Properties

Label 289.10.a.d
Level $289$
Weight $10$
Character orbit 289.a
Self dual yes
Analytic conductor $148.845$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(148.845356651\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 5 x^{11} - 4542 x^{10} + 30079 x^{9} + 7481825 x^{8} - 58373994 x^{7} - 5553197164 x^{6} + 45662797046 x^{5} + 1825619566875 x^{4} - 13493392282513 x^{3} - 214257360688838 x^{2} + 1064197876989579 x + 245077910606835\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5}\cdot 17 \)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{2} + ( -6 + \beta_{3} ) q^{3} + ( 249 - \beta_{1} + \beta_{2} ) q^{4} + ( -38 + \beta_{2} + \beta_{4} ) q^{5} + ( -219 - 7 \beta_{1} + 6 \beta_{3} + \beta_{7} ) q^{6} + ( 483 - 56 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -1087 + 200 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{8} + ( 5926 + 163 \beta_{1} - 11 \beta_{2} - 12 \beta_{3} + 4 \beta_{4} + \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{2} + ( -6 + \beta_{3} ) q^{3} + ( 249 - \beta_{1} + \beta_{2} ) q^{4} + ( -38 + \beta_{2} + \beta_{4} ) q^{5} + ( -219 - 7 \beta_{1} + 6 \beta_{3} + \beta_{7} ) q^{6} + ( 483 - 56 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -1087 + 200 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{8} + ( 5926 + 163 \beta_{1} - 11 \beta_{2} - 12 \beta_{3} + 4 \beta_{4} + \beta_{9} ) q^{9} + ( 105 + 248 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} + \beta_{10} ) q^{10} + ( -12638 - 251 \beta_{1} - 28 \beta_{2} - 62 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{9} - \beta_{11} ) q^{11} + ( -3383 - 206 \beta_{1} + 13 \beta_{2} + 67 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{12} + ( 1979 - 88 \beta_{1} + 36 \beta_{2} + 3 \beta_{3} - 15 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{13} + ( -41498 + 995 \beta_{1} - 97 \beta_{2} - 249 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 4 \beta_{6} - 9 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{14} + ( -12469 + 98 \beta_{1} + 28 \beta_{2} + 206 \beta_{3} - 35 \beta_{4} - 12 \beta_{5} + 4 \beta_{6} + 11 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{15} + ( 23190 - 1256 \beta_{1} + 319 \beta_{2} + 277 \beta_{3} - 45 \beta_{4} + 23 \beta_{5} + 3 \beta_{6} - 8 \beta_{7} + \beta_{8} - 3 \beta_{9} - 6 \beta_{10} + 6 \beta_{11} ) q^{16} + ( 134364 - 357 \beta_{1} + 78 \beta_{2} + 74 \beta_{3} + 108 \beta_{4} - 6 \beta_{5} - 24 \beta_{6} - 34 \beta_{7} - 2 \beta_{8} + 10 \beta_{9} + 7 \beta_{10} + 4 \beta_{11} ) q^{18} + ( 87781 + 185 \beta_{1} - 118 \beta_{2} - 135 \beta_{3} + 51 \beta_{4} - 16 \beta_{5} - 8 \beta_{6} + 18 \beta_{7} + 2 \beta_{8} + 19 \beta_{9} - 12 \beta_{10} + 2 \beta_{11} ) q^{19} + ( 208646 - 5095 \beta_{1} + 24 \beta_{2} + 391 \beta_{3} + 99 \beta_{4} + 51 \beta_{5} - 23 \beta_{6} + 73 \beta_{7} - \beta_{8} + 23 \beta_{9} - 5 \beta_{10} + 4 \beta_{11} ) q^{20} + ( -113046 - 7967 \beta_{1} - 169 \beta_{2} - 579 \beta_{3} - 309 \beta_{4} - 59 \beta_{5} - 16 \beta_{6} - 67 \beta_{7} + 6 \beta_{8} - 9 \beta_{9} - 26 \beta_{10} - 5 \beta_{11} ) q^{21} + ( -189031 - 24580 \beta_{1} - 265 \beta_{2} - 659 \beta_{3} - 143 \beta_{4} + 61 \beta_{5} - 68 \beta_{6} - 28 \beta_{7} - 5 \beta_{8} - 35 \beta_{9} - 10 \beta_{11} ) q^{22} + ( -166367 - 2670 \beta_{1} + 262 \beta_{2} + 2015 \beta_{3} - 159 \beta_{4} - 116 \beta_{5} - 48 \beta_{6} - 35 \beta_{7} - 2 \beta_{8} - 30 \beta_{9} + 12 \beta_{10} + 9 \beta_{11} ) q^{23} + ( -61611 + 6330 \beta_{1} - 648 \beta_{2} - 1311 \beta_{3} + 335 \beta_{4} - 103 \beta_{5} - 4 \beta_{6} - 266 \beta_{7} - 5 \beta_{8} + 15 \beta_{9} - 10 \beta_{11} ) q^{24} + ( 506184 + 10156 \beta_{1} - 1350 \beta_{2} - 1857 \beta_{3} - 155 \beta_{4} + 19 \beta_{5} - 36 \beta_{6} - 183 \beta_{7} + 6 \beta_{8} + 72 \beta_{9} + 10 \beta_{10} + 11 \beta_{11} ) q^{25} + ( -70196 + 21317 \beta_{1} + 6 \beta_{2} - 586 \beta_{3} - 744 \beta_{4} + 122 \beta_{5} + 128 \beta_{6} - 308 \beta_{7} + 6 \beta_{8} - 14 \beta_{9} - 11 \beta_{10} + 12 \beta_{11} ) q^{26} + ( -271585 + 6962 \beta_{1} - 727 \beta_{2} + 2965 \beta_{3} - 551 \beta_{4} - 13 \beta_{5} + 56 \beta_{6} + 278 \beta_{7} - 12 \beta_{8} + 10 \beta_{9} - 28 \beta_{10} + 22 \beta_{11} ) q^{27} + ( 521418 - 59003 \beta_{1} + 4314 \beta_{2} - 1015 \beta_{3} + 793 \beta_{4} + 225 \beta_{5} - 153 \beta_{6} - 287 \beta_{7} + \beta_{8} - 39 \beta_{9} - 61 \beta_{10} + 24 \beta_{11} ) q^{28} + ( -1048103 - 37430 \beta_{1} + 305 \beta_{2} + 4506 \beta_{3} + 547 \beta_{4} - 74 \beta_{5} - 148 \beta_{6} + 580 \beta_{7} + 28 \beta_{8} - 114 \beta_{9} + 32 \beta_{10} + 4 \beta_{11} ) q^{29} + ( 13509 + 4702 \beta_{1} + 1721 \beta_{2} + 8099 \beta_{3} - 1801 \beta_{4} - 645 \beta_{5} - 44 \beta_{6} + 768 \beta_{7} - 35 \beta_{8} - 