Properties

Label 27.9.d.a
Level 27
Weight 9
Character orbit 27.d
Analytic conductor 10.999
Analytic rank 0
Dimension 14
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 27.d (of order \(6\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{30} \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -2 \beta_{1} - 2 \beta_{2} + 110 \beta_{3} + \beta_{4} + \beta_{6} ) q^{4} \) \( + ( -42 + 21 \beta_{3} + \beta_{7} ) q^{5} \) \( + ( 139 + 32 \beta_{1} - 16 \beta_{2} - 139 \beta_{3} - 2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{7} \) \( + ( 688 + 107 \beta_{1} - 1376 \beta_{3} - 5 \beta_{4} - \beta_{5} - 10 \beta_{6} - 2 \beta_{10} + \beta_{11} - \beta_{13} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -2 \beta_{1} - 2 \beta_{2} + 110 \beta_{3} + \beta_{4} + \beta_{6} ) q^{4} \) \( + ( -42 + 21 \beta_{3} + \beta_{7} ) q^{5} \) \( + ( 139 + 32 \beta_{1} - 16 \beta_{2} - 139 \beta_{3} - 2 \beta_{7} - \beta_{8} - \beta_{10} ) q^{7} \) \( + ( 688 + 107 \beta_{1} - 1376 \beta_{3} - 5 \beta_{4} - \beta_{5} - 10 \beta_{6} - 2 \beta_{10} + \beta_{11} - \beta_{13} ) q^{8} \) \( + ( -7 + 86 \beta_{1} - 171 \beta_{2} + \beta_{3} + 8 \beta_{4} + 2 \beta_{5} - \beta_{6} + 9 \beta_{7} - 8 \beta_{8} - 2 \beta_{11} - \beta_{12} - \beta_{13} ) q^{10} \) \( + ( 1359 - \beta_{1} + 13 \beta_{2} + 1350 \beta_{3} + 39 \beta_{4} - 2 \beta_{5} + 21 \beta_{6} - \beta_{7} - 8 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - 7 \beta_{13} ) q^{11} \) \( + ( -1 + 367 \beta_{1} + 367 \beta_{2} + 87 \beta_{3} + 6 \beta_{4} - \beta_{5} + 6 \beta_{6} + 9 \beta_{7} + 18 \beta_{8} + 8 \beta_{9} - \beta_{10} - \beta_{11} - 8 \beta_{12} + 2 \beta_{13} ) q^{13} \) \( + ( -11552 - 358 \beta_{1} + 358 \beta_{2} + 5776 \beta_{3} - 77 \beta_{4} + \beta_{5} + 77 \beta_{6} - 10 \beta_{7} + 7 \beta_{9} - \beta_{10} + 7 \beta_{12} ) q^{14} \) \( + ( -9987 - 3202 \beta_{1} + 1611 \beta_{2} + 10012 \beta_{3} - 20 \beta_{4} + 65 \beta_{6} + 20 \beta_{7} + 20 \beta_{8} + 20 \beta_{9} + 13 \beta_{10} - 30 \beta_{11} + 25 \beta_{13} ) q^{16} \) \( + ( 1416 - 225 \beta_{1} + 14 \beta_{2} - 2818 \beta_{3} - 46 \beta_{4} + 15 \beta_{5} - 50 \beta_{6} - 33 \beta_{7} - 19 \beta_{8} + 28 \beta_{9} + 30 \beta_{10} + 23 \beta_{11} - 14 \beta_{12} - 9 \beta_{13} ) q^{17} \) \( + ( -18583 + 1384 \beta_{1} - 2771 \beta_{2} - 3 \beta_{3} - 83 \beta_{4} - 31 \beta_{5} + 3 \beta_{6} - 38 \beta_{7} + 35 \beta_{8} - 39 \beta_{11} + 3 \beta_{12} - 42 \beta_{13} ) q^{19} \) \( + ( -25671 - 4 \beta_{1} + 311 \beta_{2} - 25742 \beta_{3} - 140 \beta_{4} + 30 \beta_{5} - 64 \beta_{6} - 4 \beta_{7} - 52 \beta_{8} - 4 \beta_{9} + 15 \beta_{10} + 4 \beta_{11} + 8 \beta_{12} - 63 \beta_{13} ) q^{20} \) \( + ( -17 + 5108 \beta_{1} + 5108 \beta_{2} + 6526 \beta_{3} + 112 \beta_{4} + 17 \beta_{5} + 112 \beta_{6} - 56 \beta_{7} - 112 \beta_{8} + 46 \beta_{9} + 17 \beta_{10} - 17 \beta_{11} - 46 \beta_{12} + 34 \beta_{13} ) q^{22} \) \( + ( 95183 - 133 \beta_{1} + 133 \beta_{2} - 47595 \beta_{3} + 78 \beta_{4} - 16 \beta_{5} - 78 \beta_{6} + \beta_{7} + 28 \beta_{9} + 16 \beta_{10} + 7 \beta_{11} + 28 \beta_{12} ) q^{23} \) \( + ( 5662 - 16600 \beta_{1} + 8302 \beta_{2} - 5666 \beta_{3} - 4 \beta_{4} - 78 \beta_{6} + 286 \beta_{7} + 145 \beta_{8} + 4 \beta_{9} - 58 \beta_{10} + 12 \beta_{11} - 4 \beta_{13} ) q^{25} \) \( + ( -132933 + 1856 \beta_{1} - 21 \beta_{2} + 265845 \beta_{3} + 582 \beta_{4} - 90 \beta_{5} + 1101 \beta_{6} + \beta_{7} - 20 \beta_{8} - 42 \beta_{9} - 180 \beta_{10} + 6 \beta_{11} + 21 \beta_{12} - 27 \beta_{13} ) q^{26} \) \( + ( 84708 + 19465 \beta_{1} - 38874 \beta_{2} + 56 \beta_{3} + 588 \beta_{4} + 195 \beta_{5} - 56 \beta_{6} + 232 \beta_{7} - 176 \beta_{8} + 77 \beta_{11} - 56 \beta_{12} + 133 \beta_{13} ) q^{28} \) \( + ( -181578 - 14 \beta_{1} + 5955 \beta_{2} - 181509 \beta_{3} - 690 \beta_{4} - 178 \beta_{5} - 324 \beta_{6} - 14 \beta_{7} + 79 \beta_{8} - 14 \beta_{9} - 89 \beta_{10} + 14 \beta_{11} + 28 \beta_{12} + 97 \beta_{13} ) q^{29} \) \( + ( 105 + 19715 \beta_{1} + 19715 \beta_{2} - 31907 \beta_{3} - 318 \beta_{4} - 122 \beta_{5} - 318 \beta_{6} + 43 \beta_{7} + 86 \beta_{8} - 210 \beta_{9} - 122 \beta_{10} + 105 \beta_{11} + 210 \beta_{12} - 210 \beta_{13} ) q^{31} \) \( + ( 808197 + 13233 \beta_{1} - 13233 \beta_{2} - 404068 \beta_{3} + 1547 \beta_{4} + 105 \beta_{5} - 1547 \beta_{6} + 372 \beta_{7} - 188 \beta_{9} - 105 \beta_{10} - 61 \beta_{11} - 188 \beta_{12} ) q^{32} \) \( + ( 89162 - 21815 \beta_{1} + 10749 \beta_{2} - 89475 \beta_{3} + 317 \beta_{4} - 260 \beta_{6} - 1979 \beta_{7} - 1148 \beta_{8} - 317 \beta_{9} + 38 \beta_{10} + 