Properties

Label 9.9.d.a
Level 9
Weight 9
Character orbit 9.d
Analytic conductor 3.666
Analytic rank 0
Dimension 14
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 9.d (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(3.66640749055\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{21} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( -5 - 3 \beta_{3} + \beta_{4} + \beta_{6} ) q^{3} \) \( + ( 110 - \beta_{1} + 2 \beta_{2} - 110 \beta_{3} - \beta_{4} + \beta_{10} ) q^{4} \) \( + ( 20 + \beta_{1} + \beta_{2} + 21 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{12} ) q^{5} \) \( + ( -136 + 28 \beta_{1} + 3 \beta_{2} - 63 \beta_{3} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{6} \) \( + ( -2 - 13 \beta_{1} - 31 \beta_{2} + 140 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{7} \) \( + ( 690 - \beta_{1} + 94 \beta_{2} - 1375 \beta_{3} + 9 \beta_{4} + 2 \beta_{5} + 16 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 11 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{8} \) \( + ( 1034 - 55 \beta_{1} - 39 \beta_{2} + 524 \beta_{3} - 6 \beta_{4} - 5 \beta_{6} - 15 \beta_{7} - \beta_{8} - \beta_{9} - 9 \beta_{10} - 8 \beta_{11} - 7 \beta_{12} - 3 \beta_{13} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( -5 - 3 \beta_{3} + \beta_{4} + \beta_{6} ) q^{3} \) \( + ( 110 - \beta_{1} + 2 \beta_{2} - 110 \beta_{3} - \beta_{4} + \beta_{10} ) q^{4} \) \( + ( 20 + \beta_{1} + \beta_{2} + 21 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{12} ) q^{5} \) \( + ( -136 + 28 \beta_{1} + 3 \beta_{2} - 63 \beta_{3} - 2 \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{12} ) q^{6} \) \( + ( -2 - 13 \beta_{1} - 31 \beta_{2} + 140 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{7} \) \( + ( 690 - \beta_{1} + 94 \beta_{2} - 1375 \beta_{3} + 9 \beta_{4} + 2 \beta_{5} + 16 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} + 11 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{8} \) \( + ( 1034 - 55 \beta_{1} - 39 \beta_{2} + 524 \beta_{3} - 6 \beta_{4} - 5 \beta_{6} - 15 \beta_{7} - \beta_{8} - \beta_{9} - 9 \beta_{10} - 8 \beta_{11} - 7 \beta_{12} - 3 \beta_{13} ) q^{9} \) \( + ( 19 - 201 \beta_{1} - 94 \beta_{2} - 14 \beta_{3} + 66 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} + 7 \beta_{11} - 7 \beta_{12} + \beta_{13} ) q^{10} \) \( + ( -2737 + 2 \beta_{1} - \beta_{2} + 1372 \beta_{3} - 53 \beta_{4} + 10 \beta_{5} + 27 \beta_{6} + 11 \beta_{7} + 3 \beta_{8} - 23 \beta_{10} - \beta_{11} + 8 \beta_{13} ) q^{11} \) \( + ( -7747 + 369 \beta_{1} + 165 \beta_{2} + 7471 \beta_{3} + 89 \beta_{4} - 3 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} - 3 \beta_{8} - 63 \beta_{10} + 9 \beta_{11} - 6 \beta_{12} - 15 \beta_{13} ) q^{12} \) \( + ( 39 + 459 \beta_{1} - 290 \beta_{2} - 77 \beta_{3} - 212 \beta_{4} - 21 \beta_{5} - 129 \beta_{6} + 21 \beta_{7} - 2 \beta_{8} + \beta_{9} + 14 \beta_{10} - 16 \beta_{11} - 8 \beta_{12} + 4 \beta_{13} ) q^{13} \) \( + ( 5800 - 290 \beta_{1} - 76 \beta_{2} + 5703 \beta_{3} - 163 \beta_{4} + 37 \beta_{5} - 448 \beta_{6} - 133 \beta_{7} - 3 \beta_{9} - 56 \beta_{10} - 10 \beta_{12} + 29 \beta_{13} ) q^{14} \) \( + ( -57 - 26 \beta_{1} - 972 \beta_{2} - 10603 \beta_{3} + 20 \beta_{4} - 9 \beta_{5} + 41 \beta_{6} + 88 \beta_{7} + 15 \beta_{8} - \beta_{9} + 77 \beta_{10} + \beta_{11} + 13 \beta_{12} - 54 \beta_{13} ) q^{15} \) \( + ( -37 + 1640 \beta_{1} + 2845 \beta_{2} - 9833 \beta_{3} + 170 \beta_{4} - 47 \beta_{5} + 717 \beta_{6} + 65 \beta_{7} + 13 \beta_{8} - 26 \beta_{9} + 80 \beta_{10} + 5 \beta_{11} + 10 \beta_{12} + 15 \beta_{13} ) q^{16} \) \( + ( 1488 - 120 \beta_{1} - 482 \beta_{2} - 2732 \beta_{3} + 504 \beta_{4} + 78 \beta_{5} + 569 \beta_{6} - 33 \beta_{7} - 45 \beta_{8} + 45 \beta_{9} + 87 \beta_{10} - 10 \beta_{11} - 10 \beta_{12} + 75 \beta_{13} ) q^{17} \) \( + ( 20763 - 5010 \beta_{1} - 1464 \beta_{2} - 4719 \beta_{3} - 15 \beta_{4} - 30 \beta_{5} - 255 \beta_{6} - 66 \beta_{7} + 12 \beta_{8} + 18 \beta_{9} - 42 \beta_{10} + 72 \beta_{11} + 69 \beta_{12} - 105 \beta_{13} ) q^{18} \) \( + ( -18483 - 3228 \beta_{1} - 1668 \beta_{2} - 62 \beta_{3} + 787 \beta_{4} - 95 \beta_{5} + 27 \beta_{6} + 200 \beta_{7} - 31 \beta_{8} - 31 \beta_{9} + 117 \beta_{10} - 77 \beta_{11} + 77 \beta_{12} + 22 \beta_{13} ) q^{19} \) \( + ( 51119 + 462 \beta_{1} + 312 \beta_{2} - 25529 \beta_{3} - 1202 \beta_{4} + 129 \beta_{5} + 18 \beta_{6} - 174 \beta_{7} - 45 \beta_{8} + 21 \beta_{10} + 11 \beta_{11} + 105 \beta_{13} ) q^{20} \) \( + ( 15834 + 5505 \beta_{1} + 8499 \beta_{2} + 16498 \beta_{3} - 234 \beta_{4} - 21 \beta_{5} - 89 \beta_{6} + 63 \beta_{7} + 48 \beta_{8} - 3 \beta_{9} + 126 \beta_{10} - 69 \beta_{11} + 81 \beta_{12} - 150 \beta_{13} ) q^{21} \) \( + ( 6347 + 5583 \beta_{1} - 4682 \beta_{2} - 6628 \beta_{3} - 944 \beta_{4} - 87 \beta_{5} - 687 \beta_{6} + 87 \beta_{7} + 34 \beta_{8} - 17 \beta_{9} + 158 \beta_{10} + 146 \beta_{11} + 73 \beta_{12} + 34 \beta_{13} ) q^{22} \) \( + ( -47519 - 268 \beta_{1} + 115 \beta_{2} - 47820 \beta_{3} - 76 \beta_{4} + 101 \beta_{5} - 772 \beta_{6} + 247 \beta_{7} + 48 \beta_{9} + 176 \beta_{10} + 8 \beta_{12} + 103 \beta_{13} ) q^{23} \) \( + ( -113113 + 3336 \beta_{1} - 15477 \beta_{2} + 64051 \beta_{3} + 608 \beta_{4} - 27 \beta_{5} - 219 \beta_{6} - 401 \beta_{7} - 91 \beta_{8} + 16 \beta_{9} - 742 \beta_{10} + 11 \beta_{11} - 222 \beta_{12} - 45 \beta_{13} ) q^{24} \) \( + ( 55 + 8333 \beta_{1} + 16394 \beta_{2} + 5919 \beta_{3} + 118 \beta_{4} - 70 \beta_{5} + 746 \beta_{6} - 78 \beta_{7} - 58 \beta_{8} + 116 \beta_{9} - 102 \beta_{10} - 149 \beta_{11} - 298 \beta_{12} - 24 \beta_{13} ) q^{25} \) \( + ( -132879 - 765 \beta_{1} + 2328 \beta_{2} + 265724 \beta_{3} + 540 \beta_{4} - 9 \beta_{5} - 959 \beta_{6} - 504 \beta_{7} + 270 \beta_{8} - 270 \beta_{9} - 1116 \beta_{10} + 7 \beta_{11} + 7 \beta_{12} - 99 \beta_{13} ) q^{26} \) \( + ( 40902 - 15321 \beta_{1} - 801 \beta_{2} - 149511 \beta_{3} + 1323 \beta_{4} + 117 \beta_{5} + 1344 \beta_{6} + 1098 \beta_{7} - 39 \beta_{8} - 138 \beta_{9} - 90 \beta_{10} - 159 \beta_{11} - 228 \beta_{12} + 144 \beta_{13} ) q^{27} \) \( + ( 84758 - 37662 \beta_{1} - 18321 \beta_{2} - 51 \beta_{3} - 1782 \beta_{4} + 