 \beta_{6} - 46717885799 \beta_{5} + 48257098543 \beta_{4} + 750217122010 \beta_{3} - 506105335224 \beta_{2} - 328104652597688 \beta_{1} - 54900969805708496\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-18960265230 \beta_{15} + 1916947458 \beta_{14} + 63902236750 \beta_{13} + 85424542612 \beta_{12} + 68457631001 \beta_{11} - 101191129961 \beta_{10} + 35821370390 \beta_{9} + 363719243005 \beta_{8} - 1261116608566 \beta_{7} - 71950621610 \beta_{6} + 12283999974614 \beta_{5} - 69286013820350 \beta_{4} - 278511235871750 \beta_{3} + 3164851750303880 \beta_{2} - 57106121646628317 \beta_{1} + 116979372641391738622\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(-521684003538 \beta_{15} + 11486427320766 \beta_{14} - 573087780438 \beta_{13} - 24966914404044 \beta_{12} - 13433733301725 \beta_{11} - 29906084712339 \beta_{10} + 6470068889410 \beta_{9} - 45962473250625 \beta_{8} - 24600712683426 \beta_{7} - 485668141947982 \beta_{6} + 526989722824818 \beta_{5} + 5338708321125494 \beta_{4} - 20744332616111058 \beta_{3} - 736492389047486144 \beta_{2} + 30152256563989010133 \beta_{1} - 7337295015705543331226\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-290548649560470 \beta_{15} - 1059065818993270 \beta_{14} - 365078937159998 \beta_{13} + 987746867301032 \beta_{12} - 589366266076741 \beta_{11} + 6531336741795541 \beta_{10} - 350129990820518 \beta_{9} - 7989042956121415 \beta_{8} - 24742005175789626 \beta_{7} + 207300150961799548 \beta_{6} + 106778382750492952 \beta_{5} - 567209823690642528 \beta_{4} + 7247732911996001707 \beta_{3} - 52348635557813358881 \beta_{2} - 1977429472511683545104 \beta_{1} - 735962758336797326003649\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(203704740535313116 \beta_{15} - 455366842462657060 \beta_{14} + 695348843181429596 \beta_{13} - 1148399344625115712 \beta_{12} - 764873108938276694 \beta_{11} - 2064786517473938890 \beta_{10} - 15599305539912660 \beta_{9} + 8799784449435952627 \beta_{8} - 20416951446533730412 \beta_{7} + 32572567014303089901 \beta_{6} - 114482633639011526893 \beta_{5} - 112188685079062847683 \beta_{4} - 1948139540077710383780 \beta_{3} + 39989737823098732510580 \beta_{2} - 669798878257773844425669 \beta_{1} - 16178208115319925956433822\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(-30531198687422943342 \beta_{15} - 565022429758710807102 \beta_{14} - 162445388444376459290 \beta_{13} - 673499482041608816964 \beta_{12} + 248848003526911205 \beta_{11} + 364474620646837694475 \beta_{10} - 201873262225369648242 \beta_{9} - 1509918240430083592547 \beta_{8} + 16702250946530522443218 \beta_{7} - 31037162852797393911918 \beta_{6} - 33936385655609580637494 \beta_{5} + 59775045659350676479950 \beta_{4} - 641335986845214224926614 \beta_{3} - 28250339361839066511873384 \beta_{2} - 34519455406244747102737789 \beta_{1} + 272252999060435273843890816830\)\()/16\)
