Properties

Label 43.8.a.a
Level 43
Weight 8
Character orbit 43.a
Self dual yes
Analytic conductor 13.433
Analytic rank 1
Dimension 11
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 - \beta_{1} ) q^{2} + ( -6 - \beta_{5} ) q^{3} + ( 54 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( -69 + 2 \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{10} ) q^{5} + ( -64 + 6 \beta_{1} - 3 \beta_{2} + 8 \beta_{5} + \beta_{8} - 2 \beta_{10} ) q^{6} + ( -2 + 12 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} - 3 \beta_{8} + \beta_{9} - 3 \beta_{10} ) q^{7} + ( -348 - 8 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 18 \beta_{5} + 3 \beta_{7} - \beta_{8} - 4 \beta_{9} + 2 \beta_{10} ) q^{8} + ( 224 + 90 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + \beta_{4} + 21 \beta_{5} + 3 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{9} +O(q^{10})\) \( q + ( -2 - \beta_{1} ) q^{2} + ( -6 - \beta_{5} ) q^{3} + ( 54 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( -69 + 2 \beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{10} ) q^{5} + ( -64 + 6 \beta_{1} - 3 \beta_{2} + 8 \beta_{5} + \beta_{8} - 2 \beta_{10} ) q^{6} + ( -2 + 12 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} - 3 \beta_{8} + \beta_{9} - 3 \beta_{10} ) q^{7} + ( -348 - 8 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 18 \beta_{5} + 3 \beta_{7} - \beta_{8} - 4 \beta_{9} + 2 \beta_{10} ) q^{8} + ( 224 + 90 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + \beta_{4} + 21 \beta_{5} + 3 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} ) q^{9} + ( -149 + 182 \beta_{1} - 2 \beta_{2} - 15 \beta_{3} + 10 \beta_{4} - 14 \beta_{5} - \beta_{6} - 9 \beta_{7} + \beta_{8} + 4 \beta_{9} + 2 \beta_{10} ) q^{10} + ( 86 + 154 \beta_{1} + 17 \beta_{2} + 3 \beta_{3} - 10 \beta_{4} + 5 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} + 11 \beta_{8} - \beta_{9} ) q^{11} + ( 431 + 229 \beta_{1} + 21 \beta_{2} + 5 \beta_{3} + 6 \beta_{4} - 64 \beta_{5} - 7 \beta_{6} - 4 \beta_{7} - 10 \beta_{8} + 14 \beta_{9} + 12 \beta_{10} ) q^{12} + ( -1647 + 96 \beta_{1} + 3 \beta_{2} - 23 \beta_{3} + 8 \beta_{4} + \beta_{5} - 19 \beta_{6} + 12 \beta_{7} - 6 \beta_{8} - 19 \beta_{9} ) q^{13} + ( -2059 + 37 \beta_{1} + 24 \beta_{2} + 71 \beta_{3} - 28 \beta_{4} + 4 \beta_{5} + 39 \beta_{6} + 36 \beta_{7} + \beta_{8} - 12 \beta_{9} + 12 \beta_{10} ) q^{14} + ( -4472 - 138 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 18 \beta_{4} + 11 \beta_{5} + 9 \beta_{6} + 12 \beta_{7} - 4 \beta_{8} - 39 \beta_{9} - 22 \beta_{10} ) q^{15} + ( -3175 + 484 \beta_{1} + 7 \beta_{2} - 71 \beta_{3} + 34 \beta_{4} - 256 \beta_{5} - 29 \beta_{6} - 29 \beta_{7} - 22 \beta_{8} + 76 \beta_{9} - 26 \beta_{10} ) q^{16} + ( -5628 - 320 \beta_{1} - 38 \beta_{2} + 49 \beta_{3} - 5 \beta_{4} - 63 \beta_{5} - 18 \beta_{6} - 33 \beta_{7} + 5 \beta_{8} - 9 \beta_{9} - 40 \beta_{10} ) q^{17} + ( -14911 - 332 \beta_{1} - 132 \beta_{2} - 76 \beta_{3} + 24 \beta_{4} - 248 \beta_{5} - 25 \beta_{6} - 67 \beta_{7} - 37 \beta_{8} - 16 \beta_{9} - 6 \beta_{10} ) q^{18} + ( -5058 + 252 \beta_{1} + 40 \beta_{2} - \beta_{3} + 43 \beta_{4} + 79 \beta_{5} + 17 \beta_{6} + 7 \beta_{7} + 54 \beta_{8} + 79 \beta_{9} - 48 \beta_{10} ) q^{19} + ( -25410 - 881 \beta_{1} - 188 \beta_{2} + 50 \beta_{3} - 28 \beta_{4} + 200 \beta_{5} + 42 \beta_{6} + 45 \beta_{7} + 70 \beta_{8} - 78 \beta_{9} + 62 \beta_{10} ) q^{20} + ( -12844 - 346 \beta_{1} + 176 \beta_{2} - 69 \beta_{3} - 82 \beta_{4} + 294 \beta_{5} + 124 \beta_{6} + 28 \beta_{7} + 69 \beta_{8} + 71 \beta_{9} - 15 \beta_{10} ) q^{21} + ( -25958 - 1353 \beta_{1} - 213 \beta_{2} + 19 \beta_{3} + 68 \beta_{4} + 36 \beta_{5} - 102 \beta_{6} + 25 \beta_{7} - 45 \beta_{8} - 100 \beta_{9} + 96 \beta_{10} ) q^{22} + ( -12407 - 1490 \beta_{1} + 125 \beta_{2} + 125 \beta_{3} + 23 \beta_{4} + 396 \beta_{5} - 138 \beta_{6} + 87 \beta_{7} - 11 \beta_{8} + 75 \beta_{9} + 169 \beta_{10} ) q^{23} + ( -38587 - 2533 \beta_{1} - 199 \beta_{2} + 99 \beta_{3} - 94 \beta_{4} + 436 \beta_{5} + 127 \beta_{6} + 58 \beta_{7} - 30 \beta_{8} - 298 \beta_{9} + 120 \beta_{10} ) q^{24} + ( 614 - 1494 \beta_{1} + 261 \beta_{2} + 154 \beta_{3} + 35 \beta_{4} + 74 \beta_{5} + 37 \beta_{6} - 85 \beta_{7} - 129 \beta_{8} + 74 \beta_{9} - 126 \beta_{10} ) q^{25} + ( -12514 + 759 \beta_{1} + 445 \beta_{2} - 303 \beta_{3} - 172 \beta_{4} - 236 \beta_{5} - 174 \beta_{6} + 51 \beta_{7} + 65 \beta_{8} + 368 \beta_{9} + 76 \beta_{10} ) q^{26} + ( -21057 - 3094 \beta_{1} + 11 \beta_{2} - 174 \beta_{3} + 158 \beta_{4} + 147 \beta_{5} - 182 \beta_{6} - 152 \beta_{7} - 310 \beta_{8} - 196 \beta_{9} + 107 \beta_{10} ) q^{27} + ( -1204 + 1528 \beta_{1} - 46 \beta_{2} + 34 \beta_{3} + 76 \beta_{4} - 480 \beta_{5} + 156 \beta_{6} - 282 \beta_{7} + 86 \beta_{8} + 28 \beta_{9} - 440 \beta_{10} ) q^{28} + ( -28029 - 1554 \beta_{1} + 36 \beta_{2} + 24 \beta_{3} - 78 \beta_{4} + 957 \beta_{5} + 405 \beta_{6} + 18 \beta_{7} + 153 \beta_{8} - 144 \beta_{9} - 186 \beta_{10} ) q^{29} + ( 37316 + 3968 \beta_{1} + 915 \beta_{2} + 55 \beta_{3} - 96 \beta_{4} - 764 \beta_{5} - 122 \beta_{6} + 127 \beta_{7} - 81 \beta_{8} + 568 \beta_{9} - 28 \beta_{10} ) q^{30} + ( -19383 + 178 \beta_{1} + 10 \beta_{2} - 203 \beta_{3} + 103 \beta_{4} + 1193 \beta_{5} + 211 \beta_{6} + 199 \beta_{7} + 194 \beta_{8} - 233 \beta_{9} - 236 \beta_{10} ) q^{31} + ( -58833 + 982 \beta_{1} - 1121 \beta_{2} + 179 \beta_{3} + 86 \beta_{4} + 1184 \beta_{5} + 201 \beta_{6} + 345 \beta_{7} + 466 \beta_{8} - 672 \beta_{9} - 82 \beta_{10} ) q^{32} + ( 9886 - 826 \beta_{1} - 145 \beta_{2} - 315 \beta_{3} + 419 \beta_{4} - 2063 \beta_{5} - 734 \beta_{6} - 647 \beta_{7} - 665 \beta_{8} + 341 \beta_{9} + 97 \beta_{10} ) q^{33} + ( 68096 + 7674 \beta_{1} + 404 \beta_{2} + 934 \beta_{3} - 82 \beta_{4} - 1354 \beta_{5} + 148 \beta_{6} + 89 \beta_{7} + 210 \beta_{8} + 116 \beta_{9} - 282 \beta_{10} ) q^{34} + ( -52324 - 208 \beta_{1} - 808 \beta_{2} - 671 \beta_{3} + 284 \beta_{4} - 770 \beta_{5} - 1038 \beta_{6} - 66 \beta_{7} + 15 \beta_{8} + 355 \beta_{9} + 695 \beta_{10} ) q^{35} + ( 36862 + 9428 \beta_{1} - 15 \beta_{2} + 10 \beta_{3} - 84 \beta_{4} - 416 \beta_{5} + 304 \beta_{6} + 67 \beta_{7} + 316 \beta_{8} + 118 \beta_{9} + 90 \beta_{10} ) q^{36} + ( -28269 + 144 \beta_{1} + 719 \beta_{2} - 537 \beta_{3} - 1167 \beta_{4} + 456 \beta_{5} + 681 \beta_{6} + 663 \beta_{7} + 276 \beta_{8} - 9 \beta_{9} - 195 \beta_{10} ) q^{37} + ( -32478 + 315 \beta_{1} - 1766 \beta_{2} + 412 \beta_{3} - 190 \beta_{4} - 1806 \beta_{5} + 72 \beta_{6} - 111 \beta_{7} - 420 \beta_{8} - 1064 \beta_{9} + 246 \beta_{10} ) q^{38} + ( 25406 - 356 \beta_{1} + 437 \beta_{2} + 1801 \beta_{3} - 439 \beta_{4} + 3057 \beta_{5} + 902 \beta_{6} + 947 \beta_{7} + 425 \beta_{8} - 117 \beta_{9} + 1495 \beta_{10} ) q^{39} + ( 236268 + 21211 \beta_{1} + 1284 \beta_{2} - 334 \beta_{3} + 360 \beta_{4} - 5244 \beta_{5} - 552 \beta_{6} - 707 \beta_{7} - 292 \beta_{8} + 1602 \beta_{9} - 758 \beta_{10} ) q^{40} + ( -118942 - 9666 \beta_{1} - 3199 \beta_{2} - 198 \beta_{3} - 105 \beta_{4} - 1528 \beta_{5} + 663 \beta_{6} - 1377 \beta_{7} + 93 \beta_{8} - 1158 \beta_{9} - 386 \beta_{10} ) q^{41} + ( 107103 + 1938 \beta_{1} - 326 \beta_{2} - 536 \beta_{3} + 502 \beta_{4} + 518 \beta_{5} - 711 \beta_{6} + 546 \beta_{7} - 1359 \beta_{8} - 1804 \beta_{9} + 1292 \beta_{10} ) q^{42} + 79507 q^{43} + ( 280631 + 25063 \beta_{1} + 686 \beta_{2} - 2165 \beta_{3} + 1626 \beta_{4} - 4064 \beta_{5} - 753 \beta_{6} - 603 \beta_{7} - 452 \beta_{8} + 2780 \beta_{9} - 714 \beta_{10} ) q^{44} + ( 172805 - 10576 \beta_{1} + 1112 \beta_{2} + 1551 \beta_{3} - 355 \beta_{4} + 4412 \beta_{5} + 322 \beta_{6} + 883 \beta_{7} - 9 \beta_{8} + 119 \beta_{9} - 438 \beta_{10} ) q^{45} + ( 323799 + 18030 \beta_{1} + 1613 \beta_{2} - 2701 \beta_{3} + 184 \beta_{4} - 2456 \beta_{5} - 625 \beta_{6} - 1376 \beta_{7} - 210 \beta_{8} - 728 \beta_{9} - 1230 \beta_{10} ) q^{46} + ( 45611 - 9978 \beta_{1} - 71 \beta_{2} + 500 \beta_{3} + 289 \beta_{4} + 3542 \beta_{5} - 111 \beta_{6} + 1209 \beta_{7} + 819 \beta_{8} + 788 \beta_{9} + 1012 \beta_{10} ) q^{47} + ( 505779 + 29431 \beta_{1} + 3385 \beta_{2} - 2397 \beta_{3} + 790 \beta_{4} + 3440 \beta_{5} + 1049 \beta_{6} - 1330 \beta_{7} + 1108 \beta_{8} + 3022 \beta_{9} - 1832 \beta_{10} ) q^{48} + ( 235803 - 17144 \beta_{1} + 240 \beta_{2} + 2519 \beta_{3} + 882 \beta_{4} - 572 \beta_{5} - 2122 \beta_{6} + 360 \beta_{7} - 547 \beta_{8} - 519 \beta_{9} - 473 \beta_{10} ) q^{49} + ( 280246 - 25230 \beta_{1} + 1133 \beta_{2} + 6131 \beta_{3} - 2562 \beta_{4} + 7830 \beta_{5} + 2856 \beta_{6} + 2148 \beta_{7} - 307 \beta_{8} - 2852 \beta_{9} + 26 \beta_{10} ) q^{50} + ( 116238 - 20124 \beta_{1} + 47 \beta_{2} + 680 \beta_{3} - 346 \beta_{4} + 9246 \beta_{5} - 249 \beta_{6} - 1032 \beta_{7} + 2343 \beta_{8} + 106 \beta_{9} - 5 \beta_{10} ) q^{51} + ( 92357 - 16801 \beta_{1} - 2108 \beta_{2} - 1891 \beta_{3} - 538 \beta_{4} + 6752 \beta_{5} - 1871 \beta_{6} + 1075 \beta_{7} - 116 \beta_{8} - 4060 \beta_{9} + 1034 \beta_{10} ) q^{52} + ( -193225 - 34520 \beta_{1} - 1797 \beta_{2} - 4036 \beta_{3} - 52 \beta_{4} - 7529 \beta_{5} - 231 \beta_{6} - 1194 \beta_{7} - 1223 \beta_{8} + 1674 \beta_{9} - 2151 \beta_{10} ) q^{53} + ( 608229 + 9314 \beta_{1} + 9175 \beta_{2} + 2789 \beta_{3} - 2966 \beta_{4} + 12610 \beta_{5} + 2389 \beta_{6} + 2371 \beta_{7} + 2228 \beta_{8} + 1848 \beta_{9} + 2952 \beta_{10} ) q^{54} + ( -164247 - 37462 \beta_{1} + 371 \beta_{2} + 3351 \beta_{3} - 3809 \beta_{4} + 6415 \beta_{5} + 3959 \beta_{6} + 2061 \beta_{7} + 1582 \beta_{8} - 3383 \beta_{9} - 1165 \beta_{10} ) q^{55} + ( -34154 - 29776 \beta_{1} - 7438 \beta_{2} + 738 \beta_{3} + 1556 \beta_{4} + 296 \beta_{5} - 2334 \beta_{6} - 1866 \beta_{7} + 188 \beta_{8} - 352 \beta_{9} + 540 \beta_{10} ) q^{56} + ( -101998 - 29622 \beta_{1} - 2529 \beta_{2} - 5484 \beta_{3} + 3588 \beta_{4} - 12161 \beta_{5} - 3381 \beta_{6} - 3534 \beta_{7} - 3059 \beta_{8} + 1932 \beta_{9} - 2165 \beta_{10} ) q^{57} + ( 398880 + 14700 \beta_{1} + 2115 \beta_{2} + 2664 \beta_{3} + 648 \beta_{4} - 8772 \beta_{5} + 900 \beta_{6} - 270 \beta_{7} - 2673 \beta_{8} + 1320 \beta_{9} + 2862 \beta_{10} ) q^{58} + ( -526806 - 3530 \beta_{1} - 3048 \beta_{2} - 6035 \beta_{3} + 3432 \beta_{4} - 11440 \beta_{5} + 1740 \beta_{6} - 2214 \beta_{7} + 823 \beta_{8} + 2177 \beta_{9} + 1325 \beta_{10} ) q^{59} + ( -249299 - 72348 \beta_{1} - 9641 \beta_{2} + 1519 \beta_{3} - 542 \beta_{4} + 19208 \beta_{5} - 1737 \beta_{6} + 2163 \beta_{7} - 1084 \beta_{8} - 5968 \beta_{9} - 326 \beta_{10} ) q^{60} + ( -407087 + 19608 \beta_{1} - 1529 \beta_{2} - 3735 \beta_{3} + 4179 \beta_{4} - 11745 \beta_{5} - 1479 \beta_{6} + 3693 \beta_{7} - 420 \beta_{8} + 561 \beta_{9} + 831 \beta_{10} ) q^{61} + ( 92746 - 136 \beta_{1} + 3730 \beta_{2} - 1198 \beta_{3} - 954 \beta_{4} - 18502 \beta_{5} - 1050 \beta_{6} - 951 \beta_{7} - 2626 \beta_{8} + 4588 \beta_{9} + 3522 \beta_{10} ) q^{62} + ( -558701 - 20446 \beta_{1} - 4441 \beta_{2} - 8313 \beta_{3} + 3731 \beta_{4} - 8527 \beta_{5} - 5747 \beta_{6} - 3971 \beta_{7} - 3200 \beta_{8} + 1721 \beta_{9} + 607 \beta_{10} ) q^{63} + ( 422185 + 73866 \beta_{1} + 2415 \beta_{2} + 15 \beta_{3} + 978 \beta_{4} - 20276 \beta_{5} + 359 \beta_{6} - 6079 \beta_{7} + 1656 \beta_{8} + 6936 \beta_{9} + 1422 \beta_{10} ) q^{64} + ( -190417 - 17012 \beta_{1} + 6519 \beta_{2} + 9739 \beta_{3} - 1847 \beta_{4} - 26611 \beta_{5} + 2199 \beta_{6} + 1683 \beta_{7} - 1732 \beta_{8} + 4479 \beta_{9} - 5673 \beta_{10} ) q^{65} + ( -31907 - 19463 \beta_{1} + 760 \beta_{2} + 10329 \beta_{3} - 6128 \beta_{4} + 42472 \beta_{5} + 5771 \beta_{6} + 7532 \beta_{7} + 8013 \beta_{8} - 7100 \beta_{9} + 1932 \beta_{10} ) q^{66} + ( -624554 + 3140 \beta_{1} + 6641 \beta_{2} + 12623 \beta_{3} - 8962 \beta_{4} - 10319 \beta_{5} + 6392 \beta_{6} + 3506 \beta_{7} - 1313 \beta_{8} - 4951 \beta_{9} + 3446 \beta_{10} ) q^{67} + ( -843478 - 53242 \beta_{1} - 13851 \beta_{2} + 978 \beta_{3} - 388 \beta_{4} + 7200 \beta_{5} + 4572 \beta_{6} + 1773 \beta_{7} - 1660 \beta_{8} - 1594 \beta_{9} - 4822 \beta_{10} ) q^{68} + ( -908209 - 5196 \beta_{1} - 3128 \beta_{2} - 203 \beta_{3} + 319 \beta_{4} + 3980 \beta_{5} + 3156 \beta_{6} - 2379 \beta_{7} + 2623 \beta_{8} - 9583 \beta_{9} - 4446 \beta_{10} ) q^{69} + ( 50901 + 142680 \beta_{1} + 1200 \beta_{2} - 18540 \beta_{3} + 4422 \beta_{4} - 7450 \beta_{5} - 7889 \beta_{6} - 3428 \beta_{7} + 6189 \beta_{8} + 540 \beta_{9} + 684 \beta_{10} ) q^{70} + ( -977721 + 25174 \beta_{1} + 2790 \beta_{2} - 210 \beta_{3} - 4000 \beta_{4} - 10368 \beta_{5} - 5121 \beta_{6} - 5124 \beta_{7} + 4691 \beta_{8} + 806 \beta_{9} - 670 \beta_{10} ) q^{71} + ( 99788 + 6017 \beta_{1} - 1067 \beta_{2} + 5885 \beta_{3} + 772 \beta_{4} + 30926 \beta_{5} + 1612 \beta_{6} + 5554 \beta_{7} + 3447 \beta_{8} + 714 \beta_{9} - 8 \beta_{10} ) q^{72} + ( -425520 + 119766 \beta_{1} + 5591 \beta_{2} + 3681 \beta_{3} + 3585 \beta_{4} - 16935 \beta_{5} - 3474 \beta_{6} + 6159 \beta_{7} - 5469 \beta_{8} + 4629 \beta_{9} - 861 \beta_{10} ) q^{73} + ( 95071 + 13933 \beta_{1} + 7545 \beta_{2} - 13683 \beta_{3} + 5956 \beta_{4} - 7344 \beta_{5} - 11667 \beta_{6} + 4206 \beta_{7} - 8852 \beta_{8} - 1916 \beta_{9} - 386 \beta_{10} ) q^{74} + ( -331398 + 94564 \beta_{1} + 10884 \beta_{2} - 3235 \beta_{3} - 1377 \beta_{4} + 24548 \beta_{5} + 257 \beta_{6} - 1801 \beta_{7} + 3580 \beta_{8} + 6737 \beta_{9} - 3526 \beta_{10} ) q^{75} + ( 510282 + 129587 \beta_{1} + 3156 \beta_{2} - 604 \beta_{3} - 328 \beta_{4} - 13520 \beta_{5} + 1370 \beta_{6} - 6199 \beta_{7} - 452 \beta_{8} + 9818 \beta_{9} - 2134 \beta_{10} ) q^{76} + ( -1557383 + 64704 \beta_{1} + 14275 \beta_{2} - 5425 \beta_{3} - 3349 \beta_{4} - 26941 \beta_{5} + 5007 \beta_{6} - 1563 \beta_{7} + 3280 \beta_{8} - 7461 \beta_{9} + 4229 \beta_{10} ) q^{77} + ( 189803 + 56003 \beta_{1} - 15454 \beta_{2} - 20553 \beta_{3} + 13976 \beta_{4} - 18624 \beta_{5} + 781 \beta_{6} - 21836 \beta_{7} - 8067 \beta_{8} + 3740 \beta_{9} - 14880 \beta_{10} ) q^{78} + ( -1428975 + 91052 \beta_{1} - 3532 \beta_{2} + 7858 \beta_{3} + 597 \beta_{4} + 27231 \beta_{5} + 1590 \beta_{6} + 4229 \beta_{7} - 8084 \beta_{8} - 12828 \beta_{9} + 16649 \beta_{10} ) q^{79} + ( -1376036 - 267761 \beta_{1} - 21492 \beta_{2} + 16210 \beta_{3} - 7912 \beta_{4} + 48188 \beta_{5} + 120 \beta_{6} + 10269 \beta_{7} - 4224 \beta_{8} - 20582 \beta_{9} - 10286 \beta_{10} ) q^{80} + ( -1204643 + 646 \beta_{1} + 12561 \beta_{2} + 14855 \beta_{3} - 20124 \beta_{4} + 58117 \beta_{5} + 15341 \beta_{6} + 10016 \beta_{7} + 9618 \beta_{8} - 715 \beta_{9} - 5690 \beta_{10} ) q^{81} + ( 1782638 + 287504 \beta_{1} + 13827 \beta_{2} + 15543 \beta_{3} + 10678 \beta_{4} - 18002 \beta_{5} + 6908 \beta_{6} - 1656 \beta_{7} + 9487 \beta_{8} + 19396 \beta_{9} + 9518 \beta_{10} ) q^{82} + ( -1028644 + 45292 \beta_{1} + 13123 \beta_{2} + 16352 \beta_{3} + 3900 \beta_{4} + 22169 \beta_{5} - 2166 \beta_{6} + 4578 \beta_{7} - 2182 \beta_{8} + 4348 \beta_{9} + 22149 \beta_{10} ) q^{83} + ( 1127202 - 47470 \beta_{1} + 9242 \beta_{2} - 6174 \beta_{3} + 3356 \beta_{4} - 1072 \beta_{5} - 12026 \beta_{6} - 6488 \beta_{7} - 520 \beta_{8} + 20456 \beta_{9} + 1604 \beta_{10} ) q^{84} + ( -1166360 + 174478 \beta_{1} + 6524 \beta_{2} - 20239 \beta_{3} + 1851 \beta_{4} - 18908 \beta_{5} - 8845 \beta_{6} + 1101 \beta_{7} - 4336 \beta_{8} + 5445 \beta_{9} + 2284 \beta_{10} ) q^{85} + ( -159014 - 79507 \beta_{1} ) q^{86} + ( -1673823 - 90342 \beta_{1} - 19602 \beta_{2} - 8928 \beta_{3} + 4545 \beta_{4} + 14847 \beta_{5} - 15012 \beta_{6} - 135 \beta_{7} - 17070 \beta_{8} + 8136 \beta_{9} - 1509 \beta_{10} ) q^{87} + ( -2193103 - 256006 \beta_{1} - 32754 \beta_{2} + 13404 \beta_{3} - 24146 \beta_{4} + 78282 \beta_{5} + 22107 \beta_{6} + 15010 \beta_{7} + 10525 \beta_{8} - 28628 \beta_{9} - 412 \beta_{10} ) q^{88} + ( -1238463 + 110424 \beta_{1} + 4827 \beta_{2} - 19527 \beta_{3} + 18567 \beta_{4} - 32559 \beta_{5} - 4569 \beta_{6} - 15147 \beta_{7} + 9174 \beta_{8} + 321 \beta_{9} - 933 \beta_{10} ) q^{89} + ( 1944242 - 200631 \beta_{1} + 18229 \beta_{2} + 5428 \beta_{3} - 4118 \beta_{4} - 49082 \beta_{5} + 2724 \beta_{6} - 3375 \beta_{7} - 11837 \beta_{8} - 2188 \beta_{9} - 10512 \beta_{10} ) q^{90} + ( -1834699 + 251610 \beta_{1} - 50557 \beta_{2} - 25887 \beta_{3} + 16959 \beta_{4} + 49539 \beta_{5} + 183 \beta_{6} - 21447 \beta_{7} + 17172 \beta_{8} + 687 \beta_{9} - 14235 \beta_{10} ) q^{91} + ( -2304423 - 426576 \beta_{1} - 6228 \beta_{2} + 6111 \beta_{3} - 21846 \beta_{4} + 14472 \beta_{5} + 11853 \beta_{6} + 19398 \beta_{7} + 9204 \beta_{8} - 12918 \beta_{9} + 12204 \beta_{10} ) q^{92} + ( -1947366 - 192104 \beta_{1} - 20501 \beta_{2} + 2364 \beta_{3} + 5074 \beta_{4} - 15380 \beta_{5} - 9985 \beta_{6} + 5870 \beta_{7} - 17243 \beta_{8} + 5956 \beta_{9} + 12361 \beta_{10} ) q^{93} + ( 1855868 + 65541 \beta_{1} - 5255 \beta_{2} - 33405 \beta_{3} + 10890 \beta_{4} - 62378 \beta_{5} - 8896 \beta_{6} - 20320 \beta_{7} - 7767 \beta_{8} - 1056 \beta_{9} - 8730 \beta_{10} ) q^{94} + ( -1114227 - 106246 \beta_{1} + 7333 \beta_{2} + 2147 \beta_{3} - 989 \beta_{4} + 60236 \beta_{5} + 2547 \beta_{6} + 10953 \beta_{7} + 7452 \beta_{8} - 8815 \beta_{9} - 3401 \beta_{10} ) q^{95} + ( -1147121 - 593817 \beta_{1} - 37667 \beta_{2} + 23659 \beta_{3} - 2450 \beta_{4} - 9448 \beta_{5} - 11979 \beta_{6} + 21018 \beta_{7} - 10048 \beta_{8} - 23986 \beta_{9} + 30616 \beta_{10} ) q^{96} + ( -969344 - 26256 \beta_{1} - 11649 \beta_{2} + 40349 \beta_{3} - 15389 \beta_{4} - 81524 \beta_{5} - 7297 \beta_{6} - 5411 \beta_{7} + 1590 \beta_{8} - 11735 \beta_{9} + 20091 \beta_{10} ) q^{97} + ( 2757403 - 218251 \beta_{1} + 31704 \beta_{2} + 14500 \beta_{3} - 17654 \beta_{4} - 51022 \beta_{5} + 5715 \beta_{6} - 6414 \beta_{7} + 7661 \beta_{8} + 13380 \beta_{9} - 26952 \beta_{10} ) q^{98} + ( 2599310 + 122556 \beta_{1} + 34509 \beta_{2} + 14268 \beta_{3} - 4704 \beta_{4} + 109605 \beta_{5} + 21714 \beta_{6} + 20418 \beta_{7} + 29086 \beta_{8} + 17088 \beta_{9} - 16397 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} - 60918607872 x + 238240894976\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 178 \)
\(\beta_{3}\)\(=\)\((\)\(152109332450999 \nu^{10} + 23355659686447470 \nu^{9} - 4810250278160783 \nu^{8} - 17923617677841179548 \nu^{7} - 48030435418379425262 \nu^{6} + 4431575850596006399012 \nu^{5} + 11830191617346714444880 \nu^{4} - 410206139481361153289152 \nu^{3} - 695943005803893942892672 \nu^{2} + 10618895172606014473188608 \nu + 1592740462443118267757568\)\()/ \)\(10\!\cdots\!80\)\( \)
\(\beta_{4}\)\(=\)\((\)\(2657334781406279 \nu^{10} + 101385742130232670 \nu^{9} - 371886320705156063 \nu^{8} - 74456178762460814828 \nu^{7} - 847207609957825843822 \nu^{6} + 16345713194306033750212 \nu^{5} + 295424244486587309205200 \nu^{4} - 986641555137393675121472 \nu^{3} - 27116563156681068533668992 \nu^{2} - 24715820870674953243194112 \nu + 378973296582979192222225408\)\()/ \)\(83\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-15417932210853737 \nu^{10} - 160668271242151970 \nu^{9} + 13242754276875787409 \nu^{8} + 121276838471867429364 \nu^{7} - 3954053314307800952334 \nu^{6} - 28598369111565979213116 \nu^{5} + 506603051558042172010320 \nu^{4} + 2378682162648831159750336 \nu^{3} - 26071421681368337555849344 \nu^{2} - 45455414445115310312452864 \nu + 367311556444564633143940096\)\()/ \)\(83\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(2028489770240437 \nu^{10} - 4759521264037670 \nu^{9} - 1847853186071304349 \nu^{8} + 5686415320515638076 \nu^{7} + 583154221203139258134 \nu^{6} - 2165221007502907270644 \nu^{5} - 75782955375860487170640 \nu^{4} + 286943665573460913129024 \nu^{3} + 3693496481156841308303744 \nu^{2} - 10199339557661504272404736 \nu - 46558864347266574282407936\)\()/ \)\(10\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(995986480737037 \nu^{10} + 13910975507549194 \nu^{9} - 761862579126582421 \nu^{8} - 9741212482021031460 \nu^{7} + 187621172292291514950 \nu^{6} + 1950766727523486759276 \nu^{5} - 17529131825073983720208 \nu^{4} - 105889744764038807825856 \nu^{3} + 431080487927184205374080 \nu^{2} - 764909818781895325305088 \nu + 4558336130707968141564928\)\()/ \)\(41\!