# Properties

 Label 8.18.b.a Level 8 Weight 18 Character orbit 8.b Analytic conductor 14.658 Analytic rank 0 Dimension 16 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$18$$ Character orbit: $$[\chi]$$ = 8.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$14.6577669876$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{120}\cdot 3^{14}\cdot 7$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 17 - \beta_{1} ) q^{2}$$ $$+ ( 3 \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( -1712 - 17 \beta_{1} - \beta_{2} - \beta_{3} ) q^{4}$$ $$+ ( -6 + 63 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{5} ) q^{5}$$ $$+ ( 365010 + 59 \beta_{1} - 56 \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{6}$$ $$+ ( 721574 - 7724 \beta_{1} - 16 \beta_{2} - 11 \beta_{3} + \beta_{8} ) q^{7}$$ $$+ ( 1520837 + 2001 \beta_{1} - 406 \beta_{2} - 12 \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{6} + \beta_{7} ) q^{8}$$ $$+ ( -37672113 + 49342 \beta_{1} + 96 \beta_{2} + 118 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 17 - \beta_{1} ) q^{2}$$ $$+ ( 3 \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( -1712 - 17 \beta_{1} - \beta_{2} - \beta_{3} ) q^{4}$$ $$+ ( -6 + 63 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{5} ) q^{5}$$ $$+ ( 365010 + 59 \beta_{1} - 56 \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{6}$$ $$+ ( 721574 - 7724 \beta_{1} - 16 \beta_{2} - 11 \beta_{3} + \beta_{8} ) q^{7}$$ $$+ ( 1520837 + 2001 \beta_{1} - 406 \beta_{2} - 12 \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{6} + \beta_{7} ) q^{8}$$ $$+ ( -37672113 + 49342 \beta_{1} + 96 \beta_{2} + 118 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{9}$$ $$+ ( 8187673 + 1160 \beta_{1} - 445 \beta_{2} + 71 \beta_{3} - 5 \beta_{4} - 19 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{13} ) q^{10}$$ $$+ ( -14371 + 106244 \beta_{1} + 2365 \beta_{2} + 718 \beta_{3} + 7 \beta_{4} - 149 \beta_{5} - 10 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{12} ) q^{11}$$ $$+ ( -174642568 - 385855 \beta_{1} - 12081 \beta_{2} + 136 \beta_{3} - 31 \beta_{4} + 138 \beta_{5} + 52 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} + 4 \beta_{11} + \beta_{12} + \beta_{15} ) q^{12}$$ $$+ ( -145744 + 1180147 \beta_{1} - 4135 \beta_{2} - 1048 \beta_{3} + 75 \beta_{4} - 208 \beta_{5} + \beta_{6} + 12 \beta_{7} - \beta_{8} + 2 \beta_{11} + 4 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{13}$$ $$+ ( 1022819755 - 586731 \beta_{1} - 20305 \beta_{2} - 7894 \beta_{3} + 27 \beta_{4} + 618 \beta_{5} + 4 \beta_{6} + 22 \beta_{7} + 102 \beta_{8} - \beta_{9} - \beta_{10} - 20 \beta_{11} + 6 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} - 2 \beta_{15} ) q^{14}$$ $$+ ( -624642853 + 487825 \beta_{1} + 367 \beta_{2} + 7094 \beta_{3} - 27 \beta_{4} + 66 \beta_{5} + 369 \beta_{6} - 132 \beta_{7} + 32 \beta_{8} - \beta_{9} - 2 \beta_{10} + 12 \beta_{11} - 4 \beta_{12} + 24 \beta_{13} - 2 \beta_{14} + 4 \beta_{15} ) q^{15}$$ $$+ ( 1656484605 - 1283650 \beta_{1} - 127143 \beta_{2} + 3095 \beta_{3} + 119 \beta_{4} - 166 \beta_{5} + 316 \beta_{6} + 40 \beta_{7} - 180 \beta_{8} + 2 \beta_{10} - 64 \beta_{11} - 10 \beta_{12} + 8 \beta_{13} - 4 \beta_{14} + 6 \beta_{15} ) q^{16}$$ $$+ ( -468326394 + 2068552 \beta_{1} + 6232 \beta_{2} - 21710 \beta_{3} - 370 \beta_{4} - 165 \beta_{5} - 1766 \beta_{6} - 263 \beta_{7} - 8 \beta_{8} + 6 \beta_{9} + 4 \beta_{10} - 9 \beta_{11} + 8 \beta_{12} - 48 \beta_{13} - 12 \beta_{14} - 8 \beta_{15} ) q^{17}$$ $$+ ( -7095304781 + 36824371 \beta_{1} + 73468 \beta_{2} + 52894 \beta_{3} - 12 \beta_{4} - 9652 \beta_{5} - 132 \beta_{6} - 244 \beta_{7} - 1824 \beta_{8} - 22 \beta_{9} - 18 \beta_{10} + 216 \beta_{11} + 4 \beta_{12} + 12 \beta_{13} - 20 \beta_{14} - 12 \beta_{15} ) q^{18}$$ $$+ ( -1431619 + 11519612 \beta_{1} - 45827 \beta_{2} + 24426 \beta_{3} + 439 \beta_{4} - 6201 \beta_{5} - 4578 \beta_{6} + 762 \beta_{7} + 45 \beta_{8} - 8 \beta_{9} - 32 \beta_{10} + 48 \beta_{11} + 19 \beta_{12} - 32 \beta_{13} - 24 \beta_{14} + 16 \beta_{15} ) q^{19}$$ $$+ ( -13088871766 - 9720610 \beta_{1} - 749208 \beta_{2} + 19626 \beta_{3} + 228 \beta_{4} - 39636 \beta_{5} + 1164 \beta_{6} - 282 \beta_{7} + 3790 \beta_{8} - 8 \beta_{9} + 36 \beta_{10} + 344 \beta_{11} - 46 \beta_{12} + 48 \beta_{13} - 40 \beta_{14} - 14 \beta_{15} ) q^{20}$$ $$+ ( 4347142 - 23889874 \beta_{1} - 3681969 \beta_{2} + 109325 \beta_{3} + 481 \beta_{4} - 13069 \beta_{5} + 25851 \beta_{6} + 1732 \beta_{7} + 357 \beta_{8} + 112 \beta_{9} + 64 \beta_{10} + 86 \beta_{11} + 32 \beta_{12} - 84 \beta_{13} - 43 \beta_{14} + 42 \beta_{15} ) q^{21}$$ $$+ ( 13944711998 + 1506399 \beta_{1} + 1322124 \beta_{2} + 71683 \beta_{3} + 1156 \beta_{4} + 70168 \beta_{5} - 611 \beta_{6} - 536 \beta_{7} + 9624 \beta_{8} - 224 \beta_{9} - 136 \beta_{10} + 96 \beta_{11} - 160 \beta_{12} + 136 \beta_{13} - 48 \beta_{14} + 32 \beta_{15} ) q^{22}$$ $$+ ( 46665420895 + 99184513 \beta_{1} + 246935 \beta_{2} - 245992 \beta_{3} - 2403 \beta_{4} - 8430 \beta_{5} + 42937 \beta_{6} - 1700 \beta_{7} - 614 \beta_{8} + 55 \beta_{9} - 210 \beta_{10} - 148 \beta_{11} + 92 \beta_{12} - 552 \beta_{13} - 82 \beta_{14} - 92 \beta_{15} ) q^{23}$$ $$+ ( -68731151692 + 167832562 \beta_{1} - 8427550 \beta_{2} - 368118 \beta_{3} - 2048 \beta_{4} + 286990 \beta_{5} + 5886 \beta_{6} - 242 \beta_{7} - 19008 \beta_{8} - 176 \beta_{9} + 268 \beta_{10} + 1056 \beta_{11} + 268 \beta_{12} - 144 \beta_{13} - 88 \beta_{14} - 148 \beta_{15} ) q^{24}$$ $$+ ( -113074388407 - 241637026 \beta_{1} - 362840 \beta_{2} - 1848452 \beta_{3} + 3368 \beta_{4} - 15926 \beta_{5} - 96714 \beta_{6} - 766 \beta_{7} + 2734 \beta_{8} + 938 \beta_{9} + 412 \beta_{10} + 398 \beta_{11} - 200 \beta_{12} + 1200 \beta_{13} + 44 \beta_{14} + 200 \beta_{15} ) q^{25}$$ $$+ ( 154229378067 + 19829320 \beta_{1} + 1759441 \beta_{2} + 1304189 \beta_{3} + 441 \beta_{4} - 553385 \beta_{5} + 2038 \beta_{6} + 1459 \beta_{7} - 14678 \beta_{8} - 1408 \beta_{9} - 528 \beta_{10} - 3392 \beta_{11} - 64 \beta_{12} + 45 \beta_{13} + 160 \beta_{14} + 320 \beta_{15} ) q^{26}$$ $$+ ( -57817763 + 343812239 \beta_{1} + 37407628 \beta_{2} + 2897498 \beta_{3} + 5719 \beta_{4} - 5673 \beta_{5} - 239490 \beta_{6} - 9062 \beta_{7} - 707 \beta_{8} + 1112 \beta_{9} - 672 \beta_{10} - 272 \beta_{11} - 125 \beta_{12} + 864 \beta_{13} + 136 \beta_{14} - 432 \beta_{15} ) q^{27}$$ $$+ ( 201406681004 - 999848912 \beta_{1} + 5786804 \beta_{2} - 420412 \beta_{3} - 7444 \beta_{4} - 1039600 \beta_{5} + 24024 \beta_{6} + 9904 \beta_{7} + 15880 \beta_{8} - 1744 \beta_{9} + 984 \beta_{10} - 7136 \beta_{11} + 816 \beta_{12} - 1376 \beta_{13} + 400 \beta_{14} - 16 \beta_{15} ) q^{28}$$ $$+ ( 32427454 - 299052279 \beta_{1} + 8479012 \beta_{2} + 7397561 \beta_{3} - 1328 \beta_{4} - 28561 \beta_{5} + 199384 \beta_{6} - 18656 \beta_{7} + 4168 \beta_{8} + 4688 \beta_{9} + 1216 \beta_{10} - 1328 \beta_{11} - 928 \beta_{12} + 672 \beta_{13} + 664 \beta_{14} - 336 \beta_{15} ) q^{29}$$ $$+ ( -74381062165 + 605795397 \beta_{1} + 45026159 \beta_{2} - 725222 \beta_{3} - 24805 \beta_{4} + 2890634 \beta_{5} - 28 \beta_{6} - 2634 \beta_{7} - 65498 \beta_{8} - 6161 \beta_{9} - 881 \beta_{10} + 4396 \beta_{11} + 1894 \beta_{12} - 1254 \beta_{13} + 926 \beta_{14} - 34 \beta_{15} ) q^{30}$$ $$+ ( -19890520757 - 336321781 \beta_{1} + 867737 \beta_{2} - 15845509 \beta_{3} + 37507 \beta_{4} - 248722 \beta_{5} + 221383 \beta_{6} + 32420 \beta_{7} + 4205 \beta_{8} + 7497 \beta_{9} - 686 \beta_{10} + 2708 \beta_{11} - 860 \beta_{12} + 5160 \beta_{13} + 1490 \beta_{14} + 860 \beta_{15} ) q^{31}$$ $$+ ( 91231895062 - 1621485796 \beta_{1} - 134648090 \beta_{2} - 1025894 \beta_{3} - 54398 \beta_{4} + 4055172 \beta_{5} + 65376 \beta_{6} + 5592 \beta_{7} + 136024 \beta_{8} - 10208 \beta_{9} + 1228 \beta_{10} - 3520 \beta_{11} - 3148 \beta_{12} + 432 \beta_{13} + 2024 \beta_{14} + 1556 \beta_{15} ) q^{32}$$ $$+ ( 352314467796 - 1677091702 \beta_{1} + 129072 \beta_{2} - 38575306 \beta_{3} + 73382 \beta_{4} - 539695 \beta_{5} - 147644 \beta_{6} + 72311 \beta_{7} + 6122 \beta_{8} + 15868 \beta_{9} + 552 \beta_{10} - 6423 \beta_{11} + 2128 \beta_{12} - 12768 \beta_{13} + 1160 \beta_{14} - 2128 \beta_{15} ) q^{33}$$ $$+ ( -278806668522 + 482185556 \beta_{1} - 223848476 \beta_{2} + 5718114 \beta_{3} + 61804 \beta_{4} - 4689548 \beta_{5} - 39804 \beta_{6} + 2612 \beta_{7} + 213216 \beta_{8} - 20042 \beta_{9} + 1074 \beta_{10} + 16808 \beta_{11} + 60 \beta_{12} - 1612 \beta_{13} + 596 \beta_{14} - 3764 \beta_{15} ) q^{34}$$ $$+ ( 573428218 - 4803643674 \beta_{1} + 37050616 \beta_{2} + 53335480 \beta_{3} - 227234 \beta_{4} - 461038 \beta_{5} + 626260 \beta_{6} - 9668 \beta_{7} + 36174 \beta_{8} + 29560 \beta_{9} + 2528 \beta_{10} - 4048 \beta_{11} + 114 \beta_{12} - 10272 \beta_{13} + 2024 \beta_{14} + 5136 \beta_{15} ) q^{35}$$ $$+ ( -2069050301576 + 6742536281 \beta_{1} + 483114145 \beta_{2} + 33853505 \beta_{3} - 171128 \beta_{4} - 5635104 \beta_{5} + 120720 \beta_{6} - 75680 \beta_{7} - 549776 \beta_{8} - 38752 \beta_{9} - 4080 \beta_{10} + 54208 \beta_{11} - 7712 \beta_{12} + 17856 \beta_{13} + 352 \beta_{14} + 1632 \beta_{15} ) q^{36}$$ $$+ ( -781872320 + 5250720265 \beta_{1} + 282565939 \beta_{2} + 75199440 \beta_{3} + 228857 \beta_{4} - 444968 \beta_{5} - 1785621 \beta_{6} + 39684 \beta_{7} + 29077 \beta_{8} + 39744 \beta_{9} - 7424 \beta_{10} + 4694 \beta_{11} + 12160 \beta_{12} - 1876 \beta_{13} - 2347 \beta_{14} + 938 \beta_{15} ) q^{37}$$ $$+ ( 1504542886518 + 72765867 \beta_{1} + 574311196 \beta_{2} + 4239487 \beta_{3} + 449140 \beta_{4} + 4349240 \beta_{5} + 291841 \beta_{6} + 35272 \beta_{7} - 345800 \beta_{8} - 50016 \beta_{9} + 8856 \beta_{10} - 64800 \beta_{11} - 12832 \beta_{12} + 5352 \beta_{13} - 4464 \beta_{14} - 3168 \beta_{15} ) q^{38}$$ $$+ ( -1153296917750 - 2163956412 \beta_{1} + 12371576 \beta_{2} - 181680073 \beta_{3} - 449768 \beta_{4} - 2240048 \beta_{5} - 3531896 \beta_{6} - 129312 \beta_{7} + 181883 \beta_{8} + 77048 \beta_{9} + 11312 \beta_{10} - 48544 \beta_{11} + 3680 \beta_{12} - 22080 \beta_{13} - 7760 \beta_{14} - 3680 \beta_{15} ) q^{39}$$ $$+ ( 3787978898608 + 13929457092 \beta_{1} - 1540558852 \beta_{2} + 5133628 \beta_{3} - 352904 \beta_{4} + 2861116 \beta_{5} + 11052 \beta_{6} - 17172 \beta_{7} + 224288 \beta_{8} - 97504 \beta_{9} - 19896 \beta_{10} - 40384 \beta_{11} + 20520 \beta_{12} + 11680 \beta_{13} - 13072 \beta_{14} - 8600 \beta_{15} ) q^{40}$$ $$+ ( 468610503174 - 7424991414 \beta_{1} + 180184 \beta_{2} - 161410156 \beta_{3} + 695288 \beta_{4} - 2821314 \beta_{5} + 4452274 \beta_{6} - 576314 \beta_{7} - 39302 \beta_{8} + 78062 \beta_{9} - 23308 \beta_{10} + 41578 \beta_{11} - 11800 \beta_{12} + 70800 \beta_{13} - 10460 \beta_{14} + 11800 \beta_{15} ) q^{41}$$ $$+ ( -3225612163524 + 317421744 \beta_{1} - 3533823452 \beta_{2} + 8086804 \beta_{3} + 346180 \beta_{4} + 16736932 \beta_{5} + 2208216 \beta_{6} - 169388 \beta_{7} - 396536 \beta_{8} - 94336 \beta_{9} + 19536 \beta_{10} + 19008 \beta_{11} + 6464 \beta_{12} + 11340 \beta_{13} - 10016 \beta_{14} + 25024 \beta_{15} ) q^{42}$$ $$+ ( 924933786 - 6716603231 \beta_{1} - 359193417 \beta_{2} + 200311636 \beta_{3} + 214606 \beta_{4} - 10342098 \beta_{5} + 5924348 \beta_{6} + 472500 \beta_{7} + 157626 \beta_{8} + 134256 \beta_{9} + 19904 \beta_{10} + 49760 \beta_{11} + 2630 \beta_{12} + 68544 \beta_{13} - 24880 \beta_{14} - 34272 \beta_{15} ) q^{43}$$ $$+ ( 12103614536008 - 13256691803 \beta_{1} + 3945655611 \beta_{2} - 51904872 \beta_{3} - 2262955 \beta_{4} + 36338546 \beta_{5} - 39324 \beta_{6} + 232743 \beta_{7} + 1658553 \beta_{8} - 152000 \beta_{9} - 29856 \beta_{10} - 173932 \beta_{11} + 41925 \beta_{12} - 137088 \beta_{13} - 20416 \beta_{14} - 14651 \beta_{15} ) q^{44}$$ $$+ ( -8599941428 + 64868382179 \beta_{1} + 1088165815 \beta_{2} + 310032278 \beta_{3} + 3698893 \beta_{4} - 20802342 \beta_{5} - 6918705 \beta_{6} + 395956 \beta_{7} + 213457 \beta_{8} + 114320 \beta_{9} - 23616 \beta_{10} + 15166 \beta_{11} - 93728 \beta_{12} - 6276 \beta_{13} - 7583 \beta_{14} + 3138 \beta_{15} ) q^{45}$$ $$+ ( -12189443298943 - 49464330065 \beta_{1} + 5427175949 \beta_{2} + 55734126 \beta_{3} + 2173841 \beta_{4} - 50783522 \beta_{5} + 3481580 \beta_{6} + 101730 \beta_{7} + 2471794 \beta_{8} - 117331 \beta_{9} + 14669 \beta_{10} + 368644 \beta_{11} + 52466 \beta_{12} - 3698 \beta_{13} - 1766 \beta_{14} + 35674 \beta_{15} ) q^{46}$$ $$+ ( -23554803794147 + 88805215129 \beta_{1} + 213457323 \beta_{2} - 209803853 \beta_{3} - 5454295 \beta_{4} - 3169110 \beta_{5} - 2435995 \beta_{6} - 444244 \beta_{7} - 839887 \beta_{8} + 121803 \beta_{9} + 8342 \beta_{10} + 492156 \beta_{11} - 1748 \beta_{12} + 10488 \beta_{13} + 1686 \beta_{14} + 1748 \beta_{15} ) q^{47}$$ $$+ ( -20589530802334 + 68188583164 \beta_{1} - 9210368022 \beta_{2} + 293875318 \beta_{3} - 8039578 \beta_{4} - 100183532 \beta_{5} + 2946408 \beta_{6} - 183232 \beta_{7} - 5696040 \beta_{8} - 79552 \beta_{9} + 12212 \beta_{10} + 411904 \beta_{11} - 73508 \beta_{12} - 165936 \beta_{13} + 24472 \beta_{14} + 22140 \beta_{15} ) q^{48}$$ $$+ ( 8001034970649 - 161220715376 \beta_{1} - 329512736 \beta_{2} - 230521584 \beta_{3} + 8406352 \beta_{4} - 2152024 \beta_{5} - 6432928 \beta_{6} + 1239192 \beta_{7} + 2214096 \beta_{8} + 71840 \beta_{9} + 31296 \beta_{10} - 95640 \beta_{11} + 28800 \beta_{12} - 172800 \beta_{13} + 22080 \beta_{14} - 28800 \beta_{15} ) q^{49}$$ $$+ ( 29677294015917 + 119865995047 \beta_{1} - 13054565112 \beta_{2} - 132225228 \beta_{3} + 14509656 \beta_{4} + 110691592 \beta_{5} - 5138968 \beta_{6} + 1873928 \beta_{7} - 4551296 \beta_{8} - 15268 \beta_{9} - 36620 \beta_{10} - 548720 \beta_{11} - 71464 \beta_{12} - 26232 \beta_{13} + 25672 \beta_{14} - 97416 \beta_{15} ) q^{50}$$ $$+ ( 34857566039 - 287736795515 \beta_{1} + 3014799684 \beta_{2} - 83538278 \beta_{3} - 19102315 \beta_{4} + 35986865 \beta_{5} - 19417470 \beta_{6} - 1806106 \beta_{7} - 207053 \beta_{8} - 133536 \beta_{9} - 56960 \beta_{10} - 192576 \beta_{11} - 11571 \beta_{12} - 260736 \beta_{13} + 96288 \beta_{14} + 130368 \beta_{15} ) q^{51}$$ $$+ ( -17002477627234 - 160685848870 \beta_{1} + 18391849816 \beta_{2} - 218613602 \beta_{3} - 14225844 \beta_{4} + 149544452 \beta_{5} + 5294052 \beta_{6} - 364238 \beta_{7} + 4791594 \beta_{8} + 304488 \beta_{9} + 130732 \beta_{10} - 34040 \beta_{11} - 115466 \beta_{12} + 678800 \beta_{13} + 93448 \beta_{14} + 57174 \beta_{15} ) q^{52}$$ $$+ ( -27065118220 + 234725009873 \beta_{1} - 5338620715 \beta_{2} - 988901894 \beta_{3} + 15644503 \beta_{4} + 101464854 \beta_{5} + 38946941 \beta_{6} - 2815780 \beta_{7} - 774813 \beta_{8} - 219088 \beta_{9} + 155968 \beta_{10} - 129830 \beta_{11} + 458400 \beta_{12} + 58548 \beta_{13} + 64915 \beta_{14} - 29274 \beta_{15} ) q^{53}$$ $$+ ( 45947769015176 - 649227162 \beta_{1} + 31753017380 \beta_{2} + 60250050 \beta_{3} + 28837300 \beta_{4} - 142387528 \beta_{5} - 20200830 \beta_{6} - 2515896 \beta_{7} - 193352 \beta_{8} + 388768 \beta_{9} - 153576 \beta_{10} - 778016 \beta_{11} - 115744 \beta_{12} - 64152 \beta_{13} + 73616 \beta_{14} - 189024 \beta_{15} ) q^{54}$$ $$+ ( 138006414078426 + 461441620688 \beta_{1} + 858089700 \beta_{2} + 1600861461 \beta_{3} - 24079524 \beta_{4} + 19128888 \beta_{5} + 56040572 \beta_{6} + 5834768 \beta_{7} + 4977993 \beta_{8} - 836924 \beta_{9} - 188216 \beta_{10} - 3003184 \beta_{11} - 53360 \beta_{12} + 320160 \beta_{13} + 121928 \beta_{14} + 53360 \beta_{15} ) q^{55}$$ $$+ ( -10137782432168 - 197287843512 \beta_{1} - 27641691744 \beta_{2} - 699670768 \beta_{3} - 33522216 \beta_{4} - 54495336 \beta_{5} - 21236184 \beta_{6} + 587880 \beta_{7} + 12997440 \beta_{8} + 1126400 \beta_{9} + 264480 \beta_{10} - 1247744 \beta_{11} + 78048 \beta_{12} + 1100928 \beta_{13} + 89536 \beta_{14} + 14816 \beta_{15} ) q^{56}$$ $$+ ( -11816092446988 - 731144791830 \beta_{1} - 1816324112 \beta_{2} + 2262869998 \beta_{3} + 50700014 \beta_{4} + 36879125 \beta_{5} - 57787972 \beta_{6} + 4059171 \beta_{7} - 18157238 \beta_{8} - 957692 \beta_{9} + 290392 \beta_{10} - 219011 \beta_{11} + 42160 \beta_{12} - 252960 \beta_{13} + 84728 \beta_{14} - 42160 \beta_{15} ) q^{57}$$ $$+ ( -38943764148817 + 709086776 \beta_{1} - 27332376491 \beta_{2} + 3246193 \beta_{3} + 45855837 \beta_{4} - 3998661 \beta_{5} - 11411698 \beta_{6} - 10284857 \beta_{7} + 14246386 \beta_{8} + 1259520 \beta_{9} - 268160 \beta_{10} + 2255360 \beta_{11} + 365056 \beta_{12} - 81863 \beta_{13} + 88832 \beta_{14} + 181760 \beta_{15} ) q^{58}$$ $$+ ( 134444010324 - 1084476769513 \beta_{1} + 5002784559 \beta_{2} - 3132543020 \beta_{3} - 71308084 \beta_{4} - 40662520 \beta_{5} - 72774912 \beta_{6} - 986880 \beta_{7} - 2481616 \beta_{8} - 1954312 \beta_{9} - 270368 \beta_{10} + 88624 \beta_{11} - 8432 \beta_{12} + 398304 \beta_{13} - 44312 \beta_{14} - 199152 \beta_{15} ) q^{59}$$ $$+ ( -123373575999380 + 49592908720 \beta_{1} + 43330928244 \beta_{2} + 285188740 \beta_{3} - 102013524 \beta_{4} - 386916336 \beta_{5} - 45762728 \beta_{6} + 877488 \beta_{7} - 32618360 \beta_{8} + 2222128 \beta_{9} + 233816 \beta_{10} + 1941024 \beta_{11} - 29648 \beta_{12} - 2181984 \beta_{13} - 30064 \beta_{14} - 43792 \beta_{15} ) q^{60}$$ $$+ ( -226598464000 + 1871797658247 \beta_{1} - 17418398943 \beta_{2} - 3426912092 \beta_{3} + 116121443 \beta_{4} - 146251940 \beta_{5} + 57734705 \beta_{6} + 4822860 \beta_{7} - 420113 \beta_{8} - 2209456 \beta_{9} + 186560 \beta_{10} + 83842 \beta_{11} - 1381792 \beta_{12} - 95228 \beta_{13} - 41921 \beta_{14} + 47614 \beta_{15} ) q^{61}$$ $$+ ( 43470286343852 + 20795625828 \beta_{1} + 22135693628 \beta_{2} - 662387272 \beta_{3} + 145681836 \beta_{4} + 395041032 \beta_{5} + 25100848 \beta_{6} + 16123960 \beta_{7} - 27141032 \beta_{8} + 2516684 \beta_{9} - 141524 \beta_{10} - 1447568 \beta_{11} + 18488 \beta_{12} + 248136 \beta_{13} - 133160 \beta_{14} + 495000 \beta_{15} ) q^{62}$$ $$+ ( -508621907498271 + 3391707506971 \beta_{1} + 6607028085 \beta_{2} + 7919502862 \beta_{3} - 193408025 \beta_{4} + 