# Properties

 Label 24.8.d.a Level 24 Weight 8 Character orbit 24.d Analytic conductor 7.497 Analytic rank 0 Dimension 14 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$24 = 2^{3} \cdot 3$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 24.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.49724061162$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{37}\cdot 3^{16}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{3} q^{3} + ( -15 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{4} + ( -3 \beta_{1} - 2 \beta_{3} - \beta_{6} + \beta_{7} ) q^{5} + ( -4 + \beta_{3} + \beta_{9} ) q^{6} + ( 98 + 4 \beta_{1} + \beta_{3} - \beta_{5} + 3 \beta_{6} ) q^{7} + ( -30 - 15 \beta_{1} - 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{8} -729 q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{3} q^{3} + ( -15 - \beta_{1} + \beta_{3} + \beta_{6} ) q^{4} + ( -3 \beta_{1} - 2 \beta_{3} - \beta_{6} + \beta_{7} ) q^{5} + ( -4 + \beta_{3} + \beta_{9} ) q^{6} + ( 98 + 4 \beta_{1} + \beta_{3} - \beta_{5} + 3 \beta_{6} ) q^{7} + ( -30 - 15 \beta_{1} - 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{8} -729 q^{9} + ( 359 + 4 \beta_{1} - 2 \beta_{2} - 24 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{7} - \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - 2 \beta_{12} ) q^{10} + ( 2 + 71 \beta_{1} + 3 \beta_{3} + \beta_{4} - 4 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{11} + ( 549 + 14 \beta_{3} + 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{12} + ( 1 - 238 \beta_{1} + 3 \beta_{2} + 48 \beta_{3} + 2 \beta_{4} + 6 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - 7 \beta_{9} - 7 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} ) q^{13} + ( 336 + 102 \beta_{1} + 4 \beta_{2} + 50 \beta_{3} + 5 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} - 15 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} - 8 \beta_{10} - \beta_{11} - 2 \beta_{13} ) q^{14} + ( -965 - 144 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 18 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} - 6 \beta_{12} ) q^{15} + ( -3099 - 53 \beta_{1} + 4 \beta_{2} - 21 \beta_{3} - 7 \beta_{4} + 6 \beta_{5} - 26 \beta_{6} - 17 \beta_{7} + 2 \beta_{8} + 12 \beta_{9} + 10 \beta_{10} - 2 \beta_{11} - 11 \beta_{12} - 6 \beta_{13} ) q^{16} + ( -204 + 631 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} + 3 \beta_{4} - 14 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} - 21 \beta_{10} + 7 \beta_{11} - \beta_{12} + 4 \beta_{13} ) q^{17} + ( 729 - 729 \beta_{1} ) q^{18} + ( -10 + 471 \beta_{1} - 4 \beta_{2} + 135 \beta_{3} + \beta_{4} + 28 \beta_{6} + 75 \beta_{7} - 6 \beta_{8} - 16 \beta_{9} + 11 \beta_{10} + 11 \beta_{11} - \beta_{12} + 6 \beta_{13} ) q^{19} + ( 12520 + 438 \beta_{1} - 286 \beta_{3} + 10 \beta_{4} - 12 \beta_{5} + 44 \beta_{6} - 36 \beta_{7} - 10 \beta_{8} + 26 \beta_{9} + 32 \beta_{10} - 16 \beta_{11} + 26 \beta_{12} - 4 \beta_{13} ) q^{20} + ( 3 - 243 \beta_{1} - 15 \beta_{2} - 98 \beta_{3} - 6 \beta_{4} + 27 \beta_{6} + 9 \beta_{7} - 9 \beta_{8} + 3 \beta_{9} - 9 \beta_{10} - 3 \beta_{11} + 6 \beta_{12} + 9 \beta_{13} ) q^{21} + ( -9200 - 96 \beta_{1} - 392 \beta_{3} - 4 \beta_{4} + 20 \beta_{5} + 56 \beta_{6} + 76 \beta_{7} - 4 \beta_{8} - 16 \beta_{9} - 16 \beta_{10} - 20 \beta_{11} + 16 \beta_{12} - 24 \beta_{13} ) q^{22} + ( -10244 - 1320 \beta_{1} - 2 \beta_{3} + 32 \beta_{4} + 2 \beta_{5} + 26 \beta_{6} + 32 \beta_{7} + 32 \beta_{8} - 32 \beta_{9} + 80 \beta_{10} + 32 \beta_{12} + 32 \beta_{13} ) q^{23} + ( -2083 + 558 \beta_{1} - 12 \beta_{2} + 34 \beta_{3} - 12 \beta_{4} - 27 \beta_{5} + 9 \beta_{6} - 15 \beta_{7} + 6 \beta_{8} - 17 \beta_{9} - 9 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 18 \beta_{13} ) q^{24} + ( -14547 + 56 \beta_{1} + 36 \beta_{2} + 12 \beta_{3} - 36 \beta_{4} + 68 \beta_{5} + 36 \beta_{6} + 28 \beta_{7} + 52 \beta_{8} + 144 \beta_{9} + 12 \beta_{10} - 4 \beta_{11} + 24 \beta_{13} ) q^{25} + ( 30358 + 264 \beta_{1} - 36 \beta_{2} + 252 \beta_{3} - 8 \beta_{4} + 12 \beta_{5} - 252 \beta_{6} - 20 \beta_{7} + 24 \beta_{8} - 12 \beta_{9} - 10 \beta_{10} + 2 \beta_{11} - 28 \beta_{12} - 52 \beta_{13} ) q^{26} + 729 \beta_{3} q^{27} + ( 40484 + 652 \beta_{1} + 80 \beta_{2} - 224 \beta_{3} - 38 \beta_{4} + 56 \beta_{5} + 114 \beta_{6} - 70 \beta_{7} + 20 \beta_{8} - 52 \beta_{9} - 76 \beta_{10} + 12 \beta_{11} - 10 \beta_{12} - 60 \beta_{13} ) q^{28} + ( 78 + 3061 \beta_{1} + 50 \beta_{2} + 714 \beta_{3} - 32 \beta_{4} - 285 \beta_{6} + 105 \beta_{7} - 10 \beta_{8} - 266 \beta_{9} + 74 \beta_{10} + 14 \beta_{11} + 32 \beta_{12} + 10 \beta_{13} ) q^{29} + ( -17862 - 1170 \beta_{1} - 24 \beta_{2} - 338 \beta_{3} + 21 \beta_{4} + 27 \beta_{5} - 126 \beta_{6} - 147 \beta_{7} - 15 \beta_{8} + 60 \beta_{9} + 90 \beta_{10} - 39 \beta_{11} + 24 \beta_{12} + 18 \beta_{13} ) q^{30} + ( -6470 + 3240 \beta_{1} - 28 \beta_{2} - 49 \beta_{3} - 64 \beta_{4} + 77 \beta_{5} + 81 \beta_{6} + 120 \beta_{7} + 52 \beta_{8} - 176 \beta_{9} - 204 \beta_{10} - 40 \beta_{11} - 92 \beta_{12} ) q^{31} + ( -63818 - 3684 \beta_{1} + 8 \beta_{2} + 196 \beta_{3} + 4 \beta_{4} + 90 \beta_{5} - 42 \beta_{6} - 302 \beta_{7} - 32 \beta_{8} + 66 \beta_{9} - 34 \beta_{10} + 14 \beta_{11} + 74 \beta_{12} + 4 \beta_{13} ) q^{32} + ( 70 - 2079 \beta_{1} - \beta_{3} + 81 \beta_{4} - 54 \beta_{5} - 162 \beta_{6} - 69 \beta_{7} + 6 \beta_{8} + 74 \beta_{9} + 153 \beta_{10} + 45 \beta_{11} + 81 \beta_{12} + 36 \beta_{13} ) q^{33} + ( 79198 + 454 \beta_{1} + 40 \beta_{2} + 1328 \beta_{3} + 44 \beta_{4} + 36 \beta_{5} + 656 \beta_{6} - 92 \beta_{7} - 44 \beta_{8} + 16 \beta_{9} + 76 \beta_{10} - 96 \beta_{11} - 32 \beta_{12} ) q^{34} + ( 50 - 7349 \beta_{1} - 152 \beta_{2} - 717 \beta_{3} - 3 \beta_{4} - 436 \beta_{6} + 175 \beta_{7} - 82 \beta_{8} + 444 \beta_{9} - 121 \beta_{10} - 117 \beta_{11} + 3 \beta_{12} + 82 \beta_{13} ) q^{35} + ( 10935 + 729 \beta_{1} - 729 \beta_{3} - 729 \beta_{6} ) q^{36} + ( -27 + 3648 \beta_{1} + 111 \beta_{2} - 1900 \beta_{3} + 10 \beta_{4} + 608 \beta_{6} - 24 \beta_{7} - 15 \beta_{8} - 483 \beta_{9} + 13 \beta_{10} + 143 \beta_{11} - 10 \beta_{12} + 15 \beta_{13} ) q^{37} + ( -58840 - 160 \beta_{1} - 240 \beta_{2} + 308 \beta_{3} + 12 \beta_{4} - 236 \beta_{5} + 520 \beta_{6} - 180 \beta_{7} - 20 \beta_{8} - 52 \beta_{9} + 24 \beta_{10} + 52 \beta_{11} - 160 \beta_{12} - 40 \beta_{13} ) q^{38} + ( 34107 + 2304 \beta_{1} - 129 \beta_{2} - 99 \beta_{3} + 123 \beta_{4} - 189 \beta_{5} - 495 \beta_{6} - 141 \beta_{7} - 39 \beta_{8} - 258 \beta_{9} - 108 \beta_{10} + 93 \beta_{11} - 6 \beta_{12} ) q^{39} + ( -61554 + 12076 \beta_{1} + 280 \beta_{2} + 804 \beta_{3} - 72 \beta_{4} - 82 \beta_{5} + 238 \beta_{6} + 702 \beta_{7} - 36 \beta_{8} - 62 \beta_{9} - 166 \beta_{10} + 18 \beta_{11} + 66 \beta_{12} - 12 \beta_{13} ) q^{40} + ( -31608 - 4565 \beta_{1} + 100 \beta_{2} + 249 \beta_{3} - 161 \beta_{4} - 126 \beta_{5} + 222 \beta_{6} - 259 \beta_{7} - 26 \beta_{8} + 782 \beta_{9} + 167 \beta_{10} + 83 \beta_{11} - 61 \beta_{12} - 60 \beta_{13} ) q^{41} + ( 30691 + 180 \beta_{1} - 90 \beta_{2} - 532 \beta_{3} + 81 \beta_{4} - 243 \beta_{5} - 180 \beta_{6} + 447 \beta_{7} - 15 \beta_{8} - 22 \beta_{9} + 207 \beta_{10} - 90 \beta_{11} + 126 \beta_{12} + 36 \beta_{13} ) q^{42} + ( -418 - 2369 \beta_{1} + 36 \beta_{2} + 2887 \beta_{3} + 249 \beta_{4} + 892 \beta_{6} + 19 \beta_{7} + 114 \beta_{8} + 856 \beta_{9} + 3 \beta_{10} + 91 \beta_{11} - 249 \beta_{12} - 114 \beta_{13} ) q^{43} + ( 91112 - 8360 \beta_{1} - 4008 \beta_{3} - 88 \beta_{4} - 320 \beta_{5} + 384 \beta_{6} - 120 \beta_{7} + 16 \beta_{8} + 272 \beta_{9} - 304 \beta_{10} + 368 \beta_{11} - 72 \beta_{12} + 144 \beta_{13} ) q^{44} + ( 2187 \beta_{1} + 1458 \beta_{3} + 729 \beta_{6} - 729 \beta_{7} ) q^{45} + ( -155104 - 11916 \beta_{1} - 8 \beta_{2} - 7268 \beta_{3} + 54 \beta_{4} + 330 \beta_{5} - 1932 \beta_{6} + 1630 \beta_{7} + 198 \beta_{8} + 8 \beta_{9} - 112 \beta_{10} - 62 \beta_{11} - 384 \beta_{12} + 132 \beta_{13} ) q^{46} + ( -75296 - 9200 \beta_{1} - 188 \beta_{2} - 14 \beta_{3} + 108 \beta_{4} - 154 \beta_{5} + 1286 \beta_{6} + 508 \beta_{7} - 276 \beta_{8} - 1800 \beta_{9} + 136 \beta_{10} - 316 \beta_{11} - 80 \beta_{12} - 160 \beta_{13} ) q^{47} + ( -16437 - 2169 \beta_{1} + 204 \beta_{2} + 3791 \beta_{3} + 69 \beta_{4} - 216 \beta_{5} + 684 \beta_{6} - 855 \beta_{7} - 54 \beta_{8} - 66 \beta_{9} + 144 \beta_{10} - 60 \beta_{11} + 111 \beta_{12} + 126 \beta_{13} ) q^{48} + ( 154117 + 30858 \beta_{1} + 364 \beta_{2} + 386 \beta_{3} - 170 \beta_{4} + 32 \beta_{5} + 360 \beta_{6} - 470 \beta_{7} - 408 \beta_{8} + 1412 \beta_{9} - 906 \beta_{10} - 18 \beta_{11} + 194 \beta_{12} - 240 \beta_{13} ) q^{49} + ( 23151 - 14183 \beta_{1} - 496 \beta_{2} + 13840 \beta_{3} + 416 \beta_{4} + 320 \beta_{5} - 464 \beta_{6} + 1360 \beta_{7} - 304 \beta_{8} - 192 \beta_{9} - 248 \beta_{10} + 168 \beta_{11} - 64 \beta_{12} + 400 \beta_{13} ) q^{50} + ( -354 - 6021 \beta_{1} - 372 \beta_{2} + 537 \beta_{3} - 3 \beta_{4} + 1836 \beta_{6} - 225 \beta_{7} + 18 \beta_{8} + 816 \beta_{9} - 225 \beta_{10} - 33 \beta_{11} + 3 \beta_{12} - 18 \beta_{13} ) q^{51} + ( -147004 + 28944 \beta_{1} + 256 \beta_{2} - 13320 \beta_{3} - 304 \beta_{4} + 264 \beta_{5} + 952 \beta_{6} - 1588 \beta_{7} - 4 \beta_{8} + 196 \beta_{9} + 760 \beta_{10} + 8 \beta_{11} + 136 \beta_{12} + 336 \beta_{13} ) q^{52} + ( 1018 - 277 \beta_{1} + 470 \beta_{2} - 16786 \beta_{3} - 312 \beta_{4} - 4507 \beta_{6} - 1289 \beta_{7} + 178 \beta_{8} - 1182 \beta_{9} + 102 \beta_{10} - 414 \beta_{11} + 312 \beta_{12} - 178 \beta_{13} ) q^{53} + ( 2916 - 729 \beta_{3} - 729 \beta_{9} ) q^{54} + ( 340196 - 36716 \beta_{1} - 308 \beta_{2} - 824 \beta_{3} + 208 \beta_{4} + 380 \beta_{5} - 1356 \beta_{6} + 344 \beta_{7} - 356 \beta_{8} - 2032 \beta_{9} + 972 \beta_{10} - 232 \beta_{11} - 100 \beta_{12} - 192 \beta_{13} ) q^{55} + ( 117252 + 42942 \beta_{1} + 96 \beta_{2} + 27014 \beta_{3} - 74 \beta_{4} + 242 \beta_{5} + 346 \beta_{6} - 1984 \beta_{7} + 248 \beta_{8} + 586 \beta_{9} - 66 \beta_{10} + 814 \beta_{11} - 72 \beta_{12} + 144 \beta_{13} ) q^{56} + ( 110982 + 1845 \beta_{1} - 204 \beta_{2} - 513 \beta_{3} - 231 \beta_{4} + 54 \beta_{5} - 2790 \beta_{6} - 741 \beta_{7} - 78 \beta_{8} + 690 \beta_{9} - 495 \beta_{10} + 357 \beta_{11} - 435 \beta_{12} - 180 \beta_{13} ) q^{57} + ( -392821 - 1548 \beta_{1} - 410 \beta_{2} + 28048 \beta_{3} - 225 \beta_{4} - 201 \beta_{5} + 2168 \beta_{6} - 939 \beta_{7} + 671 \beta_{8} - 550 \beta_{9} - 1249 \beta_{10} - 196 \beta_{11} - 234 \beta_{12} - 24 \beta_{13} ) q^{58} + ( 488 + 