\(\nu^{13}\)\(=\)\((\)\(-380696555214739268216 \beta_{15} + 41463590418026211815584 \beta_{14} + 3910136793747879730754 \beta_{13} + 34808714131111997969182 \beta_{12} + 62345582795100705715137 \beta_{11} + 102085684925818138955311 \beta_{10} + 8191025309018448687978 \beta_{9} - 214768799843655835804295 \beta_{8} + 15339576750118853152694 \beta_{7} + 2992729881980550212011222 \beta_{6} + 2820536944977396673693206 \beta_{5} - 18968774714917030011624726 \beta_{4} + 7620945924028040189737746 \beta_{3} + 1460475311548866545117691778 \beta_{2} + 34510659316774651723986820521 \beta_{1} + 672245436590552209780154125626\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(1850072468264186124749292 \beta_{15} + 9049193699603647382571668 \beta_{14} + 2232011672989867144673826 \beta_{13} - 8793648264611295959132558 \beta_{12} + 806750280325613037670677 \beta_{11} - 20050979752037265351073765 \beta_{10} - 1370414710262694452355798 \beta_{9} + 7601098637591281363386286 \beta_{8} - 214220208030279166107201034 \beta_{7} - 629543512575508142002038217 \beta_{6} + 1245062976539100804311596685 \beta_{5} + 4599852059754259750117582851 \beta_{4} + 13791115983397624495300516173 \beta_{3} - 218387123138162195561853207961 \beta_{2} + 82793941083405400858278924335 \beta_{1} - 2642555275615170574522537718181815\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-1442125312308807492598167030 \beta_{15} - 7905422684471875153747879830 \beta_{14} - 1028669704520420743071762606 \beta_{13} + 3252982211161229018588618488 \beta_{12} + 836628871665743144247524931 \beta_{11} + 5274521425335804265882183117 \beta_{10} - 3219220341795311713223380694 \beta_{9} + 17276644597424505740556727796 \beta_{8} + 76159730564206564427496829174 \beta_{7} + 257030160006525637740002896735 \beta_{6} + 601036582124184592343274882413 \beta_{5} + 503739869406816442015875948683 \beta_{4} - 1801324712654039632760892272058 \beta_{3} - 41962203414453645627637375773168 \beta_{2} - 2439729706193109187971965724163022 \beta_{1} - 2162863718636745510450719480654138772\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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
−226.199 141.712i
−226.199 + 141.712i
−195.992 179.025i
−195.992 + 179.025i
−162.812 207.587i
−162.812 + 207.587i
−13.0246 256.000i
−13.0246 + 256.000i
24.2392 253.277i
24.2392 + 253.277i
135.604 208.454i
135.604 + 208.454i
202.692 137.887i
202.692 + 137.887i
237.992 50.3904i
237.992 + 50.3904i