45 \beta_{9} - 20 \beta_{10} - 6 \beta_{11} ) q^{30} + ( -476824 + 34385 \beta_{1} + 657 \beta_{2} - 4299 \beta_{3} - 1132 \beta_{4} - 143 \beta_{5} + 212 \beta_{6} - 208 \beta_{7} - 54 \beta_{8} - 73 \beta_{9} + 24 \beta_{10} + 8 \beta_{11} ) q^{31} + ( -474921 + 79822 \beta_{1} - 256 \beta_{2} - 7843 \beta_{3} - 2673 \beta_{4} - 431 \beta_{5} + 308 \beta_{6} - 260 \beta_{7} + 103 \beta_{8} - 45 \beta_{9} - 38 \beta_{10} + 2 \beta_{11} ) q^{32} + ( -1373506 - 14157 \beta_{1} - 746 \beta_{2} - 16201 \beta_{3} + 924 \beta_{4} + 413 \beta_{5} + 180 \beta_{6} - 1123 \beta_{7} - 14 \beta_{8} - 167 \beta_{9} + 54 \beta_{10} - 65 \beta_{11} ) q^{33} + ( 1985382 - 36178 \beta_{1} + 4567 \beta_{2} - 26364 \beta_{3} - 106 \beta_{4} - 611 \beta_{5} - 336 \beta_{6} - 2527 \beta_{7} + 30 \beta_{8} - 10 \beta_{9} - 16 \beta_{10} - 39 \beta_{11} ) q^{35} + ( -3169045 + 69880 \beta_{1} - 9063 \beta_{2} - 14209 \beta_{3} + 3891 \beta_{4} - 669 \beta_{5} - 55 \beta_{6} + 789 \beta_{7} - 25 \beta_{8} - \beta_{9} + 95 \beta_{10} - 36 \beta_{11} ) q^{36} + ( -54005 - 135341 \beta_{1} + 91 \beta_{2} + 16277 \beta_{3} - 1549 \beta_{4} + 813 \beta_{5} - 76 \beta_{6} + 2319 \beta_{7} + 22 \beta_{8} - 91 \beta_{9} + 58 \beta_{10} - 63 \beta_{11} ) q^{37} + ( 304553 + 19547 \beta_{1} - 8466 \beta_{2} + 10004 \beta_{3} - 3750 \beta_{4} - 702 \beta_{5} - 232 \beta_{6} - 1477 \beta_{7} - 98 \beta_{8} + 34 \beta_{9} + 248 \beta_{10} + 44 \beta_{11} ) q^{38} + ( 124684 - 19153 \beta_{1} - 16161 \beta_{2} - 18524 \beta_{3} - 6362 \beta_{4} + 1321 \beta_{5} - 280 \beta_{6} - 686 \beta_{7} + 82 \beta_{8} + 173 \beta_{9} - 52 \beta_{10} + 122 \beta_{11} ) q^{39} + ( -3756474 + 90082 \beta_{1} - 18827 \beta_{2} + 22829 \beta_{3} - 2977 \beta_{4} - 1477 \beta_{5} + 19 \beta_{6} - 706 \beta_{7} + 25 \beta_{8} + 85 \beta_{9} - 100 \beta_{10} - 134 \beta_{11} ) q^{40} + ( -3419849 + 32575 \beta_{1} - 906 \beta_{2} + 21846 \beta_{3} + 7415 \beta_{4} - 204 \beta_{5} + 32 \beta_{6} - 4170 \beta_{7} + 96 \beta_{8} - 63 \beta_{9} - 56 \beta_{10} + 146 \beta_{11} ) q^{41} + ( -6087953 - 123269 \beta_{1} - 21922 \beta_{2} - 40022 \beta_{3} - 15506 \beta_{4} - 334 \beta_{5} + 64 \beta_{6} - 2104 \beta_{7} - 98 \beta_{8} - 550 \beta_{9} - 102 \beta_{10} - 132 \beta_{11} ) q^{42} + ( -1197488 + 67637 \beta_{1} - 20189 \beta_{2} + 20087 \beta_{3} - 6754 \beta_{4} - 277 \beta_{5} - 368 \beta_{6} - 3052 \beta_{7} - 86 \beta_{8} - 53 \beta_{9} + 176 \beta_{10} - 172 \beta_{11} ) q^{43} + ( -12306233 - 81839 \beta_{1} - 54353 \beta_{2} - 29038 \beta_{3} - 2470 \beta_{4} - 690 \beta_{5} - 136 \beta_{6} + 1230 \beta_{7} + 198 \beta_{8} - 114 \beta_{9} + 162 \beta_{10} - 288 \beta_{11} ) q^{44} + ( 6495794 + 268053 \beta_{1} + 22990 \beta_{2} - 49526 \beta_{3} + 16951 \beta_{4} - 432 \beta_{5} + 508 \beta_{6} - 1884 \beta_{7} - 20 \beta_{8} + 275 \beta_{9} + 676 \beta_{10} - 228 \beta_{11} ) q^{45} + ( -2640360 - 13423 \beta_{1} - 21577 \beta_{2} + 5099 \beta_{3} + 6461 \beta_{4} - 635 \beta_{5} + 444 \beta_{6} + 4657 \beta_{7} - 293 \beta_{8} + 789 \beta_{9} - 402 \beta_{10} - 50 \beta_{11} ) q^{46} + ( 11038704 + 196269 \beta_{1} + 25308 \beta_{2} + 1882 \beta_{3} - 5406 \beta_{4} + 606 \beta_{5} + 100 \beta_{6} + 2911 \beta_{7} + 272 \beta_{8} + 215 \beta_{9} + 388 \beta_{10} - 5 \beta_{11} ) q^{47} + ( 6822225 - 338978 \beta_{1} - 10897 \beta_{2} - 162906 \beta_{3} + 1038 \beta_{4} + 700 \beta_{5} - 1225 \beta_{6} - 1420 \beta_{7} + 420 \beta_{8} + 924 \beta_{9} - 380 \beta_{10} + 112 \beta_{11} ) q^{48} + ( 15500614 - 345782 \beta_{1} + 18923 \beta_{2} - 69592 \beta_{3} - 3039 \beta_{4} + 394 \beta_{5} - 1744 \beta_{6} + 1632 \beta_{7} - 740 \beta_{8} + 650 \beta_{9} + 380 \beta_{10} + 104 \beta_{11} ) q^{49} + ( 8647063 - 114892 \beta_{1} - 12358 \beta_{2} - 109458 \beta_{3} + 14028 \beta_{4} - 330 \beta_{5} - 1232 \beta_{6} - 2586 \beta_{7} + 66 \beta_{8} + 1542 \beta_{9} - 187 \beta_{10} + 220 \beta_{11} ) q^{50} + ( 14887297 + 57030 \beta_{1} + 77675 \beta_{2} - 167109 \beta_{3} - 8089 \beta_{4} + 2143 \beta_{5} + 2585 \beta_{6} - 1483 \beta_{7} - 165 \beta_{8} - 445 \beta_{9} - 465 \beta_{10} + 68 \beta_{11} ) q^{52} + ( -25116189 + 363140 \beta_{1} - 51183 \beta_{2} - 246398 \beta_{3} + 12765 \beta_{4} + 2770 \beta_{5} + 292 \beta_{6} + 1076 \beta_{7} + 84 \beta_{8} - 1012 \beta_{9} - 64 \beta_{10} + 276 \beta_{11} ) q^{53} + ( 4422559 - 541614 \beta_{1} + 41279 \beta_{2} + 177121 \beta_{3} - 17807 \beta_{4} - 3139 \beta_{5} + 676 \beta_{6} + 762 \beta_{7} - 197 \beta_{8} - 259 \beta_{9} - 288 \beta_{10} + 438 \beta_{11} ) q^{54} + ( -7347813 - 550776 \beta_{1} - 48652 \beta_{2} + 62049 \beta_{3} - 20981 \beta_{4} + 3242 \beta_{5} + 2432 \beta_{6} + 711 \beta_{7} + 646 \beta_{8} + 332 \beta_{9} - 716 \beta_{10} - 13 \beta_{11} ) q^{55} + ( -23114192 + 2034114 \beta_{1} - 137991 \beta_{2} - 154789 \beta_{3} - 30703 \beta_{4} - 6263 \beta_{5} + 4315 \beta_{6} + 834 \beta_{7} - 541 \beta_{8} + 583 \beta_{9} + 676 \beta_{10} - 498 \beta_{11} ) q^{56} + ( -4272221 + 525213 \beta_{1} - 34192 \beta_{2} + 353691 \beta_{3} + 3116 \beta_{4} + 8409 \beta_{5} + 1220 \beta_{6} + 953 \beta_{7} - 102 \beta_{8} + 1055 \beta_{9} + 102 \beta_{10} + 835 \beta_{11} ) q^{57} + ( -30126336 - 889265 \beta_{1} - 93494 \beta_{2} + 328002 \beta_{3} + 28424 \beta_{4} + 2198 \beta_{5} - 360 \beta_{6} + 6336 \beta_{7} - 774 \beta_{8} - 162 \beta_{9} + 779 \beta_{10} - 860 \beta_{11} ) q^{58} + ( -8989231 + 251099 \beta_{1} - 75148 \beta_{2} - 224128 \beta_{3} - 15811 \beta_{4} + 1190 \beta_{5} + 856 \beta_{6} + 2300 \beta_{7} + 650 \beta_{8} + 945 \beta_{9} + 420 \beta_{10} + 516 \beta_{11} ) q^{59} + ( 8294027 + 1003573 \beta_{1} - 725 \beta_{2} + 454526 \beta_{3} - 11866 \beta_{4} + 1042 \beta_{5} - 1816 \beta_{6} + 6266 \beta_{7} + 458 \beta_{8} + 1186 \beta_{9} - 1194 \beta_{10} + 1008 \beta_{11} ) q^{60} + ( -21273672 - 1262695 \beta_{1} - 98093 \beta_{2} + 10009 \beta_{3} + 1415 \beta_{4} + 4689 \beta_{5} - 912 \beta_{6} - 475 \beta_{7} - 1330 \beta_{8} + 255 \beta_{9} + 190 \beta_{10} + 1387 \beta_{11} ) q^{61} + ( 26154209 - 78656 \beta_{1} + 171955 \beta_{2} - 71259 \beta_{3} - 17011 \beta_{4} - 607 \beta_{5} + 2372 \beta_{6} + 3682 \beta_{7} + 183 \beta_{8} + 25 \beta_{9} - 2608 \beta_{10} + 1118 \beta_{11} ) q^{62} + ( -18237710 + 38617 \beta_{1} - 99281 \beta_{2} - 279149 \beta_{3} + 11982 \beta_{4} + 6599 \beta_{5} + 3212 \beta_{6} - 19324 \beta_{7} + 1506 \beta_{8} + 87 \beta_{9} + 1424 \beta_{10} + 220 \beta_{11} ) q^{63} + ( 49467189 + 271422 \beta_{1} + 155551 \beta_{2} - 129054 \beta_{3} - 6094 \beta_{4} + 9208 \beta_{5} + 1383 \beta_{6} - 13148 \beta_{7} - 1208 \beta_{8} + 1008 \beta_{9} - 1772 \beta_{10} - 56 \beta_{11} ) q^{64} + ( -26687024 - 2323639 \beta_{1} + 11989 \beta_{2} - 756117 \beta_{3} - 6839 \beta_{4} - 1709 \beta_{5} + 420 \beta_{6} + 4609 \beta_{7} - 1778 \beta_{8} - 3801 \beta_{9} + 1234 \beta_{10} - 949 \beta_{11} ) q^{65} + ( -9190988 - 1785447 \beta_{1} + 67682 \beta_{2} - 755594 \beta_{3} + 4508 \beta_{4} + 8982 \beta_{5} - 448 \beta_{6} - 7438 \beta_{7} + 2706 \beta_{8} - 3178 \beta_{9} - 1011 \beta_{10} - 932 \beta_{11} ) q^{66} + ( -40458963 - 943697 \beta_{1} - 20013 \beta_{2} - 275695 \beta_{3} - 32953 \beta_{4} - 11273 \beta_{5} + 1640 \beta_{6} + 20543 \beta_{7} - 24 \beta_{8} - 3463 \beta_{9} + 644 \beta_{10} - 17 \beta_{11} ) q^{67} + ( 55037279 - 69814 \beta_{1} + 110404 \beta_{2} - 537208 \beta_{3} + 25216 \beta_{4} + 2664 \beta_{5} - 764 \beta_{6} + 30938 \beta_{7} + 240 \beta_{8} + 1510 \beta_{9} + 3420 \beta_{10} + 614 \beta_{11} ) q^{69} + ( -20762309 + 4190349 \beta_{1} - 273674 \beta_{2} - 1500184 \beta_{3} - 10406 \beta_{4} + 1858 \beta_{5} + 2752 \beta_{6} - 17215 \beta_{7} + 286 \beta_{8} + 3682 \beta_{9} - 1592 \beta_{10} - 1108 \beta_{11} ) q^{70} + ( -15447911 - 3699041 \beta_{1} - 23272 \beta_{2} + 167696 \beta_{3} + 2665 \beta_{4} - 7614 \beta_{5} + 4280 \beta_{6} - 34028 \beta_{7} + 878 \beta_{8} + 125 \beta_{9} + 892 \beta_{10} - 1748 \beta_{11} ) q^{71} + ( -13907097 - 8024238 \beta_{1} + 23052 \beta_{2} + 497907 \beta_{3} + 45653 \beta_{4} + 6179 \beta_{5} - 6796 \beta_{6} + 13198 \beta_{7} - 1967 \beta_{8} - 2739 \beta_{9} + 12 \beta_{10} - 1174 \beta_{11} ) q^{72} + ( -56758499 + 1303328 \beta_{1} + 20709 \beta_{2} - 505037 \beta_{3} + 22892 \beta_{4} - 7519 \beta_{5} - 8444 \beta_{6} + 6869 \beta_{7} + 1618 \beta_{8} - 5624 \beta_{9} - 3458 \beta_{10} + 127 \beta_{11} ) q^{73} + ( -106247474 + 652656 \beta_{1} - 127684 \beta_{2} + 1147428 \beta_{3} + 11672 \beta_{4} + 3036 \beta_{5} + 56 \beta_{6} + 9760 \beta_{7} + 20 \beta_{8} - 6100 \beta_{9} + 1370 \beta_{10} - 2712 \beta_{11} ) q^{74} + ( -47414850 - 3364301 \beta_{1} - 65989 \beta_{2} + 1552635 \beta_{3} - 107088 \beta_{4} + 1279 \beta_{5} + 3392 \beta_{6} + 45476 \beta_{7} + 3222 \beta_{8} + 189 \beta_{9} - 4352 \beta_{10} + 2356 \beta_{11} ) q^{75} + ( -32402775 - 3131032 \beta_{1} + 20181 \beta_{2} - 538987 \beta_{3} + 93637 \beta_{4} + 3345 \beta_{5} - 3507 \beta_{6} + 34957 \beta_{7} - 2015 \beta_{8} + 1905 \beta_{9} - 1345 \beta_{10} - 32 \beta_{11} ) q^{76} + ( -61743983 + 7703319 \beta_{1} - 70423 \beta_{2} + 1305078 \beta_{3} + 49358 \beta_{4} - 15776 \beta_{5} + 3664 \beta_{6} + 34590 \beta_{7} - 796 \beta_{8} - 5087 \beta_{9} + 4256 \beta_{10} - 2926 \beta_{11} ) q^{77} + ( -11326654 - 6011431 \beta_{1} - 185303 \beta_{2} - 823615 \beta_{3} - 8565 \beta_{4} - 8029 \beta_{5} + 300 \beta_{6} - 35715 \beta_{7} + 4237 \beta_{8} + 2083 \beta_{9} - 4902 \beta_{10} - 2126 \beta_{11} ) q^{78} + ( -35103295 - 2611553 \beta_{1} - 94798 \beta_{2} - 257582 \beta_{3} - 61371 \beta_{4} - 11532 \beta_{5} - 10168 \beta_{6} + 39406 \beta_{7} - 982 \beta_{8} - 3043 \beta_{9} + 2412 \beta_{10} - 2850 \beta_{11} ) q^{79} + ( -45992271 - 9688486 \beta_{1} + 99650 \beta_{2} - 115259 \beta_{3} - 90545 \beta_{4} - 16291 \beta_{5} - 14874 \beta_{6} - 16212 \beta_{7} - 3557 \beta_{8} - 12409 \beta_{9} - 374 \beta_{10} - 230 \beta_{11} ) q^{80} + ( -33664521 + 2795047 \beta_{1} + 251375 \beta_{2} - 97403 \beta_{3} + 81351 \beta_{4} + 7033 \beta_{5} - 7036 \beta_{6} - 28141 \beta_{7} + 3114 \beta_{8} - 9087 \beta_{9} + 1966 \beta_{10} + 1873 \beta_{11} ) q^{81} + ( 16895478 - 5272406 \beta_{1} - 53016 \beta_{2} - 1994200 \beta_{3} + 56136 \beta_{4} + 9184 \beta_{5} + 3072 \beta_{6} + 24596 \beta_{7} + 3968 \beta_{8} + 7536 \beta_{9} + 1692 \beta_{10} + 2160 \beta_{11} ) q^{82} + ( 33221712 + 2114014 \beta_{1} + 130117 \beta_{2} - 1000638 \beta_{3} + 14584 \beta_{4} - 10525 \beta_{5} + 15752 \beta_{6} + 25437 \beta_{7} - 4106 \beta_{8} + 7838 \beta_{9} - 2144 \beta_{10} + 3325 \beta_{11} ) q^{83} + ( -36109627 - 9057303 \beta_{1} + 45529 \beta_{2} - 1186014 \beta_{3} - 51878 \beta_{4} + 25794 \beta_{5} + 2242 \beta_{6} + 15238 \beta_{7} - 870 \beta_{8} - 4646 \beta_{9} - 3870 \beta_{10} - 904 \beta_{11} ) q^{84} + ( 44801509 - 9429434 \beta_{1} - 141303 \beta_{2} - 1593433 \beta_{3} + 42103 \beta_{4} + 1187 \beta_{5} - 21332 \beta_{6} + 53370 \beta_{7} + 1525 \beta_{8} + 2915 \beta_{9} - 8900 \beta_{10} - 3398 \beta_{11} ) q^{86} + ( 121291209 + 10669578 \beta_{1} + 106674 \beta_{2} - 1980231 \beta_{3} - 121283 \beta_{4} - 5084 \beta_{5} - 19712 \beta_{6} + 19709 \beta_{7} - 3082 \beta_{8} + 906 \beta_{9} + 6252 \beta_{10} + 2041 \beta_{11} ) q^{87} + ( 32114081 - 24148512 \beta_{1} + 117083 \beta_{2} + 1120470 \beta_{3} + 257854 \beta_{4} + 16376 \beta_{5} - 33443 \beta_{6} - 17756 \beta_{7} - 1584 \beta_{8} + 12736 \beta_{9} - 2196 \beta_{10} + 4184 \beta_{11} ) q^{88} + ( 81741027 - 13441449 \beta_{1} + 220732 \beta_{2} + 1357408 \beta_{3} + 147947 \beta_{4} - 29406 \beta_{5} + 8512 \beta_{6} + 52808 \beta_{7} + 1156 \beta_{8} + 8137 \beta_{9} + 1364 \beta_{10} + 5000 \beta_{11} ) q^{89} + ( 222014127 + 13486173 \beta_{1} + 598890 \beta_{2} - 814610 \beta_{3} + 419806 \beta_{4} + 55894 \beta_{5} - 11480 \beta_{6} - 2676 \beta_{7} - 566 \beta_{8} + 12638 \beta_{9} + 7916 \beta_{10} + 5316 \beta_{11} ) q^{90} + ( 44146158 + 6661439 \beta_{1} + 417530 \beta_{2} + 2261212 \beta_{3} + 28846 \beta_{4} - 45092 \beta_{5} - 21216 \beta_{6} - 22581 \beta_{7} + 924 \beta_{8} + 8733 \beta_{9} - 3784 \beta_{10} + 4283 \beta_{11} ) q^{91} + ( 76156136 - 12618847 \beta_{1} - 78944 \beta_{2} + 1713475 \beta_{3} - 62381 \beta_{4} + 3875 \beta_{5} - 16919 \beta_{6} - 10389 \beta_{7} - 6077 \beta_{8} + 8027 \beta_{9} + 11217 \beta_{10} - 1384 \beta_{11} ) q^{92} + ( -101410505 - 3289076 \beta_{1} + 420517 \beta_{2} - 1862654 \beta_{3} + 340737 \beta_{4} - 33994 \beta_{5} + 3596 \beta_{6} - 40644 \beta_{7} + 1292 \beta_{8} - 1900 \beta_{9} + 6088 \beta_{10} - 5100 \beta_{11} ) q^{93} + ( 157916759 + 23293090 \beta_{1} + 535087 \beta_{2} + 1653733 \beta_{3} + 185201 \beta_{4} + 53573 \beta_{5} + 26140 \beta_{6} - 11836 \beta_{7} - 2093 \beta_{8} + 6157 \beta_{9} - 7232 \beta_{10} + 3382 \beta_{11} ) q^{94} + ( 98693800 - 14153893 \beta_{1} + 610977 \beta_{2} - 324527 \beta_{3} + 158912 \beta_{4} - 347 \beta_{5} + 24820 \beta_{6} + 43798 \beta_{7} - 3542 \beta_{8} - 14139 \beta_{9} + 17944 \beta_{10} - 5218 \beta_{11} ) q^{95} + ( -183346592 - 709940 \beta_{1} - 862885 \beta_{2} - 1415825 \beta_{3} - 169743 \beta_{4} + 51121 \beta_{5} - 5985 \beta_{6} - 101160 \beta_{7} + 3191 \beta_{8} - 181 \beta_{9} + 9174 \beta_{10} + 2186 \beta_{11} ) q^{96} + ( 303924296 - 11857348 \beta_{1} - 46905 \beta_{2} - 783622 \beta_{3} - 60763 \beta_{4} + 18180 \beta_{5} + 7952 \beta_{6} + 3690 \beta_{7} - 680 \beta_{8} + 28712 \beta_{9} - 6920 \beta_{10} - 1122 \beta_{11} ) q^{97} + ( -233392683 + 25748310 \beta_{1} - 1478646 \beta_{2} - 300434 \beta_{3} + 99492 \beta_{4} - 133818 \beta_{5} + 10024 \beta_{6} + 9670 \beta_{7} - 2926 \beta_{8} + 20246 \beta_{9} + 6601 \beta_{10} - 10532 \beta_{11} ) q^{98} + ( -163371784 - 20639310 \beta_{1} + 547462 \beta_{2} - 2220823 \beta_{3} - 152658 \beta_{4} - 24442 \beta_{5} + 25848 \beta_{6} - 87166 \beta_{7} - 1688 \beta_{8} - 12914 \beta_{9} - 16752 \beta_{10} + 2698 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 17q^{2} - 74q^{3} + 2987q^{4} - 454q^{5} - 2674q^{6} + 5524q^{7} - 12036q^{8} + 71898q^{9} + O(q^{10}) \) \( 12q + 17q^{2} - 74q^{3} + 2987q^{4} - 454q^{5} - 2674q^{6} + 5524q^{7} - 12036q^{8} + 71898q^{9} + 2449q^{10} - 152886q^{11} - 41717q^{12} + 23478q^{13} - 