Properties

Label 29.12.a.b
Level $29$
Weight $12$
Character orbit 29.a
Self dual yes
Analytic conductor $22.282$
Analytic rank $0$
Dimension $14$
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 23517 x^{12} - 42196 x^{11} + 214206700 x^{10} + 532863376 x^{9} - 951901011680 x^{8} - 1439668569088 x^{7} + 2125122050961664 x^{6} - 3617975849298944 x^{5} - 2179155305696991232 x^{4} + 15507015861824716800 x^{3} + 709645914927917694976 x^{2} - 9478460460037847384064 x + 30797548279705794248704\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 34 - \beta_{3} ) q^{3} + ( 1312 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( 698 + 15 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{5} + ( 1319 - 65 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{6} + ( 6077 - 161 \beta_{1} + 5 \beta_{2} - 15 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{7} + ( 9049 + 1181 \beta_{1} + 17 \beta_{2} - 55 \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{8} + ( 98018 - 52 \beta_{1} + 26 \beta_{2} - 115 \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{7} + 5 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} + 9 \beta_{12} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 34 - \beta_{3} ) q^{3} + ( 1312 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( 698 + 15 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{5} + ( 1319 - 65 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{6} + ( 6077 - 161 \beta_{1} + 5 \beta_{2} - 15 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{7} + ( 9049 + 1181 \beta_{1} + 17 \beta_{2} - 55 \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} + \beta_{10} ) q^{8} + ( 98018 - 52 \beta_{1} + 26 \beta_{2} - 115 \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{7} + 5 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} + 9 \beta_{12} ) q^{9} + ( 50973 + 1777 \beta_{1} + 23 \beta_{2} - 97 \beta_{3} + 15 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} - 8 \beta_{8} - 8 \beta_{9} + 5 \beta_{10} - 3 \beta_{11} - 12 \beta_{12} ) q^{10} + ( 28433 + 3435 \beta_{1} + 11 \beta_{2} - 294 \beta_{3} + 9 \beta_{4} + 2 \beta_{5} + 6 \beta_{6} + 11 \beta_{7} + \beta_{8} + 11 \beta_{9} + 14 \beta_{11} + 8 \beta_{12} - \beta_{13} ) q^{11} + ( -287683 + 4894 \beta_{1} - 128 \beta_{2} - 1654 \beta_{3} - 28 \beta_{4} + 22 \beta_{5} - 9 \beta_{6} - 19 \beta_{7} + 2 \beta_{8} + 7 \beta_{9} - 25 \beta_{10} - \beta_{11} - 8 \beta_{12} + 8 \beta_{13} ) q^{12} + ( 162318 + 7763 \beta_{1} - 59 \beta_{2} - 464 \beta_{3} + 7 \beta_{4} - 20 \beta_{5} - 18 \beta_{6} - 13 \beta_{7} + 31 \beta_{8} + 43 \beta_{9} - 15 \beta_{10} + 4 \beta_{11} + 15 \beta_{12} - 23 \beta_{13} ) q^{13} + ( -514528 + 10522 \beta_{1} - 495 \beta_{2} - 177 \beta_{3} + 11 \beta_{4} + 115 \beta_{5} - 31 \beta_{6} - 22 \beta_{7} - 41 \beta_{8} - 40 \beta_{9} + 37 \beta_{10} + 43 \beta_{11} - 75 \beta_{12} + 22 \beta_{13} ) q^{14} + ( -340370 + 8918 \beta_{1} - 288 \beta_{2} - 77 \beta_{3} + 68 \beta_{4} - 116 \beta_{5} - 22 \beta_{6} - 3 \beta_{7} - 39 \beta_{8} - 51 \beta_{9} - 23 \beta_{10} - 49 \beta_{11} + 23 \beta_{12} + 21 \beta_{13} ) q^{15} + ( 1358608 + 33871 \beta_{1} + 982 \beta_{2} + 4162 \beta_{3} + 15 \beta_{4} - 43 \beta_{5} + 97 \beta_{6} - 24 \beta_{7} + 15 \beta_{8} - 92 \beta_{9} + 63 \beta_{10} - 67 \beta_{11} + 117 \beta_{12} - 66 \beta_{13} ) q^{16} + ( 401231 + 27783 \beta_{1} - 1201 \beta_{2} - 408 \beta_{3} - 18 \beta_{4} - 152 \beta_{5} + 38 \beta_{6} + 98 \beta_{7} + 67 \beta_{8} - 68 \beta_{9} + 78 \beta_{10} + 72 \beta_{11} - 37 \beta_{12} + 62 \beta_{13} ) q^{17} + ( -14845 + 128829 \beta_{1} - 730 \beta_{2} + 2218 \beta_{3} - 285 \beta_{4} - 124 \beta_{5} - 386 \beta_{6} - 90 \beta_{7} - \beta_{8} + 501 \beta_{9} - 83 \beta_{10} - 208 \beta_{11} + 34 \beta_{12} - 20 \beta_{13} ) q^{18} + ( 2129477 + 42673 \beta_{1} + 1303 \beta_{2} + 6707 \beta_{3} - 35 \beta_{4} - 46 \beta_{5} + 184 \beta_{6} + 87 \beta_{7} + 177 \beta_{8} - 277 \beta_{9} - 112 \beta_{10} - 22 \beta_{11} - 34 \beta_{12} - 49 \beta_{13} ) q^{19} + ( 4655690 + 72929 \beta_{1} + 2571 \beta_{2} + 9471 \beta_{3} + 108 \beta_{4} - 468 \beta_{5} + 570 \beta_{6} + 182 \beta_{7} + 64 \beta_{8} + 438 \beta_{9} + 134 \beta_{10} + 394 \beta_{11} + 216 \beta_{12} + 72 \beta_{13} ) q^{20} + ( 3660045 + 21927 \beta_{1} + 2001 \beta_{2} + 2456 \beta_{3} + 122 \beta_{4} - 422 \beta_{5} + 472 \beta_{6} + 290 \beta_{7} - 521 \beta_{8} - 576 \beta_{9} - 310 \beta_{10} + 72 \beta_{11} - 115 \beta_{12} + 50 \beta_{13} ) q^{21} + ( 11953303 + 28946 \beta_{1} + 3752 \beta_{2} + 26763 \beta_{3} + 258 \beta_{4} + 1486 \beta_{5} - 263 \beta_{6} - 55 \beta_{7} - 518 \beta_{8} - 293 \beta_{9} + 93 \beta_{10} + 144 \beta_{11} - 391 \beta_{12} - 242 \beta_{13} ) q^{22} + ( 3759645 - 31251 \beta_{1} - 4183 \beta_{2} + 4683 \beta_{3} + 193 \beta_{4} + 140 \beta_{5} - 388 \beta_{6} - 64 \beta_{7} - 2 \beta_{8} + 786 \beta_{9} + 127 \beta_{10} + 293 \beta_{11} - 979 \beta_{12} + 176 \beta_{13} ) q^{23} + ( 15821866 - 504727 \beta_{1} + 1698 \beta_{2} + 390 \beta_{3} + 367 \beta_{4} + 3773 \beta_{5} - 1353 \beta_{6} - 1182 \beta_{7} + 1755 \beta_{8} - 1184 \beta_{9} + 31 \beta_{10} - 697 \beta_{11} + 783 \beta_{12} + 346 \beta_{13} ) q^{24} + ( 13925902 - 149390 \beta_{1} - 4666 \beta_{2} - 20178 \beta_{3} + 65 \beta_{4} - 2060 \beta_{5} - 98 \beta_{6} - 335 \beta_{7} - 16 \beta_{8} + 1795 \beta_{9} + 7 \beta_{10} - 336 \beta_{11} - 622 \beta_{12} - 193 \beta_{13} ) q^{25} + ( 26683057 + 494 \beta_{1} + 4983 \beta_{2} - 10657 \beta_{3} - 1062 \beta_{4} + 3607 \beta_{5} - 927 \beta_{6} + 282 \beta_{7} - 1614 \beta_{8} - 2495 \beta_{9} + 134 \beta_{10} - 253 \beta_{11} + 1207 \beta_{12} - 614 \beta_{13} ) q^{26} + ( 28381105 - 232641 \beta_{1} - 4321 \beta_{2} - 104846 \beta_{3} - 1521 \beta_{4} - 4874 \beta_{5} + 1286 \beta_{6} + 385 \beta_{7} + 349 \beta_{8} + 697 \beta_{9} + 226 \beta_{10} - 706 \beta_{11} + 2908 \beta_{12} - 447 \beta_{13} ) q^{27} + ( 22822780 - 923774 \beta_{1} - 2486 \beta_{2} - 57614 \beta_{3} + 900 \beta_{4} - 2012 \beta_{5} + 1400 \beta_{6} - 548 \beta_{7} + 1748 \beta_{8} + 1012 \beta_{9} - 464 \beta_{10} + 376 \beta_{11} + 12 \beta_{12} + 1096 \beta_{13} ) q^{28} -20511149 q^{29} + ( 30206069 - 994704 \beta_{1} - 