# Properties

 Label 38.10.c.b Level $38$ Weight $10$ Character orbit 38.c Analytic conductor $19.571$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 38.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.5713617742$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{15} + 104960 x^{14} - 3899480 x^{13} + 8040649724 x^{12} - 270026959304 x^{11} + 275078771318560 x^{10} - 7559006944968384 x^{9} + 6786319237849450512 x^{8} - 132172428275143142112 x^{7} + 58413131602192832245632 x^{6} + 306187936352529387788160 x^{5} + 338231098965100183334201088 x^{4} + 833605976716998105836915712 x^{3} + 801869139824974932731497058304 x^{2} + 7085244797393532409215050711040 x + 1344731733940407880332419688038400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{12}\cdot 3^{5}\cdot 19^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -16 + 16 \beta_{2} ) q^{2} + ( 9 - \beta_{1} - 9 \beta_{2} ) q^{3} -256 \beta_{2} q^{4} + ( -43 + 43 \beta_{2} - \beta_{5} ) q^{5} + ( 16 \beta_{1} + 144 \beta_{2} + 16 \beta_{3} ) q^{6} + ( 230 - 5 \beta_{3} - \beta_{4} - \beta_{7} ) q^{7} + 4096 q^{8} + ( -45 \beta_{1} - 6645 \beta_{2} - 45 \beta_{3} + 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + ( -16 + 16 \beta_{2} ) q^{2} + ( 9 - \beta_{1} - 9 \beta_{2} ) q^{3} -256 \beta_{2} q^{4} + ( -43 + 43 \beta_{2} - \beta_{5} ) q^{5} + ( 16 \beta_{1} + 144 \beta_{2} + 16 \beta_{3} ) q^{6} + ( 230 - 5 \beta_{3} - \beta_{4} - \beta_{7} ) q^{7} + 4096 q^{8} + ( -45 \beta_{1} - 6645 \beta_{2} - 45 \beta_{3} + 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{9} + ( -688 \beta_{2} + 16 \beta_{6} ) q^{10} + ( 6654 - 40 \beta_{3} - 3 \beta_{4} - 8 \beta_{5} + 8 \beta_{6} - 3 \beta_{7} - \beta_{13} - \beta_{14} ) q^{11} + ( -2304 - 256 \beta_{3} ) q^{12} + ( 92 \beta_{1} - 65 \beta_{2} + 91 \beta_{3} + \beta_{5} + 7 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{13} + ( -3680 - 80 \beta_{1} + 3680 \beta_{2} + 16 \beta_{4} ) q^{14} + ( 252 \beta_{1} + 375 \beta_{2} + 250 \beta_{3} + 2 \beta_{5} - 57 \beta_{6} + 13 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{15} + ( -65536 + 65536 \beta_{2} ) q^{16} + ( -76434 + 62 \beta_{1} + 76432 \beta_{2} - \beta_{3} - 19 \beta_{4} - 15 \beta_{5} + \beta_{6} - 7 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 5 \beta_{12} - \beta_{13} + 3 \beta_{14} ) q^{17} + ( 106320 + 720 \beta_{3} + 32 \beta_{5} - 32 \beta_{6} - 16 \beta_{9} ) q^{18} + ( -85878 + 491 \beta_{1} - 29751 \beta_{2} - 377 \beta_{3} + 20 \beta_{4} + 3 \beta_{5} - 7 \beta_{6} + 11 \beta_{7} + 9 \beta_{8} - 4 \beta_{10} + 6 \beta_{11} - 5 \beta_{12} + \beta_{13} - 6 \beta_{14} - 5 \beta_{15} ) q^{19} + ( 11008 + 256 \beta_{5} - 256 \beta_{6} ) q^{20} + ( -117714 + 45 \beta_{1} + 117736 \beta_{2} + 11 \beta_{3} + 57 \beta_{4} - 242 \beta_{5} - 11 \beta_{6} - 4 \beta_{8} + 11 \beta_{9} + 22 \beta_{10} + 11 \beta_{11} + 5 \beta_{12} + 11 \beta_{13} + 7 \beta_{14} ) q^{21} + ( -106464 - 640 \beta_{1} + 106464 \beta_{2} + 48 \beta_{4} + 128 \beta_{5} + 16 \beta_{14} ) q^{22} + ( -249 \beta_{1} - 52132 \beta_{2} - 249 \beta_{3} + 222 \beta_{6} + 110 \beta_{7} - 27 \beta_{8} - 27 \beta_{9} - 7 \beta_{13} ) q^{23} + ( 36864 - 4096 \beta_{1} - 36864 \beta_{2} ) q^{24} + ( -4167 \beta_{1} - 414705 \beta_{2} - 4180 \beta_{3} + 13 \beta_{5} + 133 \beta_{6} + 44 \beta_{7} - 33 \beta_{8} - 20 \beta_{9} + 13 \beta_{10} - 13 \beta_{11} + 33 \beta_{13} + 13 \beta_{14} - 26 \beta_{15} ) q^{25} + ( 1056 + 16 \beta_{2} - 1440 \beta_{3} + 112 \beta_{5} - 128 \beta_{6} + 16 \beta_{8} + 32 \beta_{9} + 16 \beta_{10} + 32 \beta_{11} - 16 \beta_{14} ) q^{26} + ( -1058858 + 34 \beta_{2} - 9762 \beta_{3} - 86 \beta_{4} + 160 \beta_{5} - 194 \beta_{6} - 86 \beta_{7} + 34 \beta_{8} + 115 \beta_{9} + 34 \beta_{10} + 68 \beta_{11} - 20 \beta_{12} + 22 \beta_{13} - 12 \beta_{14} - 20 \beta_{15} ) q^{27} + ( 1280 \beta_{1} - 58880 \beta_{2} + 1280 \beta_{3} + 256 \beta_{7} ) q^{28} + ( -3627 \beta_{1} - 635812 \beta_{2} - 3643 \beta_{3} + 16 \beta_{5} + 210 \beta_{6} + 225 \beta_{7} + 7 \beta_{8} + 23 \beta_{9} + 16 \beta_{10} - 16 \beta_{11} - 49 \beta_{13} + 16 \beta_{14} - 15 \beta_{15} ) q^{29} + ( -5968 + 32 \beta_{2} - 3968 \beta_{3} - 208 \beta_{4} - 912 \beta_{5} + 880 \beta_{6} - 208 \beta_{7} + 32 \beta_{8} + 32 \beta_{10} + 64 \beta_{11} - 16 \beta_{12} + 32 \beta_{13} - 16 \beta_{15} ) q^{30} + ( 539861 + 34 \beta_{2} + 6006 \beta_{3} - 320 \beta_{4} - 639 \beta_{5} + 605 \beta_{6} - 320 \beta_{7} + 34 \beta_{8} - 51 \beta_{9} + 34 \beta_{10} + 68 \beta_{11} - 85 \beta_{12} + 35 \beta_{13} + \beta_{14} - 85 \beta_{15} ) q^{31} -1048576 \beta_{2} q^{32} + ( -949116 + 367 \beta_{1} + 949286 \beta_{2} + 85 \beta_{3} + 309 \beta_{4} - 307 \beta_{5} - 85 \beta_{6} - 50 \beta_{8} + 85 \beta_{9} + 170 \beta_{10} + 85 \beta_{11} - 5 \beta_{12} + 85 \beta_{13} + 131 \beta_{14} ) q^{33} + ( -992 \beta_{1} - 1222928 \beta_{2} - 1008 \beta_{3} + 16 \beta_{5} + 224 \beta_{6} - 304 \beta_{7} + 96 \beta_{8} + 112 \beta_{9} + 16 \beta_{10} - 16 \beta_{11} + 64 \beta_{13} + 16 \beta_{14} - 80 \beta_{15} ) q^{34} + ( -248980 - 23682 \beta_{1} + 249076 \beta_{2} + 48 \beta_{3} - 367 \beta_{4} - 1021 \beta_{5} - 48 \beta_{6} + 148 \beta_{8} + 48 \beta_{9} + 96 \beta_{10} + 48 \beta_{11} - 59 \beta_{12} + 48 \beta_{13} + 133 \beta_{14} ) q^{35} + ( -1701120 + 11520 \beta_{1} + 1701120 \beta_{2} - 512 \beta_{5} - 256 \beta_{8} ) q^{36} + ( -2076757 + 85 \beta_{2} - 37548 \beta_{3} + 120 \beta_{4} + 2068 \beta_{5} - 2153 \beta_{6} + 120 \beta_{7} + 85 \beta_{8} + 72 \beta_{9} + 85 \beta_{10} + 170 \beta_{11} + 20 \beta_{12} - 56 \beta_{13} - 141 \beta_{14} + 20 \beta_{15} ) q^{37} + ( 1850000 - 13984 \beta_{1} - 1374144 \beta_{2} - 8016 \beta_{3} - 176 \beta_{4} - 112 \beta_{5} + 64 \beta_{6} + 144 \beta_{7} - 144 \beta_{8} - 144 \beta_{9} - 96 \beta_{10} - 160 \beta_{11} + 80 \beta_{12} - 112 \beta_{13} - 16 \beta_{14} ) q^{38} + ( 2381048 - 60 \beta_{2} + 34536 \beta_{3} + 1409 \beta_{4} + 3748 \beta_{5} - 3688 \beta_{6} + 1409 \beta_{7} - 60 \beta_{8} - 507 \beta_{9} - 60 \beta_{10} - 120 \beta_{11} + 30 \beta_{12} + 13 \beta_{13} + 73 \beta_{14} + 30 \beta_{15} ) q^{39} + ( -176128 + 176128 \beta_{2} - 4096 \beta_{5} ) q^{40} + ( -911404 - 42001 \beta_{1} + 911404 \beta_{2} + 1104 \beta_{4} + 5644 \beta_{5} - 255 \beta_{8} - 272 \beta_{14} ) q^{41} + ( -720 \beta_{1} - 1883600 \beta_{2} - 544 \beta_{3} - 176 \beta_{5} + 4048 \beta_{6} + 912 \beta_{7} + 240 \beta_{8} + 64 \beta_{9} - 176 \beta_{10} + 176 \beta_{11} - 64 \beta_{13} - 176 \beta_{14} + 80 \beta_{15} ) q^{42} + ( -12352330 + 75008 \beta_{1} + 12352222 \beta_{2} - 54 \beta_{3} + 111 \beta_{4} + 2320 \beta_{5} + 54 \beta_{6} - 523 \beta_{8} - 54 \beta_{9} - 108 \beta_{10} - 54 \beta_{11} - 190 \beta_{12} - 54 \beta_{13} + 317 \beta_{14} ) q^{43} + ( 10240 \beta_{1} - 1703424 \beta_{2} + 10240 \beta_{3} - 2048 \beta_{6} + 768 \beta_{7} + 256 \beta_{13} ) q^{44} + ( 5574129 - 220 \beta_{2} - 12335 \beta_{3} + 2493 \beta_{4} - 771 \beta_{5} + 991 \beta_{6} + 2493 \beta_{7} - 220 \beta_{8} - 917 \beta_{9} - 220 \beta_{10} - 440 \beta_{11} + 5 \beta_{12} + 7 \beta_{13} + 227 \beta_{14} + 5 \beta_{15} ) q^{45} + ( 834112 + 3984 \beta_{3} - 1760 \beta_{4} + 3552 \beta_{5} - 3552 \beta_{6} - 1760 \beta_{7} + 432 \beta_{9} + 112 \beta_{13} + 112 \beta_{14} ) q^{46} + ( -8170 \beta_{1} + 4268059 \beta_{2} - 8400 \beta_{3} + 230 \beta_{5} - 10425 \beta_{6} + 47 \beta_{7} + 408 \beta_{8} + 638 \beta_{9} + 230 \beta_{10} - 230 \beta_{11} + 132 \beta_{13} + 230 \beta_{14} + 65 \beta_{15} ) q^{47} + ( 65536 \beta_{1} + 589824 \beta_{2} + 65536 \beta_{3} ) q^{48} + ( 4033859 - 180 \beta_{2} - 101322 \beta_{3} - 3365 \beta_{4} - 7333 \beta_{5} + 7513 \beta_{6} - 3365 \beta_{7} - 180 \beta_{8} - 626 \beta_{9} - 180 \beta_{10} - 360 \beta_{11} - 85 \beta_{12} - 595 \beta_{13} - 415 \beta_{14} - 85 \beta_{15} ) q^{49} + ( 6635488 + 208 \beta_{2} + 67088 \beta_{3} - 704 \beta_{4} + 2128 \beta_{5} - 2336 \beta_{6} - 704 \beta_{7} + 208 \beta_{8} + 528 \beta_{9} + 208 \beta_{10} + 416 \beta_{11} + 416 \beta_{12} - 320 \beta_{13} - 528 \beta_{14} + 416 \beta_{15} ) q^{50} + ( 157962 \beta_{1} + 2043828 \beta_{2} + 157632 \beta_{3} + 330 \beta_{5} - 5643 \beta_{6} - 4950 \beta_{7} - 777 \beta_{8} - 447 \beta_{9} + 330 \beta_{10} - 330 \beta_{11} + 78 \beta_{13} + 330 \beta_{14} + 165 \beta_{15} ) q^{51} + ( -16896 - 23552 \beta_{1} + 16384 \beta_{2} - 256 \beta_{3} - 2048 \beta_{5} + 256 \beta_{6} + 256 \beta_{8} - 256 \beta_{9} - 512 \beta_{10} - 256 \beta_{11} - 256 \beta_{13} ) q^{52} + ( -21265 \beta_{1} - 13773304 \beta_{2} - 21200 \beta_{3} - 65 \beta_{5} + 31220 \beta_{6} + 2253 \beta_{7} + 1507 \beta_{8} + 1442 \beta_{9} - 65 \beta_{10} + 65 \beta_{11} - 122 \beta_{13} - 65 \beta_{14} + 405 \beta_{15} ) q^{53} + ( 16941728 - 157280 \beta_{1} - 16942816 \beta_{2} - 544 \beta_{3} + 1376 \beta_{4} - 3104 \beta_{5} + 544 \beta_{6} + 1296 \beta_{8} - 544 \beta_{9} - 1088 \beta_{10} - 544 \beta_{11} + 320 \beta_{12} - 544 \beta_{13} - 352 \beta_{14} ) q^{54} + ( 17001200 - 40063 \beta_{1} - 17000736 \beta_{2} + 232 \beta_{3} + 4015 \beta_{4} - 24058 \beta_{5} - 232 \beta_{6} + 1334 \beta_{8} + 232 \beta_{9} + 464 \beta_{10} + 232 \beta_{11} + 174 \beta_{12} + 232 \beta_{13} - 506 \beta_{14} ) q^{55} + ( 942080 - 20480 \beta_{3} - 4096 \beta_{4} - 4096 \beta_{7} ) q^{56} + ( -11040698 - 14309 \beta_{1} + 23872302 \beta_{2} - 122729 \beta_{3} - 5801 \beta_{4} - 30636 \beta_{5} + 25756 \beta_{6} - 7977 \beta_{7} - 1307 \beta_{8} + 777 \beta_{9} - 365 \beta_{10} + 101 \beta_{11} - 245 \beta_{12} - 820 \beta_{13} - 684 \beta_{14} - 35 \beta_{15} ) q^{57} + ( 10173248 + 256 \beta_{2} + 58544 \beta_{3} - 3600 \beta_{4} + 3360 \beta_{5} - 3616 \beta_{6} - 3600 \beta_{7} + 256 \beta_{8} - 112 \beta_{9} + 256 \beta_{10} + 512 \beta_{11} + 240 \beta_{12} + 1040 \beta_{13} + 784 \beta_{14} + 240 \beta_{15} ) q^{58} + ( -260642 + 53054 \beta_{1} + 259422 \beta_{2} - 610 \beta_{3} - 5436 \beta_{4} + 17302 \beta_{5} + 610 \beta_{6} - 797 \beta_{8} - 610 \beta_{9} - 1220 \beta_{10} - 610 \beta_{11} - 120 \beta_{12} - 610 \beta_{13} - 508 \beta_{14} ) q^{59} + ( 95488 - 64512 \beta_{1} - 96512 \beta_{2} - 512 \beta_{3} + 3328 \beta_{4} + 14080 \beta_{5} + 512 \beta_{6} - 512 \beta_{8} - 512 \beta_{9} - 1024 \beta_{10} - 512 \beta_{11} + 256 \beta_{12} - 512 \beta_{13} - 512 \beta_{14} ) q^{60} + ( 66946 \beta_{1} - 27721501 \beta_{2} + 66350 \beta_{3} + 596 \beta_{5} + 21899 \beta_{6} + 10358 \beta_{7} + 3858 \beta_{8} + 4454 \beta_{9} + 596 \beta_{10} - 596 \beta_{11} + 962 \beta_{13} + 596 \beta_{14} - 430 \beta_{15} ) q^{61} + ( -8637776 + 95008 \beta_{1} + 8636688 \beta_{2} - 544 \beta_{3} + 5120 \beta_{4} + 9680 \beta_{5} + 544 \beta_{6} - 1360 \beta_{8} - 544 \beta_{9} - 1088 \beta_{10} - 544 \beta_{11} + 1360 \beta_{12} - 544 \beta_{13} - 560 \beta_{14} ) q^{62} + ( 307136 \beta_{1} + 7198392 \beta_{2} + 306940 \beta_{3} + 196 \beta_{5} - 48552 \beta_{6} + 8384 \beta_{7} - 2520 \beta_{8} - 2324 \beta_{9} + 196 \beta_{10} - 196 \beta_{11} + 1092 \beta_{13} + 196 \beta_{14} + 560 \beta_{15} ) q^{63} + 16777216 q^{64} + ( 16351546 + 245 \beta_{2} - 342190 \beta_{3} + 470 \beta_{4} + 59377 \beta_{5} - 59622 \beta_{6} + 470 \beta_{7} + 245 \beta_{8} + 3800 \beta_{9} + 245 \beta_{10} + 490 \beta_{11} + 620 \beta_{12} + 50 \beta_{13} - 195 \beta_{14} + 620 \beta_{15} ) q^{65} + ( -5872 \beta_{1} - 15187216 \beta_{2} - 4512 \beta_{3} - 1360 \beta_{5} + 6272 \beta_{6} + 4944 \beta_{7} + 2160 \beta_{8} + 800 \beta_{9} - 1360 \beta_{10} + 1360 \beta_{11} + 736 \beta_{13} - 1360 \beta_{14} - 80 \beta_{15} ) q^{66} + ( -65257 \beta_{1} + 29675469 \beta_{2} - 64793 \beta_{3} - 464 \beta_{5} + 12527 \beta_{6} - 4391 \beta_{7} - 7138 \beta_{8} - 7602 \beta_{9} - 464 \beta_{10} + 464 \beta_{11} - 2403 \beta_{13} - 464 \beta_{14} + 875 \beta_{15} ) q^{67} + ( 19567104 + 256 \beta_{2} + 16384 \beta_{3} + 4864 \beta_{4} + 3584 \beta_{5} - 3840 \beta_{6} + 4864 \beta_{7} + 256 \beta_{8} - 1536 \beta_{9} + 256 \beta_{10} + 512 \beta_{11} + 1280 \beta_{12} - 