492839q^{14} - 149346q^{15} + 272743q^{16} + 1610378q^{18} + 1053982q^{19} + 2477218q^{20} - 1395256q^{21} - 2391095q^{22} - 2012428q^{23} - 708393q^{24} + 6123166q^{25} - 733144q^{26} - 3231638q^{27} + 5978216q^{28} - 12772842q^{29} + 181633q^{30} - 5535814q^{31} - 5277485q^{32} - 16526054q^{33} + 23712622q^{35} - 37692723q^{36} - 1352872q^{37} + 3704404q^{38} + 1380780q^{39} - 44739331q^{40} - 40941240q^{41} - 73649073q^{42} - 14142490q^{43} - 148233417q^{44} + 79449336q^{45} - 31855859q^{46} + 133558002q^{47} + 80444894q^{48} + 184488050q^{49} + 103322437q^{50} + 179597031q^{52} - 299319258q^{53} + 50215469q^{54} - 91197532q^{55} - 267350757q^{56} - 49507694q^{57} - 367048088q^{58} - 106431852q^{59} + 103650215q^{60} - 262041240q^{61} + 314328847q^{62} - 218532626q^{63} + 595820098q^{64} - 330279796q^{65} - 117450322q^{66} - 489635100q^{67} + 661586712q^{69} - 226277420q^{70} - 204290852q^{71} - 208030791q^{72} - 673538852q^{73} - 1274510282q^{74} - 588895258q^{75} - 403517977q^{76} - 705301770q^{77} - 165043245q^{78} - 434002980q^{79} - 599590757q^{80} - 389011392q^{81} + 180073450q^{82} + 411781442q^{83} - 475971445q^{84} + 492988379q^{86} + 1513399800q^{87} + 262108460q^{88} + 911678128q^{89} + 2734590475q^{90} + 560105446q^{91} + 847049266q^{92} - 1228562570q^{93} + 2009871759q^{94} + 1116511966q^{95} - 2204198979q^{96} + 3589270998q^{97} - 2677144485q^{98} - 2056683494q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 4542 x^{10} + 30079 x^{9} + 7481825 x^{8} - 58373994 x^{7} - 5553197164 x^{6} + 45662797046 x^{5} + 1825619566875 x^{4} - 13493392282513 x^{3} - 214257360688838 x^{2} + 1064197876989579 x + 245077910606835\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \nu - 760 \)
\(\beta_{3}\)\(=\)\((\)\(-80093781578284239 \nu^{11} - 1668432767667957782 \nu^{10} + 322634003582572989064 \nu^{9} + 5896655418155773172983 \nu^{8} - 454514350612185104303014 \nu^{7} - 6948299449742092042655364 \nu^{6} + 275013179932437482820616952 \nu^{5} + 3292956958038732608477108606 \nu^{4} - 65957615113789675276347461075 \nu^{3} - 548628192819923458648856575006 \nu^{2} + 3732897309622961792946904076232 \nu + 701509941388414409433140763459\)\()/ \)\(56\!\cdots\!12\)\( \)
\(\beta_{4}\)\(=\)\((\)\(1027318723442988917 \nu^{11} + 20639885694648035978 \nu^{10} - 4134688441827015236896 \nu^{9} - 72280913916115925890469 \nu^{8} + 5809816055731970105835522 \nu^{7} + 83912267684529093307762012 \nu^{6} - 3495269006865732661897190648 \nu^{5} - 38853988970844220928548203898 \nu^{4} + 830170664689432011658635188065 \nu^{3} + 6271988632058581672051154664066 \nu^{2} - 47137456301623675542202638476736 \nu + 5255414047772828734012062681015\)\()/ \)\(94\!\cdots\!20\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-900897902017065789 \nu^{11} - 14416752093429009316 \nu^{10} + 3735766330517180425382 \nu^{9} + 50614426288897207070903 \nu^{8} - 5451799779840212435870594 \nu^{7} - 59342598944998734836732304 \nu^{6} + 3420432139412872864132007476 \nu^{5} + 27949877987499825416074974046 \nu^{4} - 847340873690671014742472820745 \nu^{3} - 4669017400385352154334333734772 \nu^{2} + 49335026553834213842769693626862 \nu - 18986969898323996210182528522605\)\()/ \)\(70\!\cdots\!40\)\( \)
\(\beta_{6}\)\(=\)\((\)\(670371815609386389 \nu^{11} + 13394373311401079756 \nu^{10} - 2697723826892045191342 \nu^{9} - 46839866164392227951623 \nu^{8} + 3789219103533746198997874 \nu^{7} + 54248826451219204927502304 \nu^{6} - 2277685280233752642897121796 \nu^{5} - 25024295530584749935814767166 \nu^{4} + 543713607213283367639166505025 \nu^{3} + 4032336743372817742689793040572 \nu^{2} - 34989710991916834158048917072022 \nu + 2359739161006408280825862502365\)\()/ \)\(35\!\cdots\!20\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-834216383833978891 \nu^{11} - 16404894253827117782 \nu^{10} + 3346313128168059926272 \nu^{9} + 57625014826991253144123 \nu^{8} - 4675560810984620609410430 \nu^{7} - 67510941766707768483402212 \nu^{6} + 2787598575616195980365790920 \nu^{5} + 31899187465172393995042989510 \nu^{4} - 649788465738338760770811241119 \nu^{3} - 5342321987414437803190005836510 \nu^{2} + 33639153956546129459008461216096 \nu + 8662793208503295878078855808663\)\()/ \)\(28\!\cdots\!56\)\( \)
\(\beta_{8}\)\(=\)\((\)\(40942056061913583085 \nu^{11} + 911977503281285182058 \nu^{10} - 162162795112870331791024 \nu^{9} - 3206928289437307271662221 \nu^{8} + 222729126452520365073830066 \nu^{7} + 3737836836902791769863653596 \nu^{6} - 129717785895488546113318450904 \nu^{5} - 1734729304056625276164990597834 \nu^{4} + 28969767384961347880393591130361 \nu^{3} + 277924747615285773947753802361058 \nu^{2} - 1256908941343437190429417964976912 \nu - 275526238849183879649282172772305\)\()/ \)\(28\!\cdots\!60\)\( \)