20208 \beta_{1} + 344 \beta_{2} - 5260 \beta_{3} - 16 \beta_{4} - 8448 \beta_{6} + 816 \beta_{7} + 696 \beta_{8} + 1592 \beta_{9} + 1280 \beta_{10} - 824 \beta_{11} + 16 \beta_{12} - 696 \beta_{13} ) q^{59} + ( -229040 - 20952 \beta_{1} + 336 \beta_{2} - 14092 \beta_{3} - 222 \beta_{4} + 432 \beta_{5} - 1674 \beta_{6} + 2730 \beta_{7} + 132 \beta_{8} - 76 \beta_{9} + 396 \beta_{10} - 300 \beta_{11} + 438 \beta_{12} + 36 \beta_{13} ) q^{60} + ( -393 - 36020 \beta_{1} + 277 \beta_{2} + 6108 \beta_{3} - 282 \beta_{4} + 3884 \beta_{6} - 1484 \beta_{7} + 507 \beta_{8} - 2065 \beta_{9} - 1609 \beta_{10} + 445 \beta_{11} + 282 \beta_{12} - 507 \beta_{13} ) q^{61} + ( 411304 - 3462 \beta_{1} - 724 \beta_{2} - 24386 \beta_{3} + 811 \beta_{4} + 21 \beta_{5} + 5546 \beta_{6} - 1841 \beta_{7} + 419 \beta_{8} + 964 \beta_{9} - 320 \beta_{10} + 233 \beta_{11} + 416 \beta_{12} + 466 \beta_{13} ) q^{62} + ( -71442 - 2916 \beta_{1} - 729 \beta_{3} + 729 \beta_{5} - 2187 \beta_{6} ) q^{63} + ( 275338 - 62318 \beta_{1} + 264 \beta_{2} + 21666 \beta_{3} - 538 \beta_{4} + 288 \beta_{5} - 5528 \beta_{6} + 4078 \beta_{7} + 460 \beta_{8} - 588 \beta_{9} + 2624 \beta_{10} + 24 \beta_{11} + 514 \beta_{12} + 228 \beta_{13} ) q^{64} + ( -181082 - 9033 \beta_{1} + 1700 \beta_{2} + 2093 \beta_{3} + 363 \beta_{4} + 826 \beta_{5} + 9094 \beta_{6} + 1793 \beta_{7} - 402 \beta_{8} + 1158 \beta_{9} + 2323 \beta_{10} - 1361 \beta_{11} + 2063 \beta_{12} + 244 \beta_{13} ) q^{65} + ( -258420 - 2232 \beta_{1} + 456 \beta_{2} + 7360 \beta_{3} - 588 \beta_{4} + 540 \beta_{5} - 4032 \beta_{6} + 2700 \beta_{7} - 36 \beta_{8} - 336 \beta_{9} + 684 \beta_{10} - 456 \beta_{11} - 672 \beta_{12} - 144 \beta_{13} ) q^{66} + ( -944 + 42884 \beta_{1} - 1848 \beta_{2} + 3496 \beta_{3} - 260 \beta_{4} + 6736 \beta_{6} - 1228 \beta_{7} - 240 \beta_{8} + 3224 \beta_{9} + 964 \beta_{10} - 404 \beta_{11} + 260 \beta_{12} + 240 \beta_{13} ) q^{67} + ( -267370 + 76858 \beta_{1} + 448 \beta_{2} - 36618 \beta_{3} + 360 \beta_{4} + 240 \beta_{5} + 1134 \beta_{6} + 1048 \beta_{7} + 208 \beta_{8} - 1600 \beta_{9} - 1536 \beta_{10} + 960 \beta_{11} + 264 \beta_{12} - 656 \beta_{13} ) q^{68} + ( -1542 + 36342 \beta_{1} - 66 \beta_{2} + 11508 \beta_{3} + 60 \beta_{4} + 7722 \beta_{6} + 1854 \beta_{7} + 306 \beta_{8} - 1542 \beta_{9} + 306 \beta_{10} + 1110 \beta_{11} - 60 \beta_{12} - 306 \beta_{13} ) q^{69} + ( 922288 + 4000 \beta_{1} - 544 \beta_{2} - 60004 \beta_{3} + 44 \beta_{4} - 892 \beta_{5} - 4552 \beta_{6} + 5212 \beta_{7} - 1428 \beta_{8} - 1164 \beta_{9} - 1056 \beta_{10} - 1300 \beta_{11} + 1072 \beta_{12} - 24 \beta_{13} ) q^{70} + ( 367888 + 50400 \beta_{1} + 364 \beta_{2} + 1662 \beta_{3} - 908 \beta_{4} - 6 \beta_{5} + 8922 \beta_{6} + 2180 \beta_{7} + 484 \beta_{8} - 1432 \beta_{9} - 2088 \beta_{10} - 1060 \beta_{11} - 544 \beta_{12} ) q^{71} + ( 21870 + 10935 \beta_{1} + 2187 \beta_{3} - 729 \beta_{4} + 729 \beta_{5} + 729 \beta_{6} + 729 \beta_{9} - 729 \beta_{10} - 729 \beta_{11} ) q^{72} + ( -387758 - 574 \beta_{1} - 1588 \beta_{2} - 2646 \beta_{3} + 110 \beta_{4} - 592 \beta_{5} - 13608 \beta_{6} - 2830 \beta_{7} + 1176 \beta_{8} + 1588 \beta_{9} - 1266 \beta_{10} + 2214 \beta_{11} - 1478 \beta_{12} + 432 \beta_{13} ) q^{73} + ( -463424 - 224 \beta_{1} - 624 \beta_{2} + 57308 \beta_{3} + 186 \beta_{4} - 226 \beta_{5} + 1172 \beta_{6} - 3134 \beta_{7} + 1498 \beta_{8} + 3648 \beta_{9} + 1656 \beta_{10} + 842 \beta_{11} - 968 \beta_{12} - 644 \beta_{13} ) q^{74} + ( -1452 - 82620 \beta_{1} + 276 \beta_{2} + 17307 \beta_{3} + 780 \beta_{4} + 10800 \beta_{6} - 1548 \beta_{7} - 324 \beta_{8} - 300 \beta_{9} - 2916 \beta_{10} + 1272 \beta_{11} - 780 \beta_{12} + 324 \beta_{13} ) q^{75} + ( -647276 - 67144 \beta_{1} + 1024 \beta_{2} - 61056 \beta_{3} + 1096 \beta_{4} - 1248 \beta_{5} + 2208 \beta_{6} - 6084 \beta_{7} - 548 \beta_{8} + 828 \beta_{9} + 3888 \beta_{10} - 1008 \beta_{11} + 1208 \beta_{12} - 240 \beta_{13} ) q^{76} + ( 2564 + 94740 \beta_{1} + 644 \beta_{2} - 22620 \beta_{3} - 28 \beta_{4} - 16576 \beta_{6} + 5668 \beta_{7} - 1068 \beta_{8} - 820 \beta_{9} + 3896 \beta_{10} - 824 \beta_{11} + 28 \beta_{12} + 1068 \beta_{13} ) q^{77} + ( 252222 + 35352 \beta_{1} + 1068 \beta_{2} - 28140 \beta_{3} - 1056 \beta_{4} + 432 \beta_{5} + 2772 \beta_{6} - 1884 \beta_{7} + 588 \beta_{8} + 1152 \beta_{9} + 2502 \beta_{10} - 1410 \beta_{11} - 312 \beta_{12} - 612 \beta_{13} ) q^{78} + ( -1023942 - 46100 \beta_{1} - 1388 \beta_{2} - 745 \beta_{3} + 2020 \beta_{4} - 3159 \beta_{5} - 9027 \beta_{6} - 3324 \beta_{7} + 716 \beta_{8} + 1256 \beta_{9} + 2448 \beta_{10} + 2428 \beta_{11} + 632 \beta_{12} + 960 \beta_{13} ) q^{79} + ( 1025690 - 47278 \beta_{1} - 1304 \beta_{2} + 101954 \beta_{3} + 182 \beta_{4} - 1696 \beta_{5} + 12376 \beta_{6} - 5378 \beta_{7} - 852 \beta_{8} - 1932 \beta_{9} - 8048 \beta_{10} + 1320 \beta_{11} - 2062 \beta_{12} - 924 \beta_{13} ) q^{80} + 531441 q^{81} + ( -562966 - 34982 \beta_{1} + 232 \beta_{2} + 100432 \beta_{3} + 892 \beta_{4} - 1068 \beta_{5} - 6928 \beta_{6} - 4556 \beta_{7} - 2908 \beta_{8} - 1968 \beta_{9} - 180 \beta_{10} - 304 \beta_{11} + 992 \beta_{12} - 288 \beta_{13} ) q^{82} + ( 4046 - 158651 \beta_{1} + 2680 \beta_{2} - 18135 \beta_{3} + 51 \beta_{4} - 19084 \beta_{6} + 6817 \beta_{7} - 1774 \beta_{8} - 7644 \beta_{9} - 