Properties

Label 43.6.a.a
Level 43
Weight 6
Character orbit 43.a
Self dual yes
Analytic conductor 6.897
Analytic rank 1
Dimension 8
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{1} ) q^{2} + ( -2 - \beta_{1} + \beta_{4} - \beta_{7} ) q^{3} + ( 15 - 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{4} + ( -28 - \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{5} + ( -5 - 10 \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{6} + ( -10 - 13 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{7} + ( -82 + 7 \beta_{1} + 5 \beta_{2} - \beta_{3} + 8 \beta_{4} + \beta_{5} + 8 \beta_{6} - 6 \beta_{7} ) q^{8} + ( 62 - 8 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} - 15 \beta_{4} + 8 \beta_{5} + \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -2 + \beta_{1} ) q^{2} + ( -2 - \beta_{1} + \beta_{4} - \beta_{7} ) q^{3} + ( 15 - 2 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{4} + ( -28 - \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{5} + ( -5 - 10 \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{6} + ( -10 - 13 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{7} + ( -82 + 7 \beta_{1} + 5 \beta_{2} - \beta_{3} + 8 \beta_{4} + \beta_{5} + 8 \beta_{6} - 6 \beta_{7} ) q^{8} + ( 62 - 8 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} - 15 \beta_{4} + 8 \beta_{5} + \beta_{6} - \beta_{7} ) q^{9} + ( -59 - 35 \beta_{1} - 13 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} - 12 \beta_{7} ) q^{10} + ( -74 + 28 \beta_{1} + 19 \beta_{2} - 9 \beta_{3} - 2 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{11} + ( -499 + 10 \beta_{1} + 19 \beta_{3} + 52 \beta_{4} - 3 \beta_{5} + 7 \beta_{6} - 20 \beta_{7} ) q^{12} + ( -292 - 4 \beta_{1} + \beta_{2} + 13 \beta_{3} + 24 \beta_{4} + 2 \beta_{5} - 19 \beta_{6} - \beta_{7} ) q^{13} + ( -534 + 24 \beta_{1} - 8 \beta_{2} - 24 \beta_{3} + 36 \beta_{4} - 18 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{14} + ( -297 + 90 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} - 54 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + 33 \beta_{7} ) q^{15} + ( 226 - 53 \beta_{1} - 9 \beta_{2} - 15 \beta_{3} - 88 \beta_{4} - 29 \beta_{5} - 36 \beta_{6} + 22 \beta_{7} ) q^{16} + ( -419 + 109 \beta_{1} - 17 \beta_{2} - 12 \beta_{3} - 67 \beta_{4} - 8 \beta_{5} + 9 \beta_{6} + 32 \beta_{7} ) q^{17} + ( -513 + 182 \beta_{1} + 71 \beta_{2} + 40 \beta_{3} + 50 \beta_{4} + 12 \beta_{5} + 63 \beta_{6} - 10 \beta_{7} ) q^{18} + ( -284 - 17 \beta_{1} - 105 \beta_{2} - 62 \beta_{3} - 9 \beta_{4} + 64 \beta_{5} + 9 \beta_{6} + 38 \beta_{7} ) q^{19} + ( -275 - 33 \beta_{1} + 5 \beta_{2} - 40 \beta_{3} + 92 \beta_{4} - 16 \beta_{5} - 15 \beta_{6} - 6 \beta_{7} ) q^{20} + ( -351 + 189 \beta_{1} + 65 \beta_{2} + 87 \beta_{3} - 4 \beta_{4} + 24 \beta_{5} + 30 \beta_{6} + 23 \beta_{7} ) q^{21} + ( 1542 - 84 \beta_{1} + 107 \beta_{2} + 37 \beta_{3} - 8 \beta_{4} + 57 \beta_{5} + 22 \beta_{6} - 76 \beta_{7} ) q^{22} + ( -329 - 2 \beta_{1} - 21 \beta_{2} + 112 \beta_{3} - 125 \beta_{4} - 70 \beta_{5} - 26 \beta_{6} + 20 \beta_{7} ) q^{23} + ( 2531 - 488 \beta_{1} - 190 \beta_{2} - 103 \beta_{3} - 268 \beta_{4} + 25 \beta_{5} - 109 \beta_{6} + 90 \beta_{7} ) q^{24} + ( 750 - 41 \beta_{1} + 158 \beta_{2} - 21 \beta_{3} + 221 \beta_{4} + 34 \beta_{5} - 26 \beta_{6} - 115 \beta_{7} ) q^{25} + ( 830 - 394 \beta_{1} - 55 \beta_{2} - 65 \beta_{3} + 168 \beta_{4} - 29 \beta_{5} - 54 \beta_{6} - 44 \beta_{7} ) q^{26} + ( -1216 + 125 \beta_{1} - 98 \beta_{2} - 74 \beta_{3} + 150 \beta_{4} + 6 \beta_{5} + 93 \beta_{6} + 