Properties

 Label 1075.6.a.b Level 1075 Weight 6 Character orbit 1075.a Self dual yes Analytic conductor 172.413 Analytic rank 1 Dimension 10 CM no Inner twists 1

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1075 = 5^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1075.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$172.412606299$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -3 - \beta_{6} ) q^{3} + ( 21 - \beta_{1} + \beta_{5} + \beta_{6} ) q^{4} + ( 9 - 11 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{6} + ( -8 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{7} + ( -37 + 23 \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} ) q^{8} + ( 133 + 6 \beta_{1} + 3 \beta_{2} - 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( -3 - \beta_{6} ) q^{3} + ( 21 - \beta_{1} + \beta_{5} + \beta_{6} ) q^{4} + ( 9 - 11 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{6} + ( -8 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{7} + ( -37 + 23 \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} ) q^{8} + ( 133 + 6 \beta_{1} + 3 \beta_{2} - 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{9} + ( 66 + 16 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} - 4 \beta_{5} - 11 \beta_{6} - 3 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} ) q^{11} + ( -464 - 16 \beta_{1} - 4 \beta_{2} - \beta_{3} + 7 \beta_{4} - 9 \beta_{5} - 8 \beta_{6} + 9 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} ) q^{12} + ( -192 - 12 \beta_{1} - \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 11 \beta_{6} + 3 \beta_{7} - 7 \beta_{8} - 6 \beta_{9} ) q^{13} + ( 193 + 23 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} - \beta_{4} + 5 \beta_{5} - 3 \beta_{6} + 14 \beta_{7} + \beta_{9} ) q^{14} + ( 567 - 5 \beta_{1} + 20 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} + 23 \beta_{5} + 24 \beta_{6} - 8 \beta_{7} + 13 \beta_{8} - 8 \beta_{9} ) q^{16} + ( -402 - 55 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} - 7 \beta_{4} - 8 \beta_{5} - 25 \beta_{6} + 4 \beta_{7} - 4 \beta_{8} + 8 \beta_{9} ) q^{17} + ( 262 + 92 \beta_{1} - 4 \beta_{2} + 20 \beta_{3} - 26 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} - 3 \beta_{7} - 12 \beta_{8} + 8 \beta_{9} ) q^{18} + ( -241 - 15 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} + 20 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} - 4 \beta_{7} + 13 \beta_{8} - 8 \beta_{9} ) q^{19} + ( -51 + 71 \beta_{1} + 5 \beta_{2} - 28 \beta_{3} - 22 \beta_{4} - 27 \beta_{5} + 32 \beta_{6} + 26 \beta_{7} + 5 \beta_{8} + 7 \beta_{9} ) q^{21} + ( 654 - 34 \beta_{1} + 32 \beta_{2} - 23 \beta_{3} + 33 \beta_{4} + 59 \beta_{5} + 70 \beta_{6} + 10 \beta_{7} + 51 \beta_{8} - 34 \beta_{9} ) q^{22} + ( -214 + 44 \beta_{1} - 28 \beta_{2} + 17 \beta_{3} - 17 \beta_{4} - 25 \beta_{5} - 23 \beta_{6} - 13 \beta_{7} - 29 \beta_{8} + 13 \beta_{9} ) q^{23} + ( -948 - 436 \beta_{1} + 20 \beta_{2} - \beta_{3} + 77 \beta_{4} + 15 \beta_{5} - 62 \beta_{6} - 37 \beta_{7} + 77 \beta_{8} - 8 \beta_{9} ) q^{24} + ( -84 - 168 \beta_{1} + 24 \beta_{2} + 31 \beta_{3} - 9 \beta_{4} + 37 \beta_{5} - 22 \beta_{6} - 26 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} ) q^{26} + ( 216 - 127 \beta_{1} - 18 \beta_{2} + 5 \beta_{3} + 40 \beta_{4} - 39 \beta_{5} - 134 \beta_{6} - 55 \beta_{7} + 17 \beta_{8} + 43 \beta_{9} ) q^{27} + ( 1472 + 136 \beta_{1} - 26 \beta_{2} - 28 \beta_{3} + 34 \beta_{4} + 4 \beta_{5} + 72 \beta_{6} - 50 \beta_{7} + 16 \beta_{8} - 36 \beta_{9} ) q^{28} + ( 749 - 2 \beta_{1} - 36 \beta_{2} + 18 \beta_{3} - 20 \beta_{4} + 102 \beta_{5} + 47 \beta_{6} - 10 \beta_{7} + 18 \beta_{8} - 36 \beta_{9} ) q^{29} + ( -500 - 89 \beta_{1} + 50 \beta_{2} - 6 \beta_{3} + 27 \beta_{4} - 16 \beta_{5} - 87 \beta_{6} - 14 \beta_{7} - 56 \beta_{8} - 2 \beta_{9} ) q^{31} + ( 577 + 481 \beta_{1} - 82 \beta_{2} + 67 \beta_{3} - 73 \beta_{4} - 99 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} - 91 \beta_{8} + 42 \beta_{9} ) q^{32} + ( 1728 - 306 \beta_{1} + 128 \beta_{2} + 39 \beta_{3} + 34 \beta_{4} + 39 \beta_{5} + 5 \beta_{6} - 69 \beta_{7} + 24 \beta_{8} - 45 \beta_{9} ) q^{33} + ( -2550 - 696 \beta_{1} + 46 \beta_{2} - 68 \beta_{3} - 56 \beta_{4} - 44 \beta_{5} + 80 \beta_{6} + \beta_{7} + 92 \beta_{8} - 58 \beta_{9} ) q^{34} + ( 448 + 686 \beta_{1} + 56 \beta_{2} + 56 \beta_{3} - 158 \beta_{4} + 87 \beta_{5} + 475 \beta_{6} - 57 \beta_{7} - 42 \beta_{8} - 40 \beta_{9} ) q^{36} + ( 87 - 286 \beta_{1} + 42 \beta_{2} - 51 \beta_{3} - 30 \beta_{4} + 127 \beta_{5} + 280 \beta_{6} + 141 \beta_{7} + 168 \beta_{8} - 63 \beta_{9} ) q^{37} + ( -1221 - 629 \beta_{1} + 70 \beta_{2} + 6 \beta_{3} + 136 \beta_{4} - 14 \beta_{5} - 218 \beta_{6} + 131 \beta_{7} + 88 \beta_{8} - 18 \beta_{9} ) q^{38} + ( -2498 + 122 \beta_{1} - 24 \beta_{2} + 89 \beta_{3} + 150 \beta_{4} + 57 \beta_{5} + 205 \beta_{6} - 3 \beta_{7} + 168 \beta_{8} - 67 \beta_{9} ) q^{39} + ( 1037 + 375 \beta_{1} + 93 \beta_{2} + 85 \beta_{3} + 37 \beta_{4} - 4 \beta_{5} - 251 \beta_{6} + 67 \beta_{7} + 110 \beta_{8} + 76 \beta_{9} ) q^{41} + ( 3902 - 626 \beta_{1} - 130 \beta_{2} + 22 \beta_{3} + 10 \beta_{4} + 162 \beta_{5} - 238 \beta_{6} - 30 \beta_{7} + 96 \beta_{8} + 58 \beta_{9} ) q^{42} -1849 q^{43} + ( -4232 + 1362 \beta_{1} - 298 \beta_{2} - 21 \beta_{3} + 87 \beta_{4} - 180 \beta_{5} - 335 \beta_{6} + 100 \beta_{7} - 239 \beta_{8} + 190 \beta_{9} ) q^{44} + ( 2267 - 625 \beta_{1} + 326 \beta_{2} + 11 \beta_{3} - 175 \beta_{4} + 165 \beta_{5} + 428 \beta_{6} + 29 \beta_{7} + 175 \beta_{8} - 140 \beta_{9} ) q^{46} + ( -4628 - 31 \beta_{1} + 251 \beta_{2} + 149 \beta_{3} + 82 \beta_{4} + 48 \beta_{5} + 106 \beta_{6} - 45 \beta_{7} + 15 \beta_{8} + 68 \beta_{9} ) q^{47} + ( -8730 - 938 \beta_{1} - 222 \beta_{2} - 77 \beta_{3} + 173 \beta_{4} - 519 \beta_{5} - 794 \beta_{6} + 127 \beta_{7} - 233 \beta_{8} + 74 \beta_{9} ) q^{48} + ( 2966 - 755 \beta_{1} - 177 \beta_{2} - 116 \beta_{3} - 74 \beta_{4} + 39 \beta_{5} - 82 \beta_{6} + 242 \beta_{7} - 45 \beta_{8} - 135 \beta_{9} ) q^{49} + ( 9449 + 229 \beta_{1} + 122 \beta_{2} - 77 \beta_{3} - 188 \beta_{4} + 171 \beta_{5} + 531 \beta_{6} + 87 \beta_{7} + 57 \beta_{8} - 11 \beta_{9} ) q^{51} + ( -1826 + 1216 \beta_{1} + 26 \beta_{2} + 173 \beta_{3} - 95 \beta_{4} - 398 \beta_{5} - 139 \beta_{6} - 180 \beta_{7} - 105 \beta_{8} + 242 \beta_{9} ) q^{52} + ( -12425 + 263 \beta_{1} + 146 \beta_{2} + 273 \beta_{3} + 101 \beta_{4} + 7 \beta_{5} + 341 \beta_{6} - 63 \beta_{7} + 186 \beta_{8} + 95 \beta_{9} ) q^{53} + ( -8989 - 1087 \beta_{1} + 26 \beta_{2} - 193 \beta_{3} + 177 \beta_{4} - 123 \beta_{5} - 194 \beta_{6} + 490 \beta_{7} + 253 \beta_{8} - 72 \beta_{9} ) q^{54} + ( -916 + 908 \beta_{1} - 42 \beta_{2} + 414 \beta_{3} + 254 \beta_{4} + 20 \beta_{5} - 548 \beta_{6} - 176 \beta_{8} + 110 \beta_{9} ) q^{56} + ( -3558 + 1859 \beta_{1} + 142 \beta_{2} + 35 \beta_{3} - 252 \beta_{4} + 195 \beta_{5} + 952 \beta_{6} + 35 \beta_{7} - 253 \beta_{8} - 215 \beta_{9} ) q^{57} + ( 189 + 3057 \beta_{1} + 273 \beta_{2} + 108 \beta_{3} - 426 \beta_{4} - 474 \beta_{5} + 219 \beta_{6} + 27 \beta_{7} - 519 \beta_{8} + 81 \beta_{9} ) q^{58} + ( 10469 - 825 \beta_{1} + 407 \beta_{2} - 28 \beta_{3} - 310 \beta_{4} - 125 \beta_{5} + 1032 \beta_{6} + 186 \beta_{7} + 185 \beta_{8} - 315 \beta_{9} ) q^{59} + ( 2554 - 2926 \beta_{1} - 66 \beta_{2} + 237 \beta_{3} + 264 \beta_{4} + 139 \beta_{5} - 5 \beta_{6} + 45 \beta_{7} + 126 \beta_{8} - 213 \beta_{9} ) q^{61} + ( -2534 - 996 \beta_{1} - 280 \beta_{2} - 102 \beta_{3} + 390 \beta_{4} + 426 \beta_{5} + 22 \beta_{6} - 119 \beta_{7} + 136 \beta_{8} - 104 \beta_{9} ) q^{62} + ( -66 + 54 \beta_{1} - 46 \beta_{2} + 117 \beta_{3} - 248 \beta_{4} + 399 \beta_{5} - 603 \beta_{6} + 117 \beta_{7} + 258 \beta_{8} - 9 \beta_{9} ) q^{63} + ( 5225 - 703 \beta_{1} + 414 \beta_{2} - 245 \beta_{3} - 541 \beta_{4} + 101 \beta_{5} + 1154 \beta_{6} - 20 \beta_{7} + 265 \beta_{8} - 226 \beta_{9} ) q^{64} + ( -16257 + 2593 \beta_{1} - 545 \beta_{2} + 397 \beta_{3} + 19 \beta_{4} - 675 \beta_{5} - 423 \beta_{6} - 122 \beta_{7} - 812 \beta_{8} + 329 \beta_{9} ) q^{66} + ( 954 + 682 \beta_{1} + 713 \beta_{2} + 333 \beta_{3} + 477 \beta_{4} + 542 \beta_{5} + 9 \beta_{6} - 317 \beta_{7} + 505 \beta_{8} - 416 \beta_{9} ) q^{67} + ( -19274 - 3060 \beta_{1} - 240 \beta_{2} + 346 \beta_{3} - 44 \beta_{4} - 891 \beta_{5} - 1445 \beta_{6} + 299 \beta_{7} - 320 \beta_{8} + 188 \beta_{9} ) q^{68} + ( 940 - 1833 \beta_{1} + 312 \beta_{2} - 402 \beta_{3} + 56 \beta_{4} + 252 \beta_{5} + 619 \beta_{6} + 376 \beta_{7} + 541 \beta_{8} - 428 \beta_{9} ) q^{69} + ( 1374 + 500 \beta_{1} + 168 \beta_{2} + 278 \beta_{3} + 482 \beta_{4} + 90 \beta_{5} + 392 \beta_{6} + 46 \beta_{7} - 70 \beta_{8} + 46 \beta_{9} ) q^{71} + ( 29314 + 3438 \beta_{1} + 800 \beta_{2} + 903 \beta_{3} - 717 \beta_{4} - 333 \beta_{5} + 852 \beta_{6} - 659 \beta_{7} - 1265 \beta_{8} + 454 \beta_{9} ) q^{72} + ( -5706 + 518 \beta_{1} + 318 \beta_{2} - 129 \beta_{3} - 312 \beta_{4} - 1115 \beta_{5} - 551 \beta_{6} - 141 \beta_{7} - 186 \beta_{8} - 135 \beta_{9} ) q^{73} + ( -14652 + 2698 \beta_{1} - 828 \beta_{2} + 97 \beta_{3} - 175 \beta_{4} - 1461 \beta_{5} - 1512 \beta_{6} - 467 \beta_{7} - 839 \beta_{8} + 614 \beta_{9} ) q^{74} + ( -25135 - 3923 \beta_{1} - 644 \beta_{2} - 984 \beta_{3} + 594 \beta_{4} - 382 \beta_{5} - 1130 \beta_{6} - 247 \beta_{7} + 386 \beta_{8} - 88 \beta_{9} ) q^{76} + ( -10614 + 1124 \beta_{1} - 604 \beta_{2} + 139 \beta_{3} + 572 \beta_{4} - 345 \beta_{5} + 13 \beta_{6} - 397 \beta_{7} - 712 \beta_{8} + 711 \beta_{9} ) q^{77} + ( 2727 - 1283 \beta_{1} + 235 \beta_{2} + 175 \beta_{3} + 421 \beta_{4} - 1077 \beta_{5} - 227 \beta_{6} + 40 \beta_{7} - 326 \beta_{8} + 161 \beta_{9} ) q^{78} + ( -9232 + 224 \beta_{1} + 183 \beta_{2} - 648 \beta_{3} + 328 \beta_{4} + 73 \beta_{5} + 1250 \beta_{6} + 8 \beta_{7} - 242 \beta_{8} + 29 \beta_{9} ) q^{79} + ( -3514 + 2984 \beta_{1} + 375 \beta_{2} + 287 \beta_{3} + 14 \beta_{4} + 96 \beta_{5} - 1140 \beta_{6} - 455 \beta_{7} - 830 \beta_{8} + 528 \beta_{9} ) q^{81} + ( 15291 - 653 \beta_{1} - 932 \beta_{2} - 899 \beta_{3} - 33 \beta_{4} - 549 \beta_{5} + 1350 \beta_{6} - 445 \beta_{7} + 223 \beta_{8} - 114 \beta_{9} ) q^{82} + ( 9905 - 757 \beta_{1} - 538 \beta_{2} + 87 \beta_{3} - 299 \beta_{4} + 9 \beta_{5} - 1701 \beta_{6} - 117 \beta_{7} - 542 \beta_{8} + 853 \beta_{9} ) q^{83} + ( -37176 + 3964 \beta_{1} + 108 \beta_{2} + 112 \beta_{3} + 152 \beta_{4} - 846 \beta_{5} - 1110 \beta_{6} - 74 \beta_{7} - 404 \beta_{8} - 304 \beta_{9} ) q^{84} + ( 1849 - 1849 \beta_{1} ) q^{86} + ( -18222 - 1890 \beta_{1} - 399 \beta_{2} - 1254 \beta_{3} - 24 \beta_{4} - 1251 \beta_{5} + 2196 \beta_{6} + 2214 \beta_{7} + 372 \beta_{8} - 963 \beta_{9} ) q^{87} + ( 48862 - 6606 \beta_{1} + 526 \beta_{2} - 1006 \beta_{3} + 284 \beta_{4} + 1952 \beta_{5} + 1034 \beta_{6} - 234 \beta_{7} + 1646 \beta_{8} - 556 \beta_{9} ) q^{88} + ( -7070 + 482 \beta_{1} - 366 \beta_{2} + 393 \beta_{3} + 230 \beta_{4} - 1305 \beta_{5} - 1601 \beta_{6} - 659 \beta_{7} + 324 \beta_{8} + 579 \beta_{9} ) q^{89} + ( -28810 - 668 \beta_{1} + 240 \beta_{2} + 1239 \beta_{3} + 636 \beta_{4} - 145 \beta_{5} - 379 \beta_{6} - 777 \beta_{7} + 48 \beta_{8} + 975 \beta_{9} ) q^{91} + ( -21241 + 5281 \beta_{1} - 846 \beta_{2} + 987 \beta_{3} - 1031 \beta_{4} - 1920 \beta_{5} - 1039 \beta_{6} - 679 \beta_{7} - 2037 \beta_{8} + 1134 \beta_{9} ) q^{92} + ( 24405 + 379 \beta_{1} + 900 \beta_{2} + 1987 \beta_{3} + 1208 \beta_{4} - 375 \beta_{5} - 2079 \beta_{6} - 2327 \beta_{7} - 95 \beta_{8} + 899 \beta_{9} ) q^{93} + ( 114 - 488 \beta_{1} - 