3543 \beta_{10} + 357 \beta_{11} - 51 \beta_{12} + 1774 \beta_{13} ) q^{83} + ( -197628 + 28170 \beta_{1} + 768 \beta_{2} - 40202 \beta_{3} - 42 \beta_{4} - 1404 \beta_{5} + 1692 \beta_{6} + 840 \beta_{7} - 786 \beta_{8} - 1278 \beta_{9} - 4536 \beta_{10} - 984 \beta_{11} - 786 \beta_{12} - 108 \beta_{13} ) q^{84} + ( 124 - 142386 \beta_{1} - 2108 \beta_{2} + 37644 \beta_{3} + 896 \beta_{4} + 6194 \beta_{6} - 3066 \beta_{7} - 2388 \beta_{8} + 6892 \beta_{9} - 3356 \beta_{10} - 740 \beta_{11} - 896 \beta_{12} + 2388 \beta_{13} ) q^{85} + ( 337528 + 224 \beta_{1} + 1008 \beta_{2} - 100388 \beta_{3} - 436 \beta_{4} + 3124 \beta_{5} + 40 \beta_{6} - 4820 \beta_{7} - 2452 \beta_{8} - 780 \beta_{9} + 1480 \beta_{10} + 2564 \beta_{11} - 2496 \beta_{12} - 1224 \beta_{13} ) q^{86} + ( 564085 + 162396 \beta_{1} - 153 \beta_{2} - 706 \beta_{3} - 549 \beta_{4} + 378 \beta_{5} - 4932 \beta_{6} - 1149 \beta_{7} + 897 \beta_{8} + 2942 \beta_{9} - 6228 \beta_{10} + 909 \beta_{11} - 702 \beta_{12} + 288 \beta_{13} ) q^{87} + ( 903072 + 101552 \beta_{1} - 1280 \beta_{2} + 134896 \beta_{3} + 1744 \beta_{4} - 1552 \beta_{5} - 11760 \beta_{6} - 5216 \beta_{7} - 1760 \beta_{8} + 4624 \beta_{9} + 4560 \beta_{10} - 2288 \beta_{11} - 448 \beta_{12} - 768 \beta_{13} ) q^{88} + ( -857326 - 166786 \beta_{1} - 1320 \beta_{2} + 2266 \beta_{3} - 3082 \beta_{4} - 1372 \beta_{5} + 14396 \beta_{6} + 4226 \beta_{7} + 2828 \beta_{8} - 5396 \beta_{9} + 2598 \beta_{10} - 1026 \beta_{11} - 4402 \beta_{12} + 200 \beta_{13} ) q^{89} + ( -261711 - 2916 \beta_{1} + 1458 \beta_{2} + 17496 \beta_{3} + 729 \beta_{4} + 729 \beta_{5} + 2187 \beta_{7} + 729 \beta_{8} - 1458 \beta_{9} + 3645 \beta_{10} + 1458 \beta_{12} ) q^{90} + ( 1278 + 223363 \beta_{1} + 1372 \beta_{2} + 34919 \beta_{3} - 395 \beta_{4} + 3916 \beta_{6} + 2695 \beta_{7} - 1422 \beta_{8} - 9568 \beta_{9} + 5031 \beta_{10} + 1911 \beta_{11} + 395 \beta_{12} + 1422 \beta_{13} ) q^{91} + ( 783352 - 153432 \beta_{1} - 5280 \beta_{2} - 180992 \beta_{3} - 564 \beta_{4} + 3472 \beta_{5} - 3620 \beta_{6} - 4084 \beta_{7} - 808 \beta_{8} + 11112 \beta_{9} - 8296 \beta_{10} + 232 \beta_{11} - 3436 \beta_{12} - 136 \beta_{13} ) q^{92} + ( -1143 + 151551 \beta_{1} - 693 \beta_{2} + 12934 \beta_{3} + 1386 \beta_{4} + 10557 \beta_{6} - 4941 \beta_{7} - 1179 \beta_{8} + 5265 \beta_{9} + 4329 \beta_{10} + 243 \beta_{11} - 1386 \beta_{12} + 1179 \beta_{13} ) q^{93} + ( -1102328 - 88556 \beta_{1} + 1544 \beta_{2} - 208948 \beta_{3} + 22 \beta_{4} - 3798 \beta_{5} - 988 \beta_{6} - 9874 \beta_{7} + 4406 \beta_{8} + 1928 \beta_{9} - 5224 \beta_{10} + 2058 \beta_{11} + 992 \beta_{12} - 1644 \beta_{13} ) q^{94} + ( -4957600 + 63400 \beta_{1} + 4744 \beta_{2} + 5300 \beta_{3} - 2176 \beta_{4} + 3556 \beta_{5} + 17572 \beta_{6} + 2736 \beta_{7} + 1736 \beta_{8} + 12544 \beta_{9} + 1272 \beta_{10} - 1840 \beta_{11} + 2568 \beta_{12} + 1120 \beta_{13} ) q^{95} + ( 95888 - 16722 \beta_{1} + 2448 \beta_{2} + 59110 \beta_{3} + 810 \beta_{4} + 1674 \beta_{5} - 6930 \beta_{6} + 4368 \beta_{7} + 780 \beta_{8} - 5786 \beta_{9} - 3474 \beta_{10} - 1098 \beta_{11} + 828 \beta_{12} - 1656 \beta_{13} ) q^{96} + ( 13474 + 292702 \beta_{1} - 1168 \beta_{2} + 850 \beta_{3} + 5870 \beta_{4} - 3748 \beta_{5} + 9780 \beta_{6} + 3610 \beta_{7} - 604 \beta_{8} - 13652 \beta_{9} - 6690 \beta_{10} - 1226 \beta_{11} + 4702 \beta_{12} + 1752 \beta_{13} ) q^{97} + ( 3787187 + 196237 \beta_{1} + 1216 \beta_{2} + 249008 \beta_{3} - 2232 \beta_{4} - 3336 \beta_{5} + 30224 \beta_{6} + 264 \beta_{7} - 4056 \beta_{8} - 6368 \beta_{9} - 2576 \beta_{10} + 2280 \beta_{11} + 1536 \beta_{12} - 1904 \beta_{13} ) q^{98} + ( -1458 - 51759 \beta_{1} - 2187 \beta_{3} - 729 \beta_{4} + 2916 \beta_{6} + 3645 \beta_{7} + 1458 \beta_{8} - 2916 \beta_{9} - 2187 \beta_{10} + 729 \beta_{11} + 729 \beta_{12} - 1458 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q - 14q^{2} - 208q^{4} - 54q^{6} + 1372q^{7} - 428q^{8} - 10206q^{9} + O(q^{10})$$ $$14q - 14q^{2} - 208q^{4} - 54q^{6} + 1372q^{7} - 428q^{8} - 10206q^{9} + 5020q^{10} + 7668q^{12} + 4636q^{14} - 13500q^{15} - 43336q^{16} - 2908q^{17} + 10206q^{18} + 175096q^{20} - 128480q^{22} - 143416q^{23} - 29268q^{24} - 202626q^{25} + 424984q^{26} + 567520q^{28} - 250668q^{30} - 89468q^{31} - 893944q^{32} + 1109820q^{34} + 151632q^{36} - 823816q^{38} + 474552q^{39} - 860888q^{40} - 441284q^{41} + 427788q^{42} + 1275264q^{44} - 2167992q^{46} - 1056408q^{47} - 233280q^{48} + 2158134q^{49} + 324610q^{50} - 2059248q^{52} + 39366q^{54} + 4757504q^{55} + 1643704q^{56} + 1551096q^{57} - 5494676q^{58} - 3203712q^{60} + 5767172q^{62} - 1000188q^{63} + 3852224q^{64} - 2520464q^{65} - 3615840q^{66} - 3735840q^{68} + 12890312q^{70} + 5172696q^{71} + 312012q^{72} - 5446196q^{73} - 6468800q^{74} - 9084624q^{76} + 3542184q^{78} - 14373548q^{79} + 14369088q^{80} + 7440174q^{81} - 7935708q^{82} - 2775816q^{84} + 4738312q^{86} + 7902036q^{87} + 12598720q^{88} - 11952620q^{89} - 3659580q^{90} + 11004480q^{92} - 15440088q^{94} - 69327376q^{95} + 1341576q^{96} + 133732q^{97} + 53030538q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + 2867784 x^{7} - 20463993 x^{6} - 78987210 x^{5} + 94608296 x^{4} - 6477861300 x^{3} - 21982316987 x^{2} + 613270661346 x + 3813237677250$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1339739 \nu^{13} - 15980711 \nu^{12} + 21034557 \nu^{11} + 215590849 \nu^{10} - 3531636254 \nu^{9} - 5161573380 \nu^{8} + 605168763306 \nu^{7} - 672810221310 \nu^{6} - 22756879607105 \nu^{5} + 59351796070289 \nu^{4} - 494035412025687 \nu^{3} - 7312415504567539 \nu^{2} + 61932245079522164 \nu + 666600389770469706$$$$)/ 26141971282132992$$ $$\beta_{2}$$ $$=$$ $$($$$$24634993 \nu^{13} - 9286397141 \nu^{12} + 125608142679 \nu^{11} - 691247115421 \nu^{10} + 736524196326 \nu^{9} + 24523366318964 \nu^{8} + 22455365532734 \nu^{7} - 2314107202284730 \nu^{6} + 2712848667100957 \nu^{5} + 42592219961590483 \nu^{4} - 1005761082390441445 \nu^{3} + 4585027918947309255 \nu^{2} + 31792968567468819164 \nu + 85126426364778113310$$$$)/ 418271540514127872$$ $$\beta_{3}$$ $$=$$ $$($$$$-45486279 \nu^{13} + 613276227 \nu^{12} - 1623882033 \nu^{11} - 9211813317 \nu^{10} + 98592061590 \nu^{9} + 139036807188 \nu^{8} - 17264509936722 \nu^{7} + 26579270759670 \nu^{6} + 763640822074437 \nu^{5} - 2197506233280213 \nu^{4} + 10592431687392435 \nu^{3} + 162964039439097279 \nu^{2} - 943117822339232580 \nu - 18530919373189822578$$$$)/ 278847693676085248$$ $$\beta_{4}$$ $$=$$ $$($$$$-232216063 \nu^{13} + 10715039579 \nu^{12} - 173010009017 \nu^{11} + 628568622867 \nu^{10} + 2578457985862 \nu^{9} - 5129030302540 \nu^{8} + 29542541170462 \nu^{7} + 3023045629067942 \nu^{6} - 27365449133286547 \nu^{5} - 117817943307082365 \nu^{4} - 994118421822911157 \nu^{3} + 432044914313970359 \nu^{2} + 51017517211941195420 \nu + 396518755201684537854$$$$)/ 836543081028255744$$ $$\beta_{5}$$ $$=$$ $$($$$$179339189 \nu^{13} + 2135131735 \nu^{12} - 31681195725 \nu^{11} + 37724754799 \nu^{10} + 395974760062 \nu^{9} + 1223639902020 \nu^{8} - 149069377626570 \nu^{7} + 884410074883422 \nu^{6} - 1904717189989103 \nu^{5} - 21651971740573537 \nu^{4} - 133198211347413753 \nu^{3} + 2414183229687104099 \nu^{2} - 6774218898148935604 \nu - 137985402012981152106$$$$)/ 418271540514127872$$ $$\beta_{6}$$ $$=$$ $$($$$$-414718539 \nu^{13} + 4476417111 \nu^{12} - 7726634189 \nu^{11} - 119116174993 \nu^{10} + 1480427111038 \nu^{9} + 8474709208900 \nu^{8} - 256524977155530 \nu^{7} + 240002029089118 \nu^{6} + 9008436356946705 \nu^{5} - 35037032272036193 \nu^{4} + 71470424723166983 \nu^{3} + 3920519013043945571 \nu^{2} - 7400815112066275252 \nu - 280149875016460942698$$$$)/ 836543081028255744$$ $$\beta_{7}$$ $$=$$ $$($$$$629872933 \nu^{13} - 7758524121 \nu^{12} + 38268518211 \nu^{11} - 384199764289 \nu^{10} + 3351357700894 \nu^{9} - 28225688237564 \nu^{8} + 284497539819862 \nu^{7} + 402175881962494 \nu^{6} - 8673948209266239 \nu^{5} - 1202971975465745 \nu^{4} - 250633584302503721 \nu^{3} + 578796584671814003 \nu^{2} - 14077712511175804788 \nu + 399396707615402476470$$$$)/ 836543081028255744$$ $$\beta_{8}$$ $$=$$ $$($$$$39005411 \nu^{13} - 428467791 \nu^{12} + 1109091573 \nu^{11} - 13635772967 \nu^{10} + 76003424018 \nu^{9} - 1166217765988 \nu^{8} + 19764103168922 \nu^{7} + 10489055910194 \nu^{6} - 590538392866905 \nu^{5} + 489206687057705 \nu^{4} - 19131109292577919 \nu^{3} - 693327584589172667 \nu^{2} + 320806492279068756 \nu + 31445535164083592250$$$$)/ 32174733885702144$$ $$\beta_{9}$$ $$=$$ $$($$$$-340358553 \nu^{13} + 3989168541 \nu^{12} - 4434070383 \nu^{11} - 52878351195 \nu^{10} + 918519179562 \nu^{9} + 1347496326252 \nu^{8} - 157024093895406 \nu^{7} + 167190072977610 \nu^{6} + 5790340504771803 \nu^{5} - 14895811034963019 \nu^{4} + 131689766976005421 \nu^{3} + 1943011625876353953 \nu^{2} - 1835593302054547260 \nu - 179863489835255413454$$$$)/ 278847693676085248$$ $$\beta_{10}$$ $$=$$ $$($$$$19011601 \nu^{13} - 238222965 \nu^{12} + 740348087 \nu^{11} + 1084756419 \nu^{10} - 45973437786 \nu^{9} - 196481597964 \nu^{8} + 6667570095358 \nu^{7} + 10043933144070 \nu^{6} - 295169943090627 \nu^{5} + 542462260830771 \nu^{4} - 4111962796493317 \nu^{3} - 93325024487801433 \nu^{2} - 129206824240582820 \nu + 8192379896740010718$$$$)/ 13070985641066496$$ $$\beta_{11}$$ $$=$$ $$($$$$39027681 \nu^{13} - 536450053 \nu^{12} + 1433725287 \nu^{11} + 10887104691 \nu^{10} - 39614348666 \nu^{9} - 1873523380684 \nu^{8} + 19724141109086 \nu^{7} - 32875120933402 \nu^{6} - 725054774155315 \nu^{5} + 3607132367066403 \nu^{4} + 6129239630334443 \nu^{3} - 131140969204625449 \nu^{2} - 886179002041893092 \nu + 15351715124178563646$$$$)/ 16087366942851072$$ $$\beta_{12}$$ $$=$$ $$($$$$2625997723 \nu^{13} - 37720840423 \nu^{12} + 153136812541 \nu^{11} + 409146235777 \nu^{10} - 7746436103838 \nu^{9} - 33014387888900 \nu^{8} + 1030946208884010 \nu^{7} - 888988640501630 \nu^{6} - 61393181077847041 \nu^{5} + 465792858486010193 \nu^{4} - 254908486107767959 \nu^{3} - 15380325852683043891 \nu^{2} + 15096736805335112820 \nu + 1053565623064455856842$$$$)/ 836543081028255744$$ $$\beta_{13}$$ $$=$$ $$($$$$-4393185439 \nu^{13} + 53314528379 \nu^{12} - 1504144921 \nu^{11} - 1197574289101 \nu^{10} - 1027431345274 \nu^{9} + 154397635214388 \nu^{8} - 2278517731731362 \nu^{7} - 2746904737671450 \nu^{6} + 138949097303831885 \nu^{5} - 73581696139208285 \nu^{4} - 1702719437059510165 \nu^{3} + 21675806870389573207 \nu^{2} - 2887637449602953956 \nu - 3098515675542038978754$$$$)/ 836543081028255744$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{9} + \beta_{3} + 27 \beta_{1} + 23$$$$)/54$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{9} - 