Properties

Label 387.8.a.d
Level 387
Weight 8
Character orbit 387.a
Self dual yes
Analytic conductor 120.893
Analytic rank 1
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(120.893004862\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{2} + ( 70 + 2 \beta_{1} + \beta_{2} ) q^{4} + ( -77 - \beta_{1} + \beta_{6} ) q^{5} + ( 101 + 16 \beta_{1} - \beta_{10} + \beta_{11} ) q^{7} + ( -277 - 89 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{11} ) q^{8} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{2} + ( 70 + 2 \beta_{1} + \beta_{2} ) q^{4} + ( -77 - \beta_{1} + \beta_{6} ) q^{5} + ( 101 + 16 \beta_{1} - \beta_{10} + \beta_{11} ) q^{7} + ( -277 - 89 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{11} ) q^{8} + ( 329 + 115 \beta_{1} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - \beta_{8} + 4 \beta_{9} + \beta_{10} - 4 \beta_{11} - 4 \beta_{12} ) q^{10} + ( -117 - 75 \beta_{1} + 11 \beta_{2} + \beta_{3} - \beta_{4} + 5 \beta_{5} + 3 \beta_{6} - \beta_{7} + 4 \beta_{9} + \beta_{10} - 5 \beta_{11} - 3 \beta_{12} ) q^{11} + ( 1035 + 83 \beta_{1} - 21 \beta_{2} + 3 \beta_{3} + 8 \beta_{4} - 2 \beta_{5} - 13 \beta_{6} - 6 \beta_{7} + 11 \beta_{8} - 6 \beta_{9} + \beta_{10} + 6 \beta_{11} - 5 \beta_{12} ) q^{13} + ( -3345 - 137 \beta_{1} - 17 \beta_{2} - 7 \beta_{3} + 4 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} - 6 \beta_{7} - 11 \beta_{8} - 5 \beta_{9} + 8 \beta_{11} + 5 \beta_{12} ) q^{14} + ( 8697 + 232 \beta_{1} + 40 \beta_{2} - 7 \beta_{3} - 6 \beta_{4} - 23 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 16 \beta_{9} + 2 \beta_{10} - 8 \beta_{11} ) q^{16} + ( -8530 + 100 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + 10 \beta_{6} + 10 \beta_{7} - 2 \beta_{8} - 31 \beta_{9} + 9 \beta_{10} + 7 \beta_{11} + 34 \beta_{12} ) q^{17} + ( 8070 + 195 \beta_{1} - 12 \beta_{2} - 10 \beta_{3} + 25 \beta_{4} - 15 \beta_{5} - 33 \beta_{6} + 7 \beta_{7} + 15 \beta_{8} + 2 \beta_{9} - 13 \beta_{10} + 12 \beta_{11} + 32 \beta_{12} ) q^{19} + ( -12866 + 221 \beta_{1} - 119 \beta_{2} + 30 \beta_{3} - 18 \beta_{4} - \beta_{5} + 4 \beta_{6} + 17 \beta_{7} - 30 \beta_{8} + 51 \beta_{9} - 11 \beta_{10} - 36 \beta_{11} + 19 \beta_{12} ) q^{20} + ( 15664 - 807 \beta_{1} + 48 \beta_{2} + 9 \beta_{3} - 21 \beta_{4} - 60 \beta_{5} + 45 \beta_{6} + 18 \beta_{7} - 49 \beta_{8} + 72 \beta_{9} - 16 \beta_{10} + 13 \beta_{11} + 24 \beta_{12} ) q^{22} + ( -12731 + 1621 \beta_{1} + 19 \beta_{2} + 26 \beta_{3} + 11 \beta_{4} - 47 \beta_{5} + 5 \beta_{6} - 56 \beta_{7} - \beta_{8} + 39 \beta_{9} - 9 \beta_{10} - \beta_{11} - 6 \beta_{12} ) q^{23} + ( 21288 - 1735 \beta_{1} - 193 \beta_{2} - 13 \beta_{3} - 19 \beta_{4} + 65 \beta_{5} - 53 \beta_{6} + 34 \beta_{7} - 29 \beta_{8} - 61 \beta_{9} + 34 \beta_{10} - 87 \beta_{11} - 29 \beta_{12} ) q^{25} + ( -20954 + 993 \beta_{1} - 112 \beta_{2} - 89 \beta_{3} + 73 \beta_{4} - 164 \beta_{5} - 117 \beta_{6} - 50 \beta_{7} - 51 \beta_{8} - 168 \beta_{9} + 76 \beta_{10} + 67 \beta_{11} + 40 \beta_{12} ) q^{26} + ( 15528 + 2603 \beta_{1} - 126 \beta_{2} + 4 \beta_{3} + 110 \beta_{4} + 21 \beta_{5} - 150 \beta_{6} - 64 \beta_{7} + 122 \beta_{8} - 80 \beta_{9} - 32 \beta_{10} + 12 \beta_{11} + 82 \beta_{12} ) q^{28} + ( -22498 + 1836 \beta_{1} + 241 \beta_{2} - 94 \beta_{3} - 115 \beta_{4} + 27 \beta_{5} + 113 \beta_{7} - 28 \beta_{8} + 14 \beta_{9} + 38 \beta_{10} - 49 \beta_{11} - 30 \beta_{12} ) q^{29} + ( -7298 + 238 \beta_{1} - 374 \beta_{2} - 98 \beta_{3} + 98 \beta_{4} - 150 \beta_{5} - 204 \beta_{6} - 59 \beta_{7} - 123 \beta_{8} + 63 \beta_{9} + 27 \beta_{10} + 122 \beta_{11} + 106 \beta_{12} ) q^{31} + ( -14341 - 4512 \beta_{1} - 370 \beta_{2} + 215 \beta_{3} + 112 \beta_{4} + 323 \beta_{5} + 121 \beta_{6} - 61 \beta_{7} + 182 \beta_{8} - 166 \beta_{9} + 82 \beta_{10} - 118 \beta_{11} - 82 \beta_{12} ) q^{32} + ( -8367 + 8965 \beta_{1} + 253 \beta_{2} - 210 \beta_{3} + 44 \beta_{4} + 178 \beta_{5} - 514 \beta_{6} + 113 \beta_{7} + 472 \beta_{8} - 361 \beta_{9} + 79 \beta_{10} - 32 \beta_{11} - 209 \beta_{12} ) q^{34} + ( 14756 + 113 \beta_{1} - 832 \beta_{2} + 109 \beta_{3} + 124 \beta_{4} - 200 \beta_{5} - 155 \beta_{6} - 201 \beta_{7} - 71 \beta_{8} - 163 \beta_{9} + 53 \beta_{10} - 137 \beta_{11} - 87 \beta_{12} ) q^{35} + ( 13471 - 1450 \beta_{1} + 413 \beta_{2} + 94 \beta_{3} - 141 \beta_{4} + 89 \beta_{5} + 106 \beta_{6} - 111 \beta_{7} + 32 \beta_{8} + 8 \beta_{9} + 45 \beta_{10} - 64 \beta_{11} - 210 \beta_{12} ) q^{37} + ( -46850 - 9676 \beta_{1} - 1061 \beta_{2} - 292 \beta_{3} + 200 \beta_{4} + 204 \beta_{5} - 534 \beta_{6} - 113 \beta_{7} - 270 \beta_{8} + 139 \beta_{9} + 261 \beta_{10} + 230 \beta_{11} + 17 \beta_{12} ) q^{38} + ( -72548 + 15166 \beta_{1} - 17 \beta_{2} + 48 \beta_{3} - 184 \beta_{4} - 158 \beta_{5} + 44 \beta_{6} - 115 \beta_{7} - 270 \beta_{8} + 641 \beta_{9} - 87 \beta_{10} - 280 \beta_{11} - 41 \beta_{12} ) q^{40} + ( 32133 + 20 \beta_{1} - 1241 \beta_{2} + 3 \beta_{3} - 214 \beta_{4} - 292 \beta_{5} - 282 \beta_{6} + 396 \beta_{7} - 231 \beta_{8} + 462 \beta_{9} - 106 \beta_{10} - 237 \beta_{11} - 235 \beta_{12} ) q^{41} -79507 q^{43} + ( 169955 - 14950 \beta_{1} + 223 \beta_{2} + 397 \beta_{3} - 724 \beta_{4} + 889 \beta_{5} + 483 \beta_{6} + 437 \beta_{7} - 18 \beta_{8} + 1106 \beta_{9} - 302 \beta_{10} - 590 \beta_{11} - 550 \beta_{12} ) q^{44} + ( -305856 + 3967 \beta_{1} - 2207 \beta_{2} + 267 \beta_{3} - 342 \beta_{4} + 231 \beta_{5} - 201 \beta_{6} - 436 \beta_{8} + 1077 \beta_{9} - 33 \beta_{10} - 216 \beta_{11} - 435 \beta_{12} ) q^{46} + ( 30783 + 9761 \beta_{1} - 679 \beta_{2} - 175 \beta_{3} - 611 \beta_{4} + 647 \beta_{5} + 29 \beta_{6} + 166 \beta_{7} - 609 \beta_{8} + 337 \beta_{9} + 428 \beta_{10} - 643 \beta_{11} - 59 \beta_{12} ) q^{47} + ( -115979 - 12977 \beta_{1} - 1996 \beta_{2} + 353 \beta_{3} + 470 \beta_{4} - 730 \beta_{5} - 1433 \beta_{6} - 279 \beta_{7} + 395 \beta_{8} + 793 \beta_{9} - 305 \beta_{10} + 271 \beta_{11} - 467 \beta_{12} ) q^{49} + ( 309710 + 11393 \beta_{1} + 947 \beta_{2} + 21 \beta_{3} + 647 \beta_{4} - 944 \beta_{5} + 205 \beta_{6} - 117 \beta_{7} + 719 \beta_{8} - 1367 \beta_{9} - 187 \beta_{10} + 561 \beta_{11} + 757 \beta_{12} ) q^{50} + ( -321731 + 2546 \beta_{1} - 377 \beta_{2} + 663 \beta_{3} + 76 \beta_{4} + 2123 \beta_{5} - 167 \beta_{6} + 95 \beta_{7} + 1858 \beta_{8} - 1858 \beta_{9} - 770 \beta_{10} + 158 \beta_{11} + 758 \beta_{12} ) q^{52} + ( -310211 + 21294 \beta_{1} - 2255 \beta_{2} - 98 \beta_{3} + 104 \beta_{4} + 370 \beta_{5} + 230 \beta_{6} + 409 \beta_{7} - 374 \beta_{8} + 847 \beta_{9} + 392 \beta_{10} + 853 \beta_{11} + 442 \beta_{12} ) q^{53} + ( 36835 - 25250 \beta_{1} + 576 \beta_{2} + 292 \beta_{3} - 815 \beta_{4} - 743 \beta_{5} + 248 \beta_{6} + 939 \beta_{7} + 19 \beta_{8} - 142 \beta_{9} - 719 \beta_{10} + 48 \beta_{11} + 872 \beta_{12} ) q^{55} + ( -117666 + 9908 \beta_{1} - 4400 \beta_{2} - 938 \beta_{3} + 1024 \beta_{4} - 1204 \beta_{5} - 3066 \beta_{6} + 68 \beta_{7} + 138 \beta_{8} - 426 \beta_{9} + 1152 \beta_{10} + 1324 \beta_{11} + 198 \beta_{12} ) q^{56} + ( -313192 - 811 \beta_{1} - 166 \beta_{2} + 1974 \beta_{3} - 100 \beta_{4} + 1582 \beta_{5} + 2714 \beta_{6} - 50 \beta_{7} + 1197 \beta_{8} - 572 \beta_{9} - 511 \beta_{10} - 332 \beta_{11} + 904 \beta_{12} ) q^{58} + ( -173540 + 12667 \beta_{1} - 3746 \beta_{2} - 441 \beta_{3} - 1184 \beta_{4} - 896 \beta_{5} - 625 \beta_{6} - 227 \beta_{7} - 2039 \beta_{8} - 161 \beta_{9} - 609 \beta_{10} + 393 \beta_{11} - 529 \beta_{12} ) q^{59} + ( 473639 + 3482 \beta_{1} + 4756 \beta_{2} - 748 \beta_{3} - 1813 \beta_{4} + 695 \beta_{5} + 136 \beta_{6} + 729 \beta_{7} + 557 \beta_{8} - 902 \beta_{9} + 295 \beta_{10} - 1544 \beta_{11} - 500 \beta_{12} ) q^{61} + ( -73999 + 33417 \beta_{1} - 3989 \beta_{2} + 260 \beta_{3} + 962 \beta_{4} + 1550 \beta_{5} - 1164 \beta_{6} - 681 \beta_{7} + 932 \beta_{8} + 1105 \beta_{9} - 1877 \beta_{10} + 482 \beta_{11} + 465 \beta_{12} ) q^{62} + ( -250169 + 62038 \beta_{1} + 1878 \beta_{2} - 3805 \beta_{3} + 1060 \beta_{4} - 4275 \beta_{5} - 2751 \beta_{6} + 1235 \beta_{7} - 978 \beta_{8} - 714 \beta_{9} + 1650 \beta_{10} + 1498 \beta_{11} - 418 \beta_{12} ) q^{64} + ( -437725 + 28860 \beta_{1} - 2308 \beta_{2} + 56 \beta_{3} - 2089 \beta_{4} + 155 \beta_{5} + 4542 \beta_{6} + 821 \beta_{7} - 3479 \beta_{8} - 462 \beta_{9} - 2893 \beta_{10} - 748 \beta_{11} + 1532 \beta_{12} ) q^{65} + ( -156771 + 20983 \beta_{1} + 213 \beta_{2} - 751 \beta_{3} - 629 \beta_{4} + 2685 \beta_{5} + 321 \beta_{6} + 287 \beta_{7} + 666 \beta_{8} + 852 \beta_{9} + 1089 \beta_{10} + 2143 \beta_{11} + 1829 \beta_{12} ) q^{67} + ( -700014 - 34286 \beta_{1} - 11196 \beta_{2} + 608 \beta_{3} + 4392 \beta_{4} - 4382 \beta_{5} - 1996 \beta_{6} - 4155 \beta_{7} + 328 \beta_{8} - 6495 \beta_{9} + 1955 \beta_{10} + 5324 \beta_{11} + 2051 \beta_{12} ) q^{68} + ( -97740 + 67916 \beta_{1} - 5820 \beta_{2} + 310 \beta_{3} + 757 \beta_{4} - 683 \beta_{5} - 2584 \beta_{6} - 581 \beta_{7} + 1521 \beta_{8} - 1260 \beta_{9} + 292 \beta_{10} + 263 \beta_{11} - 1270 \beta_{12} ) q^{70} + ( -387754 - 8102 \beta_{1} + 6106 \beta_{2} + 560 \beta_{3} - 981 \beta_{4} + 1259 \beta_{5} + 804 \beta_{6} + 1363 \beta_{7} + 1473 \beta_{8} + 552 \beta_{9} - 776 \beta_{10} - 2995 \beta_{11} - 3992 \beta_{12} ) q^{71} + ( 633591 + 30314 \beta_{1} - 7782 \beta_{2} - 24 \beta_{3} + 2300 \beta_{4} - 5360 \beta_{5} - 3346 \beta_{6} - 4816 \beta_{7} - 1190 \beta_{8} - 2272 \beta_{9} + 2305 \beta_{10} + 5131 \beta_{11} + 928 \beta_{12} ) q^{73} + ( 285237 - 52230 \beta_{1} + 6613 \beta_{2} + 1851 \beta_{3} - 1208 \beta_{4} - 2127 \beta_{5} + 2021 \beta_{6} + 180 \beta_{7} + 270 \beta_{8} - 713 \beta_{9} - 731 \beta_{10} - 264 \beta_{11} + 369 \beta_{12} ) q^{74} + ( 779618 + 163004 \beta_{1} + 5657 \beta_{2} - 206 \beta_{3} + 1156 \beta_{4} - 3404 \beta_{5} + 3878 \beta_{6} - 2219 \beta_{7} + 284 \beta_{8} - 1737 \beta_{9} - 5879 \beta_{10} + 2480 \beta_{11} + 2321 \beta_{12} ) q^{76} + ( -1341185 + 48720 \beta_{1} + 3640 \beta_{2} - 1210 \beta_{3} + 1097 \beta_{4} + 2849 \beta_{5} + 2518 \beta_{6} - 1947 \beta_{7} - 787 \beta_{8} - 3704 \beta_{9} - 845 \beta_{10} - 1834 \beta_{11} + 2442 \beta_{12} ) q^{77} + ( 502894 + 164868 \beta_{1} - 12904 \beta_{2} - 1657 \beta_{3} + 3066 \beta_{4} - 1334 \beta_{5} + 494 \beta_{6} + 1201 \beta_{7} - 1037 \beta_{8} + 497 \beta_{9} + 4646 \beta_{10} - 650 \beta_{11} - 4425 \beta_{12} ) q^{79} + ( -1226336 + 31762 \beta_{1} - 5513 \beta_{2} + 712 \beta_{3} - 906 \beta_{4} + 3898 \beta_{5} + 5588 \beta_{6} - 1655 \beta_{7} - 