768 \beta_{13} - 1024 \beta_{14} + 1280 \beta_{15} ) q^{68} + ( -8348251 - 1324 \beta_{2} + 437350 \beta_{3} - 5842 \beta_{4} + 9861 \beta_{5} - 8537 \beta_{6} - 5842 \beta_{7} - 1324 \beta_{8} - 2286 \beta_{9} - 1324 \beta_{10} - 2648 \beta_{11} - 370 \beta_{12} - 154 \beta_{13} + 1170 \beta_{14} - 370 \beta_{15} ) q^{69} + ( 378912 \beta_{1} - 3984448 \beta_{2} + 379680 \beta_{3} - 768 \beta_{5} + 17104 \beta_{6} - 5872 \beta_{7} - 1600 \beta_{8} - 2368 \beta_{9} - 768 \beta_{10} + 768 \beta_{11} + 1360 \beta_{13} - 768 \beta_{14} - 944 \beta_{15} ) q^{70} + ( -53780206 - 544442 \beta_{1} + 53779646 \beta_{2} - 280 \beta_{3} + 631 \beta_{4} - 46022 \beta_{5} + 280 \beta_{6} + 5313 \beta_{8} - 280 \beta_{9} - 560 \beta_{10} - 280 \beta_{11} + 140 \beta_{12} - 280 \beta_{13} + 1139 \beta_{14} ) q^{71} + ( -184320 \beta_{1} - 27217920 \beta_{2} - 184320 \beta_{3} + 8192 \beta_{6} + 4096 \beta_{8} + 4096 \beta_{9} ) q^{72} + ( 25017287 - 152706 \beta_{1} - 25014567 \beta_{2} + 1360 \beta_{3} + 30539 \beta_{4} - 36125 \beta_{5} - 1360 \beta_{6} + 3338 \beta_{8} + 1360 \beta_{9} + 2720 \beta_{10} + 1360 \beta_{11} - 405 \beta_{12} + 1360 \beta_{13} + 973 \beta_{14} ) q^{73} + ( 33228112 - 603488 \beta_{1} - 33230832 \beta_{2} - 1360 \beta_{3} - 1920 \beta_{4} - 34448 \beta_{5} + 1360 \beta_{6} - 208 \beta_{8} - 1360 \beta_{9} - 2720 \beta_{10} - 1360 \beta_{11} - 320 \beta_{12} - 1360 \beta_{13} + 896 \beta_{14} ) q^{74} + ( -113849646 - 1928 \beta_{2} + 304752 \beta_{3} + 30369 \beta_{4} + 3616 \beta_{5} - 1688 \beta_{6} + 30369 \beta_{7} - 1928 \beta_{8} - 468 \beta_{9} - 1928 \beta_{10} - 3856 \beta_{11} - 416 \beta_{12} - 865 \beta_{13} + 1063 \beta_{14} - 416 \beta_{15} ) q^{75} + ( -7615232 + 98048 \beta_{1} + 29602560 \beta_{2} + 224768 \beta_{3} - 2304 \beta_{4} + 1024 \beta_{5} + 768 \beta_{6} - 5120 \beta_{7} + 2304 \beta_{9} + 2560 \beta_{10} + 1024 \beta_{11} + 1536 \beta_{13} + 1792 \beta_{14} + 1280 \beta_{15} ) q^{76} + ( 125792572 - 159 \beta_{2} - 471767 \beta_{3} - 48733 \beta_{4} + 68701 \beta_{5} - 68542 \beta_{6} - 48733 \beta_{7} - 159 \beta_{8} - 8193 \beta_{9} - 159 \beta_{10} - 318 \beta_{11} - 1905 \beta_{12} - 987 \beta_{13} - 828 \beta_{14} - 1905 \beta_{15} ) q^{77} + ( -38096768 + 554496 \beta_{1} + 38098688 \beta_{2} + 960 \beta_{3} - 22544 \beta_{4} - 59008 \beta_{5} - 960 \beta_{6} - 7152 \beta_{8} + 960 \beta_{9} + 1920 \beta_{10} + 960 \beta_{11} - 480 \beta_{12} + 960 \beta_{13} - 208 \beta_{14} ) q^{78} + ( -36880336 - 632996 \beta_{1} + 36884568 \beta_{2} + 2116 \beta_{3} + 28797 \beta_{4} + 128322 \beta_{5} - 2116 \beta_{6} - 849 \beta_{8} + 2116 \beta_{9} + 4232 \beta_{10} + 2116 \beta_{11} - 1460 \beta_{12} + 2116 \beta_{13} + 1905 \beta_{14} ) q^{79} + ( -2818048 \beta_{2} + 65536 \beta_{6} ) q^{80} + ( -134275808 + 2325377 \beta_{1} + 134283608 \beta_{2} + 3900 \beta_{3} - 37142 \beta_{4} - 142452 \beta_{5} - 3900 \beta_{6} - 5377 \beta_{8} + 3900 \beta_{9} + 7800 \beta_{10} + 3900 \beta_{11} - 1950 \beta_{12} + 3900 \beta_{13} + 14 \beta_{14} ) q^{81} + ( 672016 \beta_{1} - 14582464 \beta_{2} + 672016 \beta_{3} - 90304 \beta_{6} + 17664 \beta_{7} + 4080 \beta_{8} + 4080 \beta_{9} - 4352 \beta_{13} ) q^{82} + ( 99256875 + 1590 \beta_{2} - 1032793 \beta_{3} + 37739 \beta_{4} - 10278 \beta_{5} + 8688 \beta_{6} + 37739 \beta_{7} + 1590 \beta_{8} - 437 \beta_{9} + 1590 \beta_{10} + 3180 \beta_{11} + 230 \beta_{12} - 889 \beta_{13} - 2479 \beta_{14} + 230 \beta_{15} ) q^{83} + ( 30134784 - 2816 \beta_{2} + 5888 \beta_{3} - 14592 \beta_{4} + 64768 \beta_{5} - 61952 \beta_{6} - 14592 \beta_{7} - 2816 \beta_{8} - 3840 \beta_{9} - 2816 \beta_{10} - 5632 \beta_{11} - 1280 \beta_{12} - 1792 \beta_{13} + 1024 \beta_{14} - 1280 \beta_{15} ) q^{84} + ( -546440 \beta_{1} - 36947275 \beta_{2} - 547825 \beta_{3} + 1385 \beta_{5} + 325855 \beta_{6} + 22020 \beta_{7} - 2270 \beta_{8} - 885 \beta_{9} + 1385 \beta_{10} - 1385 \beta_{11} - 3215 \beta_{13} + 1385 \beta_{14} - 860 \beta_{15} ) q^{85} + ( -1200128 \beta_{1} - 197636416 \beta_{2} - 1200992 \beta_{3} + 864 \beta_{5} - 37984 \beta_{6} + 1776 \beta_{7} + 7504 \beta_{8} + 8368 \beta_{9} + 864 \beta_{10} - 864 \beta_{11} + 5936 \beta_{13} + 864 \beta_{14} - 3040 \beta_{15} ) q^{86} + ( -103215458 + 376 \beta_{2} - 1165135 \beta_{3} + 33100 \beta_{4} + 156720 \beta_{5} - 157096 \beta_{6} + 33100 \beta_{7} + 376 \beta_{8} + 3417 \beta_{9} + 376 \beta_{10} + 752 \beta_{11} - 2750 \beta_{12} + 6331 \beta_{13} + 5955 \beta_{14} - 2750 \beta_{15} ) q^{87} + ( 27254784 - 163840 \beta_{3} - 12288 \beta_{4} - 32768 \beta_{5} + 32768 \beta_{6} - 12288 \beta_{7} - 4096 \beta_{13} - 4096 \beta_{14} ) q^{88} + ( 758608 \beta_{1} - 55422579 \beta_{2} + 761943 \beta_{3} - 3335 \beta_{5} + 140910 \beta_{6} - 41435 \beta_{7} + 13148 \beta_{8} + 9813 \beta_{9} - 3335 \beta_{10} + 3335 \beta_{11} - 7108 \beta_{13} - 3335 \beta_{14} + 695 \beta_{15} ) q^{89} + ( -89186064 - 190320 \beta_{1} + 89193104 \beta_{2} + 3520 \beta_{3} - 39888 \beta_{4} + 15856 \beta_{5} - 3520 \beta_{6} - 11152 \beta_{8} + 3520 \beta_{9} + 7040 \beta_{10} + 3520 \beta_{11} - 80 \beta_{12} + 3520 \beta_{13} - 112 \beta_{14} ) q^{90} + ( -2534884 \beta_{1} - 36158006 \beta_{2} - 2535888 \beta_{3} + 1004 \beta_{5} + 26930 \beta_{6} - 9684 \beta_{7} + 2710 \beta_{8} + 3714 \beta_{9} + 1004 \beta_{10} - 1004 \beta_{11} + 1802 \beta_{13} + 1004 \beta_{14} - 3110 \beta_{15} ) q^{91} + ( -13345792 + 63744 \beta_{1} + 13345792 \beta_{2} + 28160 \beta_{4} - 56832 \beta_{5} + 6912 \beta_{8} - 1792 \beta_{14} ) q^{92} + ( 165852573 - 2899216 \beta_{1} - 165846415 \beta_{2} + 3079 \beta_{3} + 21438 \beta_{4} - 381961 \beta_{5} - 3079 \beta_{6} - 7463 \beta_{8} + 3079 \beta_{9} + 6158 \beta_{10} + 3079 \beta_{11} - 710 \beta_{12} + 3079 \beta_{13} + 2030 \beta_{14} ) q^{93} + ( -68285264 + 3680 \beta_{2} + 138080 \beta_{3} - 752 \beta_{4} - 166800 \beta_{5} + 163120 \beta_{6} - 752 \beta_{7} + 3680 \beta_{8} - 6528 \beta_{9} + 3680 \beta_{10} + 7360 \beta_{11} - 1040 \beta_{12} + 1568 \beta_{13} - 