28 \beta_{7} ) q^{27} + ( 2668 - 678 \beta_{1} - 102 \beta_{2} + 50 \beta_{3} - 56 \beta_{4} - 30 \beta_{5} - 108 \beta_{6} - 40 \beta_{7} ) q^{28} + ( -492 + 103 \beta_{1} + 52 \beta_{2} + 54 \beta_{3} + 19 \beta_{4} - 122 \beta_{5} + 12 \beta_{6} - 139 \beta_{7} ) q^{29} + ( 3014 + 135 \beta_{1} + 65 \beta_{2} - 19 \beta_{3} - 76 \beta_{4} + 137 \beta_{5} + 176 \beta_{6} - 40 \beta_{7} ) q^{30} + ( 631 + 137 \beta_{1} + 155 \beta_{2} - 140 \beta_{3} - 83 \beta_{4} - 202 \beta_{5} + 39 \beta_{6} + 56 \beta_{7} ) q^{31} + ( -1964 + 241 \beta_{1} + 177 \beta_{2} + 247 \beta_{3} + 692 \beta_{4} - 49 \beta_{5} + 108 \beta_{6} - 204 \beta_{7} ) q^{32} + ( -1582 - 61 \beta_{1} - 161 \beta_{2} - 34 \beta_{3} + 164 \beta_{4} - 118 \beta_{5} - 146 \beta_{6} + 133 \beta_{7} ) q^{33} + ( 3749 - 92 \beta_{1} + 61 \beta_{2} + 128 \beta_{3} - 82 \beta_{4} + 172 \beta_{5} + 193 \beta_{6} + 6 \beta_{7} ) q^{34} + ( 323 + 539 \beta_{1} + 13 \beta_{2} - 323 \beta_{3} - 60 \beta_{4} + 56 \beta_{5} + 72 \beta_{6} - 29 \beta_{7} ) q^{35} + ( 7982 - 33 \beta_{1} - 5 \beta_{2} - 120 \beta_{3} - 1070 \beta_{4} - 91 \beta_{5} - 265 \beta_{6} + 467 \beta_{7} ) q^{36} + ( -176 - 422 \beta_{1} - 307 \beta_{2} + 114 \beta_{3} + 217 \beta_{4} + 280 \beta_{5} + 98 \beta_{6} - 398 \beta_{7} ) q^{37} + ( -739 - 223 \beta_{1} - 375 \beta_{2} + 378 \beta_{3} + 122 \beta_{4} + 142 \beta_{5} - 61 \beta_{6} + 74 \beta_{7} ) q^{38} + ( 778 + 1023 \beta_{1} + 445 \beta_{2} + 242 \beta_{3} - 418 \beta_{4} - 2 \beta_{5} - 30 \beta_{6} + 477 \beta_{7} ) q^{39} + ( 2189 - 149 \beta_{1} + 201 \beta_{2} + 84 \beta_{3} + 336 \beta_{4} - 104 \beta_{5} - 247 \beta_{6} + 110 \beta_{7} ) q^{40} + ( -1208 - 295 \beta_{1} - 274 \beta_{2} - 173 \beta_{3} - \beta_{4} + 138 \beta_{5} - 214 \beta_{6} - 245 \beta_{7} ) q^{41} + ( 8790 + 576 \beta_{1} - 56 \beta_{2} - 642 \beta_{3} - 1140 \beta_{4} + 150 \beta_{5} + 22 \beta_{6} + 336 \beta_{7} ) q^{42} -1849 q^{43} + ( -1275 + 1667 \beta_{1} + 73 \beta_{2} - 253 \beta_{3} - 538 \beta_{4} - 82 \beta_{5} + 260 \beta_{6} + 279 \beta_{7} ) q^{44} + ( -5010 - 232 \beta_{1} + 49 \beta_{2} + 404 \beta_{3} + 546 \beta_{4} - 26 \beta_{5} - 437 \beta_{6} - 137 \beta_{7} ) q^{45} + ( -2729 + 407 \beta_{1} + 120 \beta_{2} - 465 \beta_{3} + 350 \beta_{4} - 277 \beta_{5} + 441 \beta_{6} + 230 \beta_{7} ) q^{46} + ( -9863 + 439 \beta_{1} + 8 \beta_{2} + 193 \beta_{3} - 25 \beta_{4} - 300 \beta_{5} + 108 \beta_{6} - 99 \beta_{7} ) q^{47} + ( -15453 + 3208 \beta_{1} + 510 \beta_{2} + 349 \beta_{3} + 1916 \beta_{4} - 171 \beta_{5} + 811 \beta_{6} - 470 \beta_{7} ) q^{48} + ( 1696 - 1383 \beta_{1} - 61 \beta_{2} + 205 \beta_{3} - 254 \beta_{4} + 516 \beta_{5} + 162 \beta_{6} - 741 \beta_{7} ) q^{49} + ( 3741 - 263 \beta_{1} + 362 \beta_{2} - 189 \beta_{3} - 210 \beta_{4} - 37 \beta_{5} - 525 \beta_{6} - 278 \beta_{7} ) q^{50} + ( -9866 - 398 \beta_{1} - 80 \beta_{2} - 54 \beta_{3} + 269 \beta_{4} - 482 \beta_{5} - 21 \beta_{6} + 419 \beta_{7} ) q^{51} + ( -6723 - 1067 \beta_{1} - 485 \beta_{2} + 97 \beta_{3} + 670 \beta_{4} - 500 \beta_{5} + 4 \beta_{6} - 481 \beta_{7} ) q^{52} + ( -6787 - 1303 \beta_{1} - 48 \beta_{2} - 100 \beta_{3} + 568 \beta_{4} + 406 \beta_{5} - 423 \beta_{6} - 348 \beta_{7} ) q^{53} + ( 8084 - 2755 \beta_{1} - 1229 \beta_{2} + 465 \beta_{3} - 1512 \beta_{4} + 237 \beta_{5} - 906 \beta_{6} + 552 \beta_{7} ) q^{54} + ( -5326 - 2591 \beta_{1} - 1005 \beta_{2} + 516 \beta_{3} - 308 \beta_{4} + 322 \beta_{5} + 244 \beta_{6} - 99 \beta_{7} ) q^{55} + ( -17948 + 1818 \beta_{1} + 426 \beta_{2} + 698 \beta_{3} + 1608 \beta_{4} - 414 \beta_{5} + 220 \beta_{6} - 248 \beta_{7} ) q^{56} + ( -112 - 17 \beta_{1} + 342 \beta_{2} - 1038 \beta_{3} - 84 \beta_{4} + 350 \beta_{5} + 617 \beta_{6} - 910 \beta_{7} ) q^{57} + ( 6943 - 1424 \beta_{1} + 422 \beta_{2} - 283 \beta_{3} - 662 \beta_{4} - 353 \beta_{5} - 31 \beta_{6} + 744 \beta_{7} ) q^{58} + ( -3531 + 815 \beta_{1} + 1163 \beta_{2} - 445 \beta_{3} + 226 \beta_{4} + 728 \beta_{5} + 388 \beta_{6} + 179 \beta_{7} ) q^{59} + ( 11054 + 2135 \beta_{1} + 275 \beta_{2} - 729 \beta_{3} - 1132 \beta_{4} + 423 \beta_{5} + 98 \beta_{6} - 40 \beta_{7} ) q^{60} + ( -8856 - 1315 \beta_{1} + 1067 \beta_{2} - 296 \beta_{3} + 1356 \beta_{4} - 646 \beta_{5} + 204 \beta_{6} + 161 \beta_{7} ) q^{61} + ( 339 - 746 \beta_{1} + 689 \beta_{2} + 1106 \beta_{3} + 422 \beta_{4} + 482 \beta_{5} + 151 \beta_{6} - 1330 \beta_{7} ) q^{62} + ( -7702 - 877 \beta_{1} - 97 \beta_{2} + 484 \beta_{3} - 894 \beta_{4} - 1414 \beta_{5} - 1076 \beta_{6} + 1639 \beta_{7} ) q^{63} + ( 19272 - 3883 \beta_{1} - 1823 \beta_{2} - 1393 \beta_{3} - 1860 \beta_{4} + 395 \beta_{5} - 1500 \beta_{6} + 1328 \beta_{7} ) q^{64} + ( -2522 + 211 \beta_{1} - 1247 \beta_{2} - 796 \beta_{3} - 2214 \beta_{4} + 1154 \beta_{5} + 1134 \beta_{6} - 355 \beta_{7} ) q^{65} + ( -2634 - 4458 \beta_{1} - 1444 \beta_{2} + 880 \beta_{3} + 2476 \beta_{4} - 6 \beta_{5} - 776 \beta_{6} - 1066 \beta_{7} ) q^{66} + ( 3110 + 50 \beta_{1} + 1093 \beta_{2} + 77 \beta_{3} - 948 \beta_{4} - 874 \beta_{5} + 159 \beta_{6} + 1283 \beta_{7} ) q^{67} + ( 2796 + 3405 \beta_{1} + 429 \beta_{2} - 976 \beta_{3} - 1054 \beta_{4} + 153 \beta_{5} - 119 \beta_{6} + 259 \beta_{7} ) q^{68} + ( -12360 + 1856 \beta_{1} + 963 \beta_{2} + 1728 \beta_{3} + 142 \beta_{4} - 578 \beta_{5} + 449 \beta_{6} + 1377 \beta_{7} ) q^{69} + ( 23722 - 286 \beta_{1} + 894 \beta_{2} + 1712 \beta_{3} - 844 \beta_{4} + 1196 \beta_{5} + 194 \beta_{6} - 244 \beta_{7} ) q^{70} + ( -2506 - 838 \beta_{1} - 1682 \beta_{2} + 816 \beta_{3} - 1340 \beta_{4} + 1644 \beta_{5} + 612 \beta_{6} + 1198 \beta_{7} ) q^{71} + ( -24007 + 8416 \beta_{1} + 1064 \beta_{2} + 359 \beta_{3} + 5912 \beta_{4} + 661 \beta_{5} + 2231 \beta_{6} - 3924 \beta_{7} ) q^{72} + ( 780 + 1677 \beta_{1} - 65 \beta_{2} - 280 \beta_{3} - 908 \beta_{4} - 714 \beta_{5} - 32 \beta_{6} - 463 \beta_{7} ) q^{73} + ( -4821 + 174 \beta_{1} - 746 \beta_{2} - 1433 \beta_{3} - 2774 \beta_{4} - 1473 \beta_{5} - 693 \beta_{6} + 4022 \beta_{7} ) q^{74} + ( 21510 - 1282 \beta_{1} - 787 \beta_{2} - 276 \beta_{3} - 134 \beta_{4} + 472 \beta_{5} - 1295 \beta_{6} - 243 \beta_{7} ) q^{75} + ( 9 + 1007 \beta_{1} + 645 \beta_{2} - 58 \beta_{3} - 304 \beta_{4} - 3112 \beta_{5} - 961 \beta_{6} + 1040 \beta_{7} ) q^{76} + ( -28556 + 1371 \beta_{1} - 907 \beta_{2} - 1714 \beta_{3} + 544 \beta_{4} - 1842 \beta_{5} - 328 \beta_{6} + 935 \beta_{7} ) q^{77} + ( 29672 + 6678 \beta_{1} + 1148 \beta_{2} - 1386 \beta_{3} - 800 \beta_{4} + 1936 \beta_{5} + 1698 \beta_{6} - 2326 \beta_{7} ) q^{78} + ( 19609 + 1226 \beta_{1} + 60 \beta_{2} - 639 \beta_{3} + 1839 \beta_{4} + 1676 \beta_{5} + 195 \beta_{6} + 275 \beta_{7} ) q^{79} + ( 437 + 843 \beta_{1} - 671 \beta_{2} + 1140 \beta_{3} + 232 \beta_{4} + 512 \beta_{5} - 383 \beta_{6} - 2314 \beta_{7} ) q^{80} + ( 20406 - 1448 \beta_{1} + 249 \beta_{2} - 467 \beta_{3} - 1522 \beta_{4} - 1210 \beta_{5} - 701 \beta_{6} + 493 \beta_{7} ) q^{81} + ( -3185 - 2105 \beta_{1} + 1014 \beta_{2} + 1005 \beta_{3} + 3514 \beta_{4} - 575 \beta_{5} + 541 \beta_{6} + 402 \beta_{7} ) q^{82} + ( -7731 - 