Properties

Label 43.10.a.b
Level 43
Weight 10
Character orbit 43.a
Self dual yes
Analytic conductor 22.147
Analytic rank 0
Dimension 17
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.1465409550\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{5} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta_{1} ) q^{2} + ( 10 - \beta_{3} ) q^{3} + ( 267 - 4 \beta_{1} + \beta_{2} ) q^{4} + ( 237 - \beta_{1} + \beta_{3} - \beta_{9} ) q^{5} + ( 350 - 25 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{9} + \beta_{10} ) q^{6} + ( -9 + 31 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + \beta_{5} ) q^{7} + ( 2451 - 187 \beta_{1} + 7 \beta_{2} - 28 \beta_{3} - \beta_{7} - \beta_{9} ) q^{8} + ( 7953 - 7 \beta_{1} + 8 \beta_{2} - 18 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{14} + \beta_{15} ) q^{9} +O(q^{10})\) \( q + ( 3 - \beta_{1} ) q^{2} + ( 10 - \beta_{3} ) q^{3} + ( 267 - 4 \beta_{1} + \beta_{2} ) q^{4} + ( 237 - \beta_{1} + \beta_{3} - \beta_{9} ) q^{5} + ( 350 - 25 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{9} + \beta_{10} ) q^{6} + ( -9 + 31 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + \beta_{5} ) q^{7} + ( 2451 - 187 \beta_{1} + 7 \beta_{2} - 28 \beta_{3} - \beta_{7} - \beta_{9} ) q^{8} + ( 7953 - 7 \beta_{1} + 8 \beta_{2} - 18 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{14} + \beta_{15} ) q^{9} + ( 1473 - 405 \beta_{1} + 5 \beta_{2} - 58 \beta_{3} - 4 \beta_{5} + 3 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 3 \beta_{14} - 3 \beta_{15} ) q^{10} + ( 4687 - 359 \beta_{1} + 13 \beta_{2} - 68 \beta_{3} - \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 11 \beta_{9} - 7 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} ) q^{11} + ( 16211 - 1268 \beta_{1} + 12 \beta_{2} - 348 \beta_{3} + 5 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 10 \beta_{9} + 4 \beta_{10} + 4 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} + 3 \beta_{14} + 2 \beta_{15} - 5 \beta_{16} ) q^{12} + ( 6838 - 589 \beta_{1} - 19 \beta_{2} - 14 \beta_{3} - 7 \beta_{4} + 8 \beta_{5} + 2 \beta_{6} - 3 \beta_{8} + 9 \beta_{9} - 6 \beta_{10} - \beta_{11} + 2 \beta_{13} - 8 \beta_{14} + 2 \beta_{15} + 7 \beta_{16} ) q^{13} + ( -21945 - 1041 \beta_{1} - 41 \beta_{2} - 47 \beta_{3} - 2 \beta_{4} + 13 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} - 7 \beta_{8} + 36 \beta_{9} + 6 \beta_{10} + 8 \beta_{11} + \beta_{12} + 7 \beta_{13} - \beta_{14} - 6 \beta_{15} + \beta_{16} ) q^{14} + ( -14680 - 2195 \beta_{1} - 135 \beta_{2} - 474 \beta_{3} + 13 \beta_{4} + 14 \beta_{5} + 3 \beta_{6} + \beta_{7} + 5 \beta_{8} - 12 \beta_{9} + 5 \beta_{10} - 15 \beta_{11} + 2 \beta_{12} - 3 \beta_{13} + 8 \beta_{14} + 8 \beta_{15} - 6 \beta_{16} ) q^{15} + ( 25049 - 4307 \beta_{1} + 23 \beta_{2} - 115 \beta_{3} - 13 \beta_{4} - 10 \beta_{6} - 8 \beta_{7} + 3 \beta_{8} + \beta_{9} + 20 \beta_{10} + 5 \beta_{11} - 7 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} - 2 \beta_{16} ) q^{16} + ( 43093 - 2153 \beta_{1} - 64 \beta_{2} + 414 \beta_{3} - 6 \beta_{4} - 21 \beta_{5} - 19 \beta_{6} + 17 \beta_{7} - 5 \beta_{8} - 10 \beta_{9} + 5 \beta_{10} + 14 \beta_{11} - 5 \beta_{13} - 7 \beta_{14} - 12 \beta_{15} + 5 \beta_{16} ) q^{17} + ( 35888 - 11644 \beta_{1} - 206 \beta_{2} + 256 \beta_{3} + 36 \beta_{4} - 40 \beta_{5} + 12 \beta_{6} + 9 \beta_{7} + 42 \beta_{8} - 50 \beta_{9} + 64 \beta_{10} - 26 \beta_{11} - 16 \beta_{13} + \beta_{14} - 12 \beta_{15} + 5 \beta_{16} ) q^{18} + ( 32128 - 1424 \beta_{1} - 242 \beta_{2} + 799 \beta_{3} - 5 \beta_{4} + 22 \beta_{5} + 27 \beta_{6} + \beta_{7} - 28 \beta_{8} - 62 \beta_{9} - 27 \beta_{10} + 3 \beta_{11} - 18 \beta_{12} + 5 \beta_{13} + 6 \beta_{14} + 7 \beta_{15} - 28 \beta_{16} ) q^{19} + ( 214976 - 4745 \beta_{1} + 243 \beta_{2} + 1673 \beta_{3} - 48 \beta_{4} - 16 \beta_{5} - 2 \beta_{6} - 12 \beta_{7} - 15 \beta_{8} - 257 \beta_{9} - 12 \beta_{10} - 19 \beta_{11} + 19 \beta_{12} + 12 \beta_{13} + 35 \beta_{14} + 13 \beta_{15} + 25 \beta_{16} ) q^{20} + ( 184413 - 723 \beta_{1} - 451 \beta_{2} + 1921 \beta_{3} + 11 \beta_{4} - 35 \beta_{5} - 36 \beta_{6} + 38 \beta_{7} + 13 \beta_{8} - 47 \beta_{9} - 94 \beta_{10} + 108 \beta_{11} + 40 \beta_{12} + 44 \beta_{13} - 7 \beta_{14} - 7 \beta_{15} + 12 \beta_{16} ) q^{21} + ( 311671 - 9194 \beta_{1} + 657 \beta_{2} + 2541 \beta_{3} + 37 \beta_{5} + 24 \beta_{6} - \beta_{7} - 75 \beta_{8} + 119 \beta_{9} - 61 \beta_{10} - 102 \beta_{11} - 31 \beta_{12} + 65 \beta_{13} - 68 \beta_{14} - 30 \beta_{15} + 22 \beta_{16} ) q^{22} + ( 329638 - 10002 \beta_{1} - 214 \beta_{2} + 78 \beta_{3} - 8 \beta_{4} + 31 \beta_{5} + 55 \beta_{6} - 55 \beta_{7} + 95 \beta_{8} - 93 \beta_{9} - 31 \beta_{10} - 8 \beta_{11} - 8 \beta_{12} - 81 \beta_{13} + 3 \beta_{14} + 30 \beta_{15} - 57 \beta_{16} ) q^{23} + ( 956686 - 14047 \beta_{1} + 1246 \beta_{2} + 764 \beta_{3} + 38 \beta_{4} - 93 \beta_{5} - 30 \beta_{6} - 20 \beta_{7} - 29 \beta_{8} + 141 \beta_{9} + 133 \beta_{10} + 66 \beta_{11} - 3 \beta_{12} - 143 \beta_{13} - 42 \beta_{14} + 122 \beta_{15} - 29 \beta_{16} ) q^{24} + ( 638058 + 9120 \beta_{1} + 442 \beta_{2} + 1692 \beta_{3} + 89 \beta_{4} - 61 \beta_{5} - 91 \beta_{6} - 51 \beta_{7} - 113 \beta_{8} - 69 \beta_{9} - 83 \beta_{10} + 69 \beta_{11} + 66 \beta_{12} - 61 \beta_{13} + 47 \beta_{14} - 76 \beta_{15} + 116 \beta_{16} ) q^{25} + ( 471633 + 152 \beta_{1} + 1193 \beta_{2} + 2497 \beta_{3} - 72 \beta_{4} + 177 \beta_{5} + 8 \beta_{6} + 35 \beta_{7} + 133 \beta_{8} + 835 \beta_{9} - 73 \beta_{10} - 34 \beta_{11} - 71 \beta_{12} + 117 \beta_{13} - 48 \beta_{14} - 74 \beta_{15} - 2 \beta_{16} ) q^{26} + ( 487964 + 22331 \beta_{1} + 576 \beta_{2} - 8792 \beta_{3} - 15 \beta_{4} + 200 \beta_{5} - 36 \beta_{6} + 4 \beta_{7} + 35 \beta_{8} - 38 \beta_{9} - 146 \beta_{10} - 382 \beta_{11} - 34 \beta_{12} + 122 \beta_{13} + 132 \beta_{14} + 36 \beta_{15} - 222 \beta_{16} ) q^{27} + ( 733135 + 25935 \beta_{1} + 694 \beta_{2} + 4152 \beta_{3} - 121 \beta_{4} + 291 \beta_{5} + 38 \beta_{6} - 36 \beta_{7} - 27 \beta_{8} + 1001 \beta_{9} - 95 \beta_{10} + 314 \beta_{11} - 133 \beta_{12} + 143 \beta_{13} - 23 \beta_{14} + 80 \beta_{15} + 28 \beta_{16} ) q^{28} + ( 476500 + 37985 \beta_{1} - 1480 \beta_{2} - 1172 \beta_{3} - 68 \beta_{4} - 61 \beta_{5} - \beta_{6} + 19 \beta_{7} + 272 \beta_{8} - 294 \beta_{9} + 59 \beta_{10} + 51 \beta_{11} + 118 \beta_{12} + 25 \beta_{13} + 107 \beta_{14} - 217 \beta_{15} + 212 \beta_{16} ) q^{29} + ( 1789604 + 63683 \beta_{1} + 1717 \beta_{2} + 627 \beta_{3} + 236 \beta_{4} - 389 \beta_{5} + 150 \beta_{6} + 209 \beta_{7} - 427 \beta_{8} - 169 \beta_{9} + 695 \beta_{10} - 6 \beta_{11} + 183 \beta_{12} - 247 \beta_{13} + 114 \beta_{14} - 10 \beta_{15} - 6 \beta_{16} ) q^{30} + ( 763229 + 23852 \beta_{1} - 2906 \beta_{2} - 10987 \beta_{3} + 136 \beta_{4} + 2 \beta_{5} + 290 \beta_{6} - 182 \beta_{7} + 49 \beta_{8} - 594 \beta_{9} + 330 \beta_{10} - 301 \beta_{11} - 146 \beta_{12} - 166 \beta_{13} - 64 \beta_{14} + 7 \beta_{15} - 15 \beta_{16} ) q^{31} + ( 2173929 + 56597 \beta_{1} + 1681 \beta_{2} - 3785 \beta_{3} - \beta_{4} - 274 \beta_{5} - 242 \beta_{6} + 214 \beta_{7} + 47 \beta_{8} + 607 \beta_{9} + 140 \beta_{10} + 205 \beta_{11} + 159 \beta_{12} - 32 \beta_{13} - 20 \beta_{14} - 57 \beta_{15} - 56 \beta_{16} ) q^{32} + ( 2103241 + 85375 \beta_{1} - 3715 \beta_{2} - 7448 \beta_{3} - 402 \beta_{4} - 469 \beta_{5} - 60 \beta_{6} + 176 \beta_{7} - 63 \beta_{8} - 716 \beta_{9} + 290 \beta_{10} + 546 \beta_{11} - 34 \beta_{12} + 106 \beta_{13} - 484 \beta_{14} + 232 \beta_{15} + 118 \beta_{16} ) q^{33} + ( 1648995 - 11663 \beta_{1} - 1468 \beta_{2} - 8743 \beta_{3} - 98 \beta_{4} + 173 \beta_{5} - 84 \beta_{6} + 56 \beta_{7} + 514 \beta_{8} + 259 \beta_{9} - 483 \beta_{10} - 507 \beta_{11} - 198 \beta_{12} + 421 \beta_{13} - 348 \beta_{14} - 25 \beta_{15} - 161 \beta_{16} ) q^{34} + ( 1023377 + 134967 \beta_{1} - 5548 \beta_{2} - 11991 \beta_{3} + 259 \beta_{4} + 63 \beta_{5} - 146 \beta_{6} - 228 \beta_{7} - 190 \beta_{8} - 1041 \beta_{9} - 170 \beta_{10} + 514 \beta_{11} + 178 \beta_{12} - 20 \beta_{13} + 455 \beta_{14} - 101 \beta_{15} + 360 \beta_{16} ) q^{35} + ( 4939204 + 58814 \beta_{1} + 5032 \beta_{2} - 20323 \beta_{3} + 107 \beta_{4} - 553 \beta_{5} - 516 \beta_{6} + 60 \beta_{7} - 670 \beta_{8} - 1624 \beta_{9} + 121 \beta_{10} - 573 \beta_{11} + 238 \beta_{12} - 393 \beta_{13} + 80 \beta_{14} + 497 \beta_{15} - 499 \beta_{16} ) q^{36} + ( 2738773 + 58884 \beta_{1} - 3851 \beta_{2} - 24734 \beta_{3} - 286 \beta_{4} + 52 \beta_{5} - 91 \beta_{6} - 413 \beta_{7} - 247 \beta_{8} - 1764 \beta_{9} - 615 \beta_{10} - 351 \beta_{11} - 416 \beta_{12} + 39 \beta_{13} - 325 \beta_{14} + 377 \beta_{15} - 286 \beta_{16} ) q^{37} + ( 910416 + 68108 \beta_{1} - 4640 \beta_{2} + 21825 \beta_{3} + 604 \beta_{4} + 567 \beta_{5} + 590 \beta_{6} - 292 \beta_{7} + 266 \beta_{8} - 59 \beta_{9} - 387 \beta_{10} + 661 \beta_{11} + 280 \beta_{12} - 387 \beta_{13} + 