Properties

Label 29.12.a.a
Level $29$
Weight $12$
Character orbit 29.a
Self dual yes
Analytic conductor $22.282$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2819522362\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} - 388180519304 x^{4} + 193065378004825 x^{3} + 1279291654973975 x^{2} - 65244901875230266 x - 758324542468966858\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 + \beta_{1} ) q^{2} + ( -89 - \beta_{1} - \beta_{3} ) q^{3} + ( 832 - 6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( -247 - 22 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} ) q^{5} + ( -2568 + 16 \beta_{1} - 3 \beta_{2} + 15 \beta_{3} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{6} + ( -4506 + 119 \beta_{1} - 6 \beta_{2} + 19 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} ) q^{7} + ( -13720 + 681 \beta_{1} - 8 \beta_{2} + 100 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} ) q^{8} + ( 30119 - 846 \beta_{1} - 37 \beta_{2} + 177 \beta_{3} - \beta_{4} - 11 \beta_{5} + 9 \beta_{6} - 7 \beta_{7} + \beta_{8} ) q^{9} +O(q^{10})\) \( q + ( -3 + \beta_{1} ) q^{2} + ( -89 - \beta_{1} - \beta_{3} ) q^{3} + ( 832 - 6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{4} + ( -247 - 22 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} ) q^{5} + ( -2568 + 16 \beta_{1} - 3 \beta_{2} + 15 \beta_{3} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{6} + ( -4506 + 119 \beta_{1} - 6 \beta_{2} + 19 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} ) q^{7} + ( -13720 + 681 \beta_{1} - 8 \beta_{2} + 100 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} + 6 \beta_{8} + 2 \beta_{9} ) q^{8} + ( 30119 - 846 \beta_{1} - 37 \beta_{2} + 177 \beta_{3} - \beta_{4} - 11 \beta_{5} + 9 \beta_{6} - 7 \beta_{7} + \beta_{8} ) q^{9} + ( -62223 - 1816 \beta_{1} - 56 \beta_{2} + 238 \beta_{3} + 10 \beta_{4} + 11 \beta_{5} - 30 \beta_{6} + 5 \beta_{7} - 14 \beta_{8} + 5 \beta_{9} ) q^{10} + ( -55626 - 1433 \beta_{1} + 42 \beta_{2} + 548 \beta_{3} + 47 \beta_{5} + 10 \beta_{7} - 43 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} ) q^{11} + ( 235149 - 8707 \beta_{1} + 9 \beta_{2} + 307 \beta_{3} + 38 \beta_{4} + 12 \beta_{5} + 41 \beta_{6} - 43 \beta_{7} - 2 \beta_{8} - 19 \beta_{10} ) q^{12} + ( 138077 - 10081 \beta_{1} + 54 \beta_{2} + 683 \beta_{3} - 46 \beta_{4} - 86 \beta_{5} - 40 \beta_{6} - 74 \beta_{7} + 61 \beta_{8} + 84 \beta_{9} + 57 \beta_{10} ) q^{13} + ( 360932 - 15213 \beta_{1} + 204 \beta_{2} + 152 \beta_{3} - 144 \beta_{4} - 177 \beta_{5} + 60 \beta_{6} - 241 \beta_{7} + 56 \beta_{8} - 127 \beta_{9} - 12 \beta_{10} ) q^{14} + ( -222460 - 15936 \beta_{1} + 476 \beta_{2} - 1357 \beta_{3} + 105 \beta_{4} - 38 \beta_{5} - 65 \beta_{6} + 253 \beta_{7} + 97 \beta_{8} + 95 \beta_{9} - 230 \beta_{10} ) q^{15} + ( 277920 - 29535 \beta_{1} + 793 \beta_{2} - 501 \beta_{3} + 92 \beta_{4} + 307 \beta_{5} + 202 \beta_{6} + 73 \beta_{7} + 24 \beta_{8} - 239 \beta_{9} + 290 \beta_{10} ) q^{16} + ( -295343 - 35576 \beta_{1} + 362 \beta_{2} - 3356 \beta_{3} + 23 \beta_{4} + 475 \beta_{5} - 43 \beta_{6} + 536 \beta_{7} - 101 \beta_{8} + 146 \beta_{9} + 104 \beta_{10} ) q^{17} + ( -2529373 - 52283 \beta_{1} - 249 \beta_{2} - 4101 \beta_{3} + 16 \beta_{4} + 159 \beta_{5} - 338 \beta_{6} + 77 \beta_{7} - 168 \beta_{8} - 361 \beta_{9} + 181 \beta_{10} ) q^{18} + ( -4006999 - 42928 \beta_{1} - 168 \beta_{2} - 4841 \beta_{3} + 16 \beta_{4} - 625 \beta_{5} - 84 \beta_{6} + 740 \beta_{7} - 477 \beta_{8} + 217 \beta_{9} - 924 \beta_{10} ) q^{19} + ( -4482870 - 135454 \beta_{1} - 4151 \beta_{2} - 13013 \beta_{3} + 800 \beta_{4} - 892 \beta_{5} - 640 \beta_{6} - 308 \beta_{7} + 288 \beta_{8} + 360 \beta_{9} - 690 \beta_{10} ) q^{20} + ( -4265977 + 3596 \beta_{1} - 480 \beta_{2} + 11134 \beta_{3} - 1911 \beta_{4} + 45 \beta_{5} + 375 \beta_{6} - 1072 \beta_{7} - 367 \beta_{8} + 478 \beta_{9} + 2778 \beta_{10} ) q^{21} + ( -3940403 - 42115 \beta_{1} - 5828 \beta_{2} - 20040 \beta_{3} - 182 \beta_{4} + 287 \beta_{5} + 1664 \beta_{6} - 1863 \beta_{7} + 411 \beta_{8} + 42 \beta_{9} - 1837 \beta_{10} ) q^{22} + ( -8064406 - 10339 \beta_{1} + 2080 \beta_{2} + 14639 \beta_{3} + 2465 \beta_{4} - 347 \beta_{5} + 663 \beta_{6} + 1599 \beta_{7} + 538 \beta_{8} + 1320 \beta_{9} + 1182 \beta_{10} ) q^{23} + ( -20445792 + 230165 \beta_{1} - 3347 \beta_{2} - 5993 \beta_{3} - 1508 \beta_{4} + 1439 \beta_{5} - 1494 \beta_{6} - 2355 \beta_{7} + 580 \beta_{8} - 1223 \beta_{9} - 198 \beta_{10} ) q^{24} + ( -4047335 + 297093 \beta_{1} + 5472 \beta_{2} + 12141 \beta_{3} - 2045 \beta_{4} + 619 \beta_{5} + 3395 \beta_{6} - 1184 \beta_{7} - 896 \beta_{8} - 1850 \beta_{9} - 1875 \beta_{10} ) q^{25} + ( -29564010 + 169973 \beta_{1} - 713 \beta_{2} + 16079 \beta_{3} + 4324 \beta_{4} + 70 \beta_{5} - 4102 \beta_{6} + 1256 \beta_{7} + 2042 \beta_{8} + 462 \beta_{9} + 1855 \beta_{10} ) q^{26} + ( -21534970 + 525531 \beta_{1} + 10250 \beta_{2} + 348 \beta_{3} - 496 \beta_{4} - 317 \beta_{5} - 4032 \beta_{6} + 