# 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 $1$

# Related objects

## 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: $$\mathbb{Q}[x]/(x^{13} - \cdots)$$ Defining polynomial: $$x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660$$ 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$43$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$20\!\cdots\!08$$$$T_{2} + 689040691200$$">$$T_{2}^{13} + \cdots$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(387))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$689040691200 + 2060918673408 T - 470565043072 T^{2} - 252864488128 T^{3} + 46963036352 T^{4} + 7540604768 T^{5} - 974729640 T^{6} - 91843236 T^{7} + 6529604 T^{8} + 492122 T^{9} - 17350 T^{10} - 1165 T^{11} + 16 T^{12} + T^{13}$$
$3$ $$T^{13}$$
$5$ $$-$$$$85\!\cdots\!00$$$$-$$$$16\!\cdots\!00$$$$T +$$$$44\!\cdots\!00$$$$T^{2} +$$$$20\!\cdots\!00$$$$T^{3} +$$$$38\!\cdots\!00$$$$T^{4} -$$$$82\!\cdots\!00$$$$T^{5} - 2727917772915635500 T^{6} + 12169682680435150 T^{7} + 53890755533470 T^{8} - 51053689333 T^{9} - 399935994 T^{10} - 144885 T^{11} + 998 T^{12} + T^{13}$$
$7$ $$-$$$$15\!\cdots\!12$$$$+$$$$10\!\cdots\!20$$$$T +$$$$39\!\cdots\!00$$$$T^{2} -$$$$29\!\cdots\!08$$$$T^{3} -$$$$15\!\cdots\!44$$$$T^{4} +$$$$36\!\cdots\!48$$$$T^{5} +$$$$17\!\cdots\!04$$$$T^{6} - 155885586415368712 T^{7} - 6219022182331104 T^{8} + 3119485047796 T^{9} + 5589150136 T^{10} - 3647550 T^{11} - 1360 T^{12} + T^{13}$$
$11$ $$-$$$$10\!\cdots\!16$$$$-$$$$47\!\cdots\!60$$$$T -$$$$46\!\cdots\!08$$$$T^{2} +$$$$32\!\cdots\!84$$$$T^{3} +$$$$49\!\cdots\!92$$$$T^{4} +$$$$41\!\cdots\!53$$$$T^{5} -$$$$14\!\cdots\!72$$$$T^{6} -$$$$31\!\cdots\!60$$$$T^{7} + 12773208882222023844 T^{8} + 3966080080029806 T^{9} - 254848999740 T^{10} - 120745204 T^{11} + 1620 T^{12} + T^{13}$$
$13$ $$13\!\cdots\!60$$$$-$$$$88\!\cdots\!36$$$$T +$$$$32\!\cdots\!08$$$$T^{2} +$$$$31\!\cdots\!96$$$$T^{3} -$$$$96\!\cdots\!30$$$$T^{4} -$$$$10\!\cdots\!99$$$$T^{5} +$$$$47\!\cdots\!08$$$$T^{6} -$$$$30\!\cdots\!28$$$$T^{7} -$$$$83\!\cdots\!36$$$$T^{8} + 37743539561364398 T^{9} + 5745637147508 T^{10} - 364726604 T^{11} - 13550 T^{12} + T^{13}$$