1607 \beta_{1} - 502 \beta_{2} + 884 \beta_{3} + 384 \beta_{4} - 2078 \beta_{5} - 1885 \beta_{6} - 546 \beta_{7} ) q^{83} + ( 530 + 9052 \beta_{1} + 2280 \beta_{2} + 1138 \beta_{3} + 3888 \beta_{4} + 1858 \beta_{5} + 3314 \beta_{6} - 3924 \beta_{7} ) q^{84} + ( 23712 - 4392 \beta_{1} + 151 \beta_{2} + 1274 \beta_{3} + 3908 \beta_{4} + 666 \beta_{5} - 2305 \beta_{6} - 4121 \beta_{7} ) q^{85} + ( 3698 - 1849 \beta_{1} ) q^{86} + ( 33093 - 2474 \beta_{1} - 1040 \beta_{2} - 639 \beta_{3} - 1871 \beta_{4} - 80 \beta_{5} - 327 \beta_{6} + 1337 \beta_{7} ) q^{87} + ( 10698 + 3934 \beta_{1} - 2452 \beta_{2} + 444 \beta_{3} - 3908 \beta_{4} + 898 \beta_{5} + 640 \beta_{6} + 2618 \beta_{7} ) q^{88} + ( -35082 + 4015 \beta_{1} + 1847 \beta_{2} + 1528 \beta_{3} - 430 \beta_{4} - 1342 \beta_{5} + 1516 \beta_{6} + 1507 \beta_{7} ) q^{89} + ( 10914 - 7473 \beta_{1} - 963 \beta_{2} - 2121 \beta_{3} + 3652 \beta_{4} - 1291 \beta_{5} - 1150 \beta_{6} - 338 \beta_{7} ) q^{90} + ( 48192 + 2775 \beta_{1} - 3 \beta_{2} - 2070 \beta_{3} - 1056 \beta_{4} + 894 \beta_{5} + 2052 \beta_{6} - 2601 \beta_{7} ) q^{91} + ( 27760 - 8902 \beta_{1} - 1838 \beta_{2} - 571 \beta_{3} - 814 \beta_{4} + 3880 \beta_{5} - 1837 \beta_{6} - 1795 \beta_{7} ) q^{92} + ( -17226 + 3476 \beta_{1} - 1948 \beta_{2} - 1312 \beta_{3} + 1783 \beta_{4} - 638 \beta_{5} + 25 \beta_{6} - 829 \beta_{7} ) q^{93} + ( 34811 - 11390 \beta_{1} - 482 \beta_{2} - 773 \beta_{3} - 2710 \beta_{4} - 437 \beta_{5} - 309 \beta_{6} + 1914 \beta_{7} ) q^{94} + ( 20120 - 270 \beta_{1} + 3451 \beta_{2} + 2330 \beta_{3} + 2375 \beta_{4} + 2092 \beta_{5} - 1150 \beta_{6} - 550 \beta_{7} ) q^{95} + ( 117271 - 11172 \beta_{1} - 966 \beta_{2} - 247 \beta_{3} - 15020 \beta_{4} + 1109 \beta_{5} - 4989 \beta_{6} + 5918 \beta_{7} ) q^{96} + ( 22025 - 1322 \beta_{1} + 1891 \beta_{2} + 1116 \beta_{3} + 4143 \beta_{4} + 2578 \beta_{5} - 760 \beta_{6} - 2298 \beta_{7} ) q^{97} + ( -42998 + 8381 \beta_{1} + 3356 \beta_{2} - 3104 \beta_{3} - 2328 \beta_{4} - 2820 \beta_{5} + 1888 \beta_{6} + 4712 \beta_{7} ) q^{98} + ( 23821 + 7573 \beta_{1} + 202 \beta_{2} + 1668 \beta_{3} - 3870 \beta_{4} + 1126 \beta_{5} + 2105 \beta_{6} + 1884 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 12q^{2} - 26q^{3} + 122q^{4} - 212q^{5} - 69q^{6} - 136q^{7} - 666q^{8} + 546q^{9} + O(q^{10}) \) \( 8q - 12q^{2} - 26q^{3} + 122q^{4} - 212q^{5} - 69q^{6} - 136q^{7} - 666q^{8} + 546q^{9} - 617q^{10} - 532q^{11} - 4195q^{12} - 2492q^{13} - 4240q^{14} - 1780q^{15} + 1882q^{16} - 2534q^{17} - 3711q^{18} - 1678q^{19} - 2607q^{20} - 2256q^{21} + 11502q^{22} - 2488q^{23} + 19953q^{24} + 4378q^{25} + 4586q^{26} - 8960q^{27} + 18640q^{28} - 4360q^{29} + 25092q^{30} + 5704q^{31} - 18294q^{32} - 12852q^{33} + 30007q^{34} + 5640q^{35} + 67969q^{36} - 3772q^{37} - 6559q^{38} + 11120q^{39} + 14869q^{40} - 10698q^{41} + 78698q^{42} - 14792q^{43} - 356q^{44} - 44912q^{45} - 19389q^{46} - 77864q^{47} - 118727q^{48} + 7188q^{49} + 26877q^{50} - 80246q^{51} - 60736q^{52} - 62352q^{53} + 61026q^{54} - 49552q^{55} - 144528q^{56} - 808q^{57} + 52951q^{58} - 26224q^{59} + 101500q^{60} - 82540q^{61} - 9023q^{62} - 61768q^{63} + 153858q^{64} - 5000q^{65} - 48516q^{66} + 27784q^{67} + 40507q^{68} - 93776q^{69} + 185910q^{70} - 9504q^{71} - 186687q^{72} + 14260q^{73} - 15239q^{74} + 167420q^{75} + 1279q^{76} - 218140q^{77} + 264170q^{78} + 160248q^{79} - 1291q^{80} + 161076q^{81} - 47781q^{82} - 77176q^{83} + 16382q^{84} + 141096q^{85} + 22188q^{86} + 268136q^{87} + 129544q^{88} - 265692q^{89} + 48990q^{90} + 401148q^{91} + 190391q^{92} - 123860q^{93} + 248737q^{94} + 135884q^{95} + 950817q^{96} + 144742q^{97} - 292244q^{98} + 239516q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 8125 \nu^{7} + 510320 \nu^{6} - 2147429 \nu^{5} - 76756386 \nu^{4} + 85006462 \nu^{3} + 2758106176 \nu^{2} - 206135304 \nu - 20471935488 \)\()/ 547265856 \)