810 \beta_{8} + 6539 \beta_{9} - 1203 \beta_{10} + 1210 \beta_{11} - 3779 \beta_{12} ) q^{80} + ( -78426 + 128149 \beta_{1} + 641 \beta_{2} + 4627 \beta_{3} - 2717 \beta_{4} + 6754 \beta_{5} + 13097 \beta_{6} - 39 \beta_{7} - 2481 \beta_{8} + 6185 \beta_{9} + 769 \beta_{10} - 7965 \beta_{11} - 2005 \beta_{12} ) q^{82} + ( -1759771 - 25330 \beta_{1} + 427 \beta_{2} + 1650 \beta_{3} + 819 \beta_{4} - 295 \beta_{5} - 8232 \beta_{6} + 3422 \beta_{7} + 9297 \beta_{8} + 861 \beta_{9} - 3080 \beta_{10} + 1560 \beta_{11} - 2426 \beta_{12} ) q^{83} + ( -85570 + 217719 \beta_{1} - 783 \beta_{2} - 4698 \beta_{3} + 8427 \beta_{4} + 2241 \beta_{5} - 12351 \beta_{6} - 1959 \beta_{7} + 4324 \beta_{8} + 5424 \beta_{9} + 8515 \beta_{10} + 3242 \beta_{11} - 1554 \beta_{12} ) q^{85} + ( 79507 + 79507 \beta_{1} ) q^{86} + ( 830576 + 52713 \beta_{1} + 24458 \beta_{2} - 654 \beta_{3} - 4453 \beta_{4} - 488 \beta_{5} + 14618 \beta_{6} + 1017 \beta_{7} - 8901 \beta_{8} + 5418 \beta_{9} + 2054 \beta_{10} - 9817 \beta_{11} - 1910 \beta_{12} ) q^{88} + ( -733623 + 71388 \beta_{1} - 522 \beta_{2} + 5584 \beta_{3} + 55 \beta_{4} - 3113 \beta_{5} + 3742 \beta_{6} - 5105 \beta_{7} - 4235 \beta_{8} + 3840 \beta_{9} - 3539 \beta_{10} - 3656 \beta_{11} + 4480 \beta_{12} ) q^{89} + ( 1929315 + 67692 \beta_{1} + 21956 \beta_{2} + 3270 \beta_{3} - 10173 \beta_{4} + 6015 \beta_{5} + 15870 \beta_{6} + 8079 \beta_{7} + 203 \beta_{8} + 3732 \beta_{9} - 7281 \beta_{10} - 2724 \beta_{11} - 3750 \beta_{12} ) q^{91} + ( 999135 + 433330 \beta_{1} - 280 \beta_{2} - 1595 \beta_{3} - 5592 \beta_{4} + 651 \beta_{5} + 15771 \beta_{6} + 8644 \beta_{7} - 8650 \beta_{8} + 9549 \beta_{9} - 5181 \beta_{10} - 9014 \beta_{11} + 1119 \beta_{12} ) q^{92} + ( -1935753 + 140984 \beta_{1} - 4739 \beta_{2} + 5767 \beta_{3} - 381 \beta_{4} - 3466 \beta_{5} + 10483 \beta_{6} + 2223 \beta_{7} + 5139 \beta_{8} + 3827 \beta_{9} - 9625 \beta_{10} - 1815 \beta_{11} + 4875 \beta_{12} ) q^{94} + ( -2408905 + 154008 \beta_{1} + 20591 \beta_{2} - 5714 \beta_{3} + 1233 \beta_{4} + 11535 \beta_{5} + 2028 \beta_{6} + 6697 \beta_{7} + 5738 \beta_{8} - 2598 \beta_{9} + 10767 \beta_{10} - 2006 \beta_{11} - 1962 \beta_{12} ) q^{95} + ( 718525 + 224873 \beta_{1} + 11127 \beta_{2} - 1406 \beta_{3} + 4558 \beta_{4} - 4412 \beta_{5} - 7441 \beta_{6} - 3067 \beta_{7} + 7124 \beta_{8} - 11279 \beta_{9} + 479 \beta_{10} + 2454 \beta_{11} + 10304 \beta_{12} ) q^{97} + ( 2417455 + 290011 \beta_{1} + 5234 \beta_{2} - 750 \beta_{3} - 1055 \beta_{4} - 1803 \beta_{5} + 4232 \beta_{6} - 4079 \beta_{7} - 13875 \beta_{8} + 9146 \beta_{9} + 4626 \beta_{10} - 4281 \beta_{11} - 1172 \beta_{12} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 197 \)
\(\beta_{3}\)\(=\)\((\)\(-11644561393100943114882344733 \nu^{12} + 381172805209634330178560184566 \nu^{11} + 15501432242950985103457686236177 \nu^{10} - 451012933922670114912981301432604 \nu^{9} - 7366396189565751031090727540858762 \nu^{8} + 187359995456315918988627875206048812 \nu^{7} + 1488381862739567786482947551232950606 \nu^{6} - 31922412650444841997040848956458008016 \nu^{5} - 119536607530995582695965492425192465261 \nu^{4} + 1951414361015228070549841691077222633406 \nu^{3} + 3023434000447915424721413923158911006321 \nu^{2} - 30582172910074546841517927512651020934644 \nu - 13305316072938557965854677632794396791228\)\()/ \)\(48\!\cdots\!52\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-854459448335333719793315653 \nu^{12} + 10457577384080147006273157068 \nu^{11} + 1070609903100447582555447791491 \nu^{10} - 12835049128455947213058717042408 \nu^{9} - 480068146024777367299752835115182 \nu^{8} + 5474383383759495451200076091846676 \nu^{7} + 91822017315183812478842819965689918 \nu^{6} - 932910362369806216777926699431049588 \nu^{5} - 7049906948238183243525669990947959401 \nu^{4} + 53794078346293388896499622866286670352 \nu^{3} + 198594908373095703861244751156900597919 \nu^{2} - 932241439180452830959672033507375024020 \nu - 1848217807948777715360672110616864936964\)\()/ \)\(15\!\cdots\!36\)\( \)
\(\beta_{5}\)\(=\)\((\)\(12876917253478895406041754703 \nu^{12} - 112171272741429165204008927374 \nu^{11} - 16151996223921079364716881328719 \nu^{10} + 138083156333172137924049288596076 \nu^{9} + 7291227876748309736911681081915286 \nu^{8} - 58536204775574681992209472289885044 \nu^{7} - 1417941438393719271289280622134501562 \nu^{6} + 9680669082418377010133770359930434584 \nu^{5} + 111290604273225957547086655239166575687 \nu^{4} - 493180244251651452703266922999561331006 \nu^{3} - 2738873917407689654167043816605901830023 \nu^{2} + 4980417573119460921708552509829984918204 \nu + 5977405933856904249708870104676424130148\)\()/ \)\(90\!\cdots\!16\)\( \)
\(\beta_{6}\)\(=\)\((\)\(1056556185794670677828069741849 \nu^{12} - 8454288711877173675704940058814 \nu^{11} - 1304639684754092216888790657404541 \nu^{10} + 10601183567948437527824647131523308 \nu^{9} + 575013133954134363315255824016309826 \nu^{8} - 4583273858964789484423729678489595324 \nu^{7} - 107279627570593604628851924687235693750 \nu^{6} + 777421199738619929197771693628469390992 \nu^{5} + 7745841349489648802189471331983148765225 \nu^{4} - 41786709722639531130546181351697403195622 \nu^{3} - 163536605795427541230974759147059669963869 \nu^{2} + 482107703639224888174277992475198157930468 \nu + 193887990924386511125671837705086680822796\)\()/ \)\(43\!\cdots\!68\)\( \)
