Properties

Label 531.8.a.c
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: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 68 + \beta_{1} + \beta_{2} ) q^{4} + ( 19 - 2 \beta_{1} + \beta_{4} ) q^{5} + ( 185 - 2 \beta_{1} + \beta_{10} ) q^{7} + ( -128 - 87 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 68 + \beta_{1} + \beta_{2} ) q^{4} + ( 19 - 2 \beta_{1} + \beta_{4} ) q^{5} + ( 185 - 2 \beta_{1} + \beta_{10} ) q^{7} + ( -128 - 87 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{8} + ( 380 - 9 \beta_{1} + 7 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{16} ) q^{10} + ( 96 + 69 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{11} + ( 1076 - 107 \beta_{1} + 12 \beta_{2} + \beta_{3} - 9 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} - \beta_{16} ) q^{13} + ( 483 - 180 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 19 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{15} + \beta_{16} ) q^{14} + ( 8125 + 43 \beta_{1} + 121 \beta_{2} - 3 \beta_{3} - 23 \beta_{4} - 4 \beta_{5} - \beta_{6} - 2 \beta_{7} + 7 \beta_{9} + 6 \beta_{10} - \beta_{11} - 2 \beta_{12} - 5 \beta_{13} + 2 \beta_{14} + \beta_{15} + 6 \beta_{16} ) q^{16} + ( 913 + 94 \beta_{1} - 35 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} + 3 \beta_{6} - 12 \beta_{8} + \beta_{9} + 5 \beta_{10} + \beta_{11} + 3 \beta_{12} - 4 \beta_{13} + 7 \beta_{14} + 3 \beta_{15} + 3 \beta_{16} ) q^{17} + ( 2993 + 248 \beta_{1} + 77 \beta_{2} - 6 \beta_{3} - 16 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + \beta_{9} - 4 \beta_{10} - 5 \beta_{11} - \beta_{12} - 10 \beta_{13} + 2 \beta_{15} + \beta_{16} ) q^{19} + ( 106 - 935 \beta_{1} - 57 \beta_{2} + 15 \beta_{3} + 99 \beta_{4} - 9 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - \beta_{9} - 21 \beta_{10} - 9 \beta_{11} + 7 \beta_{12} + 4 \beta_{13} - 8 \beta_{14} + 6 \beta_{15} - 8 \beta_{16} ) q^{20} + ( -13772 - 238 \beta_{1} + 33 \beta_{2} + \beta_{3} + 2 \beta_{4} - 9 \beta_{5} - 4 \beta_{6} - 5 \beta_{7} + 12 \beta_{8} + 9 \beta_{9} + 7 \beta_{10} - 12 \beta_{11} + 7 \beta_{12} + 2 \beta_{13} - 5 \beta_{14} - 9 \beta_{15} + 7 \beta_{16} ) q^{22} + ( -3851 + 914 \beta_{1} - 68 \beta_{2} - 7 \beta_{3} + 20 \beta_{4} - \beta_{5} + 6 \beta_{6} + 8 \beta_{7} + \beta_{8} + 14 \beta_{9} + 21 \beta_{10} - 12 \beta_{11} - 10 \beta_{12} - 18 \beta_{13} + 7 \beta_{14} - 2 \beta_{15} + 24 \beta_{16} ) q^{23} + ( 11847 - 209 \beta_{1} + 145 \beta_{2} + 22 \beta_{3} + 71 \beta_{4} - 8 \beta_{5} + \beta_{6} + 8 \beta_{7} - 24 \beta_{8} - 11 \beta_{9} + 14 \beta_{10} + 13 \beta_{11} + 8 \beta_{12} + 3 \beta_{13} + 17 \beta_{14} + 19 \beta_{15} - 27 \beta_{16} ) q^{25} + ( 21933 - 2583 \beta_{1} + 53 \beta_{2} - 55 \beta_{3} - 43 \beta_{4} + \beta_{5} - 15 \beta_{6} - 7 \beta_{7} + 8 \beta_{8} + 22 \beta_{9} + 43 \beta_{10} - 4 \beta_{11} - 6 \beta_{12} + \beta_{13} - 14 \beta_{14} - 27 \beta_{15} + 5 \beta_{16} ) q^{26} + ( 10417 + 848 \beta_{1} + 395 \beta_{2} - 11 \beta_{3} - 95 \beta_{4} - 2 \beta_{5} - 12 \beta_{6} + 20 \beta_{7} + 14 \beta_{8} + 25 \beta_{9} + 49 \beta_{10} - 9 \beta_{11} - 14 \beta_{12} + 13 \beta_{13} + 28 \beta_{14} - 11 \beta_{15} + 30 \beta_{16} ) q^{28} + ( 29772 - 1986 \beta_{1} + 76 \beta_{2} + 12 \beta_{3} + 76 \beta_{4} - 17 \beta_{5} - 10 \beta_{6} + 30 \beta_{7} + 22 \beta_{8} + 7 \beta_{9} + 50 \beta_{10} + 18 \beta_{11} - 16 \beta_{12} - 10 \beta_{13} + 44 \beta_{14} + \beta_{15} - 2 \beta_{16} ) q^{29} + ( 20225 + 1722 \beta_{1} - 80 \beta_{2} - 79 \beta_{3} + 47 \beta_{4} - 4 \beta_{5} - 38 \beta_{6} + 34 \beta_{7} + 16 \beta_{8} - 51 \beta_{9} + 43 \beta_{10} + 3 \beta_{11} - 23 \beta_{12} - 13 \beta_{13} - 13 \beta_{14} - 13 \beta_{15} + 30 \beta_{16} ) q^{31} + ( 16093 - 14219 \beta_{1} - 38 \beta_{2} - 60 \beta_{3} + 199 \beta_{4} - 50 \beta_{5} + 29 \beta_{6} - 55 \beta_{7} + 62 \beta_{8} - 50 \beta_{9} + 57 \beta_{10} + 14 \beta_{11} + 11 \beta_{12} + 52 \beta_{13} + 30 \beta_{14} + 9 \beta_{15} - 14 \beta_{16} ) q^{32} + ( -19916 + 5244 \beta_{1} - 134 \beta_{2} + 10 \beta_{3} + 373 \beta_{4} + 49 \beta_{5} - 100 \beta_{6} - 22 \beta_{7} + 49 \beta_{8} + 77 \beta_{9} - 11 \beta_{10} - 45 \beta_{11} + 28 \beta_{12} + 79 \beta_{13} - 116 \beta_{14} - 56 \beta_{15} - 4 \beta_{16} ) q^{34} + ( -4155 - 9369 \beta_{1} - 341 \beta_{2} - 17 \beta_{3} + 277 \beta_{4} + 23 \beta_{5} - 47 \beta_{6} + 34 \beta_{7} - 57 \beta_{8} + \beta_{9} - 15 \beta_{10} + 31 \beta_{11} + 46 \beta_{12} + 7 \beta_{13} - 50 \beta_{14} - 29 \beta_{15} - 15 \beta_{16} ) q^{35} + ( 25685 + 3919 \beta_{1} + 265 \beta_{2} - 57 \beta_{3} + 412 \beta_{4} - 30 \beta_{5} + 69 \beta_{6} + 3 \beta_{7} - 11 \beta_{8} - 99 \beta_{9} + 167 \beta_{10} + 56 \beta_{11} - 28 \beta_{12} - 21 \beta_{13} + 37 \beta_{14} + 16 \beta_{15} + 22 \beta_{16} ) q^{37} + ( -43810 - 14578 \beta_{1} - 75 \beta_{2} - 150 \beta_{3} - 283 \beta_{4} - 120 \beta_{5} + 3 \beta_{6} - 25 \beta_{7} + 65 \beta_{8} + 40 \beta_{9} + 93 \beta_{10} - 22 \beta_{11} - 7 \beta_{12} + 54 \beta_{13} - 48 \beta_{14} - 98 \beta_{15} - 30 \beta_{16} ) q^{38} + ( 127778 + 11403 \beta_{1} + 994 \beta_{2} + 246 \beta_{3} + 235 \beta_{4} - 40 \beta_{5} + 88 \beta_{6} + 104 \beta_{7} + 19 \beta_{8} + 131 \beta_{9} + 71 \beta_{10} - 49 \beta_{11} + 43 \beta_{12} + 4 \beta_{13} + 76 \beta_{14} + 106 \beta_{15} - 56 \beta_{16} ) q^{40} + ( -54646 - 9363 \beta_{1} + 698 \beta_{2} + 4 \beta_{3} - 153 \beta_{4} + 24 \beta_{5} + 21 \beta_{6} + 85 \beta_{7} - 26 \beta_{8} + 50 \beta_{9} - 118 \beta_{10} - 95 \beta_{11} + \beta_{12} - 79 \beta_{13} - 44 \beta_{14} + 24 \beta_{15} + 83 \beta_{16} ) q^{41} + ( 26464 + 