894 \beta_{14} - 485 \beta_{15} + 317 \beta_{16} ) q^{38} + ( 496639 + 78155 \beta_{1} - 7342 \beta_{2} - 11692 \beta_{3} - 204 \beta_{4} + 803 \beta_{5} + 220 \beta_{6} - 34 \beta_{7} + 236 \beta_{8} + 1524 \beta_{9} - 6 \beta_{10} + 176 \beta_{11} - 692 \beta_{12} + 442 \beta_{13} - 368 \beta_{14} + 54 \beta_{15} + 130 \beta_{16} ) q^{39} + ( 3093661 - 129410 \beta_{1} - 1465 \beta_{2} + 6661 \beta_{3} + 325 \beta_{4} + 561 \beta_{5} + 324 \beta_{6} - 960 \beta_{7} + 744 \beta_{8} - 4332 \beta_{9} - 1151 \beta_{10} - 1163 \beta_{11} + 294 \beta_{12} - 535 \beta_{13} + 952 \beta_{14} + 95 \beta_{15} + 41 \beta_{16} ) q^{40} + ( 3135109 + 140931 \beta_{1} + 4933 \beta_{2} - 18559 \beta_{3} + 182 \beta_{4} + 529 \beta_{5} + 694 \beta_{6} + 350 \beta_{7} - 864 \beta_{8} + 2145 \beta_{9} - 62 \beta_{10} - 1621 \beta_{11} - 6 \beta_{12} + 18 \beta_{13} - 488 \beta_{14} - 492 \beta_{15} - 895 \beta_{16} ) q^{41} + ( 450605 - 13235 \beta_{1} - 4278 \beta_{2} + 93417 \beta_{3} - 992 \beta_{4} + 307 \beta_{5} + 30 \beta_{6} + 891 \beta_{7} + 792 \beta_{8} + 4763 \beta_{9} - 559 \beta_{10} + 2991 \beta_{11} - 300 \beta_{12} + 1157 \beta_{13} - 551 \beta_{14} - 1201 \beta_{15} + 1324 \beta_{16} ) q^{42} + 3418801 q^{43} + ( 4873652 - 344983 \beta_{1} + 15662 \beta_{2} + 34619 \beta_{3} - 269 \beta_{4} + 1154 \beta_{5} - 1014 \beta_{6} + 382 \beta_{7} + 1501 \beta_{8} + 5847 \beta_{9} - 2384 \beta_{10} - 1325 \beta_{11} - 215 \beta_{12} + 896 \beta_{13} - 718 \beta_{14} - 507 \beta_{15} - 168 \beta_{16} ) q^{44} + ( 7303651 + 75059 \beta_{1} + 10215 \beta_{2} + 11687 \beta_{3} - 852 \beta_{4} - 2595 \beta_{5} - 1074 \beta_{6} + 1768 \beta_{7} - 482 \beta_{8} - 1275 \beta_{9} + 2032 \beta_{10} + 680 \beta_{11} + 1068 \beta_{12} + 442 \beta_{13} - 147 \beta_{14} + 116 \beta_{15} + 694 \beta_{16} ) q^{45} + ( 8686326 - 299255 \beta_{1} + 10265 \beta_{2} + 84999 \beta_{3} - 42 \beta_{4} - 2257 \beta_{5} + 124 \beta_{6} - 604 \beta_{7} - 1617 \beta_{8} - 7889 \beta_{9} + 1439 \beta_{10} + 1614 \beta_{11} + 1321 \beta_{12} - 1949 \beta_{13} + 965 \beta_{14} + 608 \beta_{15} + 205 \beta_{16} ) q^{46} + ( 7226888 + 87773 \beta_{1} + 24496 \beta_{2} - 24337 \beta_{3} + 2041 \beta_{4} - 289 \beta_{5} - 1591 \beta_{6} + 583 \beta_{7} - 2192 \beta_{8} + 7245 \beta_{9} + 1889 \beta_{10} + 1261 \beta_{11} + 720 \beta_{12} - 1199 \beta_{13} + 1752 \beta_{14} + 884 \beta_{15} - 664 \beta_{16} ) q^{47} + ( 5221068 - 792633 \beta_{1} + 10386 \beta_{2} + 42412 \beta_{3} + 180 \beta_{4} + 91 \beta_{5} + 1738 \beta_{6} - 908 \beta_{7} - 761 \beta_{8} + 1661 \beta_{9} + 3295 \beta_{10} - 1456 \beta_{11} - 1397 \beta_{12} - 1203 \beta_{13} - 2024 \beta_{14} + 1848 \beta_{15} - 409 \beta_{16} ) q^{48} + ( 6634532 - 481071 \beta_{1} - 4306 \beta_{2} - 9811 \beta_{3} + 593 \beta_{4} - 3499 \beta_{5} - 554 \beta_{6} - 146 \beta_{7} + 2494 \beta_{8} - 6115 \beta_{9} + 2640 \beta_{10} + 1940 \beta_{11} + 1022 \beta_{12} - 1686 \beta_{13} + 385 \beta_{14} + 1667 \beta_{15} + 84 \beta_{16} ) q^{49} + ( -5538749 - 792844 \beta_{1} + 12253 \beta_{2} + 23916 \beta_{3} - 2454 \beta_{4} + 5220 \beta_{5} + 1440 \beta_{6} - 427 \beta_{7} + 2495 \beta_{8} + 5476 \beta_{9} - 3798 \beta_{10} - 4807 \beta_{11} - 3205 \beta_{12} + 3372 \beta_{13} - 134 \beta_{14} + 1035 \beta_{15} - 3135 \beta_{16} ) q^{50} + ( -10943085 - 339445 \beta_{1} - 6308 \beta_{2} - 74035 \beta_{3} + 2209 \beta_{4} + 2277 \beta_{5} + 1677 \beta_{6} - 91 \beta_{7} + 1444 \beta_{8} + 9950 \beta_{9} + 4201 \beta_{10} - 3345 \beta_{11} - 530 \beta_{12} + 1333 \beta_{13} - 72 \beta_{14} - 1693 \beta_{15} - 52 \beta_{16} ) q^{51} + ( -3110846 - 545191 \beta_{1} + 1664 \beta_{2} + 100531 \beta_{3} + 579 \beta_{4} - 502 \beta_{5} - 2422 \beta_{6} - 474 \beta_{7} - 4739 \beta_{8} + 1999 \beta_{9} - 4680 \beta_{10} + 2971 \beta_{11} - 471 \beta_{12} + 2616 \beta_{13} - 1334 \beta_{14} - 1299 \beta_{15} + 24 \beta_{16} ) q^{52} + ( -42299 - 65590 \beta_{1} + 9499 \beta_{2} + 6135 \beta_{3} - 140 \beta_{4} + 967 \beta_{5} + 3846 \beta_{6} - 712 \beta_{7} - 1131 \beta_{8} + 6426 \beta_{9} + 1372 \beta_{10} + 469 \beta_{11} - 1054 \beta_{12} - 1878 \beta_{13} - 1737 \beta_{14} + 1309 \beta_{15} - 677 \beta_{16} ) q^{53} + ( -12774461 - 850688 \beta_{1} - 8961 \beta_{2} + 64400 \beta_{3} + 1658 \beta_{4} - 4704 \beta_{5} + 276 \beta_{6} - 778 \beta_{7} - 2195 \beta_{8} - 17198 \beta_{9} + 7814 \beta_{10} - 5073 \beta_{11} + 1183 \beta_{12} - 2520 \beta_{13} + 1015 \beta_{14} - 4521 \beta_{15} + 5258 \beta_{16} ) q^{54} + ( -14599306 - 20671 \beta_{1} - 18862 \beta_{2} - 55598 \beta_{3} - 1490 \beta_{4} - 1708 \beta_{5} - 2793 \beta_{6} - 115 \beta_{7} + 5435 \beta_{8} - 10124 \beta_{9} - 4017 \beta_{10} - 867 \beta_{11} + 2056 \beta_{12} + 991 \beta_{13} + 2716 \beta_{14} - 4129 \beta_{15} + 1226 \beta_{16} ) q^{55} + ( -8249038 - 315736 \beta_{1} - 17274 \beta_{2} + 110294 \beta_{3} + 122 \beta_{4} + 856 \beta_{5} - 2340 \beta_{6} - 1328 \beta_{7} - 2410 \beta_{8} + 8666 \beta_{9} - 6318 \beta_{10} + 11720 \beta_{11} + 1284 \beta_{12} - 148 \beta_{13} - 1676 \beta_{14} + 1136 \beta_{15} + 2558 \beta_{16} ) q^{56} + ( -23849765 + 827067 \beta_{1} - 30429 \beta_{2} - 96162 \beta_{3} - 5631 \beta_{4} + 3859 \beta_{5} + 339 \beta_{6} - 305 \beta_{7} + 2241 \beta_{8} - 3198 \beta_{9} - 4375 \beta_{10} + 2769 \beta_{11} - 584 \beta_{12} + 3937 \beta_{13} + 1994 \beta_{14} - 27 \beta_{15} + 56 \beta_{16} ) q^{57} + ( -27496574 + 104197 \beta_{1} - 56594 \beta_{2} + 77956 \beta_{3} - 5592 \beta_{4} - 396 \beta_{5} - 564 \beta_{6} + 2284 \beta_{7} - 1010 \beta_{8} - 31777 \beta_{9} - 7075 \beta_{10} - 614 \beta_{11} + 1422 \beta_{12} + 4448 \beta_{13} + 1496 \beta_{14} - 482 \beta_{15} + 268 \beta_{16} ) q^{58} + ( -5664195 + 408821 \beta_{1} + 29921 \beta_{2} - 161861 \beta_{3} - 2311 \beta_{4} + 5095 \beta_{5} - 1872 \beta_{6} - 2776 \beta_{7} + 2921 \beta_{8} + 15251 \beta_{9} - 2468 \beta_{10} + 706 \beta_{11} - 1892 \beta_{12} - 5202 \beta_{13} - 2771 \beta_{14} + 3421 \beta_{15} + 328 \beta_{16} ) q^{59} + ( -36384235 - 1078129 \beta_{1} - 84283 \beta_{2} - 387029 \beta_{3} + 6063 \beta_{4} - 784 \beta_{5} + 4602 \beta_{6} - 2482 \beta_{7} + 7435 \beta_{8} - 15923 \beta_{9} + 6946 \beta_{10} - 4857 \beta_{11} - 725 \beta_{12} - 7354 \beta_{13} + 4958 \beta_{14} + 5145 \beta_{15} - 8270 \beta_{16} ) q^{60} + ( -18519986 + 966347 \beta_{1} - 42135 \beta_{2} - 82068 \beta_{3} + 8006 \beta_{4} - 7354 \beta_{5} - 617 \beta_{6} + 4523 \beta_{7} - 758 \beta_{8} - 1294 \beta_{9} - 3141 \beta_{10} + 2191 \beta_{11} + 2186 \beta_{12} - 3093 \beta_{13} - 2384 \beta_{14} - 3371 \beta_{15} - 2150 \beta_{16} ) q^{61} + ( -12552843 + 365067 \beta_{1} + 21160 \beta_{2} - 163823 \beta_{3} + 2038 \beta_{4} - 4745 \beta_{5} - 1524 \beta_{6} + 5222 \beta_{7} - 5404 \beta_{8} - 13993 \beta_{9} + 4949 \beta_{10} - 4855 \beta_{11} + 788 \beta_{12} - 4913 \beta_{13} + 2050 \beta_{14} + 6351 \beta_{15} - 5241 \beta_{16} ) q^{62} + ( -55105500 + 2787703 \beta_{1} + 10492 \beta_{2} - 284838 \beta_{3} - 1496 \beta_{4} + 18852 \beta_{5} + 2001 \beta_{6} - 455 \beta_{7} - 9863 \beta_{8} + 29708 \beta_{9} - 11701 \beta_{10} - 10199 \beta_{11} - 3240 \beta_{12} + 12751 \beta_{13} + 1470 \beta_{14} - 3817 \beta_{15} - 1574 \beta_{16} ) q^{63} + ( -48739011 - 606551 \beta_{1} - 99889 \beta_{2} - 48115 \beta_{3} + 7199 \beta_{4} + 540 \beta_{5} + 3758 \beta_{6} + 3158 \beta_{7} + 2291 \beta_{8} - 8241 \beta_{9} + 1006 \beta_{10} - 4017 \beta_{11} + 2355 \beta_{12} + 1386 \beta_{13} - 6022 \beta_{14} + 265 \beta_{15} + 1428 \beta_{16} ) q^{64} + ( -6711800 + 2036711 \beta_{1} + 49759 \beta_{2} + 169862 \beta_{3} + 2468 \beta_{4} - 5314 \beta_{5} - 1815 \beta_{6} + 2409 \beta_{7} - 2152 \beta_{8} - 13910 \beta_{9} - 11775 \beta_{10} + 985 \beta_{11} + 4366 \beta_{12} + 4081 \beta_{13} + 3196 \beta_{14} - 1793 \beta_{15} + 11050 \beta_{16} ) q^{65} + ( -57435301 - 717765 \beta_{1} - 82985 \beta_{2} - 291741 \beta_{3} + 3154 \beta_{4} - 10693 \beta_{5} - 7866 \beta_{6} + 4300 \beta_{7} + 15113 \beta_{8} + 25934 \beta_{9} + 19720 \beta_{10} + 16026 \beta_{11} + 2357 \beta_{12} - 2843 \beta_{13} - 1355 \beta_{14} - 2756 \beta_{15} + 645 \beta_{16} ) q^{66} + ( -17509661 + 1827909 \beta_{1} + 49298 \beta_{2} + 196922 \beta_{3} - 7805 \beta_{4} + 351 \beta_{5} + 4093 \beta_{6} - 949 \beta_{7} - 3351 \beta_{8} + 26955 \beta_{9} + 9145 \beta_{10} + 11432 \beta_{11} - 10958 \beta_{12} - 9079 \beta_{13} - 8186 \beta_{14} + 2787 \beta_{15} - 2665 \beta_{16} ) q^{67} + ( -5119348 - 182706 \beta_{1} + 140234 \beta_{2} + 225425 \beta_{3} - 6239 \beta_{4} - 4981 \beta_{5} - 2040 \beta_{6} + 3368 \beta_{7} - 9418 \beta_{8} + 26896 \beta_{9} - 6851 \beta_{10} - 6069 \beta_{11} - 3026 \beta_{12} + 7395 \beta_{13} - 13566 \beta_{14} - 187 \beta_{15} + 12223 \beta_{16} ) q^{68} + ( 4195235 + 2705877 \beta_{1} + 139609 \beta_{2} - 268303 \beta_{3} - 6922 \beta_{4} - 585 \beta_{5} - 3540 \beta_{6} - 13508 \beta_{7} - 16396 \beta_{8} - 28005 \beta_{9} + 