# Properties

 Label 531.8.a.d Level $531$ Weight $8$ Character orbit 531.a Self dual yes Analytic conductor $165.876$ Analytic rank $0$ Dimension $17$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$531 = 3^{2} \cdot 59$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 531.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$165.876448532$$ Analytic rank: $$0$$ Dimension: $$17$$ Coefficient field: $$\mathbb{Q}[x]/(x^{17} - \cdots)$$ Defining polynomial: $$x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: multiple of $$2^{10}\cdot 3^{5}$$ Twist minimal: no (minimal twist has level 177) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{16}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - \beta_{1} ) q^{2} + ( 69 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( 63 + 3 \beta_{1} - \beta_{3} ) q^{5} + ( -141 - 6 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{11} ) q^{7} + ( 401 - 86 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{8} + \beta_{9} ) q^{8} +O(q^{10})$$ $$q + ( 2 - \beta_{1} ) q^{2} + ( 69 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( 63 + 3 \beta_{1} - \beta_{3} ) q^{5} + ( -141 - 6 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{11} ) q^{7} + ( 401 - 86 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{8} + \beta_{9} ) q^{8} + ( -361 - 129 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} + 2 \beta_{14} - \beta_{16} ) q^{10} + ( 509 + 98 \beta_{1} + 7 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 3 \beta_{10} + 2 \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{15} - 2 \beta_{16} ) q^{11} + ( -733 - 110 \beta_{1} + 8 \beta_{2} + \beta_{3} + 7 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} - 4 \beta_{11} - 6 \beta_{12} - 6 \beta_{13} - 6 \beta_{14} + 2 \beta_{15} + \beta_{16} ) q^{13} + ( 989 + 362 \beta_{1} + 22 \beta_{2} + 10 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} + \beta_{6} + 6 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} + 6 \beta_{10} + 15 \beta_{11} - 3 \beta_{12} - 4 \beta_{13} - 3 \beta_{14} + 9 \beta_{15} + 3 \beta_{16} ) q^{14} + ( 8259 - 450 \beta_{1} + 68 \beta_{2} - 34 \beta_{3} + 9 \beta_{4} - 10 \beta_{5} + 4 \beta_{6} + 7 \beta_{7} + 8 \beta_{8} - 12 \beta_{9} + 3 \beta_{10} + 6 \beta_{11} + 4 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} - \beta_{15} + \beta_{16} ) q^{16} + ( 2109 + 154 \beta_{1} + 40 \beta_{2} + 18 \beta_{3} + 13 \beta_{4} - 4 \beta_{5} + \beta_{6} - 6 \beta_{7} + 12 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - 8 \beta_{12} - 5 \beta_{14} - 7 \beta_{15} + 6 \beta_{16} ) q^{17} + ( -4096 - 528 \beta_{1} - 61 \beta_{2} - 50 \beta_{3} + \beta_{4} - \beta_{6} - 6 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} - 7 \beta_{10} - 11 \beta_{11} + 11 \beta_{12} + 14 \beta_{13} + 15 \beta_{14} - 10 \beta_{15} - 3 \beta_{16} ) q^{19} + ( 16127 + 565 \beta_{1} + 239 \beta_{2} - 39 \beta_{3} - 11 \beta_{4} - 21 \beta_{5} + 10 \beta_{6} - 3 \beta_{7} + 7 \beta_{8} - 22 \beta_{9} - 18 \beta_{10} + 6 \beta_{11} + 22 \beta_{12} + 8 \beta_{13} + 38 \beta_{14} - 19 \beta_{15} - 15 \beta_{16} ) q^{20} + ( -19006 - 951 \beta_{1} - 177 \beta_{2} - 48 \beta_{3} + 12 \beta_{4} - \beta_{5} + 19 \beta_{6} - 19 \beta_{7} - 3 \beta_{8} + 30 \beta_{9} + 3 \beta_{10} + 20 \beta_{11} - 35 \beta_{12} - 46 \beta_{13} + 12 \beta_{14} + 15 \beta_{15} + 5 \beta_{16} ) q^{22} + ( 16045 - 1504 \beta_{1} + 48 \beta_{2} + 57 \beta_{3} - 9 \beta_{4} + 19 \beta_{5} - \beta_{6} - 7 \beta_{7} + 10 \beta_{8} - 3 \beta_{9} - 8 \beta_{10} - 8 \beta_{11} + 18 \beta_{12} + 22 \beta_{13} - 8 \beta_{14} + 6 \beta_{15} - 20 \beta_{16} ) q^{23} + ( 5551 + 1490 \beta_{1} - 37 \beta_{2} - 110 \beta_{3} - 28 \beta_{4} - 18 \beta_{5} - 29 \beta_{6} - 33 \beta_{7} + 44 \beta_{8} - 29 \beta_{9} - 22 \beta_{10} - 22 \beta_{11} - 15 \beta_{12} + 32 \beta_{13} - 4 \beta_{14} - 13 \beta_{15} - 5 \beta_{16} ) q^{25} + ( 19896 - 656 \beta_{1} + 189 \beta_{2} - 2 \beta_{3} + 14 \beta_{4} - 4 \beta_{5} - 20 \beta_{6} - 24 \beta_{7} + 27 \beta_{8} + 47 \beta_{9} + 19 \beta_{10} - 10 \beta_{11} - 69 \beta_{12} - 86 \beta_{13} - 16 \beta_{14} - 2 \beta_{15} + 13 \beta_{16} ) q^{26} + ( -52783 - 2852 \beta_{1} - 295 \beta_{2} + 200 \beta_{3} + 35 \beta_{4} + 33 \beta_{5} - 25 \beta_{6} + 126 \beta_{7} - 111 \beta_{8} + 8 \beta_{9} + 66 \beta_{10} + 192 \beta_{11} + 109 \beta_{12} - 24 \beta_{13} - 18 \beta_{14} + 60 \beta_{15} - 18 \beta_{16} ) q^{28} + ( 31813 + 1157 \beta_{1} + 132 \beta_{2} + 146 \beta_{3} + 71 \beta_{4} - 27 \beta_{5} - 54 \beta_{6} + 16 \beta_{7} - 6 \beta_{8} + 109 \beta_{9} - 9 \beta_{10} - 2 \beta_{11} - 74 \beta_{12} - 24 \beta_{13} - 104 \beta_{14} + 77 \beta_{15} + 88 \beta_{16} ) q^{29} + ( -37081 - 2275 \beta_{1} + 172 \beta_{2} - 30 \beta_{3} - 97 \beta_{4} - 107 \beta_{5} - 103 \beta_{6} + 25 \beta_{7} + 74 \beta_{8} + 62 \beta_{9} - 171 \beta_{10} - 12 \beta_{11} - 76 \beta_{12} + 41 \beta_{13} + \beta_{14} - 72 \beta_{15} - 81 \beta_{16} ) q^{31} + ( 50506 - 10510 \beta_{1} + 257 \beta_{2} - 221 \beta_{3} - 111 \beta_{4} - 64 \beta_{5} - 27 \beta_{6} + 190 \beta_{7} - 39 \beta_{8} + 150 \beta_{9} - 21 \beta_{10} + 207 \beta_{11} + 50 \beta_{12} + 39 \beta_{13} + 152 \beta_{14} + 39 \beta_{15} - 138 \beta_{16} ) q^{32} + ( -30373 - 6790 \beta_{1} + 170 \beta_{2} + 239 \beta_{3} + 149 \beta_{4} + 14 \beta_{5} + 34 \beta_{6} + 14 \beta_{7} + 57 \beta_{8} + 105 \beta_{9} + 82 \beta_{10} + 77 \beta_{11} - 229 \beta_{12} - 160 \beta_{13} - 127 \beta_{14} + 111 \beta_{15} + 78 \beta_{16} ) q^{34} + ( 