\(\beta_{7}\)\(=\)\((\)\(609628040274773765686700495791 \nu^{12} - 3301618975195359347168205809938 \nu^{11} - 760874503332197114367450231385995 \nu^{10} + 4183735150063373724133400593303092 \nu^{9} + 342034545868647564761782594665503246 \nu^{8} - 1799231755919372330810645921223096484 \nu^{7} - 66257546825269814332011305387380602618 \nu^{6} + 292455220071800822298943286296795258672 \nu^{5} + 5152786339727040474821337987184234695519 \nu^{4} - 13088519649976582512173886867071436691178 \nu^{3} - 121243361248500489325962055907659411333995 \nu^{2} + 72744182619509649251226122347394116326204 \nu + 209094036383880949965367496444269310507220\)\()/ \)\(21\!\cdots\!84\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-191590755258224372380382007533 \nu^{12} + 1337230315893986401319183245558 \nu^{11} + 237686733830652456821240679341953 \nu^{10} - 1676956685288974787039426199380508 \nu^{9} - 105769272392877638573688904761177322 \nu^{8} + 719804417434332761155324012203188844 \nu^{7} + 20139079551474772420695951582015782446 \nu^{6} - 119056858190946237429490759019074536080 \nu^{5} - 1524010461526012671138758047002558328573 \nu^{4} + 5835448902401378857536433463918115495550 \nu^{3} + 35666861434320951610929431388020248545825 \nu^{2} - 48064799726629005291385214900703202426164 \nu - 70768164591854233846067681045476161940220\)\()/ \)\(48\!\cdots\!52\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-197157336932212300949429648639 \nu^{12} + 1489494202797531478957014303986 \nu^{11} + 244936240378734369994336002847131 \nu^{10} - 1856518840997356907023668631455476 \nu^{9} - 109096333280764235689689955396852398 \nu^{8} + 794538054641015619711363656372346212 \nu^{7} + 20765716563108978761519920511040059098 \nu^{6} - 131902623640898086726073394896218826544 \nu^{5} - 1566311602910780069752641601896000980655 \nu^{4} + 6631733302204599608812505926531283486794 \nu^{3} + 36487381711712303096867598183123940979323 \nu^{2} - 61514122458320011391123199502476059671676 \nu - 83002386531772132181060198298394428646548\)\()/ \)\(48\!\cdots\!52\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(19\!\cdots\!57\)\( \nu^{12} + \)\(15\!\cdots\!02\)\( \nu^{11} + \)\(23\!\cdots\!81\)\( \nu^{10} - \)\(19\!\cdots\!12\)\( \nu^{9} - \)\(10\!\cdots\!14\)\( \nu^{8} + \)\(80\!\cdots\!24\)\( \nu^{7} + \)\(20\!\cdots\!98\)\( \nu^{6} - \)\(13\!\cdots\!12\)\( \nu^{5} - \)\(15\!\cdots\!33\)\( \nu^{4} + \)\(66\!\cdots\!06\)\( \nu^{3} + \)\(35\!\cdots\!17\)\( \nu^{2} - \)\(60\!\cdots\!64\)\( \nu - \)\(69\!\cdots\!12\)\(\)\()/ \)\(43\!\cdots\!68\)\( \)
\(\beta_{11}\)\(=\)\((\)\(51227735900484919958819335 \nu^{12} - 393398885573251513939979554 \nu^{11} - 63346131840941245303684566531 \nu^{10} + 492344960539917037776058343124 \nu^{9} + 28002219585339198717827297997758 \nu^{8} - 211890051727789670311133019457348 \nu^{7} - 5256350580797526760033192642707978 \nu^{6} + 35551160309990962142826899944133680 \nu^{5} + 384314709834907385781060560501758647 \nu^{4} - 1853405734184595970206179539280841530 \nu^{3} - 8220529527243861532312208652708433251 \nu^{2} + 21054985866693526451605060539920207004 \nu + 9181067048688447034159022829403219188\)\()/ \)\(11\!\cdots\!88\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-71671680901154440242219326113 \nu^{12} + 600872369970819842284645054670 \nu^{11} + 89003061784657421306349036424837 \nu^{10} - 744512994562053045273064523779468 \nu^{9} - 39533013498053042823158503681735570 \nu^{8} + 317284424917478634924952267643844380 \nu^{7} + 7463545253349365662894182279496712166 \nu^{6} - 52637474854881031556626220955664673232 \nu^{5} - 550184829869035215104319502734741085681 \nu^{4} + 2675924287273234938115903360706107282806 \nu^{3} + 11927375709521648868135952062918031477925 \nu^{2} - 26554970015650460611609888173168828345860 \nu - 13550425149168447806979658661906941959916\)\()/ \)\(12\!\cdots\!88\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 197\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} + 342 \beta_{1} - 59\)
\(\nu^{4}\)\(=\)\(-4 \beta_{11} + 2 \beta_{10} - 16 \beta_{9} + 2 \beta_{8} + \beta_{7} - 5 \beta_{6} - 19 \beta_{5} - 10 \beta_{4} - 3 \beta_{3} + 430 \beta_{2} - 372 \beta_{1} + 67398\)
\(\nu^{5}\)\(=\)\(82 \beta_{12} - 364 \beta_{11} - 92 \beta_{10} + 246 \beta_{9} - 694 \beta_{8} + 56 \beta_{7} + 406 \beta_{6} - 730 \beta_{5} + 440 \beta_{4} - 702 \beta_{3} - 1760 \beta_{2} + 130435 \beta_{1} - 100286\)
\(\nu^{6}\)\(=\)\(-910 \beta_{12} - 1358 \beta_{11} + 3452 \beta_{10} - 12190 \beta_{9} + 1896 \beta_{8} + 1524 \beta_{7} - 5772 \beta_{6} - 14310 \beta_{5} - 5290 \beta_{4} - 4008 \beta_{3} + 179089 \beta_{2} - 278446 \beta_{1} + 25712561\)
\(\nu^{7}\)\(=\)\(43958 \beta_{12} - 135381 \beta_{11} - 60312 \beta_{10} + 210482 \beta_{9} - 399599 \beta_{8} + 42038 \beta_{7} + 148885 \beta_{6} - 403673 \beta_{5} + 171917 \beta_{4} - 376669 \beta_{3} - 885307 \beta_{2} + 52368676 \beta_{1} - 66204871\)
\(\nu^{8}\)\(=\)\(-648870 \beta_{12} - 147986 \beta_{11} + 2602726 \beta_{10} - 7148222 \beta_{9} + 1038090 \beta_{8} + 949613 \beta_{7} - 4117249 \beta_{6} - 8029893 \beta_{5} - 2282548 \beta_{4} - 2874435 \beta_{3} + 75599778 \beta_{2} - 154079462 \beta_{1} + 10326801418\)