2112 \beta_{14} - 1040 \beta_{15} ) q^{94} + ( -6286969 - 518890 \beta_{1} + 14866756 \beta_{2} + 2192172 \beta_{3} - 58153 \beta_{4} + 192821 \beta_{5} + 179173 \beta_{6} - 54110 \beta_{7} - 7355 \beta_{8} - 10357 \beta_{9} + 442 \beta_{10} + 2624 \beta_{11} + 189 \beta_{12} + 5899 \beta_{13} - 1630 \beta_{14} - 4161 \beta_{15} ) q^{95} + ( -9437184 - 1048576 \beta_{3} ) q^{96} + ( -76585634 + 518235 \beta_{1} + 76593862 \beta_{2} + 4114 \beta_{3} - 25967 \beta_{4} + 39515 \beta_{5} - 4114 \beta_{6} + 12931 \beta_{8} + 4114 \beta_{9} + 8228 \beta_{10} + 4114 \beta_{11} + 4145 \beta_{12} + 4114 \beta_{13} + 7951 \beta_{14} ) q^{97} + ( -64541744 - 1615392 \beta_{1} + 64547504 \beta_{2} + 2880 \beta_{3} + 53840 \beta_{4} + 120208 \beta_{5} - 2880 \beta_{6} - 7136 \beta_{8} + 2880 \beta_{9} + 5760 \beta_{10} + 2880 \beta_{11} + 1360 \beta_{12} + 2880 \beta_{13} + 9520 \beta_{14} ) q^{98} + ( 1769662 \beta_{1} + 150799758 \beta_{2} + 1771718 \beta_{3} - 2056 \beta_{5} - 258845 \beta_{6} + 117007 \beta_{7} - 22162 \beta_{8} - 24218 \beta_{9} - 2056 \beta_{10} + 2056 \beta_{11} - 7731 \beta_{13} - 2056 \beta_{14} + 3055 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 128q^{2} + 70q^{3} - 2048q^{4} - 341q^{5} + 1120q^{6} + 3704q^{7} + 65536q^{8} - 53064q^{9} + O(q^{10})$$ $$16q - 128q^{2} + 70q^{3} - 2048q^{4} - 341q^{5} + 1120q^{6} + 3704q^{7} + 65536q^{8} - 53064q^{9} - 5456q^{10} + 106682q^{11} - 35840q^{12} - 683q^{13} - 29632q^{14} + 2299q^{15} - 524288q^{16} - 611267q^{17} + 1698048q^{18} - 1609677q^{19} + 174592q^{20} - 941048q^{21} - 853456q^{22} - 416119q^{23} + 286720q^{24} - 3309027q^{25} + 21856q^{26} - 16903604q^{27} - 474112q^{28} - 5079157q^{29} - 73568q^{30} + 8618324q^{31} - 8388608q^{32} - 7592195q^{33} - 9780272q^{34} - 2035632q^{35} - 13584384q^{36} - 33091520q^{37} + 18612624q^{38} + 37931126q^{39} - 1396736q^{40} - 7394646q^{41} - 15056768q^{42} - 98675599q^{43} - 13655296q^{44} + 89231852q^{45} + 13315808q^{46} + 34129475q^{47} + 4587520q^{48} + 65004400q^{49} + 105888864q^{50} + 16027749q^{51} - 174848q^{52} - 110053995q^{53} + 135228832q^{54} + 135992300q^{55} + 15171584q^{56} + 14986101q^{57} + 162533024q^{58} - 2017760q^{59} + 588544q^{60} - 221861413q^{61} - 68946592q^{62} + 56812456q^{63} + 268435456q^{64} + 262635974q^{65} - 121475120q^{66} + 237580440q^{67} + 312968704q^{68} - 135348410q^{69} - 32570112q^{70} - 431190909q^{71} - 217350144q^{72} + 199873544q^{73} + 264732160q^{74} - 1822944486q^{75} + 114275328q^{76} + 2014342624q^{77} - 303449008q^{78} - 296762835q^{79} - 22347776q^{80} - 1069077456q^{81} - 118314336q^{82} + 1592138306q^{83} + 481816576q^{84} - 293558115q^{85} - 1578809584q^{86} - 1647860854q^{87} + 436969472q^{88} - 444394631q^{89} - 713854816q^{90} - 284099544q^{91} - 106526464q^{92} + 1322110374q^{93} - 1092143200q^{94} + 8705459q^{95} - 146800640q^{96} - 611719542q^{97} - 520035200q^{98} + 1201848626q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 104960 x^{14} - 3899480 x^{13} + 8040649724 x^{12} - 270026959304 x^{11} + 275078771318560 x^{10} - 7559006944968384 x^{9} + 6786319237849450512 x^{8} - 132172428275143142112 x^{7} + 58413131602192832245632 x^{6} + 306187936352529387788160 x^{5} + 338231098965100183334201088 x^{4} + 833605976716998105836915712 x^{3} + 801869139824974932731497058304 x^{2} + 7085244797393532409215050711040 x + 1344731733940407880332419688038400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$92\!\cdots\!97$$$$\nu^{15} -$$$$11\!\cdots\!98$$$$\nu^{14} +$$$$94\!\cdots\!56$$$$\nu^{13} -$$$$15\!\cdots\!52$$$$\nu^{12} +$$$$76\!\cdots\!92$$$$\nu^{11} -$$$$11\!\cdots\!32$$$$\nu^{10} +$$$$26\!\cdots\!44$$$$\nu^{9} -$$$$35\!\cdots\!56$$$$\nu^{8} +$$$$64\!\cdots\!56$$$$\nu^{7} -$$$$81\!\cdots\!56$$$$\nu^{6} +$$$$52\!\cdots\!96$$$$\nu^{5} -$$$$48\!\cdots\!52$$$$\nu^{4} +$$$$17\!\cdots\!00$$$$\nu^{3} -$$$$36\!\cdots\!36$$$$\nu^{2} +$$$$31\!\cdots\!40$$$$\nu -$$$$18\!\cdots\!00$$$$)/$$$$66\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$33\!\cdots\!77$$$$\nu^{15} +$$$$70\!\cdots\!32$$$$\nu^{14} +$$$$34\!\cdots\!96$$$$\nu^{13} -$$$$53\!\cdots\!32$$$$\nu^{12} +$$$$25\!\cdots\!72$$$$\nu^{11} -$$$$29\!\cdots\!12$$$$\nu^{10} +$$$$84\!\cdots\!04$$$$\nu^{9} -$$$$51\!\cdots\!96$$$$\nu^{8} +$$$$20\!\cdots\!96$$$$\nu^{7} +$$$$37\!\cdots\!04$$$$\nu^{6} +$$$$15\!\cdots\!36$$$$\nu^{5} +$$$$41\!\cdots\!68$$$$\nu^{4} +$$$$11\!\cdots\!00$$$$\nu^{3} +$$$$12\!\cdots\!24$$$$\nu^{2} +$$$$72\!\cdots\!40$$$$\nu +$$$$36\!\cdots\!00$$$$)/$$$$19\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$51\!\cdots\!21$$$$\nu^{15} -$$$$24\!\cdots\!86$$$$\nu^{14} -$$$$38\!\cdots\!08$$$$\nu^{13} -$$$$26\!\cdots\!64$$$$\nu^{12} -$$$$16\!\cdots\!56$$$$\nu^{11} -$$$$21\!\cdots\!24$$$$\nu^{10} +$$$$46\!\cdots\!08$$$$\nu^{9} -$$$$81\!\cdots\!92$$$$\nu^{8} +$$$$23\!\cdots\!92$$$$\nu^{7} -$$$$21\!\cdots\!92$$$$\nu^{6} +$$$$97\!\cdots\!72$$$$\nu^{5} -$$$$24\!\cdots\!64$$$$\nu^{4} +$$$$34\!\cdots\!00$$$$\nu^{3} -$$$$92\!\cdots\!52$$$$\nu^{2} +$$$$10\!\cdots\!80$$$$\nu -$$$$21\!\cdots\!00$$$$)/$$$$30\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$55\!\cdots\!53$$$$\nu^{15} +$$$$10\!\cdots\!58$$$$\nu^{14} +$$$$50\!\cdots\!84$$$$\nu^{13} +$$$$94\!\cdots\!52$$$$\nu^{12} +$$$$32\!\cdots\!48$$$$\nu^{11} +$$$$76\!\cdots\!12$$$$\nu^{10} +$$$$64\!\cdots\!36$$$$\nu^{9} +$$$$28\!\cdots\!16$$$$\nu^{8} +$$$$10\!\cdots\!44$$$$\nu^{7} +$$$$75\!\cdots\!36$$$$\nu^{6} -$$$$28\!\cdots\!36$$$$\nu^{5} +$$$$81\!\cdots\!12$$$$\nu^{4} -$$$$10\!\cdots\!80$$$$\nu^{3} +$$$$32\!\cdots\!96$$$$\nu^{2} -$$$$12\!\cdots\!40$$$$\nu +$$$$72\!\cdots\!00$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!31$$$$\nu^{15} +$$$$91\!\cdots\!26$$$$\nu^{14} -$$$$21\!\cdots\!60$$$$\nu^{13} +$$$$16\!\cdots\!76$$$$\nu^{12} -$$$$16\!\cdots\!28$$$$\nu^{11} +$$$$12\!\cdots\!32$$$$\nu^{10} -$$$$57\!\cdots\!76$$$$\nu^{9} +$$$$38\!\cdots\!20$$$$\nu^{8} -$$$$14\!\cdots\!88$$$$\nu^{7} +$$$$81\!\cdots\!44$$$$\nu^{6} -$$$$12\!\cdots\!16$$$$\nu^{5} +$$$$33\!\cdots\!28$$$$\nu^{4} -$$$$59\!\cdots\!96$$$$\nu^{3} +$$$$14\!