102061750 \beta_{5} + 2459019 \beta_{6} - 20330092 \beta_{7} - 27572180 \beta_{8} - 2906811 \beta_{9} - 47094 \beta_{10} + 11898052 \beta_{11} + 209940 \beta_{12} - 1259640 \beta_{13} - 345846 \beta_{14} - 209940 \beta_{15} ) q^{63}$$ $$+ ( 69515438965724 - 77354880456 \beta_{1} - 25272582692 \beta_{2} - 1670032092 \beta_{3} - 193447788 \beta_{4} + 726188616 \beta_{5} + 94853088 \beta_{6} + 4831184 \beta_{7} + 35358512 \beta_{8} + 2820032 \beta_{9} - 277704 \beta_{10} - 1200768 \beta_{11} + 525192 \beta_{12} - 4325408 \beta_{13} - 452976 \beta_{14} - 277176 \beta_{15} ) q^{64}$$ $$+ ( 149593787368060 - 3744403988958 \beta_{1} - 8891094664 \beta_{2} + 7439445396 \beta_{3} + 203567688 \beta_{4} + 111920670 \beta_{5} + 110252410 \beta_{6} - 23505674 \beta_{7} + 85022098 \beta_{8} - 3531354 \beta_{9} - 504380 \beta_{10} + 1733370 \beta_{11} - 504952 \beta_{12} + 3029712 \beta_{13} - 443724 \beta_{14} + 504952 \beta_{15} ) q^{65}$$ $$+ ( 224977637985968 - 316831008170 \beta_{1} - 30441331492 \beta_{2} - 926856914 \beta_{3} + 299488532 \beta_{4} - 1682591636 \beta_{5} + 39341724 \beta_{6} + 33185324 \beta_{7} + 35805024 \beta_{8} + 3461978 \beta_{9} + 463166 \beta_{10} - 2131688 \beta_{11} - 847004 \beta_{12} + 640044 \beta_{13} - 525812 \beta_{14} + 154068 \beta_{15} ) q^{66}$$ $$+ ( 593330811121 - 4611773630412 \beta_{1} - 40293464351 \beta_{2} - 6280040242 \beta_{3} - 262205597 \beta_{4} + 442699711 \beta_{5} + 254384062 \beta_{6} + 18611802 \beta_{7} - 933299 \beta_{8} - 2545008 \beta_{9} + 638528 \beta_{10} + 1517472 \beta_{11} + 174515 \beta_{12} + 1128512 \beta_{13} - 758736 \beta_{14} - 564256 \beta_{15} ) q^{67}$$ $$+ ( 373780658976216 + 329723715902 \beta_{1} - 7383711738 \beta_{2} + 736762470 \beta_{3} - 298647304 \beta_{4} - 1470930400 \beta_{5} + 186401136 \beta_{6} - 2993760 \beta_{7} + 22267536 \beta_{8} + 1673568 \beta_{9} - 1312528 \beta_{10} - 4458432 \beta_{11} + 1411104 \beta_{12} + 4101696 \beta_{13} - 940896 \beta_{14} - 529504 \beta_{15} ) q^{68}$$ $$+ ( -868947128374 + 6991377920002 \beta_{1} - 9910921119 \beta_{2} - 5614055309 \beta_{3} + 396839759 \beta_{4} + 131865437 \beta_{5} - 291871131 \beta_{6} + 14460860 \beta_{7} - 4360325 \beta_{8} - 3343824 \beta_{9} - 1521344 \beta_{10} + 1279146 \beta_{11} + 1871520 \beta_{12} - 457644 \beta_{13} - 639573 \beta_{14} + 228822 \beta_{15} ) q^{69}$$ $$+ ( -627738254750888 - 60717523776 \beta_{1} - 101146202184 \beta_{2} - 4127840696 \beta_{3} + 374916472 \beta_{4} + 1822694992 \beta_{5} - 15245728 \beta_{6} - 63051600 \beta_{7} + 35681744 \beta_{8} + 2527168 \beta_{9} + 1406224 \beta_{10} + 10313536 \beta_{11} + 690496 \beta_{12} - 30096 \beta_{13} - 475040 \beta_{14} - 38976 \beta_{15} ) q^{70}$$ $$+ ( 563337765295705 + 6227439041663 \beta_{1} + 13920860345 \beta_{2} - 4590653228 \beta_{3} - 464911533 \beta_{4} - 7961202 \beta_{5} - 416237993 \beta_{6} + 20309092 \beta_{7} + 68389090 \beta_{8} + 689849 \beta_{9} + 1652786 \beta_{10} - 31518380 \beta_{11} - 28060 \beta_{12} + 168360 \beta_{13} - 442190 \beta_{14} + 28060 \beta_{15} ) q^{71}$$ $$+ ( -1245232394693437 + 1866693822711 \beta_{1} + 214701790406 \beta_{2} + 4576639692 \beta_{3} - 567373529 \beta_{4} + 1185953693 \beta_{5} - 481507449 \beta_{6} - 29314809 \beta_{7} - 128547968 \beta_{8} - 2118656 \beta_{9} - 1909824 \beta_{10} + 17333248 \beta_{11} - 2602432 \beta_{12} + 9539328 \beta_{13} + 129152 \beta_{14} + 665664 \beta_{15} ) q^{72}$$ $$+ ( 709196740731698 - 7522179270578 \beta_{1} - 16149462336 \beta_{2} - 564117754 \beta_{3} + 660129230 \beta_{4} - 95259095 \beta_{5} + 432153184 \beta_{6} + 17889303 \beta_{7} - 318811954 \beta_{8} + 755104 \beta_{9} - 1873344 \beta_{10} - 2725143 \beta_{11} + 1196160 \beta_{12} - 7176960 \beta_{13} + 105024 \beta_{14} - 1196160 \beta_{15} ) q^{73}$$ $$+ ( 693581972030009 + 43850062936 \beta_{1} + 220309741043 \beta_{2} + 3162456823 \beta_{3} + 693475499 \beta_{4} - 196468411 \beta_{5} - 118497566 \beta_{6} - 74947751 \beta_{7} - 140716546 \beta_{8} - 2382976 \beta_{9} + 2081616 \beta_{10} - 12224960 \beta_{11} - 645824 \beta_{12} - 945849 \beta_{13} + 148192 \beta_{14} - 1718848 \beta_{15} ) q^{74}$$ $$+ ( 2096346687386 - 16834319187803 \beta_{1} + 10429508323 \beta_{2} + 16387502728 \beta_{3} - 965293954 \beta_{4} - 2165867006 \beta_{5} + 563980020 \beta_{6} - 20356708 \beta_{7} + 11773374 \beta_{8} + 7773080 \beta_{9} + 2227808 \beta_{10} - 3893648 \beta_{11} - 266366 \beta_{12} - 7346592 \beta_{13} + 1946824 \beta_{14} + 3673296 \beta_{15} ) q^{75}$$ $$+ ( 372723897544808 - 1413876055319 \beta_{1} - 221075977993 \beta_{2} + 2492579320 \beta_{3} - 964076327 \beta_{4} + 511596570 \beta_{5} - 827120972 \beta_{6} + 4485971 \beta_{7} + 192100109 \beta_{8} - 8918720 \beta_{9} - 941088 \beta_{10} - 4323932 \beta_{11} - 4390263 \beta_{12} - 1782144 \beta_{13} + 2396992 \beta_{14} + 2115977 \beta_{15} ) q^{76}$$ $$+ ( -2154288088410 + 16899578599406 \beta_{1} + 100223529623 \beta_{2} + 16182227061 \beta_{3} + 987390905 \beta_{4} + 638725371 \beta_{5} - 486385485 \beta_{6} - 58083996 \beta_{7} - 24211 \beta_{8} + 10271696 \beta_{9} - 823616 \beta_{10} - 2701946 \beta_{11} + 3391840 \beta_{12} + 2027532 \beta_{13} + 1350973 \beta_{14} - 1013766 \beta_{15} ) q^{77}$$ $$+ ( 261540011812097 + 1296387828927 \beta_{1} - 500986832371 \beta_{2} + 3994722302 \beta_{3} + 1333919313 \beta_{4} - 2571414978 \beta_{5} - 7390964 \beta_{6} + 160780098 \beta_{7} + 182443826 \beta_{8} - 12728707 \beta_{9} + 1233277 \beta_{10} - 8458812 \beta_{11} - 2035438 \beta_{12} - 1885458 \beta_{13} + 1543930 \beta_{14} - 4137734 \beta_{15} ) q^{78}$$ $$+ ( -2833767518629296 + 20223253364204 \beta_{1} + 46922725692 \beta_{2} - 29321383114 \beta_{3} - 1212261100 \beta_{4} - 467037464 \beta_{5} + 13764692 \beta_{6} + 36155056 \beta_{7} - 82219018 \beta_{8} + 17091948 \beta_{9} + 568856 \beta_{10} + 52643056 \beta_{11} - 1873360 \beta_{12} + 11240160 \beta_{13} + 3500440 \beta_{14} + 1873360 \beta_{15} ) q^{79}$$ $$+ ( 83882174082468 - 3889337989256 \beta_{1} + 496659840756 \beta_{2} + 8391500492 \beta_{3} - 1371443348 \beta_{4} - 2736848088 \beta_{5} + 1596730000 \beta_{6} + 37255744 \beta_{7} - 107628368 \beta_{8} - 18221696 \beta_{9} + 1613480 \beta_{10} - 32619520 \beta_{11} + 3735032 \beta_{12} - 4175712 \beta_{13} + 2516144 \beta_{14} + 262968 \beta_{15} ) q^{80}$$ $$+ ( 1260531113036013 - 32972135978878 \beta_{1} - 65420117520 \beta_{2} - 67749362362 \beta_{3} + 1552258598 \beta_{4} - 752988583 \beta_{5} - 572127972 \beta_{6} + 109314463 \beta_{7} + 879568610 \beta_{8} + 24647076 \beta_{9} + 2641944 \beta_{10} - 5267455 \beta_{11} + 878640 \beta_{12} - 5271840 \beta_{13} + 2729400 \beta_{14} - 878640 \beta_{15} ) q^{81}$$ $$+ ( 977362108764898 - 448118631350 \beta_{1} + 486314875928 \beta_{2} - 14848427620 \beta_{3} + 1499352072 \beta_{4} + 8354808088 \beta_{5} + 409524664 \beta_{6} + 154459672 \beta_{7} - 172223872 \beta_{8} - 27232236 \beta_{9} - 2620964 \beta_{10} + 42667440 \beta_{11} + 9898120 \beta_{12} - 3600744 \beta_{13} + 3807960 \beta_{14} + 3657832 \beta_{15} ) q^{82}$$ $$+ ( 2738138341300 - 23116731876221 \beta_{1} + 376976131547 \beta_{2} + 42480737436 \beta_{3} - 1511998900 \beta_{4} + 7923576176 \beta_{5} - 1767255440 \beta_{6} - 81955376 \beta_{7} + 7005624 \beta_{8} + 24994696 \beta_{9} - 4044256 \beta_{10} - 974896 \beta_{11} - 1228920 \beta_{12} + 13711392 \beta_{13} + 487448 \beta_{14} - 6855696 \beta_{15} ) q^{83}$$ $$+ ( 1237216453821192 + 3602793208504 \beta_{1} - 1018345092608 \beta_{2} + 6397298952 \beta_{3} - 1946504592 \beta_{4} + 9915179312 \beta_{5} + 2785150960 \beta_{6} + 667544 \beta_{7} - 241034248 \beta_{8} - 26038688 \beta_{9} + 5934160 \beta_{10} + 29260896 \beta_{11} + 1487688 \beta_{12} - 13025856 \beta_{13} + 1584608 \beta_{14} - 1383992 \beta_{15} ) q^{84}$$ $$+ ( -3793491894746 + 30459828627268 \beta_{1} - 86618013571 \beta_{2} + 73440546873 \beta_{3} + 1970497547 \beta_{4} - 3431602449 \beta_{5} + 1466723025 \beta_{6} - 28565556 \beta_{7} + 60122223 \beta_{8} + 38466080 \beta_{9} + 8141696 \beta_{10} - 5980446 \beta_{11} - 22530112 \beta_{12} + 44996 \beta_{13} + 2990223 \beta_{14} - 22498 \beta_{15} ) q^{85}$$ $$+ ( -890449902664862 + 62273226055 \beta_{1} - 831537979632 \beta_{2} + 4212058311 \beta_{3} + 1559850088 \beta_{4} - 7186636944 \beta_{5} + 238016965 \beta_{6} - 237478256 \beta_{7} - 307409040 \beta_{8} - 39398080 \beta_{9} - 8194512 \beta_{10} - 51873344 \beta_{11} + 2437056 \beta_{12} + 3363024 \beta_{13} + 1876768 \beta_{14} + 12477248 \beta_{15} ) q^{86}$$ $$+ ( 1618492094848821 + 34839465118943 \beta_{1} + 82662225297 \beta_{2} - 66689648974 \beta_{3} - 2057566981 \beta_{4} - 1185981730 \beta_{5} + 1843870783 \beta_{6} - 17721148 \beta_{7} + 28172024 \beta_{8} + 26936401 \beta_{9} - 9290526 \beta_{10} - 26916300 \beta_{11} + 4267972 \beta_{12} - 25607832 \beta_{13} - 2548126 \beta_{14} - 4267972 \beta_{15} ) q^{87}$$ $$+ ( -4184045017547708 - 12863047473094 \beta_{1} + 934157899562 \beta_{2} - 18338262926 \beta_{3} - 2239503808 \beta_{4} - 10357748538 \beta_{5} - 3818173962 \beta_{6} + 113925446 \beta_{7} + 604048320 \beta_{8} - 26092400 \beta_{9} + 9802940 \beta_{10} - 53785440 \beta_{11} + 7676348 \beta_{12} - 43590864 \beta_{13} - 2416824 \beta_{14} - 3893860 \beta_{15} ) q^{88}$$ $$+ ( -4364095041977806 - 26566061009946 \beta_{1} - 52509070368 \beta_{2} - 54956180474 \beta_{3} + 2169504670 \beta_{4} - 782701239 \beta_{5} - 2515783448 \beta_{6} - 241220729 \beta_{7} - 1764474458 \beta_{8} + 38533528 \beta_{9} + 9388176 \beta_{10} + 25332281 \beta_{11} - 10328800 \beta_{12} + 61972800 \beta_{13} - 3137200 \beta_{14} + 10328800 \beta_{15} ) q^{89}$$ $$+ ( 8509004104887527 + 910768674312 \beta_{1} + 903190617293 \beta_{2} + 53161830729 \beta_{3} + 2185381141 \beta_{4} + 8598099563 \beta_{5} - 433505250 \beta_{6} - 284526633 \beta_{7} + 826362178 \beta_{8} - 37529216 \beta_{9} - 11332848 \beta_{10} - 26263232 \beta_{11} - 23284672 \beta_{12} + 15045705 \beta_{13} - 5921440 \beta_{14} - 1205568 \beta_{15} ) q^{90}$$ $$+ ( 3464248509774 - 29282306104094 \beta_{1} + 466361490712 \beta_{2} + 67164584056 \beta_{3} - 1817298566 \beta_{4} - 27798200218 \beta_{5} - 2869763812 \beta_{6} + 138253716 \beta_{7} + 10543898 \beta_{8} + 14626952 \beta_{9} - 12439008 \beta_{10} + 13942992 \beta_{11} + 4200102 \beta_{12} + 8358432 \beta_{13} - 6971496 \beta_{14} - 4179216 \beta_{15} ) q^{91}$$ $$+ ( 3582108719628996 + 12775193080848 \beta_{1} - 478876057380 \beta_{2} - 44450297076 \beta_{3} - 1766490492 \beta_{4} + 9710004784 \beta_{5} - 6271731448 \beta_{6} - 34378480 \beta_{7} - 915274088 \beta_{8} - 13916528 \beta_{9} + 6127368 \beta_{10} - 4395424 \beta_{11} + 25022096 \beta_{12} + 42013152 \beta_{13} - 14675536 \beta_{14} - 11255600 \beta_{15} ) q^{92}$$ $$+ ( -3332784286560 + 28155169013568 \beta_{1} - 485430414968 \beta_{2} - 13179197400 \beta_{3} + 1954734744 \beta_{4} + 2553689544 \beta_{5} + 3426966648 \beta_{6} + 329938848 \beta_{7} + 20504904 \beta_{8} + 13320288 \beta_{9} + 3725952 \beta_{10} + 23038128 \beta_{11} + 46743360 \beta_{12} - 15644832 \beta_{13} - 11519064 \beta_{14} + 7822416 \beta_{15} ) q^{93}$$ $$+ ( -12022612815409278 + 22114207787054 \beta_{1} - 290232791270 \beta_{2} + 96227373356 \beta_{3} + 2192041442 \beta_{4} - 5146403428 \beta_{5} + 56601720 \beta_{6} + 129216292 \beta_{7} - 945914460 \beta_{8} - 8280598 \beta_{9} - 9084150 \beta_{10} + 94046920 \beta_{11} - 180348 \beta_{12} + 5973244 \beta_{13} - 9776492 \beta_{14} - 7636012 \beta_{15} ) q^{94}$$ $$+ ( 5858514990578618 + 26904579000056 \beta_{1} + 57008819996 \beta_{2} + 15507403569 \beta_{3} - 1816076764 \beta_{4} + 178753704 \beta_{5} + 869328276 \beta_{6} - 263538000 \beta_{7} + 301768437 \beta_{8} - 16579796 \beta_{9} - 7433128 \beta_{10} - 156673552 \beta_{11} + 4667568 \beta_{12} - 28005408 \beta_{13} - 14563240 \beta_{14} - 4667568 \beta_{15} ) q^{95}$$ $$+ ( -21426980793655860 + 18000408149304 \beta_{1} - 626530280020 \beta_{2} + 72091004756 \beta_{3} - 1061576476 \beta_{4} + 558442376 \beta_{5} + 9573720960 \beta_{6} - 408484240 \beta_{7} + 245639600 \beta_{8} + 19964224 \beta_{9} - 3378600 \beta_{10} + 282854016 \beta_{11} - 39637208 \beta_{12} + 142785120 \beta_{13} - 10728880 \beta_{14} + 4258280 \beta_{15} ) q^{96}$$ $$+ ( 5978984889766950 - 35018754224208 \beta_{1} - 80721645224 \beta_{2} + 38391932738 \beta_{3} + 760920366 \beta_{4} + 978572835 \beta_{5} + 464593090 \beta_{6} - 71478143 \beta_{7} + 3067418336 \beta_{8} - 41314594 \beta_{9} - 4056556 \beta_{10} - 27573713 \beta_{11} + 16082984 \beta_{12} - 96497904 \beta_{13} - 9375548 \beta_{14} - 16082984 \beta_{15} ) q^{97}$$ $$+ ( 21228629879633321 - 5046839046953 \beta_{1} - 1029303868320 \beta_{2} - 148942681296 \beta_{3} + 1364481312 \beta_{4} - 20992402208 \beta_{5} - 85417888 \beta_{6} + 205171424 \beta_{7} + 657365760 \beta_{8} + 57554960 \beta_{9} + 4326320 \beta_{10} - 109358144 \beta_{11} - 5514336 \beta_{12} - 3321120 \beta_{13} - 12569120 \beta_{14} - 6984416 \beta_{15} ) q^{98}$$ $$+ ( 3728874040020 - 24914395884687 \beta_{1} - 1490565312447 \beta_{2} - 178458165828 \beta_{3} - 1402523988 \beta_{4} + 67730105520 \beta_{5} + 6497967600 \beta_{6} + 344187600 \beta_{7} + 28751928 \beta_{8} - 61214904 \beta_{9} + 13120800 \beta_{10} - 4424112 \beta_{11} + 3455112 \beta_{12} - 86937312 \beta_{13} + 2212056 \beta_{14} + 43468656 \beta_{15} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q$$ $$\mathstrut +\mathstrut 270q^{2}$$ $$\mathstrut -\mathstrut 27436q^{4}$$ $$\mathstrut +\mathstrut 5839948q^{6}$$ $$\mathstrut +\mathstrut 11529600q^{7}$$ $$\mathstrut +\mathstrut 24334920q^{8}$$ $$\mathstrut -\mathstrut 602654096q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$16q$$ $$\mathstrut +\mathstrut 270q^{2}$$ $$\mathstrut -\mathstrut 27436q^{4}$$ $$\mathstrut +\mathstrut 5839948q^{6}$$ $$\mathstrut +\mathstrut 11529600q^{7}$$ $$\mathstrut +\mathstrut 24334920q^{8}$$ $$\mathstrut -\mathstrut 602654096q^{9}$$ $$\mathstrut +\mathstrut 131002712q^{10}$$ $$\mathstrut -\mathstrut 2795125400q^{12}$$ $$\mathstrut +\mathstrut 16363788528q^{14}$$ $$\mathstrut -\mathstrut 9993282176q^{15}$$ $$\mathstrut +\mathstrut 26500434192q^{16}$$ $$\mathstrut -\mathstrut 7489125600q^{17}$$ $$\mathstrut -\mathstrut 113450563870q^{18}$$ $$\mathstrut -\mathstrut 209445719856q^{20}$$ $$\mathstrut +\mathstrut 223126527100q^{22}$$ $$\mathstrut +\mathstrut 746845345920q^{23}$$ $$\mathstrut -\mathstrut 1099415493232q^{24}$$ $$\mathstrut -\mathstrut 1809682431664q^{25}$$ $$\mathstrut +\mathstrut 2467726531080q^{26}$$ $$\mathstrut +\mathstrut 3220542267040q^{28}$$ $$\mathstrut -\mathstrut 1188624268048q^{30}$$ $$\mathstrut -\mathstrut 318979758592q^{31}$$ $$\mathstrut +\mathstrut 1455647316000q^{32}$$ $$\mathstrut +\mathstrut 5633526177600q^{33}$$ $$\mathstrut -\mathstrut 4461251980292q^{34}$$ $$\mathstrut -\mathstrut 33088278002484q^{36}$$ $$\mathstrut +\mathstrut 24076283913900q^{38}$$ $$\mathstrut -\mathstrut 18457706051456q^{39}$$ $$\mathstrut +\mathstrut 60626292962592q^{40}$$ $$\mathstrut +\mathstrut 7482251536032q^{41}$$ $$\mathstrut -\mathstrut 51630378688160q^{42}$$ $$\mathstrut +\mathstrut 193654716236040q^{44}$$ $$\mathstrut -\mathstrut 195097141003568q^{46}$$ $$\mathstrut -\mathstrut 376698804821760q^{47}$$ $$\mathstrut -\mathstrut 329350060416480q^{48}$$ $$\mathstrut +\mathstrut 127691292101520q^{49}$$ $$\mathstrut +\mathstrut 474997408872102q^{50}$$ $$\mathstrut -\mathstrut 272251877663120q^{52}$$ $$\mathstrut +\mathstrut 735354219382520q^{54}$$ $$\mathstrut +\mathstrut 2209036687713152q^{55}$$ $$\mathstrut -\mathstrut 162767516076480q^{56}$$ $$\mathstrut -\mathstrut 190521298294720q^{57}$$ $$\mathstrut -\mathstrut 623262610679960q^{58}$$ $$\mathstrut -\mathstrut 1973616194963808q^{60}$$ $$\mathstrut +\mathstrut 695695648144320q^{62}$$ $$\mathstrut -\mathstrut 8131096607338880q^{63}$$ $$\mathstrut +\mathstrut 1111931745501248q^{64}$$ $$\mathstrut +\mathstrut 2385987975356160q^{65}$$ $$\mathstrut +\mathstrut 3598826202828312q^{66}$$ $$\mathstrut +\mathstrut 5981109959771880q^{68}$$ $$\mathstrut -\mathstrut 10044559836180288q^{70}$$ $$\mathstrut +\mathstrut 9025926285576576q^{71}$$ $$\mathstrut -\mathstrut 19918679666289160q^{72}$$ $$\mathstrut +\mathstrut 11332002046118560q^{73}$$ $$\mathstrut +\mathstrut 11098735408189464q^{74}$$ $$\mathstrut +\mathstrut 5959440926938280q^{76}$$ $$\mathstrut +\mathstrut 4184252259031760q^{78}$$ $$\mathstrut -\mathstrut 45299671392008448q^{79}$$ $$\mathstrut +\mathstrut 1337342539452480q^{80}$$ $$\mathstrut +\mathstrut 20101901999290832q^{81}$$ $$\mathstrut +\mathstrut 15639739637081420q^{82}$$ $$\mathstrut +\mathstrut 19796542864700224q^{84}$$ $$\mathstrut -\mathstrut 14252032276026564q^{86}$$ $$\mathstrut +\mathstrut 25965768920837760q^{87}$$ $$\mathstrut -\mathstrut 66964872768837680q^{88}$$ $$\mathstrut -\mathstrut 69879174608766048q^{89}$$ $$\mathstrut +\mathstrut 136151511125051240q^{90}$$ $$\mathstrut +\mathstrut 57336249810701280q^{92}$$ $$\mathstrut -\mathstrut 192318922166254176q^{94}$$ $$\mathstrut +\mathstrut 93790444358203776q^{95}$$ $$\mathstrut -\mathstrut 342799224184788928q^{96}$$ $$\mathstrut +\mathstrut 95593398602180640q^{97}$$ $$\mathstrut +\mathstrut 339641261743253790q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16}\mathstrut -\mathstrut$$ $$7$$ $$x^{15}\mathstrut +\mathstrut$$ $$4022$$ $$x^{14}\mathstrut -\mathstrut$$ $$1102776$$ $$x^{13}\mathstrut -\mathstrut$$ $$373411968$$ $$x^{12}\mathstrut +\mathstrut$$ $$2100004864$$ $$x^{11}\mathstrut -\mathstrut$$ $$3763915816960$$ $$x^{10}\mathstrut +\mathstrut$$ $$7317489121656832$$ $$x^{9}\mathstrut -\mathstrut$$ $$1108241988138827776$$ $$x^{8}\mathstrut +\mathstrut$$ $$163121042717484777472$$ $$x^{7}\mathstrut +\mathstrut$$ $$5699397839986467274752$$ $$x^{6}\mathstrut +\mathstrut$$ $$1127435088957285706235904$$ $$x^{5}\mathstrut -\mathstrut$$ $$217909345031306501735579648$$ $$x^{4}\mathstrut -\mathstrut$$ $$78950720850572326734309359616$$ $$x^{3}\mathstrut +\mathstrut$$ $$13720647095471028734661620662272$$ $$x^{2}\mathstrut -\mathstrut$$ $$5242030267748791654842336509165568$$ $$x\mathstrut +\mathstrut$$ $$1286374137827816254118965326485913600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{15}\mathstrut -\mathstrut$$ $$15$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$4142$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$1135912$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$364324672$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$5014602240$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$3804032634880$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$7347921382735872$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$1167025359200714752$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$172457245591090495488$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$4319739875257743310848$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$1092877169955223759749120$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$226652362390948291813572608$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$77137501951444740399800778752$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$14337747111082586657860026892288$$ $$\nu\mathstrut -\mathstrut$$ $$5011931281375373950898113452441600$$$$)/$$$$20\!