Properties

Label 43.6.a.b
Level 43
Weight 6
Character orbit 43.a
Self dual yes
Analytic conductor 6.897
Analytic rank 0
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.89650425196\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 3 + \beta_{6} ) q^{3} + ( 21 - \beta_{1} + \beta_{5} + \beta_{6} ) q^{4} + ( 13 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{5} + ( 9 - 11 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{6} + ( 8 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{7} + ( 37 - 23 \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} ) q^{8} + ( 133 + 6 \beta_{1} + 3 \beta_{2} - 4 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} + ( 3 + \beta_{6} ) q^{3} + ( 21 - \beta_{1} + \beta_{5} + \beta_{6} ) q^{4} + ( 13 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{5} + ( 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} + ( -10 - 8 \beta_{1} - 7 \beta_{2} - \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 17 \beta_{6} + \beta_{7} - 4 \beta_{8} + 3 \beta_{9} ) q^{10} + ( 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} + ( 147 + 98 \beta_{1} - 17 \beta_{2} - \beta_{3} - 12 \beta_{4} + 34 \beta_{6} + 11 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{15} + ( 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} + ( 69 + 181 \beta_{1} - 26 \beta_{2} + 20 \beta_{3} - 36 \beta_{4} - 12 \beta_{5} + 12 \beta_{6} + 5 \beta_{7} - 50 \beta_{8} + 12 \beta_{9} ) q^{20} + ( -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} + ( 687 + 17 \beta_{1} + 7 \beta_{2} + 31 \beta_{3} - 8 \beta_{4} - 10 \beta_{5} - 78 \beta_{6} - 71 \beta_{7} - 31 \beta_{8} + 32 \beta_{9} ) q^{25} + ( -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} + ( -4919 - 317 \beta_{1} - 132 \beta_{2} - 35 \beta_{3} + 79 \beta_{4} - 105 \beta_{5} - 140 \beta_{6} + 44 \beta_{7} - 31 \beta_{8} + 18 \beta_{9} ) q^{30} + ( -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} + ( 805 + 41 \beta_{1} + 89 \beta_{2} - 40 \beta_{3} + 2 \beta_{4} + 81 \beta_{5} + 30 \beta_{6} + 62 \beta_{7} + 111 \beta_{8} - \beta_{9} ) q^{35} + ( 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} + ( -9473 - 73 \beta_{1} - 150 \beta_{2} - 6 \beta_{3} + 176 \beta_{4} - 212 \beta_{5} - 240 \beta_{6} + 157 \beta_{7} - 10 \beta_{8} + 58 \beta_{9} ) q^{40} + ( 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} + ( 7134 + 1157 \beta_{1} + 272 \beta_{2} + 254 \beta_{3} + 20 \beta_{4} - 120 \beta_{5} - 363 \beta_{6} - 376 \beta_{7} - 241 \beta_{8} + 172 \beta_{9} ) q^{45} + ( 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} + ( 89 - 875 \beta_{1} - 194 \beta_{2} - 97 \beta_{3} + 251 \beta_{4} - 35 \beta_{5} - 494 \beta_{6} - 173 \beta_{7} + 117 \beta_{8} + 16 \beta_{9} ) q^{50} + ( 