2654 \beta_{7} - 1163 \beta_{8} - 3903 \beta_{9} - 6606 \beta_{10} ) q^{27} + ( -35634728 + 580142 \beta_{1} + 4934 \beta_{2} + 74178 \beta_{3} - 3104 \beta_{4} - 1716 \beta_{5} - 142 \beta_{6} - 2714 \beta_{7} - 628 \beta_{8} + 44 \beta_{9} + 6316 \beta_{10} ) q^{28} + 20511149 q^{29} + ( -45076773 + 807618 \beta_{1} + 11840 \beta_{2} + 83856 \beta_{3} + 670 \beta_{4} + 5668 \beta_{5} + 1580 \beta_{6} + 3458 \beta_{7} + 1353 \beta_{8} + 3575 \beta_{9} + 11645 \beta_{10} ) q^{30} + ( -26657337 + 675078 \beta_{1} + 28922 \beta_{2} + 97932 \beta_{3} + 10593 \beta_{4} - 4213 \beta_{5} + 5747 \beta_{6} + 13093 \beta_{7} + 122 \beta_{8} + 6086 \beta_{9} - 4604 \beta_{10} ) q^{31} + ( -57536172 + 511047 \beta_{1} - 27266 \beta_{2} - 96970 \beta_{3} - 14140 \beta_{4} - 9104 \beta_{5} + 6059 \beta_{6} - 6553 \beta_{7} - 9842 \beta_{8} - 3400 \beta_{9} - 19732 \beta_{10} ) q^{32} + ( -98310786 + 1253947 \beta_{1} + 43116 \beta_{2} + 159857 \beta_{3} + 11215 \beta_{4} + 1125 \beta_{5} - 9953 \beta_{6} + 13756 \beta_{7} + 6860 \beta_{8} + 5712 \beta_{9} + 11887 \beta_{10} ) q^{33} + ( -101252191 + 628850 \beta_{1} - 24695 \beta_{2} - 79073 \beta_{3} - 242 \beta_{4} + 4532 \beta_{5} + 22288 \beta_{6} + 3140 \beta_{7} - 3698 \beta_{8} + 11574 \beta_{9} - 12341 \beta_{10} ) q^{34} + ( -119460204 + 1230451 \beta_{1} + 18358 \beta_{2} + 46047 \beta_{3} - 14585 \beta_{4} + 10375 \beta_{5} - 9875 \beta_{6} - 4477 \beta_{7} - 8660 \beta_{8} - 16720 \beta_{9} + 9040 \beta_{10} ) q^{35} + ( -203231568 - 535388 \beta_{1} - 68234 \beta_{2} - 312570 \beta_{3} - 4204 \beta_{4} - 8470 \beta_{5} - 15638 \beta_{6} - 11844 \beta_{7} + 5408 \beta_{8} - 1906 \beta_{9} - 17648 \beta_{10} ) q^{36} + ( -125471128 - 194092 \beta_{1} - 48996 \beta_{2} + 119670 \beta_{3} - 21940 \beta_{4} - 10514 \beta_{5} - 11932 \beta_{6} - 42778 \beta_{7} + 16142 \beta_{8} + 10006 \beta_{9} + 39750 \beta_{10} ) q^{37} + ( -110716973 - 4035701 \beta_{1} - 60825 \beta_{2} - 281175 \beta_{3} + 18006 \beta_{4} + 17793 \beta_{5} + 8696 \beta_{6} + 429 \beta_{7} - 34 \beta_{8} - 5091 \beta_{9} + 36661 \beta_{10} ) q^{38} + ( -107756971 - 639000 \beta_{1} - 5078 \beta_{2} - 513016 \beta_{3} + 27655 \beta_{4} + 46847 \beta_{5} - 1803 \beta_{6} - 657 \beta_{7} + 10408 \beta_{8} - 32672 \beta_{9} - 57828 \beta_{10} ) q^{39} + ( -246040848 - 6126465 \beta_{1} - 111732 \beta_{2} - 191360 \beta_{3} + 9020 \beta_{4} - 33286 \beta_{5} - 20755 \beta_{6} - 12493 \beta_{7} - 6366 \beta_{8} + 18070 \beta_{9} + 34120 \beta_{10} ) q^{40} + ( -96235491 - 3647630 \beta_{1} - 104 \beta_{2} - 140542 \beta_{3} + 21901 \beta_{4} - 67989 \beta_{5} - 7837 \beta_{6} - 5864 \beta_{7} + 10173 \beta_{8} - 13812 \beta_{9} - 53552 \beta_{10} ) q^{41} + ( 19486805 - 7304440 \beta_{1} + 2139 \beta_{2} - 930499 \beta_{3} + 37994 \beta_{4} - 24804 \beta_{5} + 34664 \beta_{6} + 15204 \beta_{7} + 23450 \beta_{8} - 21214 \beta_{9} - 86815 \beta_{10} ) q^{42} + ( 6578338 + 1569233 \beta_{1} + 53152 \beta_{2} + 607304 \beta_{3} - 82152 \beta_{4} + 64783 \beta_{5} + 49636 \beta_{6} - 36332 \beta_{7} - 42373 \beta_{8} + \beta_{9} + 58708 \beta_{10} ) q^{43} + ( 5628589 - 8536683 \beta_{1} + 131387 \beta_{2} - 166215 \beta_{3} + 24806 \beta_{4} + 88058 \beta_{5} - 25033 \beta_{6} + 124729 \beta_{7} - 47730 \beta_{8} + 14598 \beta_{9} + 88725 \beta_{10} ) q^{44} + ( 411514218 - 316517 \beta_{1} + 151219 \beta_{2} + 999066 \beta_{3} - 76090 \beta_{4} - 14920 \beta_{5} + 80860 \beta_{6} - 98901 \beta_{7} + 21245 \beta_{8} + 1250 \beta_{9} + 66835 \beta_{10} ) q^{45} + ( -7346060 - 6443101 \beta_{1} - 85470 \beta_{2} - 594198 \beta_{3} - 72976 \beta_{4} - 90985 \beta_{5} - 56828 \beta_{6} + 23979 \beta_{7} + 8226 \beta_{8} + 74815 \beta_{9} - 76988 \beta_{10} ) q^{46} + ( -166444564 + 5296906 \beta_{1} + 345658 \beta_{2} + 1379021 \beta_{3} + 61955 \beta_{4} - 57866 \beta_{5} - 51103 \beta_{6} + 181493 \beta_{7} - 33255 \beta_{8} + 25113 \beta_{9} - 55340 \beta_{10} ) q^{47} + ( 242932248 - 7179333 \beta_{1} + 277362 \beta_{2} + 362034 \beta_{3} - 16740 \beta_{4} + 20880 \beta_{5} - 89937 \beta_{6} + 19851 \beta_{7} - 9210 \beta_{8} - 33720 \beta_{9} + 69408 \beta_{10} ) q^{48} + ( 426751391 - 3466290 \beta_{1} - 351534 \beta_{2} + 442290 \beta_{3} + 201548 \beta_{4} + 7682 \beta_{5} - 988 \beta_{6} + 81674 \beta_{7} + 38128 \beta_{8} + 203122 \beta_{9} - 141618 \beta_{10} ) q^{49} + ( 862733174 + 3806556 \beta_{1} + 657840 \beta_{2} + 328582 \beta_{3} - 69030 \beta_{4} + 193256 \beta_{5} + 160410 \beta_{6} + 125146 \beta_{7} - 19044 \beta_{8} - 117870 \beta_{9} + 81070 \beta_{10} ) q^{50} + ( 796088256 + 8269235 \beta_{1} - 70988 \beta_{2} + 2235833 \beta_{3} - 16875 \beta_{4} + 17249 \beta_{5} - 33793 \beta_{6} - 197003 \beta_{7} + 96628 \beta_{8} - 29308 \beta_{9} + 205754 \beta_{10} ) q^{51} + ( 298077230 - 10891490 \beta_{1} - 421693 \beta_{2} - 1076247 \beta_{3} + 43976 \beta_{4} - 145446 \beta_{5} - 135246 \beta_{6} - 94020 \beta_{7} - 35048 \beta_{8} - 53866 \beta_{9} - 235494 \beta_{10} ) q^{52} + ( 709231185 + 16180109 \beta_{1} - 434402 \beta_{2} + 466915 \beta_{3} - 160022 \beta_{4} - 164208 \beta_{5} + 4648 \beta_{6} - 181782 \beta_{7} - 60739 \beta_{8} - 78890 \beta_{9} - 176385 \beta_{10} ) q^{53} + ( 1582239197 - 2620595 \beta_{1} + 400908 \beta_{2} + 1144152 \beta_{3} - 3110 \beta_{4} - 17265 \beta_{5} + 116272 \beta_{6} - 314503 \beta_{7} + 185739 \beta_{8} - 158086 \beta_{9} + 242755 \beta_{10} ) q^{54} + ( -20524739 + 34383792 \beta_{1} - 651028 \beta_{2} + 1696914 \beta_{3} + 2865 \beta_{4} + 240473 \beta_{5} + 34615 \beta_{6} - 56545 \beta_{7} - 154262 \beta_{8} - 180100 \beta_{9} + 95150 \beta_{10} ) q^{55} + ( 1024948024 - 6622432 \beta_{1} - 80206 \beta_{2} - 2919042 \beta_{3} + 309712 \beta_{4} + 137446 \beta_{5} - 121382 \beta_{6} + 359120 \beta_{7} + 84188 \beta_{8} + 153282 \beta_{9} - 181052 \beta_{10} ) q^{56} + ( 1405952650 + 26334737 \beta_{1} - 430977 \beta_{2} + 2560032 \beta_{3} - 58822 \beta_{4} - 271968 \beta_{5} + 286448 \beta_{6} - 468971 \beta_{7} - 31893 \beta_{8} - 85010 \beta_{9} + 41147 \beta_{10} ) q^{57} + ( -61533447 + 20511149 \beta_{1} ) q^{58} + ( 107609610 + 51877587 \beta_{1} - 668322 \beta_{2} - 2458741 \beta_{3} - 22727 \beta_{4} + 214513 \beta_{5} - 247117 \beta_{6} + 515345 \beta_{7} + 67008 \beta_{8} + 300312 \beta_{9} + 100740 \beta_{10} ) q^{59} + ( 2894770751 - 5673699 \beta_{1} + 7623 \beta_{2} - 1040963 \beta_{3} - 360270 \beta_{4} - 180870 \beta_{5} + 294375 \beta_{6} - 520607 \beta_{7} - 69290 \beta_{8} - 42850 \beta_{9} - 58965 \beta_{10} ) q^{60} + ( -1696882937 + 64191096 \beta_{1} - 1322730 \beta_{2} + 526454 \beta_{3} - 215097 \beta_{4} - 168823 \beta_{5} - 323175 \beta_{6} + 35444 \beta_{7} - 105395 \beta_{8} + 240364 \beta_{9} + 268024 \beta_{10} ) q^{61} + ( 2006597360 + 8514447 \beta_{1} + 1108755 \beta_{2} - 3661739 \beta_{3} - 355600 \beta_{4} - 43017 \beta_{5} + 18380 \beta_{6} + 200637 \beta_{7} + 140637 \beta_{8} + 179124 \beta_{9} + 72068 \beta_{10} ) q^{62} + ( -909642564 + 46874368 \beta_{1} + 242998 \beta_{2} - 6095532 \beta_{3} + 824180 \beta_{4} + 490694 \beta_{5} - 526556 \beta_{6} + 1227396 \beta_{7} - 258274 \beta_{8} + 103046 \beta_{9} - 833486 \beta_{10} ) q^{63} + ( 1078048152 - 39272793 \beta_{1} - 295827 \beta_{2} - 1161473 \beta_{3} + 383140 \beta_{4} + 556319 \beta_{5} - 76972 \beta_{6} + 246199 \beta_{7} - 418100 \beta_{8} - 69403 \beta_{9} + 401682 \beta_{10} ) q^{64} + ( 2930353846 + 17315104 \beta_{1} + 970635 \beta_{2} - 4064637 \beta_{3} + 118545 \beta_{4} - 626861 \beta_{5} + 645355 \beta_{6} + 303269 \beta_{7} + 451389 \beta_{8} + 364560 \beta_{9} + 176810 \beta_{10} ) q^{65} + ( 3883516149 - 43261279 \beta_{1} + 752732 \beta_{2} - 2749930 \beta_{3} - 325930 \beta_{4} - 1104482 \beta_{5} - 196194 \beta_{6} - 667228 \beta_{7} + 719128 \beta_{8} + 337044 \beta_{9} - 552342 \beta_{10} ) q^{66} + ( 2500545660 - 1583404 \beta_{1} - 1070742 \beta_{2} - 7268112 \beta_{3} - 93612 \beta_{4} + 140050 \beta_{5} + 536132 \beta_{6} - 329148 \beta_{7} - 68502 \beta_{8} - 970718 \beta_{9} + 339534 \beta_{10} ) q^{67} + ( 2695933886 - 73855472 \beta_{1} + 2732430 \beta_{2} + 2695554 \beta_{3} + 105676 \beta_{4} + 803530 \beta_{5} + 512602 \beta_{6} + 478628 \beta_{7} - 862760 \beta_{8} - 34734 \beta_{9} + 273698 \beta_{10} ) q^{68} + ( -1873668333 + 42001446 \beta_{1} + 1889926 \beta_{2} - 303828 \beta_{3} - 467809 \beta_{4} + 770021 \beta_{5} - 726923 \beta_{6} - 35108 \beta_{7} - 1036137 \beta_{8} - 603800 \beta_{9} - 399470 \beta_{10} ) q^{69} + ( 3905380574 - 91504757 \beta_{1} - 689782 \beta_{2} - 1710134 \beta_{3} - 63260 \beta_{4} - 674513 \beta_{5} + 716300 \beta_{6} - 1127585 \beta_{7} + 655052 \beta_{8} - 601635 \beta_{9} - 568970 \beta_{10} ) q^{70} + ( -1856525358 + 75896938 \beta_{1} + 3548868 \beta_{2} - 716286 \beta_{3} - 1274880 \beta_{4} - 725996 \beta_{5} - 325720 \beta_{6} - 43300 \beta_{7} + 1118956 \beta_{8} - 496884 \beta_{9} + 1170664 \beta_{10} ) q^{71} + ( 4291072180 - 174943400 \beta_{1} + 25010 \beta_{2} + 17578998 \beta_{3} + 898320 \beta_{4} + 181834 \beta_{5} - 80114 \beta_{6} - 278120 \beta_{7} - 208700 \beta_{8} + 744998 \beta_{9} + 730720 \beta_{10} ) q^{72} + ( -5248760606 + 11110136 \beta_{1} - 1030660 \beta_{2} - 4801634 \beta_{3} + 991762 \beta_{4} + 1918068 \beta_{5} - 537010 \beta_{6} + 518718 \beta_{7} + 264252 \beta_{8} + 498402 \beta_{9} + 230122 \beta_{10} ) q^{73} + ( -220924150 - 224134436 \beta_{1} + 265262 \beta_{2} + 2343438 \beta_{3} + 1128112 \beta_{4} - 613228 \beta_{5} - 1267588 \beta_{6} - 18152 \beta_{7} + 185312 \beta_{8} - 15912 \beta_{9} - 624270 \beta_{10} ) q^{74} + ( -2139958372 - 29997372 \beta_{1} + 574694 \beta_{2} + 7427206 \beta_{3} + 431050 \beta_{4} - 1264610 \beta_{5} - 269590 \beta_{6} + 334884 \beta_{7} + 531610 \beta_{8} + 498760 \beta_{9} - 556210 \beta_{10} ) q^{75} + ( -3024255746 - 89285026 \beta_{1} - 8004180 \beta_{2} + 9535748 \beta_{3} - 1100412 \beta_{4} - 371470 \beta_{5} - 