Properties

Label 12.15.d.a
Level 12
Weight 15
Character orbit 12.d
Analytic conductor 14.919
Analytic rank 0
Dimension 14
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 15 \)
Character orbit: \([\chi]\) = 12.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(14.9194761782\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{81}\cdot 3^{41} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( 13 - \beta_{1} ) q^{2} \) \( + \beta_{2} q^{3} \) \( + ( 665 - 13 \beta_{1} + \beta_{3} ) q^{4} \) \( + ( -1152 - 4 \beta_{1} + \beta_{3} - \beta_{4} ) q^{5} \) \( + ( 4062 + 13 \beta_{2} - \beta_{6} ) q^{6} \) \( + ( -4 - 888 \beta_{1} + 52 \beta_{2} - 24 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{7} \) \( + ( 310920 - 396 \beta_{1} - 128 \beta_{2} + 15 \beta_{3} + 6 \beta_{4} + \beta_{7} + \beta_{8} ) q^{8} \) \( -1594323 q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 13 - \beta_{1} ) q^{2} \) \( + \beta_{2} q^{3} \) \( + ( 665 - 13 \beta_{1} + \beta_{3} ) q^{4} \) \( + ( -1152 - 4 \beta_{1} + \beta_{3} - \beta_{4} ) q^{5} \) \( + ( 4062 + 13 \beta_{2} - \beta_{6} ) q^{6} \) \( + ( -4 - 888 \beta_{1} + 52 \beta_{2} - 24 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{7} \) \( + ( 310920 - 396 \beta_{1} - 128 \beta_{2} + 15 \beta_{3} + 6 \beta_{4} + \beta_{7} + \beta_{8} ) q^{8} \) \( -1594323 q^{9} \) \( + ( 41861 + 692 \beta_{1} + 2008 \beta_{2} - 15 \beta_{3} - 41 \beta_{4} - 11 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{13} ) q^{10} \) \( + ( 54 - 41799 \beta_{1} + 1648 \beta_{2} + 250 \beta_{3} + 7 \beta_{4} + 22 \beta_{5} - 16 \beta_{6} + 5 \beta_{7} + \beta_{8} + 8 \beta_{9} - 2 \beta_{10} + \beta_{11} - 3 \beta_{12} + 2 \beta_{13} ) q^{11} \) \( + ( 44347 - 4237 \beta_{1} + 667 \beta_{2} - 18 \beta_{3} - 34 \beta_{4} - 8 \beta_{5} - 10 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} - 2 \beta_{10} + \beta_{11} + 6 \beta_{12} + 5 \beta_{13} ) q^{12} \) \( + ( 7852367 + 207120 \beta_{1} - 483 \beta_{2} - 511 \beta_{3} - 121 \beta_{4} + 103 \beta_{5} - 58 \beta_{6} + 8 \beta_{7} + 22 \beta_{8} - 12 \beta_{9} - 11 \beta_{10} - \beta_{11} - 10 \beta_{12} - 5 \beta_{13} ) q^{13} \) \( + ( -14339567 - 9175 \beta_{1} - 10868 \beta_{2} + 877 \beta_{3} + 90 \beta_{4} - 213 \beta_{5} - 34 \beta_{6} - 45 \beta_{7} - 16 \beta_{8} + 47 \beta_{9} - 2 \beta_{10} + 18 \beta_{11} - 28 \beta_{12} - \beta_{13} ) q^{14} \) \( + ( -112 - 197510 \beta_{1} - 589 \beta_{2} - 747 \beta_{3} - 23 \beta_{4} + 92 \beta_{5} - 16 \beta_{6} + 24 \beta_{7} - 22 \beta_{8} - 44 \beta_{9} + 5 \beta_{10} + 11 \beta_{11} + 30 \beta_{12} - 8 \beta_{13} ) q^{15} \) \( + ( 27188220 - 298022 \beta_{1} + 21583 \beta_{2} + 638 \beta_{3} + 791 \beta_{4} - 179 \beta_{5} + 228 \beta_{6} + 11 \beta_{7} - 97 \beta_{8} - 40 \beta_{9} + 36 \beta_{10} + 59 \beta_{11} + 43 \beta_{12} - 46 \beta_{13} ) q^{16} \) \( + ( -20818238 + 233495 \beta_{1} - 614 \beta_{2} + 6324 \beta_{3} + 2573 \beta_{4} + 46 \beta_{5} + 224 \beta_{6} - 87 \beta_{7} - 171 \beta_{8} + 116 \beta_{9} + 82 \beta_{10} + 81 \beta_{11} - 55 \beta_{12} - 76 \beta_{13} ) q^{17} \) \( + ( -20726199 + 1594323 \beta_{1} ) q^{18} \) \( + ( 246 + 473041 \beta_{1} - 82626 \beta_{2} + 2488 \beta_{3} - 241 \beta_{4} - 490 \beta_{5} - 278 \beta_{6} - 335 \beta_{7} - 167 \beta_{8} + 218 \beta_{9} - 164 \beta_{10} + 215 \beta_{11} + 61 \beta_{12} + 106 \beta_{13} ) q^{19} \) \( + ( 379765454 + 289570 \beta_{1} - 105748 \beta_{2} - 2610 \beta_{3} - 4180 \beta_{4} + 1028 \beta_{5} - 2928 \beta_{6} + 4 \beta_{7} + 20 \beta_{8} - 480 \beta_{9} - 240 \beta_{10} + 300 \beta_{11} - 276 \beta_{12} - 120 \beta_{13} ) q^{20} \) \( + ( -79835945 - 1121002 \beta_{1} + 2721 \beta_{2} - 10116 \beta_{3} - 4822 \beta_{4} - 845 \beta_{5} - 1642 \beta_{6} - 42 \beta_{7} - 164 \beta_{8} - 148 \beta_{9} - 119 \beta_{10} + 37 \beta_{11} - 300 \beta_{12} - 229 \beta_{13} ) q^{21} \) \( + ( -669547122 - 628922 \beta_{1} + 298180 \beta_{2} + 47950 \beta_{3} + 10684 \beta_{4} + 2226 \beta_{5} - 488 \beta_{6} + 178 \beta_{7} + 160 \beta_{8} + 346 \beta_{9} - 140 \beta_{10} + 556 \beta_{11} - 392 \beta_{12} + 122 \beta_{13} ) q^{22} \) \( + ( 2448 + 5864164 \beta_{1} + 600448 \beta_{2} + 25896 \beta_{3} + 3490 \beta_{4} - 3176 \beta_{5} + 4698 \beta_{6} + 1924 \beta_{7} - 172 \beta_{8} + 1914 \beta_{9} - 216 \beta_{10} - 92 \beta_{11} + 196 \beta_{12} - 24 \beta_{13} ) q^{23} \) \( + ( 205742332 + 40346 \beta_{1} + 310309 \beta_{2} + 6579 \beta_{3} + 7199 \beta_{4} + 535 \beta_{5} + 1084 \beta_{6} + 552 \beta_{7} - 692 \beta_{8} + 2144 \beta_{9} + 316 \beta_{10} + 121 \beta_{11} - 255 \beta_{12} + 110 \beta_{13} ) q^{24} \) \( + ( 893364857 - 8866810 \beta_{1} + 21402 \beta_{2} + 48186 \beta_{3} + 18368 \beta_{4} - 2962 \beta_{5} - 8396 \beta_{6} + 874 \beta_{7} + 2694 \beta_{8} - 224 \beta_{9} - 950 \beta_{10} + 76 \beta_{11} - 1074 \beta_{12} - 942 \beta_{13} ) q^{25} \) \( + ( -3250810456 - 10347174 \beta_{1} - 441720 \beta_{2} - 220686 \beta_{3} - 51714 \beta_{4} + 674 \beta_{5} - 7084 \beta_{6} - 72 \beta_{7} - 730 \beta_{8} - 238 \beta_{9} + 28 \beta_{10} + 1246 \beta_{11} + 42 \beta_{12} - 658 \beta_{13} ) q^{26} \) \( -1594323 \beta_{2} q^{27} \) \( + ( 5969963684 + 18978044 \beta_{1} + 1206440 \beta_{2} - 15592 \beta_{3} - 23908 \beta_{4} - 1828 \beta_{5} + 10392 \beta_{6} - 3392 \beta_{7} + 4996 \beta_{8} + 2956 \beta_{9} - 520 \beta_{10} + 1008 \beta_{11} - 1764 \beta_{12} + 2020 \beta_{13} ) q^{28} \) \( + ( -866117598 + 22007966 \beta_{1} - 54486 \beta_{2} + 88301 \beta_{3} + 24121 \beta_{4} + 9758 \beta_{5} - 32860 \beta_{6} - 426 \beta_{7} - 5710 \beta_{8} + 6320 \beta_{9} + 1450 \beta_{10} - 3472 \beta_{11} + 5482 \beta_{12} + 1082 \beta_{13} ) q^{29} \) \( + ( -3210068567 - 2667907 \beta_{1} + 50846 \beta_{2} + 195273 \beta_{3} - 25822 \beta_{4} - 4217 \beta_{5} - 5308 \beta_{6} + 1911 \beta_{7} - 4112 \beta_{8} - 7261 \beta_{9} + 2278 \beta_{10} + 202 \beta_{11} + 420 \beta_{12} + 83 \beta_{13} ) q^{30} \) \( + ( -5328 - 40259314 \beta_{1} - 453284 \beta_{2} + 83900 \beta_{3} + 69117 \beta_{4} + 24452 \beta_{5} + 124229 \beta_{6} - 7458 \beta_{7} - 330 \beta_{8} - 5451 \beta_{9} - 1396 \beta_{10} - 6546 \beta_{11} + 3070 \beta_{12} - 2740 \beta_{13} ) q^{31} \) \( + ( 4855323880 - 23737092 \beta_{1} + 165042 \beta_{2} + 273432 \beta_{3} - 38342 \beta_{4} - 26842 \beta_{5} - 36392 \beta_{6} - 2730 \beta_{7} - 3394 \beta_{8} - 29888 \beta_{9} + 6872 \beta_{10} + 2266 \beta_{11} - 598 \beta_{12} + 2188 \beta_{13} ) q^{32} \) \( + ( -2453302598 + 27348203 \beta_{1} - 64104 \beta_{2} - 414162 \beta_{3} - 56689 \beta_{4} + 7048 \beta_{5} - 40972 \beta_{6} - 411 \beta_{7} - 7643 \beta_{8} + 572 \beta_{9} - 848 \beta_{10} - 4337 \beta_{11} + 3273 \beta_{12} + 1022 \beta_{13} ) q^{33} \) \( + ( -4090646020 + 18193586 \beta_{1} - 1405472 \beta_{2} - 169114 \beta_{3} + 153354 \beta_{4} - 24274 \beta_{5} + 2012 \beta_{6} - 13640 \beta_{7} + 16626 \beta_{8} - 45370 \beta_{9} + 2036 \beta_{10} + 3290 \beta_{11} - 658 \beta_{12} + 9018 \beta_{13} ) q^{34} \) \( + ( -150174 - 32367861 \beta_{1} - 14151888 \beta_{2} - 628158 \beta_{3} + 105901 \beta_{4} + 18594 \beta_{5} + 274484 \beta_{6} + 743 \beta_{7} + 4771 \beta_{8} + 13580 \beta_{9} + 3598 \beta_{10} - 19817 \beta_{11} + 4551 \beta_{12} - 9322 \beta_{13} ) q^{35} \) \( + ( -1060224795 + 20726199 \beta_{1} - 1594323 \beta_{3} ) q^{36} \) \( + ( 8526418905 + 40826900 \beta_{1} - 95445 \beta_{2} - 12107 \beta_{3} + 437127 \beta_{4} + 42225 \beta_{5} - 254406 \beta_{6} + 18460 \beta_{7} + 21822 \beta_{8} + 19676 \beta_{9} - 9837 \beta_{10} - 28795 \beta_{11} + 31534 \beta_{12} + 10645 \beta_{13} ) q^{37} \) \( + ( 7353787494 + 4214590 \beta_{1} + 7684188 \beta_{2} - 750090 \beta_{3} - 602004 \beta_{4} + 218 \beta_{5} + 38432 \beta_{6} - 4598 \beta_{7} + 7840 \beta_{8} + 94610 \beta_{9} + 23524 \beta_{10} + 12156 \beta_{11} - 17416 \beta_{12} + 16370 \beta_{13} ) q^{38} \) \( + ( 166056 + 43685436 \beta_{1} + 7748255 \beta_{2} + 1279449 \beta_{3} + 189102 \beta_{4} - 1128 \beta_{5} + 219705 \beta_{6} + 33786 \beta_{7} + 2580 \beta_{8} + 237 \beta_{9} - 2487 \beta_{10} - 15987 \beta_{11} + 2880 \beta_{12} - 5460 \beta_{13} ) q^{39} \) \( + ( -35115091536 - 418349256 \beta_{1} + 26630640 \beta_{2} - 767358 \beta_{3} - 1496796 \beta_{4} + 108304 \beta_{5} - 95296 \beta_{6} + 6446 \beta_{7} + 7342 \beta_{8} - 50112 \beta_{9} + 5568 \beta_{10} + 36016 \beta_{11} - 18576 \beta_{12} + 14176 \beta_{13} ) q^{40} \) \( + ( 13523454854 - 462633587 \beta_{1} + 1073986 \beta_{2} + 3604064 \beta_{3} - 388453 \beta_{4} - 207834 \beta_{5} - 854056 \beta_{6} + 20787 \beta_{7} + 9263 \beta_{8} + 62156 \beta_{9} - 24470 \beta_{10} - 34921 \beta_{11} + 1771 \beta_{12} - 43144 \beta_{13} ) q^{41} \) \( + ( 17181131131 + 97741310 \beta_{1} - 14542896 \beta_{2} + 689841 \beta_{3} - 1102729 \beta_{4} + 88189 \beta_{5} - 185542 \beta_{6} - 12876 \beta_{7} + 31603 \beta_{8} - 94015 \beta_{9} - 26690 \beta_{10} + 11887 \beta_{11} - 28515 \beta_{12} + 13247 \beta_{13} ) q^{42} \) \( + ( 530770 + 455131899 \beta_{1} - 23262210 \beta_{2} + 4918476 \beta_{3} + 794445 \beta_{4} - 201710 \beta_{5} + 1132082 \beta_{6} + 45043 \beta_{7} + 1171 \beta_{8} + 126802 \beta_{9} - 43368 \beta_{10} - 49855 \beta_{11} + 1487 \beta_{12} - 2658 \beta_{13} ) q^{43} \) \( + ( -71257367092 + 601613564 \beta_{1} + 69853924 \beta_{2} + 1069720 \beta_{3} + 708464 \beta_{4} + 162760 \beta_{5} - 213128 \beta_{6} - 10540 \beta_{7} + 16480 \beta_{8} - 9644 \beta_{9} + 46232 \beta_{10} + 80060 \beta_{11} - 6960 \beta_{12} + 16228 \beta_{13} ) q^{44} \) \( + ( 1836660096 + 6377292 \beta_{1} - 1594323 \beta_{3} + 1594323 \beta_{4} ) q^{45} \) \( + ( 97717662482 + 95015154 \beta_{1} - 57560656 \beta_{2} - 4991686 \beta_{3} + 2392308 \beta_{4} + 125750 \beta_{5} - 482748 \beta_{6} + 9030 \beta_{7} - 73248 \beta_{8} - 160130 \beta_{9} + 7228 \beta_{10} + 112868 \beta_{11} - 115000 \beta_{12} - 21730 \beta_{13} ) q^{46} \) \( + ( -2380392 - 218241896 \beta_{1} - 36338268 \beta_{2} - 12260644 \beta_{3} - 97590 \beta_{4} - 126192 \beta_{5} + 1258114 \beta_{6} - 40912 \beta_{7} - 127064 \beta_{8} + 192370 \beta_{9} - 8500 \beta_{10} + 31908 \beta_{11} + 170392 \beta_{12} - 43328 \beta_{13} ) q^{47} \) \( + ( -33259158436 - 241695542 \beta_{1} + 27821167 \beta_{2} + 1317834 \beta_{3} + 3231007 \beta_{4} + 40157 \beta_{5} + 52612 \beta_{6} - 59313 \beta_{7} + 26003 \beta_{8} - 93848 \beta_{9} - 37564 \beta_{10} + 47771 \beta_{11} - 66597 \beta_{12} + 35266 \beta_{13} ) q^{48} \) \( + ( -54939707405 + 968165156 \beta_{1} - 2264574 \beta_{2} - 10922226 \beta_{3} + 805598 \beta_{4} + 171174 \beta_{5} - 1405428 \beta_{6} - 76180 \beta_{7} - 307240 \beta_{8} + 17688 \beta_{9} - 22430 \beta_{10} - 12422 \beta_{11} - 113976 \beta_{12} - 137114 \beta_{13} ) q^{49} \) \( + ( 155003254611 - 746519667 \beta_{1} - 140246384 \beta_{2} + 6759788 \beta_{3} - 4476236 \beta_{4} - 394196 \beta_{5} - 703880 \beta_{6} - 68080 \beta_{7} - 87836 \beta_{8} - 113108 \beta_{9} + 85544 \beta_{10} + 208436 \beta_{11} + 119868 \beta_{12} - 43820 \beta_{13} ) q^{50} \) \( + ( 2492870 + 133464361 \beta_{1} - 21274756 \beta_{2} + 15532488 \beta_{3} + 980167 \beta_{4} + 57062 \beta_{5} + 374906 \beta_{6} - 187095 \beta_{7} - 1615 \beta_{8} - 96470 \beta_{9} - 116788 \beta_{10} + 18767 \beta_{11} - 65355 \beta_{12} + 66970 \beta_{13} ) q^{51} \) \( + ( -171383065534 + 3094916262 \beta_{1} + 192915288 \beta_{2} + 7520434 \beta_{3} - 6989096 \beta_{4} - 1153144 \beta_{5} - 330464 \beta_{6} - 93176 \beta_{7} - 158424 \beta_{8} - 215744 \beta_{9} - 13792 \beta_{10} + 56792 \beta_{11} - 167848 \beta_{12} + 12048 \beta_{13} ) q^{52} \) \( + ( 89393722386 + 518221354 \beta_{1} - 1553950 \beta_{2} + 49700773 \beta_{3} + 2451701 \beta_{4} + 670918 \beta_{5} - 288188 \beta_{6} - 2070 \beta_{7} + 117206 \beta_{8} + 571648 \beta_{9} + 180674 \beta_{10} + 52340 \beta_{11} + 164382 \beta_{12} - 68390 \beta_{13} ) q^{53} \) \( + ( -6476140026 - 20726199 \beta_{2} + 1594323 \beta_{6} ) q^{54} \) \( + ( 3550292 - 3312036762 \beta_{1} - 59326728 \beta_{2} + 17332980 \beta_{3} + 950166 \beta_{4} + 1993988 \beta_{5} - 934100 \beta_{6} + 527486 \beta_{7} - 78058 \beta_{8} - 640996 \beta_{9} + 7788 \beta_{10} + 39022 \beta_{11} + 101470 \beta_{12} - 23412 \beta_{13} ) q^{55} \) \( + ( 165653103888 - 6021311592 \beta_{1} + 516902348 \beta_{2} - 20758940 \beta_{3} - 1850876 \beta_{4} - 1486716 \beta_{5} - 574448 \beta_{6} - 171888 \beta_{7} + 66112 \beta_{8} + 1498816 \beta_{9} + 88592 \beta_{10} + 30460 \beta_{11} + 7964 \beta_{12} - 187064 \beta_{13} ) q^{56} \) \( + ( 128924750004 + 882496227 \beta_{1} - 1663614 \beta_{2} - 45566604 \beta_{3} - 2347935 \beta_{4} + 374118 \beta_{5} - 116448 \beta_{6} + 232029 \beta_{7} + 567609 \beta_{8} - 718332 \beta_{9} - 362406 \beta_{10} - 90363 \beta_{11} - 211779 \beta_{12} + 79140 \beta_{13} ) q^{57} \) \( + ( -368391277393 + 763308408 \beta_{1} - 702651624 \beta_{2} - 17023345 \beta_{3} + 11525993 \beta_{4} - 657125 \beta_{5} + 1456262 \beta_{6} + 421980 \beta_{7} + 12509 \beta_{8} + 837759 \beta_{9} - 191102 \beta_{10} - 384799 \beta_{11} + 266563 \beta_{12} + 376449 \beta_{13} ) q^{58} \) \( + ( -10524240 + 2527400872 \beta_{1} + 398724272 \beta_{2} - 68826044 \beta_{3} - 6346848 \beta_{4} - 1976560 \beta_{5} - 4282508 \beta_{6} - 21952 \beta_{7} + 480808 \beta_{8} + 1678660 \beta_{9} + 417188 \beta_{10} - 110148 \beta_{11} - 457544 \beta_{12} - 23264 \beta_{13} ) q^{59} \) \( + ( 168161866570 + 3283861682 \beta_{1} + 372765182 \beta_{2} + 6315156 \beta_{3} + 8183144 \beta_{4} + 1949788 \beta_{5} + 792996 \beta_{6} + 501510 \beta_{7} + 261232 \beta_{8} - 1228378 \beta_{9} - 238700 \beta_{10} - 292238 \beta_{11} + 312 \beta_{12} + 78494 \beta_{13} ) q^{60} \) \( + ( -563050553931 + 1067676712 \beta_{1} - 1921737 \beta_{2} - 83242403 \beta_{3} - 9583237 \beta_{4} + 306645 \beta_{5} + 6027474 \beta_{6} + 131352 \beta_{7} + 433346 \beta_{8} - 1324324 \beta_{9} - 236129 \beta_{10} - 17091 \beta_{11} - 8606 \beta_{12} + 604721 \beta_{13} ) q^{61} \) \( + ( -654380481439 - 80378263 \beta_{1} - 1835737964 \beta_{2} + 39118349 \beta_{3} - 8167974 \beta_{4} + 1313643 \beta_{5} - 2364266 \beta_{6} - 264141 \beta_{7} + 738800 \beta_{8} - 21233 \beta_{9} + 213950 \beta_{10} - 815406 \beta_{11} - 62300 \beta_{12} - 553121 \beta_{13} ) q^{62} \) \( + ( 6377292 + 1415758824 \beta_{1} - 82904796 \beta_{2} + 38263752 \beta_{3} + 1594323 \beta_{4} - 1594323 \beta_{6} - 1594323 \beta_{9} ) q^{63} \) \( + ( 1251501114624 - 4377304000 \beta_{1} + 1860940496 \beta_{2} + 27538232 \beta_{3} + 12475040 \beta_{4} + 3992144 \beta_{5} + 2895360 \beta_{6} + 982872 \beta_{7} + 336120 \beta_{8} - 1695200 \beta_{9} - 344320 \beta_{10} - 1087344 \beta_{11} + 934576 \beta_{12} + 89408 \beta_{13} ) q^{64} \) \( + ( 418449970296 - 14584951545 \beta_{1} + 33228446 \beta_{2} + 153149576 \beta_{3} - 19608503 \beta_{4} - 6235526 \beta_{5} + 12127192 \beta_{6} - 692487 \beta_{7} - 1028227 \beta_{8} + 1310788 \beta_{9} + 1306262 \beta_{10} + 617645 \beta_{11} + 843057 \beta_{12} + 542032 \beta_{13} ) q^{65} \) \( + ( -472481155394 + 2279925752 \beta_{1} - 674439312 \beta_{2} - 25276122 \beta_{3} + 1091210 \beta_{4} + 2990302 \beta_{5} + 197084 \beta_{6} + 248664 \beta_{7} - 37166 \beta_{8} + 507014 \beta_{9} - 565772 \beta_{10} - 495686 \beta_{11} - 130962 \beta_{12} + 445562 \beta_{13} ) q^{66} \) \( + ( 12452224 + 4159708608 \beta_{1} + 264031976 \beta_{2} + 55907412 \beta_{3} - 5417000 \beta_{4} - 947424 \beta_{5} - 15929212 \beta_{6} - 3073992 \beta_{7} + 888576 \beta_{8} - 2505964 \beta_{9} - 36204 \beta_{10} + 353844 \beta_{11} - 1527888 \beta_{12} + 639312 \beta_{13} ) q^{67} \) \( + ( 1339123329306 + 4586489102 \beta_{1} + 2515584264 \beta_{2} - 21135862 \beta_{3} - 1305208 \beta_{4} + 5366936 \beta_{5} + 5824864 \beta_{6} + 99224 \beta_{7} - 1048584 \beta_{8} + 3231424 \beta_{9} + 1003616 \beta_{10} - 1212792 \beta_{11} + 2472712 \beta_{12} - 576464 \beta_{13} ) q^{68} \) \( + ( -984104184346 - 6408946664 \beta_{1} + 15470106 \beta_{2} - 89050176 \beta_{3} + 7485184 \beta_{4} - 4618642 \beta_{5} + 7513036 \beta_{6} - 779520 \beta_{7} - 2009188 \beta_{8} - 669656 \beta_{9} + 464426 \beta_{10} + 358286 \beta_{11} - 166884 \beta_{12} + 275542 \beta_{13} ) q^{69} \) \( + ( -586673566646 + 593564874 \beta_{1} - 4305803528 \beta_{2} + 37716386 \beta_{3} + 10871524 \beta_{4} - 4504578 \beta_{5} + 11274364 \beta_{6} - 1692578 \beta_{7} + 370528 \beta_{8} - 2414858 \beta_{9} - 465620 \beta_{10} - 1307468 \beta_{11} - 419576 \beta_{12} - 1677226 \beta_{13} ) q^{70} \) \( + ( -38592288 - 882776068 \beta_{1} - 280662020 \beta_{2} - 263746636 \beta_{3} - 30428222 \beta_{4} - 2042744 \beta_{5} - 33786410 \beta_{6} - 108828 \beta_{7} - 1098516 \beta_{8} - 4051610 \beta_{9} + 1950548 \beta_{10} + 1957240 \beta_{11} + 1644428 \beta_{12} - 545912 \beta_{13} ) q^{71} \) \( + ( -495706907160 + 631351908 \beta_{1} + 204073344 \beta_{2} - 23914845 \beta_{3} - 9565938 \beta_{4} - 1594323 \beta_{7} - 1594323 \beta_{8} ) q^{72} \) \( + ( 2798930668672 + 32462683644 \beta_{1} - 75636546 \beta_{2} - 65822142 \beta_{3} + 35754738 \beta_{4} + 17192858 \beta_{5} + 29237236 \beta_{6} + 321716 \beta_{7} + 3684040 \beta_{8} - 2763416 \beta_{9} - 10530 \beta_{10} + 1760166 \beta_{11} - 1289640 \beta_{12} + 810266 \beta_{13} ) q^{73} \) \( + ( -549889189338 - 7695845306 \beta_{1} - 5385189624 \beta_{2} - 27589332 \beta_{3} + 33860180 \beta_{4} - 2306764 \beta_{5} + 6331320 \beta_{6} + 2152128 \beta_{7} - 4638124 \beta_{8} + 10671084 \beta_{9} + 835560 \beta_{10} - 261468 \beta_{11} + 4351452 \beta_{12} + 497556 \beta_{13} ) q^{74} \) \( + ( 33561132 + 13661140602 \beta_{1} + 855921417 \beta_{2} + 195709986 \beta_{3} + 5652318 \beta_{4} - 4740852 \beta_{5} - 9956334 \beta_{6} + 3172806 \beta_{7} + 923082 \beta_{8} + 677082 \beta_{9} - 457014 \beta_{10} - 78312 \beta_{11} - 1573974 \beta_{12} + 650892 \beta_{13} ) q^{75} \) \( + ( -658892046876 - 9852776556 \beta_{1} + 7524432796 \beta_{2} + 23335688 \beta_{3} + 93663232 \beta_{4} - 27007000 \beta_{5} + 10817576 \beta_{6} - 1878388 \beta_{7} + 1549616 \beta_{8} + 5778876 \beta_{9} - 2548856 \beta_{10} + 251684 \beta_{11} - 2823264 \beta_{12} - 593268 \beta_{13} ) q^{76} \) \( + ( -2949298952812 - 2417731266 \beta_{1} + 6391160 \beta_{2} + 4781684 \beta_{3} - 82350754 \beta_{4} - 2202008 \beta_{5} - 3615816 \beta_{6} + 136290 \beta_{7} + 1944722 \beta_{8} - 2760264 \beta_{9} - 1520664 \beta_{10} + 1656638 \beta_{11} - 4443718 \beta_{12} - 2551164 \beta_{13} ) q^{77} \) \( + ( 747943786566 + 1428948258 \beta_{1} - 3246657454 \beta_{2} - 19070910 \beta_{3} + 50981460 \beta_{4} + 10804710 \beta_{5} - 7868078 \beta_{6} + 1103166 \beta_{7} + 194592 \beta_{8} - 6023082 \beta_{9} - 344724 \beta_{10} + 357780 \beta_{11} - 1101384 \beta_{12} - 1459434 \beta_{13} ) q^{78} \) \( + ( 30739116 - 68769365464 \beta_{1} + 2920027184 \beta_{2} + 94604588 \beta_{3} - 2099725 \beta_{4} + 33249408 \beta_{5} - 20317639 \beta_{6} + 3520312 \beta_{7} - 1461344 \beta_{8} + 4391177 \beta_{9} - 1810012 \beta_{10} + 2810276 \beta_{11} - 118128 \beta_{12} + 1579472 \beta_{13} ) q^{79} \) \( + ( -4080515077240 + 28770301324 \beta_{1} + 13058113730 \beta_{2} + 402376708 \beta_{3} - 53633614 \beta_{4} - 5386106 \beta_{5} - 27105608 \beta_{6} + 4034442 \beta_{7} + 852962 \beta_{8} - 713648 \beta_{9} - 1640392 \beta_{10} + 134250 \beta_{11} - 4074614 \beta_{12} - 1256804 \beta_{13} ) q^{80} \) \( + 2541865828329 q^{81} \) \( + ( 7647904607256 - 4413071286 \beta_{1} - 13165216576 \beta_{2} + 382893786 \beta_{3} - 152704202 \beta_{4} - 27821454 \beta_{5} - 29739100 \beta_{6} + 5437064 \beta_{7} + 3240974 \beta_{8} - 2026630 \beta_{9} - 3493236 \beta_{10} + 