Properties

Label 91.10.a.a
Level $91$
Weight $10$
Character orbit 91.a
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + 13724467264 x^{5} + 1698856105344 x^{4} - 3404524011264 x^{3} - 154369782114304 x^{2} + 70325953652224 x + 170905444356096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 + \beta_{1} ) q^{2} + ( -27 - \beta_{1} + \beta_{3} ) q^{3} + ( 246 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( -432 - 8 \beta_{1} + \beta_{5} ) q^{5} + ( -814 - 41 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{6} + 2401 q^{7} + ( -2165 + 200 \beta_{1} - \beta_{2} - 17 \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} + ( 6504 + 185 \beta_{1} + 15 \beta_{2} - 25 \beta_{3} + 3 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( -2 + \beta_{1} ) q^{2} + ( -27 - \beta_{1} + \beta_{3} ) q^{3} + ( 246 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( -432 - 8 \beta_{1} + \beta_{5} ) q^{5} + ( -814 - 41 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{6} + 2401 q^{7} + ( -2165 + 200 \beta_{1} - \beta_{2} - 17 \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{8} + ( 6504 + 185 \beta_{1} + 15 \beta_{2} - 25 \beta_{3} + 3 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{9} + ( -5374 - 382 \beta_{1} - 5 \beta_{2} - 17 \beta_{3} + \beta_{4} - 13 \beta_{5} + 9 \beta_{6} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{10} + ( -6774 + 477 \beta_{1} - 23 \beta_{2} - 21 \beta_{3} + 6 \beta_{4} - 19 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - 5 \beta_{9} - 3 \beta_{11} ) q^{11} + ( -15155 - 1244 \beta_{1} - 106 \beta_{2} + 376 \beta_{3} + 12 \beta_{4} - 10 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} + 15 \beta_{8} - 2 \beta_{9} - \beta_{10} - 6 \beta_{11} ) q^{12} + 28561 q^{13} + ( -4802 + 2401 \beta_{1} ) q^{14} + ( 13483 + 209 \beta_{1} - 73 \beta_{2} - 564 \beta_{3} + 29 \beta_{4} - 20 \beta_{5} - 3 \beta_{6} + 16 \beta_{7} - 5 \beta_{8} + 4 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{15} + ( 32757 - 1036 \beta_{1} + 86 \beta_{2} - 1038 \beta_{3} + 16 \beta_{4} - 90 \beta_{5} + 9 \beta_{6} + 15 \beta_{7} - 19 \beta_{8} + 8 \beta_{9} + 19 \beta_{10} + 10 \beta_{11} ) q^{16} + ( -124405 + 1140 \beta_{1} - 234 \beta_{2} - 818 \beta_{3} - 19 \beta_{4} - 26 \beta_{5} + 4 \beta_{6} - 45 \beta_{7} - 22 \beta_{8} - 4 \beta_{9} + 11 \beta_{10} + 4 \beta_{11} ) q^{17} + ( 123951 + 13437 \beta_{1} + 246 \beta_{2} - 1565 \beta_{3} - 18 \beta_{4} - 190 \beta_{5} + 2 \beta_{6} - 5 \beta_{7} - 54 \beta_{8} - \beta_{9} - 37 \beta_{10} + 76 \beta_{11} ) q^{18} + ( -7973 - 2421 \beta_{1} - 225 \beta_{2} - 1234 \beta_{3} - 96 \beta_{4} - 55 \beta_{5} + 51 \beta_{6} + 17 \beta_{7} - 10 \beta_{8} + 61 \beta_{9} - 38 \beta_{10} + 27 \beta_{11} ) q^{19} + ( -50837 - 2901 \beta_{1} - 763 \beta_{2} - 1016 \beta_{3} + 294 \beta_{5} - 61 \beta_{6} + 51 \beta_{7} + 19 \beta_{8} + 62 \beta_{9} + 27 \beta_{10} - 38 \beta_{11} ) q^{20} + ( -64827 - 2401 \beta_{1} + 2401 \beta_{3} ) q^{21} + ( 391334 - 17735 \beta_{1} + 141 \beta_{2} - 2589 \beta_{3} + 29 \beta_{4} + 417 \beta_{5} - 189 \beta_{6} + 120 \beta_{7} - 53 \beta_{8} - 23 \beta_{9} + 92 \beta_{10} - 61 \beta_{11} ) q^{22} + ( -276706 - 13837 \beta_{1} - 1158 \beta_{2} - 1429 \beta_{3} - 42 \beta_{4} + 341 \beta_{5} - 42 \beta_{6} - 78 \beta_{7} + 11 \beta_{8} - 175 \beta_{9} - 64 \beta_{10} - 46 \beta_{11} ) q^{23} + ( -483849 - 46726 \beta_{1} - 1918 \beta_{2} - 3400 \beta_{3} - 188 \beta_{4} + 476 \beta_{5} + 97 \beta_{6} - 175 \beta_{7} + 249 \beta_{8} - 102 \beta_{9} + 129 \beta_{10} - 80 \beta_{11} ) q^{24} + ( -375 - 14575 \beta_{1} + 221 \beta_{2} + 366 \beta_{3} - 167 \beta_{4} - 771 \beta_{5} + 123 \beta_{6} + 54 \beta_{7} + 74 \beta_{8} + 53 \beta_{9} - 215 \beta_{10} + 91 \beta_{11} ) q^{25} + ( -57122 + 28561 \beta_{1} ) q^{26} + ( -668477 - 51830 \beta_{1} - 1495 \beta_{2} + 4897 \beta_{3} - 113 \beta_{4} + 222 \beta_{5} + 99 \beta_{6} + 50 \beta_{7} + 74 \beta_{8} + 211 \beta_{9} - 205 \beta_{10} - 213 \beta_{11} ) q^{27} + ( 590646 - 7203 \beta_{1} + 2401 \beta_{2} ) q^{28} + ( -1093103 - 61399 \beta_{1} - 1203 \beta_{2} - 6446 \beta_{3} + 43 \beta_{4} + 217 \beta_{5} + 307 \beta_{6} - 292 \beta_{7} + 261 \beta_{8} - 122 \beta_{9} + 169 \beta_{10} + 67 \beta_{11} ) q^{29} + ( 253153 - 12465 \beta_{1} + 2892 \beta_{2} - 22949 \beta_{3} + 213 \beta_{4} - 242 \beta_{5} + 260 \beta_{6} + 277 \beta_{7} - 300 \beta_{8} - 165 \beta_{9} + 93 \beta_{10} + 56 \beta_{11} ) q^{30} + ( -308407 - 74737 \beta_{1} + 4309 \beta_{2} - 7301 \beta_{3} + 43 \beta_{4} - 1308 \beta_{5} + 258 \beta_{6} - 377 \beta_{7} - 331 \beta_{8} - 103 \beta_{9} - \beta_{10} - \beta_{11} ) q^{31} + ( 372295 - 20142 \beta_{1} + 2726 \beta_{2} - 5472 \beta_{3} + 1028 \beta_{4} - 480 \beta_{5} - 647 \beta_{6} - 47 \beta_{7} + 337 \beta_{8} - 254 \beta_{9} + 353 \beta_{10} + 132 \beta_{11} ) q^{32} + ( -454846 - 88660 \beta_{1} + 1499 \beta_{2} - 20735 \beta_{3} - 803 \beta_{4} + 1990 \beta_{5} + 581 \beta_{6} + 82 \beta_{7} - 483 \beta_{8} - 510 \beta_{9} + 15 \beta_{10} - 367 \beta_{11} ) q^{33} + ( 1288777 - 216415 \beta_{1} + 541 \beta_{2} + 14734 \beta_{3} + 961 \beta_{4} + 1477 \beta_{5} - 621 \beta_{6} - 431 \beta_{7} + 963 \beta_{8} + 28 \beta_{9} + 237 \beta_{10} + 241 \beta_{11} ) q^{34} + ( -1037232 - 19208 \beta_{1} + 2401 \beta_{5} ) q^{35} + ( 6687341 + 147435 \beta_{1} + 20947 \beta_{2} - 15654 \beta_{3} + 832 \beta_{4} + 1140 \beta_{5} - 2189 \beta_{6} + 1737 \beta_{7} - 1937 \beta_{8} + 1540 \beta_{9} + 685 \beta_{10} + 656 \beta_{11} ) q^{36} + ( 44895 + 8876 \beta_{1} + 5655 \beta_{2} - 579 \beta_{3} + 592 \beta_{4} - 1068 \beta_{5} + 699 \beta_{6} + 1437 \beta_{7} - 1423 \beta_{8} + 1412 \beta_{9} - 656 \beta_{10} - 17 \beta_{11} ) q^{37} + ( -1584122 - 115084 \beta_{1} + 489 \beta_{2} + 58977 \beta_{3} + 1247 \beta_{4} - 583 \beta_{5} + 275 \beta_{6} - 1154 \beta_{7} + 1075 \beta_{8} + 867 \beta_{9} - 42 \beta_{10} - 105 \beta_{11} ) q^{38} + ( -771147 - 28561 \beta_{1} + 28561 \beta_{3} ) q^{39} + ( 1067100 - 192670 \beta_{1} - 4485 \beta_{2} + 43673 \beta_{3} - 524 \beta_{4} - 2461 \beta_{5} + 897 \beta_{6} - 1120 \beta_{7} + 1579 \beta_{8} - 1287 \beta_{9} - 862 \beta_{10} + 295 \beta_{11} ) q^{40} + ( -2221200 - 162523 \beta_{1} - 11234 \beta_{2} + 31317 \beta_{3} - 1825 \beta_{4} + 736 \beta_{5} - 3160 \beta_{6} + 141 \beta_{7} - 206 \beta_{8} + 408 \beta_{9} - 1951 \beta_{10} + 216 \beta_{11} ) q^{41} + ( -1954414 - 98441 \beta_{1} - 4802 \beta_{2} - 2401 \beta_{4} ) q^{42} + ( 2683357 - 235858 \beta_{1} + 4390 \beta_{2} - 45844 \beta_{3} - 2117 \beta_{4} - 7915 \beta_{5} + 936 \beta_{6} - 601 \beta_{7} + 898 \beta_{8} + 570 \beta_{9} + 1013 \beta_{10} + 2800 \beta_{11} ) q^{43} + ( -10503565 + 299574 \beta_{1} - 6844 \beta_{2} + 33132 \beta_{3} + 1048 \beta_{4} - 5962 \beta_{5} + 2497 \beta_{6} - 1443 \beta_{7} - 2023 \beta_{8} - 2010 \beta_{9} - 2531 \beta_{10} + 642 \beta_{11} ) q^{44} + ( -5066436 - 650259 \beta_{1} - 3717 \beta_{2} + 10052 \beta_{3} + 538 \beta_{4} - 8269 \beta_{5} + 2455 \beta_{6} - 405 \beta_{7} + 445 \beta_{8} - 1690 \beta_{9} + 426 \beta_{10} - 2353 \beta_{11} ) q^{45} + ( -9338116 - 709013 \beta_{1} - 22368 \beta_{2} + 49156 \beta_{3} + 4460 \beta_{4} + 2916 \beta_{5} - 2358 \beta_{6} - 230 \beta_{7} + 3802 \beta_{8} - 2372 \beta_{9} + 3774 \beta_{10} - 4236 \beta_{11} ) q^{46} + ( -1806853 - 166686 \beta_{1} + 23034 \beta_{2} + 48785 \beta_{3} - 1479 \beta_{4} - 13228 \beta_{5} - 1067 \beta_{6} + 2944 \beta_{7} - 2452 \beta_{8} - 2009 \beta_{9} + 961 \beta_{10} - 832 \beta_{11} ) q^{47} + ( -25679021 - 634374 \beta_{1} - 47048 \beta_{2} + 100762 \beta_{3} - 44 \beta_{4} + 7878 \beta_{5} + 7431 \beta_{6} - 2443 \beta_{7} + 1379 \beta_{8} - 2396 \beta_{9} - 1527 \beta_{10} - 2254 \beta_{11} ) q^{48} + 5764801 q^{49} + ( -10982979 + 41638 \beta_{1} - 4470 \beta_{2} + 68483 \beta_{3} - 544 \beta_{4} + 11762 \beta_{5} - 8486 \beta_{6} + 783 \beta_{7} - 894 \beta_{8} + 6147 \beta_{9} + 4683 \beta_{10} + 332 \beta_{11} ) q^{50} + ( -14820877 + 585202 \beta_{1} + 17194 \beta_{2} - 104180 \beta_{3} - 7084 \beta_{4} - 10270 \beta_{5} + 8978 \beta_{6} + 7664 \beta_{7} - 4613 \beta_{8} + 2797 \beta_{9} - 3522 \beta_{10} + 2122 \beta_{11} ) q^{51} + ( 7026006 - 85683 \beta_{1} + 28561 \beta_{2} ) q^{52} + ( -16476476 + 353911 \beta_{1} - 34145 \beta_{2} + 28458 \beta_{3} + 5281 \beta_{4} + 5601 \beta_{5} - 10535 \beta_{6} + 3952 \beta_{7} + 8330 \beta_{8} + 501 \beta_{9} + 2501 \beta_{10} - 1335 \beta_{11} ) q^{53} + ( -37676299 - 1402388 \beta_{1} - 71147 \beta_{2} + 96486 \beta_{3} - 11536 \beta_{4} + 6075 \beta_{5} + 15129 \beta_{6} - 4977 \beta_{7} + 6825 \beta_{8} - 2664 \beta_{9} - 3957 \beta_{10} - 3465 \beta_{11} ) q^{54} + ( -26054593 + 931464 \beta_{1} + 4870 \beta_{2} + 192032 \beta_{3} - 4701 \beta_{4} + 8310 \beta_{5} - 7698 \beta_{6} - 7039 \beta_{7} - 4818 \beta_{8} + 2034 \beta_{9} + 1541 \beta_{10} + 3464 \beta_{11} ) q^{55} + ( -5198165 + 480200 \beta_{1} - 2401 \beta_{2} - 40817 \beta_{3} + 2401 \beta_{5} + 2401 \beta_{7} - 4802 \beta_{8} + 2401 \beta_{9} - 2401 \beta_{10} + 2401 \beta_{11} ) q^{56} + ( -23333189 + 2022473 \beta_{1} + 17317 \beta_{2} - 108074 \beta_{3} - 6459 \beta_{4} - 1902 \beta_{5} - 5 \beta_{6} - 8660 \beta_{7} - 3229 \beta_{8} - 2154 \beta_{9} - 1377 \beta_{10} + 8707 \beta_{11} ) q^{57} + ( -43126977 - 1507519 \beta_{1} - 91423 \beta_{2} + 48452 \beta_{3} + 7678 \beta_{4} + 13807 \beta_{5} - 741 \beta_{6} - 4921 \beta_{7} + 4059 \beta_{8} + 1558 \beta_{9} + 175 \beta_{10} - 2185 \beta_{11} ) q^{58} + ( -18183374 + 964693 \beta_{1} + 22655 \beta_{2} - 156360 \beta_{3} + 1948 \beta_{4} + 11670 \beta_{5} - 2617 \beta_{6} - 7379 \beta_{7} + 8111 \beta_{8} - 5878 \beta_{9} + 7416 \beta_{10} + 2223 \beta_{11} ) q^{59} + ( -15249846 + 1855846 \beta_{1} + 85120 \beta_{2} + 15726 \beta_{3} + 13876 \beta_{4} + 14974 \beta_{5} - 5088 \beta_{6} + 3050 \beta_{7} - 10180 \beta_{8} + 18 \beta_{9} + 14 \beta_{10} + 4054 \beta_{11} ) q^{60} + ( -17991026 - 611099 \beta_{1} + 20135 \beta_{2} - 152561 \beta_{3} - 4597 \beta_{4} - 24667 \beta_{5} + 2808 \beta_{6} + 11101 \beta_{7} - 5112 \beta_{8} + 3932 \beta_{9} - 511 \beta_{10} - 10223 \beta_{11} ) q^{61} + ( -56538177 + 1911834 \beta_{1} - 46239 \beta_{2} - 211478 \beta_{3} + 6291 \beta_{4} + 47621 \beta_{5} - 13441 \beta_{6} + 8909 \beta_{7} - 8225 \beta_{8} + 10580 \beta_{9} - 139 \beta_{10} + 11113 \beta_{11} ) q^{62} + ( 15616104 + 444185 \beta_{1} + 36015 \beta_{2} - 60025 \beta_{3} + 7203 \beta_{4} - 2401 \beta_{5} - 9604 \beta_{6} - 2401 \beta_{7} + 2401 \beta_{8} + 2401 \beta_{9} + 7203 \beta_{10} + 2401 \beta_{11} ) q^{63} + ( -32771485 + 2312234 \beta_{1} - 35072 \beta_{2} - 146598 \beta_{3} - 8396 \beta_{4} + 33070 \beta_{5} - 15849 \beta_{6} + 1509 \beta_{7} - 7629 \beta_{8} - 3356 \beta_{9} - 5287 \beta_{10} + 3258 \beta_{11} ) q^{64} + ( -12338352 - 228488 \beta_{1} + 28561 \beta_{5} ) q^{65} + ( -65184683 + 839244 \beta_{1} - 127272 \beta_{2} + 581667 \beta_{3} + 42818 \beta_{4} + 12076 \beta_{5} + 6510 \beta_{6} + 3007 \beta_{7} + 8062 \beta_{8} - 9785 \beta_{9} - 8369 \beta_{10} - 14090 \beta_{11} ) q^{66} + ( 1109020 - 8903 \beta_{1} + 92437 \beta_{2} + 326865 \beta_{3} + 13607 \beta_{4} + 2029 \beta_{5} + 1962 \beta_{6} - 27327 \beta_{7} + 21446 \beta_{8} - 966 \beta_{9} + 7077 \beta_{10} - 8993 \beta_{11} ) q^{67} + ( -104046222 + 964782 \beta_{1} - 116372 \beta_{2} - 153166 \beta_{3} - 19288 \beta_{4} - 19458 \beta_{5} + 12132 \beta_{6} + 14362 \beta_{7} + 5184 \beta_{8} + 2062 \beta_{9} - 4218 \beta_{10} - 298 \beta_{11} ) q^{68} + ( -5838401 + 2752093 \beta_{1} + 148849 \beta_{2} - 963119 \beta_{3} - 7484 \beta_{4} - 30565 \beta_{5} + 23828 \beta_{6} + 38622 \beta_{7} - 22690 \beta_{8} + 2130 \beta_{9} - 14392 \beta_{10} - 999 \beta_{11} ) q^{69} + ( -12902974 - 917182 \beta_{1} - 12005 \beta_{2} - 40817 \beta_{3} + 2401 \beta_{4} - 31213 \beta_{5} + 21609 \beta_{6} - 4802 \beta_{7} + 2401 \beta_{8} - 7203 \beta_{9} - 4802 \beta_{10} - 7203 \beta_{11} ) q^{70} + ( -46109859 + 274769 \beta_{1} - 83285 \beta_{2} + 79406 \beta_{3} + 28261 \beta_{4} - 28972 \beta_{5} - 41241 \beta_{6} - 6276 \beta_{7} + 18559 \beta_{8} + 7200 \beta_{9} + 16499 \beta_{10} - 15391 \beta_{11} ) q^{71} + ( 28526910 + 9493118 \beta_{1} + 381119 \beta_{2} - 464671 \beta_{3} + 13612 \beta_{4} - 111997 \beta_{5} + 31351 \beta_{6} + 33854 \beta_{7} - 41487 \beta_{8} - 2355 \beta_{9} - 18192 \beta_{10} + 11647 \beta_{11} ) q^{72} + ( -28776206 - 449984 \beta_{1} - 31809 \beta_{2} - 550493 \beta_{3} - 3535 \beta_{4} - 11139 \beta_{5} + 21531 \beta_{6} - 14786 \beta_{7} + 35545 \beta_{8} + 6364 \beta_{9} + 7431 \beta_{10} - 2535 \beta_{11} ) q^{73} + ( 5886138 + 2426849 \beta_{1} + 270167 \beta_{2} - 1190925 \beta_{3} - 25549 \beta_{4} - 19185 \beta_{5} + 45583 \beta_{6} + 19708 \beta_{7} - 31601 \beta_{8} + 7297 \beta_{9} - 17728 \beta_{10} + 19137 \beta_{11} ) q^{74} + ( 26996959 + 2528598 \beta_{1} + 137275 \beta_{2} - 584261 \beta_{3} - 27767 \beta_{4} + 16164 \beta_{5} - 18159 \beta_{6} - 38032 \beta_{7} + 6062 \beta_{8} - 9017 \beta_{9} + 2829 \beta_{10} + 14345 \beta_{11} ) q^{75} + ( -85992307 - 1048517 \beta_{1} - 71087 \beta_{2} - 314696 \beta_{3} - 59152 \beta_{4} + 88130 \beta_{5} + 18441 \beta_{6} - 27239 \beta_{7} + 5817 \beta_{8} - 16870 \beta_{9} + 6833 \beta_{10} + 3070 \beta_{11} ) q^{76} + ( -16264374 + 1145277 \beta_{1} - 55223 \beta_{2} - 50421 \beta_{3} + 14406 \beta_{4} - 