204 \beta_{5} - 579 \beta_{6} - 819 \beta_{7} + 195 \beta_{8} + 195 \beta_{9} - 231 \beta_{10} + 309 \beta_{11} - 309 \beta_{12} - 27 \beta_{13} ) q^{28} \) \( + ( 363696 - 6665 \beta_{1} - 116 \beta_{2} - 181977 \beta_{3} + 1554 \beta_{4} - 325 \beta_{5} - 810 \beta_{6} - 74 \beta_{7} + 267 \beta_{8} + 491 \beta_{10} - 18 \beta_{11} - 233 \beta_{13} ) q^{29} \) \( + ( -1695 + 29769 \beta_{1} + 29031 \beta_{2} + 329715 \beta_{3} - 600 \beta_{4} + 21 \beta_{5} + 96 \beta_{6} - 765 \beta_{7} - 315 \beta_{8} + 51 \beta_{9} + 333 \beta_{10} - 6 \beta_{11} - 456 \beta_{12} + 546 \beta_{13} ) q^{30} \) \( + ( -30829 + 17530 \beta_{1} - 21177 \beta_{2} + 31800 \beta_{3} + 4296 \beta_{4} + 315 \beta_{5} + 2064 \beta_{6} - 315 \beta_{7} - 244 \beta_{8} + 122 \beta_{9} - 528 \beta_{10} - 296 \beta_{11} - 148 \beta_{12} - 193 \beta_{13} ) q^{31} \) \( + ( -404999 + 12529 \beta_{1} + 7891 \beta_{2} - 402564 \beta_{3} + 1619 \beta_{4} - 669 \beta_{5} + 8459 \beta_{6} + 2469 \beta_{7} - 315 \beta_{9} + 861 \beta_{10} + 311 \beta_{12} - 708 \beta_{13} ) q^{32} \) \( + ( -333844 + 4270 \beta_{1} - 14478 \beta_{2} - 158523 \beta_{3} - 3224 \beta_{4} + 288 \beta_{5} - 1183 \beta_{6} - 979 \beta_{7} + 263 \beta_{8} - 105 \beta_{9} + 541 \beta_{10} - 246 \beta_{11} + 976 \beta_{12} + 693 \beta_{13} ) q^{33} \) \( + ( 166 + 10216 \beta_{1} + 27527 \beta_{2} + 85735 \beta_{3} - 2525 \beta_{4} + 989 \beta_{5} - 13218 \beta_{6} - 260 \beta_{7} + 38 \beta_{8} - 76 \beta_{9} - 248 \beta_{10} + 835 \beta_{11} + 1670 \beta_{12} + 12 \beta_{13} ) q^{34} \) \( + ( -340596 + 5215 \beta_{1} - 8465 \beta_{2} + 677526 \beta_{3} - 9522 \beta_{4} - 1115 \beta_{5} - 3621 \beta_{6} + 2198 \beta_{7} - 741 \beta_{8} + 741 \beta_{9} + 2800 \beta_{10} + 321 \beta_{11} + 321 \beta_{12} - 481 \beta_{13} ) q^{35} \) \( + ( 1423088 - 32020 \beta_{1} - 4338 \beta_{2} - 982315 \beta_{3} - 5232 \beta_{4} - 246 \beta_{5} - 44 \beta_{6} - 2178 \beta_{7} - 115 \beta_{8} + 557 \beta_{9} + 2208 \beta_{10} - 395 \beta_{11} + 215 \beta_{12} + 1107 \beta_{13} ) q^{36} \) \( + ( -121638 + 5700 \beta_{1} + 447 \beta_{2} + 1939 \beta_{3} - 5546 \beta_{4} + 682 \beta_{5} + 3645 \beta_{6} + 791 \beta_{7} - 583 \beta_{8} - 583 \beta_{9} - 1089 \beta_{10} - 443 \beta_{11} + 443 \beta_{12} - 407 \beta_{13} ) q^{37} \) \( + ( 1040454 + 20592 \beta_{1} - 5033 \beta_{2} - 519921 \beta_{3} + 16491 \beta_{4} - 1267 \beta_{5} + 2808 \beta_{6} + 1984 \beta_{7} - 696 \beta_{8} - 820 \beta_{10} - 303 \beta_{11} - 1370 \beta_{13} ) q^{38} \) \( + ( -442604 - 21996 \beta_{1} + 8679 \beta_{2} + 1377845 \beta_{3} + 1732 \beta_{4} + 690 \beta_{5} + 1350 \beta_{6} + 2601 \beta_{7} + 1005 \beta_{8} - 360 \beta_{9} + 1341 \beta_{10} + 1611 \beta_{11} + 1344 \beta_{12} + 912 \beta_{13} ) q^{39} \) \( + ( 101982 - 33784 \beta_{1} + 23804 \beta_{2} - 98262 \beta_{3} + 9014 \beta_{4} + 1674 \beta_{5} + 14112 \beta_{6} - 1674 \beta_{7} + 912 \beta_{8} - 456 \beta_{9} - 1202 \beta_{10} - 1236 \beta_{11} - 618 \beta_{12} + 246 \beta_{13} ) q^{40} \) \( + ( -480348 - 47654 \beta_{1} - 43983 \beta_{2} - 482031 \beta_{3} + 6234 \beta_{4} - 949 \beta_{5} + 795 \beta_{6} - 10169 \beta_{7} + 1011 \beta_{9} - 4636 \beta_{10} - 1761 \beta_{12} - 26 \beta_{13} ) q^{41} \) \( + ( -1984947 - 4340 \beta_{1} + 39771 \beta_{2} - 1144597 \beta_{3} + 4673 \beta_{4} - 612 \beta_{5} + 8117 \beta_{6} + 8260 \beta_{7} - 240 \beta_{8} + 338 \beta_{9} + 7574 \beta_{10} + 1444 \beta_{11} - 1097 \beta_{12} + 1053 \beta_{13} ) q^{42} \) \( + ( 414 - 18872 \beta_{1} - 39471 \beta_{2} + 121987 \beta_{3} - 2619 \beta_{4} - 483 \beta_{5} + 3293 \beta_{6} - 1795 \beta_{7} + 522 \beta_{8} - 1044 \beta_{9} - 2698 \beta_{10} - 1800 \beta_{11} - 3600 \beta_{12} - 903 \beta_{13} ) q^{43} \) \( + ( -1518972 - 3671 \beta_{1} - 3301 \beta_{2} + 3039382 \beta_{3} + 585 \beta_{4} - 110 \beta_{5} - 9307 \beta_{6} + 23 \beta_{7} + 402 \beta_{8} - 402 \beta_{9} - 1442 \beta_{10} - 1768 \beta_{11} - 1768 \beta_{12} - 1378 \beta_{13} ) q^{44} \) \( + ( 2920305 + 87327 \beta_{1} - 1110 \beta_{2} - 1933455 \beta_{3} + 1122 \beta_{4} + 1317 \beta_{5} - 9549 \beta_{6} - 5001 \beta_{7} + 1170 \beta_{8} - 1059 \beta_{9} - 6078 \beta_{10} + 1914 \beta_{11} - 210 \beta_{12} - 714 \beta_{13} ) q^{45} \) \( + ( 52655 + 147021 \beta_{1} + 78623 \beta_{2} - 1711 \beta_{3} - 9914 \beta_{4} + 773 \beta_{5} - 12700 \beta_{6} + 111 \beta_{7} + 499 \beta_{8} + 499 \beta_{9} + 561 \beta_{10} - 184 \beta_{11} + 184 \beta_{12} + 1334 \beta_{13} ) q^{46} \) \( + ( 2959636 + 21321 \beta_{1} + 7461 \beta_{2} - 1482946 \beta_{3} - 20050 \beta_{4} + 1827 \beta_{5} - 7992 \beta_{6} + 3168 \beta_{7} + 135 \beta_{8} - 3600 \beta_{10} + 2071 \beta_{11} + 3222 \beta_{13} ) q^{47} \) \( + ( -1334127 - 94959 \beta_{1} - 221247 \beta_{2} + 5149964 \beta_{3} + 2187 \beta_{4} - 2421 \beta_{5} - 223 \beta_{6} - 4617 \beta_{7} - 1044 \beta_{8} + 1269 \beta_{9} - 11637 \beta_{10} - 5832 \beta_{11} - 2061 \beta_{12} - 1872 \beta_{13} ) q^{48} \) \( + ( -674058 - 35089 \beta_{1} + 57074 \beta_{2} + 662526 \beta_{3} - 16690 \beta_{4} - 4767 \beta_{5} - 42309 \beta_{6} + 4767 \beta_{7} - 1578 \beta_{8} + 789 \beta_{9} + 448 \beta_{10} + 6240 \beta_{11} + 3120 \beta_{12} - 1122 \beta_{13} ) q^{49} \) \( + ( -3052109 + 61269 \beta_{1} + 61666 \beta_{2} - 3049023 \beta_{3} - 26401 \beta_{4} + 2884 \beta_{5} - 21475 \beta_{6} + 11690 \beta_{7} - 1146 \beta_{9} + 4492 \beta_{10} + 3353 \beta_{12} + 89 \beta_{13} ) q^{50} \) \( + ( -3508680 - 23889 \beta_{1} + 198225 \beta_{2} + 174894 \beta_{3} - 8991 \beta_{4} + 1836 \beta_{5} - 11022 \beta_{6} - 14814 \beta_{7} - 588 \beta_{8} - 399 \beta_{9} - 12987 \beta_{10} - 3516 \beta_{11} - 3171 \beta_{12} - 4410 \beta_{13} ) q^{51} \) \( + ( 2239 - 177926 \beta_{1} - 367012 \beta_{2} - 710839 \beta_{3} + 26242 \beta_{4} - 883 \beta_{5} + 40398 \beta_{6} + 4666 \beta_{7} - 1567 \beta_{8} + 3134 \beta_{9} + 8113 \beta_{10} + 325 \beta_{11} + 650 \beta_{12} + 3447 \beta_{13} ) q^{52} \) \( + ( -2128860 - 3020 \beta_{1} + 100175 \beta_{2} + 4272321 \beta_{3} + 35640 \beta_{4} + 5878 \beta_{5} + 66645 \beta_{6} - 1987 \beta_{7} + 2823 \beta_{8} - 2823 \beta_{9} + 9397 \beta_{10} + 3033 \beta_{11} + 3033 \beta_{12} + 7493 \beta_{13} ) q^{53} \) \( + ( 5245119 + 355671 \beta_{1} + 157014 \beta_{2} - 5356737 \beta_{3} + 12780 \beta_{4} - 3897 \beta_{5} + 7506 \beta_{6} + 14868 \beta_{7} - 2889 \beta_{8} - 414 \beta_{9} + 9783 \beta_{10} + 117 \beta_{11} + 3276 \beta_{12} - 3825 \beta_{13} ) q^{54} \) \( + ( 566336 + 98979 \beta_{1} + 41984 \beta_{2} - 3379 \beta_{3} + 82738 \beta_{4} - 9571 \beta_{5} + 28088 \beta_{6} - 3531 \beta_{7} + 1852 \beta_{8} + 1852 \beta_{9} + 6393 \beta_{10} + 164 \beta_{11} - 164 \beta_{12} - 3178 \beta_{13} ) q^{55} \) \( + ( 11153006 - 186808 \beta_{1} - 364 \beta_{2} - 5564840 \beta_{3} - 37232 \beta_{4} + 4846 \beta_{5} + 28278 \beta_{6} - 20314 \beta_{7} + 3522 \beta_{8} + 9394 \beta_{10} - 5104 \beta_{11} - 1228 \beta_{13} ) q^{56} \) \( + ( 455147 - 290565 \beta_{1} - 231984 \beta_{2} + 4541079 \beta_{3} - 12544 \beta_{4} + 4158 \beta_{5} - 11122 \beta_{6} + 5832 \beta_{7} - 3096 \beta_{8} - 1692 \beta_{9} + 18036 \beta_{10} + 4851 \beta_{11} + 1062 \beta_{12} - 5508 \beta_{13} ) q^{57} \) \( + ( 2118303 - 209362 \beta_{1} + 258971 \beta_{2} - 2119233 \beta_{3} - 64393 \beta_{4} + 972 \beta_{5} + 17019 \beta_{6} - 972 \beta_{7} - 660 \beta_{8} + 330 \beta_{9} + 16324 \beta_{10} - 5334 \beta_{11} - 2667 \beta_{12} + 5541 \beta_{13} ) q^{58} \) \( + ( -3974167 + 45277 \beta_{1} + 81250 \beta_{2} - 4004151 \beta_{3} + 16606 \beta_{4} + 7299 \beta_{5} - 59399 \beta_{6} + 3501 \beta_{7} - 2376 \beta_{9} + 9027 \beta_{10} + 2794 \beta_{12} + 10926 \beta_{13} ) q^{59} \) \( + ( -7821971 - 157540 \beta_{1} + 5424 \beta_{2} + 139359 \beta_{3} + 43916 \beta_{4} - 6813 \beta_{5} + 12424 \beta_{6} + 8176 \beta_{7} + 1585 \beta_{8} - 672 \beta_{9} - 13147 \beta_{10} + 1455 \beta_{11} + 7592 \beta_{12} - 4599 \beta_{13} ) q^{60} \) \( + ( -20778 + 33943 \beta_{1} - 18888 \beta_{2} - 726287 \beta_{3} - 38814 \beta_{4} - 11217 \beta_{5} + 56336 \beta_{6} + 13544 \beta_{7} - 75 \beta_{8} + 150 \beta_{9} + 8405 \beta_{10} + 4422 \beta_{11} + 8844 \beta_{12} - 5139 \beta_{13} ) q^{61} \) \( + ( -7216653 - 36863 \beta_{1} - 199165 \beta_{2} + 14431475 \beta_{3} + 31896 \beta_{4} + 907 \beta_{5} - 82580 \beta_{6} - 21703 \beta_{7} - 6213 \beta_{8} + 6213 \beta_{9} - 46343 \beta_{10} + 4552 \beta_{11} + 4552 \beta_{12} - 3844 \beta_{13} ) q^{62} \) \( + ( 5245573 + 69415 \beta_{1} - 363171 \beta_{2} - 5032199 \beta_{3} + 18360 \beta_{4} + 4272 \beta_{5} + 40436 \beta_{6} + 6384 \beta_{7} - 254 \beta_{8} + 6718 \beta_{9} - 25431 \beta_{10} - 9616 \beta_{11} - 8570 \beta_{12} - 5745 \beta_{13} ) q^{63} \) \( + ( -2057050 + 672561 \beta_{1} + 357996 \beta_{2} - 13871 \beta_{3} - 52367 \beta_{4} + 8962 \beta_{5} - 55752 \beta_{6} + 14402 \beta_{7} - 5401 \beta_{8} - 5401 \beta_{9} + 843 \beta_{10} + 3223 \beta_{11} - 3223 \beta_{12} + 9805 \beta_{13} ) q^{64} \) \( + ( 6527718 + 212851 \beta_{1} + 44548 \beta_{2} - 3277251 \beta_{3} - 93264 \beta_{4} + 2333 \beta_{5} - 66114 \beta_{6} + 3838 \beta_{7} - 6393 \beta_{8} + 2573 \beta_{10} + 552 \beta_{11} + 11077 \beta_{13} ) q^{65} \) \( + ( -671772 + 122481 \beta_{1} + 130890 \beta_{2} + 6450087 \beta_{3} + 22431 \beta_{4} - 7797 \beta_{5} + 11772 \beta_{6} - 5148 \beta_{7} + 11310 \beta_{8} - 3276 \beta_{9} + 10872 \beta_{10} + 13221 \beta_{11} + 2028 \beta_{12} + 1002 \beta_{13} ) q^{66} \) \( + ( -1245958 + 141691 \beta_{1} - 182955 \beta_{2} + 1248669 \beta_{3} + 30210 \beta_{4} - 7482 \beta_{5} - 38706 \beta_{6} + 7482 \beta_{7} + 8606 \beta_{8} - 4303 \beta_{9} - 11181 \beta_{10} - 15158 \beta_{11} - 7579 \beta_{12} - 10096 \beta_{13} ) q^{67} \) \( + ( -3666967 - 253337 \beta_{1} - 286425 \beta_{2} - 3646392 \beta_{3} - 18167 \beta_{4} - 6865 \beta_{5} + 73891 \beta_{6} - 8771 \beta_{7} + 8769 \beta_{9} - 15493 \beta_{10} - 23045 \beta_{12} - 14540 \beta_{13} ) q^{68} \) \( + ( -4378263 + 246229 \beta_{1} + 183897 \beta_{2} - 248170 \beta_{3} - 60334 \beta_{4} + 10431 \beta_{5} - 101134 \beta_{6} - 17696 \beta_{7} - 627 \beta_{8} + 2498 \beta_{9} + 2021 \beta_{10} + 9085 \beta_{11} + 4177 \beta_{12} + 4725 \beta_{13} ) q^{69} \) \( + ( 33885 - 65771 \beta_{1} + 44097 \beta_{2} + 3919272 \beta_{3} + 61479 \beta_{4} + 20880 \beta_{5} - 131332 \beta_{6} - 51394 \beta_{7} + 6831 \beta_{8} - 13662 \beta_{9} - 43141 \beta_{10} - 4752 \beta_{11} - 9504 \beta_{12} + 8253 \beta_{13} ) q^{70} \) \( + ( 80742 + 47390 \beta_{1} - 32101 \beta_{2} - 151481 \beta_{3} - 56088 \beta_{4} - 5356 \beta_{5} + 110201 \beta_{6} + 14911 \beta_{7} + 471 \beta_{8} - 471 \beta_{9} + 28151 \beta_{10} - 26071 \beta_{11} - 26071 \beta_{12} + 3685 \beta_{13} ) q^{71} \) \( + ( 9439683 - 633921 \beta_{1} + 652587 \beta_{2} - 5940810 \beta_{3} - 74577 \beta_{4} - 6099 \beta_{5} - 56811 \beta_{6} - 13695 \beta_{7} + 14214 \beta_{8} - 12441 \beta_{9} + 41001 \beta_{10} + 8760 \beta_{11} - 3357 \beta_{12} + 13002 \beta_{13} ) q^{72} \) \( + ( -2357722 - 1241514 \beta_{1} - 680778 \beta_{2} + 28824 \beta_{3} + 49956 \beta_{4} - 3036 \beta_{5} + 134433 \beta_{6} + 20163 \beta_{7} + 2613 \beta_{8} + 2613 \beta_{9} - 11979 \beta_{10} - 1830 \beta_{11} + 1830 \beta_{12} - 15015 \beta_{13} ) q^{73} \) \( + ( -1004148 + 189258 \beta_{1} - 91466 \beta_{2} + 498114 \beta_{3} + 326892 \beta_{4} - 18028 \beta_{5} + 73008 \beta_{6} + 26080 \beta_{7} - 2784 \beta_{8} - 13918 \beta_{10} + 25518 \beta_{11} - 23894 \beta_{13} ) q^{74} \) \( + ( -2588727 - 84441 \beta_{1} + 233202 \beta_{2} + 7187597 \beta_{3} - 11466 \beta_{4} + 10419 \beta_{5} + 4058 \beta_{6} - 8577 \beta_{7} - 9420 \beta_{8} + 15837 \beta_{9} - 48501 \beta_{10} - 26250 \beta_{11} - 9153 \beta_{12} + 12387 \beta_{13} ) q^{75} \) \( + ( -6190603 + 182589 \beta_{1} - 329389 \beta_{2} + 6225054 \beta_{3} + 186383 \beta_{4} + 28539 \beta_{5} + 176511 \beta_{6} - 28539 \beta_{7} - 14662 \beta_{8} + 7331 \beta_{9} - 68423 \beta_{10} + 20818 \beta_{11} + 10409 \beta_{12} + 12218 \beta_{13} ) q^{76} \) \( + ( 2614350 + 353889 \beta_{1} + 306265 \beta_{2} + 2649267 \beta_{3} + 102366 \beta_{4} - 18632 \beta_{5} + 120036 \beta_{6} - 15736 \beta_{7} - 5988 \beta_{9} - 14 \beta_{10} + 30531 \beta_{12} - 1462 \beta_{13} ) q^{77} \) \( + ( 7662131 - 247593 \beta_{1} - 1282479 \beta_{2} - 11276363 \beta_{3} - 92848 \beta_{4} - 7425 \beta_{5} + 