\(\nu^{9}\)\(=\)\(16786808 \beta_{12} - 57213756 \beta_{11} - 28254332 \beta_{10} + 131196368 \beta_{9} - 210479464 \beta_{8} + 25185846 \beta_{7} + 60736046 \beta_{6} - 199812850 \beta_{5} + 64621264 \beta_{4} - 184518630 \beta_{3} - 417901804 \beta_{2} + 21744711681 \beta_{1} - 35097253198\)
\(\nu^{10}\)\(=\)\(-326459580 \beta_{12} + 189972212 \beta_{11} + 1543929920 \beta_{10} - 3809775548 \beta_{9} + 489177400 \beta_{8} + 448580004 \beta_{7} - 2475166660 \beta_{6} - 4076880528 \beta_{5} - 898705500 \beta_{4} - 1679026308 \beta_{3} + 32341046761 \beta_{2} - 76379589068 \beta_{1} + 4288361875145\)
\(\nu^{11}\)\(=\)\(5161725412 \beta_{12} - 26700665177 \beta_{11} - 11736596944 \beta_{10} + 72532507884 \beta_{9} - 105428458997 \beta_{8} + 13933907740 \beta_{7} + 27970377857 \beta_{6} - 93906894985 \beta_{5} + 23579751209 \beta_{4} - 86528972593 \beta_{3} - 190907348355 \beta_{2} + 9240954760234 \beta_{1} - 17034713528459\)
\(\nu^{12}\)\(=\)\(-139155983092 \beta_{12} + 207979112680 \beta_{11} + 821451174890 \beta_{10} - 1938413731572 \beta_{9} + 222871358386 \beta_{8} + 184449462297 \beta_{7} - 1363536557997 \beta_{6} - 1977787024591 \beta_{5} - 329457520662 \beta_{4} - 891826642051 \beta_{3} + 13978942788310 \beta_{2} - 35973707277712 \beta_{1} + 1822187281003358\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
21.1822
17.8435
16.3440
15.0703
7.08185
2.37903
−0.684341
−4.71679
−4.99525
−8.80627
−16.1540
−20.1662
−21.3781
−22.1822 0 364.050 −122.553 0 −90.8173 −5236.10 0 2718.48
1.2 −18.8435 0 227.079 −70.1157 0 1065.82 −1867.00 0 1321.23
1.3 −17.3440 0 172.814 122.945 0 247.011 −777.246 0 −2132.35
1.4 −16.0703 0 130.256 −431.884 0 218.970 −36.2566 0 6940.53
1.5 −8.08185 0 −62.6836 −164.718 0 1314.61 1541.08 0 1331.23
1.6 −3.37903 0 −116.582 241.175 0 −173.086 826.450 0 −814.939
1.7 −0.315659 0 −127.900 −531.642 0 −349.841 80.7773 0 167.818
1.8 3.71679 0 −114.185 −385.350 0 −1546.77 −900.153 0 −1432.27
1.9 3.99525 0 −112.038 383.451 0 1003.83 −959.011 0 1531.98
1.10 7.80627 0 −67.0621 402.005 0 −356.766 −1522.71 0 3138.16
1.11 15.1540 0 101.645 −210.950 0 1100.87 −399.390 0 −3196.74
1.12 19.1662 0 239.342 174.722 0 −1126.43 2133.99 0 3348.75
1.13 20.3781 0 287.267 −405.085 0 52.5965 3245.57 0 −8254.88
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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 387.8.a.d 13
3.b odd 2 1 43.8.a.b 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.8.a.b 13 3.b odd 2 1
387.8.a.d 13 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(43\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{13} + \cdots\) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(387))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 16 T + 499 T^{2} + 7226 T^{3} + 129754 T^{4} + 1623108 T^{5} + 25061980 T^{6} + 301751896 T^{7} + 4320739680 T^{8} + 53577914560 T^{9} + 686733102400 T^{10} + 8227010823296 T^{11} + 97673737613312 T^{12} + 1101231869857792 T^{13} + 12502238414503936 T^{14} + 134791345328881664 T^{15} + 1440183699164364800 T^{16} + 14382211926442639360 T^{17} + \)\(14\!\cdots\!40\)\( T^{18} + \)\(13\!\cdots\!84\)\( T^{19} + \)\(14\!\cdots\!60\)\( T^{20} + \)\(11\!\cdots\!88\)\( T^{21} + \)\(11\!\cdots\!32\)\( T^{22} + \)\(85\!\cdots\!24\)\( T^{23} + \)\(75\!\cdots\!28\)\( T^{24} + \)\(30\!\cdots\!56\)\( T^{25} + \)\(24\!\cdots\!48\)\( T^{26} \)
$3$ 1
$5$ \( 1 + 998 T + 870740 T^{2} + 535689006 T^{3} + 300509982542 T^{4} + 143467127408470 T^{5} + 64011052241216400 T^{6} + 25802469413530458250 T^{7} + \)\(98\!\cdots\!00\)\( T^{8} + \)\(34\!\cdots\!50\)\( T^{9} + \)\(11\!\cdots\!75\)\( T^{10} + \)\(37\!\cdots\!00\)\( T^{11} + \)\(11\!\cdots\!50\)\( T^{12} + \)\(32\!\cdots\!00\)\( T^{13} + \)\(88\!\cdots\!50\)\( T^{14} + \)\(22\!\cdots\!00\)\( T^{15} + \)\(56\!\cdots\!75\)\( T^{16} + \)\(12\!\cdots\!50\)\( T^{17} + \)\(28\!\cdots\!00\)\( T^{18} + \)\(58\!\cdots\!50\)\( T^{19} + \)\(11\!\cdots\!00\)\( T^{20} + \)\(19\!\cdots\!50\)\( T^{21} + \)\(32\!\cdots\!50\)\( T^{22} + \)\(45\!\cdots\!50\)\( T^{23} + \)\(57\!\cdots\!00\)\( T^{24} + \)\(51\!\cdots\!50\)\( T^{25} + \)\(40\!\cdots\!25\)\( T^{26} \)
$7$ \( 1 - 1360 T + 7058509 T^{2} - 7851071624 T^{3} + 22977827763868 T^{4} - 21067270496738864 T^{5} + 46647610898487061092 T^{6} - \)\(35\!\cdots\!92\)\( T^{7} + \)\(68\!\cdots\!45\)\( T^{8} - \)\(44\!\cdots\!60\)\( T^{9} + \)\(78\!\cdots\!53\)\( T^{10} - \)\(44\!\cdots\!96\)\( T^{11} + \)\(74\!\cdots\!48\)\( T^{12} - \)\(39\!\cdots\!32\)\( T^{13} + \)\(61\!\cdots\!64\)\( T^{14} - \)\(30\!\cdots\!04\)\( T^{15} + \)\(43\!\cdots\!71\)\( T^{16} - \)\(20\!\cdots\!60\)\( T^{17} + \)\(25\!\cdots\!35\)\( T^{18} - \)\(11\!\cdots\!08\)\( T^{19} + \)\(11\!\cdots\!44\)\( T^{20} - \)\(44\!\cdots\!64\)\( T^{21} + \)\(40\!\cdots\!24\)\( T^{22} - \)\(11\!\cdots\!76\)\( T^{23} + \)\(83\!\cdots\!63\)\( T^{24} - \)\(13\!\cdots\!60\)\( T^{25} + \)\(80\!\cdots\!43\)\( T^{26} \)
$11$ \( 1 + 1620 T + 132588019 T^{2} + 123981604500 T^{3} + 7703760283965880 T^{4} + 3713200717818796164 T^{5} + \)\(25\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!44\)\( T^{8} + \)\(83\!\cdots\!32\)\( T^{9} + \)\(85\!\cdots\!00\)\( T^{10} + \)\(37\!\cdots\!84\)\( T^{11} + \)\(11\!\cdots\!86\)\( T^{12} + \)\(96\!\cdots\!20\)\( T^{13} + \)\(22\!\cdots\!06\)\( T^{14} + \)\(14\!\cdots\!44\)\( T^{15} + \)\(62\!\cdots\!00\)\( T^{16} + \)\(11\!\cdots\!92\)\( T^{17} + \)\(15\!\cdots\!44\)\( T^{18} + \)\(68\!\cdots\!20\)\( T^{19} + \)\(27\!\cdots\!00\)\( T^{20} + \)\(77\!\cdots\!04\)\( T^{21} + \)\(31\!\cdots\!80\)\( T^{22} + \)\(97\!\cdots\!00\)\( T^{23} + \)\(20\!\cdots\!49\)\( T^{24} + \)\(48\!\cdots\!20\)\( T^{25} + \)\(58\!\cdots\!11\)\( T^{26} \)