\cdots\!72\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-690593231497871 \nu^{10} - 5944478544321660 \nu^{9} + 594993015681134987 \nu^{8} + 4459195741859647882 \nu^{7} - 172025748821259154282 \nu^{6} - 1049935282559078585768 \nu^{5} + 19039962621377461182440 \nu^{4} + 90903350288060544374368 \nu^{3} - 584805446727930828384512 \nu^{2} - 2435571525681660279660032 \nu + 1012109866517881083130368\)\()/ \)\(26\!\cdots\!20\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-31436440924802291 \nu^{10} - 261455405569143350 \nu^{9} + 28061582102792100587 \nu^{8} + 208384299365670282012 \nu^{7} - 8726499799177411121082 \nu^{6} - 52869583371496047989268 \nu^{5} + 1149056758502396716020720 \nu^{4} + 4788931906705605573799488 \nu^{3} - 60998170276882034477759872 \nu^{2} - 106780993618636806923391232 \nu + 1013826711314230671609453568\)\()/ \)\(83\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(7689755899080515 \nu^{10} + 66633255225710870 \nu^{9} - 6274381659934927163 \nu^{8} - 48993677315181693084 \nu^{7} + 1726876617122279165466 \nu^{6} + 11167341019182369500244 \nu^{5} - 196382880151662636090864 \nu^{4} - 933686904001652240422464 \nu^{3} + 8643971281507625168254336 \nu^{2} + 21874566232933717460533504 \nu - 105358334555684169749945344\)\()/ \)\(16\!\cdots\!88\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 178\)
\(\nu^{3}\)\(=\)\(-2 \beta_{10} + 4 \beta_{9} + \beta_{8} - 3 \beta_{7} - 18 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 258 \beta_{1} - 216\)
\(\nu^{4}\)\(=\)\(-10 \beta_{10} + 44 \beta_{9} - 30 \beta_{8} - 5 \beta_{7} - 29 \beta_{6} - 112 \beta_{5} + 2 \beta_{4} - 47 \beta_{3} + 391 \beta_{2} - 436 \beta_{1} + 47769\)
\(\nu^{5}\)\(=\)\(-762 \beta_{10} + 2120 \beta_{9} + 306 \beta_{8} - 1711 \beta_{7} + 89 \beta_{6} - 8560 \beta_{5} + 1782 \beta_{4} - 1125 \beta_{3} - 1213 \beta_{2} + 79074 \beta_{1} - 82473\)
\(\nu^{6}\)\(=\)\(-5154 \beta_{10} + 26856 \beta_{9} - 14456 \beta_{8} - 3327 \beta_{7} - 17529 \beta_{6} - 71796 \beta_{5} + 594 \beta_{4} - 28625 \beta_{3} + 145687 \beta_{2} - 137966 \beta_{1} + 14953881\)
\(\nu^{7}\)\(=\)\(-283250 \beta_{10} + 917648 \beta_{9} + 85284 \beta_{8} - 735667 \beta_{7} + 92087 \beta_{6} - 3463644 \beta_{5} + 684130 \beta_{4} - 359437 \beta_{3} - 379385 \beta_{2} + 26515390 \beta_{1} - 23795839\)
\(\nu^{8}\)\(=\)\(-1867314 \beta_{10} + 12553584 \beta_{9} - 5561184 \beta_{8} - 1698147 \beta_{7} - 8240085 \beta_{6} - 34146660 \beta_{5} + 387306 \beta_{4} - 13489149 \beta_{3} + 54537663 \beta_{2} - 35059170 \beta_{1} + 5074980349\)
\(\nu^{9}\)\(=\)\(-110006946 \beta_{10} + 372850128 \beta_{9} + 22506528 \beta_{8} - 292040643 \beta_{7} + 53754771 \beta_{6} - 1352198916 \beta_{5} + 255643866 \beta_{4} - 110376789 \beta_{3} - 104821521 \beta_{2} + 9361711054 \beta_{1} - 5320106523\)
\(\nu^{10}\)\(=\)\(-587476050 \beta_{10} + 5358368064 \beta_{9} - 2031998808 \beta_{8} - 802180395 \beta_{7} - 3535897605 \beta_{6} - 14704028820 \beta_{5} + 275082090 \beta_{4} - 5842500021 \beta_{3} + 20532567295 \beta_{2} - 6170261866 \beta_{1} + 1804293884701\)

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
19.7860
14.1572
13.1261
11.4671
6.31419
3.52766
−5.14919
−10.3097
−14.3182
−17.2185
−19.3827
−21.7860 54.2998 346.631 −409.924 −1182.98 476.571 −4763.10 761.467 8930.61
1.2 −16.1572 −48.3227 133.057 −272.935 780.762 1101.65 −81.7025 148.083 4409.88
1.3 −15.1261 62.8737 100.799 12.0619 −951.035 −1248.65 411.441 1766.10 −182.450
1.4 −13.4671 −87.0645 53.3630 131.138 1172.51 −712.280 1005.14 5393.23 −1766.05
1.5 −8.31419 11.6940 −58.8742 148.086 −97.2261 122.189 1553.71 −2050.25 −1231.21
1.6 −5.52766 −57.7031 −97.4450 304.619 318.963 1421.71 1246.18 1142.65 −1683.83
1.7 3.14919 34.1076 −118.083 −39.2891 107.411 434.472 −774.961 −1023.67 −123.729
1.8 8.30971 47.1073 −58.9487 −187.284 391.449 −1501.21 −1553.49 32.1014 −1556.27
1.9 12.3182 −46.7197 23.7379 330.186 −575.503 −467.397 −1284.32 −4.26512 4067.29
1.10 15.2185 −9.20709 103.603 −537.911 −140.118 1471.62 −371.288 −2102.23 −8186.21
1.11 17.3827 −29.0652 174.160 −230.747 −505.234 −1110.68 802.384 −1342.21 −4011.02
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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 43.8.a.a 11
3.b odd 2 1 387.8.a.b 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.8.a.a 11 1.a even 1 1 trivial
387.8.a.b 11 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 \(T_{2}^{11} + \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(43))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 24 T + 691 T^{2} + 12222 T^{3} + 244862 T^{4} + 3813984 T^{5} + 61591176 T^{6} + 856359456 T^{7} + 11928317344 T^{8} + 150942640896 T^{9} + 1864194305664 T^{10} + 21383753094144 T^{11} + 238616871124992 T^{12} + 2473044228440064 T^{13} + 25015494574604288 T^{14} + 229877241071271936 T^{15} + 2116256693137440768 T^{16} + 16774079024606478336 T^{17} + \)\(13\!\cdots\!44\)\( T^{18} + \)\(88\!\cdots\!92\)\( T^{19} + \)\(63\!\cdots\!28\)\( T^{20} + \)\(28\!\cdots\!76\)\( T^{21} + \)\(15\!\cdots\!72\)\( T^{22} \)