27483 \beta_{2} - 54130 \beta_{3} + 1207 \beta_{4} - 1831 \beta_{5} - 2038 \beta_{6} - 417 \beta_{7} + 915 \beta_{8} + 3047 \beta_{9} + 438 \beta_{10} + 135 \beta_{11} - 2608 \beta_{12} + 1760 \beta_{13} ) q^{30} + ( 45276315 - 247691 \beta_{1} - 27153 \beta_{2} - 92624 \beta_{3} - 1171 \beta_{4} - 1392 \beta_{5} + 1726 \beta_{6} + 3336 \beta_{7} - 5112 \beta_{8} - 3366 \beta_{9} + 1305 \beta_{10} + 3149 \beta_{11} - 4969 \beta_{12} - 1128 \beta_{13} ) q^{31} + ( 90032609 + 1136975 \beta_{1} + 16651 \beta_{2} + 7543 \beta_{3} - 2825 \beta_{4} - 12999 \beta_{5} + 4488 \beta_{6} + 4177 \beta_{7} - 5191 \beta_{8} + 5740 \beta_{9} - 1975 \beta_{10} + 1370 \beta_{11} - 756 \beta_{12} - 2552 \beta_{13} ) q^{32} + ( 84339343 - 2143246 \beta_{1} - 6678 \beta_{2} - 76394 \beta_{3} + 2327 \beta_{4} + 5026 \beta_{5} - 100 \beta_{6} - 847 \beta_{7} + 3664 \beta_{8} - 7627 \beta_{9} + 1557 \beta_{10} - 34 \beta_{11} + 3174 \beta_{12} - 77 \beta_{13} ) q^{33} + ( 94018058 - 1868945 \beta_{1} + 18021 \beta_{2} + 262729 \beta_{3} + 7517 \beta_{4} + 7543 \beta_{5} - 1287 \beta_{6} - 1660 \beta_{7} + 1081 \beta_{8} - 3319 \beta_{9} - 2782 \beta_{10} - 635 \beta_{11} - 9688 \beta_{12} + 1512 \beta_{13} ) q^{34} + ( 114284075 - 1510557 \beta_{1} - 475 \beta_{2} - 143565 \beta_{3} + 531 \beta_{4} - 14668 \beta_{5} + 4110 \beta_{6} - 1982 \beta_{7} + 11378 \beta_{8} + 9704 \beta_{9} + 1769 \beta_{10} - 2727 \beta_{11} + 9287 \beta_{12} + 2114 \beta_{13} ) q^{35} + ( 228474884 - 382730 \beta_{1} + 128566 \beta_{2} + 344142 \beta_{3} - 304 \beta_{4} + 20148 \beta_{5} + 12 \beta_{6} - 2776 \beta_{7} + 5800 \beta_{8} - 18456 \beta_{9} - 3300 \beta_{10} - 6804 \beta_{11} + 15720 \beta_{12} - 4504 \beta_{13} ) q^{36} + ( 34879270 + 546924 \beta_{1} - 49826 \beta_{2} - 178890 \beta_{3} - 1932 \beta_{4} + 7750 \beta_{5} - 12732 \beta_{6} - 3242 \beta_{7} - 10006 \beta_{8} + 8376 \beta_{9} - 6504 \beta_{10} + 722 \beta_{11} - 5024 \beta_{12} + 3482 \beta_{13} ) q^{37} + ( 135205264 + 5017751 \beta_{1} + 9252 \beta_{2} + 425428 \beta_{3} - 8902 \beta_{4} + 2280 \beta_{5} - 2954 \beta_{6} + 2052 \beta_{7} - 2718 \beta_{8} + 10409 \beta_{9} + 4281 \beta_{10} + 2082 \beta_{11} - 4537 \beta_{12} + 370 \beta_{13} ) q^{38} + ( 140854003 + 1194581 \beta_{1} - 54057 \beta_{2} + 16248 \beta_{3} - 5093 \beta_{4} + 37328 \beta_{5} - 26850 \beta_{6} - 10324 \beta_{7} + 442 \beta_{8} + 1038 \beta_{9} + 7485 \beta_{10} - 7633 \beta_{11} + 7717 \beta_{12} - 472 \beta_{13} ) q^{39} + ( 130512255 + 6262085 \beta_{1} + 128039 \beta_{2} + 836087 \beta_{3} + 2969 \beta_{4} + 7547 \beta_{5} + 17040 \beta_{6} + 9583 \beta_{7} - 14521 \beta_{8} - 6250 \beta_{9} + 1725 \beta_{10} + 13034 \beta_{11} - 7902 \beta_{12} - 532 \beta_{13} ) q^{40} + ( 14175037 + 954961 \beta_{1} + 23063 \beta_{2} - 91852 \beta_{3} + 2808 \beta_{4} - 12480 \beta_{5} + 21266 \beta_{6} - 542 \beta_{7} - 1471 \beta_{8} - 4098 \beta_{9} + 4144 \beta_{10} + 5942 \beta_{11} - 1077 \beta_{12} + 5846 \beta_{13} ) q^{41} + ( 72241178 + 7189975 \beta_{1} - 34339 \beta_{2} + 984697 \beta_{3} + 8345 \beta_{4} - 10505 \beta_{5} - 3267 \beta_{6} - 5532 \beta_{7} + 7685 \beta_{8} + 33275 \beta_{9} + 17668 \beta_{10} + 9753 \beta_{11} - 11970 \beta_{12} - 1428 \beta_{13} ) q^{42} + ( 156661479 - 2315855 \beta_{1} + 30227 \beta_{2} - 367568 \beta_{3} - 6885 \beta_{4} + 45306 \beta_{5} - 5564 \beta_{6} - 9291 \beta_{7} + 26769 \beta_{8} - 28551 \beta_{9} - 12310 \beta_{10} - 8910 \beta_{11} - 2774 \beta_{12} - 5871 \beta_{13} ) q^{43} + ( 1946827 + 16105974 \beta_{1} + 28316 \beta_{2} - 64798 \beta_{3} - 31632 \beta_{4} - 102042 \beta_{5} + 33045 \beta_{6} + 19739 \beta_{7} + 706 \beta_{8} + 21861 \beta_{9} - 6559 \beta_{10} - 6611 \beta_{11} + 14472 \beta_{12} - 8616 \beta_{13} ) q^{44} + ( -80681698 + 1850984 \beta_{1} - 40148 \beta_{2} - 492557 \beta_{3} + 21293 \beta_{4} - 28217 \beta_{5} - 4136 \beta_{6} - 97 \beta_{7} - 52373 \beta_{8} - 6515 \beta_{9} - 15213 \beta_{10} - 8138 \beta_{11} - 13613 \beta_{12} - 9759 \beta_{13} ) q^{45} + ( -111981366 - 3212398 \beta_{1} - 43457 \beta_{2} + 138347 \beta_{3} + 39305 \beta_{4} - 17743 \beta_{5} + 34541 \beta_{6} + 24892 \beta_{7} - 739 \beta_{8} - 73002 \beta_{9} + 2869 \beta_{10} + 14149 \beta_{11} - 739 \beta_{12} + 17686 \beta_{13} ) q^{46} + ( -298326384 + 2219680 \beta_{1} - 139396 \beta_{2} - 779549 \beta_{3} - 12192 \beta_{4} - 11396 \beta_{5} - 44052 \beta_{6} - 8605 \beta_{7} + 21619 \beta_{8} + 5315 \beta_{9} - 19625 \beta_{10} - 48439 \beta_{11} - 1589 \beta_{12} - 8817 \beta_{13} ) q^{47} + ( -1113375239 + 9128785 \beta_{1} - 786205 \beta_{2} - 2171777 \beta_{3} - 31679 \beta_{4} - 21073 \beta_{5} - 97514 \beta_{6} - 23191 \beta_{7} + 21459 \beta_{8} + 81680 \beta_{9} - 20841 \beta_{10} - 3864 \beta_{11} - 18682 \beta_{12} + 31652 \beta_{13} ) q^{48} + ( 79009529 - 11009606 \beta_{1} + 79738 \beta_{2} - 1625854 \beta_{3} + 32506 \beta_{4} - 29464 \beta_{5} + 53554 \beta_{6} + 31036 \beta_{7} + 29648 \beta_{8} + 36170 \beta_{9} - 5610 \beta_{10} + 62086 \beta_{11} + 21164 \beta_{12} + 1304 \beta_{13} ) q^{49} + ( -473644055 + 1241874 \beta_{1} + 21620 \beta_{2} + 322176 \beta_{3} + 29685 \beta_{4} - 27774 \beta_{5} + 50758 \beta_{6} + 30042 \beta_{7} - 38463 \beta_{8} - 137050 \beta_{9} + 35416 \beta_{10} - 8 \beta_{11} + 30797 \beta_{12} + 7254 \beta_{13} ) q^{50} + ( 235626021 - 21740509 \beta_{1} + 271475 \beta_{2} - 764345 \beta_{3} + 51705 \beta_{4} + 40182 \beta_{5} + 85090 \beta_{6} + 3114 \beta_{7} + 21596 \beta_{8} - 26938 \beta_{9} + 65523 \beta_{10} + 44323 \beta_{11} + 27859 \beta_{12} - 6870 \beta_{13} ) q^{51} + ( -325338590 + 24397391 \beta_{1} + 371577 \beta_{2} - 386347 \beta_{3} - 66192 \beta_{4} - 61512 \beta_{5} - 41002 \beta_{6} - 19946 \beta_{7} + 53108 \beta_{8} + 164702 \beta_{9} - 37494 \beta_{10} - 46338 \beta_{11} + 103492 \beta_{12} - 38272 \beta_{13} ) q^{52} + ( -944820798 - 10611429 \beta_{1} - 590827 \beta_{2} - 1266308 \beta_{3} - 50187 \beta_{4} + 198330 \beta_{5} - 49732 \beta_{6} + 2827 \beta_{7} - 50317 \beta_{8} - 34575 \beta_{9} + 35127 \beta_{10} + 72218 \beta_{11} - 173777 \beta_{12} + 14773 \beta_{13} ) q^{53} + ( -638962983 + 5929542 \beta_{1} + 378568 \beta_{2} + 1862163 \beta_{3} - 19196 \beta_{4} + 200586 \beta_{5} - 