Properties

Label 5.19.c.a
Level 5
Weight 19
Character orbit 5.c
Analytic conductor 10.269
Analytic rank 0
Dimension 16
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 5.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(10.2693068855\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{39}\cdot 3^{12}\cdot 5^{25} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 32 + \beta_{2} + 32 \beta_{4} ) q^{2} + ( -1259 - 5 \beta_{1} + 1259 \beta_{4} - \beta_{5} ) q^{3} + ( -56 \beta_{1} + 56 \beta_{2} + 81315 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{4} + ( 196599 - 145 \beta_{1} + 329 \beta_{2} - \beta_{3} - 70016 \beta_{4} + 9 \beta_{5} + 5 \beta_{6} + \beta_{7} + \beta_{11} ) q^{5} + ( -1924188 - 4912 \beta_{1} - 4912 \beta_{2} + 5 \beta_{3} - 83 \beta_{5} - 83 \beta_{6} + \beta_{10} - \beta_{11} ) q^{6} + ( 4923154 + 1948 \beta_{2} + 9 \beta_{3} + 4923154 \beta_{4} - 206 \beta_{6} - 9 \beta_{7} - \beta_{9} + \beta_{14} ) q^{7} + ( -13591052 - 57743 \beta_{1} - 5 \beta_{2} + 24 \beta_{3} + 13591046 \beta_{4} - 540 \beta_{5} + 2 \beta_{6} + 23 \beta_{7} + 5 \beta_{8} + 13 \beta_{9} + 5 \beta_{10} - \beta_{12} + \beta_{15} ) q^{8} + ( -15 + 108176 \beta_{1} - 108215 \beta_{2} - 15 \beta_{3} - 116046400 \beta_{4} + 1795 \beta_{5} - 1820 \beta_{6} + 24 \beta_{7} - 24 \beta_{8} - 72 \beta_{9} + 46 \beta_{11} - 6 \beta_{12} + \beta_{13} - 5 \beta_{14} - 5 \beta_{15} ) q^{9} +O(q^{10})\) \( q + ( 32 + \beta_{2} + 32 \beta_{4} ) q^{2} + ( -1259 - 5 \beta_{1} + 1259 \beta_{4} - \beta_{5} ) q^{3} + ( -56 \beta_{1} + 56 \beta_{2} + 81315 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{4} + ( 196599 - 145 \beta_{1} + 329 \beta_{2} - \beta_{3} - 70016 \beta_{4} + 9 \beta_{5} + 5 \beta_{6} + \beta_{7} + \beta_{11} ) q^{5} + ( -1924188 - 4912 \beta_{1} - 4912 \beta_{2} + 5 \beta_{3} - 83 \beta_{5} - 83 \beta_{6} + \beta_{10} - \beta_{11} ) q^{6} + ( 4923154 + 1948 \beta_{2} + 9 \beta_{3} + 4923154 \beta_{4} - 206 \beta_{6} - 9 \beta_{7} - \beta_{9} + \beta_{14} ) q^{7} + ( -13591052 - 57743 \beta_{1} - 5 \beta_{2} + 24 \beta_{3} + 13591046 \beta_{4} - 540 \beta_{5} + 2 \beta_{6} + 23 \beta_{7} + 5 \beta_{8} + 13 \beta_{9} + 5 \beta_{10} - \beta_{12} + \beta_{15} ) q^{8} + ( -15 + 108176 \beta_{1} - 108215 \beta_{2} - 15 \beta_{3} - 116046400 \beta_{4} + 1795 \beta_{5} - 1820 \beta_{6} + 24 \beta_{7} - 24 \beta_{8} - 72 \beta_{9} + 46 \beta_{11} - 6 \beta_{12} + \beta_{13} - 5 \beta_{14} - 5 \beta_{15} ) q^{9} + ( -39646131 - 92498 \beta_{1} + 453549 \beta_{2} + 300 \beta_{3} + 117182971 \beta_{4} - 21 \beta_{5} + 8203 \beta_{6} + 746 \beta_{7} + 40 \beta_{8} + \beta_{9} - 55 \beta_{10} + 9 \beta_{11} - 22 \beta_{12} + \beta_{13} - 5 \beta_{14} + 10 \beta_{15} ) q^{10} + ( 295088490 + 378054 \beta_{1} + 377816 \beta_{2} - 743 \beta_{3} - 210 \beta_{4} - 2076 \beta_{5} - 1656 \beta_{6} - 175 \beta_{7} + 242 \beta_{9} - 112 \beta_{10} - 542 \beta_{11} - 63 \beta_{12} + 7 \beta_{13} + 10 \beta_{14} - 10 \beta_{15} ) q^{11} + ( -1420062327 - 255 \beta_{1} - 2401515 \beta_{2} + 8754 \beta_{3} - 1420062405 \beta_{4} - 826 \beta_{5} - 73168 \beta_{6} - 9567 \beta_{7} - 165 \beta_{8} - 2020 \beta_{9} + 165 \beta_{10} - 1229 \beta_{11} - 160 \beta_{12} + 13 \beta_{13} - 13 \beta_{14} ) q^{12} + ( 542533608 - 4393893 \beta_{1} - 1740 \beta_{2} - 3357 \beta_{3} - 542535612 \beta_{4} + 61408 \beta_{5} + 493 \beta_{6} - 3516 \beta_{7} + 320 \beta_{8} + 2782 \beta_{9} + 320 \beta_{10} + 1422 \beta_{11} - 334 \beta_{12} + 35 \beta_{13} - 14 \beta_{15} ) q^{13} + ( -1650 - 2285184 \beta_{1} + 2281202 \beta_{2} - 1265 \beta_{3} + 952288028 \beta_{4} + 166053 \beta_{5} - 168033 \beta_{6} + 26230 \beta_{7} - 15 \beta_{8} - 4594 \beta_{9} + 5283 \beta_{11} - 583 \beta_{12} + 33 \beta_{13} + 135 \beta_{14} + 135 \beta_{15} ) q^{14} + ( -3733668776 - 13819409 \beta_{1} - 1994053 \beta_{2} + 67636 \beta_{3} + 4373494554 \beta_{4} - 622983 \beta_{5} + 360289 \beta_{6} + 83230 \beta_{7} - 520 \beta_{8} - 2496 \beta_{9} + 440 \beta_{10} - 761 \beta_{11} - 839 \beta_{12} + 117 \beta_{13} + 140 \beta_{14} - 305 \beta_{15} ) q^{15} + ( 656027506 - 6332430 \beta_{1} - 6335388 \beta_{2} - 126115 \beta_{3} - 2850 \beta_{4} - 1018506 \beta_{5} - 1012126 \beta_{6} - 2715 \beta_{7} + 3327 \beta_{9} + 1726 \beta_{10} - 11375 \beta_{11} - 923 \beta_{12} + 27 \beta_{13} - 315 \beta_{14} + 315 \beta_{15} ) q^{16} + ( -13855793169 - 85 \beta_{1} + 25312031 \beta_{2} + 247061 \beta_{3} - 13855793679 \beta_{4} - 1445 \beta_{5} + 975069 \beta_{6} - 248421 \beta_{7} + 3400 \beta_{8} + 816 \beta_{9} - 3400 \beta_{10} + 1190 \beta_{11} - 255 \beta_{12} + 85 \beta_{13} - 136 \beta_{14} ) q^{17} + ( 40911561122 + 126940073 \beta_{1} + 9210 \beta_{2} - 309327 \beta_{3} - 40911550742 \beta_{4} + 7044318 \beta_{5} - 2060 \beta_{6} - 308997 \beta_{7} - 8475 \beta_{8} - 20635 \beta_{9} - 8475 \beta_{10} - 1927 \beta_{11} + 1730 \beta_{12} - 280 \beta_{13} - 122 \beta_{15} ) q^{18} + ( 16590 + 72643127 \beta_{1} - 72603965 \beta_{2} + 11625 \beta_{3} + 24258506156 \beta_{4} + 3287311 \beta_{5} - 3269591 \beta_{6} + 318891 \beta_{7} + 7168 \beta_{8} + 49422 \beta_{9} - 51958 \beta_{11} + 5643 \beta_{12} - 113 \beta_{13} - 1510 \beta_{14} - 1510 \beta_{15} ) q^{19} + ( -54931441216 - 238700231 \beta_{1} - 37663185 \beta_{2} + 147354 \beta_{3} + 106916079701 \beta_{4} - 10742850 \beta_{5} + 16411976 \beta_{6} + 453934 \beta_{7} + 325 \beta_{8} + 27959 \beta_{9} + 7225 \beta_{10} - 830 \beta_{11} + 11665 \beta_{12} - 1870 \beta_{13} - 1650 \beta_{14} + 4175 \beta_{15} ) q^{20} + ( 89509450011 + 205223952 \beta_{1} + 205287345 \beta_{2} - 757974 \beta_{3} + 60825 \beta_{4} - 21609689 \beta_{5} - 21745194 \beta_{6} + 57615 \beta_{7} - 69167 \beta_{9} - 7680 \beta_{10} + 214659 \beta_{11} + 19633 \beta_{12} - 642 \beta_{13} + 4490 \beta_{14} - 4490 \beta_{15} ) q^{21} + ( 138155210786 + 32250 \beta_{1} + 483455926 \beta_{2} - 553673 \beta_{3} + 138155231894 \beta_{4} + 131636 \beta_{5} + 24648614 \beta_{6} + 681791 \beta_{7} - 28085 \beta_{8} + 252301 \beta_{9} + 28085 \beta_{10} + 149579 \beta_{11} + 24920 \beta_{12} - 3518 \beta_{13} + 4582 \beta_{14} ) q^{22} + ( -219376463710 - 1101860662 \beta_{1} + 117030 \beta_{2} + 1076375 \beta_{3} + 219376603810 \beta_{4} + 35018826 \beta_{5} - 46000 \beta_{6} + 1099025 \beta_{7} + 82400 \beta_{8} - 103560 \beta_{9} + 82400 \beta_{10} - 172775 \beta_{11} + 23350 \beta_{12} - 140 \beta_{13} + 4705 \beta_{15} ) q^{23} + ( 7140 + 2435422040 \beta_{1} - 2435382236 \beta_{2} + 33690 \beta_{3} - 416524300708 \beta_{4} + 53804476 \beta_{5} - 53739476 \beta_{6} - 2682738 \beta_{7} - 86412 \beta_{8} + 26466 \beta_{9} - 48206 \beta_{11} + 8166 \beta_{12} - 5786 \beta_{13} + 7930 \beta_{14} + 7930 \beta_{15} ) q^{24} + ( 653706923390 - 1972251660 \beta_{1} - 660111255 \beta_{2} - 4121375 \beta_{3} - 465419168355 \beta_{4} - 26541055 \beta_{5} + 86549010 \beta_{6} - 6055750 \beta_{7} + 27400 \beta_{8} - 54420 \beta_{9} - 136800 \beta_{10} + 138310 \beta_{11} - 25320 \beta_{12} + 12385 \beta_{13} + 9575 \beta_{14} - 33275 \beta_{15} ) q^{25} + ( -1457430718264 + 1593838977 \beta_{1} + 1593618091 \beta_{2} + 11060181 \beta_{3} - 245610 \beta_{4} - 107712572 \beta_{5} - 107077672 \beta_{6} - 276515 \beta_{7} + 246279 \beta_{9} - 41472 \beta_{10} - 932601 \beta_{11} - 88051 \beta_{12} - 6181 \beta_{13} - 37755 \beta_{14} + 37755 \beta_{15} ) q^{26} + ( 731159620368 - 164250 \beta_{1} + 9187719654 \beta_{2} - 10651164 \beta_{3} + 731159436876 \beta_{4} - 854064 \beta_{5} + 85450740 \beta_{6} + 9827682 \beta_{7} + 107040 \beta_{8} - 1623750 \beta_{9} - 107040 \beta_{10} - 1089096 \beta_{11} - 158580 \beta_{12} + 30582 \beta_{13} - 49842 \beta_{14} ) q^{27} + ( 504245670101 - 5039554377 \beta_{1} - 1144155 \beta_{2} + 9827041 \beta_{3} - 504247045775 \beta_{4} + 59525056 \beta_{5} + 464158 \beta_{6} + 9592162 \beta_{7} - 392905 \beta_{8} + 1448047 \beta_{9} - 392905 \beta_{10} + 1242020 \beta_{11} - 229279 \beta_{12} - 1120 \beta_{13} - 54561 \beta_{15} ) q^{28} + ( -589890 + 13048501603 \beta_{1} - 13050156305 \beta_{2} - 741125 \beta_{3} - 1520887570664 \beta_{4} - 9858897 \beta_{5} + 8573277 \beta_{6} - 5243339 \beta_{7} + 420336 \beta_{8} - 2204098 \beta_{9} + 2182818 \beta_{11} - 266203 \beta_{12} + 69573 \beta_{13} - 4290 \beta_{14} - 4290 \beta_{15} ) q^{29} + ( -5059958909682 - 26769590680 \beta_{1} - 19963764242 \beta_{2} + 6140538 \beta_{3} - 611734379812 \beta_{4} + 77035043 \beta_{5} - 299522225 \beta_{6} - 4980003 \beta_{7} - 122700 \beta_{8} - 485435 \beta_{9} + 955275 \beta_{10} - 1310888 \beta_{11} - 257015 \beta_{12} - 63755 \beta_{13} - 13725 \beta_{14} + 165575 \beta_{15} ) q^{30} + ( 3319163331828 + 17683163406 \beta_{1} + 17681936274 \beta_{2} - 618146 \beta_{3} - 698940 \beta_{4} + 309904428 \beta_{5} + 310214808 \beta_{6} - 38700 \beta_{7} + 1288213 \beta_{9} + 668784 \beta_{10} - 1057715 \beta_{11} - 100932 \beta_{12} + 132048 \beta_{13} + 203015 \beta_{14} - 203015 \beta_{15} ) q^{31} + ( 1279159783004 - 540030 \beta_{1} + 36978692602 \beta_{2} + 26183166 \beta_{3} + 1279160721344 \beta_{4} + 917580 \beta_{5} - 570418392 \beta_{6} - 25421976 \beta_{7} - 122450 \beta_{8} + 400496 \beta_{9} + 122450 \beta_{10} + 1854430 \beta_{11} + 120960 \beta_{12} - 156390 \beta_{13} + 303014 \beta_{14} ) q^{32} + ( -314769421796 - 39446167861 \beta_{1} + 2156595 \beta_{2} - 38381851 \beta_{3} + 314772334922 \beta_{4} - 898167541 \beta_{5} - 1648567 \beta_{6} - 37218805 \beta_{7} + 763720 \beta_{8} - 3933478 \beta_{9} + 763720 \beta_{10} - 722292 \beta_{11} + 485521 \beta_{12} + 135505 \beta_{13} + 357452 \beta_{15} ) q^{33} + ( 1480020 + 65173987269 \beta_{1} - 65168978457 \beta_{2} + 2930970 \beta_{3} + 7896545981192 \beta_{4} - 1137245090 \beta_{5} + 1142613690 \beta_{6} + 66740164 \beta_{7} - 408442 \beta_{8} + 10259092 \beta_{9} - 6987918 \beta_{11} + 882198 \beta_{12} - 388858 \beta_{13} - 214710 \beta_{14} - 214710 \beta_{15} ) q^{34} + ( -5162220500141 - 62984247573 \beta_{1} - 46371152186 \beta_{2} + 26851165 \beta_{3} + 670446528561 \beta_{4} + 991287169 \beta_{5} - 1234685622 \beta_{6} + 116947681 \beta_{7} - 692960 \beta_{8} + 1559526 \beta_{9} - 3323680 \beta_{10} + 5368444 \beta_{11} + 1444503 \beta_{12} + 421101 \beta_{13} - 202630 \beta_{14} - 493740 \beta_{15} ) q^{35} + ( 16494071946829 + 107156695826 \beta_{1} + 107168326676 \beta_{2} - 211096428 \beta_{3} + 7878990 \beta_{4} + 2868554528 \beta_{5} + 2859549868 \beta_{6} + 3189165 \beta_{7} - 12814505 \beta_{9} - 3659754 \beta_{10} + 16358289 \beta_{11} + 1688365 \beta_{12} - 937965 \beta_{13} - 692275 \beta_{14} + 692275 \beta_{15} ) q^{36} + ( -1817851515310 + 7889295 \beta_{1} + 83272183418 \beta_{2} + 151718267 \beta_{3} - 1817856168424 \beta_{4} + 8478212 \beta_{5} - 1315092789 \beta_{6} - 142464536 \beta_{7} + 15280 \beta_{8} + 28267880 \beta_{9} - 15280 \beta_{10} + 5846228 \beta_{11} + 2005850 \beta_{12} + 775519 \beta_{13} - 1080324 \beta_{14} ) q^{37} + ( 24464597251302 - 68838503278 \beta_{1} + 11134530 \beta_{2} - 113182340 \beta_{3} - 24464586972138 \beta_{4} - 2363600556 \beta_{5} + 2995012 \beta_{6} - 117890546 \beta_{7} + 908580 \beta_{8} - 5091292 \beta_{9} + 908580 \beta_{10} - 24605774 \beta_{11} + 1713194 \beta_{12} - 1284280 \beta_{13} - 1433058 \beta_{15} ) q^{38} + ( 8434560 + 161076262944 \beta_{1} - 161062673280 \beta_{2} - 1990680 \beta_{3} + 24767849267844 \beta_{4} - 103618098 \beta_{5} + 96825218 \beta_{6} - 110253918 \beta_{7} - 5340864 \beta_{8} - 14540571 \beta_{9} - 14696261 \beta_{11} + 1288776 \beta_{12} + 1522744 \beta_{13} + 1470955 \beta_{14} + 1470955 \beta_{15} ) q^{39} + ( -57421165795970 - 187756317025 \beta_{1} - 49102163205 \beta_{2} + 81425425 \beta_{3} + 1435147185700 \beta_{4} - 568114030 \beta_{5} - 802115100 \beta_{6} - 18107260 \beta_{7} + 9094425 \beta_{8} + 10482980 \beta_{9} + 4187775 \beta_{10} - 8972435 \beta_{11} - 596340 \beta_{12} - 2603505 \beta_{13} + 1683025 \beta_{14} + 615700 \beta_{15} ) q^{40} + ( -44830537044795 - 25767169655 \beta_{1} - 25801962630 \beta_{2} + 295162202 \beta_{3} - 18838815 \beta_{4} - 2009528666 \beta_{5} - 2005456831 \beta_{6} + 1103885 \beta_{7} + 41499530 \beta_{9} + 10957784 \beta_{10} - 9510244 \beta_{11} - 2291065 \beta_{12} + 3988540 \beta_{13} + 1270525 \beta_{14} - 1270525 \beta_{15} ) q^{41} + ( 69998157226148 - 32780640 \beta_{1} + 103398304068 \beta_{2} - 471004861 \beta_{3} + 69998182528868 \beta_{4} - 20803360 \beta_{5} + 1064136710 \beta_{6} + 445984381 \beta_{7} - 4990975 \beta_{8} - 93748511 \beta_{9} + 4990975 \beta_{10} - 6339135 \beta_{11} - 5847520 \beta_{12} - 4217120 \beta_{13} + 1718816 \beta_{14} ) q^{42} + ( 80974120542651 - 45214536891 \beta_{1} - 53289780 \beta_{2} + 598820938 \beta_{3} - 80974169439267 \beta_{4} + 6332516693 \beta_{5} - 15057628 \beta_{6} + 622028002 \beta_{7} - 4938320 \beta_{8} + 22629368 \beta_{9} - 4938320 \beta_{10} + 115889114 \beta_{11} - 8149436 \beta_{12} + 6271300 \beta_{13} + 3097450 \beta_{15} ) q^{43} + ( -54226470 - 275051657724 \beta_{1} + 274948459410 \beta_{2} - 6988335 \beta_{3} + 99931297628248 \beta_{4} + 3265865162 \beta_{5} - 3261766342 \beta_{6} + 193261705 \beta_{7} + 26939698 \beta_{8} + 9963845 \beta_{9} + 123117181 \beta_{11} - 12242961 \beta_{12} - 5832529 \beta_{13} - 4567055 \beta_{14} - 4567055 \beta_{15} ) q^{44} + ( -150822338372889 + 420749477746 \beta_{1} - 56812708705 \beta_{2} - 815202294 \beta_{3} - 300427786935146 \beta_{4} - 2319914065 \beta_{5} + 9376362134 \beta_{6} - 1102813809 \beta_{7} - 40083600 \beta_{8} - 73227084 \beta_{9} + 8659200 \beta_{10} - 1465135 \beta_{11} - 12107520 \beta_{12} + 10982435 \beta_{13} - 6262300 \beta_{14} + 1176350 \beta_{15} ) q^{45} + ( -385440418109700 - 361134606450 \beta_{1} - 361084590060 \beta_{2} + 1589469722 \beta_{3} - 559110 \beta_{4} - 23492065581 \beta_{5} - 23364322241 \beta_{6} - 63778485 \beta_{7} - 75408815 \beta_{9} - 19164541 \beta_{10} - 210579954 \beta_{11} - 12830245 \beta_{12} - 12643875 \beta_{13} + 104675 \beta_{14} - 104675 \beta_{15} ) q^{46} + ( 332764356793496 + 79625250 \beta_{1} - 286140945724 \beta_{2} - 485899033 \beta_{3} + 332764246405820 \beta_{4} - 67707092 \beta_{5} + 22768858188 \beta_{6} + 436589887 \beta_{7} + 39135120 \beta_{8} + 35624099 \beta_{9} - 39135120 \beta_{10} - 145141888 \beta_{11} - 6182240 \beta_{12} + 18397946 \beta_{13} + 2915925 \beta_{14} ) q^{47} + ( 479853216318224 + 866244173060 \beta_{1} + 26861100 \beta_{2} - 89166528 \beta_{3} - 479853234066344 \beta_{4} + 25751434832 \beta_{5} + 110044040 \beta_{6} - 196252548 \beta_{7} - 14133900 \beta_{8} + 134483860 \beta_{9} - 14133900 \beta_{10} - 223769888 \beta_{11} - 2958020 \beta_{12} - 20825600 \beta_{13} + 171332 \beta_{15} ) q^{48} + ( 102923955 - 764580505490 \beta_{1} + 764730059885 \beta_{2} - 44635905 \beta_{3} + 421313313141288 \beta_{4} + 28589238763 \beta_{5} - 28712818558 \beta_{6} - 2286803216 \beta_{7} - 43315752 \beta_{8} - 214877682 \beta_{9} - 168858540 \beta_{11} + 11657610 \beta_{12} + 22650375 \beta_{13} + 4011075 \beta_{14} + 4011075 \beta_{15} ) q^{49} + ( -641262055321560 + 1379093133640 \beta_{1} + 1696203758895 \beta_{2} + 1238776625 \beta_{3} - 207454022152580 \beta_{4} - 20727278780 \beta_{5} + 33237083210 \beta_{6} + 280027875 \beta_{7} + 78533525 \beta_{8} + 142308055 \beta_{9} - 27073425 \beta_{10} - 10512865 \beta_{11} + 19915530 \beta_{12} - 30444790 \beta_{13} + 10062950 \beta_{14} - 5600650 \beta_{15} ) q^{50} + ( -375130069034870 - 1323456084725 \beta_{1} - 1323583102061 \beta_{2} - 3202648258 \beta_{3} + 10151040 \beta_{4} - 2993905173 \beta_{5} - 3360511873 \beta_{6} + 181611510 \beta_{7} + 201966154 \beta_{9} + 23196976 \beta_{10} + 637645826 \beta_{11} + 37675774 \beta_{12} + 34292094 \beta_{13} - 6364630 \beta_{14} + 6364630 \beta_{15} ) q^{51} + ( 340636488382239 - 185270070 \beta_{1} - 3406704430878 \beta_{2} + 359695597 \beta_{3} + 340636809782163 \beta_{4} + 313541308 \beta_{5} - 13097746468 \beta_{6} - 99720943 \beta_{7} - 125445730 \beta_{8} + 176971304 \beta_{9} + 125445730 \beta_{10} + 550777102 \beta_{11} + 41281600 \beta_{12} - 53566654 \beta_{13} - 21304450 \beta_{14} ) q^{52} + ( 1563838595100270 + 1162576624994 \beta_{1} + 200660755 \beta_{2} - 553219776 \beta_{3} - 1563838220993316 \beta_{4} - 20856486873 \beta_{5} - 402439918 \beta_{6} - 213131017 \beta_{7} + 107568880 \beta_{8} - 724525512 \beta_{9} + 107568880 \beta_{10} + 286737590 \beta_{11} + 62351159 \beta_{12} + 55547520 \beta_{13} - 22859134 \beta_{15} ) q^{53} + ( -45308880 - 3535302728916 \beta_{1} + 3535476685092 \beta_{2} + 319390200 \beta_{3} + 3194657422172832 \beta_{4} - 47396564892 \beta_{5} + 48050448252 \beta_{6} + 7652950332 \beta_{7} - 46679532 \beta_{8} + 1084027260 \beta_{9} - 261470184 \beta_{11} + 54816264 \beta_{12} - 69919224 \beta_{13} + 22118520 \beta_{14} + 22118520 \beta_{15} ) q^{54} + ( -1777253756173382 + 3177026741135 \beta_{1} + 2457627293403 \beta_{2} - 1473328857 \beta_{3} - 687713719803012 \beta_{4} + 778261163 \beta_{5} - 28677827615 \beta_{6} + 7620205857 \beta_{7} + 19209000 \beta_{8} + 82306925 \beta_{9} - 43663000 \beta_{10} + 227033207 \beta_{11} + 57935675 \beta_{12} + 59577225 \beta_{13} + 13741375 \beta_{14} + 4710375 \beta_{15} ) q^{55} + ( -1430195345117476 - 922746129356 \beta_{1} - 922129687800 \beta_{2} - 4472035254 \beta_{3} + 311327820 \beta_{4} + 55629039260 \beta_{5} + 55665392020 \beta_{6} - 70064350 \beta_{7} - 781182554 \beta_{9} - 32191652 \beta_{10} - 211468950 \beta_{11} + 27497506 \beta_{12} - 76278434 \beta_{13} + 12184130 \beta_{14} - 12184130 \beta_{15} ) q^{56} + ( 1644975585175590 + 505408050 \beta_{1} - 3714601855464 \beta_{2} + 11150095176 \beta_{3} + 1644975014360562 \beta_{4} - 115949526 \beta_{5} - 51885001032 \beta_{6} - 11170908864 \beta_{7} + 137889360 \beta_{8} + 658944822 \beta_{9} - 137889360 \beta_{10} - 534010764 \beta_{11} + 14864430 \beta_{12} + 95135838 \beta_{13} + 37861050 \beta_{14} ) q^{57} + ( 4502693777003474 + 3350805015474 \beta_{1} - 14013570 \beta_{2} - 9896902974 \beta_{3} - 4502694085862222 \beta_{4} - 47042116472 \beta_{5} + 711374716 \beta_{6} - 10556801232 \beta_{7} - 169598110 \beta_{8} + 972319714 \beta_{9} - 169598110 \beta_{10} - 1021189176 \beta_{11} - 51476458 \beta_{12} - 121684360 \beta_{13} + 63519202 \beta_{15} ) q^{58} + ( 176718390 - 2480774869115 \beta_{1} + 2480608471629 \beta_{2} - 605613235 \beta_{3} + 3306850802167928 \beta_{4} - 41398115327 \beta_{5} + 40127982727 \beta_{6} - 8687282597 \beta_{7} + 243751360 \beta_{8} - 1859801392 \beta_{9} + 249519304 \beta_{11} - 85778969 \beta_{12} + 144685099 \beta_{13} - 88436120 \beta_{14} - 88436120 \beta_{15} ) q^{59} + ( -8124915099010451 + 2289987229537 \beta_{1} - 4550341685071 \beta_{2} + 13801044460 \beta_{3} - 6059723058426889 \beta_{4} + 174863496834 \beta_{5} - 228563300532 \beta_{6} - 6956162809 \beta_{7} - 444469385 \beta_{8} - 655620394 \beta_{9} + 224036045 \beta_{10} - 630855931 \beta_{11} - 159055682 \beta_{12} - 100086269 \beta_{13} - 108217155 \beta_{14} + 8315810 \beta_{15} ) q^{60} + ( -3444321162470509 + 387185924493 \beta_{1} + 385898470632 \beta_{2} + 1184334531 \beta_{3} - 854423505 \beta_{4} + 247230933822 \beta_{5} + 248142012777 \beta_{6} - 313135560 \beta_{7} + 1507272219 \beta_{9} - 40660080 \beta_{10} - 1140286323 \beta_{11} - 176550246 \beta_{12} + 108257589 \beta_{13} - 3303180 \beta_{14} + 3303180 \beta_{15} ) q^{61} + ( 6185346350550862 - 844966530 \beta_{1} + 6697424462526 \beta_{2} - 18680200831 \beta_{3} + 6185346818720842 \beta_{4} - 981005540 \beta_{5} - 152138673318 \beta_{6} + 17621166961 \beta_{7} + 320510125 \beta_{8} - 2745027101 \beta_{9} - 320510125 \beta_{10} - 419737875 \beta_{11} - 227412440 \beta_{12} - 78028330 \beta_{13} + 24474626 \beta_{14} ) q^{62} + ( 8074249704491350 - 5739699597350 \beta_{1} - 1425273870 \beta_{2} + 26829108303 \beta_{3} - 8074251021669082 \beta_{4} - 292614612102 \beta_{5} - 380005156 \beta_{6} + 27428643081 \beta_{7} - 243787440 \beta_{8} + 529572256 \beta_{9} - 243787440 \beta_{10} + 3049817725 \beta_{11} - 219529622 \beta_{12} + 163812880 \beta_{13} - 37842763 \beta_{15} ) q^{63} + ( -1155984540 + 