309 \beta_{11} - 313 \beta_{13} ) q^{34} \) \( + ( -339043 - 10554 \beta_{1} - 112 \beta_{2} + 677974 \beta_{3} - 1442 \beta_{4} + 247 \beta_{5} - 3220 \beta_{6} + 629 \beta_{7} + 517 \beta_{8} - 224 \beta_{9} + 494 \beta_{10} - 308 \beta_{11} + 112 \beta_{12} + 196 \beta_{13} ) q^{35} \) \( + ( -118948 - 1895 \beta_{1} + 3460 \beta_{2} - 330 \beta_{3} - 1880 \beta_{4} - 583 \beta_{5} + 330 \beta_{6} - 806 \beta_{7} + 476 \beta_{8} + 363 \beta_{11} + 330 \beta_{12} + 33 \beta_{13} ) q^{37} \) \( + ( -519060 + 241 \beta_{1} - 30701 \beta_{2} - 517467 \beta_{3} + 2175 \beta_{4} + 464 \beta_{5} + 726 \beta_{6} + 241 \beta_{7} + 808 \beta_{8} + 241 \beta_{9} + 232 \beta_{10} - 241 \beta_{11} - 482 \beta_{12} + 1111 \beta_{13} ) q^{38} \) \( + ( -234 - 30758 \beta_{1} - 30758 \beta_{2} + 101796 \beta_{3} - 878 \beta_{4} + 456 \beta_{5} - 878 \beta_{6} + 852 \beta_{7} + 1704 \beta_{8} - 324 \beta_{9} + 456 \beta_{10} - 234 \beta_{11} + 324 \beta_{12} + 468 \beta_{13} ) q^{40} \) \( + ( 966299 - 48729 \beta_{1} + 48729 \beta_{2} - 483297 \beta_{3} - 5238 \beta_{4} - 337 \beta_{5} + 5238 \beta_{6} - 2056 \beta_{7} + 4 \beta_{9} + 337 \beta_{10} + 295 \beta_{11} + 4 \beta_{12} ) q^{41} \) \( + ( 120056 + 38683 \beta_{1} - 19174 \beta_{2} - 120022 \beta_{3} - 335 \beta_{4} - 1795 \beta_{6} + 3333 \beta_{7} + 1834 \beta_{8} + 335 \beta_{9} + 522 \beta_{10} + 267 \beta_{11} + 34 \beta_{13} ) q^{43} \) \( + ( -1520878 - 5798 \beta_{1} + 84 \beta_{2} + 3041840 \beta_{3} + 305 \beta_{4} - 134 \beta_{5} + 862 \beta_{6} - 1104 \beta_{7} - 1020 \beta_{8} + 168 \beta_{9} - 268 \beta_{10} - 664 \beta_{11} - 84 \beta_{12} + 748 \beta_{13} ) q^{44} \) \( + ( 53659 - 71281 \beta_{1} + 143173 \beta_{2} + 611 \beta_{3} + 450 \beta_{4} + 499 \beta_{5} - 611 \beta_{6} + 3 \beta_{7} + 608 \beta_{8} - 187 \beta_{11} - 611 \beta_{12} + 424 \beta_{13} ) q^{46} \) \( + ( -1481307 - 528 \beta_{1} - 14728 \beta_{2} - 1484091 \beta_{3} + 7440 \beta_{4} - 90 \beta_{5} + 4512 \beta_{6} - 528 \beta_{7} + 343 \beta_{8} - 528 \beta_{9} - 45 \beta_{10} + 528 \beta_{11} + 1056 \beta_{12} - 1728 \beta_{13} ) q^{47} \) \( + ( 637 - 37293 \beta_{1} - 37293 \beta_{2} - 675614 \beta_{3} - 504 \beta_{4} - 789 \beta_{5} - 504 \beta_{6} - 3757 \beta_{7} - 7514 \beta_{8} + 952 \beta_{9} - 789 \beta_{10} + 637 \beta_{11} - 952 \beta_{12} - 1274 \beta_{13} ) q^{49} \) \( + ( 6090589 + 55892 \beta_{1} - 55892 \beta_{2} - 3044742 \beta_{3} + 6093 \beta_{4} + 382 \beta_{5} - 6093 \beta_{6} + 4458 \beta_{7} + 203 \beta_{9} - 382 \beta_{10} - 1105 \beta_{11} + 203 \beta_{12} ) q^{50} \) \( + ( -717569 + 346930 \beta_{1} - 173579 \beta_{2} + 718490 \beta_{3} + 228 \beta_{4} + 4666 \beta_{6} + 1420 \beta_{7} + 596 \beta_{8} - 228 \beta_{9} - 1567 \beta_{10} - 2070 \beta_{11} + 921 \beta_{13} ) q^{52} \) \( + ( -2129146 + 131573 \beta_{1} + 160 \beta_{2} + 4258452 \beta_{3} - 2790 \beta_{4} - 941 \beta_{5} - 5100 \beta_{6} - 1104 \beta_{7} - 944 \beta_{8} + 320 \beta_{9} - 1882 \beta_{10} + 4137 \beta_{11} - 160 \beta_{12} - 3977 \beta_{13} ) q^{53} \) \( + ( 547471 - 66325 \beta_{1} + 132208 \beta_{2} - 442 \beta_{3} + 9924 \beta_{4} + 1852 \beta_{5} + 442 \beta_{6} + 2295 \beta_{7} - 2737 \beta_{8} - 2131 \beta_{11} + 442 \beta_{12} - 2573 \beta_{13} ) q^{55} \) \( + ( -5580864 + 420 \beta_{1} + 225948 \beta_{2} - 5582404 \beta_{3} - 26488 \beta_{4} - 2348 \beta_{5} - 13874 \beta_{6} + 420 \beta_{7} - 7484 \beta_{8} + 420 \beta_{9} - 1174 \beta_{10} - 420 \beta_{11} - 840 \beta_{12} - 2380 \beta_{13} ) q^{56} \) \( + ( -1737 - 255527 \beta_{1} - 255527 \beta_{2} + 2137704 \beta_{3} + 14911 \beta_{4} - 330 \beta_{5} + 14911 \beta_{6} + 4404 \beta_{7} + 8808 \beta_{8} + 1413 \beta_{9} - 330 \beta_{10} - 1737 \beta_{11} - 1413 \beta_{12} + 3474 \beta_{13} ) q^{58} \) \( + ( 7971420 + 60885 \beta_{1} - 60885 \beta_{2} - 3987255 \beta_{3} - 2436 \beta_{4} + 792 \beta_{5} + 2436 \beta_{6} - 296 \beta_{7} + 1761 \beta_{9} - 792 \beta_{10} + 3090 \beta_{11} + 1761 \beta_{12} ) q^{59} \) \( + ( -743162 - 4760 \beta_{1} + 4237 \beta_{2} + 745163 \beta_{3} - 3714 \beta_{4} + 13544 \beta_{6} - 8556 \beta_{7} - 2421 \beta_{8} + 3714 \beta_{9} - 75 \beta_{10} - 288 \beta_{11} + 2001 \beta_{13} ) q^{61} \) \( + ( -7227247 - 246627 \beta_{1} + 1319 \beta_{2} + 14455813 \beta_{3} + 20044 \beta_{4} + 2071 \beta_{5} + 44045 \beta_{6} + 8169 \beta_{7} + 9488 \beta_{8} + 2638 \beta_{9} + 4142 \beta_{10} - 3617 \beta_{11} - 1319 \beta_{12} + 4936 \beta_{13} ) q^{62} \) \( + ( -2054190 - 330747 \beta_{1} + 662962 \beta_{2} + 1468 \beta_{3} - 13559 \beta_{4} - 5401 \beta_{5} - 1468 \beta_{6} + 3504 \beta_{7} - 2036 \beta_{8} - 281 \beta_{11} - 1468 \beta_{12} + 1187 \beta_{13} ) q^{64} \) \( + ( -3270288 - 1492 \beta_{1} - 170803 \beta_{2} - 3275611 \beta_{3} + 620 \beta_{4} + 4262 \beta_{5} + 2548 \beta_{6} - 1492 \beta_{7} - 1787 \beta_{8} - 1492 \beta_{9} + 2131 \beta_{10} + 1492 \beta_{11} + 2984 \beta_{12} - 2339 \beta_{13} ) q^{65} \) \( + ( 1931 + 180246 \beta_{1} + 180246 \beta_{2} - 1256319 \beta_{3} - 11744 \beta_{4} + 4303 \beta_{5} - 11744 \beta_{6} + 5648 \beta_{7} + 11296 \beta_{8} + 563 \beta_{9} + 4303 \beta_{10} + 1931 \beta_{11} - 563 \beta_{12} - 3862 \beta_{13} ) q^{67} \) \( + ( 7332401 - 268655 \beta_{1} + 268655 \beta_{2} - 3663200 \beta_{3} + 721 \beta_{4} - 2923 \beta_{5} - 721 \beta_{6} - 17044 \beta_{7} - 1404 \beta_{9} + 2923 \beta_{10} - 6001 \beta_{11} - 1404 \beta_{12} ) q^{68} \) \( + ( 3951891 + 19555 \beta_{1} - 12119 \beta_{2} - 3953823 \beta_{3} + 4683 \beta_{4} - 51394 \beta_{6} + 10323 \beta_{7} + 2820 \beta_{8} - 4683 \beta_{9} + 6831 \beta_{10} - 819 \beta_{11} - 1932 \beta_{13} ) q^{70} \) \( + ( 99538 + 16303 \beta_{1} - 1652 \beta_{2} - 200728 \beta_{3} - 11050 \beta_{4} - 157 \beta_{5} - 27056 \beta_{6} - 28818 \beta_{7} - 30470 \beta_{8} - 3304 \beta_{9} - 314 \beta_{10} + 2747 \beta_{11} + 1652 \beta_{12} - 4399 \beta_{13} ) q^{71} \) \( + ( -2345260 + 605673 \beta_{1} - 1215480 \beta_{2} - 4134 \beta_{3} - 32142 \beta_{4} + 2613 \beta_{5} + 4134 \beta_{6} - 5823 \beta_{7} + 1689 \beta_{8} + 3993 \beta_{11} + 4134 \beta_{12} - 141 \beta_{13} ) q^{73} \) \( + ( 535794 + 3616 \beta_{1} - 376976 \beta_{2} + 557928 \beta_{3} + 32328 \beta_{4} + 1856 \beta_{5} + 10740 \beta_{6} + 3616 \beta_{7} + 40420 \beta_{8} + 3616 \beta_{9} + 928 \beta_{10} - 3616 \beta_{11} - 7232 \beta_{12} + 14902 \beta_{13} ) q^{74} \) \( + ( -1629 + 265293 \beta_{1} + 265293 \beta_{2} - 6208112 \beta_{3} - 60539 \beta_{4} - 7331 \beta_{5} - 60539 \beta_{6} - 8780 \beta_{7} - 17560 \beta_{8} - 7884 \beta_{9} - 7331 \beta_{10} - 1629 \beta_{11} + 7884 \beta_{12} + 3258 \beta_{13} ) q^{76} \) \( + ( -5244416 + 370712 \beta_{1} - 370712 \beta_{2} + 2618141 \beta_{3} - 7588 \beta_{4} + 1996 \beta_{5} + 7588 \beta_{6} + 22397 \beta_{7} - 2898 \beta_{9} - 1996 \beta_{10} + 8134 \beta_{11} - 2898 \beta_{12} ) q^{77} \) \( + ( -5042334 - 1845552 \beta_{1} + 921707 \beta_{2} + 5034977 \beta_{3} + 2138 \beta_{4} + 68932 \beta_{6} - 19144 \beta_{7} - 10641 \beta_{8} - 2138 \beta_{9} - 6812 \beta_{10} + 12576 \beta_{11} - 7357 \beta_{13} ) q^{79} \) \( + ( 4565788 + 161264 \beta_{1} - 2384 \beta_{2} - 9133960 \beta_{3} - 50014 \beta_{4} - 2668 \beta_{5} - 107180 \beta_{6} + 40656 \beta_{7} + 38272 \beta_{8} - 4768 \beta_{9} - 5336 \beta_{10} - 14356 \beta_{11} + 2384 \beta_{12} + 11972 \beta_{13} ) q^{80} \) \( + ( 17113224 + 696563 \beta_{1} - 1389658 \beta_{2} + 3468 \beta_{3} + 116644 \beta_{4} + 6636 \beta_{5} - 3468 \beta_{6} - 24792 \beta_{7} + 28260 \beta_{8} + 8706 \beta_{11} - 3468 \beta_{12} + 12174 \beta_{13} ) q^{82} \) \( + ( 9392061 - 1918 \beta_{1} + 443564 \beta_{2} + 9391131 \beta_{3} + 44910 \beta_{4} - 9134 \beta_{5} + 25332 \beta_{6} - 1918 \beta_{7} - 48901 \beta_{8} - 1918 \beta_{9} - 4567 \beta_{10} + 1918 \beta_{11} + 3836 \beta_{12} + 2906 \beta_{13} ) q^{83} \) \( + ( 6007 + 1091281 \beta_{1} + 1091281 \beta_{2} + 13385986 \beta_{3} + 130734 \beta_{4} + 1241 \beta_{5} + 130734 \beta_{6} - 19706 \beta_{7} - 39412 \beta_{8} - 5018 \beta_{9} + 1241 \beta_{10} + 6007 \beta_{11} + 5018 \beta_{12} - 12014 \beta_{13} ) q^{85} \) \( + ( -14054833 - 402708 \beta_{1} + 402708 \beta_{2} + 7031174 \beta_{3} + 9158 \beta_{4} + 2555 \beta_{5} - 9158 \beta_{6} + 29564 \beta_{7} - 5704 \beta_{9} - 2555 \beta_{10} - 7515 \beta_{11} - 5704 \beta_{12} ) q^{86} \) \( + ( 3681177 + 666770 \beta_{1} - 338557 \beta_{2} - 3687492 \beta_{3} + 10344 \beta_{4} - 38387 \beta_{6} + 36120 \beta_{7} + 12888 \beta_{8} - 10344 \beta_{9} - 11685 \beta_{10} + 2286 \beta_{11} - 6315 \beta_{13} ) q^{88} \) \( + ( 14870494 + 762511 \beta_{1} - 3636 \beta_{2} - 29744624 \beta_{3} + 14662 \beta_{4} - 1735 \beta_{5} + 18416 \beta_{6} + 37020 \beta_{7} + 33384 \beta_{8} - 7272 \beta_{9} - 3470 \beta_{10} + 9331 \beta_{11} + 3636 \beta_{12} - 12967 \beta_{13} ) q^{89} \) \( + ( -13891749 + 1082369 \beta_{1} - 2166054 \beta_{2} - 1316 \beta_{3} - 107324 \beta_{4} - 3566 \beta_{5} + 1316 \beta_{6} + 29221 \beta_{7} - 30537 \beta_{8} - 4613 \beta_{11} + 1316 \beta_{12} - 5929 \beta_{13} ) q^{91} \) \( + ( 14088117 + 1496 \beta_{1} - 225609 \beta_{2} + 14089062 \beta_{3} + 44940 \beta_{4} - 5414 \beta_{5} + 20226 \beta_{6} + 1496 \beta_{7} + 