179538 \beta_{6} + 45157 \beta_{7} - 37 \beta_{8} - 1397 \beta_{9} + 83597 \beta_{10} - 11536 \beta_{11} - 12228 \beta_{12} + 23742 \beta_{13} ) q^{78} \) \( + ( -19339 + 961377 \beta_{1} + 1880317 \beta_{2} - 5075193 \beta_{3} - 138238 \beta_{4} - 398 \beta_{5} - 124042 \beta_{6} + 68932 \beta_{7} - 6812 \beta_{8} + 13624 \beta_{9} + 53275 \beta_{10} + 3284 \beta_{11} + 6568 \beta_{12} - 15657 \beta_{13} ) q^{79} \) \( + ( 4564308 + 67046 \beta_{1} + 244626 \beta_{2} - 9225740 \beta_{3} - 205398 \beta_{4} - 23608 \beta_{5} - 276718 \beta_{6} + 73906 \beta_{7} + 8004 \beta_{8} - 8004 \beta_{9} + 90440 \beta_{10} + 26300 \beta_{11} + 26300 \beta_{12} - 33764 \beta_{13} ) q^{80} \) \( + ( -2560770 - 695475 \beta_{1} - 872937 \beta_{2} + 3054798 \beta_{3} + 38124 \beta_{4} + 18684 \beta_{5} - 99756 \beta_{6} - 53460 \beta_{7} - 21312 \beta_{8} - 522 \beta_{9} + 4806 \beta_{10} + 16560 \beta_{11} + 36495 \beta_{12} + 23814 \beta_{13} ) q^{81} \) \( + ( 17012874 - 1088544 \beta_{1} - 453067 \beta_{2} + 38436 \beta_{3} - 493120 \beta_{4} + 29886 \beta_{5} - 248236 \beta_{6} - 142762 \beta_{7} + 6636 \beta_{8} + 6636 \beta_{9} - 26118 \beta_{10} - 16086 \beta_{11} + 16086 \beta_{12} + 3768 \beta_{13} ) q^{82} \) \( + ( -18813510 - 486861 \beta_{1} - 21847 \beta_{2} + 9417888 \beta_{3} + 7044 \beta_{4} - 16133 \beta_{5} - 39960 \beta_{6} + 38816 \beta_{7} + 13701 \beta_{8} - 12836 \beta_{10} - 51807 \beta_{11} - 6286 \beta_{13} ) q^{83} \) \( + ( -7153516 + 3136122 \beta_{1} + 2136069 \beta_{2} - 13767495 \beta_{3} - 27376 \beta_{4} - 5526 \beta_{5} + 18857 \beta_{6} + 54243 \beta_{7} - 13905 \beta_{8} - 16857 \beta_{9} + 25137 \beta_{10} + 1683 \beta_{11} + 27873 \beta_{12} + 27153 \beta_{13} ) q^{84} \) \( + ( 13458281 + 1107904 \beta_{1} - 1100461 \beta_{2} - 13458652 \beta_{3} + 53030 \beta_{4} - 2967 \beta_{5} - 64947 \beta_{6} + 2967 \beta_{7} + 2482 \beta_{8} - 1241 \beta_{9} + 125716 \beta_{10} + 27398 \beta_{11} + 13699 \beta_{12} - 19262 \beta_{13} ) q^{85} \) \( + ( 6967657 - 334657 \beta_{1} - 434456 \beta_{2} + 7073034 \beta_{3} - 110794 \beta_{4} - 15895 \beta_{5} + 96095 \beta_{6} - 6311 \beta_{7} - 7665 \beta_{9} - 22984 \beta_{10} + 22049 \beta_{12} - 27776 \beta_{13} ) q^{86} \) \( + ( 5011505 + 1492975 \beta_{1} + 278346 \beta_{2} + 5773626 \beta_{3} + 345490 \beta_{4} + 14418 \beta_{5} + 212777 \beta_{6} + 31025 \beta_{7} - 4771 \beta_{8} - 1719 \beta_{9} + 14878 \beta_{10} - 5733 \beta_{11} - 30365 \beta_{12} + 8370 \beta_{13} ) q^{87} \) \( + ( 50637 - 399940 \beta_{1} - 614781 \beta_{2} + 3661143 \beta_{3} + 104070 \beta_{4} + 19347 \beta_{5} - 80915 \beta_{6} - 38387 \beta_{7} - 11685 \beta_{8} + 23370 \beta_{9} - 26300 \beta_{10} - 19203 \beta_{11} - 38406 \beta_{12} + 12087 \beta_{13} ) q^{88} \) \( + ( 14830008 + 58880 \beta_{1} + 772917 \beta_{2} - 29727709 \beta_{3} - 53640 \beta_{4} - 7114 \beta_{5} + 48283 \beta_{6} - 9449 \beta_{7} + 5205 \beta_{8} - 5205 \beta_{9} - 12721 \beta_{10} + 46351 \beta_{11} + 46351 \beta_{12} + 13291 \beta_{13} ) q^{89} \) \( + ( -33256839 - 1820091 \beta_{1} + 180810 \beta_{2} + 35627352 \beta_{3} + 16344 \beta_{4} - 17685 \beta_{5} + 323301 \beta_{6} - 13032 \beta_{7} - 10254 \beta_{8} + 31650 \beta_{9} - 78516 \beta_{10} - 20337 \beta_{11} - 29685 \beta_{12} + 1017 \beta_{13} ) q^{90} \) \( + ( -13808544 - 2396883 \beta_{1} - 1263926 \beta_{2} - 38065 \beta_{3} + 355098 \beta_{4} - 14221 \beta_{5} + 194788 \beta_{6} + 121163 \beta_{7} - 3566 \beta_{8} - 3566 \beta_{9} + 13839 \beta_{10} + 24608 \beta_{11} - 24608 \beta_{12} - 382 \beta_{13} ) q^{91} \) \( + ( -28168815 + 162470 \beta_{1} - 67048 \beta_{2} + 14102283 \beta_{3} + 107478 \beta_{4} + 5875 \beta_{5} + 119988 \beta_{6} + 10148 \beta_{7} + 8121 \beta_{8} - 29753 \beta_{10} + 5439 \beta_{11} - 7855 \beta_{13} ) q^{92} \) \( + ( 23241337 - 1649673 \beta_{1} + 744717 \beta_{2} - 32315770 \beta_{3} + 27076 \beta_{4} + 16629 \beta_{5} - 26694 \beta_{6} - 109314 \beta_{7} + 29949 \beta_{8} - 19782 \beta_{9} - 33975 \beta_{10} - 4023 \beta_{11} - 45249 \beta_{12} - 16053 \beta_{13} ) q^{93} \) \( + ( -4984040 - 620858 \beta_{1} + 642198 \beta_{2} + 4972831 \beta_{3} - 66993 \beta_{4} - 3261 \beta_{5} + 11694 \beta_{6} + 3261 \beta_{7} + 20282 \beta_{8} - 10141 \beta_{9} - 43182 \beta_{10} - 6368 \beta_{11} - 3184 \beta_{12} + 7871 \beta_{13} ) q^{94} \) \( + ( 18483160 + 272082 \beta_{1} + 383404 \beta_{2} + 18280716 \beta_{3} + 175652 \beta_{4} + 37102 \beta_{5} - 38716 \beta_{6} + 103130 \beta_{7} - 768 \beta_{9} + 68464 \beta_{10} - 96094 \beta_{12} + 35450 \beta_{13} ) q^{95} \) \( + ( 45887412 - 1703625 \beta_{1} - 2732346 \beta_{2} + 18507081 \beta_{3} - 313497 \beta_{4} - 17460 \beta_{5} - 714990 \beta_{6} - 126036 \beta_{7} + 3963 \beta_{8} - 4587 \beta_{9} - 153423 \beta_{10} - 16797 \beta_{11} + 27975 \beta_{12} - 32427 \beta_{13} ) q^{96} \) \( + ( -61797 + 1207924 \beta_{1} + 2320494 \beta_{2} + 17674018 \beta_{3} - 58188 \beta_{4} - 36183 \beta_{5} + 81080 \beta_{6} - 175228 \beta_{7} + 13359 \beta_{8} - 26718 \beta_{9} - 184645 \beta_{10} + 16197 \beta_{11} + 32394 \beta_{12} - 9417 \beta_{13} ) q^{97} \) \( + ( 14120907 - 198919 \beta_{1} - 1464423 \beta_{2} - 27996448 \beta_{3} + 372132 \beta_{4} + 42005 \beta_{5} + 310315 \beta_{6} - 101696 \beta_{7} - 10338 \beta_{8} + 10338 \beta_{9} - 137788 \beta_{10} - 119351 \beta_{11} - 119351 \beta_{12} + 23599 \beta_{13} ) q^{98} \) \( + ( -45183069 + 371559 \beta_{1} - 1022793 \beta_{2} + 17114874 \beta_{3} - 361761 \beta_{4} - 3108 \beta_{5} - 461145 \beta_{6} + 113010 \beta_{7} + 44400 \beta_{8} - 30843 \beta_{9} + 79257 \beta_{10} - 2358 \beta_{11} - 21477 \beta_{12} - 53970 \beta_{13} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 93q^{3} \) \(\mathstrut +\mathstrut 767q^{4} \) \(\mathstrut +\mathstrut 438q^{5} \) \(\mathstrut -\mathstrut 2259q^{6} \) \(\mathstrut +\mathstrut 922q^{7} \) \(\mathstrut +\mathstrut 17973q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut -\mathstrut 93q^{3} \) \(\mathstrut +\mathstrut 767q^{4} \) \(\mathstrut +\mathstrut 438q^{5} \) \(\mathstrut -\mathstrut 2259q^{6} \) \(\mathstrut +\mathstrut 922q^{7} \) \(\mathstrut +\mathstrut 17973q^{9} \) \(\mathstrut -\mathstrut 