Properties

Label 12.10.b.a
Level 12
Weight 10
Character orbit 12.b
Analytic conductor 6.180
Analytic rank 0
Dimension 16
CM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 12.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(6.18043003397\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{68}\cdot 3^{24} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\beta_{2} q^{2} \) \( -\beta_{4} q^{3} \) \( + ( -21 - \beta_{1} ) q^{4} \) \( + ( -2 \beta_{2} - \beta_{8} ) q^{5} \) \( + ( -100 - \beta_{11} ) q^{6} \) \( + ( \beta_{4} - \beta_{5} ) q^{7} \) \( + ( 22 \beta_{2} - \beta_{6} + \beta_{8} ) q^{8} \) \( + ( -868 + 2 \beta_{1} + 32 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{10} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{2} q^{2} \) \( -\beta_{4} q^{3} \) \( + ( -21 - \beta_{1} ) q^{4} \) \( + ( -2 \beta_{2} - \beta_{8} ) q^{5} \) \( + ( -100 - \beta_{11} ) q^{6} \) \( + ( \beta_{4} - \beta_{5} ) q^{7} \) \( + ( 22 \beta_{2} - \beta_{6} + \beta_{8} ) q^{8} \) \( + ( -868 + 2 \beta_{1} + 32 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{10} ) q^{9} \) \( + ( -996 - \beta_{1} + 2 \beta_{2} + 10 \beta_{4} - 2 \beta_{5} + \beta_{10} + \beta_{12} ) q^{10} \) \( + ( 4 - \beta_{1} - 42 \beta_{2} - 14 \beta_{4} - 2 \beta_{6} - 5 \beta_{11} + \beta_{14} + \beta_{15} ) q^{11} \) \( + ( 2662 + \beta_{1} + 104 \beta_{2} + 21 \beta_{4} - 6 \beta_{5} - \beta_{7} - 12 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{12} \) \( + ( -6061 - 33 \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 15 \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{15} ) q^{13} \) \( + ( -2 \beta_{1} - 17 \beta_{2} + \beta_{3} - 79 \beta_{4} + 2 \beta_{6} - 33 \beta_{8} - 3 \beta_{9} - 4 \beta_{10} + \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{14} \) \( + ( 52 - 99 \beta_{1} - 409 \beta_{2} - \beta_{3} - 14 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{11} - 4 \beta_{12} + \beta_{14} + 5 \beta_{15} ) q^{15} \) \( + ( 10300 + 8 \beta_{1} - 44 \beta_{2} - 5 \beta_{3} - 277 \beta_{4} + 20 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 10 \beta_{9} - 6 \beta_{10} + 6 \beta_{11} + 6 \beta_{12} + 4 \beta_{13} - 2 \beta_{15} ) q^{16} \) \( + ( -35 + 15 \beta_{1} - 1624 \beta_{2} + 3 \beta_{3} - 11 \beta_{4} + 11 \beta_{6} - 8 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} + 59 \beta_{11} - \beta_{12} + \beta_{13} + 8 \beta_{14} - 3 \beta_{15} ) q^{17} \) \( + ( 16046 + 23 \beta_{1} + 893 \beta_{2} + 13 \beta_{3} + 97 \beta_{4} + 30 \beta_{5} + 6 \beta_{6} - 12 \beta_{7} + 172 \beta_{8} - 20 \beta_{9} - 3 \beta_{10} + 12 \beta_{11} + 7 \beta_{12} + 2 \beta_{13} - 6 \beta_{14} - 4 \beta_{15} ) q^{18} \) \( + ( -264 + 412 \beta_{1} - 49 \beta_{2} - 2 \beta_{3} - 212 \beta_{4} + 22 \beta_{5} - 8 \beta_{6} + 20 \beta_{7} + 8 \beta_{8} + \beta_{9} + 8 \beta_{10} + 96 \beta_{11} - 16 \beta_{12} - 8 \beta_{13} + 16 \beta_{15} ) q^{19} \) \( + ( -14 - 4 \beta_{1} + 1312 \beta_{2} - 11 \beta_{3} + 1323 \beta_{4} + 8 \beta_{6} + 302 \beta_{8} + 48 \beta_{9} - 12 \beta_{10} + 16 \beta_{11} + 6 \beta_{12} - 6 \beta_{13} - 14 \beta_{14} - 8 \beta_{15} ) q^{20} \) \( + ( 21837 + 891 \beta_{1} + 2956 \beta_{2} - 16 \beta_{3} + 146 \beta_{4} + \beta_{6} - 22 \beta_{7} - 7 \beta_{8} - 9 \beta_{9} - 2 \beta_{10} - 7 \beta_{11} - 19 \beta_{12} - \beta_{13} + 8 \beta_{14} - 9 \beta_{15} ) q^{21} \) \( + ( 20118 + 14 \beta_{1} + 437 \beta_{2} - 28 \beta_{3} + 2474 \beta_{4} - 32 \beta_{5} + 16 \beta_{6} + 24 \beta_{7} + 5 \beta_{8} - 89 \beta_{9} - 32 \beta_{10} - 3 \beta_{11} + 16 \beta_{12} + 16 \beta_{13} - 11 \beta_{15} ) q^{22} \) \( + ( 104 - 24 \beta_{1} + 4946 \beta_{2} + 44 \beta_{3} - 192 \beta_{4} + 24 \beta_{6} - 32 \beta_{8} + 6 \beta_{9} + 16 \beta_{10} - 152 \beta_{11} - 8 \beta_{12} + 8 \beta_{13} + 16 \beta_{14} + 24 \beta_{15} ) q^{23} \) \( + ( 10408 + 32 \beta_{1} - 2622 \beta_{2} + 111 \beta_{3} + 3 \beta_{4} + 12 \beta_{5} + 7 \beta_{6} - 58 \beta_{7} - 585 \beta_{8} + 166 \beta_{9} - 2 \beta_{10} + 10 \beta_{11} + 30 \beta_{12} - 8 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} ) q^{24} \) \( + ( -170360 - 2217 \beta_{1} - 64 \beta_{2} - 23 \beta_{3} - 291 \beta_{4} - 15 \beta_{6} + 76 \beta_{7} - 18 \beta_{8} - 3 \beta_{9} + 36 \beta_{10} - 117 \beta_{11} - 9 \beta_{12} - 15 \beta_{13} - 3 \beta_{15} ) q^{25} \) \( + ( -20 + 8 \beta_{1} + 4714 \beta_{2} - 142 \beta_{3} - 6562 \beta_{4} - 60 \beta_{6} - 696 \beta_{8} - 264 \beta_{9} + 24 \beta_{10} - 8 \beta_{11} - 12 \beta_{12} + 12 \beta_{13} - 20 \beta_{14} - 8 \beta_{15} ) q^{26} \) \( + ( 1556 - 2983 \beta_{1} - 21748 \beta_{2} - 124 \beta_{3} + 439 \beta_{4} - 66 \beta_{5} - 86 \beta_{6} - 96 \beta_{7} + 32 \beta_{8} - 10 \beta_{9} - 16 \beta_{10} - 3 \beta_{11} + 8 \beta_{12} - 8 \beta_{13} + 15 \beta_{14} + 7 \beta_{15} ) q^{27} \) \( + ( -90358 - 150 \beta_{1} - 2152 \beta_{2} - 45 \beta_{3} - 11821 \beta_{4} - 180 \beta_{5} - 12 \beta_{6} + 114 \beta_{7} - 18 \beta_{8} + 414 \beta_{9} + 46 \beta_{10} - 126 \beta_{11} + 10 \beta_{12} - 12 \beta_{13} - 6 \beta_{15} ) q^{28} \) \( + ( 86 - 30 \beta_{1} - 33230 \beta_{2} + 290 \beta_{3} - 210 \beta_{4} - 134 \beta_{6} - 95 \beta_{8} + 6 \beta_{9} + 28 \beta_{10} - 182 \beta_{11} - 14 \beta_{12} + 14 \beta_{13} - 32 \beta_{14} + 6 \beta_{15} ) q^{29} \) \( + ( 205584 - 186 \beta_{1} - 329 \beta_{2} + 449 \beta_{3} + 729 \beta_{4} - 288 \beta_{5} - 86 \beta_{6} - 136 \beta_{7} + 575 \beta_{8} - 627 \beta_{9} - 20 \beta_{10} - 49 \beta_{11} - 22 \beta_{12} - 10 \beta_{13} + 38 \beta_{14} + 15 \beta_{15} ) q^{30} \) \( + ( -2264 + 4884 \beta_{1} - 54 \beta_{2} - 102 \beta_{3} - 259 \beta_{4} - 219 \beta_{5} + 40 \beta_{6} + 124 \beta_{7} - 40 \beta_{8} + 10 \beta_{9} - 40 \beta_{10} - 480 \beta_{11} + 80 \beta_{12} + 40 \beta_{13} - 80 \beta_{15} ) q^{31} \) \( + ( 200 - 16 \beta_{1} - 5272 \beta_{2} - 636 \beta_{3} + 22460 \beta_{4} - 28 \beta_{6} + 84 \beta_{8} + 864 \beta_{9} + 80 \beta_{10} - 288 \beta_{11} - 40 \beta_{12} + 40 \beta_{13} + 72 \beta_{14} + 64 \beta_{15} ) q^{32} \) \( + ( 292156 + 5407 \beta_{1} + 97148 \beta_{2} - 613 \beta_{3} + 1420 \beta_{4} + 86 \beta_{6} - 196 \beta_{7} - 316 \beta_{8} + 63 \beta_{9} + 9 \beta_{10} + 89 \beta_{11} + 149 \beta_{12} - 13 \beta_{13} - 40 \beta_{14} + 63 \beta_{15} ) q^{33} \) \( + ( -824576 - 956 \beta_{1} + 5456 \beta_{2} - 12 \beta_{3} + 30588 \beta_{4} + 504 \beta_{5} - 112 \beta_{6} + 248 \beta_{7} - 32 \beta_{8} - 1168 \beta_{9} + 212 \beta_{10} + 48 \beta_{11} - 124 \beta_{12} - 112 \beta_{13} + 80 \beta_{15} ) q^{34} \) \( + ( -860 + 203 \beta_{1} + 144743 \beta_{2} + 1144 \beta_{3} - 288 \beta_{4} - 26 \beta_{6} + 192 \beta_{8} - 69 \beta_{9} - 96 \beta_{10} + 1207 \beta_{11} + 48 \beta_{12} - 48 \beta_{13} - 155 \beta_{14} - 203 \beta_{15} ) q^{35} \) \( + ( -426951 + 423 \beta_{1} - 16368 \beta_{2} + 1321 \beta_{3} - 2673 \beta_{4} + 336 \beta_{5} - 56 \beta_{6} - 120 \beta_{7} + 1462 \beta_{8} + 1432 \beta_{9} - 4 \beta_{10} - 24 \beta_{11} - 242 \beta_{12} + 26 \beta_{13} + 66 \beta_{14} - 16 \beta_{15} ) q^{36} \) \( + ( 689033 - 4659 \beta_{1} - 154 \beta_{2} - 186 \beta_{3} - 758 \beta_{4} + 169 \beta_{6} + 34 \beta_{7} + 218 \beta_{8} + 49 \beta_{9} - 360 \beta_{10} + 1455 \beta_{11} + 147 \beta_{12} + 169 \beta_{13} + 49 \beta_{15} ) q^{37} \) \( + ( 178 + 6 \beta_{1} - 9791 \beta_{2} - 1846 \beta_{3} - 48076 \beta_{4} + 532 \beta_{6} + 2601 \beta_{8} - 1869 \beta_{9} - 40 \beta_{10} - \beta_{11} + 20 \beta_{12} - 20 \beta_{13} + 236 \beta_{14} + 105 \beta_{15} ) q^{38} \) \( + ( 796 - 2015 \beta_{1} - 251747 \beta_{2} - 2117 \beta_{3} + 4995 \beta_{4} + 642 \beta_{5} + 422 \beta_{6} - 114 \beta_{7} - 68 \beta_{8} + 91 \beta_{9} - 44 \beta_{10} + 195 \beta_{11} + 100 \beta_{12} + 56 \beta_{13} - 171 \beta_{14} - 271 \beta_{15} ) q^{39} \) \( + ( 1462552 - 208 \beta_{1} - 11672 \beta_{2} + 10 \beta_{3} - 64278 \beta_{4} + 440 \beta_{5} - 72 \beta_{6} + 124 \beta_{7} + 108 \beta_{8} + 2172 \beta_{9} - 164 \beta_{10} + 1188 \beta_{11} - 380 \beta_{12} - 72 \beta_{13} + 180 \beta_{15} ) q^{40} \) \( + ( 968 - 480 \beta_{1} - 401764 \beta_{2} + 3076 \beta_{3} - 3428 \beta_{4} + 712 \beta_{6} + 2102 \beta_{8} + 96 \beta_{9} - 368 \beta_{10} - 1280 \beta_{11} + 184 \beta_{12} - 184 \beta_{13} - 104 \beta_{14} + 96 \beta_{15} ) q^{41} \) \( + ( 1495932 + 3381 \beta_{1} - 25736 \beta_{2} + 3304 \beta_{3} - 1282 \beta_{4} + 762 \beta_{5} + 600 \beta_{6} + 152 \beta_{7} - 6224 \beta_{8} - 2464 \beta_{9} + 323 \beta_{10} + 32 \beta_{11} - 149 \beta_{12} - 24 \beta_{13} - 40 \beta_{14} + 64 \beta_{15} ) q^{42} \) \( + ( 2872 - 4868 \beta_{1} + 1487 \beta_{2} + 46 \beta_{3} + 6368 \beta_{4} + 1298 \beta_{5} + 56 \beta_{6} - 204 \beta_{7} - 56 \beta_{8} - 223 \beta_{9} - 56 \beta_{10} - 672 \beta_{11} + 112 \beta_{12} + 56 \beta_{13} - 112 \beta_{15} ) q^{43} \) \( + ( -1446 + 492 \beta_{1} - 2520 \beta_{2} - 4367 \beta_{3} + 75215 \beta_{4} + 28 \beta_{6} - 7326 \beta_{8} + 2592 \beta_{9} - 60 \beta_{10} + 2304 \beta_{11} + 30 \beta_{12} - 30 \beta_{13} + 90 \beta_{14} - 216 \beta_{15} ) q^{44} \) \( + ( -10084 - 11842 \beta_{1} + 702194 \beta_{2} - 5310 \beta_{3} + 3736 \beta_{4} - 640 \beta_{6} + 360 \beta_{7} + 4293 \beta_{8} - 90 \beta_{9} - 242 \beta_{10} - 342 \beta_{11} - 342 \beta_{12} + 198 \beta_{13} - 72 \beta_{14} - 90 \beta_{15} ) q^{45} \) \( + ( -2581748 + 7212 \beta_{1} + 14298 \beta_{2} + 160 \beta_{3} + 81580 \beta_{4} - 2048 \beta_{5} + 128 \beta_{6} - 576 \beta_{7} + 10 \beta_{8} - 2770 \beta_{9} - 384 \beta_{10} - 294 \beta_{11} + 128 \beta_{13} - 118 \beta_{15} ) q^{46} \) \( + ( 1000 - 254 \beta_{1} + 748226 \beta_{2} + 6248 \beta_{3} + 1208 \beta_{4} - 652 \beta_{6} + 64 \beta_{8} + 426 \beta_{9} - 32 \beta_{10} - 1206 \beta_{11} + 16 \beta_{12} - 16 \beta_{13} + 270 \beta_{14} + 254 \beta_{15} ) q^{47} \) \( + ( -5481452 - 2216 \beta_{1} - 14436 \beta_{2} + 6719 \beta_{3} - 12649 \beta_{4} - 1932 \beta_{5} + 88 \beta_{6} + 658 \beta_{7} + 8102 \beta_{8} + 2842 \beta_{9} + 474 \beta_{10} - 74 \beta_{11} + 542 \beta_{12} - 36 \beta_{13} - 104 \beta_{14} + 398 \beta_{15} ) q^{48} \) \( + ( -1347012 + 27717 \beta_{1} + 1096 \beta_{2} + 767 \beta_{3} + 4155 \beta_{4} - 453 \beta_{6} - 628 \beta_{7} - 438 \beta_{8} + 15 \beta_{9} + 1404 \beta_{10} - 2583 \beta_{11} + 45 \beta_{12} - 453 \beta_{13} + 15 \beta_{15} ) q^{49} \) \( + ( -444 + 24 \beta_{1} + 166995 \beta_{2} - 8522 \beta_{3} - 44230 \beta_{4} - 2228 \beta_{6} + 7064 \beta_{8} - 2136 \beta_{9} + 72 \beta_{10} + 168 \beta_{11} - 36 \beta_{12} + 36 \beta_{13} - 444 \beta_{14} - 216 \beta_{15} ) q^{50} \) \( + ( -16596 + 34409 \beta_{1} - 1179929 \beta_{2} - 10070 \beta_{3} + 1514 \beta_{4} - 3564 \beta_{5} - 518 \beta_{6} + 1204 \beta_{7} - 152 \beta_{8} - 765 \beta_{9} + 216 \beta_{10} - 1567 \beta_{11} - 248 \beta_{12} - 32 \beta_{13} + 427 \beta_{14} + 675 \beta_{15} ) q^{51} \) \( + ( 8260538 + 2930 \beta_{1} - 2848 \beta_{2} + 552 \beta_{3} - 23128 \beta_{4} + 352 \beta_{5} + 224 \beta_{6} - 1552 \beta_{7} - 656 \beta_{8} + 1328 \beta_{9} + 560 \beta_{10} - 6576 \beta_{11} + 1232 \beta_{12} + 224 \beta_{13} - 880 \beta_{15} ) q^{52} \) \( + ( -3984 + 1920 \beta_{1} - 1501434 \beta_{2} + 10648 \beta_{3} - 9688 \beta_{4} - 2064 \beta_{6} - 12941 \beta_{8} - 384 \beta_{9} + 1248 \beta_{10} + 5568 \beta_{11} - 624 \beta_{12} + 624 \beta_{13} + 528 \beta_{14} - 384 \beta_{15} ) q^{53} \) \( + ( 11244178 - 23270 \beta_{1} - 7997 \beta_{2} + 11326 \beta_{3} - 22300 \beta_{4} + 1056 \beta_{5} - 2572 \beta_{6} + 1320 \beta_{7} + 1027 \beta_{8} - 191 \beta_{9} - 872 \beta_{10} + 318 \beta_{11} + 628 \beta_{12} + 236 \beta_{13} - 132 \beta_{14} - 301 \beta_{15} ) q^{54} \) \( + ( 25832 - 56716 \beta_{1} - 7958 \beta_{2} + 1226 \beta_{3} - 36848 \beta_{4} - 5054 \beta_{5} - 536 \beta_{6} - 1380 \beta_{7} + 536 \beta_{8} + 1546 \beta_{9} + 536 \beta_{10} + 6432 \beta_{11} - 1072 \beta_{12} - 536 \beta_{13} + 1072 \beta_{15} ) q^{55} \) \( + ( 6856 - 2832 \beta_{1} + 91412 \beta_{2} - 13660 \beta_{3} - 25060 \beta_{4} + 290 \beta_{6} + 4822 \beta_{8} - 864 \beta_{9} - 688 \beta_{10} - 10720 \beta_{11} + 344 \beta_{12} - 344 \beta_{13} - 952 \beta_{14} + 768 \beta_{15} ) q^{56} \) \( + ( -5498379 - 59544 \beta_{1} + 1851732 \beta_{2} - 15498 \beta_{3} + 5805 \beta_{4} + 1791 \beta_{6} + 2160 \beta_{7} - 22590 \beta_{8} - 234 \beta_{9} + 1071 \beta_{10} - 486 \beta_{11} - 54 \beta_{12} - 810 \beta_{13} + 648 \beta_{14} - 234 \beta_{15} ) q^{57} \) \( + ( -16938204 - 34743 \beta_{1} - 13890 \beta_{2} + 408 \beta_{3} - 84386 \beta_{4} + 946 \beta_{5} + 736 \beta_{6} - 2288 \beta_{7} + 320 \beta_{8} + 