6852 \beta_{10} - 1896 \beta_{11} + 308 \beta_{12} - 5398 \beta_{13} + 6735 \beta_{14} + 8116 \beta_{15} - 614 \beta_{16} ) q^{69} + ( -97357439 + 1117527 \beta_{1} - 88318 \beta_{2} + 198755 \beta_{3} - 5368 \beta_{4} + 9573 \beta_{5} + 7250 \beta_{6} + 559 \beta_{7} + 1424 \beta_{8} - 8433 \beta_{9} + 9393 \beta_{10} + 9113 \beta_{11} + 144 \beta_{12} + 8347 \beta_{13} + 10261 \beta_{14} - 3675 \beta_{15} - 5610 \beta_{16} ) q^{70} + ( -1206509 + 2672552 \beta_{1} + 195340 \beta_{2} + 177950 \beta_{3} - 13560 \beta_{4} + 14803 \beta_{5} + 3397 \beta_{6} - 8655 \beta_{7} - 859 \beta_{8} + 32696 \beta_{9} - 8689 \beta_{10} - 2753 \beta_{11} + 2760 \beta_{12} + 9379 \beta_{13} - 4738 \beta_{14} - 499 \beta_{15} - 896 \beta_{16} ) q^{71} + ( -41241924 - 1462279 \beta_{1} + 113006 \beta_{2} - 772305 \beta_{3} - 857 \beta_{4} + 161 \beta_{5} - 4416 \beta_{6} - 12007 \beta_{7} + 7070 \beta_{8} + 10143 \beta_{9} + 17371 \beta_{10} - 19211 \beta_{11} - 2742 \beta_{12} - 11027 \beta_{13} + 8548 \beta_{14} + 3463 \beta_{15} - 9785 \beta_{16} ) q^{72} + ( -29531011 - 64877 \beta_{1} - 6071 \beta_{2} + 478648 \beta_{3} + 11598 \beta_{4} - 5855 \beta_{5} + 12506 \beta_{6} + 3388 \beta_{7} + 8243 \beta_{8} - 30890 \beta_{9} - 4482 \beta_{10} - 25932 \beta_{11} - 9230 \beta_{12} + 4494 \beta_{13} - 964 \beta_{14} - 5140 \beta_{15} - 352 \beta_{16} ) q^{73} + ( -28753089 - 1801516 \beta_{1} + 147289 \beta_{2} + 205789 \beta_{3} - 618 \beta_{4} - 17395 \beta_{5} - 7338 \beta_{6} + 1774 \beta_{7} + 9391 \beta_{8} + 22705 \beta_{9} + 24997 \beta_{10} - 2238 \beta_{11} - 2605 \beta_{12} - 4385 \beta_{13} + 303 \beta_{14} - 10088 \beta_{15} + 4433 \beta_{16} ) q^{74} + ( -37828254 + 297934 \beta_{1} - 87348 \beta_{2} - 137234 \beta_{3} + 19391 \beta_{4} - 13930 \beta_{5} - 1603 \beta_{6} - 7179 \beta_{7} + 29522 \beta_{8} + 31162 \beta_{9} + 18449 \beta_{10} + 18541 \beta_{11} + 10962 \beta_{12} - 12419 \beta_{13} + 5848 \beta_{14} - 12745 \beta_{15} + 3866 \beta_{16} ) q^{75} + ( -73886751 + 2062668 \beta_{1} - 87691 \beta_{2} + 543475 \beta_{3} - 5485 \beta_{4} + 20091 \beta_{5} + 8472 \beta_{6} - 5448 \beta_{7} - 7080 \beta_{8} - 37950 \beta_{9} - 17875 \beta_{10} + 16233 \beta_{11} + 15556 \beta_{12} + 5667 \beta_{13} + 17130 \beta_{14} + 3087 \beta_{15} + 7023 \beta_{16} ) q^{76} + ( 46139922 + 915473 \beta_{1} + 84159 \beta_{2} + 567906 \beta_{3} + 23622 \beta_{4} - 19552 \beta_{5} - 3915 \beta_{6} + 13391 \beta_{7} + 7418 \beta_{8} + 4156 \beta_{9} + 2727 \beta_{10} + 1463 \beta_{11} + 4770 \beta_{12} + 2891 \beta_{13} - 7050 \beta_{14} + 1515 \beta_{15} - 10210 \beta_{16} ) q^{77} + ( -56169153 + 2750705 \beta_{1} - 43981 \beta_{2} + 146623 \beta_{3} - 6086 \beta_{4} + 4915 \beta_{5} - 6618 \beta_{6} + 8028 \beta_{7} - 27175 \beta_{8} + 70468 \beta_{9} - 7310 \beta_{10} + 29706 \beta_{11} + 2941 \beta_{12} + 13261 \beta_{13} - 16259 \beta_{14} - 3588 \beta_{15} + 7957 \beta_{16} ) q^{78} + ( 43099347 + 2413053 \beta_{1} + 52299 \beta_{2} + 502298 \beta_{3} - 7541 \beta_{4} + 4434 \beta_{5} - 23860 \beta_{6} + 3802 \beta_{7} - 4905 \beta_{8} - 53770 \beta_{9} - 7842 \beta_{10} + 2386 \beta_{11} - 324 \beta_{12} - 8776 \beta_{13} + 7897 \beta_{14} + 19099 \beta_{15} + 302 \beta_{16} ) q^{79} + ( -1443307 - 968998 \beta_{1} + 140181 \beta_{2} + 251729 \beta_{3} + 14271 \beta_{4} - 24059 \beta_{5} + 11740 \beta_{6} + 3382 \beta_{7} - 1408 \beta_{8} + 9106 \beta_{9} - 14893 \beta_{10} - 10241 \beta_{11} + 116 \beta_{12} - 16303 \beta_{13} + 14254 \beta_{14} - 17159 \beta_{15} - 13243 \beta_{16} ) q^{80} + ( 92397811 + 3636412 \beta_{1} - 99949 \beta_{2} - 369835 \beta_{3} + 3304 \beta_{4} - 32116 \beta_{5} - 24651 \beta_{6} + 13533 \beta_{7} + 8685 \beta_{8} - 75903 \beta_{9} - 28687 \beta_{10} + 26361 \beta_{11} + 20262 \beta_{12} - 10273 \beta_{13} + 7729 \beta_{14} + 21078 \beta_{15} - 26908 \beta_{16} ) q^{81} + ( -93339909 - 4844354 \beta_{1} - 174661 \beta_{2} + 982 \beta_{3} + 12052 \beta_{4} + 43684 \beta_{5} + 8294 \beta_{6} + 4467 \beta_{7} + 10713 \beta_{8} - 19430 \beta_{9} + 10334 \beta_{10} - 59635 \beta_{11} - 20409 \beta_{12} - 10146 \beta_{13} - 7584 \beta_{14} + 13275 \beta_{15} + 1907 \beta_{16} ) q^{82} + ( 43866924 + 2227182 \beta_{1} - 15324 \beta_{2} + 1419007 \beta_{3} - 23970 \beta_{4} + 28406 \beta_{5} + 30665 \beta_{6} - 2163 \beta_{7} - 5775 \beta_{8} - 51984 \beta_{9} - 6751 \beta_{10} - 17978 \beta_{11} - 7928 \beta_{12} + 10007 \beta_{13} - 15867 \beta_{14} + 8416 \beta_{15} + 19809 \beta_{16} ) q^{83} + ( -115590094 + 3317766 \beta_{1} - 156344 \beta_{2} + 112754 \beta_{3} - 43448 \beta_{4} + 82588 \beta_{5} + 14572 \beta_{6} - 19524 \beta_{7} - 29382 \beta_{8} + 140206 \beta_{9} - 84476 \beta_{10} + 29882 \beta_{11} - 20006 \beta_{12} + 38780 \beta_{13} - 27794 \beta_{14} - 5054 \beta_{15} + 14182 \beta_{16} ) q^{84} + ( 62757932 + 737519 \beta_{1} - 111751 \beta_{2} + 1357005 \beta_{3} - 3534 \beta_{4} - 17626 \beta_{5} - 4909 \beta_{6} + 24531 \beta_{7} - 27401 \beta_{8} - 105311 \beta_{9} - 17215 \beta_{10} - 18875 \beta_{11} - 19408 \beta_{12} + 4763 \beta_{13} - 3787 \beta_{14} - 3967 \beta_{15} - 10570 \beta_{16} ) q^{85} + ( 10256403 - 3418801 \beta_{1} ) q^{86} + ( 10217349 + 3418443 \beta_{1} - 343290 \beta_{2} + 320202 \beta_{3} + 24741 \beta_{4} - 4644 \beta_{5} + 23586 \beta_{6} + 17946 \beta_{7} - 24546 \beta_{8} + 1656 \beta_{9} + 8844 \beta_{10} - 4470 \beta_{11} + 2970 \beta_{12} - 3978 \beta_{13} - 1107 \beta_{14} - 18753 \beta_{15} + 22038 \beta_{16} ) q^{87} + ( 110794080 - 6397882 \beta_{1} + 150258 \beta_{2} + 511267 \beta_{3} - 42411 \beta_{4} - 21052 \beta_{5} - 33550 \beta_{6} + 8947 \beta_{7} - 5971 \beta_{8} + 48318 \beta_{9} - 47878 \beta_{10} + 27681 \beta_{11} - 6563 \beta_{12} + 5542 \beta_{13} - 36110 \beta_{14} + 1071 \beta_{15} + 29580 \beta_{16} ) q^{88} + ( 87252900 - 4763837 \beta_{1} - 133775 \beta_{2} + 493002 \beta_{3} - 10710 \beta_{4} + 27942 \beta_{5} + 19931 \beta_{6} - 35919 \beta_{7} + 2914 \beta_{8} - 87614 \beta_{9} + 33125 \beta_{10} - 3877 \beta_{11} - 15266 \beta_{12} + 14925 \beta_{13} - 2364 \beta_{14} + 10537 \beta_{15} - 16952 \beta_{16} ) q^{89} + ( -39126324 - 11059283 \beta_{1} - 440280 \beta_{2} - 2488053 \beta_{3} + 27622 \beta_{4} - 35775 \beta_{5} - 9918 \beta_{6} - 2812 \beta_{7} + 71028 \beta_{8} - 67864 \beta_{9} + 28292 \beta_{10} - 23635 \beta_{11} + 9218 \beta_{12} - 12389 \beta_{13} + 11774 \beta_{14} - 6925 \beta_{15} - 7297 \beta_{16} ) q^{90} + ( 171592432 - 7272097 \beta_{1} - 124070 \beta_{2} - 798108 \beta_{3} + 23424 \beta_{4} - 12052 \beta_{5} - 31013 \beta_{6} - 12081 \beta_{7} - 307 \beta_{8} - 14768 \beta_{9} + 46073 \beta_{10} - 713 \beta_{11} + 43532 \beta_{12} - 15619 \beta_{13} + 5190 \beta_{14} + 4463 \beta_{15} + 2246 \beta_{16} ) q^{91} + ( 62745205 - 7981269 \beta_{1} + 148527 \beta_{2} - 1591990 \beta_{3} + 41430 \beta_{4} - 23073 \beta_{5} + 422 \beta_{6} - 16586 \beta_{7} + 48419 \beta_{8} - 61781 \beta_{9} + 11767 \beta_{10} - 12884 \beta_{11} + 19039 \beta_{12} - 7687 \beta_{13} + 86972 \beta_{14} - 3604 \beta_{15} - 22819 \beta_{16} ) q^{92} + ( 267855616 - 4950525 \beta_{1} - 115446 \beta_{2} + 159921 \beta_{3} - 21984 \beta_{4} - 80795 \beta_{5} + 4897 \beta_{6} + 4171 \beta_{7} + 1729 \beta_{8} - 153241 \beta_{9} + 39433 \beta_{10} + 4127 \beta_{11} - 6800 \beta_{12} - 28167 \beta_{13} - 10352 \beta_{14} - 531 \beta_{15} + 46274 \beta_{16} ) q^{93} + ( -38705850 - 15869437 \beta_{1} - 295441 \beta_{2} - 2312728 \beta_{3} + 52026 \beta_{4} + 48398 \beta_{5} + 38436 \beta_{6} - 48977 \beta_{7} - 7461 \beta_{8} + 72250 \beta_{9} + 114652 \beta_{10} - 14201 \beta_{11} - 2761 \beta_{12} - 18462 \beta_{13} + 582 \beta_{14} + 32501 \beta_{15} - 42563 \beta_{16} ) q^{94} + ( 194060621 - 136336 \beta_{1} + 172864 \beta_{2} - 55908 \beta_{3} - 17688 \beta_{4} - 54640 \beta_{5} - 35165 \beta_{6} - 6477 \beta_{7} + 3210 \beta_{8} + 134522 \beta_{9} + 73487 \beta_{10} + 109411 \beta_{11} + 17594 \beta_{12} + 9741 \beta_{13} + 39727 \beta_{14} - 57269 \beta_{15} - 30972 \beta_{16} ) q^{95} + ( 123374150 - 1372609 \beta_{1} + 307610 \beta_{2} - 3339770 \beta_{3} + 29616 \beta_{4} + 7811 \beta_{5} - 7938 \beta_{6} + 12266 \beta_{7} - 6671 \beta_{8} + 178741 \beta_{9} - 83225 \beta_{10} - 37566 \beta_{11} + 6913 \beta_{12} + 12773 \beta_{13} + 9446 \beta_{14} - 7734 \beta_{15} - 27311 \beta_{16} ) q^{96} + ( 116051726 - 8691948 \beta_{1} - 154018 \beta_{2} + 1953796 \beta_{3} - 65025 \beta_{4} + 30774 \beta_{5} + 3528 \beta_{6} + 23198 \beta_{7} + 61537 \beta_{8} + 22870 \beta_{9} + 60486 \beta_{10} + 1539 \beta_{11} - 13702 \beta_{12} + 40508 \beta_{13} - 12229 \beta_{14} - 34977 \beta_{15} + 29255 \beta_{16} ) q^{97} + ( 395138466 - 6420060 \beta_{1} + 443954 \beta_{2} - 1296629 \beta_{3} + 23820 \beta_{4} - 159183 \beta_{5} - 27782 \beta_{6} + 9273 \beta_{7} - 1496 \beta_{8} - 158501 \beta_{9} + 107373 \beta_{10} + 74485 \beta_{11} + 50748 \beta_{12} - 64297 \beta_{13} + 24391 \beta_{14} + 23565 \beta_{15} - 21660 \beta_{16} ) q^{98} + ( 72589748 + 1307864 \beta_{1} - 188078 \beta_{2} - 3453443 \beta_{3} - 10064 \beta_{4} + 138018 \beta_{5} + 48597 \beta_{6} - 40311 \beta_{7} - 11259 \beta_{8} + 171440 \beta_{9} - 33131 \beta_{10} - 96586 \beta_{11} - 19224 \beta_{12} + 27843 \beta_{13} + 8037 \beta_{14} + 2568 \beta_{15} - 30435 \beta_{16} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 48q^{2} + 169q^{3} + 4522q^{4} + 4033q^{5} + 5871q^{6} - 76q^{7} + 41046q^{8} + 135126q^{9} + O(q^{10}) \) \( 17q + 48q^{2} + 169q^{3} + 4522q^{4} + 4033q^{5} + 5871q^{6} - 76q^{7} + 41046q^{8} + 135126q^{9} + 23763q^{10} + 78370q^{11} + 271339q^{12} + 114452q^{13} - 376208q^{14} - 255820q^{15} + 412586q^{16} + 726937q^{17} + 577055q^{18} + 544263q^{19} + 3642183q^{20} + 3137394q^{21} + 5269148q^{22} + 5575241q^{23} + 16215113q^{24} + 10874708q^{25} + 8009180q^{26} + 8350126q^{27} + 12534764q^{28} + 8223345q^{29} + 30612012q^{30} + 13054147q^{31} + 37111710q^{32} + 36024808q^{33} + 27991291q^{34} + 17826330q^{35} + 84105953q^{36} + 46733879q^{37} + 15733789q^{38} + 8689898q^{39} + 52241669q^{40} + 53667013q^{41} + 7708286q^{42} + 58119617q^{43} + 81727236q^{44} + 124361968q^{45} + 146859355q^{46} + 122945511q^{47} + 86356095q^{48} + 111396073q^{49} - 96642133q^{50} - 187132423q^{51} - 54447944q^{52} - 993146q^{53} - 219468490q^{54} - 248155792q^{55} - 141048116q^{56} - 402917960q^{57} - 466599837q^{58} - 95519644q^{59} - 621611940q^{60} - 311752038q^{61} - 212471691q^{62} - 928966350q^{63} - 829842590q^{64} - 107969830q^{65} - 978530932q^{66} - 292438130q^{67} - 88281129q^{68} + 78577726q^{69} - 1650972530q^{70} - 13576908q^{71} - 706943493q^{72} - 501490738q^{73} - 494831691q^{74} - 641914030q^{75} - 1248630771q^{76} + 787365348q^{77} - 946670550q^{78} + 740350275q^{79} - 27802861q^{80} + 1582210525q^{81} - 1600400057q^{82} + 754109940q^{83} - 1955423842q^{84} + 1071609956q^{85} + 164102448q^{86} + 186301257q^{87} + 1863375104q^{88} + 1470581868q^{89} - 698098630q^{90} + 2895349644q^{91} + 1041082071q^{92} + 4540331515q^{93} - 706582361q^{94} + 3297255729q^{95} + 2087289393q^{96} + 1949310583q^{97} + 6695989160q^{98} + 1234191326q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17} - 3 x^{16} - 6541 x^{15} + 10299 x^{14} + 17445509 x^{13} - 2347983 x^{12} - 24275147889 x^{11} - 32817396993 x^{10} + 18668870018715 x^{9} + 54284997199479 x^{8} - 7688031373215127 x^{7} - 34414384567646799 x^{6} + 1472871456600457271 x^{5} + 8285298256630038219 x^{4} - 82173286944147552299 x^{3} - 301942019130667260459 x^{2} + 866271940633182378600 x - 378136193010971707440\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 770 \)
\(\beta_{3}\)\(=\)\((\)\(\)\(12\!\cdots\!87\)\( \nu^{16} - \)\(36\!\cdots\!52\)\( \nu^{15} - \)\(71\!\cdots\!07\)\( \nu^{14} + \)\(20\!\cdots\!48\)\( \nu^{13} + \)\(16\!\cdots\!91\)\( \nu^{12} - \)\(44\!\cdots\!76\)\( \nu^{11} - \)\(20\!\cdots\!59\)\( \nu^{10} + \)\(48\!\cdots\!84\)\( \nu^{9} + \)\(12\!\cdots\!09\)\( \nu^{8} - \)\(28\!\cdots\!16\)\( \nu^{7} - \)\(42\!\cdots\!73\)\( \nu^{6} + \)\(79\!\cdots\!64\)\( \nu^{5} + \)\(59\!\cdots\!05\)\( \nu^{4} - \)\(87\!\cdots\!52\)\( \nu^{3} - \)\(16\!\cdots\!37\)\( \nu^{2} + \)\(17\!\cdots\!48\)\( \nu + \)\(62\!\cdots\!44\)\(\)\()/ \)\(28\!\cdots\!64\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(10\!\cdots\!89\)\( \nu^{16} - \)\(67\!\cdots\!92\)\( \nu^{15} + \)\(62\!\cdots\!45\)\( \nu^{14} + \)\(44\!\cdots\!88\)\( \nu^{13} - \)\(13\!\cdots\!73\)\( \nu^{12} - \)\(11\!\cdots\!32\)\( \nu^{11} + \)\(12\!\cdots\!65\)\( \nu^{10} + \)\(16\!\cdots\!36\)\( \nu^{9} - \)\(36\!\cdots\!19\)\( \nu^{8} - \)\(11\!\cdots\!92\)\( \nu^{7} - \)\(20\!\cdots\!81\)\( \nu^{6} + \)\(44\!\cdots\!84\)\( \nu^{5} + \)\(16\!\cdots\!17\)\( \nu^{4} - \)\(72\!\cdots\!72\)\( \nu^{3} - \)\(26\!\cdots\!05\)\( \nu^{2} + \)\(30\!\cdots\!00\)\( \nu - \)\(10\!\cdots\!60\)\(\)\()/ \)\(70\!\cdots\!60\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(22\!\cdots\!69\)\( \nu^{16} + \)\(24\!\cdots\!40\)\( \nu^{15} - \)\(18\!\cdots\!09\)\( \nu^{14} - \)\(16\!\cdots\!92\)\( \nu^{13} + \)\(65\!\cdots\!17\)\( \nu^{12} + \)\(42\!\cdots\!56\)\( \nu^{11} - \)\(11\!\cdots\!93\)\( \nu^{10} - \)\(57\!\cdots\!72\)\( \nu^{9} + \)\(11\!\cdots\!03\)\( \nu^{8} + \)\(42\!\cdots\!92\)\( \nu^{7} - \)\(64\!\cdots\!35\)\( \nu^{6} - \)\(16\!\cdots\!80\)\( \nu^{5} + \)\(15\!\cdots\!35\)\( \nu^{4} + \)\(28\!\cdots\!68\)\( \nu^{3} - \)\(10\!\cdots\!43\)\( \nu^{2} - \)\(11\!\cdots\!64\)\( \nu + \)\(11\!\cdots\!76\)\(\)\()/ \)\(14\!\cdots\!32\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(33\!\cdots\!79\)\( \nu^{16} + \)\(35\!\cdots\!48\)\( \nu^{15} + \)\(10\!\cdots\!15\)\( \nu^{14} - \)\(18\!\cdots\!92\)\( \nu^{13} - \)\(11\!\cdots\!23\)\( \nu^{12} + \)\(37\!\cdots\!28\)\( \nu^{11} - \)\(25\!\cdots\!65\)\( \nu^{10} - \)\(37\!\cdots\!64\)\( \nu^{9} + \)\(30\!\cdots\!51\)\( \nu^{8} + \)\(19\!\cdots\!48\)\( \nu^{7} - \)\(11\!\cdots\!51\)\( \nu^{6} - \)\(52\!\cdots\!76\)\( \nu^{5} + \)\(78\!\cdots\!47\)\( \nu^{4} + \)\(58\!\cdots\!68\)\( \nu^{3} + \)\(20\!\cdots\!65\)\( \nu^{2} - \)\(11\!\cdots\!40\)\( \nu - \)\(89\!\cdots\!40\)\(\)\()/ \)\(70\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(42\!\cdots\!41\)\( \nu^{16} + \)\(32\!\cdots\!32\)\( \nu^{15} + \)\(19\!\cdots\!05\)\( \nu^{14} - \)\(18\!\cdots\!88\)\( \nu^{13} - \)\(30\!\cdots\!37\)\( \nu^{12} + \)\(42\!\cdots\!52\)\( \nu^{11} + \)\(12\!\cdots\!45\)\( \nu^{10} - \)\(49\!\cdots\!16\)\( \nu^{9} + \)\(13\!\cdots\!09\)\( \nu^{8} + \)\(31\!\cdots\!52\)\( \nu^{7} - \)\(15\!\cdots\!89\)\( \nu^{6} - \)\(10\!\cdots\!24\)\( \nu^{5} + \)\(50\!\cdots\!93\)\( \nu^{4} + \)\(14\!\cdots\!92\)\( \nu^{3} - \)\(38\!\cdots\!25\)\( \nu^{2} - \)\(52\!\cdots\!80\)\( \nu + \)\(25\!\cdots\!00\)\(\)\()/ \)\(87\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(20\!\cdots\!83\)\( \nu^{16} - \)\(92\!\cdots\!88\)\( \nu^{15} - \)\(10\!\cdots\!91\)\( \nu^{14} + \)\(54\!\cdots\!04\)\( \nu^{13} + \)\(21\!\cdots\!35\)\( \nu^{12} - \)\(12\!\cdots\!40\)\( \nu^{11} - \)\(20\!\cdots\!11\)\( \nu^{10} + \)\(15\!\cdots\!68\)\( \nu^{9} + \)\(79\!\cdots\!61\)\( \nu^{8} - \)\(98\!\cdots\!68\)\( \nu^{7} + \)\(60\!\cdots\!31\)\( \nu^{6} + \)\(32\!\cdots\!44\)\( \nu^{5} - \)\(10\!\cdots\!67\)\( \nu^{4} - \)\(43\!\cdots\!60\)\( \nu^{3} + \)\(16\!\cdots\!59\)\( \nu^{2} + \)\(10\!\cdots\!40\)\( \nu - \)\(32\!\cdots\!40\)\(\)\()/ \)\(35\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(26\!\cdots\!81\)\( \nu^{16} - \)\(17\!\cdots\!08\)\( \nu^{15} + \)\(17\!\cdots\!25\)\( \nu^{14} + \)\(29\!\cdots\!72\)\( \nu^{13} - \)\(46\!\cdots\!37\)\( \nu^{12} - \)\(13\!\cdots\!48\)\( \nu^{11} + \)\(65\!\cdots\!85\)\( \nu^{10} + \)\(25\!\cdots\!24\)\( \nu^{9} - \)\(50\!\cdots\!51\)\( \nu^{8} - \)\(25\!\cdots\!48\)\( \nu^{7} + \)\(21\!\cdots\!11\)\( \nu^{6} + \)\(13\!\cdots\!56\)\( \nu^{5} - \)\(40\!\cdots\!47\)\( \nu^{4} - \)\(28\!\cdots\!68\)\( \nu^{3} + \)\(21\!\cdots\!35\)\( \nu^{2} + \)\(10\!\cdots\!00\)\( \nu - \)\(12\!\cdots\!60\)\(\)\()/ \)\(35\!\cdots\!80\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(45\!\cdots\!19\)\( \nu^{16} + \)\(73\!\cdots\!88\)\( \nu^{15} + \)\(27\!\cdots\!35\)\( \nu^{14} - \)\(38\!\cdots\!52\)\( \nu^{13} - \)\(69\!\cdots\!63\)\( \nu^{12} + \)\(75\!\cdots\!68\)\( \nu^{11} + \)\(89\!\cdots\!35\)\( \nu^{10} - \)\(69\!\cdots\!84\)\( \nu^{9} - \)\(64\!\cdots\!49\)\( \nu^{8} + \)\(28\!\cdots\!28\)\( \nu^{7} + \)\(24\!\cdots\!49\)\( \nu^{6} - \)\(19\!\cdots\!76\)\( \nu^{5} - \)\(43\!\cdots\!73\)\( \nu^{4} - \)\(11\!\cdots\!72\)\( \nu^{3} + \)\(18\!\cdots\!65\)\( \nu^{2} + \)\(61\!\cdots\!60\)\( \nu - \)\(48\!\cdots\!60\)\(\)\()/ \)\(23\!\cdots\!20\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(30\!\cdots\!53\)\( \nu^{16} + \)\(96\!\cdots\!32\)\( \nu^{15} - \)\(20\!\cdots\!61\)\( \nu^{14} - \)\(31\!\cdots\!56\)\( \nu^{13} + \)\(57\!\cdots\!05\)\( \nu^{12} + \)\(20\!\cdots\!80\)\( \nu^{11} - \)\(80\!\cdots\!01\)\( \nu^{10} - \)\(62\!\cdots\!32\)\( \nu^{9} + \)\(62\!\cdots\!11\)\( \nu^{8} + \)\(97\!\cdots\!72\)\( \nu^{7} - \)\(25\!\cdots\!79\)\( \nu^{6} - \)\(72\!\cdots\!36\)\( \nu^{5} + \)\(46\!\cdots\!23\)\( \nu^{4} + \)\(20\!\cdots\!80\)\( \nu^{3} - \)\(22\!\cdots\!11\)\( \nu^{2} - \)\(75\!\cdots\!20\)\( \nu + \)\(99\!\cdots\!80\)\(\)\()/ \)\(14\!\cdots\!20\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(30\!\cdots\!99\)\( \nu^{16} + \)\(23\!\cdots\!08\)\( \nu^{15} + \)\(19\!\cdots\!15\)\( \nu^{14} - \)\(10\!\cdots\!32\)\( \nu^{13} - \)\(52\!\cdots\!43\)\( \nu^{12} + \)\(16\!\cdots\!88\)\( \nu^{11} + \)\(72\!\cdots\!55\)\( \nu^{10} - \)\(69\!\cdots\!24\)\( \nu^{9} - \)\(55\!\cdots\!89\)\( \nu^{8} - \)\(71\!\cdots\!92\)\( \nu^{7} + \)\(22\!\cdots\!89\)\( \nu^{6} + \)\(77\!\cdots\!64\)\( \nu^{5} - \)\(43\!\cdots\!93\)\( \nu^{4} - \)\(21\!\cdots\!72\)\( \nu^{3} + \)\(23\!\cdots\!45\)\( \nu^{2} + \)\(58\!\cdots\!40\)\( \nu - \)\(18\!\cdots\!20\)\(\)\()/ \)\(70\!\cdots\!60\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(79\!\cdots\!01\)\( \nu^{16} + \)\(10\!