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} + ( 10936 - 1986 \beta_{1} - 34 \beta_{2} - \beta_{3} - 96 \beta_{4} - 43 \beta_{5} - 331 \beta_{6} - 269 \beta_{7} + 14 \beta_{8} + 165 \beta_{9} ) q^{55} + ( -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} + ( 9427 + 5467 \beta_{1} + 50 \beta_{2} + 709 \beta_{3} - 511 \beta_{4} - 531 \beta_{5} - 596 \beta_{6} - 802 \beta_{7} - 1321 \beta_{8} + 714 \beta_{9} ) q^{60} + ( 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} + ( 5378 - 128 \beta_{1} + 302 \beta_{2} + 465 \beta_{3} + 296 \beta_{4} - 629 \beta_{5} - 829 \beta_{6} - 375 \beta_{7} - 60 \beta_{8} + 191 \beta_{9} ) q^{65} + ( -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} + ( -1702 - 2102 \beta_{1} + 1092 \beta_{2} + 182 \beta_{3} - 86 \beta_{4} + 670 \beta_{5} + 772 \beta_{6} + 194 \beta_{7} + 590 \beta_{8} - 464 \beta_{9} ) q^{70} + ( 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} + ( -4084 + 2151 \beta_{1} - 754 \beta_{2} - 12 \beta_{3} - 394 \beta_{4} + 1320 \beta_{5} + 2387 \beta_{6} + 512 \beta_{7} + 257 \beta_{8} - 454 \beta_{9} ) q^{75} + ( -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} + ( -1249 + 10751 \beta_{1} + 512 \beta_{2} + 874 \beta_{3} - 676 \beta_{4} - 746 \beta_{5} - 548 \beta_{6} - 213 \beta_{7} - 1172 \beta_{8} + 808 \beta_{9} ) q^{80} + ( -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} + ( -9894 + 4487 \beta_{1} + 258 \beta_{2} + 358 \beta_{3} - 818 \beta_{4} - 634 \beta_{5} + 13 \beta_{6} - 196 \beta_{7} - 513 \beta_{8} + 414 \beta_{9} ) q^{85} + ( 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} + ( -52298 - 8214 \beta_{1} - 183 \beta_{2} - 786 \beta_{3} + 734 \beta_{4} - 1752 \beta_{5} - 4609 \beta_{6} + 364 \beta_{7} + 763 \beta_{8} + 25 \beta_{9} ) q^{90} + ( -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} + ( -31135 - 3002 \beta_{1} - 1766 \beta_{2} - 761 \beta_{3} + 420 \beta_{4} - 1083 \beta_{5} + 22 \beta_{6} + 683 \beta_{7} + 510 \beta_{8} + 235 \beta_{9} ) q^{95} + ( -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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 8q^{2} + 28q^{3} + 202q^{4} + 138q^{5} + 75q^{6} + 60q^{7} + 294q^{8} + 1356q^{9} + O(q^{10}) \) \( 10q + 8q^{2} + 28q^{3} + 202q^{4} + 138q^{5} + 75q^{6} + 60q^{7} + 294q^{8} + 1356q^{9} - 17q^{10} + 745q^{11} + 4627q^{12} + 1917q^{13} + 1936q^{14} + 1688q^{15} + 5354q^{16} + 4017q^{17} - 2725q^{18} - 2404q^{19} + 1311q^{20} - 228q^{21} - 5836q^{22} + 1733q^{23} - 10711q^{24} + 7120q^{25} - 1484q^{26} - 2324q^{27} - 15028q^{28} + 6996q^{29} - 48420q^{30} - 4899q^{31} - 7554q^{32} - 15734q^{33} - 27033q^{34} + 7084q^{35} + 4433q^{36} + 1466q^{37} + 13905q^{38} - 26542q^{39} - 93211q^{40} + 10297q^{41} - 