105315 \beta_{6} - 115203 \beta_{7} - 30252 \beta_{8} - 32681 \beta_{9} + 14437 \beta_{10} - 93304 \beta_{11} + 15227 \beta_{12} - 66230 \beta_{13} ) q^{54} + ( -194375681 - 9729059 \beta_{1} + 893369 \beta_{2} + 971776 \beta_{3} - 9481 \beta_{4} + 27100 \beta_{5} + 12472 \beta_{6} + 8362 \beta_{7} - 49738 \beta_{8} + 91432 \beta_{9} - 17587 \beta_{10} - 65103 \beta_{11} + 6961 \beta_{12} + 894 \beta_{13} ) q^{55} + ( -1967100730 - 16448798 \beta_{1} - 278058 \beta_{2} - 689818 \beta_{3} - 4950 \beta_{4} - 24002 \beta_{5} - 17760 \beta_{6} - 58266 \beta_{7} + 54 \beta_{8} - 56676 \beta_{9} - 34990 \beta_{10} - 56940 \beta_{11} - 47324 \beta_{12} + 52568 \beta_{13} ) q^{56} + ( -1748182224 - 32992740 \beta_{1} + 514258 \beta_{2} - 3167737 \beta_{3} - 63027 \beta_{4} + 132405 \beta_{5} + 70614 \beta_{6} - 34937 \beta_{7} - 43125 \beta_{8} - 14315 \beta_{9} - 73213 \beta_{10} + 102898 \beta_{11} - 13423 \beta_{12} + 25189 \beta_{13} ) q^{57} -20511149 \beta_{1} q^{58} + ( 25144121 - 22144799 \beta_{1} - 90041 \beta_{2} + 2424447 \beta_{3} - 15531 \beta_{4} - 250060 \beta_{5} - 34502 \beta_{6} + 119050 \beta_{7} - 16042 \beta_{8} + 36436 \beta_{9} + 79851 \beta_{10} - 88205 \beta_{11} + 80185 \beta_{12} + 22778 \beta_{13} ) q^{59} + ( -2574963937 - 48732008 \beta_{1} - 980442 \beta_{2} - 1361452 \beta_{3} + 46252 \beta_{4} + 326522 \beta_{5} + 26221 \beta_{6} - 14105 \beta_{7} + 106142 \beta_{8} - 210215 \beta_{9} + 41025 \beta_{10} + 106357 \beta_{11} - 144852 \beta_{12} + 86528 \beta_{13} ) q^{60} + ( -546830859 - 41037941 \beta_{1} + 466719 \beta_{2} + 5760280 \beta_{3} + 48982 \beta_{4} + 124110 \beta_{5} - 110440 \beta_{6} + 69396 \beta_{7} + 106611 \beta_{8} + 154020 \beta_{9} - 22570 \beta_{10} + 109702 \beta_{11} - 13753 \beta_{12} + 12152 \beta_{13} ) q^{61} + ( -715395813 - 9259977 \beta_{1} + 841643 \beta_{2} + 10559894 \beta_{3} + 267483 \beta_{4} - 447773 \beta_{5} + 116044 \beta_{6} + 17633 \beta_{7} + 91063 \beta_{8} + 73462 \beta_{9} + 2627 \beta_{10} + 107223 \beta_{11} + 92811 \beta_{12} - 143142 \beta_{13} ) q^{62} + ( -1645127110 - 51947964 \beta_{1} + 872190 \beta_{2} - 1648698 \beta_{3} - 23146 \beta_{4} - 347806 \beta_{5} + 189920 \beta_{6} + 92518 \beta_{7} - 145424 \beta_{8} - 305804 \beta_{9} - 145364 \beta_{10} - 132836 \beta_{11} + 127266 \beta_{12} + 21774 \beta_{13} ) q^{63} + ( 1051882516 + 49439283 \beta_{1} + 1437734 \beta_{2} + 12116434 \beta_{3} + 139751 \beta_{4} + 222765 \beta_{5} + 126437 \beta_{6} + 266988 \beta_{7} - 108417 \beta_{8} - 150292 \beta_{9} + 181543 \beta_{10} + 62649 \beta_{11} + 97249 \beta_{12} - 112890 \beta_{13} ) q^{64} + ( 83063435 - 33301540 \beta_{1} + 598834 \beta_{2} + 5816253 \beta_{3} + 27460 \beta_{4} - 882519 \beta_{5} + 32996 \beta_{6} - 19258 \beta_{7} + 234337 \beta_{8} + 258434 \beta_{9} - 112036 \beta_{10} - 213798 \beta_{11} + 222229 \beta_{12} - 96314 \beta_{13} ) q^{65} + ( -7108828433 + 58019683 \beta_{1} - 3570044 \beta_{2} - 11208756 \beta_{3} - 203269 \beta_{4} - 81530 \beta_{5} - 316614 \beta_{6} - 207486 \beta_{7} - 70481 \beta_{8} + 472556 \beta_{9} - 210702 \beta_{10} + 77324 \beta_{11} - 305691 \beta_{12} + 45094 \beta_{13} ) q^{66} + ( 1555939850 - 46627916 \beta_{1} + 372734 \beta_{2} + 456382 \beta_{3} - 289362 \beta_{4} - 31254 \beta_{5} - 69468 \beta_{6} - 111634 \beta_{7} + 303496 \beta_{8} - 136792 \beta_{9} + 67456 \beta_{10} - 272068 \beta_{11} + 263914 \beta_{12} + 18774 \beta_{13} ) q^{67} + ( -7442154226 + 87579662 \beta_{1} - 2508666 \beta_{2} - 5538206 \beta_{3} + 30220 \beta_{4} - 423208 \beta_{5} + 158982 \beta_{6} + 111622 \beta_{7} - 213736 \beta_{8} + 168486 \beta_{9} - 39678 \beta_{10} + 218982 \beta_{11} - 164224 \beta_{12} + 222048 \beta_{13} ) q^{68} + ( -1043604257 - 8845719 \beta_{1} - 1201375 \beta_{2} + 5566520 \beta_{3} + 117728 \beta_{4} + 341794 \beta_{5} + 48244 \beta_{6} - 103538 \beta_{7} - 196283 \beta_{8} + 50366 \beta_{9} + 335256 \beta_{10} + 121094 \beta_{11} - 346299 \beta_{12} - 30434 \beta_{13} ) q^{69} + ( -4854700484 + 71477458 \beta_{1} - 1466415 \beta_{2} - 4398317 \beta_{3} - 176269 \beta_{4} + 1058067 \beta_{5} - 153003 \beta_{6} - 181058 \beta_{7} - 486673 \beta_{8} - 674248 \beta_{9} + 83797 \beta_{10} - 198013 \beta_{11} - 518875 \beta_{12} + 199206 \beta_{13} ) q^{70} + ( -398316696 + 32443528 \beta_{1} - 16960 \beta_{2} + 7725822 \beta_{3} + 6552 \beta_{4} - 51240 \beta_{5} - 37640 \beta_{6} - 19448 \beta_{7} - 182680 \beta_{8} - 350808 \beta_{9} + 93152 \beta_{10} + 143264 \beta_{11} + 193112 \beta_{12} - 54664 \beta_{13} ) q^{71} + ( -1719129054 + 240547498 \beta_{1} - 923454 \beta_{2} - 26446142 \beta_{3} - 612998 \beta_{4} + 122758 \beta_{5} + 271668 \beta_{6} + 98058 \beta_{7} + 52654 \beta_{8} + 754900 \beta_{9} - 86758 \beta_{10} + 302976 \beta_{11} + 113320 \beta_{12} - 160944 \beta_{13} ) q^{72} + ( 2833908108 + 1418178 \beta_{1} + 2100304 \beta_{2} + 2206530 \beta_{3} - 264436 \beta_{4} + 569238 \beta_{5} - 43444 \beta_{6} - 410142 \beta_{7} + 201072 \beta_{8} + 89136 \beta_{9} - 203864 \beta_{10} - 121906 \beta_{11} - 268418 \beta_{12} - 95386 \beta_{13} ) q^{73} + ( 2035715722 - 53095418 \beta_{1} - 487988 \beta_{2} - 5979604 \beta_{3} + 514618 \beta_{4} + 721768 \beta_{5} - 465916 \beta_{6} - 333412 \beta_{7} + 786178 \beta_{8} - 391438 \beta_{9} - 29102 \beta_{10} - 215784 \beta_{11} + 598040 \beta_{12} + 50304 \beta_{13} ) q^{74} + ( 5844373968 - 28112656 \beta_{1} - 3319742 \beta_{2} + 1033772 \beta_{3} + 398706 \beta_{4} + 258792 \beta_{5} - 417158 \beta_{6} + 123192 \beta_{7} - 199424 \beta_{8} - 199684 \beta_{9} + 473522 \beta_{10} - 346434 \beta_{11} - 228500 \beta_{12} + 50980 \beta_{13} ) q^{75} + ( 11927337388 + 137359820 \beta_{1} + 6295924 \beta_{2} + 883948 \beta_{3} + 338900 \beta_{4} + 72764 \beta_{5} + 498104 \beta_{6} + 431988 \beta_{7} - 116148 \beta_{8} - 32640 \beta_{9} + 223844 \beta_{10} - 202216 \beta_{11} + 643056 \beta_{12} - 88368 \beta_{13} ) q^{76} + ( 2769084273 - 41874403 \beta_{1} - 1635335 \beta_{2} - 2158712 \beta_{3} + 500854 \beta_{4} - 148334 \beta_{5} - 174260 \beta_{6} - 13916 \beta_{7} - 224311 \beta_{8} - 81632 \beta_{9} - 517330 \beta_{10} + 766478 \beta_{11} - 215571 \beta_{12} - 58072 \beta_{13} ) q^{77} + ( 3875762617 + 95463409 \beta_{1} + 