15468 \beta_{1} - 271 \beta_{2} - 6 \beta_{3} + 222 \beta_{4} - 48 \beta_{5} + 85 \beta_{6} - 37 \beta_{7} - 103 \beta_{8} + 8 \beta_{9} + 234 \beta_{10} + 89 \beta_{11} - 20 \beta_{12} - 61 \beta_{13} + 102 \beta_{14} - 35 \beta_{15} - 60 \beta_{16} ) q^{43} + ( 34438 + 366 \beta_{1} + 1067 \beta_{2} + 137 \beta_{3} + 95 \beta_{4} - 106 \beta_{5} + 51 \beta_{6} + 64 \beta_{7} - 77 \beta_{8} - 49 \beta_{9} + 32 \beta_{10} + 38 \beta_{11} - 94 \beta_{12} - 20 \beta_{13} + 65 \beta_{14} + 197 \beta_{15} - 54 \beta_{16} ) q^{44} + ( -181798 + 14854 \beta_{1} - 1805 \beta_{2} + 87 \beta_{3} + 1154 \beta_{4} - 80 \beta_{5} - 111 \beta_{6} - 81 \beta_{7} + 220 \beta_{8} + 5 \beta_{9} - 144 \beta_{10} - 112 \beta_{11} - 25 \beta_{12} + 108 \beta_{13} - 165 \beta_{14} - 147 \beta_{15} - 22 \beta_{16} ) q^{46} + ( -42560 + 1891 \beta_{1} + 1731 \beta_{2} - 10 \beta_{3} + 606 \beta_{4} - 153 \beta_{5} + 162 \beta_{6} + 53 \beta_{7} - 26 \beta_{8} - 118 \beta_{9} + 19 \beta_{10} + 86 \beta_{11} - 264 \beta_{12} - 77 \beta_{13} + 123 \beta_{14} + 90 \beta_{15} + 66 \beta_{16} ) q^{47} + ( 111047 + 7723 \beta_{1} - 1471 \beta_{2} - 54 \beta_{3} - 475 \beta_{4} + 218 \beta_{5} + 56 \beta_{6} + 33 \beta_{7} - 218 \beta_{8} - 41 \beta_{9} + 256 \beta_{10} - 48 \beta_{11} + 150 \beta_{12} - 93 \beta_{13} - 99 \beta_{14} + 133 \beta_{15} ) q^{49} + ( 54788 - 31363 \beta_{1} + 1336 \beta_{2} + 19 \beta_{3} + 1084 \beta_{4} + 92 \beta_{5} - 325 \beta_{6} - 28 \beta_{8} + 110 \beta_{9} - 79 \beta_{10} - 229 \beta_{11} + 296 \beta_{12} + 161 \beta_{13} - 459 \beta_{14} + 13 \beta_{15} + 19 \beta_{16} ) q^{50} + ( 370016 - 16333 \beta_{1} + 2350 \beta_{2} - 562 \beta_{3} - 174 \beta_{4} - 12 \beta_{5} + 251 \beta_{6} - 131 \beta_{7} - 214 \beta_{8} - 235 \beta_{9} + 140 \beta_{10} + 326 \beta_{11} - 204 \beta_{12} - 153 \beta_{13} + 135 \beta_{14} + 21 \beta_{15} + 158 \beta_{16} ) q^{52} + ( 61018 - 21688 \beta_{1} + 2415 \beta_{2} + 397 \beta_{3} + 1979 \beta_{4} + 324 \beta_{5} + 133 \beta_{6} - 181 \beta_{7} - 67 \beta_{8} - 33 \beta_{9} - 8 \beta_{10} + 120 \beta_{11} + 349 \beta_{12} + 114 \beta_{13} + 231 \beta_{14} - 76 \beta_{15} - 238 \beta_{16} ) q^{53} + ( 307782 - 1245 \beta_{1} - 1193 \beta_{2} + 30 \beta_{3} - 277 \beta_{4} - 388 \beta_{5} + 241 \beta_{6} - 174 \beta_{7} + 8 \beta_{8} - 55 \beta_{9} - 558 \beta_{10} + 101 \beta_{11} + 100 \beta_{12} + 225 \beta_{13} - 187 \beta_{14} + 141 \beta_{15} - 411 \beta_{16} ) q^{55} + ( -195943 - 43936 \beta_{1} + 650 \beta_{2} - 211 \beta_{3} + 1566 \beta_{4} - 238 \beta_{5} + 68 \beta_{6} - 333 \beta_{7} + 37 \beta_{8} - 141 \beta_{9} - 699 \beta_{10} + 300 \beta_{11} + 147 \beta_{12} + 154 \beta_{13} - 403 \beta_{14} - 210 \beta_{15} - 116 \beta_{16} ) q^{56} + ( 398168 - 36279 \beta_{1} + 2196 \beta_{2} - 16 \beta_{3} + 2244 \beta_{4} + 164 \beta_{5} - 350 \beta_{6} - 194 \beta_{7} + 260 \beta_{8} - 250 \beta_{9} - 380 \beta_{10} - 269 \beta_{11} + 337 \beta_{12} + 160 \beta_{13} - 569 \beta_{14} - 21 \beta_{15} + 163 \beta_{16} ) q^{58} + 205379 q^{59} + ( 671823 + 35121 \beta_{1} + 2144 \beta_{2} + 208 \beta_{3} - 967 \beta_{4} - 11 \beta_{5} - 167 \beta_{6} + 153 \beta_{7} - 136 \beta_{8} - 129 \beta_{9} + 61 \beta_{10} + 309 \beta_{11} - 105 \beta_{12} - 131 \beta_{13} + 264 \beta_{14} - 19 \beta_{15} - 139 \beta_{16} ) q^{61} + ( -353359 - 10573 \beta_{1} + 2963 \beta_{2} - 483 \beta_{3} - 642 \beta_{4} + 340 \beta_{5} + 67 \beta_{6} - 81 \beta_{7} - 149 \beta_{8} - 237 \beta_{9} + 992 \beta_{10} + 537 \beta_{11} - 578 \beta_{12} - 428 \beta_{13} + 592 \beta_{14} - \beta_{15} + 205 \beta_{16} ) q^{62} + ( 1729020 - 7545 \beta_{1} + 13932 \beta_{2} - 855 \beta_{3} - 1810 \beta_{4} - 381 \beta_{5} - 452 \beta_{6} + 210 \beta_{7} + 144 \beta_{8} + 458 \beta_{9} + 122 \beta_{10} + 122 \beta_{11} - 746 \beta_{12} - 562 \beta_{13} + 292 \beta_{14} - 20 \beta_{15} + 588 \beta_{16} ) q^{64} + ( -655145 + 14584 \beta_{1} + 902 \beta_{2} + 781 \beta_{3} + 2434 \beta_{4} + 1174 \beta_{5} + 240 \beta_{6} - 161 \beta_{7} - 553 \beta_{8} - 600 \beta_{9} - 28 \beta_{10} - 173 \beta_{11} + 260 \beta_{12} - 30 \beta_{13} + 308 \beta_{14} + 399 \beta_{15} - 17 \beta_{16} ) q^{65} + ( 819163 + 38412 \beta_{1} + 1223 \beta_{2} - 545 \beta_{3} - 2027 \beta_{4} + 85 \beta_{5} + 13 \beta_{6} - 336 \beta_{7} + 158 \beta_{8} + 563 \beta_{9} - 307 \beta_{10} - 304 \beta_{11} - 420 \beta_{12} + 441 \beta_{13} + 306 \beta_{14} - 143 \beta_{15} + 205 \beta_{16} ) q^{67} + ( -1159755 + 23364 \beta_{1} - 4399 \beta_{2} + 1319 \beta_{3} + 1091 \beta_{4} + 391 \beta_{5} + 418 \beta_{6} + 827 \beta_{7} + 180 \beta_{8} - 566 \beta_{9} + 604 \beta_{10} - 132 \beta_{11} - 167 \beta_{12} - 48 \beta_{13} + 948 \beta_{14} + 801 \beta_{15} - 510 \beta_{16} ) q^{68} + ( 1805728 + 63232 \beta_{1} + 9667 \beta_{2} + 1454 \beta_{3} - 64 \beta_{4} + 1326 \beta_{5} + 266 \beta_{6} + 415 \beta_{7} - 944 \beta_{8} - 507 \beta_{9} + 233 \beta_{10} + 265 \beta_{11} + 639 \beta_{12} + 123 \beta_{13} + 576 \beta_{14} + 572 \beta_{15} - 543 \beta_{16} ) q^{70} + ( -307006 - 9945 \beta_{1} + 2337 \beta_{2} - 943 \beta_{3} + 368 \beta_{4} - 105 \beta_{5} - 398 \beta_{6} - 653 \beta_{7} - 341 \beta_{8} + 823 \beta_{9} + 663 \beta_{10} - 114 \beta_{11} - 510 \beta_{12} + 265 \beta_{13} - 568 \beta_{14} - 225 \beta_{15} - 138 \beta_{16} ) q^{71} + ( 316127 + 33918 \beta_{1} + 751 \beta_{2} + 1243 \beta_{3} - 306 \beta_{4} + 393 \beta_{5} + 839 \beta_{6} + 128 \beta_{7} + 365 \beta_{8} - 791 \beta_{9} - 575 \beta_{10} - 93 \beta_{11} + 381 \beta_{12} - 214 \beta_{13} + 390 \beta_{14} + 27 \beta_{15} - 287 \beta_{16} ) q^{73} + ( -748606 - 