197128 \beta_{6} - 1156574 \beta_{7} + 150212 \beta_{8} + 474242 \beta_{9} - 372026 \beta_{10} ) q^{76} + ( 94462419 - 142219720 \beta_{1} + 59354 \beta_{2} - 14446342 \beta_{3} - 132483 \beta_{4} - 434253 \beta_{5} + 1402923 \beta_{6} - 1954896 \beta_{7} - 208229 \beta_{8} - 690428 \beta_{9} - 1864080 \beta_{10} ) q^{77} + ( -1403156766 - 20854473 \beta_{1} + 876417 \beta_{2} + 40285319 \beta_{3} - 1168964 \beta_{4} + 1823187 \beta_{5} + 337756 \beta_{6} + 166577 \beta_{7} - 1502211 \beta_{8} + 447608 \beta_{9} + 1393498 \beta_{10} ) q^{78} + ( -10943698544 + 147150496 \beta_{1} + 223356 \beta_{2} - 6531229 \beta_{3} + 323313 \beta_{4} + 233514 \beta_{5} + 2140095 \beta_{6} + 1231009 \beta_{7} + 419105 \beta_{8} - 1486561 \beta_{9} - 1887594 \beta_{10} ) q^{79} + ( -7654496488 - 142258421 \beta_{1} - 4904921 \beta_{2} + 14091733 \beta_{3} - 682060 \beta_{4} - 646339 \beta_{5} - 961630 \beta_{6} - 34205 \beta_{7} - 672224 \beta_{8} + 86735 \beta_{9} + 85510 \beta_{10} ) q^{80} + ( -4454097091 + 50097957 \beta_{1} + 3122711 \beta_{2} + 37621050 \beta_{3} - 554326 \beta_{4} + 611494 \beta_{5} + 1717284 \beta_{6} - 1078303 \beta_{7} + 67345 \beta_{8} + 2390832 \beta_{9} + 3353763 \beta_{10} ) q^{81} + ( -10132522847 - 54491058 \beta_{1} - 5302541 \beta_{2} + 19293489 \beta_{3} - 272890 \beta_{4} - 703406 \beta_{5} - 2198708 \beta_{6} - 296666 \beta_{7} + 513802 \beta_{8} - 807888 \beta_{9} + 2854509 \beta_{10} ) q^{82} + ( -12933467766 - 161295045 \beta_{1} - 2841940 \beta_{2} - 22153545 \beta_{3} - 822671 \beta_{4} - 2872037 \beta_{5} - 1488733 \beta_{6} - 565939 \beta_{7} + 54258 \beta_{8} + 1809966 \beta_{9} + 26598 \beta_{10} ) q^{83} + ( -12215166342 + 161418044 \beta_{1} - 1431284 \beta_{2} + 25112004 \beta_{3} + 2214212 \beta_{4} + 2322806 \beta_{5} - 1296002 \beta_{6} + 5520596 \beta_{7} - 1281080 \beta_{8} - 845634 \beta_{9} - 1944266 \beta_{10} ) q^{84} + ( -16509585335 + 22615906 \beta_{1} - 2815936 \beta_{2} - 23991708 \beta_{3} + 1349485 \beta_{4} - 83947 \beta_{5} - 791225 \beta_{6} + 1790732 \beta_{7} - 170117 \beta_{8} + 795330 \beta_{9} - 68180 \beta_{10} ) q^{85} + ( 4356488227 + 2092355 \beta_{1} + 5764394 \beta_{2} - 29412630 \beta_{3} + 678678 \beta_{4} + 3040473 \beta_{5} + 4395912 \beta_{6} + 1465603 \beta_{7} - 167929 \beta_{8} - 1295380 \beta_{9} - 2697267 \beta_{10} ) q^{86} + ( -1825492261 - 20511149 \beta_{1} - 20511149 \beta_{3} ) q^{87} + ( -16449011748 + 320731829 \beta_{1} - 7421233 \beta_{2} - 4876779 \beta_{3} - 1422476 \beta_{4} - 5409591 \beta_{5} + 703912 \beta_{6} + 1159157 \beta_{7} + 802088 \beta_{8} + 2214343 \beta_{9} - 2916078 \beta_{10} ) q^{88} + ( -8753227027 - 314512296 \beta_{1} + 3813862 \beta_{2} - 49778846 \beta_{3} + 591533 \beta_{4} - 1518741 \beta_{5} - 774977 \beta_{6} - 6073398 \beta_{7} + 410435 \beta_{8} - 1751452 \beta_{9} + 747446 \beta_{10} ) q^{89} + ( -2374729663 + 567272019 \beta_{1} + 11212799 \beta_{2} + 10557693 \beta_{3} - 504370 \beta_{4} + 2544301 \beta_{5} + 1235820 \beta_{6} + 4154345 \beta_{7} + 753676 \beta_{8} - 2186745 \beta_{9} - 504105 \beta_{10} ) q^{90} + ( -32315555448 + 399235807 \beta_{1} + 5792756 \beta_{2} - 44644429 \beta_{3} - 262237 \beta_{4} + 1356961 \beta_{5} - 3629991 \beta_{6} + 2103275 \beta_{7} + 1232522 \beta_{8} - 1051478 \beta_{9} + 2619378 \beta_{10} ) q^{91} + ( -2019070714 - 101104836 \beta_{1} + 1474868 \beta_{2} - 38334660 \beta_{3} + 1877892 \beta_{4} + 5558092 \beta_{5} - 1775024 \beta_{6} - 2067260 \beta_{7} - 966416 \beta_{8} - 3368364 \beta_{9} + 4484650 \beta_{10} ) q^{92} + ( -15711320453 + 207220538 \beta_{1} + 7290739 \beta_{2} + 10738103 \beta_{3} - 2639278 \beta_{4} + 1230080 \beta_{5} - 1439558 \beta_{6} - 4085957 \beta_{7} - 4868820 \beta_{8} - 1719314 \beta_{9} - 1391420 \beta_{10} ) q^{93} + ( 15681121199 + 246965936 \beta_{1} + 1069102 \beta_{2} - 60884582 \beta_{3} - 1431122 \beta_{4} - 4554286 \beta_{5} + 1803396 \beta_{6} - 6093708 \beta_{7} + 6040767 \beta_{8} - 748917 \beta_{9} + 1069361 \beta_{10} ) q^{94} + ( -17816809515 + 147471674 \beta_{1} + 15045226 \beta_{2} - 55126707 \beta_{3} + 1431190 \beta_{4} + 845327 \beta_{5} + 1514150 \beta_{6} + 2753318 \beta_{7} + 3848037 \beta_{8} + 153735 \beta_{9} - 42370 \beta_{10} ) q^{95} + ( 20552062680 + 256650539 \beta_{1} - 11636303 \beta_{2} + 22974859 \beta_{3} + 2364628 \beta_{4} - 9831973 \beta_{5} + 4146252 \beta_{6} - 1332549 \beta_{7} + 4267900 \beta_{8} + 1939081 \beta_{9} - 3049158 \beta_{10} ) q^{96} + ( -27581067407 + 179003206 \beta_{1} + 4417834 \beta_{2} - 3593354 \beta_{3} - 602383 \beta_{4} + 1706673 \beta_{5} - 328513 \beta_{6} - 3935736 \beta_{7} - 5609989 \beta_{8} + 2986050 \beta_{9} - 2934054 \beta_{10} ) q^{97} + ( -11227470195 - 258903107 \beta_{1} + 7343904 \beta_{2} - 20878288 \beta_{3} + 3747536 \beta_{4} + 7858452 \beta_{5} - 8044672 \beta_{6} + 3849828 \beta_{7} - 7245232 \beta_{8} + 8695740 \beta_{9} + 4359384 \beta_{10} ) q^{98} + ( -12690985185 + 382487252 \beta_{1} - 6188866 \beta_{2} + 