16724 + 816 \beta_{1} - 1104 \beta_{2} + 180 \beta_{3} + 106 \beta_{4} + 199 \beta_{5} - 74 \beta_{6} + 25 \beta_{7} + 52 \beta_{8} + 23 \beta_{9} + 122 \beta_{10} - 84 \beta_{11} - 94 \beta_{12} - 25 \beta_{13} - 266 \beta_{14} + 95 \beta_{15} + 85 \beta_{16} ) q^{35} + ( -50877 - 2358 \beta_{1} + 16 \beta_{2} + 317 \beta_{3} - 37 \beta_{4} + 152 \beta_{5} - 72 \beta_{6} - 163 \beta_{7} + 9 \beta_{8} + 7 \beta_{9} - 7 \beta_{10} - 100 \beta_{11} - 34 \beta_{12} - 8 \beta_{13} - 89 \beta_{14} - 6 \beta_{15} + 121 \beta_{16} ) q^{37} + ( 100775 + 9236 \beta_{1} + 413 \beta_{2} - 811 \beta_{3} - 219 \beta_{4} + 3 \beta_{5} + 18 \beta_{6} - 86 \beta_{7} - 36 \beta_{8} - 92 \beta_{9} - 131 \beta_{10} - 44 \beta_{11} + 196 \beta_{12} + 89 \beta_{13} + 367 \beta_{14} - 106 \beta_{15} - 164 \beta_{16} ) q^{38} + ( -44013 - 32402 \beta_{1} - 379 \beta_{2} - 468 \beta_{3} - 325 \beta_{4} + 29 \beta_{5} + 230 \beta_{6} - 109 \beta_{7} + 156 \beta_{8} + 295 \beta_{9} - 126 \beta_{10} + 44 \beta_{11} + 112 \beta_{12} + 26 \beta_{13} + 342 \beta_{14} - 67 \beta_{15} - 161 \beta_{16} ) q^{40} + ( 83236 + 7199 \beta_{1} - 394 \beta_{2} - 754 \beta_{3} - 284 \beta_{4} - 179 \beta_{5} + 102 \beta_{6} - 251 \beta_{7} + 91 \beta_{8} - 44 \beta_{9} - 192 \beta_{10} + \beta_{11} + 64 \beta_{12} + 19 \beta_{13} + 44 \beta_{14} - 232 \beta_{15} - 38 \beta_{16} ) q^{41} + ( -26897 - 8743 \beta_{1} + 254 \beta_{2} - 197 \beta_{3} - 120 \beta_{4} + 51 \beta_{5} - 203 \beta_{6} - 104 \beta_{7} - 211 \beta_{8} + 219 \beta_{10} - 130 \beta_{11} + 462 \beta_{12} + 110 \beta_{13} + 153 \beta_{14} - 132 \beta_{15} - 53 \beta_{16} ) q^{43} + ( 94604 + 33319 \beta_{1} + 444 \beta_{2} - 1298 \beta_{3} - 114 \beta_{4} - 37 \beta_{5} + 132 \beta_{6} - 482 \beta_{7} + 41 \beta_{8} - 276 \beta_{9} - 132 \beta_{10} - 164 \beta_{11} + 185 \beta_{12} - 68 \beta_{13} + 198 \beta_{14} - 154 \beta_{15} + 169 \beta_{16} ) q^{44} + ( 314992 - 21162 \beta_{1} + 2885 \beta_{2} - 441 \beta_{3} + \beta_{4} + 57 \beta_{5} + 219 \beta_{6} + 191 \beta_{7} - 168 \beta_{8} - 582 \beta_{9} + 246 \beta_{10} + 274 \beta_{11} + 492 \beta_{12} + 369 \beta_{13} + 191 \beta_{14} - 125 \beta_{15} - 146 \beta_{16} ) q^{46} + ( 71747 + 743 \beta_{1} - 738 \beta_{2} - 669 \beta_{3} + 106 \beta_{4} + 184 \beta_{5} - 87 \beta_{6} + 69 \beta_{7} + 123 \beta_{8} - 408 \beta_{9} + 316 \beta_{10} - 40 \beta_{11} + 210 \beta_{12} + 353 \beta_{13} - 189 \beta_{14} - 406 \beta_{15} - 240 \beta_{16} ) q^{47} + ( 259383 - 26635 \beta_{1} + 2478 \beta_{2} - 849 \beta_{3} - 421 \beta_{4} - 99 \beta_{5} + 175 \beta_{6} + 11 \beta_{7} + 59 \beta_{8} - 93 \beta_{9} - 281 \beta_{10} + 37 \beta_{11} + 78 \beta_{12} + 501 \beta_{13} + 251 \beta_{14} + 107 \beta_{15} - 210 \beta_{16} ) q^{49} + ( -267332 - 6848 \beta_{1} - 1850 \beta_{2} - 514 \beta_{3} - 124 \beta_{4} - 68 \beta_{5} + 265 \beta_{6} + 19 \beta_{7} + 228 \beta_{8} + 697 \beta_{9} - 118 \beta_{10} - 147 \beta_{11} - 542 \beta_{12} - 233 \beta_{13} + 206 \beta_{14} - 68 \beta_{15} - 231 \beta_{16} ) q^{50} + ( 249294 - 28397 \beta_{1} + 1586 \beta_{2} - 729 \beta_{3} - 153 \beta_{4} - 140 \beta_{5} + 135 \beta_{6} - 666 \beta_{7} + 688 \beta_{8} + 283 \beta_{9} - 418 \beta_{10} - 425 \beta_{11} - 523 \beta_{12} - 165 \beta_{13} + 242 \beta_{14} - 133 \beta_{15} + 591 \beta_{16} ) q^{52} + ( 206001 - 3664 \beta_{1} - 670 \beta_{2} - 722 \beta_{3} + 17 \beta_{4} + 264 \beta_{5} + 248 \beta_{6} - 275 \beta_{7} + 655 \beta_{8} + 704 \beta_{9} - 444 \beta_{10} - 333 \beta_{11} - 256 \beta_{12} + 233 \beta_{13} + 96 \beta_{14} + 260 \beta_{15} + 480 \beta_{16} ) q^{53} + ( -55943 + 3428 \beta_{1} + 3834 \beta_{2} - 1025 \beta_{3} + 993 \beta_{4} + 46 \beta_{5} + 201 \beta_{6} + 444 \beta_{7} - 200 \beta_{8} - 420 \beta_{9} - 22 \beta_{10} - 364 \beta_{11} - 414 \beta_{12} - 131 \beta_{13} - 272 \beta_{14} - 397 \beta_{15} + 117 \beta_{16} ) q^{55} + ( 310035 + 49634 \beta_{1} + 3240 \beta_{2} + 373 \beta_{3} + 1133 \beta_{4} + 837 \beta_{5} - 15 \beta_{6} + 1714 \beta_{7} - 1482 \beta_{8} - 1282 \beta_{9} + 1365 \beta_{10} + 1481 \beta_{11} + 1647 \beta_{12} + 385 \beta_{13} - 970 \beta_{14} + 1001 \beta_{15} + 707 \beta_{16} ) q^{56} + ( -175983 - 53071 \beta_{1} - 937 \beta_{2} + 1748 \beta_{3} + 1058 \beta_{4} + 164 \beta_{5} - 366 \beta_{6} + 971 \beta_{7} - 871 \beta_{8} + 36 \beta_{9} + 1571 \beta_{10} + 1480 \beta_{11} - 516 \beta_{12} - 1447 \beta_{13} - 1469 \beta_{14} + 473 \beta_{15} + 499 \beta_{16} ) q^{58} + 205379 q^{59} + ( 57226 + 21118 \beta_{1} + 1486 \beta_{2} - 2767 \beta_{3} + 243 \beta_{4} + 740 \beta_{5} + 522 \beta_{6} - 1055 \beta_{7} + 789 \beta_{8} + 119 \beta_{9} - 861 \beta_{10} - 962 \beta_{11} - 1060 \beta_{12} - 603 \beta_{13} - 834 \beta_{14} + 389 \beta_{15} + 356 \beta_{16} ) q^{61} + ( 336595 + 13220 \beta_{1} + 3793 \beta_{2} + 617 \beta_{3} + 891 \beta_{4} - 1388 \beta_{5} + 156 \beta_{6} + 400 \beta_{7} + 762 \beta_{8} + 640 \beta_{9} + 923 \beta_{10} - 789 \beta_{11} - 2472 \beta_{12} - 614 \beta_{13} - 955 \beta_{14} + 239 \beta_{15} + 137 \beta_{16} ) q^{62} + ( 1037361 - 41996 \beta_{1} + 12555 \beta_{2} - 4447 \beta_{3} - 1392 \beta_{4} - 1638 \beta_{5} - 756 \beta_{6} + 1240 \beta_{7} - 449 \beta_{8} - 1427 \beta_{9} - 1108 \beta_{10} + 1788 \beta_{11} + 2946 \beta_{12} + 2968 \beta_{13} + 1956 \beta_{14} - 744 \beta_{15} - 1590 \beta_{16} ) q^{64} + ( 360883 - 28275 \beta_{1} + 2677 \beta_{2} + 879 \beta_{3} - 331 \beta_{4} - 712 \beta_{5} + 73 \beta_{6} - 462 \beta_{7} + 331 \beta_{8} + 160 \beta_{9} - 