# Properties

 Label 8.17.d.b Level 8 Weight 17 Character orbit 8.d Analytic conductor 12.986 Analytic rank 0 Dimension 14 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$17$$ Character orbit: $$[\chi]$$ = 8.d (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$12.9859635085$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{91}\cdot 3^{10}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -25 - \beta_{1} ) q^{2}$$ $$+ ( 855 - 6 \beta_{1} + \beta_{3} ) q^{3}$$ $$+ ( -3873 + 27 \beta_{1} + \beta_{2} ) q^{4}$$ $$+ ( 1 + 95 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{5}$$ $$+ ( 347700 - 979 \beta_{1} + 6 \beta_{2} - 51 \beta_{3} + \beta_{5} ) q^{6}$$ $$+ ( -10 - 2937 \beta_{1} - 24 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{7}$$ $$+ ( 778563 + 3569 \beta_{1} - 28 \beta_{2} + 109 \beta_{3} - \beta_{4} - 3 \beta_{6} - \beta_{8} ) q^{8}$$ $$+ ( 6172027 + 5481 \beta_{1} - 163 \beta_{2} + 1795 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} + 5 \beta_{6} + \beta_{7} - \beta_{10} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -25 - \beta_{1} ) q^{2}$$ $$+ ( 855 - 6 \beta_{1} + \beta_{3} ) q^{3}$$ $$+ ( -3873 + 27 \beta_{1} + \beta_{2} ) q^{4}$$ $$+ ( 1 + 95 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{5}$$ $$+ ( 347700 - 979 \beta_{1} + 6 \beta_{2} - 51 \beta_{3} + \beta_{5} ) q^{6}$$ $$+ ( -10 - 2937 \beta_{1} - 24 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{7}$$ $$+ ( 778563 + 3569 \beta_{1} - 28 \beta_{2} + 109 \beta_{3} - \beta_{4} - 3 \beta_{6} - \beta_{8} ) q^{8}$$ $$+ ( 6172027 + 5481 \beta_{1} - 163 \beta_{2} + 1795 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} + 5 \beta_{6} + \beta_{7} - \beta_{10} ) q^{9}$$ $$+ ( 6225105 - 2212 \beta_{1} - 90 \beta_{2} - 664 \beta_{3} + 2 \beta_{4} - \beta_{5} - 11 \beta_{6} + \beta_{7} + \beta_{9} - \beta_{12} + \beta_{13} ) q^{10}$$ $$+ ( -3507114 + 55855 \beta_{1} - 119 \beta_{2} + 6787 \beta_{3} + 10 \beta_{4} + 32 \beta_{5} + 30 \beta_{6} - 2 \beta_{7} + \beta_{10} - 3 \beta_{11} - \beta_{13} ) q^{11}$$ $$+ ( 2248978 - 344911 \beta_{1} + 1065 \beta_{2} - 21557 \beta_{3} - 74 \beta_{4} - 82 \beta_{5} - 60 \beta_{6} - 2 \beta_{7} - 6 \beta_{8} + 14 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} + 4 \beta_{13} ) q^{12}$$ $$+ ( -2452 - 568320 \beta_{1} + 1539 \beta_{2} - 5026 \beta_{3} + 61 \beta_{4} + 110 \beta_{5} + 151 \beta_{6} - 4 \beta_{7} + 19 \beta_{8} - 7 \beta_{10} + \beta_{11} + 8 \beta_{12} + 2 \beta_{13} ) q^{13}$$ $$+ ( -191088515 + 76459 \beta_{1} + 2610 \beta_{2} - 7011 \beta_{3} - 236 \beta_{4} - 10 \beta_{5} - 35 \beta_{6} - 34 \beta_{7} + 14 \beta_{8} - 90 \beta_{9} + 28 \beta_{10} - 14 \beta_{11} - 6 \beta_{12} + 10 \beta_{13} ) q^{14}$$ $$+ ( -1900 - 399187 \beta_{1} + 1820 \beta_{2} - 3974 \beta_{3} + 251 \beta_{4} - 476 \beta_{5} + 47 \beta_{6} + 8 \beta_{7} + 90 \beta_{8} - 19 \beta_{9} + 14 \beta_{10} - 34 \beta_{11} - 16 \beta_{12} + 28 \beta_{13} ) q^{15}$$ $$+ ( 694048236 - 1069318 \beta_{1} - 2330 \beta_{2} + 51782 \beta_{3} + 2422 \beta_{4} + 156 \beta_{5} - 192 \beta_{6} + 68 \beta_{7} + 2 \beta_{8} - 364 \beta_{9} - 30 \beta_{10} - 40 \beta_{11} + 12 \beta_{12} + 32 \beta_{13} ) q^{16}$$ $$+ ( -713187718 - 872749 \beta_{1} - 21257 \beta_{2} + 76217 \beta_{3} - 36 \beta_{4} - 1096 \beta_{5} - 401 \beta_{6} - 69 \beta_{7} - 208 \beta_{8} - 35 \beta_{10} + 8 \beta_{11} + 72 \beta_{13} ) q^{17}$$ $$+ ( -549327193 - 6099849 \beta_{1} - 3588 \beta_{2} + 258934 \beta_{3} + 6416 \beta_{4} + 2384 \beta_{5} + 3570 \beta_{6} - 64 \beta_{7} + 132 \beta_{8} + 1072 \beta_{9} - 48 \beta_{10} - 20 \beta_{11} + 16 \beta_{12} + 56 \beta_{13} ) q^{18}$$ $$+ ( 4494158110 - 11440793 \beta_{1} - 1327 \beta_{2} - 651613 \beta_{3} + 410 \beta_{4} + 2400 \beta_{5} - 4194 \beta_{6} + 142 \beta_{7} - 608 \beta_{8} - 23 \beta_{10} - 155 \beta_{11} + 151 \beta_{13} ) q^{19}$$ $$+ ( 2282301836 - 7101682 \beta_{1} - 2646 \beta_{2} - 231418 \beta_{3} - 25252 \beta_{4} - 1996 \beta_{5} + 1548 \beta_{6} + 132 \beta_{7} - 20 \beta_{8} + 2436 \beta_{9} + 442 \beta_{10} - 412 \beta_{11} - 36 \beta_{12} + 72 \beta_{13} ) q^{20}$$ $$+ ( -76705 - 7244875 \beta_{1} + 157962 \beta_{2} - 104515 \beta_{3} + 2528 \beta_{4} + 18498 \beta_{5} + 4353 \beta_{6} - 412 \beta_{7} + 933 \beta_{8} - 128 \beta_{9} - 465 \beta_{10} + 103 \beta_{11} - 200 \beta_{12} + 206 \beta_{13} ) q^{21}$$ $$+ ( -3677400984 + 4449201 \beta_{1} - 96066 \beta_{2} + 1485545 \beta_{3} - 62432 \beta_{4} + 9025 \beta_{5} + 5364 \beta_{6} - 128 \beta_{7} + 296 \beta_{8} - 4128 \beta_{9} - 64 \beta_{10} + 632 \beta_{11} + 160 \beta_{12} + 112 \beta_{13} ) q^{22}$$ $$+ ( 44320 + 23158935 \beta_{1} + 155148 \beta_{2} + 121650 \beta_{3} + 5389 \beta_{4} - 37836 \beta_{5} - 10491 \beta_{6} + 808 \beta_{7} - 1150 \beta_{8} + 123 \beta_{9} - 1146 \beta_{10} - 362 \beta_{11} + 432 \beta_{12} - 244 \beta_{13} ) q^{23}$$ $$+ ( 3925120854 - 3149514 \beta_{1} + 417836 \beta_{2} - 1530938 \beta_{3} + 143298 \beta_{4} - 23960 \beta_{5} - 22558 \beta_{6} + 120 \beta_{7} - 870 \beta_{8} - 4328 \beta_{9} + 3836 \beta_{10} - 800 \beta_{11} - 344 \beta_{12} - 160 \beta_{13} ) q^{24}$$ $$+ ( -30704880831 + 69566950 \beta_{1} - 1273834 \beta_{2} + 5956954 \beta_{3} + 13608 \beta_{4} - 20400 \beta_{5} + 24134 \beta_{6} - 1514 \beta_{7} + 4240 \beta_{8} - 3918 \beta_{10} + 1048 \beta_{11} - 1064 \beta_{13} ) q^{25}$$ $$+ ( -37022759465 + 16726172 \beta_{1} + 675626 \beta_{2} - 6996512 \beta_{3} + 272990 \beta_{4} - 1711 \beta_{5} + 9523 \beta_{6} - 2609 \beta_{7} - 464 \beta_{8} - 753 \beta_{9} + 5536 \beta_{10} + 2640 \beta_{11} - 15 \beta_{12} - 721 \beta_{13} ) q^{26}$$ $$+ ( 46851667593 + 2180925 \beta_{1} - 1809123 \beta_{2} - 2729604 \beta_{3} + 35538 \beta_{4} + 39648 \beta_{5} - 2490 \beta_{6} + 2742 \beta_{7} + 6048 \beta_{8} - 7467 \beta_{10} - 351 \beta_{11} - 2133 \beta_{13} ) q^{27}$$ $$+ ( 7319218872 + 193909668 \beta_{1} - 284996 \beta_{2} - 12930860 \beta_{3} - 580568 \beta_{4} - 28744 \beta_{5} - 42632 \beta_{6} + 4952 \beta_{7} - 3224 \beta_{8} - 16424 \beta_{9} + 9148 \beta_{10} + 2104 \beta_{11} + 104 \beta_{12} - 1552 \beta_{13} ) q^{28}$$ $$+ ( 66403 + 282855581 \beta_{1} + 3105493 \beta_{2} + 1248499 \beta_{3} + 28197 \beta_{4} + 40560 \beta_{5} - 60424 \beta_{6} - 544 \beta_{7} - 12776 \beta_{8} + 4224 \beta_{9} - 17592 \beta_{10} + 4232 \beta_{11} + 2112 \beta_{12} - 3824 \beta_{13} ) q^{29}$$ $$+ ( -26010931639 + 617695 \beta_{1} - 75286 \beta_{2} + 28832201 \beta_{3} - 965500 \beta_{4} + 4670 \beta_{5} + 114857 \beta_{6} - 2362 \beta_{7} + 150 \beta_{8} + 46190 \beta_{9} + 24236 \beta_{10} - 406 \beta_{11} - 1870 \beta_{12} - 3902 \beta_{13} ) q^{30}$$ $$+ ( -375262 + 382719546 \beta_{1} + 5751268 \beta_{2} + 1174638 \beta_{3} + 71524 \beta_{4} + 65756 \beta_{5} - 79502 \beta_{6} - 520 \beta_{7} - 7898 \beta_{8} - 76 \beta_{9} - 25998 \beta_{10} + 1186 \beta_{11} - 5104 \beta_{12} - 796 \beta_{13} ) q^{31}$$ $$+ ( 6868452784 - 724524100 \beta_{1} + 1245676 \beta_{2} + 69237524 \beta_{3} + 790996 \beta_{4} + 127544 \beta_{5} + 224392 \beta_{6} + 4552 \beta_{7} + 1292 \beta_{8} + 82920 \beta_{9} + 23908 \beta_{10} + 6864 \beta_{11} + 4440 \beta_{12} - 3712 \beta_{13} ) q^{32}$$ $$+ ( 288147051324 - 1064834883 \beta_{1} - 12349359 \beta_{2} - 42040145 \beta_{3} + 174132 \beta_{4} + 61032 \beta_{5} - 402663 \beta_{6} + 13173 \beta_{7} - 16416 \beta_{8} - 47109 \beta_{10} + 720 \beta_{11} + 5712 \beta_{13} ) q^{33}$$ $$+ ( 72524827540 + 717036188 \beta_{1} + 1542548 \beta_{2} - 77101582 \beta_{3} + 843952 \beta_{4} + 43504 \beta_{5} + 210310 \beta_{6} + 22848 \beta_{7} + 12268 \beta_{8} - 104176 \beta_{9} + 54768 \beta_{10} - 17948 \beta_{11} - 1616 \beta_{12} - 24 \beta_{13} ) q^{34}$$ $$+ ( -218392578598 + 558489150 \beta_{1} - 24872602 \beta_{2} + 17695932 \beta_{3} + 228444 \beta_{4} - 423680 \beta_{5} + 41572 \beta_{6} - 31692 \beta_{7} - 22880 \beta_{8} - 69994 \beta_{10} + 7774 \beta_{11} + 10218 \beta_{13} ) q^{35}$$ $$+ ( -683586160739 + 866844729 \beta_{1} + 5680747 \beta_{2} - 150951960 \beta_{3} - 294128 \beta_{4} + 253008 \beta_{5} + 272320 \beta_{6} - 55472 \beta_{7} + 28464 \beta_{8} - 74288 \beta_{9} + 102280 \beta_{10} - 3344 \beta_{11} + 2992 \beta_{12} + 6368 \beta_{13} ) q^{36}$$ $$+ ( 3083532 + 2636571336 \beta_{1} + 21036053 \beta_{2} + 13739258 \beta_{3} + 703803 \beta_{4} - 1216590 \beta_{5} - 759687 \beta_{6} + 24132 \beta_{7} + 69693 \beta_{8} - 65024 \beta_{9} - 81673 \beta_{10} - 42897 \beta_{11} - 11400 \beta_{12} + 24798 \beta_{13} ) q^{37}$$ $$+ ( 643676885576 - 4840542767 \beta_{1} + 12691582 \beta_{2} + 187305289 \beta_{3} + 1599264 \beta_{4} - 624127 \beta_{5} + 238548 \beta_{6} + 29568 \beta_{7} - 8984 \beta_{8} - 54048 \beta_{9} + 134080 \beta_{10} - 24136 \beta_{11} + 12192 \beta_{12} + 32496 \beta_{13} ) q^{38}$$ $$+ ( -30767246 - 5394692987 \beta_{1} + 39509408 \beta_{2} - 55743384 \beta_{3} + 1044733 \beta_{4} + 1742944 \beta_{5} + 1512179 \beta_{6} - 46400 \beta_{7} + 135408 \beta_{8} - 2877 \beta_{9} - 170800 \beta_{10} + 18000 \beta_{11} + 33408 \beta_{12} + 16800 \beta_{13} ) q^{39}$$ $$+ ( -287455492624 - 2379135848 \beta_{1} + 1906184 \beta_{2} + 587936200 \beta_{3} - 5518984 \beta_{4} + 162256 \beta_{5} - 810176 \beta_{6} - 67984 \beta_{7} + 46440 \beta_{8} - 297296 \beta_{9} + 174264 \beta_{10} - 9984 \beta_{11} - 33584 \beta_{12} + 47936 \beta_{13} ) q^{40}$$ $$+ ( 622884039810 - 5578871598 \beta_{1} - 37204638 \beta_{2} + 114258798 \beta_{3} + 808952 \beta_{4} + 1288176 \beta_{5} - 1347726 \beta_{6} - 21790 \beta_{7} - 62800 \beta_{8} - 96906 \beta_{10} - 99256 \beta_{11} - 12152 \beta_{13} ) q^{41}$$ $$+ ( -475764464356 + 550698424 \beta_{1} + 7394248 \beta_{2} - 1177763048 \beta_{3} - 11497432 \beta_{4} - 1290900 \beta_{5} + 1127980 \beta_{6} - 29068 \beta_{7} - 190896 \beta_{8} + 603444 \beta_{9} + 193888 \beta_{10} + 5424 \beta_{11} + 17868 \beta_{12} + 40980 \beta_{13} ) q^{42}$$ $$+ ( 1194102016621 + 9335286124 \beta_{1} - 53465446 \beta_{2} + 170355189 \beta_{3} + 893284 \beta_{4} + 775104 \beta_{5} + 2795596 \beta_{6} + 109996 \beta_{7} + 3136 \beta_{8} - 230902 \beta_{10} + 6946 \beta_{11} + 1270 \beta_{13} ) q^{43}$$ $$+ ( -311356584302 + 4203451425 \beta_{1} + 3967257 \beta_{2} - 1160304325 \beta_{3} + 14432022 \beta_{4} + 728206 \beta_{5} - 2660156 \beta_{6} + 174302 \beta_{7} - 34022 \beta_{8} + 893934 \beta_{9} - 32845 \beta_{10} + 37794 \beta_{11} - 41694 \beta_{12} + 32068 \beta_{13} ) q^{44}$$ $$+ ( 54720198 + 16114577166 \beta_{1} + 1570187 \beta_{2} + 128852180 \beta_{3} - 1018671 \beta_{4} + 1605778 \beta_{5} - 3315031 \beta_{6} - 30204 \beta_{7} - 154515 \beta_{8} + 618112 \beta_{9} - 43897 \beta_{10} + 73087 \beta_{11} + 24568 \beta_{12} - 50434 \beta_{13} ) q^{45}$$ $$+ ( 1504035397087 - 1257407495 \beta_{1} - 19427290 \beta_{2} + 2165618847 \beta_{3} + 21473692 \beta_{4} + 2581682 \beta_{5} + 5269887 \beta_{6} + 10986 \beta_{7} - 148774 \beta_{8} - 983166 \beta_{9} - 190156 \beta_{10} + 24614 \beta_{11} - 44962 \beta_{12} - 112050 \beta_{13} ) q^{46}$$ $$+ ( -32569502 - 17417875156 \beta_{1} - 82636724 \beta_{2} - 114233270 \beta_{3} - 4690134 \beta_{4} - 5681676 \beta_{5} + 3131336 \beta_{6} + 154280 \beta_{7} - 484382 \beta_{8} + 11742 \beta_{9} + 329382 \beta_{10} - 53578 \beta_{11} - 120144 \beta_{12} - 62132 \beta_{13} ) q^{47}$$ $$+ ( 3326923551960 - 6010447364 \beta_{1} - 14068268 \beta_{2} + 2411871268 \beta_{3} - 20383164 \beta_{4} - 738680 \beta_{5} - 10466096 \beta_{6} + 118584 \beta_{7} - 192852 \beta_{8} - 620392 \beta_{9} - 340964 \beta_{10} + 11408 \beta_{11} + 160808 \beta_{12} - 234176 \beta_{13} ) q^{48}$$ $$+ ( -7481402721727 - 34574795560 \beta_{1} + 122878264 \beta_{2} + 219586888 \beta_{3} - 3327392 \beta_{4} - 7192896 \beta_{5} - 9475464 \beta_{6} + 61592 \beta_{7} + 468224 \beta_{8} + 281192 \beta_{10} + 538880 \beta_{11} + 23552 \beta_{13} ) q^{49}$$ $$+ ( -3892679080525 + 33208374523 \beta_{1} - 68076760 \beta_{2} - 2331247004 \beta_{3} - 13882016 \beta_{4} + 5708384 \beta_{5} + 18935308 \beta_{6} - 175488 \beta_{7} + 1175000 \beta_{8} - 206304 \beta_{9} - 911648 \beta_{10} + 185288 \beta_{11} - 91296 \beta_{12} - 253488 \beta_{13} ) q^{50}$$ $$+ ( 2921012490151 + 57622339239 \beta_{1} + 484530239 \beta_{2} - 2697603952 \beta_{3} - 6309818 \beta_{4} + 88544 \beta_{5} + 13179410 \beta_{6} - 369806 \beta_{7} + 408192 \beta_{8} + 1808135 \beta_{10} - 139413 \beta_{11} - 182535 \beta_{13} ) q^{51}$$ $$+ ( -5233343569228 + 40308936274 \beta_{1} - 14743306 \beta_{2} - 2914911270 \beta_{3} + 9500772 \beta_{4} - 10028276 \beta_{5} - 26475468 \beta_{6} - 43588 \beta_{7} - 303660 \beta_{8} - 1650500 \beta_{9} - 941338 \beta_{10} - 159012 \beta_{11} + 269284 \beta_{12} - 385480 \beta_{13} ) q^{52}$$ $$+ ( 437680658 + 103403274458 \beta_{1} - 294609203 \beta_{2} + 933759652 \beta_{3} - 3942233 \beta_{4} + 16431822 \beta_{5} - 23038201 \beta_{6} - 338244 \beta_{7} - 430077 \beta_{8} - 4045440 \beta_{9} + 1281225 \beta_{10} + 440913 \beta_{11} + 65160 \beta_{12} - 187230 \beta_{13} ) q^{53}$$ $$+ ( -1287386640276 - 47233175634 \beta_{1} + 1737252 \beta_{2} + 1312994022 \beta_{3} - 19802976 \beta_{4} - 1357566 \beta_{5} + 34503876 \beta_{6} - 721536 \beta_{7} + 1736136 \beta_{8} + 3739488 \beta_{9} - 1634112 \beta_{10} + 530520 \beta_{11} + 66336 \beta_{12} + 50736 \beta_{13} ) q^{54}$$ $$+ ( -476020282 - 178255009197 \beta_{1} - 479592368 \beta_{2} - 1271484896 \beta_{3} + 14416883 \beta_{4} - 15083856 \beta_{5} + 38100997 \beta_{6} + 318048 \beta_{7} - 261960 \beta_{8} + 14061 \beta_{9} + 2218024 \beta_{10} - 199704 \beta_{11} + 131904 \beta_{12} - 38832 \beta_{13} ) q^{55}$$ $$+ ( 7921700014304 - 12036892336 \beta_{1} - 187072720 \beta_{2} + 4992796016 \beta_{3} + 66161424 \beta_{4} - 12594464 \beta_{5} - 51787200 \beta_{6} + 1146528 \beta_{7} - 212304 \beta_{8} + 5472288 \beta_{9} - 2651056 \beta_{10} - 178688 \beta_{11} - 477984 \beta_{12} + 478080 \beta_{13} ) q^{56}$$ $$+ ( -22193252671492 - 165288011811 \beta_{1} + 718572817 \beta_{2} + 4402414255 \beta_{3} - 8163724 \beta_{4} + 5382376 \beta_{5} - 35536295 \beta_{6} - 856171 \beta_{7} - 1632 \beta_{8} + 2932795 \beta_{10} - 764688 \beta_{11} - 254352 \beta_{13} ) q^{57}$$ $$+ ( 18328652478699 - 5723536524 \beta_{1} - 181059390 \beta_{2} - 4297103432 \beta_{3} + 125853558 \beta_{4} + 