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 \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(16.4308910168\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{120}\cdot 3^{13}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

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

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 yes

Hecke kernels

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