37642q^{42} + 18490q^{43} - 36140q^{44} + 73822q^{45} + 17991q^{46} + 48592q^{47} + 83607q^{48} + 29458q^{49} + 983q^{50} + 92972q^{51} + 14232q^{52} + 127165q^{53} - 92002q^{54} + 106672q^{55} - 7780q^{56} + 34060q^{57} - 10305q^{58} + 99372q^{59} + 111372q^{60} + 17408q^{61} + 28265q^{62} + 2244q^{63} + 47202q^{64} + 54484q^{65} - 150292q^{66} - 2021q^{67} + 192151q^{68} + 1654q^{69} - 33194q^{70} + 11286q^{71} - 298365q^{72} + 49892q^{73} - 125431q^{74} - 44662q^{75} - 249803q^{76} + 98144q^{77} - 28494q^{78} - 91524q^{79} + 12251q^{80} - 26450q^{81} - 158909q^{82} - 105203q^{83} - 357682q^{84} - 87212q^{85} + 14792q^{86} + 181200q^{87} - 461824q^{88} - 62682q^{89} - 522670q^{90} - 295304q^{91} + 183783q^{92} - 238430q^{93} + 7259q^{94} - 305340q^{95} - 162399q^{96} + 108383q^{97} + 354656q^{98} - 270499q^{99} + O(q^{100}) \)

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
11.5305
9.86547
5.31531
3.50018
2.86024
−1.48720
−4.38824
−6.91219
−8.57770
−9.70631
−10.5305 27.4953 78.8905 86.8464 −289.538 −19.8137 −493.778 512.989 −914.532
1.2 −8.86547 1.50169 46.5966 −37.8251 −13.3132 −124.747 −129.406 −240.745 335.337
1.3 −4.31531 −23.8469 −13.3781 −52.5837 102.907 −174.859 195.821 325.673 226.915
1.4 −2.50018 16.8892 −25.7491 47.4635 −42.2260 67.4603 144.383 42.2439 −118.667
1.5 −1.86024 −14.8716 −28.5395 −42.2365 27.6647 202.971 112.618 −21.8357 78.5700
1.6 2.48720 −27.5943 −25.8138 101.308 −68.6327 15.7005 −143.795 518.448 251.974
1.7 5.38824 25.0462 −2.96684 −0.456695 134.955 166.517 −188.410 384.312 −2.46078
1.8 7.91219 12.8799 30.6028 79.5677 101.908 −172.354 −11.0549 −77.1083 629.555
1.9 9.57770 −7.84343 59.7324 28.1028 −75.1221 195.604 265.613 −181.481 269.160
1.10 10.7063 18.3440 82.6251 −72.1865 196.397 −96.4803 542.008 93.5034 −772.851
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.6.a.b 10
3.b odd 2 1 387.6.a.e 10
4.b odd 2 1 688.6.a.h 10
5.b even 2 1 1075.6.a.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.b 10 1.a even 1 1 trivial
387.6.a.e 10 3.b odd 2 1
688.6.a.h 10 4.b odd 2 1
1075.6.a.b 10 5.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(43))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8 T + 91 T^{2} - 570 T^{3} + 4178 T^{4} - 19212 T^{5} + 126300 T^{6} - 603512 T^{7} + 4655376 T^{8} - 24295776 T^{9} + 168985984 T^{10} - 777464832 T^{11} + 4767105024 T^{12} - 19775881216 T^{13} + 132435148800 T^{14} - 644647747584 T^{15} + 4486093340672 T^{16} - 19585050869760 T^{17} + 100055558127616 T^{18} - 281474976710656 T^{19} + 1125899906842624 T^{20} \)