1123 \beta_{10} - 109 \beta_{11} + 615 \beta_{12} + 1176 \beta_{13} + 1719 \beta_{14} - 943 \beta_{15} - 94 \beta_{16} ) q^{65} + ( -21056 + 992 \beta_{1} + 3408 \beta_{2} + 1115 \beta_{3} + 307 \beta_{4} - 412 \beta_{5} - 28 \beta_{6} + 333 \beta_{7} - 402 \beta_{8} + 770 \beta_{9} + 803 \beta_{10} + 152 \beta_{11} - 600 \beta_{12} + 501 \beta_{13} - 1116 \beta_{14} + 274 \beta_{15} + 99 \beta_{16} ) q^{67} + ( 950194 - 2945 \beta_{1} + 6740 \beta_{2} - 1860 \beta_{3} - 203 \beta_{4} - 193 \beta_{5} - 742 \beta_{6} + 709 \beta_{7} - 829 \beta_{8} - 1033 \beta_{9} + 264 \beta_{10} + 997 \beta_{11} + 405 \beta_{12} - 565 \beta_{13} - 900 \beta_{14} + 588 \beta_{15} - 113 \beta_{16} ) q^{68} + ( -46828 + 126315 \beta_{1} + 3471 \beta_{2} + 4251 \beta_{3} + 4211 \beta_{4} + 1705 \beta_{5} - 126 \beta_{6} + 1614 \beta_{7} - 1406 \beta_{8} - 2487 \beta_{9} + 2958 \beta_{10} - 503 \beta_{11} - 726 \beta_{12} - 560 \beta_{13} - 4067 \beta_{14} + 1243 \beta_{15} + 2247 \beta_{16} ) q^{70} + ( 760714 - 30630 \beta_{1} + 1440 \beta_{2} - 2309 \beta_{3} - 736 \beta_{4} - 458 \beta_{5} - 940 \beta_{6} + 14 \beta_{7} - 1113 \beta_{8} - 476 \beta_{9} - 319 \beta_{10} + 1129 \beta_{11} + 650 \beta_{12} - 1279 \beta_{13} + 167 \beta_{14} + 497 \beta_{15} - 886 \beta_{16} ) q^{71} + ( -371179 + 78450 \beta_{1} + 2408 \beta_{2} - 5212 \beta_{3} - 1108 \beta_{4} - 67 \beta_{5} - 14 \beta_{6} - 663 \beta_{7} - 582 \beta_{8} + 1332 \beta_{9} + 53 \beta_{10} - 72 \beta_{11} + 328 \beta_{12} - 2020 \beta_{13} + 539 \beta_{14} + 1515 \beta_{15} - 36 \beta_{16} ) q^{73} + ( 343792 + 61974 \beta_{1} - 2468 \beta_{2} + 6136 \beta_{3} + 3340 \beta_{4} + 2813 \beta_{5} + 938 \beta_{6} - 1303 \beta_{7} - 1058 \beta_{8} + 206 \beta_{9} + 2307 \beta_{10} - 1666 \beta_{11} - 2343 \beta_{12} - 3271 \beta_{13} - 3375 \beta_{14} + 1477 \beta_{15} + 2758 \beta_{16} ) q^{74} + ( -1024385 - 101695 \beta_{1} - 7531 \beta_{2} - 5510 \beta_{3} - 4379 \beta_{4} - 242 \beta_{5} + 933 \beta_{6} - 1986 \beta_{7} + 1004 \beta_{8} + 416 \beta_{9} - 2556 \beta_{10} - 420 \beta_{11} + 2016 \beta_{12} - 220 \beta_{13} + 4800 \beta_{14} - 932 \beta_{15} - 2287 \beta_{16} ) q^{76} + ( -3148 - 104137 \beta_{1} - 5098 \beta_{2} + 4280 \beta_{3} - 371 \beta_{4} + 536 \beta_{5} + 302 \beta_{6} - 371 \beta_{7} - 1051 \beta_{8} + 1567 \beta_{9} - 1443 \beta_{10} - 936 \beta_{11} - 1388 \beta_{12} + 94 \beta_{13} + 331 \beta_{14} - 730 \beta_{15} - 791 \beta_{16} ) q^{77} + ( -1116093 + 42168 \beta_{1} + 2850 \beta_{2} + 3858 \beta_{3} + 3194 \beta_{4} + 990 \beta_{5} - 400 \beta_{6} + 162 \beta_{7} + 952 \beta_{8} - 1346 \beta_{9} + 2202 \beta_{10} - 542 \beta_{11} - 1130 \beta_{12} - 1329 \beta_{13} - 2347 \beta_{14} - 1123 \beta_{15} + 1809 \beta_{16} ) q^{79} + ( 4146789 + 18962 \beta_{1} + 29150 \beta_{2} - 11469 \beta_{3} - 2252 \beta_{4} - 2927 \beta_{5} - 758 \beta_{6} - 2278 \beta_{7} + 1803 \beta_{8} - 3776 \beta_{9} - 1991 \beta_{10} - 1774 \beta_{11} + 2054 \beta_{12} + 4286 \beta_{13} + 2212 \beta_{14} - 1836 \beta_{15} - 1074 \beta_{16} ) q^{80} + ( -1136526 - 49121 \beta_{1} - 4030 \beta_{2} + 9881 \beta_{3} - 1765 \beta_{4} - 3234 \beta_{5} + 154 \beta_{6} - 2700 \beta_{7} + 1889 \beta_{8} + 4484 \beta_{9} - 2059 \beta_{10} - 2392 \beta_{11} - 6763 \beta_{12} - 2355 \beta_{13} + 541 \beta_{14} + 1362 \beta_{15} + 61 \beta_{16} ) q^{82} + ( 1347614 - 59151 \beta_{1} - 5574 \beta_{2} - 1710 \beta_{3} - 2126 \beta_{4} + 1157 \beta_{5} - 1926 \beta_{6} - 393 \beta_{7} - 2925 \beta_{8} - 3298 \beta_{9} + 716 \beta_{10} + 89 \beta_{11} + 1504 \beta_{12} - 965 \beta_{13} - 774 \beta_{14} - 672 \beta_{15} - 2966 \beta_{16} ) q^{83} + ( -473690 + 57539 \beta_{1} + 14214 \beta_{2} - 532 \beta_{3} + 2373 \beta_{4} + 746 \beta_{5} + 791 \beta_{6} + 662 \beta_{7} - 581 \beta_{8} - 2307 \beta_{9} - 256 \beta_{10} - 242 \beta_{11} + 776 \beta_{12} + 115 \beta_{13} - 130 \beta_{14} - 986 \beta_{15} - 1302 \beta_{16} ) q^{85} + ( 1627821 - 47441 \beta_{1} + 9235 \beta_{2} - 7918 \beta_{3} - 280 \beta_{4} + 2443 \beta_{5} + 2170 \beta_{6} - 592 \beta_{7} + 317 \beta_{8} + 2036 \beta_{9} - 3246 \beta_{10} - 1568 \beta_{11} + 721 \beta_{12} - 572 \beta_{13} + 1920 \beta_{14} + 2136 \beta_{15} + 1433 \beta_{16} ) q^{86} + ( -3675454 - 66578 \beta_{1} - 28442 \beta_{2} + 10461 \beta_{3} - 5660 \beta_{4} + 2547 \beta_{5} - 1578 \beta_{6} - 3418 \beta_{7} + 876 \beta_{8} + 6177 \beta_{9} - 2974 \beta_{10} - 4798 \beta_{11} + 589 \beta_{12} - 1514 \beta_{13} + 3414 \beta_{14} - 688 \beta_{15} - 1143 \beta_{16} ) q^{88} + ( 1821018 - 167159 \beta_{1} - 2827 \beta_{2} - 210 \beta_{3} + 2820 \beta_{4} + 1713 \beta_{5} + 1165 \beta_{6} + 1082 \beta_{7} - 409 \beta_{8} + 244 \beta_{9} + 3249 \beta_{10} + 1013 \beta_{11} - 5305 \beta_{12} - 5111 \beta_{13} - 1718 \beta_{14} + 2298 \beta_{15} + 1689 \beta_{16} ) q^{89} + ( -2142205 + 63921 \beta_{1} - 3698 \beta_{2} + 3712 \beta_{3} + 280 \beta_{4} + 3963 \beta_{5} - 1704 \beta_{6} + 1837 \beta_{7} - 3301 \beta_{8} - 429 \beta_{9} + 867 \beta_{10} - 3135 \beta_{11} + 4098 \beta_{12} + 2490 \beta_{13} + 245 \beta_{14} + 244 \beta_{15} + 17 \beta_{16} ) q^{91} + ( 2506081 - 601315 \beta_{1} + 7165 \beta_{2} + 212 \beta_{3} - 2201 \beta_{4} - 892 \beta_{5} + 423 \beta_{6} + 5110 \beta_{7} - 100 \beta_{8} + 3932 \beta_{9} - 282 \beta_{10} + 5166 \beta_{11} + 3506 \beta_{12} + 4074 \beta_{13} + 4312 \beta_{14} + 722 \beta_{15} - 377 \beta_{16} ) q^{92} + ( 66225 - 50701 \beta_{1} + 7567 \beta_{2} + 9031 \beta_{3} + 4349 \beta_{4} + 851 \beta_{5} + 1067 \beta_{6} + 