Properties

Label 14.13.d.a
Level $14$
Weight $13$
Character orbit 14.d
Analytic conductor $12.796$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 14.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7959134419\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 527434 x^{14} - 31307480 x^{13} + 193554267483 x^{12} - 12267558721140 x^{11} + 36740634631350658 x^{10} - 2684653740993140180 x^{9} + 5051027688394458110177 x^{8} - 329042050982173184619740 x^{7} + 328571000441888143595366884 x^{6} - 20799163177686535050419379920 x^{5} + 15169931128056380497155715556632 x^{4} - 685746033739471361446313307383040 x^{3} + 159607415711477686151635843287775328 x^{2} + 6015487435743293961505549572273215360 x + 735121207672317095821497912841120438336\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{8}\cdot 7^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{4} ) q^{2} + ( 5 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{3} + ( -2048 + 2048 \beta_{1} ) q^{4} + ( 1512 - 756 \beta_{1} + 45 \beta_{2} - 3 \beta_{3} - 45 \beta_{4} - 3 \beta_{5} - \beta_{7} - \beta_{8} ) q^{5} + ( 4928 - 9856 \beta_{1} - \beta_{3} - \beta_{7} - \beta_{9} ) q^{6} + ( -32005 + 5295 \beta_{1} - 181 \beta_{2} - 40 \beta_{3} - 418 \beta_{4} - 18 \beta_{5} + \beta_{11} + 3 \beta_{13} ) q^{7} + 2048 \beta_{2} q^{8} + ( 295281 \beta_{1} - 299 \beta_{2} - 75 \beta_{3} - 292 \beta_{4} + 76 \beta_{5} - 5 \beta_{6} + 12 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 7 \beta_{10} - 10 \beta_{11} - 3 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{4} ) q^{2} + ( 5 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{3} + ( -2048 + 2048 \beta_{1} ) q^{4} + ( 1512 - 756 \beta_{1} + 45 \beta_{2} - 3 \beta_{3} - 45 \beta_{4} - 3 \beta_{5} - \beta_{7} - \beta_{8} ) q^{5} + ( 4928 - 9856 \beta_{1} - \beta_{3} - \beta_{7} - \beta_{9} ) q^{6} + ( -32005 + 5295 \beta_{1} - 181 \beta_{2} - 40 \beta_{3} - 418 \beta_{4} - 18 \beta_{5} + \beta_{11} + 3 \beta_{13} ) q^{7} + 2048 \beta_{2} q^{8} + ( 295281 \beta_{1} - 299 \beta_{2} - 75 \beta_{3} - 292 \beta_{4} + 76 \beta_{5} - 5 \beta_{6} + 12 \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - 7 \beta_{10} - 10 \beta_{11} - 3 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} ) q^{9} + ( -95424 - 95424 \beta_{1} - 1586 \beta_{2} - 623 \beta_{4} - 325 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 24 \beta_{8} - 4 \beta_{9} - 11 \beta_{10} - 8 \beta_{11} - 10 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{10} + ( 259011 - 259011 \beta_{1} + 328 \beta_{2} + 666 \beta_{3} + 1674 \beta_{4} + 326 \beta_{5} - 9 \beta_{6} + 81 \beta_{7} + 40 \beta_{8} + 11 \beta_{9} - \beta_{10} + \beta_{11} - 4 \beta_{12} + 11 \beta_{13} - 8 \beta_{15} ) q^{11} + ( -6144 \beta_{2} - 2048 \beta_{3} + 6144 \beta_{4} - 2048 \beta_{5} ) q^{12} + ( 168245 - 336490 \beta_{1} - 28710 \beta_{2} + 1096 \beta_{3} - 58536 \beta_{4} + 25 \beta_{5} - 5 \beta_{6} + 110 \beta_{7} + 30 \beta_{8} - 55 \beta_{9} - 5 \beta_{10} - 40 \beta_{11} - 30 \beta_{12} - 50 \beta_{13} - 10 \beta_{14} + 10 \beta_{15} ) q^{13} + ( -884736 + 331776 \beta_{1} + 31401 \beta_{2} - 842 \beta_{3} + 27376 \beta_{4} - 1292 \beta_{5} + 13 \beta_{6} - 102 \beta_{7} - 25 \beta_{8} - 8 \beta_{9} - 48 \beta_{10} - 10 \beta_{11} + 28 \beta_{12} - 2 \beta_{13} - 15 \beta_{14} - 17 \beta_{15} ) q^{14} + ( -1697733 + 133388 \beta_{2} + 2595 \beta_{3} - 2747 \beta_{4} + 5341 \beta_{5} + 108 \beta_{6} - 252 \beta_{7} - 331 \beta_{8} + \beta_{9} - 68 \beta_{10} - 177 \beta_{11} + 43 \beta_{12} + 44 \beta_{13} - 24 \beta_{14} - 24 \beta_{15} ) q^{15} -4194304 \beta_{1} q^{16} + ( -4231710 - 4231710 \beta_{1} - 115973 \beta_{2} - 21 \beta_{3} - 56544 \beta_{4} - 2671 \beta_{5} - 82 \beta_{6} + 82 \beta_{7} - 116 \beta_{8} - 3 \beta_{9} - 232 \beta_{10} - 230 \beta_{11} + 21 \beta_{12} - 88 \beta_{13} - 100 \beta_{14} - 50 \beta_{15} ) q^{17} + ( -574080 + 574080 \beta_{1} + 6794 \beta_{2} + 13834 \beta_{3} - 300782 \beta_{4} + 7093 \beta_{5} + 100 \beta_{6} - 60 \beta_{7} + 172 \beta_{8} - 158 \beta_{9} - 105 \beta_{10} - 404 \beta_{11} - 152 \beta_{12} - 230 \beta_{13} + 35 \beta_{15} ) q^{18} + ( 14577654 - 7288827 \beta_{1} - 12568 \beta_{2} - 25986 \beta_{3} + 12688 \beta_{4} - 26056 \beta_{5} - 70 \beta_{6} + 622 \beta_{7} + 629 \beta_{8} - 169 \beta_{9} - 358 \beta_{10} - 308 \beta_{11} - 77 \beta_{12} - 259 \beta_{13} - 112 \beta_{14} - 224 \beta_{15} ) q^{19} + ( -1548288 + 3096576 \beta_{1} + 98304 \beta_{2} + 6144 \beta_{3} + 190464 \beta_{4} + 2048 \beta_{7} ) q^{20} + ( -27642606 + 14818791 \beta_{1} + 362 \beta_{2} - 15733 \beta_{3} - 339858 \beta_{4} - 78451 \beta_{5} - 45 \beta_{6} + 4174 \beta_{7} + 2319 \beta_{8} - 542 \beta_{9} - 879 \beta_{10} - 918 \beta_{11} + 77 \beta_{12} - 906 \beta_{13} + 102 \beta_{14} + 208 \beta_{15} ) q^{21} + ( 3923520 - 246583 \beta_{2} + 16244 \beta_{3} - 16191 \beta_{4} + 31445 \beta_{5} - 559 \beta_{6} + 724 \beta_{7} + 814 \beta_{8} + 990 \beta_{9} + 701 \beta_{10} + 582 \beta_{11} - 484 \beta_{12} + 818 \beta_{13} - 435 \beta_{14} - 435 \beta_{15} ) q^{22} + ( 3203820 \beta_{1} + 429065 \beta_{2} - 36853 \beta_{3} + 430800 \beta_{4} + 36604 \beta_{5} - 1055 \beta_{6} - 165 \beta_{7} + 1623 \beta_{8} + 68 \beta_{9} - 1735 \beta_{10} - 3334 \beta_{11} - 403 \beta_{12} - 1418 \beta_{13} - 40 \beta_{14} ) q^{23} + ( 10092544 + 10092544 \beta_{1} + 2048 \beta_{4} - 2048 \beta_{5} - 2048 \beta_{8} - 2048 \beta_{10} ) q^{24} + ( 65150726 - 65150726 \beta_{1} + 62988 \beta_{2} + 123634 \beta_{3} - 3078764 \beta_{4} + 61332 \beta_{5} - 628 \beta_{6} - 8516 \beta_{7} - 4286 \beta_{8} - 2286 \beta_{9} - 1600 \beta_{10} + 56 \beta_{11} - 286 \beta_{12} + 1092 \beta_{13} - 396 \beta_{15} ) q^{25} + ( -119556864 + 59778432 \beta_{1} - 205385 \beta_{2} - 60389 \beta_{3} + 199589 \beta_{4} - 58854 \beta_{5} + 1535 \beta_{6} - 4099 \beta_{7} - 1984 \beta_{8} + 2561 \beta_{9} + 2396 \beta_{10} - 1410 \beta_{11} - 580 \beta_{12} + 2280 \beta_{13} + 415 \beta_{14} + 830 \beta_{15} ) q^{26} + ( -57968505 + 115937010 \beta_{1} + 2056772 \beta_{2} - 221402 \beta_{3} + 4323552 \beta_{4} - 1898 \beta_{5} + 1599 \beta_{6} + 81 \beta_{7} - 3497 \beta_{8} - 6298 \beta_{9} + 1599 \beta_{10} + 6695 \beta_{11} + 3497 \beta_{12} + 3796 \beta_{13} - 400 \beta_{14} + 400 \beta_{15} ) q^{27} + ( 54702080 - 65546240 \beta_{1} + 892928 \beta_{2} + 36864 \beta_{3} + 567296 \beta_{4} - 45056 \beta_{5} + 4096 \beta_{11} - 2048 \beta_{13} ) q^{28} + ( 33272559 - 1715156 \beta_{2} + 117488 \beta_{3} - 126078 \beta_{4} + 234191 \beta_{5} - 245 \beta_{6} - 5232 \beta_{7} - 10414 \beta_{8} + 9375 \beta_{9} + 18951 \beta_{10} + 2656 \beta_{11} - 540 \beta_{12} + 1670 \beta_{13} + 518 \beta_{14} + 518 \beta_{15} ) q^{29} + ( -268989696 \beta_{1} + 1719943 \beta_{2} + 110522 \beta_{3} + 1714666 \beta_{4} - 107021 \beta_{5} + 3765 \beta_{6} - 16972 \beta_{7} + 13075 \beta_{8} + 220 \beta_{9} + 5277 \beta_{10} + 10994 \beta_{11} + 132 \beta_{12} + 1996 \beta_{13} + 1901 \beta_{14} ) q^{30} + ( 190992431 + 190992431 \beta_{1} - 12057128 \beta_{2} - 1211 \beta_{3} - 6039822 \beta_{4} + 38138 \beta_{5} + 1957 \beta_{6} - 1957 \beta_{7} - 2759 \beta_{8} - 2806 \beta_{9} - 14027 \beta_{10} - 2525 \beta_{11} + 1211 \beta_{12} - 3655 \beta_{13} + 3552 \beta_{14} + 1776 \beta_{15} ) q^{31} + 4194304 \beta_{4} q^{32} + ( 504579810 - 252289905 \beta_{1} - 5112920 \beta_{2} + 598176 \beta_{3} + 5123952 \beta_{4} + 591200 \beta_{5} - 6976 \beta_{6} + 15976 \beta_{7} + 4788 \beta_{8} + 7212 \beta_{9} + 11504 \beta_{10} + 14272 \beta_{11} + 4212 \beta_{12} - 5260 \beta_{13} + 80 \beta_{14} + 160 \beta_{15} ) q^{33} + ( -118164480 + 236328960 \beta_{1} + 4339548 \beta_{2} + 268025 \beta_{3} + 8414703 \beta_{4} + 6079 \beta_{5} + 2595 \beta_{6} - 3819 \beta_{7} + 3484 \beta_{8} + 2743 \beta_{9} + 2595 \beta_{10} + 1706 \beta_{11} - 3484 \beta_{12} - 12158 \beta_{13} - 417 \beta_{14} + 417 \beta_{15} ) q^{34} + ( -226155699 + 250743843 \beta_{1} + 14618982 \beta_{2} - 519953 \beta_{3} + 6968517 \beta_{4} + 721620 \beta_{5} - 45 \beta_{6} + 21849 \beta_{7} + 26980 \beta_{8} + 41395 \beta_{9} + 10475 \beta_{10} + 3375 \beta_{11} + 6160 \beta_{12} + 7605 \beta_{13} + 1880 \beta_{14} + 2000 \beta_{15} ) q^{35} + ( -604735488 + 442368 \beta_{2} - 159744 \beta_{3} + 139264 \beta_{4} - 309248 \beta_{5} + 10240 \beta_{6} + 4096 \beta_{7} + 28672 \beta_{8} + 10240 \beta_{9} + 18432 \beta_{10} + 8192 \beta_{11} + 20480 \beta_{13} + 4096 \beta_{14} + 4096 \beta_{15} ) q^{36} + ( 720565505 \beta_{1} + 428074 \beta_{2} + 484200 \beta_{3} + 409205 \beta_{4} - 492531 \beta_{5} - 3721 \beta_{6} + 24181 \beta_{7} - 22765 \beta_{8} - 28286 \beta_{9} + 18869 \beta_{10} - 18834 \beta_{11} + 2305 \beta_{12} - 1086 \beta_{13} - 330 \beta_{14} ) q^{37} + ( -6229440 - 6229440 \beta_{1} - 14618092 \beta_{2} + 11872 \beta_{3} - 7137039 \beta_{4} - 325946 \beta_{5} - 4914 \beta_{6} + 4914 \beta_{7} - 9493 \beta_{8} + 11228 \beta_{9} - 17424 \beta_{10} + 37016 \beta_{11} - 11872 \beta_{12} + 17542 \beta_{13} - 4270 \beta_{14} - 2135 \beta_{15} ) q^{38} + ( 1315760409 - 1315760409 \beta_{1} - 1260897 \beta_{2} - 2687588 \beta_{3} - 22924818 \beta_{4} - 1350183 \beta_{5} - 175 \beta_{6} - 170625 \beta_{7} - 97828 \beta_{8} - 140763 \beta_{9} - 64255 \beta_{10} + 25031 \beta_{11} + 12428 \beta_{12} + 9901 \beta_{13} + 4688 \beta_{15} ) q^{39} + ( 390856704 - 195428352 \beta_{1} + 1941504 \beta_{2} + 669696 \beta_{3} - 1974272 \beta_{4} + 667648 \beta_{5} - 2048 \beta_{6} - 47104 \beta_{7} - 40960 \beta_{8} + 26624 \beta_{9} + 16384 \beta_{10} - 4096 \beta_{11} - 8192 \beta_{12} + 16384 \beta_{13} - 2048 \beta_{14} - 4096 \beta_{15} ) q^{40} + ( -354831435 + 709662870 \beta_{1} - 6273042 \beta_{2} - 170722 \beta_{3} - 12339726 \beta_{4} - 4929 \beta_{5} - 22775 \beta_{6} - 79328 \beta_{7} + 17846 \beta_{8} - 4985 \beta_{9} - 22775 \beta_{10} - 63396 \beta_{11} - 17846 \beta_{12} + 9858 \beta_{13} + 5858 \beta_{14} - 5858 \beta_{15} ) q^{41} + ( -770371520 - 44892800 \beta_{1} + 28450550 \beta_{2} + 1426611 \beta_{3} + 12134491 \beta_{4} + 1092502 \beta_{5} + 12823 \beta_{6} + 72117 \beta_{7} - 138716 \beta_{8} + 73737 \beta_{9} + 15128 \beta_{10} - 12306 \beta_{11} - 9604 \beta_{12} + 22904 \beta_{13} + 1175 \beta_{14} - 2810 \beta_{15} ) q^{42} + ( -4183089500 + 1654808 \beta_{2} - 720212 \beta_{3} + 714608 \beta_{4} - 1446830 \beta_{5} - 21678 \beta_{6} + 175938 \beta_{7} + 293248 \beta_{8} + 12010 \beta_{9} + 62562 \beta_{10} - 59496 \beta_{11} + 15272 \beta_{12} - 104444 \beta_{13} - 6840 \beta_{14} - 6840 \beta_{15} ) q^{43} + ( 530454528 \beta_{1} - 4096000 \beta_{2} - 667648 \beta_{3} - 4104192 \beta_{4} + 679936 \beta_{5} - 8192 \beta_{6} - 65536 \beta_{7} + 83968 \beta_{8} + 2048 \beta_{9} + 8192 \beta_{10} + 20480 \beta_{11} - 10240 \beta_{12} - 2048 \beta_{13} - 16384 \beta_{14} ) q^{44} + ( 2385556341 + 2385556341 \beta_{1} + 25683017 \beta_{2} - 19775 \beta_{3} + 13070904 \beta_{4} - 197861 \beta_{5} - 15978 \beta_{6} + 15978 \beta_{7} - 502136 \beta_{8} - 9661 \beta_{9} - 271044 \beta_{10} - 100942 \beta_{11} + 19775 \beta_{12} - 35300 \beta_{13} - 26092 \beta_{14} - 13046 \beta_{15} ) q^{45} + ( 900401664 - 900401664 \beta_{1} + 422832 \beta_{2} + 784202 \beta_{3} - 3416293 \beta_{4} + 420693 \beta_{5} + 17344 \beta_{6} + 202682 \beta_{7} + 132509 \beta_{8} - 123798 \beta_{9} - 64475 \beta_{10} - 62336 \beta_{11} - 22496 \beta_{12} - 61296 \beta_{13} - 4632 \beta_{15} ) q^{46} + ( 1509680970 - 754840485 \beta_{1} + 40087086 \beta_{2} - 1478820 \beta_{3} - 40222743 \beta_{4} - 1421121 \beta_{5} + 57699 \beta_{6} + 497381 \beta_{7} + 558140 \beta_{8} + 21159 \beta_{9} + 22059 \beta_{10} - 99558 \beta_{11} - 3060 \beta_{12} - 28719 \beta_{13} + 3960 \beta_{14} + 7920 \beta_{15} ) q^{47} + ( -8388608 \beta_{2} + 4194304 \beta_{3} - 20971520 \beta_{4} ) q^{48} + ( 2767084900 - 6781351308 \beta_{1} - 143972688 \beta_{2} + 2134120 \beta_{3} - 82325546 \beta_{4} - 2654239 \beta_{5} - 45951 \beta_{6} + 103980 \beta_{7} + 170942 \beta_{8} + 224529 \beta_{9} + 17013 \beta_{10} - 75864 \beta_{11} - 61376 \beta_{12} - 49666 \beta_{13} - 39742 \beta_{14} - 38558 \beta_{15} ) q^{49} + ( -6203005056 - 68212374 \beta_{2} - 3758780 \beta_{3} + 3765456 \beta_{4} - 7550528 \beta_{5} - 31144 \beta_{6} + 229316 \beta_{7} + 398168 \beta_{8} + 26292 \beta_{9} + 57024 \beta_{10} - 17584 \beta_{11} - 1824 \beta_{12} - 54992 \beta_{13} - 11528 \beta_{14} - 11528 \beta_{15} ) q^{50} + ( -2912407047 \beta_{1} + 71359512 \beta_{2} - 1305902 \beta_{3} + 71204242 \beta_{4} + 1251198 \beta_{5} + 59854 \beta_{6} + 237878 \beta_{7} - 355011 \beta_{8} - 85905 \beta_{9} + 155270 \beta_{10} + 138730 \beta_{11} + 57279 \beta_{12} + 124069 \beta_{13} - 6936 \beta_{14} ) q^{51} + ( 344565760 + 344565760 \beta_{1} + 119830528 \beta_{2} - 71680 \beta_{3} + 58818560 \beta_{4} + 2213888 \beta_{5} + 102400 \beta_{6} - 102400 \beta_{7} + 143360 \beta_{8} - 10240 \beta_{9} - 81920 \beta_{10} - 20480 \beta_{11} + 71680 \beta_{12} + 81920 \beta_{13} + 40960 \beta_{14} + 20480 \beta_{15} ) q^{52} + ( 4891770225 - 4891770225 \beta_{1} + 6506112 \beta_{2} + 12886881 \beta_{3} + 80778234 \beta_{4} + 6458448 \beta_{5} - 8508 \beta_{6} - 569612 \beta_{7} - 242029 \beta_{8} - 210897 \beta_{9} - 133218 \beta_{10} - 85554 \beta_{11} - 47031 \beta_{12} - 12936 \beta_{13} - 19230 \beta_{15} ) q^{53} + ( 8762683776 - 4381341888 \beta_{1} + 52625227 \beta_{2} - 9411018 \beta_{3} - 52390176 \beta_{4} - 9434545 \beta_{5} - 23527 \beta_{6} - 917660 \beta_{7} - 994023 \beta_{8} - 145920 \beta_{9} - 103843 \beta_{10} + 4018 \beta_{11} + 52836 \beta_{12} - 188776 \beta_{13} - 10759 \beta_{14} - 21518 \beta_{15} ) q^{54} + ( -12093817386 + 24187634772 \beta_{1} - 77260386 \beta_{2} - 8359306 \beta_{3} - 146168802 \beta_{4} + 26379 \beta_{5} + 29503 \beta_{6} - 256387 \beta_{7} - 3124 \beta_{8} + 25291 \beta_{9} + 29503 \beta_{10} + 62130 \beta_{11} + 3124 \beta_{12} - 52758 \beta_{13} - 19576 \beta_{14} + 19576 \beta_{15} ) q^{55} + ( 1132462080 - 1811939328 \beta_{1} - 53420032 \beta_{2} + 2615296 \beta_{3} + 10002432 \beta_{4} + 886784 \beta_{5} - 61440 \beta_{6} + 55296 \beta_{7} - 188416 \beta_{8} + 67584 \beta_{9} + 47104 \beta_{10} + 16384 \beta_{11} + 20480 \beta_{13} - 4096 \beta_{14} + 30720 \beta_{15} ) q^{56} + ( -20515247505 + 185552312 \beta_{2} + 12567764 \beta_{3} - 12795412 \beta_{4} + 25225856 \beta_{5} + 113384 \beta_{6} - 747004 \beta_{7} - 1244184 \beta_{8} + 137320 \beta_{9} + 220904 \beta_{10} + 174928 \beta_{11} - 23056 \beta_{12} + 318992 \beta_{13} + 52880 \beta_{14} + 52880 \beta_{15} ) q^{57} + ( 3704674560 \beta_{1} - 36784247 \beta_{2} - 14200497 \beta_{3} - 36695305 \beta_{4} + 14157484 \beta_{5} - 85837 \beta_{6} - 395483 \beta_{7} + 502732 \beta_{8} - 94755 \beta_{9} - 88942 \beta_{10} - 367394 \beta_{11} - 21412 \beta_{12} - 140684 \beta_{13} + 33435 \beta_{14} ) q^{58} + ( 8410227462 + 8410227462 \beta_{1} - 410165 \beta_{2} + 209146 \beta_{3} + 2144082 \beta_{4} - 4888573 \beta_{5} - 94130 \beta_{6} + 94130 \beta_{7} + 1044582 \beta_{8} + 74744 \beta_{9} + 324646 \beta_{10} + 513922 \beta_{11} - 209146 \beta_{12} + 55358 \beta_{13} + 40272 \beta_{14} + 20136 \beta_{15} ) q^{59} + ( 3476957184 - 3476957184 \beta_{1} - 5312512 \beta_{2} - 10987520 \beta_{3} + 273539072 \beta_{4} - 5672960 \beta_{5} - 88064 \beta_{6} + 677888 \beta_{7} + 112640 \beta_{8} + 90112 \beta_{9} + 92160 \beta_{10} + 452608 \beta_{11} + 182272 \beta_{12} + 362496 \beta_{13} + 49152 \beta_{15} ) q^{60} + ( 28065021194 - 14032510597 \beta_{1} - 87006558 \beta_{2} + 2002038 \beta_{3} + 86899579 \beta_{4} + 1768063 \beta_{5} - 233975 \beta_{6} + 629531 \beta_{7} + 487823 \beta_{8} + 487520 \beta_{9} + 400111 \beta_{10} + 487382 \beta_{11} - 92267 \beta_{12} + 535066 \beta_{13} + 4858 \beta_{14} + 9716 \beta_{15} ) q^{61} + ( -12350378880 + 24700757760 \beta_{1} - 180767147 \beta_{2} + 24472180 \beta_{3} - 386240629 \beta_{4} - 99111 \beta_{5} - 37771 \beta_{6} + 933496 \beta_{7} - 61340 \beta_{8} - 210586 \beta_{9} - 37771 \beta_{10} - 14202 \beta_{11} + 61340 \beta_{12} + 198222 \beta_{13} + 38425 \beta_{14} - 38425 \beta_{15} ) q^{62} + ( -5586677160 - 37463532495 \beta_{1} - 133666973 \beta_{2} - 34518195 \beta_{3} + 661321753 \beta_{4} - 34257345 \beta_{5} + 384042 \beta_{6} - 477382 \beta_{7} + 779293 \beta_{8} - 717065 \beta_{9} - 50758 \beta_{10} + 606882 \beta_{11} + 396627 \beta_{12} + 215917 \beta_{13} + 263040 \beta_{14} + 178160 \beta_{15} ) q^{63} + 8589934592 q^{64} + ( 19790337987 \beta_{1} + 321917249 \beta_{2} + 37516241 \beta_{3} + 322473288 \beta_{4} - 36876968 \beta_{5} - 87605 \beta_{6} - 429896 \beta_{7} + 880940 \beta_{8} + 610670 \beta_{9} - 556039 \beta_{10} + 109262 \beta_{11} - 363439 \beta_{12} - 584642 \beta_{13} + 133598 \beta_{14} ) q^{65} + ( 11203503360 + 11203503360 \beta_{1} - 499544522 \beta_{2} + 124544 \beta_{3} - 262713417 \beta_{4} + 25794848 \beta_{5} - 455832 \beta_{6} + 455832 \beta_{7} - 506500 \beta_{8} - 210992 \beta_{9} + 423000 \beta_{10} - 749024 \beta_{11} - 124544 \beta_{12} - 877816 \beta_{13} - 120296 \beta_{14} - 60148 \beta_{15} ) q^{66} + ( 13470889990 - 13470889990 \beta_{1} + 7232477 \beta_{2} + 16679882 \beta_{3} + 25389564 \beta_{4} + 8459289 \beta_{5} + 133250 \beta_{6} - 927650 \beta_{7} - 425004 \beta_{8} + 2137286 \beta_{9} + 1149170 \beta_{10} - 77642 \beta_{11} + 27804 \beta_{12} - 313830 \beta_{13} + 42960 \beta_{15} ) q^{67} + ( 17333084160 - 8666542080 \beta_{1} + 121272320 \beta_{2} + 5265408 \beta_{3} - 121565184 \beta_{4} + 5324800 \beta_{5} + 59392 \beta_{6} + 135168 \beta_{7} + 200704 \beta_{8} + 270336 \beta_{9} + 366592 \beta_{10} + 290816 \beta_{11} - 6144 \beta_{12} + 471040 \beta_{13} + 102400 \beta_{14} + 204800 \beta_{15} ) q^{68} + ( -31016712258 + 62033424516 \beta_{1} + 138231552 \beta_{2} - 76857311 \beta_{3} + 352295583 \beta_{4} + 176688 \beta_{5} + 436160 \beta_{6} + 386911 \beta_{7} - 259472 \beta_{8} - 329200 \beta_{9} + 436160 \beta_{10} + 1131792 \beta_{11} + 259472 \beta_{12} - 353376 \beta_{13} - 55208 \beta_{14} + 55208 \beta_{15} ) q^{69} + ( 14697302592 - 30154332096 \beta_{1} + 263956567 \beta_{2} + 52152863 \beta_{3} - 67233996 \beta_{4} + 83130254 \beta_{5} + 20731 \beta_{6} - 42817 \beta_{7} - 776095 \beta_{8} - 1775763 \beta_{9} - 671738 \beta_{10} - 191358 \beta_{11} + 283444 \beta_{12} - 724378 \beta_{13} - 25329 \beta_{14} - 72753 \beta_{15} ) q^{70} + ( -71891189334 - 564485056 \beta_{2} + 28926802 \beta_{3} - 28535130 \beta_{4} + 57450886 \beta_{5} - 58012 \beta_{6} - 931012 \beta_{7} - 1633342 \beta_{8} + 11046 \beta_{9} - 981324 \beta_{10} + 662102 \beta_{11} - 344706 \beta_{12} + 1262800 \beta_{13} - 186008 \beta_{14} - 186008 \beta_{15} ) q^{71} + ( -1175715840 \beta_{1} + 601475072 \beta_{2} - 14526464 \beta_{3} + 601511936 \beta_{4} + 13877248 \beta_{5} + 382976 \beta_{6} - 423936 \beta_{7} - 475136 \beta_{8} + 215040 \beta_{9} - 36864 \beta_{10} + 356352 \beta_{11} + 516096 \beta_{12} + 827392 \beta_{13} + 71680 \beta_{14} ) q^{72} + ( -30767261805 - 30767261805 \beta_{1} - 1088367954 \beta_{2} - 706132 \beta_{3} - 513701944 \beta_{4} - 64837050 \beta_{5} + 1171112 \beta_{6} - 1171112 \beta_{7} + 1634018 \beta_{8} + 160100 \beta_{9} + 3006752 \beta_{10} + 383928 \beta_{11} + 706132 \beta_{12} + 1491312 \beta_{13} + 304880 \beta_{14} + 152440 \beta_{15} ) q^{73} + ( 612847296 - 612847296 \beta_{1} - 38692536 \beta_{2} - 74813496 \beta_{3} - 692423825 \beta_{4} - 37240236 \beta_{5} - 84048 \beta_{6} - 571888 \beta_{7} + 173152 \beta_{8} + 1653384 \beta_{9} + 534108 \beta_{10} - 918192 \beta_{11} - 501120 \beta_{12} - 296472 \beta_{13} - 274308 \beta_{15} ) q^{74} + ( 128085720000 - 64042860000 \beta_{1} + 1694250814 \beta_{2} + 121897602 \beta_{3} - 1692543192 \beta_{4} + 122572204 \beta_{5} + 674602 \beta_{6} - 4592362 \beta_{7} - 4114298 \beta_{8} - 3031216 \beta_{9} - 3005606 \beta_{10} - 2032916 \beta_{11} + 196538 \beta_{12} - 2118856 \beta_{13} - 170928 \beta_{14} - 341856 \beta_{15} ) q^{75} + ( -14927517696 + 29855035392 \beta_{1} + 27377664 \beta_{2} + 53133312 \beta_{3} + 1953792 \beta_{4} - 315392 \beta_{5} - 71680 \beta_{6} - 1058816 \beta_{7} - 243712 \beta_{8} + 503808 \beta_{9} - 71680 \beta_{10} + 100352 \beta_{11} + 243712 \beta_{12} + 630784 \beta_{13} - 229376 \beta_{14} + 229376 \beta_{15} ) q^{76} + ( 2579501295 - 45309421710 \beta_{1} + 75895981 \beta_{2} - 90357736 \beta_{3} + 901313160 \beta_{4} - 64964080 \beta_{5} - 1327969 \beta_{6} - 186686 \beta_{7} + 800335 \beta_{8} - 2348359 \beta_{9} - 732369 \beta_{10} - 1683176 \beta_{11} - 1769530 \beta_{12} - 733846 \beta_{13} - 844290 \beta_{14} - 276910 \beta_{15} ) q^{77} + ( -49130555520 - 1427308301 \beta_{2} - 144340525 \beta_{3} + 145921669 \beta_{4} - 288453027 \beta_{5} - 140637 \beta_{6} - 1854733 \beta_{7} - 4359400 \beta_{8} - 1809167 \beta_{9} - 2700267 \beta_{10} - 1065870 \beta_{11} + 368660 \beta_{12} - 1755914 \beta_{13} + 20055 \beta_{14} + 20055 \beta_{15} ) q^{78} + ( 28454008096 \beta_{1} + 1659724445 \beta_{2} + 73800387 \beta_{3} + 1661352910 \beta_{4} - 74902698 \beta_{5} - 651641 \beta_{6} + 4102373 \beta_{7} - 3676067 \beta_{8} + 770720 \beta_{9} - 1628465 \beta_{10} - 1715490 \beta_{11} + 225335 \beta_{12} + 244566 \beta_{13} - 670872 \beta_{14} ) q^{79} + ( -3170893824 - 3170893824 \beta_{1} - 390070272 \beta_{2} - 201326592 \beta_{4} + 12582912 \beta_{5} + 4194304 \beta_{8} ) q^{80} + ( -48998022300 + 48998022300 \beta_{1} + 22725729 \beta_{2} + 49835823 \beta_{3} - 3108473328 \beta_{4} + 24862713 \beta_{5} - 329592 \beta_{6} + 18296118 \beta_{7} + 9532050 \beta_{8} + 3616383 \beta_{9} + 1369002 \beta_{10} - 767982 \beta_{11} - 548787 \beta_{12} + 747468 \beta_{13} - 120654 \beta_{15} ) q^{81} + ( -25214837760 + 12607418880 \beta_{1} + 321290167 \beta_{2} - 54184997 \beta_{3} - 320514647 \beta_{4} - 54270210 \beta_{5} - 85213 \beta_{6} + 7737973 \beta_{7} + 7968972 \beta_{8} - 1172951 \beta_{9} - 1740808 \beta_{10} - 836154 \beta_{11} - 316212 \beta_{12} - 224376 \beta_{13} - 251645 \beta_{14} - 503290 \beta_{15} ) q^{82} + ( -22318410870 + 44636821740 \beta_{1} - 12733692 \beta_{2} - 127306668 \beta_{3} + 107950320 \beta_{4} - 399474 \beta_{5} - 1536912 \beta_{6} + 6092252 \beta_{7} + 1137438 \beta_{8} + 3436686 \beta_{9} - 1536912 \beta_{10} - 4211262 \beta_{11} - 1137438 \beta_{12} + 798948 \beta_{13} + 484920 \beta_{14} - 484920 \beta_{15} ) q^{83} + ( 26263173120 - 56612057088 \beta_{1} + 856696832 \beta_{2} + 160876544 \beta_{3} + 728565760 \beta_{4} + 128872448 \beta_{5} + 59392 \beta_{6} - 4966400 \beta_{7} + 4007936 \beta_{8} + 2009088 \beta_{9} + 1116160 \beta_{10} + 24576 \beta_{11} - 458752 \beta_{12} + 1880064 \beta_{13} + 217088 \beta_{14} - 208896 \beta_{15} ) q^{84} + ( 44388106005 + 605647246 \beta_{2} + 108223037 \beta_{3} - 102310619 \beta_{4} + 216732845 \beta_{5} - 638997 \beta_{6} + 4201827 \beta_{7} + 6199892 \beta_{8} - 6199189 \beta_{9} - 10046553 \beta_{10} - 2916012 \beta_{11} + 925768 \beta_{12} - 4981066 \beta_{13} - 69354 \beta_{14} - 69354 \beta_{15} ) q^{85} + ( -4763149056 \beta_{1} + 4159862340 \beta_{2} - 127158386 \beta_{3} + 4159208846 \beta_{4} + 129293342 \beta_{5} - 436084 \beta_{6} + 6930918 \beta_{7} - 5209314 \beta_{8} - 47354 \beta_{9} + 653494 \beta_{10} + 1212280 \beta_{11} - 1285520 \beta_{12} - 1528816 \beta_{13} - 192788 \beta_{14} ) q^{86} + ( 81542403495 + 81542403495 \beta_{1} - 10745508391 \beta_{2} + 1099672 \beta_{3} - 5395864524 \beta_{4} + 45818443 \beta_{5} - 3454353 \beta_{6} + 3454353 \beta_{7} + 3815752 \beta_{8} - 591241 \beta_{9} + 2093127 \beta_{10} - 4200931 \beta_{11} - 1099672 \beta_{12} - 4636835 \beta_{13} - 1763440 \beta_{14} - 881720 \beta_{15} ) q^{87} + ( -8035368960 + 8035368960 \beta_{1} - 31240192 \beta_{2} - 65290240 \beta_{3} - 504219648 \beta_{4} - 32022528 \beta_{5} + 991232 \beta_{6} - 1667072 \beta_{7} - 1075200 \beta_{8} - 2326528 \beta_{9} - 299008 \beta_{10} + 483328 \beta_{11} + 737280 \beta_{12} - 1191936 \beta_{13} + 890880 \beta_{15} ) q^{88} + ( -207083953566 + 103541976783 \beta_{1} + 2784164596 \beta_{2} - 353083492 \beta_{3} - 2783618670 \beta_{4} - 353864758 \beta_{5} - 781266 \beta_{6} - 7186998 \beta_{7} - 7924222 \beta_{8} + 1576192 \beta_{9} + 2135778 \beta_{10} + 3977044 \beta_{11} - 44042 \beta_{12} + 3931532 \beta_{13} + 603628 \beta_{14} + 1207256 \beta_{15} ) q^{89} + ( 26241444096 - 52482888192 \beta_{1} - 2159994275 \beta_{2} + 562969155 \beta_{3} - 4879537593 \beta_{4} + 2342117 \beta_{5} + 176369 \beta_{6} - 7368905 \beta_{7} + 2165748 \beta_{8} + 1430733 \beta_{9} + 176369 \beta_{10} - 1813010 \beta_{11} - 2165748 \beta_{12} - 4684234 \beta_{13} + 363077 \beta_{14} - 363077 \beta_{15} ) q^{90} + ( 164006165005 - 102606288272 \beta_{1} + 1204800208 \beta_{2} - 122124941 \beta_{3} + 7294160231 \beta_{4} - 305155788 \beta_{5} + 3386969 \beta_{6} - 14281625 \beta_{7} - 25516317 \beta_{8} - 129208 \beta_{9} + 2420945 \beta_{10} + 1065665 \beta_{11} + 4079985 \beta_{12} + 2565126 \beta_{13} + 1325616 \beta_{14} - 14328 \beta_{15} ) q^{91} + ( -6561423360 - 957243392 \beta_{2} - 75046912 \beta_{3} + 71921664 \beta_{4} - 150439936 \beta_{5} + 907264 \beta_{6} - 3405824 \beta_{7} - 3743744 \beta_{8} + 3471360 \beta_{9} + 3692544 \beta_{10} + 3923968 \beta_{11} - 1253376 \beta_{12} + 6828032 \beta_{13} + 81920 \beta_{14} + 81920 \beta_{15} ) q^{92} + ( -57333596085 \beta_{1} + 10846946390 \beta_{2} - 112977008 \beta_{3} + 10841928911 \beta_{4} + 111422183 \beta_{5} + 2726349 \beta_{6} - 17137329 \beta_{7} + 12270393 \beta_{8} - 2741658 \beta_{9} + 5017479 \beta_{10} + 4551642 \beta_{11} + 2140587 \beta_{12} + 3830646 \beta_{13} + 1036290 \beta_{14} ) q^{93} + ( -84220778880 - 84220778880 \beta_{1} - 1464508628 \beta_{2} - 604800 \beta_{3} - 837540539 \beta_{4} + 202766144 \beta_{5} + 3755962 \beta_{6} - 3755962 \beta_{7} - 12712233 \beta_{8} + 2863028 \beta_{9} + 4338478 \beta_{10} + 8560552 \beta_{11} + 604800 \beta_{12} + 9482018 \beta_{13} + 288134 \beta_{14} + 144067 \beta_{15} ) q^{94} + ( -23312982870 + 23312982870 \beta_{1} + 175128835 \beta_{2} + 332157285 \beta_{3} - 1055551890 \beta_{4} + 164672705 \beta_{5} + 651030 \beta_{6} - 16406590 \beta_{7} - 11991715 \beta_{8} - 10523545 \beta_{9} - 2879290 \beta_{10} + 7576840 \beta_{11} + 4113935 \beta_{12} - 1223650 \beta_{13} + 364720 \beta_{15} ) q^{95} + ( -41339060224 + 20669530112 \beta_{1} + 4194304 \beta_{3} - 4194304 \beta_{4} + 4194304 \beta_{5} + 4194304 \beta_{7} + 4194304 \beta_{8} + 4194304 \beta_{9} + 4194304 \beta_{10} ) q^{96} + ( 179404856785 - 358809713570 \beta_{1} - 3797152346 \beta_{2} - 346922682 \beta_{3} - 7253888574 \beta_{4} - 856701 \beta_{5} + 367441 \beta_{6} - 19494916 \beta_{7} - 1224142 \beta_{8} - 4914981 \beta_{9} + 367441 \beta_{10} + 1959024 \beta_{11} + 1224142 \beta_{12} + 1713402 \beta_{13} - 752310 \beta_{14} + 752310 \beta_{15} ) q^{97} + ( -169891430400 + 295236403200 \beta_{1} - 2610070716 \beta_{2} + 237394841 \beta_{3} + 3832457666 \beta_{4} + 342131300 \beta_{5} + 647993 \beta_{6} + 9592987 \beta_{7} + 14224292 \beta_{8} + 4347395 \beta_{9} + 1542678 \beta_{10} + 4184794 \beta_{11} - 1373932 \beta_{12} + 1028252 \beta_{13} - 533775 \beta_{14} + 1002080 \beta_{15} ) q^{98} + ( 245073772161 - 6652761168 \beta_{2} + 372546321 \beta_{3} - 385707723 \beta_{4} + 750017112 \beta_{5} + 2699445 \beta_{6} + 8547351 \beta_{7} + 20268567 \beta_{8} + 8236932 \beta_{9} + 23745957 \beta_{10} - 1153587 \beta_{11} + 2225025 \beta_{12} - 3501210 \beta_{13} + 2760744 \beta_{14} + 2760744 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16384q^{4} + 18144q^{5} - 469720q^{7} + 2362248q^{9} + O(q^{10}) \) \( 16q - 16384q^{4} + 18144q^{5} - 469720q^{7} + 2362248q^{9} - 2290176q^{10} + 2072088q^{11} - 11501568q^{14} - 27163728q^{15} - 33554432q^{16} - 101561040q^{17} - 4592640q^{18} + 174931848q^{19} - 323731368q^{21} + 62776320q^{22} + 25630560q^{23} + 242221056q^{24} + 521205808q^{25} - 1434682368q^{26} + 350863360q^{28} + 532360944q^{29} - 2151917568q^{30} + 4583818344q^{31} + 6054957720q^{33} - 1612540440q^{35} - 9675767808q^{36} + 5764524040q^{37} - 149506560q^{38} + 10526083272q^{39} + 4690280448q^{40} - 12685086720q^{42} - 66929432000q^{43} + 4243636224q^{44} + 57253352184q^{45} + 7203213312q^{46} + 18116171640q^{47} - 9977452064q^{49} - 99248080896q^{50} - 23299256376q^{51} + 8269578240q^{52} + 39134161800q^{53} + 105152205312q^{54} + 3623878656q^{56} - 328243960080q^{57} + 29637396480q^{58} + 201845459088q^{59} + 27815657472q^{60} + 336780254328q^{61} - 389095094520q^{63} + 137438953472q^{64} + 158322703896q^{65} + 268884080640q^{66} + 107767119920q^{67} + 207997009920q^{68} - 6077815296q^{70} - 1150259029344q^{71} - 9405726720q^{72} - 738414283320q^{73} + 4902778368q^{74} + 1537028640000q^{75} - 321203352960q^{77} - 786088888320q^{78} + 227632064768q^{79} - 76101451776q^{80} - 391984178400q^{81} - 302578053120q^{82} - 32685686784q^{84} + 710209696080q^{85} - 38105192448q^{86} + 1957017683880q^{87} - 64282951680q^{88} - 2485007442792q^{89} + 1803248333904q^{91} - 104982773760q^{92} - 458668768680q^{93} - 2021298693120q^{94} - 186503862960q^{95} - 496068722688q^{96} - 356371660800q^{98} + 3921180354576q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 527434 x^{14} - 31307480 x^{13} + 193554267483 x^{12} - 12267558721140 x^{11} + 36740634631350658 x^{10} - 2684653740993140180 x^{9} + 5051027688394458110177 x^{8} - 329042050982173184619740 x^{7} + 328571000441888143595366884 x^{6} - 20799163177686535050419379920 x^{5} + 15169931128056380497155715556632 x^{4} - 685746033739471361446313307383040 x^{3} + 159607415711477686151635843287775328 x^{2} + 6015487435743293961505549572273215360 x + 735121207672317095821497912841120438336\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(56\!\cdots\!65\)\( \nu^{15} + \)\(59\!