Properties

Label 91.8.a.e
Level $91$
Weight $8$
Character orbit 91.a
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + 186018392 x^{5} + 10752389517 x^{4} + 491049966 x^{3} - 344602049215 x^{2} - 835334324470 x + 59402280000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 7 + \beta_{3} ) q^{3} + ( 82 + \beta_{2} ) q^{4} + ( 87 - 3 \beta_{1} - \beta_{3} - \beta_{4} ) q^{5} + ( 28 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{6} + 343 q^{7} + ( 60 - 86 \beta_{1} - 15 \beta_{3} - \beta_{4} - \beta_{8} ) q^{8} + ( 901 + 20 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} - 3 \beta_{4} + \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 7 + \beta_{3} ) q^{3} + ( 82 + \beta_{2} ) q^{4} + ( 87 - 3 \beta_{1} - \beta_{3} - \beta_{4} ) q^{5} + ( 28 - 5 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{6} + 343 q^{7} + ( 60 - 86 \beta_{1} - 15 \beta_{3} - \beta_{4} - \beta_{8} ) q^{8} + ( 901 + 20 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} - 3 \beta_{4} + \beta_{11} ) q^{9} + ( 609 - 106 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} + \beta_{9} - \beta_{11} ) q^{10} + ( 992 + 46 \beta_{1} + 8 \beta_{2} + 16 \beta_{3} + 3 \beta_{4} + \beta_{6} + \beta_{8} ) q^{11} + ( -106 + 202 \beta_{1} + 17 \beta_{2} + 107 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{8} - 3 \beta_{9} - \beta_{11} ) q^{12} + 2197 q^{13} + ( 343 - 343 \beta_{1} ) q^{14} + ( -2335 - 76 \beta_{1} + 5 \beta_{2} + 162 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{15} + ( 7163 - \beta_{1} + 110 \beta_{2} + 117 \beta_{3} + \beta_{4} + 13 \beta_{5} - 5 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + 4 \beta_{10} + 4 \beta_{11} ) q^{16} + ( 7062 - 302 \beta_{1} - 2 \beta_{2} + 52 \beta_{3} - 6 \beta_{4} + 12 \beta_{5} + \beta_{6} + 4 \beta_{7} + \beta_{8} - 4 \beta_{10} + 3 \beta_{11} ) q^{17} + ( -3158 - 1140 \beta_{1} - 37 \beta_{2} + 69 \beta_{3} + 10 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{7} - 10 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{18} + ( -293 - 1093 \beta_{1} + 9 \beta_{2} + 52 \beta_{3} + 20 \beta_{4} + 22 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} - 9 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 9 \beta_{11} ) q^{19} + ( 12344 - 1156 \beta_{1} + 90 \beta_{2} - 56 \beta_{3} - 2 \beta_{4} + 16 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} - 2 \beta_{8} + 8 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{20} + ( 2401 + 343 \beta_{3} ) q^{21} + ( -7096 - 2051 \beta_{1} - 249 \beta_{2} - 162 \beta_{3} + 29 \beta_{4} - 19 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + \beta_{8} - 8 \beta_{9} - 13 \beta_{10} - 7 \beta_{11} ) q^{22} + ( 7833 + 719 \beta_{1} - 62 \beta_{2} + 99 \beta_{3} + 42 \beta_{4} - 8 \beta_{5} - 15 \beta_{6} - 10 \beta_{7} + 11 \beta_{8} - 16 \beta_{9} + 12 \beta_{10} + 10 \beta_{11} ) q^{23} + ( -42479 - 1789 \beta_{1} - 364 \beta_{2} - 214 \beta_{3} + 147 \beta_{4} - 100 \beta_{5} + 17 \beta_{6} - 15 \beta_{7} - 25 \beta_{8} - 6 \beta_{9} + 5 \beta_{10} - 22 \beta_{11} ) q^{24} + ( 23564 - 1729 \beta_{1} - 277 \beta_{2} - 558 \beta_{3} - 128 \beta_{4} - 14 \beta_{5} - 11 \beta_{6} + 38 \beta_{7} + 20 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - 8 \beta_{11} ) q^{25} + ( 2197 - 2197 \beta_{1} ) q^{26} + ( 26005 - 726 \beta_{1} - 25 \beta_{2} + 390 \beta_{3} + 55 \beta_{4} + 50 \beta_{5} + 7 \beta_{6} - 30 \beta_{7} + 2 \beta_{8} + 10 \beta_{9} - 5 \beta_{10} + 32 \beta_{11} ) q^{27} + ( 28126 + 343 \beta_{2} ) q^{28} + ( 29316 - 131 \beta_{1} + 213 \beta_{2} + 401 \beta_{3} - 101 \beta_{4} + 50 \beta_{5} - 18 \beta_{6} - 12 \beta_{7} + 39 \beta_{8} + 10 \beta_{9} + 7 \beta_{10} - 34 \beta_{11} ) q^{29} + ( 18511 + 1626 \beta_{1} - 573 \beta_{2} - 22 \beta_{3} - 150 \beta_{4} - 163 \beta_{5} - 5 \beta_{6} + 39 \beta_{7} + 70 \beta_{8} - 24 \beta_{9} - 26 \beta_{10} - 5 \beta_{11} ) q^{30} + ( 5013 - 1259 \beta_{1} - 146 \beta_{2} - 1333 \beta_{3} + 64 \beta_{4} - 24 \beta_{5} + 30 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 16 \beta_{9} + 20 \beta_{10} + 15 \beta_{11} ) q^{31} + ( 13875 - 9387 \beta_{1} - 360 \beta_{2} - 2543 \beta_{3} + 83 \beta_{4} - 37 \beta_{5} + 7 \beta_{6} - 49 \beta_{7} - 104 \beta_{8} + 55 \beta_{9} + 14 \beta_{10} - 10 \beta_{11} ) q^{32} + ( 50088 + 1892 \beta_{1} + 305 \beta_{2} + 999 \beta_{3} - 510 \beta_{4} + 90 \beta_{5} - 50 \beta_{6} + 24 \beta_{7} + 7 \beta_{8} - 30 \beta_{9} - 5 \beta_{10} + 4 \beta_{11} ) q^{33} + ( 70361 - 6042 \beta_{1} + 40 \beta_{2} - 2395 \beta_{3} + 155 \beta_{4} + 27 \beta_{5} + 81 \beta_{6} + 23 \beta_{7} - \beta_{8} + 72 \beta_{9} - 11 \beta_{10} + 16 \beta_{11} ) q^{34} + ( 29841 - 1029 \beta_{1} - 343 \beta_{3} - 343 \beta_{4} ) q^{35} + ( 118931 + 6845 \beta_{1} + 1715 \beta_{2} + 86 \beta_{3} + 41 \beta_{4} + 62 \beta_{5} - 43 \beta_{6} + 21 \beta_{7} + 89 \beta_{8} - 26 \beta_{9} + 55 \beta_{10} - 24 \beta_{11} ) q^{36} + ( 30816 + 1514 \beta_{1} - 831 \beta_{2} + 135 \beta_{3} - 368 \beta_{4} - 26 \beta_{5} + 25 \beta_{6} - 40 \beta_{7} - 16 \beta_{8} - 10 \beta_{9} - 59 \beta_{10} - 11 \beta_{11} ) q^{37} + ( 233611 + 2516 \beta_{1} + 1415 \beta_{2} - 2407 \beta_{3} + 249 \beta_{4} + 228 \beta_{5} - 48 \beta_{6} - 44 \beta_{7} - 68 \beta_{8} - 15 \beta_{9} - 16 \beta_{10} + 75 \beta_{11} ) q^{38} + ( 15379 + 2197 \beta_{3} ) q^{39} + ( 184449 - 8633 \beta_{1} + 804 \beta_{2} - 4928 \beta_{3} - 330 \beta_{4} + 359 \beta_{5} - 7 \beta_{6} + 41 \beta_{7} - 163 \beta_{8} + \beta_{9} + 4 \beta_{10} + 124 \beta_{11} ) q^{40} + ( 114183 + 2580 \beta_{1} + 766 \beta_{2} - 525 \beta_{3} + 392 \beta_{4} + 188 \beta_{5} + 51 \beta_{6} + 20 \beta_{7} - 81 \beta_{8} + 64 \beta_{9} + 32 \beta_{10} - 55 \beta_{11} ) q^{41} + ( 9604 - 1715 \beta_{1} - 686 \beta_{2} - 343 \beta_{3} - 343 \beta_{5} ) q^{42} + ( 47309 - 167 \beta_{1} + 544 \beta_{2} - 719 \beta_{3} + 147 \beta_{4} + 272 \beta_{5} - \beta_{6} - 52 \beta_{7} - 139 \beta_{8} - 44 \beta_{9} - 2 \beta_{10} - 143 \beta_{11} ) q^{43} + ( 265682 + 34366 \beta_{1} + 1367 \beta_{2} + 4245 \beta_{3} + 538 \beta_{4} - 209 \beta_{5} - 6 \beta_{6} + 52 \beta_{7} + 309 \beta_{8} - 85 \beta_{9} + 28 \beta_{10} + 73 \beta_{11} ) q^{44} + ( 293312 + 4989 \beta_{1} - 623 \beta_{2} - 4189 \beta_{3} - 207 \beta_{4} - 46 \beta_{5} - 75 \beta_{6} + 156 \beta_{7} - 4 \beta_{8} - 50 \beta_{9} - 47 \beta_{10} + 65 \beta_{11} ) q^{45} + ( -145525 + 1533 \beta_{1} - 1638 \beta_{2} + 1409 \beta_{3} + 1256 \beta_{4} - 461 \beta_{5} - 44 \beta_{6} - 222 \beta_{7} - 219 \beta_{8} - 113 \beta_{9} + 27 \beta_{10} - 80 \beta_{11} ) q^{46} + ( 108490 + 10363 \beta_{1} - 785 \beta_{2} + 1419 \beta_{3} + 506 \beta_{4} + 258 \beta_{5} + 89 \beta_{6} - 24 \beta_{7} + 24 \beta_{8} + 78 \beta_{9} - 51 \beta_{10} - 240 \beta_{11} ) q^{47} + ( 315269 + 63745 \beta_{1} + 2496 \beta_{2} + 13832 \beta_{3} + 1251 \beta_{4} - 242 \beta_{5} - 211 \beta_{6} - 3 \beta_{7} + 471 \beta_{8} - 370 \beta_{9} - 5 \beta_{10} + 216 \beta_{11} ) q^{48} + 117649 q^{49} + ( 327747 + 9473 \beta_{1} + 2733 \beta_{2} + 3712 \beta_{3} - 546 \beta_{4} + 215 \beta_{5} + 193 \beta_{6} + 155 \beta_{7} + 251 \beta_{8} + 379 \beta_{9} - 19 \beta_{10} - 319 \beta_{11} ) q^{50} + ( 211327 + 32442 \beta_{1} - 594 \beta_{2} + 10057 \beta_{3} - 177 \beta_{4} - 256 \beta_{5} + 99 \beta_{6} - 114 \beta_{7} + 141 \beta_{8} + 240 \beta_{9} + 84 \beta_{10} + 52 \beta_{11} ) q^{51} + ( 180154 + 2197 \beta_{2} ) q^{52} + ( 169261 + 7163 \beta_{1} - 4083 \beta_{2} + 2234 \beta_{3} + 50 \beta_{4} - 622 \beta_{5} + 15 \beta_{6} + 198 \beta_{7} - 56 \beta_{8} + 214 \beta_{9} + \beta_{10} + 80 \beta_{11} ) q^{53} + ( 190897 - 15845 \beta_{1} - 2522 \beta_{2} - 6580 \beta_{3} + 3 \beta_{4} + 854 \beta_{5} + 141 \beta_{6} - 93 \beta_{7} - 243 \beta_{8} + 42 \beta_{9} - 45 \beta_{10} + 300 \beta_{11} ) q^{54} + ( -180228 + 13018 \beta_{1} - 818 \beta_{2} + 14776 \beta_{3} + 384 \beta_{4} - 416 \beta_{5} + 37 \beta_{6} + 16 \beta_{7} + 25 \beta_{8} + 240 \beta_{9} + 92 \beta_{10} - 395 \beta_{11} ) q^{55} + ( 20580 - 29498 \beta_{1} - 5145 \beta_{3} - 343 \beta_{4} - 343 \beta_{8} ) q^{56} + ( 183691 + 29928 \beta_{1} - 7493 \beta_{2} + 4674 \beta_{3} - 210 \beta_{4} - 562 \beta_{5} - 28 \beta_{6} - 240 \beta_{7} - 413 \beta_{8} - 170 \beta_{9} - 51 \beta_{10} + 662 \beta_{11} ) q^{57} + ( 83642 - 59368 \beta_{1} - 5374 \beta_{2} - 21491 \beta_{3} - 2803 \beta_{4} - 181 \beta_{5} + 209 \beta_{6} + 73 \beta_{7} - 344 \beta_{8} + 405 \beta_{9} + 80 \beta_{10} - 126 \beta_{11} ) q^{58} + ( 19019 + 27100 \beta_{1} - 1753 \beta_{2} + 848 \beta_{3} + 260 \beta_{4} - 1134 \beta_{5} - 235 \beta_{6} + 92 \beta_{7} + 46 \beta_{8} - 366 \beta_{9} - 65 \beta_{10} - 237 \beta_{11} ) q^{59} + ( -112067 + 52717 \beta_{1} - 6973 \beta_{2} + 11969 \beta_{3} + 997 \beta_{4} - 1437 \beta_{5} - 103 \beta_{6} + 171 \beta_{7} + 694 \beta_{8} - 309 \beta_{9} - 95 \beta_{10} - 157 \beta_{11} ) q^{60} + ( 17199 + 42358 \beta_{1} - 1510 \beta_{2} + 3389 \beta_{3} - 2780 \beta_{4} + 796 \beta_{5} - 167 \beta_{6} - 298 \beta_{7} - 103 \beta_{8} - 372 \beta_{9} + 238 \beta_{10} + 495 \beta_{11} ) q^{61} + ( 231382 + 14229 \beta_{1} + 3494 \beta_{2} + 13004 \beta_{3} + 3117 \beta_{4} + 504 \beta_{5} - 353 \beta_{6} - 105 \beta_{7} + 336 \beta_{8} - 495 \beta_{9} - 396 \beta_{10} - 384 \beta_{11} ) q^{62} + ( 309043 + 6860 \beta_{1} + 686 \beta_{2} + 4116 \beta_{3} - 1029 \beta_{4} + 343 \beta_{11} ) q^{63} + ( 983619 + 41413 \beta_{1} + 10390 \beta_{2} + 11955 \beta_{3} - 4849 \beta_{4} + 2797 \beta_{5} - 257 \beta_{6} + 147 \beta_{7} + 222 \beta_{8} - 255 \beta_{9} - 42 \beta_{10} + 344 \beta_{11} ) q^{64} + ( 191139 - 6591 \beta_{1} - 2197 \beta_{3} - 2197 \beta_{4} ) q^{65} + ( -339989 - 96437 \beta_{1} - 1145 \beta_{2} - 27311 \beta_{3} + 20 \beta_{4} - 1978 \beta_{5} + 483 \beta_{6} - 213 \beta_{7} - 852 \beta_{8} + 1084 \beta_{9} + 356 \beta_{10} - 395 \beta_{11} ) q^{66} + ( -266779 + 21612 \beta_{1} - 5342 \beta_{2} + 4077 \beta_{3} + 642 \beta_{4} + 1052 \beta_{5} + 107 \beta_{6} + 578 \beta_{7} + 287 \beta_{8} - 604 \beta_{9} - 2 \beta_{10} + 189 \beta_{11} ) q^{67} + ( 404801 - 31611 \beta_{1} + 6983 \beta_{2} - 2361 \beta_{3} - 2243 \beta_{4} + 3353 \beta_{5} - 223 \beta_{6} + 155 \beta_{7} + 432 \beta_{8} - 59 \beta_{9} - 175 \beta_{10} - 319 \beta_{11} ) q^{68} + ( 325067 - 9782 \beta_{1} + 11750 \beta_{2} + 14569 \beta_{3} + 4828 \beta_{4} + 1100 \beta_{5} - 192 \beta_{6} - 174 \beta_{7} - 430 \beta_{8} - 120 \beta_{9} + 152 \beta_{10} + 516 \beta_{11} ) q^{69} + ( 208887 - 36358 \beta_{1} + 2401 \beta_{2} + 1715 \beta_{3} - 1029 \beta_{4} + 343 \beta_{9} - 343 \beta_{11} ) q^{70} + ( 481431 - 15448 \beta_{1} - 239 \beta_{2} - 11872 \beta_{3} + 270 \beta_{4} + 1342 \beta_{5} + 502 \beta_{6} - 320 \beta_{7} - 865 \beta_{8} + 26 \beta_{9} - 101 \beta_{10} + 496 \beta_{11} ) q^{71} + ( -721319 - 204133 \beta_{1} - 10042 \beta_{2} - 60301 \beta_{3} - 2046 \beta_{4} - 1012 \beta_{5} - 67 \beta_{6} + 129 \beta_{7} - 1132 \beta_{8} - 204 \beta_{9} + 47 \beta_{10} - 502 \beta_{11} ) q^{72} + ( 536053 + 95355 \beta_{1} + 7401 \beta_{2} - 5200 \beta_{3} + 1649 \beta_{4} - 1658 \beta_{5} - 254 \beta_{6} - 420 \beta_{7} + 1047 \beta_{8} - 62 \beta_{9} - 21 \beta_{10} - 662 \beta_{11} ) q^{73} + ( -399364 + 65677 \beta_{1} - 2103 \beta_{2} + 19420 \beta_{3} + 569 \beta_{4} - 559 \beta_{5} + 460 \beta_{6} + 158 \beta_{7} + 1537 \beta_{8} + 88 \beta_{9} + 167 \beta_{10} + 311 \beta_{11} ) q^{74} + ( -1325593 - 61598 \beta_{1} - 2085 \beta_{2} + 10412 \beta_{3} + 4941 \beta_{4} - 1718 \beta_{5} + 1103 \beta_{6} + 510 \beta_{7} + 858 \beta_{8} + 1538 \beta_{9} + 175 \beta_{10} - 2286 \beta_{11} ) q^{75} + ( -131208 - 266336 \beta_{1} + 7036 \beta_{2} - 63268 \beta_{3} - 1142 \beta_{4} + 1876 \beta_{5} - 14 \beta_{6} + 188 \beta_{7} - 1402 \beta_{8} + 252 \beta_{9} + 307 \beta_{10} + 225 \beta_{11} ) q^{76} + ( 340256 + 15778 \beta_{1} + 2744 \beta_{2} + 5488 \beta_{3} + 1029 \beta_{4} + 343 \beta_{6} + 343 \beta_{8} ) q^{77} + ( 61516 - 10985 \beta_{1} - 4394 \beta_{2} - 2197 \beta_{3} - 2197 \beta_{5} ) q^{78} + ( 362150 - 10071 \beta_{1} - 226 \beta_{2} - 1080 \beta_{3} - 537 \beta_{4} - 1744 \beta_{5} - 850 \beta_{6} - 208 \beta_{7} + 392 \beta_{8} - 724 \beta_{9} - 794 \beta_{10} + 258 \beta_{11} ) q^{79} + ( 369148 - 123078 \beta_{1} + 25670 \beta_{2} - 44558 \beta_{3} + 3300 \beta_{4} + 5202 \beta_{5} + 296 \beta_{6} - 1468 \beta_{7} - 1632 \beta_{8} + 384 \beta_{9} - 14 \beta_{10} + 510 \beta_{11} ) q^{80} + ( -566636 + 38642 \beta_{1} - 19112 \beta_{2} + 42191 \beta_{3} + 4009 \beta_{4} + 752 \beta_{5} - 281 \beta_{6} - 942 \beta_{7} - 217 \beta_{8} + 460 \beta_{9} - 470 \beta_{10} + 1314 \beta_{11} ) q^{81} + ( -297523 - 205655 \beta_{1} + 3604 \beta_{2} - 42096 \beta_{3} - 3673 \beta_{4} + 2488 \beta_{5} - 665 \beta_{6} + 489 \beta_{7} - 123 \beta_{8} - 66 \beta_{9} - 129 \beta_{10} + 566 \beta_{11} ) q^{82} + ( 208126 + 29793 \beta_{1} + 1348 \beta_{2} + 11664 \beta_{3} + 15028 \beta_{4} - 3896 \beta_{5} - 81 \beta_{6} + 770 \beta_{7} + 471 \beta_{8} - 532 \beta_{9} - 126 \beta_{10} + 520 \beta_{11} ) q^{83} + ( -36358 + 69286 \beta_{1} + 5831 \beta_{2} + 36701 \beta_{3} - 686 \beta_{4} - 1029 \beta_{5} - 686 \beta_{6} + 1029 \beta_{8} - 1029 \beta_{9} - 343 \beta_{11} ) q^{84} + ( 785437 - 18292 \beta_{1} + 2785 \beta_{2} - 4686 \beta_{3} - 13988 \beta_{4} - 1302 \beta_{5} - 444 \beta_{6} + 1192 \beta_{7} - 131 \beta_{8} + 418 \beta_{9} - 157 \beta_{10} + 112 \beta_{11} ) q^{85} + ( 170096 - 116512 \beta_{1} + 13607 \beta_{2} - 64122 \beta_{3} + 2020 \beta_{4} - 1285 \beta_{5} - 275 \beta_{6} - 81 \beta_{7} + 685 \beta_{8} - 209 \beta_{9} + 919 \beta_{10} + 1223 \beta_{11} ) q^{86} + ( 1207768 + 287748 \beta_{1} + 11354 \beta_{2} + 9416 \beta_{3} - 3420 \beta_{4} - 1808 \beta_{5} - 353 \beta_{6} + 1800 \beta_{7} + 3023 \beta_{8} - 520 \beta_{9} - 576 \beta_{10} + 289 \beta_{11} ) q^{87} + ( -5759949 - 207999 \beta_{1} - 43436 \beta_{2} + 12390 \beta_{3} + 4365 \beta_{4} - 4544 \beta_{5} + 763 \beta_{6} - 821 \beta_{7} - 2387 \beta_{8} + 178 \beta_{9} + 35 \beta_{10} - 1170 \beta_{11} ) q^{88} + ( -983622 + 129535 \beta_{1} + 17781 \beta_{2} + 11401 \beta_{3} - 3399 \beta_{4} - 2458 \beta_{5} - 951 \beta_{6} - 156 \beta_{7} + 2136 \beta_{8} - 866 \beta_{9} + 1033 \beta_{10} + 83 \beta_{11} ) q^{89} + ( -955895 - 226070 \beta_{1} + 14300 \beta_{2} + 19480 \beta_{3} + 5024 \beta_{4} + 2598 \beta_{5} + 948 \beta_{6} - 18 \beta_{7} - 127 \beta_{8} + 1163 \beta_{9} + 387 \beta_{10} - 368 \beta_{11} ) q^{90} + 753571 q^{91} + ( -1504628 + 278248 \beta_{1} + 26745 \beta_{2} + 111987 \beta_{3} + 4122 \beta_{4} - 4767 \beta_{5} - 478 \beta_{6} - 170 \beta_{7} + 1267 \beta_{8} - 2475 \beta_{9} + 119 \beta_{10} + 1218 \beta_{11} ) q^{92} + ( -3838968 - 194442 \beta_{1} - 4212 \beta_{2} - 13154 \beta_{3} - 3719 \beta_{4} + 2072 \beta_{5} - 411 \beta_{6} + 138 \beta_{7} - 2955 \beta_{8} - 628 \beta_{9} + 202 \beta_{10} - 2280 \beta_{11} ) q^{93} + ( -2022684 - 15623 \beta_{1} - 26273 \beta_{2} - 63905 \beta_{3} - 6624 \beta_{4} - 138 \beta_{5} + 639 \beta_{6} + 1533 \beta_{7} + 3375 \beta_{8} + 29 \beta_{9} - 87 \beta_{10} + 1141 \beta_{11} ) q^{94} + ( -1824167 + 1267 \beta_{1} + 26280 \beta_{2} - 2435 \beta_{3} + 16683 \beta_{4} + 7716 \beta_{5} + 301 \beta_{6} - 2620 \beta_{7} - 4277 \beta_{8} + 1164 \beta_{9} + 1174 \beta_{10} - 303 \beta_{11} ) q^{95} + ( -6988629 - 430469 \beta_{1} - 87810 \beta_{2} - 27010 \beta_{3} + 11735 \beta_{4} - 9622 \beta_{5} + 971 \beta_{6} - 729 \beta_{7} - 2913 \beta_{8} - 786 \beta_{9} - 1401 \beta_{10} + 1110 \beta_{11} ) q^{96} + ( -1958825 - 145141 \beta_{1} - 4413 \beta_{2} + 19600 \beta_{3} - 12721 \beta_{4} + 894 \beta_{5} + 1080 \beta_{6} - 1132 \beta_{7} - 1249 \beta_{8} + 1102 \beta_{9} - 167 \beta_{10} - 1266 \beta_{11} ) q^{97} + ( 117649 - 117649 \beta_{1} ) q^{98} + ( 803049 + 328498 \beta_{1} + 25917 \beta_{2} + 66706 \beta_{3} + 8806 \beta_{4} - 898 \beta_{5} + 158 \beta_{6} - 456 \beta_{7} + 951 \beta_{8} + 1018 \beta_{9} + 779 \beta_{10} - 1598 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{2} + 82q^{3} + 986q^{4} + 1026q^{5} + 309q^{6} + 4116q^{7} + 228q^{8} + 10902q^{9} + O(q^{10}) \) \( 12q + 6q^{2} + 82q^{3} + 986q^{4} + 1026q^{5} + 309q^{6} + 4116q^{7} + 228q^{8} + 10902q^{9} + 6668q^{10} + 12168q^{11} - 183q^{12} + 26364q^{13} + 2058q^{14} - 28790q^{15} + 85914q^{16} + 82710q^{17} - 44965q^{18} - 10302q^{19} + 141318q^{20} + 28126q^{21} - 97457q^{22} + 98376q^{23} - 519981q^{24} + 272736q^{25} + 13182q^{26} + 306652q^{27} + 338198q^{28} + 350592q^{29} + 231528q^{30} + 55092q^{31} + 114420q^{32} + 609912q^{33} + 812002q^{34} + 351918q^{35} + 1472143q^{36} + 376310q^{37} + 2825424q^{38} + 180154q^{39} + 2169290q^{40} + 1387272q^{41} + 105987q^{42} + 568708q^{43} + 3392031q^{44} + 3556226q^{45} - 1736829q^{46} + 1359444q^{47} + 4151249q^{48} + 1411788q^{49} + 3983712q^{50} + 2709260q^{51} + 2166242q^{52} + 2061780q^{53} + 2196651q^{54} - 2112846q^{55} + 78204q^{56} + 2359902q^{57} + 670268q^{58} + 395964q^{59} - 1052376q^{60} + 444006q^{61} + 2854353q^{62} + 