45619 \beta_{5} + 4802 \beta_{6} + 9604 \beta_{7} + 7203 \beta_{8} - 12005 \beta_{9} - 7203 \beta_{11} ) q^{77} + ( -23248654 - 1171001 \beta_{1} - 57122 \beta_{2} - 28561 \beta_{4} ) q^{78} + ( 79546989 + 1642999 \beta_{1} + 120356 \beta_{2} + 655971 \beta_{3} + 19298 \beta_{4} - 36203 \beta_{5} - 45474 \beta_{6} - 29508 \beta_{7} + 7202 \beta_{8} - 10254 \beta_{9} - 5778 \beta_{10} - 12044 \beta_{11} ) q^{79} + ( -124873682 + 10758 \beta_{1} + 25234 \beta_{2} + 1044040 \beta_{3} - 28700 \beta_{4} - 15088 \beta_{5} - 39538 \beta_{6} - 30058 \beta_{7} + 6918 \beta_{8} - 4452 \beta_{9} + 32102 \beta_{10} - 2216 \beta_{11} ) q^{80} + ( 71942374 + 4306726 \beta_{1} + 327392 \beta_{2} - 742828 \beta_{3} + 37162 \beta_{4} + 38798 \beta_{5} - 9610 \beta_{6} - 188 \beta_{7} - 16749 \beta_{8} - 18115 \beta_{9} - 22924 \beta_{10} + 7060 \beta_{11} ) q^{81} + ( -116574047 - 7481848 \beta_{1} - 90797 \beta_{2} + 1221240 \beta_{3} - 30146 \beta_{4} - 118415 \beta_{5} - 26409 \beta_{6} - 17973 \beta_{7} + 41655 \beta_{8} - 11258 \beta_{9} + 47327 \beta_{10} - 17367 \beta_{11} ) q^{82} + ( -168270277 - 3722972 \beta_{1} + 93850 \beta_{2} - 812714 \beta_{3} - 37548 \beta_{4} + 153901 \beta_{5} + 43182 \beta_{6} + 20414 \beta_{7} - 24305 \beta_{8} + 25417 \beta_{9} + 1998 \beta_{10} - 16034 \beta_{11} ) q^{83} + ( -36387155 - 2986844 \beta_{1} - 254506 \beta_{2} + 902776 \beta_{3} + 28812 \beta_{4} - 24010 \beta_{5} - 21609 \beta_{6} - 21609 \beta_{7} + 36015 \beta_{8} - 4802 \beta_{9} - 2401 \beta_{10} - 14406 \beta_{11} ) q^{84} + ( -35104005 + 4129617 \beta_{1} + 124867 \beta_{2} - 250386 \beta_{3} + 24681 \beta_{4} - 326858 \beta_{5} + 36519 \beta_{6} + 29994 \beta_{7} + 34105 \beta_{8} - 3166 \beta_{9} + 26239 \beta_{10} - 20959 \beta_{11} ) q^{85} + ( -179938511 + 4522325 \beta_{1} - 198332 \beta_{2} + 1668355 \beta_{3} + 81220 \beta_{4} - 9360 \beta_{5} - 141576 \beta_{6} - 16563 \beta_{7} - 15216 \beta_{8} + 71351 \beta_{9} + 27337 \beta_{10} + 39226 \beta_{11} ) q^{86} + ( -64338883 + 5623160 \beta_{1} + 214166 \beta_{2} - 1255502 \beta_{3} + 1923 \beta_{4} - 67716 \beta_{5} + 77360 \beta_{6} + 64085 \beta_{7} - 88670 \beta_{8} + 23622 \beta_{9} - 44199 \beta_{10} + 18320 \beta_{11} ) q^{87} + ( 47993939 - 4738522 \beta_{1} + 552712 \beta_{2} - 1032390 \beta_{3} - 33908 \beta_{4} + 90266 \beta_{5} - 24081 \beta_{6} - 1491 \beta_{7} + 14155 \beta_{8} + 63108 \beta_{9} + 57681 \beta_{10} + 26486 \beta_{11} ) q^{88} + ( -24278072 + 3032121 \beta_{1} - 86019 \beta_{2} + 337282 \beta_{3} - 72254 \beta_{4} + 68195 \beta_{5} - 70187 \beta_{6} - 41029 \beta_{7} - 11367 \beta_{8} - 9068 \beta_{9} - 19370 \beta_{10} + 54757 \beta_{11} ) q^{89} + ( -478979790 - 6235232 \beta_{1} - 1042574 \beta_{2} + 384552 \beta_{3} - 13063 \beta_{4} + 333044 \beta_{5} - 41586 \beta_{6} + 27234 \beta_{7} + 16566 \beta_{8} - 2600 \beta_{9} - 11210 \beta_{10} - 9404 \beta_{11} ) q^{90} + 68574961 q^{91} + ( -371256184 - 12007101 \beta_{1} - 1009245 \beta_{2} + 708404 \beta_{3} - 65656 \beta_{4} - 143148 \beta_{5} + 122814 \beta_{6} + 14858 \beta_{7} + 48246 \beta_{8} - 34760 \beta_{9} - 35774 \beta_{10} - 24244 \beta_{11} ) q^{92} + ( -142236288 - 7287951 \beta_{1} - 399958 \beta_{2} + 158583 \beta_{3} + 73820 \beta_{4} - 84440 \beta_{5} + 40052 \beta_{6} - 410 \beta_{7} + 77529 \beta_{8} - 2937 \beta_{9} - 15862 \beta_{10} - 36006 \beta_{11} ) q^{93} + ( -135259765 + 8173180 \beta_{1} - 272150 \beta_{2} + 1415849 \beta_{3} + 29801 \beta_{4} + 83152 \beta_{5} - 192906 \beta_{6} + 102293 \beta_{7} - 69274 \beta_{8} + 9009 \beta_{9} + 4769 \beta_{10} + 34446 \beta_{11} ) q^{94} + ( -106876871 - 410316 \beta_{1} + 482714 \beta_{2} - 394126 \beta_{3} - 14627 \beta_{4} + 330171 \beta_{5} + 36912 \beta_{6} - 46531 \beta_{7} - 31676 \beta_{8} - 73542 \beta_{9} - 3329 \beta_{10} + 4028 \beta_{11} ) q^{95} + ( -173943233 - 23683858 \beta_{1} - 205216 \beta_{2} + 2126078 \beta_{3} + 3500 \beta_{4} + 146522 \beta_{5} + 58039 \beta_{6} + 33505 \beta_{7} + 8363 \beta_{8} + 13360 \beta_{9} - 39683 \beta_{10} - 87898 \beta_{11} ) q^{96} + ( -177909038 + 5780446 \beta_{1} - 21237 \beta_{2} + 776129 \beta_{3} + 43661 \beta_{4} - 110973 \beta_{5} + 105927 \beta_{6} + 30608 \beta_{7} + 21631 \beta_{8} - 57434 \beta_{9} - 22493 \beta_{10} - 899 \beta_{11} ) q^{97} + ( -11529602 + 5764801 \beta_{1} ) q^{98} + ( -301333593 + 10000495 \beta_{1} - 581603 \beta_{2} - 2292664 \beta_{3} + 13235 \beta_{4} + 117840 \beta_{5} + 79975 \beta_{6} - 14314 \beta_{7} - 61347 \beta_{8} + 33794 \beta_{9} - 76807 \beta_{10} + 43503 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + 13724467264 x^{5} + 1698856105344 x^{4} - 3404524011264 x^{3} - 154369782114304 x^{2} + 70325953652224 x + 170905444356096\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 754 \)