979889 \beta_{2} - 49452648 \beta_{3} - 331809 \beta_{4} - 543567 \beta_{5} - 418294 \beta_{6} - 79551 \beta_{7} + 857627 \beta_{8} + 640120 \beta_{9} - 968801 \beta_{10} - 503499 \beta_{11} + 1097201 \beta_{12} - 159730 \beta_{13} ) q^{78} + ( 7541192582 + 69202122 \beta_{1} + 1655604 \beta_{2} + 11991767 \beta_{3} + 116362 \beta_{4} - 498016 \beta_{5} - 23182 \beta_{6} + 237067 \beta_{7} + 230285 \beta_{8} + 844819 \beta_{9} + 345943 \beta_{10} + 434135 \beta_{11} + 472399 \beta_{12} + 305879 \beta_{13} ) q^{79} + ( 10447616062 + 327106491 \beta_{1} + 3913372 \beta_{2} + 12771944 \beta_{3} - 58437 \beta_{4} - 78895 \beta_{5} + 434535 \beta_{6} + 637094 \beta_{7} - 455273 \beta_{8} - 274900 \beta_{9} + 114687 \beta_{10} - 205033 \beta_{11} - 797149 \beta_{12} - 371950 \beta_{13} ) q^{80} + ( 12183682329 - 168707202 \beta_{1} - 1336484 \beta_{2} - 16796069 \beta_{3} + 425417 \beta_{4} + 358191 \beta_{5} - 56076 \beta_{6} + 275441 \beta_{7} + 361077 \beta_{8} - 501731 \beta_{9} - 128553 \beta_{10} - 62436 \beta_{11} + 204851 \beta_{12} + 44791 \beta_{13} ) q^{81} + ( 3383301068 + 28999499 \beta_{1} + 2455907 \beta_{2} + 6471087 \beta_{3} + 163241 \beta_{4} - 173683 \beta_{5} + 485267 \beta_{6} - 293696 \beta_{7} - 672331 \beta_{8} - 103369 \beta_{9} + 293650 \beta_{10} + 486347 \beta_{11} - 1464262 \beta_{12} + 356612 \beta_{13} ) q^{82} + ( 9132405151 + 17319037 \beta_{1} - 784383 \beta_{2} + 8048689 \beta_{3} - 333413 \beta_{4} - 1485494 \beta_{5} + 646346 \beta_{6} - 208200 \beta_{7} + 149670 \beta_{8} + 7756 \beta_{9} + 49139 \beta_{10} + 85743 \beta_{11} - 273697 \beta_{12} + 93160 \beta_{13} ) q^{83} + ( 15381858734 + 74623644 \beta_{1} + 6302560 \beta_{2} - 15049316 \beta_{3} + 114964 \beta_{4} - 89920 \beta_{5} + 1908878 \beta_{6} + 1485526 \beta_{7} - 329856 \beta_{8} - 1076658 \beta_{9} + 613298 \beta_{10} + 335182 \beta_{11} - 219152 \beta_{12} + 93792 \beta_{13} ) q^{84} + ( 5991383433 - 82651159 \beta_{1} - 272449 \beta_{2} + 6963980 \beta_{3} - 849276 \beta_{4} + 1064322 \beta_{5} - 352700 \beta_{6} - 1059116 \beta_{7} - 1413727 \beta_{8} - 721654 \beta_{9} - 656936 \beta_{10} - 273312 \beta_{11} - 1956297 \beta_{12} + 13832 \beta_{13} ) q^{85} + ( -7371942929 + 186695416 \beta_{1} - 7442546 \beta_{2} - 19215851 \beta_{3} - 1378192 \beta_{4} + 98592 \beta_{5} - 2242933 \beta_{6} - 1231939 \beta_{7} + 444256 \beta_{8} + 1325981 \beta_{9} - 438467 \beta_{10} + 48646 \beta_{11} + 367965 \beta_{12} + 221062 \beta_{13} ) q^{86} + ( -697379066 + 20511149 \beta_{3} ) q^{87} + ( 29883132558 - 37906273 \beta_{1} + 20288062 \beta_{2} + 68719226 \beta_{3} - 135175 \beta_{4} - 355109 \beta_{5} + 1287721 \beta_{6} + 408966 \beta_{7} + 587517 \beta_{8} - 1485288 \beta_{9} + 1241457 \beta_{10} - 1552663 \beta_{11} + 1693529 \beta_{12} - 540522 \beta_{13} ) q^{88} + ( 13413415261 + 166735373 \beta_{1} - 4131441 \beta_{2} + 4390610 \beta_{3} - 822378 \beta_{4} + 332762 \beta_{5} - 494582 \beta_{6} + 171612 \beta_{7} - 258885 \beta_{8} + 832568 \beta_{9} - 71074 \beta_{10} - 960782 \beta_{11} + 1486585 \beta_{12} - 70220 \beta_{13} ) q^{89} + ( 6881571654 - 205402825 \beta_{1} - 1532248 \beta_{2} + 4751260 \beta_{3} + 794636 \beta_{4} - 3630004 \beta_{5} + 502774 \beta_{6} + 457216 \beta_{7} + 299044 \beta_{8} + 980023 \beta_{9} + 889235 \beta_{10} + 732274 \beta_{11} + 1350183 \beta_{12} - 435054 \beta_{13} ) q^{90} + ( 4168240201 - 230649441 \beta_{1} - 9035729 \beta_{2} + 58928237 \beta_{3} + 526617 \beta_{4} - 1752294 \beta_{5} - 2058866 \beta_{6} - 833180 \beta_{7} - 444682 \beta_{8} + 2505236 \beta_{9} - 750475 \beta_{10} + 1780977 \beta_{11} - 787903 \beta_{12} - 359436 \beta_{13} ) q^{91} + ( -18549199314 - 186389346 \beta_{1} - 11602630 \beta_{2} + 15773990 \beta_{3} + 233812 \beta_{4} - 1197416 \beta_{5} - 251770 \beta_{6} - 653682 \beta_{7} - 534048 \beta_{8} + 1861010 \beta_{9} - 401162 \beta_{10} + 928502 \beta_{11} - 2687076 \beta_{12} + 752408 \beta_{13} ) q^{92} + ( 28185658434 - 910784257 \beta_{1} - 6858487 \beta_{2} - 16031379 \beta_{3} + 2860112 \beta_{4} + 751021 \beta_{5} + 920354 \beta_{6} + 1552990 \beta_{7} + 1521724 \beta_{8} - 730904 \beta_{9} + 210436 \beta_{10} - 514190 \beta_{11} + 1340440 \beta_{12} + 252390 \beta_{13} ) q^{93} + ( 8403645265 - 637660826 \beta_{1} - 4862787 \beta_{2} - 2148588 \beta_{3} - 629667 \beta_{4} - 386991 \beta_{5} - 2658988 \beta_{6} - 549163 \beta_{7} + 1725409 \beta_{8} - 711081 \beta_{9} - 8074 \beta_{10} - 1588781 \beta_{11} + 3385842 \beta_{12} + 114524 \beta_{13} ) q^{94} + ( 4995699595 + 133332461 \beta_{1} + 1247929 \beta_{2} + 36717949 \beta_{3} - 1403583 \beta_{4} + 2032088 \beta_{5} + 2524000 \beta_{6} - 147887 \beta_{7} + 868571 \beta_{8} - 1126633 \beta_{9} + 87076 \beta_{10} + 492816 \beta_{11} - 715566 \beta_{12} - 426907 \beta_{13} ) q^{95} + ( 892148478 - 1668905551 \beta_{1} - 4851428 \beta_{2} - 25207712 \beta_{3} + 3027005 \beta_{4} + 2712727 \beta_{5} - 2455195 \beta_{6} - 1027530 \beta_{7} + 654217 \beta_{8} - 4579932 \beta_{9} - 1429231 \beta_{10} - 1564851 \beta_{11} + 308201 \beta_{12} + 520502 \beta_{13} ) q^{96} + ( 9804399069 - 202835745 \beta_{1} + 2579087 \beta_{2} - 1105460 \beta_{3} - 812844 \beta_{4} + 3992576 \beta_{5} + 176182 \beta_{6} - 307576 \beta_{7} - 129953 \beta_{8} - 1623310 \beta_{9} - 239132 \beta_{10} + 723560 \beta_{11} + 2023603 \beta_{12} - 584696 \beta_{13} ) q^{97} + ( -34643281724 - 40062165 \beta_{1} - 10176346 \beta_{2} + 79849338 \beta_{3} + 572322 \beta_{4} + 7508146 \beta_{5} - 1482718 \beta_{6} - 1476500 \beta_{7} - 3470614 \beta_{8} - 2407670 \beta_{9} + 962456 \beta_{10} + 1258030 \beta_{11} - 4148104 \beta_{12} - 32320 \beta_{13} ) q^{98} + ( 16813511439 + 247762407 \beta_{1} - 6897793 \beta_{2} - 104876381 \beta_{3} - 2938921 \beta_{4} - 2337366 \beta_{5} + 2246206 \beta_{6} + 1618805 \beta_{7} + 729725 \beta_{8} + 705053 \beta_{9} + 481058 \beta_{10} - 1940958 \beta_{11} + 444280 \beta_{12} + 261501 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q + 476q^{3} + 18362q^{4} + 9760q^{5} + 18454q^{6} + 85024q^{7} + 126588q^{8} + 1372146q^{9} + O(q^{10}) \) \( 14q + 476q^{3} + 18362q^{4} + 9760q^{5} + 18454q^{6} + 85024q^{7} + 126588q^{8} + 1372146q^{9} + 713576q^{10} + 398020q^{11} - 4026800q^{12} + 