3616 \beta_{9} - 889 \beta_{10} + 672 \beta_{11} + 1319 \beta_{12} + 736 \beta_{13} - 416 \beta_{15} ) q^{58} \) \( + ( 5776 - 1292 \beta_{1} + 1988701 \beta_{2} + 16272 \beta_{3} - 88946 \beta_{4} + 2888 \beta_{6} - 2432 \beta_{8} - 2193 \beta_{9} + 1216 \beta_{10} - 8892 \beta_{11} - 608 \beta_{12} + 608 \beta_{13} + 684 \beta_{14} + 1292 \beta_{15} ) q^{59} \) \( + ( -25903754 + 2892 \beta_{1} - 217496 \beta_{2} + 17203 \beta_{3} - 8611 \beta_{4} + 2112 \beta_{5} + 468 \beta_{6} + 1712 \beta_{7} - 17930 \beta_{8} - 6496 \beta_{9} - 2260 \beta_{10} + 384 \beta_{11} + 394 \beta_{12} - 138 \beta_{13} - 514 \beta_{14} - 2312 \beta_{15} ) q^{60} \) \( + ( 3139897 + 61965 \beta_{1} + 558 \beta_{2} + 746 \beta_{3} + 8206 \beta_{4} - 47 \beta_{6} - 1398 \beta_{7} - 934 \beta_{8} - 887 \beta_{9} - 2520 \beta_{10} - 8265 \beta_{11} - 2661 \beta_{12} - 47 \beta_{13} - 887 \beta_{15} ) q^{61} \) \( + ( -620 - 338 \beta_{1} + 61363 \beta_{2} - 19333 \beta_{3} + 260111 \beta_{4} + 4726 \beta_{6} - 23725 \beta_{8} + 9033 \beta_{9} - 236 \beta_{10} - 291 \beta_{11} + 118 \beta_{12} - 118 \beta_{13} - 1398 \beta_{14} - 589 \beta_{15} ) q^{62} \) \( + ( -26288 + 46480 \beta_{1} - 2512760 \beta_{2} - 20666 \beta_{3} - 18403 \beta_{4} + 11973 \beta_{5} - 2344 \beta_{6} + 1932 \beta_{7} + 1048 \beta_{8} + 4204 \beta_{9} + 232 \beta_{10} + 5052 \beta_{11} - 872 \beta_{12} - 640 \beta_{13} + 444 \beta_{14} + 1316 \beta_{15} ) q^{63} \) \( + ( 46411760 - 3552 \beta_{1} + 57168 \beta_{2} + 348 \beta_{3} + 323868 \beta_{4} - 1264 \beta_{5} + 592 \beta_{6} - 1880 \beta_{7} + 2792 \beta_{8} - 11000 \beta_{9} - 1336 \beta_{10} + 23352 \beta_{11} + 440 \beta_{12} + 592 \beta_{13} + 2200 \beta_{15} ) q^{64} \) \( + ( -882 + 210 \beta_{1} - 2905964 \beta_{2} + 21622 \beta_{3} - 23302 \beta_{4} + 2898 \beta_{6} + 45882 \beta_{8} - 42 \beta_{9} - 756 \beta_{10} + 2394 \beta_{11} + 378 \beta_{12} - 378 \beta_{13} + 504 \beta_{14} - 42 \beta_{15} ) q^{65} \) \( + ( 49366474 + 90685 \beta_{1} - 302170 \beta_{2} + 21771 \beta_{3} - 17609 \beta_{4} - 7782 \beta_{5} + 6642 \beta_{6} + 548 \beta_{7} + 36996 \beta_{8} + 15220 \beta_{9} - 1369 \beta_{10} - 748 \beta_{11} + 549 \beta_{12} - 314 \beta_{13} + 62 \beta_{14} - 316 \beta_{15} ) q^{66} \) \( + ( -1488 + 3480 \beta_{1} + 33057 \beta_{2} - 84 \beta_{3} + 171536 \beta_{4} + 13288 \beta_{5} + 48 \beta_{6} + 72 \beta_{7} - 48 \beta_{8} - 6597 \beta_{9} - 48 \beta_{10} - 576 \beta_{11} + 96 \beta_{12} + 48 \beta_{13} - 96 \beta_{15} ) q^{67} \) \( + ( -21912 + 7856 \beta_{1} + 744384 \beta_{2} - 22812 \beta_{3} - 430692 \beta_{4} - 1216 \beta_{6} + 51832 \beta_{8} - 19776 \beta_{9} + 1552 \beta_{10} + 31808 \beta_{11} - 776 \beta_{12} + 776 \beta_{13} + 104 \beta_{14} - 3488 \beta_{15} ) q^{68} \) \( + ( 3969866 + 18740 \beta_{1} + 3152320 \beta_{2} - 22508 \beta_{3} + 25058 \beta_{4} - 518 \beta_{6} - 1136 \beta_{7} + 67744 \beta_{8} + 360 \beta_{9} - 666 \beta_{10} + 8056 \beta_{11} + 976 \beta_{12} + 976 \beta_{13} - 104 \beta_{14} + 360 \beta_{15} ) q^{69} \) \( + ( -74365252 + 136324 \beta_{1} - 111406 \beta_{2} + 692 \beta_{3} - 634720 \beta_{4} + 14176 \beta_{5} - 1712 \beta_{6} + 2040 \beta_{7} - 454 \beta_{8} + 21838 \beta_{9} + 4192 \beta_{10} + 1050 \beta_{11} - 944 \beta_{12} - 1712 \beta_{13} + 1258 \beta_{15} ) q^{70} \) \( + ( -10992 + 2578 \beta_{1} + 2760364 \beta_{2} + 22564 \beta_{3} + 262280 \beta_{4} - 964 \beta_{6} + 2720 \beta_{8} + 9672 \beta_{9} - 1360 \beta_{10} + 15610 \beta_{11} + 680 \beta_{12} - 680 \beta_{13} - 1898 \beta_{14} - 2578 \beta_{15} ) q^{71} \) \( + ( -78132680 - 5648 \beta_{1} + 399574 \beta_{2} + 20634 \beta_{3} - 15910 \beta_{4} + 9144 \beta_{5} - 1805 \beta_{6} - 2052 \beta_{7} - 41583 \beta_{8} - 23556 \beta_{9} + 1436 \beta_{10} + 36 \beta_{11} - 2460 \beta_{12} + 984 \beta_{13} + 1440 \beta_{14} + 5556 \beta_{15} ) q^{72} \) \( + ( 1495474 - 85140 \beta_{1} - 860 \beta_{2} - 1882 \beta_{3} - 12186 \beta_{4} + 1320 \beta_{6} + 1124 \beta_{7} + 2904 \beta_{8} + 1584 \beta_{9} + 792 \beta_{10} + 22176 \beta_{11} + 4752 \beta_{12} + 1320 \beta_{13} + 1584 \beta_{15} ) q^{73} \) \( + ( 4820 - 392 \beta_{1} - 536606 \beta_{2} - 18578 \beta_{3} + 673474 \beta_{4} - 3204 \beta_{6} - 30792 \beta_{8} + 26376 \beta_{9} - 1176 \beta_{10} - 1528 \beta_{11} + 588 \beta_{12} - 588 \beta_{13} + 4820 \beta_{14} + 2312 \beta_{15} ) q^{74} \) \( + ( 65476 - 129431 \beta_{1} - 1822943 \beta_{2} - 15314 \beta_{3} + 137999 \beta_{4} - 22506 \beta_{5} + 7634 \beta_{6} - 5388 \beta_{7} - 1592 \beta_{8} - 15311 \beta_{9} - 1208 \beta_{10} - 1983 \beta_{11} + 2608 \beta_{12} + 1400 \beta_{13} - 2925 \beta_{14} - 5533 \beta_{15} ) q^{75} \) \( + ( 102843966 + 1902 \beta_{1} + 138808 \beta_{2} - 2075 \beta_{3} + 800677 \beta_{4} - 7052 \beta_{5} - 1812 \beta_{6} + 7774 \beta_{7} - 6606 \beta_{8} - 31710 \beta_{9} - 1454 \beta_{10} - 54018 \beta_{11} - 6890 \beta_{12} - 1812 \beta_{13} - 4794 \beta_{15} ) q^{76} \) \( + ( 19864 - 9000 \beta_{1} - 1769668 \beta_{2} + 15008 \beta_{3} - 14752 \beta_{4} + 1352 \beta_{6} - 120114 \beta_{8} + 1800 \beta_{9} - 3472 \beta_{10} - 30856 \beta_{11} + 1736 \beta_{12} - 1736 \beta_{13} - 3664 \beta_{14} + 1800 \beta_{15} ) q^{77} \) \( + ( 130111260 - 272368 \beta_{1} + 6464 \beta_{2} + 14258 \beta_{3} - 27578 \beta_{4} + 4704 \beta_{5} - 7316 \beta_{6} - 4584 \beta_{7} + 7880 \beta_{8} + 29144 \beta_{9} + 8936 \beta_{10} - 374 \beta_{11} - 4564 \beta_{12} - 1100 \beta_{13} + 36 \beta_{14} + 2296 \beta_{15} ) q^{78} \) \( + ( -90952 + 213532 \beta_{1} - 102862 \beta_{2} - 5186 \beta_{3} - 572455 \beta_{4} - 22163 \beta_{5} + 3000 \beta_{6} + 4372 \beta_{7} - 3000 \beta_{8} + 21498 \beta_{9} - 3000 \beta_{10} - 36000 \beta_{11} + 6000 \beta_{12} + 3000 \beta_{13} - 6000 \beta_{15} ) q^{79} \) \( + ( 44272 - 12512 \beta_{1} - 1645680 \beta_{2} - 6728 \beta_{3} - 832056 \beta_{4} - 344 \beta_{6} - 37016 \beta_{8} - 27072 \beta_{9} + 1120 \beta_{10} - 61632 \beta_{11} - 560 \beta_{12} + 560 \beta_{13} + 5616 \beta_{14} + 9344 \beta_{15} ) q^{80} \) \( + ( 21739524 + 268413 \beta_{1} + 512988 \beta_{2} + 2241 \beta_{3} + 43635 \beta_{4} - 8859 \beta_{6} - 8856 \beta_{7} - 146610 \beta_{8} + 999 \beta_{9} - 4614 \beta_{10} - 23679 \beta_{11} - 1323 \beta_{12} + 1755 \beta_{13} - 4320 \beta_{14} + 999 \beta_{15} ) q^{81} \) \( + ( -204497768 - 386346 \beta_{1} - 147884 \beta_{2} - 3264 \beta_{3} - 766236 \beta_{4} - 33108 \beta_{5} - 1280 \beta_{6} + 9088 \beta_{7} - 1408 \beta_{8} + 25216 \beta_{9} - 3158 \beta_{10} - 8832 \beta_{11} - 6998 \beta_{12} - 1280 \beta_{13} - 128 \beta_{15} ) q^{82} \) \( + ( -18700 + 3927 \beta_{1} - 443508 \beta_{2} - 4424 \beta_{3} - 793250 \beta_{4} - 19074 \beta_{6} + 11968 \beta_{8} - 32334 \beta_{9} - 5984 \beta_{10} + 31603 \beta_{11} + 2992 \beta_{12} - 2992 \beta_{13} - 935 \beta_{14} - 3927 \beta_{15} ) q^{83} \) \( + ( -233435368 + 12302 \beta_{1} - 1487520 \beta_{2} - 5825 \beta_{3} + 80193 \beta_{4} - 22080 \beta_{5} - 136 \beta_{6} - 8608 \beta_{7} + 52138 \beta_{8} - 24976 \beta_{9} + 12540 \beta_{10} - 2224 \beta_{11} + 658 \beta_{12} - 1458 \beta_{13} + 758 \beta_{14} - 5624 \beta_{15} ) q^{84} \) \( + ( -15263260 - 406596 \beta_{1} - 7832 \beta_{2} - 5384 \beta_{3} - 54488 \beta_{4} - 20 \beta_{6} + 10808 \beta_{7} + 2936 \beta_{8} + 2956 \beta_{9} + 8928 \beta_{10} + 26484 \beta_{11} + 8868 \beta_{12} - 20 \beta_{13} + 2956 \beta_{15} ) q^{85} \) \( + ( -5134 + 4158 \beta_{1} + 90205 \beta_{2} + 19662 \beta_{3} + 447800 \beta_{4} - 8676 \beta_{6} + 31109 \beta_{8} + 17223 \beta_{9} + 6088 \beta_{10} + 4387 \beta_{11} - 3044 \beta_{12} + 3044 \beta_{13} + 1252 \beta_{14} + 69 \beta_{15} ) q^{86} \) \( + ( 180676 - 311199 \beta_{1} + 3077243 \beta_{2} + 31205 \beta_{3} - 25262 \beta_{4} + 11109 \beta_{5} + 822 \beta_{6} - 11558 \beta_{7} - 3340 \beta_{8} + 42025 \beta_{9} - 1172 \beta_{10} - 34993 \beta_{11} + 3428 \beta_{12} + 2256 \beta_{13} + 1801 \beta_{14} - 1627 \beta_{15} ) q^{87} \) \( + ( 279175816 - 22256 \beta_{1} + 35448 \beta_{2} - 1882 \beta_{3} + 195654 \beta_{4} + 10824 \beta_{5} - 3064 \beta_{6} + 9892 \beta_{7} + 7444 \beta_{8} - 6460 \beta_{9} + 14180 \beta_{10} + 76188 \beta_{11} + 4988 \beta_{12} - 3064 \beta_{13} + 10508 \beta_{15} ) q^{88} \) \( + ( -3367 + 2475 \beta_{1} + 6334196 \beta_{2} - 41261 \beta_{3} + 49573 \beta_{4} - 15041 \beta_{6} + 245654 \beta_{8} - 495 \beta_{9} + 5146 \beta_{10} + 103 \beta_{11} - 2573 \beta_{12} + 2573 \beta_{13} - 1088 \beta_{14} - 495 \beta_{15} ) q^{89} \) \( + ( 357570800 + 657563 \beta_{1} + 21674 \beta_{2} - 46238 \beta_{3} - 91064 \beta_{4} + 26166 \beta_{5} - 10740 \beta_{6} - 11208 \beta_{7} - 135080 \beta_{8} + 952 \beta_{9} - 5475 \beta_{10} + 2424 \beta_{11} + 817 \beta_{12} + 3908 \beta_{13} + 3540 \beta_{14} + 536 \beta_{15} ) q^{90} \) \( + ( -117656 + 213332 \beta_{1} + 276304 \beta_{2} - 2758 \beta_{3} + 1551820 \beta_{4} + 13310 \beta_{5} - 1176 \beta_{6} + 7868 \beta_{7} + 1176 \beta_{8} - 57540 \beta_{9} + 1176 \beta_{10} + 14112 \beta_{11} - 2352 \beta_{12} - 1176 \beta_{13} + 2352 \beta_{15} ) q^{91} \) \( + ( -47180 + 12760 \beta_{1} + 2592464 \beta_{2} + 66914 \beta_{3} + 229406 \beta_{4} + 9080 \beta_{6} - 183676 \beta_{8} + 6720 \beta_{9} - 6776 \beta_{10} + 73728 \beta_{11} + 3388 \beta_{12} - 3388 \beta_{13} - 2124 \beta_{14} - 9136 \beta_{15} ) q^{92} \) \( + ( -16548051 + 252513 \beta_{1} - 9797320 \beta_{2} + 78322 \beta_{3} - 46418 \beta_{4} + 16733 \beta_{6} - 6254 \beta_{7} + 277051 \beta_{8} + 189 \beta_{9} + 7376 \beta_{10} + 5251 \beta_{11} + 4471 \beta_{12} - 5915 \beta_{13} + 3904 \beta_{14} + 189 \beta_{15} ) q^{93} \) \( + ( -384592712 + 798184 \beta_{1} + 101188 \beta_{2} - 8776 \beta_{3} + 574352 \beta_{4} - 5568 \beta_{5} + 4576 \beta_{6} + 8400 \beta_{7} + 2804 \beta_{8} - 20900 \beta_{9} - 8896 \beta_{10} + 11508 \beta_{11} + 4832 \beta_{12} + 4576 \beta_{13} - 1772 \beta_{15} ) q^{94} \) \( + ( 30072 - 6160 \beta_{1} - 11586754 \beta_{2} - 93004 \beta_{3} + 2107504 \beta_{4} + 36568 \beta_{6} - 21728 \beta_{8} + 79962 \beta_{9} + 10864 \beta_{10} - 52528 \beta_{11} - 5432 \beta_{12} + 5432 \beta_{13} + 728 \beta_{14} + 6160 \beta_{15} ) q^{95} \) \( + ( -434059016 + 9104 \beta_{1} + 5557048 \beta_{2} - 88928 \beta_{3} + 55640 \beta_{4} - 11376 \beta_{5} + 10612 \beta_{6} - 4760 \beta_{7} + 185212 \beta_{8} + 37032 \beta_{9} - 25896 \beta_{10} + 984 \beta_{11} - 752 \beta_{12} - 4040 \beta_{13} - 5240 \beta_{14} + 2712 \beta_{15} ) q^{96} \) \( + ( 1951603 - 2415 \beta_{1} - 14828 \beta_{2} - 3367 \beta_{3} - 4299 \beta_{4} + 1227 \beta_{6} + 4280 \beta_{7} - 8094 \beta_{8} - 9321 \beta_{9} - 31644 \beta_{10} - 76527 \beta_{11} - 27963 \beta_{12} + 1227 \beta_{13} - 9321 \beta_{15} ) q^{97} \) \( + ( -14676 - 120 \beta_{1} + 1020845 \beta_{2} + 112370 \beta_{3} - 1284770 \beta_{4} + 27524 \beta_{6} + 260872 \beta_{8} - 48456 \beta_{9} - 360 \beta_{10} + 7608 \beta_{11} + 180 \beta_{12} - 180 \beta_{13} - 14676 \beta_{14} - 7368 \beta_{15} ) q^{98} \) \( + ( -47344 - 3346 \beta_{1} + 15375593 \beta_{2} + 128240 \beta_{3} - 251540 \beta_{4} + 47850 \beta_{5} - 27884 \beta_{6} + 6312 \beta_{7} + 16112 \beta_{8} - 98689 \beta_{9} + 3392 \beta_{10} + 97134 \beta_{11} - 13144 \beta_{12} - 9752 \beta_{13} + 2706 \beta_{14} + 15850 \beta_{15} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut -\mathstrut 344q^{4} \) \(\mathstrut -\mathstrut 1608q^{6} \) \(\mathstrut -\mathstrut 13872q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut -\mathstrut 344q^{4} \) \(\mathstrut -\mathstrut 1608q^{6} \) \(\mathstrut -\mathstrut 13872q^{9} \) \(\mathstrut -\mathstrut 15952q^{10} \) \(\mathstrut +\mathstrut 42600q^{12} \) \(\mathstrut -\mathstrut 97312q^{13} \) \(\mathstrut +\mathstrut 164896q^{16} \) \(\mathstrut +\mathstrut 256944q^{18} \) \(\mathstrut +\mathstrut 356448q^{21} \) \(\mathstrut +\mathstrut 322128q^{22} \) \(\mathstrut +\mathstrut 166176q^{24} \) \(\mathstrut -\mathstrut 2743728q^{25} \) \(\mathstrut -\mathstrut 1447056q^{28} \) \(\mathstrut +\mathstrut 3286128q^{30} \) \(\mathstrut +\mathstrut 4715520q^{33} \) \(\mathstrut -\mathstrut 13198144q^{34} \) \(\mathstrut -\mathstrut 6827448q^{36} \) \(\mathstrut +\mathstrut 10997600q^{37} \) \(\mathstrut +\mathstrut 23411264q^{40} \) \(\mathstrut +\mathstrut 23964432q^{42} \) \(\mathstrut -\mathstrut 251904q^{45} \) \(\mathstrut -\mathstrut 41256288q^{46} \) \(\mathstrut -\mathstrut 87722976q^{48} \) \(\mathstrut -\mathstrut 21356624q^{49} \) \(\mathstrut +\mathstrut 132124208q^{52} \) \(\mathstrut +\mathstrut 179732232q^{54} \) \(\mathstrut -\mathstrut 88439904q^{57} \) \(\mathstrut -\mathstrut 