3 \beta_{8} + 3 \beta_{7} + 16 \beta_{3} + 54 \beta_{1} + 541$$$$)/54$$ $$\nu^{3}$$ $$=$$ $$($$$$-18 \beta_{13} + 3 \beta_{12} - 24 \beta_{11} - 36 \beta_{10} - 2 \beta_{9} - 12 \beta_{8} + 3 \beta_{7} + 36 \beta_{6} - 39 \beta_{4} + 205 \beta_{3} - 12 \beta_{2} + 1125 \beta_{1} + 1943$$$$)/108$$ $$\nu^{4}$$ $$=$$ $$($$$$30 \beta_{13} + 103 \beta_{12} - 14 \beta_{11} - 138 \beta_{10} - 104 \beta_{9} - 22 \beta_{8} + 145 \beta_{7} + 282 \beta_{6} - 54 \beta_{5} + 11 \beta_{4} + 401 \beta_{3} - 52 \beta_{2} + 1833 \beta_{1} + 28343$$$$)/36$$ $$\nu^{5}$$ $$=$$ $$($$$$702 \beta_{13} + 51 \beta_{12} + 195 \beta_{11} + 243 \beta_{10} + 216 \beta_{9} - 114 \beta_{8} + 2049 \beta_{7} + 3951 \beta_{6} - 1431 \beta_{5} + 48 \beta_{4} - 13963 \beta_{3} - 996 \beta_{2} + 37575 \beta_{1} + 615006$$$$)/54$$ $$\nu^{6}$$ $$=$$ $$($$$$2952 \beta_{13} + 4176 \beta_{12} - 5130 \beta_{11} + 21078 \beta_{10} + 15751 \beta_{9} - 4965 \beta_{8} + 17637 \beta_{7} + 15138 \beta_{6} - 12690 \beta_{5} + 5202 \beta_{4} - 25546 \beta_{3} - 6264 \beta_{2} + 131472 \beta_{1} - 1406845$$$$)/54$$ $$\nu^{7}$$ $$=$$ $$($$$$26082 \beta_{13} + 669 \beta_{12} - 98736 \beta_{11} - 59508 \beta_{10} - 142050 \beta_{9} - 122844 \beta_{8} + 347781 \beta_{7} - 448596 \beta_{6} - 187488 \beta_{5} + 20679 \beta_{4} - 1171129 \beta_{3} - 67908 \beta_{2} - 2900889 \beta_{1} - 122247195$$$$)/108$$ $$\nu^{8}$$ $$=$$ $$($$$$11706 \beta_{13} + 103325 \beta_{12} - 536842 \beta_{11} - 584718 \beta_{10} - 774740 \beta_{9} - 110510 \beta_{8} + 568103 \beta_{7} - 368442 \beta_{6} - 230058 \beta_{5} - 42431 \beta_{4} - 11907421 \beta_{3} - 165644 \beta_{2} - 26885853 \beta_{1} - 122156023$$$$)/36$$ $$\nu^{9}$$ $$=$$ $$($$$$1105452 \beta_{13} + 968922 \beta_{12} - 2879379 \beta_{11} - 11779551 \beta_{10} + 3108490 \beta_{9} + 2491686 \beta_{8} + 10485252 \beta_{7} - 3306663 \beta_{6} - 2664765 \beta_{5} + 1491723 \beta_{4} - 239975882 \beta_{3} - 2616840 \beta_{2} - 376847100 \beta_{1} - 930607885$$$$)/54$$ $$\nu^{10}$$ $$=$$ $$($$$$18378774 \beta_{13} - 7602453 \beta_{12} - 3286872 \beta_{11} - 16841340 \beta_{10} + 117660149 \beta_{9} + 2829867 \beta_{8} + 31068372 \beta_{7} - 193933368 \beta_{6} - 23323356 \beta_{5} + 27920637 \beta_{4} - 2433412241 \beta_{3} - 19043748 \beta_{2} - 3991740795 \beta_{1} + 7709620384$$$$)/54$$ $$\nu^{11}$$ $$=$$ $$($$$$104200830 \beta_{13} - 238923429 \beta_{12} - 432406644 \beta_{11} + 372886416 \beta_{10} + 3131694970 \beta_{9} - 56223516 \beta_{8} + 569598699 \beta_{7} - 7655061384 \beta_{6} + 279561564 \beta_{5} + 33364869 \beta_{4} - 29226710111 \beta_{3} + 119201172 \beta_{2} - 66626154855 \beta_{1} - 86001757345$$$$)/108$$ $$\nu^{12}$$ $$=$$ $$($$$$107793438 \beta_{13} - 712698473 \beta_{12} - 1777191854 \beta_{11} - 2574032586 \beta_{10} + 5355211600 \beta_{9} - 499965694 \beta_{8} + 2440982377 \beta_{7} - 28625019414 \beta_{6} + 3216865626 \beta_{5} - 687925045 \beta_{4} - 54326213391 \beta_{3} + 263892716 \beta_{2} - 165373754727 \beta_{1} - 262461379393$$$$)/36$$ $$\nu^{13}$$ $$=$$ $$($$$$3338497530 \beta_{13} - 9980609295 \beta_{12} - 19026211413 \beta_{11} - 50017662477 \beta_{10} + 54534300432 \beta_{9} + 3160073334 \beta_{8} + 37532836959 \beta_{7} - 267783284745 \beta_{6} + 66015083673 \beta_{5} - 22540793478 \beta_{4} - 985655509789 \beta_{3} - 4488661068 \beta_{2} - 1396944410787 \beta_{1} + 2425490692320$$$$)/54$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$13$$ $$17$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −4.99225 + 5.30027i −4.99225 − 5.30027i −6.99438 − 0.299706i −6.99438 + 0.299706i 2.71713 − 7.81354i 2.71713 + 7.81354i −5.80663 − 4.20354i −5.80663 + 4.20354i 7.97707 − 3.91414i 7.97707 + 3.91414i 1.24645 − 7.99620i 1.24645 + 7.99620i 8.85262 + 1.52851i 8.85262 − 1.52851i
−11.2925 0.691979i 27.0000i 127.042 + 15.6284i 124.215i −18.6834 + 304.898i 646.373 −1423.81 264.394i −729.000 85.9543 1402.70i
13.2 −11.2925 + 0.691979i 27.0000i 127.042 15.6284i 124.215i −18.6834 304.898i 646.373 −1423.81 + 264.394i −729.000 85.9543 + 1402.70i
13.3 −7.69468 8.29409i 27.0000i −9.58387 + 127.641i 455.347i −223.940 + 207.756i −743.502 1132.41 902.665i −729.000 −3776.69 + 3503.75i
13.4 −7.69468 + 8.29409i 27.0000i −9.58387 127.641i 455.347i −223.940 207.756i −743.502 1132.41 + 902.665i −729.000 −3776.69 3503.75i
13.5 −6.09641 9.53067i 27.0000i −53.6675 + 116.206i 137.155i 257.328 164.603i 808.153 1434.70 196.951i −729.000 −1307.18 + 836.155i
13.6 −6.09641 + 9.53067i 27.0000i −53.6675 116.206i 137.155i 257.328 + 164.603i 808.153 1434.70 + 196.951i −729.000 −1307.18 836.155i
13.7 −2.60309 11.0102i 27.0000i −114.448 + 57.3210i 468.400i −297.275 + 70.2836i 81.2421 929.033 + 1110.88i −729.000 5157.17 1219.29i
13.8 −2.60309 + 11.0102i 27.0000i −114.448 57.3210i 468.400i −297.275 70.2836i 81.2421 929.033 1110.88i −729.000 5157.17 + 1219.29i
13.9 3.06293 10.8912i 27.0000i −109.237 66.7181i 23.0228i 294.063 + 82.6992i −1547.56 −1061.23 + 985.368i −729.000 250.747 + 70.5175i
13.10 3.06293 + 10.8912i 27.0000i −109.237 + 66.7181i 23.0228i 294.063 82.6992i −1547.56 −1061.23 985.368i −729.000 250.747 70.5175i
13.11 8.24265 7.74976i 27.0000i 7.88255 127.757i 76.0929i −209.243 222.552i −222.735 −925.113 1114.14i −729.000 −589.701 627.207i
13.12 8.24265 + 7.74976i 27.0000i 7.88255 + 127.757i 76.0929i −209.243 + 222.552i −222.735 −925.113 + 1114.14i −729.000 −589.701 + 627.207i