\(\beta_{9}\)\(=\)\((\)\(2723145386634939529 \nu^{11} + 57664270417532406786 \nu^{10} - 10919117377568207322192 \nu^{9} - 204676297414330842841673 \nu^{8} + 15248916558194711495917834 \nu^{7} + 242917792176288379859012204 \nu^{6} - 9069757075530163387757868216 \nu^{5} - 116468461958859652257548430466 \nu^{4} + 2091695195385783247141789314085 \nu^{3} + 19831728501896212918730559222682 \nu^{2} - 100970716386893921192277805163952 \nu - 82703258449303166667783527020605\)\()/ \)\(14\!\cdots\!40\)\( \)
\(\beta_{10}\)\(=\)\((\)\(31760766357754896357 \nu^{11} + 658974985043379944162 \nu^{10} - 126988627419959499901288 \nu^{9} - 2331463978248493959698317 \nu^{8} + 176792970112448756785443490 \nu^{7} + 2754592608886413282124299660 \nu^{6} - 105181967259297219159697054280 \nu^{5} - 1309508399666758654746154761290 \nu^{4} + 24648431426288915422541328320273 \nu^{3} + 218075646266251424503402000602490 \nu^{2} - 1330219526532410422035111109153512 \nu - 522763747975699712741618218931985\)\()/ \)\(14\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(41393373062460576063 \nu^{11} + 837346354213994513342 \nu^{10} - 166686752186959380648064 \nu^{9} - 2960241598858705747480111 \nu^{8} + 234354542123187532885421638 \nu^{7} + 3492878412547823975070240468 \nu^{6} - 141066887150315321200515021992 \nu^{5} - 1656373188096760002436379918702 \nu^{4} + 33472142953519634570880217210115 \nu^{3} + 273838509849131543772109488838054 \nu^{2} - 1837825967478208821502769004876384 \nu + 276147846796300786476702103266405\)\()/ \)\(94\!\cdots\!20\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3 \beta_{1} + 760\)
\(\nu^{3}\)\(=\)\(\beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 1230 \beta_{1} - 2344\)
\(\nu^{4}\)\(=\)\(6 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} + \beta_{8} - 8 \beta_{7} - \beta_{6} + 23 \beta_{5} - 37 \beta_{4} + 285 \beta_{3} + 1865 \beta_{2} - 7698 \beta_{1} + 934757\)
\(\nu^{5}\)\(=\)\(-28 \beta_{11} - 8 \beta_{10} - 30 \beta_{9} + 98 \beta_{8} - 220 \beta_{7} + 2351 \beta_{6} - 546 \beta_{5} - 6564 \beta_{4} - 13344 \beta_{3} - 11599 \beta_{2} + 1826357 \beta_{1} - 6048323\)
\(\nu^{6}\)\(=\)\(15382 \beta_{11} - 16994 \beta_{10} - 6447 \beta_{9} + 749 \beta_{8} - 32188 \beta_{7} - 5048 \beta_{6} + 71019 \beta_{5} - 81315 \beta_{4} + 655895 \beta_{3} + 3373171 \beta_{2} - 16170467 \beta_{1} + 1389690995\)
\(\nu^{7}\)\(=\)\(-123158 \beta_{11} + 35494 \beta_{10} - 121637 \beta_{9} + 299351 \beta_{8} - 619944 \beta_{7} + 4657224 \beta_{6} - 1831727 \beta_{5} - 15170097 \beta_{4} - 42652739 \beta_{3} - 28155575 \beta_{2} + 3014741172 \beta_{1} - 12695977074\)
\(\nu^{8}\)\(=\)\(31949910 \beta_{11} - 37552994 \beta_{10} - 11776623 \beta_{9} - 855251 \beta_{8} - 79699916 \beta_{7} - 18182128 \beta_{6} + 164525691 \beta_{5} - 123148739 \beta_{4} + 1330244439 \beta_{3} + 6164298792 \beta_{2} - 32993868842 \beta_{1} + 2298052582854\)
\(\nu^{9}\)\(=\)\(-353044934 \beta_{11} + 198388582 \beta_{10} - 307497373 \beta_{9} + 681811423 \beta_{8} - 1291060968 \beta_{7} + 8852554421 \beta_{6} - 4491899815 \beta_{5} - 31098788531 \beta_{4} - 101496417429 \beta_{3} - 64494185675 \beta_{2} + 5278206650269 \beta_{1} - 25858085245810\)
\(\nu^{10}\)\(=\)\(62576155124 \beta_{11} - 76179819744 \beta_{10} - 20611068286 \beta_{9} - 4985190606 \beta_{8} - 165523073956 \beta_{7} - 52362321877 \beta_{6} + 345184535614 \beta_{5} - 145731852556 \beta_{4} + 2667121631736 \beta_{3} + 11389952971032 \beta_{2} - 67809675665853 \beta_{1} + 4029359487789231\)
\(\nu^{11}\)\(=\)\(-858432302130 \beta_{11} + 620020240718 \beta_{10} - 653127262579 \beta_{9} + 1401239006697 \beta_{8} - 2394196332052 \beta_{7} + 16629126582819 \beta_{6} - 9867679427873 \beta_{5} - 60574383797949 \beta_{4} - 213851914930155 \beta_{3} - 142790641051714 \beta_{2} + 9548799387596088 \beta_{1} - 53025082773303641\)

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
−44.0937
−33.3505
−31.5756
−20.4468
−14.2275
−0.220633
4.74483
21.0274
25.3822
25.6779
30.1791
41.9032
−43.0937 38.5560 1345.07 1118.08 −1661.52 12272.9 −35899.9 −18196.4 −48182.3
1.2 −32.3505 71.1986 534.552 −1755.25 −2303.31 −7326.34 −729.582 −14613.8 56783.1
1.3 −30.5756 −194.921 422.865 941.135 5959.81 −4513.44 2725.35 18311.1 −28775.7
1.4 −19.4468 193.562 −133.821 1266.43 −3764.16 1846.43 12559.2 17783.1 −24628.1
1.5 −13.2275 −162.576 −337.033 −1690.93 2150.48 10426.0 11230.6 6748.01 22366.8
1.6 0.779367 −32.1989 −511.393 894.723 −25.0947 −2726.00 −797.598 −18646.2 697.318
1.7 5.74483 214.211 −478.997 −2392.81 1230.60 2771.45 −5693.11 26203.3 −13746.3
1.8 22.0274 −192.112 −26.7930 −2093.86 −4231.72 −11187.9 −11868.2 17223.9 −46122.3
1.9 26.3822 66.0249 184.023 −164.991 1741.89 5776.54 −8652.77 −15323.7 −4352.83
1.10 26.6779 −248.720 199.710 2486.89 −6635.33 7643.33 −8331.23 42178.6 66345.1