69643 \beta_{1} + 4707 \beta_{2} - 1640 \beta_{3} - 521 \beta_{4} - 845 \beta_{5} - 1122 \beta_{6} + 121 \beta_{7} + 609 \beta_{8} + 1008 \beta_{9} + 324 \beta_{10} - 419 \beta_{11} - 1228 \beta_{12} - 694 \beta_{13} - 219 \beta_{14} - 381 \beta_{15} + 1274 \beta_{16} ) q^{74} + ( 2454679 + 16214 \beta_{1} + 21000 \beta_{2} - 286 \beta_{3} - 2472 \beta_{4} - 46 \beta_{5} + 113 \beta_{6} - 438 \beta_{7} + 37 \beta_{8} + 338 \beta_{9} - 887 \beta_{10} + 447 \beta_{11} - 714 \beta_{12} - 705 \beta_{13} + 821 \beta_{14} - 130 \beta_{15} + 680 \beta_{16} ) q^{76} + ( -584622 - 15505 \beta_{1} + 5069 \beta_{2} + 573 \beta_{3} - 2299 \beta_{4} - 360 \beta_{5} + 623 \beta_{6} + 721 \beta_{7} + 801 \beta_{8} - 211 \beta_{9} - 527 \beta_{10} + 844 \beta_{11} + 76 \beta_{12} - 825 \beta_{13} + 687 \beta_{14} - 628 \beta_{15} - 256 \beta_{16} ) q^{77} + ( 894176 + 50886 \beta_{1} - 1613 \beta_{2} + 135 \beta_{3} - 6055 \beta_{4} - 41 \beta_{5} - 544 \beta_{6} + 134 \beta_{7} + 679 \beta_{8} + 1904 \beta_{9} - 426 \beta_{10} - 873 \beta_{11} + 249 \beta_{12} - 236 \beta_{13} - 263 \beta_{14} - 1529 \beta_{15} - 309 \beta_{16} ) q^{79} + ( -2125919 - 156019 \beta_{1} - 10727 \beta_{2} + 1714 \beta_{3} + 6266 \beta_{4} - 763 \beta_{5} - 967 \beta_{6} + 62 \beta_{7} + 753 \beta_{8} + 450 \beta_{9} - 4261 \beta_{10} - 1234 \beta_{11} + 2235 \beta_{12} + 1049 \beta_{13} - 3054 \beta_{14} - 235 \beta_{15} - 494 \beta_{16} ) q^{80} + ( 1888080 - 32772 \beta_{1} + 3886 \beta_{2} + 1524 \beta_{3} + 7344 \beta_{4} - 1155 \beta_{5} + 221 \beta_{6} - 72 \beta_{7} + 534 \beta_{8} - 368 \beta_{9} - 2227 \beta_{10} + 150 \beta_{11} + 860 \beta_{12} + 1092 \beta_{13} - 926 \beta_{14} - 314 \beta_{15} - 1807 \beta_{16} ) q^{82} + ( -458094 - 29771 \beta_{1} + 5174 \beta_{2} - 588 \beta_{3} + 725 \beta_{4} + 1244 \beta_{5} + 1253 \beta_{6} - 915 \beta_{7} - 394 \beta_{8} + 198 \beta_{9} + 1154 \beta_{10} + 529 \beta_{11} + 609 \beta_{12} + 1009 \beta_{13} + 1052 \beta_{14} - 792 \beta_{15} - 145 \beta_{16} ) q^{83} + ( 718832 - 208403 \beta_{1} + 2026 \beta_{2} + 421 \beta_{3} + 7041 \beta_{4} + 2747 \beta_{5} + 1579 \beta_{6} - 825 \beta_{7} - 1891 \beta_{8} - 2026 \beta_{9} + 1562 \beta_{10} + 215 \beta_{11} + 1477 \beta_{12} - 657 \beta_{13} + 541 \beta_{14} + 1142 \beta_{15} + 81 \beta_{16} ) q^{85} + ( -3030466 - 3870 \beta_{1} - 16541 \beta_{2} - 1896 \beta_{3} - 3119 \beta_{4} + 955 \beta_{5} - 1076 \beta_{6} - 405 \beta_{7} + 405 \beta_{8} + 444 \beta_{9} + 168 \beta_{10} - 1572 \beta_{11} - 109 \beta_{12} + 46 \beta_{13} - 1446 \beta_{14} - 1104 \beta_{15} + 1309 \beta_{16} ) q^{86} + ( 1798422 - 148654 \beta_{1} - 4474 \beta_{2} - 180 \beta_{3} + 11995 \beta_{4} - 435 \beta_{5} - 841 \beta_{6} + 366 \beta_{7} - 177 \beta_{8} - 1157 \beta_{9} - 2410 \beta_{10} + 252 \beta_{11} - 4 \beta_{12} + 1352 \beta_{13} - 2337 \beta_{14} + 1219 \beta_{15} - 918 \beta_{16} ) q^{88} + ( 359525 + 105088 \beta_{1} + 14126 \beta_{2} + 309 \beta_{3} - 5027 \beta_{4} - 192 \beta_{5} - 220 \beta_{6} + 181 \beta_{7} + 478 \beta_{8} + 1024 \beta_{9} - 2458 \beta_{10} - 324 \beta_{11} - 613 \beta_{12} - 551 \beta_{13} + 832 \beta_{14} + 419 \beta_{15} + 761 \beta_{16} ) q^{89} + ( 700466 - 155896 \beta_{1} + 2670 \beta_{2} + 154 \beta_{3} - 4270 \beta_{4} + 1830 \beta_{5} - 2681 \beta_{6} + 387 \beta_{7} + 2628 \beta_{8} + 676 \beta_{9} + 1124 \beta_{10} - 1873 \beta_{11} - 710 \beta_{12} + 642 \beta_{13} - 2 \beta_{14} - 374 \beta_{15} + 883 \beta_{16} ) q^{91} + ( -2588769 + 313138 \beta_{1} - 3210 \beta_{2} + 3988 \beta_{3} - 6732 \beta_{4} + 582 \beta_{5} + 293 \beta_{6} + 2238 \beta_{7} - 391 \beta_{8} + 1070 \beta_{9} + 2405 \beta_{10} - 695 \beta_{11} - 1208 \beta_{12} - 197 \beta_{13} + 2731 \beta_{14} + 1128 \beta_{15} - 32 \beta_{16} ) q^{92} + ( -241576 - 205666 \beta_{1} + 3169 \beta_{2} - 3811 \beta_{3} + 5795 \beta_{4} - 2864 \beta_{5} - 1700 \beta_{6} - 37 \beta_{7} + 2629 \beta_{8} + 1124 \beta_{9} - 298 \beta_{10} - 1957 \beta_{11} - 2195 \beta_{12} + 509 \beta_{13} - 3099 \beta_{14} - 2063 \beta_{15} + 2771 \beta_{16} ) q^{94} + ( -1409890 + 131351 \beta_{1} - 4623 \beta_{2} + 822 \beta_{3} + 2835 \beta_{4} - 1177 \beta_{5} + 1597 \beta_{6} - 339 \beta_{7} - 340 \beta_{8} - 1161 \beta_{9} - 3427 \beta_{10} - 786 \beta_{11} + 696 \beta_{12} - 1149 \beta_{13} + 150 \beta_{14} + 134 \beta_{15} + 340 \beta_{16} ) q^{95} + ( 1586738 - 319410 \beta_{1} + 5446 \beta_{2} - 493 \beta_{3} - 2282 \beta_{4} - 460 \beta_{5} - 477 \beta_{6} + 176 \beta_{7} - 2479 \beta_{8} + 1128 \beta_{9} + 331 \beta_{10} + 770 \beta_{11} + 712 \beta_{12} - 110 \beta_{13} - 898 \beta_{14} - 805 \beta_{15} + 870 \beta_{16} ) q^{97} + ( -1559125 + 97851 \beta_{1} - 20907 \beta_{2} + 2786 \beta_{3} - 68 \beta_{4} + 1891 \beta_{5} - 725 \beta_{6} - 903 \beta_{7} - 1133 \beta_{8} - 1575 \beta_{9} - 1972 \beta_{10} + 1741 \beta_{11} + 1843 \beta_{12} + 1319 \beta_{13} - 432 \beta_{14} + 449 \beta_{15} - 3846 \beta_{16} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + O(q^{10}) \) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + 6521q^{10} + 1764q^{11} + 18192q^{13} + 7827q^{14} + 139226q^{16} + 15507q^{17} + 52083q^{19} - 721q^{20} - 234434q^{22} - 63823q^{23} + 202153q^{25} + 367956q^{26} + 182306q^{28} + 502955q^{29} + 347531q^{31} + 243908q^{32} - 330872q^{34} - 92641q^{35} + 447615q^{37} - 775669q^{38} + 2203270q^{40} - 940335q^{41} + 478562q^{43} + 596924q^{44} - 3078663q^{46} - 703121q^{47} + 1895082q^{49} + 876967q^{50} + 6278296q^{52} + 1005974q^{53} + 5212846q^{55} - 3425294q^{56} + 6710166q^{58} + 3491443q^{59} + 11510749q^{61} - 5996234q^{62} + 29496941q^{64} - 11094180q^{65} + 14007144q^{67} - 19688159q^{68} + 30909708q^{70} - 5229074q^{71} + 5452211q^{73} - 12819662q^{74} + 41929340q^{76} - 9930777q^{77} + 15275654q^{79} - 36576105q^{80} + 32025935q^{82} - 7826609q^{83} + 11836945q^{85} - 51649136q^{86} + 30223741q^{88} + 6436185q^{89} + 11633535q^{91} - 43357972q^{92} - 4494252q^{94} - 23741055q^{95} + 26377540q^{97} - 26517816q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 196 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(25\!