$17$ $$61\!\cdots\!38$$$$+$$$$28\!\cdots\!05$$$$T +$$$$38\!\cdots\!88$$$$T^{2} +$$$$10\!\cdots\!15$$$$T^{3} -$$$$34\!\cdots\!76$$$$T^{4} -$$$$18\!\cdots\!13$$$$T^{5} +$$$$62\!\cdots\!84$$$$T^{6} +$$$$68\!\cdots\!90$$$$T^{7} +$$$$52\!\cdots\!34$$$$T^{8} - 7951108877889687249 T^{9} - 195671920944168 T^{10} + 1934838631 T^{11} + 110880 T^{12} + T^{13}$$
$19$ $$68\!\cdots\!00$$$$-$$$$29\!\cdots\!00$$$$T -$$$$17\!\cdots\!60$$$$T^{2} +$$$$14\!\cdots\!40$$$$T^{3} +$$$$69\!\cdots\!32$$$$T^{4} -$$$$94\!\cdots\!40$$$$T^{5} +$$$$87\!\cdots\!80$$$$T^{6} +$$$$16\!\cdots\!36$$$$T^{7} -$$$$44\!\cdots\!48$$$$T^{8} - 8418946658473423163 T^{9} + 397826424172450 T^{10} - 725403573 T^{11} - 105058 T^{12} + T^{13}$$
$23$ $$25\!\cdots\!60$$$$+$$$$12\!\cdots\!21$$$$T -$$$$24\!\cdots\!92$$$$T^{2} -$$$$73\!\cdots\!75$$$$T^{3} +$$$$89\!\cdots\!96$$$$T^{4} +$$$$98\!\cdots\!47$$$$T^{5} -$$$$12\!\cdots\!88$$$$T^{6} -$$$$49\!\cdots\!18$$$$T^{7} +$$$$72\!\cdots\!68$$$$T^{8} + 19083155410974949779 T^{9} - 1791690390393400 T^{10} - 7283028983 T^{11} + 160184 T^{12} + T^{13}$$
$29$ $$71\!\cdots\!00$$$$+$$$$74\!\cdots\!60$$$$T +$$$$53\!\cdots\!20$$$$T^{2} -$$$$11\!\cdots\!52$$$$T^{3} +$$$$41\!\cdots\!08$$$$T^{4} -$$$$60\!\cdots\!52$$$$T^{5} -$$$$15\!\cdots\!04$$$$T^{6} +$$$$61\!\cdots\!70$$$$T^{7} +$$$$24\!\cdots\!98$$$$T^{8} -$$$$27\!\cdots\!69$$$$T^{9} - 18094253665681302 T^{10} - 43643538617 T^{11} + 285546 T^{12} + T^{13}$$
$31$ $$-$$$$19\!\cdots\!60$$$$-$$$$32\!\cdots\!51$$$$T +$$$$98\!\cdots\!80$$$$T^{2} -$$$$67\!\cdots\!07$$$$T^{3} +$$$$13\!\cdots\!00$$$$T^{4} +$$$$18\!\cdots\!43$$$$T^{5} -$$$$56\!\cdots\!36$$$$T^{6} -$$$$45\!\cdots\!98$$$$T^{7} +$$$$50\!\cdots\!96$$$$T^{8} +$$$$37\!\cdots\!31$$$$T^{9} - 14550013917777136 T^{10} - 116049569979 T^{11} + 99616 T^{12} + T^{13}$$
$37$ $$-$$$$89\!\cdots\!00$$$$-$$$$66\!\cdots\!60$$$$T -$$$$16\!\cdots\!72$$$$T^{2} -$$$$13\!\cdots\!36$$$$T^{3} +$$$$11\!\cdots\!80$$$$T^{4} +$$$$19\!\cdots\!32$$$$T^{5} -$$$$13\!\cdots\!24$$$$T^{6} -$$$$91\!\cdots\!12$$$$T^{7} -$$$$12\!\cdots\!18$$$$T^{8} +$$$$20\!\cdots\!81$$$$T^{9} + 27917997870380038 T^{10} - 231521506147 T^{11} - 176038 T^{12} + T^{13}$$