−426.398 283.424i 35846.9 101486. + 241703.i 3.47680e6i −1.52851e7 1.01599e7i 1.08296e7i 2.52310e7 1.31825e8i 8.97583e8 −9.85407e8 + 1.48250e9i
3.2 −426.398 + 283.424i 35846.9 101486. 241703.i 3.47680e6i −1.52851e7 + 1.01599e7i 1.08296e7i 2.52310e7 + 1.31825e8i 8.97583e8 −9.85407e8 1.48250e9i
3.3 −365.984 358.050i −33476.4 5744.83 + 262081.i 436282.i 1.22518e7 + 1.19862e7i 5.56713e7i 9.17355e7 9.79745e7i 7.33247e8 −1.56211e8 + 1.59672e8i
3.4 −365.984 + 358.050i −33476.4 5744.83 262081.i 436282.i 1.22518e7 1.19862e7i 5.56713e7i 9.17355e7 + 9.79745e7i 7.33247e8 −1.56211e8 1.59672e8i
3.5 −299.623 415.174i 5937.76 −82595.7 + 248792.i 2.28157e6i −1.77909e6 2.46521e6i 3.15838e7i 1.28040e8 4.02522e7i −3.52164e8 9.47248e8 6.83611e8i
3.6 −299.623 + 415.174i 5937.76 −82595.7 248792.i 2.28157e6i −1.77909e6 + 2.46521e6i 3.15838e7i 1.28040e8 + 4.02522e7i −3.52164e8 9.47248e8 + 6.83611e8i
3.7 −0.0491315 512.000i −9978.36 −262144. + 50.3107i 3.00573e6i 490.252 + 5.10892e6i 5.44315e7i 38638.6 + 1.34218e8i −2.87853e8 −1.53893e9 + 147676.i
3.8 −0.0491315 + 512.000i −9978.36 −262144. 50.3107i 3.00573e6i 490.252 5.10892e6i 5.44315e7i 38638.6 1.34218e8i −2.87853e8 −1.53893e9 147676.i
3.9 74.4785 506.554i 19252.8 −251050. 75454.8i 901770.i 1.43392e6 9.75260e6i 6.41362e7i −5.69197e7 + 1.21551e8i −1.67489e7 4.56795e8 + 6.71625e7i
3.10 74.4785 + 506.554i 19252.8 −251050. + 75454.8i 901770.i 1.43392e6 + 9.75260e6i 6.41362e7i −5.69197e7 1.21551e8i −1.67489e7 4.56795e8 6.71625e7i
3.11 297.207 416.907i −25925.7 −85479.4 247816.i 1.90314e6i −7.70530e6 + 1.08086e7i 9.33592e6i −1.28721e8 3.80157e7i 2.84720e8 7.93431e8 + 5.65626e8i
3.12 297.207 + 416.907i −25925.7 −85479.4 + 247816.i 1.90314e6i −7.70530e6 1.08086e7i 9.33592e6i −1.28721e8 + 3.80157e7i 2.84720e8 7.93431e8 5.65626e8i
3.13 431.385 275.773i 22640.1 110042. 237929.i 113148.i 9.76661e6 6.24355e6i 3.55480e7i −1.81440e7 1.32986e8i 1.25155e8 −3.12032e7 4.88104e7i
3.14 431.385 + 275.773i 22640.1 110042. + 237929.i 113148.i 9.76661e6 + 6.24355e6i 3.55480e7i −1.81440e7 + 1.32986e8i 1.25155e8 −3.12032e7 + 4.88104e7i
3.15 501.983 100.781i −12665.3 241830. 101181.i 2.68543e6i −6.35775e6 + 1.27642e6i 4.44900e7i 1.11198e8 7.51629e7i −2.27011e8 −2.70640e8 1.34804e9i
3.16 501.983 + 100.781i −12665.3 241830. + 101181.i 2.68543e6i −6.35775e6 1.27642e6i 4.44900e7i 1.11198e8 + 7.51629e7i −2.27011e8 −2.70640e8 + 1.34804e9i
\(n\): 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
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.19.d.b 16
3.b odd 2 1 72.19.b.b 16
4.b odd 2 1 32.19.d.b 16
8.b even 2 1 32.19.d.b 16
8.d odd 2 1 inner 8.19.d.b 16