3392 \beta_{8} + 1496 \beta_{9} - 2707 \beta_{10} - 1496 \beta_{11} - 2992 \beta_{12} - 2047 \beta_{13} ) q^{92} \) \( + ( -6004 - 633828 \beta_{1} - 633828 \beta_{2} - 4936750 \beta_{3} - 50273 \beta_{4} + 10141 \beta_{5} - 50273 \beta_{6} + 9188 \beta_{7} + 18376 \beta_{8} + 7091 \beta_{9} + 10141 \beta_{10} - 6004 \beta_{11} - 7091 \beta_{12} + 12008 \beta_{13} ) q^{94} \) \( + ( -36719198 + 428906 \beta_{1} - 428906 \beta_{2} + 18357954 \beta_{3} + 37956 \beta_{4} + 256 \beta_{5} - 37956 \beta_{6} - 99384 \beta_{7} + 7976 \beta_{9} - 256 \beta_{10} + 3290 \beta_{11} + 7976 \beta_{12} ) q^{95} \) \( + ( 17600516 - 2250596 \beta_{1} + 1133555 \beta_{2} - 17587141 \beta_{3} - 16514 \beta_{4} - 175228 \beta_{6} - 22158 \beta_{7} - 2822 \beta_{8} + 16514 \beta_{9} + 13359 \beta_{10} - 10236 \beta_{11} + 13375 \beta_{13} ) q^{97} \) \( + ( 14058781 - 1333743 \beta_{1} + 7861 \beta_{2} - 28109701 \beta_{3} + 64106 \beta_{4} + 3446 \beta_{5} + 151795 \beta_{6} - 125497 \beta_{7} - 117636 \beta_{8} + 15722 \beta_{9} + 6892 \beta_{10} + 6146 \beta_{11} - 7861 \beta_{12} + 1715 \beta_{13} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 767q^{4} \) \(\mathstrut -\mathstrut 438q^{5} \) \(\mathstrut +\mathstrut 922q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 767q^{4} \) \(\mathstrut -\mathstrut 438q^{5} \) \(\mathstrut +\mathstrut 922q^{7} \) \(\mathstrut -\mathstrut 516q^{10} \) \(\mathstrut +\mathstrut 28677q^{11} \) \(\mathstrut +\mathstrut 1684q^{13} \) \(\mathstrut -\mathstrut 120966q^{14} \) \(\mathstrut -\mathstrut 65281q^{16} \) \(\mathstrut -\mathstrut 269630q^{19} \) \(\mathstrut -\mathstrut 539454q^{20} \) \(\mathstrut +\mathstrut 61311q^{22} \) \(\mathstrut +\mathstrut 1000452q^{23} \) \(\mathstrut +\mathstrut 65177q^{25} \) \(\mathstrut +\mathstrut 1075708q^{28} \) \(\mathstrut -\mathstrut 3797682q^{29} \) \(\mathstrut -\mathstrut 164132q^{31} \) \(\mathstrut +\mathstrut 8461881q^{32} \) \(\mathstrut +\mathstrut 654993q^{34} \) \(\mathstrut -\mathstrut 1671668q^{37} \) \(\mathstrut -\mathstrut 10967691q^{38} \) \(\mathstrut +\mathstrut 613326q^{40} \) \(\mathstrut +\mathstrut 10239447q^{41} \) \(\mathstrut +\mathstrut 791815q^{43} \) \(\mathstrut +\mathstrut 1189536q^{46} \) \(\mathstrut -\mathstrut 31148628q^{47} \) \(\mathstrut -\mathstrut 4826637q^{49} \) \(\mathstrut +\mathstrut 63849453q^{50} \) \(\mathstrut -\mathstrut 5552720q^{52} \) \(\mathstrut +\mathstrut 8107476q^{55} \) \(\mathstrut -\mathstrut 116638674q^{56} \) \(\mathstrut +\mathstrut 14211822q^{58} \) \(\mathstrut +\mathstrut 83493795q^{59} \) \(\mathstrut -\mathstrut 5255600q^{61} \) \(\mathstrut -\mathstrut 26813830q^{64} \) \(\mathstrut -\mathstrut 69232992q^{65} \) \(\mathstrut -\mathstrut 8288855q^{67} \) \(\mathstrut +\mathstrut 77746743q^{68} \) \(\mathstrut +\mathstrut 27813756q^{70} \) \(\mathstrut -\mathstrut 36721682q^{73} \) \(\mathstrut +\mathstrut 10383450q^{74} \) \(\mathstrut -\mathstrut 42822959q^{76} \) \(\mathstrut -\mathstrut 56158710q^{77} \) \(\mathstrut -\mathstrut 32771822q^{79} \) \(\mathstrut +\mathstrut 236099418q^{82} \) \(\mathstrut +\mathstrut 198915996q^{83} \) \(\mathstrut +\mathstrut 97486146q^{85} \) \(\mathstrut -\mathstrut 146190669q^{86} \) \(\mathstrut +\mathstrut 24955827q^{88} \) \(\mathstrut -\mathstrut 201514504q^{91} \) \(\mathstrut +\mathstrut 295365804q^{92} \) \(\mathstrut -\mathstrut 36698244q^{94} \) \(\mathstrut -\mathstrut 386813838q^{95} \) \(\mathstrut +\mathstrut 127049161q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(442054787914350989925983\) \(\nu^{13}\mathstrut -\mathstrut \) \(23628808584322386004267421\) \(\nu^{12}\mathstrut +\mathstrut \) \(50640806940675695590816011\) \(\nu^{11}\mathstrut -\mathstrut \) \(9110652565048392910003247466\) \(\nu^{10}\mathstrut +\mathstrut \) \(57543855647402141458544986806\) \(\nu^{9}\mathstrut -\mathstrut \) \(3101563233324860607105549277932\) \(\nu^{8}\mathstrut +\mathstrut \) \(27246232517001751558799077332700\) \(\nu^{7}\mathstrut -\mathstrut \) \(408260669537384694699705658647872\) \(\nu^{6}\mathstrut +\mathstrut \) \(3036725485917744621466950043094736\) \(\nu^{5}\mathstrut -\mathstrut \) \(37304937996215854140380362605307776\) \(\nu^{4}\mathstrut +\mathstrut \) \(371785821517531872040637539606808832\) \(\nu^{3}\mathstrut -\mathstrut \) \(962102681229186015894305741252662272\) \(\nu^{2}\mathstrut +\mathstrut \) \(20848898132329312254552493742295294976\) \(\nu\mathstrut +\mathstrut \) \(8669628126778214704913602579186622464\)\()/\)\(10\!\cdots\!64\)