3221 \beta_{7} + 3296 \beta_{8} + 793 \beta_{9} - 1245 \beta_{10} - 3531 \beta_{11} - 4486 \beta_{12} + 5148 \beta_{13} - 3627 \beta_{14} + 4358 \beta_{15} + 2065 \beta_{16} ) q^{94} + ( 1755208 - 298967 \beta_{1} - 5588 \beta_{2} + 8450 \beta_{3} + 3218 \beta_{4} - 3043 \beta_{5} - 2497 \beta_{6} + 538 \beta_{7} + 3465 \beta_{8} + 3396 \beta_{9} - 2261 \beta_{10} - 1821 \beta_{11} - 7754 \beta_{12} - 2726 \beta_{13} - 1973 \beta_{14} - 852 \beta_{15} - 2147 \beta_{16} ) q^{95} + ( -1540008 - 75686 \beta_{1} + 19244 \beta_{2} + 5871 \beta_{3} - 3310 \beta_{4} - 5077 \beta_{5} - 3389 \beta_{6} - 1876 \beta_{7} - 66 \beta_{8} + 3570 \beta_{9} - 1928 \beta_{10} + 707 \beta_{11} - 4694 \beta_{12} - 2271 \beta_{13} - 80 \beta_{14} - 2245 \beta_{15} - 2371 \beta_{16} ) q^{97} + ( 5557208 - 673602 \beta_{1} + 24776 \beta_{2} - 8308 \beta_{3} - 10272 \beta_{4} - 4728 \beta_{5} - 493 \beta_{6} + 1336 \beta_{7} + 1912 \beta_{8} - 2297 \beta_{9} - 2857 \beta_{10} + 3302 \beta_{11} + 9699 \beta_{12} + 10091 \beta_{13} + 9435 \beta_{14} - 7652 \beta_{15} - 9124 \beta_{16} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10})$$ $$17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 193$$ $$\beta_{3}$$ $$=$$ $$($$$$59\!\cdots\!07$$$$\nu^{16} +$$$$79\!\cdots\!90$$$$\nu^{15} -$$$$10\!\cdots\!69$$$$\nu^{14} -$$$$18\!\cdots\!57$$$$\nu^{13} +$$$$67\!\cdots\!34$$$$\nu^{12} +$$$$14\!\cdots\!07$$$$\nu^{11} -$$$$23\!\cdots\!72$$$$\nu^{10} -$$$$49\!\cdots\!64$$$$\nu^{9} +$$$$43\!\cdots\!44$$$$\nu^{8} +$$$$85\!\cdots\!16$$$$\nu^{7} -$$$$42\!\cdots\!64$$$$\nu^{6} -$$$$68\!\cdots\!84$$$$\nu^{5} +$$$$19\!\cdots\!20$$$$\nu^{4} +$$$$19\!\cdots\!88$$$$\nu^{3} -$$$$38\!\cdots\!44$$$$\nu^{2} -$$$$18\!\cdots\!16$$$$\nu +$$$$21\!\cdots\!32$$$$)/$$$$86\!\cdots\!08$$ $$\beta_{4}$$ $$=$$ $$($$$$48\!\cdots\!11$$$$\nu^{16} -$$$$53\!\cdots\!10$$$$\nu^{15} -$$$$79\!\cdots\!25$$$$\nu^{14} -$$$$85\!\cdots\!17$$$$\nu^{13} +$$$$51\!\cdots\!94$$$$\nu^{12} +$$$$97\!\cdots\!67$$$$\nu^{11} -$$$$16\!\cdots\!04$$$$\nu^{10} -$$$$42\!\cdots\!08$$$$\nu^{9} +$$$$29\!\cdots\!60$$$$\nu^{8} +$$$$85\!\cdots\!04$$$$\nu^{7} -$$$$27\!\cdots\!20$$$$\nu^{6} -$$$$80\!\cdots\!76$$$$\nu^{5} +$$$$12\!\cdots\!04$$$$\nu^{4} +$$$$28\!\cdots\!80$$$$\nu^{3} -$$$$21\!\cdots\!08$$$$\nu^{2} -$$$$28\!\cdots\!28$$$$\nu +$$$$12\!\cdots\!20$$$$)/$$$$13\!\cdots\!72$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$35\!\cdots\!73$$$$\nu^{16} -$$$$20\!\cdots\!74$$$$\nu^{15} +$$$$58\!\cdots\!03$$$$\nu^{14} +$$$$89\!\cdots\!79$$$$\nu^{13} -$$$$38\!\cdots\!62$$$$\nu^{12} -$$$$83\!\cdots\!49$$$$\nu^{11} +$$$$12\!\cdots\!04$$$$\nu^{10} +$$$$33\!\cdots\!44$$$$\nu^{9} -$$$$22\!\cdots\!92$$$$\nu^{8} -$$$$64\!\cdots\!40$$$$\nu^{7} +$$$$20\!\cdots\!84$$$$\nu^{6} +$$$$57\!\cdots\!64$$$$\nu^{5} -$$$$90\!\cdots\!08$$$$\nu^{4} -$$$$19\!\cdots\!28$$$$\nu^{3} +$$$$15\!\cdots\!20$$$$\nu^{2} +$$$$18\!\cdots\!72$$$$\nu -$$$$84\!\cdots\!00$$$$)/$$$$86\!\cdots\!08$$ $$\beta_{6}$$ $$=$$ $$($$$$45\!\cdots\!93$$$$\nu^{16} +$$$$26\!\cdots\!78$$$$\nu^{15} -$$$$74\!\cdots\!39$$$$\nu^{14} -$$$$11\!\cdots\!55$$$$\nu^{13} +$$$$48\!\cdots\!86$$$$\nu^{12} +$$$$10\!\cdots\!97$$$$\nu^{11} -$$$$16\!\cdots\!20$$$$\nu^{10} -$$$$43\!\cdots\!76$$$$\nu^{9} +$$$$28\!\cdots\!64$$$$\nu^{8} +$$$$83\!\cdots\!96$$$$\nu^{7} -$$$$26\!\cdots\!08$$$$\nu^{6} -$$$$75\!\cdots\!84$$$$\nu^{5} +$$$$11\!\cdots\!24$$$$\nu^{4} +$$$$25\!\cdots\!44$$$$\nu^{3} -$$$$19\!\cdots\!68$$$$\nu^{2} -$$$$24\!\cdots\!88$$$$\nu +$$$$11\!\cdots\!36$$$$)/$$$$86\!\cdots\!08$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!07$$$$\nu^{16} -$$$$47\!\cdots\!74$$$$\nu^{15} +$$$$21\!\cdots\!37$$$$\nu^{14} +$$$$30\!\cdots\!89$$$$\nu^{13} -$$$$14\!\cdots\!98$$$$\nu^{12} -$$$$30\!\cdots\!95$$$$\nu^{11} +$$$$46\!\cdots\!84$$$$\nu^{10} +$$$$12\!\cdots\!76$$$$\nu^{9} -$$$$82\!\cdots\!72$$$$\nu^{8} -$$$$24\!\cdots\!76$$$$\nu^{7} +$$$$76\!\cdots\!76$$$$\nu^{6} +$$$$22\!\cdots\!00$$$$\nu^{5} -$$$$32\!\cdots\!84$$$$\nu^{4} -$$$$77\!\cdots\!12$$$$\nu^{3} +$$$$56\!\cdots\!52$$$$\nu^{2} +$$$$76\!\cdots\!68$$$$\nu -$$$$31\!\cdots\!48$$$$)/$$$$21\!\cdots\!52$$ $$\beta_{8}$$ $$=$$ $$($$$$70\!\cdots\!35$$$$\nu^{16} +$$$$28\!\cdots\!98$$$$\nu^{15} -$$$$11\!\cdots\!97$$$$\nu^{14} -$$$$16\!\cdots\!41$$$$\nu^{13} +$$$$74\!\cdots\!78$$$$\nu^{12} +$$$$16\!\cdots\!11$$$$\nu^{11} -$$$$24\!\cdots\!24$$$$\nu^{10} -$$$$65\!\cdots\!60$$$$\nu^{9} +$$$$43\!\cdots\!24$$$$\nu^{8} +$$$$12\!\cdots\!40$$$$\nu^{7} -$$$$40\!\cdots\!28$$$$\nu^{6} -$$$$11\!\cdots\!76$$$$\nu^{5} +$$$$17\!\cdots\!76$$$$\nu^{4} +$$$$39\!\cdots\!28$$$$\nu^{3} -$$$$29\!\cdots\!80$$$$\nu^{2} -$$$$38\!\cdots\!28$$$$\nu +$$$$16\!\cdots\!92$$$$)/$$$$10\!\cdots\!76$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$56\!\cdots\!87$$$$\nu^{16} -$$$$23\!\cdots\!74$$$$\nu^{15} +$$$$92\!\cdots\!45$$$$\nu^{14} +$$$$13\!\cdots\!85$$$$\nu^{13} -$$$$60\!\cdots\!58$$$$\nu^{12} -$$$$13\!\cdots\!95$$$$\nu^{11} +$$$$19\!\cdots\!64$$$$\nu^{10} +$$$$53\!\cdots\!44$$$$\nu^{9} -$$$$35\!\cdots\!36$$$$\nu^{8} -$$$$10\!\cdots\!36$$$$\nu^{7} +$$$$32\!\cdots\!88$$$$\nu^{6} +$$$$94\!\cdots\!92$$$$\nu^{5} -$$$$14\!\cdots\!28$$$$\nu^{4} -$$$$31\!\cdots\!20$$$$\nu^{3} +$$$$24\!\cdots\!08$$$$\nu^{2} +$$$$31\!\cdots\!04$$$$\nu -$$$$13\!\cdots\!76$$$$)/$$$$86\!\cdots\!08$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$31\!\cdots\!73$$$$\nu^{16} -$$$$83\!\cdots\!90$$$$\nu^{15} +$$$$52\!\cdots\!19$$$$\nu^{14} +$$$$66\!\cdots\!03$$$$\nu^{13} -$$$$33\!\cdots\!50$$$$\nu^{12} -$$$$68\!\cdots\!01$$$$\nu^{11} +$$$$11\!\cdots\!64$$$$\nu^{10} +$$$$28\!\cdots\!32$$$$\nu^{9} -$$$$19\!