2978598 \beta_{11} + 2221138 \beta_{12} + 3439750 \beta_{13} ) q^{82} \) \( + ( 13764438 + 24060875737 \beta_{1} - 1012672300 \beta_{2} + 110042518 \beta_{3} + 2763175 \beta_{4} - 16276810 \beta_{5} + 5473956 \beta_{6} - 2203379 \beta_{7} - 580063 \beta_{8} + 13555692 \beta_{9} - 3211878 \beta_{10} + 1959133 \beta_{11} - 1715411 \beta_{12} + 2295474 \beta_{13} ) q^{83} \) \( + ( -2007422947940 - 20431414852 \beta_{1} + 6028750764 \beta_{2} - 138703284 \beta_{3} - 127204756 \beta_{4} + 22463812 \beta_{5} + 2623376 \beta_{6} + 5280708 \beta_{7} + 3084244 \beta_{8} + 5697824 \beta_{9} - 1562096 \beta_{10} - 328724 \beta_{11} - 1730388 \beta_{12} - 936184 \beta_{13} ) q^{84} \) \( + ( -13491389095480 + 57729228984 \beta_{1} - 139600032 \beta_{2} + 199973182 \beta_{3} + 13901362 \beta_{4} + 23825952 \beta_{5} - 30592704 \beta_{6} - 2456656 \beta_{7} - 7775696 \beta_{8} + 6042944 \beta_{9} + 1798976 \beta_{10} + 448720 \beta_{11} - 905744 \beta_{12} - 3653664 \beta_{13} ) q^{85} \) \( + ( 7317417051658 + 10140221554 \beta_{1} - 16313276748 \beta_{2} - 409121318 \beta_{3} + 116482164 \beta_{4} + 35416502 \beta_{5} + 16949200 \beta_{6} - 1334234 \beta_{7} + 5907808 \beta_{8} + 640606 \beta_{9} + 792572 \beta_{10} + 2843940 \beta_{11} - 9433208 \beta_{12} - 4277378 \beta_{13} ) q^{86} \) \( + ( -85220188 + 69255871828 \beta_{1} - 1060153765 \beta_{2} - 395179551 \beta_{3} - 15300299 \beta_{4} - 42183208 \beta_{5} + 26514242 \beta_{6} - 6274326 \beta_{7} - 2202172 \beta_{8} + 5491318 \beta_{9} + 400169 \beta_{10} + 569213 \beta_{11} + 3218736 \beta_{12} - 1016564 \beta_{13} ) q^{87} \) \( + ( -5947368801296 + 62945092968 \beta_{1} + 18008180308 \beta_{2} - 568707220 \beta_{3} + 94914556 \beta_{4} - 37930980 \beta_{5} - 44017424 \beta_{6} - 2425152 \beta_{7} - 4696368 \beta_{8} - 1690880 \beta_{9} + 5945584 \beta_{10} + 3433636 \beta_{11} + 5388356 \beta_{12} + 942264 \beta_{13} ) q^{88} \) \( + ( 15980176834306 - 87182954506 \beta_{1} + 207220372 \beta_{2} - 137709232 \beta_{3} + 232931082 \beta_{4} - 41708932 \beta_{5} - 32750496 \beta_{6} + 601962 \beta_{7} - 3055358 \beta_{8} + 4156296 \beta_{9} + 238212 \beta_{10} - 3877622 \beta_{11} + 5114314 \beta_{12} + 1553304 \beta_{13} ) q^{89} \) \( + ( -66739955103 - 1103271516 \beta_{1} - 3201400584 \beta_{2} + 23914845 \beta_{3} + 65367243 \beta_{4} + 17537553 \beta_{5} + 9565938 \beta_{6} - 6377292 \beta_{7} - 4782969 \beta_{8} - 1594323 \beta_{9} + 3188646 \beta_{10} + 1594323 \beta_{11} + 4782969 \beta_{12} + 1594323 \beta_{13} ) q^{90} \) \( + ( -140746686 - 102573136349 \beta_{1} - 8913681770 \beta_{2} - 756731984 \beta_{3} + 40346173 \beta_{4} + 51973026 \beta_{5} + 139363126 \beta_{6} + 5708915 \beta_{7} - 2290117 \beta_{8} - 10260314 \beta_{9} + 5659228 \beta_{10} - 9185123 \beta_{11} + 10857351 \beta_{12} - 8567234 \beta_{13} ) q^{91} \) \( + ( -5706802207184 - 106934588384 \beta_{1} + 14696480952 \beta_{2} - 49158080 \beta_{3} + 14369096 \beta_{4} + 93097592 \beta_{5} + 62952704 \beta_{6} - 15919960 \beta_{7} - 3681000 \beta_{8} - 41609680 \beta_{9} + 3334528 \beta_{10} + 9418136 \beta_{11} + 2485000 \beta_{12} + 8396768 \beta_{13} ) q^{92} \) \( + ( 906397032479 - 180302488400 \beta_{1} + 424313781 \beta_{2} + 766343448 \beta_{3} - 16957100 \beta_{4} - 75487825 \beta_{5} - 45227930 \beta_{6} + 4508556 \beta_{7} + 10948970 \beta_{8} + 5481220 \beta_{9} - 2079763 \beta_{10} - 2767501 \beta_{11} + 2717850 \beta_{12} - 847013 \beta_{13} ) q^{93} \) \( + ( -3763860366514 + 1356616190 \beta_{1} - 17819544728 \beta_{2} + 84292822 \beta_{3} - 289097620 \beta_{4} - 107293414 \beta_{5} - 11741692 \beta_{6} - 14395222 \beta_{7} - 17557984 \beta_{8} + 4822578 \beta_{9} + 14907684 \beta_{10} + 1781308 \beta_{11} - 11366984 \beta_{12} - 1681134 \beta_{13} ) q^{94} \) \( + ( 329393952 + 229390371088 \beta_{1} + 1580856584 \beta_{2} + 2502610144 \beta_{3} + 275784752 \beta_{4} - 74679232 \beta_{5} + 277631648 \beta_{6} + 5534256 \beta_{7} + 7790832 \beta_{8} - 33791200 \beta_{9} - 6339104 \beta_{10} - 22279504 \beta_{11} - 3716048 \beta_{12} - 4074784 \beta_{13} ) q^{95} \) \( + ( -161641525096 + 32199583876 \beta_{1} + 4799196574 \beta_{2} + 291205296 \beta_{3} + 74017222 \beta_{4} + 92409578 \beta_{5} - 8374872 \beta_{6} - 10649550 \beta_{7} - 819286 \beta_{8} + 7964320 \beta_{9} + 8063912 \beta_{10} + 5078582 \beta_{11} + 7531590 \beta_{12} + 3841780 \beta_{13} ) q^{96} \) \( + ( 20207295792302 + 296730561386 \beta_{1} - 706780920 \beta_{2} - 271091844 \beta_{3} - 275559286 \beta_{4} + 137236248 \beta_{5} - 253677144 \beta_{6} + 6148022 \beta_{7} - 9533050 \beta_{8} + 16326504 \beta_{9} - 6223976 \beta_{10} - 20514038 \beta_{11} + 16184382 \beta_{12} - 389108 \beta_{13} ) q^{97} \) \( + ( -16331663298555 + 49191847551 \beta_{1} - 25362695632 \beta_{2} - 1024393400 \beta_{3} - 307923528 \beta_{4} + 101569144 \beta_{5} - 68444976 \beta_{6} - 19178624 \beta_{7} + 16644280 \beta_{8} - 64165944 \beta_{9} - 16070928 \beta_{10} + 1705368 \beta_{11} - 8010648 \beta_{12} + 21915960 \beta_{13} ) q^{98} \) \( + ( -86093442 + 66641107077 \beta_{1} - 2627444304 \beta_{2} - 398580750 \beta_{3} - 11160261 \beta_{4} - 35075106 \beta_{5} + 25509168 \beta_{6} - 7971615 \beta_{7} - 1594323 \beta_{8} - 12754584 \beta_{9} + 3188646 \beta_{10} - 1594323 \beta_{11} + 4782969 \beta_{12} - 3188646 \beta_{13} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut +\mathstrut 182q^{2} \) \(\mathstrut +\mathstrut 9308q^{4} \) \(\mathstrut -\mathstrut 16124q^{5} \) \(\mathstrut +\mathstrut 56862q^{6} \) \(\mathstrut +\mathstrut 4352816q^{8} \) \(\mathstrut -\mathstrut 22320522q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut +\mathstrut 182q^{2} \) \(\mathstrut +\mathstrut 9308q^{4} \) \(\mathstrut -\mathstrut 16124q^{5} \) \(\mathstrut +\mathstrut 56862q^{6} \) \(\mathstrut +\mathstrut 4352816q^{8} \) \(\mathstrut -\mathstrut 22320522q^{9} \) \(\mathstrut +\mathstrut 586324q^{10} \) \(\mathstrut +\mathstrut 621108q^{12} \) \(\mathstrut +\mathstrut 109934140q^{13} \) \(\mathstrut -\mathstrut 200755992q^{14} \) \(\mathstrut +\mathstrut 380631536q^{16} \) \(\mathstrut -\mathstrut 291483380q^{17} \) \(\mathstrut -\mathstrut 290166786q^{18} \) \(\mathstrut +\mathstrut 5316726088q^{20} \) \(\mathstrut -\mathstrut 1117661976q^{21} \) \(\mathstrut -\mathstrut 9373833288q^{22} \) \(\mathstrut +\mathstrut 2880331488q^{24} \) \(\mathstrut +\mathstrut 12506859258q^{25} \) \(\mathstrut -\mathstrut 45510637748q^{26} \) \(\mathstrut +\mathstrut 83579713776q^{28} \) \(\mathstrut -\mathstrut 12126204812q^{29} \) \(\mathstrut -\mathstrut 