64229373 \beta_{3} - 1661024 \beta_{4} + 1446655 \beta_{5} - 684760 \beta_{6} - 811038 \beta_{7} + 1243309 \beta_{8} - 363395 \beta_{9} + 3083534 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} + O(q^{10}) \) \( 11q - 32q^{2} - 982q^{3} + 9146q^{4} - 2740q^{5} - 28202q^{6} - 49432q^{7} - 150054q^{8} + 330749q^{9} - 685834q^{10} - 612246q^{11} + 2578538q^{12} + 1510364q^{13} + 3955400q^{14} - 2462818q^{15} + 3024818q^{16} - 3291098q^{17} - 27885614q^{18} - 44121388q^{19} - 49472662q^{20} - 46916800q^{21} - 43435618q^{22} - 88684076q^{23} - 224700678q^{24} - 44195521q^{25} - 324999762q^{26} - 236304286q^{27} - 391274848q^{28} + 225622639q^{29} - 494910382q^{30} - 292235934q^{31} - 632542514q^{32} - 1079766410q^{33} - 1113307936q^{34} - 1312820120q^{35} - 2236726492q^{36} - 1380429338q^{37} - 1222857284q^{38} - 1186931090q^{39} - 2713154106q^{40} - 1062067494q^{41} + 205598960q^{42} + 74588594q^{43} + 52891466q^{44} + 4527996830q^{45} - 87670324q^{46} - 1821239394q^{47} + 2666035542q^{48} + 4692522003q^{49} + 9494259926q^{50} + 8768158380q^{51} + 3266669866q^{52} + 7818635688q^{53} + 17402728558q^{54} - 191002682q^{55} + 11263587512q^{56} + 15495358340q^{57} - 656356768q^{58} + 1230002712q^{59} + 31834046430q^{60} - 18602654230q^{61} + 22075953162q^{62} - 9964531456q^{63} + 11813658086q^{64} + 32245789334q^{65} + 42677188354q^{66} + 27481284652q^{67} + 29588811820q^{68} - 20565315068q^{69} + 42862666712q^{70} - 20347168516q^{71} + 47061083616q^{72} - 57740010478q^{73} - 2640709564q^{74} - 23544691000q^{75} - 33350650772q^{76} + 871959792q^{77} - 15384525342q^{78} - 120245016462q^{79} - 84319695274q^{80} - 48880047865q^{81} - 111495532412q^{82} - 142463983824q^{83} - 134146226376q^{84} - 181628566552q^{85} + 47870165542q^{86} - 20141948318q^{87} - 180608014462q^{88} - 96700717270q^{89} - 25522461244q^{90} - 355162031176q^{91} - 22429477796q^{92} - 172582115142q^{93} + 172608565078q^{94} - 195922150708q^{95} + 226391047758q^{96} - 303190852014q^{97} - 123776497136q^{98} - 139125462440q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - x^{10} - 15790 x^{9} + 14666 x^{8} + 87206462 x^{7} - 14008334 x^{6} - 203974096304 x^{5} - 388180519304 x^{4} + 193065378004825 x^{3} + 1279291654973975 x^{2} - 65244901875230266 x - 758324542468966858\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(565550727753207886291601 \nu^{10} - 12036436634048687917592128 \nu^{9} - 8679501631223090279693232390 \nu^{8} + 184357050811285113015962991580 \nu^{7} + 45484105320154672732669722806418 \nu^{6} - 930882828848581596917379957802196 \nu^{5} - 95992437020855812880527329597716284 \nu^{4} + 1730622925673756035085534196275520532 \nu^{3} + 73201787045201984178508972133624628997 \nu^{2} - 769094726496573889613325678204209667292 \nu - 20927726404008919633603863043721530286566\)\()/ \)\(44\!\cdots\!80\)\( \)
\(\beta_{3}\)\(=\)\((\)\(565550727753207886291601 \nu^{10} - 12036436634048687917592128 \nu^{9} - 8679501631223090279693232390 \nu^{8} + 184357050811285113015962991580 \nu^{7} + 45484105320154672732669722806418 \nu^{6} - 930882828848581596917379957802196 \nu^{5} - 95992437020855812880527329597716284 \nu^{4} + 1730622925673756035085534196275520532 \nu^{3} + 73156959763450908017776739864204986117 \nu^{2} - 769094726496573889613325678204209667292 \nu - 20799027278101579976141624198217735578086\)\()/ \)\(44\!\cdots\!80\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-8617734043333765972515011 \nu^{10} + 188341917972914172692555076 \nu^{9} + 132130528398360410442463524558 \nu^{8} - 2882017436490520158113317655848 \nu^{7} - 691197098684240636958802794767090 \nu^{6} + 14536110000730915264602865705858432 \nu^{5} + 1453615787430263809984772774473389120 \nu^{4} - 26974973355615002420363496826985036440 \nu^{3} - 1099383289792373686347690632223703779603 \nu^{2} + 11925121211898856721840144023427382234364 \nu + 312559249114659251991080429495174026305978\)\()/ \)\(22\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-18253212080138340670261163 \nu^{10} + 398630822978303275149536136 \nu^{9} + 280129695981365998688208550362 \nu^{8} - 6101878256249561270150643949852 \nu^{7} - 1467847187148692280207295574986142 \nu^{6} + 30798356972142673983308324764715812 \nu^{5} + 3097069696455885041998598320472670636 \nu^{4} - 57250931664527986263205560544621895188 \nu^{3} - 2360309853726266833540730627197739464335 \nu^{2} + 25473320247181162976551503226618208561284 \nu + 675599119758130342212747344724863912013906\)\()/ \)\(44\!\cdots\!80\)\( \)