\(\beta_{2}\)\(=\)\((\)\(442054787914350989925983\) \(\nu^{13}\mathstrut +\mathstrut \) \(23628808584322386004267421\) \(\nu^{12}\mathstrut -\mathstrut \) \(50640806940675695590816011\) \(\nu^{11}\mathstrut +\mathstrut \) \(9110652565048392910003247466\) \(\nu^{10}\mathstrut -\mathstrut \) \(57543855647402141458544986806\) \(\nu^{9}\mathstrut +\mathstrut \) \(3101563233324860607105549277932\) \(\nu^{8}\mathstrut -\mathstrut \) \(27246232517001751558799077332700\) \(\nu^{7}\mathstrut +\mathstrut \) \(408260669537384694699705658647872\) \(\nu^{6}\mathstrut -\mathstrut \) \(3036725485917744621466950043094736\) \(\nu^{5}\mathstrut +\mathstrut \) \(37304937996215854140380362605307776\) \(\nu^{4}\mathstrut -\mathstrut \) \(371785821517531872040637539606808832\) \(\nu^{3}\mathstrut +\mathstrut \) \(962102681229186015894305741252662272\) \(\nu^{2}\mathstrut +\mathstrut \) \(11673431310210505523707812991412566016\) \(\nu\mathstrut -\mathstrut \) \(8669628126778214704913602579186622464\)\()/\)\(10\!\cdots\!64\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(14\!\cdots\!61\) \(\nu^{13}\mathstrut +\mathstrut \) \(13\!\cdots\!59\) \(\nu^{12}\mathstrut -\mathstrut \) \(62\!\cdots\!21\) \(\nu^{11}\mathstrut +\mathstrut \) \(19\!\cdots\!16\) \(\nu^{10}\mathstrut -\mathstrut \) \(19\!\cdots\!86\) \(\nu^{9}\mathstrut +\mathstrut \) \(54\!\cdots\!48\) \(\nu^{8}\mathstrut -\mathstrut \) \(26\!\cdots\!96\) \(\nu^{7}\mathstrut +\mathstrut \) \(46\!\cdots\!96\) \(\nu^{6}\mathstrut -\mathstrut \) \(26\!\cdots\!08\) \(\nu^{5}\mathstrut +\mathstrut \) \(25\!\cdots\!36\) \(\nu^{4}\mathstrut -\mathstrut \) \(94\!\cdots\!84\) \(\nu^{3}\mathstrut -\mathstrut \) \(21\!\cdots\!68\) \(\nu^{2}\mathstrut -\mathstrut \) \(19\!\cdots\!52\) \(\nu\mathstrut -\mathstrut \) \(10\!\cdots\!32\)\()/\)\(15\!\cdots\!16\)
\(\beta_{4}\)\(=\)\((\)\(12477486474032719487022685\) \(\nu^{13}\mathstrut -\mathstrut \) \(96070292105729398140337955\) \(\nu^{12}\mathstrut +\mathstrut \) \(4805724786153193783550402145\) \(\nu^{11}\mathstrut -\mathstrut \) \(49668396317066737366434897960\) \(\nu^{10}\mathstrut +\mathstrut \) \(1575292854857265821546270468586\) \(\nu^{9}\mathstrut -\mathstrut \) \(14557846548144127130964618369600\) \(\nu^{8}\mathstrut +\mathstrut \) \(183371860995283563400371771260980\) \(\nu^{7}\mathstrut -\mathstrut \) \(1498465735540954330277243125606856\) \(\nu^{6}\mathstrut +\mathstrut \) \(15932902446084986471210290476666480\) \(\nu^{5}\mathstrut -\mathstrut \) \(162211742872683955400526492077001600\) \(\nu^{4}\mathstrut +\mathstrut \) \(68683801135688474012987887944771840\) \(\nu^{3}\mathstrut -\mathstrut \) \(4319327656161451359073182846686530560\) \(\nu^{2}\mathstrut -\mathstrut \) \(3740510205416776797840204990428318720\) \(\nu\mathstrut -\mathstrut \) \(670149814637827415587204671958154273792\)\()/\)\(18\!\cdots\!44\)
\(\beta_{5}\)\(=\)\((\)\(36\!\cdots\!33\) \(\nu^{13}\mathstrut -\mathstrut \) \(34\!\cdots\!29\) \(\nu^{12}\mathstrut +\mathstrut \) \(15\!\cdots\!63\) \(\nu^{11}\mathstrut -\mathstrut \) \(18\!\cdots\!82\) \(\nu^{10}\mathstrut +\mathstrut \) \(50\!\cdots\!54\) \(\nu^{9}\mathstrut -\mathstrut \) \(56\!\cdots\!72\) \(\nu^{8}\mathstrut +\mathstrut \) \(67\!\cdots\!88\) \(\nu^{7}\mathstrut -\mathstrut \) \(72\!\cdots\!08\) \(\nu^{6}\mathstrut +\mathstrut \) \(60\!\cdots\!64\) \(\nu^{5}\mathstrut -\mathstrut \) \(64\!\cdots\!76\) \(\nu^{4}\mathstrut +\mathstrut \) \(12\!\cdots\!32\) \(\nu^{3}\mathstrut -\mathstrut \) \(17\!\cdots\!88\) \(\nu^{2}\mathstrut -\mathstrut \) \(14\!\cdots\!64\) \(\nu\mathstrut -\mathstrut \) \(26\!\cdots\!32\)\()/\)\(21\!\cdots\!64\)
\(\beta_{6}\)\(=\)\((\)\(88\!\cdots\!19\) \(\nu^{13}\mathstrut -\mathstrut \) \(87\!\cdots\!73\) \(\nu^{12}\mathstrut +\mathstrut \) \(37\!\cdots\!15\) \(\nu^{11}\mathstrut -\mathstrut \) \(12\!\cdots\!80\) \(\nu^{10}\mathstrut +\mathstrut \) \(12\!\cdots\!10\) \(\nu^{9}\mathstrut -\mathstrut \) \(33\!\cdots\!36\) \(\nu^{8}\mathstrut +\mathstrut \) \(16\!\cdots\!56\) \(\nu^{7}\mathstrut -\mathstrut \) \(28\!\cdots\!24\) \(\nu^{6}\mathstrut +\mathstrut \) \(15\!\cdots\!72\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!40\) \(\nu^{4}\mathstrut +\mathstrut \) \(58\!\cdots\!32\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!88\) \(\nu^{2}\mathstrut +\mathstrut \) \(12\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!44\)\()/\)\(26\!\cdots\!36\)