$3$ \( 1 - 28 T + 929 T^{2} - 15652 T^{3} + 384094 T^{4} - 5703294 T^{5} + 127217574 T^{6} - 1773759114 T^{7} + 38322042597 T^{8} - 511917861924 T^{9} + 10155906521514 T^{10} - 124396040447532 T^{11} + 2262878293310253 T^{12} - 25451504567188398 T^{13} + 443580252556263174 T^{14} - 4832336042504605242 T^{15} + 79081548490762113006 T^{16} - \)\(78\!\cdots\!64\)\( T^{17} + \)\(11\!\cdots\!29\)\( T^{18} - \)\(82\!\cdots\!04\)\( T^{19} + \)\(71\!\cdots\!49\)\( T^{20} \)
$5$ \( 1 - 138 T + 21587 T^{2} - 2230608 T^{3} + 229459202 T^{4} - 19157670180 T^{5} + 1558167162738 T^{6} - 109829488059312 T^{7} + 7514260269046033 T^{8} - 457924168440376602 T^{9} + 27012319396517106654 T^{10} - \)\(14\!\cdots\!50\)\( T^{11} + \)\(73\!\cdots\!25\)\( T^{12} - \)\(33\!\cdots\!00\)\( T^{13} + \)\(14\!\cdots\!50\)\( T^{14} - \)\(57\!\cdots\!00\)\( T^{15} + \)\(21\!\cdots\!50\)\( T^{16} - \)\(64\!\cdots\!00\)\( T^{17} + \)\(19\!\cdots\!75\)\( T^{18} - \)\(39\!\cdots\!50\)\( T^{19} + \)\(88\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 - 60 T + 71106 T^{2} - 5672772 T^{3} + 2998880769 T^{4} - 247503266272 T^{5} + 92903828588736 T^{6} - 7232195163049824 T^{7} + 2203025939434531374 T^{8} - \)\(15\!\cdots\!56\)\( T^{9} + \)\(41\!\cdots\!04\)\( T^{10} - \)\(26\!\cdots\!92\)\( T^{11} + \)\(62\!\cdots\!26\)\( T^{12} - \)\(34\!\cdots\!32\)\( T^{13} + \)\(74\!\cdots\!36\)\( T^{14} - \)\(33\!\cdots\!04\)\( T^{15} + \)\(67\!\cdots\!81\)\( T^{16} - \)\(21\!\cdots\!96\)\( T^{17} + \)\(45\!\cdots\!06\)\( T^{18} - \)\(64\!\cdots\!20\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} \)
$11$ \( 1 - 745 T + 1092200 T^{2} - 566790565 T^{3} + 495359905314 T^{4} - 194173124255379 T^{5} + 135383021483146674 T^{6} - 43098674076488688185 T^{7} + \)\(27\!\cdots\!81\)\( T^{8} - \)\(77\!\cdots\!58\)\( T^{9} + \)\(47\!\cdots\!16\)\( T^{10} - \)\(12\!\cdots\!58\)\( T^{11} + \)\(71\!\cdots\!81\)\( T^{12} - \)\(18\!\cdots\!35\)\( T^{13} + \)\(91\!\cdots\!74\)\( T^{14} - \)\(21\!\cdots\!29\)\( T^{15} + \)\(86\!\cdots\!14\)\( T^{16} - \)\(15\!\cdots\!15\)\( T^{17} + \)\(49\!\cdots\!00\)\( T^{18} - \)\(54\!\cdots\!95\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 - 1917 T + 3889346 T^{2} - 4848596699 T^{3} + 5963954609742 T^{4} - 5730569373162787 T^{5} + 5360926546177278684 T^{6} - \)\(42\!\cdots\!79\)\( T^{7} + \)\(32\!\cdots\!21\)\( T^{8} - \)\(21\!\cdots\!94\)\( T^{9} + \)\(14\!\cdots\!96\)\( T^{10} - \)\(81\!\cdots\!42\)\( T^{11} + \)\(44\!\cdots\!29\)\( T^{12} - \)\(21\!\cdots\!03\)\( T^{13} + \)\(10\!\cdots\!84\)\( T^{14} - \)\(40\!\cdots\!91\)\( T^{15} + \)\(15\!\cdots\!58\)\( T^{16} - \)\(47\!\cdots\!43\)\( T^{17} + \)\(14\!\cdots\!46\)\( T^{18} - \)\(25\!\cdots\!81\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} \)