942 \beta_{2} + 295 \beta_{3} + 143 \beta_{4} - 627 \beta_{5} + 1326 \beta_{6} - 1301 \beta_{7} - 947 \beta_{8} + 464 \beta_{9} ) q^{94} + ( -12386 - 12410 \beta_{1} + 74 \beta_{2} - 2261 \beta_{3} + 245 \beta_{4} + 2205 \beta_{5} + 1522 \beta_{6} + 2063 \beta_{7} + 2639 \beta_{8} - 2166 \beta_{9} ) q^{96} + ( -10134 + 1604 \beta_{1} + 448 \beta_{2} + 157 \beta_{3} + 1075 \beta_{4} - 13 \beta_{5} + 771 \beta_{6} + 529 \beta_{7} + 1145 \beta_{8} + 605 \beta_{9} ) q^{97} + ( -35573 - 43 \beta_{1} + 292 \beta_{2} - 930 \beta_{3} + 230 \beta_{4} - 34 \beta_{5} - 1116 \beta_{6} - 172 \beta_{7} + 1158 \beta_{8} - 800 \beta_{9} ) q^{98} + ( -26859 - 3077 \beta_{1} + 1130 \beta_{2} + 1291 \beta_{3} + 243 \beta_{4} - 1467 \beta_{5} - 2577 \beta_{6} - 1481 \beta_{7} - 740 \beta_{8} + 677 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 8q^{2} - 28q^{3} + 202q^{4} + 75q^{6} - 60q^{7} - 294q^{8} + 1356q^{9} + O(q^{10})$$ $$10q - 8q^{2} - 28q^{3} + 202q^{4} + 75q^{6} - 60q^{7} - 294q^{8} + 1356q^{9} + 745q^{11} - 4627q^{12} - 1917q^{13} + 1936q^{14} + 5354q^{16} - 4017q^{17} + 2725q^{18} - 2404q^{19} - 228q^{21} + 5836q^{22} - 1733q^{23} - 10711q^{24} - 1484q^{26} + 2324q^{27} + 15028q^{28} + 6996q^{29} - 4899q^{31} + 7554q^{32} + 15734q^{33} - 27033q^{34} + 4433q^{36} - 1466q^{37} - 13905q^{38} - 26542q^{39} + 10297q^{41} + 37642q^{42} - 18490q^{43} - 36140q^{44} + 17991q^{46} - 48592q^{47} - 83607q^{48} + 29458q^{49} + 92972q^{51} - 14232q^{52} - 127165q^{53} - 92002q^{54} - 7780q^{56} - 34060q^{57} + 10305q^{58} + 99372q^{59} + 17408q^{61} - 28265q^{62} - 2244q^{63} + 47202q^{64} - 150292q^{66} + 2021q^{67} - 192151q^{68} + 1654q^{69} + 11286q^{71} + 298365q^{72} - 49892q^{73} - 125431q^{74} - 249803q^{76} - 98144q^{77} + 28494q^{78} - 91524q^{79} - 26450q^{81} + 158909q^{82} + 105203q^{83} - 357682q^{84} + 14792q^{86} - 181200q^{87} + 461824q^{88} - 62682q^{89} - 295304q^{91} - 183783q^{92} + 238430q^{93} + 7259q^{94} - 162399q^{96} - 108383q^{97} - 354656q^{98} - 270499q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$18033 \nu^{9} - 1188163 \nu^{8} + 3451179 \nu^{7} + 239748231 \nu^{6} - 1005840413 \nu^{5} - 14422814397 \nu^{4} + 59540816201 \nu^{3} + 224666692021 \nu^{2} - 996337120808 \nu - 40045363612$$$$)/ 24633272448$$ $$\beta_{3}$$ $$=$$ $$($$$$-24107 \nu^{9} - 594245 \nu^{8} + 7832247 \nu^{7} + 116173337 \nu^{6} - 397671097 \nu^{5} - 6546797939 \nu^{4} - 20833059243 \nu^{3} + 64822510771 \nu^{2} + 762684015384 \nu - 192111505364$$$$)/ 24633272448$$ $$\beta_{4}$$ $$=$$ $$($$$$-144543 \nu^{9} - 820999 \nu^{8} + 35600547 \nu^{7} + 213763899 \nu^{6} - 2591602757 \nu^{5} - 15768164889 \nu^{4} + 54235511681 \nu^{3} + 287212616929 \nu^{2} - 418161182816 \nu - 912070606876$$$$)/ 24633272448$$ $$\beta_{5}$$ $$=$$ $$($$$$474917 \nu^{9} + 1144977 \nu^{8} - 116809809 \nu^{7} - 421392509 \nu^{6} + 8979038055 \nu^{5} + 39727817087 \nu^{4} - 220849424171 \nu^{3} - 937883085431 \nu^{2} + 1590463778000 \nu + 2423789997252$$$$)/ 65688726528$$ $$\beta_{6}$$ $$=$$ $$($$$$-474917 \nu^{9} - 1144977 \nu^{8} + 116809809 \nu^{7} + 421392509 \nu^{6} - 8979038055 \nu^{5} - 39727817087 \nu^{4} + 220849424171 \nu^{3} + 1003571811959 \nu^{2} - 1656152504528 \nu - 5839603776708$$$$)/ 65688726528$$ $$\beta_{7}$$ $$=$$ $$($$$$808641 \nu^{9} - 7898035 \nu^{8} - 155012637 \nu^{7} + 1407118071 \nu^{6} + 9333970747 \nu^{5} - 78913681917 \nu^{4} - 180644132719 \nu^{3} + 1606414562725 \nu^{2} + 768287694064 \nu - 7896805510540$$$$)/ 98533089792$$ $$\beta_{8}$$ $$=$$ $$($$$$-5132245 \nu^{9} + 307343 \nu^{8} + 1269871233 \nu^{7} + 1411070749 \nu^{6} - 100861721975 \nu^{5} - 189530468575 \nu^{4} + 2673777652347 \nu^{3} + 5229545463959 \nu^{2} - 17120027667600 \nu - 30463422181444$$$$)/ 197066179584$$ $$\beta_{9}$$ $$=$$ $$($$$$-2389487 \nu^{9} - 2695583 \nu^{8} + 612520467 \nu^{7} + 1209002243 \nu^{6} - 50464104709 \nu^{5} - 