705605 \beta_{5} + 38476343 \beta_{6} + 319163 \beta_{7} - 3991424 \beta_{8} - 6652741 \beta_{9} - 3343616 \beta_{10} - 21632 \beta_{11} + 211781 \beta_{12} + 487355 \beta_{13} ) q^{58}$$ $$+ ( 24787242511287 + 298913196986 \beta_{1} + 397987376 \beta_{2} + 2490999473 \beta_{3} - 1802976 \beta_{4} + 18062656 \beta_{5} + 89879280 \beta_{6} + 2490048 \beta_{7} - 1591520 \beta_{8} + 282128 \beta_{10} - 82832 \beta_{11} + 502896 \beta_{13} ) q^{59}$$ $$+ ( -5986597815720 + 28032661076 \beta_{1} - 135033460 \beta_{2} + 1548060612 \beta_{3} - 140707320 \beta_{4} + 27535448 \beta_{5} - 103811560 \beta_{6} - 502920 \beta_{7} + 726600 \beta_{8} - 6627848 \beta_{9} - 2402164 \beta_{10} - 453416 \beta_{11} - 998072 \beta_{12} + 1371440 \beta_{13} ) q^{60}$$ $$+ ( 1340436696 + 335200927092 \beta_{1} - 630972305 \beta_{2} + 2874611994 \beta_{3} - 21354055 \beta_{4} - 31266274 \beta_{5} - 73743825 \beta_{6} + 565340 \beta_{7} + 3738187 \beta_{8} + 19220608 \beta_{9} + 3811233 \beta_{10} - 1603607 \beta_{11} - 556216 \beta_{12} + 1179602 \beta_{13} ) q^{61}$$ $$+ ( 24761449391444 - 7449065044 \beta_{1} - 370988664 \beta_{2} - 7221640204 \beta_{3} - 195174640 \beta_{4} - 6960264 \beta_{5} + 86216276 \beta_{6} + 1354712 \beta_{7} - 7708104 \beta_{8} + 2515704 \beta_{9} - 2701584 \beta_{10} - 1417528 \beta_{11} + 168840 \beta_{12} + 782728 \beta_{13} ) q^{62}$$ $$+ ( -2375004348 - 717451090101 \beta_{1} - 210365932 \beta_{2} - 5667577234 \beta_{3} - 125293515 \beta_{4} + 65982892 \beta_{5} + 177060713 \beta_{6} - 1705320 \beta_{7} + 5059374 \beta_{8} - 197885 \beta_{9} + 1020554 \beta_{10} + 536314 \beta_{11} + 809680 \beta_{12} + 742676 \beta_{13} ) q^{63}$$ $$+ ( -23728632064576 + 3786122664 \beta_{1} + 717644904 \beta_{2} - 3995293576 \beta_{3} + 234111480 \beta_{4} + 74126096 \beta_{5} - 124247344 \beta_{6} - 5033488 \beta_{7} + 2251144 \beta_{8} - 3826256 \beta_{9} - 376840 \beta_{10} - 360992 \beta_{11} + 699344 \beta_{12} + 144896 \beta_{13} ) q^{64}$$ $$+ ( 15776424193216 - 792423315910 \beta_{1} + 621371754 \beta_{2} - 1006752154 \beta_{3} - 2148968 \beta_{4} + 32616240 \beta_{5} - 216916294 \beta_{6} + 2206474 \beta_{7} - 5076560 \beta_{8} + 1522638 \beta_{10} - 1569848 \beta_{11} + 1168904 \beta_{13} ) q^{65}$$ $$+ ( 62564074176630 - 314386862338 \beta_{1} + 1341005100 \beta_{2} - 859272594 \beta_{3} + 194123856 \beta_{4} - 48181616 \beta_{5} + 207982362 \beta_{6} + 619200 \beta_{7} + 10003284 \beta_{8} + 11795952 \beta_{9} + 3433488 \beta_{10} - 1806372 \beta_{11} + 164688 \beta_{12} + 1034136 \beta_{13} ) q^{66}$$ $$+ ( -43880676271546 + 881540186367 \beta_{1} - 242334279 \beta_{2} - 11986490493 \beta_{3} - 20116374 \beta_{4} - 99783264 \beta_{5} + 210323358 \beta_{6} - 9452706 \beta_{7} - 933696 \beta_{8} + 3247857 \beta_{10} + 1393389 \beta_{11} + 775695 \beta_{13} ) q^{67}$$ $$+ ( -88250333745922 - 54226047890 \beta_{1} - 1060981702 \beta_{2} + 46665959672 \beta_{3} - 145512016 \beta_{4} - 43536528 \beta_{5} - 159386816 \beta_{6} + 229488 \beta_{7} + 1297168 \beta_{8} + 25077744 \beta_{9} - 1426728 \beta_{10} + 3060816 \beta_{11} + 1843344 \beta_{12} - 1341024 \beta_{13} ) q^{68}$$ $$+ ( 5127732617 + 1560757268099 \beta_{1} + 882407670 \beta_{2} + 12205786331 \beta_{3} + 87160976 \beta_{4} - 112492530 \beta_{5} - 395604057 \beta_{6} + 2526524 \beta_{7} - 5821149 \beta_{8} - 67960448 \beta_{9} - 5712855 \beta_{10} - 643919 \beta_{11} + 1077640 \beta_{12} - 1250974 \beta_{13} ) q^{69}$$ $$+ ( -31600852937160 + 245093614744 \beta_{1} - 310360240 \beta_{2} - 39930588632 \beta_{3} + 75868352 \beta_{4} + 8870912 \beta_{5} + 358489784 \beta_{6} + 4644096 \beta_{7} + 20572400 \beta_{8} - 32737472 \beta_{9} + 7549056 \beta_{10} - 2852656 \beta_{11} - 1004608 \beta_{12} - 1867104 \beta_{13} ) q^{70}$$ $$+ ( -5187867064 - 1222100553323 \beta_{1} + 2645884692 \beta_{2} - 10594581538 \beta_{3} + 557284943 \beta_{4} + 75226284 \beta_{5} + 276252239 \beta_{6} - 872296 \beta_{7} - 7185746 \beta_{8} + 240873 \beta_{9} - 9323766 \beta_{10} + 1621370 \beta_{11} - 3971376 \beta_{12} - 967148 \beta_{13} ) q^{71}$$ $$+ ( -93740834263543 + 733441294323 \beta_{1} - 1106114036 \beta_{2} - 24146525017 \beta_{3} - 314465251 \beta_{4} - 189056576 \beta_{5} - 387229961 \beta_{6} + 2358080 \beta_{7} - 1929123 \beta_{8} - 33392064 \beta_{9} + 10565536 \beta_{10} + 5012736 \beta_{11} + 544704 \beta_{12} - 2141952 \beta_{13} ) q^{72}$$ $$+ ( -38777824511494 - 1636448971167 \beta_{1} - 4513796811 \beta_{2} - 1651350613 \beta_{3} + 53209828 \beta_{4} - 34717752 \beta_{5} - 471370723 \beta_{6} + 4763577 \beta_{7} + 6871296 \beta_{8} - 18469561 \beta_{10} + 5410560 \beta_{11} - 486912 \beta_{13} ) q^{73}$$ $$+ ( 171073614154481 - 86662184508 \beta_{1} - 2798549626 \beta_{2} + 62854702304 \beta_{3} - 722174670 \beta_{4} + 52366055 \beta_{5} + 474865589 \beta_{6} + 3065 \beta_{7} - 22973744 \beta_{8} + 35799225 \beta_{9} + 19486816 \beta_{10} - 50000 \beta_{11} - 2510265 \beta_{12} - 6433127 \beta_{13} ) q^{74}$$ $$+ ( 204426146975209 + 1837714737120 \beta_{1} - 7087137874 \beta_{2} - 5356085491 \beta_{3} + 126015148 \beta_{4} + 108538880 \beta_{5} + 497878484 \beta_{6} + 10873636 \beta_{7} + 18717600 \beta_{8} - 29542498 \beta_{10} + 992838 \beta_{11} - 5908254 \beta_{13} ) q^{75}$$ $$+ ( -43889445251566 - 649867751071 \beta_{1} + 4595290201 \beta_{2} + 59152264187 \beta_{3} + 739442838 \beta_{4} + 248398862 \beta_{5} - 478436412 \beta_{6} - 1446818 \beta_{7} - 3829350 \beta_{8} + 13663086 \beta_{9} + 30315123 \beta_{10} - 74718 \beta_{11} + 993698 \beta_{12} - 4678076 \beta_{13} ) q^{76}$$ $$+ ( 5555351339 + 2125706878905 \beta_{1} + 5908237762 \beta_{2} + 15053689665 \beta_{3} - 125642960 \beta_{4} + 205525002 \beta_{5} - 404104475 \beta_{6} - 3741324 \beta_{7} - 19747047 \beta_{8} + 179439744 \beta_{9} - 31660149 \beta_{10} + 8934819 \beta_{11} + 1962264 \beta_{12} - 6128826 \beta_{13} ) q^{77}$$ $$+ ( -352064627398985 + 175936189889 \beta_{1} + 6119184150 \beta_{2} - 122563768809 \beta_{3} + 925554748 \beta_{4} - 84542878 \beta_{5} + 442132695 \beta_{6} - 15086054 \beta_{7} - 40255062 \beta_{8} + 21546866 \beta_{9} + 35350356 \beta_{10} + 14181974 \beta_{11} + 1539950 \beta_{12} - 815714 \beta_{13} ) q^{78}$$ $$+ ( -6486169308 - 1825831736826 \beta_{1} + 3518515056 \beta_{2} - 16087399144 \beta_{3} - 1749211570 \beta_{4} - 362986096 \beta_{5} + 409109082 \beta_{6} + 7789088 \beta_{7} - 14967320 \beta_{8} + 1613138 \beta_{9} - 30343240 \beta_{10} - 2218760 \beta_{11} + 5366720 \beta_{12} - 3623056 \beta_{13} ) q^{79}$$ $$+ ( 457260415694144 + 143358092592 \beta_{1} + 1980739696 \beta_{2} - 91888223472 \beta_{3} - 1584293744 \beta_{4} + 440775776 \beta_{5} - 625185760 \beta_{6} + 22514080 \beta_{7} - 7972880 \beta_{8} + 41961504 \beta_{9} + 28783312 \beta_{10} - 3840192 \beta_{11} - 4824864 \beta_{12} + 1264640 \beta_{13} ) q^{80}$$ $$+ ( -381684704622723 - 1451351229939 \beta_{1} - 12313294911 \beta_{2} + 115800420735 \beta_{3} + 165660084 \beta_{4} - 52461720 \beta_{5} - 246485367 \beta_{6} - 23896251 \beta_{7} + 27830304 \beta_{8} - 30721077 \beta_{10} - 5190480 \beta_{11} - 11006928 \beta_{13} ) q^{81}$$ $$+ ( 343645292460338 - 776865500686 \beta_{1} + 5374144248 \beta_{2} + 52830905580 \beta_{3} - 1418628576 \beta_{4} + 133060896 \beta_{5} + 743610724 \beta_{6} - 1716352 \beta_{7} + 36228104 \beta_{8} - 99718048 \beta_{9} + 11159200 \beta_{10} + 14663384 \beta_{11} + 6003744 \beta_{12} + 7530608 \beta_{13} ) q^{82}$$ $$+ ( 525867047257663 + 1431278972818 \beta_{1} - 6203603128 \beta_{2} - 48878600639 \beta_{3} + 142947088 \beta_{4} + 281414848 \beta_{5} + 321409376 \beta_{6} + 22345552 \beta_{7} - 22279072 \beta_{8} - 28715992 \beta_{10} - 9200536 \beta_{11} + 4359512 \beta_{13} ) q^{83}$$ $$+ ( -665453287298832 + 713544996440 \beta_{1} - 760641912 \beta_{2} + 73017630136 \beta_{3} + 1363218096 \beta_{4} - 1140981104 \beta_{5} - 979743504 \beta_{6} + 5129424 \beta_{7} - 4899600 \beta_{8} - 145423152 \beta_{9} + 49343688 \beta_{10} - 19156656 \beta_{11} - 14859600 \beta_{12} + 14922912 \beta_{13} ) q^{84}$$ $$+ ( 6500781811 + 2523995292105 \beta_{1} + 6514648824 \beta_{2} + 18395539501 \beta_{3} + 523133594 \beta_{4} + 582027430 \beta_{5} - 592352845 \beta_{6} - 14150420 \beta_{7} + 72836255 \beta_{8} - 344559360 \beta_{9} - 11361635 \beta_{10} - 8074555 \beta_{11} - 12526040 \beta_{12} + 18687370 \beta_{13} ) q^{85}$$ $$+ ( -640227790743876 - 960925166827 \beta_{1} - 9398152266 \beta_{2} - 1356498011 \beta_{3} + 835126592 \beta_{4} + 317454785 \beta_{5} + 963518216 \beta_{6} - 6714624 \beta_{7} + 44834448 \beta_{8} + 123044544 \beta_{9} - 5678720 \beta_{10} - 3302992 \beta_{11} + 1315392 \beta_{12} + 5574240 \beta_{13} ) q^{86}$$ $$+ ( -13364623796 - 2829761725841 \beta_{1} + 5781836372 \beta_{2} - 24248241618 \beta_{3} + 5447680153 \beta_{4} - 464218772 \beta_{5} + 481450469 \beta_{6} + 6985240 \beta_{7} + 38150286 \beta_{8} - 4690401 \beta_{9} - 841558 \beta_{10} - 13105638 \beta_{11} + 11922384 \beta_{12} + 7866708 \beta_{13} ) q^{87}$$ $$+ ( 242676711094742 + 190224454390 \beta_{1} - 3454504852 \beta_{2} + 92656086278 \beta_{3} + 54396802 \beta_{4} - 1249519384 \beta_{5} + 123146466 \beta_{6} - 43239432 \beta_{7} + 16142810 \beta_{8} + 153964696 \beta_{9} + 19105468 \beta_{10} - 37197088 \beta_{11} + 9959720 \beta_{12} + 3347808 \beta_{13} ) q^{88}$$ $$+ ( -29199620227590 - 1216159596695 \beta_{1} + 6741379453 \beta_{2} - 108892541021 \beta_{3} - 186048572 \beta_{4} - 289206904 \beta_{5} - 557863515 \beta_{6} + 7110145 \beta_{7} - 72550464 \beta_{8} + 19765087 \beta_{10} + 12561696 \beta_{11} + 28370720 \beta_{13} ) q^{89}$$ $$+ ( 1048549394612667 - 374737973796 \beta_{1} - 14271106622 \beta_{2} - 103312939280 \beta_{3} + 1396237446 \beta_{4} - 60326083 \beta_{5} - 217608089 \beta_{6} - 15633501 \beta_{7} + 1252560 \beta_{8} - 125894557 \beta_{9} - 27336608 \beta_{10} - 7546192 \beta_{11} + 955037 \beta_{12} + 15370819 \beta_{13} ) q^{90}$$ $$+ ( -511678034967290 - 1367822719006 \beta_{1} + 22576527994 \beta_{2} - 94494965308 \beta_{3} - 474774684 \beta_{4} - 623771136 \beta_{5} - 580710500 \beta_{6} - 73375348 \beta_{7} - 98102560 \beta_{8} + 113968842 \beta_{10} - 9964798 \beta_{11} + 29379254 \beta_{13} ) q^{91}$$ $$+ ( -658197561496472 - 1217386119860 \beta_{1} - 600130604 \beta_{2} - 151324836004 \beta_{3} - 2003446728 \beta_{4} + 2362336360 \beta_{5} + 967419560 \beta_{6} + 22065736 \beta_{7} + 9305592 \beta_{8} + 44125128 \beta_{9} - 140000044 \beta_{10} + 12812264 \beta_{11} + 34790136 \beta_{12} - 11556784 \beta_{13} ) q^{92}$$ $$+ ( -10719700808 - 4842222082136 \beta_{1} - 19557949184 \beta_{2} - 32687873848 \beta_{3} - 704737712 \beta_{4} - 594405136 \beta_{5} + 1166945272 \beta_{6} + 11035360 \beta_{7} + 6218328 \beta_{8} + 428969728 \beta_{9} + 86175496 \beta_{10} - 8734904 \beta_{11} + 12661312 \beta_{12} + 458384 \beta_{13} ) q^{93}$$ $$+ ( -1131182922184650 + 321607750170 \beta_{1} + 13143842172 \beta_{2} + 371395464022 \beta_{3} - 3403237800 \beta_{4} + 268926164 \beta_{5} - 716468746 \beta_{6} + 55855556 \beta_{7} + 71085124 \beta_{8} - 72263500 \beta_{9} - 127802360 \beta_{10} - 50463044 \beta_{11} - 6647284 \beta_{12} + 6849068 \beta_{13} ) q^{94}$$ $$+ ( 35649623094 + 5846457094115 \beta_{1} - 37781027664 \beta_{2} + 58639498864 \beta_{3} - 13394957709 \beta_{4} + 1402608720 \beta_{5} - 882947435 \beta_{6} - 25089120 \beta_{7} + 25193160 \beta_{8} - 6506835 \beta_{9} + 145267800 \beta_{10} + 3532440 \beta_{11} - 54761280 \beta_{12} + 15284400 \beta_{13} ) q^{95}$$ $$+ ( 3494181075612512 - 4745899852376 \beta_{1} + 10704795656 \beta_{2} + 150514845880 \beta_{3} + 6149857592 \beta_{4} + 2438458448 \beta_{5} + 694109168 \beta_{6} + 14984624 \beta_{7} - 11874168 \beta_{8} - 184520080 \beta_{9} - 176521768 \beta_{10} + 60798944 \beta_{11} - 11012848 \beta_{12} + 15162112 \beta_{13} ) q^{96}$$ $$+ ( 1312259460143930 + 4117687417539 \beta_{1} + 57395487495 \beta_{2} - 64862652183 \beta_{3} - 621669156 \beta_{4} + 728810424 \beta_{5} + 1284584319 \beta_{6} + 61852971 \beta_{7} - 63465360 \beta_{8} + 171156621 \beta_{10} - 22557336 \beta_{11} + 13636008 \beta_{13} ) q^{97}$$ $$+ ( 2428807224885271 + 6743831418095 \beta_{1} + 40510442912 \beta_{2} - 358387814896 \beta_{3} + 6725738880 \beta_{4} - 477732992 \beta_{5} - 2754838608 \beta_{6} + 2087424 \beta_{7} - 124240288 \beta_{8} + 502316160 \beta_{9} - 89087104 \beta_{10} - 67615712 \beta_{11} - 33502848 \beta_{12} - 46776000 \beta_{13} ) q^{98}$$ $$+ ( -1352255859553763 - 4424321731050 \beta_{1} + 25612262936 \beta_{2} + 527434772323 \beta_{3} - 177408464 \beta_{4} + 473285312 \beta_{5} - 114410560 \beta_{6} + 89614000 \beta_{7} + 225071712 \beta_{8} + 31276664 \beta_{10} + 43360632 \beta_{11} - 60570360 \beta_{13} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q$$ $$\mathstrut -\mathstrut 350q^{2}$$ $$\mathstrut +\mathstrut 11964q^{3}$$ $$\mathstrut -\mathstrut 54220q^{4}$$ $$\mathstrut +\mathstrut 4868124q^{6}$$ $$\mathstrut +\mathstrut 10899160q^{8}$$ $$\mathstrut +\mathstrut 86397354q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$14q$$ $$\mathstrut -\mathstrut 350q^{2}$$ $$\mathstrut +\mathstrut 11964q^{3}$$ $$\mathstrut -\mathstrut 54220q^{4}$$ $$\mathstrut +\mathstrut 4868124q^{6}$$ $$\mathstrut +\mathstrut 10899160q^{8}$$ $$\mathstrut +\mathstrut 86397354q^{9}$$ $$\mathstrut +\mathstrut 87155280q^{10}$$ $$\mathstrut -\mathstrut 49140292q^{11}$$ $$\mathstrut +\mathstrut 31616184q^{12}$$ $$\mathstrut -\mathstrut 2675193504q^{14}$$ $$\mathstrut +\mathstrut 9716374544q^{16}$$ $$\mathstrut -\mathstrut 9985136356q^{17}$$ $$\mathstrut -\mathstrut 7692086106q^{18}$$ $$\mathstrut +\mathstrut 62922133180q^{19}$$ $$\mathstrut +\mathstrut 31953450720q^{20}$$ $$\mathstrut -\mathstrut 51493048676q^{22}$$ $$\mathstrut +\mathstrut 54962419344q^{24}$$ $$\mathstrut -\mathstrut 429906632050q^{25}$$ $$\mathstrut -\mathstrut 518273679696q^{26}$$ $$\mathstrut +\mathstrut 655936606584q^{27}$$ $$\mathstrut +\mathstrut 102542376000q^{28}$$ $$\mathstrut -\mathstrut 364331826720q^{30}$$ $$\mathstrut +\mathstrut 95751080800q^{32}$$ $$\mathstrut +\mathstrut 4034287537464q^{33}$$ $$\mathstrut +\mathstrut 1015818512644q^{34}$$ $$\mathstrut -\mathstrut 3057653406720q^{35}$$ $$\mathstrut -\mathstrut 9569289445092q^{36}$$ $$\mathstrut +\mathstrut 9010383442844q^{38}$$ $$\mathstrut -\mathstrut 4027933291200q^{40}$$ $$\mathstrut +\mathstrut 8719629072668q^{41}$$ $$\mathstrut -\mathstrut 6653698130880q^{42}$$ $$\mathstrut +\mathstrut 16716309178300q^{43}$$ $$\mathstrut -\mathstrut 4351929975880q^{44}$$ $$\mathstrut +\mathstrut 21043605267744q^{46}$$ $$\mathstrut +\mathstrut 46562301973344q^{48}$$ $$\mathstrut -\mathstrut 104740771400434q^{49}$$ $$\mathstrut -\mathstrut 54483696687710q^{50}$$ $$\mathstrut +\mathstrut 40911296041848q^{51}$$ $$\mathstrut -\mathstrut 73249356722400q^{52}$$ $$\mathstrut -\mathstrut 18031398151752q^{54}$$ $$\mathstrut +\mathstrut 110873799752064q^{56}$$ $$\mathstrut -\mathstrut 310730528383176q^{57}$$ $$\mathstrut +\mathstrut 256627273576560q^{58}$$ $$\mathstrut +\mathstrut 347007330293180q^{59}$$ $$\mathstrut -\mathstrut 83822619107520q^{60}$$ $$\mathstrut +\mathstrut 346701622780800q^{62}$$ $$\mathstrut -\mathstrut 332173589517760q^{64}$$ $$\mathstrut +\mathstrut 220877370432000q^{65}$$ $$\mathstrut +\mathstrut 875905832132808q^{66}$$ $$\mathstrut -\mathstrut 614258765968196q^{67}$$ $$\mathstrut -\mathstrut 1235787871965976q^{68}$$ $$\mathstrut -\mathstrut 442172390987520q^{70}$$ $$\mathstrut -\mathstrut 1312232111609976q^{72}$$ $$\mathstrut -\mathstrut 542888518370276q^{73}$$ $$\mathstrut +\mathstrut 2394643836493776q^{74}$$ $$\mathstrut +\mathstrut 2861985572983740q^{75}$$ $$\mathstrut -\mathstrut 614792000883272q^{76}$$ $$\mathstrut -\mathstrut 4928152344599520q^{78}$$ $$\mathstrut +\mathstrut 6402194304908160q^{80}$$ $$\mathstrut -\mathstrut 5344304371237818q^{81}$$ $$\mathstrut +\mathstrut 4810720063300804q^{82}$$ $$\mathstrut +\mathstrut 7362421930746044q^{83}$$ $$\mathstrut -\mathstrut 9316784627856000q^{84}$$ $$\mathstrut -\mathstrut 8963192329247716q^{86}$$ $$\mathstrut +\mathstrut 3396904358318224q^{88}$$ $$\mathstrut -\mathstrut 408131327364580q^{89}$$ $$\mathstrut +\mathstrut 14680290550241520q^{90}$$ $$\mathstrut -\mathstrut 7162887702930432q^{91}$$ $$\mathstrut -\mathstrut 9213856910218560q^{92}$$ $$\mathstrut -\mathstrut 15838781433150144q^{94}$$ $$\mathstrut +\mathstrut 48917704300183104q^{96}$$ $$\mathstrut +\mathstrut 18372136634257180q^{97}$$ $$\mathstrut +\mathstrut 34005570793293730q^{98}$$ $$\mathstrut -\mathstrut 18934692015468492q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14}\mathstrut -\mathstrut$$ $$7$$ $$x^{13}\mathstrut +\mathstrut$$ $$7894$$ $$x^{12}\mathstrut +\mathstrut$$ $$219192$$ $$x^{11}\mathstrut -\mathstrut$$ $$135772320$$ $$x^{10}\mathstrut +\mathstrut$$ $$11331786624$$ $$x^{9}\mathstrut -\mathstrut$$ $$652293490688$$ $$x^{8}\mathstrut +\mathstrut$$ $$465478371454976$$ $$x^{7}\mathstrut -\mathstrut$$ $$697940891942912$$ $$x^{6}\mathstrut +\mathstrut$$ $$5355071973862932480$$ $$x^{5}\mathstrut +\mathstrut$$ $$64434406660761845760$$ $$x^{4}\mathstrut -\mathstrut$$ $$122098039773726666915840$$ $$x^{3}\mathstrut +\mathstrut$$ $$21723325027733376807731200$$ $$x^{2}\mathstrut -\mathstrut$$ $$2811006993630615015868334080$$ $$x\mathstrut +\mathstrut$$ $$279841437428593117864520581120$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{2} + 42 \nu + 4476$$ $$\beta_{3}$$ $$=$$ $$($$$$58320936057$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$5986305819397$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$2567640267290346$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$338450263655874432$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$12941406522678329952$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$514365470864709861120$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$20269713654432500382720$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$8384552064063068792651776$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$2693005054430854357422981120$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$715300597594788165511777812480$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$70452696231227959291828369489920$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$741381185833276749197284453908480$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$1513638432009977863658379407964241920$$ $$\nu\mathstrut -\mathstrut$$ $$176481310109098931717428953956766187520$$$$)/$$$$35\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$59559682613073$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$3679768792675075$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$134304160947187674$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$13059367493115071616$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$17516388906524481238176$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$2711837486550076197460224$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$445367533666365059689552896$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$25240273199306757761974435840$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$1729543258117384764998431948800$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$183813749390261554716393891102720$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$4366867283294468571413400132648960$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$8975155651649775233886645322084515840$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$607477185539139873028601405720399708160$$ $$\nu\mathstrut +\mathstrut$$ $$270101432507374827453346061180382761451520$$$$)/$$$$46\!\cdots\!40$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$11214439470053$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$4052879338989057$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$582007274862497634$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$32581393512709134720$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$642447840602638989600$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$21657438899412178858752$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$36977882278288483605302272$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$5530983882330275337754804224$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$878686711322623677776799252480$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$114076526505930585119890224906240$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$13822283514080509977104087012474880$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$473410272226725214191675938785198080$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$75695509955635579329771774691948625920$$ $$\nu\mathstrut +\mathstrut$$ $$13680408558541930348351581163360270417920$$$$)/$$$$35\!\cdots\!80$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$375239388039$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$14223951977221$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$953736685017366$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$285673832524316544$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$72440214758722847328$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$5112633309255982039296$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$330062848935416240440320$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$197632392274712785729126400$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$81797756945076138688364544$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1639743776687142447551323373568$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$70777026487837759674608820289536$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$53674354918537191358353350200393728$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$7980980499461337222692955919365439488$$ $$\nu\mathstrut +\mathstrut$$ $$1057810030778773363076297235168973815808$$$$)/$$$$10\!