\cdots\!60$$$$\nu^{2} -$$$$13\!\cdots\!00$$$$\nu +$$$$44\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!61$$$$\nu^{15} +$$$$32\!\cdots\!14$$$$\nu^{14} -$$$$29\!\cdots\!68$$$$\nu^{13} +$$$$44\!\cdots\!76$$$$\nu^{12} -$$$$23\!\cdots\!36$$$$\nu^{11} +$$$$32\!\cdots\!36$$$$\nu^{10} -$$$$85\!\cdots\!52$$$$\nu^{9} +$$$$10\!\cdots\!68$$$$\nu^{8} -$$$$21\!\cdots\!28$$$$\nu^{7} +$$$$23\!\cdots\!48$$$$\nu^{6} -$$$$19\!\cdots\!08$$$$\nu^{5} +$$$$13\!\cdots\!16$$$$\nu^{4} -$$$$63\!\cdots\!20$$$$\nu^{3} +$$$$50\!\cdots\!08$$$$\nu^{2} -$$$$12\!\cdots\!20$$$$\nu +$$$$48\!\cdots\!00$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$18\!\cdots\!41$$$$\nu^{15} -$$$$31\!\cdots\!14$$$$\nu^{14} +$$$$19\!\cdots\!88$$$$\nu^{13} -$$$$39\!\cdots\!56$$$$\nu^{12} +$$$$16\!\cdots\!96$$$$\nu^{11} -$$$$29\!\cdots\!56$$$$\nu^{10} +$$$$58\!\cdots\!72$$$$\nu^{9} -$$$$97\!\cdots\!88$$$$\nu^{8} +$$$$14\!\cdots\!68$$$$\nu^{7} -$$$$22\!\cdots\!28$$$$\nu^{6} +$$$$13\!\cdots\!68$$$$\nu^{5} -$$$$15\!\cdots\!76$$$$\nu^{4} +$$$$45\!\cdots\!60$$$$\nu^{3} -$$$$10\!\cdots\!28$$$$\nu^{2} +$$$$14\!\cdots\!20$$$$\nu -$$$$23\!\cdots\!00$$$$)/$$$$65\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$57\!\cdots\!16$$$$\nu^{15} -$$$$84\!\cdots\!59$$$$\nu^{14} +$$$$59\!\cdots\!08$$$$\nu^{13} -$$$$29\!\cdots\!56$$$$\nu^{12} +$$$$45\!\cdots\!16$$$$\nu^{11} -$$$$20\!\cdots\!16$$$$\nu^{10} +$$$$14\!\cdots\!12$$$$\nu^{9} -$$$$53\!\cdots\!08$$$$\nu^{8} +$$$$36\!\cdots\!68$$$$\nu^{7} -$$$$95\!\cdots\!88$$$$\nu^{6} +$$$$25\!\cdots\!48$$$$\nu^{5} +$$$$46\!\cdots\!04$$$$\nu^{4} +$$$$12\!\cdots\!20$$$$\nu^{3} +$$$$16\!\cdots\!52$$$$\nu^{2} +$$$$11\!\cdots\!20$$$$\nu +$$$$53\!\cdots\!00$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$91\!\cdots\!13$$$$\nu^{15} +$$$$70\!\cdots\!78$$$$\nu^{14} +$$$$90\!\cdots\!04$$$$\nu^{13} +$$$$37\!\cdots\!92$$$$\nu^{12} +$$$$65\!\cdots\!48$$$$\nu^{11} +$$$$32\!\cdots\!32$$$$\nu^{10} +$$$$19\!\cdots\!36$$$$\nu^{9} +$$$$11\!\cdots\!96$$$$\nu^{8} +$$$$44\!\cdots\!24$$$$\nu^{7} +$$$$32\!\cdots\!36$$$$\nu^{6} +$$$$14\!\cdots\!04$$$$\nu^{5} +$$$$31\!\cdots\!12$$$$\nu^{4} +$$$$10\!\cdots\!40$$$$\nu^{3} +$$$$61\!\cdots\!76$$$$\nu^{2} -$$$$64\!\cdots\!40$$$$\nu -$$$$15\!\cdots\!00$$$$)/$$$$93\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$94\!\cdots\!57$$$$\nu^{15} +$$$$29\!\cdots\!62$$$$\nu^{14} +$$$$10\!\cdots\!36$$$$\nu^{13} -$$$$19\!\cdots\!12$$$$\nu^{12} +$$$$77\!\cdots\!52$$$$\nu^{11} +$$$$75\!\cdots\!08$$$$\nu^{10} +$$$$27\!\cdots\!64$$$$\nu^{9} +$$$$25\!\cdots\!64$$$$\nu^{8} +$$$$68\!\cdots\!36$$$$\nu^{7} +$$$$12\!\cdots\!64$$$$\nu^{6} +$$$$66\!\cdots\!76$$$$\nu^{5} +$$$$32\!\cdots\!88$$$$\nu^{4} +$$$$40\!\cdots\!00$$$$\nu^{3} +$$$$21\!\cdots\!84$$$$\nu^{2} +$$$$12\!\cdots\!40$$$$\nu +$$$$50\!\cdots\!00$$$$)/$$$$93\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$14\!\cdots\!89$$$$\nu^{15} +$$$$30\!\cdots\!14$$$$\nu^{14} +$$$$12\!\cdots\!32$$$$\nu^{13} +$$$$27\!\cdots\!76$$$$\nu^{12} +$$$$82\!\cdots\!64$$$$\nu^{11} +$$$$22\!\cdots\!36$$$$\nu^{10} +$$$$14\!\cdots\!48$$$$\nu^{9} +$$$$84\!\cdots\!68$$$$\nu^{8} +$$$$19\!\cdots\!72$$$$\nu^{7} +$$$$21\!\cdots\!48$$$$\nu^{6} -$$$$87\!\cdots\!08$$$$\nu^{5} +$$$$23\!\cdots\!16$$$$\nu^{4} -$$$$32\!\cdots\!20$$$$\nu^{3} +$$$$94\!\cdots\!08$$$$\nu^{2} -$$$$76\!\cdots\!20$$$$\nu +$$$$21\!\cdots\!00$$$$)/$$$$12\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$35\!\cdots\!67$$$$\nu^{15} -$$$$27\!\cdots\!14$$$$\nu^{14} -$$$$38\!\cdots\!24$$$$\nu^{13} -$$$$15\!\cdots\!08$$$$\nu^{12} -$$$$28\!\cdots\!80$$$$\nu^{11} -$$$$12\!\cdots\!04$$$$\nu^{10} -$$$$98\!\cdots\!80$$$$\nu^{9} -$$$$46\!\cdots\!36$$$$\nu^{8} -$$$$24\!\cdots\!16$$$$\nu^{7} -$$$$12\!\cdots\!00$$$$\nu^{6} -$$$$23\!\cdots\!68$$$$\nu^{5} -$$$$14\!\cdots\!80$$$$\nu^{4} -$$$$16\!\cdots\!44$$$$\nu^{3} -$$$$71\!\cdots\!56$$$$\nu^{2} -$$$$40\!\cdots\!60$$$$\nu -$$$$89\!\cdots\!00$$$$)/$$$$22\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$21\!\cdots\!45$$$$\nu^{15} +$$$$26\!\cdots\!14$$$$\nu^{14} -$$$$21\!\cdots\!44$$$$\nu^{13} +$$$$11\!\cdots\!80$$$$\nu^{12} -$$$$16\!\cdots\!84$$$$\nu^{11} +$$$$77\!\cdots\!32$$$$\nu^{10} -$$$$50\!\cdots\!48$$$$\nu^{9} +$$$$24\!\cdots\!64$$$$\nu^{8} -$$$$11\!\cdots\!60$$$$\nu^{7} +$$$$49\!\cdots\!92$$$$\nu^{6} -$$$$55\!\cdots\!36$$$$\nu^{5} +$$$$26\!\cdots\!24$$$$\nu^{4} -$$$$16\!\cdots\!12$$$$\nu^{3} +$$$$25\!\cdots\!64$$$$\nu^{2} +$$$$14\!\cdots\!40$$$$\nu +$$$$59\!\cdots\!00$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$65\!\cdots\!77$$$$\nu^{15} -$$$$84\!\cdots\!78$$$$\nu^{14} +$$$$70\!\cdots\!56$$$$\nu^{13} -$$$$11\!\cdots\!32$$$$\nu^{12} +$$$$57\!\cdots\!32$$$$\nu^{11} -$$$$83\!\cdots\!92$$$$\nu^{10} +$$$$21\!\cdots\!24$$$$\nu^{9} -$$$$26\!\cdots\!56$$$$\nu^{8} +$$$$52\!\cdots\!96$$$$\nu^{7} -$$$$60\!\cdots\!76$$$$\nu^{6} +$$$$53\!\cdots\!76$$$$\nu^{5} -$$$$34\!\cdots\!92$$$$\nu^{4} +$$$$15\!\cdots\!80$$$$\nu^{3} -$$$$13\!\cdots\!36$$$$\nu^{2} +$$$$30\!\cdots\!40$$$$\nu -$$$$12\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{8} + 2 \beta_{6} - 27 \beta_{3} - 26247 \beta_{2} - 27 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$20 \beta_{15} + 12 \beta_{14} - 22 \beta_{13} + 20 \beta_{12} - 68 \beta_{11} - 34 \beta_{10} - 88 \beta_{9} - 34 \beta_{8} + 86 \beta_{7} + 248 \beta_{6} - 214 \beta_{5} + 86 \beta_{4} + 48156 \beta_{3} - 34 \beta_{2} + 703754$$ $$\nu^{4}$$ $$=$$ $$-778 \beta_{14} + 2676 \beta_{13} - 1230 \beta_{12} + 2676 \beta_{11} + 5352 \beta_{10} + 2676 \beta_{9} - 61996 \beta_{8} - 2676 \beta_{6} - 268506 \beta_{5} - 34046 \beta_{4} + 2676 \beta_{3} + 1263404996 \beta_{2} + 3230480 \beta_{1} - 1263399644$$ $$\nu^{5}$$ $$=$$ $$-1309040 \beta_{15} - 2907028 \beta_{14} - 919192 \beta_{13} + 2907028 \beta_{11} - 2907028 \beta_{10} + 5536824 \beta_{9} + 8443852 \beta_{8} - 7739524 \beta_{7} + 17156748 \beta_{6} - 2907028 \beta_{5} - 2722161024 \beta_{3} - 83999431580 \beta_{2} - 2725068052 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$143536500 \beta_{15} + 330697892 \beta_{14} + 5797748 \beta_{13} + 143536500 \beta_{12} - 649800288 \beta_{11} - 