$41$ $$-$$$$11\!\cdots\!50$$$$+$$$$14\!\cdots\!37$$$$T +$$$$80\!\cdots\!64$$$$T^{2} -$$$$74\!\cdots\!29$$$$T^{3} -$$$$21\!\cdots\!52$$$$T^{4} +$$$$13\!\cdots\!03$$$$T^{5} +$$$$26\!\cdots\!84$$$$T^{6} -$$$$12\!\cdots\!82$$$$T^{7} -$$$$15\!\cdots\!30$$$$T^{8} +$$$$57\!\cdots\!95$$$$T^{9} + 438232010289172568 T^{10} - 1256031205409 T^{11} - 410260 T^{12} + T^{13}$$
$43$ $$( 79507 + T )^{13}$$
$47$ $$51\!\cdots\!40$$$$-$$$$93\!\cdots\!56$$$$T -$$$$15\!\cdots\!24$$$$T^{2} +$$$$10\!\cdots\!68$$$$T^{3} +$$$$14\!\cdots\!04$$$$T^{4} -$$$$13\!\cdots\!44$$$$T^{5} -$$$$39\!\cdots\!16$$$$T^{6} -$$$$82\!\cdots\!72$$$$T^{7} +$$$$71\!\cdots\!04$$$$T^{8} +$$$$28\!\cdots\!41$$$$T^{9} + 160493353645897332 T^{10} - 2897847882579 T^{11} - 424556 T^{12} + T^{13}$$
$53$ $$-$$$$23\!\cdots\!20$$$$+$$$$14\!\cdots\!36$$$$T +$$$$35\!\cdots\!64$$$$T^{2} +$$$$76\!\cdots\!96$$$$T^{3} -$$$$47\!\cdots\!58$$$$T^{4} -$$$$67\!\cdots\!67$$$$T^{5} +$$$$12\!\cdots\!96$$$$T^{6} +$$$$33\!\cdots\!60$$$$T^{7} +$$$$13\!\cdots\!12$$$$T^{8} -$$$$19\!\cdots\!18$$$$T^{9} - 17196848260208368448 T^{10} + 318005317936 T^{11} + 3992458 T^{12} + T^{13}$$
$59$ $$25\!\cdots\!00$$$$+$$$$11\!\cdots\!00$$$$T -$$$$49\!\cdots\!80$$$$T^{2} -$$$$61\!\cdots\!36$$$$T^{3} +$$$$19\!\cdots\!32$$$$T^{4} +$$$$11\!\cdots\!92$$$$T^{5} -$$$$25\!\cdots\!52$$$$T^{6} -$$$$12\!\cdots\!64$$$$T^{7} +$$$$13\!\cdots\!92$$$$T^{8} +$$$$61\!\cdots\!44$$$$T^{9} - 29581938298284196464 T^{10} - 13366280084476 T^{11} + 2248836 T^{12} + T^{13}$$
$61$ $$-$$$$15\!\cdots\!00$$$$-$$$$40\!\cdots\!44$$$$T +$$$$22\!\cdots\!04$$$$T^{2} +$$$$44\!\cdots\!40$$$$T^{3} -$$$$43\!\cdots\!08$$$$T^{4} -$$$$74\!\cdots\!24$$$$T^{5} +$$$$49\!\cdots\!24$$$$T^{6} +$$$$43\!\cdots\!28$$$$T^{7} -$$$$29\!\cdots\!48$$$$T^{8} -$$$$82\!\cdots\!40$$$$T^{9} + 77922113851176841212 T^{10} - 2428990365270 T^{11} - 6210394 T^{12} + T^{13}$$
$67$ $$43\!\cdots\!84$$$$+$$$$59\!\cdots\!04$$$$T -$$$$11\!\cdots\!80$$$$T^{2} -$$$$16\!\cdots\!44$$$$T^{3} +$$$$90\!\cdots\!92$$$$T^{4} +$$$$13\!\cdots\!81$$$$T^{5} -$$$$34\!\cdots\!60$$$$T^{6} -$$$$43\!\cdots\!28$$$$T^{7} +$$$$72\!\cdots\!80$$$$T^{8} +$$$$64\!\cdots\!62$$$$T^{9} - 64626086782936447884 T^{10} - 42456751386156 T^{11} + 1993648 T^{12} + T^{13}$$