\(\beta_{3}\)\(=\)\((\)\( 12949 \nu^{7} - 681148 \nu^{6} + 2601727 \nu^{5} + 80176782 \nu^{4} - 378607946 \nu^{3} - 1837549088 \nu^{2} + 7674423864 \nu + 1709840448 \)\()/ 547265856 \)
\(\beta_{4}\)\(=\)\((\)\( -17485 \nu^{7} + 269956 \nu^{6} + 1611305 \nu^{5} - 37491046 \nu^{4} + 6237658 \nu^{3} + 1274965584 \nu^{2} - 1052986520 \nu - 6710235008 \)\()/ 364843904 \)
\(\beta_{5}\)\(=\)\((\)\( 38033 \nu^{7} - 126518 \nu^{6} - 5886547 \nu^{5} + 8711358 \nu^{4} + 252968474 \nu^{3} + 152252096 \nu^{2} - 3214801896 \nu - 7712512416 \)\()/ 273632928 \)
\(\beta_{6}\)\(=\)\((\)\( -83131 \nu^{7} + 19780 \nu^{6} + 13177295 \nu^{5} + 14631678 \nu^{4} - 470063386 \nu^{3} - 1263726976 \nu^{2} + 1067316504 \nu + 17642049216 \)\()/ 547265856 \)
\(\beta_{7}\)\(=\)\((\)\( -128521 \nu^{7} + 1062904 \nu^{6} + 16607009 \nu^{5} - 129895854 \nu^{4} - 487223974 \nu^{3} + 4067658416 \nu^{2} + 2176112520 \nu - 28238112000 \)\()/ 547265856 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{5} - 2 \beta_{4} + 2 \beta_{1} + 43\)
\(\nu^{3}\)\(=\)\(8 \beta_{6} + 7 \beta_{5} - 4 \beta_{4} - \beta_{3} + 5 \beta_{2} + 71 \beta_{1} + 56\)
\(\nu^{4}\)\(=\)\(94 \beta_{7} + 28 \beta_{6} + 99 \beta_{5} - 264 \beta_{4} - 23 \beta_{3} + 31 \beta_{2} + 307 \beta_{1} + 3114\)
\(\nu^{5}\)\(=\)\(48 \beta_{7} + 1092 \beta_{6} + 869 \beta_{5} - 924 \beta_{4} - 71 \beta_{3} + 927 \beta_{2} + 6567 \beta_{1} + 9672\)
\(\nu^{6}\)\(=\)\(8760 \beta_{7} + 5444 \beta_{6} + 10339 \beta_{5} - 29780 \beta_{4} - 3425 \beta_{3} + 6801 \beta_{2} + 40661 \beta_{1} + 285312\)
\(\nu^{7}\)\(=\)\(11036 \beta_{7} + 127500 \beta_{6} + 102849 \beta_{5} - 146996 \beta_{4} - 10463 \beta_{3} + 125743 \beta_{2} + 685627 \beta_{1} + 1391004\)

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
−9.16809
−6.09504
−5.06235
−2.08717
2.58275
5.65705
7.21373
10.9591
−11.1681 −28.1764 92.7262 −10.1483 314.677 135.849 −678.196 550.912 113.337
1.2 −8.09504 11.1803 33.5297 −63.3756 −90.5052 223.489 −12.3830 −118.000 513.028
1.3 −7.06235 −9.19190 17.8767 73.4416 64.9164 −4.24720 99.7434 −158.509 −518.670
1.4 −4.08717 25.6605 −15.2950 −61.4284 −104.879 −184.774 193.303 415.460 251.069
1.5 0.582753 3.05838 −31.6604 27.7074 1.78228 −103.690 −37.0983 −233.646 16.1466
1.6 3.65705 7.84314 −18.6260 −107.102 28.6827 −25.5214 −185.142 −181.485 −391.677
1.7 5.21373 −11.2683 −4.81697 −9.80186 −58.7500 −11.1041 −191.954 −116.025 −51.1043
1.8 8.95911 −25.1057 48.2657 −61.2927 −224.924 −166.001 145.726 387.294 −549.129
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.a 8
3.b odd 2 1 387.6.a.c 8
4.b odd 2 1 688.6.a.e 8
5.b even 2 1 1075.6.a.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.a 8 1.a even 1 1 trivial
387.6.a.c 8 3.b odd 2 1
688.6.a.e 8 4.b odd 2 1
1075.6.a.a 8 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 \(T_{2}^{8} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(43))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 12 T + 139 T^{2} + 1186 T^{3} + 9566 T^{4} + 66832 T^{5} + 428248 T^{6} + 2656128 T^{7} + 15088832 T^{8} + 84996096 T^{9} + 438525952 T^{10} + 2189950976 T^{11} + 10030678016 T^{12} + 39795556352 T^{13} + 149250113536 T^{14} + 412316860416 T^{15} + 1099511627776 T^{16} \)