\cdots\!23\)\( \nu^{16} + \)\(51\!\cdots\!52\)\( \nu^{15} + \)\(39\!\cdots\!71\)\( \nu^{14} - \)\(78\!\cdots\!57\)\( \nu^{13} - \)\(23\!\cdots\!78\)\( \nu^{12} + \)\(45\!\cdots\!89\)\( \nu^{11} + \)\(73\!\cdots\!34\)\( \nu^{10} - \)\(13\!\cdots\!12\)\( \nu^{9} - \)\(13\!\cdots\!36\)\( \nu^{8} + \)\(19\!\cdots\!40\)\( \nu^{7} + \)\(13\!\cdots\!88\)\( \nu^{6} - \)\(14\!\cdots\!64\)\( \nu^{5} - \)\(82\!\cdots\!00\)\( \nu^{4} + \)\(47\!\cdots\!92\)\( \nu^{3} + \)\(24\!\cdots\!52\)\( \nu^{2} - \)\(42\!\cdots\!56\)\( \nu - \)\(18\!\cdots\!68\)\(\)\()/ \)\(20\!\cdots\!72\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(14\!\cdots\!41\)\( \nu^{16} + \)\(78\!\cdots\!69\)\( \nu^{15} + \)\(23\!\cdots\!07\)\( \nu^{14} - \)\(11\!\cdots\!44\)\( \nu^{13} - \)\(15\!\cdots\!25\)\( \nu^{12} + \)\(64\!\cdots\!51\)\( \nu^{11} + \)\(50\!\cdots\!11\)\( \nu^{10} - \)\(17\!\cdots\!80\)\( \nu^{9} - \)\(91\!\cdots\!20\)\( \nu^{8} + \)\(24\!\cdots\!04\)\( \nu^{7} + \)\(92\!\cdots\!84\)\( \nu^{6} - \)\(17\!\cdots\!92\)\( \nu^{5} - \)\(48\!\cdots\!44\)\( \nu^{4} + \)\(51\!\cdots\!24\)\( \nu^{3} + \)\(11\!\cdots\!76\)\( \nu^{2} - \)\(32\!\cdots\!44\)\( \nu - \)\(85\!\cdots\!48\)\(\)\()/ \)\(10\!\cdots\!36\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(30\!\cdots\!41\)\( \nu^{16} - \)\(20\!\cdots\!76\)\( \nu^{15} + \)\(52\!\cdots\!57\)\( \nu^{14} + \)\(32\!\cdots\!81\)\( \nu^{13} - \)\(36\!\cdots\!22\)\( \nu^{12} - \)\(19\!\cdots\!85\)\( \nu^{11} + \)\(12\!\cdots\!10\)\( \nu^{10} + \)\(59\!\cdots\!92\)\( \nu^{9} - \)\(23\!\cdots\!44\)\( \nu^{8} - \)\(95\!\cdots\!24\)\( \nu^{7} + \)\(23\!\cdots\!48\)\( \nu^{6} + \)\(77\!\cdots\!96\)\( \nu^{5} - \)\(11\!\cdots\!76\)\( \nu^{4} - \)\(27\!\cdots\!24\)\( \nu^{3} + \)\(21\!\cdots\!52\)\( \nu^{2} + \)\(21\!\cdots\!84\)\( \nu - \)\(15\!\cdots\!40\)\(\)\()/ \)\(20\!\cdots\!72\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(12\!\cdots\!95\)\( \nu^{16} + \)\(10\!\cdots\!06\)\( \nu^{15} + \)\(18\!\cdots\!67\)\( \nu^{14} - \)\(13\!\cdots\!43\)\( \nu^{13} - \)\(10\!\cdots\!52\)\( \nu^{12} + \)\(65\!\cdots\!65\)\( \nu^{11} + \)\(28\!\cdots\!96\)\( \nu^{10} - \)\(12\!\cdots\!24\)\( \nu^{9} - \)\(38\!\cdots\!08\)\( \nu^{8} + \)\(73\!\cdots\!84\)\( \nu^{7} + \)\(20\!\cdots\!76\)\( \nu^{6} + \)\(47\!\cdots\!24\)\( \nu^{5} + \)\(92\!\cdots\!76\)\( \nu^{4} - \)\(53\!\cdots\!60\)\( \nu^{3} - \)\(41\!\cdots\!72\)\( \nu^{2} + \)\(84\!\cdots\!92\)\( \nu + \)\(62\!\cdots\!00\)\(\)\()/ \)\(82\!\cdots\!88\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(21\!\cdots\!01\)\( \nu^{16} + \)\(44\!\cdots\!90\)\( \nu^{15} + \)\(35\!\cdots\!97\)\( \nu^{14} - \)\(61\!\cdots\!29\)\( \nu^{13} - \)\(23\!\cdots\!88\)\( \nu^{12} + \)\(30\!\cdots\!87\)\( \nu^{11} + \)\(80\!\cdots\!68\)\( \nu^{10} - \)\(68\!\cdots\!80\)\( \nu^{9} - \)\(14\!\cdots\!92\)\( \nu^{8} + \)\(61\!\cdots\!64\)\( \nu^{7} + \)\(15\!\cdots\!48\)\( \nu^{6} - \)\(35\!\cdots\!64\)\( \nu^{5} - \)\(79\!\cdots\!92\)\( \nu^{4} - \)\(23\!\cdots\!72\)\( \nu^{3} + \)\(19\!\cdots\!72\)\( \nu^{2} + \)\(88\!\cdots\!96\)\( \nu - \)\(16\!\cdots\!76\)\(\)\()/ \)\(82\!\cdots\!88\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(22\!\cdots\!77\)\( \nu^{16} - \)\(18\!\cdots\!96\)\( \nu^{15} - \)\(36\!\cdots\!97\)\( \nu^{14} + \)\(28\!\cdots\!27\)\( \nu^{13} + \)\(23\!\cdots\!86\)\( \nu^{12} - \)\(16\!\cdots\!67\)\( \nu^{11} - \)\(76\!\cdots\!14\)\( \nu^{10} + \)\(47\!\cdots\!12\)\( \nu^{9} + \)\(13\!\cdots\!16\)\( \nu^{8} - \)\(71\!\cdots\!24\)\( \nu^{7} - \)\(13\!\cdots\!24\)\( \nu^{6} + \)\(55\!\cdots\!20\)\( \nu^{5} + \)\(69\!\cdots\!52\)\( \nu^{4} - \)\(19\!\cdots\!72\)\( \nu^{3} - \)\(16\!\cdots\!68\)\( \nu^{2} + \)\(16\!\cdots\!36\)\( \nu + \)\(11\!\cdots\!44\)\(\)\()/ \)\(41\!\cdots\!44\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(55\!\cdots\!51\)\( \nu^{16} + \)\(26\!\cdots\!86\)\( \nu^{15} + \)\(92\!\cdots\!51\)\( \nu^{14} - \)\(39\!\cdots\!11\)\( \nu^{13} - \)\(61\!\cdots\!40\)\( \nu^{12} + \)\(22\!\cdots\!01\)\( \nu^{11} + \)\(20\!\cdots\!76\)\( \nu^{10} - \)\(63\!\cdots\!24\)\( \nu^{9} - \)\(37\!\cdots\!16\)\( \nu^{8} + \)\(92\!\cdots\!52\)\( \nu^{7} + \)\(38\!\cdots\!28\)\( \nu^{6} - \)\(66\!\cdots\!96\)\( \nu^{5} - \)\(19\!\cdots\!72\)\( \nu^{4} + \)\(20\!\cdots\!00\)\( \nu^{3} + \)\(45\!\cdots\!84\)\( \nu^{2} - \)\(17\!\cdots\!00\)\( \nu - \)\(31\!\cdots\!96\)\(\)\()/ \)\(82\!\cdots\!88\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(72\!\cdots\!87\)\( \nu^{16} - \)\(41\!\cdots\!26\)\( \nu^{15} - \)\(11\!\cdots\!11\)\( \nu^{14} + \)\(61\!\cdots\!27\)\( \nu^{13} + \)\(77\!\cdots\!32\)\( \nu^{12} - \)\(35\!\cdots\!77\)\( \nu^{11} - \)\(25\!\cdots\!04\)\( \nu^{10} + \)\(99\!\cdots\!68\)\( \nu^{9} + \)\(46\!\cdots\!44\)\( \nu^{8} - \)\(14\!\cdots\!88\)\( \nu^{7} - \)\(47\!\cdots\!08\)\( \nu^{6} + \)\(10\!\cdots\!28\)\( \nu^{5} + \)\(24\!\cdots\!20\)\( \nu^{4} - \)\(34\!\cdots\!04\)\( \nu^{3} - \)\(59\!\cdots\!44\)\( \nu^{2} + \)\(29\!\cdots\!72\)\( \nu + \)\(45\!\cdots\!88\)\(\)\()/ \)\(82\!\cdots\!88\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(10\!\cdots\!01\)\( \nu^{16} + \)\(59\!\cdots\!34\)\( \nu^{15} + \)\(17\!\cdots\!41\)\( \nu^{14} - \)\(90\!