2272440q^{13} - 7199712q^{14} - 4763864q^{15} + 19015138q^{16} + 5623508q^{17} - 204156q^{18} + 29803300q^{19} + 65161006q^{20} + 51227832q^{21} + 167334266q^{22} + 52654304q^{23} + 221514842q^{24} + 194970462q^{25} + 373581536q^{26} + 397348256q^{27} + 319501772q^{28} - 287156086q^{29} + 423014226q^{30} + 634041348q^{31} + 1260290884q^{32} + 1180833420q^{33} + 1316105060q^{34} + 1599853768q^{35} + 3198076132q^{36} + 488665204q^{37} + 1892845072q^{38} + 1972619104q^{39} + 1826486880q^{40} + 198215164q^{41} + 1011384468q^{42} + 2193188100q^{43} + 26522720q^{44} - 1129321956q^{45} - 1567525268q^{46} - 4175934476q^{47} - 15582938120q^{48} + 1105222462q^{49} - 6630582612q^{50} + 3297462720q^{51} - 4557341374q^{52} - 13223081840q^{53} - 8946135054q^{54} - 2726359424q^{55} - 27538267872q^{56} - 24477013312q^{57} + 352219640q^{59} - 36042747924q^{60} - 7658546476q^{61} - 10024135594q^{62} - 23037581736q^{63} + 14721327762q^{64} + 1152802884q^{65} - 99505241364q^{66} + 21781534280q^{67} - 104178000188q^{68} - 14601399408q^{69} - 67948872984q^{70} - 5573287168q^{71} - 24062143544q^{72} + 39661511924q^{73} + 28506052056q^{74} + 81845109044q^{75} + 166950090320q^{76} + 38773567192q^{77} + 54249159006q^{78} + 105565209020q^{79} + 146242150550q^{80} + 170581084750q^{81} + 47345182756q^{82} + 127846064024q^{83} + 215311861496q^{84} + 83883234552q^{85} - 103162039382q^{86} - 9763306924q^{87} + 418253082102q^{88} + 187826099404q^{89} + 96335639960q^{90} + 58390389864q^{91} - 259645875396q^{92} + 394641636020q^{93} + 117694719934q^{94} + 69935059424q^{95} + 12533631786q^{96} + 137285937500q^{97} - 484896369168q^{98} + 235419947204q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 23517 x^{12} - 42196 x^{11} + 214206700 x^{10} + 532863376 x^{9} - 951901011680 x^{8} - 1439668569088 x^{7} + 2125122050961664 x^{6} - 3617975849298944 x^{5} - 2179155305696991232 x^{4} + 15507015861824716800 x^{3} + 709645914927917694976 x^{2} - 9478460460037847384064 x + 30797548279705794248704\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(65\!\cdots\!65\)\( \nu^{13} + \)\(36\!\cdots\!56\)\( \nu^{12} + \)\(12\!\cdots\!25\)\( \nu^{11} - \)\(67\!\cdots\!80\)\( \nu^{10} - \)\(93\!\cdots\!76\)\( \nu^{9} + \)\(45\!\cdots\!00\)\( \nu^{8} + \)\(30\!\cdots\!84\)\( \nu^{7} - \)\(13\!\cdots\!60\)\( \nu^{6} - \)\(39\!\cdots\!40\)\( \nu^{5} + \)\(17\!\cdots\!24\)\( \nu^{4} + \)\(71\!\cdots\!36\)\( \nu^{3} - \)\(55\!\cdots\!64\)\( \nu^{2} + \)\(65\!\cdots\!52\)\( \nu - \)\(18\!\cdots\!92\)\(\)\()/ \)\(44\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(65\!\cdots\!65\)\( \nu^{13} - \)\(36\!\cdots\!56\)\( \nu^{12} - \)\(12\!\cdots\!25\)\( \nu^{11} + \)\(67\!\cdots\!80\)\( \nu^{10} + \)\(93\!\cdots\!76\)\( \nu^{9} - \)\(45\!\cdots\!00\)\( \nu^{8} - \)\(30\!\cdots\!84\)\( \nu^{7} + \)\(13\!\cdots\!60\)\( \nu^{6} + \)\(39\!\cdots\!40\)\( \nu^{5} - \)\(17\!\cdots\!24\)\( \nu^{4} - \)\(71\!\cdots\!36\)\( \nu^{3} + \)\(59\!\cdots\!64\)\( \nu^{2} - \)\(66\!\cdots\!52\)\( \nu + \)\(36\!\cdots\!92\)\(\)\()/ \)\(44\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(10\!\cdots\!87\)\( \nu^{13} - \)\(39\!\cdots\!29\)\( \nu^{12} + \)\(44\!\cdots\!15\)\( \nu^{11} + \)\(81\!\cdots\!57\)\( \nu^{10} - \)\(60\!\cdots\!36\)\( \nu^{9} - \)\(62\!\cdots\!08\)\( \nu^{8} + \)\(36\!\cdots\!64\)\( \nu^{7} + \)\(21\!\cdots\!00\)\( \nu^{6} - \)\(10\!\cdots\!08\)\( \nu^{5} - \)\(30\!\cdots\!28\)\( \nu^{4} + \)\(13\!\cdots\!84\)\( \nu^{3} + \)\(10\!\cdots\!12\)\( \nu^{2} - \)\(51\!\cdots\!64\)\( \nu + \)\(25\!\cdots\!28\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(40\!\cdots\!27\)\( \nu^{13} - \)\(45\!\cdots\!28\)\( \nu^{12} - \)\(69\!\cdots\!15\)\( \nu^{11} + \)\(88\!\cdots\!68\)\( \nu^{10} + \)\(36\!\cdots\!08\)\( \nu^{9} - \)\(63\!\cdots\!52\)\( \nu^{8} - \)\(29\!\cdots\!12\)\( \nu^{7} + \)\(20\!\cdots\!20\)\( \nu^{6} - \)\(27\!\cdots\!72\)\( \nu^{5} - \)\(27\!\cdots\!40\)\( \nu^{4} + \)\(70\!\cdots\!84\)\( \nu^{3} + \)\(91\!\cdots\!16\)\( \nu^{2} - \)\(36\!\cdots\!00\)\( \nu + \)\(18\!\cdots\!96\)\(\)\()/ \)\(21\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(16\!\cdots\!69\)\( \nu^{13} + \)\(37\!\cdots\!60\)\( \nu^{12} + \)\(35\!\cdots\!05\)\( \nu^{11} - \)\(62\!\cdots\!76\)\( \nu^{10} - \)\(30\!\cdots\!00\)\( \nu^{9} + \)\(36\!\cdots\!84\)\( \nu^{8} + \)\(12\!\cdots\!80\)\( \nu^{7} - \)\(94\!\cdots\!80\)\( \nu^{6} - \)\(25\!\cdots\!36\)\( \nu^{5} + \)\(13\!\cdots\!16\)\( \nu^{4} + \)\(24\!\cdots\!76\)\( \nu^{3} - \)\(14\!\cdots\!68\)\( \nu^{2} - \)\(80\!\cdots\!72\)\( \nu + \)\(67\!\cdots\!80\)\(\)\()/ \)\(44\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(36\!\cdots\!97\)\( \nu^{13} + \)\(36\!\cdots\!98\)\( \nu^{12} + \)\(63\!\cdots\!65\)\( \nu^{11} - \)\(69\!\cdots\!98\)\( \nu^{10} - \)\(35\!\cdots\!28\)\( \nu^{9} + \)\(49\!\cdots\!72\)\( \nu^{8} + \)\(50\!\cdots\!92\)\( \nu^{7} - \)\(15\!\cdots\!20\)\( \nu^{6} + \)\(15\!\cdots\!92\)\( \nu^{5} + \)\(21\!\cdots\!00\)\( \nu^{4} - \)\(47\!\cdots\!84\)\( \nu^{3} - \)\(66\!\cdots\!36\)\( \nu^{2} + \)\(24\!\cdots\!80\)\( \nu - \)\(15\!\cdots\!36\)\(\)\()/ \)\(74\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(20\!\cdots\!69\)\( \nu^{13} + \)\(21\!\cdots\!44\)\( \nu^{12} - \)\(47\!\cdots\!05\)\( \nu^{11} - \)\(57\!\cdots\!04\)\( \nu^{10} + \)\(43\!\cdots\!16\)\( \nu^{9} + \)\(53\!\cdots\!56\)\( \nu^{8} - \)\(19\!\cdots\!24\)\( \nu^{7} - \)\(20\!\cdots\!60\)\( \nu^{6} + \)\(41\!\cdots\!16\)\( \nu^{5} + \)\(29\!\cdots\!60\)\( \nu^{4} - \)\(41\!\cdots\!92\)\( \nu^{3} - \)\(29\!\cdots\!88\)\( \nu^{2} + \)\(13\!\cdots\!20\)\( \nu - \)\(80\!\cdots\!08\)\(\)\()/ \)\(27\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(40\!\cdots\!83\)\( \nu^{13} - \)\(39\!\cdots\!60\)\( \nu^{12} - \)\(70\!\cdots\!35\)\( \nu^{11} + \)\(76\!\cdots\!32\)\( \nu^{10} + \)\(41\!\cdots\!40\)\( \nu^{9} - \)\(54\!\cdots\!88\)\( \nu^{8} - \)\(74\!\cdots\!20\)\( \nu^{7} + \)\(17\!\cdots\!60\)\( \nu^{6} - \)\(11\!\cdots\!48\)\( \nu^{5} - \)\(24\!\cdots\!72\)\( \nu^{4} + \)\(43\!\cdots\!28\)\( \nu^{3} + \)\(87\!\cdots\!36\)\( \nu^{2} - \)\(24\!\cdots\!76\)\( \nu + \)\(11\!\cdots\!20\)\(\)\()/ \)\(44\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(24\!\cdots\!83\)\( \nu^{13} - \)\(15\!\cdots\!36\)\( \nu^{12} - \)\(49\!\cdots\!35\)\( \nu^{11} + \)\(28\!