\cdots\!08$$ $$\beta_{7}$$ $$=$$ $$($$$$22798506203329$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$53794823890515539$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$8735973951543936198$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$963298369771195712640$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$60863620998736426826400$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$5606724662346279236318976$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$596764021078698023431742464$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$119123179786183557735667761152$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$36907461707714430355325396336640$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$560944245809642659568346380697600$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$116870868096182953733431300752015360$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$21774355382132346478128392138328637440$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$4982396811329695285770436994328313200640$$ $$\nu\mathstrut -\mathstrut$$ $$322209802111205245917055032357473610629120$$$$)/$$$$35\!\cdots\!80$$ $$\beta_{8}$$ $$=$$ $$($$$$218134071226359$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$4870955965165397$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$1691500331582104182$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$25526624903855638656$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$32686436336515324766304$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$1665254284266544386461952$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$308208502237179693644301312$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$85469617510151187745823555584$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$1885728452825311439592137932800$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$1145214983786675275384713152102400$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$1241725971346736615918569702311198720$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$263004655702424731611774721279918080$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$6011052032619844281886301526957993492480$$ $$\nu\mathstrut -\mathstrut$$ $$551679602029196350940807242550260331970560$$$$)/$$$$15\!\cdots\!80$$ $$\beta_{9}$$ $$=$$ $$($$$$238078010965047$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$26908102121867243$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$347060958724085514$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$479709212226828720000$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$80588539711877262741600$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$6556701619342514477559552$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$137915000797290506345659392$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$102758743819068287844839358464$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$2222732640607341472268550389760$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$1646824685553866520545900916572160$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$100272256112897076060289698526003200$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$37875980785742335857103888733070950400$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$400332714841088082495775252295209451520$$ $$\nu\mathstrut +\mathstrut$$ $$18177411131067093194996903043799061626880$$$$)/$$$$11\!\cdots\!60$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$4186634072503101$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$320633945769169529$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$21960089176649252178$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$4157265813116128890240$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$585128904208779297783840$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$15359708300982390039618816$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$962106828521238672684880896$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$1360506502701944376368809213952$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$162474436447823333694130623528960$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$17593305208907891447070576807444480$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$2296697739545688153720844107965399040$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$461863725615516519923961233747906396160$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$43983203169215878242860042808914419384320$$ $$\nu\mathstrut +\mathstrut$$ $$3795165327056209117152118129046015589744640$$$$)/$$$$46\!\cdots\!40$$ $$\beta_{11}$$ $$=$$ $$($$$$4994608270611919$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$229383117872071075$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$40109149264408807782$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$12224144162338428796032$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$1405653423925877946171552$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$211455767470386056182249728$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$287578160767890582529088512$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$325306134897063931254293954560$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$185338972362262178255694258094080$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$2414605791274134937888865244610560$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$3670021536411728944401198060217958400$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$122247022119048274622513657726191534080$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$124857911880152686291094996342290492948480$$ $$\nu\mathstrut -\mathstrut$$ $$4628522820869557735066896221439839940116480$$$$)/$$$$23\!\cdots\!20$$ $$\beta_{12}$$ $$=$$ $$($$$$3532537240168851$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$275912899806284311$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$23737172275955491278$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$1943541086894356116864$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$1474984638296136506548704$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$171477744490864343215614720$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$28168883036313927292802749440$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$40177498144879219065499844608$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$85150724912387887699323095531520$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$23920337102188935774732225351843840$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$1124665942085803264011735225078251520$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$639545684755019567276063937602802155520$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$4807123178791455698851154304238564147200$$ $$\nu\mathstrut -\mathstrut$$ $$12365866495923744547152099580355678012702720$$$$)/$$$$11\!