44941179132q^{30} \) \(\mathstrut +\mathstrut 67974212192q^{32} \) \(\mathstrut -\mathstrut 34345330344q^{33} \) \(\mathstrut -\mathstrut 57269346212q^{34} \) \(\mathstrut -\mathstrut 14839958484q^{36} \) \(\mathstrut +\mathstrut 119365701580q^{37} \) \(\mathstrut +\mathstrut 102957884712q^{38} \) \(\mathstrut -\mathstrut 491601579872q^{40} \) \(\mathstrut +\mathstrut 189318893932q^{41} \) \(\mathstrut +\mathstrut 240539889384q^{42} \) \(\mathstrut -\mathstrut 997611383472q^{44} \) \(\mathstrut +\mathstrut 25706864052q^{45} \) \(\mathstrut +\mathstrut 1368039641184q^{46} \) \(\mathstrut -\mathstrut 465649986384q^{48} \) \(\mathstrut -\mathstrut 769149171250q^{49} \) \(\mathstrut +\mathstrut 2170057449522q^{50} \) \(\mathstrut -\mathstrut 2399333559176q^{52} \) \(\mathstrut +\mathstrut 1251391890964q^{53} \) \(\mathstrut -\mathstrut 90656394426q^{54} \) \(\mathstrut +\mathstrut 2319191796096q^{56} \) \(\mathstrut +\mathstrut 1805052294792q^{57} \) \(\mathstrut -\mathstrut 5157502168892q^{58} \) \(\mathstrut +\mathstrut 2354207329944q^{60} \) \(\mathstrut -\mathstrut 7882441676660q^{61} \) \(\mathstrut -\mathstrut 9161379391272q^{62} \) \(\mathstrut +\mathstrut 17520900128384q^{64} \) \(\mathstrut +\mathstrut 5858206778312q^{65} \) \(\mathstrut -\mathstrut 6614704234440q^{66} \) \(\mathstrut +\mathstrut 18747786717976q^{68} \) \(\mathstrut -\mathstrut 13777261381728q^{69} \) \(\mathstrut -\mathstrut 8213486211792q^{70} \) \(\mathstrut -\mathstrut 6939794663568q^{72} \) \(\mathstrut +\mathstrut 39185062250428q^{73} \) \(\mathstrut -\mathstrut 7698562888484q^{74} \) \(\mathstrut -\mathstrut 9224963770896q^{76} \) \(\mathstrut -\mathstrut 41289727781472q^{77} \) \(\mathstrut +\mathstrut 10470873014172q^{78} \) \(\mathstrut -\mathstrut 57127847610848q^{80} \) \(\mathstrut +\mathstrut 35586121596606q^{81} \) \(\mathstrut +\mathstrut 107070799921084q^{82} \) \(\mathstrut -\mathstrut 28102976768880q^{84} \) \(\mathstrut -\mathstrut 188880254078680q^{85} \) \(\mathstrut +\mathstrut 102443851819896q^{86} \) \(\mathstrut -\mathstrut 83262676567680q^{88} \) \(\mathstrut +\mathstrut 223721333984572q^{89} \) \(\mathstrut -\mathstrut 934789838652q^{90} \) \(\mathstrut -\mathstrut 79895035003584q^{92} \) \(\mathstrut +\mathstrut 12688158423960q^{93} \) \(\mathstrut -\mathstrut 52692266305296q^{94} \) \(\mathstrut -\mathstrut 2264434006752q^{96} \) \(\mathstrut +\mathstrut 282902280361756q^{97} \) \(\mathstrut -\mathstrut 228639957171082q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(x^{13}\mathstrut +\mathstrut \) \(9158\) \(x^{12}\mathstrut +\mathstrut \) \(65217\) \(x^{11}\mathstrut +\mathstrut \) \(61148515\) \(x^{10}\mathstrut +\mathstrut \) \(439019974\) \(x^{9}\mathstrut +\mathstrut \) \(189458968156\) \(x^{8}\mathstrut +\mathstrut \) \(1788546506656\) \(x^{7}\mathstrut +\mathstrut \) \(430738312102192\) \(x^{6}\mathstrut +\mathstrut \) \(3436431888153888\) \(x^{5}\mathstrut +\mathstrut \) \(232550799864813632\) \(x^{4}\mathstrut +\mathstrut \) \(2976833070807329792\) \(x^{3}\mathstrut +\mathstrut \) \(104055634594691484928\) \(x^{2}\mathstrut +\mathstrut \) \(901843441512469175808\) \(x\mathstrut +\mathstrut \) \(8926647999096479376384\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-\)\(19\!\cdots\!83\) \(\nu^{13}\mathstrut -\mathstrut \) \(89\!\cdots\!57\) \(\nu^{12}\mathstrut -\mathstrut \) \(14\!\cdots\!62\) \(\nu^{11}\mathstrut -\mathstrut \) \(97\!\cdots\!07\) \(\nu^{10}\mathstrut -\mathstrut \) \(99\!\cdots\!85\) \(\nu^{9}\mathstrut -\mathstrut \) \(63\!\cdots\!34\) \(\nu^{8}\mathstrut -\mathstrut \) \(25\!\cdots\!84\) \(\nu^{7}\mathstrut -\mathstrut \) \(20\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(54\!\cdots\!76\) \(\nu^{5}\mathstrut -\mathstrut \) \(43\!\cdots\!84\) \(\nu^{4}\mathstrut +\mathstrut \) \(20\!\cdots\!88\) \(\nu^{3}\mathstrut -\mathstrut \) \(22\!\cdots\!68\) \(\nu^{2}\mathstrut -\mathstrut \) \(14\!\cdots\!92\) \(\nu\mathstrut -\mathstrut \) \(58\!\cdots\!76\)\()/\)\(25\!\cdots\!00\)
\(\beta_{2}\)\(=\)\((\)\(44\!\cdots\!16\) \(\nu^{13}\mathstrut +\mathstrut \) \(23\!\cdots\!57\) \(\nu^{12}\mathstrut +\mathstrut \) \(40\!\cdots\!45\) \(\nu^{11}\mathstrut +\mathstrut \) \(32\!\cdots\!88\) \(\nu^{10}\mathstrut +\mathstrut \) \(26\!\cdots\!33\) \(\nu^{9}\mathstrut +\mathstrut \) \(21\!\cdots\!57\) \(\nu^{8}\mathstrut +\mathstrut \) \(82\!\cdots\!16\) \(\nu^{7}\mathstrut +\mathstrut \) \(77\!\cdots\!92\) \(\nu^{6}\mathstrut +\mathstrut \) \(18\!\cdots\!36\) \(\nu^{5}\mathstrut +\mathstrut \) \(14\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(92\!\cdots\!24\) \(\nu^{3}\mathstrut +\mathstrut \) \(50\!\cdots\!72\) \(\nu^{2}\mathstrut +\mathstrut \) \(40\!\cdots\!80\) \(\nu\mathstrut +\mathstrut \) \(15\!\cdots\!04\)\()/\)\(26\!\cdots\!40\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(52\!\cdots\!63\) \(\nu^{13}\mathstrut +\mathstrut \) \(69\!\cdots\!19\) \(\nu^{12}\mathstrut -\mathstrut \) \(47\!\cdots\!70\) \(\nu^{11}\mathstrut +\mathstrut \) \(20\!\cdots\!33\) \(\nu^{10}\mathstrut -\mathstrut \) \(31\!\cdots\!21\) \(\nu^{9}\mathstrut +\mathstrut \) \(12\!\cdots\!98\) \(\nu^{8}\mathstrut -\mathstrut \) \(93\!\cdots\!20\) \(\nu^{7}\mathstrut +\mathstrut \) \(57\!\cdots\!00\) \(\nu^{6}\mathstrut -\mathstrut \) \(20\!\cdots\!36\) \(\nu^{5}\mathstrut +\mathstrut \) \(33\!\cdots\!88\) \(\nu^{4}\mathstrut -\mathstrut \) \(85\!\cdots\!08\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!72\) \(\nu^{2}\mathstrut -\mathstrut \) \(31\!\cdots\!28\) \(\nu\mathstrut +\mathstrut \) \(18\!\cdots\!80\)\()/\)\(25\!\cdots\!00\)
\(\beta_{4}\)\(=\)\((\)\(11\!\cdots\!43\) \(\nu^{13}\mathstrut +\mathstrut \) \(17\!\cdots\!89\) \(\nu^{12}\mathstrut +\mathstrut \) \(93\!\cdots\!26\) \(\nu^{11}\mathstrut +\mathstrut \) \(16\!\cdots\!67\) \(\nu^{10}\mathstrut +\mathstrut \) \(72\!\cdots\!13\) \(\nu^{9}\mathstrut +\mathstrut \) \(10\!\cdots\!58\) \(\nu^{8}\mathstrut +\mathstrut \) \(21\!\cdots\!72\) \(\nu^{7}\mathstrut +\mathstrut \) \(33\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(56\!\cdots\!96\) \(\nu^{5}\mathstrut +\mathstrut \) \(74\!\cdots\!88\) \(\nu^{4}\mathstrut +\mathstrut \) \(25\!\cdots\!00\) \(\nu^{3}\mathstrut +\mathstrut \) \(36\!\cdots\!80\) \(\nu^{2}\mathstrut +\mathstrut \) \(28\!\cdots\!00\) \(\nu\mathstrut +\mathstrut \) \(95\!\cdots\!28\)\()/\)\(50\!\cdots\!00\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(49\!\cdots\!15\) \(\nu^{13}\mathstrut -\mathstrut \) \(17\!\cdots\!93\) \(\nu^{12}\mathstrut -\mathstrut \) \(40\!\cdots\!86\) \(\nu^{11}\mathstrut -\mathstrut \) \(19\!