324900144 \beta_{10} - 4138301224 \beta_{9} - 324900144 \beta_{8} + 3728526172 \beta_{7} - 21269634124 \beta_{6} + 21594534268 \beta_{5} + 3728526172 \beta_{4} + 291728860600 \beta_{3} - 324900144 \beta_{2} + 71321822224984$$ $$\nu^{7}$$ $$=$$ $$139606515776 \beta_{14} + 211189868296 \beta_{13} - 81567747560 \beta_{12} + 211189868296 \beta_{11} + 422379736592 \beta_{10} + 211189868296 \beta_{9} - 477141519696 \beta_{8} - 211189868296 \beta_{6} - 2454004282104 \beta_{5} - 622107523776 \beta_{4} + 211189868296 \beta_{3} + 7601690316434528 \beta_{2} + 166063017877656 \beta_{1} - 7601267936697936$$ $$\nu^{8}$$ $$=$$ $$-12601294630920 \beta_{15} - 29424601708560 \beta_{14} - 25368495885000 \beta_{13} + 29424601708560 \beta_{11} - 29424601708560 \beta_{10} + 243230787299728 \beta_{9} + 272655389008288 \beta_{8} - 298857727314216 \beta_{7} + 1544370094344296 \beta_{6} - 29424601708560 \beta_{5} - 23859520187947632 \beta_{3} - 4343015124287732064 \beta_{2} - 23888944789656192 \beta_{1}$$ $$\nu^{9}$$ $$=$$ $$5266263452152160 \beta_{15} + 5424588245271360 \beta_{14} - 9220560875433328 \beta_{13} + 5266263452152160 \beta_{12} - 29290298241409376 \beta_{11} - 14645149120704688 \beta_{10} - 52786768266182320 \beta_{9} - 14645149120704688 \beta_{8} + 48835480263528272 \beta_{7} - 233198487042456640 \beta_{6} + 247843636163161328 \beta_{5} + 48835480263528272 \beta_{4} + 10594216081000607856 \beta_{3} - 14645149120704688 \beta_{2} + 621404562549728935808$$ $$\nu^{10}$$ $$=$$ $$567647592367004816 \beta_{14} + 2394630001463592000 \beta_{13} - 994749332008262640 \beta_{12} + 2394630001463592000 \beta_{11} + 4789260002927184000 \beta_{10} + 2394630001463592000 \beta_{9} - 15970923892938950752 \beta_{8} - 2394630001463592000 \beta_{6} - 104743641247754862096 \beta_{5} - 21652749334560233168 \beta_{4} + 2394630001463592000 \beta_{3} + 277664791619610154513376 \beta_{2} + 1851220367892271777376 \beta_{1} - 277660002359607227329376$$ $$\nu^{11}$$ $$=$$ $$-350900836200937420640 \beta_{15} - 1001924484016839380704 \beta_{14} - 402020008869882205408 \beta_{13} + 1001924484016839380704 \beta_{11} - 1001924484016839380704 \beta_{10} + 2922144494859172526784 \beta_{9} + 3924068978876011907488 \beta_{8} - 3760552674857383962112 \beta_{7} + 20017821328995950747904 \beta_{6} - 1001924484016839380704 \beta_{5} - 700930409566166095483584 \beta_{3} - 48184991157427080250824608 \beta_{2} - 701932334050182934864288 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$74774792097954971485920 \beta_{15} + 128716485029624825306336 \beta_{14} - 56276673586278957534688 \beta_{13} + 74774792097954971485920 \beta_{12} - 369986317231807565682048 \beta_{11} - 184993158615903782841024 \beta_{10} - 1255795106893379917515136 \beta_{9} - 184993158615903782841024 \beta_{8} + 1511785482410818711565152 \beta_{7} - 7194910015449248270968480 \beta_{6} + 7379903174065152053809504 \beta_{5} + 1511785482410818711565152 \beta_{4} + 138675738185218326753360256 \beta_{3} - 184993158615903782841024 \beta_{2} + 18336773129811286054018741696$$ $$\nu^{13}$$ $$=$$ $$39114999154649437266691136 \beta_{14} + 68485728950109868703647552 \beta_{13} - 23879105356219282562314880 \beta_{12} + 68485728950109868703647552 \beta_{11} + 136971457900219737407295104 \beta_{10} + 68485728950109868703647552 \beta_{9} - 218095587824474857461877248 \beta_{8} - 68485728950109868703647552 \beta_{6} - 1456231587543999265822864128 \beta_{5} - 284262250869598281245213376 \beta_{4} + 68485728950109868703647552 \beta_{3} + 3621143040417472631274706696832 \beta_{2} + 47382984461921733683005352640 \beta_{1} - 3621006068959572411537299401728$$ $$\nu^{14}$$ $$=$$ $$-5481832242444660767403432000 \beta_{15} - 13875201308468045727569675520 \beta_{14} - 9004873797246764864051078976 \beta_{13} + 13875201308468045727569675520 \beta_{11} - 13875201308468045727569675520 \beta_{10} + 72869063617592557029819187840 \beta_{9} + 86744264926060602757388863360 \beta_{8} - 104297823187561117181866458816 \beta_{7} + 508949056408521815058589692608 \beta_{6} - 13875201308468045727569675520 \beta_{5} - 10227278819923191104912316696960 \beta_{3} - 1237324662677251337608531037815680 \beta_{2} - 10241154021231659150639886372480 \beta_{1}$$ $$\nu^{15}$$ $$=$$ $$1646815618617180934100092688000 \beta_{15} + 2125651009998177578845931336832 \beta_{14} - 2574268147499343165916447922176 \beta_{13} + 1646815618617180934100092688000 \beta_{12} - 9399838314995041489524758518016 \beta_{11} - 4699919157497520744762379259008 \beta_{10} - 20705052193077432397863618298240 \beta_{9} - 4699919157497520744762379259008 \beta_{8} + 21146878347820898863899486672896 \beta_{7} - 108101925164797661641890577311616 \beta_{6} + 112801844322295182386652956570624 \beta_{5} + 21146878347820898863899486672896 \beta_{4} + 3235024940882599615421220518428800 \beta_{3} - 4699919157497520744762379259008 \beta_{2} + 266806917626312341543501433521067264$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 113.220 + 196.103i 95.0097 + 164.562i 48.3490 + 83.7429i 26.7983 + 46.4161i −24.6030 − 42.6137i −34.9668 − 60.5642i −89.8616 − 155.645i −132.946 − 230.268i 113.220 − 196.103i 95.0097 − 164.562i 48.3490 − 83.7429i 26.7983 − 46.4161i −24.6030 + 42.6137i −34.9668 + 60.5642i −89.8616 + 155.645i −132.946 + 230.268i
−8.00000 13.8564i −108.720 188.309i −128.000 + 221.703i −480.308 831.918i −1739.52 + 3012.94i −4308.78 4096.00 −13798.6 + 23899.8i −7684.93 + 13310.7i
7.2 −8.00000 13.8564i −90.5097 156.767i −128.000 + 221.703i −153.299 265.521i −1448.15 + 2508.28i 12144.4 4096.00 −6542.50 + 11331.9i −2452.78 + 4248.34i
7.3 −8.00000 13.8564i −43.8490 75.9487i −128.000 + 221.703i 993.746 + 1721.22i −701.584 + 1215.18i −8244.38 4096.00 5996.03 10385.4i 15899.9 27539.5i
7.4 −8.00000 13.8564i −22.2983 38.6218i −128.000 + 221.703i 397.433 + 688.374i −356.773 + 617.949i 2271.77 4096.00 8847.07 15323.6i 6358.93 11014.0i
7.5 −8.00000 13.8564i 29.1030 + 50.4079i −128.000 + 221.703i −1358.85 2353.60i 465.648 806.526i 3587.38 4096.00 8147.53 14111.9i −21741.6 + 37657.5i