$3$ \( 1 + 68 T + 12980 T^{2} + 808376 T^{3} + 89267473 T^{4} + 5191228722 T^{5} + 419071780872 T^{6} + 22651159895028 T^{7} + 1473168146266863 T^{8} + 73195215317998578 T^{9} + 4049205818337122625 T^{10} + \)\(18\!\cdots\!08\)\( T^{11} + \)\(88\!\cdots\!75\)\( T^{12} + \)\(35\!\cdots\!82\)\( T^{13} + \)\(15\!\cdots\!89\)\( T^{14} + \)\(51\!\cdots\!08\)\( T^{15} + \)\(20\!\cdots\!04\)\( T^{16} + \)\(56\!\cdots\!98\)\( T^{17} + \)\(21\!\cdots\!59\)\( T^{18} + \)\(42\!\cdots\!96\)\( T^{19} + \)\(14\!\cdots\!60\)\( T^{20} + \)\(17\!\cdots\!32\)\( T^{21} + \)\(54\!\cdots\!63\)\( T^{22} \)
$5$ \( 1 + 752 T + 710712 T^{2} + 398620948 T^{3} + 228483912909 T^{4} + 103902073006684 T^{5} + 45575868805383884 T^{6} + 17514273523939793160 T^{7} + \)\(63\!\cdots\!75\)\( T^{8} + \)\(21\!\cdots\!00\)\( T^{9} + \)\(65\!\cdots\!75\)\( T^{10} + \)\(18\!\cdots\!00\)\( T^{11} + \)\(51\!\cdots\!75\)\( T^{12} + \)\(12\!\cdots\!00\)\( T^{13} + \)\(30\!\cdots\!75\)\( T^{14} + \)\(65\!\cdots\!00\)\( T^{15} + \)\(13\!\cdots\!00\)\( T^{16} + \)\(23\!\cdots\!00\)\( T^{17} + \)\(40\!\cdots\!25\)\( T^{18} + \)\(55\!\cdots\!00\)\( T^{19} + \)\(77\!\cdots\!00\)\( T^{20} + \)\(63\!\cdots\!00\)\( T^{21} + \)\(66\!\cdots\!25\)\( T^{22} \)
$7$ \( 1 + 12 T + 3257277 T^{2} - 112854536 T^{3} + 6210749066091 T^{4} - 424770058939316 T^{5} + 8920530970834454311 T^{6} - \)\(59\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} - \)\(63\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!38\)\( T^{10} - \)\(56\!\cdots\!64\)\( T^{11} + \)\(83\!\cdots\!34\)\( T^{12} - \)\(43\!\cdots\!80\)\( T^{13} + \)\(58\!\cdots\!70\)\( T^{14} - \)\(27\!\cdots\!92\)\( T^{15} + \)\(33\!\cdots\!73\)\( T^{16} - \)\(13\!\cdots\!84\)\( T^{17} + \)\(15\!\cdots\!37\)\( T^{18} - \)\(23\!\cdots\!36\)\( T^{19} + \)\(56\!\cdots\!11\)\( T^{20} + \)\(17\!\cdots\!88\)\( T^{21} + \)\(11\!\cdots\!07\)\( T^{22} \)
$11$ \( 1 - 1333 T + 101138498 T^{2} - 129892434295 T^{3} + 5204321338235473 T^{4} - 5619737451849796480 T^{5} + \)\(18\!\cdots\!69\)\( T^{6} - \)\(16\!\cdots\!73\)\( T^{7} + \)\(54\!\cdots\!02\)\( T^{8} - \)\(41\!\cdots\!79\)\( T^{9} + \)\(12\!\cdots\!37\)\( T^{10} - \)\(91\!\cdots\!80\)\( T^{11} + \)\(24\!\cdots\!27\)\( T^{12} - \)\(15\!\cdots\!39\)\( T^{13} + \)\(40\!\cdots\!22\)\( T^{14} - \)\(23\!\cdots\!13\)\( T^{15} + \)\(53\!\cdots\!19\)\( T^{16} - \)\(30\!\cdots\!80\)\( T^{17} + \)\(55\!\cdots\!43\)\( T^{18} - \)\(27\!\cdots\!95\)\( T^{19} + \)\(40\!\cdots\!38\)\( T^{20} - \)\(10\!\cdots\!33\)\( T^{21} + \)\(15\!\cdots\!71\)\( T^{22} \)
$13$ \( 1 + 17967 T + 384990808 T^{2} + 3743076892617 T^{3} + 49853715106570123 T^{4} + \)\(36\!\cdots\!76\)\( T^{5} + \)\(48\!\cdots\!61\)\( T^{6} + \)\(35\!\cdots\!27\)\( T^{7} + \)\(44\!\cdots\!64\)\( T^{8} + \)\(29\!\cdots\!17\)\( T^{9} + \)\(32\!\cdots\!67\)\( T^{10} + \)\(18\!\cdots\!72\)\( T^{11} + \)\(20\!\cdots\!39\)\( T^{12} + \)\(11\!\cdots\!13\)\( T^{13} + \)\(11\!\cdots\!32\)\( T^{14} + \)\(54\!\cdots\!67\)\( T^{15} + \)\(46\!\cdots\!77\)\( T^{16} + \)\(22\!\cdots\!44\)\( T^{17} + \)\(19\!\cdots\!79\)\( T^{18} + \)\(89\!\cdots\!97\)\( T^{19} + \)\(58\!\cdots\!76\)\( T^{20} + \)\(17\!\cdots\!83\)\( T^{21} + \)\(59\!\cdots\!33\)\( T^{22} \)
$17$ \( 1 + 63095 T + 4595440161 T^{2} + 183012158048722 T^{3} + 7845892593369027108 T^{4} + \)\(23\!\cdots\!66\)\( T^{5} + \)\(75\!\cdots\!23\)\( T^{6} + \)\(18\!\cdots\!63\)\( T^{7} + \)\(50\!\cdots\!42\)\( T^{8} + \)\(10\!\cdots\!53\)\( T^{9} + \)\(25\!\cdots\!74\)\( T^{10} + \)\(48\!\cdots\!85\)\( T^{11} + \)\(10\!\cdots\!02\)\( T^{12} + \)\(17\!\cdots\!37\)\( T^{13} + \)\(34\!\cdots\!14\)\( T^{14} + \)\(52\!\cdots\!83\)\( T^{15} + \)\(88\!\cdots\!39\)\( T^{16} + \)\(11\!\cdots\!74\)\( T^{17} + \)\(15\!\cdots\!76\)\( T^{18} + \)\(14\!\cdots\!82\)\( T^{19} + \)\(15\!\cdots\!93\)\( T^{20} + \)\(85\!\cdots\!55\)\( T^{21} + \)\(55\!\cdots\!77\)\( T^{22} \)
$19$ \( 1 + 54524 T + 7869332012 T^{2} + 357391219306324 T^{3} + 28295251291043717253 T^{4} + \)\(11\!\cdots\!62\)\( T^{5} + \)\(62\!\cdots\!16\)\( T^{6} + \)\(21\!\cdots\!64\)\( T^{7} + \)\(98\!\cdots\!19\)\( T^{8} + \)\(30\!\cdots\!82\)\( T^{9} + \)\(11\!\cdots\!53\)\( T^{10} + \)\(31\!\cdots\!48\)\( T^{11} + \)\(10\!\cdots\!67\)\( T^{12} + \)\(24\!\cdots\!22\)\( T^{13} + \)\(70\!\cdots\!61\)\( T^{14} + \)\(13\!\cdots\!24\)\( T^{15} + \)\(35\!\cdots\!84\)\( T^{16} + \)\(56\!\cdots\!82\)\( T^{17} + \)\(12\!\cdots\!87\)\( T^{18} + \)\(14\!\cdots\!44\)\( T^{19} + \)\(28\!\cdots\!08\)\( T^{20} + \)\(17\!\cdots\!24\)\( T^{21} + \)\(29\!\cdots\!39\)\( T^{22} \)
$23$ \( 1 + 138139 T + 27512602431 T^{2} + 2853019807837640 T^{3} + \)\(33\!\cdots\!86\)\( T^{4} + \)\(28\!\cdots\!30\)\( T^{5} + \)\(25\!\cdots\!31\)\( T^{6} + \)\(18\!\cdots\!43\)\( T^{7} + \)\(13\!\cdots\!66\)\( T^{8} + \)\(86\!\cdots\!13\)\( T^{9} + \)\(56\!\cdots\!14\)\( T^{10} + \)\(32\!\cdots\!79\)\( T^{11} + \)\(19\!\cdots\!58\)\( T^{12} + \)\(10\!\cdots\!17\)\( T^{13} + \)\(53\!\cdots\!18\)\( T^{14} + \)\(24\!\cdots\!83\)\( T^{15} + \)\(11\!\cdots\!17\)\( T^{16} + \)\(43\!\cdots\!70\)\( T^{17} + \)\(17\!\cdots\!18\)\( T^{18} + \)\(51\!\cdots\!40\)\( T^{19} + \)\(16\!\cdots\!77\)\( T^{20} + \)\(28\!\cdots\!11\)\( T^{21} + \)\(71\!\cdots\!03\)\( T^{22} \)