24.f even 2 1 72.19.b.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.19.d.b 16 1.a even 1 1 trivial
8.19.d.b 16 8.d odd 2 1 inner
32.19.d.b 16 4.b odd 2 1
32.19.d.b 16 8.b even 2 1
72.19.b.b 16 3.b odd 2 1
72.19.b.b 16 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{19}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 426 T + 312904 T^{2} - 209165760 T^{3} + 137587907584 T^{4} - 94074125451264 T^{5} + 56978306756509696 T^{6} - 27162432886580183040 T^{7} + \)\(12\!\cdots\!40\)\( T^{8} - \)\(71\!\cdots\!60\)\( T^{9} + \)\(39\!\cdots\!56\)\( T^{10} - \)\(16\!\cdots\!76\)\( T^{11} + \)\(64\!\cdots\!64\)\( T^{12} - \)\(25\!\cdots\!40\)\( T^{13} + \)\(10\!\cdots\!24\)\( T^{14} - \)\(36\!\cdots\!64\)\( T^{15} + \)\(22\!\cdots\!16\)\( T^{16} \)
$3$ \( ( 1 - 1632 T + 972549624 T^{2} - 3302603583264 T^{3} + 575343743591230812 T^{4} - \)\(28\!\cdots\!92\)\( T^{5} + \)\(25\!\cdots\!80\)\( T^{6} - \)\(19\!\cdots\!44\)\( T^{7} + \)\(10\!\cdots\!86\)\( T^{8} - \)\(75\!\cdots\!16\)\( T^{9} + \)\(37\!\cdots\!80\)\( T^{10} - \)\(16\!\cdots\!48\)\( T^{11} + \)\(12\!\cdots\!92\)\( T^{12} - \)\(28\!\cdots\!36\)\( T^{13} + \)\(32\!\cdots\!64\)\( T^{14} - \)\(21\!\cdots\!28\)\( T^{15} + \)\(50\!\cdots\!81\)\( T^{16} )^{2} \)
$5$ \( 1 - 22857323554960 T^{2} + \)\(27\!\cdots\!00\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{8} - \)\(93\!\cdots\!00\)\( T^{10} + \)\(47\!\cdots\!00\)\( T^{12} - \)\(21\!\cdots\!00\)\( T^{14} + \)\(84\!\cdots\!50\)\( T^{16} - \)\(30\!\cdots\!00\)\( T^{18} + \)\(99\!\cdots\!00\)\( T^{20} - \)\(28\!\cdots\!00\)\( T^{22} + \)\(73\!\cdots\!00\)\( T^{24} - \)\(15\!\cdots\!00\)\( T^{26} + \)\(26\!\cdots\!00\)\( T^{28} - \)\(31\!\cdots\!00\)\( T^{30} + \)\(20\!\cdots\!25\)\( T^{32} \)
$7$ \( 1 - 11434100931217168 T^{2} + \)\(70\!\cdots\!64\)\( T^{4} - \)\(30\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!24\)\( T^{8} - \)\(29\!\cdots\!36\)\( T^{10} + \)\(69\!\cdots\!20\)\( T^{12} - \)\(13\!\cdots\!12\)\( T^{14} + \)\(24\!\cdots\!42\)\( T^{16} - \)\(36\!\cdots\!12\)\( T^{18} + \)\(48\!\cdots\!20\)\( T^{20} - \)\(54\!\cdots\!36\)\( T^{22} + \)\(52\!\cdots\!24\)\( T^{24} - \)\(40\!\cdots\!04\)\( T^{26} + \)\(24\!\cdots\!64\)\( T^{28} - \)\(10\!\cdots\!68\)\( T^{30} + \)\(24\!\cdots\!01\)\( T^{32} \)
$11$ \( ( 1 + 1235785632 T + 21880493500193963128 T^{2} + \)\(34\!\cdots\!88\)\( T^{3} + \)\(26\!\cdots\!48\)\( T^{4} + \)\(45\!\cdots\!28\)\( T^{5} + \)\(23\!\cdots\!08\)\( T^{6} + \)\(37\!\cdots\!28\)\( T^{7} + \)\(14\!\cdots\!78\)\( T^{8} + \)\(20\!\cdots\!68\)\( T^{9} + \)\(71\!\cdots\!88\)\( T^{10} + \)\(78\!\cdots\!48\)\( T^{11} + \)\(25\!\cdots\!08\)\( T^{12} + \)\(18\!\cdots\!88\)\( T^{13} + \)\(64\!\cdots\!68\)\( T^{14} + \)\(20\!\cdots\!52\)\( T^{15} + \)\(91\!\cdots\!41\)\( T^{16} )^{2} \)
