# Properties

 Label 8.14.b.b Level 8 Weight 14 Character orbit 8.b Analytic conductor 8.578 Analytic rank 0 Dimension 10 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$14$$ Character orbit: $$[\chi]$$ = 8.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$8.57847431615$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{48}\cdot 3^{4}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 11 + \beta_{1} ) q^{2}$$ $$+ ( 3 \beta_{1} + \beta_{3} ) q^{3}$$ $$+ ( -472 + 12 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{4}$$ $$+ ( 53 \beta_{1} + \beta_{2} + 10 \beta_{3} ) q^{5}$$ $$+ ( -26769 + 26 \beta_{1} - \beta_{2} + 26 \beta_{3} + 5 \beta_{4} + \beta_{5} + \beta_{6} ) q^{6}$$ $$+ ( 58697 - 62 \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{7}$$ $$+ ( -27077 - 442 \beta_{1} + \beta_{2} - 10 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{8}$$ $$+ ( 201402 - 2129 \beta_{1} - 4 \beta_{2} + 50 \beta_{3} + 20 \beta_{4} + 11 \beta_{5} - 5 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 11 + \beta_{1} ) q^{2}$$ $$+ ( 3 \beta_{1} + \beta_{3} ) q^{3}$$ $$+ ( -472 + 12 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{4}$$ $$+ ( 53 \beta_{1} + \beta_{2} + 10 \beta_{3} ) q^{5}$$ $$+ ( -26769 + 26 \beta_{1} - \beta_{2} + 26 \beta_{3} + 5 \beta_{4} + \beta_{5} + \beta_{6} ) q^{6}$$ $$+ ( 58697 - 62 \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{7}$$ $$+ ( -27077 - 442 \beta_{1} + \beta_{2} - 10 \beta_{3} + 10 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{8}$$ $$+ ( 201402 - 2129 \beta_{1} - 4 \beta_{2} + 50 \beta_{3} + 20 \beta_{4} + 11 \beta_{5} - 5 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{9}$$ $$+ ( -454220 + 750 \beta_{1} - 20 \beta_{2} - 478 \beta_{3} + 47 \beta_{4} + 22 \beta_{5} - 3 \beta_{6} - 8 \beta_{7} + 8 \beta_{8} + 2 \beta_{9} ) q^{10}$$ $$+ ( 44 + 2651 \beta_{1} + 22 \beta_{2} - 1265 \beta_{3} - 99 \beta_{4} - 22 \beta_{5} - 11 \beta_{6} - 22 \beta_{7} ) q^{11}$$ $$+ ( 2798750 - 30960 \beta_{1} - 126 \beta_{2} - 204 \beta_{3} + 116 \beta_{4} + 28 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} - 16 \beta_{8} - 10 \beta_{9} ) q^{12}$$ $$+ ( -248 - 48385 \beta_{1} - 161 \beta_{2} + 6042 \beta_{3} + 494 \beta_{4} + 4 \beta_{5} + 134 \beta_{6} - 28 \beta_{7} - 16 \beta_{9} ) q^{13}$$ $$+ ( 141124 + 61292 \beta_{1} + 444 \beta_{2} - 5588 \beta_{3} - 494 \beta_{4} + 136 \beta_{5} - 50 \beta_{6} - 48 \beta_{7} - 48 \beta_{8} + 4 \beta_{9} ) q^{14}$$ $$+ ( -14592559 + 97418 \beta_{1} - 192 \beta_{2} - 928 \beta_{3} + 2822 \beta_{4} + 425 \beta_{5} + 240 \beta_{6} - 40 \beta_{7} + 31 \beta_{8} + 72 \beta_{9} ) q^{15}$$ $$+ ( 5662330 - 43092 \beta_{1} + 894 \beta_{2} + 24044 \beta_{3} + 488 \beta_{4} + 28 \beta_{5} - 136 \beta_{6} + 52 \beta_{7} + 96 \beta_{8} - 78 \beta_{9} ) q^{16}$$ $$+ ( 21730719 - 108095 \beta_{1} + 260 \beta_{2} + 3854 \beta_{3} + 4188 \beta_{4} - 1115 \beta_{5} - 91 \beta_{6} + 81 \beta_{7} - 66 \beta_{8} - 17 \beta_{9} ) q^{17}$$ $$+ ( -14759899 + 239599 \beta_{1} - 1912 \beta_{2} - 41528 \beta_{3} - 3860 \beta_{4} + 48 \beta_{5} - 236 \beta_{6} + 32 \beta_{7} - 32 \beta_{8} - 72 \beta_{9} ) q^{18}$$ $$+ ( 6716 + 230823 \beta_{1} + 990 \beta_{2} + 27867 \beta_{3} - 16455 \beta_{4} - 478 \beta_{5} - 527 \beta_{6} + 290 \beta_{7} + 384 \beta_{9} ) q^{19}$$ $$+ ( 2167396 - 419952 \beta_{1} - 4292 \beta_{2} - 131112 \beta_{3} - 4312 \beta_{4} - 472 \beta_{5} - 1036 \beta_{6} + 440 \beta_{7} + 96 \beta_{8} - 396 \beta_{9} ) q^{20}$$ $$+ ( 9560 - 702076 \beta_{1} - 440 \beta_{2} + 144284 \beta_{3} - 25926 \beta_{4} - 340 \beta_{5} + 1730 \beta_{6} + 844 \beta_{7} + 592 \beta_{9} ) q^{21}$$ $$+ ( -17794645 + 84370 \beta_{1} + 7227 \beta_{2} - 178574 \beta_{3} - 9823 \beta_{4} + 693 \beta_{5} - 803 \beta_{6} + 704 \beta_{7} + 704 \beta_{8} - 528 \beta_{9} ) q^{22}$$ $$+ ( -7892749 - 708114 \beta_{1} + 832 \beta_{2} + 36960 \beta_{3} + 58658 \beta_{4} - 6469 \beta_{5} - 16 \beta_{6} + 408 \beta_{7} - 435 \beta_{8} + 392 \beta_{9} ) q^{23}$$ $$+ ( 32033310 + 2752716 \beta_{1} + 9562 \beta_{2} + 173356 \beta_{3} - 26948 \beta_{4} + 1668 \beta_{5} - 6196 \beta_{6} + 44 \beta_{7} - 1600 \beta_{8} - 890 \beta_{9} ) q^{24}$$ $$+ ( -307689609 + 4426842 \beta_{1} - 4184 \beta_{2} - 45428 \beta_{3} + 115912 \beta_{4} + 4866 \beta_{5} + 11314 \beta_{6} - 678 \beta_{7} + 1004 \beta_{8} + 2150 \beta_{9} ) q^{25}$$ $$+ ( 373492924 - 705206 \beta_{1} - 1692 \beta_{2} + 637286 \beta_{3} - 13747 \beta_{4} + 6242 \beta_{5} + 1271 \beta_{6} + 488 \beta_{7} - 744 \beta_{8} - 2106 \beta_{9} ) q^{26}$$ $$+ ( 956 - 748054 \beta_{1} - 18530 \beta_{2} + 995676 \beta_{3} - 8871 \beta_{4} - 1438 \beta_{5} + 6545 \beta_{6} - 1694 \beta_{7} - 128 \beta_{9} ) q^{27}$$ $$+ ( -165361968 + 871616 \beta_{1} + 23600 \beta_{2} - 1579184 \beta_{3} - 17400 \beta_{4} + 14176 \beta_{5} - 21504 \beta_{6} - 4576 \beta_{7} + 1280 \beta_{8} - 176 \beta_{9} ) q^{28}$$ $$+ ( 53440 - 9290205 \beta_{1} + 11415 \beta_{2} - 443578 \beta_{3} - 149296 \beta_{4} - 12320 \beta_{5} + 14736 \beta_{6} - 8480 \beta_{7} + 1920 \beta_{9} ) q^{29}$$ $$+ ( 633816900 - 14826388 \beta_{1} - 52676 \beta_{2} - 2643540 \beta_{3} - 10670 \beta_{4} - 3704 \beta_{5} + 24974 \beta_{6} - 5168 \beta_{7} - 3632 \beta_{8} - 4732 \beta_{9} ) q^{30}$$ $$+ ( 64793900 - 20926432 \beta_{1} - 6976 \beta_{2} + 468896 \beta_{3} + 126736 \beta_{4} + 15212 \beta_{5} - 43504 \beta_{6} - 1560 \beta_{7} + 3596 \beta_{8} + 2296 \beta_{9} ) q^{31}$$ $$+ ( -1129872988 + 6405080 \beta_{1} - 94644 \beta_{2} + 2620760 \beta_{3} + 142208 \beta_{4} + 21784 \beta_{5} - 40224 \beta_{6} - 5624 \beta_{7} + 10560 \beta_{8} + 1620 \beta_{9} ) q^{32}$$ $$+ ( 1548463631 + 26283367 \beta_{1} + 4444 \beta_{2} - 606078 \beta_{3} - 193644 \beta_{4} + 2211 \beta_{5} + 54131 \beta_{6} + 759 \beta_{7} - 9262 \beta_{8} - 2167 \beta_{9} ) q^{33}$$ $$+ ( -609724706 + 21655034 \beta_{1} + 116344 \beta_{2} + 3892024 \beta_{3} - 8556 \beta_{4} - 41520 \beta_{5} + 97644 \beta_{6} - 6176 \beta_{7} + 7200 \beta_{8} + 1352 \beta_{9} ) q^{34}$$ $$+ ( -2648 + 75559352 \beta_{1} + 88468 \beta_{2} - 1354980 \beta_{3} + 157190 \beta_{4} + 4204 \beta_{5} - 150762 \beta_{6} + 4972 \beta_{7} + 384 \beta_{9} ) q^{35}$$ $$+ ( 400375928 - 15910060 \beta_{1} + 74656 \beta_{2} + 792978 \beta_{3} + 286167 \beta_{4} - 45760 \beta_{5} - 81664 \beta_{6} + 20416 \beta_{7} - 16896 \beta_{8} + 352 \beta_{9} ) q^{36}$$ $$+ ( -317480 - 93260595 \beta_{1} - 56499 \beta_{2} - 933618 \beta_{3} + 618906 \beta_{4} + 63340 \beta_{5} + 181154 \beta_{6} + 37900 \beta_{7} - 12720 \beta_{9} ) q^{37}$$ $$+ ( -1876499561 + 4914394 \beta_{1} - 24985 \beta_{2} - 7228870 \beta_{3} - 187227 \beta_{4} + 46793 \beta_{5} + 221521 \beta_{6} + 26560 \beta_{7} + 4032 \beta_{8} + 34864 \beta_{9} ) q^{38}$$ $$+ ( -6349205925 - 93530234 \beta_{1} + 35840 \beta_{2} + 1436672 \beta_{3} - 1164358 \beta_{4} - 12925 \beta_{5} - 217856 \beta_{6} + 4928 \beta_{7} - 18883 \beta_{8} - 21056 \beta_{9} ) q^{39}$$ $$+ ( 746682892 - 2861192 \beta_{1} + 107940 \beta_{2} + 12579384 \beta_{3} + 175864 \beta_{4} - 217368 \beta_{5} - 308072 \beta_{6} + 40184 \beta_{7} - 30592 \beta_{8} + 6300 \beta_{9} ) q^{40}$$ $$+ ( 5933317952 + 146320078 \beta_{1} + 69624 \beta_{2} - 3847900 \beta_{3} - 2484520 \beta_{4} - 137274 \beta_{5} + 256726 \beta_{6} + 10446 \beta_{7} + 57156 \beta_{8} - 38286 \beta_{9} ) q^{41}$$ $$+ ( 5389922032 - 6465896 \beta_{1} - 255152 \beta_{2} - 1713880 \beta_{3} - 797140 \beta_{4} + 142744 \beta_{5} + 478340 \beta_{6} + 40800 \beta_{7} - 19040 \beta_{8} + 60136 \beta_{9} ) q^{42}$$ $$+ ( -379016 + 334310023 \beta_{1} - 210628 \beta_{2} - 9561015 \beta_{3} + 1655986 \beta_{4} + 41668 \beta_{5} - 698590 \beta_{6} + 2244 \beta_{7} - 19712 \beta_{9} ) q^{43}$$ $$+ ( 1332537910 - 21952304 \beta_{1} - 437910 \beta_{2} + 5775044 \beta_{3} + 754820 \beta_{4} - 149556 \beta_{5} - 714494 \beta_{6} - 30844 \beta_{7} + 73392 \beta_{8} + 36718 \beta_{9} ) q^{44}$$ $$+ ( -596104 - 383910733 \beta_{1} + 45451 \beta_{2} - 19558710 \beta_{3} + 848370 \beta_{4} + 21692 \beta_{5} + 660314 \beta_{6} - 52004 \beta_{7} - 36848 \beta_{9} ) q^{45}$$ $$+ ( -5504873204 - 4840700 \beta_{1} + 662132 \beta_{2} + 14726596 \beta_{3} - 723178 \beta_{4} - 378600 \beta_{5} + 1036938 \beta_{6} - 97168 \beta_{7} + 31344 \beta_{8} - 23348 \beta_{9} ) q^{46}$$ $$+ ( -1016950802 - 563994860 \beta_{1} - 85952 \beta_{2} + 11213792 \beta_{3} - 330884 \beta_{4} + 349022 \beta_{5} - 1236048 \beta_{6} - 27720 \beta_{7} + 61194 \beta_{8} + 2792 \beta_{9} ) q^{47}$$ $$+ ( -30184952396 + 71893528 \beta_{1} + 526268 \beta_{2} + 839320 \beta_{3} + 3032304 \beta_{4} + 598840 \beta_{5} - 764528 \beta_{6} - 102040 \beta_{7} + 5568 \beta_{8} + 44324 \beta_{9} ) q^{48}$$ $$+ ( 18270820881 + 652777880 \beta_{1} - 142240 \beta_{2} - 13661872 \beta_{3} - 941792 \beta_{4} + 668408 \beta_{5} + 1399160 \beta_{6} - 33768 \beta_{7} - 243760 \beta_{8} + 40936 \beta_{9} ) q^{49}$$ $$+ ( 32643907847 - 339422027 \beta_{1} - 1045840 \beta_{2} - 57050064 \beta_{3} + 2099656 \beta_{4} - 484832 \beta_{5} + 1518136 \beta_{6} - 183616 \beta_{7} - 38592 \beta_{8} - 141616 \beta_{9} ) q^{50}$$ $$+ ( -933116 + 680235082 \beta_{1} + 1005698 \beta_{2} + 65691264 \beta_{3} + 3209271 \beta_{4} + 32638 \beta_{5} - 847553 \beta_{6} - 83074 \beta_{7} - 57856 \beta_{9} ) q^{51}$$ $$+ ( -5330279700 + 366987952 \beta_{1} - 878092 \beta_{2} + 56565256 \beta_{3} + 3218936 \beta_{4} + 973304 \beta_{5} - 1128932 \beta_{6} - 71768 \beta_{7} - 84192 \beta_{8} - 13796 \beta_{9} ) q^{52}$$ $$+ ( 2709608 - 855035743 \beta_{1} + 544009 \beta_{2} - 12830354 \beta_{3} - 8358570 \beta_{4} - 455884 \beta_{5} + 1524622 \beta_{6} - 216172 \beta_{7} + 119856 \beta_{9} ) q^{53}$$ $$+ ( 3543962566 - 18872460 \beta_{1} - 437466 \beta_{2} + 41775316 \beta_{3} + 1581034 \beta_{4} + 965642 \beta_{5} + 1293714 \beta_{6} + 200640 \beta_{7} - 108608 \beta_{8} - 89552 \beta_{9} ) q^{54}$$ $$+ ( -12305069843 - 928200086 \beta_{1} + 244992 \beta_{2} + 22940544 \beta_{3} + 10952854 \beta_{4} - 1393403 \beta_{5} - 1691712 \beta_{6} + 93984 \beta_{7} - 90277 \beta_{8} + 36960 \beta_{9} ) q^{55}$$ $$+ ( -46211452808 - 58941968 \beta_{1} + 389864 \beta_{2} - 136026256 \beta_{3} - 6205872 \beta_{4} - 667248 \beta_{5} - 3623664 \beta_{6} - 68816 \beta_{7} + 201216 \beta_{8} - 100840 \beta_{9} ) q^{56}$$ $$+ ( -51144839445 + 1218278435 \beta_{1} - 399284 \beta_{2} - 18101558 \beta_{3} + 16122916 \beta_{4} - 521009 \beta_{5} + 2786975 \beta_{6} - 46733 \beta_{7} + 703834 \beta_{8} + 259085 \beta_{9} ) q^{57}$$ $$+ ( 76654198700 - 70779742 \beta_{1} + 2960308 \beta_{2} - 85819954 \beta_{3} - 15738407 \beta_{4} + 716602 \beta_{5} + 1001627 \beta_{6} + 478280 \beta_{7} + 310200 \beta_{8} - 70930 \beta_{9} ) q^{58}$$ $$+ ( 5925120 + 549676647 \beta_{1} - 6205632 \beta_{2} + 5423117 \beta_{3} - 13419904 \beta_{4} - 425280 \beta_{5} - 1562048 \beta_{6} + 251328 \beta_{7} + 338304 \beta_{9} ) q^{59}$$ $$+ ( 181315037392 + 384106176 \beta_{1} + 6153776 \beta_{2} + 56893008 \beta_{3} - 14161272 \beta_{4} - 2159776 \beta_{5} - 4238336 \beta_{6} + 338464 \beta_{7} - 416512 \beta_{8} - 334512 \beta_{9} ) q^{60}$$ $$+ ( 3922824 - 1276639589 \beta_{1} - 1251637 \beta_{2} + 75366930 \beta_{3} - 12567986 \beta_{4} + 403908 \beta_{5} + 2603238 \beta_{6} + 1034660 \beta_{7} + 315376 \beta_{9} ) q^{61}$$ $$+ ( -166520982448 + 314321008 \beta_{1} - 3146960 \beta_{2} - 31498000 \beta_{3} - 25385816 \beta_{4} - 272480 \beta_{5} + 503512 \beta_{6} - 21440 \beta_{7} - 27072 \beta_{8} - 161328 \beta_{9} ) q^{62}$$ $$+ ( -89907478473 + 397056550 \beta_{1} - 227392 \beta_{2} - 623584 \beta_{3} + 19508234 \beta_{4} - 496993 \beta_{5} + 1281616 \beta_{6} + 9416 \beta_{7} - 155527 \beta_{8} + 255640 \beta_{9} ) q^{63}$$ $$+ ( -218060179128 - 865322704 \beta_{1} - 8099432 \beta_{2} + 37970992 \beta_{3} + 10143168 \beta_{4} + 666928 \beta_{5} + 2628992 \beta_{6} + 980240 \beta_{7} - 432000 \beta_{8} - 539992 \beta_{9} ) q^{64}$$ $$+ ( 157722994006 - 1043371650 \beta_{1} + 1301432 \beta_{2} + 21040452 \beta_{3} + 369112 \beta_{4} - 3335882 \beta_{5} - 1835258 \beta_{6} + 339966 \beta_{7} - 1200540 \beta_{8} - 281534 \beta_{9} ) q^{65}$$ $$+ ( 226942670020 + 1265834284 \beta_{1} + 4799432 \beta_{2} - 51943672 \beta_{3} + 27399372 \beta_{4} + 1557424 \beta_{5} - 1830092 \beta_{6} - 215776 \beta_{7} - 347424 \beta_{8} + 175736 \beta_{9} ) q^{66}$$ $$+ ( 8425980 + 781961615 \beta_{1} + 21797758 \beta_{2} + 29706099 \beta_{3} - 20584951 \beta_{4} - 747390 \beta_{5} - 711039 \beta_{6} + 176770 \beta_{7} + 462080 \beta_{9} ) q^{67}$$ $$+ ( 233813222896 - 976593000 \beta_{1} - 4994976 \beta_{2} + 79067292 \beta_{3} + 32163090 \beta_{4} + 4524736 \beta_{5} + 7433984 \beta_{6} - 313280 \beta_{7} + 1638912 \beta_{8} + 12960 \beta_{9} ) q^{68}$$ $$+ ( -10085720 + 2314320460 \beta_{1} - 2106808 \beta_{2} + 207661572 \beta_{3} + 28413510 \beta_{4} + 188500 \beta_{5} - 2875586 \beta_{6} - 1105996 \beta_{7} - 647248 \beta_{9} ) q^{69}$$ $$+ ( -607035216816 + 886920896 \beta_{1} + 693200 \beta_{2} - 276100352 \beta_{3} + 63856608 \beta_{4} - 2172592 \beta_{5} - 3983488 \beta_{6} - 860544 \beta_{7} + 591488 \beta_{8} + 331552 \beta_{9} ) q^{70}$$ $$+ ( 72648964185 + 2923311210 \beta_{1} - 2964288 \beta_{2} - 71313504 \beta_{3} - 30956154 \beta_{4} + 11007537 \beta_{5} + 4967952 \beta_{6} - 858648 \beta_{7} + 1095495 \beta_{8} + 388344 \beta_{9} ) q^{71}$$ $$+ ( -360070773683 + 840966474 \beta_{1} + 8813847 \beta_{2} + 196613146 \beta_{3} - 15564698 \beta_{4} - 1389706 \beta_{5} + 4688814 \beta_{6} - 1829934 \beta_{7} - 41984 \beta_{8} + 782569 \beta_{9} ) q^{72}$$ $$+ ( -63300715771 - 5238266937 \beta_{1} + 237148 \beta_{2} + 88267906 \beta_{3} - 44908748 \beta_{4} + 2828163 \beta_{5} - 12222989 \beta_{6} - 125289 \beta_{7} + 277202 \beta_{8} - 613015 \beta_{9} ) q^{73}$$ $$+ ( 752874413684 - 1228039234 \beta_{1} - 13673492 \beta_{2} + 715892050 \beta_{3} - 51022761 \beta_{4} - 4103546 \beta_{5} - 7256011 \beta_{6} - 2590408 \beta_{7} - 1463352 \beta_{8} + 32882 \beta_{9} ) q^{74}$$ $$+ ( -37591448 - 10424934857 \beta_{1} - 44716300 \beta_{2} - 608964015 \beta_{3} + 78747990 \beta_{4} + 2075404 \beta_{5} + 16345318 \beta_{6} - 2383348 \beta_{7} - 2229376 \beta_{9} ) q^{75}$$ $$+ ( 1033849336974 - 3022119920 \beta_{1} - 6310510 \beta_{2} - 867908908 \beta_{3} - 35134572 \beta_{4} - 8398340 \beta_{5} + 13575050 \beta_{6} - 371052 \beta_{7} - 1116816 \beta_{8} + 1170598 \beta_{9} ) q^{76}$$ $$+ ( -19687272 + 9840301460 \beta_{1} + 5847160 \beta_{2} - 959839716 \beta_{3} + 73231466 \beta_{4} - 305844 \beta_{5} - 23336654 \beta_{6} - 3012372 \beta_{7} - 1353264 \beta_{9} ) q^{77}$$ $$+ ( -825161592628 - 5455859900 \beta_{1} + 12398132 \beta_{2} + 532579204 \beta_{3} - 80693834 \beta_{4} + 7184280 \beta_{5} - 17867798 \beta_{6} + 1473392 \beta_{7} - 275600 \beta_{8} + 1423116 \beta_{9} ) q^{78}$$ $$+ ( 544561634522 + 8223218036 \beta_{1} + 7924480 \beta_{2} - 222874496 \beta_{3} - 154133556 \beta_{4} - 16020598 \beta_{5} + 15582784 \beta_{6} + 1616736 \beta_{7} - 2081898 \beta_{8} - 3074272 \beta_{9} ) q^{79}$$ $$+ ( -1540702585336 + 2590746096 \beta_{1} + 19221080 \beta_{2} + 816388848 \beta_{3} + 11831008 \beta_{4} + 4414512 \beta_{5} + 30548768 \beta_{6} - 1039856 \beta_{7} + 685440 \beta_{8} + 2319464 \beta_{9} ) q^{80}$$ $$+ ( -967434570912 - 8273452465 \beta_{1} - 5636612 \beta_{2} + 155994034 \beta_{3} - 27024716 \beta_{4} + 18754603 \beta_{5} - 19836997 \beta_{6} - 1639457 \beta_{7} + 4400770 \beta_{8} + 718241 \beta_{9} ) q^{81}$$ $$+ ( 1227249279038 + 3968133690 \beta_{1} - 16462320 \beta_{2} + 885031568 \beta_{3} + 233468504 \beta_{4} - 425376 \beta_{5} - 25704024 \beta_{6} + 6975552 \beta_{7} + 4080576 \beta_{8} + 2221680 \beta_{9} ) q^{82}$$ $$+ ( -34502432 - 12072829957 \beta_{1} + 58375600 \beta_{2} + 320134105 \beta_{3} + 58109256 \beta_{4} + 5728336 \beta_{5} + 28915016 \beta_{6} + 2655568 \beta_{7} - 1536384 \beta_{9} ) q^{83}$$ $$+ ( 2036275648528 + 3115986752 \beta_{1} - 11114384 \beta_{2} - 1248455072 \beta_{3} - 59372896 \beta_{4} + 986016 \beta_{5} + 32338640 \beta_{6} + 282592 \beta_{7} - 5078656 \beta_{8} + 1787472 \beta_{9} ) q^{84}$$ $$+ ( 22052104 + 24794090250 \beta_{1} + 9744798 \beta_{2} + 1530600720 \beta_{3} - 15668370 \beta_{4} + 5743108 \beta_{5} - 40579834 \beta_{6} + 10214884 \beta_{7} + 2235888 \beta_{9} ) q^{85}$$ $$+ ( -2679572959125 + 3695196066 \beta_{1} + 13539515 \beta_{2} - 564299166 \beta_{3} + 311314345 \beta_{4} - 21111387 \beta_{5} - 25348939 \beta_{6} - 770176 \beta_{7} - 1896576 \beta_{8} - 1663904 \beta_{9} ) q^{86}$$ $$+ ( 763225810151 + 12283473382 \beta_{1} + 5765824 \beta_{2} - 262690656 \beta_{3} - 45730806 \beta_{4} - 18241713 \beta_{5} + 26075792 \beta_{6} + 1430952 \beta_{7} - 276631 \beta_{8} - 1472968 \beta_{9} ) q^{87}$$ $$+ ( -2767741945738 + 5194601148 \beta_{1} - 14405534 \beta_{2} - 115046756 \beta_{3} - 67283348 \beta_{4} - 17079084 \beta_{5} + 35548348 \beta_{6} + 8820636 \beta_{7} + 828608 \beta_{8} - 1414402 \beta_{9} ) q^{88}$$ $$+ ( 550637710917 - 17626489609 \beta_{1} + 8312860 \beta_{2} + 357989794 \beta_{3} + 15525684 \beta_{4} - 17101453 \beta_{5} - 34139741 \beta_{6} + 2168391 \beta_{7} - 11809038 \beta_{8} - 1807687 \beta_{9} ) q^{89}$$ $$+ ( 3145563778252 - 4383604542 \beta_{1} + 18452820 \beta_{2} + 807197998 \beta_{3} - 357936407 \beta_{4} - 11609542 \beta_{5} - 32926965 \beta_{6} - 2705208 \beta_{7} + 1316920 \beta_{8} - 3693170 \beta_{9} ) q^{90}$$ $$+ ( 105789112 - 15362091032 \beta_{1} - 54485860 \beta_{2} - 953856396 \beta_{3} - 284560862 \beta_{4} - 3653276 \beta_{5} + 16805330 \beta_{6} + 9477732 \beta_{7} + 6565504 \beta_{9} ) q^{91}$$ $$+ ( 3397128056688 - 10620099520 \beta_{1} + 4142480 \beta_{2} + 1145177584 \beta_{3} + 85968152 \beta_{4} + 27777056 \beta_{5} + 30190592 \beta_{6} + 434272 \beta_{7} + 11321088 \beta_{8} - 1891088 \beta_{9} ) q^{92}$$ $$+ ( 47094720 - 1894796592 \beta_{1} - 12318288 \beta_{2} - 1945353280 \beta_{3} - 109297008 \beta_{4} - 10140960 \beta_{5} - 9333552 \beta_{6} - 6565920 \beta_{7} + 1787520 \beta_{9} ) q^{93}$$ $$+ ( -4535337247272 + 5362366024 \beta_{1} - 57150808 \beta_{2} + 205803336 \beta_{3} - 621379124 \beta_{4} + 10120368 \beta_{5} - 34971276 \beta_{6} + 4354016 \beta_{7} - 610848 \beta_{8} - 423464 \beta_{9} ) q^{94}$$ $$+ ( -1421631768543 + 17309214802 \beta_{1} - 31674112 \beta_{2} - 201341056 \beta_{3} + 401030574 \beta_{4} + 71677849 \beta_{5} + 40231616 \beta_{6} - 6826272 \beta_{7} + 8325639 \beta_{8} + 11195296 \beta_{9} ) q^{95}$$ $$+ ( -3539806193272 - 24598612432 \beta_{1} - 42293416 \beta_{2} - 3662008016 \beta_{3} - 114637952 \beta_{4} + 13970864 \beta_{5} - 26872256 \beta_{6} - 10691952 \beta_{7} + 920192 \beta_{8} - 5181976 \beta_{9} ) q^{96}$$ $$+ ( 135946286999 - 8942656439 \beta_{1} + 9838116 \beta_{2} + 341106174 \beta_{3} + 392173180 \beta_{4} - 71273651 \beta_{5} - 11423859 \beta_{6} + 3692505 \beta_{7} + 13407278 \beta_{8} + 1239399 \beta_{9} ) q^{97}$$ $$+ ( 5432359791379 + 12421831961 \beta_{1} + 74109760 \beta_{2} - 4805274304 \beta_{3} + 505846240 \beta_{4} + 41204608 \beta_{5} - 15276512 \beta_{6} - 14779136 \beta_{7} - 16022784 \beta_{8} - 1644864 \beta_{9} ) q^{98}$$ $$+ ( 77120736 - 2787378231 \beta_{1} + 27909552 \beta_{2} + 1836562035 \beta_{3} - 210226104 \beta_{4} - 25223088 \beta_{5} + 16535112 \beta_{6} - 21666480 \beta_{7} + 1778304 \beta_{9} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q$$ $$\mathstrut +\mathstrut 110q^{2}$$ $$\mathstrut -\mathstrut 4716q^{4}$$ $$\mathstrut -\mathstrut 267668q^{6}$$ $$\mathstrut +\mathstrut 586960q^{7}$$ $$\mathstrut -\mathstrut 270712q^{8}$$ $$\mathstrut +\mathstrut 2014054q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$10q$$ $$\mathstrut +\mathstrut 110q^{2}$$ $$\mathstrut -\mathstrut 4716q^{4}$$ $$\mathstrut -\mathstrut 267668q^{6}$$ $$\mathstrut +\mathstrut 586960q^{7}$$ $$\mathstrut -\mathstrut 270712q^{8}$$ $$\mathstrut +\mathstrut 2014054q^{9}$$ $$\mathstrut -\mathstrut 4542088q^{10}$$ $$\mathstrut +\mathstrut 27987880q^{12}$$ $$\mathstrut +\mathstrut 1408688q^{14}$$ $$\mathstrut -\mathstrut 145914416q^{15}$$ $$\mathstrut +\mathstrut 56624912q^{16}$$ $$\mathstrut +\mathstrut 217326004q^{17}$$ $$\mathstrut -\mathstrut 147615262q^{18}$$ $$\mathstrut +\mathstrut 21655184q^{20}$$ $$\mathstrut -\mathstrut 177987876q^{22}$$ $$\mathstrut -\mathstrut 78679952q^{23}$$ $$\mathstrut +\mathstrut 320199056q^{24}$$ $$\mathstrut -\mathstrut 3076402574q^{25}$$ $$\mathstrut +\mathstrut 3734872040q^{26}$$ $$\mathstrut -\mathstrut 1653812448q^{28}$$ $$\mathstrut +\mathstrut 6338232752q^{30}$$ $$\mathstrut +\mathstrut 648233792q^{31}$$ $$\mathstrut -\mathstrut 11298380000q^{32}$$ $$\mathstrut +\mathstrut 15484079688q^{33}$$ $$\mathstrut -\mathstrut 6096822724q^{34}$$ $$\mathstrut +\mathstrut 4004708940q^{36}$$ $$\mathstrut -\mathstrut 18764968628q^{38}$$ $$\mathstrut -\mathstrut 63497510288q^{39}$$ $$\mathstrut +\mathstrut 7466802592q^{40}$$ $$\mathstrut +\mathstrut 59324640356q^{41}$$ $$\mathstrut +\mathstrut 53897620960q^{42}$$ $$\mathstrut +\mathstrut 13325704392q^{44}$$ $$\mathstrut -\mathstrut 55046867440q^{46}$$ $$\mathstrut -\mathstrut 10176534816q^{47}$$ $$\mathstrut -\mathstrut 301841943264q^{48}$$ $$\mathstrut +\mathstrut 182708552058q^{49}$$ $$\mathstrut +\mathstrut 326454435302q^{50}$$ $$\mathstrut -\mathstrut 53296499536q^{52}$$ $$\mathstrut +\mathstrut 35449773752q^{54}$$ $$\mathstrut -\mathstrut 123010753008q^{55}$$ $$\mathstrut -\mathstrut 462152447680q^{56}$$ $$\mathstrut -\mathstrut 511372324504q^{57}$$ $$\mathstrut +\mathstrut 766482705096q^{58}$$ $$\mathstrut +\mathstrut 1813082440992q^{60}$$ $$\mathstrut -\mathstrut 1665308528960q^{62}$$ $$\mathstrut -\mathstrut 898991123792q^{63}$$ $$\mathstrut -\mathstrut 2180548996032q^{64}$$ $$\mathstrut +\mathstrut 1577231990240q^{65}$$ $$\mathstrut +\mathstrut 2269525079448q^{66}$$ $$\mathstrut +\mathstrut 2338280915304q^{68}$$ $$\mathstrut -\mathstrut 6070110714688q^{70}$$ $$\mathstrut +\mathstrut 726361179984q^{71}$$ $$\mathstrut -\mathstrut 3600753685960q^{72}$$ $$\mathstrut -\mathstrut 633240365532q^{73}$$ $$\mathstrut +\mathstrut 7528513982264q^{74}$$ $$\mathstrut +\mathstrut 10338420845032q^{76}$$ $$\mathstrut -\mathstrut 8252024440816q^{78}$$ $$\mathstrut +\mathstrut 5445103565344q^{79}$$ $$\mathstrut -\mathstrut 15406871881920q^{80}$$ $$\mathstrut -\mathstrut 9674575380574q^{81}$$ $$\mathstrut +\mathstrut 12273334206796q^{82}$$ $$\mathstrut +\mathstrut 20362643366464q^{84}$$ $$\mathstrut -\mathstrut 26794541719396q^{86}$$ $$\mathstrut +\mathstrut 7632221772720q^{87}$$ $$\mathstrut -\mathstrut 27677491769136q^{88}$$ $$\mathstrut +\mathstrut 5506344808004q^{89}$$ $$\mathstrut +\mathstrut 31454099524040q^{90}$$ $$\mathstrut +\mathstrut 33971694298464q^{92}$$ $$\mathstrut -\mathstrut 45356008560096q^{94}$$ $$\mathstrut -\mathstrut 14214732035504q^{95}$$ $$\mathstrut -\mathstrut 35398666935232q^{96}$$ $$\mathstrut +\mathstrut 1361133320788q^{97}$$ $$\mathstrut +\mathstrut 54325451514942q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10}\mathstrut -\mathstrut$$ $$5$$ $$x^{9}\mathstrut +\mathstrut$$ $$752$$ $$x^{8}\mathstrut +\mathstrut$$ $$708$$ $$x^{7}\mathstrut -\mathstrut$$ $$743866$$ $$x^{6}\mathstrut +\mathstrut$$ $$96647426$$ $$x^{5}\mathstrut +\mathstrut$$ $$2540283092$$ $$x^{4}\mathstrut -\mathstrut$$ $$180067834748$$ $$x^{3}\mathstrut +\mathstrut$$ $$15101451375489$$ $$x^{2}\mathstrut -\mathstrut$$ $$802096030557765$$ $$x\mathstrut +\mathstrut$$ $$31616997813655668$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$4673$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$7354742$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$521346978$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$35423339110$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$3119603134524$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$27286958100142$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$135650155419758$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$72918037609395038$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$438921715422012651$$ $$\nu\mathstrut -\mathstrut$$ $$240168537249774828060$$$$)/$$$$17492130486288384$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$53727$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$600394$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$135456610$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$4741322330$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$316521478980$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$11136753949934$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$26417243554130$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$1817150962183070$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$42069080694288501$$ $$\nu\mathstrut +\mathstrut$$ $$17999672649818073444$$$$)/$$$$69968521945153536$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$53727$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$600394$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$135456610$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$4741322330$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$316521478980$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$11136753949934$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$26417243554130$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$141754194852490142$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$517679094866939787$$ $$\nu\mathstrut +\mathstrut$$ $$38430481057802905956$$$$)/$$$$34984260972576768$$ $$\beta_{5}$$ $$=$$ $$($$$$1098741$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$90806382$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$2440766858$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$54449976802$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$1297531495828$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$78627040368902$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$1466142409638458$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$36672486504925558$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$1802119555982310951$$ $$\nu\mathstrut +\mathstrut$$ $$45775038388159863348$$$$)/$$$$17492130486288384$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$393119$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$2793526$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$136735774$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$5861994714$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$563380286532$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$22898903292818$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$718390716500434$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$89713441546092318$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$5000468461985800117$$ $$\nu\mathstrut +\mathstrut$$ $$287833874783644461668$$$$)/$$$$5830710162096128$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$712491$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$83201106$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$2967283894$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$68951707810$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$515357778668$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$149753232903494$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$14258798073026042$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$318818021959084106$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$1662894743218891719$$ $$\nu\mathstrut +\mathstrut$$ $$80019370682910065076$$$$)/$$$$8746065243144192$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$1412391$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$53876346$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$6991130158$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$262411250934$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$3603178854948$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$358142733536254$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$3965347224942526$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$110417944407472430$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$33149970901662232461$$ $$\nu\mathstrut +\mathstrut$$ $$774492335446627231044$$$$)/$$$$8746065243144192$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$12461681$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$378837910$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$5471783426$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$1181809514106$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$9343551156356$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$2306079501218638$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$156691468785680654$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$543216588390237954$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$140585877661687524603$$ $$\nu\mathstrut +\mathstrut$$ $$1111730392134764711772$$$$)/$$$$34984260972576768$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1}\mathstrut +\mathstrut$$ $$1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$8$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$592$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$\beta_{9}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$20$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$50$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$502$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$10617$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$19$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$48$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$66$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$108$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$134$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$344$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$11622$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$427$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$11106$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$3214105$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$565$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$720$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$1278$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$3928$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1798$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$14726$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$178570$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$17293$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1005960$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$179362061$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-$$$$45887$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$57600$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$89730$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