\cdots\!67\)\( \nu^{14} - \)\(29\!\cdots\!30\)\( \nu^{13} + \)\(48\!\cdots\!04\)\( \nu^{12} - \)\(10\!\cdots\!95\)\( \nu^{11} + \)\(17\!\cdots\!25\)\( \nu^{10} - \)\(20\!\cdots\!10\)\( \nu^{9} + \)\(35\!\cdots\!76\)\( \nu^{8} - \)\(28\!\cdots\!65\)\( \nu^{7} + \)\(45\!\cdots\!67\)\( \nu^{6} - \)\(18\!\cdots\!60\)\( \nu^{5} + \)\(27\!\cdots\!70\)\( \nu^{4} - \)\(84\!\cdots\!00\)\( \nu^{3} + \)\(11\!\cdots\!64\)\( \nu^{2} - \)\(72\!\cdots\!60\)\( \nu + \)\(63\!\cdots\!68\)\(\)\()/ \)\(84\!\cdots\!60\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(11\!\cdots\!34\)\( \nu^{15} + \)\(16\!\cdots\!50\)\( \nu^{14} + \)\(65\!\cdots\!72\)\( \nu^{13} + \)\(76\!\cdots\!40\)\( \nu^{12} + \)\(19\!\cdots\!50\)\( \nu^{11} + \)\(27\!\cdots\!30\)\( \nu^{10} + \)\(33\!\cdots\!12\)\( \nu^{9} + \)\(43\!\cdots\!00\)\( \nu^{8} + \)\(36\!\cdots\!06\)\( \nu^{7} + \)\(53\!\cdots\!70\)\( \nu^{6} + \)\(19\!\cdots\!60\)\( \nu^{5} + \)\(14\!\cdots\!40\)\( \nu^{4} + \)\(18\!\cdots\!88\)\( \nu^{3} + \)\(19\!\cdots\!00\)\( \nu^{2} + \)\(90\!\cdots\!44\)\( \nu - \)\(34\!\cdots\!40\)\(\)\()/ \)\(78\!\cdots\!95\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(13\!\cdots\!71\)\( \nu^{15} + \)\(56\!\cdots\!75\)\( \nu^{14} + \)\(11\!\cdots\!92\)\( \nu^{13} + \)\(23\!\cdots\!54\)\( \nu^{12} + \)\(44\!\cdots\!85\)\( \nu^{11} + \)\(67\!\cdots\!85\)\( \nu^{10} + \)\(11\!\cdots\!48\)\( \nu^{9} + \)\(10\!\cdots\!50\)\( \nu^{8} + \)\(18\!\cdots\!51\)\( \nu^{7} + \)\(13\!\cdots\!87\)\( \nu^{6} + \)\(21\!\cdots\!10\)\( \nu^{5} + \)\(91\!\cdots\!80\)\( \nu^{4} + \)\(10\!\cdots\!92\)\( \nu^{3} + \)\(32\!\cdots\!80\)\( \nu^{2} + \)\(25\!\cdots\!84\)\( \nu + \)\(78\!\cdots\!88\)\(\)\()/ \)\(78\!\cdots\!20\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(15\!\cdots\!64\)\( \nu^{15} - \)\(22\!\cdots\!20\)\( \nu^{14} + \)\(80\!\cdots\!60\)\( \nu^{13} - \)\(16\!\cdots\!40\)\( \nu^{12} + \)\(29\!\cdots\!20\)\( \nu^{11} - \)\(62\!\cdots\!00\)\( \nu^{10} + \)\(56\!\cdots\!72\)\( \nu^{9} - \)\(12\!\cdots\!60\)\( \nu^{8} + \)\(78\!\cdots\!00\)\( \nu^{7} - \)\(15\!\cdots\!20\)\( \nu^{6} + \)\(49\!\cdots\!00\)\( \nu^{5} - \)\(95\!\cdots\!00\)\( \nu^{4} + \)\(23\!\cdots\!28\)\( \nu^{3} - \)\(33\!\cdots\!40\)\( \nu^{2} + \)\(17\!\cdots\!20\)\( \nu + \)\(42\!\cdots\!00\)\(\)\()/ \)\(48\!\cdots\!45\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(17\!\cdots\!54\)\( \nu^{15} + \)\(89\!\cdots\!77\)\( \nu^{14} + \)\(89\!\cdots\!74\)\( \nu^{13} - \)\(10\!\cdots\!28\)\( \nu^{12} + \)\(32\!\cdots\!10\)\( \nu^{11} - \)\(37\!\cdots\!25\)\( \nu^{10} + \)\(58\!\cdots\!82\)\( \nu^{9} - \)\(13\!\cdots\!84\)\( \nu^{8} + \)\(76\!\cdots\!82\)\( \nu^{7} - \)\(95\!\cdots\!59\)\( \nu^{6} + \)\(40\!\cdots\!10\)\( \nu^{5} - \)\(94\!\cdots\!50\)\( \nu^{4} + \)\(14\!\cdots\!88\)\( \nu^{3} + \)\(88\!\cdots\!44\)\( \nu^{2} - \)\(72\!\cdots\!52\)\( \nu + \)\(62\!\cdots\!04\)\(\)\()/ \)\(78\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(96\!\cdots\!09\)\( \nu^{15} - \)\(50\!\cdots\!06\)\( \nu^{14} - \)\(46\!\cdots\!72\)\( \nu^{13} - \)\(26\!\cdots\!32\)\( \nu^{12} + \)\(16\!\cdots\!85\)\( \nu^{11} - \)\(93\!\cdots\!90\)\( \nu^{10} + \)\(47\!\cdots\!68\)\( \nu^{9} - \)\(16\!\cdots\!28\)\( \nu^{8} + \)\(13\!\cdots\!19\)\( \nu^{7} - \)\(21\!\cdots\!46\)\( \nu^{6} + \)\(20\!\cdots\!90\)\( \nu^{5} - \)\(11\!\cdots\!40\)\( \nu^{4} + \)\(13\!\cdots\!12\)\( \nu^{3} - \)\(37\!\cdots\!12\)\( \nu^{2} + \)\(43\!\cdots\!96\)\( \nu + \)\(45\!\cdots\!56\)\(\)\()/ \)\(68\!\cdots\!40\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(35\!\cdots\!61\)\( \nu^{15} - \)\(33\!\cdots\!45\)\( \nu^{14} - \)\(18\!\cdots\!44\)\( \nu^{13} - \)\(44\!\cdots\!94\)\( \nu^{12} - \)\(66\!\cdots\!15\)\( \nu^{11} - \)\(12\!\cdots\!95\)\( \nu^{10} - \)\(12\!\cdots\!48\)\( \nu^{9} - \)\(33\!\cdots\!70\)\( \nu^{8} - \)\(16\!\cdots\!57\)\( \nu^{7} - \)\(17\!\cdots\!97\)\( \nu^{6} - \)\(10\!\cdots\!70\)\( \nu^{5} - \)\(39\!\cdots\!00\)\( \nu^{4} - \)\(49\!\cdots\!12\)\( \nu^{3} - \)\(15\!\cdots\!80\)\( \nu^{2} - \)\(11\!\cdots\!48\)\( \nu - \)\(27\!\cdots\!08\)\(\)\()/ \)\(97\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(43\!\cdots\!07\)\( \nu^{15} - \)\(17\!\cdots\!34\)\( \nu^{14} - \)\(21\!\cdots\!88\)\( \nu^{13} + \)\(40\!\cdots\!36\)\( \nu^{12} - \)\(79\!\cdots\!65\)\( \nu^{11} + \)\(13\!\cdots\!50\)\( \nu^{10} - \)\(14\!\cdots\!16\)\( \nu^{9} + \)\(34\!\cdots\!08\)\( \nu^{8} - \)\(19\!\cdots\!39\)\( \nu^{7} + \)\(20\!\cdots\!98\)\( \nu^{6} - \)\(11\!\cdots\!70\)\( \nu^{5} + \)\(33\!\cdots\!20\)\( \nu^{4} - \)\(55\!\cdots\!44\)\( \nu^{3} - \)\(88\!\cdots\!08\)\( \nu^{2} + \)\(44\!\cdots\!04\)\( \nu - \)\(23\!\cdots\!68\)\(\)\()/ \)\(97\!\cdots\!20\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(27\!\cdots\!19\)\( \nu^{15} + \)\(23\!\cdots\!75\)\( \nu^{14} + \)\(14\!\cdots\!82\)\( \nu^{13} + \)\(28\!\cdots\!00\)\( \nu^{12} + \)\(51\!\cdots\!65\)\( \nu^{11} + \)\(83\!\cdots\!85\)\( \nu^{10} + \)\(93\!\cdots\!42\)\( \nu^{9} + \)\(23\!\cdots\!60\)\( \nu^{8} + \)\(12\!\cdots\!31\)\( \nu^{7} + \)\(13\!\cdots\!15\)\( \nu^{6} + \)\(75\!\cdots\!60\)\( \nu^{5} + \)\(30\!\cdots\!70\)\( \nu^{4} + \)\(37\!\cdots\!48\)\( \nu^{3} + \)\(10\!\cdots\!60\)\( \nu^{2} + \)\(86\!\cdots\!84\)\( \nu + \)\(19\!\cdots\!20\)\(\)\()/ \)\(34\!\cdots\!20\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(59\!\cdots\!75\)\( \nu^{15} + \)\(34\!\cdots\!19\)\( \nu^{14} + \)\(30\!\cdots\!30\)\( \nu^{13} - \)\(72\!\cdots\!36\)\( \nu^{12} + \)\(11\!\cdots\!65\)\( \nu^{11} - \)\(36\!\cdots\!95\)\( \nu^{10} + \)\(20\!\cdots\!10\)\( \nu^{9} - \)\(25\!\cdots\!08\)\( \nu^{8} + \)\(27\!\cdots\!15\)\( \nu^{7} - \)\(45\!\cdots\!13\)\( \nu^{6} + \)\(16\!\cdots\!80\)\( \nu^{5} + \)\(16\!\cdots\!50\)\( \nu^{4} + \)\(80\!\cdots\!60\)\( \nu^{3} + \)\(14\!\cdots\!88\)\( \nu^{2} - \)\(67\!\cdots\!60\)\( \nu + \)\(36\!\cdots\!68\)\(\)\()/ \)\(68\!\cdots\!40\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(83\!\cdots\!69\)\( \nu^{15} + \)\(15\!\cdots\!18\)\( \nu^{14} - \)\(39\!\cdots\!74\)\( \nu^{13} + \)\(10\!\cdots\!46\)\( \nu^{12} - \)\(14\!\cdots\!55\)\( \nu^{11} + \)\(37\!\cdots\!90\)\( \nu^{10} - \)\(24\!\cdots\!02\)\( \nu^{9} + \)\(67\!\cdots\!34\)\( \nu^{8} - \)\(32\!\cdots\!17\)\( \nu^{7} + \)\(80\!\cdots\!58\)\( \nu^{6} - \)\(13\!\cdots\!80\)\( \nu^{5} + \)\(39\!\cdots\!50\)\( \nu^{4} - \)\(57\!\cdots\!68\)\( \nu^{3} + \)\(14\!\cdots\!56\)\( \nu^{2} + \)\(20\!\cdots\!72\)\( \nu - \)\(99\!\cdots\!88\)\(\)\()/ \)\(68\!\cdots\!40\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(16\!\cdots\!70\)\( \nu^{15} + \)\(73\!\cdots\!85\)\( \nu^{14} - \)\(82\!\cdots\!40\)\( \nu^{13} + \)\(45\!\cdots\!82\)\( \nu^{12} - \)\(31\!\cdots\!30\)\( \nu^{11} + \)\(16\!\cdots\!15\)\( \nu^{10} - \)\(61\!\cdots\!40\)\( \nu^{9} + \)\(32\!\cdots\!30\)\( \nu^{8} - \)\(88\!\cdots\!10\)\( \nu^{7} + \)\(43\!\cdots\!41\)\( \nu^{6} - \)\(60\!\cdots\!00\)\( \nu^{5} + \)\(27\!\cdots\!80\)\( \nu^{4} - \)\(32\!\cdots\!80\)\( \nu^{3} + \)\(11\!\cdots\!60\)\( \nu^{2} - \)\(39\!\cdots\!80\)\( \nu + \)\(26\!\cdots\!24\)\(\)\()/ \)\(68\!\cdots\!40\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(26\!\cdots\!45\)\( \nu^{15} - \)\(12\!\cdots\!46\)\( \nu^{14} - \)\(13\!\cdots\!08\)\( \nu^{13} + \)\(26\!\cdots\!68\)\( \nu^{12} - \)\(50\!\cdots\!75\)\( \nu^{11} + \)\(13\!\cdots\!90\)\( \nu^{10} - \)\(95\!\cdots\!00\)\( \nu^{9} + \)\(39\!\cdots\!32\)\( \nu^{8} - \)\(13\!\cdots\!89\)\( \nu^{7} + \)\(47\!\cdots\!14\)\( \nu^{6} - \)\(83\!\cdots\!70\)\( \nu^{5} + \)\(39\!