\cdots\!01\)\( \nu^{13} - \)\(11\!\cdots\!92\)\( \nu^{12} + \)\(53\!\cdots\!07\)\( \nu^{11} + \)\(39\!\cdots\!04\)\( \nu^{10} - \)\(15\!\cdots\!40\)\( \nu^{9} - \)\(74\!\cdots\!88\)\( \nu^{8} + \)\(22\!\cdots\!16\)\( \nu^{7} + \)\(76\!\cdots\!92\)\( \nu^{6} - \)\(17\!\cdots\!48\)\( \nu^{5} - \)\(40\!\cdots\!96\)\( \nu^{4} + \)\(52\!\cdots\!80\)\( \nu^{3} + \)\(99\!\cdots\!56\)\( \nu^{2} - \)\(35\!\cdots\!76\)\( \nu - \)\(74\!\cdots\!20\)\(\)\()/ \)\(82\!\cdots\!88\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(12\!\cdots\!33\)\( \nu^{16} + \)\(47\!\cdots\!50\)\( \nu^{15} + \)\(20\!\cdots\!21\)\( \nu^{14} - \)\(70\!\cdots\!17\)\( \nu^{13} - \)\(13\!\cdots\!20\)\( \nu^{12} + \)\(41\!\cdots\!59\)\( \nu^{11} + \)\(46\!\cdots\!88\)\( \nu^{10} - \)\(11\!\cdots\!20\)\( \nu^{9} - \)\(87\!\cdots\!44\)\( \nu^{8} + \)\(17\!\cdots\!92\)\( \nu^{7} + \)\(90\!\cdots\!80\)\( \nu^{6} - \)\(12\!\cdots\!88\)\( \nu^{5} - \)\(48\!\cdots\!32\)\( \nu^{4} + \)\(39\!\cdots\!76\)\( \nu^{3} + \)\(12\!\cdots\!56\)\( \nu^{2} - \)\(23\!\cdots\!40\)\( \nu - \)\(91\!\cdots\!52\)\(\)\()/ \)\(82\!\cdots\!88\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(12\!\cdots\!47\)\( \nu^{16} + \)\(87\!\cdots\!22\)\( \nu^{15} + \)\(21\!\cdots\!43\)\( \nu^{14} - \)\(13\!\cdots\!11\)\( \nu^{13} - \)\(13\!\cdots\!00\)\( \nu^{12} + \)\(75\!\cdots\!49\)\( \nu^{11} + \)\(44\!\cdots\!48\)\( \nu^{10} - \)\(21\!\cdots\!32\)\( \nu^{9} - \)\(81\!\cdots\!52\)\( \nu^{8} + \)\(31\!\cdots\!76\)\( \nu^{7} + \)\(80\!\cdots\!96\)\( \nu^{6} - \)\(23\!\cdots\!96\)\( \nu^{5} - \)\(41\!\cdots\!60\)\( \nu^{4} + \)\(79\!\cdots\!92\)\( \nu^{3} + \)\(97\!\cdots\!60\)\( \nu^{2} - \)\(72\!\cdots\!52\)\( \nu - \)\(72\!\cdots\!96\)\(\)\()/ \)\(82\!\cdots\!88\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(37\!\cdots\!15\)\( \nu^{16} + \)\(21\!\cdots\!28\)\( \nu^{15} + \)\(62\!\cdots\!59\)\( \nu^{14} - \)\(31\!\cdots\!85\)\( \nu^{13} - \)\(40\!\cdots\!82\)\( \nu^{12} + \)\(18\!\cdots\!61\)\( \nu^{11} + \)\(13\!\cdots\!14\)\( \nu^{10} - \)\(51\!\cdots\!64\)\( \nu^{9} - \)\(25\!\cdots\!48\)\( \nu^{8} + \)\(75\!\cdots\!64\)\( \nu^{7} + \)\(26\!\cdots\!40\)\( \nu^{6} - \)\(57\!\cdots\!68\)\( \nu^{5} - \)\(13\!\cdots\!48\)\( \nu^{4} + \)\(18\!\cdots\!60\)\( \nu^{3} + \)\(34\!\cdots\!40\)\( \nu^{2} - \)\(16\!\cdots\!80\)\( \nu - \)\(26\!\cdots\!04\)\(\)\()/ \)\(20\!\cdots\!72\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(43\!\cdots\!78\)\( \nu^{16} - \)\(29\!\cdots\!35\)\( \nu^{15} - \)\(71\!\cdots\!18\)\( \nu^{14} + \)\(43\!\cdots\!53\)\( \nu^{13} + \)\(46\!\cdots\!11\)\( \nu^{12} - \)\(24\!\cdots\!84\)\( \nu^{11} - \)\(15\!\cdots\!23\)\( \nu^{10} + \)\(69\!\cdots\!50\)\( \nu^{9} + \)\(28\!\cdots\!36\)\( \nu^{8} - \)\(10\!\cdots\!68\)\( \nu^{7} - \)\(29\!\cdots\!08\)\( \nu^{6} + \)\(73\!\cdots\!28\)\( \nu^{5} + \)\(15\!\cdots\!68\)\( \nu^{4} - \)\(23\!\cdots\!32\)\( \nu^{3} - \)\(39\!\cdots\!48\)\( \nu^{2} + \)\(17\!\cdots\!68\)\( \nu + \)\(30\!\cdots\!64\)\(\)\()/ \)\(20\!\cdots\!72\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(10\!\cdots\!61\)\( \nu^{16} + \)\(66\!\cdots\!42\)\( \nu^{15} + \)\(17\!\cdots\!85\)\( \nu^{14} - \)\(98\!\cdots\!41\)\( \nu^{13} - \)\(11\!\cdots\!04\)\( \nu^{12} + \)\(56\!\cdots\!67\)\( \nu^{11} + \)\(38\!\cdots\!68\)\( \nu^{10} - \)\(15\!\cdots\!12\)\( \nu^{9} - \)\(69\!\cdots\!88\)\( \nu^{8} + \)\(22\!\cdots\!04\)\( \nu^{7} + \)\(70\!\cdots\!04\)\( \nu^{6} - \)\(16\!\cdots\!36\)\( \nu^{5} - \)\(36\!\cdots\!08\)\( \nu^{4} + \)\(52\!\cdots\!08\)\( \nu^{3} + \)\(88\!\cdots\!24\)\( \nu^{2} - \)\(40\!\cdots\!88\)\( \nu - \)\(65\!\cdots\!52\)\(\)\()/ \)\(41\!\cdots\!44\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 196\)
\(\nu^{3}\)\(=\)\(\beta_{5} - 2 \beta_{4} + \beta_{3} + 343 \beta_{1} + 128\)
\(\nu^{4}\)\(=\)\(6 \beta_{16} + \beta_{15} + 2 \beta_{14} - 5 \beta_{13} - 2 \beta_{12} - \beta_{11} + 6 \beta_{10} + 7 \beta_{9} - 2 \beta_{7} - \beta_{6} - 4 \beta_{5} - 23 \beta_{4} - 3 \beta_{3} + 505 \beta_{2} + 427 \beta_{1} + 67005\)
\(\nu^{5}\)\(=\)\(14 \beta_{16} - 9 \beta_{15} - 30 \beta_{14} - 52 \beta_{13} - 11 \beta_{12} - 14 \beta_{11} - 57 \beta_{10} + 50 \beta_{9} - 62 \beta_{8} + 55 \beta_{7} - 29 \beta_{6} + 562 \beta_{5} - 1223 \beta_{4} + 572 \beta_{3} + 38 \beta_{2} + 140683 \beta_{1} + 49443\)
\(\nu^{6}\)\(=\)\(4428 \beta_{16} + 620 \beta_{15} + 1572 \beta_{14} - 3762 \beta_{13} - 2026 \beta_{12} - 518 \beta_{11} + 3962 \beta_{10} + 4938 \beta_{9} + 144 \beta_{8} - 1070 \beta_{7} - 1092 \beta_{6} - 2941 \beta_{5} - 16530 \beta_{4} - 2775 \beta_{3} + 238828 \beta_{2} + 167431 \beta_{1} + 27441788\)
\(\nu^{7}\)\(=\)\(9070 \beta_{16} - 4023 \beta_{15} - 20770 \beta_{14} - 39287 \beta_{13} - 7888 \beta_{12} - 14535 \beta_{11} - 48956 \beta_{10} + 45493 \beta_{9} - 43296 \beta_{8} + 42764 \beta_{7} - 23413 \beta_{6} + 272324 \beta_{5} - 613131 \beta_{4} + 291223 \beta_{3} + 8031 \beta_{2} + 62130819 \beta_{1} + 16725561\)
\(\nu^{8}\)\(=\)\(2479010 \beta_{16} + 290925 \beta_{15} + 963786 \beta_{14} - 2165060 \beta_{13} - 1340093 \beta_{12} - 165778 \beta_{11} + 2151401 \beta_{10} + 2680162 \beta_{9} + 67906 \beta_{8} - 524431 \beta_{7} - 659127 \beta_{6} - 1615928 \beta_{5} - 9448433 \beta_{4} - 1825206 \beta_{3} + 112359374 \beta_{2} + 64621131 \beta_{1} + 12112167141\)
\(\nu^{9}\)\(=\)\(4315240 \beta_{16} - 408182 \beta_{15} - 10904788 \beta_{14} - 22246216 \beta_{13} - 3753658 \beta_{12} - 9996164 \beta_{11} - 30004726 \beta_{10} + 28726936 \beta_{9} - 23442252 \beta_{8} + 25587866 \beta_{7} - 13707258 \beta_{6} + 127967121 \beta_{5} - 285868736 \beta_{4} + 145292169 \beta_{3} - 3604234 \beta_{2} + 28395203415 \beta_{1} + 5128297790\)