271309360q^{58} \) \(\mathstrut -\mathstrut 414400704q^{60} \) \(\mathstrut +\mathstrut 50685152q^{61} \) \(\mathstrut +\mathstrut 742710400q^{64} \) \(\mathstrut +\mathstrut 790585104q^{66} \) \(\mathstrut +\mathstrut 63713280q^{69} \) \(\mathstrut -\mathstrut 1188731232q^{70} \) \(\mathstrut -\mathstrut 1250220480q^{72} \) \(\mathstrut +\mathstrut 23382176q^{73} \) \(\mathstrut +\mathstrut 1645242192q^{76} \) \(\mathstrut +\mathstrut 2079579408q^{78} \) \(\mathstrut +\mathstrut 349756560q^{81} \) \(\mathstrut -\mathstrut 3274996000q^{82} \) \(\mathstrut -\mathstrut 3734920464q^{84} \) \(\mathstrut -\mathstrut 247261184q^{85} \) \(\mathstrut +\mathstrut 4467199680q^{88} \) \(\mathstrut +\mathstrut 5726283888q^{90} \) \(\mathstrut -\mathstrut 262825248q^{93} \) \(\mathstrut -\mathstrut 6146963136q^{94} \) \(\mathstrut -\mathstrut 6944875392q^{96} \) \(\mathstrut +\mathstrut 30926624q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut +\mathstrut \) \(43\) \(x^{14}\mathstrut -\mathstrut \) \(1652\) \(x^{12}\mathstrut -\mathstrut \) \(2031680\) \(x^{10}\mathstrut +\mathstrut \) \(40456192\) \(x^{8}\mathstrut -\mathstrut \) \(33287045120\) \(x^{6}\mathstrut -\mathstrut \) \(443455373312\) \(x^{4}\mathstrut +\mathstrut \) \(189115999977472\) \(x^{2}\mathstrut +\mathstrut \) \(72057594037927936\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{14} + 43 \nu^{12} - 1652 \nu^{10} - 2031680 \nu^{8} + 40456192 \nu^{6} - 33287045120 \nu^{4} - 443455373312 \nu^{2} + 166026255794176 \)\()/\)\(1099511627776\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{15} - 43 \nu^{13} + 1652 \nu^{11} + 2031680 \nu^{9} - 40456192 \nu^{7} + 33287045120 \nu^{5} + 443455373312 \nu^{3} - 189115999977472 \nu \)\()/\)\(281474976710656\)
\(\beta_{3}\)\(=\)\((\)\(637169\) \(\nu^{15}\mathstrut +\mathstrut \) \(9143936\) \(\nu^{14}\mathstrut -\mathstrut \) \(22335365\) \(\nu^{13}\mathstrut +\mathstrut \) \(425170816\) \(\nu^{12}\mathstrut -\mathstrut \) \(8643744564\) \(\nu^{11}\mathstrut -\mathstrut \) \(212909683200\) \(\nu^{10}\mathstrut +\mathstrut \) \(674888696768\) \(\nu^{9}\mathstrut +\mathstrut \) \(1804532080640\) \(\nu^{8}\mathstrut -\mathstrut \) \(44517732626432\) \(\nu^{7}\mathstrut -\mathstrut \) \(1382370481012736\) \(\nu^{6}\mathstrut -\mathstrut \) \(15891641341575168\) \(\nu^{5}\mathstrut -\mathstrut \) \(78059002977583104\) \(\nu^{4}\mathstrut -\mathstrut \) \(509119879538802688\) \(\nu^{3}\mathstrut -\mathstrut \) \(26119808701129818112\) \(\nu^{2}\mathstrut +\mathstrut \) \(4345864626033657905152\) \(\nu\mathstrut +\mathstrut \) \(8931773249656767643648\)\()/\)\(15116050674292359168\)
\(\beta_{4}\)\(=\)\((\)\(959387\) \(\nu^{15}\mathstrut +\mathstrut \) \(9143936\) \(\nu^{14}\mathstrut -\mathstrut \) \(8479991\) \(\nu^{13}\mathstrut +\mathstrut \) \(425170816\) \(\nu^{12}\mathstrut -\mathstrut \) \(9176048700\) \(\nu^{11}\mathstrut -\mathstrut \) \(212909683200\) \(\nu^{10}\mathstrut +\mathstrut \) \(20244830528\) \(\nu^{9}\mathstrut +\mathstrut \) \(1804532080640\) \(\nu^{8}\mathstrut -\mathstrut \) \(31482019352576\) \(\nu^{7}\mathstrut -\mathstrut \) \(1382370481012736\) \(\nu^{6}\mathstrut -\mathstrut \) \(26617326446051328\) \(\nu^{5}\mathstrut -\mathstrut \) \(78059002977583104\) \(\nu^{4}\mathstrut -\mathstrut \) \(652009183016648704\) \(\nu^{3}\mathstrut -\mathstrut \) \(26119808701129818112\) \(\nu^{2}\mathstrut +\mathstrut \) \(537092232695555031040\) \(\nu\mathstrut +\mathstrut \) \(8931773249656767643648\)\()/\)\(15116050674292359168\)
\(\beta_{5}\)\(=\)\((\)\(959387\) \(\nu^{15}\mathstrut -\mathstrut \) \(378805888\) \(\nu^{14}\mathstrut -\mathstrut \) \(8479991\) \(\nu^{13}\mathstrut -\mathstrut \) \(34687491968\) \(\nu^{12}\mathstrut -\mathstrut \) \(9176048700\) \(\nu^{11}\mathstrut +\mathstrut \) \(4766870327808\) \(\nu^{10}\mathstrut +\mathstrut \) \(20244830528\) \(\nu^{9}\mathstrut -\mathstrut \) \(183488206102528\) \(\nu^{8}\mathstrut -\mathstrut \) \(31482019352576\) \(\nu^{7}\mathstrut +\mathstrut \) \(43307163991932928\) \(\nu^{6}\mathstrut -\mathstrut \) \(26617326446051328\) \(\nu^{5}\mathstrut +\mathstrut \) \(2355279720918024192\) \(\nu^{4}\mathstrut -\mathstrut \) \(652009183016648704\) \(\nu^{3}\mathstrut +\mathstrut \) \(1397128505694416273408\) \(\nu^{2}\mathstrut +\mathstrut \) \(537092232695555031040\) \(\nu\mathstrut -\mathstrut \) \(251842991552892445392896\)\()/\)\(15116050674292359168\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(1429\) \(\nu^{15}\mathstrut -\mathstrut \) \(69639\) \(\nu^{13}\mathstrut -\mathstrut \) \(148986492\) \(\nu^{11}\mathstrut +\mathstrut \) \(5013955904\) \(\nu^{9}\mathstrut +\mathstrut \) \(1677154086912\) \(\nu^{7}\mathstrut +\mathstrut \) \(70438257426432\) \(\nu^{5}\mathstrut +\mathstrut \) \(5541582627405824\) \(\nu^{3}\mathstrut +\mathstrut \) \(2799334614085140480\) \(\nu\)\()/\)\(5488762045857792\)
\(\beta_{7}\)\(=\)\((\)\(4582123\) \(\nu^{15}\mathstrut +\mathstrut \) \(59467648\) \(\nu^{14}\mathstrut -\mathstrut \) \(51636871\) \(\nu^{13}\mathstrut +\mathstrut \) \(2717016704\) \(\nu^{12}\mathstrut -\mathstrut \) \(45525374076\) \(\nu^{11}\mathstrut -\mathstrut \) \(1087260059136\) \(\nu^{10}\mathstrut +\mathstrut \) \(537653396800\) \(\nu^{9}\mathstrut -\mathstrut \) \(18908811223040\) \(\nu^{8}\mathstrut -\mathstrut \) \(166100572278784\) \(\nu^{7}\mathstrut -\mathstrut \) \(6355661972045824\) \(\nu^{6}\mathstrut -\mathstrut \) \(125936175493939200\) \(\nu^{5}\mathstrut -\mathstrut \) \(847924246012231680\) \(\nu^{4}\mathstrut -\mathstrut \) \(3164786379431346176\) \(\nu^{3}\mathstrut +\mathstrut \) \(1798158832522051452928\) \(\nu^{2}\mathstrut +\mathstrut \) \(4579691263600036413440\) \(\nu\mathstrut +\mathstrut \) \(57341232764015127560192\)\()/\)\(15116050674292359168\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(2177\) \(\nu^{15}\mathstrut +\mathstrut \) \(541269\) \(\nu^{13}\mathstrut -\mathstrut \) \(44601228\) \(\nu^{11}\mathstrut +\mathstrut \) \(4422721600\) \(\nu^{9}\mathstrut -\mathstrut \) \(518784897024\) \(\nu^{7}\mathstrut +\mathstrut \) \(109751967940608\) \(\nu^{5}\mathstrut -\mathstrut \) \(17850278145425408\) \(\nu^{3}\mathstrut +\mathstrut \) \(839727816458108928\) \(\nu\)\()/\)\(2744381022928896\)
