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 $1$

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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:

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

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