\cdots\!52\)\( \nu^{15} + \)\(53\!\cdots\!85\)\( \nu^{14} - \)\(21\!\cdots\!68\)\( \nu^{13} - \)\(14\!\cdots\!37\)\( \nu^{12} - \)\(94\!\cdots\!48\)\( \nu^{11} + \)\(20\!\cdots\!05\)\( \nu^{10} + \)\(39\!\cdots\!04\)\( \nu^{9} - \)\(16\!\cdots\!31\)\( \nu^{8} - \)\(55\!\cdots\!28\)\( \nu^{7} + \)\(66\!\cdots\!31\)\( \nu^{6} + \)\(34\!\cdots\!36\)\( \nu^{5} - \)\(12\!\cdots\!27\)\( \nu^{4} - \)\(86\!\cdots\!08\)\( \nu^{3} + \)\(58\!\cdots\!35\)\( \nu^{2} + \)\(35\!\cdots\!40\)\( \nu - \)\(39\!\cdots\!80\)\(\)\()/ \)\(14\!\cdots\!20\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(31\!\cdots\!37\)\( \nu^{16} + \)\(30\!\cdots\!88\)\( \nu^{15} + \)\(19\!\cdots\!97\)\( \nu^{14} - \)\(14\!\cdots\!36\)\( \nu^{13} - \)\(50\!\cdots\!17\)\( \nu^{12} + \)\(24\!\cdots\!12\)\( \nu^{11} + \)\(68\!\cdots\!17\)\( \nu^{10} - \)\(15\!\cdots\!32\)\( \nu^{9} - \)\(50\!\cdots\!63\)\( \nu^{8} - \)\(21\!\cdots\!72\)\( \nu^{7} + \)\(19\!\cdots\!39\)\( \nu^{6} + \)\(62\!\cdots\!88\)\( \nu^{5} - \)\(35\!\cdots\!83\)\( \nu^{4} - \)\(20\!\cdots\!68\)\( \nu^{3} + \)\(16\!\cdots\!67\)\( \nu^{2} + \)\(88\!\cdots\!80\)\( \nu - \)\(65\!\cdots\!20\)\(\)\()/ \)\(46\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(51\!\cdots\!77\)\( \nu^{16} + \)\(25\!\cdots\!56\)\( \nu^{15} + \)\(32\!\cdots\!41\)\( \nu^{14} - \)\(77\!\cdots\!16\)\( \nu^{13} - \)\(86\!\cdots\!93\)\( \nu^{12} - \)\(23\!\cdots\!92\)\( \nu^{11} + \)\(11\!\cdots\!21\)\( \nu^{10} + \)\(33\!\cdots\!28\)\( \nu^{9} - \)\(92\!\cdots\!55\)\( \nu^{8} - \)\(46\!\cdots\!24\)\( \nu^{7} + \)\(38\!\cdots\!03\)\( \nu^{6} + \)\(26\!\cdots\!20\)\( \nu^{5} - \)\(74\!\cdots\!31\)\( \nu^{4} - \)\(59\!\cdots\!92\)\( \nu^{3} + \)\(39\!\cdots\!31\)\( \nu^{2} + \)\(22\!\cdots\!80\)\( \nu - \)\(29\!\cdots\!00\)\(\)\()/ \)\(70\!\cdots\!60\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(10\!\cdots\!11\)\( \nu^{16} + \)\(84\!\cdots\!08\)\( \nu^{15} + \)\(67\!\cdots\!83\)\( \nu^{14} + \)\(17\!\cdots\!52\)\( \nu^{13} - \)\(17\!\cdots\!39\)\( \nu^{12} - \)\(24\!\cdots\!56\)\( \nu^{11} + \)\(24\!\cdots\!63\)\( \nu^{10} + \)\(61\!\cdots\!84\)\( \nu^{9} - \)\(18\!\cdots\!25\)\( \nu^{8} - \)\(70\!\cdots\!72\)\( \nu^{7} + \)\(74\!\cdots\!69\)\( \nu^{6} + \)\(38\!\cdots\!20\)\( \nu^{5} - \)\(13\!\cdots\!73\)\( \nu^{4} - \)\(87\!\cdots\!96\)\( \nu^{3} + \)\(69\!\cdots\!93\)\( \nu^{2} + \)\(35\!\cdots\!60\)\( \nu - \)\(49\!\cdots\!40\)\(\)\()/ \)\(14\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 770\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{7} + 28 \beta_{3} + 2 \beta_{2} + 1202 \beta_{1} + 1434\)
\(\nu^{4}\)\(=\)\(-2 \beta_{16} + 3 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 7 \beta_{12} + 5 \beta_{11} + 20 \beta_{10} + 13 \beta_{9} + 3 \beta_{8} + 4 \beta_{7} - 10 \beta_{6} - 13 \beta_{4} + 221 \beta_{3} + 1529 \beta_{2} + 3973 \beta_{1} + 934996\)
\(\nu^{5}\)\(=\)\(26 \beta_{16} + 102 \beta_{15} + 50 \beta_{14} + 2 \beta_{13} - 264 \beta_{12} - 130 \beta_{11} + 160 \beta_{10} + 1546 \beta_{9} - 2 \beta_{8} + 1804 \beta_{7} + 92 \beta_{6} + 274 \beta_{5} - 194 \beta_{4} + 61924 \beta_{3} + 7008 \beta_{2} + 1588649 \beta_{1} + 2978286\)
\(\nu^{6}\)\(=\)\(-2954 \beta_{16} + 9376 \beta_{15} - 272 \beta_{14} - 3428 \beta_{13} - 19372 \beta_{12} + 5768 \beta_{11} + 52386 \beta_{10} + 20932 \beta_{9} + 9530 \beta_{8} + 15150 \beta_{7} - 18836 \beta_{6} + 5472 \beta_{5} - 27818 \beta_{4} + 757402 \beta_{3} + 2237881 \beta_{2} + 7637780 \beta_{1} + 1243629870\)
\(\nu^{7}\)\(=\)\(75738 \beta_{16} + 248410 \beta_{15} + 175918 \beta_{14} + 15470 \beta_{13} - 695800 \beta_{12} - 364318 \beta_{11} + 694500 \beta_{10} + 1512695 \beta_{9} + 187710 \beta_{8} + 2764657 \beta_{7} + 236476 \beta_{6} + 839974 \beta_{5} - 547210 \beta_{4} + 113805020 \beta_{3} + 16168438 \beta_{2} + 2192195998 \beta_{1} + 5907635424\)
\(\nu^{8}\)\(=\)\(-4266988 \beta_{16} + 20339991 \beta_{15} - 5735802 \beta_{14} - 4914850 \beta_{13} - 40665675 \beta_{12} + 851169 \beta_{11} + 106660130 \beta_{10} + 27682609 \beta_{9} + 22436933 \beta_{8} + 33923282 \beta_{7} - 28185070 \beta_{6} + 20514220 \beta_{5} - 47149303 \beta_{4} + 1790255507 \beta_{3} + 3279850721 \beta_{2} + 14649375031 \beta_{1} + 1724545573164\)
\(\nu^{9}\)\(=\)\(155887148 \beta_{16} + 456856200 \beta_{15} + 411347284 \beta_{14} + 54156836 \beta_{13} - 1413355764 \beta_{12} - 779829936 \beta_{11} + 1826672072 \beta_{10} + 832524952 \beta_{9} + 705620736 \beta_{8} + 4105356320 \beta_{7} + 473918656 \beta_{6} + 1835028012 \beta_{5} - 1127823472 \beta_{4} + 195827596700 \beta_{3} + 32320123388 \beta_{2} + 3105166479781 \beta_{1} + 11540559794232\)
\(\nu^{10}\)\(=\)\(-6603044108 \beta_{16} + 37986491776 \beta_{15} - 14755785840 \beta_{14} - 6722872760 \beta_{13} - 76793722664 \beta_{12} - 12345376976 \beta_{11} + 197177081212 \beta_{10} + 36770809704 \beta_{9} + 47488207036 \beta_{8} + 64672403748 \beta_{7} - 38974960792 \beta_{6} + 51074966176 \beta_{5} - 75039980092 \beta_{4} + 3629789676092 \beta_{3} + 4850813924545 \beta_{2} + 27482992983134 \beta_{1} + 2452669042615802\)
\(\nu^{11}\)\(=\)\(268469732908 \beta_{16} + 765650812108 \beta_{15} + 823343670868 \beta_{14} + 126316708260 \beta_{13} - 2626855335952 \beta_{12} - 1518965758724 \beta_{11} + 3949307251704 \beta_{10} - 636456870459 \beta_{9} + 1797232780948 \beta_{8} + 6088719238689 \beta_{7} + 865227385864 \beta_{6} + 3530418697556 \beta_{5} - 2065703094268 \beta_{4} + 326113232795596 \beta_{3} + 60483039070906 \beta_{2} + 4483208893806458 \beta_{1} + 21865007225311854\)
\(\nu^{12}\)\(=\)\(-10857460993998 \beta_{16} + 65580253566683 \beta_{15} - 28024090051030 \beta_{14} - 9055306897026 \beta_{13} - 137430827493887 \beta_{12} - 38799541474355 \beta_{11} + 346536005589880 \beta_{10} + 53107820227525 \beta_{9} + 94893689149103 \beta_{8} + 115302144678136 \beta_{7} - 51829293819522 \beta_{6} + 107390666136392 \beta_{5} - 117097385260617 \beta_{4} + 6795877674312417 \beta_{3} + 7250364448428041 \beta_{2} + 50231438407297489 \beta_{1} + 3553210500214484452\)
\(\nu^{13}\)\(=\)\(408107417553334 \beta_{16} + 1236268066607954 \beta_{15} + 1527522386638846 \beta_{14} + 240628173133406 \beta_{13} - 4684643511461600 \beta_{12} - 2820658066185494 \beta_{11} + 7733637378588688 \beta_{10} - 3128781124322050 \beta_{9} + 3905725954336890 \beta_{8} + 9101449715054324 \beta_{7} + 1501847887604596 \beta_{6} + 6383827493019326 \beta_{5} - 3570016879699846 \beta_{4} + 532867216409299252 \beta_{3} + 109059527390725480 \beta_{2} + 6574701861063116177 \beta_{1} + 40172207663164553658\)
\(\nu^{14}\)\(=\)\(-18627285085566118 \beta_{16} + 108164475044284592 \beta_{15} - 46176006420363680 \beta_{14} - 12159184729403452 \beta_{13} - 238505323478218404 \beta_{12} - 87424192763710680 \beta_{11} + 590809087808200558 \beta_{10} + 85240666155860188 \beta_{9} + 182555368002325110 \beta_{8} + 198789927970557378 \beta_{7} - 67006655962853676 \beta_{6} + 207080030858655328 \beta_{5} - 181753708672068870 \beta_{4} + 12143883479478215382 \beta_{3} + 10949641958355434857 \beta_{2} + 89727704141319491408 \beta_{1} + 5225683209950009694198\)
\(\nu^{15}\)\(=\)\(551963102682748822 \beta_{16} + 1966658615593422422 \beta_{15} + 2716760955874158386 \beta_{14} + 408086009915817698 \beta_{13} - 8165861644829589896 \beta_{12} - 5087362022946187314 \beta_{11} + 14305610796335793564 \beta_{10} - 6963906723014460821 \beta_{9} + 7811665855653348130 \beta_{8} + 13746857642273502513 \beta_{7} + 2526977589569922596 \beta_{6} + 11155157073558197514 \beta_{5} - 5971404500362146422 \beta_{4} + 860450607279843465356 \beta_{3} + 192000666648321708942 \beta_{2} + 9772802978254083681574 \beta_{1} + 71955481447223620831812\)
\(\nu^{16}\)\(=\)\(-32849229247184260424 \beta_{16} + 173470783136975634863 \beta_{15} - 69268200742891860402 \beta_{14} - 16365812325032203778 \beta_{13} - 406250695263354375715 \beta_{12} - 173623348015198217815 \beta_{11} + 987848622680015392726 \beta_{10} + 148598467413368066473 \beta_{9} + 341709850913969539873 \beta_{8} + 336228138437168029430 \beta_{7} - 84104663732850350982 \beta_{6} + 380024934721055044980 \beta_{5} - 282103494088711659939 \beta_{4} + 21071488845426120753927 \beta_{3} + 16695729864792723480881 \beta_{2} + 157342504881947186948371 \beta_{1} + 7785730688589175230168348\)

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
40.5953
39.1708
34.9313
33.5331
25.5994
23.8827
7.34621
1.53684
0.562738
−5.60390
−14.8584
−16.0958
−22.6959
−34.7389
−35.3457
−37.0877
−37.7321
−37.5953 −105.944 901.404 −205.302 3983.01 7435.13 −14639.7 −8458.78 7718.39
1.2 −36.1708 −228.603 796.325 2349.81 8268.76 528.884 −10284.3 32576.5 −84994.4
1.3 −31.9313 275.950 507.605 217.489 −8811.43 4698.28 140.338 56465.5 −6944.68
1.4 −30.5331 134.360 420.267 −2538.51 −4102.43 −3047.70 2800.88 −1630.30 77508.3
1.5 −22.5994 −82.1072 −1.26822 454.995 1855.57 −6480.48 11599.5 −12941.4 −10282.6
1.6 −20.8827 85.1471 −75.9135 2696.38 −1778.10 8355.76 12277.2 −12433.0 −56307.6
1.7 −4.34621 171.146 −493.110 −694.593 −743.834 −7145.50 4368.42 9607.83 3018.84