13.13 9.38113 6.32411i 27.0000i 48.0111 118.655i 425.308i 170.751 + 253.290i 1664.03 −299.987 1416.74i −729.000 2689.70 + 3989.87i
13.14 9.38113 + 6.32411i 27.0000i 48.0111 + 118.655i 425.308i 170.751 253.290i 1664.03 −299.987 + 1416.74i −729.000 2689.70 3989.87i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.8.d.a 14
3.b odd 2 1 72.8.d.d 14
4.b odd 2 1 96.8.d.a 14
8.b even 2 1 inner 24.8.d.a 14
8.d odd 2 1 96.8.d.a 14
12.b even 2 1 288.8.d.d 14
24.f even 2 1 288.8.d.d 14
24.h odd 2 1 72.8.d.d 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.d.a 14 1.a even 1 1 trivial
24.8.d.a 14 8.b even 2 1 inner
72.8.d.d 14 3.b odd 2 1
72.8.d.d 14 24.h odd 2 1
96.8.d.a 14 4.b odd 2 1
96.8.d.a 14 8.d odd 2 1
288.8.d.d 14 12.b even 2 1
288.8.d.d 14 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(24, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 14 T + 202 T^{2} + 2056 T^{3} + 30032 T^{4} + 487040 T^{5} + 5227008 T^{6} + 74227712 T^{7} + 669057024 T^{8} + 7979663360 T^{9} + 62981668864 T^{10} + 551903297536 T^{11} + 6940667150336 T^{12} + 61572651155456 T^{13} + 562949953421312 T^{14}$$
$3$ $$( 1 + 729 T^{2} )^{7}$$
$5$ $$1 - 445562 T^{2} + 95581517971 T^{4} - 14439744600730244 T^{6} +$$$$18\!\cdots\!25$$$$T^{8} -$$$$20\!\cdots\!50$$$$T^{10} +$$$$19\!\cdots\!75$$$$T^{12} -$$$$15\!\cdots\!00$$$$T^{14} +$$$$11\!\cdots\!75$$$$T^{16} -$$$$76\!\cdots\!50$$$$T^{18} +$$$$41\!\cdots\!25$$$$T^{20} -$$$$20\!\cdots\!00$$$$T^{22} +$$$$80\!\cdots\!75$$$$T^{24} -$$$$23\!\cdots\!50$$$$T^{26} +$$$$31\!\cdots\!25$$$$T^{28}$$
$7$ $$( 1 - 686 T + 2578165 T^{2} - 1521968292 T^{3} + 2893978871217 T^{4} - 1670463194093586 T^{5} + 2150164822127173261 T^{6} -$$$$14\!\cdots\!28$$$$T^{7} +$$$$17\!\cdots\!23$$$$T^{8} -$$$$11\!\cdots\!14$$$$T^{9} +$$$$16\!\cdots\!19$$$$T^{10} -$$$$70\!\cdots\!92$$$$T^{11} +$$$$97\!\cdots\!95$$$$T^{12} -$$$$21\!\cdots\!14$$$$T^{13} +$$$$25\!\cdots\!07$$$$T^{14} )^{2}$$
$11$ $$1 - 126612650 T^{2} + 7858670884605619 T^{4} -$$$$31\!\cdots\!48$$$$T^{6} +$$$$91\!\cdots\!17$$$$T^{8} -$$$$20\!\cdots\!90$$$$T^{10} +$$$$40\!\cdots\!27$$$$T^{12} -$$$$76\!\cdots\!72$$$$T^{14} +$$$$15\!\cdots\!07$$$$T^{16} -$$$$29\!\cdots\!90$$$$T^{18} +$$$$50\!\cdots\!57$$$$T^{20} -$$$$65\!\cdots\!28$$$$T^{22} +$$$$62\!\cdots\!19$$$$T^{24} -$$$$37\!\cdots\!50$$$$T^{26} +$$$$11\!\cdots\!81$$$$T^{28}$$
$13$ $$1 - 387011990 T^{2} + 68908126519682275 T^{4} -$$$$76\!\cdots\!28$$$$T^{6} +$$$$63\!\cdots\!97$$$$T^{8} -$$$$48\!\cdots\!86$$$$T^{10} +$$$$37\!\cdots\!83$$$$T^{12} -$$$$25\!\cdots\!72$$$$T^{14} +$$$$14\!\cdots\!87$$$$T^{16} -$$$$75\!\cdots\!06$$$$T^{18} +$$$$38\!\cdots\!93$$$$T^{20} -$$$$18\!\cdots\!48$$$$T^{22} +$$$$65\!\cdots\!75$$$$T^{24} -$$$$14\!\cdots\!90$$$$T^{26} +$$$$14\!\cdots\!29$$$$T^{28}$$
$17$ $$( 1 + 1454 T + 1716640747 T^{2} + 5874996114796 T^{3} + 1576054531620987017 T^{4} +$$$$56\!\cdots\!70$$$$T^{5} +$$$$93\!\cdots\!91$$$$T^{6} +$$$$30\!\cdots\!24$$$$T^{7} +$$$$38\!\cdots\!43$$$$T^{8} +$$$$95\!\cdots\!30$$$$T^{9} +$$$$10\!\cdots\!89$$$$T^{10} +$$$$16\!\cdots\!36$$$$T^{11} +$$$$19\!\cdots\!71$$$$T^{12} +$$$$69\!\cdots\!06$$$$T^{13} +$$$$19\!\cdots\!97$$$$T^{14} )^{2}$$
$19$ $$1 - 6974011850 T^{2} + 24092658824990296867 T^{4} -$$$$55\!\cdots\!68$$$$T^{6} +$$$$96\!\cdots\!09$$$$T^{8} -$$$$13\!\cdots\!10$$$$T^{10} +$$$$15\!\cdots\!79$$$$T^{12} -$$$$15\!\cdots\!56$$$$T^{14} +$$$$12\!\cdots\!59$$$$T^{16} -$$$$85\!\cdots\!10$$$$T^{18} +$$$$49\!\cdots\!49$$$$T^{20} -$$$$22\!\cdots\!08$$$$T^{22} +$$$$78\!\cdots\!67$$$$T^{24} -$$$$18\!\cdots\!50$$$$T^{26} +$$$$20\!\cdots\!41$$$$T^{28}$$
$23$ $$( 1 + 71708 T + 8440179361 T^{2} + 617221190675032 T^{3} + 59565824725230322613 T^{4} +$$$$38\!\cdots\!00$$$$T^{5} +$$$$26\!\cdots\!93$$$$T^{6} +$$$$15\!\cdots\!24$$$$T^{7} +$$$$90\!\cdots\!71$$$$T^{8} +$$$$44\!\cdots\!00$$$$T^{9} +$$$$23\!\cdots\!99$$$$T^{10} +$$$$82\!\cdots\!92$$$$T^{11} +$$$$38\!\cdots\!27$$$$T^{12} +$$$$11\!\cdots\!32$$$$T^{13} +$$$$53\!\cdots\!63$$$$T^{14} )^{2}$$
$29$ $$1 - 131972594954 T^{2} +$$$$90\!\cdots\!15$$$$T^{4} -$$$$42\!\cdots\!72$$$$T^{6} +$$$$15\!\cdots\!49$$$$T^{8} -$$$$42\!\cdots\!06$$$$T^{10} +$$$$96\!\cdots\!91$$$$T^{12} -$$$$18\!\cdots\!28$$$$T^{14} +$$$$28\!\cdots\!71$$$$T^{16} -$$$$37\!\cdots\!66$$$$T^{18} +$$$$39\!\cdots\!09$$$$T^{20} -$$$$33\!\cdots\!12$$$$T^{22} +$$$$21\!\cdots\!15$$$$T^{24} -$$$$91\!\cdots\!74$$$$T^{26} +$$$$20\!\cdots\!61$$$$T^{28}$$
$31$ $$( 1 + 44734 T + 103891511821 T^{2} + 1878217501397540 T^{3} +$$$$54\!\cdots\!97$$$$T^{4} -$$$$16\!\cdots\!38$$$$T^{5} +$$$$19\!\cdots\!65$$$$T^{6} -$$$$22\!\cdots\!20$$$$T^{7} +$$$$53\!\cdots\!15$$$$T^{8} -$$$$12\!\cdots\!98$$$$T^{9} +$$$$11\!\cdots\!07$$$$T^{10} +$$$$10\!\cdots\!40$$$$T^{11} +$$$$16\!\cdots\!71$$$$T^{12} +$$$$19\!\cdots\!74$$$$T^{13} +$$$$11\!\cdots\!71$$$$T^{14} )^{2}$$
$37$ $$1 - 856038434294 T^{2} +$$$$35\!\cdots\!75$$$$T^{4} -$$$$98\!\cdots\!16$$$$T^{6} +$$$$19\!\cdots\!85$$$$T^{8} -$$$$31\!\cdots\!86$$$$T^{10} +$$$$39\!\cdots\!27$$$$T^{12} -$$$$41\!\cdots\!48$$$$T^{14} +$$$$35\!\cdots\!03$$$$T^{16} -$$$$25\!\cdots\!06$$$$T^{18} +$$$$14\!\cdots\!65$$$$T^{20} -$$$$64\!\cdots\!56$$$$T^{22} +$$$$21\!\cdots\!75$$$$T^{24} -$$$$45\!\cdots\!34$$$$T^{26} +$$$$48\!\cdots\!29$$$$T^{28}$$