3739386q^{63} + 12026858q^{64} + 2254122q^{65} - 4605681q^{66} - 3094010q^{67} + 4668954q^{68} + 3839892q^{69} + 2287124q^{70} + 5694366q^{71} - 9780585q^{72} + 7052346q^{73} - 4436259q^{74} - 16288696q^{75} - 3051830q^{76} + 4173624q^{77} + 678873q^{78} + 4304160q^{79} + 3807018q^{80} - 6689556q^{81} - 4733665q^{82} + 2704554q^{83} - 62769q^{84} + 9301878q^{85} + 1510998q^{86} + 16231802q^{87} - 70453923q^{88} - 10986042q^{89} - 12851300q^{90} + 9042852q^{91} - 16505451q^{92} - 47230934q^{93} - 24306151q^{94} - 21839424q^{95} - 86512741q^{96} - 24462382q^{97} + 705894q^{98} + 11555078q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + 186018392 x^{5} + 10752389517 x^{4} + 491049966 x^{3} - 344602049215 x^{2} - 835334324470 x + 59402280000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 209 \)
\(\beta_{3}\)\(=\)\((\)\(116429657912251546397 \nu^{11} - 1138016347474817517247 \nu^{10} - 139797096343619060307636 \nu^{9} + 1178692702718681078134506 \nu^{8} + 60895066450672328807569768 \nu^{7} - 432682308525274351930308880 \nu^{6} - 11698074756976265164966939470 \nu^{5} + 66349192792533587939476333054 \nu^{4} + 951549587174881606995743281219 \nu^{3} - 3683784470633856804373932985433 \nu^{2} - 23436379137177392576984407981510 \nu + 1679148611865916382256857279040\)\()/ \)\(25\!\cdots\!80\)\( \)
\(\beta_{4}\)\(=\)\((\)\(141031799835156954133 \nu^{11} - 1321368208330912848743 \nu^{10} - 170525503250158274999604 \nu^{9} + 1376189494391125315953594 \nu^{8} + 74888085287797374072248072 \nu^{7} - 510859156602059802197533040 \nu^{6} - 14512440979280355872025645630 \nu^{5} + 79911352078036297609303854446 \nu^{4} + 1187070934316815045629774870251 \nu^{3} - 4571284511998824301781850392977 \nu^{2} - 28939668772391818786596548000150 \nu + 3446270623281713659494357549120\)\()/ \)\(25\!\cdots\!80\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-672297715825811331659 \nu^{11} + 7201001136259246898329 \nu^{10} + 806513670413135042019372 \nu^{9} - 7542437021495047512502182 \nu^{8} - 351936065708782709495083096 \nu^{7} + 2799096589087306991508056560 \nu^{6} + 68087284560539008509126878370 \nu^{5} - 433045831613775202229351305138 \nu^{4} - 5644056424542822771207113820373 \nu^{3} + 24048047849053299449847971927951 \nu^{2} + 145824718509625566957099538645930 \nu - 9168059273764298644104549965760\)\()/ \)\(25\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-444215779571142725249 \nu^{11} + 5757076556160489338539 \nu^{10} + 532198440675710379396612 \nu^{9} - 6171321615378891390862002 \nu^{8} - 233456620972939522657384456 \nu^{7} + 2343747323291820081413066320 \nu^{6} + 46003512553343583784819821990 \nu^{5} - 370139735993344499256041019478 \nu^{4} - 3991627290446877087145103977183 \nu^{3} + 20745653678817745884840227142221 \nu^{2} + 114045612110390167858989308805310 \nu - 11995431178475375449376816873280\)\()/ \)\(12\!\cdots\!40\)\( \)
\(\beta_{7}\)\(=\)\((\)\(45868994553761432239 \nu^{11} - 469732263709633455509 \nu^{10} - 54989797161806797472892 \nu^{9} + 489020560147228722315822 \nu^{8} + 23928732437573602807670936 \nu^{7} - 180343773913668276713264720 \nu^{6} - 4597163497409732536978558170 \nu^{5} + 27744496546709916819675850058 \nu^{4} + 375001404612388190581224707153 \nu^{3} - 1540810224746301919403569695571 \nu^{2} - 9314994513411804651098692684610 \nu + 1048418219356569337167844866240\)\()/ \)\(93\!\cdots\!40\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-235934583564866268761 \nu^{11} + 2298951677556646950931 \nu^{10} + 283435243550555522451768 \nu^{9} - 2382072504396417685996398 \nu^{8} - 123539260255985288273224324 \nu^{7} + 875136723060146885144020780 \nu^{6} + 23747945291740541668316217210 \nu^{5} - 134393655495755014587681106282 \nu^{4} - 1932224301429983448516094511707 \nu^{3} + 7477561322500770210009917648729 \nu^{2} + 47453870473811818471895339682310 \nu - 3479949461714552153204912293440\)\()/ \)\(31\!\cdots\!60\)\( \)