\(\beta_{3}\)\(=\)\((\)\(28959119358379533950236039 \nu^{11} + 44813643235319440102637021 \nu^{10} - 128968740688594353361553988576 \nu^{9} - 106024704081776252608824929766 \nu^{8} + 209483272925449879527551404421732 \nu^{7} - 161680092804149587624267845505000 \nu^{6} - 152356821298278893211170884091920848 \nu^{5} + 357613458619353585854417378044259680 \nu^{4} + 46966743298983114045741004843514219456 \nu^{3} - 126274473565456795357541678015697924224 \nu^{2} - 4358858340130304226983124265704804464384 \nu + 2005615752447508085165491930072472598528\)\()/ \)\(33\!\cdots\!80\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-1229546043228316234214551 \nu^{11} - 31579496058787671353468429 \nu^{10} + 2822661547774168258827259488 \nu^{9} + 110978695645830003033126614406 \nu^{8} + 825946280706616553732377754332 \nu^{7} - 119598507214076975647281338359448 \nu^{6} - 4414643593126046241486133731659568 \nu^{5} + 49097672427071405727727510627059872 \nu^{4} + 2026932382712662975897386947447193664 \nu^{3} - 6620282006106866910343190082662750080 \nu^{2} - 152505698266172130170502492847828334848 \nu + 502550967550064922742251130323293030400\)\()/ \)\(55\!\cdots\!48\)\( \)
\(\beta_{5}\)\(=\)\((\)\(107501518699207524941105977 \nu^{11} - 287955302000261592559320397 \nu^{10} - 592209950860615301842756549968 \nu^{9} + 2106433881729141027341453885382 \nu^{8} + 1153745530317033734083994414297276 \nu^{7} - 4304551952339932911789454605909400 \nu^{6} - 956980158288443218264800857464381104 \nu^{5} + 3223437185347306982978292430650086560 \nu^{4} + 314150249816464146358756125664501011008 \nu^{3} - 618049548645598850584859447998883460992 \nu^{2} - 28473763563108957792101024356075514761472 \nu - 12469303178629367487541590579118253438976\)\()/ \)\(27\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(112150029100055927067458261 \nu^{11} - 1742899282993964572345028521 \nu^{10} - 505884589801419649050952593824 \nu^{9} + 6270517354452962171383796632526 \nu^{8} + 839578171422971887172742933748268 \nu^{7} - 8055825499117979349820750574351800 \nu^{6} - 617495355068433318461037426995196272 \nu^{5} + 4444826104686373102818193163229851680 \nu^{4} + 188524991555118377291380623509229351744 \nu^{3} - 933911399718861670636954029447285993856 \nu^{2} - 16339156634845108523319076421747726935296 \nu + 27133126428553561213418320715937814751232\)\()/ \)\(61\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-2041023485832654233649762719 \nu^{11} - 16549340152137427805330617141 \nu^{10} + 8895403231906357944103612868496 \nu^{9} + 76662448387990794702969752957046 \nu^{8} - 14034877411750363638812866693486372 \nu^{7} - 112819249874082576481762168497203800 \nu^{6} + 9851970689276045421630842561784942288 \nu^{5} + 64827875831093372423981061824298942880 \nu^{4} - 2939846640488186290120469559492123606976 \nu^{3} - 12849146172883181076185939609141557678976 \nu^{2} + 266007670228747402081370302650255701104384 \nu + 418891062669198402781296578196410652444672\)\()/ \)\(82\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-4763779993224170209392751031 \nu^{11} - 233874184580772978665443309 \nu^{10} + 22380996121388578423554619617504 \nu^{9} + 5424653891454522151280750797254 \nu^{8} - 38767891277514066415843098869483428 \nu^{7} + 1546771457841552243694930646609000 \nu^{6} + 30303491791603636615468088390921736912 \nu^{5} - 11131144638826022961475278649418480480 \nu^{4} - 10177914240858312666772666970672628389824 \nu^{3} + 3693231189678000957362859616517513320576 \nu^{2} + 1063332231927460104207784108728498650703616 \nu + 39333426728784333107144697891098669795328\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-5923199989776505127138730737 \nu^{11} + 112263785275744533522482896757 \nu^{10} + 25228291024548894031205573863008 \nu^{9} - 395360108835133383342666722805942 \nu^{8} - 39725048681996042014665014353482556 \nu^{7} + 481717337664296990076279149603898200 \nu^{6} + 28269836658663407122486742869229506224 \nu^{5} - 226762415101817552849887290523648441760 \nu^{4} - 8638478932672301513331165222374072707648 \nu^{3} + 29729716025767710070704645757836973101952 \nu^{2} + 761823937716723606547559800662870104088832 \nu + 