\(\nu^{10}\)\(=\)\(1271346030 \beta_{16} + 122290725 \beta_{15} + 543072090 \beta_{14} - 1138061973 \beta_{13} - 765429318 \beta_{12} - 22065377 \beta_{11} + 1112657602 \beta_{10} + 1329275823 \beta_{9} + 15505352 \beta_{8} - 264272094 \beta_{7} - 327400813 \beta_{6} - 808074216 \beta_{5} - 5066755123 \beta_{4} - 1064278631 \beta_{3} + 52932783385 \beta_{2} + 24974261195 \beta_{1} + 5534138272313\)
\(\nu^{11}\)\(=\)\(1772327478 \beta_{16} + 788518275 \beta_{15} - 5296131630 \beta_{14} - 11335515344 \beta_{13} - 1346923211 \beta_{12} - 6019672906 \beta_{11} - 16303973993 \beta_{10} + 15919748238 \beta_{9} - 11755882902 \beta_{8} + 14036242503 \beta_{7} - 7207662569 \beta_{6} + 59784914446 \beta_{5} - 129108515139 \beta_{4} + 72138409452 \beta_{3} - 5712820678 \beta_{2} + 13198072482363 \beta_{1} + 1353066234647\)
\(\nu^{12}\)\(=\)\(630160762572 \beta_{16} + 47674381828 \beta_{15} + 293440025772 \beta_{14} - 574597910894 \beta_{13} - 410079019782 \beta_{12} + 19303497830 \beta_{11} + 566964971334 \beta_{10} + 633748513166 \beta_{9} - 2541329360 \beta_{8} - 136313704498 \beta_{7} - 149068384980 \beta_{6} - 389795004229 \beta_{5} - 2648559262818 \beta_{4} - 588186166067 \beta_{3} + 24987034668304 \beta_{2} + 9625626696375 \beta_{1} + 2571972234616652\)
\(\nu^{13}\)\(=\)\(638016799126 \beta_{16} + 872848948757 \beta_{15} - 2522693475330 \beta_{14} - 5490034897119 \beta_{13} - 299342507884 \beta_{12} - 3418695594055 \beta_{11} - 8400885191744 \beta_{10} + 8323885412845 \beta_{9} - 5721572233440 \beta_{8} + 7412533742680 \beta_{7} - 3626518145881 \beta_{6} + 27951184443440 \beta_{5} - 57407559796975 \beta_{4} + 35746919056187 \beta_{3} - 4930435995513 \beta_{2} + 6189182100167011 \beta_{1} + 217406968471469\)
\(\nu^{14}\)\(=\)\(307896248891050 \beta_{16} + 17087373773281 \beta_{15} + 154652656782170 \beta_{14} - 284255776124720 \beta_{13} - 212808407443653 \beta_{12} + 23537792313026 \beta_{11} + 287456275880049 \beta_{10} + 296482508196990 \beta_{9} - 5691777953782 \beta_{8} - 70952238778263 \beta_{7} - 64756229233451 \beta_{6} - 185497454650628 \beta_{5} - 1365385818714677 \beta_{4} - 316090134987742 \beta_{3} + 11817064331166946 \beta_{2} + 3625937040746763 \beta_{1} + 1206066635695471665\)
\(\nu^{15}\)\(=\)\(186458253689432 \beta_{16} + 647735166648634 \beta_{15} - 1202315594658524 \beta_{14} - 2585315683076348 \beta_{13} + 56998028479866 \beta_{12} - 1878792212084400 \beta_{11} - 4218883330406426 \beta_{10} + 4227374554871652 \beta_{9} - 2747643952479660 \beta_{8} + 3838474415465446 \beta_{7} - 1788506669154242 \beta_{6} + 13098553113737905 \beta_{5} - 25302954079545768 \beta_{4} + 17688323419832013 \beta_{3} - 3599271350485134 \beta_{2} + 2917174028025045959 \beta_{1} - 77733083442393626\)
\(\nu^{16}\)\(=\)\(149500171473508822 \beta_{16} + 5348269201255457 \beta_{15} + 80199792741029114 \beta_{14} - 139080443721718085 \beta_{13} - 108523860979066426 \beta_{12} + 17908342392161799 \beta_{11} + 145395416058407318 \beta_{10} + 137403730224670415 \beta_{9} - 4573434518632008 \beta_{8} - 36942710291819514 \beta_{7} - 27270507225586001 \beta_{6} - 87948680322322668 \beta_{5} - 697283431518726263 \beta_{4} - 167027393319373115 \beta_{3} + 5597797898811901241 \beta_{2} + 1284550941666577771 \beta_{1} + 568465574333221591325\)

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
21.8139
21.5848
16.5905
15.1891
11.5967
9.14085
7.49302
3.56813
−3.62331
−3.88629
−5.66556
−8.14464
−12.3679
−13.8304
−14.3045
−20.9857
−22.1687
−21.8139 0 347.844 −424.937 0 1192.89 −4795.65 0 9269.52
1.2 −21.5848 0 337.903 113.334 0 −647.523 −4530.72 0 −2446.30
1.3 −16.5905 0 147.246 −174.998 0 1111.07 −319.307 0 2903.31
1.4 −15.1891 0 102.710 418.807 0 −814.653 384.131 0 −6361.32
1.5 −11.5967 0 6.48306 401.104 0 213.573 1409.19 0 −4651.48
1.6 −9.14085 0 −44.4448 27.9722 0 −1415.76 1576.29 0 −255.690
1.7 −7.49302 0 −71.8546 −400.807 0 −272.200 1497.52 0 3003.25
1.8 −3.56813 0 −115.268 210.232 0 1544.27 868.013 0 −750.136
1.9 3.62331 0 −114.872 −300.817 0 1353.78 −880.000 0 −1089.96
1.10 3.88629 0 −112.897 26.2639 0 −644.460 −936.194 0 102.069
1.11 5.66556 0 −95.9014 −117.894 0 1197.66 −1268.53 0 −667.937
1.12 8.14464 0 −61.6648 54.3103 0 −413.400 −1544.75 0 442.338
1.13 12.3679 0 24.9655 477.980 0 1344.57 −1274.32 0 5911.62
1.14 13.8304 0 63.2803 149.469 0 14.2725 −895.100 0 2067.22
1.15 14.3045 0 76.6177 −348.885 0 −1316.66 −734.996 0 −4990.62
1.16 20.9857 0 312.401 465.631 0 286.579 3869.79 0 9771.60
1.17 22.1687 0 363.451 −258.765 0 411.003 5219.64 0 −5736.49
\(n\): 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.c 17
3.b odd 2 1 177.8.a.c 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.8.a.c 17 3.b odd 2 1
531.8.a.c 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(74\!\cdots\!60\)\( T_{2}^{7} + \)\(10\!\cdots\!72\)\( T_{2}^{6} + \)\(41\!\cdots\!20\)\( T_{2}^{5} - \)\(88\!\cdots\!76\)\( T_{2}^{4} - \)\(10\!\cdots\!64\)\( T_{2}^{3} + \)\(29\!\cdots\!76\)\( T_{2}^{2} + \)\(79\!\cdots\!32\)\( T_{2} - \)\(24\!\cdots\!16\)\( \)">\(T_{2}^{17} + \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(531))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -248723246810300416 + 79787300207378432 T + 29180140031461376 T^{2} - 10454479722123264 T^{3} - 886760582981376 T^{4} + 413177144536320 T^{5} + 10751966150272 T^{6} - 7480805165760 T^{7} - 50741131904 T^{8} + 71331230512 T^{9} - 12724944 T^{10} - 377920980 T^{11} + 848131 T^{12} + 1108684 T^{13} - 2385 T^{14} - 1669 T^{15} + 2 T^{16} + T^{17} \)
$3$ \( T^{17} \)