1.8 1.46316 −83.8389 −509.859 −1977.91 −122.670 −7406.23 −1495.14 −12654.0 −2894.00
1.9 2.43726 −44.8177 −506.060 1836.28 −109.232 1435.14 −2481.28 −17674.4 4475.49
1.10 8.60390 −44.7892 −437.973 −1151.27 −385.362 6348.30 −8173.47 −17676.9 −9905.41
1.11 17.8584 −263.946 −193.078 542.767 −4713.65 −12329.9 −12591.6 49984.5 9692.93
1.12 19.0958 219.847 −147.352 1171.72 4198.14 4845.77 −12590.8 28649.6 22374.8
1.13 25.6959 −189.280 148.278 −1738.18 −4863.71 2771.25 −9346.16 16143.9 −44664.0
1.14 37.7389 83.4314 912.221 1398.64 3148.61 8642.65 15103.9 −12722.2 52783.1
1.15 38.3457 −137.794 958.392 2131.30 −5283.82 −2993.43 17117.2 −695.689 81726.4
1.16 40.0877 243.316 1095.02 1211.89 9753.98 −12008.0 23372.0 39519.8 48581.8
1.17 40.7321 136.923 1147.10 −1672.50 5577.18 6274.02 25869.0 −934.956 −68124.5
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

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 43.10.a.b 17
3.b odd 2 1 387.10.a.e 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.10.a.b 17 1.a even 1 1 trivial
387.10.a.e 17 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(43\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{17} - \cdots\) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(43))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 48 T + 3243 T^{2} - 124290 T^{3} + 5555346 T^{4} - 189454860 T^{5} + 7059572252 T^{6} - 218457960984 T^{7} + 7191773541168 T^{8} - 204994833215136 T^{9} + 6150447292466944 T^{10} - 164005367565160704 T^{11} + 4550818290404407296 T^{12} - \)\(11\!\cdots\!76\)\( T^{13} + \)\(29\!\cdots\!64\)\( T^{14} - \)\(69\!\cdots\!24\)\( T^{15} + \)\(17\!\cdots\!40\)\( T^{16} - \)\(37\!\cdots\!40\)\( T^{17} + \)\(87\!\cdots\!80\)\( T^{18} - \)\(18\!\cdots\!56\)\( T^{19} + \)\(39\!\cdots\!92\)\( T^{20} - \)\(78\!\cdots\!36\)\( T^{21} + \)\(16\!\cdots\!72\)\( T^{22} - \)\(29\!\cdots\!36\)\( T^{23} + \)\(56\!\cdots\!52\)\( T^{24} - \)\(96\!\cdots\!56\)\( T^{25} + \)\(17\!\cdots\!36\)\( T^{26} - \)\(27\!\cdots\!16\)\( T^{27} + \)\(44\!\cdots\!76\)\( T^{28} - \)\(61\!\cdots\!60\)\( T^{29} + \)\(92\!\cdots\!12\)\( T^{30} - \)\(10\!\cdots\!60\)\( T^{31} + \)\(14\!\cdots\!24\)\( T^{32} - \)\(10\!\cdots\!68\)\( T^{33} + \)\(11\!\cdots\!92\)\( T^{34} \)
$3$ \( 1 - 169 T + 114023 T^{2} - 19335517 T^{3} + 6985015048 T^{4} - 1127506999959 T^{5} + 291313273320243 T^{6} - 44888081573471535 T^{7} + 9367133380691135499 T^{8} - \)\(13\!\cdots\!66\)\( T^{9} + \)\(25\!\cdots\!29\)\( T^{10} - \)\(35\!\cdots\!41\)\( T^{11} + \)\(58\!\cdots\!21\)\( T^{12} - \)\(83\!\cdots\!65\)\( T^{13} + \)\(12\!\cdots\!58\)\( T^{14} - \)\(17\!\cdots\!91\)\( T^{15} + \)\(25\!\cdots\!92\)\( T^{16} - \)\(35\!\cdots\!90\)\( T^{17} + \)\(50\!\cdots\!36\)\( T^{18} - \)\(68\!\cdots\!99\)\( T^{19} + \)\(96\!\cdots\!46\)\( T^{20} - \)\(12\!\cdots\!65\)\( T^{21} + \)\(17\!\cdots\!03\)\( T^{22} - \)\(20\!\cdots\!29\)\( T^{23} + \)\(28\!\cdots\!83\)\( T^{24} - \)\(31\!\cdots\!06\)\( T^{25} + \)\(41\!\cdots\!97\)\( T^{26} - \)\(39\!\cdots\!15\)\( T^{27} + \)\(50\!\cdots\!81\)\( T^{28} - \)\(38\!\cdots\!99\)\( T^{29} + \)\(46\!\cdots\!24\)\( T^{30} - \)\(25\!\cdots\!93\)\( T^{31} + \)\(29\!\cdots\!61\)\( T^{32} - \)\(85\!\cdots\!89\)\( T^{33} + \)\(99\!\cdots\!23\)\( T^{34} \)
$5$ \( 1 - 4033 T + 19296753 T^{2} - 54396147383 T^{3} + 161720019955300 T^{4} - 370675403719895889 T^{5} + \)\(86\!\cdots\!47\)\( T^{6} - \)\(17\!\cdots\!07\)\( T^{7} + \)\(34\!\cdots\!97\)\( T^{8} - \)\(61\!\cdots\!90\)\( T^{9} + \)\(11\!\cdots\!25\)\( T^{10} - \)\(18\!\cdots\!25\)\( T^{11} + \)\(29\!\cdots\!75\)\( T^{12} - \)\(45\!\cdots\!75\)\( T^{13} + \)\(69\!\cdots\!50\)\( T^{14} - \)\(10\!\cdots\!25\)\( T^{15} + \)\(14\!\cdots\!00\)\( T^{16} - \)\(20\!\cdots\!50\)\( T^{17} + \)\(28\!\cdots\!00\)\( T^{18} - \)\(38\!\cdots\!25\)\( T^{19} + \)\(51\!\cdots\!50\)\( T^{20} - \)\(65\!\cdots\!75\)\( T^{21} + \)\(84\!\cdots\!75\)\( T^{22} - \)\(10\!\cdots\!25\)\( T^{23} + \)\(11\!\cdots\!25\)\( T^{24} - \)\(13\!\cdots\!50\)\( T^{25} + \)\(14\!\cdots\!25\)\( T^{26} - \)\(13\!\cdots\!75\)\( T^{27} + \)\(13\!\cdots\!75\)\( T^{28} - \)\(11\!\cdots\!25\)\( T^{29} + \)\(97\!\cdots\!00\)\( T^{30} - \)\(63\!\cdots\!75\)\( T^{31} + \)\(44\!\cdots\!25\)\( T^{32} - \)\(18\!\cdots\!25\)\( T^{33} + \)\(87\!\cdots\!25\)\( T^{34} \)
$7$ \( 1 + 76 T + 287310511 T^{2} + 791973280144 T^{3} + 42360281382215704 T^{4} + \)\(22\!\cdots\!52\)\( T^{5} + \)\(44\!\cdots\!44\)\( T^{6} + \)\(31\!\cdots\!56\)\( T^{7} + \)\(38\!\cdots\!24\)\( T^{8} + \)\(30\!\cdots\!64\)\( T^{9} + \)\(28\!\cdots\!04\)\( T^{10} + \)\(22\!\cdots\!56\)\( T^{11} + \)\(18\!\cdots\!60\)\( T^{12} + \)\(13\!\cdots\!72\)\( T^{13} + \)\(10\!\cdots\!20\)\( T^{14} + \)\(68\!\cdots\!56\)\( T^{15} + \)\(48\!\cdots\!34\)\( T^{16} + \)\(29\!\cdots\!72\)\( T^{17} + \)\(19\!\cdots\!38\)\( T^{18} + \)\(11\!\cdots\!44\)\( T^{19} + \)\(67\!\cdots\!60\)\( T^{20} + \)\(36\!\cdots\!72\)\( T^{21} + \)\(19\!\cdots\!20\)\( T^{22} + \)\(98\!\cdots\!44\)\( T^{23} + \)\(49\!\cdots\!72\)\( T^{24} + \)\(21\!\cdots\!64\)\( T^{25} + \)\(10\!\cdots\!68\)\( T^{26} + \)\(36\!\cdots\!44\)\( T^{27} + \)\(20\!\cdots\!92\)\( T^{28} + \)\(41\!\cdots\!52\)\( T^{29} + \)\(31\!\cdots\!28\)\( T^{30} + \)\(24\!\cdots\!56\)\( T^{31} + \)\(35\!\cdots\!73\)\( T^{32} + \)\(37\!\cdots\!76\)\( T^{33} + \)\(19\!\cdots\!07\)\( T^{34} \)
$11$ \( 1 - 78370 T + 21685692632 T^{2} - 1559331069335344 T^{3} + \)\(23\!\cdots\!83\)\( T^{4} - \)\(15\!\cdots\!72\)\( T^{5} + \)\(17\!\cdots\!57\)\( T^{6} - \)\(10\!\cdots\!04\)\( T^{7} + \)\(98\!\cdots\!92\)\( T^{8} - \)\(57\!\cdots\!70\)\( T^{9} + \)\(44\!\cdots\!31\)\( T^{10} - \)\(24\!\cdots\!00\)\( T^{11} + \)\(16\!\cdots\!90\)\( T^{12} - \)\(85\!\cdots\!28\)\( T^{13} + \)\(53\!\cdots\!36\)\( T^{14} - \)\(25\!\cdots\!64\)\( T^{15} + \)\(14\!\cdots\!18\)\( T^{16} - \)\(64\!\cdots\!36\)\( T^{17} + \)\(34\!\cdots\!38\)\( T^{18} - \)\(14\!\cdots\!84\)\( T^{19} + \)\(70\!\cdots\!56\)\( T^{20} - \)\(26\!\cdots\!08\)\( T^{21} + \)\(12\!\cdots\!90\)\( T^{22} - \)\(41\!\cdots\!00\)\( T^{23} + \)\(18\!\cdots\!61\)\( T^{24} - \)\(54\!\cdots\!70\)\( T^{25} + \)\(22\!\cdots\!12\)\( T^{26} - \)\(58\!\cdots\!04\)\( T^{27} + \)\(21\!\cdots\!87\)\( T^{28} - \)\(46\!\cdots\!32\)\( T^{29} + \)\(16\!\cdots\!93\)\( T^{30} - \)\(25\!\cdots\!84\)\( T^{31} + \)\(83\!\cdots\!32\)\( T^{32} - \)\(71\!\cdots\!70\)\( T^{33} + \)\(21\!\cdots\!31\)\( T^{34} \)
$13$ \( 1 - 114452 T + 103365458738 T^{2} - 10716014970850574 T^{3} + \)\(52\!\cdots\!01\)\( T^{4} - \)\(49\!\cdots\!44\)\( T^{5} + \)\(17\!\cdots\!73\)\( T^{6} - \)\(15\!\cdots\!02\)\( T^{7} + \)\(42\!\cdots\!86\)\( T^{8} - \)\(35\!\cdots\!84\)\( T^{9} + \)\(82\!\cdots\!45\)\( T^{10} - \)\(67\!\cdots\!24\)\( T^{11} + \)\(13\!\cdots\!78\)\( T^{12} - \)\(10\!\cdots\!32\)\( T^{13} + \)\(18\!\cdots\!28\)\( T^{14} - \)\(13\!\cdots\!72\)\( T^{15} + \)\(22\!\cdots\!46\)\( T^{16} - \)\(15\!\cdots\!96\)\( T^{17} + \)\(23\!\cdots\!58\)\( T^{18} - \)\(15\!\cdots\!88\)\( T^{19} + \)\(22\!\cdots\!76\)\( T^{20} - \)\(13\!\cdots\!12\)\( T^{21} + \)\(17\!\cdots\!54\)\( T^{22} - \)\(95\!\cdots\!36\)\( T^{23} + \)\(12\!\cdots\!65\)\( T^{24} - \)\(57\!\cdots\!04\)\( T^{25} + \)\(71\!\cdots\!18\)\( T^{26} - \)\(27\!\cdots\!98\)\( T^{27} + \)\(32\!\cdots\!21\)\( T^{28} - \)\(10\!\cdots\!24\)\( T^{29} + \)\(11\!\cdots\!33\)\( T^{30} - \)\(24\!\cdots\!66\)\( T^{31} + \)\(24\!\cdots\!66\)\( T^{32} - \)\(29\!\cdots\!72\)\( T^{33} + \)\(27\!\cdots\!53\)\( T^{34} \)
$17$ \( 1 - 726937 T + 1348735407438 T^{2} - 817643622619815812 T^{3} + \)\(85\!\cdots\!39\)\( T^{4} - \)\(45\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!24\)\( T^{6} - \)\(16\!\cdots\!41\)\( T^{7} + \)\(10\!\cdots\!62\)\( T^{8} - \)\(43\!\cdots\!21\)\( T^{9} + \)\(23\!\cdots\!51\)\( T^{10} - \)\(92\!\cdots\!39\)\( T^{11} + \)\(43\!\cdots\!19\)\( T^{12} - \)\(16\!\cdots\!38\)\( T^{13} + \)\(69\!\cdots\!95\)\( T^{14} - \)\(24\!\cdots\!73\)\( T^{15} + \)\(94\!\cdots\!40\)\( T^{16} - \)\(30\!\cdots\!08\)\( T^{17} + \)\(11\!\cdots\!80\)\( T^{18} - \)\(34\!\cdots\!57\)\( T^{19} + \)\(11\!\cdots\!35\)\( T^{20} - \)\(32\!\cdots\!78\)\( T^{21} + \)\(10\!\cdots\!83\)\( T^{22} - \)\(25\!\cdots\!31\)\( T^{23} + \)\(76\!\cdots\!63\)\( T^{24} - \)\(17\!\cdots\!81\)\( T^{25} + \)\(47\!\cdots\!54\)\( T^{26} - \)\(89\!\cdots\!09\)\( T^{27} + \)\(22\!\cdots\!72\)\( T^{28} - \)\(34\!\cdots\!00\)\( T^{29} + \)\(78\!\cdots\!03\)\( T^{30} - \)\(88\!\cdots\!28\)\( T^{31} + \)\(17\!\cdots\!34\)\( T^{32} - \)\(11\!\cdots\!77\)\( T^{33} + \)\(18\!\cdots\!37\)\( T^{34} \)