\cdots\!16$$$$\nu^{8} -$$$$56\!\cdots\!20$$$$\nu^{7} +$$$$18\!\cdots\!20$$$$\nu^{6} +$$$$51\!\cdots\!80$$$$\nu^{5} -$$$$79\!\cdots\!04$$$$\nu^{4} -$$$$17\!\cdots\!56$$$$\nu^{3} +$$$$13\!\cdots\!36$$$$\nu^{2} +$$$$17\!\cdots\!56$$$$\nu -$$$$77\!\cdots\!04$$$$)/$$$$43\!\cdots\!04$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$66\!\cdots\!61$$$$\nu^{16} -$$$$22\!\cdots\!54$$$$\nu^{15} +$$$$10\!\cdots\!47$$$$\nu^{14} +$$$$14\!\cdots\!47$$$$\nu^{13} -$$$$70\!\cdots\!10$$$$\nu^{12} -$$$$14\!\cdots\!65$$$$\nu^{11} +$$$$23\!\cdots\!52$$$$\nu^{10} +$$$$60\!\cdots\!12$$$$\nu^{9} -$$$$41\!\cdots\!44$$$$\nu^{8} -$$$$11\!\cdots\!44$$$$\nu^{7} +$$$$38\!\cdots\!80$$$$\nu^{6} +$$$$10\!\cdots\!56$$$$\nu^{5} -$$$$16\!\cdots\!68$$$$\nu^{4} -$$$$36\!\cdots\!80$$$$\nu^{3} +$$$$28\!\cdots\!88$$$$\nu^{2} +$$$$36\!\cdots\!96$$$$\nu -$$$$15\!\cdots\!32$$$$)/$$$$86\!\cdots\!08$$ $$\beta_{12}$$ $$=$$ $$($$$$31\!\cdots\!53$$$$\nu^{16} +$$$$82\!\cdots\!82$$$$\nu^{15} -$$$$52\!\cdots\!75$$$$\nu^{14} -$$$$66\!\cdots\!91$$$$\nu^{13} +$$$$33\!\cdots\!82$$$$\nu^{12} +$$$$68\!\cdots\!33$$$$\nu^{11} -$$$$11\!\cdots\!20$$$$\nu^{10} -$$$$28\!\cdots\!28$$$$\nu^{9} +$$$$19\!\cdots\!64$$$$\nu^{8} +$$$$56\!\cdots\!72$$$$\nu^{7} -$$$$18\!\cdots\!24$$$$\nu^{6} -$$$$51\!\cdots\!56$$$$\nu^{5} +$$$$78\!\cdots\!64$$$$\nu^{4} +$$$$17\!\cdots\!20$$$$\nu^{3} -$$$$13\!\cdots\!92$$$$\nu^{2} -$$$$17\!\cdots\!24$$$$\nu +$$$$75\!\cdots\!48$$$$)/$$$$21\!\cdots\!52$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$66\!\cdots\!77$$$$\nu^{16} -$$$$18\!\cdots\!42$$$$\nu^{15} +$$$$10\!\cdots\!43$$$$\nu^{14} +$$$$14\!\cdots\!79$$$$\nu^{13} -$$$$70\!\cdots\!42$$$$\nu^{12} -$$$$14\!\cdots\!01$$$$\nu^{11} +$$$$23\!\cdots\!80$$$$\nu^{10} +$$$$59\!\cdots\!64$$$$\nu^{9} -$$$$41\!\cdots\!60$$$$\nu^{8} -$$$$11\!\cdots\!08$$$$\nu^{7} +$$$$38\!\cdots\!96$$$$\nu^{6} +$$$$10\!\cdots\!88$$$$\nu^{5} -$$$$16\!\cdots\!60$$$$\nu^{4} -$$$$36\!\cdots\!36$$$$\nu^{3} +$$$$28\!\cdots\!36$$$$\nu^{2} +$$$$35\!\cdots\!84$$$$\nu -$$$$15\!\cdots\!20$$$$)/$$$$43\!\cdots\!04$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!89$$$$\nu^{16} -$$$$54\!\cdots\!98$$$$\nu^{15} +$$$$17\!\cdots\!47$$$$\nu^{14} +$$$$26\!\cdots\!91$$$$\nu^{13} -$$$$11\!\cdots\!82$$$$\nu^{12} -$$$$25\!\cdots\!57$$$$\nu^{11} +$$$$37\!\cdots\!88$$$$\nu^{10} +$$$$10\!\cdots\!52$$$$\nu^{9} -$$$$66\!\cdots\!36$$$$\nu^{8} -$$$$19\!\cdots\!40$$$$\nu^{7} +$$$$61\!\cdots\!52$$$$\nu^{6} +$$$$17\!\cdots\!44$$$$\nu^{5} -$$$$26\!\cdots\!08$$$$\nu^{4} -$$$$60\!\cdots\!20$$$$\nu^{3} +$$$$45\!\cdots\!96$$$$\nu^{2} +$$$$59\!\cdots\!12$$$$\nu -$$$$25\!\cdots\!20$$$$)/$$$$54\!\cdots\!88$$ $$\beta_{15}$$ $$=$$ $$($$$$21\!\cdots\!51$$$$\nu^{16} +$$$$79\!\cdots\!98$$$$\nu^{15} -$$$$36\!\cdots\!29$$$$\nu^{14} -$$$$49\!\cdots\!89$$$$\nu^{13} +$$$$23\!\cdots\!86$$$$\nu^{12} +$$$$49\!\cdots\!87$$$$\nu^{11} -$$$$77\!\cdots\!24$$$$\nu^{10} -$$$$20\!\cdots\!76$$$$\nu^{9} +$$$$13\!\cdots\!68$$$$\nu^{8} +$$$$40\!\cdots\!80$$$$\nu^{7} -$$$$12\!\cdots\!88$$$$\nu^{6} -$$$$36\!\cdots\!48$$$$\nu^{5} +$$$$54\!\cdots\!00$$$$\nu^{4} +$$$$12\!\cdots\!12$$$$\nu^{3} -$$$$93\!\cdots\!24$$$$\nu^{2} -$$$$12\!\cdots\!64$$$$\nu +$$$$52\!\cdots\!68$$$$)/$$$$86\!\cdots\!08$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$37\!\cdots\!17$$$$\nu^{16} -$$$$15\!\cdots\!98$$$$\nu^{15} +$$$$61\!\cdots\!79$$$$\nu^{14} +$$$$88\!\cdots\!39$$$$\nu^{13} -$$$$40\!\cdots\!02$$$$\nu^{12} -$$$$86\!\cdots\!29$$$$\nu^{11} +$$$$13\!\cdots\!72$$$$\nu^{10} +$$$$35\!\cdots\!48$$$$\nu^{9} -$$$$23\!\cdots\!56$$$$\nu^{8} -$$$$69\!\cdots\!52$$$$\nu^{7} +$$$$21\!\cdots\!00$$$$\nu^{6} +$$$$62\!\cdots\!84$$$$\nu^{5} -$$$$93\!\cdots\!96$$$$\nu^{4} -$$$$21\!\cdots\!68$$$$\nu^{3} +$$$$16\!\cdots\!64$$$$\nu^{2} +$$$$21\!\cdots\!20$$$$\nu -$$$$91\!\cdots\!84$$$$)/$$$$86\!\cdots\!08$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 193$$ $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{8} - \beta_{3} + 3 \beta_{2} + 336 \beta_{1} + 253$$ $$\nu^{4}$$ $$=$$ $$\beta_{16} - \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + 6 \beta_{11} + 3 \beta_{10} - 20 \beta_{9} + 7 \beta_{7} + 4 \beta_{6} - 10 \beta_{5} + 9 \beta_{4} - 42 \beta_{3} + 452 \beta_{2} + 1094 \beta_{1} + 64899$$ $$\nu^{5}$$ $$=$$ $$148 \beta_{16} - 49 \beta_{15} - 132 \beta_{14} + \beta_{13} - 10 \beta_{12} - 147 \beta_{11} + 51 \beta_{10} - 822 \beta_{9} - 433 \beta_{8} - 120 \beta_{7} + 67 \beta_{6} - 36 \beta_{5} + 201 \beta_{4} - 671 \beta_{3} + 2687 \beta_{2} + 133962 \beta_{1} + 234684$$ $$\nu^{6}$$ $$=$$ $$766 \beta_{16} - 1912 \beta_{15} + 1532 \beta_{14} + 5300 \beta_{13} + 5146 \beta_{12} + 3504 \beta_{11} + 1244 \beta_{10} - 17931 \beta_{9} - 685 \beta_{8} + 3860 \beta_{7} + 2368 \beta_{6} - 7870 \beta_{5} + 6240 \beta_{4} - 31899 \beta_{3} + 208895 \beta_{2} + 823252 \beta_{1} + 25899709$$ $$\nu^{7}$$ $$=$$ $$114021 \beta_{16} - 57379 \beta_{15} - 98702 \beta_{14} + 27464 \beta_{13} + 12614 \beta_{12} - 117618 \beta_{11} + 21245 \beta_{10} - 532076 \beta_{9} - 170026 \beta_{8} - 94623 \beta_{7} + 37144 \beta_{6} - 60144 \beta_{5} + 163691 \beta_{4} - 454278 \beta_{3} + 1935338 \beta_{2} + 59445674 \beta_{1} + 168642875$$ $$\nu^{8}$$ $$=$$ $$635630 \beta_{16} - 1725907 \beta_{15} + 716880 \beta_{14} + 3990095 \beta_{13} + 3826244 \beta_{12} + 1480107 \beta_{11} + 273189 \beta_{10} - 12300698 \beta_{9} - 644885 \beta_{8} + 1573860 \beta_{7} + 1108317 \beta_{6} - 4832486 \beta_{5} + 3680059 \beta_{4} - 19282533 \beta_{3} + 102204187 \beta_{2} + 556690654 \beta_{1} + 11512527046$$ $$\nu^{9}$$ $$=$$ $$66441380 \beta_{16} - 