351344235263709836042633608637961914873856\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(6332051864678522046966713 \nu^{11} + 34311699928680747572416582 \nu^{10} - 30121926154191952079392630917 \nu^{9} - 97868802083951174151453348192 \nu^{8} + 51901044664715114305259994950794 \nu^{7} + 46485541551405996262405542762400 \nu^{6} - 39504260436804758263939086590519976 \nu^{5} + 43285509788829923765417594335581440 \nu^{4} + 12378482588354749809370562804117011552 \nu^{3} - 25451297441922960260020137302436245248 \nu^{2} - 1053009792456149418619759823145977738368 \nu + 1066401275716760011955874207186025337856\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(4287799229010791141183580431 \nu^{11} - 20029377762188100156617615291 \nu^{10} - 17773569621954848879868299981904 \nu^{9} + 67073982518727722922624077368746 \nu^{8} + 26314231137313533603380183941369828 \nu^{7} - 97393566977883550905192346064220200 \nu^{6} - 16881734833970795773000272555889359312 \nu^{5} + 64420060371529629677012292853485968480 \nu^{4} + 4357318490023947549801113372234159141824 \nu^{3} - 14872757759494465324075478189586997326976 \nu^{2} - 306126740809198602454030966796397563881216 \nu + 382037438811819631502381494825538199631872\)\()/ \)\(82\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 754\)
\(\nu^{3}\)\(=\)\(\beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{5} - 17 \beta_{3} + 5 \beta_{2} + 1218 \beta_{1} + 319\)
\(\nu^{4}\)\(=\)\(18 \beta_{11} + 11 \beta_{10} + 16 \beta_{9} - 35 \beta_{8} + 23 \beta_{7} + 9 \beta_{6} - 82 \beta_{5} + 16 \beta_{4} - 1174 \beta_{3} + 1638 \beta_{2} + 4108 \beta_{1} + 919341\)
\(\nu^{5}\)\(=\)\(2320 \beta_{11} - 1545 \beta_{10} + 1914 \beta_{9} - 4029 \beta_{8} + 2191 \beta_{7} - 557 \beta_{6} + 708 \beta_{5} + 1188 \beta_{4} - 51348 \beta_{3} + 16938 \beta_{2} + 1692538 \beta_{1} + 2557937\)
\(\nu^{6}\)\(=\)\(55778 \beta_{11} + 23993 \beta_{10} + 39292 \beta_{9} - 102837 \beta_{8} + 64981 \beta_{7} - 33 \beta_{6} - 183754 \beta_{5} + 45860 \beta_{4} - 3352334 \beta_{3} + 2649920 \beta_{2} + 12840994 \beta_{1} + 1277966691\)
\(\nu^{7}\)\(=\)\(4457714 \beta_{11} - 2074921 \beta_{10} + 3248736 \beta_{9} - 7455887 \beta_{8} + 4273827 \beta_{7} - 1730347 \beta_{6} - 1288946 \beta_{5} + 3558228 \beta_{4} - 114920854 \beta_{3} + 42822344 \beta_{2} + 2554261954 \beta_{1} + 9019267565\)
\(\nu^{8}\)\(=\)\(130964522 \beta_{11} + 41125549 \beta_{10} + 82029132 \beta_{9} - 233995353 \beta_{8} + 148474009 \beta_{7} - 31978053 \beta_{6} - 343274322 \beta_{5} + 103371660 \beta_{4} - 7236395734 \beta_{3} + 4421731516 \beta_{2} + 33046605054 \beta_{1} + 1927600064999\)
\(\nu^{9}\)\(=\)\(8260490214 \beta_{11} - 2734686897 \beta_{10} + 5531449548 \beta_{9} - 13717491851 \beta_{8} + 8133064099 \beta_{7} - 3949217119 \beta_{6} - 6177381838 \beta_{5} + 7875047924 \beta_{4} - 236174040810 \beta_{3} + 96182267412 \beta_{2} + 4092816760146 \beta_{1} + 24056315139757\)
\(\nu^{10}\)\(=\)\(277436523322 \beta_{11} + 65113911469 \beta_{10} + 164667623548 \beta_{9} - 486291501609 \beta_{8} + 310342450681 \beta_{7} - 104520176021 \beta_{6} - 620277418178 \beta_{5} + 217262136828 \beta_{4} - 14267473359622 \beta_{3} + 7609640124260 \beta_{2} + 76006768096614 \beta_{1} + 3085524578364247\)
\(\nu^{11}\)\(=\)\(15265079776830 \beta_{11} - 3628033451641 \beta_{10} + 9648812818724 \beta_{9} - 25355854032779 \beta_{8} + 15351923348875 \beta_{7} - 8067923010927 \beta_{6} - 16394088690678 \beta_{5} + 15682871427796 \beta_{4} - 470551905184994 \beta_{3} + 202922858146252 \beta_{2} + 6858782180589698 \beta_{1} + 56114336356097541\)

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
−37.6810
−35.8265
−33.9466
−22.2119
−11.0043
−0.857959
1.29652
13.5332
24.5414
29.7766
31.8951
43.4855
−39.6810 −201.837 1062.58 764.411 8009.09 2401.00 −21847.5 21055.2 −30332.6
1.2 −37.8265 32.9997 918.841 −2427.98 −1248.26 2401.00 −15389.3 −18594.0 91841.7
1.3 −35.9466 230.724 780.160 434.116 −8293.74 2401.00 −9639.45 33550.4 −15605.0
1.4 −24.2119 6.29566 74.2177 2011.41 −152.430 2401.00 10599.6 −19643.4 −48700.2
1.5 −13.0043 −81.8707 −342.889 −1042.73 1064.67 2401.00 11117.2 −12980.2 13559.9
1.6 −2.85796 143.668 −503.832 −15.3125 −410.598 2401.00 2903.21 957.575 43.7625
1.7 −0.703483 −141.700 −511.505 −2244.71 99.6835 2401.00 720.018 395.868 1579.12
1.8 11.5332 −234.910 −378.986 −380.245 −2709.25 2401.00 −10275.9 35499.6 −4385.43
1.9 22.5414 66.2971 −3.88662 −391.554 1494.43 2401.00 −11628.8 −15287.7 −8826.17
1.10 27.7766 192.937 259.538 −1979.40 5359.14 2401.00 −7012.53 17541.9 −54980.9