\cdots\!20\)\( \nu^{4} - \)\(37\!\cdots\!00\)\( \nu^{3} + \)\(85\!\cdots\!68\)\( \nu^{2} - \)\(36\!\cdots\!56\)\( \nu + \)\(35\!\cdots\!96\)\(\)\()/ \)\(68\!\cdots\!40\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(42\!\cdots\!13\)\( \nu^{15} + \)\(13\!\cdots\!62\)\( \nu^{14} - \)\(24\!\cdots\!20\)\( \nu^{13} + \)\(79\!\cdots\!16\)\( \nu^{12} - \)\(96\!\cdots\!95\)\( \nu^{11} + \)\(29\!\cdots\!30\)\( \nu^{10} - \)\(20\!\cdots\!24\)\( \nu^{9} + \)\(55\!\cdots\!16\)\( \nu^{8} - \)\(31\!\cdots\!65\)\( \nu^{7} + \)\(73\!\cdots\!18\)\( \nu^{6} - \)\(26\!\cdots\!30\)\( \nu^{5} + \)\(43\!\cdots\!00\)\( \nu^{4} - \)\(13\!\cdots\!76\)\( \nu^{3} + \)\(19\!\cdots\!64\)\( \nu^{2} - \)\(33\!\cdots\!80\)\( \nu + \)\(66\!\cdots\!52\)\(\)\()/ \)\(68\!\cdots\!40\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(22\!\cdots\!67\)\( \nu^{15} + \)\(17\!\cdots\!62\)\( \nu^{14} + \)\(11\!\cdots\!00\)\( \nu^{13} - \)\(59\!\cdots\!84\)\( \nu^{12} + \)\(43\!\cdots\!45\)\( \nu^{11} - \)\(24\!\cdots\!70\)\( \nu^{10} + \)\(81\!\cdots\!36\)\( \nu^{9} - \)\(53\!\cdots\!84\)\( \nu^{8} + \)\(11\!\cdots\!35\)\( \nu^{7} - \)\(67\!\cdots\!82\)\( \nu^{6} + \)\(73\!\cdots\!50\)\( \nu^{5} - \)\(44\!\cdots\!00\)\( \nu^{4} + \)\(33\!\cdots\!24\)\( \nu^{3} - \)\(16\!\cdots\!36\)\( \nu^{2} + \)\(34\!\cdots\!40\)\( \nu + \)\(95\!\cdots\!52\)\(\)\()/ \)\(68\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + 4 \beta_{4} + \beta_{3} + 3 \beta_{2}\)\()/192\)
\(\nu^{2}\)\(=\)\((\)\(32 \beta_{15} + 64 \beta_{13} - 48 \beta_{12} - 96 \beta_{11} - 45 \beta_{10} - 42 \beta_{9} + 83 \beta_{8} + 70 \beta_{7} - 1309 \beta_{5} - 3482 \beta_{4} - 2666 \beta_{3} - 1360 \beta_{2} + 12658416 \beta_{1} - 12658416\)\()/96\)
\(\nu^{3}\)\(=\)\((\)\(-9457 \beta_{15} - 9457 \beta_{14} - 10586 \beta_{13} - 6828 \beta_{12} + 1570 \beta_{11} + 368741 \beta_{10} + 191181 \beta_{9} + 606880 \beta_{8} + 318975 \beta_{7} - 18949 \beta_{6} + 4659341 \beta_{5} - 2507963 \beta_{4} + 2342559 \beta_{3} - 30625511 \beta_{2} + 1127069280\)\()/192\)
\(\nu^{4}\)\(=\)\((\)\(4256109 \beta_{14} + 12951212 \beta_{13} + 6520092 \beta_{12} + 21196514 \beta_{11} + 2595814 \beta_{10} + 8002443 \beta_{9} - 3834012 \beta_{8} - 13373309 \beta_{7} + 10687229 \beta_{6} - 291788044 \beta_{5} + 7110768841 \beta_{4} + 289435089 \beta_{3} + 7113364655 \beta_{2} - 1307062927320 \beta_{1}\)\()/48\)
\(\nu^{5}\)\(=\)\((\)\(3308728467 \beta_{15} - 1189281510 \beta_{13} - 2473590136 \beta_{12} - 7549426420 \beta_{11} - 43448989807 \beta_{10} - 87026635626 \beta_{9} - 68351845770 \beta_{8} - 144253117960 \beta_{7} + 2602246148 \beta_{6} - 936011006577 \beta_{5} - 13510438098722 \beta_{4} - 1879700095586 \beta_{3} - 900111443190 \beta_{2} + 509408482221600 \beta_{1} - 509408482221600\)\()/192\)
\(\nu^{6}\)\(=\)\((\)\(-2198533145185 \beta_{15} - 2198533145185 \beta_{14} - 10205420386202 \beta_{13} - 410522769996 \beta_{12} - 3165497980382 \beta_{11} + 5189098547677 \beta_{10} + 2241727322473 \beta_{9} + 264010072536 \beta_{8} + 5850499384363 \beta_{7} - 5923755733093 \beta_{6} + 414080200870949 \beta_{5} - 206114688506403 \beta_{4} + 210207239687019 \beta_{3} - 5564584969535935 \beta_{2} + 609414648938123856\)\()/96\)
\(\nu^{7}\)\(=\)\((\)\(1106779165906591 \beta_{14} + 3006717608467908 \beta_{13} + 1983971549940396 \beta_{12} + 2336599466042438 \beta_{11} - 9798215457464560 \beta_{10} + 10966515190485779 \beta_{9} - 18296304216364702 \beta_{8} + 14182807441990203 \beta_{7} + 2129525224434103 \beta_{6} - 310182111273461374 \beta_{5} + 5153603856323020215 \beta_{4} + 308343693398014685 \beta_{3} + 5143805640865555655 \beta_{2} - 190856064903555162720 \beta_{1}\)\()/192\)
\(\nu^{8}\)\(=\)\((\)\(144095019732918505 \beta_{15} + 132087363123581366 \beta_{13} - 217289622080101756 \beta_{12} - 486613469607622060 \beta_{11} - 404678688827102717 \beta_{10} - 644101981021522226 \beta_{9} + 133023428273993346 \beta_{8} - 220566613059635368 \beta_{7} + 52034225447418548 \beta_{6} - 16949642781743038231 \beta_{5} - 459992376350847153318 \beta_{4} - 34220643636461015314 \beta_{3} - 17031577562523557574 \beta_{2} + 37711244292591893750988 \beta_{1} - 37711244292591893750988\)\()/24\)
\(\nu^{9}\)\(=\)\((\)\(-356969562122233861767 \beta_{15} - 356969562122233861767 \beta_{14} - 787631257683855173974 \beta_{13} - 325971771756104061428 \beta_{12} + 263976191023844460750 \beta_{11} + 5393916651011855877003 \beta_{10} + 2857020073207198006775 \beta_{9} + 7984332027625967344696 \beta_{8} + 4874939300289067351477 \beta_{7} - 1045759172354135709843 \beta_{6} + 191149857105928922205603 \beta_{5} - 97746083154116539223941 \beta_{4} + 96260794025019580988437 \beta_{3} - 1599053065830676205968177 \beta_{2} + 65864450137541175725133600\)\()/192\)
\(\nu^{10}\)\(=\)\((\)\(154798330391547194656863 \beta_{14} + 543412051743827637845444 \beta_{13} + 311116590850075767148956 \beta_{12} + 616545322657950337801766 \beta_{11} - 193359046781460114409824 \beta_{10} + 501631708110435283310707 \beta_{9} - 493082009945738709448126 \beta_{8} - 205128372189636123054181 \beta_{7} + 387093791285299065353351 \beta_{6} - 21669435023707136550153022 \beta_{5} + 590582251865778049010819511 \beta_{4} + 21434295633292284081208461 \beta_{3} + 590388892818996588896409687 \beta_{2} - 38881129381359388588960917456 \beta_{1}\)\()/96\)
\(\nu^{11}\)\(=\)\((\)\(112007071131656606504810971 \beta_{15} - 88852029438062312148638710 \beta_{13} - 121129707356165143206766072 \beta_{12} - 346548138612788794799619028 \beta_{11} - 786516820510535330677429279 \beta_{10} - 1556192657565364026534179370 \beta_{9} - 806924898919599432138017474 \beta_{8} - 1960397936451987659075653976 \beta_{7} + 104288723900458508386086884 \beta_{6} - 28614012913817027925829095185 \beta_{5} - 506368539982692175268735061618 \beta_{4} - 57557732982791138011637130210 \beta_{3} - 28174044231919281389951284934 \beta_{2} + 21623421310367447674746187137120 \beta_{1} - 21623421310367447674746187137120\)\()/192\)
\(\nu^{12}\)\(=\)\((\)\(-21263290507533857546323867875 \beta_{15} - 21263290507533857546323867875 \beta_{14} - 79160633184736454144588796702 \beta_{13} - 13268239732060728396175580436 \beta_{12} - 2721861818885529904120994442 \beta_{11} + 113992851359037096674293494687 \beta_{10} + 58469557890868292347323416043 \beta_{9} + 85674827955559183756190435736 \beta_{8} + 102320090168238911544652986873 \beta_{7} - 66116796056489683864645559223 \beta_{6} + 6658235939279363465307668537479 \beta_{5} - 3347895009636274818870747114953 \beta_{4} + 3368810487533956938784244838569 \beta_{3} - 85479862536147462192566377844645 \beta_{2} + 5170134649846911390116007785804760\)\()/48\)
\(\nu^{13}\)\(=\)\((\)\(34479268823926248336942519400507 \beta_{14} + 110338797546032100615474727678036 \beta_{13} + 71289657349538962575570487708572 \beta_{12} + 82575783734747123682361023261726 \beta_{11} - 177103989680870156981595123465328 \beta_{10} + 218391881548243718822775635096191 \beta_{9} - 304155313815951159256629353906262 \beta_{8} + 159337247445992810304212106827719 \beta_{7} + 73528409020419386376846759369971 \beta_{6} - 8548687898722116084418335837571126 \beta_{5} + 165398232281275830065490575745173267 \beta_{4} + 8479636993043457545644041621523953 \beta_{3} + 165221128291594959908508980621707939 \beta_{2} - 6871399615189589134819847146881449760 \beta_{1}\)\()/192\)