$19$ \( 1 - 544263 T + 2661892561361 T^{2} - 1569862869589391887 T^{3} + \)\(35\!\cdots\!36\)\( T^{4} - \)\(22\!\cdots\!05\)\( T^{5} + \)\(33\!\cdots\!09\)\( T^{6} - \)\(22\!\cdots\!53\)\( T^{7} + \)\(23\!\cdots\!47\)\( T^{8} - \)\(16\!\cdots\!14\)\( T^{9} + \)\(14\!\cdots\!77\)\( T^{10} - \)\(92\!\cdots\!83\)\( T^{11} + \)\(71\!\cdots\!39\)\( T^{12} - \)\(44\!\cdots\!75\)\( T^{13} + \)\(30\!\cdots\!22\)\( T^{14} - \)\(17\!\cdots\!49\)\( T^{15} + \)\(11\!\cdots\!98\)\( T^{16} - \)\(62\!\cdots\!62\)\( T^{17} + \)\(37\!\cdots\!42\)\( T^{18} - \)\(18\!\cdots\!09\)\( T^{19} + \)\(10\!\cdots\!58\)\( T^{20} - \)\(48\!\cdots\!75\)\( T^{21} + \)\(25\!\cdots\!61\)\( T^{22} - \)\(10\!\cdots\!43\)\( T^{23} + \)\(51\!\cdots\!43\)\( T^{24} - \)\(18\!\cdots\!54\)\( T^{25} + \)\(90\!\cdots\!93\)\( T^{26} - \)\(27\!\cdots\!53\)\( T^{27} + \)\(13\!\cdots\!11\)\( T^{28} - \)\(29\!\cdots\!05\)\( T^{29} + \)\(14\!\cdots\!04\)\( T^{30} - \)\(20\!\cdots\!47\)\( T^{31} + \)\(11\!\cdots\!39\)\( T^{32} - \)\(75\!\cdots\!23\)\( T^{33} + \)\(44\!\cdots\!59\)\( T^{34} \)
$23$ \( 1 - 5575241 T + 27839624620374 T^{2} - 86029260328417440598 T^{3} + \)\(24\!\cdots\!63\)\( T^{4} - \)\(52\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!50\)\( T^{6} - \)\(17\!\cdots\!83\)\( T^{7} + \)\(30\!\cdots\!50\)\( T^{8} - \)\(43\!\cdots\!35\)\( T^{9} + \)\(69\!\cdots\!93\)\( T^{10} - \)\(90\!\cdots\!91\)\( T^{11} + \)\(14\!\cdots\!41\)\( T^{12} - \)\(18\!\cdots\!10\)\( T^{13} + \)\(29\!\cdots\!55\)\( T^{14} - \)\(34\!\cdots\!33\)\( T^{15} + \)\(55\!\cdots\!52\)\( T^{16} - \)\(63\!\cdots\!46\)\( T^{17} + \)\(99\!\cdots\!76\)\( T^{18} - \)\(11\!\cdots\!77\)\( T^{19} + \)\(17\!\cdots\!85\)\( T^{20} - \)\(19\!\cdots\!10\)\( T^{21} + \)\(27\!\cdots\!63\)\( T^{22} - \)\(30\!\cdots\!19\)\( T^{23} + \)\(42\!\cdots\!31\)\( T^{24} - \)\(47\!\cdots\!35\)\( T^{25} + \)\(61\!\cdots\!50\)\( T^{26} - \)\(64\!\cdots\!67\)\( T^{27} + \)\(70\!\cdots\!50\)\( T^{28} - \)\(61\!\cdots\!88\)\( T^{29} + \)\(51\!\cdots\!89\)\( T^{30} - \)\(32\!\cdots\!22\)\( T^{31} + \)\(18\!\cdots\!18\)\( T^{32} - \)\(68\!\cdots\!81\)\( T^{33} + \)\(22\!\cdots\!83\)\( T^{34} \)
$29$ \( 1 - 8223345 T + 175375794027265 T^{2} - \)\(11\!\cdots\!43\)\( T^{3} + \)\(14\!\cdots\!20\)\( T^{4} - \)\(83\!\cdots\!13\)\( T^{5} + \)\(75\!\cdots\!47\)\( T^{6} - \)\(38\!\cdots\!51\)\( T^{7} + \)\(28\!\cdots\!61\)\( T^{8} - \)\(12\!\cdots\!74\)\( T^{9} + \)\(82\!\cdots\!29\)\( T^{10} - \)\(33\!\cdots\!93\)\( T^{11} + \)\(19\!\cdots\!47\)\( T^{12} - \)\(73\!\cdots\!71\)\( T^{13} + \)\(38\!\cdots\!46\)\( T^{14} - \)\(13\!\cdots\!45\)\( T^{15} + \)\(64\!\cdots\!96\)\( T^{16} - \)\(20\!\cdots\!70\)\( T^{17} + \)\(93\!\cdots\!24\)\( T^{18} - \)\(28\!\cdots\!45\)\( T^{19} + \)\(11\!\cdots\!14\)\( T^{20} - \)\(32\!\cdots\!91\)\( T^{21} + \)\(12\!\cdots\!03\)\( T^{22} - \)\(31\!\cdots\!33\)\( T^{23} + \)\(11\!\cdots\!81\)\( T^{24} - \)\(25\!\cdots\!34\)\( T^{25} + \)\(80\!\cdots\!69\)\( T^{26} - \)\(15\!\cdots\!51\)\( T^{27} + \)\(45\!\cdots\!43\)\( T^{28} - \)\(72\!\cdots\!93\)\( T^{29} + \)\(18\!\cdots\!80\)\( T^{30} - \)\(21\!\cdots\!03\)\( T^{31} + \)\(46\!\cdots\!85\)\( T^{32} - \)\(31\!\cdots\!45\)\( T^{33} + \)\(55\!\cdots\!89\)\( T^{34} \)
$31$ \( 1 - 13054147 T + 332047812251192 T^{2} - \)\(37\!\cdots\!90\)\( T^{3} + \)\(54\!\cdots\!81\)\( T^{4} - \)\(54\!\cdots\!64\)\( T^{5} + \)\(58\!\cdots\!48\)\( T^{6} - \)\(52\!\cdots\!05\)\( T^{7} + \)\(46\!\cdots\!46\)\( T^{8} - \)\(36\!\cdots\!33\)\( T^{9} + \)\(27\!\cdots\!43\)\( T^{10} - \)\(19\!\cdots\!97\)\( T^{11} + \)\(13\!\cdots\!01\)\( T^{12} - \)\(84\!\cdots\!54\)\( T^{13} + \)\(51\!\cdots\!11\)\( T^{14} - \)\(29\!\cdots\!79\)\( T^{15} + \)\(16\!\cdots\!34\)\( T^{16} - \)\(86\!\cdots\!26\)\( T^{17} + \)\(43\!\cdots\!14\)\( T^{18} - \)\(20\!\cdots\!39\)\( T^{19} + \)\(95\!\cdots\!21\)\( T^{20} - \)\(41\!\cdots\!74\)\( T^{21} + \)\(17\!\cdots\!51\)\( T^{22} - \)\(66\!\cdots\!37\)\( T^{23} + \)\(25\!\cdots\!13\)\( T^{24} - \)\(86\!\cdots\!13\)\( T^{25} + \)\(29\!\cdots\!26\)\( T^{26} - \)\(86\!\cdots\!05\)\( T^{27} + \)\(26\!\cdots\!08\)\( T^{28} - \)\(63\!\cdots\!24\)\( T^{29} + \)\(16\!\cdots\!91\)\( T^{30} - \)\(30\!\cdots\!90\)\( T^{31} + \)\(71\!\cdots\!92\)\( T^{32} - \)\(74\!\cdots\!87\)\( T^{33} + \)\(15\!\cdots\!91\)\( T^{34} \)
$37$ \( 1 - 46733879 T + 2441525929829305 T^{2} - \)\(79\!\cdots\!53\)\( T^{3} + \)\(25\!\cdots\!72\)\( T^{4} - \)\(65\!\cdots\!83\)\( T^{5} + \)\(16\!\cdots\!95\)\( T^{6} - \)\(34\!\cdots\!97\)\( T^{7} + \)\(70\!\cdots\!85\)\( T^{8} - \)\(13\!\cdots\!30\)\( T^{9} + \)\(22\!\cdots\!73\)\( T^{10} - \)\(37\!\cdots\!15\)\( T^{11} + \)\(57\!\cdots\!51\)\( T^{12} - \)\(83\!\cdots\!45\)\( T^{13} + \)\(11\!\cdots\!98\)\( T^{14} - \)\(14\!\cdots\!67\)\( T^{15} + \)\(18\!\cdots\!00\)\( T^{16} - \)\(21\!\cdots\!42\)\( T^{17} + \)\(23\!\cdots\!00\)\( T^{18} - \)\(25\!\cdots\!43\)\( T^{19} + \)\(25\!\cdots\!34\)\( T^{20} - \)\(23\!\cdots\!45\)\( T^{21} + \)\(21\!\cdots\!07\)\( T^{22} - \)\(17\!\cdots\!35\)\( T^{23} + \)\(14\!\cdots\!69\)\( T^{24} - \)\(10\!\cdots\!30\)\( T^{25} + \)\(74\!\cdots\!45\)\( T^{26} - \)\(47\!\cdots\!53\)\( T^{27} + \)\(28\!\cdots\!35\)\( T^{28} - \)\(15\!\cdots\!43\)\( T^{29} + \)\(77\!\cdots\!24\)\( T^{30} - \)\(31\!\cdots\!77\)\( T^{31} + \)\(12\!\cdots\!65\)\( T^{32} - \)\(30\!\cdots\!19\)\( T^{33} + \)\(86\!\cdots\!97\)\( T^{34} \)
$41$ \( 1 - 53667013 T + 3048579858190106 T^{2} - \)\(97\!\cdots\!80\)\( T^{3} + \)\(34\!\cdots\!55\)\( T^{4} - \)\(85\!\cdots\!68\)\( T^{5} + \)\(25\!\cdots\!44\)\( T^{6} - \)\(54\!\cdots\!73\)\( T^{7} + \)\(14\!\cdots\!14\)\( T^{8} - \)\(29\!\cdots\!57\)\( T^{9} + \)\(75\!\cdots\!43\)\( T^{10} - \)\(14\!\cdots\!11\)\( T^{11} + \)\(33\!\cdots\!07\)\( T^{12} - \)\(59\!\cdots\!14\)\( T^{13} + \)\(13\!\cdots\!59\)\( T^{14} - \)\(22\!\cdots\!53\)\( T^{15} + \)\(48\!\cdots\!40\)\( T^{16} - \)\(77\!\cdots\!20\)\( T^{17} + \)\(15\!\cdots\!40\)\( T^{18} - \)\(24\!\cdots\!13\)\( T^{19} + \)\(47\!\cdots\!79\)\( T^{20} - \)\(68\!\cdots\!74\)\( T^{21} + \)\(12\!\cdots\!07\)\( T^{22} - \)\(17\!\cdots\!71\)\( T^{23} + \)\(30\!\cdots\!03\)\( T^{24} - \)\(38\!\cdots\!17\)\( T^{25} + \)\(63\!\cdots\!74\)\( T^{26} - \)\(77\!\cdots\!73\)\( T^{27} + \)\(11\!\cdots\!84\)\( T^{28} - \)\(13\!\cdots\!28\)\( T^{29} + \)\(17\!\cdots\!55\)\( T^{30} - \)\(15\!\cdots\!80\)\( T^{31} + \)\(16\!\cdots\!06\)\( T^{32} - \)\(93\!\cdots\!93\)\( T^{33} + \)\(57\!\cdots\!21\)\( T^{34} \)
$43$ \( ( 1 - 3418801 T )^{17} \)
$47$ \( 1 - 122945511 T + 14520062102156677 T^{2} - \)\(10\!\cdots\!83\)\( T^{3} + \)\(75\!\cdots\!60\)\( T^{4} - \)\(40\!\cdots\!77\)\( T^{5} + \)\(22\!\cdots\!49\)\( T^{6} - \)\(10\!\cdots\!01\)\( T^{7} + \)\(48\!\cdots\!63\)\( T^{8} - \)\(19\!\cdots\!50\)\( T^{9} + \)\(83\!\cdots\!85\)\( T^{10} - \)\(30\!\cdots\!23\)\( T^{11} + \)\(11\!\cdots\!59\)\( T^{12} - \)\(40\!\cdots\!07\)\( T^{13} + \)\(15\!\cdots\!38\)\( T^{14} - \)\(48\!\cdots\!89\)\( T^{15} + \)\(17\!\cdots\!14\)\( T^{16} - \)\(55\!\cdots\!42\)\( T^{17} + \)\(20\!\cdots\!38\)\( T^{18} - \)\(61\!\cdots\!21\)\( T^{19} + \)\(21\!\cdots\!94\)\( T^{20} - \)\(63\!\cdots\!47\)\( T^{21} + \)\(20\!\cdots\!13\)\( T^{22} - \)\(59\!\cdots\!87\)\( T^{23} + \)\(18\!\cdots\!55\)\( T^{24} - \)\(47\!\cdots\!50\)\( T^{25} + \)\(13\!\cdots\!61\)\( T^{26} - \)\(31\!\cdots\!49\)\( T^{27} + \)\(77\!\cdots\!67\)\( T^{28} - \)\(15\!\cdots\!97\)\( T^{29} + \)\(32\!\cdots\!20\)\( T^{30} - \)\(50\!\cdots\!07\)\( T^{31} + \)\(78\!\cdots\!11\)\( T^{32} - \)\(74\!\cdots\!91\)\( T^{33} + \)\(67\!\cdots\!27\)\( T^{34} \)
$53$ \( 1 + 993146 T + 37697586152317598 T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + \)\(69\!\cdots\!01\)\( T^{4} + \)\(32\!\cdots\!28\)\( T^{5} + \)\(82\!\cdots\!53\)\( T^{6} + \)\(52\!\cdots\!12\)\( T^{7} + \)\(71\!\cdots\!06\)\( T^{8} + \)\(56\!\cdots\!58\)\( T^{9} + \)\(49\!\cdots\!09\)\( T^{10} + \)\(43\!\cdots\!20\)\( T^{11} + \)\(27\!\cdots\!22\)\( T^{12} + \)\(25\!\cdots\!16\)\( T^{13} + \)\(12\!\cdots\!76\)\( T^{14} + \)\(11\!\cdots\!52\)\( T^{15} + \)\(49\!\cdots\!38\)\( T^{16} + \)\(42\!\cdots\!04\)\( T^{17} + \)\(16\!\cdots\!54\)\( T^{18} + \)\(12\!\cdots\!28\)\( T^{19} + \)\(45\!\cdots\!12\)\( T^{20} + \)\(29\!\cdots\!36\)\( T^{21} + \)\(10\!\cdots\!46\)\( T^{22} + \)\(55\!\cdots\!80\)\( T^{23} + \)\(20\!\cdots\!93\)\( T^{24} + \)\(78\!\cdots\!78\)\( T^{25} + \)\(33\!\cdots\!18\)\( T^{26} + \)\(81\!\cdots\!88\)\( T^{27} + \)\(41\!\cdots\!01\)\( T^{28} + \)\(54\!\cdots\!08\)\( T^{29} + \)\(37\!\cdots\!13\)\( T^{30} + \)\(19\!\cdots\!60\)\( T^{31} + \)\(22\!\cdots\!86\)\( T^{32} + \)\(19\!\cdots\!26\)\( T^{33} + \)\(65\!\cdots\!73\)\( T^{34} \)
$59$ \( 1 + 95519644 T + 84047810362809759 T^{2} + \)\(80\!\cdots\!84\)\( T^{3} + \)\(35\!\cdots\!84\)\( T^{4} + \)\(34\!\cdots\!80\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} + \)\(96\!\cdots\!00\)\( T^{7} + \)\(22\!\cdots\!88\)\( T^{8} + \)\(20\!\cdots\!84\)\( T^{9} + \)\(38\!\cdots\!24\)\( T^{10} + \)\(33\!\cdots\!32\)\( T^{11} + \)\(54\!\cdots\!24\)\( T^{12} + \)\(45\!\cdots\!76\)\( T^{13} + \)\(64\!\cdots\!24\)\( T^{14} + \)\(51\!\cdots\!52\)\( T^{15} + \)\(65\!\cdots\!98\)\( T^{16} + \)\(48\!\cdots\!76\)\( T^{17} + \)\(57\!\cdots\!22\)\( T^{18} + \)\(38\!\cdots\!92\)\( T^{19} + \)\(42\!\cdots\!56\)\( T^{20} + \)\(25\!\cdots\!16\)\( T^{21} + \)\(26\!\cdots\!76\)\( T^{22} + \)\(14\!\cdots\!52\)\( T^{23} + \)\(13\!\cdots\!96\)\( T^{24} + \)\(64\!\cdots\!04\)\( T^{25} + \)\(60\!\cdots\!92\)\( T^{26} + \)\(23\!\cdots\!00\)\( T^{27} + \)\(21\!\cdots\!64\)\( T^{28} + \)\(61\!\cdots\!80\)\( T^{29} + \)\(55\!\cdots\!96\)\( T^{30} + \)\(10\!\cdots\!44\)\( T^{31} + \)\(97\!\cdots\!41\)\( T^{32} + \)\(96\!\cdots\!84\)\( T^{33} + \)\(87\!\cdots\!79\)\( T^{34} \)