46530816 \beta_{15} - 54081184 \beta_{14} + 34859488 \beta_{13} + 23733736 \beta_{12} - 69574116 \beta_{11} + 2638544 \beta_{10} - 323608777 \beta_{9} - 68189073 \beta_{8} - 56625720 \beta_{7} + 16872652 \beta_{6} - 54482596 \beta_{5} + 100926008 \beta_{4} - 301029881 \beta_{3} + 1274711627 \beta_{2} + 28523377628 \beta_{1} + 111819940985$$ $$\nu^{10}$$ $$=$$ $$501465529 \beta_{16} - 1237972625 \beta_{15} + 233253826 \beta_{14} + 2511200180 \beta_{13} + 2389824196 \beta_{12} + 503051806 \beta_{11} - 39641013 \beta_{10} - 7689673076 \beta_{9} - 455032840 \beta_{8} + 527747223 \beta_{7} + 492524380 \beta_{6} - 2767466570 \beta_{5} + 2128323305 \beta_{4} - 11027353730 \beta_{3} + 52428277236 \beta_{2} + 357043723790 \beta_{1} + 5535112643107$$ $$\nu^{11}$$ $$=$$ $$35736675116 \beta_{16} - 32314549633 \beta_{15} - 26860437140 \beta_{14} + 29741506937 \beta_{13} + 22882238054 \beta_{12} - 37289224731 \beta_{11} - 3193459157 \beta_{10} - 192124557958 \beta_{9} - 28967589585 \beta_{8} - 30930398288 \beta_{7} + 7650146203 \beta_{6} - 39795597420 \beta_{5} + 57637687609 \beta_{4} - 192577166479 \beta_{3} + 800858285671 \beta_{2} + 14487549145530 \beta_{1} + 71002791721804$$ $$\nu^{12}$$ $$=$$ $$361105418406 \beta_{16} - 804046573904 \beta_{15} + 30850932748 \beta_{14} + 1473767444980 \beta_{13} + 1394674156634 \beta_{12} + 115674091168 \beta_{11} - 97900651060 \beta_{10} - 4617688891091 \beta_{9} - 292190770973 \beta_{8} + 124997453404 \beta_{7} + 221651564864 \beta_{6} - 1553901022966 \beta_{5} + 1231670769656 \beta_{4} - 6227336288043 \beta_{3} + 27864472537351 \beta_{2} + 221664860348788 \beta_{1} + 2816880273754165$$ $$\nu^{13}$$ $$=$$ $$18880479302309 \beta_{16} - 20707039000419 \beta_{15} - 12995109669646 \beta_{14} + 21435326874856 \beta_{13} + 17532114461750 \beta_{12} - 19313083144786 \beta_{11} - 3698963227643 \beta_{10} - 112727941468092 \beta_{9} - 13197273940146 \beta_{8} - 16334138192703 \beta_{7} + 3655945268920 \beta_{6} - 26288789426696 \beta_{5} + 32323772521195 \beta_{4} - 119416720335534 \beta_{3} + 489247664844250 \beta_{2} + 7661493335974906 \beta_{1} + 43832377051773091$$ $$\nu^{14}$$ $$=$$ $$241898320413574 \beta_{16} - 496953590726531 \beta_{15} - 31391010068768 \beta_{14} + 841704157680503 \beta_{13} + 792723427066324 \beta_{12} - 12606051406781 \beta_{11} - 83767497777507 \beta_{10} - 2720372235757786 \beta_{9} - 179989902332949 \beta_{8} - 6774735830852 \beta_{7} + 103874593049941 \beta_{6} - 870690670192766 \beta_{5} + 713532777800107 \beta_{4} - 3517801159349237 \beta_{3} + 15190734218406003 \beta_{2} + 134662620540970350 \beta_{1} + 1492178253392643894$$ $$\nu^{15}$$ $$=$$ $$10036960860727548 \beta_{16} - 12684333178374264 \beta_{15} - 6330442398863280 \beta_{14} + 14150371142284144 \beta_{13} + 11999865734781128 \beta_{12} - 9938264213595812 \beta_{11} - 2788367475148896 \beta_{10} - 65715614135554449 \beta_{9} - 6431988297408113 \beta_{8} - 8553315234597312 \beta_{7} + 1855669246394652 \beta_{6} - 16461370647050844 \beta_{5} + 18151159772838256 \beta_{4} - 72348159979624697 \beta_{3} + 293603308263291683 \beta_{2} + 4166551145495440220 \beta_{1} + 26537911593091200545$$ $$\nu^{16}$$ $$=$$ $$154209500770224953 \beta_{16} - 299008966998001473 \beta_{15} - 39438974200862334 \beta_{14} + 476562338666043236 \beta_{13} + 446856473440287476 \beta_{12} - 41961170240042290 \beta_{11} - 58242401952447453 \beta_{10} - 1587285673634174612 \beta_{9} - 108377377751011568 \beta_{8} - 37558631020600425 \beta_{7} + 51129416628082604 \beta_{6} - 489846436908785026 \beta_{5} + 413236512480235225 \beta_{4} - 1994990087758290122 \beta_{3} + 8430399536766469124 \beta_{2} + 80592841391241170750 \beta_{1} + 812600091187178438779$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 24.0278 18.2619 17.5255 15.6681 12.7630 7.01814 4.11298 2.41303 2.39686 −4.01497 −4.85375 −10.1391 −11.3335 −15.8998 −15.9892 −19.6388 −20.3182
−22.0278 0 357.226 492.460 0 −1209.81 −5049.35 0 −10847.8
1.2 −16.2619 0 136.450 30.6671 0 −1356.62 −137.421 0 −498.706
1.3 −15.5255 0 113.041 141.845 0 105.285 232.240 0 −2202.21
1.4 −13.6681 0 58.8158 207.404 0 882.793 945.614 0 −2834.81
1.5 −10.7630 0 −12.1577 −451.863 0 −504.672 1508.52 0 4863.40
1.6 −5.01814 0 −102.818 77.5687 0 −216.829 1158.28 0 −389.251
1.7 −2.11298 0 −123.535 537.118 0 1259.15 531.490 0 −1134.92
1.8 −0.413031 0 −127.829 −231.152 0 302.150 105.665 0 95.4729
1.9 −0.396855 0 −127.843 −247.233 0 −652.929 101.532 0 98.1157
1.10 6.01497 0 −91.8201 385.807 0 847.649 −1322.21 0 2320.62
1.11 6.85375 0 −81.0262 190.727 0 −799.288 −1432.61 0 1307.20
1.12 12.1391 0 19.3578 −236.334 0 1426.66 −1318.82 0 −2868.89
1.13 13.3335 0 49.7814 −152.093 0 −1328.47 −1042.93 0 −2027.93
1.14 17.8998 0 192.403 −255.355 0 −1072.82 1152.80 0 −4570.81
1.15 17.9892 0 195.612 98.9095 0 159.201 1216.28 0 1779.30
1.16 21.6388 0 340.238 399.107 0 −1780.57 4592.58 0 8636.21
1.17 22.3182 0 370.104 84.4166 0 1532.13 5403.34 0 1884.03
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.17 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$59$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.8.a.d 17
3.b odd 2 1 177.8.a.b 17

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.b 17 3.b odd 2 1