516q^{10} \) \(\mathstrut -\mathstrut 28677q^{11} \) \(\mathstrut -\mathstrut 55110q^{12} \) \(\mathstrut +\mathstrut 1684q^{13} \) \(\mathstrut +\mathstrut 120966q^{14} \) \(\mathstrut -\mathstrut 75276q^{15} \) \(\mathstrut -\mathstrut 65281q^{16} \) \(\mathstrut +\mathstrut 243324q^{18} \) \(\mathstrut -\mathstrut 269630q^{19} \) \(\mathstrut +\mathstrut 539454q^{20} \) \(\mathstrut +\mathstrut 354054q^{21} \) \(\mathstrut +\mathstrut 61311q^{22} \) \(\mathstrut -\mathstrut 1000452q^{23} \) \(\mathstrut -\mathstrut 1125513q^{24} \) \(\mathstrut +\mathstrut 65177q^{25} \) \(\mathstrut -\mathstrut 524826q^{27} \) \(\mathstrut +\mathstrut 1075708q^{28} \) \(\mathstrut +\mathstrut 3797682q^{29} \) \(\mathstrut +\mathstrut 2372562q^{30} \) \(\mathstrut -\mathstrut 164132q^{31} \) \(\mathstrut -\mathstrut 8461881q^{32} \) \(\mathstrut -\mathstrut 5761521q^{33} \) \(\mathstrut +\mathstrut 654993q^{34} \) \(\mathstrut +\mathstrut 12958047q^{36} \) \(\mathstrut -\mathstrut 1671668q^{37} \) \(\mathstrut +\mathstrut 10967691q^{38} \) \(\mathstrut +\mathstrut 3383976q^{39} \) \(\mathstrut +\mathstrut 613326q^{40} \) \(\mathstrut -\mathstrut 10239447q^{41} \) \(\mathstrut -\mathstrut 35844228q^{42} \) \(\mathstrut +\mathstrut 791815q^{43} \) \(\mathstrut +\mathstrut 27631638q^{45} \) \(\mathstrut +\mathstrut 1189536q^{46} \) \(\mathstrut +\mathstrut 31148628q^{47} \) \(\mathstrut +\mathstrut 17088189q^{48} \) \(\mathstrut -\mathstrut 4826637q^{49} \) \(\mathstrut -\mathstrut 63849453q^{50} \) \(\mathstrut -\mathstrut 47946573q^{51} \) \(\mathstrut -\mathstrut 5552720q^{52} \) \(\mathstrut +\mathstrut 36967995q^{54} \) \(\mathstrut +\mathstrut 8107476q^{55} \) \(\mathstrut +\mathstrut 116638674q^{56} \) \(\mathstrut +\mathstrut 37276797q^{57} \) \(\mathstrut +\mathstrut 14211822q^{58} \) \(\mathstrut -\mathstrut 83493795q^{59} \) \(\mathstrut -\mathstrut 109012050q^{60} \) \(\mathstrut -\mathstrut 5255600q^{61} \) \(\mathstrut +\mathstrut 38322432q^{63} \) \(\mathstrut -\mathstrut 26813830q^{64} \) \(\mathstrut +\mathstrut 69232992q^{65} \) \(\mathstrut +\mathstrut 36133866q^{66} \) \(\mathstrut -\mathstrut 8288855q^{67} \) \(\mathstrut -\mathstrut 77746743q^{68} \) \(\mathstrut -\mathstrut 62108640q^{69} \) \(\mathstrut +\mathstrut 27813756q^{70} \) \(\mathstrut +\mathstrut 88821171q^{72} \) \(\mathstrut -\mathstrut 36721682q^{73} \) \(\mathstrut -\mathstrut 10383450q^{74} \) \(\mathstrut +\mathstrut 13792011q^{75} \) \(\mathstrut -\mathstrut 42822959q^{76} \) \(\mathstrut +\mathstrut 56158710q^{77} \) \(\mathstrut +\mathstrut 27355140q^{78} \) \(\mathstrut -\mathstrut 32771822q^{79} \) \(\mathstrut -\mathstrut 16336971q^{81} \) \(\mathstrut +\mathstrut 236099418q^{82} \) \(\mathstrut -\mathstrut 198915996q^{83} \) \(\mathstrut -\mathstrut 187060614q^{84} \) \(\mathstrut +\mathstrut 97486146q^{85} \) \(\mathstrut +\mathstrut 146190669q^{86} \) \(\mathstrut +\mathstrut 114294006q^{87} \) \(\mathstrut +\mathstrut 24955827q^{88} \) \(\mathstrut -\mathstrut 222035850q^{90} \) \(\mathstrut -\mathstrut 201514504q^{91} \) \(\mathstrut -\mathstrut 295365804q^{92} \) \(\mathstrut +\mathstrut 94187784q^{93} \) \(\mathstrut -\mathstrut 36698244q^{94} \) \(\mathstrut +\mathstrut 386813838q^{95} \) \(\mathstrut +\mathstrut 768275928q^{96} \) \(\mathstrut +\mathstrut 127049161q^{97} \) \(\mathstrut -\mathstrut 511060752q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(x^{13}\mathstrut +\mathstrut \) \(427\) \(x^{12}\mathstrut -\mathstrut \) \(1362\) \(x^{11}\mathstrut +\mathstrut \) \(135762\) \(x^{10}\mathstrut -\mathstrut \) \(371244\) \(x^{9}\mathstrut +\mathstrut \) \(18261508\) \(x^{8}\mathstrut -\mathstrut \) \(29352736\) \(x^{7}\mathstrut +\mathstrut \) \(1757083840\) \(x^{6}\mathstrut -\mathstrut \) \(1434930432\) \(x^{5}\mathstrut +\mathstrut \) \(61638378240\) \(x^{4}\mathstrut +\mathstrut \) \(183604924416\) \(x^{3}\mathstrut +\mathstrut \) \(1254910111744\) \(x^{2}\mathstrut +\mathstrut \) \(1078377512960\) \(x\mathstrut +\mathstrut \) \(872385888256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(442054787914350989925983\) \(\nu^{13}\mathstrut +\mathstrut \) \(23628808584322386004267421\) \(\nu^{12}\mathstrut -\mathstrut \) \(50640806940675695590816011\) \(\nu^{11}\mathstrut +\mathstrut \) \(9110652565048392910003247466\) \(\nu^{10}\mathstrut -\mathstrut \) \(57543855647402141458544986806\) \(\nu^{9}\mathstrut +\mathstrut \) \(3101563233324860607105549277932\) \(\nu^{8}\mathstrut -\mathstrut \) \(27246232517001751558799077332700\) \(\nu^{7}\mathstrut +\mathstrut \) \(408260669537384694699705658647872\) \(\nu^{6}\mathstrut -\mathstrut \) \(3036725485917744621466950043094736\) \(\nu^{5}\mathstrut +\mathstrut \) \(37304937996215854140380362605307776\) \(\nu^{4}\mathstrut -\mathstrut \) \(371785821517531872040637539606808832\) \(\nu^{3}\mathstrut +\mathstrut \) \(962102681229186015894305741252662272\) \(\nu^{2}\mathstrut +\mathstrut \) \(11673431310210505523707812991412566016\) \(\nu\mathstrut -\mathstrut \) \(8669628126778214704913602579186622464\)\()/\)\(10\!\cdots\!64\)
\(\beta_{2}\)\(=\)\((\)\(442054787914350989925983\) \(\nu^{13}\mathstrut +\mathstrut \) \(23628808584322386004267421\) \(\nu^{12}\mathstrut -\mathstrut \) \(50640806940675695590816011\) \(\nu^{11}\mathstrut +\mathstrut \) \(9110652565048392910003247466\) \(\nu^{10}\mathstrut -\mathstrut \) \(57543855647402141458544986806\) \(\nu^{9}\mathstrut +\mathstrut \) \(3101563233324860607105549277932\) \(\nu^{8}\mathstrut -\mathstrut \) \(27246232517001751558799077332700\) \(\nu^{7}\mathstrut +\mathstrut \) \(408260669537384694699705658647872\) \(\nu^{6}\mathstrut -\mathstrut \) \(3036725485917744621466950043094736\) \(\nu^{5}\mathstrut +\mathstrut \) \(37304937996215854140380362605307776\) \(\nu^{4}\mathstrut -\mathstrut \) \(371785821517531872040637539606808832\) \(\nu^{3}\mathstrut +\mathstrut \) \(962102681229186015894305741252662272\) \(\nu^{2}\mathstrut -\mathstrut \) \(20848898132329312254552493742295294976\) \(\nu\mathstrut -\mathstrut \) \(8669628126778214704913602579186622464\)\()/\)\(10\!\cdots\!64\)