$17$ \( 1 - 4017 T + 16731071 T^{2} - 43371481422 T^{3} + 109028323359989 T^{4} - 214433083727073801 T^{5} + \)\(40\!\cdots\!11\)\( T^{6} - \)\(64\!\cdots\!84\)\( T^{7} + \)\(98\!\cdots\!87\)\( T^{8} - \)\(13\!\cdots\!91\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} - \)\(18\!\cdots\!87\)\( T^{11} + \)\(19\!\cdots\!63\)\( T^{12} - \)\(18\!\cdots\!12\)\( T^{13} + \)\(16\!\cdots\!11\)\( T^{14} - \)\(12\!\cdots\!57\)\( T^{15} + \)\(89\!\cdots\!61\)\( T^{16} - \)\(50\!\cdots\!46\)\( T^{17} + \)\(27\!\cdots\!71\)\( T^{18} - \)\(94\!\cdots\!69\)\( T^{19} + \)\(33\!\cdots\!49\)\( T^{20} \)
$19$ \( 1 + 2404 T + 15222557 T^{2} + 35388594664 T^{3} + 123780260921510 T^{4} + 260083719347815286 T^{5} + \)\(66\!\cdots\!54\)\( T^{6} + \)\(12\!\cdots\!42\)\( T^{7} + \)\(25\!\cdots\!37\)\( T^{8} + \)\(41\!\cdots\!16\)\( T^{9} + \)\(73\!\cdots\!66\)\( T^{10} + \)\(10\!\cdots\!84\)\( T^{11} + \)\(15\!\cdots\!37\)\( T^{12} + \)\(18\!\cdots\!58\)\( T^{13} + \)\(25\!\cdots\!54\)\( T^{14} + \)\(24\!\cdots\!14\)\( T^{15} + \)\(28\!\cdots\!10\)\( T^{16} + \)\(20\!\cdots\!36\)\( T^{17} + \)\(21\!\cdots\!57\)\( T^{18} + \)\(84\!\cdots\!96\)\( T^{19} + \)\(86\!\cdots\!01\)\( T^{20} \)
$23$ \( 1 - 1733 T + 47557957 T^{2} - 72067471074 T^{3} + 1063199488621929 T^{4} - 1395811354288602239 T^{5} + \)\(14\!\cdots\!11\)\( T^{6} - \)\(16\!\cdots\!50\)\( T^{7} + \)\(14\!\cdots\!39\)\( T^{8} - \)\(14\!\cdots\!37\)\( T^{9} + \)\(10\!\cdots\!02\)\( T^{10} - \)\(93\!\cdots\!91\)\( T^{11} + \)\(61\!\cdots\!11\)\( T^{12} - \)\(45\!\cdots\!50\)\( T^{13} + \)\(25\!\cdots\!11\)\( T^{14} - \)\(15\!\cdots\!77\)\( T^{15} + \)\(75\!\cdots\!21\)\( T^{16} - \)\(32\!\cdots\!18\)\( T^{17} + \)\(14\!\cdots\!57\)\( T^{18} - \)\(32\!\cdots\!19\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} \)
$29$ \( 1 - 6996 T + 88174667 T^{2} - 354425388876 T^{3} + 2685110189553942 T^{4} - 3595496283776034360 T^{5} + \)\(29\!\cdots\!74\)\( T^{6} + \)\(15\!\cdots\!88\)\( T^{7} - \)\(13\!\cdots\!47\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{9} - \)\(82\!\cdots\!38\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} - \)\(58\!\cdots\!47\)\( T^{12} + \)\(13\!\cdots\!12\)\( T^{13} + \)\(52\!\cdots\!74\)\( T^{14} - \)\(13\!\cdots\!40\)\( T^{15} + \)\(19\!\cdots\!42\)\( T^{16} - \)\(54\!\cdots\!24\)\( T^{17} + \)\(27\!\cdots\!67\)\( T^{18} - \)\(44\!\cdots\!04\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \)
$31$ \( 1 + 4899 T + 169335305 T^{2} + 687234491898 T^{3} + 14314848605964873 T^{4} + 50115254632047551877 T^{5} + \)\(80\!\cdots\!39\)\( T^{6} + \)\(24\!\cdots\!98\)\( T^{7} + \)\(33\!\cdots\!11\)\( T^{8} + \)\(91\!\cdots\!15\)\( T^{9} + \)\(10\!\cdots\!46\)\( T^{10} + \)\(26\!\cdots\!65\)\( T^{11} + \)\(27\!\cdots\!11\)\( T^{12} + \)\(58\!\cdots\!98\)\( T^{13} + \)\(54\!\cdots\!39\)\( T^{14} + \)\(96\!\cdots\!27\)\( T^{15} + \)\(78\!\cdots\!73\)\( T^{16} + \)\(10\!\cdots\!98\)\( T^{17} + \)\(76\!\cdots\!05\)\( T^{18} + \)\(63\!\cdots\!49\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \)