\cdots\!16$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$202665796662607$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$3579309129031585$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$1990638038811968562$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$1751547800883521228632$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$73336152594651528733504$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$16587271450622714462478848$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$2879891060237120944769241088$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$280741080760090838567177879552$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$81854179062030283121283913220096$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$27672768791325398176464031063212032$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$5888327403336175402603233454457880576$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1603238282147148494127706912682382721024$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$316935254505413433227207893255938658795520$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$8388768239043845953173344711830221929578496$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$7185287161653279738519647172874520152177115136$$ $$\nu\mathstrut -\mathstrut$$ $$1307918326365733351197882895792625243033117392896$$$$)/$$$$56\!\cdots\!28$$ $$\beta_{3}$$ $$=$$ $$($$$$113144783158307$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$20981730382621171$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$89379930216345162$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$220289109152326083704$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$49449917647326078950848$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$9707113364261484455623168$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$204604437956683790291259392$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$522182811371385852168883994624$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$55131719807716946056351234654208$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$9285428078136832336174350401011712$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$3789155521825886529756236065988935680$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$361463220342815868770366645375810404352$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$35181402519514491276159315828300165677056$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$11075180427959602430732688041681686202679296$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$1350933380440929000176333980599806887496515584$$ $$\nu\mathstrut +\mathstrut$$ $$18845741216881567653212805560342540108427165696$$$$)/$$$$70\!\cdots\!16$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$6904520336896303$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$303875298361647553$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$71924520144668149362$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$60355599933393004743256$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$2701371333148351121886016$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$499305254572666965941182976$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$96434234990842499939510554624$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$8739527854070694564618943791104$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$4141978656388196889091461967511552$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$1166374515623648011467010342438043648$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$233310146532465547264623475010178121728$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$54004544780781897263939623376473978568704$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$10973650936597761955187703075305555455115264$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$326948898370078637694841793787316059452211200$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$2075806875838358259300866372084555064669697474560$$ $$\nu\mathstrut -\mathstrut$$ $$42842242313433409157198097861383892281595179040768$$$$)/$$$$56\!\cdots\!28$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$23764046739574521$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$420160533246763159$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$43\!\cdots\!82$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$21\!\cdots\!92$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$34\!\cdots\!44$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$28\!\cdots\!72$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$46\!\cdots\!36$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$32\!\cdots\!08$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$18\!\cdots\!44$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$54\!\cdots\!68$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$32\!\cdots\!80$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$32\!\cdots\!44$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$54\!\cdots\!08$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$44\!\cdots\!48$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$89\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$16\!\cdots\!04$$$$)/$$$$16\!\cdots\!84$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$79475357111706657$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$14505856015322668527$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$73\!\cdots\!82$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$31\!\cdots\!56$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$95\!\cdots\!40$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$63\!\cdots\!92$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$10\!\cdots\!60$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$28\!\cdots\!28$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$63\!\cdots\!04$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$18\!\cdots\!32$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$19\!\cdots\!00$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$56\!\cdots\!56$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$14\!\cdots\!76$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$16\!\cdots\!48$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$17\!\cdots\!84$$ $$\nu\mathstrut -\mathstrut$$ $$21\!\cdots\!28$$$$)/$$$$56\!\cdots\!28$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$139983251471800851$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$88115910645405087971$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$17\!\cdots\!58$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$13\!\cdots\!48$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$50\!\cdots\!40$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$52\!\cdots\!32$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$48\!\cdots\!64$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$74\!\cdots\!60$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$35\!\cdots\!96$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$62\!\cdots\!76$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$10\!\cdots\!40$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$45\!\cdots\!32$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$14\!\cdots\!08$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$14\!\cdots\!08$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$11\!\cdots\!92$$ $$\nu\mathstrut -\mathstrut$$ $$18\!\cdots\!64$$$$)/$$$$84\!\cdots\!92$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$90518228670632477$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$3604090098844291149$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$65\!