\(\beta_{9}\)\(=\)\((\)\(257443220169661359413 \nu^{11} - 2625872673146483638435 \nu^{10} - 309088744126651132111836 \nu^{9} + 2736443413617789664855170 \nu^{8} + 134793767492426742679126360 \nu^{7} - 1010754805819621938004777792 \nu^{6} - 25989152392685012223846362958 \nu^{5} + 155842610229818604650945057350 \nu^{4} + 2133923567171396736435749146939 \nu^{3} - 8666771897379350497491877262117 \nu^{2} - 53820802387928806951216943909278 \nu + 3937168746418691106708027855168\)\()/ \)\(12\!\cdots\!44\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-105097028809302622289 \nu^{11} + 1091928376244149378735 \nu^{10} + 126396278362775742892884 \nu^{9} - 1143559512197128739100690 \nu^{8} - 55287477018474159628548928 \nu^{7} + 424956174241481052496767688 \nu^{6} + 10714468327424970704488511406 \nu^{5} - 65979650366847306993437667382 \nu^{4} - 887222437218943234060039931767 \nu^{3} + 3688010565269392815003437362961 \nu^{2} + 22703000562199630555553247973894 \nu - 1404670124409058383656692565568\)\()/ \)\(42\!\cdots\!48\)\( \)
\(\beta_{11}\)\(=\)\((\)\(3239997280258850960593 \nu^{11} - 33202403758684091513123 \nu^{10} - 3896559705367117031102244 \nu^{9} + 34694386975894305181994514 \nu^{8} + 1703570694832613150614662392 \nu^{7} - 12861517571267114511289085120 \nu^{6} - 329657296467709068917868022230 \nu^{5} + 1992415007509288983913547114486 \nu^{4} + 27197561210274725383179521872991 \nu^{3} - 111315596224730355848768113995877 \nu^{2} - 689711203535751280755640835858990 \nu + 47423945977332090939100555448640\)\()/ \)\(12\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 209\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{4} + 15 \beta_{3} + 3 \beta_{2} + 345 \beta_{1} + 312\)
\(\nu^{4}\)\(=\)\(4 \beta_{11} + 4 \beta_{10} - \beta_{9} + 6 \beta_{8} - \beta_{7} - 5 \beta_{6} + 13 \beta_{5} + 5 \beta_{4} + 177 \beta_{3} + 500 \beta_{2} + 1371 \beta_{1} + 71412\)
\(\nu^{5}\)\(=\)\(30 \beta_{11} + 6 \beta_{10} - 60 \beta_{9} + 636 \beta_{8} + 44 \beta_{7} - 32 \beta_{6} + 102 \beta_{5} + 444 \beta_{4} + 10958 \beta_{3} + 2840 \beta_{2} + 138759 \beta_{1} + 229516\)
\(\nu^{6}\)\(=\)\(3024 \beta_{11} + 2494 \beta_{10} - 1240 \beta_{9} + 5248 \beta_{8} - 214 \beta_{7} - 3574 \beta_{6} + 11534 \beta_{5} - 1600 \beta_{4} + 150228 \beta_{3} + 237831 \beta_{2} + 859638 \beta_{1} + 28454111\)
\(\nu^{7}\)\(=\)\(30290 \beta_{11} + 8156 \beta_{10} - 57708 \beta_{9} + 337579 \beta_{8} + 31150 \beta_{7} - 37726 \beta_{6} + 111982 \beta_{5} + 146303 \beta_{4} + 6630709 \beta_{3} + 1943243 \beta_{2} + 60576833 \beta_{1} + 151525518\)
\(\nu^{8}\)\(=\)\(1739412 \beta_{11} + 1232150 \beta_{10} - 962609 \beta_{9} + 3561774 \beta_{8} + 125333 \beta_{7} - 2084039 \beta_{6} + 7571523 \beta_{5} - 2609795 \beta_{4} + 99880753 \beta_{3} + 114036086 \beta_{2} + 513961203 \beta_{1} + 12316639262\)
\(\nu^{9}\)\(=\)\(20964960 \beta_{11} + 6343594 \beta_{10} - 39535284 \beta_{9} + 172280246 \beta_{8} + 17399558 \beta_{7} - 29329626 \beta_{6} + 85364640 \beta_{5} + 38780526 \beta_{4} + 3755062144 \beta_{3} + 1201115812 \beta_{2} + 27824748107 \beta_{1} + 93516704558\)
\(\nu^{10}\)\(=\)\(920147760 \beta_{11} + 573136640 \beta_{10} - 634693664 \beta_{9} + 2189379316 \beta_{8} + 166855192 \beta_{7} - 1151225688 \beta_{6} + 4445406756 \beta_{5} - 1917706276 \beta_{4} + 60822683000 \beta_{3} + 55616776629 \beta_{2} + 298263665558 \beta_{1} + 5613426256161\)
\(\nu^{11}\)\(=\)\(12687632436 \beta_{11} + 4070868840 \beta_{10} - 23812818640 \beta_{9} + 87449732997 \beta_{8} + 9157045524 \beta_{7} - 19286669780 \beta_{6} + 56431034028 \beta_{5} + 5362870693 \beta_{4} + 2059885657403 \beta_{3} + 706351191563 \beta_{2} + 13238321430409 \beta_{1} + 55368631574220\)

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
23.0670
19.0596
15.8833
10.7733
10.5927
0.0691404
−4.06937
−4.08138
−12.7078
−13.1706
−18.9614
−20.4545
−22.0670 71.8770 358.952 25.2300 −1586.11 343.000 −5096.40 2979.31 −556.750
1.2 −18.0596 −64.0898 198.149 13.5880 1157.44 343.000 −1266.87 1920.50 −245.394
1.3 −14.8833 −5.24717 93.5131 −195.689 78.0952 343.000 513.280 −2159.47 2912.49
1.4 −9.77328 −73.6575 −32.4830 542.246 719.876 343.000 1568.45 3238.43 −5299.52
1.5 −9.59274 50.3591 −35.9794 255.731 −483.082 343.000 1573.01 349.039 −2453.16
1.6 0.930860 23.4478 −127.134 −494.625 21.8266 343.000 −237.493 −1637.20 −460.427
1.7 5.06937 91.5295 −102.301 −15.8892 463.997 343.000 −1167.48 6190.65 −80.5484
1.8 5.08138 −25.2372 −102.180 250.262 −128.240 343.000 −1169.63 −1550.08 1271.68
1.9 13.7078 50.2321 59.9043 544.453 688.573 343.000 −933.443 336.267 7463.26
1.10 14.1706 −35.5734 72.8048 −268.857 −504.095 343.000 −782.147 −921.532 −3809.85