$13$ \( 1 - \)\(77\!\cdots\!48\)\( T^{2} + \)\(28\!\cdots\!44\)\( T^{4} - \)\(63\!\cdots\!24\)\( T^{6} + \)\(89\!\cdots\!44\)\( T^{8} - \)\(67\!\cdots\!16\)\( T^{10} - \)\(22\!\cdots\!80\)\( T^{12} + \)\(14\!\cdots\!28\)\( T^{14} - \)\(22\!\cdots\!98\)\( T^{16} + \)\(18\!\cdots\!48\)\( T^{18} - \)\(35\!\cdots\!80\)\( T^{20} - \)\(13\!\cdots\!36\)\( T^{22} + \)\(22\!\cdots\!84\)\( T^{24} - \)\(20\!\cdots\!24\)\( T^{26} + \)\(11\!\cdots\!04\)\( T^{28} - \)\(40\!\cdots\!88\)\( T^{30} + \)\(65\!\cdots\!21\)\( T^{32} \)
$17$ \( ( 1 + 88219650672 T + \)\(67\!\cdots\!44\)\( T^{2} + \)\(44\!\cdots\!44\)\( T^{3} + \)\(19\!\cdots\!52\)\( T^{4} + \)\(89\!\cdots\!12\)\( T^{5} + \)\(32\!\cdots\!80\)\( T^{6} + \)\(10\!\cdots\!64\)\( T^{7} + \)\(44\!\cdots\!66\)\( T^{8} + \)\(14\!\cdots\!76\)\( T^{9} + \)\(64\!\cdots\!80\)\( T^{10} + \)\(24\!\cdots\!48\)\( T^{11} + \)\(75\!\cdots\!72\)\( T^{12} + \)\(24\!\cdots\!56\)\( T^{13} + \)\(52\!\cdots\!04\)\( T^{14} + \)\(95\!\cdots\!68\)\( T^{15} + \)\(15\!\cdots\!21\)\( T^{16} )^{2} \)
$19$ \( ( 1 + 416682817184 T + \)\(50\!\cdots\!04\)\( T^{2} + \)\(11\!\cdots\!92\)\( T^{3} + \)\(97\!\cdots\!44\)\( T^{4} + \)\(12\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!12\)\( T^{6} + \)\(99\!\cdots\!16\)\( T^{7} + \)\(14\!\cdots\!18\)\( T^{8} + \)\(10\!\cdots\!56\)\( T^{9} + \)\(13\!\cdots\!72\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{11} + \)\(11\!\cdots\!84\)\( T^{12} + \)\(14\!\cdots\!92\)\( T^{13} + \)\(64\!\cdots\!64\)\( T^{14} + \)\(55\!\cdots\!04\)\( T^{15} + \)\(13\!\cdots\!21\)\( T^{16} )^{2} \)
$23$ \( 1 - \)\(24\!\cdots\!68\)\( T^{2} + \)\(30\!\cdots\!84\)\( T^{4} - \)\(25\!\cdots\!24\)\( T^{6} + \)\(16\!\cdots\!24\)\( T^{8} - \)\(87\!\cdots\!76\)\( T^{10} + \)\(39\!\cdots\!20\)\( T^{12} - \)\(15\!\cdots\!32\)\( T^{14} + \)\(53\!\cdots\!02\)\( T^{16} - \)\(16\!\cdots\!52\)\( T^{18} + \)\(43\!\cdots\!20\)\( T^{20} - \)\(10\!\cdots\!56\)\( T^{22} + \)\(20\!\cdots\!84\)\( T^{24} - \)\(33\!\cdots\!24\)\( T^{26} + \)\(41\!\cdots\!24\)\( T^{28} - \)\(34\!\cdots\!28\)\( T^{30} + \)\(15\!\cdots\!81\)\( T^{32} \)
$29$ \( 1 - \)\(15\!\cdots\!16\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{4} - \)\(69\!\cdots\!40\)\( T^{6} + \)\(27\!\cdots\!80\)\( T^{8} - \)\(88\!\cdots\!08\)\( T^{10} + \)\(23\!\cdots\!68\)\( T^{12} - \)\(55\!\cdots\!60\)\( T^{14} + \)\(12\!\cdots\!30\)\( T^{16} - \)\(24\!\cdots\!60\)\( T^{18} + \)\(46\!\cdots\!88\)\( T^{20} - \)\(76\!\cdots\!88\)\( T^{22} + \)\(10\!\cdots\!80\)\( T^{24} - \)\(11\!\cdots\!40\)\( T^{26} + \)\(97\!\cdots\!80\)\( T^{28} - \)\(53\!\cdots\!56\)\( T^{30} + \)\(14\!\cdots\!61\)\( T^{32} \)