3955704725368 \beta_{1} - 3957727901308 \beta_{2} + 71995290 \beta_{3} + 12457977684461284 \beta_{4} - 229534060068 \beta_{5} + 230063378828 \beta_{6} + 13775840150 \beta_{7} - 48371804 \beta_{8} + 188027826 \beta_{9} + 2556620690 \beta_{11} - 216797850 \beta_{12} - 168530330 \beta_{13} + 99849050 \beta_{14} + 99849050 \beta_{15} ) q^{64} + ( -9113700390544566 - 14438447087879 \beta_{1} - 1446487436308 \beta_{2} - 36712377084 \beta_{3} - 5955198695872426 \beta_{4} - 83611745488 \beta_{5} - 129774306091 \beta_{6} - 7733839275 \beta_{7} + 827791280 \beta_{8} + 408168664 \beta_{9} + 59466840 \beta_{10} + 260336104 \beta_{11} - 123436579 \beta_{12} + 161848012 \beta_{13} + 216653915 \beta_{14} + 17561645 \beta_{15} ) q^{65} + ( -13611035924201112 + 3833808498420 \beta_{1} + 3833836707852 \beta_{2} + 51055227066 \beta_{3} - 121923480 \beta_{4} - 294694914754 \beta_{5} - 294035094354 \beta_{6} - 309589620 \beta_{7} - 117325948 \beta_{9} + 634336518 \beta_{10} - 1771037442 \beta_{11} - 78174388 \beta_{12} - 37533228 \beta_{13} + 14050060 \beta_{14} - 14050060 \beta_{15} ) q^{66} + ( 8288986896602115 - 289755180 \beta_{1} + 9143865372965 \beta_{2} - 7207011058 \beta_{3} + 8288987301836907 \beta_{4} + 254929464 \beta_{5} + 195459652289 \beta_{6} + 7394401390 \beta_{7} - 1381345440 \beta_{8} - 1162316842 \beta_{9} + 1381345440 \beta_{10} - 697210944 \beta_{11} + 23970240 \beta_{12} - 67539132 \beta_{13} - 201542246 \beta_{14} ) q^{67} + ( 18209614273754989 - 21100998623686 \beta_{1} + 930362910 \beta_{2} - 11476080639 \beta_{3} - 18209612981337673 \beta_{4} + 282839271060 \beta_{5} - 797434572 \beta_{6} - 10894048953 \beta_{7} + 1105176970 \beta_{8} - 1058000678 \beta_{9} + 1105176970 \beta_{10} - 1274853800 \beta_{11} + 215402886 \beta_{12} + 73325760 \beta_{13} - 187974406 \beta_{15} ) q^{68} + ( 1068157125 + 39139953165953 \beta_{1} - 39137358641444 \beta_{2} + 839802105 \beta_{3} + 14142411669742953 \beta_{4} + 767013988546 \beta_{5} - 765690436711 \beta_{6} - 60208066474 \beta_{7} - 762489968 \beta_{8} + 2017691005 \beta_{9} - 3185812071 \beta_{11} + 381591846 \beta_{12} - 25539471 \beta_{13} + 171896130 \beta_{14} + 171896130 \beta_{15} ) q^{69} + ( -21551995609304222 - 18976337156262 \beta_{1} - 15215512875750 \beta_{2} + 20484507048 \beta_{3} - 16143966090896058 \beta_{4} + 199688198795 \beta_{5} + 836917151477 \beta_{6} - 42919853987 \beta_{7} - 6707525 \beta_{8} + 1081948963 \beta_{9} - 1329138200 \beta_{10} + 1501463410 \beta_{11} + 500289670 \beta_{12} - 123937760 \beta_{13} - 26664200 \beta_{14} - 156298350 \beta_{15} ) q^{70} + ( -9260865938805636 + 28821669655334 \beta_{1} + 28824450785026 \beta_{2} - 56324075082 \beta_{3} + 2285135100 \beta_{4} - 80585668804 \beta_{5} - 84677664224 \beta_{6} + 1665141860 \beta_{7} - 2869292923 \beta_{9} - 1810590640 \beta_{10} + 8117277741 \beta_{11} + 637713052 \beta_{12} - 123998648 \beta_{13} - 159834065 \beta_{14} + 159834065 \beta_{15} ) q^{71} + ( 26780181120345090 + 3370300605 \beta_{1} + 53698948227081 \beta_{2} - 33472431897 \beta_{3} + 26780179079436780 \beta_{4} + 3493556430 \beta_{5} + 469779341316 \beta_{6} + 37306139712 \beta_{7} + 1975595475 \beta_{8} + 13163794128 \beta_{9} - 1975595475 \beta_{10} + 4263015435 \beta_{11} + 834771840 \beta_{12} + 340151385 \beta_{13} + 189668967 \beta_{14} ) q^{72} + ( 22367828069985631 - 6424536711312 \beta_{1} + 5373393930 \beta_{2} + 23866135920 \beta_{3} - 22367823655843147 \beta_{4} + 453760972692 \beta_{5} + 2765973822 \beta_{6} + 20364471684 \beta_{7} - 833513520 \beta_{8} - 1634211252 \beta_{9} - 833513520 \beta_{10} - 11841732834 \beta_{11} + 735690414 \beta_{12} - 847470930 \beta_{13} + 437166462 \beta_{15} ) q^{73} + ( 4417170750 + 37018216169471 \beta_{1} - 37010821200509 \beta_{2} - 694922985 \beta_{3} + 28142046053605072 \beta_{4} - 434657684742 \beta_{5} + 431795448522 \beta_{6} + 61411002099 \beta_{7} - 136017878 \beta_{8} - 2274521889 \beta_{9} - 9541564853 \beta_{11} + 744449553 \beta_{12} + 727940697 \beta_{13} - 657696785 \beta_{14} - 657696785 \beta_{15} ) q^{74} + ( -41066808438952845 - 49754662782695 \beta_{1} - 20212231847260 \beta_{2} - 17751706500 \beta_{3} - 12911228406649835 \beta_{4} - 1826362999985 \beta_{5} + 719387386770 \beta_{6} + 103366145250 \beta_{7} - 1863235200 \beta_{8} - 432896590 \beta_{9} + 1206656400 \beta_{10} - 2009786630 \beta_{11} + 509790110 \beta_{12} - 470319980 \beta_{13} - 741788350 \beta_{14} + 114235950 \beta_{15} ) q^{75} + ( -15905370340202852 + 20664809089668 \beta_{1} + 20664778866840 \beta_{2} - 51612178358 \beta_{3} + 758231820 \beta_{4} - 1760010538020 \beta_{5} - 1763750882220 \beta_{6} + 1743800130 \beta_{7} + 147471142 \beta_{9} + 891963516 \beta_{10} + 5159286730 \beta_{11} + 449857602 \beta_{12} + 197113662 \beta_{13} + 276979010 \beta_{14} - 276979010 \beta_{15} ) q^{76} + ( 12583766990815864 - 3090118575 \beta_{1} + 24102964304569 \beta_{2} + 154687127161 \beta_{3} + 12583770688096534 \beta_{4} + 1209798515 \beta_{5} + 1117741057197 \beta_{6} - 154093542091 \beta_{7} - 697906400 \beta_{8} - 3669177294 \beta_{9} + 697906400 \beta_{10} + 4204647110 \beta_{11} - 4525675 \beta_{12} - 616213445 \beta_{13} + 452109014 \beta_{14} ) q^{77} + ( 54185569081537190 + 4673127031482 \beta_{1} - 4916955210 \beta_{2} - 167075792485 \beta_{3} - 54185571154302482 \beta_{4} + 2799260130218 \beta_{5} - 7283205236 \beta_{6} - 159447126367 \beta_{7} - 552176465 \beta_{8} - 7357441809 \beta_{9} - 552176465 \beta_{10} + 21072043267 \beta_{11} - 345460882 \beta_{12} + 1594825400 \beta_{13} + 118474 \beta_{15} ) q^{78} + ( -10086853020 - 15029001346388 \beta_{1} + 15014521563704 \beta_{2} + 4595690940 \beta_{3} + 47631853993232564 \beta_{4} + 1554238533864 \beta_{5} - 1541684867644 \beta_{6} + 113555574024 \beta_{7} + 4789406128 \beta_{8} + 21494261960 \beta_{9} + 17188335936 \beta_{11} - 1098232416 \beta_{12} - 2264051924 \beta_{13} + 512088420 \beta_{14} + 512088420 \beta_{15} ) q^{79} + ( -38038972604526266 - 8535362325500 \beta_{1} - 22247039456426 \beta_{2} + 146484149799 \beta_{3} - 4263416181472106 \beta_{4} + 1485560015754 \beta_{5} + 665557799410 \beta_{6} + 65827266721 \beta_{7} + 1444501350 \beta_{8} - 2074869045 \beta_{9} + 3357510800 \beta_{10} + 894695051 \beta_{11} - 1093030055 \beta_{12} + 2131681065 \beta_{13} + 1443837175 \beta_{14} + 851368775 \beta_{15} ) q^{80} + ( -20674766142265884 - 56623373477913 \beta_{1} - 56623742383836 \beta_{2} - 35341531740 \beta_{3} - 3666820635 \beta_{4} + 1127647461912 \beta_{5} + 1144448163507 \beta_{6} - 7789214025 \beta_{7} - 1864990518 \beta_{9} + 5377182840 \beta_{10} - 34247747124 \beta_{11} - 2046752223 \beta_{12} - 824478678 \beta_{13} + 584939085 \beta_{14} - 584939085 \beta_{15} ) q^{81} + ( -10510687702705686 - 2044589070 \beta_{1} - 121680548487522 \beta_{2} + 63976077387 \beta_{3} - 10510691405514930 \beta_{4} - 14059928348 \beta_{5} - 4087726783378 \beta_{6} - 77418870861 \beta_{7} - 2697922945 \beta_{8} - 30088985415 \beta_{9} + 2697922945 \beta_{10} - 23025792257 \beta_{11} - 2565131720 \beta_{12} + 617134874 \beta_{13} - 921978674 \beta_{14} ) q^{82} + ( 19631200354630777 + 51032053233097 \beta_{1} - 11987809510 \beta_{2} + 133594597670 \beta_{3} - 19631216984517997 \beta_{4} - 4025799442473 \beta_{5} + 10219370340 \beta_{6} + 126146875200 \beta_{7} - 1772393200 \beta_{8} + 26012079280 \beta_{9} - 1772393200 \beta_{10} + 491438790 \beta_{11} - 2771647870 \beta_{12} - 935214920 \beta_{13} - 1144966850 \beta_{15} ) q^{83} + ( 4068260130 - 123770987525036 \beta_{1} + 123765967401146 \beta_{2} - 15428740155 \beta_{3} + 16436937438239512 \beta_{4} - 3692831709646 \beta_{5} + 3660618142626 \beta_{6} - 87548907171 \beta_{7} - 4967268294 \beta_{8} - 61516679319 \beta_{9} + 10299045025 \beta_{11} - 2272096005 \beta_{12} + 3628182715 \beta_{13} + 734729125 \beta_{14} + 734729125 \beta_{15} ) q^{84} + ( -2330564016021361 + 102482468901727 \beta_{1} + 145479334332419 \beta_{2} - 25210099255 \beta_{3} - 24380935878356059 \beta_{4} - 2122532596251 \beta_{5} - 5262165095097 \beta_{6} - 19516051799 \beta_{7} + 2682483040 \beta_{8} - 626005399 \beta_{9} - 3994133680 \beta_{10} - 8973546611 \beta_{11} - 3122404297 \beta_{12} - 4593469649 \beta_{13} - 643320630 \beta_{14} - 1983920740 \beta_{15} ) q^{85} + ( -9361808772613068 - 31245104474820 \beta_{1} - 31279634730256 \beta_{2} + 122728814035 \beta_{3} - 18380824500 \beta_{4} + 2811316004313 \beta_{5} + 2813831017473 \beta_{6} + 1805964170 \beta_{7} + 45347747134 \beta_{9} - 9095404523 \beta_{10} + 17114263465 \beta_{11} - 2089583766 \beta_{12} + 4037357734 \beta_{13} - 2742776230 \beta_{14} + 2742776230 \beta_{15} ) q^{86} + ( 22443047977773446 - 7336998120 \beta_{1} - 8060433108154 \beta_{2} - 360923485230 \beta_{3} + 22443053914581158 \beta_{4} - 3995209996 \beta_{5} + 4071890786334 \beta_{6} + 355938807282 \beta_{7} + 10176234960 \beta_{8} - 7287588274 \beta_{9} - 10176234960 \beta_{10} + 10923003496 \beta_{11} - 1194829180 \beta_{12} - 989467952 \beta_{13} - 831998734 \beta_{14} ) q^{87} + ( -60427034600542008 + 4651968558848 \beta_{1} + 2980925760 \beta_{2} + 222806570552 \beta_{3} + 60427035898648248 \beta_{4} - 3498723780800 \beta_{5} + 4315224320 \beta_{6} + 218274995192 \beta_{7} + 2664115600 \beta_{8} + 6630818960 \beta_{9} + 2664115600 \beta_{10} - 14220490768 \beta_{11} + 216351040 \beta_{12} - 949585280 \beta_{13} + 704052352 \beta_{15} ) q^{88} + ( 680055720 - 116738594106122 \beta_{1} + 116744062864994 \beta_{2} + 5305823220 \beta_{3} - 89379225492974340 \beta_{4} + 2999533435898 \beta_{5} - 2989148474698 \beta_{6} - 284398547360 \beta_{7} - 9832409552 \beta_{8} + 10504998002 \beta_{9} - 9144891458 \beta_{11} + 1197175788 \beta_{12} - 970490548 \beta_{13} - 1281781010 \beta_{14} - 1281781010 \beta_{15} ) q^{89} + ( 148107731938302229 + 506856347876311 \beta_{1} + 62998872598362 \beta_{2} - 414888078069 \beta_{3} - 32565186988321591 \beta_{4} + 12729534834207 \beta_{5} - 570655420081 \beta_{6} - 477159050595 \beta_{7} - 4673147670 \beta_{8} - 3925871066 \beta_{9} - 10982645385 \beta_{10} + 23664072694 \beta_{11} + 3319708531 \beta_{12} + 5513246632 \beta_{13} - 1457766060 \beta_{14} - 533169655 \beta_{15} ) q^{90} + ( 31087436501792786 - 191229133942051 \beta_{1} - 191152333586455 \beta_{2} + 419828434356 \beta_{3} + 40810001580 \beta_{4} + 7427527618145 \beta_{5} + 7422280166165 \beta_{6} - 4177940940 \beta_{7} - 97269764834 \beta_{9} - 5102782992 \beta_{10} - 9691369850 \beta_{11} + 4605745356 \beta_{12} - 8997588504 \beta_{13} + 2474232230 \beta_{14} - 2474232230 \beta_{15} ) q^{91} + ( -81310449807530231 + 41217380545 \beta_{1} - 632025535525419 \beta_{2} - 23791010078 \beta_{3} - 81310480951373765 \beta_{4} + 27779162822 \beta_{5} - 10984348124592 \beta_{6} + 56760813489 \beta_{7} - 20697487445 \beta_{8} + 88391051500 \beta_{9} + 20697487445 \beta_{10} - 16430079357 \beta_{11} + 7632088800 \beta_{12} + 5190640589 \beta_{13} + 3259089011 \beta_{14} ) q^{92} + ( -189104677251032680 + 150999200336465 \beta_{1} + 36108849645 \beta_{2} - 201100414725 \beta_{3} + 189104722035898642 \beta_{4} - 9719764276019 \beta_{5} - 17957968629 \beta_{6} - 190606590423 \beta_{7} + 21063587040 \beta_{8} - 42285183546 \beta_{9} + 21063587040 \beta_{10} - 40853289072 \beta_{11} + 7464144327 \beta_{12} + 605935995 \beta_{13} + 2795643336 \beta_{15} ) q^{93} + ( 1065690210 - 511173257918796 \beta_{1} + 511204177377642 \beta_{2} + 36251520585 \beta_{3} - 73304166126346900 \beta_{4} - 11417784126307 \beta_{5} + 11489931937407 \beta_{6} - 133007867308 \beta_{7} + 21628771513 \beta_{8} + 152009066660 \beta_{9} - 45792210219 \beta_{11} + 7463442159 \beta_{12} - 7108212089 \beta_{13} + 54132945 \beta_{14} + 54132945 \beta_{15} ) q^{94} + ( 186695008901625570 + 66775379425955 \beta_{1} + 234825512591555 \beta_{2} - 78074472385 \beta_{3} + 182183480419870680 \beta_{4} - 7689919588495 \beta_{5} - 6891724734905 \beta_{6} - 344560028865 \beta_{7} - 1075836200 \beta_{8} + 23268287260 \beta_{9} + 13432971400 \beta_{10} + 1344867030 \beta_{11} + 9511754785 \beta_{12} + 1409593395 \beta_{13} + 2693940900 \beta_{14} + 6539120450 \beta_{15} ) q^{95} + ( 220740825955221576 - 13878789033592 \beta_{1} - 13896640060912 \beta_{2} - 203987311084 \beta_{3} + 8541395640 \beta_{4} - 8379645130856 \beta_{5} - 8465556111416 \beta_{6} + 41531924340 \beta_{7} + 26233446300 \beta_{9} + 19755422008 \beta_{10} + 125515139492 \beta_{11} + 9445237620 \beta_{12} + 6598105740 \beta_{13} + 4813792500 \beta_{14} - 4813792500 \beta_{15} ) q^{96} + ( -105523095621830487 - 57280974480 \beta_{1} + 285108607454356 \beta_{2} - 618495039814 \beta_{3} - 105523016337032559 \beta_{4} + 48402492126 \beta_{5} + 16430651807314 \beta_{6} + 653683398952 \beta_{7} + 8190630240 \beta_{8} + 7641396210 \beta_{9} - 8190630240 \beta_{10} + 140272765524 \beta_{11} + 4394845230 \beta_{12} - 13214132988 \beta_{13} + 430844838 \beta_{14} ) q^{97} + ( -270630508143685846 + 230079332547723 \beta_{1} + 6791156650 \beta_{2} + 511164470667 \beta_{3} + 270630530484498178 \beta_{4} + 17506042985394 \beta_{5} - 37012404844 \beta_{6} + 544453406789 \beta_{7} - 46306025785 \beta_{8} - 111572336441 \beta_{9} - 46306025785 \beta_{10} + 98692544295 \beta_{11} + 3723468722 \beta_{12} + 5913093480 \beta_{13} - 3676728362 \beta_{15} ) q^{98} + ( 37515209970 + 289297747333700 \beta_{1} - 289250574941342 \beta_{2} - 25443731985 \beta_{3} - 382264743858744790 \beta_{4} + 13409895445840 \beta_{5} - 13473287979800 \beta_{6} + 1044368227293 \beta_{7} - 8761111296 \beta_{8} - 101345062950 \beta_{9} - 53515481042 \beta_{11} + 2414295597 \beta_{12} + 10090774393 \beta_{13} - 1514497490 \beta_{14} - 1514497490 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 510q^{2} - 20130q^{3} + 3145170q^{5} - 30766728q^{6} + 78767350q^{7} - 217339260q^{8} + O(q^{10}) \) \( 16q + 510q^{2} - 20130q^{3} + 3145170q^{5} - 30766728q^{6} + 78767350q^{7} - 217339260q^{8} - 635093930q^{10} + 4719921012q^{11} - 22715951880q^{12} + 8689116940q^{13} - 59706310410q^{15} + 10530345976q^{16} - 221748184620q^{17} + 654304109970q^{18} - 878373508140q^{20} + 1431507507492q^{21} + 2209421844320q^{22} - 3507966822690q^{23} + 10464353188900q^{25} - 23324455191468q^{26} + 11679867134280q^{27} + 8077754165240q^{28} - 80865025845360q^{30} + 53033402941772q^{31} + 20394786468360q^{32} - 4953707733660q^{33} - 82375887392730q^{35} + 263454471040956q^{36} - 29247362581160q^{37} + 391580910829560q^{38} - 918259855204500q^{40} - 717170436410748q^{41} + 1119760897992480q^{42} + 1295649718427950q^{43} - 2413910465914410q^{45} - 6165422784262408q^{46} + 5324712362382270q^{47} + 7675815427929360q^{48} - 10266398091964350q^{50} - 5996743906200468q^{51} + 5457051468349900q^{52} + 25019175284457720q^{53} - 28447209909214060q^{55} - 22879869030140400q^{56} + 26327201869272960q^{57} + 72036625362175440q^{58} - 129993967558538280q^{60} - 55112667981951388q^{61} + 98952816959518920q^{62} + 129200567352114030q^{63} - 145786436727900960q^{65} - 217789774433419296q^{66} + 132604737823930030q^{67} + 291394918710173220q^{68} - 344767752141139080q^{70} - 148288220084109108q^{71} + 428373837658219140q^{72} + 357896091854250160q^{73} - 656924515001553450q^{75} - 254554257141486000q^{76} + 201286979542193700q^{77} + 866949362074211400q^{78} - 608571216720410880q^{80} - 330578957555881164q^{81} - 167911799138733280q^{82} + 314012877447835830q^{83} - 37755360901186580q^{85} - 149686440788368488q^{86} + 359090097519437040q^{87} - 966828856938988320q^{88} + 2368537241357586390q^{90} + 498102030062089052q^{91} - 1299658879961370120q^{92} - 3025937916654554460q^{93} + 2986575636848949000q^{95} + 3531977749160666592q^{96} - 1689002222572272080q^{97} - 4330619740553781090q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 1365649 x^{14} + 735328563216 x^{12} + 196541937723225664 x^{10} + 26804517896227604816896 x^{8} + 1698608176414048875695308800 x^{6} + 38033703783600508125786931200000 x^{4} + 174469353400289826571825446912000000 x^{2} + 223860255706571447791285331558400000000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(22292606489531908874689735627 \nu^{14} + 23356272799594353923523185468768123 \nu^{12} + 8975025045192404615456651156184813405232 \nu^{10} + 1543233340472868448056103094818686959098510528 \nu^{8} + 120630995604654821623399081304987836945915176430592 \nu^{6} + 5012460970256744580044873856038177381955841532792012800 \nu^{4} + 135266770138096464873296005487049653878182619410679726080000 \nu^{2} + 11597634486408557928173975443803950322432298786278604800000000 \nu - 271188562430968403694896158204604015666260913801215868928000000\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(22292606489531908874689735627 \nu^{14} + 23356272799594353923523185468768123 \nu^{12} + 8975025045192404615456651156184813405232 \nu^{10} + 1543233340472868448056103094818686959098510528 \nu^{8} + 120630995604654821623399081304987836945915176430592 \nu^{6} + 5012460970256744580044873856038177381955841532792012800 \nu^{4} + 135266770138096464873296005487049653878182619410679726080000 \nu^{2} - 11597634486408557928173975443803950322432298786278604800000000 \nu - 271188562430968403694896158204604015666260913801215868928000000\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(47\!\cdots\!61\)\( \nu^{14} - \)\(64\!\cdots\!89\)\( \nu^{12} - \)\(34\!\cdots\!76\)\( \nu^{10} - \)\(91\!\cdots\!04\)\( \nu^{8} - \)\(12\!\cdots\!56\)\( \nu^{6} - \)\(74\!\cdots\!00\)\( \nu^{4} - \)\(14\!\cdots\!00\)\( \nu^{2} + \)\(50\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(17\!\cdots\!01\)\( \nu^{15} + \)\(24\!\cdots\!49\)\( \nu^{13} + \)\(13\!\cdots\!16\)\( \nu^{11} + \)\(36\!\cdots\!64\)\( \nu^{9} + \)\(49\!\cdots\!96\)\( \nu^{7} + \)\(31\!\cdots\!00\)\( \nu^{5} + \)\(74\!\cdots\!00\)\( \nu^{3} + \)\(50\!\cdots\!00\)\( \nu\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(12\!\cdots\!31\)\( \nu^{15} - \)\(71\!\cdots\!00\)\( \nu^{14} + \)\(16\!\cdots\!19\)\( \nu^{13} - \)\(97\!\cdots\!00\)\( \nu^{12} + \)\(88\!\cdots\!96\)\( \nu^{11} - \)\(52\!\cdots\!00\)\( \nu^{10} + \)\(23\!\cdots\!84\)\( \nu^{9} - \)\(13\!\cdots\!00\)\( \nu^{8} + \)\(31\!\cdots\!76\)\( \nu^{7} - \)\(18\!\cdots\!00\)\( \nu^{6} + \)\(19\!\cdots\!00\)\( \nu^{5} - \)\(11\!\cdots\!00\)\( \nu^{4} + \)\(39\!\cdots\!00\)\( \nu^{3} - \)\(22\!\cdots\!00\)\( \nu^{2} + \)\(81\!\cdots\!00\)\( \nu - \)\(46\!\cdots\!00\)\(\)\()/ \)\(64\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(12\!\cdots\!31\)\( \nu^{15} - \)\(71\!\cdots\!00\)\( \nu^{14} - \)\(16\!\cdots\!19\)\( \nu^{13} - \)\(97\!\cdots\!00\)\( \nu^{12} - \)\(88\!\cdots\!96\)\( \nu^{11} - \)\(52\!\cdots\!00\)\( \nu^{10} - \)\(23\!\cdots\!84\)\( \nu^{9} - \)\(13\!\cdots\!00\)\( \nu^{8} - \)\(31\!\cdots\!76\)\( \nu^{7} - \)\(18\!\cdots\!00\)\( \nu^{6} - \)\(19\!\cdots\!00\)\( \nu^{5} - \)\(11\!\cdots\!00\)\( \nu^{4} - \)\(39\!\cdots\!00\)\( \nu^{3} - \)\(22\!\cdots\!00\)\( \nu^{2} - \)\(81\!\cdots\!00\)\( \nu - \)\(46\!\cdots\!00\)\(\)\()/ \)\(64\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(\)\(25\!\cdots\!79\)\( \nu^{15} + \)\(35\!\cdots\!71\)\( \nu^{13} + \)\(19\!\cdots\!64\)\( \nu^{11} + \)\(50\!\cdots\!56\)\( \nu^{9} + \)\(67\!\cdots\!84\)\( \nu^{7} + \)\(40\!\cdots\!00\)\( \nu^{5} + \)\(65\!\cdots\!00\)\( \nu^{3} - \)\(36\!\cdots\!00\)\( \nu\)\()/ \)\(71\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(47\!\cdots\!13\)\( \nu^{15} + \)\(23\!\cdots\!20\)\( \nu^{14} + \)\(64\!\cdots\!37\)\( \nu^{13} + \)\(31\!\cdots\!80\)\( \nu^{12} + \)\(34\!\cdots\!08\)\( \nu^{11} + \)\(16\!\cdots\!20\)\( \nu^{10} + \)\(89\!\cdots\!32\)\( \nu^{9} + \)\(45\!\cdots\!80\)\( \nu^{8} + \)\(11\!\cdots\!48\)\( \nu^{7} + \)\(60\!\cdots\!20\)\( \nu^{6} + \)\(64\!\cdots\!00\)\( \nu^{5} + \)\(37\!\cdots\!00\)\( \nu^{4} + \)\(88\!\cdots\!00\)\( \nu^{3} + \)\(78\!\cdots\!00\)\( \nu^{2} - \)\(15\!\cdots\!00\)\( \nu + \)\(20\!\cdots\!00\)\(\)\()/ \)\(12\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(12\!\cdots\!47\)\( \nu^{15} - \)\(64\!\cdots\!00\)\( \nu^{14} + \)\(16\!\cdots\!03\)\( \nu^{13} - \)\(87\!\cdots\!00\)\( \nu^{12} + \)\(88\!\cdots\!52\)\( \nu^{11} - \)\(47\!\cdots\!00\)\( \nu^{10} + \)\(23\!