\(\beta_{7}\)\(=\)\((\)\(53\!\cdots\!05\) \(\nu^{13}\mathstrut +\mathstrut \) \(10\!\cdots\!83\) \(\nu^{12}\mathstrut +\mathstrut \) \(22\!\cdots\!59\) \(\nu^{11}\mathstrut -\mathstrut \) \(41\!\cdots\!10\) \(\nu^{10}\mathstrut +\mathstrut \) \(72\!\cdots\!22\) \(\nu^{9}\mathstrut -\mathstrut \) \(93\!\cdots\!88\) \(\nu^{8}\mathstrut +\mathstrut \) \(95\!\cdots\!80\) \(\nu^{7}\mathstrut +\mathstrut \) \(13\!\cdots\!48\) \(\nu^{6}\mathstrut +\mathstrut \) \(92\!\cdots\!36\) \(\nu^{5}\mathstrut +\mathstrut \) \(89\!\cdots\!92\) \(\nu^{4}\mathstrut +\mathstrut \) \(31\!\cdots\!80\) \(\nu^{3}\mathstrut +\mathstrut \) \(17\!\cdots\!12\) \(\nu^{2}\mathstrut +\mathstrut \) \(78\!\cdots\!12\) \(\nu\mathstrut +\mathstrut \) \(12\!\cdots\!44\)\()/\)\(13\!\cdots\!72\)
\(\beta_{8}\)\(=\)\((\)\(11\!\cdots\!37\) \(\nu^{13}\mathstrut -\mathstrut \) \(37\!\cdots\!29\) \(\nu^{12}\mathstrut +\mathstrut \) \(47\!\cdots\!19\) \(\nu^{11}\mathstrut -\mathstrut \) \(26\!\cdots\!98\) \(\nu^{10}\mathstrut +\mathstrut \) \(15\!\cdots\!38\) \(\nu^{9}\mathstrut -\mathstrut \) \(76\!\cdots\!72\) \(\nu^{8}\mathstrut +\mathstrut \) \(20\!\cdots\!12\) \(\nu^{7}\mathstrut -\mathstrut \) \(80\!\cdots\!84\) \(\nu^{6}\mathstrut +\mathstrut \) \(19\!\cdots\!24\) \(\nu^{5}\mathstrut -\mathstrut \) \(59\!\cdots\!48\) \(\nu^{4}\mathstrut +\mathstrut \) \(69\!\cdots\!48\) \(\nu^{3}\mathstrut +\mathstrut \) \(60\!\cdots\!24\) \(\nu^{2}\mathstrut +\mathstrut \) \(95\!\cdots\!80\) \(\nu\mathstrut -\mathstrut \) \(79\!\cdots\!12\)\()/\)\(18\!\cdots\!96\)
\(\beta_{9}\)\(=\)\((\)\(41\!\cdots\!81\) \(\nu^{13}\mathstrut -\mathstrut \) \(95\!\cdots\!05\) \(\nu^{12}\mathstrut +\mathstrut \) \(17\!\cdots\!47\) \(\nu^{11}\mathstrut -\mathstrut \) \(76\!\cdots\!86\) \(\nu^{10}\mathstrut +\mathstrut \) \(54\!\cdots\!46\) \(\nu^{9}\mathstrut -\mathstrut \) \(21\!\cdots\!12\) \(\nu^{8}\mathstrut +\mathstrut \) \(67\!\cdots\!28\) \(\nu^{7}\mathstrut -\mathstrut \) \(17\!\cdots\!24\) \(\nu^{6}\mathstrut +\mathstrut \) \(57\!\cdots\!12\) \(\nu^{5}\mathstrut -\mathstrut \) \(11\!\cdots\!32\) \(\nu^{4}\mathstrut +\mathstrut \) \(10\!\cdots\!84\) \(\nu^{3}\mathstrut +\mathstrut \) \(59\!\cdots\!04\) \(\nu^{2}\mathstrut -\mathstrut \) \(23\!\cdots\!48\) \(\nu\mathstrut -\mathstrut \) \(12\!\cdots\!24\)\()/\)\(62\!\cdots\!32\)
\(\beta_{10}\)\(=\)\((\)\(24\!\cdots\!79\) \(\nu^{13}\mathstrut -\mathstrut \) \(39\!\cdots\!91\) \(\nu^{12}\mathstrut +\mathstrut \) \(10\!\cdots\!57\) \(\nu^{11}\mathstrut -\mathstrut \) \(38\!\cdots\!50\) \(\nu^{10}\mathstrut +\mathstrut \) \(34\!\cdots\!86\) \(\nu^{9}\mathstrut -\mathstrut \) \(10\!\cdots\!00\) \(\nu^{8}\mathstrut +\mathstrut \) \(46\!\cdots\!60\) \(\nu^{7}\mathstrut -\mathstrut \) \(84\!\cdots\!16\) \(\nu^{6}\mathstrut +\mathstrut \) \(44\!\cdots\!72\) \(\nu^{5}\mathstrut -\mathstrut \) \(44\!\cdots\!24\) \(\nu^{4}\mathstrut +\mathstrut \) \(16\!\cdots\!84\) \(\nu^{3}\mathstrut +\mathstrut \) \(52\!\cdots\!96\) \(\nu^{2}\mathstrut +\mathstrut \) \(33\!\cdots\!08\) \(\nu\mathstrut +\mathstrut \) \(28\!\cdots\!16\)\()/\)\(21\!\cdots\!12\)
\(\beta_{11}\)\(=\)\((\)\(43\!\cdots\!09\) \(\nu^{13}\mathstrut +\mathstrut \) \(34\!\cdots\!35\) \(\nu^{12}\mathstrut +\mathstrut \) \(18\!\cdots\!27\) \(\nu^{11}\mathstrut -\mathstrut \) \(25\!\cdots\!10\) \(\nu^{10}\mathstrut +\mathstrut \) \(56\!\cdots\!02\) \(\nu^{9}\mathstrut -\mathstrut \) \(52\!\cdots\!16\) \(\nu^{8}\mathstrut +\mathstrut \) \(72\!\cdots\!92\) \(\nu^{7}\mathstrut +\mathstrut \) \(21\!\cdots\!00\) \(\nu^{6}\mathstrut +\mathstrut \) \(68\!\cdots\!32\) \(\nu^{5}\mathstrut +\mathstrut \) \(79\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(21\!\cdots\!76\) \(\nu^{3}\mathstrut +\mathstrut \) \(13\!\cdots\!00\) \(\nu^{2}\mathstrut +\mathstrut \) \(54\!\cdots\!56\) \(\nu\mathstrut +\mathstrut \) \(86\!\cdots\!48\)\()/\)\(21\!\cdots\!12\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(26\!\cdots\!37\) \(\nu^{13}\mathstrut +\mathstrut \) \(64\!\cdots\!09\) \(\nu^{12}\mathstrut -\mathstrut \) \(11\!\cdots\!51\) \(\nu^{11}\mathstrut +\mathstrut \) \(52\!\cdots\!70\) \(\nu^{10}\mathstrut -\mathstrut \) \(37\!\cdots\!18\) \(\nu^{9}\mathstrut +\mathstrut \) \(15\!\cdots\!16\) \(\nu^{8}\mathstrut -\mathstrut \) \(53\!\cdots\!00\) \(\nu^{7}\mathstrut +\mathstrut \) \(15\!\cdots\!16\) \(\nu^{6}\mathstrut -\mathstrut \) \(51\!\cdots\!40\) \(\nu^{5}\mathstrut +\mathstrut \) \(10\!\cdots\!04\) \(\nu^{4}\mathstrut -\mathstrut \) \(19\!\cdots\!76\) \(\nu^{3}\mathstrut -\mathstrut \) \(32\!\cdots\!32\) \(\nu^{2}\mathstrut -\mathstrut \) \(28\!\cdots\!20\) \(\nu\mathstrut -\mathstrut \) \(29\!\cdots\!44\)\()/\)\(13\!\cdots\!72\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(54\!\cdots\!03\) \(\nu^{13}\mathstrut +\mathstrut \) \(17\!\cdots\!87\) \(\nu^{12}\mathstrut -\mathstrut \) \(23\!\cdots\!05\) \(\nu^{11}\mathstrut +\mathstrut \) \(12\!\cdots\!74\) \(\nu^{10}\mathstrut -\mathstrut \) \(76\!