\cdots\!52\)\( \nu^{10} + \)\(35\!\cdots\!36\)\( \nu^{9} - \)\(19\!\cdots\!48\)\( \nu^{8} - \)\(11\!\cdots\!84\)\( \nu^{7} + \)\(62\!\cdots\!20\)\( \nu^{6} + \)\(15\!\cdots\!32\)\( \nu^{5} - \)\(87\!\cdots\!16\)\( \nu^{4} - \)\(31\!\cdots\!68\)\( \nu^{3} + \)\(38\!\cdots\!00\)\( \nu^{2} - \)\(25\!\cdots\!88\)\( \nu - \)\(72\!\cdots\!48\)\(\)\()/ \)\(22\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(66\!\cdots\!51\)\( \nu^{13} + \)\(53\!\cdots\!20\)\( \nu^{12} + \)\(12\!\cdots\!35\)\( \nu^{11} - \)\(10\!\cdots\!04\)\( \nu^{10} - \)\(85\!\cdots\!20\)\( \nu^{9} + \)\(74\!\cdots\!16\)\( \nu^{8} + \)\(23\!\cdots\!80\)\( \nu^{7} - \)\(23\!\cdots\!00\)\( \nu^{6} - \)\(15\!\cdots\!04\)\( \nu^{5} + \)\(33\!\cdots\!84\)\( \nu^{4} - \)\(27\!\cdots\!56\)\( \nu^{3} - \)\(12\!\cdots\!32\)\( \nu^{2} + \)\(21\!\cdots\!52\)\( \nu - \)\(84\!\cdots\!60\)\(\)\()/ \)\(55\!\cdots\!60\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(44\!\cdots\!71\)\( \nu^{13} + \)\(19\!\cdots\!68\)\( \nu^{12} + \)\(91\!\cdots\!95\)\( \nu^{11} - \)\(36\!\cdots\!44\)\( \nu^{10} - \)\(70\!\cdots\!88\)\( \nu^{9} + \)\(24\!\cdots\!36\)\( \nu^{8} + \)\(25\!\cdots\!12\)\( \nu^{7} - \)\(76\!\cdots\!00\)\( \nu^{6} - \)\(40\!\cdots\!64\)\( \nu^{5} + \)\(10\!\cdots\!76\)\( \nu^{4} + \)\(22\!\cdots\!72\)\( \nu^{3} - \)\(47\!\cdots\!04\)\( \nu^{2} - \)\(25\!\cdots\!12\)\( \nu + \)\(32\!\cdots\!24\)\(\)\()/ \)\(19\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(11\!\cdots\!45\)\( \nu^{13} + \)\(46\!\cdots\!44\)\( \nu^{12} + \)\(24\!\cdots\!25\)\( \nu^{11} - \)\(84\!\cdots\!60\)\( \nu^{10} - \)\(19\!\cdots\!24\)\( \nu^{9} + \)\(56\!\cdots\!60\)\( \nu^{8} + \)\(71\!\cdots\!16\)\( \nu^{7} - \)\(17\!\cdots\!40\)\( \nu^{6} - \)\(12\!\cdots\!00\)\( \nu^{5} + \)\(23\!\cdots\!16\)\( \nu^{4} + \)\(77\!\cdots\!04\)\( \nu^{3} - \)\(11\!\cdots\!56\)\( \nu^{2} - \)\(13\!\cdots\!32\)\( \nu + \)\(11\!\cdots\!92\)\(\)\()/ \)\(44\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 3 \beta_{1} + 3360\)
\(\nu^{3}\)\(=\)\(\beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - 55 \beta_{3} + 17 \beta_{2} + 5277 \beta_{1} + 9049\)
\(\nu^{4}\)\(=\)\(-66 \beta_{13} + 117 \beta_{12} - 67 \beta_{11} + 63 \beta_{10} - 92 \beta_{9} + 15 \beta_{8} - 24 \beta_{7} + 97 \beta_{6} - 43 \beta_{5} + 15 \beta_{4} + 10306 \beta_{3} + 7126 \beta_{2} + 52303 \beta_{1} + 17808144\)
\(\nu^{5}\)\(=\)\(-2552 \beta_{13} - 756 \beta_{12} + 1370 \beta_{11} + 6217 \beta_{10} + 22124 \beta_{9} - 13383 \beta_{8} + 12369 \beta_{7} + 20872 \beta_{6} - 21191 \beta_{5} - 27401 \beta_{4} - 443017 \beta_{3} + 155915 \beta_{2} + 31783247 \beta_{1} + 164162017\)
\(\nu^{6}\)\(=\)\(-788730 \beta_{13} + 1295329 \beta_{12} - 623431 \beta_{11} + 826663 \beta_{10} - 1092372 \beta_{9} + 45183 \beta_{8} + 21228 \beta_{7} + 1119717 \beta_{6} - 217555 \beta_{5} + 293351 \beta_{4} + 92484050 \beta_{3} + 49242150 \beta_{2} + 509524531 \beta_{1} + 107440043028\)
\(\nu^{7}\)\(=\)\(-35663488 \beta_{13} - 2910048 \beta_{12} + 18768470 \beta_{11} + 37096933 \beta_{10} + 197875700 \beta_{9} - 119160531 \beta_{8} + 125465889 \beta_{7} + 186167300 \beta_{6} - 226172571 \beta_{5} - 208665269 \beta_{4} - 2715378969 \beta_{3} + 1249172571 \beta_{2} + 206509520851 \beta_{1} + 1603640220849\)
\(\nu^{8}\)\(=\)\(-7278434650 \beta_{13} + 11269781201 \beta_{12} - 4903315199 \beta_{11} + 7944783867 \beta_{10} - 9911687332 \beta_{9} - 649831789 \beta_{8} + 2282827744 \beta_{7} + 10574689093 \beta_{6} - 895856575 \beta_{5} + 3026351059 \beta_{4} + 789776047054 \beta_{3} + 345909220722 \beta_{2} + 4260603186287 \beta_{1} + 698080000353096\)
\(\nu^{9}\)\(=\)\(-366122300872 \beta_{13} + 28779169172 \beta_{12} + 178183947178 \beta_{11} + 247620648481 \beta_{10} + 1625477942660 \beta_{9} - 953793551703 \beta_{8} + 1147640813217 \beta_{7} + 1578416288616 \beta_{6} - 2041131691591 \beta_{5} - 1521098982969 \beta_{4} - 14013194977057 \beta_{3} + 9699803731939 \beta_{2} + 1410535940045311 \beta_{1} + 13368597364865745\)
\(\nu^{10}\)\(=\)\(-61606633335418 \beta_{13} + 90523428541985 \beta_{12} - 36795179509175 \beta_{11} + 68453113754479 \beta_{10} - 81088724248148 \beta_{9} - 12925576498649 \beta_{8} + 33040030494356 \beta_{7} + 93601109864997 \beta_{6} - 5892543698283 \beta_{5} + 26074459866751 \beta_{4} + 6508819401594202 \beta_{3} + 2478880161780942 \beta_{2} + 33819163535196475 \beta_{1} + 4763409666412737468\)
\(\nu^{11}\)\(=\)\(-3341702997035152 \beta_{13} + 628476013502792 \beta_{12} + 1508726674783998 \beta_{11} + 1861148762386989 \beta_{10} + 12784409532874228 \beta_{9} - 7412999372542571 \beta_{8} + 9896973681109553 \beta_{7} + 13032276836608364 \beta_{6} - 17173505622839555 \beta_{5} - 10968582645423213 \beta_{4} - 55611583264842713 \beta_{3} + 75064970014112187 \beta_{2} + 9966684101214984699 \beta_{1} + 105729837428853835905\)
\(\nu^{12}\)\(=\)\(-501925791142133946 \beta_{13} + 702635689638602561 \beta_{12} - 269914934005287839 \beta_{11} + 561038148943300371 \beta_{10} - 629367997791997124 \beta_{9} - 154274727147977493 \beta_{8} + 357821473402499976 \beta_{7} + 802009159414421141 \beta_{6} - 69158364403613319 \beta_{5} + 208488252166094555 \beta_{4} + 52366183298455507014 \beta_{3} + 18069796113850714058 \beta_{2} + 264579314564760708023 \beta_{1} + 33607823661155523235056\)
\(\nu^{13}\)\(=\)\(-28774296703904137304 \beta_{13} + 7534851026752235388 \beta_{12} + 12207690160739868338 \beta_{11} + 15092625253731246729 \beta_{10} + 98130940990267319844 \beta_{9} - 57267048610740123695 \beta_{8} + 82386147259083827537 \beta_{7} + 105846175603287002288 \beta_{6} - 139308617791769864175 \beta_{5} - 79080187334212623473 \beta_{4} - 61299174272126474529 \beta_{3} + 584196506542199597379 \beta_{2} + 72086822421661096058535 \beta_{1} + 824988149723128901642433\)

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
−85.4920
−67.9786
−62.7554
−61.8148
−55.2092
−26.9644
6.06439
8.22600
20.2914
35.2896
51.1787
73.5071
77.1568
88.5004
−85.4920 −725.551 5260.89 −1144.42 62028.9 2737.98 −274676. 349278. 97838.7
1.2 −67.9786 −249.255 2573.09 12131.4 16944.0 74935.0 −35694.6 −115019. −824677.
1.3 −62.7554 761.773 1890.24 1400.24 −47805.4 77426.1 9900.16 403151. −87872.8
1.4 −61.8148 211.836 1773.06 −11752.2 −13094.6 18695.9 16995.1 −132273. 726462.