$13$ \( 1 - 13550 T + 451004117 T^{2} - 4457271716692 T^{3} + 93112309019999992 T^{4} - \)\(74\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!04\)\( T^{6} - \)\(87\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!96\)\( T^{8} - \)\(81\!\cdots\!96\)\( T^{9} + \)\(11\!\cdots\!04\)\( T^{10} - \)\(63\!\cdots\!80\)\( T^{11} + \)\(82\!\cdots\!86\)\( T^{12} - \)\(42\!\cdots\!16\)\( T^{13} + \)\(51\!\cdots\!62\)\( T^{14} - \)\(24\!\cdots\!20\)\( T^{15} + \)\(28\!\cdots\!52\)\( T^{16} - \)\(12\!\cdots\!16\)\( T^{17} + \)\(12\!\cdots\!72\)\( T^{18} - \)\(53\!\cdots\!12\)\( T^{19} + \)\(48\!\cdots\!92\)\( T^{20} - \)\(17\!\cdots\!16\)\( T^{21} + \)\(14\!\cdots\!24\)\( T^{22} - \)\(42\!\cdots\!08\)\( T^{23} + \)\(26\!\cdots\!61\)\( T^{24} - \)\(50\!\cdots\!50\)\( T^{25} + \)\(23\!\cdots\!37\)\( T^{26} \)
$17$ \( 1 + 110880 T + 7269241380 T^{2} + 350308303802712 T^{3} + 13915691873194028506 T^{4} + \)\(48\!\cdots\!14\)\( T^{5} + \)\(15\!\cdots\!04\)\( T^{6} + \)\(43\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!91\)\( T^{8} + \)\(30\!\cdots\!04\)\( T^{9} + \)\(72\!\cdots\!29\)\( T^{10} + \)\(16\!\cdots\!04\)\( T^{11} + \)\(35\!\cdots\!06\)\( T^{12} + \)\(74\!\cdots\!34\)\( T^{13} + \)\(14\!\cdots\!38\)\( T^{14} + \)\(27\!\cdots\!16\)\( T^{15} + \)\(49\!\cdots\!93\)\( T^{16} + \)\(85\!\cdots\!64\)\( T^{17} + \)\(13\!\cdots\!63\)\( T^{18} + \)\(20\!\cdots\!00\)\( T^{19} + \)\(29\!\cdots\!88\)\( T^{20} + \)\(38\!\cdots\!34\)\( T^{21} + \)\(45\!\cdots\!78\)\( T^{22} + \)\(47\!\cdots\!88\)\( T^{23} + \)\(40\!\cdots\!60\)\( T^{24} + \)\(25\!\cdots\!80\)\( T^{25} + \)\(93\!\cdots\!33\)\( T^{26} \)
$19$ \( 1 - 105058 T + 10894929034 T^{2} - 729074101697894 T^{3} + 46770979546573980358 T^{4} - \)\(24\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!42\)\( T^{6} - \)\(53\!\cdots\!86\)\( T^{7} + \)\(22\!\cdots\!20\)\( T^{8} - \)\(85\!\cdots\!82\)\( T^{9} + \)\(31\!\cdots\!91\)\( T^{10} - \)\(10\!\cdots\!04\)\( T^{11} + \)\(35\!\cdots\!64\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{13} + \)\(31\!\cdots\!96\)\( T^{14} - \)\(85\!\cdots\!84\)\( T^{15} + \)\(22\!\cdots\!29\)\( T^{16} - \)\(54\!\cdots\!62\)\( T^{17} + \)\(12\!\cdots\!80\)\( T^{18} - \)\(27\!\cdots\!46\)\( T^{19} + \)\(55\!\cdots\!18\)\( T^{20} - \)\(99\!\cdots\!16\)\( T^{21} + \)\(17\!\cdots\!22\)\( T^{22} - \)\(23\!\cdots\!94\)\( T^{23} + \)\(31\!\cdots\!26\)\( T^{24} - \)\(27\!\cdots\!18\)\( T^{25} + \)\(23\!\cdots\!19\)\( T^{26} \)
$23$ \( 1 + 160184 T + 36979701828 T^{2} + 4753092322433576 T^{3} + \)\(65\!\cdots\!70\)\( T^{4} + \)\(68\!\cdots\!64\)\( T^{5} + \)\(71\!\cdots\!92\)\( T^{6} + \)\(64\!\cdots\!20\)\( T^{7} + \)\(55\!\cdots\!51\)\( T^{8} + \)\(42\!\cdots\!96\)\( T^{9} + \)\(31\!\cdots\!15\)\( T^{10} + \)\(21\!\cdots\!56\)\( T^{11} + \)\(13\!\cdots\!02\)\( T^{12} + \)\(83\!\cdots\!40\)\( T^{13} + \)\(47\!\cdots\!94\)\( T^{14} + \)\(24\!\cdots\!04\)\( T^{15} + \)\(12\!\cdots\!45\)\( T^{16} + \)\(57\!\cdots\!76\)\( T^{17} + \)\(25\!\cdots\!57\)\( T^{18} + \)\(10\!\cdots\!80\)\( T^{19} + \)\(38\!\cdots\!96\)\( T^{20} + \)\(12\!\cdots\!04\)\( T^{21} + \)\(40\!\cdots\!90\)\( T^{22} + \)\(99\!\cdots\!24\)\( T^{23} + \)\(26\!\cdots\!84\)\( T^{24} + \)\(38\!\cdots\!44\)\( T^{25} + \)\(82\!\cdots\!27\)\( T^{26} \)
$29$ \( 1 + 285546 T + 180604853400 T^{2} + 41013344500675266 T^{3} + \)\(14\!\cdots\!66\)\( T^{4} + \)\(27\!\cdots\!34\)\( T^{5} + \)\(71\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!42\)\( T^{7} + \)\(24\!\cdots\!24\)\( T^{8} + \)\(32\!\cdots\!66\)\( T^{9} + \)\(61\!\cdots\!47\)\( T^{10} + \)\(73\!\cdots\!48\)\( T^{11} + \)\(12\!\cdots\!62\)\( T^{12} + \)\(13\!\cdots\!56\)\( T^{13} + \)\(21\!\cdots\!58\)\( T^{14} + \)\(21\!\cdots\!88\)\( T^{15} + \)\(31\!\cdots\!63\)\( T^{16} + \)\(29\!\cdots\!26\)\( T^{17} + \)\(36\!\cdots\!76\)\( T^{18} + \)\(29\!\cdots\!22\)\( T^{19} + \)\(32\!\cdots\!60\)\( T^{20} + \)\(21\!\cdots\!14\)\( T^{21} + \)\(19\!\cdots\!74\)\( T^{22} + \)\(95\!\cdots\!66\)\( T^{23} + \)\(72\!\cdots\!00\)\( T^{24} + \)\(19\!\cdots\!26\)\( T^{25} + \)\(11\!\cdots\!29\)\( T^{26} \)
$31$ \( 1 + 99616 T + 241614413464 T^{2} + 18338344889599376 T^{3} + \)\(27\!\cdots\!10\)\( T^{4} + \)\(14\!\cdots\!12\)\( T^{5} + \)\(20\!\cdots\!92\)\( T^{6} + \)\(65\!\cdots\!12\)\( T^{7} + \)\(10\!\cdots\!63\)\( T^{8} + \)\(16\!\cdots\!72\)\( T^{9} + \)\(42\!\cdots\!31\)\( T^{10} + \)\(15\!\cdots\!08\)\( T^{11} + \)\(14\!\cdots\!42\)\( T^{12} - \)\(52\!\cdots\!48\)\( T^{13} + \)\(38\!\cdots\!62\)\( T^{14} + \)\(11\!\cdots\!68\)\( T^{15} + \)\(89\!\cdots\!61\)\( T^{16} + \)\(92\!\cdots\!52\)\( T^{17} + \)\(16\!\cdots\!13\)\( T^{18} + \)\(28\!\cdots\!32\)\( T^{19} + \)\(24\!\cdots\!32\)\( T^{20} + \)\(48\!\cdots\!72\)\( T^{21} + \)\(25\!\cdots\!10\)\( T^{22} + \)\(45\!\cdots\!76\)\( T^{23} + \)\(16\!\cdots\!04\)\( T^{24} + \)\(18\!\cdots\!36\)\( T^{25} + \)\(51\!\cdots\!31\)\( T^{26} \)