\cdots\!15\) \(\nu^{10}\mathstrut -\mathstrut \) \(27\!\cdots\!97\) \(\nu^{9}\mathstrut -\mathstrut \) \(12\!\cdots\!26\) \(\nu^{8}\mathstrut -\mathstrut \) \(73\!\cdots\!12\) \(\nu^{7}\mathstrut -\mathstrut \) \(41\!\cdots\!40\) \(\nu^{6}\mathstrut -\mathstrut \) \(16\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(88\!\cdots\!96\) \(\nu^{4}\mathstrut -\mathstrut \) \(30\!\cdots\!12\) \(\nu^{3}\mathstrut -\mathstrut \) \(48\!\cdots\!88\) \(\nu^{2}\mathstrut -\mathstrut \) \(59\!\cdots\!92\) \(\nu\mathstrut -\mathstrut \) \(12\!\cdots\!48\)\()/\)\(25\!\cdots\!00\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(16\!\cdots\!07\) \(\nu^{13}\mathstrut +\mathstrut \) \(15\!\cdots\!55\) \(\nu^{12}\mathstrut -\mathstrut \) \(14\!\cdots\!22\) \(\nu^{11}\mathstrut +\mathstrut \) \(13\!\cdots\!77\) \(\nu^{10}\mathstrut -\mathstrut \) \(98\!\cdots\!93\) \(\nu^{9}\mathstrut +\mathstrut \) \(79\!\cdots\!70\) \(\nu^{8}\mathstrut -\mathstrut \) \(29\!\cdots\!44\) \(\nu^{7}\mathstrut -\mathstrut \) \(54\!\cdots\!80\) \(\nu^{6}\mathstrut -\mathstrut \) \(65\!\cdots\!04\) \(\nu^{5}\mathstrut -\mathstrut \) \(31\!\cdots\!60\) \(\nu^{4}\mathstrut -\mathstrut \) \(28\!\cdots\!96\) \(\nu^{3}\mathstrut -\mathstrut \) \(28\!\cdots\!24\) \(\nu^{2}\mathstrut -\mathstrut \) \(87\!\cdots\!36\) \(\nu\mathstrut -\mathstrut \) \(40\!\cdots\!36\)\()/\)\(31\!\cdots\!00\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(12\!\cdots\!19\) \(\nu^{13}\mathstrut +\mathstrut \) \(19\!\cdots\!99\) \(\nu^{12}\mathstrut -\mathstrut \) \(12\!\cdots\!66\) \(\nu^{11}\mathstrut +\mathstrut \) \(77\!\cdots\!49\) \(\nu^{10}\mathstrut -\mathstrut \) \(80\!\cdots\!05\) \(\nu^{9}\mathstrut +\mathstrut \) \(44\!\cdots\!38\) \(\nu^{8}\mathstrut -\mathstrut \) \(25\!\cdots\!12\) \(\nu^{7}\mathstrut +\mathstrut \) \(39\!\cdots\!60\) \(\nu^{6}\mathstrut -\mathstrut \) \(59\!\cdots\!68\) \(\nu^{5}\mathstrut +\mathstrut \) \(82\!\cdots\!88\) \(\nu^{4}\mathstrut -\mathstrut \) \(43\!\cdots\!16\) \(\nu^{3}\mathstrut -\mathstrut \) \(50\!\cdots\!24\) \(\nu^{2}\mathstrut -\mathstrut \) \(25\!\cdots\!56\) \(\nu\mathstrut -\mathstrut \) \(17\!\cdots\!68\)\()/\)\(16\!\cdots\!00\)
\(\beta_{8}\)\(=\)\((\)\(89\!\cdots\!25\) \(\nu^{13}\mathstrut +\mathstrut \) \(45\!\cdots\!43\) \(\nu^{12}\mathstrut +\mathstrut \) \(78\!\cdots\!46\) \(\nu^{11}\mathstrut +\mathstrut \) \(11\!\cdots\!05\) \(\nu^{10}\mathstrut +\mathstrut \) \(51\!\cdots\!87\) \(\nu^{9}\mathstrut +\mathstrut \) \(75\!\cdots\!26\) \(\nu^{8}\mathstrut +\mathstrut \) \(14\!\cdots\!32\) \(\nu^{7}\mathstrut +\mathstrut \) \(28\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(32\!\cdots\!00\) \(\nu^{5}\mathstrut +\mathstrut \) \(59\!\cdots\!96\) \(\nu^{4}\mathstrut +\mathstrut \) \(63\!\cdots\!92\) \(\nu^{3}\mathstrut +\mathstrut \) \(52\!\cdots\!08\) \(\nu^{2}\mathstrut +\mathstrut \) \(69\!\cdots\!72\) \(\nu\mathstrut +\mathstrut \) \(12\!\cdots\!28\)\()/\)\(50\!\cdots\!00\)
\(\beta_{9}\)\(=\)\((\)\(31\!\cdots\!05\) \(\nu^{13}\mathstrut -\mathstrut \) \(31\!\cdots\!04\) \(\nu^{12}\mathstrut +\mathstrut \) \(28\!\cdots\!67\) \(\nu^{11}\mathstrut -\mathstrut \) \(45\!\cdots\!95\) \(\nu^{10}\mathstrut +\mathstrut \) \(18\!\cdots\!84\) \(\nu^{9}\mathstrut -\mathstrut \) \(27\!\cdots\!53\) \(\nu^{8}\mathstrut +\mathstrut \) \(57\!\cdots\!64\) \(\nu^{7}\mathstrut +\mathstrut \) \(73\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(12\!\cdots\!60\) \(\nu^{5}\mathstrut +\mathstrut \) \(87\!\cdots\!12\) \(\nu^{4}\mathstrut +\mathstrut \) \(63\!\cdots\!64\) \(\nu^{3}\mathstrut +\mathstrut \) \(60\!\cdots\!36\) \(\nu^{2}\mathstrut +\mathstrut \) \(24\!\cdots\!24\) \(\nu\mathstrut +\mathstrut \) \(13\!\cdots\!56\)\()/\)\(12\!\cdots\!00\)
\(\beta_{10}\)\(=\)\((\)\(13\!\cdots\!76\) \(\nu^{13}\mathstrut -\mathstrut \) \(98\!\cdots\!91\) \(\nu^{12}\mathstrut +\mathstrut \) \(11\!\cdots\!49\) \(\nu^{11}\mathstrut +\mathstrut \) \(22\!\cdots\!04\) \(\nu^{10}\mathstrut +\mathstrut \) \(78\!\cdots\!65\) \(\nu^{9}\mathstrut +\mathstrut \) \(17\!\cdots\!33\) \(\nu^{8}\mathstrut +\mathstrut \) \(23\!\cdots\!68\) \(\nu^{7}\mathstrut +\mathstrut \) \(14\!\cdots\!60\) \(\nu^{6}\mathstrut +\mathstrut \) \(50\!\cdots\!72\) \(\nu^{5}\mathstrut +\mathstrut \) \(30\!\cdots\!08\) \(\nu^{4}\mathstrut +\mathstrut \) \(17\!\cdots\!84\) \(\nu^{3}\mathstrut +\mathstrut \) \(63\!\cdots\!76\) \(\nu^{2}\mathstrut +\mathstrut \) \(18\!\cdots\!44\) \(\nu\mathstrut +\mathstrut \) \(13\!\cdots\!52\)\()/\)\(41\!\cdots\!00\)
\(\beta_{11}\)\(=\)\((\)\(30\!\cdots\!69\) \(\nu^{13}\mathstrut -\mathstrut \) \(25\!\cdots\!81\) \(\nu^{12}\mathstrut +\mathstrut \) \(27\!\cdots\!62\) \(\nu^{11}\mathstrut +\mathstrut \) \(18\!\cdots\!81\) \(\nu^{10}\mathstrut +\mathstrut \) \(18\!\cdots\!67\) \(\nu^{9}\mathstrut +\mathstrut \) \(12\!\cdots\!38\) \(\nu^{8}\mathstrut +\mathstrut \) \(54\!\cdots\!44\) \(\nu^{7}\mathstrut +\mathstrut \) \(46\!\cdots\!80\) \(\nu^{6}\mathstrut +\mathstrut \) \(12\!\cdots\!68\) \(\nu^{5}\mathstrut +\mathstrut \) \(83\!\cdots\!08\) \(\nu^{4}\mathstrut +\mathstrut \) \(51\!\cdots\!08\) \(\nu^{3}\mathstrut +\mathstrut \) \(44\!\cdots\!32\) \(\nu^{2}\mathstrut +\mathstrut \) \(23\!\cdots\!28\) \(\nu\mathstrut +\mathstrut \) \(15\!\cdots\!96\)\()/\)\(83\!\cdots\!00\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(68\!\cdots\!87\) \(\nu^{13}\mathstrut +\mathstrut \) \(14\!\cdots\!75\) \(\nu^{12}\mathstrut -\mathstrut \) \(62\!\cdots\!62\) \(\nu^{11}\mathstrut +\mathstrut \) \(81\!\cdots\!57\) \(\nu^{10}\mathstrut -\mathstrut \) \(40\!\cdots\!33\) \(\nu^{9}\mathstrut +\mathstrut \) \(53\!\cdots\!10\) \(\nu^{8}\mathstrut -\mathstrut \) \(12\!\cdots\!24\) \(\nu^{7}\mathstrut +\mathstrut \) \(12\!\cdots\!20\) \(\nu^{6}\mathstrut -\mathstrut \) \(27\!\cdots\!64\) \(\nu^{5}\mathstrut +\mathstrut \) \(32\!\cdots\!80\) \(\nu^{4}\mathstrut -\mathstrut \) \(11\!\cdots\!56\) \(\nu^{3}\mathstrut -\mathstrut \) \(11\!\cdots\!64\) \(\nu^{2}\mathstrut -\mathstrut \) \(23\!\cdots\!96\) \(\nu\mathstrut +\mathstrut \) \(27\!\cdots\!44\)\()/\)\(16\!\cdots\!00\)