\(\beta_{9}\)\(=\)\((\)\(2045717\) \(\nu^{15}\mathstrut -\mathstrut \) \(18287872\) \(\nu^{14}\mathstrut -\mathstrut \) \(19152761\) \(\nu^{13}\mathstrut -\mathstrut \) \(850341632\) \(\nu^{12}\mathstrut -\mathstrut \) \(19729675140\) \(\nu^{11}\mathstrut +\mathstrut \) \(425819366400\) \(\nu^{10}\mathstrut +\mathstrut \) \(85568600768\) \(\nu^{9}\mathstrut -\mathstrut \) \(3609064161280\) \(\nu^{8}\mathstrut -\mathstrut \) \(68643048943616\) \(\nu^{7}\mathstrut +\mathstrut \) \(2764740962025472\) \(\nu^{6}\mathstrut -\mathstrut \) \(56642082274541568\) \(\nu^{5}\mathstrut +\mathstrut \) \(156118005955166208\) \(\nu^{4}\mathstrut -\mathstrut \) \(1395167900377022464\) \(\nu^{3}\mathstrut +\mathstrut \) \(52239617402259636224\) \(\nu^{2}\mathstrut +\mathstrut \) \(1152907848672429998080\) \(\nu\mathstrut -\mathstrut \) \(17863546499313535287296\)\()/\)\(1162773128791719936\)
\(\beta_{10}\)\(=\)\((\)\(26465965\) \(\nu^{15}\mathstrut -\mathstrut \) \(2152808576\) \(\nu^{14}\mathstrut -\mathstrut \) \(3621851377\) \(\nu^{13}\mathstrut +\mathstrut \) \(280027605632\) \(\nu^{12}\mathstrut +\mathstrut \) \(537397513116\) \(\nu^{11}\mathstrut -\mathstrut \) \(11545847514624\) \(\nu^{10}\mathstrut -\mathstrut \) \(28258362075968\) \(\nu^{9}\mathstrut +\mathstrut \) \(2390953246597120\) \(\nu^{8}\mathstrut +\mathstrut \) \(7226284146102272\) \(\nu^{7}\mathstrut -\mathstrut \) \(277462607502770176\) \(\nu^{6}\mathstrut -\mathstrut \) \(1083780345646546944\) \(\nu^{5}\mathstrut +\mathstrut \) \(72188605948785328128\) \(\nu^{4}\mathstrut +\mathstrut \) \(237644228987340193792\) \(\nu^{3}\mathstrut -\mathstrut \) \(8536207668654821605376\) \(\nu^{2}\mathstrut -\mathstrut \) \(10847676109347144859648\) \(\nu\mathstrut -\mathstrut \) \(135591028028648036761600\)\()/\)\(15116050674292359168\)
\(\beta_{11}\)\(=\)\((\)\(31732033\) \(\nu^{15}\mathstrut -\mathstrut \) \(245603072\) \(\nu^{14}\mathstrut -\mathstrut \) \(976370197\) \(\nu^{13}\mathstrut +\mathstrut \) \(2170877696\) \(\nu^{12}\mathstrut -\mathstrut \) \(161265047412\) \(\nu^{11}\mathstrut +\mathstrut \) \(2349068467200\) \(\nu^{10}\mathstrut -\mathstrut \) \(9964457906240\) \(\nu^{9}\mathstrut -\mathstrut \) \(5182676615168\) \(\nu^{8}\mathstrut +\mathstrut \) \(821797006954496\) \(\nu^{7}\mathstrut +\mathstrut \) \(8059396954259456\) \(\nu^{6}\mathstrut -\mathstrut \) \(702378771081068544\) \(\nu^{5}\mathstrut +\mathstrut \) \(6814035570189139968\) \(\nu^{4}\mathstrut +\mathstrut \) \(5911364222297571328\) \(\nu^{3}\mathstrut +\mathstrut \) \(166914350852262068224\) \(\nu^{2}\mathstrut +\mathstrut \) \(12687706179602374197248\) \(\nu\mathstrut -\mathstrut \) \(139007216637491323863040\)\()/\)\(15116050674292359168\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(34033655\) \(\nu^{15}\mathstrut -\mathstrut \) \(1779665024\) \(\nu^{14}\mathstrut +\mathstrut \) \(3694309763\) \(\nu^{13}\mathstrut +\mathstrut \) \(408969810560\) \(\nu^{12}\mathstrut -\mathstrut \) \(464166558228\) \(\nu^{11}\mathstrut -\mathstrut \) \(39658475693568\) \(\nu^{10}\mathstrut +\mathstrut \) \(27878188809664\) \(\nu^{9}\mathstrut +\mathstrut \) \(3488206238654464\) \(\nu^{8}\mathstrut -\mathstrut \) \(6970082753523712\) \(\nu^{7}\mathstrut -\mathstrut \) \(354167156615151616\) \(\nu^{6}\mathstrut +\mathstrut \) \(1293143728846798848\) \(\nu^{5}\mathstrut +\mathstrut \) \(88515715644669296640\) \(\nu^{4}\mathstrut -\mathstrut \) \(232475785291032952832\) \(\nu^{3}\mathstrut -\mathstrut \) \(13571989621064131936256\) \(\nu^{2}\mathstrut +\mathstrut \) \(6571250440876284968960\) \(\nu\mathstrut +\mathstrut \) \(738782029258875811659776\)\()/\)\(15116050674292359168\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(45803147\) \(\nu^{15}\mathstrut +\mathstrut \) \(1228670272\) \(\nu^{14}\mathstrut -\mathstrut \) \(3923470681\) \(\nu^{13}\mathstrut -\mathstrut \) \(77113794112\) \(\nu^{12}\mathstrut +\mathstrut \) \(1599836006652\) \(\nu^{11}\mathstrut -\mathstrut \) \(7633305460992\) \(\nu^{10}\mathstrut -\mathstrut \) \(31156196711744\) \(\nu^{9}\mathstrut -\mathstrut \) \(638297987698688\) \(\nu^{8}\mathstrut +\mathstrut \) \(7422759422234624\) \(\nu^{7}\mathstrut +\mathstrut \) \(104248045421723648\) \(\nu^{6}\mathstrut +\mathstrut \) \(35237391340929024\) \(\nu^{5}\mathstrut -\mathstrut \) \(27467756501955772416\) \(\nu^{4}\mathstrut +\mathstrut \) \(379040491276343443456\) \(\nu^{3}\mathstrut +\mathstrut \) \(1831620589367005806592\) \(\nu^{2}\mathstrut -\mathstrut \) \(57528434259161943900160\) \(\nu\mathstrut +\mathstrut \) \(477053019108543712722944\)\()/\)\(7558025337146179584\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(94030897\) \(\nu^{15}\mathstrut +\mathstrut \) \(475188224\) \(\nu^{14}\mathstrut -\mathstrut \) \(1771244603\) \(\nu^{13}\mathstrut -\mathstrut \) \(5062507520\) \(\nu^{12}\mathstrut +\mathstrut \) \(1983977581620\) \(\nu^{11}\mathstrut -\mathstrut \) \(4473871429632\) \(\nu^{10}\mathstrut +\mathstrut \) \(10805594623040\) \(\nu^{9}\mathstrut +\mathstrut \) \(22526556962816\) \(\nu^{8}\mathstrut +\mathstrut \) \(12851483062546432\) \(\nu^{7}\mathstrut -\mathstrut \) \(15014518644015104\) \(\nu^{6}\mathstrut +\mathstrut \) \(1102756943301181440\) \(\nu^{5}\mathstrut -\mathstrut \) \(13321197521838538752\) \(\nu^{4}\mathstrut +\mathstrut \) \(48943808136316190720\) \(\nu^{3}\mathstrut -\mathstrut \) \(304660587862533603328\) \(\nu^{2}\mathstrut -\mathstrut \) \(85106232536718198702080\) \(\nu\mathstrut +\mathstrut \) \(267956514250091099324416\)\()/\)\(7558025337146179584\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(275516155\) \(\nu^{15}\mathstrut -\mathstrut \) \(2036628736\) \(\nu^{14}\mathstrut +\mathstrut \) \(5822982103\) \(\nu^{13}\mathstrut +\mathstrut \) \(27522957568\) \(\nu^{12}\mathstrut +\mathstrut \) \(1715401087932\) \(\nu^{11}\mathstrut +\mathstrut \) \(17689637987328\) \(\nu^{10}\mathstrut +\mathstrut \) \(65279280922304\) \(\nu^{9}\mathstrut -\mathstrut \) \(73634519498752\) \(\nu^{8}\mathstrut -\mathstrut \) \(4475741811077120\) \(\nu^{7}\mathstrut +\mathstrut \) \(51529025758167040\) \(\nu^{6}\mathstrut +\mathstrut \) \(5770129753204850688\) \(\nu^{5}\mathstrut +\mathstrut \) \(59162117621812297728\) \(\nu^{4}\mathstrut +\mathstrut \) \(46421072390358302720\) \(\nu^{3}\mathstrut +\mathstrut \) \(1072118997888838664192\) \(\nu^{2}\mathstrut -\mathstrut \) \(119888760894863741812736\) \(\nu\mathstrut -\mathstrut \) \(1103682226743323094155264\)\()/\)\(15116050674292359168\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(6\) \(\beta_{2}\)\()/256\)