$3$ \( 1 + 26 T + 1037 T^{2} + 21984 T^{3} + 515401 T^{4} + 8844604 T^{5} + 164541593 T^{6} + 2412229350 T^{7} + 42389674344 T^{8} + 586171732050 T^{9} + 9716016525057 T^{10} + 126910400247828 T^{11} + 1797092167059801 T^{12} + 18626792789994912 T^{13} + 213509103982151013 T^{14} + 1300820172573992382 T^{15} + 12157665459056928801 T^{16} \)
$5$ \( 1 + 212 T + 32783 T^{2} + 3621372 T^{3} + 337987365 T^{4} + 26716499536 T^{5} + 1886184834567 T^{6} + 120009344570848 T^{7} + 7002693082424752 T^{8} + 375029201783900000 T^{9} + 18419773775068359375 T^{10} + \)\(81\!\cdots\!00\)\( T^{11} + \)\(32\!\cdots\!25\)\( T^{12} + \)\(10\!\cdots\!00\)\( T^{13} + \)\(30\!\cdots\!75\)\( T^{14} + \)\(61\!\cdots\!00\)\( T^{15} + \)\(90\!\cdots\!25\)\( T^{16} \)
$7$ \( 1 + 136 T + 72882 T^{2} + 6236384 T^{3} + 2273171472 T^{4} + 116085882896 T^{5} + 47825461769326 T^{6} + 1608183672345720 T^{7} + 855930962462754846 T^{8} + 27028742981114516040 T^{9} + \)\(13\!\cdots\!74\)\( T^{10} + \)\(55\!\cdots\!28\)\( T^{11} + \)\(18\!\cdots\!72\)\( T^{12} + \)\(83\!\cdots\!88\)\( T^{13} + \)\(16\!\cdots\!18\)\( T^{14} + \)\(51\!\cdots\!48\)\( T^{15} + \)\(63\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 + 532 T + 730390 T^{2} + 277407524 T^{3} + 233648426873 T^{4} + 75202699287360 T^{5} + 55193097116019702 T^{6} + 16538641578891295312 T^{7} + \)\(10\!\cdots\!12\)\( T^{8} + \)\(26\!\cdots\!12\)\( T^{9} + \)\(14\!\cdots\!02\)\( T^{10} + \)\(31\!\cdots\!60\)\( T^{11} + \)\(15\!\cdots\!73\)\( T^{12} + \)\(30\!\cdots\!24\)\( T^{13} + \)\(12\!\cdots\!90\)\( T^{14} + \)\(14\!\cdots\!32\)\( T^{15} + \)\(45\!\cdots\!01\)\( T^{16} \)
$13$ \( 1 + 2492 T + 4165454 T^{2} + 5101254988 T^{3} + 5076664946873 T^{4} + 4290072329984560 T^{5} + 3197308617811643622 T^{6} + \)\(21\!\cdots\!28\)\( T^{7} + \)\(13\!\cdots\!48\)\( T^{8} + \)\(80\!\cdots\!04\)\( T^{9} + \)\(44\!\cdots\!78\)\( T^{10} + \)\(21\!\cdots\!20\)\( T^{11} + \)\(96\!\cdots\!73\)\( T^{12} + \)\(35\!\cdots\!84\)\( T^{13} + \)\(10\!\cdots\!46\)\( T^{14} + \)\(24\!\cdots\!44\)\( T^{15} + \)\(36\!\cdots\!01\)\( T^{16} \)
$17$ \( 1 + 2534 T + 10088213 T^{2} + 18871627772 T^{3} + 43872695037958 T^{4} + 64773449524240486 T^{5} + \)\(11\!\cdots\!72\)\( T^{6} + \)\(13\!\cdots\!98\)\( T^{7} + \)\(19\!\cdots\!73\)\( T^{8} + \)\(19\!\cdots\!86\)\( T^{9} + \)\(22\!\cdots\!28\)\( T^{10} + \)\(18\!\cdots\!98\)\( T^{11} + \)\(17\!\cdots\!58\)\( T^{12} + \)\(10\!\cdots\!04\)\( T^{13} + \)\(82\!\cdots\!37\)\( T^{14} + \)\(29\!\cdots\!62\)\( T^{15} + \)\(16\!\cdots\!01\)\( T^{16} \)
$19$ \( 1 + 1678 T + 4805907 T^{2} + 3688923072 T^{3} + 6298623964783 T^{4} + 4799890940703308 T^{5} + 9212340221857328609 T^{6} + \)\(43\!\cdots\!14\)\( T^{7} + \)\(59\!\cdots\!68\)\( T^{8} + \)\(10\!\cdots\!86\)\( T^{9} + \)\(56\!\cdots\!09\)\( T^{10} + \)\(72\!\cdots\!92\)\( T^{11} + \)\(23\!\cdots\!83\)\( T^{12} + \)\(34\!\cdots\!28\)\( T^{13} + \)\(11\!\cdots\!07\)\( T^{14} + \)\(95\!\cdots\!22\)\( T^{15} + \)\(14\!\cdots\!01\)\( T^{16} \)
$23$ \( 1 + 2488 T + 19807447 T^{2} + 41115937228 T^{3} + 232966325434474 T^{4} + 412893749849663568 T^{5} + \)\(20\!\cdots\!08\)\( T^{6} + \)\(32\!\cdots\!88\)\( T^{7} + \)\(14\!\cdots\!09\)\( T^{8} + \)\(20\!\cdots\!84\)\( T^{9} + \)\(85\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!76\)\( T^{11} + \)\(39\!\cdots\!74\)\( T^{12} + \)\(45\!\cdots\!04\)\( T^{13} + \)\(14\!\cdots\!03\)\( T^{14} + \)\(11\!\cdots\!16\)\( T^{15} + \)\(29\!\cdots\!01\)\( T^{16} \)