\(\nu^{14}\)\(=\)\((\)\(\)\(11\!\cdots\!61\)\( \beta_{15} - \)\(14\!\cdots\!90\)\( \beta_{13} - \)\(17\!\cdots\!12\)\( \beta_{12} - \)\(43\!\cdots\!28\)\( \beta_{11} - \)\(46\!\cdots\!41\)\( \beta_{10} - \)\(84\!\cdots\!74\)\( \beta_{9} - \)\(81\!\cdots\!58\)\( \beta_{8} - \)\(59\!\cdots\!44\)\( \beta_{7} + \)\(84\!\cdots\!04\)\( \beta_{6} - \)\(20\!\cdots\!35\)\( \beta_{5} - \)\(50\!\cdots\!30\)\( \beta_{4} - \)\(40\!\cdots\!90\)\( \beta_{3} - \)\(20\!\cdots\!22\)\( \beta_{2} + \)\(28\!\cdots\!52\)\( \beta_{1} - \)\(28\!\cdots\!52\)\(\)\()/96\)
\(\nu^{15}\)\(=\)\((\)\(-\)\(10\!\cdots\!75\)\( \beta_{15} - \)\(10\!\cdots\!75\)\( \beta_{14} - \)\(25\!\cdots\!66\)\( \beta_{13} - \)\(97\!\cdots\!20\)\( \beta_{12} + \)\(83\!\cdots\!10\)\( \beta_{11} + \)\(11\!\cdots\!19\)\( \beta_{10} + \)\(59\!\cdots\!43\)\( \beta_{9} + \)\(13\!\cdots\!12\)\( \beta_{8} + \)\(92\!\cdots\!69\)\( \beta_{7} - \)\(32\!\cdots\!23\)\( \beta_{6} + \)\(50\!\cdots\!19\)\( \beta_{5} - \)\(25\!\cdots\!81\)\( \beta_{4} + \)\(25\!\cdots\!81\)\( \beta_{3} - \)\(47\!\cdots\!37\)\( \beta_{2} + \)\(21\!\cdots\!00\)\(\)\()/192\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−213.975 370.616i
−134.707 233.319i
127.460 + 220.767i
222.636 + 385.618i
234.838 + 406.752i
65.8177 + 114.000i
−30.8925 53.5073i
−271.178 469.694i
−213.975 + 370.616i
−134.707 + 233.319i
127.460 220.767i
222.636 385.618i
234.838 406.752i
65.8177 114.000i
−30.8925 + 53.5073i
−271.178 + 469.694i
−22.6274 39.1918i −745.979 430.691i −1024.00 + 1773.62i −7532.28 + 4348.76i 38981.7i −93774.2 + 71047.1i 92681.9 105269. + 182331.i 340872. + 196803.i
3.2 −22.6274 39.1918i −409.670 236.523i −1024.00 + 1773.62i 18219.6 10519.1i 21407.6i 61842.4 100084.i 92681.9 −153834. 266449.i −824525. 476040.i
3.3 −22.6274 39.1918i 702.607 + 405.650i −1024.00 + 1773.62i −18305.2 + 10568.5i 36715.3i −9080.55 117298.i 92681.9 63384.0 + 109784.i 828399. + 478276.i
3.4 −22.6274 39.1918i 1106.41 + 638.785i −1024.00 + 1773.62i 24805.4 14321.4i 57816.2i −66642.6 + 96953.8i 92681.9 550372. + 953273.i −1.12257e6 648114.i
3.5 22.6274 + 39.1918i −1161.18 670.406i −1024.00 + 1773.62i 14223.7 8212.03i 60678.2i −96002.4 68006.1i −92681.9 633167. + 1.09668e6i 643689. + 371634.i
3.6 22.6274 + 39.1918i −444.082 256.391i −1024.00 + 1773.62i −12687.9 + 7325.35i 23205.9i 117217. 10074.6i −92681.9 −134248. 232524.i −574187. 331507.i
3.7 22.6274 + 39.1918i −33.7761 19.5006i −1024.00 + 1773.62i 2481.87 1432.91i 1764.99i −44556.8 + 108885.i −92681.9 −264960. 458924.i 112317. + 64846.2i
3.8 22.6274 + 39.1918i 985.668 + 569.076i −1024.00 + 1773.62i −12133.2 + 7005.11i 51506.9i −103863. + 55261.5i −92681.9 381974. + 661598.i −549087. 317015.i
5.1 −22.6274 + 39.1918i −745.979 + 430.691i −1024.00 1773.62i −7532.28 4348.76i 38981.7i −93774.2 71047.1i 92681.9 105269. 182331.i 340872. 196803.i
5.2 −22.6274 + 39.1918i −409.670 + 236.523i −1024.00 1773.62i 18219.6 + 10519.1i 21407.6i 61842.4 + 100084.i 92681.9 −153834. + 266449.i −824525. + 476040.i
5.3 −22.6274 + 39.1918i 702.607 405.650i −1024.00 1773.62i −18305.2 10568.5i 36715.3i −9080.55 + 117298.i 92681.9 63384.0 109784.i 828399. 478276.i
5.4 −22.6274 + 39.1918i 1106.41 638.785i −1024.00 1773.62i 24805.4 + 14321.4i 57816.2i −66642.6 96953.8i 92681.9 550372. 953273.i −1.12257e6 + 648114.i
5.5 22.6274 39.1918i −1161.18 + 670.406i −1024.00 1773.62i 14223.7 + 8212.03i 60678.2i −96002.4 + 68006.1i −92681.9 633167. 1.09668e6i 643689. 371634.i
5.6 22.6274 39.1918i −444.082 + 256.391i −1024.00 1773.62i −12687.9 7325.35i 23205.9i 117217. + 10074.6i −92681.9 −134248. + 232524.i −574187. + 331507.i
5.7 22.6274 39.1918i −33.7761 + 19.5006i −1024.00 1773.62i 2481.87 + 1432.91i 1764.99i −44556.8 108885.i −92681.9 −264960. + 458924.i 112317. 64846.2i
5.8 22.6274 39.1918i 985.668 569.076i −1024.00 1773.62i −12133.2 7005.11i 51506.9i −103863. 55261.5i −92681.9 381974. 661598.i −549087. + 317015.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.13.d.a 16
3.b odd 2 1 126.13.n.a 16
4.b odd 2 1 112.13.s.c 16
7.b odd 2 1 98.13.d.a 16
7.c even 3 1 98.13.b.c 16
7.c even 3 1 98.13.d.a 16
7.d odd 6 1 inner 14.13.d.a 16
7.d odd 6 1 98.13.b.c 16
21.g even 6 1 126.13.n.a 16
28.f even 6 1 112.13.s.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.13.d.a 16 1.a even 1 1 trivial
14.13.d.a 16 7.d odd 6 1 inner
98.13.b.c 16 7.c even 3 1
98.13.b.c 16 7.d odd 6 1
98.13.d.a 16 7.b odd 2 1
98.13.d.a 16 7.c even 3 1
112.13.s.c 16 4.b odd 2 1
112.13.s.c 16 28.f even 6 1
126.13.n.a 16 3.b odd 2 1
126.13.n.a 16 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4194304 + 2048 T^{2} + T^{4} )^{4} \)
$3$ \( \)\(16\!\cdots\!41\)\( + \)\(82\!\cdots\!60\)\( T + \)\(14\!\cdots\!56\)\( T^{2} + \)\(63\!\cdots\!60\)\( T^{3} + \)\(76\!\cdots\!58\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{5} - \)\(29\!\cdots\!88\)\( T^{6} + \)\(20\!\cdots\!80\)\( T^{7} + \)\(74\!\cdots\!87\)\( T^{8} - \)\(88\!\cdots\!20\)\( T^{9} - 8848813170319704864 T^{10} + 276426941444640 T^{11} + 7637014892322 T^{12} - 3306888 T^{14} + T^{16} \)
$5$ \( \)\(11\!\cdots\!25\)\( - \)\(37\!\cdots\!00\)\( T - \)\(26\!\cdots\!00\)\( T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!50\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} - \)\(25\!\cdots\!00\)\( T^{6} - \)\(39\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!75\)\( T^{8} + \)\(37\!\cdots\!00\)\( T^{9} - \)\(39\!\cdots\!00\)\( T^{10} - \)\(11\!\cdots\!00\)\( T^{11} + 874098909300438954 T^{12} + 21451613968512 T^{13} - 1072563036 T^{14} - 18144 T^{15} + T^{16} \)
$7$ \( \)\(13\!\cdots\!01\)\( + \)\(45\!\cdots\!20\)\( T + \)\(81\!\cdots\!32\)\( T^{2} + \)\(85\!\cdots\!40\)\( T^{3} + \)\(50\!\cdots\!52\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} - \)\(39\!\cdots\!80\)\( T^{6} - \)\(54\!\cdots\!20\)\( T^{7} - \)\(51\!\cdots\!66\)\( T^{8} - \)\(39\!\cdots\!20\)\( T^{9} - \)\(20\!\cdots\!80\)\( T^{10} - \)\(50\!\cdots\!80\)\( T^{11} + \)\(13\!\cdots\!52\)\( T^{12} + 16762599503754440 T^{13} + 115307165232 T^{14} + 469720 T^{15} + T^{16} \)
$11$ \( \)\(29\!\cdots\!21\)\( + \)\(46\!\cdots\!16\)\( T + \)\(22\!\cdots\!84\)\( T^{2} + \)\(19\!\cdots\!76\)\( T^{3} + \)\(91\!\cdots\!34\)\( T^{4} + \)\(73\!\cdots\!00\)\( T^{5} + \)\(19\!\cdots\!32\)\( T^{6} + \)\(30\!\cdots\!52\)\( T^{7} + \)\(14\!\cdots\!27\)\( T^{8} - \)\(20\!\cdots\!44\)\( T^{9} + \)\(10\!\cdots\!76\)\( T^{10} - \)\(30\!\cdots\!80\)\( T^{11} + \)\(21\!\cdots\!78\)\( T^{12} - 28285267118686720704 T^{13} + 17930838817944 T^{14} - 2072088 T^{15} + T^{16} \)
$13$ \( \)\(20\!\cdots\!56\)\( + \)\(19\!\cdots\!48\)\( T^{2} + \)\(53\!\cdots\!12\)\( T^{4} + \)\(68\!\cdots\!48\)\( T^{6} + \)\(44\!\cdots\!48\)\( T^{8} + \)\(15\!\cdots\!84\)\( T^{10} + \)\(28\!\cdots\!68\)\( T^{12} + 265309230378576 T^{14} + T^{16} \)
$17$ \( \)\(73\!\cdots\!41\)\( - \)\(72\!\cdots\!00\)\( T - \)\(50\!\cdots\!08\)\( T^{2} + \)\(51\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!90\)\( T^{4} - \)\(88\!\cdots\!40\)\( T^{5} + \)\(75\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!20\)\( T^{7} - \)\(33\!\cdots\!49\)\( T^{8} + \)\(51\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!24\)\( T^{10} - \)\(13\!\cdots\!00\)\( T^{11} - \)\(22\!\cdots\!70\)\( T^{12} - \)\(31\!\cdots\!20\)\( T^{13} + 3407109798755892 T^{14} + 101561040 T^{15} + T^{16} \)
$19$ \( \)\(72\!\cdots\!25\)\( + \)\(90\!\cdots\!00\)\( T + \)\(16\!\cdots\!00\)\( T^{2} - \)\(25\!\cdots\!00\)\( T^{3} - \)\(75\!\cdots\!50\)\( T^{4} + \)\(57\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!00\)\( T^{6} - \)\(44\!\cdots\!00\)\( T^{7} - \)\(24\!\cdots\!69\)\( T^{8} + \)\(32\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!36\)\( T^{10} - \)\(22\!\cdots\!16\)\( T^{11} - \)\(38\!\cdots\!42\)\( T^{12} + \)\(68\!\cdots\!72\)\( T^{13} + 6271093431881304 T^{14} - 174931848 T^{15} + T^{16} \)