$31$ \( 1 - \)\(41\!\cdots\!76\)\( T^{2} + \)\(98\!\cdots\!00\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(22\!\cdots\!00\)\( T^{8} - \)\(25\!\cdots\!88\)\( T^{10} + \)\(24\!\cdots\!68\)\( T^{12} - \)\(20\!\cdots\!00\)\( T^{14} + \)\(15\!\cdots\!10\)\( T^{16} - \)\(99\!\cdots\!00\)\( T^{18} + \)\(57\!\cdots\!48\)\( T^{20} - \)\(29\!\cdots\!08\)\( T^{22} + \)\(12\!\cdots\!00\)\( T^{24} - \)\(46\!\cdots\!80\)\( T^{26} + \)\(13\!\cdots\!00\)\( T^{28} - \)\(27\!\cdots\!36\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \)
$37$ \( 1 - \)\(12\!\cdots\!28\)\( T^{2} + \)\(79\!\cdots\!84\)\( T^{4} - \)\(31\!\cdots\!64\)\( T^{6} + \)\(85\!\cdots\!04\)\( T^{8} - \)\(16\!\cdots\!36\)\( T^{10} + \)\(22\!\cdots\!20\)\( T^{12} - \)\(21\!\cdots\!72\)\( T^{14} + \)\(22\!\cdots\!22\)\( T^{16} - \)\(61\!\cdots\!52\)\( T^{18} + \)\(18\!\cdots\!20\)\( T^{20} - \)\(38\!\cdots\!56\)\( T^{22} + \)\(56\!\cdots\!44\)\( T^{24} - \)\(59\!\cdots\!64\)\( T^{26} + \)\(42\!\cdots\!44\)\( T^{28} - \)\(19\!\cdots\!68\)\( T^{30} + \)\(43\!\cdots\!21\)\( T^{32} \)
$41$ \( ( 1 + 297465781222512 T + \)\(49\!\cdots\!68\)\( T^{2} + \)\(96\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!48\)\( T^{4} + \)\(17\!\cdots\!88\)\( T^{5} + \)\(18\!\cdots\!88\)\( T^{6} + \)\(26\!\cdots\!68\)\( T^{7} + \)\(23\!\cdots\!98\)\( T^{8} + \)\(28\!\cdots\!28\)\( T^{9} + \)\(21\!\cdots\!08\)\( T^{10} + \)\(21\!\cdots\!68\)\( T^{11} + \)\(15\!\cdots\!88\)\( T^{12} + \)\(13\!\cdots\!28\)\( T^{13} + \)\(74\!\cdots\!28\)\( T^{14} + \)\(48\!\cdots\!92\)\( T^{15} + \)\(17\!\cdots\!61\)\( T^{16} )^{2} \)
$43$ \( ( 1 + 1270077508788896 T + \)\(19\!\cdots\!04\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!64\)\( T^{4} + \)\(87\!\cdots\!64\)\( T^{5} + \)\(59\!\cdots\!96\)\( T^{6} + \)\(29\!\cdots\!20\)\( T^{7} + \)\(17\!\cdots\!70\)\( T^{8} + \)\(74\!\cdots\!80\)\( T^{9} + \)\(38\!\cdots\!96\)\( T^{10} + \)\(14\!\cdots\!36\)\( T^{11} + \)\(59\!\cdots\!64\)\( T^{12} + \)\(16\!\cdots\!60\)\( T^{13} + \)\(51\!\cdots\!04\)\( T^{14} + \)\(83\!\cdots\!04\)\( T^{15} + \)\(16\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( 1 - \)\(74\!\cdots\!28\)\( T^{2} + \)\(29\!\cdots\!24\)\( T^{4} - \)\(82\!\cdots\!84\)\( T^{6} + \)\(18\!\cdots\!64\)\( T^{8} - \)\(33\!\cdots\!16\)\( T^{10} + \)\(53\!\cdots\!60\)\( T^{12} - \)\(76\!\cdots\!92\)\( T^{14} + \)\(10\!\cdots\!82\)\( T^{16} - \)\(12\!\cdots\!32\)\( T^{18} + \)\(13\!\cdots\!60\)\( T^{20} - \)\(12\!\cdots\!76\)\( T^{22} + \)\(11\!\cdots\!84\)\( T^{24} - \)\(78\!\cdots\!84\)\( T^{26} + \)\(43\!\cdots\!04\)\( T^{28} - \)\(17\!\cdots\!48\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$53$ \( 1 - \)\(13\!