\(\beta_{6}\)\(=\)\((\)\(841395584194378181759585 \nu^{10} - 18245755519380040681256638 \nu^{9} - 12909656201176443534612689028 \nu^{8} + 279395024272136394566958154498 \nu^{7} + 67617752793492556772000337827952 \nu^{6} - 1410586119553191073447280915606786 \nu^{5} - 142555398591710649231641994417033586 \nu^{4} + 2621954921286447650030743738715890678 \nu^{3} + 108416900378812514797243463964927169471 \nu^{2} - 1164447572682571827309503098448553126712 \nu - 30923893245825780985494928113309422281882\)\()/ \)\(14\!\cdots\!40\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-925017894146270556324001 \nu^{10} + 20020628115330855489561920 \nu^{9} + 14189084609441700734893823398 \nu^{8} - 306532457196252549964930263196 \nu^{7} - 74284093661778417907203960210674 \nu^{6} + 1547182122942695217095069951118356 \nu^{5} + 156457328858201747863434643585273916 \nu^{4} - 2874126879942643802269885113303822420 \nu^{3} - 118709055269937646660049496797552834581 \nu^{2} + 1273304945394343953001065199684882896796 \nu + 33771289290772363759198711089555493152710\)\()/ \)\(14\!\cdots\!96\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-27908032678943417691002759 \nu^{10} + 605844463382525696019675600 \nu^{9} + 428170820570740581400081085178 \nu^{8} - 9274926479518936488813420862228 \nu^{7} - 2242474653313153941718504125018254 \nu^{6} + 46812781280887723864888563727951900 \nu^{5} + 4727302729085860425305643186737329812 \nu^{4} - 86983674497444326713246768304045471036 \nu^{3} - 3595184258614367514173647357237006385859 \nu^{2} + 38605650332799862741851949261582228692580 \nu + 1025593800786234040434960173932114721796426\)\()/ \)\(44\!\cdots\!80\)\( \)
\(\beta_{9}\)\(=\)\((\)\(33973970326349424743113419 \nu^{10} - 737609653830949686946857848 \nu^{9} - 521121425152810216331919598506 \nu^{8} + 11293380223682446673257828724556 \nu^{7} + 2728174423872430669394479171963886 \nu^{6} - 57005462222711052851407187921556116 \nu^{5} - 5746363770084293528113765517228744828 \nu^{4} + 105919637239176542630277984205034178084 \nu^{3} + 4361635127250046445927378755230283850975 \nu^{2} - 46975640694751754085393580157732622901092 \nu - 1242622320057593890625300420336697336483698\)\()/ \)\(44\!\cdots\!80\)\( \)
\(\beta_{10}\)\(=\)\((\)\(5345188872548264358430849 \nu^{10} - 116488587218229602938650760 \nu^{9} - 81994662642565837937773565118 \nu^{8} + 1783081175637602770090930982468 \nu^{7} + 429302622373383239489315089198154 \nu^{6} - 8998290706457080140595401225067740 \nu^{5} - 904392561662989340527170290955030772 \nu^{4} + 16716368121011272585734443649221894156 \nu^{3} + 686686412326105872966554449189302470269 \nu^{2} - 7414761537943440259758156759650525036780 \nu - 195792859844202499960390007863937164600726\)\()/ \)\(56\!\cdots\!60\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + 2871\)
\(\nu^{3}\)\(=\)\(2 \beta_{9} + 6 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 91 \beta_{3} + \beta_{2} + 4750 \beta_{1} - 142\)
\(\nu^{4}\)\(=\)\(290 \beta_{10} - 215 \beta_{9} + 96 \beta_{8} + 133 \beta_{7} + 238 \beta_{6} + 283 \beta_{5} + 44 \beta_{4} - 5499 \beta_{3} + 6895 \beta_{2} - 9291 \beta_{1} + 13621517\)
\(\nu^{5}\)\(=\)\(-15382 \beta_{10} + 9579 \beta_{9} + 40210 \beta_{8} + 35952 \beta_{7} + 33935 \beta_{6} - 21063 \beta_{5} - 45888 \beta_{4} + 631285 \beta_{3} + 10803 \beta_{2} + 26494049 \beta_{1} - 27734024\)
\(\nu^{6}\)\(=\)\(3055256 \beta_{10} - 2314236 \beta_{9} + 541720 \beta_{8} + 1625600 \beta_{7} + 2571828 \beta_{6} + 3281580 \beta_{5} + 491136 \beta_{4} - 31884599 \beta_{3} + 44836183 \beta_{2} - 87489984 \beta_{1} + 75938887363\)
\(\nu^{7}\)\(=\)\(-179336896 \beta_{10} + 49362454 \beta_{9} + 251488850 \beta_{8} + 235468575 \beta_{7} + 290285801 \beta_{6} - 166788950 \beta_{5} - 375470924 \beta_{4} + 3882662425 \beta_{3} + 30194683 \beta_{2} + 159332975544 \beta_{1} - 258006933466\)
\(\nu^{8}\)\(=\)\(24799739046 \beta_{10} - 18301020869 \beta_{9} + 1812340208 \beta_{8} + 14744018455 \beta_{7} + 20669727602 \beta_{6} + 28312895697 \beta_{5} + 4291306084 \beta_{4} - 196607594119 \beta_{3} + 294310414339 \beta_{2} - 710843139993 \beta_{1} + 456567119229865\)
\(\nu^{9}\)\(=\)\(-1548427764434 \beta_{10} + 309797115173 \beta_{9} + 1598854878186 \beta_{8} + 1528245562510 \beta_{7} + 2220410819223 \beta_{6} - 1212888034969 \beta_{5} - 2754803366360 \beta_{4} + 23727201352025 \beta_{3} - 183123620537 \beta_{2} + 1002262429256911 \beta_{1} - 2084577083203884\)
\(\nu^{10}\)\(=\)\(184293327453748 \beta_{10} - 131088051301278 \beta_{9} + 413168836200 \beta_{8} + 117756171358734 \beta_{7} + 150082167079296 \beta_{6} + 218639774363846 \beta_{5} + 34302882819528 \beta_{4} - 1257028984535199 \beta_{3} + 1952786614226359 \beta_{2} - 5625002780772818 \beta_{1} + 2871526413758164991\)

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
−82.7218
−65.3340
−56.7551
−20.7638
−20.7036
−18.9987
30.4581
33.9131
54.9918
65.2743
81.6399
−85.7218 234.308 5300.23 −1506.26 −20085.3 −7069.49 −278787. −122247. 129120.
1.2 −68.3340 537.644 2621.54 8265.40 −36739.4 −68659.4 −39192.0 111914. −564808.
1.3 −59.7551 −392.705 1522.68 −6379.08 23466.1 −23005.8 31390.8 −22929.7 381183.
1.4 −23.7638 −595.785 −1483.28 −5297.67 14158.1 80499.9 83916.8 177813. 125893.
1.5 −23.7036 −804.018 −1486.14 6643.74 19058.1 −61145.1 83771.9 469298. −157481.
1.6 −21.9987 310.394 −1564.06 −886.321 −6828.28 44263.6 79460.6 −80802.4 19497.9
1.7 27.4581 −457.582 −1294.05 9990.24 −12564.3 28164.0 −91766.4 32234.4 274313.