1.5 −55.2092 −283.770 1000.05 5132.07 15666.7 −31764.7 57856.2 −96621.5 −283337.
1.6 −26.9644 724.855 −1320.92 −9762.32 −19545.3 −73001.2 90841.0 348267. 263235.
1.7 6.06439 −282.467 −2011.22 −8656.32 −1712.99 −42605.5 −24616.7 −97359.2 −52495.3
1.8 8.22600 −223.329 −1980.33 8469.53 −1837.11 2360.62 −33137.1 −127271. 69670.4
1.9 20.2914 719.778 −1636.26 12035.6 14605.3 8773.61 −74758.8 340933. 244220.
1.10 35.2896 118.146 −802.642 −5887.06 4169.31 66890.4 −100598. −163189. −207752.
1.11 51.1787 −728.829 571.255 −4173.93 −37300.5 −8856.29 −75577.8 354044. −213616.
1.12 73.5071 274.559 3355.30 7553.66 20182.1 46751.0 96095.7 −101764. 555248.
1.13 77.1568 692.210 3905.18 −3176.79 53408.8 −18569.7 143294. 302008. −245111.
1.14 88.5004 −533.955 5784.31 7590.53 −47255.2 −38749.2 330665. 107961. 671765.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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.b 14
3.b odd 2 1 261.12.a.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.12.a.b 14 1.a even 1 1 trivial
261.12.a.e 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(14\!\cdots\!88\)\( T_{2}^{7} + \)\(21\!\cdots\!64\)\( T_{2}^{6} - \)\(36\!\cdots\!44\)\( T_{2}^{5} - \)\(21\!\cdots\!32\)\( T_{2}^{4} + \)\(15\!\cdots\!00\)\( T_{2}^{3} + \)\(70\!\cdots\!76\)\( T_{2}^{2} - \)\(94\!\cdots\!64\)\( T_{2} + \)\(30\!\cdots\!04\)\( \)">\(T_{2}^{14} - \cdots\) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(29))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(30\!\cdots\!04\)\( - \)\(94\!\cdots\!64\)\( T + \)\(70\!\cdots\!76\)\( T^{2} + 15507015861824716800 T^{3} - 2179155305696991232 T^{4} - 3617975849298944 T^{5} + 2125122050961664 T^{6} - 1439668569088 T^{7} - 951901011680 T^{8} + 532863376 T^{9} + 214206700 T^{10} - 42196 T^{11} - 23517 T^{12} + T^{14} \)
$3$ \( -\)\(23\!\cdots\!20\)\( + \)\(51\!\cdots\!48\)\( T + \)\(21\!\cdots\!09\)\( T^{2} - \)\(60\!\cdots\!60\)\( T^{3} - \)\(63\!\cdots\!26\)\( T^{4} - \)\(37\!\cdots\!84\)\( T^{5} + \)\(79\!\cdots\!11\)\( T^{6} + 72678371903563652064 T^{7} - 448558652197825824 T^{8} - 365976293784724 T^{9} + 1284056989003 T^{10} + 710185456 T^{11} - 1812814 T^{12} - 476 T^{13} + T^{14} \)
$5$ \( -\)\(45\!\cdots\!00\)\( - \)\(20\!\cdots\!00\)\( T + \)\(31\!\cdots\!25\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} - \)\(33\!\cdots\!50\)\( T^{4} - \)\(89\!\cdots\!00\)\( T^{5} + \)\(16\!\cdots\!75\)\( T^{6} + \)\(30\!\cdots\!00\)\( T^{7} - \)\(42\!\cdots\!20\)\( T^{8} - \)\(48\!\cdots\!04\)\( T^{9} + 58632762499756023 T^{10} + 3589058440724 T^{11} - 391653306 T^{12} - 9760 T^{13} + T^{14} \)
$7$ \( \)\(12\!\cdots\!08\)\( - \)\(91\!\cdots\!68\)\( T + \)\(13\!\cdots\!56\)\( T^{2} + \)\(21\!\cdots\!48\)\( T^{3} - \)\(26\!\cdots\!00\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{5} + \)\(88\!\cdots\!32\)\( T^{6} + \)\(37\!\cdots\!72\)\( T^{7} - \)\(87\!\cdots\!24\)\( T^{8} - \)\(29\!\cdots\!84\)\( T^{9} + 42558232188289544032 T^{10} + 878071751921440 T^{11} - 10779358144 T^{12} - 85024 T^{13} + T^{14} \)
$11$ \( \)\(12\!\cdots\!04\)\( - \)\(24\!\cdots\!96\)\( T - \)\(35\!\cdots\!11\)\( T^{2} + \)\(67\!\cdots\!84\)\( T^{3} + \)\(25\!\cdots\!10\)\( T^{4} - \)\(56\!\cdots\!16\)\( T^{5} + \)\(14\!\cdots\!67\)\( T^{6} + \)\(13\!\cdots\!80\)\( T^{7} - \)\(24\!\cdots\!88\)\( T^{8} - \)\(54\!\cdots\!44\)\( T^{9} + \)\(12\!\cdots\!75\)\( T^{10} + 787522250657795064 T^{11} - 1975678260326 T^{12} - 398020 T^{13} + T^{14} \)
$13$ \( \)\(17\!\cdots\!76\)\( - \)\(61\!\cdots\!08\)\( T - \)\(28\!\cdots\!87\)\( T^{2} + \)\(82\!\cdots\!04\)\( T^{3} + \)\(95\!\cdots\!10\)\( T^{4} - \)\(30\!\cdots\!96\)\( T^{5} - \)\(15\!\cdots\!41\)\( T^{6} + \)\(33\!\cdots\!40\)\( T^{7} - \)\(81\!\cdots\!00\)\( T^{8} - \)\(15\!\cdots\!64\)\( T^{9} + \)\(56\!\cdots\!19\)\( T^{10} + 32590633231860117092 T^{11} - 13291506188706 T^{12} - 2272440 T^{13} + T^{14} \)
$17$ \( -\)\(35\!\cdots\!36\)\( + \)\(66\!\cdots\!36\)\( T + \)\(11\!\cdots\!40\)\( T^{2} - \)\(93\!\cdots\!16\)\( T^{3} - \)\(36\!\cdots\!76\)\( T^{4} - \)\(33\!\cdots\!20\)\( T^{5} + \)\(31\!\cdots\!76\)\( T^{6} + \)\(34\!\cdots\!00\)\( T^{7} - \)\(13\!\cdots\!24\)\( T^{8} - \)\(11\!\cdots\!68\)\( T^{9} + \)\(27\!\cdots\!84\)\( T^{10} + \)\(13\!\cdots\!28\)\( T^{11} - 275385619571492 T^{12} - 5623508 T^{13} + T^{14} \)
$19$ \( \)\(15\!\cdots\!80\)\( + \)\(17\!\cdots\!76\)\( T - \)\(10\!\cdots\!16\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!60\)\( T^{4} - \)\(22\!\cdots\!68\)\( T^{5} - \)\(87\!\cdots\!24\)\( T^{6} + \)\(13\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!44\)\( T^{8} - \)\(24\!\cdots\!72\)\( T^{9} - \)\(35\!\cdots\!68\)\( T^{10} + \)\(16\!\cdots\!72\)\( T^{11} - 327573849020988 T^{12} - 29803300 T^{13} + T^{14} \)
$23$ \( -\)\(23\!\cdots\!24\)\( + \)\(28\!\cdots\!08\)\( T - \)\(65\!\cdots\!56\)\( T^{2} - \)\(44\!\cdots\!68\)\( T^{3} + \)\(99\!\cdots\!12\)\( T^{4} - \)\(47\!\cdots\!56\)\( T^{5} - \)\(33\!\cdots\!24\)\( T^{6} + \)\(24\!\cdots\!16\)\( T^{7} + \)\(24\!\cdots\!96\)\( T^{8} - \)\(41\!\cdots\!32\)\( T^{9} + \)\(32\!\cdots\!52\)\( T^{10} + \)\(26\!\cdots\!32\)\( T^{11} - 4067166575811448 T^{12} - 52654304 T^{13} + T^{14} \)
$29$ \( ( 20511149 + T )^{14} \)
$31$ \( \)\(97\!\cdots\!68\)\( - \)\(31\!\cdots\!88\)\( T - \)\(20\!\cdots\!39\)\( T^{2} + \)\(82\!\cdots\!92\)\( T^{3} + \)\(16\!\cdots\!78\)\( T^{4} - \)\(81\!\cdots\!24\)\( T^{5} - \)\(64\!\cdots\!81\)\( T^{6} + \)\(39\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!24\)\( T^{8} - \)\(98\!\cdots\!96\)\( T^{9} - \)\(68\!\cdots\!13\)\( T^{10} + \)\(12\!\cdots\!16\)\( T^{11} - 88556349391993574 T^{12} - 634041348 T^{13} + T^{14} \)
$37$ \( -\)\(35\!\cdots\!80\)\( - \)\(13\!\cdots\!76\)\( T + \)\(60\!\cdots\!64\)\( T^{2} - \)\(91\!\cdots\!16\)\( T^{3} + \)\(55\!\cdots\!84\)\( T^{4} - \)\(10\!\cdots\!64\)\( T^{5} - \)\(32\!\cdots\!04\)\( T^{6} + \)\(15\!\cdots\!36\)\( T^{7} - \)\(48\!\cdots\!44\)\( T^{8} - \)\(59\!\cdots\!72\)\( T^{9} + \)\(64\!\cdots\!48\)\( T^{10} + \)\(91\!\cdots\!56\)\( T^{11} - 1423293177504358684 T^{12} - 488665204 T^{13} + T^{14} \)