$5$ \( \)\(50\!\cdots\!00\)\( - \)\(46\!\cdots\!00\)\( T + \)\(13\!\cdots\!00\)\( T^{2} - \)\(57\!\cdots\!00\)\( T^{3} - \)\(20\!\cdots\!00\)\( T^{4} + \)\(15\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!00\)\( T^{6} - \)\(94\!\cdots\!25\)\( T^{7} - \)\(34\!\cdots\!50\)\( T^{8} + \)\(24\!\cdots\!65\)\( T^{9} + 5497595372591662592 T^{10} - 30585865266283218 T^{11} - 46656463139080 T^{12} + 206573821760 T^{13} + 195521044 T^{14} - 714577 T^{15} - 318 T^{16} + T^{17} \)
$7$ \( \)\(11\!\cdots\!12\)\( - \)\(83\!\cdots\!08\)\( T + \)\(24\!\cdots\!92\)\( T^{2} + \)\(24\!\cdots\!12\)\( T^{3} - \)\(64\!\cdots\!08\)\( T^{4} - \)\(24\!\cdots\!03\)\( T^{5} + \)\(52\!\cdots\!13\)\( T^{6} + \)\(10\!\cdots\!12\)\( T^{7} - \)\(18\!\cdots\!91\)\( T^{8} - \)\(20\!\cdots\!80\)\( T^{9} + \)\(35\!\cdots\!45\)\( T^{10} + 17216346882916775274 T^{11} - 36012958273255561 T^{12} - 2556310608916 T^{13} + 17413046775 T^{14} - 3002144 T^{15} - 3145 T^{16} + T^{17} \)
$11$ \( \)\(47\!\cdots\!12\)\( + \)\(23\!\cdots\!96\)\( T + \)\(27\!\cdots\!84\)\( T^{2} - \)\(14\!\cdots\!28\)\( T^{3} - \)\(30\!\cdots\!56\)\( T^{4} + \)\(61\!\cdots\!53\)\( T^{5} + \)\(11\!\cdots\!72\)\( T^{6} - \)\(27\!\cdots\!66\)\( T^{7} - \)\(16\!\cdots\!92\)\( T^{8} + \)\(43\!\cdots\!27\)\( T^{9} + \)\(98\!\cdots\!96\)\( T^{10} - \)\(29\!\cdots\!48\)\( T^{11} - 28885615630901412048 T^{12} + 10020278686455103 T^{13} + 387644300708 T^{14} - 162104594 T^{15} - 1764 T^{16} + T^{17} \)
$13$ \( \)\(46\!\cdots\!52\)\( + \)\(63\!\cdots\!20\)\( T - \)\(26\!\cdots\!32\)\( T^{2} - \)\(49\!\cdots\!58\)\( T^{3} + \)\(41\!\cdots\!48\)\( T^{4} - \)\(24\!\cdots\!39\)\( T^{5} + \)\(98\!\cdots\!36\)\( T^{6} + \)\(22\!\cdots\!96\)\( T^{7} - \)\(37\!\cdots\!44\)\( T^{8} - \)\(58\!\cdots\!01\)\( T^{9} + \)\(14\!\cdots\!60\)\( T^{10} + \)\(16\!\cdots\!34\)\( T^{11} - \)\(18\!\cdots\!28\)\( T^{12} + 52563923636508735 T^{13} + 9797315279188 T^{14} - 444575444 T^{15} - 18192 T^{16} + T^{17} \)
$17$ \( -\)\(10\!\cdots\!08\)\( - \)\(65\!\cdots\!52\)\( T + \)\(18\!\cdots\!60\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(32\!\cdots\!50\)\( T^{4} + \)\(11\!\cdots\!39\)\( T^{5} - \)\(22\!\cdots\!13\)\( T^{6} - \)\(17\!\cdots\!48\)\( T^{7} + \)\(42\!\cdots\!81\)\( T^{8} + \)\(65\!\cdots\!46\)\( T^{9} - \)\(99\!\cdots\!21\)\( T^{10} - \)\(11\!\cdots\!82\)\( T^{11} - \)\(51\!\cdots\!61\)\( T^{12} + 10607927051441943286 T^{13} + 52877285545297 T^{14} - 5095751792 T^{15} - 15507 T^{16} + T^{17} \)
$19$ \( -\)\(41\!\cdots\!20\)\( + \)\(26\!\cdots\!40\)\( T - \)\(54\!\cdots\!64\)\( T^{2} + \)\(29\!\cdots\!56\)\( T^{3} + \)\(33\!\cdots\!76\)\( T^{4} - \)\(44\!\cdots\!88\)\( T^{5} + \)\(37\!\cdots\!24\)\( T^{6} + \)\(14\!\cdots\!43\)\( T^{7} - \)\(52\!\cdots\!75\)\( T^{8} - \)\(17\!\cdots\!05\)\( T^{9} + \)\(11\!\cdots\!70\)\( T^{10} + \)\(21\!\cdots\!59\)\( T^{11} - \)\(98\!\cdots\!93\)\( T^{12} + 9179288419761720141 T^{13} + 387083839671454 T^{14} - 5874357535 T^{15} - 52083 T^{16} + T^{17} \)
$23$ \( -\)\(98\!\cdots\!72\)\( - \)\(16\!\cdots\!44\)\( T + \)\(31\!\cdots\!36\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(51\!\cdots\!72\)\( T^{4} - \)\(38\!\cdots\!56\)\( T^{5} - \)\(19\!\cdots\!68\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{7} + \)\(21\!\cdots\!15\)\( T^{8} + \)\(96\!\cdots\!87\)\( T^{9} - \)\(95\!\cdots\!46\)\( T^{10} - \)\(94\!\cdots\!35\)\( T^{11} + \)\(19\!\cdots\!17\)\( T^{12} + \)\(25\!\cdots\!65\)\( T^{13} - 1860781459343038 T^{14} - 26935892401 T^{15} + 63823 T^{16} + T^{17} \)
$29$ \( \)\(12\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T + \)\(94\!\cdots\!20\)\( T^{2} + \)\(67\!\cdots\!60\)\( T^{3} - \)\(60\!\cdots\!24\)\( T^{4} - \)\(99\!\cdots\!46\)\( T^{5} + \)\(16\!\cdots\!90\)\( T^{6} + \)\(29\!\cdots\!35\)\( T^{7} - \)\(27\!\cdots\!79\)\( T^{8} - \)\(35\!\cdots\!19\)\( T^{9} + \)\(30\!\cdots\!00\)\( T^{10} + \)\(18\!\cdots\!03\)\( T^{11} - \)\(17\!\cdots\!89\)\( T^{12} - \)\(30\!\cdots\!97\)\( T^{13} + 49264913342324700 T^{14} - 31634147807 T^{15} - 502955 T^{16} + T^{17} \)
$31$ \( -\)\(52\!\cdots\!48\)\( - \)\(50\!\cdots\!48\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!20\)\( T^{3} - \)\(13\!\cdots\!76\)\( T^{4} - \)\(58\!\cdots\!94\)\( T^{5} + \)\(70\!\cdots\!16\)\( T^{6} + \)\(58\!\cdots\!27\)\( T^{7} - \)\(89\!\cdots\!35\)\( T^{8} - \)\(13\!\cdots\!55\)\( T^{9} + \)\(44\!\cdots\!18\)\( T^{10} - \)\(38\!\cdots\!29\)\( T^{11} - \)\(96\!\cdots\!25\)\( T^{12} + \)\(18\!\cdots\!47\)\( T^{13} + 95263535126494646 T^{14} - 239329320307 T^{15} - 347531 T^{16} + T^{17} \)
$37$ \( -\)\(52\!\cdots\!64\)\( + \)\(76\!\cdots\!44\)\( T - \)\(15\!\cdots\!20\)\( T^{2} - \)\(13\!\cdots\!10\)\( T^{3} + \)\(28\!\cdots\!26\)\( T^{4} + \)\(95\!\cdots\!07\)\( T^{5} - \)\(48\!\cdots\!43\)\( T^{6} - \)\(16\!\cdots\!38\)\( T^{7} - \)\(15\!\cdots\!23\)\( T^{8} + \)\(83\!\cdots\!74\)\( T^{9} + \)\(20\!\cdots\!37\)\( T^{10} - \)\(14\!\cdots\!60\)\( T^{11} - \)\(37\!\cdots\!29\)\( T^{12} + \)\(13\!\cdots\!94\)\( T^{13} + 232286733255409327 T^{14} - 597232647388 T^{15} - 447615 T^{16} + T^{17} \)
$41$ \( \)\(44\!\cdots\!88\)\( + \)\(30\!\cdots\!92\)\( T - \)\(38\!\cdots\!60\)\( T^{2} - \)\(86\!\cdots\!92\)\( T^{3} - \)\(70\!\cdots\!26\)\( T^{4} - \)\(17\!\cdots\!09\)\( T^{5} + \)\(83\!\cdots\!57\)\( T^{6} + \)\(59\!\cdots\!96\)\( T^{7} + \)\(56\!\cdots\!43\)\( T^{8} - \)\(44\!\cdots\!70\)\( T^{9} - \)\(11\!\cdots\!87\)\( T^{10} + \)\(93\!\cdots\!22\)\( T^{11} + \)\(59\!\cdots\!41\)\( T^{12} + \)\(21\!\cdots\!62\)\( T^{13} - 1250865607341246041 T^{14} - 1036432501916 T^{15} + 940335 T^{16} + T^{17} \)