122440956545 \nu^{4} + 1432035207537 \nu^{3} + 3165045014041 \nu^{2} - 11776641778800 \nu - 18517909728476$$$$)/ 49266544896$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{1} + 52$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{9} - 5 \beta_{8} - 2 \beta_{7} + 3 \beta_{6} - \beta_{4} + \beta_{3} + 87 \beta_{1} + 56$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{8} - 16 \beta_{7} + 126 \beta_{6} + 113 \beta_{5} - 7 \beta_{4} + 13 \beta_{3} + 20 \beta_{2} + 245 \beta_{1} + 4542$$ $$\nu^{5}$$ $$=$$ $$278 \beta_{9} - 716 \beta_{8} - 308 \beta_{7} + 608 \beta_{6} + 92 \beta_{5} - 226 \beta_{4} + 250 \beta_{3} + 18 \beta_{2} + 8905 \beta_{1} + 13392$$ $$\nu^{6}$$ $$=$$ $$202 \beta_{9} - 1946 \beta_{8} - 2948 \beta_{7} + 16013 \beta_{6} + 11839 \beta_{5} - 2292 \beta_{4} + 2520 \beta_{3} + 3422 \beta_{2} + 40767 \beta_{1} + 465882$$ $$\nu^{7}$$ $$=$$ $$34356 \beta_{9} - 90155 \beta_{8} - 42446 \beta_{7} + 97339 \beta_{6} + 22646 \beta_{5} - 41129 \beta_{4} + 42101 \beta_{3} + 9022 \beta_{2} + 980257 \beta_{1} + 2212914$$ $$\nu^{8}$$ $$=$$ $$68250 \beta_{9} - 383649 \beta_{8} - 450492 \beta_{7} + 2030502 \beta_{6} + 1254459 \beta_{5} - 456771 \beta_{4} + 406485 \beta_{3} + 469962 \beta_{2} + 5987031 \beta_{1} + 51485248$$ $$\nu^{9}$$ $$=$$ $$4138872 \beta_{9} - 11178606 \beta_{8} - 5738016 \beta_{7} + 14588868 \beta_{6} + 3971370 \beta_{5} - 6655062 \beta_{4} + 6262002 \beta_{3} + 2108976 \beta_{2} + 113066491 \beta_{1} + 324019770$$

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
 −9.70631 −8.57770 −6.91219 −4.38824 −1.48720 2.86024 3.50018 5.31531 9.86547 11.5305
−10.7063 −18.3440 82.6251 0 196.397 96.4803 −542.008 93.5034 0
1.2 −9.57770 7.84343 59.7324 0 −75.1221 −195.604 −265.613 −181.481 0
1.3 −7.91219 −12.8799 30.6028 0 101.908 172.354 11.0549 −77.1083 0
1.4 −5.38824 −25.0462 −2.96684 0 134.955 −166.517 188.410 384.312 0
1.5 −2.48720 27.5943 −25.8138 0 −68.6327 −15.7005 143.795 518.448 0
1.6 1.86024 14.8716 −28.5395 0 27.6647 −202.971 −112.618 −21.8357 0
1.7 2.50018 −16.8892 −25.7491 0 −42.2260 −67.4603 −144.383 42.2439 0
1.8 4.31531 23.8469 −13.3781 0 102.907 174.859 −195.821 325.673 0
1.9 8.86547 −1.50169 46.5966 0 −13.3132 124.747 129.406 −240.745 0
1.10 10.5305 −27.4953 78.8905 0 −289.538 19.8137 493.778 512.989 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 1075.6.a.b 10
5.b even 2 1 43.6.a.b 10
15.d odd 2 1 387.6.a.e 10
20.d odd 2 1 688.6.a.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.b 10 5.b even 2 1
387.6.a.e 10 15.d odd 2 1
688.6.a.h 10 20.d odd 2 1
1075.6.a.b 10 1.a even 1 1 trivial

Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$43$$ $$1$$

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + \cdots$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(1075))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T + 91 T^{2} + 570 T^{3} + 4178 T^{4} + 19212 T^{5} + 126300 T^{6} + 603512 T^{7} + 4655376 T^{8} + 24295776 T^{9} + 168985984 T^{10} + 777464832 T^{11} + 4767105024 T^{12} + 19775881216 T^{13} + 132435148800 T^{14} + 644647747584 T^{15} + 4486093340672 T^{16} + 19585050869760 T^{17} + 100055558127616 T^{18} + 281474976710656 T^{19} + 1125899906842624 T^{20}$$
$3$ $$1 + 28 T + 929 T^{2} + 15652 T^{3} + 384094 T^{4} + 5703294 T^{5} + 127217574 T^{6} + 1773759114 T^{7} + 38322042597 T^{8} + 511917861924 T^{9} + 10155906521514 T^{10} + 124396040447532 T^{11} + 2262878293310253 T^{12} + 25451504567188398 T^{13} + 443580252556263174 T^{14} + 4832336042504605242 T^{15} + 79081548490762113006 T^{16} +$$$$78\!\cdots\!64$$$$T^{17} +$$$$11\!\cdots\!29$$$$T^{18} +$$$$82\!\cdots\!04$$$$T^{19} +$$$$71\!\cdots\!49$$$$T^{20}$$
$5$ 1