1.11 29.8951 −70.1432 381.717 1113.83 −2096.94 2401.00 −3894.82 −14762.9 33298.0
1.12 41.4855 −265.461 1209.05 −1043.84 −11012.8 2401.00 28917.3 50786.7 −43304.2
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.a.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.a.a 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(15\!\cdots\!32\)\( T_{2}^{4} + \)\(98\!\cdots\!84\)\( T_{2}^{3} - \)\(13\!\cdots\!16\)\( T_{2}^{2} - \)\(53\!\cdots\!20\)\( T_{2} - \)\(30\!\cdots\!00\)\( \)">\(T_{2}^{12} + \cdots\) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(91))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -305889376665600 - 533602237317120 T - 134244814012416 T^{2} + 9858775209984 T^{3} + 1509453567232 T^{4} - 50455845888 T^{5} - 4925448672 T^{6} + 98462880 T^{7} + 6860708 T^{8} - 78246 T^{9} - 4324 T^{10} + 21 T^{11} + T^{12} \)
$3$ \( \)\(90\!\cdots\!08\)\( - \)\(15\!\cdots\!92\)\( T + \)\(15\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!48\)\( T^{3} - 375789502634239584 T^{4} - 22490946373830276 T^{5} - 15006998579788 T^{6} + 1506189409228 T^{7} + 3170934724 T^{8} - 38252043 T^{9} - 105193 T^{10} + 323 T^{11} + T^{12} \)
$5$ \( \)\(19\!\cdots\!00\)\( + \)\(13\!\cdots\!00\)\( T + \)\(43\!\cdots\!00\)\( T^{2} - \)\(83\!\cdots\!50\)\( T^{3} - \)\(35\!\cdots\!55\)\( T^{4} + \)\(11\!\cdots\!08\)\( T^{5} + 69166841593135853128 T^{6} + 31553605178314464 T^{7} - 38790052460370 T^{8} - 28792978620 T^{9} + 1837392 T^{10} + 5202 T^{11} + T^{12} \)
$7$ \( ( -2401 + T )^{12} \)
$11$ \( -\)\(90\!\cdots\!00\)\( - \)\(25\!\cdots\!80\)\( T + \)\(39\!\cdots\!72\)\( T^{2} + \)\(17\!\cdots\!80\)\( T^{3} - \)\(15\!\cdots\!00\)\( T^{4} - \)\(72\!\cdots\!40\)\( T^{5} + \)\(55\!\cdots\!60\)\( T^{6} + \)\(59\!\cdots\!48\)\( T^{7} + 39395823301413806964 T^{8} - 1294863367205769 T^{9} - 13684050197 T^{10} + 80061 T^{11} + T^{12} \)
$13$ \( ( -28561 + T )^{12} \)
$17$ \( -\)\(31\!\cdots\!32\)\( - \)\(14\!\cdots\!32\)\( T - \)\(16\!\cdots\!56\)\( T^{2} + \)\(51\!\cdots\!80\)\( T^{3} + \)\(24\!\cdots\!48\)\( T^{4} + \)\(18\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!64\)\( T^{6} - \)\(15\!\cdots\!08\)\( T^{7} - \)\(39\!\cdots\!44\)\( T^{8} - 658789072077948624 T^{9} + 258959176808 T^{10} + 1493598 T^{11} + T^{12} \)
$19$ \( \)\(88\!\cdots\!60\)\( + \)\(10\!\cdots\!64\)\( T - \)\(59\!\cdots\!08\)\( T^{2} - \)\(54\!\cdots\!42\)\( T^{3} + \)\(24\!\cdots\!13\)\( T^{4} + \)\(56\!\cdots\!68\)\( T^{5} - \)\(29\!\cdots\!68\)\( T^{6} - \)\(14\!\cdots\!68\)\( T^{7} + \)\(12\!\cdots\!82\)\( T^{8} - 19625574417079524 T^{9} - 1997133896228 T^{10} + 109038 T^{11} + T^{12} \)
$23$ \( -\)\(17\!\cdots\!00\)\( - \)\(31\!\cdots\!99\)\( T + \)\(72\!\cdots\!31\)\( T^{2} + \)\(14\!\cdots\!71\)\( T^{3} - \)\(71\!\cdots\!99\)\( T^{4} - \)\(20\!\cdots\!02\)\( T^{5} + \)\(93\!\cdots\!62\)\( T^{6} + \)\(12\!\cdots\!10\)\( T^{7} + \)\(17\!\cdots\!86\)\( T^{8} - 33859161793762723575 T^{9} - 7982701437005 T^{10} + 3367443 T^{11} + T^{12} \)
$29$ \( \)\(10\!\cdots\!64\)\( + \)\(36\!\cdots\!80\)\( T + \)\(40\!\cdots\!72\)\( T^{2} - \)\(23\!\cdots\!18\)\( T^{3} - \)\(36\!\cdots\!75\)\( T^{4} + \)\(15\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!92\)\( T^{6} + \)\(14\!\cdots\!00\)\( T^{7} - \)\(11\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!76\)\( T^{9} + 29378401154644 T^{10} + 13333098 T^{11} + T^{12} \)
$31$ \( -\)\(81\!\cdots\!00\)\( - \)\(16\!\cdots\!65\)\( T - \)\(19\!\cdots\!01\)\( T^{2} + \)\(62\!\cdots\!77\)\( T^{3} + \)\(92\!\cdots\!01\)\( T^{4} - \)\(73\!\cdots\!10\)\( T^{5} - \)\(13\!\cdots\!58\)\( T^{6} + \)\(36\!\cdots\!58\)\( T^{7} + \)\(70\!\cdots\!22\)\( T^{8} - \)\(72\!\cdots\!05\)\( T^{9} - 152845361931177 T^{10} + 3954765 T^{11} + T^{12} \)
$37$ \( \)\(75\!\cdots\!80\)\( - \)\(56\!\cdots\!68\)\( T - \)\(23\!\cdots\!84\)\( T^{2} + \)\(33\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!48\)\( T^{4} - \)\(41\!\cdots\!40\)\( T^{5} - \)\(29\!\cdots\!52\)\( T^{6} + \)\(37\!\cdots\!48\)\( T^{7} + \)\(24\!\cdots\!46\)\( T^{8} + \)\(72\!\cdots\!15\)\( T^{9} - 876679734084199 T^{10} - 580535 T^{11} + T^{12} \)
$41$ \( -\)\(15\!\cdots\!00\)\( - \)\(78\!\cdots\!28\)\( T + \)\(79\!\cdots\!32\)\( T^{2} + \)\(64\!\cdots\!16\)\( T^{3} + \)\(27\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{5} - \)\(33\!\cdots\!12\)\( T^{6} + \)\(39\!\cdots\!80\)\( T^{7} + \)\(14\!\cdots\!26\)\( T^{8} - \)\(56\!\cdots\!75\)\( T^{9} - 2133423617317363 T^{10} + 27018171 T^{11} + T^{12} \)