$29$ \( 1 + 4360 T + 123420923 T^{2} + 388928438620 T^{3} + 6671975392676125 T^{4} + 14971808484524322944 T^{5} + \)\(21\!\cdots\!99\)\( T^{6} + \)\(36\!\cdots\!76\)\( T^{7} + \)\(51\!\cdots\!68\)\( T^{8} + \)\(75\!\cdots\!24\)\( T^{9} + \)\(91\!\cdots\!99\)\( T^{10} + \)\(12\!\cdots\!56\)\( T^{11} + \)\(11\!\cdots\!25\)\( T^{12} + \)\(14\!\cdots\!80\)\( T^{13} + \)\(91\!\cdots\!23\)\( T^{14} + \)\(66\!\cdots\!40\)\( T^{15} + \)\(31\!\cdots\!01\)\( T^{16} \)
$31$ \( 1 - 5704 T + 154537635 T^{2} - 536298509268 T^{3} + 10208569817532730 T^{4} - 23021651844205871336 T^{5} + \)\(44\!\cdots\!56\)\( T^{6} - \)\(75\!\cdots\!68\)\( T^{7} + \)\(14\!\cdots\!33\)\( T^{8} - \)\(21\!\cdots\!68\)\( T^{9} + \)\(36\!\cdots\!56\)\( T^{10} - \)\(54\!\cdots\!36\)\( T^{11} + \)\(68\!\cdots\!30\)\( T^{12} - \)\(10\!\cdots\!68\)\( T^{13} + \)\(85\!\cdots\!35\)\( T^{14} - \)\(89\!\cdots\!04\)\( T^{15} + \)\(45\!\cdots\!01\)\( T^{16} \)
$37$ \( 1 + 3772 T + 191073025 T^{2} - 279875011868 T^{3} + 17131267855690943 T^{4} - 65662976081645242692 T^{5} + \)\(16\!\cdots\!59\)\( T^{6} - \)\(36\!\cdots\!12\)\( T^{7} + \)\(13\!\cdots\!56\)\( T^{8} - \)\(25\!\cdots\!84\)\( T^{9} + \)\(80\!\cdots\!91\)\( T^{10} - \)\(21\!\cdots\!56\)\( T^{11} + \)\(39\!\cdots\!43\)\( T^{12} - \)\(44\!\cdots\!76\)\( T^{13} + \)\(21\!\cdots\!25\)\( T^{14} + \)\(29\!\cdots\!96\)\( T^{15} + \)\(53\!\cdots\!01\)\( T^{16} \)
$41$ \( 1 + 10698 T + 718082789 T^{2} + 6825101409152 T^{3} + 244397855377030766 T^{4} + \)\(20\!\cdots\!98\)\( T^{5} + \)\(51\!\cdots\!60\)\( T^{6} + \)\(36\!\cdots\!02\)\( T^{7} + \)\(71\!\cdots\!21\)\( T^{8} + \)\(42\!\cdots\!02\)\( T^{9} + \)\(68\!\cdots\!60\)\( T^{10} + \)\(31\!\cdots\!98\)\( T^{11} + \)\(44\!\cdots\!66\)\( T^{12} + \)\(14\!\cdots\!52\)\( T^{13} + \)\(17\!\cdots\!89\)\( T^{14} + \)\(29\!\cdots\!98\)\( T^{15} + \)\(32\!\cdots\!01\)\( T^{16} \)
$43$ \( ( 1 + 1849 T )^{8} \)
$47$ \( 1 + 77864 T + 4209868025 T^{2} + 159292711218168 T^{3} + 4925605431015049275 T^{4} + \)\(12\!\cdots\!88\)\( T^{5} + \)\(26\!\cdots\!31\)\( T^{6} + \)\(50\!\cdots\!80\)\( T^{7} + \)\(81\!\cdots\!28\)\( T^{8} + \)\(11\!\cdots\!60\)\( T^{9} + \)\(14\!\cdots\!19\)\( T^{10} + \)\(14\!\cdots\!84\)\( T^{11} + \)\(13\!\cdots\!75\)\( T^{12} + \)\(10\!\cdots\!76\)\( T^{13} + \)\(61\!\cdots\!25\)\( T^{14} + \)\(25\!\cdots\!52\)\( T^{15} + \)\(76\!\cdots\!01\)\( T^{16} \)
$53$ \( 1 + 62352 T + 4104997514 T^{2} + 164980080105492 T^{3} + 6411159574721863281 T^{4} + \)\(19\!\cdots\!48\)\( T^{5} + \)\(54\!\cdots\!26\)\( T^{6} + \)\(12\!\cdots\!52\)\( T^{7} + \)\(28\!\cdots\!76\)\( T^{8} + \)\(53\!\cdots\!36\)\( T^{9} + \)\(95\!\cdots\!74\)\( T^{10} + \)\(13\!\cdots\!36\)\( T^{11} + \)\(19\!\cdots\!81\)\( T^{12} + \)\(21\!\cdots\!56\)\( T^{13} + \)\(21\!\cdots\!86\)\( T^{14} + \)\(13\!\cdots\!64\)\( T^{15} + \)\(93\!\cdots\!01\)\( T^{16} \)
$59$ \( 1 + 26224 T + 2774515012 T^{2} + 45419810356528 T^{3} + 3251157373342627972 T^{4} + \)\(32\!\cdots\!68\)\( T^{5} + \)\(23\!\cdots\!96\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!70\)\( T^{8} + \)\(79\!\cdots\!60\)\( T^{9} + \)\(12\!\cdots\!96\)\( T^{10} + \)\(12\!\cdots\!32\)\( T^{11} + \)\(84\!\cdots\!72\)\( T^{12} + \)\(84\!\cdots\!72\)\( T^{13} + \)\(37\!\cdots\!12\)\( T^{14} + \)\(25\!\cdots\!76\)\( T^{15} + \)\(68\!\cdots\!01\)\( T^{16} \)
$61$ \( 1 + 82540 T + 7066815930 T^{2} + 341557091503436 T^{3} + 16786861393791028176 T^{4} + \)\(57\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!70\)\( T^{6} + \)\(60\!\cdots\!16\)\( T^{7} + \)\(19\!\cdots\!30\)\( T^{8} + \)\(50\!\cdots\!16\)\( T^{9} + \)\(15\!\cdots\!70\)\( T^{10} + \)\(34\!\cdots\!44\)\( T^{11} + \)\(85\!\cdots\!76\)\( T^{12} + \)\(14\!\cdots\!36\)\( T^{13} + \)\(25\!\cdots\!30\)\( T^{14} + \)\(25\!\cdots\!40\)\( T^{15} + \)\(25\!\cdots\!01\)\( T^{16} \)