\cdots\!60$$ $$\beta_{13}$$ $$=$$ $$($$$$16391956323533121$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$2461547649861093421$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$259734533691930798138$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$29094314084099191798656$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$2854910424521539609539936$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$118070268701451279959996160$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$2807143998932373401198484480$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$4450982558133706602982411829248$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$412503246278464397842506368532480$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$116453463217412575234282771510394880$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$14203836602766104161246265900215173120$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$216678429748838589614150887684775608320$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$128434611235071249437177721649156231004160$$ $$\nu\mathstrut -\mathstrut$$ $$4304384418588685625869253151553186451947520$$$$)/$$$$46\!\cdots\!40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1}\mathstrut +\mathstrut$$ $$1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2}\mathstrut -\mathstrut$$ $$21$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$4497$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{8}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$109$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$44$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$3785$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$470331$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$16$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$20$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$182$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$47$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$34$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$240$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$78$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$1163$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$31123$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$781$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$344339$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$377177286$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$32$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$1470$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$516$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$5077$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$9810$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1057$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$3178$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$46018$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$36566$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$268969$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$19019801$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$235759$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$207552805$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$23517462772$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$14144$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$180298$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$35228$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$350839$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$621158$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$272249$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$473810$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$11802902$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$11730034$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$46083063$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$806555335$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$109265781$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$13588863075$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2068502352968$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$12430464$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$34697334$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$41746180$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$74505833$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$216037158$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$4456353$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$65137518$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$964836082$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$103054606$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1890507969$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$264786052367$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$10992243371$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$293805027611$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1732695170738992$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-$$$$3852040192$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$2045631546$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$5476328156$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$2541798177$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$31518770954$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$3607670743$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$971333630$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$236575420174$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$165861538590$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$738835320983$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$25834433487385$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$423252192173$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$879392463287155$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$61343035082431360$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$891841605376$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$327529170870$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$171527044420$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$2564970646839$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1076943891558$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$179746802993$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$301205906770$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$414382110690$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$15350891370546$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$84950953494193$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1772488612857057$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$498935191148235$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$48119748599130549$