$71$ $$23\!\cdots\!92$$$$-$$$$50\!\cdots\!20$$$$T -$$$$40\!\cdots\!64$$$$T^{2} +$$$$20\!\cdots\!12$$$$T^{3} +$$$$12\!\cdots\!04$$$$T^{4} +$$$$23\!\cdots\!44$$$$T^{5} -$$$$71\!\cdots\!96$$$$T^{6} -$$$$17\!\cdots\!72$$$$T^{7} +$$$$16\!\cdots\!88$$$$T^{8} +$$$$35\!\cdots\!24$$$$T^{9} -$$$$15\!\cdots\!48$$$$T^{10} - 31033949865424 T^{11} + 4978064 T^{12} + T^{13}$$
$73$ $$-$$$$46\!\cdots\!64$$$$-$$$$12\!\cdots\!88$$$$T +$$$$28\!\cdots\!04$$$$T^{2} +$$$$14\!\cdots\!64$$$$T^{3} -$$$$60\!\cdots\!52$$$$T^{4} -$$$$24\!\cdots\!20$$$$T^{5} +$$$$14\!\cdots\!56$$$$T^{6} +$$$$10\!\cdots\!52$$$$T^{7} -$$$$13\!\cdots\!80$$$$T^{8} +$$$$55\!\cdots\!76$$$$T^{9} +$$$$55\!\cdots\!80$$$$T^{10} - 51134830526422 T^{11} - 8224814 T^{12} + T^{13}$$
$79$ $$-$$$$16\!\cdots\!80$$$$-$$$$19\!\cdots\!00$$$$T +$$$$74\!\cdots\!84$$$$T^{2} +$$$$31\!\cdots\!08$$$$T^{3} -$$$$13\!\cdots\!96$$$$T^{4} +$$$$30\!\cdots\!92$$$$T^{5} +$$$$12\!\cdots\!76$$$$T^{6} -$$$$88\!\cdots\!36$$$$T^{7} -$$$$53\!\cdots\!32$$$$T^{8} +$$$$57\!\cdots\!85$$$$T^{9} +$$$$10\!\cdots\!36$$$$T^{10} - 132529679983015 T^{11} - 6945708 T^{12} + T^{13}$$
$83$ $$23\!\cdots\!36$$$$-$$$$63\!\cdots\!52$$$$T -$$$$56\!\cdots\!84$$$$T^{2} -$$$$56\!\cdots\!68$$$$T^{3} -$$$$75\!\cdots\!92$$$$T^{4} +$$$$11\!\cdots\!25$$$$T^{5} +$$$$62\!\cdots\!28$$$$T^{6} +$$$$11\!\cdots\!36$$$$T^{7} -$$$$35\!\cdots\!80$$$$T^{8} -$$$$26\!\cdots\!86$$$$T^{9} -$$$$27\!\cdots\!32$$$$T^{10} + 51079549480080 T^{11} + 22937328 T^{12} + T^{13}$$
$89$ $$-$$$$15\!\cdots\!20$$$$-$$$$27\!\cdots\!00$$$$T +$$$$59\!\cdots\!96$$$$T^{2} -$$$$19\!\cdots\!04$$$$T^{3} -$$$$62\!\cdots\!00$$$$T^{4} +$$$$37\!\cdots\!96$$$$T^{5} -$$$$11\!\cdots\!64$$$$T^{6} -$$$$14\!\cdots\!32$$$$T^{7} +$$$$10\!\cdots\!24$$$$T^{8} +$$$$26\!\cdots\!76$$$$T^{9} -$$$$17\!\cdots\!92$$$$T^{10} - 251600807547778 T^{11} + 9291302 T^{12} + T^{13}$$
$97$ $$-$$$$57\!\cdots\!66$$$$-$$$$24\!\cdots\!03$$$$T +$$$$15\!\cdots\!36$$$$T^{2} +$$$$36\!\cdots\!63$$$$T^{3} -$$$$18\!\cdots\!16$$$$T^{4} -$$$$49\!\cdots\!57$$$$T^{5} +$$$$73\!\cdots\!60$$$$T^{6} -$$$$30\!\cdots\!58$$$$T^{7} -$$$$10\!\cdots\!02$$$$T^{8} +$$$$75\!\cdots\!43$$$$T^{9} +$$$$58\!\cdots\!04$$$$T^{10} - 498796147392473 T^{11} - 10001852 T^{12} + T^{13}$$