$7$ $$1 + 60 T + 71106 T^{2} + 5672772 T^{3} + 2998880769 T^{4} + 247503266272 T^{5} + 92903828588736 T^{6} + 7232195163049824 T^{7} + 2203025939434531374 T^{8} +$$$$15\!\cdots\!56$$$$T^{9} +$$$$41\!\cdots\!04$$$$T^{10} +$$$$26\!\cdots\!92$$$$T^{11} +$$$$62\!\cdots\!26$$$$T^{12} +$$$$34\!\cdots\!32$$$$T^{13} +$$$$74\!\cdots\!36$$$$T^{14} +$$$$33\!\cdots\!04$$$$T^{15} +$$$$67\!\cdots\!81$$$$T^{16} +$$$$21\!\cdots\!96$$$$T^{17} +$$$$45\!\cdots\!06$$$$T^{18} +$$$$64\!\cdots\!20$$$$T^{19} +$$$$17\!\cdots\!49$$$$T^{20}$$
$11$ $$1 - 745 T + 1092200 T^{2} - 566790565 T^{3} + 495359905314 T^{4} - 194173124255379 T^{5} + 135383021483146674 T^{6} - 43098674076488688185 T^{7} +$$$$27\!\cdots\!81$$$$T^{8} -$$$$77\!\cdots\!58$$$$T^{9} +$$$$47\!\cdots\!16$$$$T^{10} -$$$$12\!\cdots\!58$$$$T^{11} +$$$$71\!\cdots\!81$$$$T^{12} -$$$$18\!\cdots\!35$$$$T^{13} +$$$$91\!\cdots\!74$$$$T^{14} -$$$$21\!\cdots\!29$$$$T^{15} +$$$$86\!\cdots\!14$$$$T^{16} -$$$$15\!\cdots\!15$$$$T^{17} +$$$$49\!\cdots\!00$$$$T^{18} -$$$$54\!\cdots\!95$$$$T^{19} +$$$$11\!\cdots\!01$$$$T^{20}$$
$13$ $$1 + 1917 T + 3889346 T^{2} + 4848596699 T^{3} + 5963954609742 T^{4} + 5730569373162787 T^{5} + 5360926546177278684 T^{6} +$$$$42\!\cdots\!79$$$$T^{7} +$$$$32\!\cdots\!21$$$$T^{8} +$$$$21\!\cdots\!94$$$$T^{9} +$$$$14\!\cdots\!96$$$$T^{10} +$$$$81\!\cdots\!42$$$$T^{11} +$$$$44\!\cdots\!29$$$$T^{12} +$$$$21\!\cdots\!03$$$$T^{13} +$$$$10\!\cdots\!84$$$$T^{14} +$$$$40\!\cdots\!91$$$$T^{15} +$$$$15\!\cdots\!58$$$$T^{16} +$$$$47\!\cdots\!43$$$$T^{17} +$$$$14\!\cdots\!46$$$$T^{18} +$$$$25\!\cdots\!81$$$$T^{19} +$$$$49\!\cdots\!49$$$$T^{20}$$
$17$ $$1 + 4017 T + 16731071 T^{2} + 43371481422 T^{3} + 109028323359989 T^{4} + 214433083727073801 T^{5} +$$$$40\!\cdots\!11$$$$T^{6} +$$$$64\!\cdots\!84$$$$T^{7} +$$$$98\!\cdots\!87$$$$T^{8} +$$$$13\!\cdots\!91$$$$T^{9} +$$$$16\!\cdots\!00$$$$T^{10} +$$$$18\!\cdots\!87$$$$T^{11} +$$$$19\!\cdots\!63$$$$T^{12} +$$$$18\!\cdots\!12$$$$T^{13} +$$$$16\!\cdots\!11$$$$T^{14} +$$$$12\!\cdots\!57$$$$T^{15} +$$$$89\!\cdots\!61$$$$T^{16} +$$$$50\!\cdots\!46$$$$T^{17} +$$$$27\!\cdots\!71$$$$T^{18} +$$$$94\!\cdots\!69$$$$T^{19} +$$$$33\!\cdots\!49$$$$T^{20}$$
$19$ $$1 + 2404 T + 15222557 T^{2} + 35388594664 T^{3} + 123780260921510 T^{4} + 260083719347815286 T^{5} +$$$$66\!\cdots\!54$$$$T^{6} +$$$$12\!\cdots\!42$$$$T^{7} +$$$$25\!\cdots\!37$$$$T^{8} +$$$$41\!\cdots\!16$$$$T^{9} +$$$$73\!\cdots\!66$$$$T^{10} +$$$$10\!\cdots\!84$$$$T^{11} +$$$$15\!\cdots\!37$$$$T^{12} +$$$$18\!\cdots\!58$$$$T^{13} +$$$$25\!\cdots\!54$$$$T^{14} +$$$$24\!\cdots\!14$$$$T^{15} +$$$$28\!\cdots\!10$$$$T^{16} +$$$$20\!\cdots\!36$$$$T^{17} +$$$$21\!\cdots\!57$$$$T^{18} +$$$$84\!\cdots\!96$$$$T^{19} +$$$$86\!\cdots\!01$$$$T^{20}$$
$23$ $$1 + 1733 T + 47557957 T^{2} + 72067471074 T^{3} + 1063199488621929 T^{4} + 1395811354288602239 T^{5} +$$$$14\!\cdots\!11$$$$T^{6} +$$$$16\!\cdots\!50$$$$T^{7} +$$$$14\!\cdots\!39$$$$T^{8} +$$$$14\!\cdots\!37$$$$T^{9} +$$$$10\!\cdots\!02$$$$T^{10} +$$$$93\!\cdots\!91$$$$T^{11} +$$$$61\!\cdots\!11$$$$T^{12} +$$$$45\!\cdots\!50$$$$T^{13} +$$$$25\!\cdots\!11$$$$T^{14} +$$$$15\!\cdots\!77$$$$T^{15} +$$$$75\!\cdots\!21$$$$T^{16} +$$$$32\!\cdots\!18$$$$T^{17} +$$$$14\!\cdots\!57$$$$T^{18} +$$$$32\!\cdots\!19$$$$T^{19} +$$$$12\!\cdots\!49$$$$T^{20}$$
$29$ $$1 - 6996 T + 88174667 T^{2} - 354425388876 T^{3} + 2685110189553942 T^{4} - 3595496283776034360 T^{5} +$$$$29\!\cdots\!74$$$$T^{6} +$$$$15\!\cdots\!88$$$$T^{7} -$$$$13\!\cdots\!47$$$$T^{8} +$$$$63\!\cdots\!00$$$$T^{9} -$$$$82\!\cdots\!38$$$$T^{10} +$$$$13\!\cdots\!00$$$$T^{11} -$$$$58\!\cdots\!47$$$$T^{12} +$$$$13\!\cdots\!12$$$$T^{13} +$$$$52\!\cdots\!74$$$$T^{14} -$$$$13\!\cdots\!40$$$$T^{15} +$$$$19\!\cdots\!42$$$$T^{16} -$$$$54\!\cdots\!24$$$$T^{17} +$$$$27\!\cdots\!67$$$$T^{18} -$$$$44\!\cdots\!04$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$