$299902$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$29882$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$143293$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$5206788$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$68737$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$81514094$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$8831822803$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$2487037$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$3107424$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$5870294$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$24474454$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$5731714$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$9846400$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1790387310$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$21306707$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$10335222690$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1283839484109$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-$$$$51318481$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$98556528$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$12162394$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$1717394936$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1245638962$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$7853870680$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$28600747182$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$633178633$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$798648764194$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$69547145106093$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$488837006$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$167230080$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$194693340$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$24404269452$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$4566636116$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$115043898562$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$537655247824$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$20306671118$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$10811163478243$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$912576881846255$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −46.7129 − 17.5509i −46.7129 + 17.5509i −17.5296 − 43.4857i −17.5296 + 43.4857i 1.33949 − 44.8086i 1.33949 + 44.8086i 27.9424 − 31.0290i 27.9424 + 31.0290i 37.4606 − 15.6556i 37.4606 + 15.6556i
−83.4258 35.1018i 622.159i 5727.73 + 5856.79i 26675.4i −21838.9 + 51904.1i 50480.5 −272257. 689661.i 1.20724e6 936352. 2.22541e6i
5.2 −83.4258 + 35.1018i 622.159i 5727.73 5856.79i 26675.4i −21838.9 51904.1i 50480.5 −272257. + 689661.i 1.20724e6 936352. + 2.22541e6i
5.3 −25.0592 86.9715i 1746.24i −6936.08 + 4358.86i 64905.5i −151873. + 43759.4i 201238. 552909. + 494011.i −1.45504e6 −5.64492e6 + 1.62648e6i
5.4 −25.0592 + 86.9715i 1746.24i −6936.08 4358.86i 64905.5i −151873. 43759.4i 201238. 552909. 494011.i −1.45504e6 −5.64492e6 1.62648e6i
5.5 12.6790 89.6172i 86.8898i −7870.49 2272.51i 45531.7i 7786.82 + 1101.67i −249036. −303446. + 676518.i 1.58677e6 4.08042e6 + 577295.i
5.6 12.6790 + 89.6172i 86.8898i −7870.49 + 2272.51i 45531.7i 7786.82 1101.67i −249036. −303446. 676518.i 1.58677e6 4.08042e6 577295.i
5.7 65.8848 62.0580i 1231.41i 489.620 8177.36i 25270.7i 76418.8 + 81131.3i 608245. −475211. 569148.i 77950.5 −1.56825e6 1.66495e6i
5.8 65.8848 + 62.0580i 1231.41i 489.620 + 8177.36i 25270.7i 76418.8 81131.3i 608245. −475211. + 569148.i 77950.5 −1.56825e6 + 1.66495e6i
5.9 84.9212 31.3112i 1415.71i 6231.21 5317.98i 2384.10i −44327.5 120223.i −317448. 362649. 646716.i −409898. −74649.0 202460.i
5.10 84.9212 + 31.3112i 1415.71i 6231.21 + 5317.98i 2384.10i −44327.5 + 120223.i −317448. 362649. + 646716.i −409898. −74649.0 + 202460.i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.10 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.b Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{10}$$ $$\mathstrut +\mathstrut 6964588 T_{3}^{8}$$ $$\mathstrut +\mathstrut$$$$16\!\cdots\!52$$$$T_{3}^{6}$$ $$\mathstrut +\mathstrut$$$$14\!\cdots\!28$$$$T_{3}^{4}$$ $$\mathstrut +\mathstrut$$$$36\!\cdots\!44$$$$T_{3}^{2}$$ $$\mathstrut +\mathstrut$$$$27\!\cdots\!16$$ acting on $$S_{14}^{\mathrm{new}}(8, [\chi])$$.