\(\beta_{3}\)\(=\)\((\)\(14\!\cdots\!61\) \(\nu^{13}\mathstrut -\mathstrut \) \(13\!\cdots\!59\) \(\nu^{12}\mathstrut +\mathstrut \) \(62\!\cdots\!21\) \(\nu^{11}\mathstrut -\mathstrut \) \(19\!\cdots\!16\) \(\nu^{10}\mathstrut +\mathstrut \) \(19\!\cdots\!86\) \(\nu^{9}\mathstrut -\mathstrut \) \(54\!\cdots\!48\) \(\nu^{8}\mathstrut +\mathstrut \) \(26\!\cdots\!96\) \(\nu^{7}\mathstrut -\mathstrut \) \(46\!\cdots\!96\) \(\nu^{6}\mathstrut +\mathstrut \) \(26\!\cdots\!08\) \(\nu^{5}\mathstrut -\mathstrut \) \(25\!\cdots\!36\) \(\nu^{4}\mathstrut +\mathstrut \) \(94\!\cdots\!84\) \(\nu^{3}\mathstrut +\mathstrut \) \(21\!\cdots\!68\) \(\nu^{2}\mathstrut +\mathstrut \) \(19\!\cdots\!52\) \(\nu\mathstrut +\mathstrut \) \(16\!\cdots\!48\)\()/\)\(15\!\cdots\!16\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(13\!\cdots\!63\) \(\nu^{13}\mathstrut -\mathstrut \) \(16\!\cdots\!45\) \(\nu^{12}\mathstrut -\mathstrut \) \(52\!\cdots\!05\) \(\nu^{11}\mathstrut -\mathstrut \) \(55\!\cdots\!34\) \(\nu^{10}\mathstrut -\mathstrut \) \(13\!\cdots\!22\) \(\nu^{9}\mathstrut -\mathstrut \) \(19\!\cdots\!28\) \(\nu^{8}\mathstrut -\mathstrut \) \(10\!\cdots\!52\) \(\nu^{7}\mathstrut -\mathstrut \) \(28\!\cdots\!76\) \(\nu^{6}\mathstrut -\mathstrut \) \(77\!\cdots\!68\) \(\nu^{5}\mathstrut -\mathstrut \) \(29\!\cdots\!84\) \(\nu^{4}\mathstrut +\mathstrut \) \(43\!\cdots\!92\) \(\nu^{3}\mathstrut -\mathstrut \) \(12\!\cdots\!68\) \(\nu^{2}\mathstrut -\mathstrut \) \(11\!\cdots\!32\) \(\nu\mathstrut -\mathstrut \) \(10\!\cdots\!00\)\()/\)\(13\!\cdots\!72\)
\(\beta_{5}\)\(=\)\((\)\(12\!\cdots\!89\) \(\nu^{13}\mathstrut +\mathstrut \) \(77\!\cdots\!67\) \(\nu^{12}\mathstrut +\mathstrut \) \(31\!\cdots\!15\) \(\nu^{11}\mathstrut +\mathstrut \) \(30\!\cdots\!34\) \(\nu^{10}\mathstrut +\mathstrut \) \(15\!\cdots\!86\) \(\nu^{9}\mathstrut +\mathstrut \) \(98\!\cdots\!96\) \(\nu^{8}\mathstrut -\mathstrut \) \(16\!\cdots\!44\) \(\nu^{7}\mathstrut +\mathstrut \) \(12\!\cdots\!88\) \(\nu^{6}\mathstrut -\mathstrut \) \(41\!\cdots\!76\) \(\nu^{5}\mathstrut +\mathstrut \) \(11\!\cdots\!12\) \(\nu^{4}\mathstrut -\mathstrut \) \(26\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(37\!\cdots\!24\) \(\nu^{2}\mathstrut +\mathstrut \) \(24\!\cdots\!88\) \(\nu\mathstrut +\mathstrut \) \(66\!\cdots\!16\)\()/\)\(65\!\cdots\!36\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(28\!\cdots\!05\) \(\nu^{13}\mathstrut +\mathstrut \) \(74\!\cdots\!53\) \(\nu^{12}\mathstrut -\mathstrut \) \(12\!\cdots\!23\) \(\nu^{11}\mathstrut +\mathstrut \) \(58\!\cdots\!58\) \(\nu^{10}\mathstrut -\mathstrut \) \(39\!\cdots\!78\) \(\nu^{9}\mathstrut +\mathstrut \) \(16\!\cdots\!36\) \(\nu^{8}\mathstrut -\mathstrut \) \(55\!\cdots\!76\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!04\) \(\nu^{6}\mathstrut -\mathstrut \) \(53\!\cdots\!48\) \(\nu^{5}\mathstrut +\mathstrut \) \(12\!\cdots\!52\) \(\nu^{4}\mathstrut -\mathstrut \) \(19\!\cdots\!68\) \(\nu^{3}\mathstrut -\mathstrut \) \(29\!\cdots\!20\) \(\nu^{2}\mathstrut -\mathstrut \) \(29\!\cdots\!64\) \(\nu\mathstrut -\mathstrut \) \(95\!\cdots\!88\)\()/\)\(31\!\cdots\!16\)
\(\beta_{7}\)\(=\)\((\)\(12\!\cdots\!61\) \(\nu^{13}\mathstrut -\mathstrut \) \(73\!\cdots\!25\) \(\nu^{12}\mathstrut +\mathstrut \) \(53\!\cdots\!55\) \(\nu^{11}\mathstrut -\mathstrut \) \(15\!\cdots\!86\) \(\nu^{10}\mathstrut +\mathstrut \) \(16\!\cdots\!54\) \(\nu^{9}\mathstrut -\mathstrut \) \(40\!\cdots\!12\) \(\nu^{8}\mathstrut +\mathstrut \) \(23\!\cdots\!04\) \(\nu^{7}\mathstrut -\mathstrut \) \(29\!\cdots\!52\) \(\nu^{6}\mathstrut +\mathstrut \) \(21\!\cdots\!88\) \(\nu^{5}\mathstrut -\mathstrut \) \(82\!\cdots\!04\) \(\nu^{4}\mathstrut +\mathstrut \) \(77\!\cdots\!72\) \(\nu^{3}\mathstrut +\mathstrut \) \(28\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(13\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(59\!\cdots\!48\)\()/\)\(13\!\cdots\!72\)
\(\beta_{8}\)\(=\)\((\)\(57\!\cdots\!61\) \(\nu^{13}\mathstrut -\mathstrut \) \(51\!\cdots\!03\) \(\nu^{12}\mathstrut +\mathstrut \) \(24\!\cdots\!65\) \(\nu^{11}\mathstrut -\mathstrut \) \(27\!\cdots\!12\) \(\nu^{10}\mathstrut +\mathstrut \) \(82\!\cdots\!06\) \(\nu^{9}\mathstrut -\mathstrut \) \(82\!\cdots\!92\) \(\nu^{8}\mathstrut +\mathstrut \) \(11\!\cdots\!64\) \(\nu^{7}\mathstrut -\mathstrut \) \(97\!\cdots\!16\) \(\nu^{6}\mathstrut +\mathstrut \) \(10\!\cdots\!08\) \(\nu^{5}\mathstrut -\mathstrut \) \(79\!\cdots\!36\) \(\nu^{4}\mathstrut +\mathstrut \) \(32\!\cdots\!96\) \(\nu^{3}\mathstrut -\mathstrut \) \(97\!\cdots\!52\) \(\nu^{2}\mathstrut +\mathstrut \) \(13\!\cdots\!92\) \(\nu\mathstrut -\mathstrut \) \(64\!\cdots\!52\)\()/\)\(47\!\cdots\!24\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(73\!\cdots\!67\) \(\nu^{13}\mathstrut +\mathstrut \) \(73\!\cdots\!61\) \(\nu^{12}\mathstrut -\mathstrut \) \(31\!\cdots\!29\) \(\nu^{11}\mathstrut +\mathstrut \) \(89\!\cdots\!74\) \(\nu^{10}\mathstrut -\mathstrut \) \(10\!\cdots\!32\) \(\nu^{9}\mathstrut +\mathstrut \) \(23\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(13\!\cdots\!96\) \(\nu^{7}\mathstrut +\mathstrut \) \(83\!\cdots\!28\) \(\nu^{6}\mathstrut -\mathstrut \) \(12\!\cdots\!76\) \(\nu^{5}\mathstrut -\mathstrut \) \(26\!\cdots\!96\) \(\nu^{4}\mathstrut -\mathstrut \) \(46\!\cdots\!00\) \(\nu^{3}\mathstrut -\mathstrut \) \(23\!\cdots\!64\) \(\nu^{2}\mathstrut -\mathstrut \) \(91\!\cdots\!40\) \(\nu\mathstrut -\mathstrut \) \(67\!\cdots\!04\)\()/\)\(32\!\cdots\!68\)