\(\beta_{13}\)\(=\)\((\)\(46\!\cdots\!01\) \(\nu^{13}\mathstrut -\mathstrut \) \(21\!\cdots\!45\) \(\nu^{12}\mathstrut +\mathstrut \) \(41\!\cdots\!86\) \(\nu^{11}\mathstrut +\mathstrut \) \(33\!\cdots\!89\) \(\nu^{10}\mathstrut +\mathstrut \) \(27\!\cdots\!79\) \(\nu^{9}\mathstrut +\mathstrut \) \(22\!\cdots\!30\) \(\nu^{8}\mathstrut +\mathstrut \) \(83\!\cdots\!72\) \(\nu^{7}\mathstrut +\mathstrut \) \(90\!\cdots\!40\) \(\nu^{6}\mathstrut +\mathstrut \) \(18\!\cdots\!72\) \(\nu^{5}\mathstrut +\mathstrut \) \(17\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(80\!\cdots\!08\) \(\nu^{3}\mathstrut +\mathstrut \) \(15\!\cdots\!52\) \(\nu^{2}\mathstrut +\mathstrut \) \(32\!\cdots\!28\) \(\nu\mathstrut +\mathstrut \) \(25\!\cdots\!68\)\()/\)\(50\!\cdots\!00\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(22\) \(\beta_{13}\mathstrut -\mathstrut \) \(33\) \(\beta_{12}\mathstrut +\mathstrut \) \(53\) \(\beta_{11}\mathstrut +\mathstrut \) \(20\) \(\beta_{10}\mathstrut -\mathstrut \) \(68\) \(\beta_{9}\mathstrut +\mathstrut \) \(11\) \(\beta_{8}\mathstrut -\mathstrut \) \(21\) \(\beta_{7}\mathstrut +\mathstrut \) \(868\) \(\beta_{6}\mathstrut -\mathstrut \) \(1099\) \(\beta_{5}\mathstrut -\mathstrut \) \(641\) \(\beta_{4}\mathstrut +\mathstrut \) \(1170\) \(\beta_{3}\mathstrut -\mathstrut \) \(6801\) \(\beta_{2}\mathstrut +\mathstrut \) \(2164894\) \(\beta_{1}\mathstrut +\mathstrut \) \(639050\)\()/8957952\)
\(\nu^{2}\)\(=\)\((\)\(608\) \(\beta_{13}\mathstrut +\mathstrut \) \(2688\) \(\beta_{12}\mathstrut +\mathstrut \) \(3808\) \(\beta_{11}\mathstrut -\mathstrut \) \(704\) \(\beta_{10}\mathstrut +\mathstrut \) \(9824\) \(\beta_{9}\mathstrut -\mathstrut \) \(3296\) \(\beta_{8}\mathstrut +\mathstrut \) \(1632\) \(\beta_{7}\mathstrut -\mathstrut \) \(21664\) \(\beta_{6}\mathstrut -\mathstrut \) \(13499\) \(\beta_{5}\mathstrut -\mathstrut \) \(24928\) \(\beta_{4}\mathstrut +\mathstrut \) \(475542\) \(\beta_{3}\mathstrut -\mathstrut \) \(16092153\) \(\beta_{2}\mathstrut +\mathstrut \) \(1056485\) \(\beta_{1}\mathstrut -\mathstrut \) \(11718857408\)\()/8957952\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(185654\) \(\beta_{13}\mathstrut +\mathstrut \) \(253281\) \(\beta_{12}\mathstrut -\mathstrut \) \(487381\) \(\beta_{11}\mathstrut -\mathstrut \) \(275860\) \(\beta_{10}\mathstrut +\mathstrut \) \(486628\) \(\beta_{9}\mathstrut +\mathstrut \) \(218069\) \(\beta_{8}\mathstrut +\mathstrut \) \(293397\) \(\beta_{7}\mathstrut -\mathstrut \) \(7818788\) \(\beta_{6}\mathstrut -\mathstrut \) \(54490\) \(\beta_{5}\mathstrut +\mathstrut \) \(2407105\) \(\beta_{4}\mathstrut +\mathstrut \) \(11443896\) \(\beta_{3}\mathstrut +\mathstrut \) \(1153194\) \(\beta_{2}\mathstrut -\mathstrut \) \(511649027\) \(\beta_{1}\mathstrut -\mathstrut \) \(142760673034\)\()/8957952\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(7848214\) \(\beta_{13}\mathstrut -\mathstrut \) \(2674911\) \(\beta_{12}\mathstrut -\mathstrut \) \(20098805\) \(\beta_{11}\mathstrut +\mathstrut \) \(13092652\) \(\beta_{10}\mathstrut -\mathstrut \) \(9172348\) \(\beta_{9}\mathstrut +\mathstrut \) \(17886709\) \(\beta_{8}\mathstrut -\mathstrut \) \(16412811\) \(\beta_{7}\mathstrut -\mathstrut \) \(9830404\) \(\beta_{6}\mathstrut +\mathstrut \) \(69272437\) \(\beta_{5}\mathstrut -\mathstrut \) \(119825695\) \(\beta_{4}\mathstrut +\mathstrut \) \(2079166842\) \(\beta_{3}\mathstrut +\mathstrut \) \(68142447807\) \(\beta_{2}\mathstrut -\mathstrut \) \(154376523028\) \(\beta_{1}\mathstrut -\mathstrut \) \(49366760957642\)\()/8957952\)
\(\nu^{5}\)\(=\)\((\)\(479570632\) \(\beta_{13}\mathstrut -\mathstrut \) \(918301308\) \(\beta_{12}\mathstrut +\mathstrut \) \(1121930060\) \(\beta_{11}\mathstrut +\mathstrut \) \(531040112\) \(\beta_{10}\mathstrut -\mathstrut \) \(2728788176\) \(\beta_{9}\mathstrut -\mathstrut \) \(754469452\) \(\beta_{8}\mathstrut -\mathstrut \) \(2832132108\) \(\beta_{7}\mathstrut +\mathstrut \) \(18785243536\) \(\beta_{6}\mathstrut +\mathstrut \) \(20795851355\) \(\beta_{5}\mathstrut -\mathstrut \) \(2210761628\) \(\beta_{4}\mathstrut -\mathstrut \) \(70135673670\) \(\beta_{3}\mathstrut +\mathstrut \) \(908089756281\) \(\beta_{2}\mathstrut -\mathstrut \) \(41465946378473\) \(\beta_{1}\mathstrut +\mathstrut \) \(578681322136472\)\()/8957952\)
\(\nu^{6}\)\(=\)\((\)\(18248617406\) \(\beta_{13}\mathstrut -\mathstrut \) \(111457505325\) \(\beta_{12}\mathstrut +\mathstrut \) \(50007661969\) \(\beta_{11}\mathstrut -\mathstrut \) \(34562591612\) \(\beta_{10}\mathstrut -\mathstrut \) \(240723258388\) \(\beta_{9}\mathstrut -\mathstrut \) \(50613141905\) \(\beta_{8}\mathstrut -\mathstrut \) \(32173890513\) \(\beta_{7}\mathstrut +\mathstrut \) \(880633274708\) \(\beta_{6}\mathstrut +\mathstrut \) \(7049699770\) \(\beta_{5}\mathstrut +\mathstrut \) \(1954731268979\) \(\beta_{4}\mathstrut -\mathstrut \) \(19964705616936\) \(\beta_{3}\mathstrut -\mathstrut \) \(724888279914\) \(\beta_{2}\mathstrut +\mathstrut \) \(366022122826895\) \(\beta_{1}\mathstrut +\mathstrut \) \(450947966544170242\)\()/8957952\)
\(\nu^{7}\)\(=\)\((\)\(910487315038\) \(\beta_{13}\mathstrut -\mathstrut \) \(2070097496205\) \(\beta_{12}\mathstrut +\mathstrut \) \(6230242798289\) \(\beta_{11}\mathstrut +\mathstrut \) \(3612612715076\) \(\beta_{10}\mathstrut +\mathstrut \) \(6057008701132\) \(\beta_{9}\mathstrut -\mathstrut \) \(6805885257169\) \(\beta_{8}\mathstrut +\mathstrut \) \(8137355562159\) \(\beta_{7}\mathstrut +\mathstrut \) \(93217263762100\) \(\beta_{6}\mathstrut -\mathstrut \) \(110795265307945\) \(\beta_{5}\mathstrut -\mathstrut \) \(10445403881549\) \(\beta_{4}\mathstrut -\mathstrut \) \(254706297531714\) \(\beta_{3}\mathstrut -\mathstrut \) \(6178075256925339\) \(\beta_{2}\mathstrut +\mathstrut \) \(193868273634425260\) \(\beta_{1}\mathstrut +\mathstrut \) \(4113454736669419778\)\()/8957952\)
\(\nu^{8}\)\(=\)\((\)\(6258388039304\) \(\beta_{13}\mathstrut +\mathstrut \) \(739415649927876\) \(\beta_{12}\mathstrut +\mathstrut \) \(212701351239244\) \(\beta_{11}\mathstrut -\mathstrut \) \(46467118653968\) \(\beta_{10}\mathstrut +\mathstrut \) \(1857263669510000\) \(\beta_{9}\mathstrut -\mathstrut \) \(232306975668044\) \(\beta_{8}\mathstrut +\mathstrut \) \(854177828479860\) \(\beta_{7}\mathstrut -\mathstrut \) \(3373301665118512\) \(\beta_{6}\mathstrut -\mathstrut \) \(3202447509106415\) \(\beta_{5}\mathstrut -\mathstrut \) \(7253387624973724\) \(\beta_{4}\mathstrut +\mathstrut \) \(30763130452909326\) \(\beta_{3}\mathstrut -\mathstrut \) \(1478037632103365589\) \(\beta_{2}\mathstrut +\mathstrut \) \(4313820532290245453\) \(\beta_{1}\mathstrut -\mathstrut \) \(1067796306483015833960\)\()/8957952\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(12782420390200502\) \(\beta_{13}\mathstrut +\mathstrut \) \(44737081051525473\) \(\beta_{12}\mathstrut -\mathstrut \) \(59067784658873173\) \(\beta_{11}\mathstrut -\mathstrut \) \(30658497786650260\) \(\beta_{10}\mathstrut +\mathstrut \) \(70740677348784484\) \(\beta_{9}\mathstrut +\mathstrut \) \(65583400735582037\) \(\beta_{8}\mathstrut +\mathstrut \) \(46212150357379221\) \(\beta_{7}\mathstrut -\mathstrut \) \(866960731503417764\) \(\beta_{6}\mathstrut +\mathstrut \) \(98391686284908230\) \(\beta_{5}\mathstrut -\mathstrut \) \(51419333580497855\) \(\beta_{4}\mathstrut +\mathstrut \) \(3756185086727326968\) \(\beta_{3}\mathstrut -\mathstrut \) \(191353104317648118\) \(\beta_{2}\mathstrut +\mathstrut \) \(72786195589423120669\) \(\beta_{1}\mathstrut -\mathstrut \) \(53918413220874576086026\)\()/8957952\)