$37$ \( 1 - 1466 T + 314595515 T^{2} - 1000159162152 T^{3} + 48977417173952218 T^{4} - \)\(27\!\cdots\!48\)\( T^{5} + \)\(50\!\cdots\!42\)\( T^{6} - \)\(41\!\cdots\!16\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} - \)\(41\!\cdots\!94\)\( T^{9} + \)\(29\!\cdots\!30\)\( T^{10} - \)\(28\!\cdots\!58\)\( T^{11} + \)\(19\!\cdots\!49\)\( T^{12} - \)\(13\!\cdots\!88\)\( T^{13} + \)\(11\!\cdots\!42\)\( T^{14} - \)\(43\!\cdots\!36\)\( T^{15} + \)\(54\!\cdots\!82\)\( T^{16} - \)\(77\!\cdots\!36\)\( T^{17} + \)\(16\!\cdots\!15\)\( T^{18} - \)\(54\!\cdots\!62\)\( T^{19} + \)\(25\!\cdots\!49\)\( T^{20} \)
$41$ \( 1 - 10297 T + 512459663 T^{2} - 5996761727014 T^{3} + 151679237600092005 T^{4} - \)\(17\!\cdots\!49\)\( T^{5} + \)\(31\!\cdots\!39\)\( T^{6} - \)\(34\!\cdots\!68\)\( T^{7} + \)\(50\!\cdots\!19\)\( T^{8} - \)\(50\!\cdots\!39\)\( T^{9} + \)\(65\!\cdots\!96\)\( T^{10} - \)\(58\!\cdots\!39\)\( T^{11} + \)\(68\!\cdots\!19\)\( T^{12} - \)\(52\!\cdots\!68\)\( T^{13} + \)\(57\!\cdots\!39\)\( T^{14} - \)\(36\!\cdots\!49\)\( T^{15} + \)\(36\!\cdots\!05\)\( T^{16} - \)\(16\!\cdots\!14\)\( T^{17} + \)\(16\!\cdots\!63\)\( T^{18} - \)\(38\!\cdots\!97\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} \)
$43$ \( ( 1 - 1849 T )^{10} \)
$47$ \( 1 - 48592 T + 2515660075 T^{2} - 84810946501416 T^{3} + 2666904651640316270 T^{4} - \)\(68\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!42\)\( T^{6} - \)\(33\!\cdots\!20\)\( T^{7} + \)\(64\!\cdots\!57\)\( T^{8} - \)\(11\!\cdots\!92\)\( T^{9} + \)\(17\!\cdots\!66\)\( T^{10} - \)\(25\!\cdots\!44\)\( T^{11} + \)\(34\!\cdots\!93\)\( T^{12} - \)\(40\!\cdots\!60\)\( T^{13} + \)\(45\!\cdots\!42\)\( T^{14} - \)\(43\!\cdots\!64\)\( T^{15} + \)\(38\!\cdots\!30\)\( T^{16} - \)\(28\!\cdots\!88\)\( T^{17} + \)\(19\!\cdots\!75\)\( T^{18} - \)\(85\!\cdots\!44\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} \)
$53$ \( 1 - 127165 T + 9906770390 T^{2} - 550920294329671 T^{3} + 24581308628857110150 T^{4} - \)\(91\!\cdots\!35\)\( T^{5} + \)\(29\!\cdots\!20\)\( T^{6} - \)\(83\!\cdots\!31\)\( T^{7} + \)\(21\!\cdots\!05\)\( T^{8} - \)\(49\!\cdots\!78\)\( T^{9} + \)\(10\!\cdots\!76\)\( T^{10} - \)\(20\!\cdots\!54\)\( T^{11} + \)\(37\!\cdots\!45\)\( T^{12} - \)\(61\!\cdots\!67\)\( T^{13} + \)\(90\!\cdots\!20\)\( T^{14} - \)\(11\!\cdots\!55\)\( T^{15} + \)\(13\!\cdots\!50\)\( T^{16} - \)\(12\!\cdots\!47\)\( T^{17} + \)\(92\!\cdots\!90\)\( T^{18} - \)\(49\!\cdots\!45\)\( T^{19} + \)\(16\!\cdots\!49\)\( T^{20} \)