\(\nu^{2}\)\(=\)\((\)\(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut -\mathstrut \) \(1376\)\()/256\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(4\) \(\beta_{14}\mathstrut +\mathstrut \) \(4\) \(\beta_{13}\mathstrut -\mathstrut \) \(4\) \(\beta_{12}\mathstrut -\mathstrut \) \(16\) \(\beta_{11}\mathstrut +\mathstrut \) \(8\) \(\beta_{10}\mathstrut +\mathstrut \) \(16\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut +\mathstrut \) \(19\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(646\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\)\()/256\)
\(\nu^{4}\)\(=\)\((\)\(64\) \(\beta_{15}\mathstrut +\mathstrut \) \(16\) \(\beta_{13}\mathstrut +\mathstrut \) \(672\) \(\beta_{11}\mathstrut -\mathstrut \) \(48\) \(\beta_{10}\mathstrut -\mathstrut \) \(128\) \(\beta_{9}\mathstrut +\mathstrut \) \(80\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(16\) \(\beta_{6}\mathstrut -\mathstrut \) \(320\) \(\beta_{5}\mathstrut +\mathstrut \) \(4047\) \(\beta_{4}\mathstrut -\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(638\) \(\beta_{2}\mathstrut -\mathstrut \) \(2546\) \(\beta_{1}\mathstrut +\mathstrut \) \(165872\)\()/256\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(704\) \(\beta_{15}\mathstrut -\mathstrut \) \(308\) \(\beta_{14}\mathstrut -\mathstrut \) \(12\) \(\beta_{13}\mathstrut +\mathstrut \) \(12\) \(\beta_{12}\mathstrut +\mathstrut \) \(5104\) \(\beta_{11}\mathstrut -\mathstrut \) \(24\) \(\beta_{10}\mathstrut -\mathstrut \) \(7104\) \(\beta_{9}\mathstrut +\mathstrut \) \(2000\) \(\beta_{8}\mathstrut -\mathstrut \) \(780\) \(\beta_{6}\mathstrut -\mathstrut \) \(174245\) \(\beta_{4}\mathstrut +\mathstrut \) \(117\) \(\beta_{3}\mathstrut +\mathstrut \) \(688422\) \(\beta_{2}\mathstrut +\mathstrut \) \(1088\) \(\beta_{1}\mathstrut -\mathstrut \) \(3596\)\()/256\)
\(\nu^{6}\)\(=\)\((\)\(-\)\(3520\) \(\beta_{15}\mathstrut +\mathstrut \) \(8528\) \(\beta_{13}\mathstrut +\mathstrut \) \(14080\) \(\beta_{12}\mathstrut +\mathstrut \) \(19488\) \(\beta_{11}\mathstrut -\mathstrut \) \(11504\) \(\beta_{10}\mathstrut +\mathstrut \) \(49280\) \(\beta_{9}\mathstrut +\mathstrut \) \(5008\) \(\beta_{8}\mathstrut -\mathstrut \) \(5950\) \(\beta_{7}\mathstrut +\mathstrut \) \(8528\) \(\beta_{6}\mathstrut +\mathstrut \) \(5056\) \(\beta_{5}\mathstrut -\mathstrut \) \(1278105\) \(\beta_{4}\mathstrut -\mathstrut \) \(5553\) \(\beta_{3}\mathstrut -\mathstrut \) \(220818\) \(\beta_{2}\mathstrut +\mathstrut \) \(194622\) \(\beta_{1}\mathstrut +\mathstrut \) \(185578800\)\()/256\)
\(\nu^{7}\)\(=\)\((\)\(-\)\(22976\) \(\beta_{15}\mathstrut +\mathstrut \) \(65004\) \(\beta_{14}\mathstrut +\mathstrut \) \(6100\) \(\beta_{13}\mathstrut -\mathstrut \) \(6100\) \(\beta_{12}\mathstrut +\mathstrut \) \(466800\) \(\beta_{11}\mathstrut +\mathstrut \) \(12200\) \(\beta_{10}\mathstrut -\mathstrut \) \(149184\) \(\beta_{9}\mathstrut +\mathstrut \) \(272208\) \(\beta_{8}\mathstrut +\mathstrut \) \(411860\) \(\beta_{6}\mathstrut -\mathstrut \) \(3928557\) \(\beta_{4}\mathstrut +\mathstrut \) \(583997\) \(\beta_{3}\mathstrut -\mathstrut \) \(62557738\) \(\beta_{2}\mathstrut +\mathstrut \) \(117056\) \(\beta_{1}\mathstrut -\mathstrut \) \(273964\)\()/256\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(693184\) \(\beta_{15}\mathstrut +\mathstrut \) \(134864\) \(\beta_{13}\mathstrut +\mathstrut \) \(344832\) \(\beta_{12}\mathstrut -\mathstrut \) \(5429472\) \(\beta_{11}\mathstrut -\mathstrut \) \(59760\) \(\beta_{10}\mathstrut +\mathstrut \) \(4386944\) \(\beta_{9}\mathstrut -\mathstrut \) \(558320\) \(\beta_{8}\mathstrut +\mathstrut \) \(1093010\) \(\beta_{7}\mathstrut +\mathstrut \) \(134864\) \(\beta_{6}\mathstrut -\mathstrut \) \(2515520\) \(\beta_{5}\mathstrut -\mathstrut \) \(124269761\) \(\beta_{4}\mathstrut -\mathstrut \) \(681369\) \(\beta_{3}\mathstrut -\mathstrut \) \(22472930\) \(\beta_{2}\mathstrut -\mathstrut \) \(103579794\) \(\beta_{1}\mathstrut -\mathstrut \) \(15631928912\)\()/256\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(437184\) \(\beta_{15}\mathstrut +\mathstrut \) \(4953292\) \(\beta_{14}\mathstrut +\mathstrut \) \(5354228\) \(\beta_{13}\mathstrut -\mathstrut \) \(5354228\) \(\beta_{12}\mathstrut +\mathstrut \) \(23747824\) \(\beta_{11}\mathstrut +\mathstrut \) \(10708456\) \(\beta_{10}\mathstrut +\mathstrut \) \(45797184\) \(\beta_{9}\mathstrut +\mathstrut \) \(76010448\) \(\beta_{8}\mathstrut +\mathstrut \) \(2366452\) \(\beta_{6}\mathstrut +\mathstrut \) \(1291366091\) \(\beta_{4}\mathstrut -\mathstrut \) \(51612411\) \(\beta_{3}\mathstrut +\mathstrut \) \(26458049478\) \(\beta_{2}\mathstrut +\mathstrut \) \(11181888\) \(\beta_{1}\mathstrut -\mathstrut \) \(17883916\)\()/256\)
\(\nu^{10}\)\(=\)\((\)\(32384576\) \(\beta_{15}\mathstrut -\mathstrut \) \(43090096\) \(\beta_{13}\mathstrut -\mathstrut \) \(88036608\) \(\beta_{12}\mathstrut +\mathstrut \) \(32920608\) \(\beta_{11}\mathstrut +\mathstrut \) \(41233680\) \(\beta_{10}\mathstrut +\mathstrut \) \(240891008\) \(\beta_{9}\mathstrut -\mathstrut \) \(10705520\) \(\beta_{8}\mathstrut -\mathstrut \) \(65521438\) \(\beta_{7}\mathstrut -\mathstrut \) \(43090096\) \(\beta_{6}\mathstrut -\mathstrut \) \(676224064\) \(\beta_{5}\mathstrut -\mathstrut \) \(6284131113\) \(\beta_{4}\mathstrut +\mathstrut \) \(75850815\) \(\beta_{3}\mathstrut -\mathstrut \) \(1271916722\) \(\beta_{2}\mathstrut -\mathstrut \) \(1905939426\) \(\beta_{1}\mathstrut +\mathstrut \) \(6698777337136\)\()/256\)