$29$ \( 1 + 308658 T + 170705721136 T^{2} + 42113858722339938 T^{3} + \)\(13\!\cdots\!57\)\( T^{4} + \)\(27\!\cdots\!56\)\( T^{5} + \)\(63\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!28\)\( T^{7} + \)\(20\!\cdots\!95\)\( T^{8} + \)\(31\!\cdots\!42\)\( T^{9} + \)\(48\!\cdots\!31\)\( T^{10} + \)\(63\!\cdots\!16\)\( T^{11} + \)\(84\!\cdots\!79\)\( T^{12} + \)\(93\!\cdots\!02\)\( T^{13} + \)\(10\!\cdots\!55\)\( T^{14} + \)\(99\!\cdots\!08\)\( T^{15} + \)\(97\!\cdots\!80\)\( T^{16} + \)\(72\!\cdots\!96\)\( T^{17} + \)\(60\!\cdots\!33\)\( T^{18} + \)\(33\!\cdots\!98\)\( T^{19} + \)\(23\!\cdots\!04\)\( T^{20} + \)\(72\!\cdots\!58\)\( T^{21} + \)\(40\!\cdots\!09\)\( T^{22} \)
$31$ \( 1 + 209523 T + 225723508099 T^{2} + 41207632211224460 T^{3} + \)\(24\!\cdots\!70\)\( T^{4} + \)\(38\!\cdots\!54\)\( T^{5} + \)\(16\!\cdots\!95\)\( T^{6} + \)\(22\!\cdots\!67\)\( T^{7} + \)\(77\!\cdots\!90\)\( T^{8} + \)\(94\!\cdots\!37\)\( T^{9} + \)\(27\!\cdots\!02\)\( T^{10} + \)\(29\!\cdots\!19\)\( T^{11} + \)\(75\!\cdots\!22\)\( T^{12} + \)\(71\!\cdots\!77\)\( T^{13} + \)\(16\!\cdots\!90\)\( T^{14} + \)\(12\!\cdots\!47\)\( T^{15} + \)\(25\!\cdots\!45\)\( T^{16} + \)\(16\!\cdots\!94\)\( T^{17} + \)\(28\!\cdots\!70\)\( T^{18} + \)\(13\!\cdots\!60\)\( T^{19} + \)\(20\!\cdots\!09\)\( T^{20} + \)\(52\!\cdots\!23\)\( T^{21} + \)\(68\!\cdots\!11\)\( T^{22} \)
$37$ \( 1 + 298472 T + 366143447012 T^{2} + 60971230634752572 T^{3} + \)\(76\!\cdots\!45\)\( T^{4} + \)\(12\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!44\)\( T^{6} + \)\(19\!\cdots\!72\)\( T^{7} + \)\(17\!\cdots\!31\)\( T^{8} + \)\(22\!\cdots\!76\)\( T^{9} + \)\(20\!\cdots\!99\)\( T^{10} + \)\(25\!\cdots\!56\)\( T^{11} + \)\(19\!\cdots\!67\)\( T^{12} + \)\(20\!\cdots\!64\)\( T^{13} + \)\(15\!\cdots\!47\)\( T^{14} + \)\(15\!\cdots\!12\)\( T^{15} + \)\(10\!\cdots\!92\)\( T^{16} + \)\(93\!\cdots\!28\)\( T^{17} + \)\(53\!\cdots\!65\)\( T^{18} + \)\(40\!\cdots\!52\)\( T^{19} + \)\(22\!\cdots\!36\)\( T^{20} + \)\(17\!\cdots\!28\)\( T^{21} + \)\(56\!\cdots\!17\)\( T^{22} \)
$41$ \( 1 + 1346735 T + 1406042675633 T^{2} + 1173279952392795098 T^{3} + \)\(86\!\cdots\!00\)\( T^{4} + \)\(57\!\cdots\!74\)\( T^{5} + \)\(35\!\cdots\!19\)\( T^{6} + \)\(19\!\cdots\!47\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} + \)\(53\!\cdots\!53\)\( T^{9} + \)\(25\!\cdots\!02\)\( T^{10} + \)\(11\!\cdots\!49\)\( T^{11} + \)\(49\!\cdots\!62\)\( T^{12} + \)\(20\!\cdots\!33\)\( T^{13} + \)\(78\!\cdots\!66\)\( T^{14} + \)\(28\!\cdots\!87\)\( T^{15} + \)\(98\!\cdots\!19\)\( T^{16} + \)\(31\!\cdots\!94\)\( T^{17} + \)\(92\!\cdots\!00\)\( T^{18} + \)\(24\!\cdots\!18\)\( T^{19} + \)\(56\!\cdots\!93\)\( T^{20} + \)\(10\!\cdots\!35\)\( T^{21} + \)\(15\!\cdots\!81\)\( T^{22} \)
$43$ \( ( 1 - 79507 T )^{11} \)
$47$ \( 1 - 499284 T + 4243273022254 T^{2} - 1863388313046890988 T^{3} + \)\(85\!\cdots\!55\)\( T^{4} - \)\(33\!\cdots\!36\)\( T^{5} + \)\(11\!\cdots\!96\)\( T^{6} - \)\(39\!\cdots\!48\)\( T^{7} + \)\(99\!\cdots\!23\)\( T^{8} - \)\(32\!\cdots\!36\)\( T^{9} + \)\(66\!\cdots\!41\)\( T^{10} - \)\(18\!\cdots\!84\)\( T^{11} + \)\(33\!\cdots\!83\)\( T^{12} - \)\(82\!\cdots\!84\)\( T^{13} + \)\(12\!\cdots\!81\)\( T^{14} - \)\(26\!\cdots\!28\)\( T^{15} + \)\(36\!\cdots\!28\)\( T^{16} - \)\(57\!\cdots\!24\)\( T^{17} + \)\(73\!\cdots\!85\)\( T^{18} - \)\(80\!\cdots\!48\)\( T^{19} + \)\(93\!\cdots\!42\)\( T^{20} - \)\(55\!\cdots\!16\)\( T^{21} + \)\(56\!\cdots\!87\)\( T^{22} \)
$53$ \( 1 + 2210495 T + 10440160443764 T^{2} + 20051339651230402277 T^{3} + \)\(51\!\cdots\!19\)\( T^{4} + \)\(86\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!29\)\( T^{6} + \)\(23\!\cdots\!67\)\( T^{7} + \)\(34\!\cdots\!68\)\( T^{8} + \)\(43\!\cdots\!97\)\( T^{9} + \)\(54\!\cdots\!43\)\( T^{10} + \)\(59\!\cdots\!12\)\( T^{11} + \)\(64\!\cdots\!91\)\( T^{12} + \)\(60\!\cdots\!93\)\( T^{13} + \)\(56\!\cdots\!04\)\( T^{14} + \)\(44\!\cdots\!87\)\( T^{15} + \)\(35\!\cdots\!53\)\( T^{16} + \)\(22\!\cdots\!16\)\( T^{17} + \)\(15\!\cdots\!27\)\( T^{18} + \)\(72\!\cdots\!17\)\( T^{19} + \)\(44\!\cdots\!28\)\( T^{20} + \)\(11\!\cdots\!55\)\( T^{21} + \)\(58\!\cdots\!13\)\( T^{22} \)
$59$ \( 1 + 5824216 T + 29093848330233 T^{2} + 94393128322079117696 T^{3} + \)\(28\!\cdots\!95\)\( T^{4} + \)\(68\!\cdots\!00\)\( T^{5} + \)\(15\!\cdots\!23\)\( T^{6} + \)\(32\!\cdots\!56\)\( T^{7} + \)\(63\!\cdots\!06\)\( T^{8} + \)\(11\!\cdots\!52\)\( T^{9} + \)\(19\!\cdots\!34\)\( T^{10} + \)\(31\!\cdots\!80\)\( T^{11} + \)\(49\!\cdots\!46\)\( T^{12} + \)\(70\!\cdots\!72\)\( T^{13} + \)\(98\!\cdots\!54\)\( T^{14} + \)\(12\!\cdots\!76\)\( T^{15} + \)\(15\!\cdots\!77\)\( T^{16} + \)\(16\!\cdots\!00\)\( T^{17} + \)\(16\!\cdots\!05\)\( T^{18} + \)\(13\!\cdots\!36\)\( T^{19} + \)\(10\!\cdots\!07\)\( T^{20} + \)\(53\!\cdots\!16\)\( T^{21} + \)\(22\!\cdots\!19\)\( T^{22} \)
$61$ \( 1 + 4453034 T + 27496254230435 T^{2} + 84090456915815697292 T^{3} + \)\(29\!\cdots\!55\)\( T^{4} + \)\(71\!\cdots\!94\)\( T^{5} + \)\(19\!\cdots\!57\)\( T^{6} + \)\(39\!\cdots\!64\)\( T^{7} + \)\(92\!\cdots\!06\)\( T^{8} + \)\(17\!\cdots\!72\)\( T^{9} + \)\(36\!\cdots\!46\)\( T^{10} + \)\(63\!\cdots\!88\)\( T^{11} + \)\(11\!\cdots\!66\)\( T^{12} + \)\(17\!\cdots\!52\)\( T^{13} + \)\(28\!\cdots\!66\)\( T^{14} + \)\(38\!\cdots\!84\)\( T^{15} + \)\(58\!\cdots\!57\)\( T^{16} + \)\(68\!\cdots\!74\)\( T^{17} + \)\(90\!\cdots\!55\)\( T^{18} + \)\(80\!\cdots\!12\)\( T^{19} + \)\(82\!\cdots\!35\)\( T^{20} + \)\(41\!\cdots\!34\)\( T^{21} + \)\(29\!\cdots\!21\)\( T^{22} \)