\cdots\!78$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$42\!\cdots\!80$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$85\!\cdots\!20$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$18\!\cdots\!76$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$63\!\cdots\!56$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$33\!\cdots\!12$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$96\!\cdots\!76$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$40\!\cdots\!56$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$11\!\cdots\!44$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$21\!\cdots\!16$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$39\!\cdots\!84$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$42\!\cdots\!60$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$21\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$33\!\cdots\!28$$$$)/$$$$42\!\cdots\!96$$ $$\beta_{9}$$ $$=$$ $$($$$$63696133524337513$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$40453859256237604057$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$16\!\cdots\!02$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$77\!\cdots\!60$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$25\!\cdots\!08$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$36\!\cdots\!12$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$17\!\cdots\!28$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$58\!\cdots\!88$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$80\!\cdots\!40$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$37\!\cdots\!12$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$30\!\cdots\!92$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$68\!\cdots\!32$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$17\!\cdots\!04$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$12\!\cdots\!04$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$53\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$17\!\cdots\!80$$$$)/$$$$21\!\cdots\!48$$ $$\beta_{10}$$ $$=$$ $$($$$$639837761265273443$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$15\!\cdots\!31$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$22\!\cdots\!82$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$25\!\cdots\!64$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$23\!\cdots\!36$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$53\!\cdots\!24$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$27\!\cdots\!68$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$27\!\cdots\!20$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$11\!\cdots\!36$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$98\!\cdots\!56$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$22\!\cdots\!68$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$40\!\cdots\!68$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$36\!\cdots\!68$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$15\!\cdots\!24$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$38\!\cdots\!72$$ $$\nu\mathstrut -\mathstrut$$ $$44\!\cdots\!16$$$$)/$$$$16\!\cdots\!84$$ $$\beta_{11}$$ $$=$$ $$($$$$1403534495675861519$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$84577590717130483999$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$73\!\cdots\!46$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$17\!\cdots\!16$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$93\!\cdots\!84$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$14\!\cdots\!36$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$76\!\cdots\!96$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$67\!\cdots\!76$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$13\!\cdots\!00$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$19\!\cdots\!32$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$38\!\cdots\!40$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$14\!\cdots\!36$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$82\!\cdots\!08$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$10\!\cdots\!16$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$73\!\cdots\!92$$ $$\nu\mathstrut -\mathstrut$$ $$12\!\cdots\!32$$$$)/$$$$16\!\cdots\!84$$ $$\beta_{12}$$ $$=$$ $$($$$$4416227129565238249$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$45\!\cdots\!75$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$66\!\cdots\!34$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$54\!\cdots\!48$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$12\!\cdots\!44$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$56\!\cdots\!56$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$82\!\cdots\!20$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$14\!\cdots\!04$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$28\!\cdots\!12$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$46\!\cdots\!24$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$31\!\cdots\!08$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$49\!\cdots\!68$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$10\!\cdots\!52$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$58\!\cdots\!24$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$53\!\cdots\!20$$ $$\nu\mathstrut -\mathstrut$$ $$13\!\cdots\!88$$$$)/$$$$16\!\cdots\!84$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$6728427736961304819$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$16\!\cdots\!73$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$68\!\cdots\!26$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$36\!\cdots\!48$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$97\!\cdots\!92$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$22\!\cdots\!52$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$16\!\cdots\!28$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$25\!\cdots\!96$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$59\!\cdots\!60$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$24\!\cdots\!60$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$32\!\cdots\!92$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$22\!\cdots\!44$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$21\!\cdots\!80$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$28\!\cdots\!80$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$16\!\cdots\!00$$ $$\nu\mathstrut -\mathstrut$$ $$14\!\cdots\!28$$$$)/$$$$16\!\cdots\!84$$ $$\beta_{14}$$ $$=$$ $$($$$$335557145064303491$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$38657826300279070547$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$28\!\cdots\!70$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$36\!\cdots\!88$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$16\!\cdots\!52$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$58\!\cdots\!48$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$18\!\cdots\!36$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$22\!\cdots\!80$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$34\!\cdots\!48$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$12\!\cdots\!88$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$19\!\cdots\!32$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$50\!\cdots\!16$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$84\!\cdots\!56$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$11\!\cdots\!44$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$38\!\cdots\!44$$ $$\nu\mathstrut +\mathstrut$$ $$37\!\cdots\!20$$$$)/$$$$52\!\cdots\!12$$ $$\beta_{15}$$ $$=$$ $$($$$$4596605843708830027$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$30\!\cdots\!45$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$51\!\cdots\!50$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$70\!\cdots\!56$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$12\!\cdots\!72$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$34\!\cdots\!84$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$40\!\cdots\!64$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$33\!\cdots\!88$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$14\!\cdots\!44$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$30\!\cdots\!36$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$80\!\cdots\!60$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$10\!\cdots\!04$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$27\!\cdots\!04$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$53\!\cdots\!64$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$62\!\cdots\!00$$ $$\nu\mathstrut +\mathstrut$$ $$18\!\cdots\!96$$$$)/$$$$42\!