$23$ \( \)\(45\!\cdots\!41\)\( - \)\(39\!\cdots\!60\)\( T + \)\(37\!\cdots\!32\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(35\!\cdots\!82\)\( T^{4} - \)\(62\!\cdots\!20\)\( T^{5} + \)\(89\!\cdots\!32\)\( T^{6} - \)\(88\!\cdots\!20\)\( T^{7} + \)\(73\!\cdots\!03\)\( T^{8} - \)\(43\!\cdots\!00\)\( T^{9} + \)\(24\!\cdots\!16\)\( T^{10} - \)\(10\!\cdots\!20\)\( T^{11} + \)\(58\!\cdots\!58\)\( T^{12} - \)\(15\!\cdots\!60\)\( T^{13} + 86173454317903704 T^{14} - 25630560 T^{15} + T^{16} \)
$29$ \( ( -\)\(24\!\cdots\!84\)\( - \)\(33\!\cdots\!76\)\( T - \)\(10\!\cdots\!24\)\( T^{2} + \)\(17\!\cdots\!56\)\( T^{3} + \)\(86\!\cdots\!68\)\( T^{4} + \)\(37\!\cdots\!28\)\( T^{5} - 1674687874740006456 T^{6} - 266180472 T^{7} + T^{8} )^{2} \)
$31$ \( \)\(11\!\cdots\!21\)\( + \)\(20\!\cdots\!36\)\( T + \)\(12\!\cdots\!88\)\( T^{2} + \)\(17\!\cdots\!36\)\( T^{3} - \)\(18\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(25\!\cdots\!52\)\( T^{6} + \)\(17\!\cdots\!48\)\( T^{7} - \)\(58\!\cdots\!65\)\( T^{8} - \)\(38\!\cdots\!72\)\( T^{9} + \)\(10\!\cdots\!92\)\( T^{10} + \)\(19\!\cdots\!00\)\( T^{11} - \)\(97\!\cdots\!58\)\( T^{12} + \)\(73\!\cdots\!36\)\( T^{13} + 6843143641236994968 T^{14} - 4583818344 T^{15} + T^{16} \)
$37$ \( \)\(14\!\cdots\!61\)\( - \)\(50\!\cdots\!40\)\( T + \)\(12\!\cdots\!92\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!90\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{5} + \)\(11\!\cdots\!84\)\( T^{6} - \)\(71\!\cdots\!00\)\( T^{7} + \)\(41\!\cdots\!11\)\( T^{8} - \)\(19\!\cdots\!00\)\( T^{9} + \)\(82\!\cdots\!44\)\( T^{10} - \)\(25\!\cdots\!00\)\( T^{11} + \)\(74\!\cdots\!70\)\( T^{12} - \)\(15\!\cdots\!60\)\( T^{13} + 39324386602505835972 T^{14} - 5764524040 T^{15} + T^{16} \)
$41$ \( \)\(33\!\cdots\!76\)\( + \)\(30\!\cdots\!64\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{4} + \)\(21\!\cdots\!64\)\( T^{6} + \)\(24\!\cdots\!04\)\( T^{8} + \)\(15\!\cdots\!56\)\( T^{10} + \)\(59\!\cdots\!08\)\( T^{12} + \)\(12\!\cdots\!80\)\( T^{14} + T^{16} \)
$43$ \( ( \)\(41\!\cdots\!04\)\( + \)\(21\!\cdots\!40\)\( T + \)\(24\!\cdots\!68\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} - \)\(32\!\cdots\!32\)\( T^{4} - \)\(95\!\cdots\!40\)\( T^{5} + \)\(31\!\cdots\!56\)\( T^{6} + 33464716000 T^{7} + T^{8} )^{2} \)
$47$ \( \)\(90\!\cdots\!01\)\( + \)\(15\!\cdots\!00\)\( T - \)\(64\!\cdots\!36\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!74\)\( T^{4} + \)\(78\!\cdots\!80\)\( T^{5} - \)\(58\!\cdots\!64\)\( T^{6} - \)\(18\!\cdots\!80\)\( T^{7} + \)\(94\!\cdots\!79\)\( T^{8} + \)\(28\!\cdots\!20\)\( T^{9} - \)\(10\!\cdots\!36\)\( T^{10} - \)\(25\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!74\)\( T^{12} + \)\(85\!\cdots\!60\)\( T^{13} - \)\(36\!\cdots\!64\)\( T^{14} - 18116171640 T^{15} + T^{16} \)
$53$ \( \)\(53\!\cdots\!61\)\( + \)\(22\!\cdots\!20\)\( T + \)\(10\!\cdots\!32\)\( T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(27\!\cdots\!80\)\( T^{5} + \)\(43\!\cdots\!16\)\( T^{6} - \)\(45\!\cdots\!80\)\( T^{7} + \)\(54\!\cdots\!07\)\( T^{8} - \)\(45\!\cdots\!40\)\( T^{9} + \)\(44\!\cdots\!52\)\( T^{10} - \)\(20\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!18\)\( T^{12} - \)\(28\!\cdots\!40\)\( T^{13} + \)\(19\!\cdots\!36\)\( T^{14} - 39134161800 T^{15} + T^{16} \)
$59$ \( \)\(19\!\cdots\!61\)\( + \)\(15\!\cdots\!64\)\( T + \)\(35\!\cdots\!28\)\( T^{2} - \)\(50\!\cdots\!16\)\( T^{3} - \)\(94\!\cdots\!70\)\( T^{4} + \)\(12\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!76\)\( T^{6} - \)\(35\!\cdots\!04\)\( T^{7} + \)\(25\!\cdots\!35\)\( T^{8} - \)\(85\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!48\)\( T^{10} + \)\(17\!\cdots\!96\)\( T^{11} - \)\(26\!\cdots\!54\)\( T^{12} - \)\(14\!\cdots\!08\)\( T^{13} + \)\(14\!\cdots\!64\)\( T^{14} - 201845459088 T^{15} + T^{16} \)
$61$ \( \)\(56\!\cdots\!41\)\( - \)\(17\!\cdots\!08\)\( T + \)\(12\!\cdots\!12\)\( T^{2} + \)\(21\!\cdots\!28\)\( T^{3} - \)\(11\!\cdots\!90\)\( T^{4} - \)\(25\!\cdots\!28\)\( T^{5} + \)\(33\!\cdots\!36\)\( T^{6} - \)\(16\!\cdots\!52\)\( T^{7} + \)\(34\!\cdots\!03\)\( T^{8} - \)\(15\!\cdots\!92\)\( T^{9} - \)\(10\!\cdots\!24\)\( T^{10} - \)\(13\!\cdots\!08\)\( T^{11} + \)\(15\!\cdots\!10\)\( T^{12} - \)\(38\!\cdots\!72\)\( T^{13} + \)\(49\!\cdots\!52\)\( T^{14} - 336780254328 T^{15} + T^{16} \)
$67$ \( \)\(97\!\cdots\!01\)\( - \)\(75\!\cdots\!60\)\( T + \)\(57\!\cdots\!80\)\( T^{2} - \)\(94\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!94\)\( T^{4} - \)\(45\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!52\)\( T^{6} - \)\(14\!\cdots\!40\)\( T^{7} + \)\(32\!\cdots\!67\)\( T^{8} - \)\(27\!\cdots\!60\)\( T^{9} + \)\(52\!\cdots\!44\)\( T^{10} - \)\(36\!\cdots\!60\)\( T^{11} + \)\(51\!\cdots\!02\)\( T^{12} - \)\(23\!\cdots\!20\)\( T^{13} + \)\(30\!\cdots\!36\)\( T^{14} - 107767119920 T^{15} + T^{16} \)
$71$ \( ( \)\(25\!\cdots\!56\)\( + \)\(61\!\cdots\!92\)\( T - \)\(17\!\cdots\!64\)\( T^{2} - \)\(64\!\cdots\!48\)\( T^{3} - \)\(39\!\cdots\!36\)\( T^{4} + \)\(67\!\cdots\!12\)\( T^{5} + \)\(11\!\cdots\!96\)\( T^{6} + 575129514672 T^{7} + T^{8} )^{2} \)
$73$ \( \)\(80\!\cdots\!01\)\( - \)\(65\!\cdots\!40\)\( T + \)\(16\!\cdots\!12\)\( T^{2} + \)\(47\!\cdots\!20\)\( T^{3} - \)\(45\!\cdots\!82\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{5} + \)\(98\!\cdots\!52\)\( T^{6} - \)\(25\!\cdots\!40\)\( T^{7} - \)\(70\!\cdots\!33\)\( T^{8} + \)\(15\!\cdots\!40\)\( T^{9} + \)\(38\!\cdots\!04\)\( T^{10} + \)\(22\!\cdots\!40\)\( T^{11} - \)\(90\!\cdots\!74\)\( T^{12} - \)\(15\!\cdots\!00\)\( T^{13} + \)\(16\!\cdots\!20\)\( T^{14} + 738414283320 T^{15} + T^{16} \)
$79$ \( \)\(68\!\cdots\!25\)\( + \)\(88\!\cdots\!00\)\( T + \)\(14\!\cdots\!00\)\( T^{2} - \)\(38\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!50\)\( T^{4} + \)\(25\!\cdots\!00\)\( T^{5} + \)\(75\!\cdots\!00\)\( T^{6} + \)\(21\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!71\)\( T^{8} + \)\(40\!\cdots\!16\)\( T^{9} + \)\(32\!\cdots\!76\)\( T^{10} - \)\(18\!\cdots\!36\)\( T^{11} + \)\(35\!\cdots\!78\)\( T^{12} - \)\(22\!\cdots\!08\)\( T^{13} + \)\(25\!\cdots\!84\)\( T^{14} - 227632064768 T^{15} + T^{16} \)
$83$ \( \)\(52\!\cdots\!96\)\( + \)\(12\!\cdots\!00\)\( T^{2} + \)\(26\!\cdots\!88\)\( T^{4} + \)\(13\!\cdots\!24\)\( T^{6} + \)\(32\!\cdots\!44\)\( T^{8} + \)\(39\!\cdots\!36\)\( T^{10} + \)\(24\!\cdots\!56\)\( T^{12} + \)\(79\!\cdots\!16\)\( T^{14} + T^{16} \)
$89$ \( \)\(78\!\cdots\!01\)\( - \)\(42\!\cdots\!64\)\( T + \)\(94\!\cdots\!08\)\( T^{2} + \)\(38\!\cdots\!36\)\( T^{3} + \)\(50\!\cdots\!58\)\( T^{4} - \)\(37\!\cdots\!04\)\( T^{5} - \)\(23\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!28\)\( T^{7} + \)\(20\!\cdots\!75\)\( T^{8} + \)\(68\!\cdots\!44\)\( T^{9} + \)\(10\!\cdots\!40\)\( T^{10} + \)\(23\!\cdots\!12\)\( T^{11} - \)\(95\!\cdots\!82\)\( T^{12} - \)\(28\!\cdots\!32\)\( T^{13} + \)\(19\!\cdots\!92\)\( T^{14} + 2485007442792 T^{15} + T^{16} \)
$97$ \( \)\(16\!\cdots\!36\)\( + \)\(51\!\cdots\!56\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{4} + \)\(46\!\cdots\!04\)\( T^{6} + \)\(18\!\cdots\!24\)\( T^{8} + \)\(42\!\cdots\!44\)\( T^{10} + \)\(54\!\cdots\!00\)\( T^{12} + \)\(36\!\cdots\!36\)\( T^{14} + T^{16} \)
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