1.11 19.9614 56.3144 270.459 1.53059 1124.12 343.000 2843.68 984.313 30.5528
1.12 21.4545 −57.9549 332.295 368.020 −1243.39 343.000 4383.05 1171.77 7895.68
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(-1\)

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 91.8.a.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.a.e 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(14\!\cdots\!24\)\( T_{2} - \)\(11\!\cdots\!76\)\( \)">\(T_{2}^{12} - \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1109222885376 + 1479835713024 T - 278531100864 T^{2} - 42990372480 T^{3} + 9905539840 T^{4} + 520830672 T^{5} - 113159964 T^{6} - 2852052 T^{7} + 561506 T^{8} + 6942 T^{9} - 1243 T^{10} - 6 T^{11} + T^{12} \)
$3$ \( 28321761301743651840 + 5090548910858118144 T - 146387880398635200 T^{2} - 15850307902561632 T^{3} + 258366974437080 T^{4} + 15663977057208 T^{5} - 217125703708 T^{6} - 6529375952 T^{7} + 85197562 T^{8} + 1209318 T^{9} - 15211 T^{10} - 82 T^{11} + T^{12} \)
$5$ \( \)\(15\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( T + \)\(46\!\cdots\!00\)\( T^{2} + \)\(43\!\cdots\!00\)\( T^{3} - \)\(19\!\cdots\!75\)\( T^{4} + 459814742659523550 T^{5} + 7738779605134480 T^{6} - 27550420035510 T^{7} - 68834709894 T^{8} + 355794954 T^{9} - 78780 T^{10} - 1026 T^{11} + T^{12} \)
$7$ \( ( -343 + T )^{12} \)
$11$ \( -\)\(14\!\cdots\!44\)\( + \)\(18\!\cdots\!16\)\( T + \)\(10\!\cdots\!88\)\( T^{2} - \)\(52\!\cdots\!24\)\( T^{3} - \)\(47\!\cdots\!28\)\( T^{4} + \)\(16\!\cdots\!84\)\( T^{5} + \)\(45\!\cdots\!08\)\( T^{6} - 18182623694426127432 T^{7} + 224922654805818 T^{8} + 803286489774 T^{9} - 40936193 T^{10} - 12168 T^{11} + T^{12} \)
$13$ \( ( -2197 + T )^{12} \)
$17$ \( -\)\(63\!\cdots\!24\)\( + \)\(55\!\cdots\!12\)\( T - \)\(17\!\cdots\!48\)\( T^{2} + \)\(20\!\cdots\!28\)\( T^{3} - \)\(22\!\cdots\!80\)\( T^{4} - \)\(16\!\cdots\!48\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(30\!\cdots\!76\)\( T^{7} - 3030957981722111140 T^{8} + 91398519416112 T^{9} + 985416278 T^{10} - 82710 T^{11} + T^{12} \)
$19$ \( \)\(62\!\cdots\!00\)\( + \)\(46\!\cdots\!00\)\( T - \)\(33\!\cdots\!80\)\( T^{2} - \)\(76\!\cdots\!52\)\( T^{3} + \)\(48\!\cdots\!33\)\( T^{4} - \)\(64\!\cdots\!14\)\( T^{5} - \)\(13\!\cdots\!96\)\( T^{6} + \)\(15\!\cdots\!74\)\( T^{7} + 15448832558807008998 T^{8} - 78742659076206 T^{9} - 6788478584 T^{10} + 10302 T^{11} + T^{12} \)
$23$ \( -\)\(21\!\cdots\!80\)\( + \)\(32\!\cdots\!36\)\( T + \)\(14\!\cdots\!31\)\( T^{2} - \)\(18\!\cdots\!44\)\( T^{3} - \)\(23\!\cdots\!19\)\( T^{4} + \)\(28\!\cdots\!84\)\( T^{5} - \)\(93\!\cdots\!58\)\( T^{6} - \)\(12\!\cdots\!76\)\( T^{7} + 92718358515826084330 T^{8} + 1931036373458376 T^{9} - 18272191805 T^{10} - 98376 T^{11} + T^{12} \)
$29$ \( \)\(20\!\cdots\!00\)\( - \)\(25\!\cdots\!80\)\( T - \)\(12\!\cdots\!44\)\( T^{2} - \)\(37\!\cdots\!68\)\( T^{3} + \)\(41\!\cdots\!01\)\( T^{4} + \)\(39\!\cdots\!60\)\( T^{5} - \)\(38\!\cdots\!60\)\( T^{6} - \)\(47\!\cdots\!12\)\( T^{7} + \)\(16\!\cdots\!42\)\( T^{8} + 21216283931518596 T^{9} - 51985825316 T^{10} - 350592 T^{11} + T^{12} \)
$31$ \( -\)\(45\!\cdots\!28\)\( + \)\(36\!\cdots\!60\)\( T - \)\(43\!\cdots\!99\)\( T^{2} - \)\(13\!\cdots\!12\)\( T^{3} + \)\(21\!\cdots\!37\)\( T^{4} + \)\(16\!\cdots\!44\)\( T^{5} - \)\(28\!\cdots\!74\)\( T^{6} - \)\(70\!\cdots\!44\)\( T^{7} + \)\(11\!\cdots\!14\)\( T^{8} + 11596057724983208 T^{9} - 189670382283 T^{10} - 55092 T^{11} + T^{12} \)
$37$ \( -\)\(54\!\cdots\!40\)\( + \)\(67\!\cdots\!40\)\( T + \)\(34\!\cdots\!92\)\( T^{2} - \)\(62\!\cdots\!12\)\( T^{3} - \)\(21\!\cdots\!64\)\( T^{4} + \)\(21\!\cdots\!88\)\( T^{5} - \)\(25\!\cdots\!56\)\( T^{6} - \)\(31\!\cdots\!96\)\( T^{7} + \)\(60\!\cdots\!34\)\( T^{8} + 192801515670638506 T^{9} - 452213860951 T^{10} - 376310 T^{11} + T^{12} \)
$41$ \( -\)\(22\!\cdots\!20\)\( - \)\(27\!\cdots\!40\)\( T + \)\(57\!\cdots\!36\)\( T^{2} + \)\(54\!\cdots\!56\)\( T^{3} - \)\(58\!\cdots\!20\)\( T^{4} - \)\(23\!\cdots\!64\)\( T^{5} + \)\(27\!\cdots\!72\)\( T^{6} - \)\(29\!\cdots\!16\)\( T^{7} - \)\(22\!\cdots\!80\)\( T^{8} + 539846216924170416 T^{9} + 161094817451 T^{10} - 1387272 T^{11} + T^{12} \)