\cdots\!96$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$35$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$258$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$14317$$$$)/32768$$ $$\nu^{2}$$ $$=$$ $$($$$$4$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$20$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$128$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$11$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$821$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$9113$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$103010$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$16364119$$$$)/32768$$ $$\nu^{3}$$ $$=$$ $$($$$$128$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$256$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$32$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$68$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$420$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$320$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$3148$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$22592$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$347$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$16581$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1367193$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$48985066$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$6608796329$$$$)/32768$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$2048$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$6784$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$45056$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$10240$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$133888$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1824$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$52$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$438228$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$293696$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$1927260$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$280896$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$147149$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$9665069$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$205517825$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$565721526$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3186731777169$$$$)/32768$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$604160$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$132224$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$3198976$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$1906688$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$6703360$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$92320$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$96820$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$163078828$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$54697408$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$78364380$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$3784395328$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$69131493$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$22863130683$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$121744273849$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$547064973594$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$38440802803255$$$$)/32768$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$451729408$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$98987136$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$1729851392$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$254359552$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$1048176384$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$202798240$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$158597460$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$7893919628$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$6888467008$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$7102298308$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$249950553664$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$8506992227$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$4199372844669$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$10400170500561$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1503173436768182$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$34110964690075489$$$$)/32768$$ $$\nu^{7}$$ $$=$$ $$($$$$-$$$$37029533696$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$7822938240$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$44582662144$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$3116251136$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$94106181888$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$13215980960$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$10902806452$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$3718767840300$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1032392262720$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$18035250379356$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$88984876013504$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1463596641707$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$507714052131275$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$216872409168791$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$280412581544496058$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$98405263718249572487$$$$)/32768$$ $$\nu^{8}$$ $$=$$ $$($$$$10629839611904$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$3261967807360$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$2968419635200$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$19240247429120$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$26279845963520$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$5132016916320$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$888403563628$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$354110019971404$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$125465874567744$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$780324735221756$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$231412915643968$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$3237293196678707$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$73160671913439571$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1232129556048451009$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$35122333057803930326$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$18292731488810532072017$$$$)/32768$$ $$\nu^{9}$$ $$=$$ $$($$$$-$$$$306966240221184$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$117737012369536$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$2419622030831616$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$759878494373888$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$12786976133844736$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$2044795122085792$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$13625942846195060$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$109354179529741204$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$49910906205964224$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$144609598573177956$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$850054650820752320$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$687037299929336709$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$12028608394204889755$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$24256774217363588185$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$5469029942834183223270$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2306002048959217278005655$$$$)/32768$$ $$\nu^{10}$$ $$=$$ $$($$$$-$$$$48779111639488512$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$395226916791772288$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$1604443345719267328$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$318909086823581696$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$2364663660453234944$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$189877667954247328$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$2487008973597462956$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$9236938314769223820$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$604663832880480704$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$64838072930316258364$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$34733096812168688192$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$85493880836179507075$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$3122027562187575027549$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1009353963401637003471$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$744575174526235769608310$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$359021982397392654641028415$$$$)/32768$$ $$\nu^{11}$$ $$=$$ $$($$$$23252619687135193088$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$103411603887587061632$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$156232368345296957440$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$91246125808150923264$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$365521662092680978688$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$40788887358904396384$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$397222325491665166388$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$1592058906598558461012$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$632490964841657158592$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2072512437127020281892$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$66729679884072919910464$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$7751084665043360951115$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$20339430783684267944085$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$3848356448433488971030089$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$213616346666357612545543546$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$48824395202628267803142832295$$$$)/32768$$ $$\nu^{12}$$ $$=$$ $$($$$$-$$$$9438240261610110294016$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$18104592178017910512768$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$4483617970120908533760$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$14783599290846259509248$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$54345313111398248679168$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$11950146946218744769696$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$29128022009003310167828$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$1433704145153816157156916$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$296468273291049131424320$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$898535989172376907633276$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$16712600784283865902797248$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1083786485477793570018771$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$45297030337428777859631885$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$64408956750247388985742431$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1615839076994557018348367382$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$13896268217657979618516689505841$$$$)/32768$$ $$\nu^{13}$$ $$=$$ $$($$$$2142307057288753667909632$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$83188593598025905747840$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$9724480056028140140924928$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$960990864533635664431104$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$38235051799536955352469760$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$94083107337738233271392$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$3928546958090313330946804$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$178974659226792856019210988$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$25869960184194944283586496$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$40314493815464203765441564$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$649409123328078790549185600$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$389099485929436148738776805$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$23395020125719487610758937403$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$14515875073943593102535664953$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2637964616947575809801813657818$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$99063035825749045589277521384119$$$$)/32768$$ $$\nu^{14}$$ $$=$$ $$($$$$-$$$$21110744957054154894481408$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$82556808534583226412815488$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$1843739481871643044467699712$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$462455367179192860446459904$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$163731383597211438462605056$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$93205745332935581383578272$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1481346425685801838935434284$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$30879230295175828639239952628$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$8343853410111823257729496640$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$138467194789428955835068379972$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$512014871091618300950144494144$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$16691616863522950081653584093$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$3353479092625193311134421111741$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$9094169331706076623802223763729$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1531514129860700046483361007788470$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$172212605373144081222071995293291169$$$$)/32768$$ $$\nu^{15}$$ $$=$$ $$($$$$-$$$$50932300788844449362355050496$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$38030839124142733823371181952$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$14228020182636174109851357184$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$61020809728138092331908773888$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$393147105991191792627311919360$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$15841936059401190337849532832$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$98094806608216022542701950540$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$3511008587108057692092471693780$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$910464897966166046276558869440$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$13496515097967453067332104672348$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$153645525838365775981785734725568$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$5448016947716271031315431218709$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$493543262459748086065665922140683$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$8833616927690629257078355378137815$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$225967100966105780873785237874019898$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$100174263885948853220545384864892857287$$$$)/32768$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −189.013 + 1.56793i −189.013 − 1.56793i −136.895 + 127.099i −136.895 − 127.099i −108.197 + 150.760i −108.197 − 150.760i 1.16697 + 180.787i 1.16697 − 180.787i 13.1473 + 179.780i 13.1473 − 179.780i 99.0623 + 145.965i 99.0623 − 145.965i 156.321 + 75.9390i 156.321 − 75.9390i 167.908 + 42.7144i 167.908 − 42.7144i
−362.025 3.13586i 13867.2i 131052. + 2270.52i 665059.i 43485.7 5.02028e6i −7.57536e6 −4.74371e7 1.23295e6i −6.31593e7 −2.08553e6 + 2.40768e8i
5.2 −362.025 + 3.13586i 13867.2i 131052. 2270.52i 665059.i 43485.7 + 5.02028e6i −7.57536e6 −4.74371e7 + 1.23295e6i −6.31593e7 −2.08553e6 2.40768e8i
5.3 −257.790 254.197i 4834.14i 1839.67 + 131059.i 524871.i −1.22882e6 + 1.24619e6i 1.57495e7 3.28406e7 3.42534e7i 1.05771e8 −1.33421e8 + 1.35307e8i
5.4 −257.790 + 254.197i 4834.14i 1839.67 131059.i 524871.i −1.22882e6 1.24619e6i 1.57495e7 3.28406e7 + 3.42534e7i 1.05771e8 −1.33421e8 1.35307e8i
5.5 −200.394 301.520i 13481.8i −50756.5 + 120846.i 1.59197e6i 4.06504e6 2.70168e6i −1.66055e7 4.66086e7 8.91263e6i −5.26193e7 4.80011e8 3.19021e8i
5.6 −200.394 + 301.520i 13481.8i −50756.5 120846.i 1.59197e6i 4.06504e6 + 2.70168e6i −1.66055e7 4.66086e7 + 8.91263e6i −5.26193e7 4.80011e8 + 3.19021e8i
5.7 18.3339 361.574i 13786.7i −130400. 13258.2i 96356.3i −4.98490e6 252764.i −1.47728e7 −7.18455e6 + 4.69061e7i −6.09318e7 −3.48399e7 1.76659e6i
5.8 18.3339 + 361.574i 13786.7i −130400. + 13258.2i 96356.3i −4.98490e6 + 252764.i −1.47728e7 −7.18455e6 4.69061e7i −6.09318e7 −3.48399e7 + 1.76659e6i
5.9 42.2945 359.560i 16002.3i −127494. 30414.8i 1.27253e6i 5.75379e6 + 676810.i 5.50569e6 −1.63282e7 + 4.45555e7i −1.26934e8 −4.57551e8 5.38211e7i
5.10 42.2945 + 359.560i 16002.3i −127494. + 30414.8i 1.27253e6i 5.75379e6 676810.i 5.50569e6 −1.63282e7 4.45555e7i −1.26934e8 −4.57551e8 + 5.38211e7i
5.11 214.125 291.929i 1638.94i −39373.4 125018.i 1.21254e6i 478455. + 350938.i 1.76580e7 −4.49273e7 1.52753e7i 1.26454e8 3.53976e8 + 2.59635e8i
5.12 214.125 + 291.929i 1638.94i −39373.4 + 125018.i 1.21254e6i 478455. 350938.i 1.76580e7 −4.49273e7 + 1.52753e7i 1.26454e8 3.53976e8 2.59635e8i
5.13 328.641 151.878i 4248.51i 84938.1 99826.8i 663971.i 645256. + 1.39624e6i −1.66742e7 1.27527e7 4.57074e7i 1.11090e8 −1.00843e8 2.18208e8i
5.14 328.641 + 151.878i 4248.51i 84938.1 + 99826.8i 663971.i 645256. 1.39624e6i −1.66742e7 1.27527e7 + 4.57074e7i 1.11090e8 −1.00843e8 + 2.18208e8i
5.15 351.815 85.4288i 21682.7i 116476. 60110.3i 465248.i −1.85232e6 7.62829e6i 2.24795e7 3.58428e7 3.10981e7i −3.40998e8 −3.97456e7 1.63681e8i
5.16 351.815 + 85.4288i 21682.7i 116476. + 60110.3i 465248.i −1.85232e6 + 7.62829e6i 2.24795e7 3.58428e7 + 3.10981e7i −3.40998e8 −3.97456e7 + 1.63681e8i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Hecke kernels

There are no other newforms in $$S_{18}^{\mathrm{new}}(8, [\chi])$$.