1.11 31.1791 218.085 460.136 1545.34 6799.71 −9980.37 −1617.07 27878.3 48182.2
1.12 42.9032 −45.1099 1328.68 −608.765 −1935.36 521.377 35038.3 −17648.1 −26118.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)

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 289.10.a.d 12
17.b even 2 1 289.10.a.e yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.10.a.d 12 1.a even 1 1 trivial
289.10.a.e yes 12 17.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(289))\):

\(15\!\cdots\!96\)\( T_{2}^{4} - \)\(20\!\cdots\!32\)\( T_{2}^{3} - \)\(16\!\cdots\!88\)\( T_{2}^{2} + \)\(14\!\cdots\!36\)\( T_{2} - \)\(10\!\cdots\!00\)\( \)">\(T_{2}^{12} - \cdots\)
\(19\!\cdots\!29\)\( T_{3}^{6} - \)\(55\!\cdots\!60\)\( T_{3}^{5} + \)\(18\!\cdots\!28\)\( T_{3}^{4} + \)\(33\!\cdots\!38\)\( T_{3}^{3} - \)\(48\!\cdots\!45\)\( T_{3}^{2} + \)\(15\!\cdots\!52\)\( T_{3} + \)\(36\!\cdots\!39\)\( \)">\(T_{3}^{12} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1018109465395200 + 1445191108263936 T - 163361958309888 T^{2} - 20230637637632 T^{3} + 1516569700096 T^{4} + 77342075392 T^{5} - 4938565328 T^{6} - 116603152 T^{7} + 7008044 T^{8} + 75004 T^{9} - 4421 T^{10} - 17 T^{11} + T^{12} \)
$3$ \( \)\(36\!\cdots\!39\)\( + \)\(15\!\cdots\!52\)\( T - \)\(48\!\cdots\!45\)\( T^{2} + 336930405417511938 T^{3} + 1857421710435299328 T^{4} - 5576249697859260 T^{5} - 197610553835829 T^{6} + 392253170814 T^{7} + 8307631332 T^{8} - 9283700 T^{9} - 151309 T^{10} + 74 T^{11} + T^{12} \)
$5$ \( \)\(68\!\cdots\!25\)\( + \)\(33\!\cdots\!50\)\( T - \)\(73\!\cdots\!75\)\( T^{2} - \)\(10\!\cdots\!50\)\( T^{3} + \)\(21\!\cdots\!50\)\( T^{4} + \)\(56\!\cdots\!90\)\( T^{5} - \)\(19\!\cdots\!07\)\( T^{6} - 1897290945110882 T^{7} + 79285126861786 T^{8} - 3593646942 T^{9} - 14677275 T^{10} + 454 T^{11} + T^{12} \)
$7$ \( -\)\(15\!\cdots\!85\)\( + \)\(36\!\cdots\!34\)\( T - \)\(99\!\cdots\!35\)\( T^{2} - \)\(85\!\cdots\!84\)\( T^{3} + \)\(25\!\cdots\!76\)\( T^{4} + \)\(60\!\cdots\!26\)\( T^{5} - \)\(15\!\cdots\!23\)\( T^{6} - \)\(15\!\cdots\!36\)\( T^{7} + 35206332140679480 T^{8} + 1569279332898 T^{9} - 319108379 T^{10} - 5524 T^{11} + T^{12} \)
$11$ \( -\)\(12\!\cdots\!69\)\( + \)\(15\!\cdots\!34\)\( T + \)\(14\!\cdots\!16\)\( T^{2} - \)\(35\!\cdots\!62\)\( T^{3} - \)\(30\!\cdots\!77\)\( T^{4} - \)\(44\!\cdots\!60\)\( T^{5} + \)\(21\!\cdots\!92\)\( T^{6} + \)\(48\!\cdots\!88\)\( T^{7} - 41735692965175670975 T^{8} - 1671584668963062 T^{9} - 4509931128 T^{10} + 152886 T^{11} + T^{12} \)
$13$ \( -\)\(16\!\cdots\!47\)\( - \)\(16\!\cdots\!62\)\( T - \)\(26\!\cdots\!35\)\( T^{2} + \)\(15\!\cdots\!70\)\( T^{3} + \)\(35\!\cdots\!86\)\( T^{4} - \)\(42\!\cdots\!82\)\( T^{5} - \)\(12\!\cdots\!71\)\( T^{6} + \)\(30\!\cdots\!42\)\( T^{7} + \)\(14\!\cdots\!74\)\( T^{8} - 99275772257746 T^{9} - 65372574523 T^{10} - 23478 T^{11} + T^{12} \)
$17$ \( T^{12} \)
$19$ \( -\)\(42\!\cdots\!37\)\( + \)\(54\!\cdots\!00\)\( T + \)\(48\!\cdots\!67\)\( T^{2} - \)\(50\!\cdots\!22\)\( T^{3} - \)\(39\!\cdots\!24\)\( T^{4} + \)\(90\!\cdots\!44\)\( T^{5} - \)\(32\!\cdots\!21\)\( T^{6} - \)\(59\!\cdots\!62\)\( T^{7} + \)\(40\!\cdots\!44\)\( T^{8} + 1473562676495830316 T^{9} - 1264755176789 T^{10} - 1053982 T^{11} + T^{12} \)
$23$ \( -\)\(67\!\cdots\!33\)\( - \)\(14\!\cdots\!86\)\( T + \)\(24\!\cdots\!29\)\( T^{2} + \)\(67\!\cdots\!12\)\( T^{3} - \)\(18\!\cdots\!76\)\( T^{4} - \)\(10\!\cdots\!94\)\( T^{5} - \)\(15\!\cdots\!27\)\( T^{6} + \)\(73\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!80\)\( T^{8} - 21201612875707713186 T^{9} - 9185036452575 T^{10} + 2012428 T^{11} + T^{12} \)
$29$ \( \)\(59\!\cdots\!37\)\( - \)\(35\!\cdots\!38\)\( T - \)\(61\!\cdots\!71\)\( T^{2} + \)\(46\!\cdots\!58\)\( T^{3} + \)\(10\!\cdots\!22\)\( T^{4} - \)\(12\!\cdots\!78\)\( T^{5} - \)\(34\!\cdots\!87\)\( T^{6} + \)\(12\!\cdots\!82\)\( T^{7} - \)\(26\!\cdots\!98\)\( T^{8} - \)\(64\!\cdots\!74\)\( T^{9} - 19362813411511 T^{10} + 12772842 T^{11} + T^{12} \)
$31$ \( \)\(61\!\cdots\!75\)\( + \)\(15\!\cdots\!30\)\( T - \)\(33\!\cdots\!64\)\( T^{2} + \)\(98\!\cdots\!42\)\( T^{3} + \)\(75\!\cdots\!03\)\( T^{4} - \)\(27\!\cdots\!24\)\( T^{5} - \)\(76\!\cdots\!72\)\( T^{6} + \)\(23\!\cdots\!88\)\( T^{7} + \)\(45\!\cdots\!49\)\( T^{8} - \)\(68\!\cdots\!78\)\( T^{9} - 123758158181048 T^{10} + 5535814 T^{11} + T^{12} \)
$37$ \( -\)\(24\!\cdots\!00\)\( + \)\(48\!\cdots\!60\)\( T - \)\(22\!\cdots\!96\)\( T^{2} - \)\(25\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!60\)\( T^{4} + \)\(44\!\cdots\!16\)\( T^{5} - \)\(24\!\cdots\!88\)\( T^{6} - \)\(26\!\cdots\!64\)\( T^{7} + \)\(20\!\cdots\!44\)\( T^{8} + \)\(14\!\cdots\!32\)\( T^{9} - 770898061478804 T^{10} + 1352872 T^{11} + T^{12} \)