\(\beta_{10}\)\(=\)\((\)\(14\!\cdots\!07\) \(\nu^{13}\mathstrut -\mathstrut \) \(22\!\cdots\!63\) \(\nu^{12}\mathstrut +\mathstrut \) \(62\!\cdots\!21\) \(\nu^{11}\mathstrut -\mathstrut \) \(23\!\cdots\!14\) \(\nu^{10}\mathstrut +\mathstrut \) \(20\!\cdots\!18\) \(\nu^{9}\mathstrut -\mathstrut \) \(66\!\cdots\!12\) \(\nu^{8}\mathstrut +\mathstrut \) \(27\!\cdots\!00\) \(\nu^{7}\mathstrut -\mathstrut \) \(61\!\cdots\!36\) \(\nu^{6}\mathstrut +\mathstrut \) \(26\!\cdots\!48\) \(\nu^{5}\mathstrut -\mathstrut \) \(40\!\cdots\!04\) \(\nu^{4}\mathstrut +\mathstrut \) \(98\!\cdots\!04\) \(\nu^{3}\mathstrut +\mathstrut \) \(17\!\cdots\!24\) \(\nu^{2}\mathstrut +\mathstrut \) \(19\!\cdots\!80\) \(\nu\mathstrut -\mathstrut \) \(25\!\cdots\!12\)\()/\)\(43\!\cdots\!24\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(75\!\cdots\!37\) \(\nu^{13}\mathstrut +\mathstrut \) \(29\!\cdots\!73\) \(\nu^{12}\mathstrut -\mathstrut \) \(32\!\cdots\!19\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!50\) \(\nu^{10}\mathstrut -\mathstrut \) \(10\!\cdots\!98\) \(\nu^{9}\mathstrut +\mathstrut \) \(57\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(14\!\cdots\!48\) \(\nu^{7}\mathstrut +\mathstrut \) \(61\!\cdots\!64\) \(\nu^{6}\mathstrut -\mathstrut \) \(13\!\cdots\!24\) \(\nu^{5}\mathstrut +\mathstrut \) \(47\!\cdots\!40\) \(\nu^{4}\mathstrut -\mathstrut \) \(49\!\cdots\!28\) \(\nu^{3}\mathstrut -\mathstrut \) \(16\!\cdots\!20\) \(\nu^{2}\mathstrut -\mathstrut \) \(63\!\cdots\!44\) \(\nu\mathstrut +\mathstrut \) \(77\!\cdots\!56\)\()/\)\(13\!\cdots\!72\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(80\!\cdots\!19\) \(\nu^{13}\mathstrut +\mathstrut \) \(19\!\cdots\!63\) \(\nu^{12}\mathstrut -\mathstrut \) \(34\!\cdots\!45\) \(\nu^{11}\mathstrut +\mathstrut \) \(78\!\cdots\!78\) \(\nu^{10}\mathstrut -\mathstrut \) \(10\!\cdots\!42\) \(\nu^{9}\mathstrut +\mathstrut \) \(19\!\cdots\!84\) \(\nu^{8}\mathstrut -\mathstrut \) \(14\!\cdots\!68\) \(\nu^{7}\mathstrut +\mathstrut \) \(69\!\cdots\!24\) \(\nu^{6}\mathstrut -\mathstrut \) \(13\!\cdots\!48\) \(\nu^{5}\mathstrut -\mathstrut \) \(48\!\cdots\!88\) \(\nu^{4}\mathstrut -\mathstrut \) \(46\!\cdots\!36\) \(\nu^{3}\mathstrut -\mathstrut \) \(22\!\cdots\!40\) \(\nu^{2}\mathstrut -\mathstrut \) \(10\!\cdots\!72\) \(\nu\mathstrut -\mathstrut \) \(14\!\cdots\!20\)\()/\)\(13\!\cdots\!72\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(15\!\cdots\!33\) \(\nu^{13}\mathstrut +\mathstrut \) \(32\!\cdots\!59\) \(\nu^{12}\mathstrut -\mathstrut \) \(66\!\cdots\!61\) \(\nu^{11}\mathstrut +\mathstrut \) \(16\!\cdots\!40\) \(\nu^{10}\mathstrut -\mathstrut \) \(21\!\cdots\!06\) \(\nu^{9}\mathstrut +\mathstrut \) \(42\!\cdots\!00\) \(\nu^{8}\mathstrut -\mathstrut \) \(27\!\cdots\!84\) \(\nu^{7}\mathstrut +\mathstrut \) \(25\!\cdots\!24\) \(\nu^{6}\mathstrut -\mathstrut \) \(26\!\cdots\!36\) \(\nu^{5}\mathstrut +\mathstrut \) \(43\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(84\!\cdots\!36\) \(\nu^{3}\mathstrut -\mathstrut \) \(33\!\cdots\!44\) \(\nu^{2}\mathstrut -\mathstrut \) \(19\!\cdots\!32\) \(\nu\mathstrut -\mathstrut \) \(12\!\cdots\!72\)\()/\)\(15\!\cdots\!08\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(366\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(3\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(18\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(608\) \(\beta_{2}\mathstrut -\mathstrut \) \(1219\) \(\beta_{1}\mathstrut +\mathstrut \) \(2060\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(62\) \(\beta_{13}\mathstrut -\mathstrut \) \(5\) \(\beta_{12}\mathstrut -\mathstrut \) \(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(833\) \(\beta_{10}\mathstrut +\mathstrut \) \(13\) \(\beta_{9}\mathstrut -\mathstrut \) \(26\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut -\mathstrut \) \(183\) \(\beta_{6}\mathstrut -\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(1289\) \(\beta_{4}\mathstrut +\mathstrut \) \(225422\) \(\beta_{3}\mathstrut -\mathstrut \) \(3163\) \(\beta_{2}\mathstrut +\mathstrut \) \(1999\) \(\beta_{1}\mathstrut -\mathstrut \) \(225385\)\()/9\)
\(\nu^{5}\)\(=\)\((\)\(775\) \(\beta_{13}\mathstrut +\mathstrut \) \(890\) \(\beta_{12}\mathstrut +\mathstrut \) \(445\) \(\beta_{11}\mathstrut +\mathstrut \) \(7254\) \(\beta_{10}\mathstrut -\mathstrut \) \(2258\) \(\beta_{9}\mathstrut +\mathstrut \) \(1129\) \(\beta_{8}\mathstrut +\mathstrut \) \(6479\) \(\beta_{7}\mathstrut +\mathstrut \) \(11315\) \(\beta_{6}\mathstrut +\mathstrut \) \(565\) \(\beta_{5}\mathstrut +\mathstrut \) \(5874\) \(\beta_{4}\mathstrut -\mathstrut \) \(1108229\) \(\beta_{3}\mathstrut +\mathstrut \) \(289667\) \(\beta_{2}\mathstrut +\mathstrut \) \(147612\) \(\beta_{1}\mathstrut +\mathstrut \) \(895\)\()/9\)
\(\nu^{6}\)\(=\)\((\)\(5595\) \(\beta_{13}\mathstrut -\mathstrut \) \(353\) \(\beta_{12}\mathstrut +\mathstrut \) \(353\) \(\beta_{11}\mathstrut -\mathstrut \) \(2227\) \(\beta_{10}\mathstrut +\mathstrut \) \(2449\) \(\beta_{9}\mathstrut +\mathstrut \) \(2449\) \(\beta_{8}\mathstrut -\mathstrut \) \(78514\) \(\beta_{7}\mathstrut -\mathstrut \) \(135288\) \(\beta_{6}\mathstrut +\mathstrut \) \(7822\) \(\beta_{5}\mathstrut -\mathstrut \) \(157993\) \(\beta_{4}\mathstrut -\mathstrut \) \(3721\) \(\beta_{3}\mathstrut -\mathstrut \) \(278540\) \(\beta_{2}\mathstrut -\mathstrut \) \(679657\) \(\beta_{1}\mathstrut +\mathstrut \) \(18161930\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(40474\) \(\beta_{13}\mathstrut -\mathstrut \) \(48791\) \(\beta_{12}\mathstrut -\mathstrut \) \(97582\) \(\beta_{11}\mathstrut -\mathstrut \) \(775309\) \(\beta_{10}\mathstrut +\mathstrut \) \(115135\) \(\beta_{9}\mathstrut -\mathstrut \) \(230270\) \(\beta_{8}\mathstrut +\mathstrut \) \(65685\) \(\beta_{7}\mathstrut -\mathstrut \) \(632957\) \(\beta_{6}\mathstrut -\mathstrut \) \(65685\) \(\beta_{5}\mathstrut +\mathstrut \) \(697333\) \(\beta_{4}\mathstrut +\mathstrut \) \(140845590\) \(\beta_{3}\mathstrut -\mathstrut \) \(12934989\) \(\beta_{2}\mathstrut +\mathstrut \) \(12385607\) \(\beta_{1}\mathstrut -\mathstrut \) \(140887695\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(797311\) \(\beta_{13}\mathstrut +\mathstrut \) \(257290\) \(\beta_{12}\mathstrut +\mathstrut \) \(128645\) \(\beta_{11}\mathstrut +\mathstrut \) \(22245998\) \(\beta_{10}\mathstrut -\mathstrut \) \(1957994\) \(\beta_{9}\mathstrut +\mathstrut \) \(978997\) \(\beta_{8}\mathstrut +\mathstrut \) \(21448687\) \(\beta_{7}\mathstrut +\mathstrut \) \(48760555\) \(\beta_{6}\mathstrut -\mathstrut \) \(1593735\) \(\beta_{5}\mathstrut +\mathstrut \) \(8693930\) \(\beta_{4}\mathstrut -\mathstrut \) \(4788193501\) \(\beta_{3}\mathstrut +\mathstrut \) \(236354687\) \(\beta_{2}\mathstrut +\mathstrut \) \(135857412\) \(\beta_{1}\mathstrut -\mathstrut \) \(925069\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(3055185\) \(\beta_{13}\mathstrut -\mathstrut \) \(42911629\) \(\beta_{12}\mathstrut +\mathstrut \) \(42911629\) \(\beta_{11}\mathstrut -\mathstrut \) \(53614551\) \(\beta_{10}\mathstrut +\mathstrut \) \(100529997\) \(\beta_{9}\mathstrut +\mathstrut \) \(100529997\) \(\beta_{8}\mathstrut -\mathstrut \) \(827606118\) \(\beta_{7}\mathstrut -\mathstrut \) \(968960740\) \(\beta_{6}\mathstrut +\mathstrut \) \(50559366\) \(\beta_{5}\mathstrut -\mathstrut \) \(1308998705\) \(\beta_{4}\mathstrut +\mathstrut \) \(13758107\) \(\beta_{3}\mathstrut -\mathstrut \) \(9983867864\) \(\beta_{2}\mathstrut -\mathstrut \) \(20975015545\) \(\beta_{1}\mathstrut +\mathstrut \) \(145236939578\)\()/9\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(2074068722\) \(\beta_{13}\mathstrut -\mathstrut \) \(170625935\) \(\beta_{12}\mathstrut -\mathstrut \) \(341251870\) \(\beta_{11}\mathstrut -\mathstrut \) \(18456045845\) \(\beta_{10}\mathstrut +\mathstrut \) \(1034796031\) \(\beta_{9}\mathstrut -\mathstrut \) \(2069592062\) \(\beta_{8}\mathstrut +\mathstrut \) \(779841981\) \(\beta_{7}\mathstrut -\mathstrut \) \(9261302781\) \(\beta_{6}\mathstrut -\mathstrut \) \(779841981\) \(\beta_{5}\mathstrut +\mathstrut \) \(32357314637\) \(\beta_{4}\mathstrut +\mathstrut \) \(3979279076558\) \(\beta_{3}\mathstrut -\mathstrut \) \(131713712797\) \(\beta_{2}\mathstrut +\mathstrut \) \(104453033815\) \(\beta_{1}\mathstrut -\mathstrut \) \(3978594065863\)\()/9\)
\(\nu^{11}\)\(=\)\((\)\(15336140671\) \(\beta_{13}\mathstrut +\mathstrut \) \(24106321802\) \(\beta_{12}\mathstrut +\mathstrut \) \(12053160901\) \(\beta_{11}\mathstrut +\mathstrut \) \(263284992822\) \(\beta_{10}\mathstrut -\mathstrut \) \(57990091370\) \(\beta_{9}\mathstrut +\mathstrut \) \(28995045685\) \(\beta_{8}\mathstrut +\mathstrut \) \(247948852151\) \(\beta_{7}\mathstrut +\mathstrut \) \(500118246707\) \(\beta_{6}\mathstrut -\mathstrut \) \(3104044871\) \(\beta_{5}\mathstrut +\mathstrut \) \(130681894122\) \(\beta_{4}\mathstrut -\mathstrut \) \(47554710318797\) \(\beta_{3}\mathstrut +\mathstrut \) \(5884514743319\) \(\beta_{2}\mathstrut +\mathstrut \) \(3113759070804\) \(\beta_{1}\mathstrut +\mathstrut \) \(178934899\)\()/9\)
\(\nu^{12}\)\(=\)\((\)\(116297982111\) \(\beta_{13}\mathstrut -\mathstrut \) \(23192247469\) \(\beta_{12}\mathstrut +\mathstrut \) \(23192247469\) \(\beta_{11}\mathstrut -\mathstrut \) \(81279180215\) \(\beta_{10}\mathstrut +\mathstrut \) \(114453756029\) \(\beta_{9}\mathstrut +\mathstrut \) \(114453756029\) \(\beta_{8}\mathstrut -\mathstrut \) \(1871589674246\) \(\beta_{7}\mathstrut -\mathstrut \) \(3182073710580\) \(\beta_{6}\mathstrut +\mathstrut \) \(197577162326\) \(\beta_{5}\mathstrut -\mathstrut \) \(3902220639473\) \(\beta_{4}\mathstrut -\mathstrut \) \(58211049365\) \(\beta_{3}\mathstrut -\mathstrut \) \(11248730704024\) \(\beta_{2}\mathstrut -\mathstrut \) \(25412044469129\) \(\beta_{1}\mathstrut +\mathstrut \) \(378122825249866\)\()/3\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(2093005439570\) \(\beta_{13}\mathstrut -\mathstrut \) \(1117773709951\) \(\beta_{12}\mathstrut -\mathstrut \) \(2235547419902\) \(\beta_{11}\mathstrut -\mathstrut \) \(25924029256565\) \(\beta_{10}\mathstrut +\mathstrut \) \(2798160962351\) \(\beta_{9}\mathstrut -\mathstrut \) \(5596321924702\) \(\beta_{8}\mathstrut +\mathstrut \) \(1502979304461\) \(\beta_{7}\mathstrut -\mathstrut \) \(15974635075597\) \(\beta_{6}\mathstrut -\mathstrut \) \(1502979304461\) \(\beta_{5}\mathstrut +\mathstrut \) \(34802836623917\) \(\beta_{4}\mathstrut +\mathstrut \) \(5049248915624526\) \(\beta_{3}\mathstrut -\mathstrut \) \(299004121887821\) \(\beta_{2}\mathstrut +\mathstrut \) \(270245091343383\) \(\beta_{1}\mathstrut -\mathstrut \) \(5049044095083927\)\()/3\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 - \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
7.38374 + 12.7890i
5.69757 + 9.86849i
4.05115 + 7.01679i
−0.447645 0.775344i
−2.00397 3.47098i
−5.49482 9.51731i
−8.68602 15.0446i
7.38374 12.7890i
5.69757 9.86849i
4.05115 7.01679i
−0.447645 + 0.775344i
−2.00397 + 3.47098i
−5.49482 + 9.51731i
−8.68602 + 15.0446i
−22.1512 12.7890i −80.9424 3.05302i 199.117 + 344.881i 570.352 329.293i 1753.93 + 1102.80i −1512.41 + 2619.58i 3638.08i 6542.36 + 494.238i −16845.3
2.2 −17.0927 9.86849i 51.0315 62.9030i 66.7741 + 115.656i −896.557 + 517.627i −1493.02 + 571.581i −21.9132 + 37.9547i 2416.83i −1352.58 6420.07i 20432.8
2.3 −12.1534 7.01679i 54.3339 + 60.0735i −29.5292 51.1461i 676.216 390.413i −238.821 1111.35i 2168.61 3756.14i 4421.40i −656.650 + 6528.06i −10957.8
2.4 1.34294 + 0.775344i −44.1317 + 67.9220i −126.798 219.620i −604.549 + 349.037i −111.929 + 56.9976i −1124.45 + 1947.61i 790.223i −2665.79 5995.02i −1082.49
2.5 6.01192 + 3.47098i −29.7082 75.3553i −103.905 179.968i 331.396 191.331i 82.9534 556.147i 467.516 809.762i 3219.75i −4795.84 + 4477.35i 2656.43
2.6 16.4845 + 9.51731i 80.5605 + 8.42672i 53.1583 + 92.0729i −46.9888 + 27.1290i 1247.80 + 905.629i −921.012 + 1595.24i 2849.17i 6418.98 + 1357.72i −1032.78
2.7 26.0581 + 15.0446i −77.6435 + 23.0757i 324.682 + 562.365i 189.131 109.195i −2370.40 566.810i 1404.67 2432.96i 11836.0i 5496.03 3583.35i 6571.17
5.1 −22.1512 + 12.7890i −80.9424 + 3.05302i 199.117 344.881i 570.352 + 329.293i 1753.93 1102.80i −1512.41 2619.58i 3638.08i 6542.36 494.238i −16845.3
5.2 −17.0927 + 9.86849i 51.0315 + 62.9030i 66.7741 115.656i −896.557 517.627i −1493.02 571.581i −21.9132 37.9547i 2416.83i −1352.58 + 6420.07i 20432.8
5.3 −12.1534 + 7.01679i 54.3339 60.0735i −29.5292 + 51.1461i 676.216 + 390.413i −238.821 + 1111.35i 2168.61 + 3756.14i 4421.40i −656.650 6528.06i −10957.8
5.4 1.34294 0.775344i −44.1317 67.9220i −126.798 + 219.620i −604.549 349.037i −111.929 56.9976i −1124.45 1947.61i 790.223i −2665.79 + 5995.02i −1082.49
5.5 6.01192 3.47098i −29.7082 + 75.3553i −103.905 + 179.968i 331.396 + 191.331i 82.9534 + 556.147i 467.516 + 809.762i 3219.75i −4795.84 4477.35i 2656.43
5.6 16.4845 9.51731i 80.5605 8.42672i 53.1583 92.0729i −46.9888 27.1290i 1247.80 905.629i −921.012 1595.24i 2849.17i 6418.98 1357.72i −1032.78
5.7 26.0581 15.0446i −77.6435 23.0757i 324.682 562.365i 189.131 + 109.195i −2370.40 + 566.810i 1404.67 + 2432.96i 11836.0i 5496.03 + 3583.35i 6571.17
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.7
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{9}^{\mathrm{new}}(9, [\chi])\).