\(\nu^{10}\)\(=\)\((\)\(-\)\(55915014421492486\) \(\beta_{13}\mathstrut -\mathstrut \) \(332856654321409527\) \(\beta_{12}\mathstrut -\mathstrut \) \(3178369093610898941\) \(\beta_{11}\mathstrut +\mathstrut \) \(414832648576701196\) \(\beta_{10}\mathstrut -\mathstrut \) \(3947349815324168668\) \(\beta_{9}\mathstrut +\mathstrut \) \(3651511822672960765\) \(\beta_{8}\mathstrut -\mathstrut \) \(3419082125208905091\) \(\beta_{7}\mathstrut -\mathstrut \) \(15519244066599935908\) \(\beta_{6}\mathstrut +\mathstrut \) \(22081526549756367709\) \(\beta_{5}\mathstrut -\mathstrut \) \(19027781975791346743\) \(\beta_{4}\mathstrut +\mathstrut \) \(382693048544449779786\) \(\beta_{3}\mathstrut +\mathstrut \) \(7234704373481408164983\) \(\beta_{2}\mathstrut -\mathstrut \) \(33180017124279166217812\) \(\beta_{1}\mathstrut -\mathstrut \) \(5206266268869807741743066\)\()/8957952\)
\(\nu^{11}\)\(=\)\((\)\(48189863998485272680\) \(\beta_{13}\mathstrut -\mathstrut \) \(212171393812564361580\) \(\beta_{12}\mathstrut +\mathstrut \) \(127269047562182788604\) \(\beta_{11}\mathstrut +\mathstrut \) \(71165943813286363184\) \(\beta_{10}\mathstrut -\mathstrut \) \(716360140970060310416\) \(\beta_{9}\mathstrut -\mathstrut \) \(121871648630204895484\) \(\beta_{8}\mathstrut -\mathstrut \) \(535370989488176427708\) \(\beta_{7}\mathstrut +\mathstrut \) \(1776207094739212461136\) \(\beta_{6}\mathstrut +\mathstrut \) \(2327612450974011313043\) \(\beta_{5}\mathstrut +\mathstrut \) \(598732703135866436212\) \(\beta_{4}\mathstrut -\mathstrut \) \(8742200546174830793334\) \(\beta_{3}\mathstrut +\mathstrut \) \(244546665207099695080929\) \(\beta_{2}\mathstrut -\mathstrut \) \(5062656961161483845282369\) \(\beta_{1}\mathstrut +\mathstrut \) \(168031464997031684771127032\)\()/8957952\)
\(\nu^{12}\)\(=\)\((\)\(105034322834019964046\) \(\beta_{13}\mathstrut -\mathstrut \) \(21811447878390130070757\) \(\beta_{12}\mathstrut +\mathstrut \) \(15138188627791176073129\) \(\beta_{11}\mathstrut +\mathstrut \) \(577729611182495827684\) \(\beta_{10}\mathstrut -\mathstrut \) \(34528269987258243968500\) \(\beta_{9}\mathstrut -\mathstrut \) \(17488554374545818690473\) \(\beta_{8}\mathstrut -\mathstrut \) \(11217865259680432633833\) \(\beta_{7}\mathstrut +\mathstrut \) \(199182236622933369296180\) \(\beta_{6}\mathstrut -\mathstrut \) \(24030671881163865065510\) \(\beta_{5}\mathstrut +\mathstrut \) \(312827113171667385107387\) \(\beta_{4}\mathstrut -\mathstrut \) \(2662192004683390776200520\) \(\beta_{3}\mathstrut -\mathstrut \) \(10240881337065955962570\) \(\beta_{2}\mathstrut +\mathstrut \) \(11969794052427822883324199\) \(\beta_{1}\mathstrut +\mathstrut \) \(52059656754733065681703867954\)\()/8957952\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(4127240439873178008818\) \(\beta_{13}\mathstrut -\mathstrut \) \(184166569335113120295573\) \(\beta_{12}\mathstrut +\mathstrut \) \(988081019159086942065017\) \(\beta_{11}\mathstrut +\mathstrut \) \(384688126576641976580324\) \(\beta_{10}\mathstrut +\mathstrut \) \(1999394189969520236390572\) \(\beta_{9}\mathstrut -\mathstrut \) \(1447016734427145921905785\) \(\beta_{8}\mathstrut +\mathstrut \) \(1634146063553443716541959\) \(\beta_{7}\mathstrut +\mathstrut \) \(13937595355734364181897428\) \(\beta_{6}\mathstrut -\mathstrut \) \(15445174114305964566216289\) \(\beta_{5}\mathstrut +\mathstrut \) \(1838195230288865614980523\) \(\beta_{4}\mathstrut -\mathstrut \) \(87891702128156908132367346\) \(\beta_{3}\mathstrut -\mathstrut \) \(1455220802111301456743126211\) \(\beta_{2}\mathstrut +\mathstrut \) \(23099955103391770451701484476\) \(\beta_{1}\mathstrut +\mathstrut \) \(1013608449385227486324981089618\)\()/8957952\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−9.38489 + 16.2551i
−9.38489 16.2551i
29.9153 + 51.8148i
29.9153 51.8148i
−33.2709 + 57.6269i
−33.2709 57.6269i
37.1678 + 64.3766i
37.1678 64.3766i
−30.4205 + 52.6899i
−30.4205 52.6899i
12.2149 + 21.1568i
12.2149 21.1568i
−5.72169 + 9.91026i
−5.72169 9.91026i
−122.841 35.9743i 1262.67i 13795.7 + 8838.22i 49804.1 45423.5 155107.i 91750.3i −1.37672e6 1.58198e6i −1.59432e6 −6.11797e6 1.79167e6i
7.2 −122.841 + 35.9743i 1262.67i 13795.7 8838.22i 49804.1 45423.5 + 155107.i 91750.3i −1.37672e6 + 1.58198e6i −1.59432e6 −6.11797e6 + 1.79167e6i
7.3 −79.6923 100.166i 1262.67i −3682.27 + 15964.8i 53637.2 −126476. + 100625.i 1.43231e6i 1.89258e6 903439.i −1.59432e6 −4.27447e6 5.37260e6i
7.4 −79.6923 + 100.166i 1262.67i −3682.27 15964.8i 53637.2 −126476. 100625.i 1.43231e6i 1.89258e6 + 903439.i −1.59432e6 −4.27447e6 + 5.37260e6i
7.5 −47.8547 118.718i 1262.67i −11803.8 + 11362.4i −132280. 149901. 60424.5i 304975.i 1.91379e6 + 857581.i −1.59432e6 6.33024e6 + 1.57040e7i
7.6 −47.8547 + 118.718i 1262.67i −11803.8 11362.4i −132280. 149901. + 60424.5i 304975.i 1.91379e6 857581.i −1.59432e6 6.33024e6 1.57040e7i
7.7 26.2042 125.289i 1262.67i −15010.7 6566.21i −40325.1 −158198. 33087.2i 421333.i −1.21602e6 + 1.70861e6i −1.59432e6 −1.05669e6 + 5.05229e6i
7.8 26.2042 + 125.289i 1262.67i −15010.7 + 6566.21i −40325.1 −158198. + 33087.2i 421333.i −1.21602e6 1.70861e6i −1.59432e6 −1.05669e6 5.05229e6i
7.9 67.3573 108.844i 1262.67i −7309.99 14662.9i 30766.3 137433. + 85049.7i 368697.i −2.08834e6 192002.i −1.59432e6 2.07234e6 3.34873e6i
7.10 67.3573 + 108.844i 1262.67i −7309.99 + 14662.9i 30766.3 137433. 85049.7i 368697.i −2.08834e6 + 192002.i −1.59432e6 2.07234e6 + 3.34873e6i
7.11 121.962 38.8495i 1262.67i 13365.4 9476.31i 122615. −49053.8 153997.i 1.05178e6i 1.26193e6 1.67499e6i −1.59432e6 1.49544e7 4.76352e6i
7.12 121.962 + 38.8495i 1262.67i 13365.4 + 9476.31i 122615. −49053.8 + 153997.i 1.05178e6i 1.26193e6 + 1.67499e6i −1.59432e6 1.49544e7 + 4.76352e6i
7.13 125.864 23.2846i 1262.67i 15299.7 5861.41i −92279.1 29400.7 + 158924.i 1.24880e6i 1.78920e6 1.09399e6i −1.59432e6 −1.16146e7 + 2.14868e6i
7.14 125.864 + 23.2846i 1262.67i 15299.7 + 5861.41i −92279.1 29400.7 158924.i 1.24880e6i 1.78920e6 + 1.09399e6i −1.59432e6 −1.16146e7 2.14868e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.14
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{15}^{\mathrm{new}}(12, [\chi])\).