$37$ \( 1 - 176038 T + 1002592896582 T^{2} - 172621415570488610 T^{3} + \)\(48\!\cdots\!62\)\( T^{4} - \)\(79\!\cdots\!90\)\( T^{5} + \)\(14\!\cdots\!62\)\( T^{6} - \)\(22\!\cdots\!06\)\( T^{7} + \)\(31\!\cdots\!64\)\( T^{8} - \)\(45\!\cdots\!18\)\( T^{9} + \)\(50\!\cdots\!45\)\( T^{10} - \)\(65\!\cdots\!96\)\( T^{11} + \)\(61\!\cdots\!44\)\( T^{12} - \)\(71\!\cdots\!12\)\( T^{13} + \)\(58\!\cdots\!52\)\( T^{14} - \)\(59\!\cdots\!44\)\( T^{15} + \)\(43\!\cdots\!65\)\( T^{16} - \)\(36\!\cdots\!78\)\( T^{17} + \)\(24\!\cdots\!52\)\( T^{18} - \)\(16\!\cdots\!14\)\( T^{19} + \)\(10\!\cdots\!74\)\( T^{20} - \)\(52\!\cdots\!90\)\( T^{21} + \)\(30\!\cdots\!86\)\( T^{22} - \)\(10\!\cdots\!90\)\( T^{23} + \)\(56\!\cdots\!94\)\( T^{24} - \)\(94\!\cdots\!18\)\( T^{25} + \)\(50\!\cdots\!13\)\( T^{26} \)
$41$ \( 1 - 410260 T + 1275774355044 T^{2} - 520566650539856152 T^{3} + \)\(84\!\cdots\!34\)\( T^{4} - \)\(33\!\cdots\!10\)\( T^{5} + \)\(37\!\cdots\!04\)\( T^{6} - \)\(14\!\cdots\!96\)\( T^{7} + \)\(12\!\cdots\!59\)\( T^{8} - \)\(44\!\cdots\!08\)\( T^{9} + \)\(32\!\cdots\!41\)\( T^{10} - \)\(11\!\cdots\!44\)\( T^{11} + \)\(73\!\cdots\!58\)\( T^{12} - \)\(23\!\cdots\!38\)\( T^{13} + \)\(14\!\cdots\!98\)\( T^{14} - \)\(42\!\cdots\!84\)\( T^{15} + \)\(24\!\cdots\!81\)\( T^{16} - \)\(63\!\cdots\!68\)\( T^{17} + \)\(34\!\cdots\!59\)\( T^{18} - \)\(76\!\cdots\!76\)\( T^{19} + \)\(39\!\cdots\!44\)\( T^{20} - \)\(68\!\cdots\!10\)\( T^{21} + \)\(33\!\cdots\!14\)\( T^{22} - \)\(40\!\cdots\!52\)\( T^{23} + \)\(19\!\cdots\!64\)\( T^{24} - \)\(12\!\cdots\!60\)\( T^{25} + \)\(57\!\cdots\!41\)\( T^{26} \)
$43$ \( ( 1 + 79507 T )^{13} \)
$47$ \( 1 - 424556 T + 3688252683440 T^{2} - 2420585272729575804 T^{3} + \)\(67\!\cdots\!76\)\( T^{4} - \)\(56\!\cdots\!60\)\( T^{5} + \)\(83\!\cdots\!12\)\( T^{6} - \)\(77\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!18\)\( T^{8} - \)\(74\!\cdots\!56\)\( T^{9} + \)\(64\!\cdots\!19\)\( T^{10} - \)\(53\!\cdots\!64\)\( T^{11} + \)\(40\!\cdots\!52\)\( T^{12} - \)\(30\!\cdots\!32\)\( T^{13} + \)\(20\!\cdots\!76\)\( T^{14} - \)\(13\!\cdots\!16\)\( T^{15} + \)\(83\!\cdots\!93\)\( T^{16} - \)\(48\!\cdots\!16\)\( T^{17} + \)\(27\!\cdots\!74\)\( T^{18} - \)\(13\!\cdots\!20\)\( T^{19} + \)\(71\!\cdots\!04\)\( T^{20} - \)\(24\!\cdots\!60\)\( T^{21} + \)\(14\!\cdots\!48\)\( T^{22} - \)\(26\!\cdots\!96\)\( T^{23} + \)\(20\!\cdots\!80\)\( T^{24} - \)\(12\!\cdots\!36\)\( T^{25} + \)\(14\!\cdots\!03\)\( T^{26} \)
$53$ \( 1 + 3992458 T + 15589250135817 T^{2} + 39082970394967823704 T^{3} + \)\(92\!\cdots\!16\)\( T^{4} + \)\(17\!\cdots\!84\)\( T^{5} + \)\(31\!\cdots\!44\)\( T^{6} + \)\(49\!\cdots\!88\)\( T^{7} + \)\(73\!\cdots\!96\)\( T^{8} + \)\(10\!\cdots\!68\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} + \)\(16\!\cdots\!28\)\( T^{11} + \)\(19\!\cdots\!82\)\( T^{12} + \)\(21\!\cdots\!60\)\( T^{13} + \)\(22\!\cdots\!34\)\( T^{14} + \)\(22\!\cdots\!32\)\( T^{15} + \)\(21\!\cdots\!32\)\( T^{16} + \)\(19\!\cdots\!48\)\( T^{17} + \)\(16\!\cdots\!72\)\( T^{18} + \)\(12\!\cdots\!92\)\( T^{19} + \)\(96\!\cdots\!52\)\( T^{20} + \)\(63\!\cdots\!64\)\( T^{21} + \)\(39\!\cdots\!32\)\( T^{22} + \)\(19\!\cdots\!96\)\( T^{23} + \)\(91\!\cdots\!21\)\( T^{24} + \)\(27\!\cdots\!98\)\( T^{25} + \)\(81\!\cdots\!97\)\( T^{26} \)
$59$ \( 1 + 2248836 T + 18986189218171 T^{2} + 37576890307888851744 T^{3} + \)\(17\!\cdots\!18\)\( T^{4} + \)\(31\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!54\)\( T^{6} + \)\(17\!\cdots\!32\)\( T^{7} + \)\(51\!\cdots\!59\)\( T^{8} + \)\(76\!\cdots\!40\)\( T^{9} + \)\(18\!\cdots\!17\)\( T^{10} + \)\(25\!\cdots\!88\)\( T^{11} + \)\(56\!\cdots\!40\)\( T^{12} + \)\(70\!\cdots\!64\)\( T^{13} + \)\(14\!\cdots\!60\)\( T^{14} + \)\(15\!\cdots\!68\)\( T^{15} + \)\(29\!\cdots\!03\)\( T^{16} + \)\(29\!\cdots\!40\)\( T^{17} + \)\(49\!\cdots\!41\)\( T^{18} + \)\(42\!\cdots\!92\)\( T^{19} + \)\(65\!\cdots\!06\)\( T^{20} + \)\(46\!\cdots\!88\)\( T^{21} + \)\(65\!\cdots\!22\)\( T^{22} + \)\(34\!\cdots\!44\)\( T^{23} + \)\(43\!\cdots\!49\)\( T^{24} + \)\(12\!\cdots\!96\)\( T^{25} + \)\(14\!\cdots\!59\)\( T^{26} \)
$61$ \( 1 - 6210394 T + 38426666503003 T^{2} - \)\(15\!\cdots\!76\)\( T^{3} + \)\(60\!\cdots\!88\)\( T^{4} - \)\(18\!\cdots\!92\)\( T^{5} + \)\(56\!\cdots\!64\)\( T^{6} - \)\(14\!\cdots\!80\)\( T^{7} + \)\(36\!\cdots\!17\)\( T^{8} - \)\(83\!\cdots\!78\)\( T^{9} + \)\(18\!\cdots\!55\)\( T^{10} - \)\(36\!\cdots\!04\)\( T^{11} + \)\(70\!\cdots\!72\)\( T^{12} - \)\(12\!\cdots\!32\)\( T^{13} + \)\(22\!\cdots\!12\)\( T^{14} - \)\(35\!\cdots\!64\)\( T^{15} + \)\(56\!\cdots\!55\)\( T^{16} - \)\(81\!\cdots\!18\)\( T^{17} + \)\(11\!\cdots\!17\)\( T^{18} - \)\(14\!\cdots\!80\)\( T^{19} + \)\(17\!\cdots\!24\)\( T^{20} - \)\(18\!\cdots\!12\)\( T^{21} + \)\(18\!\cdots\!28\)\( T^{22} - \)\(14\!\cdots\!76\)\( T^{23} + \)\(11\!\cdots\!63\)\( T^{24} - \)\(57\!\cdots\!54\)\( T^{25} + \)\(29\!\cdots\!61\)\( T^{26} \)
$67$ \( 1 + 1993648 T + 36332499483043 T^{2} + 80369020063411411764 T^{3} + \)\(68\!\cdots\!56\)\( T^{4} + \)\(16\!\cdots\!32\)\( T^{5} + \)\(89\!\cdots\!48\)\( T^{6} + \)\(22\!\cdots\!60\)\( T^{7} + \)\(90\!\cdots\!36\)\( T^{8} + \)\(22\!\cdots\!72\)\( T^{9} + \)\(75\!\cdots\!16\)\( T^{10} + \)\(18\!\cdots\!92\)\( T^{11} + \)\(53\!\cdots\!18\)\( T^{12} + \)\(12\!\cdots\!16\)\( T^{13} + \)\(32\!\cdots\!14\)\( T^{14} + \)\(69\!\cdots\!68\)\( T^{15} + \)\(16\!\cdots\!72\)\( T^{16} + \)\(30\!\cdots\!52\)\( T^{17} + \)\(73\!\cdots\!48\)\( T^{18} + \)\(11\!\cdots\!40\)\( T^{19} + \)\(26\!\cdots\!56\)\( T^{20} + \)\(29\!\cdots\!92\)\( T^{21} + \)\(75\!\cdots\!28\)\( T^{22} + \)\(53\!\cdots\!36\)\( T^{23} + \)\(14\!\cdots\!61\)\( T^{24} + \)\(48\!\cdots\!08\)\( T^{25} + \)\(14\!\cdots\!83\)\( T^{26} \)