$59$ \( 1 - 99372 T + 7563799766 T^{2} - 407898209580180 T^{3} + 19000827281933813957 T^{4} - \)\(74\!\cdots\!72\)\( T^{5} + \)\(26\!\cdots\!44\)\( T^{6} - \)\(87\!\cdots\!68\)\( T^{7} + \)\(26\!\cdots\!22\)\( T^{8} - \)\(76\!\cdots\!76\)\( T^{9} + \)\(20\!\cdots\!96\)\( T^{10} - \)\(54\!\cdots\!24\)\( T^{11} + \)\(13\!\cdots\!22\)\( T^{12} - \)\(31\!\cdots\!32\)\( T^{13} + \)\(70\!\cdots\!44\)\( T^{14} - \)\(13\!\cdots\!28\)\( T^{15} + \)\(25\!\cdots\!57\)\( T^{16} - \)\(38\!\cdots\!20\)\( T^{17} + \)\(51\!\cdots\!66\)\( T^{18} - \)\(48\!\cdots\!28\)\( T^{19} + \)\(34\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 - 17408 T + 3107203610 T^{2} - 59824183442672 T^{3} + 5726081070022748105 T^{4} - \)\(10\!\cdots\!64\)\( T^{5} + \)\(76\!\cdots\!32\)\( T^{6} - \)\(13\!\cdots\!44\)\( T^{7} + \)\(84\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!80\)\( T^{9} + \)\(77\!\cdots\!00\)\( T^{10} - \)\(11\!\cdots\!80\)\( T^{11} + \)\(60\!\cdots\!70\)\( T^{12} - \)\(78\!\cdots\!44\)\( T^{13} + \)\(38\!\cdots\!32\)\( T^{14} - \)\(44\!\cdots\!64\)\( T^{15} + \)\(20\!\cdots\!05\)\( T^{16} - \)\(18\!\cdots\!72\)\( T^{17} + \)\(80\!\cdots\!10\)\( T^{18} - \)\(38\!\cdots\!08\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 + 2021 T + 3389673908 T^{2} - 95129046044495 T^{3} + 5951903574650911354 T^{4} - \)\(26\!\cdots\!45\)\( T^{5} + \)\(15\!\cdots\!46\)\( T^{6} - \)\(38\!\cdots\!75\)\( T^{7} + \)\(27\!\cdots\!93\)\( T^{8} - \)\(78\!\cdots\!74\)\( T^{9} + \)\(36\!\cdots\!28\)\( T^{10} - \)\(10\!\cdots\!18\)\( T^{11} + \)\(50\!\cdots\!57\)\( T^{12} - \)\(94\!\cdots\!25\)\( T^{13} + \)\(51\!\cdots\!46\)\( T^{14} - \)\(11\!\cdots\!15\)\( T^{15} + \)\(36\!\cdots\!46\)\( T^{16} - \)\(77\!\cdots\!85\)\( T^{17} + \)\(37\!\cdots\!08\)\( T^{18} + \)\(30\!\cdots\!47\)\( T^{19} + \)\(20\!\cdots\!49\)\( T^{20} \)
$71$ \( 1 - 11286 T + 12540359910 T^{2} - 44404438984650 T^{3} + 71277357843109037437 T^{4} + \)\(33\!\cdots\!36\)\( T^{5} + \)\(24\!\cdots\!24\)\( T^{6} + \)\(30\!\cdots\!96\)\( T^{7} + \)\(61\!\cdots\!22\)\( T^{8} + \)\(10\!\cdots\!68\)\( T^{9} + \)\(12\!\cdots\!84\)\( T^{10} + \)\(18\!\cdots\!68\)\( T^{11} + \)\(20\!\cdots\!22\)\( T^{12} + \)\(17\!\cdots\!96\)\( T^{13} + \)\(26\!\cdots\!24\)\( T^{14} + \)\(63\!\cdots\!36\)\( T^{15} + \)\(24\!\cdots\!37\)\( T^{16} - \)\(27\!\cdots\!50\)\( T^{17} + \)\(14\!\cdots\!10\)\( T^{18} - \)\(22\!\cdots\!86\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \)
$73$ \( 1 - 49892 T + 9676771082 T^{2} - 435365481214596 T^{3} + 51788415419782211185 T^{4} - \)\(22\!\cdots\!88\)\( T^{5} + \)\(19\!\cdots\!88\)\( T^{6} - \)\(79\!\cdots\!24\)\( T^{7} + \)\(57\!\cdots\!14\)\( T^{8} - \)\(21\!\cdots\!32\)\( T^{9} + \)\(13\!\cdots\!60\)\( T^{10} - \)\(44\!\cdots\!76\)\( T^{11} + \)\(24\!\cdots\!86\)\( T^{12} - \)\(71\!\cdots\!68\)\( T^{13} + \)\(36\!\cdots\!88\)\( T^{14} - \)\(85\!\cdots\!84\)\( T^{15} + \)\(41\!\cdots\!65\)\( T^{16} - \)\(71\!\cdots\!72\)\( T^{17} + \)\(33\!\cdots\!82\)\( T^{18} - \)\(35\!\cdots\!56\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} \)