1.8 30.9131 543.939 −1092.38 −6165.86 16814.8 8452.88 −97078.9 118722. −190606.
1.9 51.9918 135.432 655.142 3682.79 7041.33 −83752.5 −72417.1 −158805. 191474.
1.10 62.2743 −384.675 1830.09 1377.47 −23955.4 33050.3 −13570.0 −29172.4 85780.8
1.11 78.6399 −108.951 4136.23 −12464.4 −8567.92 −230.276 164218. −165277. −980202.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(-1\)

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 29.12.a.a 11
3.b odd 2 1 261.12.a.a 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.12.a.a 11 1.a even 1 1 trivial
261.12.a.a 11 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(33\!\cdots\!44\)\( T_{2}^{4} + \)\(17\!\cdots\!40\)\( T_{2}^{3} + \)\(29\!\cdots\!12\)\( T_{2}^{2} - \)\(52\!\cdots\!44\)\( T_{2} - \)\(93\!\cdots\!40\)\( \)">\(T_{2}^{11} + \cdots\) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(29))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -937413686027223040 - 52480495829254144 T + 2941272354394112 T^{2} + 170288442403840 T^{3} - 3367672715744 T^{4} - 187882927184 T^{5} + 1785306736 T^{6} + 82465976 T^{7} - 407614 T^{8} - 15325 T^{9} + 32 T^{10} + T^{11} \)
$3$ \( -\)\(10\!\cdots\!92\)\( - \)\(89\!\cdots\!81\)\( T + \)\(10\!\cdots\!78\)\( T^{2} - \)\(15\!\cdots\!55\)\( T^{3} - 23030522354389071960 T^{4} - 4806592605798138 T^{5} + 207342942225876 T^{6} + 125589329298 T^{7} - 766600680 T^{8} - 657521 T^{9} + 982 T^{10} + T^{11} \)
$5$ \( -\)\(96\!\cdots\!50\)\( - \)\(97\!\cdots\!75\)\( T + \)\(69\!\cdots\!00\)\( T^{2} + \)\(60\!\cdots\!25\)\( T^{3} - \)\(90\!\cdots\!00\)\( T^{4} - \)\(57\!\cdots\!50\)\( T^{5} + 31578273416969866000 T^{6} + 18865845213698850 T^{7} - 446617757630 T^{8} - 242703127 T^{9} + 2740 T^{10} + T^{11} \)
$7$ \( -\)\(36\!\cdots\!16\)\( - \)\(16\!\cdots\!56\)\( T - \)\(66\!\cdots\!68\)\( T^{2} + \)\(31\!\cdots\!28\)\( T^{3} - \)\(37\!\cdots\!20\)\( T^{4} - \)\(72\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} + 46152539677908554560 T^{7} - 471818114692928 T^{8} - 11999796776 T^{9} + 49432 T^{10} + T^{11} \)
$11$ \( -\)\(73\!\cdots\!76\)\( - \)\(30\!\cdots\!69\)\( T + \)\(92\!\cdots\!02\)\( T^{2} + \)\(72\!\cdots\!01\)\( T^{3} - \)\(32\!\cdots\!80\)\( T^{4} - \)\(13\!\cdots\!22\)\( T^{5} + \)\(27\!\cdots\!92\)\( T^{6} + \)\(73\!\cdots\!58\)\( T^{7} - 748874025379876176 T^{8} - 1490380632137 T^{9} + 612246 T^{10} + T^{11} \)
$13$ \( \)\(12\!\cdots\!62\)\( - \)\(31\!\cdots\!03\)\( T - \)\(23\!\cdots\!12\)\( T^{2} + \)\(26\!\cdots\!65\)\( T^{3} + \)\(54\!\cdots\!40\)\( T^{4} - \)\(48\!\cdots\!78\)\( T^{5} - \)\(42\!\cdots\!92\)\( T^{6} + \)\(33\!\cdots\!62\)\( T^{7} + 13701895344310761834 T^{8} - 9745795139479 T^{9} - 1510364 T^{10} + T^{11} \)
$17$ \( \)\(18\!\cdots\!04\)\( - \)\(38\!\cdots\!52\)\( T - \)\(41\!\cdots\!92\)\( T^{2} + \)\(12\!\cdots\!32\)\( T^{3} + \)\(18\!\cdots\!28\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!16\)\( T^{6} + \)\(72\!\cdots\!48\)\( T^{7} - \)\(16\!\cdots\!96\)\( T^{8} - 152135401069884 T^{9} + 3291098 T^{10} + T^{11} \)
$19$ \( \)\(81\!\cdots\!72\)\( - \)\(45\!\cdots\!36\)\( T - \)\(30\!\cdots\!96\)\( T^{2} - \)\(18\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!08\)\( T^{4} + \)\(16\!\cdots\!52\)\( T^{5} - \)\(23\!\cdots\!52\)\( T^{6} - \)\(18\!\cdots\!64\)\( T^{7} - \)\(95\!\cdots\!56\)\( T^{8} + 379131733968340 T^{9} + 44121388 T^{10} + T^{11} \)
$23$ \( -\)\(23\!\cdots\!08\)\( + \)\(39\!\cdots\!44\)\( T + \)\(81\!\cdots\!52\)\( T^{2} - \)\(10\!\cdots\!04\)\( T^{3} - \)\(13\!\cdots\!20\)\( T^{4} + \)\(21\!\cdots\!72\)\( T^{5} + \)\(33\!\cdots\!40\)\( T^{6} - \)\(23\!\cdots\!28\)\( T^{7} - \)\(30\!\cdots\!52\)\( T^{8} - 1974969876252652 T^{9} + 88684076 T^{10} + T^{11} \)
$29$ \( ( -20511149 + T )^{11} \)
$31$ \( \)\(15\!\cdots\!48\)\( - \)\(40\!\cdots\!17\)\( T + \)\(20\!\cdots\!02\)\( T^{2} + \)\(12\!\cdots\!17\)\( T^{3} - \)\(22\!\cdots\!32\)\( T^{4} - \)\(29\!\cdots\!10\)\( T^{5} + \)\(45\!\cdots\!72\)\( T^{6} + \)\(25\!\cdots\!18\)\( T^{7} - \)\(21\!\cdots\!60\)\( T^{8} - 85188350181927473 T^{9} + 292235934 T^{10} + T^{11} \)
$37$ \( -\)\(31\!\cdots\!88\)\( - \)\(92\!\cdots\!92\)\( T - \)\(93\!\cdots\!92\)\( T^{2} - \)\(33\!\cdots\!92\)\( T^{3} + \)\(52\!\cdots\!80\)\( T^{4} + \)\(63\!\cdots\!00\)\( T^{5} + \)\(94\!\cdots\!40\)\( T^{6} - \)\(26\!\cdots\!16\)\( T^{7} - \)\(77\!\cdots\!20\)\( T^{8} - 34631105003304288 T^{9} + 1380429338 T^{10} + T^{11} \)