\cdots\!54\) \(\nu^{9}\mathstrut +\mathstrut \) \(37\!\cdots\!36\) \(\nu^{8}\mathstrut -\mathstrut \) \(10\!\cdots\!04\) \(\nu^{7}\mathstrut +\mathstrut \) \(38\!\cdots\!44\) \(\nu^{6}\mathstrut -\mathstrut \) \(10\!\cdots\!84\) \(\nu^{5}\mathstrut +\mathstrut \) \(28\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(36\!\cdots\!28\) \(\nu^{3}\mathstrut -\mathstrut \) \(34\!\cdots\!56\) \(\nu^{2}\mathstrut -\mathstrut \) \(46\!\cdots\!08\) \(\nu\mathstrut +\mathstrut \) \(38\!\cdots\!32\)\()/\)\(21\!\cdots\!12\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6}\mathstrut +\mathstrut \) \(366\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(366\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut -\mathstrut \) \(1238\) \(\beta_{2}\mathstrut +\mathstrut \) \(619\) \(\beta_{1}\mathstrut +\mathstrut \) \(2062\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(50\) \(\beta_{13}\mathstrut +\mathstrut \) \(20\) \(\beta_{12}\mathstrut +\mathstrut \) \(25\) \(\beta_{11}\mathstrut -\mathstrut \) \(13\) \(\beta_{10}\mathstrut -\mathstrut \) \(20\) \(\beta_{9}\mathstrut -\mathstrut \) \(40\) \(\beta_{8}\mathstrut -\mathstrut \) \(20\) \(\beta_{7}\mathstrut -\mathstrut \) \(813\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut -\mathstrut \) \(813\) \(\beta_{4}\mathstrut -\mathstrut \) \(225584\) \(\beta_{3}\mathstrut +\mathstrut \) \(3127\) \(\beta_{2}\mathstrut +\mathstrut \) \(3127\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\)\()/9\)
\(\nu^{5}\)\(=\)\((\)\(1339\) \(\beta_{13}\mathstrut -\mathstrut \) \(2114\) \(\beta_{11}\mathstrut +\mathstrut \) \(3387\) \(\beta_{10}\mathstrut +\mathstrut \) \(564\) \(\beta_{9}\mathstrut +\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(556\) \(\beta_{7}\mathstrut +\mathstrut \) \(19437\) \(\beta_{6}\mathstrut -\mathstrut \) \(564\) \(\beta_{4}\mathstrut +\mathstrut \) \(3326304\) \(\beta_{3}\mathstrut +\mathstrut \) \(450669\) \(\beta_{2}\mathstrut -\mathstrut \) \(900774\) \(\beta_{1}\mathstrut -\mathstrut \) \(3324965\)\()/27\)
\(\nu^{6}\)\(=\)\((\)\(10271\) \(\beta_{13}\mathstrut -\mathstrut \) \(8044\) \(\beta_{12}\mathstrut +\mathstrut \) \(2227\) \(\beta_{11}\mathstrut +\mathstrut \) \(9212\) \(\beta_{8}\mathstrut -\mathstrut \) \(1168\) \(\beta_{7}\mathstrut -\mathstrut \) \(8044\) \(\beta_{6}\mathstrut +\mathstrut \) \(7347\) \(\beta_{5}\mathstrut +\mathstrut \) \(228861\) \(\beta_{4}\mathstrut +\mathstrut \) \(8044\) \(\beta_{3}\mathstrut -\mathstrut \) \(2373606\) \(\beta_{2}\mathstrut +\mathstrut \) \(1190825\) \(\beta_{1}\mathstrut +\mathstrut \) \(54524090\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(311218\) \(\beta_{13}\mathstrut +\mathstrut \) \(89924\) \(\beta_{12}\mathstrut +\mathstrut \) \(155609\) \(\beta_{11}\mathstrut -\mathstrut \) \(345405\) \(\beta_{10}\mathstrut -\mathstrut \) \(89924\) \(\beta_{9}\mathstrut -\mathstrut \) \(18472\) \(\beta_{8}\mathstrut -\mathstrut \) \(9236\) \(\beta_{7}\mathstrut -\mathstrut \) \(2236003\) \(\beta_{6}\mathstrut -\mathstrut \) \(345405\) \(\beta_{5}\mathstrut -\mathstrut \) \(2236003\) \(\beta_{4}\mathstrut -\mathstrut \) \(423119252\) \(\beta_{3}\mathstrut +\mathstrut \) \(39639299\) \(\beta_{2}\mathstrut +\mathstrut \) \(39639299\) \(\beta_{1}\mathstrut +\mathstrut \) \(155609\)\()/9\)
\(\nu^{8}\)\(=\)\((\)\(3370043\) \(\beta_{13}\mathstrut -\mathstrut \) \(4167354\) \(\beta_{11}\mathstrut +\mathstrut \) \(2936991\) \(\beta_{10}\mathstrut +\mathstrut \) \(2572732\) \(\beta_{9}\mathstrut +\mathstrut \) \(2984108\) \(\beta_{8}\mathstrut +\mathstrut \) \(3395484\) \(\beta_{7}\mathstrut +\mathstrut \) \(64346061\) \(\beta_{6}\mathstrut -\mathstrut \) \(2572732\) \(\beta_{4}\mathstrut +\mathstrut \) \(14386259040\) \(\beta_{3}\mathstrut +\mathstrut \) \(407553221\) \(\beta_{2}\mathstrut -\mathstrut \) \(812533710\) \(\beta_{1}\mathstrut -\mathstrut \) \(14382888997\)\()/9\)
\(\nu^{9}\)\(=\)\((\)\(50363121\) \(\beta_{13}\mathstrut -\mathstrut \) \(32491604\) \(\beta_{12}\mathstrut +\mathstrut \) \(17871517\) \(\beta_{11}\mathstrut +\mathstrut \) \(7451492\) \(\beta_{8}\mathstrut +\mathstrut \) \(25040112\) \(\beta_{7}\mathstrut -\mathstrut \) \(32491604\) \(\beta_{6}\mathstrut +\mathstrut \) \(100529997\) \(\beta_{5}\mathstrut +\mathstrut \) \(773991567\) \(\beta_{4}\mathstrut +\mathstrut \) \(32491604\) \(\beta_{3}\mathstrut -\mathstrut \) \(22019565866\) \(\beta_{2}\mathstrut +\mathstrut \) \(11026028735\) \(\beta_{1}\mathstrut +\mathstrut \) \(145305653358\)\()/9\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(2072576502\) \(\beta_{13}\mathstrut +\mathstrut \) \(776340924\) \(\beta_{12}\mathstrut +\mathstrut \) \(1036288251\) \(\beta_{11}\mathstrut -\mathstrut \) \(1034796031\) \(\beta_{10}\mathstrut -\mathstrut \) \(776340924\) \(\beta_{9}\mathstrut -\mathstrut \) \(1731324632\) \(\beta_{8}\mathstrut -\mathstrut \) \(865662316\) \(\beta_{7}\mathstrut -\mathstrut \) \(17679704921\) \(\beta_{6}\mathstrut -\mathstrut \) \(1034796031\) \(\beta_{5}\mathstrut -\mathstrut \) \(17679704921\) \(\beta_{4}\mathstrut -\mathstrut \) \(3985676941068\) \(\beta_{3}\mathstrut +\mathstrut \) \(132325454297\) \(\beta_{2}\mathstrut +\mathstrut \) \(132325454297\) \(\beta_{1}\mathstrut +\mathstrut \) \(1036288251\)\()/9\)