$31$ $$1 + 4899 T + 169335305 T^{2} + 687234491898 T^{3} + 14314848605964873 T^{4} + 50115254632047551877 T^{5} +$$$$80\!\cdots\!39$$$$T^{6} +$$$$24\!\cdots\!98$$$$T^{7} +$$$$33\!\cdots\!11$$$$T^{8} +$$$$91\!\cdots\!15$$$$T^{9} +$$$$10\!\cdots\!46$$$$T^{10} +$$$$26\!\cdots\!65$$$$T^{11} +$$$$27\!\cdots\!11$$$$T^{12} +$$$$58\!\cdots\!98$$$$T^{13} +$$$$54\!\cdots\!39$$$$T^{14} +$$$$96\!\cdots\!27$$$$T^{15} +$$$$78\!\cdots\!73$$$$T^{16} +$$$$10\!\cdots\!98$$$$T^{17} +$$$$76\!\cdots\!05$$$$T^{18} +$$$$63\!\cdots\!49$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$
$37$ $$1 + 1466 T + 314595515 T^{2} + 1000159162152 T^{3} + 48977417173952218 T^{4} +$$$$27\!\cdots\!48$$$$T^{5} +$$$$50\!\cdots\!42$$$$T^{6} +$$$$41\!\cdots\!16$$$$T^{7} +$$$$40\!\cdots\!01$$$$T^{8} +$$$$41\!\cdots\!94$$$$T^{9} +$$$$29\!\cdots\!30$$$$T^{10} +$$$$28\!\cdots\!58$$$$T^{11} +$$$$19\!\cdots\!49$$$$T^{12} +$$$$13\!\cdots\!88$$$$T^{13} +$$$$11\!\cdots\!42$$$$T^{14} +$$$$43\!\cdots\!36$$$$T^{15} +$$$$54\!\cdots\!82$$$$T^{16} +$$$$77\!\cdots\!36$$$$T^{17} +$$$$16\!\cdots\!15$$$$T^{18} +$$$$54\!\cdots\!62$$$$T^{19} +$$$$25\!\cdots\!49$$$$T^{20}$$
$41$ $$1 - 10297 T + 512459663 T^{2} - 5996761727014 T^{3} + 151679237600092005 T^{4} -$$$$17\!\cdots\!49$$$$T^{5} +$$$$31\!\cdots\!39$$$$T^{6} -$$$$34\!\cdots\!68$$$$T^{7} +$$$$50\!\cdots\!19$$$$T^{8} -$$$$50\!\cdots\!39$$$$T^{9} +$$$$65\!\cdots\!96$$$$T^{10} -$$$$58\!\cdots\!39$$$$T^{11} +$$$$68\!\cdots\!19$$$$T^{12} -$$$$52\!\cdots\!68$$$$T^{13} +$$$$57\!\cdots\!39$$$$T^{14} -$$$$36\!\cdots\!49$$$$T^{15} +$$$$36\!\cdots\!05$$$$T^{16} -$$$$16\!\cdots\!14$$$$T^{17} +$$$$16\!\cdots\!63$$$$T^{18} -$$$$38\!\cdots\!97$$$$T^{19} +$$$$43\!\cdots\!01$$$$T^{20}$$
$43$ $$( 1 + 1849 T )^{10}$$
$47$ $$1 + 48592 T + 2515660075 T^{2} + 84810946501416 T^{3} + 2666904651640316270 T^{4} +$$$$68\!\cdots\!52$$$$T^{5} +$$$$16\!\cdots\!42$$$$T^{6} +$$$$33\!\cdots\!20$$$$T^{7} +$$$$64\!\cdots\!57$$$$T^{8} +$$$$11\!\cdots\!92$$$$T^{9} +$$$$17\!\cdots\!66$$$$T^{10} +$$$$25\!\cdots\!44$$$$T^{11} +$$$$34\!\cdots\!93$$$$T^{12} +$$$$40\!\cdots\!60$$$$T^{13} +$$$$45\!\cdots\!42$$$$T^{14} +$$$$43\!\cdots\!64$$$$T^{15} +$$$$38\!\cdots\!30$$$$T^{16} +$$$$28\!\cdots\!88$$$$T^{17} +$$$$19\!\cdots\!75$$$$T^{18} +$$$$85\!\cdots\!44$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20}$$
$53$ $$1 + 127165 T + 9906770390 T^{2} + 550920294329671 T^{3} + 24581308628857110150 T^{4} +$$$$91\!\cdots\!35$$$$T^{5} +$$$$29\!\cdots\!20$$$$T^{6} +$$$$83\!\cdots\!31$$$$T^{7} +$$$$21\!\cdots\!05$$$$T^{8} +$$$$49\!\cdots\!78$$$$T^{9} +$$$$10\!\cdots\!76$$$$T^{10} +$$$$20\!\cdots\!54$$$$T^{11} +$$$$37\!\cdots\!45$$$$T^{12} +$$$$61\!\cdots\!67$$$$T^{13} +$$$$90\!\cdots\!20$$$$T^{14} +$$$$11\!\cdots\!55$$$$T^{15} +$$$$13\!\cdots\!50$$$$T^{16} +$$$$12\!\cdots\!47$$$$T^{17} +$$$$92\!\cdots\!90$$$$T^{18} +$$$$49\!\cdots\!45$$$$T^{19} +$$$$16\!\cdots\!49$$$$T^{20}$$
$59$ $$1 - 99372 T + 7563799766 T^{2} - 407898209580180 T^{3} + 19000827281933813957 T^{4} -$$$$74\!\cdots\!72$$$$T^{5} +$$$$26\!\cdots\!44$$$$T^{6} -$$$$87\!\cdots\!68$$$$T^{7} +$$$$26\!\cdots\!22$$$$T^{8} -$$$$76\!\cdots\!76$$$$T^{9} +$$$$20\!\cdots\!96$$$$T^{10} -$$$$54\!\cdots\!24$$$$T^{11} +$$$$13\!\cdots\!22$$$$T^{12} -$$$$31\!\cdots\!32$$$$T^{13} +$$$$70\!\cdots\!44$$$$T^{14} -$$$$13\!\cdots\!28$$$$T^{15} +$$$$25\!\cdots\!57$$$$T^{16} -$$$$38\!\cdots\!20$$$$T^{17} +$$$$51\!\cdots\!66$$$$T^{18} -$$$$48\!\cdots\!28$$$$T^{19} +$$$$34\!\cdots\!01$$$$T^{20}$$
$61$ $$1 - 17408 T + 3107203610 T^{2} - 59824183442672 T^{3} + 5726081070022748105 T^{4} -$$$$10\!\cdots\!64$$$$T^{5} +$$$$76\!\cdots\!32$$$$T^{6} -$$$$13\!\cdots\!44$$$$T^{7} +$$$$84\!\cdots\!70$$$$T^{8} -$$$$13\!\cdots\!80$$$$T^{9} +$$$$77\!\cdots\!00$$$$T^{10} -$$$$11\!\cdots\!80$$$$T^{11} +$$$$60\!\cdots\!70$$$$T^{12} -$$$$78\!\cdots\!44$$$$T^{13} +$$$$38\!\cdots\!32$$$$T^{14} -$$$$44\!\cdots\!64$$$$T^{15} +$$$$20\!\cdots\!05$$$$T^{16} -$$$$18\!\cdots\!72$$$$T^{17} +$$$$80\!\cdots\!10$$$$T^{18} -$$$$38\!\cdots\!08$$$$T^{19} +$$$$18\!\cdots\!01$$$$T^{20}$$