$43$ \( -\)\(19\!\cdots\!32\)\( - \)\(14\!\cdots\!12\)\( T + \)\(33\!\cdots\!24\)\( T^{2} - \)\(16\!\cdots\!24\)\( T^{3} - \)\(39\!\cdots\!51\)\( T^{4} + \)\(43\!\cdots\!00\)\( T^{5} - \)\(25\!\cdots\!20\)\( T^{6} - \)\(32\!\cdots\!52\)\( T^{7} + \)\(42\!\cdots\!06\)\( T^{8} + 810934633473502788 T^{9} - 1282369377300 T^{10} - 568708 T^{11} + T^{12} \)
$47$ \( -\)\(42\!\cdots\!72\)\( + \)\(82\!\cdots\!40\)\( T + \)\(14\!\cdots\!93\)\( T^{2} - \)\(35\!\cdots\!56\)\( T^{3} - \)\(13\!\cdots\!11\)\( T^{4} + \)\(42\!\cdots\!44\)\( T^{5} + \)\(36\!\cdots\!82\)\( T^{6} - \)\(18\!\cdots\!84\)\( T^{7} + \)\(40\!\cdots\!98\)\( T^{8} + 2849170083513028200 T^{9} - 1658302898991 T^{10} - 1359444 T^{11} + T^{12} \)
$53$ \( -\)\(49\!\cdots\!00\)\( + \)\(73\!\cdots\!00\)\( T - \)\(10\!\cdots\!20\)\( T^{2} - \)\(71\!\cdots\!32\)\( T^{3} + \)\(61\!\cdots\!17\)\( T^{4} + \)\(20\!\cdots\!80\)\( T^{5} - \)\(12\!\cdots\!84\)\( T^{6} - \)\(24\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!38\)\( T^{8} + 11948977228784070828 T^{9} - 5909280052976 T^{10} - 2061780 T^{11} + T^{12} \)
$59$ \( \)\(96\!\cdots\!40\)\( + \)\(48\!\cdots\!76\)\( T - \)\(27\!\cdots\!72\)\( T^{2} - \)\(67\!\cdots\!40\)\( T^{3} + \)\(92\!\cdots\!16\)\( T^{4} + \)\(10\!\cdots\!60\)\( T^{5} - \)\(40\!\cdots\!28\)\( T^{6} - \)\(29\!\cdots\!20\)\( T^{7} + \)\(45\!\cdots\!20\)\( T^{8} + 8184821701447735248 T^{9} - 12943066856352 T^{10} - 395964 T^{11} + T^{12} \)
$61$ \( -\)\(35\!\cdots\!00\)\( - \)\(29\!\cdots\!00\)\( T - \)\(49\!\cdots\!60\)\( T^{2} + \)\(25\!\cdots\!24\)\( T^{3} + \)\(88\!\cdots\!24\)\( T^{4} - \)\(39\!\cdots\!80\)\( T^{5} - \)\(54\!\cdots\!88\)\( T^{6} - \)\(34\!\cdots\!32\)\( T^{7} + \)\(15\!\cdots\!16\)\( T^{8} + 7538711362096875788 T^{9} - 20450832316447 T^{10} - 444006 T^{11} + T^{12} \)
$67$ \( -\)\(14\!\cdots\!40\)\( - \)\(29\!\cdots\!60\)\( T + \)\(51\!\cdots\!52\)\( T^{2} + \)\(38\!\cdots\!12\)\( T^{3} + \)\(91\!\cdots\!92\)\( T^{4} - \)\(14\!\cdots\!80\)\( T^{5} - \)\(43\!\cdots\!88\)\( T^{6} + \)\(21\!\cdots\!40\)\( T^{7} + \)\(68\!\cdots\!84\)\( T^{8} - \)\(13\!\cdots\!88\)\( T^{9} - 44329114442307 T^{10} + 3094010 T^{11} + T^{12} \)
$71$ \( -\)\(16\!\cdots\!28\)\( + \)\(23\!\cdots\!36\)\( T + \)\(34\!\cdots\!28\)\( T^{2} - \)\(19\!\cdots\!08\)\( T^{3} - \)\(13\!\cdots\!96\)\( T^{4} + \)\(11\!\cdots\!12\)\( T^{5} + \)\(52\!\cdots\!96\)\( T^{6} - \)\(61\!\cdots\!32\)\( T^{7} - \)\(22\!\cdots\!48\)\( T^{8} + \)\(10\!\cdots\!56\)\( T^{9} - 11450317136750 T^{10} - 5694366 T^{11} + T^{12} \)
$73$ \( -\)\(57\!\cdots\!88\)\( + \)\(26\!\cdots\!28\)\( T - \)\(23\!\cdots\!61\)\( T^{2} - \)\(50\!\cdots\!90\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} + \)\(25\!\cdots\!48\)\( T^{5} - \)\(29\!\cdots\!78\)\( T^{6} - \)\(44\!\cdots\!36\)\( T^{7} + \)\(47\!\cdots\!70\)\( T^{8} + \)\(31\!\cdots\!12\)\( T^{9} - 38765498692829 T^{10} - 7052346 T^{11} + T^{12} \)
$79$ \( \)\(98\!\cdots\!40\)\( + \)\(23\!\cdots\!56\)\( T - \)\(13\!\cdots\!93\)\( T^{2} - \)\(38\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!65\)\( T^{4} + \)\(13\!\cdots\!12\)\( T^{5} - \)\(40\!\cdots\!38\)\( T^{6} - \)\(12\!\cdots\!56\)\( T^{7} + \)\(35\!\cdots\!14\)\( T^{8} + \)\(43\!\cdots\!60\)\( T^{9} - 107659935730785 T^{10} - 4304160 T^{11} + T^{12} \)
$83$ \( \)\(12\!\cdots\!40\)\( - \)\(19\!\cdots\!64\)\( T - \)\(42\!\cdots\!92\)\( T^{2} + \)\(60\!\cdots\!72\)\( T^{3} + \)\(37\!\cdots\!29\)\( T^{4} + \)\(29\!\cdots\!46\)\( T^{5} - \)\(36\!\cdots\!68\)\( T^{6} - \)\(21\!\cdots\!78\)\( T^{7} + \)\(13\!\cdots\!94\)\( T^{8} + \)\(43\!\cdots\!34\)\( T^{9} - 192982941065108 T^{10} - 2704554 T^{11} + T^{12} \)
$89$ \( -\)\(86\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( T + \)\(53\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} - \)\(75\!\cdots\!67\)\( T^{4} - \)\(28\!\cdots\!54\)\( T^{5} + \)\(20\!\cdots\!68\)\( T^{6} + \)\(14\!\cdots\!34\)\( T^{7} + \)\(42\!\cdots\!14\)\( T^{8} - \)\(25\!\cdots\!18\)\( T^{9} - 182708471450276 T^{10} + 10986042 T^{11} + T^{12} \)
$97$ \( -\)\(41\!\cdots\!60\)\( - \)\(77\!\cdots\!92\)\( T + \)\(63\!\cdots\!91\)\( T^{2} + \)\(18\!\cdots\!62\)\( T^{3} - \)\(11\!\cdots\!63\)\( T^{4} - \)\(20\!\cdots\!00\)\( T^{5} + \)\(70\!\cdots\!94\)\( T^{6} + \)\(12\!\cdots\!76\)\( T^{7} - \)\(14\!\cdots\!66\)\( T^{8} - \)\(30\!\cdots\!20\)\( T^{9} + 26006677670011 T^{10} + 24462382 T^{11} + T^{12} \)
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