$43$ \( \)\(11\!\cdots\!88\)\( - \)\(45\!\cdots\!48\)\( T + \)\(64\!\cdots\!40\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} - \)\(14\!\cdots\!19\)\( T^{4} + \)\(14\!\cdots\!68\)\( T^{5} - \)\(87\!\cdots\!76\)\( T^{6} - \)\(22\!\cdots\!24\)\( T^{7} + \)\(44\!\cdots\!50\)\( T^{8} + \)\(14\!\cdots\!56\)\( T^{9} - 3773334439479420 T^{10} - 31237588 T^{11} + T^{12} \)
$47$ \( \)\(85\!\cdots\!52\)\( - \)\(98\!\cdots\!73\)\( T - \)\(10\!\cdots\!33\)\( T^{2} + \)\(19\!\cdots\!85\)\( T^{3} + \)\(27\!\cdots\!73\)\( T^{4} - \)\(25\!\cdots\!94\)\( T^{5} - \)\(12\!\cdots\!66\)\( T^{6} + \)\(14\!\cdots\!34\)\( T^{7} + \)\(15\!\cdots\!18\)\( T^{8} - \)\(11\!\cdots\!05\)\( T^{9} - 6979829350681257 T^{10} + 21983709 T^{11} + T^{12} \)
$53$ \( \)\(20\!\cdots\!00\)\( - \)\(55\!\cdots\!60\)\( T + \)\(23\!\cdots\!68\)\( T^{2} + \)\(12\!\cdots\!42\)\( T^{3} - \)\(56\!\cdots\!99\)\( T^{4} - \)\(14\!\cdots\!28\)\( T^{5} + \)\(41\!\cdots\!08\)\( T^{6} + \)\(99\!\cdots\!80\)\( T^{7} - \)\(83\!\cdots\!62\)\( T^{8} - \)\(27\!\cdots\!56\)\( T^{9} - 4398646042862228 T^{10} + 196548234 T^{11} + T^{12} \)
$59$ \( -\)\(74\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( T - \)\(86\!\cdots\!76\)\( T^{2} - \)\(15\!\cdots\!48\)\( T^{3} - \)\(85\!\cdots\!96\)\( T^{4} - \)\(25\!\cdots\!72\)\( T^{5} + \)\(87\!\cdots\!52\)\( T^{6} + \)\(14\!\cdots\!76\)\( T^{7} - \)\(19\!\cdots\!80\)\( T^{8} - \)\(48\!\cdots\!92\)\( T^{9} - 10904652220405956 T^{10} + 215907906 T^{11} + T^{12} \)
$61$ \( \)\(53\!\cdots\!00\)\( - \)\(11\!\cdots\!00\)\( T + \)\(25\!\cdots\!60\)\( T^{2} + \)\(11\!\cdots\!84\)\( T^{3} - \)\(10\!\cdots\!36\)\( T^{4} - \)\(70\!\cdots\!28\)\( T^{5} - \)\(50\!\cdots\!32\)\( T^{6} + \)\(13\!\cdots\!48\)\( T^{7} + \)\(45\!\cdots\!18\)\( T^{8} - \)\(95\!\cdots\!13\)\( T^{9} - 40198509451997287 T^{10} + 218340705 T^{11} + T^{12} \)
$67$ \( -\)\(25\!\cdots\!00\)\( - \)\(18\!\cdots\!08\)\( T - \)\(19\!\cdots\!64\)\( T^{2} + \)\(37\!\cdots\!16\)\( T^{3} + \)\(48\!\cdots\!80\)\( T^{4} - \)\(23\!\cdots\!64\)\( T^{5} - \)\(37\!\cdots\!48\)\( T^{6} + \)\(55\!\cdots\!72\)\( T^{7} + \)\(12\!\cdots\!76\)\( T^{8} - \)\(32\!\cdots\!85\)\( T^{9} - 184784273760581889 T^{10} - 14544775 T^{11} + T^{12} \)
$71$ \( \)\(28\!\cdots\!88\)\( - \)\(35\!\cdots\!76\)\( T + \)\(33\!\cdots\!64\)\( T^{2} + \)\(12\!\cdots\!96\)\( T^{3} - \)\(71\!\cdots\!16\)\( T^{4} - \)\(14\!\cdots\!08\)\( T^{5} + \)\(26\!\cdots\!32\)\( T^{6} + \)\(63\!\cdots\!92\)\( T^{7} + \)\(99\!\cdots\!40\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{9} - 135491200511853200 T^{10} + 552451776 T^{11} + T^{12} \)
$73$ \( \)\(28\!\cdots\!62\)\( - \)\(15\!\cdots\!87\)\( T + \)\(30\!\cdots\!97\)\( T^{2} + \)\(75\!\cdots\!99\)\( T^{3} + \)\(53\!\cdots\!37\)\( T^{4} - \)\(43\!\cdots\!94\)\( T^{5} - \)\(99\!\cdots\!54\)\( T^{6} + \)\(96\!\cdots\!98\)\( T^{7} + \)\(26\!\cdots\!00\)\( T^{8} - \)\(95\!\cdots\!75\)\( T^{9} - 273200368638664943 T^{10} + 349395159 T^{11} + T^{12} \)
$79$ \( \)\(16\!\cdots\!56\)\( - \)\(12\!\cdots\!53\)\( T + \)\(11\!\cdots\!95\)\( T^{2} + \)\(45\!\cdots\!49\)\( T^{3} - \)\(76\!\cdots\!99\)\( T^{4} + \)\(10\!\cdots\!78\)\( T^{5} + \)\(13\!\cdots\!18\)\( T^{6} - \)\(39\!\cdots\!18\)\( T^{7} - \)\(65\!\cdots\!58\)\( T^{8} + \)\(35\!\cdots\!71\)\( T^{9} - 108862976155091013 T^{10} - 962249727 T^{11} + T^{12} \)
$83$ \( \)\(37\!\cdots\!12\)\( + \)\(61\!\cdots\!24\)\( T + \)\(23\!\cdots\!40\)\( T^{2} + \)\(33\!\cdots\!72\)\( T^{3} + \)\(22\!\cdots\!13\)\( T^{4} + \)\(75\!\cdots\!16\)\( T^{5} + \)\(57\!\cdots\!04\)\( T^{6} - \)\(38\!\cdots\!20\)\( T^{7} - \)\(11\!\cdots\!02\)\( T^{8} - \)\(10\!\cdots\!36\)\( T^{9} + 837095301958829200 T^{10} + 2032575912 T^{11} + T^{12} \)
$89$ \( \)\(13\!\cdots\!00\)\( + \)\(22\!\cdots\!00\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{3} - \)\(19\!\cdots\!03\)\( T^{4} - \)\(50\!\cdots\!32\)\( T^{5} - \)\(71\!\cdots\!76\)\( T^{6} + \)\(43\!\cdots\!52\)\( T^{7} + \)\(85\!\cdots\!14\)\( T^{8} - \)\(69\!\cdots\!96\)\( T^{9} - 1754781247089026576 T^{10} + 280821684 T^{11} + T^{12} \)
$97$ \( -\)\(36\!\cdots\!30\)\( + \)\(20\!\cdots\!67\)\( T - \)\(30\!\cdots\!75\)\( T^{2} + \)\(14\!\cdots\!37\)\( T^{3} + \)\(21\!\cdots\!13\)\( T^{4} - \)\(23\!\cdots\!02\)\( T^{5} + \)\(19\!\cdots\!82\)\( T^{6} + \)\(12\!\cdots\!58\)\( T^{7} - \)\(12\!\cdots\!56\)\( T^{8} - \)\(27\!\cdots\!97\)\( T^{9} - 501198506799496235 T^{10} + 2115165937 T^{11} + T^{12} \)
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