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

Related objects

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{10}^{\mathrm{new}}(12, [\chi])$$.