$67$ \( 1 + 6859513 T + 54299130836534 T^{2} + \)\(22\!\cdots\!67\)\( T^{3} + \)\(10\!\cdots\!85\)\( T^{4} + \)\(33\!\cdots\!04\)\( T^{5} + \)\(13\!\cdots\!77\)\( T^{6} + \)\(37\!\cdots\!61\)\( T^{7} + \)\(13\!\cdots\!50\)\( T^{8} + \)\(32\!\cdots\!91\)\( T^{9} + \)\(97\!\cdots\!37\)\( T^{10} + \)\(21\!\cdots\!44\)\( T^{11} + \)\(59\!\cdots\!51\)\( T^{12} + \)\(11\!\cdots\!39\)\( T^{13} + \)\(28\!\cdots\!50\)\( T^{14} + \)\(50\!\cdots\!01\)\( T^{15} + \)\(10\!\cdots\!11\)\( T^{16} + \)\(16\!\cdots\!56\)\( T^{17} + \)\(31\!\cdots\!95\)\( T^{18} + \)\(40\!\cdots\!27\)\( T^{19} + \)\(59\!\cdots\!42\)\( T^{20} + \)\(45\!\cdots\!37\)\( T^{21} + \)\(40\!\cdots\!27\)\( T^{22} \)
$71$ \( 1 + 10726554 T + 108657874183401 T^{2} + \)\(72\!\cdots\!96\)\( T^{3} + \)\(44\!\cdots\!47\)\( T^{4} + \)\(22\!\cdots\!78\)\( T^{5} + \)\(10\!\cdots\!71\)\( T^{6} + \)\(43\!\cdots\!60\)\( T^{7} + \)\(16\!\cdots\!66\)\( T^{8} + \)\(59\!\cdots\!56\)\( T^{9} + \)\(20\!\cdots\!70\)\( T^{10} + \)\(61\!\cdots\!00\)\( T^{11} + \)\(18\!\cdots\!70\)\( T^{12} + \)\(48\!\cdots\!36\)\( T^{13} + \)\(12\!\cdots\!86\)\( T^{14} + \)\(29\!\cdots\!60\)\( T^{15} + \)\(65\!\cdots\!21\)\( T^{16} + \)\(12\!\cdots\!98\)\( T^{17} + \)\(23\!\cdots\!57\)\( T^{18} + \)\(33\!\cdots\!16\)\( T^{19} + \)\(46\!\cdots\!11\)\( T^{20} + \)\(41\!\cdots\!54\)\( T^{21} + \)\(35\!\cdots\!91\)\( T^{22} \)
$73$ \( 1 + 4456898 T + 74251679354831 T^{2} + \)\(30\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!59\)\( T^{4} + \)\(10\!\cdots\!46\)\( T^{5} + \)\(67\!\cdots\!09\)\( T^{6} + \)\(23\!\cdots\!28\)\( T^{7} + \)\(12\!\cdots\!46\)\( T^{8} + \)\(39\!\cdots\!92\)\( T^{9} + \)\(17\!\cdots\!82\)\( T^{10} + \)\(49\!\cdots\!16\)\( T^{11} + \)\(19\!\cdots\!54\)\( T^{12} + \)\(48\!\cdots\!28\)\( T^{13} + \)\(16\!\cdots\!58\)\( T^{14} + \)\(35\!\cdots\!68\)\( T^{15} + \)\(11\!\cdots\!13\)\( T^{16} + \)\(18\!\cdots\!34\)\( T^{17} + \)\(55\!\cdots\!67\)\( T^{18} + \)\(67\!\cdots\!64\)\( T^{19} + \)\(18\!\cdots\!27\)\( T^{20} + \)\(12\!\cdots\!02\)\( T^{21} + \)\(29\!\cdots\!53\)\( T^{22} \)
$79$ \( 1 + 15541320 T + 201904805873602 T^{2} + \)\(17\!\cdots\!84\)\( T^{3} + \)\(13\!\cdots\!91\)\( T^{4} + \)\(89\!\cdots\!40\)\( T^{5} + \)\(51\!\cdots\!16\)\( T^{6} + \)\(25\!\cdots\!76\)\( T^{7} + \)\(12\!\cdots\!47\)\( T^{8} + \)\(52\!\cdots\!60\)\( T^{9} + \)\(22\!\cdots\!41\)\( T^{10} + \)\(95\!\cdots\!60\)\( T^{11} + \)\(42\!\cdots\!19\)\( T^{12} + \)\(19\!\cdots\!60\)\( T^{13} + \)\(85\!\cdots\!13\)\( T^{14} + \)\(35\!\cdots\!36\)\( T^{15} + \)\(13\!\cdots\!84\)\( T^{16} + \)\(44\!\cdots\!40\)\( T^{17} + \)\(13\!\cdots\!29\)\( T^{18} + \)\(33\!\cdots\!64\)\( T^{19} + \)\(71\!\cdots\!78\)\( T^{20} + \)\(10\!\cdots\!20\)\( T^{21} + \)\(13\!\cdots\!59\)\( T^{22} \)
$83$ \( 1 + 11146767 T + 230549739082006 T^{2} + \)\(20\!\cdots\!69\)\( T^{3} + \)\(24\!\cdots\!77\)\( T^{4} + \)\(17\!\cdots\!40\)\( T^{5} + \)\(16\!\cdots\!13\)\( T^{6} + \)\(10\!\cdots\!27\)\( T^{7} + \)\(79\!\cdots\!66\)\( T^{8} + \)\(44\!\cdots\!81\)\( T^{9} + \)\(28\!\cdots\!49\)\( T^{10} + \)\(13\!\cdots\!48\)\( T^{11} + \)\(77\!\cdots\!23\)\( T^{12} + \)\(32\!\cdots\!49\)\( T^{13} + \)\(15\!\cdots\!78\)\( T^{14} + \)\(56\!\cdots\!07\)\( T^{15} + \)\(24\!\cdots\!91\)\( T^{16} + \)\(71\!\cdots\!60\)\( T^{17} + \)\(26\!\cdots\!31\)\( T^{18} + \)\(58\!\cdots\!89\)\( T^{19} + \)\(18\!\cdots\!22\)\( T^{20} + \)\(24\!\cdots\!83\)\( T^{21} + \)\(58\!\cdots\!23\)\( T^{22} \)
$89$ \( 1 + 13531356 T + 311067335905063 T^{2} + \)\(31\!\cdots\!84\)\( T^{3} + \)\(45\!\cdots\!03\)\( T^{4} + \)\(39\!\cdots\!56\)\( T^{5} + \)\(45\!\cdots\!69\)\( T^{6} + \)\(34\!\cdots\!88\)\( T^{7} + \)\(32\!\cdots\!22\)\( T^{8} + \)\(21\!\cdots\!92\)\( T^{9} + \)\(18\!\cdots\!18\)\( T^{10} + \)\(10\!\cdots\!68\)\( T^{11} + \)\(80\!\cdots\!22\)\( T^{12} + \)\(42\!\cdots\!72\)\( T^{13} + \)\(28\!\cdots\!58\)\( T^{14} + \)\(13\!\cdots\!28\)\( T^{15} + \)\(76\!\cdots\!81\)\( T^{16} + \)\(29\!\cdots\!76\)\( T^{17} + \)\(15\!\cdots\!27\)\( T^{18} + \)\(46\!\cdots\!24\)\( T^{19} + \)\(20\!\cdots\!47\)\( T^{20} + \)\(38\!\cdots\!56\)\( T^{21} + \)\(12\!\cdots\!29\)\( T^{22} \)
$97$ \( 1 + 10999901 T + 423520564906337 T^{2} + \)\(30\!\cdots\!30\)\( T^{3} + \)\(85\!\cdots\!48\)\( T^{4} + \)\(46\!\cdots\!82\)\( T^{5} + \)\(12\!\cdots\!91\)\( T^{6} + \)\(53\!\cdots\!77\)\( T^{7} + \)\(14\!\cdots\!14\)\( T^{8} + \)\(50\!\cdots\!59\)\( T^{9} + \)\(13\!\cdots\!10\)\( T^{10} + \)\(41\!\cdots\!35\)\( T^{11} + \)\(10\!\cdots\!30\)\( T^{12} + \)\(33\!\cdots\!71\)\( T^{13} + \)\(75\!\cdots\!58\)\( T^{14} + \)\(22\!\cdots\!97\)\( T^{15} + \)\(42\!\cdots\!63\)\( T^{16} + \)\(12\!\cdots\!38\)\( T^{17} + \)\(19\!\cdots\!16\)\( T^{18} + \)\(54\!\cdots\!30\)\( T^{19} + \)\(62\!\cdots\!01\)\( T^{20} + \)\(13\!\cdots\!49\)\( T^{21} + \)\(95\!\cdots\!37\)\( T^{22} \)
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