\(\nu^{11}\)\(=\)\((\)\(-\)\(375718336\) \(\beta_{15}\mathstrut +\mathstrut \) \(638421932\) \(\beta_{14}\mathstrut +\mathstrut \) \(264729108\) \(\beta_{13}\mathstrut -\mathstrut \) \(264729108\) \(\beta_{12}\mathstrut +\mathstrut \) \(5935152752\) \(\beta_{11}\mathstrut +\mathstrut \) \(529458216\) \(\beta_{10}\mathstrut -\mathstrut \) \(2153270976\) \(\beta_{9}\mathstrut +\mathstrut \) \(4892266576\) \(\beta_{8}\mathstrut -\mathstrut \) \(4523343084\) \(\beta_{6}\mathstrut -\mathstrut \) \(73268241085\) \(\beta_{4}\mathstrut +\mathstrut \) \(26159907597\) \(\beta_{3}\mathstrut +\mathstrut \) \(350245603382\) \(\beta_{2}\mathstrut +\mathstrut \) \(1654587712\) \(\beta_{1}\mathstrut -\mathstrut \) \(3795882988\)\()/256\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(3688452032\) \(\beta_{15}\mathstrut +\mathstrut \) \(1574668240\) \(\beta_{13}\mathstrut +\mathstrut \) \(7191487232\) \(\beta_{12}\mathstrut -\mathstrut \) \(23748058848\) \(\beta_{11}\mathstrut +\mathstrut \) \(2467482512\) \(\beta_{10}\mathstrut +\mathstrut \) \(59238828160\) \(\beta_{9}\mathstrut -\mathstrut \) \(2113783792\) \(\beta_{8}\mathstrut +\mathstrut \) \(51085120690\) \(\beta_{7}\mathstrut +\mathstrut \) \(1574668240\) \(\beta_{6}\mathstrut -\mathstrut \) \(37759194688\) \(\beta_{5}\mathstrut -\mathstrut \) \(1805380047825\) \(\beta_{4}\mathstrut -\mathstrut \) \(27117228585\) \(\beta_{3}\mathstrut -\mathstrut \) \(334470955266\) \(\beta_{2}\mathstrut +\mathstrut \) \(1047053211726\) \(\beta_{1}\mathstrut +\mathstrut \) \(95451675052208\)\()/256\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(2765060032\) \(\beta_{15}\mathstrut -\mathstrut \) \(7889651316\) \(\beta_{14}\mathstrut +\mathstrut \) \(143488735796\) \(\beta_{13}\mathstrut -\mathstrut \) \(143488735796\) \(\beta_{12}\mathstrut -\mathstrut \) \(6673064976\) \(\beta_{11}\mathstrut +\mathstrut \) \(286977471592\) \(\beta_{10}\mathstrut +\mathstrut \) \(152554238784\) \(\beta_{9}\mathstrut +\mathstrut \) \(2019433312464\) \(\beta_{8}\mathstrut +\mathstrut \) \(100373238580\) \(\beta_{6}\mathstrut +\mathstrut \) \(4540588122235\) \(\beta_{4}\mathstrut +\mathstrut \) \(315216158805\) \(\beta_{3}\mathstrut -\mathstrut \) \(272384146215386\) \(\beta_{2}\mathstrut +\mathstrut \) \(141129204544\) \(\beta_{1}\mathstrut -\mathstrut \) \(144299793356\)\()/256\)
\(\nu^{14}\)\(=\)\((\)\(1076551371328\) \(\beta_{15}\mathstrut +\mathstrut \) \(322687235152\) \(\beta_{13}\mathstrut -\mathstrut \) \(323705332992\) \(\beta_{12}\mathstrut +\mathstrut \) \(11625085752864\) \(\beta_{11}\mathstrut -\mathstrut \) \(1291767038448\) \(\beta_{10}\mathstrut +\mathstrut \) \(509125803136\) \(\beta_{9}\mathstrut +\mathstrut \) \(1399238606480\) \(\beta_{8}\mathstrut +\mathstrut \) \(577351408898\) \(\beta_{7}\mathstrut +\mathstrut \) \(322687235152\) \(\beta_{6}\mathstrut -\mathstrut \) \(15460609400896\) \(\beta_{5}\mathstrut -\mathstrut \) \(2797596053689\) \(\beta_{4}\mathstrut -\mathstrut \) \(611362939601\) \(\beta_{3}\mathstrut -\mathstrut \) \(4093078325714\) \(\beta_{2}\mathstrut -\mathstrut \) \(70647311820802\) \(\beta_{1}\mathstrut -\mathstrut \) \(69896458090999760\)\()/256\)
\(\nu^{15}\)\(=\)\((\)\(-\)\(23894565395904\) \(\beta_{15}\mathstrut -\mathstrut \) \(3198613366164\) \(\beta_{14}\mathstrut +\mathstrut \) \(6272988970836\) \(\beta_{13}\mathstrut -\mathstrut \) \(6272988970836\) \(\beta_{12}\mathstrut +\mathstrut \) \(202256635098480\) \(\beta_{11}\mathstrut +\mathstrut \) \(12545977941672\) \(\beta_{10}\mathstrut -\mathstrut \) \(147507565116096\) \(\beta_{9}\mathstrut +\mathstrut \) \(138332176041296\) \(\beta_{8}\mathstrut -\mathstrut \) \(47833099771820\) \(\beta_{6}\mathstrut -\mathstrut \) \(3136266832354701\) \(\beta_{4}\mathstrut -\mathstrut \) \(285376107236643\) \(\beta_{3}\mathstrut +\mathstrut \) \(20855370155822486\) \(\beta_{2}\mathstrut +\mathstrut \) \(50863506396480\) \(\beta_{1}\mathstrut -\mathstrut \) \(143243154613932\)\()/256\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
11.2559 1.14184i
11.2559 + 1.14184i
9.08786 6.73875i
9.08786 + 6.73875i
4.36842 10.4363i
4.36842 + 10.4363i
4.10871 10.5413i
4.10871 + 10.5413i
−4.10871 10.5413i
−4.10871 + 10.5413i
−4.36842 10.4363i
−4.36842 + 10.4363i
−9.08786 6.73875i
−9.08786 + 6.73875i
−11.2559 1.14184i
−11.2559 + 1.14184i
−22.5119 2.28368i 69.9307 121.625i 501.570 + 102.820i 852.450i −1852.02 + 2578.31i 9640.14i −11056.5 3460.10i −9902.41 17010.7i 1946.73 19190.3i
11.2 −22.5119 + 2.28368i 69.9307 + 121.625i 501.570 102.820i 852.450i −1852.02 2578.31i 9640.14i −11056.5 + 3460.10i −9902.41 + 17010.7i 1946.73 + 19190.3i
11.3 −18.1757 13.4775i −122.433 + 68.5072i 148.714 + 489.927i 8.50078i 3148.61 + 404.916i 3636.61i 3899.99 10909.1i 10296.5 16775.0i 114.569 154.508i
11.4 −18.1757 + 13.4775i −122.433 68.5072i 148.714 489.927i 8.50078i 3148.61 404.916i 3636.61i 3899.99 + 10909.1i 10296.5 + 16775.0i 114.569 + 154.508i
11.5 −8.73683 20.8727i 132.615 + 45.7844i −359.336 + 364.722i 1832.33i −202.995 3168.04i 1303.88i 10752.2 + 4313.78i 15490.6 + 12143.4i 38245.5 16008.7i
11.6 −8.73683 + 20.8727i 132.615 45.7844i −359.336 364.722i 1832.33i −202.995 + 3168.04i 1303.88i 10752.2 4313.78i 15490.6 12143.4i 38245.5 + 16008.7i
11.7 −8.21741 21.0826i −12.8513 139.706i −376.948 + 346.488i 2101.02i −2839.76 + 1418.96i 7674.38i 10402.4 + 5099.80i −19352.7 + 3590.81i −44294.8 + 17264.9i
11.8 −8.21741 + 21.0826i −12.8513 + 139.706i −376.948 346.488i 2101.02i −2839.76 1418.96i 7674.38i 10402.4 5099.80i −19352.7 3590.81i −44294.8 17264.9i
11.9 8.21741 21.0826i 12.8513 + 139.706i −376.948 346.488i 2101.02i 3050.97 + 877.086i 7674.38i −10402.4 + 5099.80i −19352.7 + 3590.81i −44294.8 17264.9i
11.10 8.21741 + 21.0826i 12.8513 139.706i −376.948 + 346.488i 2101.02i 3050.97 877.086i 7674.38i −10402.4 5099.80i −19352.7 3590.81i −44294.8 + 17264.9i
11.11 8.73683 20.8727i −132.615 45.7844i −359.336 364.722i 1832.33i −2114.28 + 2368.02i 1303.88i −10752.2 + 4313.78i 15490.6 + 12143.4i 38245.5 + 16008.7i
11.12 8.73683 + 20.8727i −132.615 + 45.7844i −359.336 + 364.722i 1832.33i −2114.28 2368.02i 1303.88i −10752.2 4313.78i 15490.6 12143.4i 38245.5 16008.7i
11.13 18.1757 13.4775i 122.433 68.5072i 148.714 489.927i 8.50078i 1302.00 2895.25i 3636.61i −3899.99 10909.1i 10296.5 16775.0i 114.569 + 154.508i
11.14 18.1757 + 13.4775i 122.433 + 68.5072i 148.714 + 489.927i 8.50078i 1302.00 + 2895.25i 3636.61i −3899.99 + 10909.1i 10296.5 + 16775.0i 114.569 154.508i
11.15 22.5119 2.28368i −69.9307 + 121.625i 501.570 102.820i 852.450i −1296.52 + 2897.71i 9640.14i 11056.5 3460.10i −9902.41 17010.7i 1946.73 + 19190.3i
11.16 22.5119 + 2.28368i −69.9307 121.625i 501.570 + 102.820i 852.450i −1296.52 2897.71i 9640.14i 11056.5 + 3460.10i −9902.41 + 17010.7i 1946.73 19190.3i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.16
Significant digits:
Format:

Inner twists

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

Hecke kernels

There are no other newforms in \(S_{10}^{\mathrm{new}}(12, [\chi])\).