7.6 −8.00000 13.8564i 39.4668 + 68.3585i −128.000 + 221.703i −134.126 232.313i 631.468 1093.74i −5487.23 4096.00 6726.25 11650.2i −2146.01 + 3717.01i
7.7 −8.00000 13.8564i 94.3616 + 163.439i −128.000 + 221.703i 1091.71 + 1890.90i 1509.79 2615.03i 6918.71 4096.00 −7966.73 + 13798.8i 17467.4 30254.4i
7.8 −8.00000 13.8564i 137.446 + 238.063i −128.000 + 221.703i −526.809 912.460i 2199.13 3809.00i −5029.91 4096.00 −27941.1 + 48395.4i −8428.95 + 14599.4i
11.1 −8.00000 + 13.8564i −108.720 + 188.309i −128.000 221.703i −480.308 + 831.918i −1739.52 3012.94i −4308.78 4096.00 −13798.6 23899.8i −7684.93 13310.7i
11.2 −8.00000 + 13.8564i −90.5097 + 156.767i −128.000 221.703i −153.299 + 265.521i −1448.15 2508.28i 12144.4 4096.00 −6542.50 11331.9i −2452.78 4248.34i
11.3 −8.00000 + 13.8564i −43.8490 + 75.9487i −128.000 221.703i 993.746 1721.22i −701.584 1215.18i −8244.38 4096.00 5996.03 + 10385.4i 15899.9 + 27539.5i
11.4 −8.00000 + 13.8564i −22.2983 + 38.6218i −128.000 221.703i 397.433 688.374i −356.773 617.949i 2271.77 4096.00 8847.07 + 15323.6i 6358.93 + 11014.0i
11.5 −8.00000 + 13.8564i 29.1030 50.4079i −128.000 221.703i −1358.85 + 2353.60i 465.648 + 806.526i 3587.38 4096.00 8147.53 + 14111.9i −21741.6 37657.5i
11.6 −8.00000 + 13.8564i 39.4668 68.3585i −128.000 221.703i −134.126 + 232.313i 631.468 + 1093.74i −5487.23 4096.00 6726.25 + 11650.2i −2146.01 3717.01i
11.7 −8.00000 + 13.8564i 94.3616 163.439i −128.000 221.703i 1091.71 1890.90i 1509.79 + 2615.03i 6918.71 4096.00 −7966.73 13798.8i 17467.4 + 30254.4i
11.8 −8.00000 + 13.8564i 137.446 238.063i −128.000 221.703i −526.809 + 912.460i 2199.13 + 3809.00i −5029.91 4096.00 −27941.1 48395.4i −8428.95 14599.4i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.10.c.b 16
19.c even 3 1 inner 38.10.c.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.c.b 16 1.a even 1 1 trivial
38.10.c.b 16 19.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$27\!\cdots\!92$$$$T_{3}^{10} +$$$$37\!\cdots\!26$$$$T_{3}^{9} +$$$$65\!\cdots\!65$$$$T_{3}^{8} +$$$$84\!\cdots\!38$$$$T_{3}^{7} +$$$$55\!\cdots\!52$$$$T_{3}^{6} -$$$$11\!\cdots\!62$$$$T_{3}^{5} +$$$$34\!\cdots\!76$$$$T_{3}^{4} -$$$$99\!\cdots\!00$$$$T_{3}^{3} +$$$$81\!\cdots\!86$$$$T_{3}^{2} +$$$$69\!\cdots\!50$$$$T_{3} +$$$$13\!\cdots\!25$$">$$T_{3}^{16} - \cdots$$ acting on $$S_{10}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 256 + 16 T + T^{2} )^{8}$$
$3$ $$13\!\cdots\!25$$$$+$$$$69\!\cdots\!50$$$$T +$$$$81\!\cdots\!86$$$$T^{2} -$$$$99\!\cdots\!00$$$$T^{3} +$$$$34\!\cdots\!76$$$$T^{4} -$$$$11\!\cdots\!62$$$$T^{5} +$$$$55\!\cdots\!52$$$$T^{6} + 8439829122596611338 T^{7} + 6575991702018991065 T^{8} + 3749972139745026 T^{9} + 270278184819892 T^{10} + 3529975322 T^{11} + 7876434392 T^{12} + 45968 T^{13} + 107714 T^{14} - 70 T^{15} + T^{16}$$
$5$ $$60\!\cdots\!00$$$$+$$$$43\!\cdots\!00$$$$T +$$$$24\!\cdots\!00$$$$T^{2} +$$$$64\!\cdots\!00$$$$T^{3} +$$$$14\!\cdots\!00$$$$T^{4} +$$$$14\!\cdots\!00$$$$T^{5} +$$$$28\!\cdots\!00$$$$T^{6} +$$$$22\!\cdots\!00$$$$T^{7} +$$$$37\!\cdots\!00$$$$T^{8} +$$$$14\!\cdots\!80$$$$T^{9} +$$$$17\!\cdots\!61$$$$T^{10} + 15675661944806237 T^{11} + 68431767251398 T^{12} + 958430885 T^{13} + 9525154 T^{14} + 341 T^{15} + T^{16}$$
$7$ $$($$$$67\!\cdots\!28$$$$-$$$$14\!\cdots\!64$$$$T -$$$$13\!\cdots\!08$$$$T^{2} + 11771965033735755408 T^{3} + 8561529974613476 T^{4} - 87818691968 T^{5} - 175950576 T^{6} - 1852 T^{7} + T^{8} )^{2}$$
$11$ $$( -$$$$60\!\cdots\!00$$$$+$$$$81\!\cdots\!60$$$$T -$$$$13\!\cdots\!04$$$$T^{2} -$$$$12\!\cdots\!17$$$$T^{3} + 26207342961644421785 T^{4} + 535140848091238 T^{5} - 10558263818 T^{6} - 53341 T^{7} + T^{8} )^{2}$$
$13$ $$29\!\cdots\!04$$$$+$$$$12\!\cdots\!36$$$$T +$$$$50\!\cdots\!96$$$$T^{2} +$$$$14\!\cdots\!00$$$$T^{3} +$$$$57\!\cdots\!12$$$$T^{4} +$$$$70\!\cdots\!84$$$$T^{5} +$$$$22\!\cdots\!32$$$$T^{6} +$$$$23\!\cdots\!08$$$$T^{7} +$$$$59\!\cdots\!80$$$$T^{8} +$$$$36\!\cdots\!06$$$$T^{9} +$$$$78\!\cdots\!27$$$$T^{10} +$$$$32\!\cdots\!01$$$$T^{11} +$$$$75\!\cdots\!04$$$$T^{12} + 1443631306796365 T^{13} + 31956088828 T^{14} + 683 T^{15} + T^{16}$$
$17$ $$10\!\cdots\!00$$$$+$$$$99\!\cdots\!00$$$$T +$$$$11\!\cdots\!00$$$$T^{2} +$$$$43\!\cdots\!00$$$$T^{3} +$$$$38\!\cdots\!00$$$$T^{4} +$$$$13\!\cdots\!00$$$$T^{5} +$$$$82\!\cdots\!00$$$$T^{6} +$$$$21\!\cdots\!00$$$$T^{7} +$$$$96\!\cdots\!00$$$$T^{8} +$$$$21\!\cdots\!50$$$$T^{9} +$$$$77\!\cdots\!11$$$$T^{10} +$$$$13\!\cdots\!73$$$$T^{11} +$$$$32\!\cdots\!16$$$$T^{12} + 344187637553261901 T^{13} + 773078639572 T^{14} + 611267 T^{15} + T^{16}$$
$19$ $$11\!\cdots\!61$$$$+$$$$58\!\cdots\!43$$$$T +$$$$12\!\cdots\!50$$$$T^{2} -$$$$35\!\cdots\!19$$$$T^{3} -$$$$64\!\cdots\!31$$$$T^{4} -$$$$16\!\cdots\!84$$$$T^{5} -$$$$12\!\cdots\!90$$$$T^{6} +$$$$33\!\cdots\!66$$$$T^{7} +$$$$10\!\cdots\!92$$$$T^{8} +$$$$10\!\cdots\!54$$$$T^{9} -$$$$11\!\cdots\!90$$$$T^{10} -$$$$48\!\cdots\!56$$$$T^{11} -$$$$59\!\cdots\!51$$$$T^{12} - 10274081541671781 T^{13} + 1082254997650 T^{14} + 1609677 T^{15} + T^{16}$$
$23$ $$60\!\cdots\!64$$$$-$$$$13\!\cdots\!28$$$$T +$$$$31\!\cdots\!24$$$$T^{2} +$$$$11\!\cdots\!00$$$$T^{3} +$$$$49\!\cdots\!16$$$$T^{4} +$$$$43\!\cdots\!28$$$$T^{5} +$$$$13\!\cdots\!72$$$$T^{6} +$$$$38\!\cdots\!60$$$$T^{7} +$$$$27\!\cdots\!92$$$$T^{8} +$$$$10\!\cdots\!22$$$$T^{9} +$$$$24\!\cdots\!71$$$$T^{10} +$$$$22\!\cdots\!25$$$$T^{11} +$$$$15\!\cdots\!04$$$$T^{12} + 753112377919171341 T^{13} + 4769308157472 T^{14} + 416119 T^{15} + T^{16}$$
$29$ $$51\!\cdots\!00$$$$+$$$$82\!\cdots\!00$$$$T +$$$$13\!\cdots\!00$$$$T^{2} +$$$$53\!\cdots\!00$$$$T^{3} +$$$$49\!\cdots\!00$$$$T^{4} +$$$$17\!\cdots\!60$$$$T^{5} +$$$$12\!\cdots\!76$$$$T^{6} +$$$$31\!\cdots\!40$$$$T^{7} +$$$$13\!\cdots\!44$$$$T^{8} +$$$$22\!\cdots\!84$$$$T^{9} +$$$$89\!\cdots\!09$$$$T^{10} +$$$$11\!\cdots\!09$$$$T^{11} +$$$$35\!\cdots\!62$$$$T^{12} +$$$$28\!\cdots\!09$$$$T^{13} + 78604266980026 T^{14} + 5079157 T^{15} + T^{16}$$
$31$ $$( -$$$$14\!\cdots\!08$$$$+$$$$58\!\cdots\!24$$$$T -$$$$16\!\cdots\!68$$$$T^{2} -$$$$35\!\cdots\!60$$$$T^{3} +$$$$97\!\cdots\!12$$$$T^{4} +$$$$69\!\cdots\!72$$$$T^{5} - 174817951381608 T^{6} - 4309162 T^{7} + T^{8} )^{2}$$
