# 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

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{9}^{\mathrm{new}}(27, [\chi])$$.