531.8.a.d 17 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$17\!\cdots\!40$$$$T_{2}^{7} +$$$$47\!\cdots\!08$$$$T_{2}^{6} -$$$$13\!\cdots\!60$$$$T_{2}^{5} -$$$$12\!\cdots\!84$$$$T_{2}^{4} +$$$$21\!\cdots\!96$$$$T_{2}^{3} +$$$$11\!\cdots\!44$$$$T_{2}^{2} +$$$$77\!\cdots\!12$$$$T_{2} +$$$$14\!\cdots\!16$$">$$T_{2}^{17} - \cdots$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(531))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1475467232854016 + 7788757416435712 T + 11284014941625344 T^{2} + 2120229732944896 T^{3} - 1232918500989184 T^{4} - 131417789647360 T^{5} + 47889020315008 T^{6} + 1766762759040 T^{7} - 750286858976 T^{8} - 4350215008 T^{9} + 5698856604 T^{10} - 60303860 T^{11} - 22317501 T^{12} + 456514 T^{13} + 43065 T^{14} - 1159 T^{15} - 32 T^{16} + T^{17}$$
$3$ $$T^{17}$$
$5$ $$-$$$$10\!\cdots\!00$$$$+$$$$73\!\cdots\!00$$$$T -$$$$15\!\cdots\!00$$$$T^{2} +$$$$84\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!00$$$$T^{4} -$$$$15\!\cdots\!50$$$$T^{5} -$$$$13\!\cdots\!00$$$$T^{6} +$$$$75\!\cdots\!25$$$$T^{7} -$$$$13\!\cdots\!60$$$$T^{8} -$$$$16\!\cdots\!19$$$$T^{9} + 5048689959553278496 T^{10} + 17934286341578002 T^{11} - 72897507384604 T^{12} - 68887937908 T^{13} + 471032456 T^{14} - 138135 T^{15} - 1072 T^{16} + T^{17}$$
$7$ $$24\!\cdots\!92$$$$-$$$$25\!\cdots\!64$$$$T -$$$$86\!\cdots\!96$$$$T^{2} +$$$$96\!\cdots\!08$$$$T^{3} +$$$$15\!\cdots\!88$$$$T^{4} -$$$$88\!\cdots\!27$$$$T^{5} -$$$$15\!\cdots\!55$$$$T^{6} +$$$$23\!\cdots\!40$$$$T^{7} +$$$$51\!\cdots\!89$$$$T^{8} -$$$$20\!\cdots\!32$$$$T^{9} -$$$$74\!\cdots\!79$$$$T^{10} - 1509583187617520794 T^{11} + 53136293400057767 T^{12} + 12104685446904 T^{13} - 18230095433 T^{14} - 6278660 T^{15} + 2407 T^{16} + T^{17}$$
$11$ $$-$$$$63\!\cdots\!60$$$$+$$$$30\!\cdots\!40$$$$T +$$$$21\!\cdots\!08$$$$T^{2} -$$$$79\!\cdots\!04$$$$T^{3} -$$$$58\!\cdots\!72$$$$T^{4} +$$$$28\!\cdots\!21$$$$T^{5} +$$$$21\!\cdots\!68$$$$T^{6} -$$$$45\!\cdots\!10$$$$T^{7} -$$$$30\!\cdots\!84$$$$T^{8} +$$$$43\!\cdots\!95$$$$T^{9} +$$$$21\!\cdots\!48$$$$T^{10} -$$$$25\!\cdots\!00$$$$T^{11} - 76421139661163220848 T^{12} + 8544088115237191 T^{13} + 1339552706104 T^{14} - 148928546 T^{15} - 8888 T^{16} + T^{17}$$
$13$ $$12\!\cdots\!88$$$$-$$$$43\!\cdots\!00$$$$T -$$$$91\!\cdots\!08$$$$T^{2} -$$$$71\!\cdots\!18$$$$T^{3} +$$$$43\!\cdots\!72$$$$T^{4} +$$$$84\!\cdots\!91$$$$T^{5} -$$$$68\!\cdots\!46$$$$T^{6} -$$$$10\!\cdots\!54$$$$T^{7} +$$$$37\!\cdots\!54$$$$T^{8} +$$$$43\!\cdots\!99$$$$T^{9} -$$$$91\!\cdots\!40$$$$T^{10} -$$$$90\!\cdots\!66$$$$T^{11} +$$$$10\!\cdots\!08$$$$T^{12} + 95813951578561825 T^{13} - 6013088859298 T^{14} - 497691754 T^{15} + 12702 T^{16} + T^{17}$$
$17$ $$-$$$$39\!\cdots\!68$$$$+$$$$16\!\cdots\!68$$$$T +$$$$52\!\cdots\!04$$$$T^{2} -$$$$39\!\cdots\!92$$$$T^{3} +$$$$76\!\cdots\!02$$$$T^{4} -$$$$29\!\cdots\!63$$$$T^{5} -$$$$81\!\cdots\!25$$$$T^{6} +$$$$10\!\cdots\!02$$$$T^{7} -$$$$53\!\cdots\!99$$$$T^{8} -$$$$56\!\cdots\!68$$$$T^{9} +$$$$27\!\cdots\!35$$$$T^{10} +$$$$81\!\cdots\!40$$$$T^{11} -$$$$89\!\cdots\!73$$$$T^{12} + 661179196028435860 T^{13} + 101134516252165 T^{14} - 2055029030 T^{15} - 36167 T^{16} + T^{17}$$
$19$ $$27\!\cdots\!00$$$$+$$$$11\!\cdots\!60$$$$T -$$$$67\!\cdots\!64$$$$T^{2} -$$$$29\!\cdots\!44$$$$T^{3} -$$$$36\!\cdots\!24$$$$T^{4} -$$$$88\!\cdots\!68$$$$T^{5} +$$$$18\!\cdots\!04$$$$T^{6} +$$$$15\!\cdots\!03$$$$T^{7} +$$$$13\!\cdots\!85$$$$T^{8} -$$$$27\!\cdots\!45$$$$T^{9} -$$$$91\!\cdots\!30$$$$T^{10} +$$$$13\!\cdots\!39$$$$T^{11} +$$$$10\!\cdots\!27$$$$T^{12} + 3590559653165734181 T^{13} - 451198576183666 T^{14} - 4748132835 T^{15} + 71037 T^{16} + T^{17}$$
$23$ $$15\!\cdots\!72$$$$-$$$$17\!\cdots\!00$$$$T +$$$$38\!\cdots\!40$$$$T^{2} -$$$$20\!\cdots\!92$$$$T^{3} -$$$$91\!\cdots\!48$$$$T^{4} +$$$$12\!\cdots\!72$$$$T^{5} -$$$$28\!\cdots\!32$$$$T^{6} -$$$$80\!\cdots\!35$$$$T^{7} +$$$$50\!\cdots\!41$$$$T^{8} -$$$$64\!\cdots\!93$$$$T^{9} -$$$$10\!\cdots\!14$$$$T^{10} +$$$$32\!\cdots\!77$$$$T^{11} -$$$$12\!\cdots\!45$$$$T^{12} -$$$$40\!\cdots\!55$$$$T^{13} + 4401398963753790 T^{14} + 6359155007 T^{15} - 269995 T^{16} + T^{17}$$
$29$ $$15\!\cdots\!00$$$$+$$$$59\!\cdots\!00$$$$T -$$$$10\!\cdots\!80$$$$T^{2} -$$$$42\!\cdots\!60$$$$T^{3} +$$$$10\!\cdots\!76$$$$T^{4} +$$$$31\!\cdots\!74$$$$T^{5} -$$$$33\!\cdots\!90$$$$T^{6} +$$$$27\!\cdots\!65$$$$T^{7} +$$$$13\!\cdots\!31$$$$T^{8} -$$$$26\!\cdots\!59$$$$T^{9} +$$$$50\!\cdots\!40$$$$T^{10} +$$$$84\!\cdots\!93$$$$T^{11} -$$$$39\!\cdots\!19$$$$T^{12} -$$$$81\!\cdots\!97$$$$T^{13} + 80929169301311500 T^{14} - 54576459477 T^{15} - 543825 T^{16} + T^{17}$$
$31$ $$83\!\cdots\!92$$$$+$$$$14\!\cdots\!52$$$$T +$$$$48\!\cdots\!00$$$$T^{2} +$$$$63\!\cdots\!60$$$$T^{3} -$$$$19\!\cdots\!56$$$$T^{4} -$$$$21\!\cdots\!14$$$$T^{5} +$$$$14\!\cdots\!96$$$$T^{6} +$$$$31\!\cdots\!77$$$$T^{7} +$$$$30\!\cdots\!45$$$$T^{8} -$$$$15\!\cdots\!55$$$$T^{9} -$$$$58\!\cdots\!02$$$$T^{10} +$$$$26\!\cdots\!21$$$$T^{11} +$$$$16\!\cdots\!35$$$$T^{12} -$$$$79\!\cdots\!93$$$$T^{13} - 173711654977129714 T^{14} - 156853176577 T^{15} + 633109 T^{16} + T^{17}$$
$37$ $$-$$$$36\!\cdots\!24$$$$-$$$$68\!\cdots\!40$$$$T +$$$$30\!\cdots\!04$$$$T^{2} +$$$$60\!\cdots\!06$$$$T^{3} -$$$$43\!\cdots\!30$$$$T^{4} -$$$$11\!\cdots\!27$$$$T^{5} -$$$$25\!\cdots\!57$$$$T^{6} +$$$$76\!\cdots\!08$$$$T^{7} +$$$$21\!\cdots\!99$$$$T^{8} -$$$$17\!\cdots\!76$$$$T^{9} -$$$$73\!\cdots\!61$$$$T^{10} +$$$$14\!\cdots\!34$$$$T^{11} +$$$$94\!\cdots\!17$$$$T^{12} -$$$$11\!\cdots\!56$$$$T^{13} - 508971507672490843 T^{14} - 384927448474 T^{15} + 867607 T^{16} + T^{17}$$
$41$ $$57\!\cdots\!00$$$$+$$$$24\!\cdots\!60$$$$T -$$$$10\!\cdots\!88$$$$T^{2} +$$$$60\!\cdots\!56$$$$T^{3} +$$$$12\!\cdots\!86$$$$T^{4} -$$$$14\!\cdots\!03$$$$T^{5} +$$$$11\!\cdots\!75$$$$T^{6} +$$$$11\!\cdots\!06$$$$T^{7} -$$$$20\!\cdots\!23$$$$T^{8} -$$$$39\!\cdots\!52$$$$T^{9} +$$$$10\!\cdots\!11$$$$T^{10} +$$$$46\!\cdots\!32$$$$T^{11} -$$$$26\!\cdots\!09$$$$T^{12} +$$$$50\!\cdots\!84$$$$T^{13} + 3164201715533339881 T^{14} - 1611382106054 T^{15} - 1428939 T^{16} + T^{17}$$