\cdots\!88\)\( T^{2} + \)\(92\!\cdots\!64\)\( T^{4} - \)\(39\!\cdots\!84\)\( T^{6} + \)\(12\!\cdots\!04\)\( T^{8} - \)\(28\!\cdots\!76\)\( T^{10} + \)\(51\!\cdots\!40\)\( T^{12} - \)\(76\!\cdots\!12\)\( T^{14} + \)\(92\!\cdots\!82\)\( T^{16} - \)\(90\!\cdots\!52\)\( T^{18} + \)\(72\!\cdots\!40\)\( T^{20} - \)\(46\!\cdots\!36\)\( T^{22} + \)\(23\!\cdots\!24\)\( T^{24} - \)\(92\!\cdots\!84\)\( T^{26} + \)\(25\!\cdots\!44\)\( T^{28} - \)\(45\!\cdots\!08\)\( T^{30} + \)\(39\!\cdots\!61\)\( T^{32} \)
$59$ \( ( 1 - 278403943157088 T + \)\(27\!\cdots\!64\)\( T^{2} + \)\(74\!\cdots\!76\)\( T^{3} + \)\(38\!\cdots\!24\)\( T^{4} + \)\(21\!\cdots\!84\)\( T^{5} + \)\(36\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!08\)\( T^{7} + \)\(28\!\cdots\!02\)\( T^{8} + \)\(21\!\cdots\!68\)\( T^{9} + \)\(20\!\cdots\!00\)\( T^{10} + \)\(88\!\cdots\!24\)\( T^{11} + \)\(12\!\cdots\!44\)\( T^{12} + \)\(17\!\cdots\!76\)\( T^{13} + \)\(49\!\cdots\!44\)\( T^{14} - \)\(37\!\cdots\!08\)\( T^{15} + \)\(10\!\cdots\!61\)\( T^{16} )^{2} \)
$61$ \( 1 - \)\(11\!\cdots\!36\)\( T^{2} + \)\(64\!\cdots\!00\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{6} + \)\(69\!\cdots\!00\)\( T^{8} - \)\(15\!\cdots\!08\)\( T^{10} + \)\(30\!\cdots\!28\)\( T^{12} - \)\(49\!\cdots\!00\)\( T^{14} + \)\(71\!\cdots\!50\)\( T^{16} - \)\(92\!\cdots\!00\)\( T^{18} + \)\(10\!\cdots\!88\)\( T^{20} - \)\(10\!\cdots\!48\)\( T^{22} + \)\(84\!\cdots\!00\)\( T^{24} - \)\(56\!\cdots\!00\)\( T^{26} + \)\(27\!\cdots\!00\)\( T^{28} - \)\(91\!\cdots\!56\)\( T^{30} + \)\(14\!\cdots\!81\)\( T^{32} \)
$67$ \( ( 1 - 81214287467306848 T + \)\(59\!\cdots\!64\)\( T^{2} - \)\(25\!\cdots\!76\)\( T^{3} + \)\(10\!\cdots\!12\)\( T^{4} - \)\(30\!\cdots\!28\)\( T^{5} + \)\(95\!\cdots\!80\)\( T^{6} - \)\(22\!\cdots\!76\)\( T^{7} + \)\(69\!\cdots\!06\)\( T^{8} - \)\(16\!\cdots\!84\)\( T^{9} + \)\(52\!\cdots\!80\)\( T^{10} - \)\(12\!\cdots\!12\)\( T^{11} + \)\(31\!\cdots\!32\)\( T^{12} - \)\(56\!\cdots\!24\)\( T^{13} + \)\(98\!\cdots\!24\)\( T^{14} - \)\(98\!\cdots\!12\)\( T^{15} + \)\(90\!\cdots\!21\)\( T^{16} )^{2} \)
$71$ \( 1 - \)\(15\!\cdots\!76\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{4} - \)\(74\!\cdots\!80\)\( T^{6} + \)\(31\!\cdots\!00\)\( T^{8} - \)\(10\!\cdots\!68\)\( T^{10} + \)\(31\!\cdots\!88\)\( T^{12} - \)\(79\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!10\)\( T^{16} - \)\(35\!\cdots\!00\)\( T^{18} + \)\(61\!\cdots\!08\)\( T^{20} - \)\(94\!\cdots\!48\)\( T^{22} + \)\(12\!\cdots\!00\)\( T^{24} - \)\(12\!\cdots\!80\)\( T^{26} + \)\(97\!\cdots\!00\)\( T^{28} - \)\(52\!\cdots\!16\)\( T^{30} + \)\(14\!\cdots\!61\)\( T^{32} \)