$41$ \( \)\(42\!\cdots\!00\)\( - \)\(21\!\cdots\!00\)\( T - \)\(83\!\cdots\!00\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!00\)\( T^{4} + \)\(39\!\cdots\!00\)\( T^{5} - \)\(55\!\cdots\!36\)\( T^{6} - \)\(90\!\cdots\!32\)\( T^{7} - \)\(11\!\cdots\!52\)\( T^{8} + \)\(47\!\cdots\!52\)\( T^{9} + \)\(10\!\cdots\!00\)\( T^{10} - \)\(48\!\cdots\!80\)\( T^{11} - 2018509380850096508 T^{12} - 198215164 T^{13} + T^{14} \)
$43$ \( \)\(63\!\cdots\!08\)\( - \)\(14\!\cdots\!24\)\( T - \)\(12\!\cdots\!79\)\( T^{2} + \)\(34\!\cdots\!04\)\( T^{3} + \)\(54\!\cdots\!58\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{5} - \)\(55\!\cdots\!73\)\( T^{6} + \)\(42\!\cdots\!04\)\( T^{7} - \)\(74\!\cdots\!84\)\( T^{8} - \)\(38\!\cdots\!28\)\( T^{9} + \)\(13\!\cdots\!71\)\( T^{10} + \)\(15\!\cdots\!24\)\( T^{11} - 6391878436353327686 T^{12} - 2193188100 T^{13} + T^{14} \)
$47$ \( -\)\(13\!\cdots\!48\)\( - \)\(67\!\cdots\!72\)\( T - \)\(95\!\cdots\!87\)\( T^{2} + \)\(31\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!06\)\( T^{4} + \)\(72\!\cdots\!16\)\( T^{5} - \)\(10\!\cdots\!69\)\( T^{6} - \)\(80\!\cdots\!76\)\( T^{7} + \)\(23\!\cdots\!12\)\( T^{8} + \)\(32\!\cdots\!24\)\( T^{9} + \)\(46\!\cdots\!31\)\( T^{10} - \)\(59\!\cdots\!44\)\( T^{11} - 9357439440633742958 T^{12} + 4175934476 T^{13} + T^{14} \)
$53$ \( \)\(45\!\cdots\!68\)\( - \)\(22\!\cdots\!52\)\( T - \)\(24\!\cdots\!03\)\( T^{2} + \)\(52\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} + \)\(68\!\cdots\!44\)\( T^{5} - \)\(15\!\cdots\!89\)\( T^{6} - \)\(28\!\cdots\!28\)\( T^{7} + \)\(97\!\cdots\!76\)\( T^{8} + \)\(26\!\cdots\!28\)\( T^{9} - \)\(19\!\cdots\!53\)\( T^{10} - \)\(99\!\cdots\!32\)\( T^{11} - 26159337280054394354 T^{12} + 13223081840 T^{13} + T^{14} \)
$59$ \( -\)\(76\!\cdots\!00\)\( + \)\(75\!\cdots\!00\)\( T + \)\(87\!\cdots\!00\)\( T^{2} - \)\(16\!\cdots\!00\)\( T^{3} - \)\(47\!\cdots\!00\)\( T^{4} + \)\(51\!\cdots\!00\)\( T^{5} + \)\(79\!\cdots\!00\)\( T^{6} - \)\(39\!\cdots\!72\)\( T^{7} - \)\(54\!\cdots\!92\)\( T^{8} + \)\(80\!\cdots\!08\)\( T^{9} + \)\(15\!\cdots\!52\)\( T^{10} - \)\(25\!\cdots\!08\)\( T^{11} - \)\(20\!\cdots\!24\)\( T^{12} - 352219640 T^{13} + T^{14} \)
$61$ \( -\)\(31\!\cdots\!36\)\( + \)\(29\!\cdots\!68\)\( T + \)\(13\!\cdots\!24\)\( T^{2} - \)\(12\!\cdots\!44\)\( T^{3} - \)\(27\!\cdots\!20\)\( T^{4} + \)\(16\!\cdots\!16\)\( T^{5} + \)\(26\!\cdots\!12\)\( T^{6} - \)\(98\!\cdots\!48\)\( T^{7} - \)\(13\!\cdots\!48\)\( T^{8} + \)\(25\!\cdots\!68\)\( T^{9} + \)\(32\!\cdots\!44\)\( T^{10} - \)\(24\!\cdots\!16\)\( T^{11} - \)\(31\!\cdots\!72\)\( T^{12} + 7658546476 T^{13} + T^{14} \)
$67$ \( \)\(49\!\cdots\!32\)\( + \)\(40\!\cdots\!60\)\( T - \)\(49\!\cdots\!24\)\( T^{2} + \)\(49\!\cdots\!40\)\( T^{3} + \)\(45\!\cdots\!04\)\( T^{4} - \)\(85\!\cdots\!40\)\( T^{5} - \)\(11\!\cdots\!80\)\( T^{6} + \)\(29\!\cdots\!60\)\( T^{7} + \)\(48\!\cdots\!28\)\( T^{8} - \)\(35\!\cdots\!00\)\( T^{9} + \)\(74\!\cdots\!16\)\( T^{10} + \)\(16\!\cdots\!80\)\( T^{11} - \)\(61\!\cdots\!52\)\( T^{12} - 21781534280 T^{13} + T^{14} \)
$71$ \( -\)\(17\!\cdots\!36\)\( - \)\(22\!\cdots\!00\)\( T + \)\(10\!\cdots\!32\)\( T^{2} + \)\(23\!\cdots\!36\)\( T^{3} - \)\(13\!\cdots\!44\)\( T^{4} - \)\(37\!\cdots\!92\)\( T^{5} + \)\(47\!\cdots\!92\)\( T^{6} + \)\(22\!\cdots\!16\)\( T^{7} - \)\(65\!\cdots\!44\)\( T^{8} + \)\(96\!\cdots\!20\)\( T^{9} + \)\(41\!\cdots\!60\)\( T^{10} - \)\(19\!\cdots\!56\)\( T^{11} - \)\(11\!\cdots\!16\)\( T^{12} + 5573287168 T^{13} + T^{14} \)
$73$ \( \)\(32\!\cdots\!00\)\( - \)\(24\!\cdots\!60\)\( T - \)\(37\!\cdots\!84\)\( T^{2} + \)\(19\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!64\)\( T^{4} - \)\(49\!\cdots\!80\)\( T^{5} - \)\(21\!\cdots\!24\)\( T^{6} + \)\(57\!\cdots\!44\)\( T^{7} + \)\(67\!\cdots\!76\)\( T^{8} - \)\(30\!\cdots\!52\)\( T^{9} + \)\(30\!\cdots\!84\)\( T^{10} + \)\(64\!\cdots\!44\)\( T^{11} - \)\(13\!\cdots\!52\)\( T^{12} - 39661511924 T^{13} + T^{14} \)
$79$ \( -\)\(38\!\cdots\!00\)\( + \)\(55\!\cdots\!80\)\( T - \)\(60\!\cdots\!47\)\( T^{2} - \)\(35\!\cdots\!04\)\( T^{3} + \)\(42\!\cdots\!66\)\( T^{4} + \)\(40\!\cdots\!40\)\( T^{5} - \)\(48\!\cdots\!41\)\( T^{6} - \)\(37\!\cdots\!20\)\( T^{7} + \)\(14\!\cdots\!24\)\( T^{8} - \)\(30\!\cdots\!88\)\( T^{9} - \)\(96\!\cdots\!81\)\( T^{10} + \)\(36\!\cdots\!24\)\( T^{11} - \)\(21\!\cdots\!46\)\( T^{12} - 105565209020 T^{13} + T^{14} \)
$83$ \( \)\(62\!\cdots\!88\)\( - \)\(46\!\cdots\!36\)\( T - \)\(39\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!28\)\( T^{3} + \)\(80\!\cdots\!40\)\( T^{4} - \)\(17\!\cdots\!64\)\( T^{5} - \)\(71\!\cdots\!32\)\( T^{6} + \)\(99\!\cdots\!96\)\( T^{7} + \)\(29\!\cdots\!12\)\( T^{8} - \)\(42\!\cdots\!52\)\( T^{9} - \)\(55\!\cdots\!52\)\( T^{10} + \)\(11\!\cdots\!00\)\( T^{11} + \)\(36\!\cdots\!92\)\( T^{12} - 127846064024 T^{13} + T^{14} \)
$89$ \( \)\(91\!\cdots\!40\)\( + \)\(25\!\cdots\!24\)\( T - \)\(14\!\cdots\!56\)\( T^{2} - \)\(55\!\cdots\!28\)\( T^{3} + \)\(32\!\cdots\!32\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} - \)\(11\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!16\)\( T^{7} + \)\(16\!\cdots\!60\)\( T^{8} - \)\(11\!\cdots\!28\)\( T^{9} - \)\(70\!\cdots\!20\)\( T^{10} + \)\(11\!\cdots\!24\)\( T^{11} + \)\(43\!\cdots\!32\)\( T^{12} - 187826099404 T^{13} + T^{14} \)
$97$ \( -\)\(35\!\cdots\!48\)\( + \)\(10\!\cdots\!40\)\( T - \)\(10\!\cdots\!12\)\( T^{2} + \)\(30\!\cdots\!20\)\( T^{3} + \)\(47\!\cdots\!80\)\( T^{4} - \)\(25\!\cdots\!40\)\( T^{5} - \)\(79\!\cdots\!92\)\( T^{6} + \)\(73\!\cdots\!40\)\( T^{7} - \)\(28\!\cdots\!80\)\( T^{8} - \)\(97\!\cdots\!20\)\( T^{9} + \)\(56\!\cdots\!32\)\( T^{10} + \)\(59\!\cdots\!40\)\( T^{11} - \)\(40\!\cdots\!56\)\( T^{12} - 137285937500 T^{13} + T^{14} \)
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