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

Downloads

Learn more about

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:

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 \(S_{18}^{\mathrm{new}}(8, [\chi])\).