$73$ \( ( 1 + 3827856426246128 T + \)\(13\!\cdots\!04\)\( T^{2} + \)\(29\!\cdots\!36\)\( T^{3} + \)\(75\!\cdots\!32\)\( T^{4} + \)\(40\!\cdots\!48\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!56\)\( T^{7} + \)\(89\!\cdots\!86\)\( T^{8} + \)\(87\!\cdots\!64\)\( T^{9} + \)\(32\!\cdots\!00\)\( T^{10} + \)\(16\!\cdots\!32\)\( T^{11} + \)\(10\!\cdots\!72\)\( T^{12} + \)\(14\!\cdots\!64\)\( T^{13} + \)\(23\!\cdots\!24\)\( T^{14} + \)\(22\!\cdots\!92\)\( T^{15} + \)\(20\!\cdots\!41\)\( T^{16} )^{2} \)
$79$ \( 1 - \)\(79\!\cdots\!96\)\( T^{2} + \)\(37\!\cdots\!40\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!40\)\( T^{8} - \)\(81\!\cdots\!88\)\( T^{10} + \)\(16\!\cdots\!28\)\( T^{12} - \)\(27\!\cdots\!80\)\( T^{14} + \)\(42\!\cdots\!50\)\( T^{16} - \)\(56\!\cdots\!80\)\( T^{18} + \)\(68\!\cdots\!48\)\( T^{20} - \)\(71\!\cdots\!68\)\( T^{22} + \)\(64\!\cdots\!40\)\( T^{24} - \)\(48\!\cdots\!00\)\( T^{26} + \)\(29\!\cdots\!40\)\( T^{28} - \)\(12\!\cdots\!36\)\( T^{30} + \)\(32\!\cdots\!61\)\( T^{32} \)
$83$ \( ( 1 - 433178668069257312 T + \)\(21\!\cdots\!64\)\( T^{2} - \)\(54\!\cdots\!64\)\( T^{3} + \)\(15\!\cdots\!32\)\( T^{4} - \)\(25\!\cdots\!52\)\( T^{5} + \)\(52\!\cdots\!60\)\( T^{6} - \)\(61\!\cdots\!24\)\( T^{7} + \)\(14\!\cdots\!66\)\( T^{8} - \)\(21\!\cdots\!16\)\( T^{9} + \)\(63\!\cdots\!60\)\( T^{10} - \)\(10\!\cdots\!08\)\( T^{11} + \)\(22\!\cdots\!52\)\( T^{12} - \)\(28\!\cdots\!36\)\( T^{13} + \)\(38\!\cdots\!24\)\( T^{14} - \)\(27\!\cdots\!28\)\( T^{15} + \)\(22\!\cdots\!21\)\( T^{16} )^{2} \)
$89$ \( ( 1 + 441823275232142064 T + \)\(55\!\cdots\!64\)\( T^{2} + \)\(21\!\cdots\!12\)\( T^{3} + \)\(16\!\cdots\!64\)\( T^{4} + \)\(57\!\cdots\!68\)\( T^{5} + \)\(32\!\cdots\!32\)\( T^{6} + \)\(10\!\cdots\!96\)\( T^{7} + \)\(46\!\cdots\!18\)\( T^{8} + \)\(12\!\cdots\!76\)\( T^{9} + \)\(48\!\cdots\!52\)\( T^{10} + \)\(10\!\cdots\!88\)\( T^{11} + \)\(37\!\cdots\!44\)\( T^{12} + \)\(60\!\cdots\!12\)\( T^{13} + \)\(19\!\cdots\!84\)\( T^{14} + \)\(18\!\cdots\!04\)\( T^{15} + \)\(51\!\cdots\!41\)\( T^{16} )^{2} \)
$97$ \( ( 1 + 1307447429263390064 T + \)\(36\!\cdots\!84\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} + \)\(54\!\cdots\!44\)\( T^{4} + \)\(36\!\cdots\!56\)\( T^{5} + \)\(51\!\cdots\!16\)\( T^{6} + \)\(29\!\cdots\!20\)\( T^{7} + \)\(35\!\cdots\!30\)\( T^{8} + \)\(17\!\cdots\!80\)\( T^{9} + \)\(17\!\cdots\!36\)\( T^{10} + \)\(70\!\cdots\!64\)\( T^{11} + \)\(60\!\cdots\!04\)\( T^{12} + \)\(20\!\cdots\!20\)\( T^{13} + \)\(13\!\cdots\!24\)\( T^{14} + \)\(28\!\cdots\!56\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} )^{2} \)
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