$79$ \( 1 + 91524 T + 16857154847 T^{2} + 842558519651240 T^{3} + 99200238774687370926 T^{4} + \)\(23\!\cdots\!36\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} + \)\(59\!\cdots\!56\)\( T^{7} + \)\(16\!\cdots\!17\)\( T^{8} + \)\(38\!\cdots\!04\)\( T^{9} + \)\(63\!\cdots\!18\)\( T^{10} + \)\(11\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!17\)\( T^{12} + \)\(17\!\cdots\!44\)\( T^{13} + \)\(33\!\cdots\!90\)\( T^{14} + \)\(65\!\cdots\!64\)\( T^{15} + \)\(84\!\cdots\!26\)\( T^{16} + \)\(22\!\cdots\!60\)\( T^{17} + \)\(13\!\cdots\!47\)\( T^{18} + \)\(22\!\cdots\!76\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} \)
$83$ \( 1 + 105203 T + 28621889044 T^{2} + 2443253098090515 T^{3} + \)\(38\!\cdots\!90\)\( T^{4} + \)\(27\!\cdots\!61\)\( T^{5} + \)\(31\!\cdots\!66\)\( T^{6} + \)\(19\!\cdots\!11\)\( T^{7} + \)\(18\!\cdots\!57\)\( T^{8} + \)\(99\!\cdots\!22\)\( T^{9} + \)\(83\!\cdots\!32\)\( T^{10} + \)\(39\!\cdots\!46\)\( T^{11} + \)\(29\!\cdots\!93\)\( T^{12} + \)\(11\!\cdots\!77\)\( T^{13} + \)\(76\!\cdots\!66\)\( T^{14} + \)\(25\!\cdots\!23\)\( T^{15} + \)\(14\!\cdots\!10\)\( T^{16} + \)\(35\!\cdots\!05\)\( T^{17} + \)\(16\!\cdots\!44\)\( T^{18} + \)\(24\!\cdots\!29\)\( T^{19} + \)\(89\!\cdots\!49\)\( T^{20} \)
$89$ \( 1 + 62682 T + 34400806898 T^{2} + 1591902274833594 T^{3} + \)\(57\!\cdots\!17\)\( T^{4} + \)\(20\!\cdots\!96\)\( T^{5} + \)\(63\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!12\)\( T^{7} + \)\(50\!\cdots\!94\)\( T^{8} + \)\(10\!\cdots\!68\)\( T^{9} + \)\(31\!\cdots\!32\)\( T^{10} + \)\(60\!\cdots\!32\)\( T^{11} + \)\(15\!\cdots\!94\)\( T^{12} + \)\(29\!\cdots\!88\)\( T^{13} + \)\(61\!\cdots\!68\)\( T^{14} + \)\(10\!\cdots\!04\)\( T^{15} + \)\(17\!\cdots\!17\)\( T^{16} + \)\(26\!\cdots\!06\)\( T^{17} + \)\(32\!\cdots\!98\)\( T^{18} + \)\(33\!\cdots\!18\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} \)
$97$ \( 1 - 108383 T + 55022053811 T^{2} - 4727913772355050 T^{3} + \)\(13\!\cdots\!17\)\( T^{4} - \)\(99\!\cdots\!31\)\( T^{5} + \)\(22\!\cdots\!35\)\( T^{6} - \)\(13\!\cdots\!96\)\( T^{7} + \)\(26\!\cdots\!83\)\( T^{8} - \)\(14\!\cdots\!33\)\( T^{9} + \)\(25\!\cdots\!28\)\( T^{10} - \)\(12\!\cdots\!81\)\( T^{11} + \)\(19\!\cdots\!67\)\( T^{12} - \)\(88\!\cdots\!28\)\( T^{13} + \)\(12\!\cdots\!35\)\( T^{14} - \)\(46\!\cdots\!67\)\( T^{15} + \)\(55\!\cdots\!33\)\( T^{16} - \)\(16\!\cdots\!50\)\( T^{17} + \)\(16\!\cdots\!11\)\( T^{18} - \)\(27\!\cdots\!31\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} \)
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