\cdots\!08\)\( \nu^{9} - \)\(12\!\cdots\!00\)\( \nu^{8} + \)\(31\!\cdots\!12\)\( \nu^{7} - \)\(16\!\cdots\!00\)\( \nu^{6} + \)\(19\!\cdots\!00\)\( \nu^{5} - \)\(10\!\cdots\!00\)\( \nu^{4} + \)\(40\!\cdots\!00\)\( \nu^{3} - \)\(21\!\cdots\!00\)\( \nu^{2} + \)\(94\!\cdots\!00\)\( \nu - \)\(55\!\cdots\!00\)\(\)\()/ \)\(35\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(37\!\cdots\!73\)\( \nu^{15} - \)\(44\!\cdots\!00\)\( \nu^{14} + \)\(51\!\cdots\!77\)\( \nu^{13} - \)\(60\!\cdots\!00\)\( \nu^{12} + \)\(27\!\cdots\!68\)\( \nu^{11} - \)\(33\!\cdots\!00\)\( \nu^{10} + \)\(73\!\cdots\!72\)\( \nu^{9} - \)\(90\!\cdots\!00\)\( \nu^{8} + \)\(99\!\cdots\!08\)\( \nu^{7} - \)\(12\!\cdots\!00\)\( \nu^{6} + \)\(61\!\cdots\!00\)\( \nu^{5} - \)\(67\!\cdots\!00\)\( \nu^{4} + \)\(12\!\cdots\!00\)\( \nu^{3} - \)\(87\!\cdots\!00\)\( \nu^{2} + \)\(33\!\cdots\!00\)\( \nu - \)\(35\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(37\!\cdots\!73\)\( \nu^{15} + \)\(19\!\cdots\!00\)\( \nu^{14} + \)\(51\!\cdots\!77\)\( \nu^{13} + \)\(27\!\cdots\!00\)\( \nu^{12} + \)\(27\!\cdots\!68\)\( \nu^{11} + \)\(14\!\cdots\!00\)\( \nu^{10} + \)\(73\!\cdots\!72\)\( \nu^{9} + \)\(38\!\cdots\!00\)\( \nu^{8} + \)\(99\!\cdots\!08\)\( \nu^{7} + \)\(51\!\cdots\!00\)\( \nu^{6} + \)\(61\!\cdots\!00\)\( \nu^{5} + \)\(31\!\cdots\!00\)\( \nu^{4} + \)\(12\!\cdots\!00\)\( \nu^{3} + \)\(64\!\cdots\!00\)\( \nu^{2} + \)\(33\!\cdots\!00\)\( \nu + \)\(15\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(18\!\cdots\!79\)\( \nu^{15} + \)\(19\!\cdots\!00\)\( \nu^{14} - \)\(25\!\cdots\!71\)\( \nu^{13} + \)\(25\!\cdots\!00\)\( \nu^{12} - \)\(13\!\cdots\!64\)\( \nu^{11} + \)\(13\!\cdots\!00\)\( \nu^{10} - \)\(36\!\cdots\!56\)\( \nu^{9} + \)\(36\!\cdots\!00\)\( \nu^{8} - \)\(49\!\cdots\!84\)\( \nu^{7} + \)\(49\!\cdots\!00\)\( \nu^{6} - \)\(30\!\cdots\!00\)\( \nu^{5} + \)\(30\!\cdots\!00\)\( \nu^{4} - \)\(63\!\cdots\!00\)\( \nu^{3} + \)\(62\!\cdots\!00\)\( \nu^{2} - \)\(16\!\cdots\!00\)\( \nu + \)\(15\!\cdots\!00\)\(\)\()/ \)\(64\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(59\!\cdots\!91\)\( \nu^{15} - \)\(15\!\cdots\!00\)\( \nu^{14} - \)\(80\!\cdots\!59\)\( \nu^{13} - \)\(21\!\cdots\!00\)\( \nu^{12} - \)\(43\!\cdots\!56\)\( \nu^{11} - \)\(11\!\cdots\!00\)\( \nu^{10} - \)\(11\!\cdots\!24\)\( \nu^{9} - \)\(30\!\cdots\!00\)\( \nu^{8} - \)\(15\!\cdots\!36\)\( \nu^{7} - \)\(40\!\cdots\!00\)\( \nu^{6} - \)\(96\!\cdots\!00\)\( \nu^{5} - \)\(25\!\cdots\!00\)\( \nu^{4} - \)\(19\!\cdots\!00\)\( \nu^{3} - \)\(53\!\cdots\!00\)\( \nu^{2} - \)\(48\!\cdots\!00\)\( \nu - \)\(13\!\cdots\!00\)\(\)\()/ \)\(10\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(51\!\cdots\!17\)\( \nu^{15} - \)\(28\!\cdots\!00\)\( \nu^{14} - \)\(69\!\cdots\!33\)\( \nu^{13} - \)\(38\!\cdots\!00\)\( \nu^{12} - \)\(37\!\cdots\!72\)\( \nu^{11} - \)\(20\!\cdots\!00\)\( \nu^{10} - \)\(99\!\cdots\!88\)\( \nu^{9} - \)\(55\!\cdots\!00\)\( \nu^{8} - \)\(13\!\cdots\!32\)\( \nu^{7} - \)\(75\!\cdots\!00\)\( \nu^{6} - \)\(83\!\cdots\!00\)\( \nu^{5} - \)\(47\!\cdots\!00\)\( \nu^{4} - \)\(17\!\cdots\!00\)\( \nu^{3} - \)\(98\!\cdots\!00\)\( \nu^{2} - \)\(41\!\cdots\!00\)\( \nu - \)\(23\!\cdots\!00\)\(\)\()/ \)\(64\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(55\!\cdots\!01\)\( \nu^{15} + \)\(25\!\cdots\!00\)\( \nu^{14} - \)\(75\!\cdots\!49\)\( \nu^{13} + \)\(35\!\cdots\!00\)\( \nu^{12} - \)\(40\!\cdots\!16\)\( \nu^{11} + \)\(19\!\cdots\!00\)\( \nu^{10} - \)\(10\!\cdots\!64\)\( \nu^{9} + \)\(50\!\cdots\!00\)\( \nu^{8} - \)\(14\!\cdots\!96\)\( \nu^{7} + \)\(69\!\cdots\!00\)\( \nu^{6} - \)\(90\!\cdots\!00\)\( \nu^{5} + \)\(43\!\cdots\!00\)\( \nu^{4} - \)\(18\!\cdots\!00\)\( \nu^{3} + \)\(90\!\cdots\!00\)\( \nu^{2} - \)\(45\!\cdots\!00\)\( \nu + \)\(21\!\cdots\!00\)\(\)\()/ \)\(64\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{6} - 2 \beta_{5} + \beta_{3} + 8 \beta_{2} + 8 \beta_{1} - 341411\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + 13 \beta_{11} + 13 \beta_{9} + 10 \beta_{8} - 145 \beta_{7} + 158 \beta_{6} - 154 \beta_{5} - 4945454 \beta_{4} + \beta_{3} + 577418 \beta_{2} - 577428 \beta_{1} - 6\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-187 \beta_{15} + 187 \beta_{14} + 101 \beta_{13} + 795 \beta_{12} + 9711 \beta_{11} - 446 \beta_{10} - 1663 \beta_{9} + 2587 \beta_{7} + 2540702 \beta_{6} + 2546570 \beta_{5} + 2082 \beta_{4} - 666589 \beta_{3} - 23699044 \beta_{2} - 23700722 \beta_{1} + 197165209486\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-332621 \beta_{15} - 332621 \beta_{14} + 248173 \beta_{13} + 512493 \beta_{12} - 5689577 \beta_{11} - 4001401 \beta_{9} - 5191570 \beta_{8} + 55380693 \beta_{7} - 158534662 \beta_{6} + 158334930 \beta_{5} + 7752469577678 \beta_{4} + 280467 \beta_{3} - 181455068418 \beta_{2} + 181459400388 \beta_{1} + 2281998\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(89122963 \beta_{15} - 89122963 \beta_{14} - 188206861 \beta_{13} - 460116339 \beta_{12} - 6434756455 \beta_{11} + 145151662 \beta_{10} - 224389769 \beta_{9} - 1756762739 \beta_{7} - 1097425826670 \beta_{6} - 1101211261626 \beta_{5} - 815728434 \beta_{4} + 220156854773 \beta_{3} + 11407641387844 \beta_{2} + 11407704288834 \beta_{1} - 61975245811240718\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(91332043533 \beta_{15} + 91332043533 \beta_{14} - 28750839341 \beta_{13} - 209834854061 \beta_{12} + 2066098248105 \beta_{11} + 952373243833 \beta_{9} + 2196290033042 \beta_{8} - 20681078286101 \beta_{7} + 70012048957190 \beta_{6} - 70440297643986 \beta_{5} - 3746985659046117454 \beta_{4} - 333417190099 \beta_{3} + 59277393372766338 \beta_{2} - 59278948469262788 \beta_{1} - 715757080206\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-33766232882451 \beta_{15} + 33766232882451 \beta_{14} + 117458971654797 \beta_{13} + 205578935426547 \beta_{12} + 3063287081830375 \beta_{11} - 13472223437230 \beta_{10} + 406627705731081 \beta_{9} + 851654749589235 \beta_{7} + 432461512758555502 \beta_{6} + 434252942221505722 \beta_{5} + 264359891315250 \beta_{4} - 73745089298423797 \beta_{3} - 4585427673535297348 \beta_{2} - 4585222197539993410 \beta_{1} + 20248563702670651592078\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-23558881473765837 \beta_{15} - 23558881473765837 \beta_{14} - 9558091503710995 \beta_{13} + 81447666909123949 \beta_{12} - 727913609144748393 \beta_{11} - 165509406359187321 \beta_{9} - 864340962737641746 \beta_{8} + 7683152174779385301 \beta_{7} - 24102214713229128070 \beta_{6} + 24413464354482476882 \beta_{5} + 1507006224375452491596494 \beta_{4} + 191569608329380883 \beta_{3} - 19834491174828189326338 \beta_{2} + 19835032634222042060996 \beta_{1} + 215668726216238862\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(12284087299133161811 \beta_{15} - 12284087299133161811 \beta_{14} - 56053430482826841677 \beta_{13} - 83858932033038920243 \beta_{12} - 1289430262551283708711 \beta_{11} - 13107915892799143890 \beta_{10} - 240619990947191652553 \beta_{9} - 363683657064770444083 \beta_{7} - 163687683455312589941230 \beta_{6} - 164442856270992342907962 \beta_{5} - 83416504650636235698 \beta_{4} + 25056675548797764920757 \beta_{3} + 1749270792459779810435652 \beta_{2} + 1749129995242499139304642 \beta_{1} - 6775380639812614428110409742\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(5867207036204593860493 \beta_{15} + 5867207036204593860493 \beta_{14} + 9383254602640109858131 \beta_{13} - 31165872638046653823789 \beta_{12} + 258210296970704148700201 \beta_{11} + 926385451617674915385 \beta_{9} + 328170501583190053981842 \beta_{8} - 2848153876688970396465557 \beta_{7} + 7221986289967245250137606 \beta_{6} - 7381166690099865980615890 \beta_{5} - 575004990294313682824229967182 \beta_{4} - 90481509084013637221971 \beta_{3} + 6748130329596446347784217986 \beta_{2} - 6748320340941104754031410116 \beta_{1} - 65347854106219631896974\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(-4463751346002819507674003 \beta_{15} + 4463751346002819507674003 \beta_{14} + 23815395801515580805671949 \beta_{13} + 32715664698960503979437171 \beta_{12} + 510700801847726488429050983 \beta_{11} + 11415529105445301476787538 \beta_{10} + 111727816770755895805062025 \beta_{9} + 145777785699912673549655411 \beta_{7} + 60600257929679495829187163246 \beta_{6} + 60900713769976766099460239290 \beta_{5} + 26700806692334769521295666 \beta_{4} - 8615383842001635759238680949 \beta_{3} - 657113859119942177163076543812 \beta_{2} - 657045298343428449609375151682 \beta_{1} + 2305005863464805619746330548294286\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(-1410380059661762334970416461 \beta_{15} - 1410380059661762334970416461 \beta_{14} - 5107749062998694207030843283 \beta_{13} + 11844603080598797440071151853 \beta_{12} - 92688560596054856674518253289 \beta_{11} + 18465584829054123964182242055 \beta_{9} - 122042256487784856728625513490 \beta_{8} + 1056643581176571869853426179413 \beta_{7} - 1891578681013438949954113038982 \beta_{6} + 1962866733696226201723542397522 \beta_{5} + 216104747329436891929250549219346382 \beta_{4} + 39012453350193677501234833555 \beta_{3} - 2323927299787637588787178124529410 \beta_{2} + 2323994888762012784286637530062532 \beta_{1} + 20210562052800309699120925710\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(1628437563219706481371552363987 \beta_{15} - 1628437563219706481371552363987 \beta_{14} - 9520071937264313415233099254221 \beta_{13} - 12435372061697881749937055755955 \beta_{12} - 195365671372161228407890413913511 \beta_{11} - 6435487304145958266677341817554 \beta_{10} - 46746093687065469005915173656137 \beta_{9} - 56346260059622272080277365776307 \beta_{7} - 22137144661864688180170268026617582 \beta_{6} - 22252752482108366292665526714671930 \beta_{5} - 8745900373300705004111869505202 \beta_{4} + 2990131389303212274568075103739189 \beta_{3} + 246183533772495026569095943881435204 \beta_{2} + 246154199385119270020439123353923522 \beta_{1} - 793720234603178962556823477468005774606\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(\)\(32\!\cdots\!01\)\( \beta_{15} + \)\(32\!\cdots\!01\)\( \beta_{14} + \)\(23\!\cdots\!87\)\( \beta_{13} - \)\(44\!\cdots\!53\)\( \beta_{12} + \)\(33\!\cdots\!17\)\( \beta_{11} - \)\(12\!\cdots\!79\)\( \beta_{9} + \)\(44\!\cdots\!50\)\( \beta_{8} - \)\(39\!\cdots\!93\)\( \beta_{7} + \)\(39\!\cdots\!62\)\( \beta_{6} - \)\(42\!\cdots\!30\)\( \beta_{5} - \)\(81\!\cdots\!38\)\( \beta_{4} - \)\(15\!\cdots\!67\)\( \beta_{3} + \)\(80\!\cdots\!98\)\( \beta_{2} - \)\(80\!\cdots\!08\)\( \beta_{1} - \)\(63\!\cdots\!98\)\(\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
601.773i
521.397i
180.361i
55.4817i
49.9217i
335.182i
482.444i
590.308i
601.773i
521.397i
180.361i
55.4817i
49.9217i
335.182i
482.444i
590.308i
−569.773 569.773i −12889.4 + 12889.4i 387139.i 1.81741e6 715334.i 1.46881e7 8.52951e6 + 8.52951e6i 7.12189e7 7.12189e7i 5.51474e7i −1.44309e9 6.27936e8i
2.2 −489.397 489.397i 9614.07 9614.07i 216875.i −1.89157e6 + 486466.i −9.41019e6 −2.11052e7 2.11052e7i −2.21547e7 + 2.21547e7i 2.02560e8i 1.16380e9 + 6.87655e8i
2.3 −148.361 148.361i 24463.5 24463.5i 218122.i 1.64948e6 1.04590e6i −7.25885e6 3.87838e7 + 3.87838e7i −7.12526e7 + 7.12526e7i 8.09507e8i −3.99889e8 8.95466e7i
2.4 −23.4817 23.4817i −19449.6 + 19449.6i 261041.i −1.65425e6 1.03834e6i 913420. 3.92374e7 + 3.92374e7i −1.22853e7 + 1.22853e7i 3.69155e8i 1.44628e7 + 6.32266e7i
2.5 −17.9217 17.9217i −4350.59 + 4350.59i 261502.i 823487. + 1.77104e6i 155940. −2.28757e7 2.28757e7i −9.38460e6 + 9.38460e6i 3.49565e8i 1.69817e7 4.64982e7i
2.6 367.182 + 367.182i 5232.51 5232.51i 7501.19i −39872.2 1.95272e6i 3.84257e6 −4.50520e7 4.50520e7i 9.35002e7 9.35002e7i 3.32662e8i 7.02362e8 7.31643e8i
2.7 514.444 + 514.444i 11708.2 11708.2i 267162.i −1.06274e6 + 1.63869e6i 1.20464e7 4.54981e7 + 4.54981e7i −2.58155e6 + 2.58155e6i 1.13259e8i −1.38973e9 + 2.96294e8i
2.8 622.308 + 622.308i −24393.6 + 24393.6i 512390.i 1.93064e6 + 295511.i −3.03607e7 −3.63215e6 3.63215e6i −1.55730e8 + 1.55730e8i 8.02680e8i 1.01755e9 + 1.38535e9i
3.1 −569.773 + 569.773i −12889.4 12889.4i 387139.i 1.81741e6 + 715334.i 1.46881e7 8.52951e6 8.52951e6i 7.12189e7 + 7.12189e7i 5.51474e7i −1.44309e9 + 6.27936e8i
3.2 −489.397 + 489.397i 9614.07 + 9614.07i 216875.i −1.89157e6 486466.i −9.41019e6 −2.11052e7 + 2.11052e7i −2.21547e7 2.21547e7i 2.02560e8i 1.16380e9 6.87655e8i
3.3 −148.361 + 148.361i 24463.5 + 24463.5i 218122.i 1.64948e6 + 1.04590e6i −7.25885e6 3.87838e7 3.87838e7i −7.12526e7 7.12526e7i 8.09507e8i −3.99889e8 + 8.95466e7i
3.4 −23.4817 + 23.4817i −19449.6 19449.6i 261041.i −1.65425e6 + 1.03834e6i 913420. 3.92374e7 3.92374e7i −1.22853e7 1.22853e7i 3.69155e8i 1.44628e7 6.32266e7i
3.5 −17.9217 + 17.9217i −4350.59 4350.59i 261502.i 823487. 1.77104e6i 155940. −2.28757e7 + 2.28757e7i −9.38460e6 9.38460e6i 3.49565e8i 1.69817e7 + 4.64982e7i
3.6 367.182 367.182i 5232.51 + 5232.51i 7501.19i −39872.2 + 1.95272e6i 3.84257e6 −4.50520e7 + 4.50520e7i 9.35002e7 + 9.35002e7i 3.32662e8i 7.02362e8 + 7.31643e8i
3.7 514.444 514.444i 11708.2 + 11708.2i 267162.i −1.06274e6 1.63869e6i 1.20464e7 4.54981e7 4.54981e7i −2.58155e6 2.58155e6i 1.13259e8i −1.38973e9 2.96294e8i
3.8 622.308 622.308i −24393.6 24393.6i 512390.i 1.93064e6 295511.i −3.03607e7 −3.63215e6 + 3.63215e6i −1.55730e8 1.55730e8i 8.02680e8i 1.01755e9 1.38535e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.c Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{19}^{\mathrm{new}}(5, [\chi])\).