$67$ \( 1 - 27784 T + 7986133086 T^{2} - 191439161437872 T^{3} + 30490262918267386393 T^{4} - \)\(62\!\cdots\!04\)\( T^{5} + \)\(72\!\cdots\!82\)\( T^{6} - \)\(12\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!76\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(13\!\cdots\!18\)\( T^{10} - \)\(15\!\cdots\!72\)\( T^{11} + \)\(10\!\cdots\!93\)\( T^{12} - \)\(85\!\cdots\!04\)\( T^{13} + \)\(48\!\cdots\!14\)\( T^{14} - \)\(22\!\cdots\!12\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \)
$71$ \( 1 + 9504 T + 6761837728 T^{2} + 123767778948544 T^{3} + 23139305175095415788 T^{4} + \)\(61\!\cdots\!44\)\( T^{5} + \)\(57\!\cdots\!36\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!86\)\( T^{8} + \)\(31\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!36\)\( T^{10} + \)\(35\!\cdots\!44\)\( T^{11} + \)\(24\!\cdots\!88\)\( T^{12} + \)\(23\!\cdots\!44\)\( T^{13} + \)\(23\!\cdots\!28\)\( T^{14} + \)\(59\!\cdots\!04\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \)
$73$ \( 1 - 14260 T + 14183423290 T^{2} - 183318998319396 T^{3} + 92012288750177195552 T^{4} - \)\(10\!\cdots\!48\)\( T^{5} + \)\(35\!\cdots\!10\)\( T^{6} - \)\(34\!\cdots\!56\)\( T^{7} + \)\(90\!\cdots\!14\)\( T^{8} - \)\(72\!\cdots\!08\)\( T^{9} + \)\(15\!\cdots\!90\)\( T^{10} - \)\(93\!\cdots\!36\)\( T^{11} + \)\(16\!\cdots\!52\)\( T^{12} - \)\(70\!\cdots\!28\)\( T^{13} + \)\(11\!\cdots\!10\)\( T^{14} - \)\(23\!\cdots\!20\)\( T^{15} + \)\(34\!\cdots\!01\)\( T^{16} \)
$79$ \( 1 - 160248 T + 28127571285 T^{2} - 2813803255473368 T^{3} + \)\(29\!\cdots\!07\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!27\)\( T^{6} - \)\(10\!\cdots\!60\)\( T^{7} + \)\(62\!\cdots\!72\)\( T^{8} - \)\(31\!\cdots\!40\)\( T^{9} + \)\(15\!\cdots\!27\)\( T^{10} - \)\(63\!\cdots\!00\)\( T^{11} + \)\(26\!\cdots\!07\)\( T^{12} - \)\(77\!\cdots\!32\)\( T^{13} + \)\(23\!\cdots\!85\)\( T^{14} - \)\(41\!\cdots\!52\)\( T^{15} + \)\(80\!\cdots\!01\)\( T^{16} \)
$83$ \( 1 + 77176 T + 20087963970 T^{2} + 1576267525049076 T^{3} + \)\(20\!\cdots\!93\)\( T^{4} + \)\(15\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!78\)\( T^{6} + \)\(91\!\cdots\!20\)\( T^{7} + \)\(61\!\cdots\!64\)\( T^{8} + \)\(35\!\cdots\!60\)\( T^{9} + \)\(20\!\cdots\!22\)\( T^{10} + \)\(93\!\cdots\!88\)\( T^{11} + \)\(48\!\cdots\!93\)\( T^{12} + \)\(14\!\cdots\!68\)\( T^{13} + \)\(75\!\cdots\!30\)\( T^{14} + \)\(11\!\cdots\!32\)\( T^{15} + \)\(57\!\cdots\!01\)\( T^{16} \)
$89$ \( 1 + 265692 T + 60287015686 T^{2} + 9508315264777636 T^{3} + \)\(13\!\cdots\!36\)\( T^{4} + \)\(15\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!78\)\( T^{6} + \)\(13\!\cdots\!44\)\( T^{7} + \)\(10\!\cdots\!34\)\( T^{8} + \)\(76\!\cdots\!56\)\( T^{9} + \)\(48\!\cdots\!78\)\( T^{10} + \)\(26\!\cdots\!40\)\( T^{11} + \)\(12\!\cdots\!36\)\( T^{12} + \)\(51\!\cdots\!64\)\( T^{13} + \)\(18\!\cdots\!86\)\( T^{14} + \)\(44\!\cdots\!08\)\( T^{15} + \)\(94\!\cdots\!01\)\( T^{16} \)
$97$ \( 1 - 144742 T + 53143560149 T^{2} - 6952055211960972 T^{3} + \)\(13\!\cdots\!46\)\( T^{4} - \)\(15\!\cdots\!66\)\( T^{5} + \)\(20\!\cdots\!88\)\( T^{6} - \)\(20\!\cdots\!62\)\( T^{7} + \)\(21\!\cdots\!53\)\( T^{8} - \)\(17\!\cdots\!34\)\( T^{9} + \)\(15\!\cdots\!12\)\( T^{10} - \)\(96\!\cdots\!38\)\( T^{11} + \)\(73\!\cdots\!46\)\( T^{12} - \)\(32\!\cdots\!04\)\( T^{13} + \)\(21\!\cdots\!01\)\( T^{14} - \)\(49\!\cdots\!06\)\( T^{15} + \)\(29\!\cdots\!01\)\( T^{16} \)
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