$67$ $$1 - 2021 T + 3389673908 T^{2} + 95129046044495 T^{3} + 5951903574650911354 T^{4} +$$$$26\!\cdots\!45$$$$T^{5} +$$$$15\!\cdots\!46$$$$T^{6} +$$$$38\!\cdots\!75$$$$T^{7} +$$$$27\!\cdots\!93$$$$T^{8} +$$$$78\!\cdots\!74$$$$T^{9} +$$$$36\!\cdots\!28$$$$T^{10} +$$$$10\!\cdots\!18$$$$T^{11} +$$$$50\!\cdots\!57$$$$T^{12} +$$$$94\!\cdots\!25$$$$T^{13} +$$$$51\!\cdots\!46$$$$T^{14} +$$$$11\!\cdots\!15$$$$T^{15} +$$$$36\!\cdots\!46$$$$T^{16} +$$$$77\!\cdots\!85$$$$T^{17} +$$$$37\!\cdots\!08$$$$T^{18} -$$$$30\!\cdots\!47$$$$T^{19} +$$$$20\!\cdots\!49$$$$T^{20}$$
$71$ $$1 - 11286 T + 12540359910 T^{2} - 44404438984650 T^{3} + 71277357843109037437 T^{4} +$$$$33\!\cdots\!36$$$$T^{5} +$$$$24\!\cdots\!24$$$$T^{6} +$$$$30\!\cdots\!96$$$$T^{7} +$$$$61\!\cdots\!22$$$$T^{8} +$$$$10\!\cdots\!68$$$$T^{9} +$$$$12\!\cdots\!84$$$$T^{10} +$$$$18\!\cdots\!68$$$$T^{11} +$$$$20\!\cdots\!22$$$$T^{12} +$$$$17\!\cdots\!96$$$$T^{13} +$$$$26\!\cdots\!24$$$$T^{14} +$$$$63\!\cdots\!36$$$$T^{15} +$$$$24\!\cdots\!37$$$$T^{16} -$$$$27\!\cdots\!50$$$$T^{17} +$$$$14\!\cdots\!10$$$$T^{18} -$$$$22\!\cdots\!86$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$
$73$ $$1 + 49892 T + 9676771082 T^{2} + 435365481214596 T^{3} + 51788415419782211185 T^{4} +$$$$22\!\cdots\!88$$$$T^{5} +$$$$19\!\cdots\!88$$$$T^{6} +$$$$79\!\cdots\!24$$$$T^{7} +$$$$57\!\cdots\!14$$$$T^{8} +$$$$21\!\cdots\!32$$$$T^{9} +$$$$13\!\cdots\!60$$$$T^{10} +$$$$44\!\cdots\!76$$$$T^{11} +$$$$24\!\cdots\!86$$$$T^{12} +$$$$71\!\cdots\!68$$$$T^{13} +$$$$36\!\cdots\!88$$$$T^{14} +$$$$85\!\cdots\!84$$$$T^{15} +$$$$41\!\cdots\!65$$$$T^{16} +$$$$71\!\cdots\!72$$$$T^{17} +$$$$33\!\cdots\!82$$$$T^{18} +$$$$35\!\cdots\!56$$$$T^{19} +$$$$14\!\cdots\!49$$$$T^{20}$$
$79$ $$1 + 91524 T + 16857154847 T^{2} + 842558519651240 T^{3} + 99200238774687370926 T^{4} +$$$$23\!\cdots\!36$$$$T^{5} +$$$$37\!\cdots\!90$$$$T^{6} +$$$$59\!\cdots\!56$$$$T^{7} +$$$$16\!\cdots\!17$$$$T^{8} +$$$$38\!\cdots\!04$$$$T^{9} +$$$$63\!\cdots\!18$$$$T^{10} +$$$$11\!\cdots\!96$$$$T^{11} +$$$$15\!\cdots\!17$$$$T^{12} +$$$$17\!\cdots\!44$$$$T^{13} +$$$$33\!\cdots\!90$$$$T^{14} +$$$$65\!\cdots\!64$$$$T^{15} +$$$$84\!\cdots\!26$$$$T^{16} +$$$$22\!\cdots\!60$$$$T^{17} +$$$$13\!\cdots\!47$$$$T^{18} +$$$$22\!\cdots\!76$$$$T^{19} +$$$$76\!\cdots\!01$$$$T^{20}$$
$83$ $$1 - 105203 T + 28621889044 T^{2} - 2443253098090515 T^{3} +$$$$38\!\cdots\!90$$$$T^{4} -$$$$27\!\cdots\!61$$$$T^{5} +$$$$31\!\cdots\!66$$$$T^{6} -$$$$19\!\cdots\!11$$$$T^{7} +$$$$18\!\cdots\!57$$$$T^{8} -$$$$99\!\cdots\!22$$$$T^{9} +$$$$83\!\cdots\!32$$$$T^{10} -$$$$39\!\cdots\!46$$$$T^{11} +$$$$29\!\cdots\!93$$$$T^{12} -$$$$11\!\cdots\!77$$$$T^{13} +$$$$76\!\cdots\!66$$$$T^{14} -$$$$25\!\cdots\!23$$$$T^{15} +$$$$14\!\cdots\!10$$$$T^{16} -$$$$35\!\cdots\!05$$$$T^{17} +$$$$16\!\cdots\!44$$$$T^{18} -$$$$24\!\cdots\!29$$$$T^{19} +$$$$89\!\cdots\!49$$$$T^{20}$$
$89$ $$1 + 62682 T + 34400806898 T^{2} + 1591902274833594 T^{3} +$$$$57\!\cdots\!17$$$$T^{4} +$$$$20\!\cdots\!96$$$$T^{5} +$$$$63\!\cdots\!68$$$$T^{6} +$$$$16\!\cdots\!12$$$$T^{7} +$$$$50\!\cdots\!94$$$$T^{8} +$$$$10\!\cdots\!68$$$$T^{9} +$$$$31\!\cdots\!32$$$$T^{10} +$$$$60\!\cdots\!32$$$$T^{11} +$$$$15\!\cdots\!94$$$$T^{12} +$$$$29\!\cdots\!88$$$$T^{13} +$$$$61\!\cdots\!68$$$$T^{14} +$$$$10\!\cdots\!04$$$$T^{15} +$$$$17\!\cdots\!17$$$$T^{16} +$$$$26\!\cdots\!06$$$$T^{17} +$$$$32\!\cdots\!98$$$$T^{18} +$$$$33\!\cdots\!18$$$$T^{19} +$$$$29\!\cdots\!01$$$$T^{20}$$
$97$ $$1 + 108383 T + 55022053811 T^{2} + 4727913772355050 T^{3} +$$$$13\!\cdots\!17$$$$T^{4} +$$$$99\!\cdots\!31$$$$T^{5} +$$$$22\!\cdots\!35$$$$T^{6} +$$$$13\!\cdots\!96$$$$T^{7} +$$$$26\!\cdots\!83$$$$T^{8} +$$$$14\!\cdots\!33$$$$T^{9} +$$$$25\!\cdots\!28$$$$T^{10} +$$$$12\!\cdots\!81$$$$T^{11} +$$$$19\!\cdots\!67$$$$T^{12} +$$$$88\!\cdots\!28$$$$T^{13} +$$$$12\!\cdots\!35$$$$T^{14} +$$$$46\!\cdots\!67$$$$T^{15} +$$$$55\!\cdots\!33$$$$T^{16} +$$$$16\!\cdots\!50$$$$T^{17} +$$$$16\!\cdots\!11$$$$T^{18} +$$$$27\!\cdots\!31$$$$T^{19} +$$$$21\!\cdots\!49$$$$T^{20}$$