$37$ $$($$$$96\!\cdots\!52$$$$-$$$$18\!\cdots\!68$$$$T -$$$$98\!\cdots\!28$$$$T^{2} +$$$$26\!\cdots\!56$$$$T^{3} +$$$$31\!\cdots\!52$$$$T^{4} -$$$$46\!\cdots\!16$$$$T^{5} - 335427937186520 T^{6} + 16545760 T^{7} + T^{8} )^{2}$$
$41$ $$50\!\cdots\!61$$$$-$$$$20\!\cdots\!30$$$$T +$$$$10\!\cdots\!30$$$$T^{2} -$$$$17\!\cdots\!72$$$$T^{3} +$$$$12\!\cdots\!76$$$$T^{4} +$$$$22\!\cdots\!46$$$$T^{5} +$$$$83\!\cdots\!40$$$$T^{6} +$$$$71\!\cdots\!78$$$$T^{7} +$$$$38\!\cdots\!21$$$$T^{8} +$$$$20\!\cdots\!10$$$$T^{9} +$$$$99\!\cdots\!44$$$$T^{10} +$$$$32\!\cdots\!74$$$$T^{11} +$$$$18\!\cdots\!24$$$$T^{12} +$$$$28\!\cdots\!80$$$$T^{13} + 1656809373603058 T^{14} + 7394646 T^{15} + T^{16}$$
$43$ $$13\!\cdots\!84$$$$-$$$$17\!\cdots\!12$$$$T +$$$$23\!\cdots\!64$$$$T^{2} +$$$$68\!\cdots\!92$$$$T^{3} +$$$$20\!\cdots\!04$$$$T^{4} +$$$$68\!\cdots\!00$$$$T^{5} +$$$$12\!\cdots\!52$$$$T^{6} +$$$$74\!\cdots\!08$$$$T^{7} +$$$$61\!\cdots\!24$$$$T^{8} +$$$$24\!\cdots\!80$$$$T^{9} +$$$$11\!\cdots\!41$$$$T^{10} +$$$$37\!\cdots\!51$$$$T^{11} +$$$$13\!\cdots\!50$$$$T^{12} +$$$$34\!\cdots\!95$$$$T^{13} + 7747113679589818 T^{14} + 98675599 T^{15} + T^{16}$$
$47$ $$21\!\cdots\!00$$$$-$$$$42\!\cdots\!00$$$$T +$$$$72\!\cdots\!00$$$$T^{2} -$$$$16\!\cdots\!20$$$$T^{3} +$$$$27\!\cdots\!64$$$$T^{4} -$$$$27\!\cdots\!28$$$$T^{5} +$$$$22\!\cdots\!24$$$$T^{6} -$$$$12\!\cdots\!84$$$$T^{7} +$$$$63\!\cdots\!08$$$$T^{8} -$$$$24\!\cdots\!68$$$$T^{9} +$$$$11\!\cdots\!85$$$$T^{10} -$$$$31\!\cdots\!79$$$$T^{11} +$$$$97\!\cdots\!10$$$$T^{12} -$$$$13\!\cdots\!35$$$$T^{13} + 3578757933345606 T^{14} - 34129475 T^{15} + T^{16}$$
$53$ $$72\!\cdots\!00$$$$+$$$$43\!\cdots\!00$$$$T +$$$$30\!\cdots\!00$$$$T^{2} +$$$$77\!\cdots\!40$$$$T^{3} +$$$$34\!\cdots\!24$$$$T^{4} +$$$$66\!\cdots\!88$$$$T^{5} +$$$$27\!\cdots\!44$$$$T^{6} +$$$$36\!\cdots\!24$$$$T^{7} +$$$$10\!\cdots\!88$$$$T^{8} +$$$$12\!\cdots\!38$$$$T^{9} +$$$$29\!\cdots\!15$$$$T^{10} +$$$$25\!\cdots\!69$$$$T^{11} +$$$$37\!\cdots\!80$$$$T^{12} +$$$$20\!\cdots\!45$$$$T^{13} + 26172216036336616 T^{14} + 110053995 T^{15} + T^{16}$$
$59$ $$14\!\cdots\!25$$$$-$$$$36\!\cdots\!20$$$$T +$$$$65\!\cdots\!96$$$$T^{2} -$$$$58\!\cdots\!00$$$$T^{3} +$$$$40\!\cdots\!94$$$$T^{4} -$$$$13\!\cdots\!28$$$$T^{5} +$$$$47\!\cdots\!12$$$$T^{6} -$$$$48\!\cdots\!72$$$$T^{7} +$$$$32\!\cdots\!47$$$$T^{8} -$$$$14\!\cdots\!36$$$$T^{9} +$$$$12\!\cdots\!80$$$$T^{10} +$$$$66\!\cdots\!92$$$$T^{11} +$$$$28\!\cdots\!62$$$$T^{12} +$$$$67\!\cdots\!92$$$$T^{13} + 18593184224735596 T^{14} + 2017760 T^{15} + T^{16}$$
$61$ $$40\!\cdots\!00$$$$+$$$$87\!\cdots\!60$$$$T +$$$$21\!\cdots\!64$$$$T^{2} +$$$$24\!\cdots\!84$$$$T^{3} +$$$$38\!\cdots\!68$$$$T^{4} +$$$$33\!\cdots\!52$$$$T^{5} +$$$$44\!\cdots\!76$$$$T^{6} +$$$$30\!\cdots\!92$$$$T^{7} +$$$$31\!\cdots\!88$$$$T^{8} +$$$$17\!\cdots\!28$$$$T^{9} +$$$$15\!\cdots\!61$$$$T^{10} +$$$$67\!\cdots\!01$$$$T^{11} +$$$$49\!\cdots\!74$$$$T^{12} +$$$$15\!\cdots\!53$$$$T^{13} + 99305326515642126 T^{14} + 221861413 T^{15} + T^{16}$$
$67$ $$35\!\cdots\!29$$$$-$$$$27\!\cdots\!92$$$$T +$$$$95\!\cdots\!00$$$$T^{2} -$$$$12\!\cdots\!64$$$$T^{3} +$$$$23\!\cdots\!18$$$$T^{4} -$$$$23\!\cdots\!68$$$$T^{5} +$$$$21\!\cdots\!96$$$$T^{6} -$$$$13\!\cdots\!00$$$$T^{7} +$$$$79\!\cdots\!23$$$$T^{8} -$$$$33\!\cdots\!16$$$$T^{9} +$$$$16\!\cdots\!76$$$$T^{10} -$$$$54\!\cdots\!76$$$$T^{11} +$$$$22\!\cdots\!50$$$$T^{12} -$$$$48\!\cdots\!40$$$$T^{13} + 180585237910526100 T^{14} - 237580440 T^{15} + T^{16}$$
$71$ $$13\!\cdots\!16$$$$-$$$$15\!\cdots\!60$$$$T +$$$$24\!\cdots\!60$$$$T^{2} +$$$$10\!\cdots\!68$$$$T^{3} +$$$$37\!\cdots\!84$$$$T^{4} +$$$$45\!\cdots\!04$$$$T^{5} +$$$$55\!\cdots\!16$$$$T^{6} +$$$$38\!\cdots\!92$$$$T^{7} +$$$$34\!\cdots\!68$$$$T^{8} +$$$$19\!\cdots\!34$$$$T^{9} +$$$$13\!\cdots\!55$$$$T^{10} +$$$$49\!\cdots\!87$$$$T^{11} +$$$$22\!\cdots\!32$$$$T^{12} +$$$$55\!\cdots\!03$$$$T^{13} + 242243856104935156 T^{14} + 431190909 T^{15} + T^{16}$$
$73$ $$46\!\cdots\!81$$$$-$$$$91\!\cdots\!36$$$$T +$$$$17\!\cdots\!08$$$$T^{2} -$$$$14\!\cdots\!56$$$$T^{3} +$$$$15\!\cdots\!78$$$$T^{4} -$$$$88\!\cdots\!28$$$$T^{5} +$$$$79\!\cdots\!76$$$$T^{6} -$$$$35\!\cdots\!76$$$$T^{7} +$$$$23\!\cdots\!03$$$$T^{8} -$$$$71\!\cdots\!92$$$$T^{9} +$$$$42\!\cdots\!88$$$$T^{10} -$$$$99\!\cdots\!64$$$$T^{11} +$$$$48\!\cdots\!10$$$$T^{12} -$$$$57\!\cdots\!80$$$$T^{13} + 260854994412815356 T^{14} - 199873544 T^{15} + T^{16}$$
$79$ $$23\!\cdots\!00$$$$-$$$$78\!\cdots\!00$$$$T +$$$$16\!\cdots\!00$$$$T^{2} -$$$$16\!\cdots\!00$$$$T^{3} +$$$$65\!\cdots\!00$$$$T^{4} -$$$$47\!\cdots\!20$$$$T^{5} +$$$$14\!\cdots\!76$$$$T^{6} -$$$$17\!\cdots\!20$$$$T^{7} +$$$$21\!\cdots\!40$$$$T^{8} +$$$$22\!\cdots\!80$$$$T^{9} +$$$$20\!\cdots\!25$$$$T^{10} +$$$$70\!\cdots\!51$$$$T^{11} +$$$$13\!\cdots\!70$$$$T^{12} +$$$$39\!\cdots\!55$$$$T^{13} + 494288687051321890 T^{14} + 296762835 T^{15} + T^{16}$$
$83$ $$( -$$$$65\!\cdots\!92$$$$-$$$$73\!\cdots\!60$$$$T -$$$$10\!\cdots\!96$$$$T^{2} -$$$$21\!\cdots\!57$$$$T^{3} +$$$$31\!\cdots\!37$$$$T^{4} +$$$$13\!\cdots\!18$$$$T^{5} - 171086311162799142 T^{6} - 796069153 T^{7} + T^{8} )^{2}$$
$89$ $$43\!\cdots\!00$$$$+$$$$23\!\cdots\!00$$$$T +$$$$48\!\cdots\!00$$$$T^{2} -$$$$70\!\cdots\!00$$$$T^{3} +$$$$50\!\cdots\!00$$$$T^{4} -$$$$25\!\cdots\!00$$$$T^{5} +$$$$20\!\cdots\!00$$$$T^{6} +$$$$11\!\cdots\!80$$$$T^{7} +$$$$16\!\cdots\!16$$$$T^{8} +$$$$60\!\cdots\!06$$$$T^{9} +$$$$77\!\cdots\!59$$$$T^{10} +$$$$61\!\cdots\!25$$$$T^{11} +$$$$17\!\cdots\!52$$$$T^{12} +$$$$62\!\cdots\!25$$$$T^{13} + 1550801526312250864 T^{14} + 444394631 T^{15} + T^{16}$$
$97$ $$27\!\cdots\!25$$$$-$$$$61\!\cdots\!10$$$$T +$$$$14\!\cdots\!26$$$$T^{2} -$$$$36\!\cdots\!60$$$$T^{3} +$$$$53\!\cdots\!96$$$$T^{4} -$$$$35\!\cdots\!82$$$$T^{5} +$$$$56\!\cdots\!52$$$$T^{6} -$$$$13\!\cdots\!30$$$$T^{7} +$$$$40\!\cdots\!81$$$$T^{8} -$$$$12\!\cdots\!90$$$$T^{9} +$$$$12\!\cdots\!68$$$$T^{10} +$$$$17\!\cdots\!14$$$$T^{11} +$$$$22\!\cdots\!40$$$$T^{12} +$$$$16\!\cdots\!76$$$$T^{13} + 1986247292418557770 T^{14} + 611719542 T^{15} + T^{16}$$