$71$ \( 1 + 4978064 T + 87202612193659 T^{2} + \)\(39\!\cdots\!40\)\( T^{3} + \)\(37\!\cdots\!18\)\( T^{4} + \)\(15\!\cdots\!52\)\( T^{5} + \)\(10\!\cdots\!70\)\( T^{6} + \)\(37\!\cdots\!88\)\( T^{7} + \)\(20\!\cdots\!19\)\( T^{8} + \)\(66\!\cdots\!92\)\( T^{9} + \)\(29\!\cdots\!13\)\( T^{10} + \)\(88\!\cdots\!48\)\( T^{11} + \)\(34\!\cdots\!08\)\( T^{12} + \)\(91\!\cdots\!08\)\( T^{13} + \)\(31\!\cdots\!28\)\( T^{14} + \)\(73\!\cdots\!88\)\( T^{15} + \)\(22\!\cdots\!23\)\( T^{16} + \)\(45\!\cdots\!12\)\( T^{17} + \)\(12\!\cdots\!69\)\( T^{18} + \)\(21\!\cdots\!08\)\( T^{19} + \)\(52\!\cdots\!70\)\( T^{20} + \)\(70\!\cdots\!92\)\( T^{21} + \)\(15\!\cdots\!98\)\( T^{22} + \)\(15\!\cdots\!40\)\( T^{23} + \)\(30\!\cdots\!69\)\( T^{24} + \)\(15\!\cdots\!84\)\( T^{25} + \)\(29\!\cdots\!71\)\( T^{26} \)
$73$ \( 1 - 8224814 T + 92481350221839 T^{2} - \)\(53\!\cdots\!16\)\( T^{3} + \)\(38\!\cdots\!04\)\( T^{4} - \)\(18\!\cdots\!96\)\( T^{5} + \)\(10\!\cdots\!88\)\( T^{6} - \)\(45\!\cdots\!64\)\( T^{7} + \)\(22\!\cdots\!37\)\( T^{8} - \)\(85\!\cdots\!50\)\( T^{9} + \)\(36\!\cdots\!23\)\( T^{10} - \)\(12\!\cdots\!68\)\( T^{11} + \)\(49\!\cdots\!68\)\( T^{12} - \)\(15\!\cdots\!60\)\( T^{13} + \)\(54\!\cdots\!96\)\( T^{14} - \)\(15\!\cdots\!12\)\( T^{15} + \)\(49\!\cdots\!79\)\( T^{16} - \)\(12\!\cdots\!50\)\( T^{17} + \)\(36\!\cdots\!09\)\( T^{18} - \)\(83\!\cdots\!56\)\( T^{19} + \)\(21\!\cdots\!44\)\( T^{20} - \)\(41\!\cdots\!56\)\( T^{21} + \)\(94\!\cdots\!68\)\( T^{22} - \)\(14\!\cdots\!84\)\( T^{23} + \)\(27\!\cdots\!67\)\( T^{24} - \)\(27\!\cdots\!74\)\( T^{25} + \)\(36\!\cdots\!77\)\( T^{26} \)
$79$ \( 1 - 6945708 T + 117121136837052 T^{2} - \)\(56\!\cdots\!28\)\( T^{3} + \)\(65\!\cdots\!68\)\( T^{4} - \)\(24\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!68\)\( T^{6} - \)\(74\!\cdots\!48\)\( T^{7} + \)\(72\!\cdots\!74\)\( T^{8} - \)\(19\!\cdots\!48\)\( T^{9} + \)\(18\!\cdots\!35\)\( T^{10} - \)\(46\!\cdots\!04\)\( T^{11} + \)\(42\!\cdots\!92\)\( T^{12} - \)\(97\!\cdots\!28\)\( T^{13} + \)\(80\!\cdots\!28\)\( T^{14} - \)\(17\!\cdots\!24\)\( T^{15} + \)\(13\!\cdots\!65\)\( T^{16} - \)\(26\!\cdots\!28\)\( T^{17} + \)\(18\!\cdots\!26\)\( T^{18} - \)\(37\!\cdots\!68\)\( T^{19} + \)\(23\!\cdots\!92\)\( T^{20} - \)\(45\!\cdots\!60\)\( T^{21} + \)\(23\!\cdots\!52\)\( T^{22} - \)\(38\!\cdots\!28\)\( T^{23} + \)\(15\!\cdots\!68\)\( T^{24} - \)\(17\!\cdots\!48\)\( T^{25} + \)\(48\!\cdots\!79\)\( T^{26} \)
$83$ \( 1 + 22937328 T + 403848212345231 T^{2} + \)\(46\!\cdots\!40\)\( T^{3} + \)\(46\!\cdots\!36\)\( T^{4} + \)\(35\!\cdots\!72\)\( T^{5} + \)\(24\!\cdots\!76\)\( T^{6} + \)\(14\!\cdots\!68\)\( T^{7} + \)\(83\!\cdots\!60\)\( T^{8} + \)\(44\!\cdots\!24\)\( T^{9} + \)\(27\!\cdots\!48\)\( T^{10} + \)\(15\!\cdots\!32\)\( T^{11} + \)\(97\!\cdots\!38\)\( T^{12} + \)\(51\!\cdots\!32\)\( T^{13} + \)\(26\!\cdots\!26\)\( T^{14} + \)\(11\!\cdots\!28\)\( T^{15} + \)\(54\!\cdots\!84\)\( T^{16} + \)\(24\!\cdots\!84\)\( T^{17} + \)\(12\!\cdots\!20\)\( T^{18} + \)\(57\!\cdots\!52\)\( T^{19} + \)\(26\!\cdots\!28\)\( T^{20} + \)\(10\!\cdots\!32\)\( T^{21} + \)\(37\!\cdots\!32\)\( T^{22} + \)\(10\!\cdots\!60\)\( T^{23} + \)\(23\!\cdots\!13\)\( T^{24} + \)\(36\!\cdots\!88\)\( T^{25} + \)\(43\!\cdots\!67\)\( T^{26} \)
$89$ \( 1 + 9291302 T + 323406546094099 T^{2} + \)\(31\!\cdots\!04\)\( T^{3} + \)\(56\!\cdots\!92\)\( T^{4} + \)\(51\!\cdots\!56\)\( T^{5} + \)\(68\!\cdots\!68\)\( T^{6} + \)\(56\!\cdots\!24\)\( T^{7} + \)\(61\!\cdots\!61\)\( T^{8} + \)\(45\!\cdots\!46\)\( T^{9} + \)\(42\!\cdots\!63\)\( T^{10} + \)\(28\!\cdots\!48\)\( T^{11} + \)\(23\!\cdots\!60\)\( T^{12} + \)\(14\!\cdots\!00\)\( T^{13} + \)\(10\!\cdots\!40\)\( T^{14} + \)\(55\!\cdots\!68\)\( T^{15} + \)\(36\!\cdots\!07\)\( T^{16} + \)\(17\!\cdots\!26\)\( T^{17} + \)\(10\!\cdots\!89\)\( T^{18} + \)\(42\!\cdots\!04\)\( T^{19} + \)\(22\!\cdots\!12\)\( T^{20} + \)\(75\!\cdots\!16\)\( T^{21} + \)\(36\!\cdots\!48\)\( T^{22} + \)\(90\!\cdots\!04\)\( T^{23} + \)\(40\!\cdots\!71\)\( T^{24} + \)\(52\!\cdots\!82\)\( T^{25} + \)\(24\!\cdots\!89\)\( T^{26} \)
$97$ \( 1 - 10001852 T + 551581550822996 T^{2} - \)\(38\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!86\)\( T^{4} - \)\(69\!\cdots\!90\)\( T^{5} + \)\(23\!\cdots\!80\)\( T^{6} - \)\(80\!\cdots\!08\)\( T^{7} + \)\(31\!\cdots\!27\)\( T^{8} - \)\(74\!\cdots\!80\)\( T^{9} + \)\(34\!\cdots\!81\)\( T^{10} - \)\(66\!\cdots\!32\)\( T^{11} + \)\(32\!\cdots\!38\)\( T^{12} - \)\(56\!\cdots\!82\)\( T^{13} + \)\(26\!\cdots\!94\)\( T^{14} - \)\(43\!\cdots\!08\)\( T^{15} + \)\(18\!\cdots\!57\)\( T^{16} - \)\(31\!\cdots\!80\)\( T^{17} + \)\(10\!\cdots\!11\)\( T^{18} - \)\(22\!\cdots\!72\)\( T^{19} + \)\(53\!\cdots\!60\)\( T^{20} - \)\(12\!\cdots\!90\)\( T^{21} + \)\(20\!\cdots\!78\)\( T^{22} - \)\(46\!\cdots\!92\)\( T^{23} + \)\(52\!\cdots\!52\)\( T^{24} - \)\(77\!\cdots\!12\)\( T^{25} + \)\(62\!\cdots\!53\)\( T^{26} \)
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