$43$ \( \)\(11\!\cdots\!12\)\( - \)\(85\!\cdots\!80\)\( T + \)\(82\!\cdots\!64\)\( T^{2} + \)\(62\!\cdots\!12\)\( T^{3} - \)\(60\!\cdots\!04\)\( T^{4} - \)\(43\!\cdots\!85\)\( T^{5} + \)\(58\!\cdots\!54\)\( T^{6} + \)\(39\!\cdots\!52\)\( T^{7} - \)\(17\!\cdots\!02\)\( T^{8} + \)\(22\!\cdots\!35\)\( T^{9} + \)\(15\!\cdots\!20\)\( T^{10} - \)\(26\!\cdots\!00\)\( T^{11} - \)\(54\!\cdots\!28\)\( T^{12} + \)\(10\!\cdots\!73\)\( T^{13} + 848009342433042170 T^{14} - 1721208163556 T^{15} - 478562 T^{16} + T^{17} \)
$47$ \( \)\(12\!\cdots\!64\)\( - \)\(19\!\cdots\!68\)\( T - \)\(26\!\cdots\!28\)\( T^{2} + \)\(21\!\cdots\!36\)\( T^{3} + \)\(22\!\cdots\!68\)\( T^{4} + \)\(10\!\cdots\!96\)\( T^{5} - \)\(27\!\cdots\!92\)\( T^{6} - \)\(35\!\cdots\!17\)\( T^{7} + \)\(13\!\cdots\!17\)\( T^{8} + \)\(20\!\cdots\!55\)\( T^{9} - \)\(34\!\cdots\!86\)\( T^{10} - \)\(53\!\cdots\!69\)\( T^{11} + \)\(44\!\cdots\!07\)\( T^{12} + \)\(69\!\cdots\!81\)\( T^{13} - 2869752674198266322 T^{14} - 4286528686671 T^{15} + 703121 T^{16} + T^{17} \)
$53$ \( -\)\(68\!\cdots\!96\)\( + \)\(59\!\cdots\!04\)\( T + \)\(49\!\cdots\!88\)\( T^{2} - \)\(42\!\cdots\!16\)\( T^{3} - \)\(82\!\cdots\!52\)\( T^{4} + \)\(75\!\cdots\!66\)\( T^{5} + \)\(48\!\cdots\!72\)\( T^{6} - \)\(41\!\cdots\!17\)\( T^{7} - \)\(12\!\cdots\!90\)\( T^{8} + \)\(10\!\cdots\!17\)\( T^{9} + \)\(14\!\cdots\!36\)\( T^{10} - \)\(11\!\cdots\!06\)\( T^{11} - \)\(66\!\cdots\!12\)\( T^{12} + \)\(56\!\cdots\!28\)\( T^{13} + 13666138672137601876 T^{14} - 12472741052597 T^{15} - 1005974 T^{16} + T^{17} \)
$59$ \( ( -205379 + T )^{17} \)
$61$ \( \)\(61\!\cdots\!84\)\( + \)\(80\!\cdots\!48\)\( T + \)\(12\!\cdots\!60\)\( T^{2} - \)\(13\!\cdots\!72\)\( T^{3} - \)\(13\!\cdots\!32\)\( T^{4} + \)\(95\!\cdots\!46\)\( T^{5} + \)\(84\!\cdots\!62\)\( T^{6} - \)\(30\!\cdots\!35\)\( T^{7} + \)\(17\!\cdots\!11\)\( T^{8} + \)\(38\!\cdots\!49\)\( T^{9} - \)\(40\!\cdots\!12\)\( T^{10} - \)\(12\!\cdots\!75\)\( T^{11} + \)\(31\!\cdots\!33\)\( T^{12} - \)\(86\!\cdots\!65\)\( T^{13} - 62269411716043146804 T^{14} + 47026866151899 T^{15} - 11510749 T^{16} + T^{17} \)
$67$ \( -\)\(99\!\cdots\!68\)\( - \)\(64\!\cdots\!92\)\( T - \)\(10\!\cdots\!96\)\( T^{2} + \)\(59\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!20\)\( T^{4} - \)\(48\!\cdots\!36\)\( T^{5} - \)\(15\!\cdots\!80\)\( T^{6} + \)\(55\!\cdots\!27\)\( T^{7} + \)\(44\!\cdots\!88\)\( T^{8} - \)\(23\!\cdots\!83\)\( T^{9} - \)\(36\!\cdots\!44\)\( T^{10} + \)\(38\!\cdots\!42\)\( T^{11} - \)\(28\!\cdots\!08\)\( T^{12} - \)\(23\!\cdots\!54\)\( T^{13} + \)\(47\!\cdots\!64\)\( T^{14} + 26158930658183 T^{15} - 14007144 T^{16} + T^{17} \)
$71$ \( -\)\(46\!\cdots\!96\)\( - \)\(33\!\cdots\!16\)\( T + \)\(27\!\cdots\!12\)\( T^{2} + \)\(44\!\cdots\!62\)\( T^{3} + \)\(69\!\cdots\!94\)\( T^{4} - \)\(24\!\cdots\!95\)\( T^{5} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!04\)\( T^{7} + \)\(29\!\cdots\!70\)\( T^{8} - \)\(92\!\cdots\!53\)\( T^{9} - \)\(33\!\cdots\!32\)\( T^{10} - \)\(25\!\cdots\!98\)\( T^{11} + \)\(18\!\cdots\!58\)\( T^{12} + \)\(24\!\cdots\!43\)\( T^{13} - \)\(50\!\cdots\!48\)\( T^{14} - 84831558041452 T^{15} + 5229074 T^{16} + T^{17} \)
$73$ \( \)\(15\!\cdots\!00\)\( - \)\(32\!\cdots\!00\)\( T + \)\(78\!\cdots\!40\)\( T^{2} + \)\(10\!\cdots\!44\)\( T^{3} - \)\(21\!\cdots\!04\)\( T^{4} - \)\(71\!\cdots\!48\)\( T^{5} + \)\(18\!\cdots\!18\)\( T^{6} + \)\(33\!\cdots\!89\)\( T^{7} - \)\(64\!\cdots\!31\)\( T^{8} + \)\(71\!\cdots\!51\)\( T^{9} + \)\(10\!\cdots\!28\)\( T^{10} - \)\(16\!\cdots\!87\)\( T^{11} - \)\(87\!\cdots\!77\)\( T^{12} + \)\(15\!\cdots\!17\)\( T^{13} + \)\(35\!\cdots\!24\)\( T^{14} - 64211707021029 T^{15} - 5452211 T^{16} + T^{17} \)
$79$ \( -\)\(22\!\cdots\!60\)\( + \)\(15\!\cdots\!60\)\( T + \)\(28\!\cdots\!28\)\( T^{2} + \)\(45\!\cdots\!48\)\( T^{3} - \)\(12\!\cdots\!84\)\( T^{4} - \)\(36\!\cdots\!85\)\( T^{5} + \)\(24\!\cdots\!30\)\( T^{6} + \)\(69\!\cdots\!92\)\( T^{7} - \)\(29\!\cdots\!22\)\( T^{8} - \)\(58\!\cdots\!37\)\( T^{9} + \)\(22\!\cdots\!16\)\( T^{10} + \)\(21\!\cdots\!36\)\( T^{11} - \)\(93\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!51\)\( T^{13} + \)\(19\!\cdots\!90\)\( T^{14} - 67281665098976 T^{15} - 15275654 T^{16} + T^{17} \)
$83$ \( -\)\(45\!\cdots\!68\)\( + \)\(59\!\cdots\!28\)\( T + \)\(60\!\cdots\!60\)\( T^{2} - \)\(30\!\cdots\!68\)\( T^{3} + \)\(36\!\cdots\!12\)\( T^{4} + \)\(60\!\cdots\!33\)\( T^{5} - \)\(10\!\cdots\!51\)\( T^{6} - \)\(61\!\cdots\!06\)\( T^{7} + \)\(10\!\cdots\!35\)\( T^{8} + \)\(35\!\cdots\!56\)\( T^{9} - \)\(50\!\cdots\!59\)\( T^{10} - \)\(11\!\cdots\!32\)\( T^{11} + \)\(12\!\cdots\!53\)\( T^{12} + \)\(23\!\cdots\!40\)\( T^{13} - \)\(16\!\cdots\!85\)\( T^{14} - 238523939774914 T^{15} + 7826609 T^{16} + T^{17} \)
$89$ \( \)\(30\!\cdots\!00\)\( - \)\(13\!\cdots\!40\)\( T + \)\(11\!\cdots\!08\)\( T^{2} + \)\(13\!\cdots\!44\)\( T^{3} - \)\(19\!\cdots\!92\)\( T^{4} - \)\(34\!\cdots\!76\)\( T^{5} + \)\(93\!\cdots\!58\)\( T^{6} - \)\(54\!\cdots\!85\)\( T^{7} - \)\(18\!\cdots\!53\)\( T^{8} + \)\(28\!\cdots\!19\)\( T^{9} + \)\(15\!\cdots\!32\)\( T^{10} - \)\(29\!\cdots\!45\)\( T^{11} - \)\(59\!\cdots\!43\)\( T^{12} + \)\(11\!\cdots\!53\)\( T^{13} + \)\(10\!\cdots\!32\)\( T^{14} - 185232892536939 T^{15} - 6436185 T^{16} + T^{17} \)
$97$ \( -\)\(44\!\cdots\!88\)\( - \)\(52\!\cdots\!24\)\( T + \)\(74\!\cdots\!24\)\( T^{2} + \)\(53\!\cdots\!72\)\( T^{3} - \)\(72\!\cdots\!00\)\( T^{4} - \)\(64\!\cdots\!00\)\( T^{5} + \)\(51\!\cdots\!78\)\( T^{6} + \)\(33\!\cdots\!65\)\( T^{7} - \)\(29\!\cdots\!88\)\( T^{8} - \)\(85\!\cdots\!43\)\( T^{9} + \)\(97\!\cdots\!28\)\( T^{10} + \)\(97\!\cdots\!38\)\( T^{11} - \)\(15\!\cdots\!24\)\( T^{12} - \)\(27\!\cdots\!66\)\( T^{13} + \)\(10\!\cdots\!66\)\( T^{14} - 204858865374459 T^{15} - 26377540 T^{16} + T^{17} \)
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