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$16562748859338969888$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$-$$$$83704357255936$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$40286808739482$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$68537342963612$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$416080977625361$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$387304173057130$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$252330624959239$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$44517146460130$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$4522769214106930$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1531395591950274$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$6467983503954183$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$401017618752659799$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$32110846512217821$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$8908734385886039395$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$165728889045496813408$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$-$$$$90574744598272$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$7541136722191638$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$6513867373394052$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$44641169945674663$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$69866229704094918$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$29825018257882625$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$7810391151260786$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$676605755526618942$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$313350484818503854$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1086491343912284417$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$66418042634796973519$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$5079844211521146491$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$203487764166013956667$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$528489269519788353385248$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$176712273341139200$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$1185945136297923770$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$623871817250639452$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$981818636487274849$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1100984323919815798$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$3898037513030110391$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$820529187230290814$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$67251989155354581074$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$14019862275274049950$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$264146974714746154935$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$4704406853417894934265$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$14050276866797714157$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$306152688699759659251245$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$98913684977183867449231584$$$$)/8$$ $$\nu^{13}$$ $$=$$ $$($$$$116316899511052747008$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$148328262197844197622$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$16665253899987801148$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$123521927538661583785$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$862695920010818789286$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$77050213296354750255$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$103256175490515869934$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$2140112057097104175262$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$36740647845562220530$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$39654215858485171427793$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$384627670167296108589569$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$149729202520309150537877$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$56737931661586284296058923$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$9751818703784608320954592800$$$$)/8$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 109.380 + 40.6323i 109.380 − 40.6323i 101.867 + 58.4667i 101.867 − 58.4667i 41.8643 + 116.115i 41.8643 − 116.115i 12.1331 + 125.704i 12.1331 − 125.704i −30.5698 + 126.646i −30.5698 − 126.646i −93.5934 + 98.6231i −93.5934 − 98.6231i −137.581 + 24.7690i −137.581 − 24.7690i
−242.759 81.2647i −7667.81 52328.1 + 39455.5i 557358.i 1.86143e6 + 623122.i 8.55454e6i −9.49679e6 1.38306e7i 1.57485e7 −4.52935e7 + 1.35304e8i
3.2 −242.759 + 81.2647i −7667.81 52328.1 39455.5i 557358.i 1.86143e6 623122.i 8.55454e6i −9.49679e6 + 1.38306e7i 1.57485e7 −4.52935e7 1.35304e8i
3.3 −227.734 116.933i 6146.29 38189.2 + 53259.3i 351107.i −1.39972e6 718707.i 1.24931e6i −2.46917e6 1.65945e7i −5.26981e6 4.10562e7 7.99589e7i
3.4 −227.734 + 116.933i 6146.29 38189.2 53259.3i 351107.i −1.39972e6 + 718707.i 1.24931e6i −2.46917e6 + 1.65945e7i −5.26981e6 4.10562e7 + 7.99589e7i
3.5 −107.729 232.229i 693.074 −42325.1 + 50035.5i 333015.i −74663.9 160952.i 1.01327e7i 1.61794e7 + 4.43887e6i −4.25664e7 −7.73360e7 + 3.58753e7i
3.6 −107.729 + 232.229i 693.074 −42325.1 50035.5i 333015.i −74663.9 + 160952.i 1.01327e7i 1.61794e7 4.43887e6i −4.25664e7 −7.73360e7 3.58753e7i
3.7 −48.2661 251.409i −8648.56 −60876.8 + 24269.1i 474083.i 417432. + 2.17432e6i 5.25940e6i 9.03974e6 + 1.41336e7i 3.17508e7 1.19189e8 2.28822e7i
3.8 −48.2661 + 251.409i −8648.56 −60876.8 24269.1i 474083.i 417432. 2.17432e6i 5.25940e6i 9.03974e6 1.41336e7i 3.17508e7 1.19189e8 + 2.28822e7i
3.9 37.1396 253.292i 11400.0 −62777.3 18814.3i 8041.89i 423391. 2.88752e6i 7.58607e6i −7.09702e6 + 1.52022e7i 8.69132e7 2.03694e6 + 298672.i
3.10 37.1396 + 253.292i 11400.0 −62777.3 + 18814.3i 8041.89i 423391. + 2.88752e6i 7.58607e6i −7.09702e6 1.52022e7i 8.69132e7 2.03694e6 298672.i
3.11 163.187 197.246i −2122.49 −12276.1 64376.0i 155753.i −346362. + 418652.i 829393.i −1.47012e7 8.08389e6i −3.85418e7 −3.07217e7 2.54169e7i
3.12 163.187 + 197.246i −2122.49 −12276.1 + 64376.0i 155753.i −346362. 418652.i 829393.i −1.47012e7 + 8.08389e6i −3.85418e7 −3.07217e7 + 2.54169e7i
3.13 251.161 49.5380i 6181.49 60628.0 24884.1i 699404.i 1.55255e6 306219.i 4.65744e6i 1.39947e7 9.25331e6i −4.83590e6 3.46471e7 + 1.75663e8i
3.14 251.161 + 49.5380i 6181.49 60628.0 + 24884.1i 699404.i 1.55255e6 + 306219.i 4.65744e6i 1.39947e7 + 9.25331e6i −4.83590e6 3.46471e7 1.75663e8i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.14 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
8.d Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{7}$$ $$\mathstrut -\mathstrut 5982 T_{3}^{6}$$ $$\mathstrut -\mathstrut 154370700 T_{3}^{5}$$ $$\mathstrut +\mathstrut 713806378344 T_{3}^{4}$$ $$\mathstrut +\mathstrut$$$$62\!\cdots\!32$$$$T_{3}^{3}$$ $$\mathstrut -\mathstrut$$$$23\!\cdots\!92$$$$T_{3}^{2}$$ $$\mathstrut -\mathstrut$$$$48\!\cdots\!28$$$$T_{3}$$ $$\mathstrut +\mathstrut$$$$42\!\cdots\!80$$ acting on $$S_{17}^{\mathrm{new}}(8, [\chi])$$.