$41$ $$( 1 + 220642 T + 805010558083 T^{2} + 256070152586388116 T^{3} +$$$$34\!\cdots\!41$$$$T^{4} +$$$$10\!\cdots\!18$$$$T^{5} +$$$$99\!\cdots\!91$$$$T^{6} +$$$$26\!\cdots\!36$$$$T^{7} +$$$$19\!\cdots\!71$$$$T^{8} +$$$$41\!\cdots\!98$$$$T^{9} +$$$$25\!\cdots\!81$$$$T^{10} +$$$$36\!\cdots\!36$$$$T^{11} +$$$$22\!\cdots\!83$$$$T^{12} +$$$$12\!\cdots\!02$$$$T^{13} +$$$$10\!\cdots\!61$$$$T^{14} )^{2}$$
$43$ $$1 - 1167136384250 T^{2} +$$$$84\!\cdots\!19$$$$T^{4} -$$$$44\!\cdots\!04$$$$T^{6} +$$$$19\!\cdots\!85$$$$T^{8} -$$$$71\!\cdots\!86$$$$T^{10} +$$$$23\!\cdots\!47$$$$T^{12} -$$$$67\!\cdots\!20$$$$T^{14} +$$$$17\!\cdots\!03$$$$T^{16} -$$$$39\!\cdots\!86$$$$T^{18} +$$$$77\!\cdots\!65$$$$T^{20} -$$$$13\!\cdots\!04$$$$T^{22} +$$$$18\!\cdots\!31$$$$T^{24} -$$$$18\!\cdots\!50$$$$T^{26} +$$$$12\!\cdots\!49$$$$T^{28}$$
$47$ $$( 1 + 528204 T + 1304181982841 T^{2} + 1396775026647992952 T^{3} +$$$$11\!\cdots\!01$$$$T^{4} +$$$$11\!\cdots\!48$$$$T^{5} +$$$$10\!\cdots\!97$$$$T^{6} +$$$$58\!\cdots\!04$$$$T^{7} +$$$$51\!\cdots\!11$$$$T^{8} +$$$$29\!\cdots\!12$$$$T^{9} +$$$$15\!\cdots\!47$$$$T^{10} +$$$$92\!\cdots\!72$$$$T^{11} +$$$$43\!\cdots\!63$$$$T^{12} +$$$$89\!\cdots\!36$$$$T^{13} +$$$$85\!\cdots\!67$$$$T^{14} )^{2}$$
$53$ $$1 - 6871654175354 T^{2} +$$$$23\!\cdots\!55$$$$T^{4} -$$$$56\!\cdots\!16$$$$T^{6} +$$$$10\!\cdots\!25$$$$T^{8} -$$$$16\!\cdots\!86$$$$T^{10} +$$$$23\!\cdots\!07$$$$T^{12} -$$$$30\!\cdots\!88$$$$T^{14} +$$$$33\!\cdots\!83$$$$T^{16} -$$$$32\!\cdots\!46$$$$T^{18} +$$$$27\!\cdots\!25$$$$T^{20} -$$$$20\!\cdots\!36$$$$T^{22} +$$$$11\!\cdots\!95$$$$T^{24} -$$$$47\!\cdots\!74$$$$T^{26} +$$$$95\!\cdots\!89$$$$T^{28}$$
$59$ $$1 - 9783224696858 T^{2} +$$$$58\!\cdots\!71$$$$T^{4} -$$$$24\!\cdots\!08$$$$T^{6} +$$$$78\!\cdots\!05$$$$T^{8} -$$$$20\!\cdots\!62$$$$T^{10} +$$$$47\!\cdots\!59$$$$T^{12} -$$$$11\!\cdots\!36$$$$T^{14} +$$$$29\!\cdots\!99$$$$T^{16} -$$$$78\!\cdots\!02$$$$T^{18} +$$$$18\!\cdots\!05$$$$T^{20} -$$$$35\!\cdots\!28$$$$T^{22} +$$$$53\!\cdots\!71$$$$T^{24} -$$$$55\!\cdots\!38$$$$T^{26} +$$$$34\!\cdots\!21$$$$T^{28}$$
$61$ $$1 - 26713863605222 T^{2} +$$$$33\!\cdots\!19$$$$T^{4} -$$$$26\!\cdots\!04$$$$T^{6} +$$$$14\!\cdots\!33$$$$T^{8} -$$$$67\!\cdots\!58$$$$T^{10} +$$$$25\!\cdots\!03$$$$T^{12} -$$$$83\!\cdots\!44$$$$T^{14} +$$$$24\!\cdots\!23$$$$T^{16} -$$$$65\!\cdots\!98$$$$T^{18} +$$$$14\!\cdots\!93$$$$T^{20} -$$$$25\!\cdots\!44$$$$T^{22} +$$$$31\!\cdots\!19$$$$T^{24} -$$$$24\!\cdots\!02$$$$T^{26} +$$$$91\!\cdots\!81$$$$T^{28}$$
$67$ $$1 - 42926594518442 T^{2} +$$$$97\!\cdots\!99$$$$T^{4} -$$$$14\!\cdots\!60$$$$T^{6} +$$$$17\!\cdots\!33$$$$T^{8} -$$$$16\!\cdots\!82$$$$T^{10} +$$$$12\!\cdots\!95$$$$T^{12} -$$$$81\!\cdots\!52$$$$T^{14} +$$$$45\!\cdots\!55$$$$T^{16} -$$$$21\!\cdots\!62$$$$T^{18} +$$$$86\!\cdots\!37$$$$T^{20} -$$$$27\!\cdots\!60$$$$T^{22} +$$$$65\!\cdots\!51$$$$T^{24} -$$$$10\!\cdots\!82$$$$T^{26} +$$$$90\!\cdots\!09$$$$T^{28}$$
$71$ $$( 1 - 2586348 T + 45775649603153 T^{2} -$$$$10\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4} -$$$$21\!\cdots\!72$$$$T^{5} +$$$$14\!\cdots\!09$$$$T^{6} -$$$$24\!\cdots\!88$$$$T^{7} +$$$$13\!\cdots\!19$$$$T^{8} -$$$$17\!\cdots\!32$$$$T^{9} +$$$$78\!\cdots\!91$$$$T^{10} -$$$$75\!\cdots\!00$$$$T^{11} +$$$$28\!\cdots\!03$$$$T^{12} -$$$$14\!\cdots\!68$$$$T^{13} +$$$$51\!\cdots\!31$$$$T^{14} )^{2}$$
$73$ $$( 1 + 2723098 T + 38312704682563 T^{2} + 83515159987399400068 T^{3} +$$$$83\!\cdots\!09$$$$T^{4} +$$$$14\!\cdots\!06$$$$T^{5} +$$$$12\!\cdots\!59$$$$T^{6} +$$$$19\!\cdots\!00$$$$T^{7} +$$$$13\!\cdots\!23$$$$T^{8} +$$$$17\!\cdots\!54$$$$T^{9} +$$$$11\!\cdots\!57$$$$T^{10} +$$$$12\!\cdots\!08$$$$T^{11} +$$$$63\!\cdots\!91$$$$T^{12} +$$$$49\!\cdots\!42$$$$T^{13} +$$$$20\!\cdots\!13$$$$T^{14} )^{2}$$
$79$ $$( 1 + 7186774 T + 105650603277469 T^{2} +$$$$49\!\cdots\!68$$$$T^{3} +$$$$43\!\cdots\!13$$$$T^{4} +$$$$14\!\cdots\!10$$$$T^{5} +$$$$10\!\cdots\!25$$$$T^{6} +$$$$30\!\cdots\!36$$$$T^{7} +$$$$20\!\cdots\!75$$$$T^{8} +$$$$55\!\cdots\!10$$$$T^{9} +$$$$30\!\cdots\!27$$$$T^{10} +$$$$67\!\cdots\!48$$$$T^{11} +$$$$27\!\cdots\!31$$$$T^{12} +$$$$36\!\cdots\!34$$$$T^{13} +$$$$96\!\cdots\!19$$$$T^{14} )^{2}$$
$83$ $$1 - 148527772994618 T^{2} +$$$$11\!\cdots\!55$$$$T^{4} -$$$$65\!\cdots\!68$$$$T^{6} +$$$$28\!\cdots\!13$$$$T^{8} -$$$$10\!\cdots\!42$$$$T^{10} +$$$$33\!\cdots\!55$$$$T^{12} -$$$$95\!\cdots\!24$$$$T^{14} +$$$$24\!\cdots\!95$$$$T^{16} -$$$$56\!\cdots\!22$$$$T^{18} +$$$$11\!\cdots\!57$$$$T^{20} -$$$$19\!\cdots\!08$$$$T^{22} +$$$$25\!\cdots\!95$$$$T^{24} -$$$$23\!\cdots\!78$$$$T^{26} +$$$$11\!\cdots\!09$$$$T^{28}$$
$89$ $$( 1 + 5976310 T + 88567149413395 T^{2} +$$$$34\!\cdots\!52$$$$T^{3} +$$$$24\!\cdots\!97$$$$T^{4} -$$$$58\!\cdots\!66$$$$T^{5} -$$$$32\!\cdots\!17$$$$T^{6} -$$$$90\!\cdots\!92$$$$T^{7} -$$$$14\!\cdots\!93$$$$T^{8} -$$$$11\!\cdots\!06$$$$T^{9} +$$$$20\!\cdots\!33$$$$T^{10} +$$$$13\!\cdots\!12$$$$T^{11} +$$$$14\!\cdots\!55$$$$T^{12} +$$$$44\!\cdots\!10$$$$T^{13} +$$$$33\!\cdots\!09$$$$T^{14} )^{2}$$
$97$ $$( 1 - 66866 T + 239738062759867 T^{2} - 35467286602626108564 T^{3} +$$$$36\!\cdots\!25$$$$T^{4} +$$$$10\!\cdots\!02$$$$T^{5} +$$$$38\!\cdots\!03$$$$T^{6} +$$$$18\!\cdots\!76$$$$T^{7} +$$$$30\!\cdots\!39$$$$T^{8} +$$$$67\!\cdots\!38$$$$T^{9} +$$$$19\!\cdots\!25$$$$T^{10} -$$$$15\!\cdots\!04$$$$T^{11} +$$$$82\!\cdots\!31$$$$T^{12} -$$$$18\!\cdots\!94$$$$T^{13} +$$$$22\!\cdots\!17$$$$T^{14} )^{2}$$