$61$ \( 1 + 311752038 T + 147844486653604641 T^{2} + \)\(35\!\cdots\!00\)\( T^{3} + \)\(98\!\cdots\!00\)\( T^{4} + \)\(19\!\cdots\!56\)\( T^{5} + \)\(41\!\cdots\!72\)\( T^{6} + \)\(68\!\cdots\!96\)\( T^{7} + \)\(12\!\cdots\!20\)\( T^{8} + \)\(18\!\cdots\!52\)\( T^{9} + \)\(28\!\cdots\!92\)\( T^{10} + \)\(37\!\cdots\!80\)\( T^{11} + \)\(52\!\cdots\!64\)\( T^{12} + \)\(64\!\cdots\!60\)\( T^{13} + \)\(81\!\cdots\!80\)\( T^{14} + \)\(92\!\cdots\!44\)\( T^{15} + \)\(11\!\cdots\!10\)\( T^{16} + \)\(11\!\cdots\!08\)\( T^{17} + \)\(12\!\cdots\!10\)\( T^{18} + \)\(12\!\cdots\!64\)\( T^{19} + \)\(13\!\cdots\!80\)\( T^{20} + \)\(11\!\cdots\!60\)\( T^{21} + \)\(11\!\cdots\!64\)\( T^{22} + \)\(95\!\cdots\!80\)\( T^{23} + \)\(84\!\cdots\!52\)\( T^{24} + \)\(63\!\cdots\!92\)\( T^{25} + \)\(50\!\cdots\!20\)\( T^{26} + \)\(32\!\cdots\!96\)\( T^{27} + \)\(23\!\cdots\!52\)\( T^{28} + \)\(12\!\cdots\!36\)\( T^{29} + \)\(75\!\cdots\!00\)\( T^{30} + \)\(31\!\cdots\!00\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} + \)\(38\!\cdots\!58\)\( T^{33} + \)\(14\!\cdots\!81\)\( T^{34} \)
$67$ \( 1 + 292438130 T + 120978400070314508 T^{2} + \)\(28\!\cdots\!68\)\( T^{3} + \)\(80\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!72\)\( T^{5} + \)\(39\!\cdots\!65\)\( T^{6} + \)\(76\!\cdots\!16\)\( T^{7} + \)\(16\!\cdots\!16\)\( T^{8} + \)\(29\!\cdots\!82\)\( T^{9} + \)\(60\!\cdots\!63\)\( T^{10} + \)\(10\!\cdots\!88\)\( T^{11} + \)\(19\!\cdots\!86\)\( T^{12} + \)\(30\!\cdots\!92\)\( T^{13} + \)\(57\!\cdots\!32\)\( T^{14} + \)\(89\!\cdots\!12\)\( T^{15} + \)\(16\!\cdots\!18\)\( T^{16} + \)\(24\!\cdots\!68\)\( T^{17} + \)\(44\!\cdots\!46\)\( T^{18} + \)\(66\!\cdots\!08\)\( T^{19} + \)\(11\!\cdots\!36\)\( T^{20} + \)\(16\!\cdots\!52\)\( T^{21} + \)\(28\!\cdots\!02\)\( T^{22} + \)\(41\!\cdots\!52\)\( T^{23} + \)\(66\!\cdots\!69\)\( T^{24} + \)\(89\!\cdots\!02\)\( T^{25} + \)\(13\!\cdots\!72\)\( T^{26} + \)\(16\!\cdots\!84\)\( T^{27} + \)\(24\!\cdots\!95\)\( T^{28} + \)\(27\!\cdots\!52\)\( T^{29} + \)\(36\!\cdots\!65\)\( T^{30} + \)\(34\!\cdots\!92\)\( T^{31} + \)\(40\!\cdots\!44\)\( T^{32} + \)\(26\!\cdots\!30\)\( T^{33} + \)\(24\!\cdots\!87\)\( T^{34} \)
$71$ \( 1 + 13576908 T + 315744496029563231 T^{2} - \)\(13\!\cdots\!88\)\( T^{3} + \)\(54\!\cdots\!88\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(66\!\cdots\!72\)\( T^{6} - \)\(20\!\cdots\!76\)\( T^{7} + \)\(63\!\cdots\!52\)\( T^{8} - \)\(25\!\cdots\!56\)\( T^{9} + \)\(50\!\cdots\!44\)\( T^{10} - \)\(23\!\cdots\!32\)\( T^{11} + \)\(34\!\cdots\!92\)\( T^{12} - \)\(16\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!16\)\( T^{14} - \)\(99\!\cdots\!68\)\( T^{15} + \)\(10\!\cdots\!26\)\( T^{16} - \)\(49\!\cdots\!64\)\( T^{17} + \)\(49\!\cdots\!06\)\( T^{18} - \)\(20\!\cdots\!48\)\( T^{19} + \)\(19\!\cdots\!56\)\( T^{20} - \)\(73\!\cdots\!28\)\( T^{21} + \)\(70\!\cdots\!92\)\( T^{22} - \)\(21\!\cdots\!92\)\( T^{23} + \)\(21\!\cdots\!84\)\( T^{24} - \)\(50\!\cdots\!96\)\( T^{25} + \)\(56\!\cdots\!92\)\( T^{26} - \)\(85\!\cdots\!76\)\( T^{27} + \)\(12\!\cdots\!32\)\( T^{28} - \)\(91\!\cdots\!00\)\( T^{29} + \)\(21\!\cdots\!08\)\( T^{30} - \)\(25\!\cdots\!48\)\( T^{31} + \)\(26\!\cdots\!81\)\( T^{32} + \)\(51\!\cdots\!48\)\( T^{33} + \)\(17\!\cdots\!11\)\( T^{34} \)
$73$ \( 1 + 501490738 T + 550612673986219501 T^{2} + \)\(26\!\cdots\!76\)\( T^{3} + \)\(16\!\cdots\!84\)\( T^{4} + \)\(71\!\cdots\!20\)\( T^{5} + \)\(33\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!84\)\( T^{7} + \)\(50\!\cdots\!76\)\( T^{8} + \)\(17\!\cdots\!68\)\( T^{9} + \)\(61\!\cdots\!60\)\( T^{10} + \)\(19\!\cdots\!52\)\( T^{11} + \)\(60\!\cdots\!36\)\( T^{12} + \)\(17\!\cdots\!80\)\( T^{13} + \)\(49\!\cdots\!28\)\( T^{14} + \)\(13\!\cdots\!72\)\( T^{15} + \)\(34\!\cdots\!50\)\( T^{16} + \)\(84\!\cdots\!08\)\( T^{17} + \)\(20\!\cdots\!50\)\( T^{18} + \)\(45\!\cdots\!68\)\( T^{19} + \)\(10\!\cdots\!16\)\( T^{20} + \)\(20\!\cdots\!80\)\( T^{21} + \)\(42\!\cdots\!48\)\( T^{22} + \)\(80\!\cdots\!68\)\( T^{23} + \)\(14\!\cdots\!20\)\( T^{24} + \)\(25\!\cdots\!28\)\( T^{25} + \)\(43\!\cdots\!48\)\( T^{26} + \)\(64\!\cdots\!16\)\( T^{27} + \)\(97\!\cdots\!60\)\( T^{28} + \)\(12\!\cdots\!20\)\( T^{29} + \)\(16\!\cdots\!52\)\( T^{30} + \)\(16\!\cdots\!64\)\( T^{31} + \)\(19\!\cdots\!57\)\( T^{32} + \)\(10\!\cdots\!58\)\( T^{33} + \)\(12\!\cdots\!33\)\( T^{34} \)
$79$ \( 1 - 740350275 T + 1257538224143455107 T^{2} - \)\(71\!\cdots\!27\)\( T^{3} + \)\(70\!\cdots\!00\)\( T^{4} - \)\(32\!\cdots\!53\)\( T^{5} + \)\(23\!\cdots\!71\)\( T^{6} - \)\(89\!\cdots\!33\)\( T^{7} + \)\(55\!\cdots\!59\)\( T^{8} - \)\(17\!\cdots\!42\)\( T^{9} + \)\(10\!\cdots\!89\)\( T^{10} - \)\(29\!\cdots\!71\)\( T^{11} + \)\(16\!\cdots\!61\)\( T^{12} - \)\(43\!\cdots\!59\)\( T^{13} + \)\(25\!\cdots\!38\)\( T^{14} - \)\(63\!\cdots\!09\)\( T^{15} + \)\(34\!\cdots\!84\)\( T^{16} - \)\(81\!\cdots\!42\)\( T^{17} + \)\(41\!\cdots\!96\)\( T^{18} - \)\(90\!\cdots\!49\)\( T^{19} + \)\(43\!\cdots\!42\)\( T^{20} - \)\(90\!\cdots\!39\)\( T^{21} + \)\(41\!\cdots\!39\)\( T^{22} - \)\(86\!\cdots\!51\)\( T^{23} + \)\(36\!\cdots\!71\)\( T^{24} - \)\(75\!\cdots\!22\)\( T^{25} + \)\(28\!\cdots\!61\)\( T^{26} - \)\(54\!\cdots\!33\)\( T^{27} + \)\(17\!\cdots\!49\)\( T^{28} - \)\(28\!\cdots\!33\)\( T^{29} + \)\(73\!\cdots\!00\)\( T^{30} - \)\(90\!\cdots\!67\)\( T^{31} + \)\(19\!\cdots\!93\)\( T^{32} - \)\(13\!\cdots\!75\)\( T^{33} + \)\(21\!\cdots\!39\)\( T^{34} \)
$83$ \( 1 - 754109940 T + 1777938959892655188 T^{2} - \)\(10\!\cdots\!46\)\( T^{3} + \)\(14\!\cdots\!03\)\( T^{4} - \)\(78\!\cdots\!84\)\( T^{5} + \)\(82\!\cdots\!17\)\( T^{6} - \)\(39\!\cdots\!38\)\( T^{7} + \)\(34\!\cdots\!52\)\( T^{8} - \)\(15\!\cdots\!72\)\( T^{9} + \)\(11\!\cdots\!19\)\( T^{10} - \)\(48\!\cdots\!28\)\( T^{11} + \)\(32\!\cdots\!30\)\( T^{12} - \)\(13\!\cdots\!00\)\( T^{13} + \)\(78\!\cdots\!32\)\( T^{14} - \)\(30\!\cdots\!52\)\( T^{15} + \)\(16\!\cdots\!62\)\( T^{16} - \)\(60\!\cdots\!08\)\( T^{17} + \)\(31\!\cdots\!86\)\( T^{18} - \)\(10\!\cdots\!68\)\( T^{19} + \)\(51\!\cdots\!64\)\( T^{20} - \)\(15\!\cdots\!00\)\( T^{21} + \)\(74\!\cdots\!90\)\( T^{22} - \)\(20\!\cdots\!12\)\( T^{23} + \)\(91\!\cdots\!53\)\( T^{24} - \)\(22\!\cdots\!92\)\( T^{25} + \)\(95\!\cdots\!16\)\( T^{26} - \)\(20\!\cdots\!62\)\( T^{27} + \)\(80\!\cdots\!99\)\( T^{28} - \)\(14\!\cdots\!44\)\( T^{29} + \)\(50\!\cdots\!69\)\( T^{30} - \)\(68\!\cdots\!74\)\( T^{31} + \)\(21\!\cdots\!16\)\( T^{32} - \)\(16\!\cdots\!40\)\( T^{33} + \)\(41\!\cdots\!63\)\( T^{34} \)
$89$ \( 1 - 1470581868 T + 5106897759371855857 T^{2} - \)\(64\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!00\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{5} + \)\(17\!\cdots\!96\)\( T^{6} - \)\(17\!\cdots\!76\)\( T^{7} + \)\(18\!\cdots\!04\)\( T^{8} - \)\(15\!\cdots\!28\)\( T^{9} + \)\(14\!\cdots\!56\)\( T^{10} - \)\(10\!\cdots\!32\)\( T^{11} + \)\(84\!\cdots\!40\)\( T^{12} - \)\(59\!\cdots\!88\)\( T^{13} + \)\(41\!\cdots\!56\)\( T^{14} - \)\(27\!\cdots\!72\)\( T^{15} + \)\(17\!\cdots\!46\)\( T^{16} - \)\(10\!\cdots\!80\)\( T^{17} + \)\(60\!\cdots\!14\)\( T^{18} - \)\(33\!\cdots\!32\)\( T^{19} + \)\(17\!\cdots\!24\)\( T^{20} - \)\(90\!\cdots\!68\)\( T^{21} + \)\(44\!\cdots\!60\)\( T^{22} - \)\(20\!\cdots\!12\)\( T^{23} + \)\(90\!\cdots\!64\)\( T^{24} - \)\(35\!\cdots\!88\)\( T^{25} + \)\(14\!\cdots\!56\)\( T^{26} - \)\(47\!\cdots\!76\)\( T^{27} + \)\(17\!\cdots\!64\)\( T^{28} - \)\(45\!\cdots\!40\)\( T^{29} + \)\(14\!\cdots\!00\)\( T^{30} - \)\(26\!\cdots\!56\)\( T^{31} + \)\(75\!\cdots\!93\)\( T^{32} - \)\(75\!\cdots\!88\)\( T^{33} + \)\(18\!\cdots\!69\)\( T^{34} \)
$97$ \( 1 - 1949310583 T + 7466841361727255818 T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!27\)\( T^{4} - \)\(33\!\cdots\!20\)\( T^{5} + \)\(54\!\cdots\!52\)\( T^{6} - \)\(60\!\cdots\!27\)\( T^{7} + \)\(79\!\cdots\!90\)\( T^{8} - \)\(74\!\cdots\!91\)\( T^{9} + \)\(83\!\cdots\!31\)\( T^{10} - \)\(65\!\cdots\!81\)\( T^{11} + \)\(64\!\cdots\!39\)\( T^{12} - \)\(39\!\cdots\!90\)\( T^{13} + \)\(37\!\cdots\!95\)\( T^{14} - \)\(16\!\cdots\!99\)\( T^{15} + \)\(18\!\cdots\!76\)\( T^{16} - \)\(70\!\cdots\!80\)\( T^{17} + \)\(14\!\cdots\!92\)\( T^{18} - \)\(93\!\cdots\!11\)\( T^{19} + \)\(16\!\cdots\!35\)\( T^{20} - \)\(13\!\cdots\!90\)\( T^{21} + \)\(16\!\cdots\!23\)\( T^{22} - \)\(12\!\cdots\!89\)\( T^{23} + \)\(12\!\cdots\!63\)\( T^{24} - \)\(82\!\cdots\!31\)\( T^{25} + \)\(67\!\cdots\!30\)\( T^{26} - \)\(38\!\cdots\!23\)\( T^{27} + \)\(26\!\cdots\!16\)\( T^{28} - \)\(12\!\cdots\!20\)\( T^{29} + \)\(72\!\cdots\!99\)\( T^{30} - \)\(25\!\cdots\!40\)\( T^{31} + \)\(12\!\cdots\!74\)\( T^{32} - \)\(24\!\cdots\!23\)\( T^{33} + \)\(94\!\cdots\!77\)\( T^{34} \)
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