\(\nu^{11}\)\(=\)\((\)\(47435231227\) \(\beta_{13}\mathstrut -\mathstrut \) \(62771371898\) \(\beta_{11}\mathstrut +\mathstrut \) \(86985137055\) \(\beta_{10}\mathstrut +\mathstrut \) \(32099090556\) \(\beta_{9}\mathstrut +\mathstrut \) \(11275748524\) \(\beta_{8}\mathstrut -\mathstrut \) \(9547593508\) \(\beta_{7}\mathstrut +\mathstrut \) \(743846556453\) \(\beta_{6}\mathstrut -\mathstrut \) \(32099090556\) \(\beta_{4}\mathstrut +\mathstrut \) \(142834909858608\) \(\beta_{3}\mathstrut +\mathstrut \) \(9442708485621\) \(\beta_{2}\mathstrut -\mathstrut \) \(18853317880686\) \(\beta_{1}\mathstrut -\mathstrut \) \(142787474627381\)\()/27\)
\(\nu^{12}\)\(=\)\((\)\(312030918355\) \(\beta_{13}\mathstrut -\mathstrut \) \(230751738140\) \(\beta_{12}\mathstrut +\mathstrut \) \(81279180215\) \(\beta_{11}\mathstrut +\mathstrut \) \(242454175948\) \(\beta_{8}\mathstrut -\mathstrut \) \(11702437808\) \(\beta_{7}\mathstrut -\mathstrut \) \(230751738140\) \(\beta_{6}\mathstrut +\mathstrut \) \(343361268087\) \(\beta_{5}\mathstrut +\mathstrut \) \(5370931482093\) \(\beta_{4}\mathstrut +\mathstrut \) \(230751738140\) \(\beta_{3}\mathstrut -\mathstrut \) \(84426036161310\) \(\beta_{2}\mathstrut +\mathstrut \) \(42328393949725\) \(\beta_{1}\mathstrut +\mathstrut \) \(1135377922210762\)\()/9\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(9782332803842\) \(\beta_{13}\mathstrut +\mathstrut \) \(3388187097460\) \(\beta_{12}\mathstrut +\mathstrut \) \(4891166401921\) \(\beta_{11}\mathstrut -\mathstrut \) \(8394482887053\) \(\beta_{10}\mathstrut -\mathstrut \) \(3388187097460\) \(\beta_{9}\mathstrut -\mathstrut \) \(3075690544136\) \(\beta_{8}\mathstrut -\mathstrut \) \(1537845272068\) \(\beta_{7}\mathstrut -\mathstrut \) \(74383900672235\) \(\beta_{6}\mathstrut -\mathstrut \) \(8394482887053\) \(\beta_{5}\mathstrut -\mathstrut \) \(74383900672235\) \(\beta_{4}\mathstrut -\mathstrut \) \(15169889745149188\) \(\beta_{3}\mathstrut +\mathstrut \) \(910919433534235\) \(\beta_{2}\mathstrut +\mathstrut \) \(910919433534235\) \(\beta_{1}\mathstrut +\mathstrut \) \(4891166401921\)\()/9\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1
−8.68602 15.0446i
−5.49482 9.51731i
−2.00397 3.47098i
−0.447645 0.775344i
4.05115 + 7.01679i
5.69757 + 9.86849i
7.38374 + 12.7890i
−8.68602 + 15.0446i
−5.49482 + 9.51731i
−2.00397 + 3.47098i
−0.447645 + 0.775344i
4.05115 7.01679i
5.69757 9.86849i
7.38374 12.7890i
−26.0581 15.0446i 0 324.682 + 562.365i −189.131 + 109.195i 0 1404.67 2432.96i 11836.0i 0 6571.17
8.2 −16.4845 9.51731i 0 53.1583 + 92.0729i 46.9888 27.1290i 0 −921.012 + 1595.24i 2849.17i 0 −1032.78
8.3 −6.01192 3.47098i 0 −103.905 179.968i −331.396 + 191.331i 0 467.516 809.762i 3219.75i 0 2656.43
8.4 −1.34294 0.775344i 0 −126.798 219.620i 604.549 349.037i 0 −1124.45 + 1947.61i 790.223i 0 −1082.49
8.5 12.1534 + 7.01679i 0 −29.5292 51.1461i −676.216 + 390.413i 0 2168.61 3756.14i 4421.40i 0 −10957.8
8.6 17.0927 + 9.86849i 0 66.7741 + 115.656i 896.557 517.627i 0 −21.9132 + 37.9547i 2416.83i 0 20432.8
8.7 22.1512 + 12.7890i 0 199.117 + 344.881i −570.352 + 329.293i 0 −1512.41 + 2619.58i 3638.08i 0 −16845.3
17.1 −26.0581 + 15.0446i 0 324.682 562.365i −189.131 109.195i 0 1404.67 + 2432.96i 11836.0i 0 6571.17
17.2 −16.4845 + 9.51731i 0 53.1583 92.0729i 46.9888 + 27.1290i 0 −921.012 1595.24i 2849.17i 0 −1032.78
17.3 −6.01192 + 3.47098i 0 −103.905 + 179.968i −331.396 191.331i 0 467.516 + 809.762i 3219.75i 0 2656.43
17.4 −1.34294 + 0.775344i 0 −126.798 + 219.620i 604.549 + 349.037i 0 −1124.45 1947.61i 790.223i 0 −1082.49
17.5 12.1534 7.01679i 0 −29.5292 + 51.1461i −676.216 390.413i 0 2168.61 + 3756.14i 4421.40i 0 −10957.8
17.6 17.0927 9.86849i 0 66.7741 115.656i 896.557 + 517.627i 0 −21.9132 37.9547i 2416.83i 0 20432.8
17.7 22.1512 12.7890i 0 199.117 344.881i −570.352 329.293i 0 −1512.41 2619.58i 3638.08i 0 −16845.3
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.7
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 \(S_{9}^{\mathrm{new}}(27, [\chi])\).