$43$ $$42\!\cdots\!92$$$$+$$$$33\!\cdots\!72$$$$T -$$$$31\!\cdots\!64$$$$T^{2} -$$$$24\!\cdots\!88$$$$T^{3} +$$$$77\!\cdots\!40$$$$T^{4} +$$$$58\!\cdots\!97$$$$T^{5} -$$$$98\!\cdots\!12$$$$T^{6} -$$$$66\!\cdots\!50$$$$T^{7} +$$$$71\!\cdots\!96$$$$T^{8} +$$$$39\!\cdots\!03$$$$T^{9} -$$$$30\!\cdots\!84$$$$T^{10} -$$$$13\!\cdots\!48$$$$T^{11} +$$$$75\!\cdots\!36$$$$T^{12} +$$$$24\!\cdots\!07$$$$T^{13} - 949572962808570580 T^{14} - 2453152665830 T^{15} + 477060 T^{16} + T^{17}$$
$47$ $$-$$$$19\!\cdots\!80$$$$-$$$$98\!\cdots\!20$$$$T +$$$$36\!\cdots\!56$$$$T^{2} +$$$$17\!\cdots\!44$$$$T^{3} -$$$$74\!\cdots\!80$$$$T^{4} -$$$$10\!\cdots\!48$$$$T^{5} +$$$$34\!\cdots\!16$$$$T^{6} +$$$$17\!\cdots\!51$$$$T^{7} -$$$$60\!\cdots\!17$$$$T^{8} -$$$$70\!\cdots\!97$$$$T^{9} +$$$$36\!\cdots\!46$$$$T^{10} -$$$$88\!\cdots\!01$$$$T^{11} -$$$$68\!\cdots\!67$$$$T^{12} +$$$$41\!\cdots\!13$$$$T^{13} + 4933084879786006586 T^{14} - 3740881796475 T^{15} - 1217849 T^{16} + T^{17}$$
$53$ $$-$$$$92\!\cdots\!84$$$$+$$$$33\!\cdots\!16$$$$T -$$$$36\!\cdots\!08$$$$T^{2} +$$$$11\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!76$$$$T^{4} -$$$$11\!\cdots\!30$$$$T^{5} +$$$$46\!\cdots\!28$$$$T^{6} +$$$$38\!\cdots\!53$$$$T^{7} -$$$$32\!\cdots\!44$$$$T^{8} -$$$$56\!\cdots\!07$$$$T^{9} +$$$$65\!\cdots\!00$$$$T^{10} +$$$$35\!\cdots\!14$$$$T^{11} -$$$$58\!\cdots\!64$$$$T^{12} -$$$$44\!\cdots\!24$$$$T^{13} + 23532998832058313232 T^{14} - 3673882215883 T^{15} - 3487068 T^{16} + T^{17}$$
$59$ $$( -205379 + T )^{17}$$
$61$ $$46\!\cdots\!96$$$$+$$$$20\!\cdots\!36$$$$T -$$$$79\!\cdots\!56$$$$T^{2} +$$$$12\!\cdots\!44$$$$T^{3} -$$$$39\!\cdots\!64$$$$T^{4} -$$$$35\!\cdots\!38$$$$T^{5} +$$$$18\!\cdots\!90$$$$T^{6} -$$$$97\!\cdots\!65$$$$T^{7} -$$$$12\!\cdots\!73$$$$T^{8} +$$$$89\!\cdots\!05$$$$T^{9} +$$$$33\!\cdots\!56$$$$T^{10} -$$$$29\!\cdots\!17$$$$T^{11} -$$$$47\!\cdots\!27$$$$T^{12} +$$$$45\!\cdots\!87$$$$T^{13} + 33905578224492256776 T^{14} - 34115605213239 T^{15} - 998917 T^{16} + T^{17}$$
$67$ $$41\!\cdots\!12$$$$+$$$$21\!\cdots\!40$$$$T +$$$$37\!\cdots\!12$$$$T^{2} +$$$$23\!\cdots\!88$$$$T^{3} +$$$$58\!\cdots\!48$$$$T^{4} +$$$$43\!\cdots\!12$$$$T^{5} -$$$$36\!\cdots\!56$$$$T^{6} -$$$$57\!\cdots\!43$$$$T^{7} +$$$$43\!\cdots\!98$$$$T^{8} +$$$$27\!\cdots\!65$$$$T^{9} +$$$$10\!\cdots\!68$$$$T^{10} -$$$$65\!\cdots\!90$$$$T^{11} -$$$$14\!\cdots\!24$$$$T^{12} +$$$$79\!\cdots\!66$$$$T^{13} - 223013770731489972 T^{14} - 45615831238195 T^{15} + 356026 T^{16} + T^{17}$$
$71$ $$88\!\cdots\!16$$$$+$$$$14\!\cdots\!92$$$$T -$$$$45\!\cdots\!40$$$$T^{2} -$$$$18\!\cdots\!58$$$$T^{3} +$$$$37\!\cdots\!78$$$$T^{4} +$$$$28\!\cdots\!99$$$$T^{5} -$$$$75\!\cdots\!50$$$$T^{6} -$$$$14\!\cdots\!90$$$$T^{7} +$$$$58\!\cdots\!32$$$$T^{8} -$$$$18\!\cdots\!53$$$$T^{9} -$$$$16\!\cdots\!72$$$$T^{10} +$$$$10\!\cdots\!94$$$$T^{11} +$$$$26\!\cdots\!94$$$$T^{12} -$$$$13\!\cdots\!71$$$$T^{13} +$$$$24\!\cdots\!74$$$$T^{14} + 31346019006810 T^{15} - 12879428 T^{16} + T^{17}$$
$73$ $$-$$$$48\!\cdots\!00$$$$-$$$$86\!\cdots\!00$$$$T +$$$$31\!\cdots\!00$$$$T^{2} -$$$$54\!\cdots\!80$$$$T^{3} -$$$$10\!\cdots\!80$$$$T^{4} +$$$$21\!\cdots\!56$$$$T^{5} -$$$$41\!\cdots\!50$$$$T^{6} -$$$$73\!\cdots\!41$$$$T^{7} +$$$$20\!\cdots\!09$$$$T^{8} +$$$$10\!\cdots\!31$$$$T^{9} -$$$$28\!\cdots\!88$$$$T^{10} -$$$$83\!\cdots\!17$$$$T^{11} +$$$$18\!\cdots\!07$$$$T^{12} +$$$$38\!\cdots\!13$$$$T^{13} -$$$$54\!\cdots\!88$$$$T^{14} - 95507635887571 T^{15} + 6176157 T^{16} + T^{17}$$
$79$ $$-$$$$80\!\cdots\!28$$$$+$$$$26\!\cdots\!64$$$$T +$$$$11\!\cdots\!64$$$$T^{2} +$$$$53\!\cdots\!32$$$$T^{3} -$$$$10\!\cdots\!64$$$$T^{4} -$$$$88\!\cdots\!57$$$$T^{5} +$$$$13\!\cdots\!50$$$$T^{6} +$$$$26\!\cdots\!56$$$$T^{7} +$$$$15\!\cdots\!74$$$$T^{8} -$$$$34\!\cdots\!29$$$$T^{9} -$$$$44\!\cdots\!96$$$$T^{10} +$$$$20\!\cdots\!36$$$$T^{11} +$$$$37\!\cdots\!72$$$$T^{12} -$$$$56\!\cdots\!35$$$$T^{13} -$$$$13\!\cdots\!10$$$$T^{14} + 25214882606872 T^{15} + 18886490 T^{16} + T^{17}$$
$83$ $$-$$$$56\!\cdots\!08$$$$+$$$$11\!\cdots\!08$$$$T -$$$$18\!\cdots\!84$$$$T^{2} -$$$$46\!\cdots\!36$$$$T^{3} +$$$$19\!\cdots\!48$$$$T^{4} +$$$$44\!\cdots\!41$$$$T^{5} -$$$$39\!\cdots\!01$$$$T^{6} +$$$$29\!\cdots\!06$$$$T^{7} +$$$$25\!\cdots\!41$$$$T^{8} -$$$$56\!\cdots\!08$$$$T^{9} -$$$$35\!\cdots\!13$$$$T^{10} +$$$$22\!\cdots\!72$$$$T^{11} -$$$$14\!\cdots\!73$$$$T^{12} -$$$$26\!\cdots\!08$$$$T^{13} +$$$$38\!\cdots\!17$$$$T^{14} - 70629000162 T^{15} - 22824893 T^{16} + T^{17}$$
$89$ $$36\!\cdots\!88$$$$-$$$$83\!\cdots\!32$$$$T +$$$$96\!\cdots\!68$$$$T^{2} +$$$$12\!\cdots\!72$$$$T^{3} -$$$$14\!\cdots\!52$$$$T^{4} +$$$$68\!\cdots\!08$$$$T^{5} -$$$$10\!\cdots\!70$$$$T^{6} -$$$$23\!\cdots\!25$$$$T^{7} +$$$$11\!\cdots\!41$$$$T^{8} -$$$$97\!\cdots\!25$$$$T^{9} -$$$$23\!\cdots\!76$$$$T^{10} +$$$$50\!\cdots\!99$$$$T^{11} -$$$$64\!\cdots\!37$$$$T^{12} -$$$$63\!\cdots\!35$$$$T^{13} +$$$$51\!\cdots\!00$$$$T^{14} + 131680142526477 T^{15} - 30609647 T^{16} + T^{17}$$
$97$ $$80\!\cdots\!88$$$$+$$$$10\!\cdots\!64$$$$T +$$$$35\!\cdots\!96$$$$T^{2} -$$$$95\!\cdots\!60$$$$T^{3} -$$$$83\!\cdots\!24$$$$T^{4} -$$$$11\!\cdots\!20$$$$T^{5} +$$$$31\!\cdots\!66$$$$T^{6} +$$$$99\!\cdots\!59$$$$T^{7} +$$$$19\!\cdots\!66$$$$T^{8} -$$$$21\!\cdots\!15$$$$T^{9} -$$$$18\!\cdots\!20$$$$T^{10} +$$$$16\!\cdots\!86$$$$T^{11} +$$$$25\!\cdots\!68$$$$T^{12} -$$$$14\!\cdots\!58$$$$T^{13} -$$$$13\!\cdots\!82$$$$T^{14} - 332004026069621 T^{15} + 26249806 T^{16} + T^{17}$$