$41$ \( -\)\(36\!\cdots\!28\)\( + \)\(35\!\cdots\!80\)\( T + \)\(32\!\cdots\!64\)\( T^{2} - \)\(20\!\cdots\!32\)\( T^{3} - \)\(39\!\cdots\!32\)\( T^{4} - \)\(14\!\cdots\!16\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} + \)\(11\!\cdots\!52\)\( T^{7} - \)\(23\!\cdots\!72\)\( T^{8} - 2036343913280571352 T^{9} + 1062067494 T^{10} + T^{11} \)
$43$ \( -\)\(14\!\cdots\!60\)\( - \)\(18\!\cdots\!29\)\( T - \)\(22\!\cdots\!46\)\( T^{2} + \)\(30\!\cdots\!29\)\( T^{3} + \)\(19\!\cdots\!40\)\( T^{4} - \)\(10\!\cdots\!66\)\( T^{5} - \)\(26\!\cdots\!64\)\( T^{6} + \)\(13\!\cdots\!98\)\( T^{7} + \)\(99\!\cdots\!28\)\( T^{8} - 6328771775300000993 T^{9} - 74588594 T^{10} + T^{11} \)
$47$ \( \)\(21\!\cdots\!68\)\( - \)\(70\!\cdots\!89\)\( T - \)\(23\!\cdots\!62\)\( T^{2} + \)\(78\!\cdots\!73\)\( T^{3} + \)\(13\!\cdots\!24\)\( T^{4} - \)\(11\!\cdots\!26\)\( T^{5} + \)\(29\!\cdots\!68\)\( T^{6} + \)\(57\!\cdots\!10\)\( T^{7} - \)\(14\!\cdots\!96\)\( T^{8} - 12243512776017368689 T^{9} + 1821239394 T^{10} + T^{11} \)
$53$ \( \)\(17\!\cdots\!82\)\( + \)\(13\!\cdots\!45\)\( T - \)\(41\!\cdots\!20\)\( T^{2} - \)\(19\!\cdots\!35\)\( T^{3} - \)\(32\!\cdots\!36\)\( T^{4} + \)\(19\!\cdots\!14\)\( T^{5} - \)\(13\!\cdots\!44\)\( T^{6} - \)\(69\!\cdots\!94\)\( T^{7} + \)\(27\!\cdots\!70\)\( T^{8} - 14495590711615911727 T^{9} - 7818635688 T^{10} + T^{11} \)
$59$ \( \)\(21\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( T - \)\(16\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!76\)\( T^{3} + \)\(99\!\cdots\!44\)\( T^{4} - \)\(21\!\cdots\!44\)\( T^{5} - \)\(10\!\cdots\!04\)\( T^{6} + \)\(10\!\cdots\!48\)\( T^{7} + \)\(92\!\cdots\!88\)\( T^{8} - \)\(18\!\cdots\!16\)\( T^{9} - 1230002712 T^{10} + T^{11} \)
$61$ \( \)\(21\!\cdots\!20\)\( - \)\(25\!\cdots\!52\)\( T + \)\(26\!\cdots\!44\)\( T^{2} + \)\(60\!\cdots\!16\)\( T^{3} - \)\(17\!\cdots\!00\)\( T^{4} - \)\(25\!\cdots\!48\)\( T^{5} + \)\(10\!\cdots\!32\)\( T^{6} + \)\(48\!\cdots\!48\)\( T^{7} - \)\(23\!\cdots\!12\)\( T^{8} - 84689162462461737068 T^{9} + 18602654230 T^{10} + T^{11} \)
$67$ \( -\)\(21\!\cdots\!80\)\( - \)\(41\!\cdots\!92\)\( T + \)\(15\!\cdots\!48\)\( T^{2} - \)\(93\!\cdots\!44\)\( T^{3} + \)\(37\!\cdots\!60\)\( T^{4} + \)\(67\!\cdots\!16\)\( T^{5} - \)\(11\!\cdots\!24\)\( T^{6} - \)\(21\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!00\)\( T^{8} - \)\(24\!\cdots\!72\)\( T^{9} - 27481284652 T^{10} + T^{11} \)
$71$ \( \)\(26\!\cdots\!92\)\( + \)\(13\!\cdots\!36\)\( T + \)\(17\!\cdots\!80\)\( T^{2} - \)\(14\!\cdots\!32\)\( T^{3} - \)\(40\!\cdots\!28\)\( T^{4} - \)\(86\!\cdots\!28\)\( T^{5} + \)\(21\!\cdots\!76\)\( T^{6} + \)\(85\!\cdots\!68\)\( T^{7} - \)\(37\!\cdots\!60\)\( T^{8} - \)\(17\!\cdots\!96\)\( T^{9} + 20347168516 T^{10} + T^{11} \)
$73$ \( -\)\(11\!\cdots\!48\)\( + \)\(56\!\cdots\!88\)\( T + \)\(59\!\cdots\!16\)\( T^{2} - \)\(23\!\cdots\!40\)\( T^{3} - \)\(32\!\cdots\!60\)\( T^{4} + \)\(31\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!12\)\( T^{6} - \)\(72\!\cdots\!44\)\( T^{7} - \)\(60\!\cdots\!48\)\( T^{8} - \)\(15\!\cdots\!84\)\( T^{9} + 57740010478 T^{10} + T^{11} \)
$79$ \( -\)\(12\!\cdots\!80\)\( + \)\(10\!\cdots\!39\)\( T - \)\(17\!\cdots\!58\)\( T^{2} - \)\(12\!\cdots\!07\)\( T^{3} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(95\!\cdots\!90\)\( T^{5} - \)\(11\!\cdots\!24\)\( T^{6} - \)\(14\!\cdots\!30\)\( T^{7} - \)\(30\!\cdots\!44\)\( T^{8} + \)\(18\!\cdots\!67\)\( T^{9} + 120245016462 T^{10} + T^{11} \)
$83$ \( -\)\(45\!\cdots\!64\)\( + \)\(15\!\cdots\!84\)\( T + \)\(65\!\cdots\!60\)\( T^{2} - \)\(24\!\cdots\!12\)\( T^{3} - \)\(96\!\cdots\!60\)\( T^{4} + \)\(42\!\cdots\!88\)\( T^{5} + \)\(22\!\cdots\!40\)\( T^{6} - \)\(77\!\cdots\!08\)\( T^{7} - \)\(26\!\cdots\!64\)\( T^{8} + \)\(33\!\cdots\!80\)\( T^{9} + 142463983824 T^{10} + T^{11} \)
$89$ \( -\)\(45\!\cdots\!40\)\( + \)\(17\!\cdots\!68\)\( T + \)\(24\!\cdots\!12\)\( T^{2} - \)\(15\!\cdots\!08\)\( T^{3} - \)\(65\!\cdots\!04\)\( T^{4} + \)\(47\!\cdots\!68\)\( T^{5} + \)\(55\!\cdots\!12\)\( T^{6} + \)\(30\!\cdots\!32\)\( T^{7} - \)\(13\!\cdots\!24\)\( T^{8} - \)\(11\!\cdots\!48\)\( T^{9} + 96700717270 T^{10} + T^{11} \)
$97$ \( -\)\(18\!\cdots\!04\)\( - \)\(43\!\cdots\!56\)\( T - \)\(14\!\cdots\!20\)\( T^{2} + \)\(90\!\cdots\!92\)\( T^{3} + \)\(61\!\cdots\!28\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} - \)\(80\!\cdots\!24\)\( T^{6} - \)\(39\!\cdots\!92\)\( T^{7} - \)\(29\!\cdots\!28\)\( T^{8} + \)\(17\!\cdots\!80\)\( T^{9} + 303190852014 T^{10} + T^{11} \)
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