$41$ \( \)\(10\!\cdots\!36\)\( + \)\(75\!\cdots\!60\)\( T + \)\(69\!\cdots\!48\)\( T^{2} - \)\(55\!\cdots\!08\)\( T^{3} - \)\(14\!\cdots\!12\)\( T^{4} - \)\(98\!\cdots\!64\)\( T^{5} + \)\(33\!\cdots\!60\)\( T^{6} + \)\(53\!\cdots\!24\)\( T^{7} + \)\(48\!\cdots\!92\)\( T^{8} - \)\(83\!\cdots\!52\)\( T^{9} - 1667840440513000 T^{10} + 40941240 T^{11} + T^{12} \)
$43$ \( \)\(86\!\cdots\!59\)\( - \)\(27\!\cdots\!34\)\( T - \)\(12\!\cdots\!68\)\( T^{2} + \)\(33\!\cdots\!82\)\( T^{3} + \)\(18\!\cdots\!51\)\( T^{4} - \)\(38\!\cdots\!36\)\( T^{5} - \)\(10\!\cdots\!88\)\( T^{6} + \)\(15\!\cdots\!32\)\( T^{7} + \)\(24\!\cdots\!81\)\( T^{8} - \)\(24\!\cdots\!22\)\( T^{9} - 2689831814675444 T^{10} + 14142490 T^{11} + T^{12} \)
$47$ \( -\)\(69\!\cdots\!65\)\( + \)\(12\!\cdots\!98\)\( T - \)\(30\!\cdots\!12\)\( T^{2} + \)\(26\!\cdots\!54\)\( T^{3} - \)\(52\!\cdots\!81\)\( T^{4} - \)\(44\!\cdots\!40\)\( T^{5} + \)\(24\!\cdots\!00\)\( T^{6} - \)\(96\!\cdots\!24\)\( T^{7} - \)\(19\!\cdots\!23\)\( T^{8} + \)\(39\!\cdots\!42\)\( T^{9} + 2414488734172740 T^{10} - 133558002 T^{11} + T^{12} \)
$53$ \( -\)\(52\!\cdots\!95\)\( - \)\(10\!\cdots\!66\)\( T - \)\(72\!\cdots\!15\)\( T^{2} - \)\(21\!\cdots\!94\)\( T^{3} - \)\(65\!\cdots\!94\)\( T^{4} + \)\(10\!\cdots\!54\)\( T^{5} + \)\(20\!\cdots\!37\)\( T^{6} - \)\(31\!\cdots\!54\)\( T^{7} - \)\(43\!\cdots\!66\)\( T^{8} - \)\(33\!\cdots\!54\)\( T^{9} + 16487961258010953 T^{10} + 299319258 T^{11} + T^{12} \)
$59$ \( \)\(21\!\cdots\!88\)\( + \)\(14\!\cdots\!24\)\( T - \)\(21\!\cdots\!20\)\( T^{2} - \)\(19\!\cdots\!48\)\( T^{3} + \)\(44\!\cdots\!76\)\( T^{4} - \)\(35\!\cdots\!84\)\( T^{5} - \)\(20\!\cdots\!56\)\( T^{6} + \)\(17\!\cdots\!24\)\( T^{7} + \)\(38\!\cdots\!28\)\( T^{8} - \)\(24\!\cdots\!36\)\( T^{9} - 31984605246873424 T^{10} + 106431852 T^{11} + T^{12} \)
$61$ \( -\)\(36\!\cdots\!87\)\( - \)\(42\!\cdots\!84\)\( T + \)\(33\!\cdots\!86\)\( T^{2} + \)\(81\!\cdots\!44\)\( T^{3} - \)\(42\!\cdots\!57\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} + \)\(66\!\cdots\!00\)\( T^{6} + \)\(28\!\cdots\!08\)\( T^{7} + \)\(28\!\cdots\!15\)\( T^{8} - \)\(17\!\cdots\!96\)\( T^{9} - 52743815666737078 T^{10} + 262041240 T^{11} + T^{12} \)
$67$ \( \)\(41\!\cdots\!16\)\( - \)\(94\!\cdots\!08\)\( T + \)\(18\!\cdots\!48\)\( T^{2} + \)\(33\!\cdots\!80\)\( T^{3} - \)\(26\!\cdots\!40\)\( T^{4} - \)\(11\!\cdots\!84\)\( T^{5} + \)\(50\!\cdots\!72\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} - \)\(16\!\cdots\!76\)\( T^{8} - \)\(40\!\cdots\!24\)\( T^{9} - 29254876355354292 T^{10} + 489635100 T^{11} + T^{12} \)
$71$ \( \)\(52\!\cdots\!80\)\( + \)\(59\!\cdots\!44\)\( T - \)\(78\!\cdots\!84\)\( T^{2} - \)\(20\!\cdots\!20\)\( T^{3} + \)\(57\!\cdots\!16\)\( T^{4} + \)\(63\!\cdots\!40\)\( T^{5} + \)\(94\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!68\)\( T^{7} + \)\(51\!\cdots\!32\)\( T^{8} - \)\(72\!\cdots\!12\)\( T^{9} - 219544971544944416 T^{10} + 204290852 T^{11} + T^{12} \)
$73$ \( -\)\(47\!\cdots\!83\)\( - \)\(13\!\cdots\!56\)\( T + \)\(14\!\cdots\!86\)\( T^{2} + \)\(33\!\cdots\!68\)\( T^{3} - \)\(13\!\cdots\!93\)\( T^{4} - \)\(26\!\cdots\!64\)\( T^{5} + \)\(47\!\cdots\!60\)\( T^{6} + \)\(86\!\cdots\!32\)\( T^{7} - \)\(30\!\cdots\!77\)\( T^{8} - \)\(12\!\cdots\!20\)\( T^{9} - 92228348419834306 T^{10} + 673538852 T^{11} + T^{12} \)
$79$ \( \)\(21\!\cdots\!92\)\( + \)\(48\!\cdots\!08\)\( T - \)\(10\!\cdots\!80\)\( T^{2} + \)\(12\!\cdots\!08\)\( T^{3} + \)\(11\!\cdots\!60\)\( T^{4} - \)\(23\!\cdots\!68\)\( T^{5} - \)\(43\!\cdots\!84\)\( T^{6} + \)\(10\!\cdots\!32\)\( T^{7} + \)\(60\!\cdots\!04\)\( T^{8} - \)\(12\!\cdots\!16\)\( T^{9} - 383670423544602192 T^{10} + 434002980 T^{11} + T^{12} \)
$83$ \( -\)\(11\!\cdots\!73\)\( + \)\(25\!\cdots\!40\)\( T + \)\(28\!\cdots\!63\)\( T^{2} - \)\(67\!\cdots\!74\)\( T^{3} - \)\(15\!\cdots\!04\)\( T^{4} + \)\(19\!\cdots\!28\)\( T^{5} - \)\(24\!\cdots\!49\)\( T^{6} - \)\(16\!\cdots\!70\)\( T^{7} + \)\(28\!\cdots\!08\)\( T^{8} + \)\(46\!\cdots\!68\)\( T^{9} - 972043373956081121 T^{10} - 411781442 T^{11} + T^{12} \)
$89$ \( \)\(48\!\cdots\!08\)\( + \)\(14\!\cdots\!24\)\( T + \)\(18\!\cdots\!64\)\( T^{2} - \)\(52\!\cdots\!08\)\( T^{3} + \)\(15\!\cdots\!76\)\( T^{4} + \)\(50\!\cdots\!72\)\( T^{5} - \)\(41\!\cdots\!00\)\( T^{6} - \)\(18\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!80\)\( T^{8} + \)\(22\!\cdots\!16\)\( T^{9} - 2478590210474411448 T^{10} - 911678128 T^{11} + T^{12} \)
$97$ \( -\)\(78\!\cdots\!39\)\( - \)\(16\!\cdots\!86\)\( T - \)\(11\!\cdots\!63\)\( T^{2} + \)\(70\!\cdots\!14\)\( T^{3} - \)\(18\!\cdots\!70\)\( T^{4} - \)\(80\!\cdots\!98\)\( T^{5} + \)\(12\!\cdots\!89\)\( T^{6} + \)\(17\!\cdots\!06\)\( T^{7} - \)\(49\!\cdots\!26\)\( T^{8} + \)\(19\!\cdots\!30\)\( T^{9} + 3381683864484627797 T^{10} - 3589270998 T^{11} + T^{12} \)
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