# Properties

 Label 8.8.b.a Level 8 Weight 8 Character orbit 8.b Analytic conductor 2.499 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$2.49908020387$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{15}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 1 + \beta_{1} ) q^{2}$$ $$+ \beta_{4} q^{3}$$ $$+ ( 19 + \beta_{1} + \beta_{3} ) q^{4}$$ $$+ ( 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{5}$$ $$+ ( 44 + 2 \beta_{1} - \beta_{2} + 6 \beta_{4} + 2 \beta_{5} ) q^{6}$$ $$+ ( -112 + 16 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{7}$$ $$+ ( 250 + 10 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} ) q^{8}$$ $$+ ( -481 - 112 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 1 + \beta_{1} ) q^{2}$$ $$+ \beta_{4} q^{3}$$ $$+ ( 19 + \beta_{1} + \beta_{3} ) q^{4}$$ $$+ ( 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{5}$$ $$+ ( 44 + 2 \beta_{1} - \beta_{2} + 6 \beta_{4} + 2 \beta_{5} ) q^{6}$$ $$+ ( -112 + 16 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{7}$$ $$+ ( 250 + 10 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} ) q^{8}$$ $$+ ( -481 - 112 \beta_{1} - 4 \beta_{2} - 12 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{9}$$ $$+ ( -280 - 12 \beta_{1} - 10 \beta_{2} + 8 \beta_{3} - 52 \beta_{4} + 4 \beta_{5} ) q^{10}$$ $$+ ( -208 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} - 11 \beta_{4} - 8 \beta_{5} ) q^{11}$$ $$+ ( -682 + 74 \beta_{1} + 20 \beta_{2} - 6 \beta_{3} + 88 \beta_{4} + 8 \beta_{5} ) q^{12}$$ $$+ ( 438 \beta_{1} - 32 \beta_{2} - 5 \beta_{3} + 37 \beta_{4} + 5 \beta_{5} ) q^{13}$$ $$+ ( 2008 - 88 \beta_{1} - 40 \beta_{2} + 16 \beta_{3} + 80 \beta_{4} - 16 \beta_{5} ) q^{14}$$ $$+ ( 2960 + 1040 \beta_{1} + 50 \beta_{2} + 42 \beta_{3} - 14 \beta_{4} + 14 \beta_{5} ) q^{15}$$ $$+ ( 5908 + 164 \beta_{1} + 76 \beta_{2} - 12 \beta_{3} - 56 \beta_{4} - 40 \beta_{5} ) q^{16}$$ $$+ ( 194 - 1296 \beta_{1} - 92 \beta_{2} + 108 \beta_{3} - 36 \beta_{4} + 36 \beta_{5} ) q^{17}$$ $$+ ( -14801 - 305 \beta_{1} - 80 \beta_{2} - 96 \beta_{3} + 160 \beta_{4} - 32 \beta_{5} ) q^{18}$$ $$+ ( -1296 \beta_{1} + 80 \beta_{2} - 88 \beta_{3} + 65 \beta_{4} + 88 \beta_{5} ) q^{19}$$ $$+ ( -19100 - 292 \beta_{1} + 120 \beta_{2} - 4 \beta_{3} - 368 \beta_{4} - 80 \beta_{5} ) q^{20}$$ $$+ ( 1320 \beta_{1} - 96 \beta_{2} - 12 \beta_{3} - 404 \beta_{4} + 12 \beta_{5} ) q^{21}$$ $$+ ( 25532 - 358 \beta_{1} - 77 \beta_{2} - 192 \beta_{3} - 434 \beta_{4} + 26 \beta_{5} ) q^{22}$$ $$+ ( -208 + 1200 \beta_{1} + 70 \beta_{2} - 18 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{23}$$ $$+ ( 47044 - 764 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} + 856 \beta_{4} + 136 \beta_{5} ) q^{24}$$ $$+ ( -6435 - 160 \beta_{1} + 40 \beta_{2} - 264 \beta_{3} + 88 \beta_{4} - 88 \beta_{5} ) q^{25}$$ $$+ ( -53000 + 636 \beta_{1} + 18 \beta_{2} + 472 \beta_{3} + 452 \beta_{4} + 44 \beta_{5} ) q^{26}$$ $$+ ( 1392 \beta_{1} - 48 \beta_{2} + 360 \beta_{3} - 286 \beta_{4} - 360 \beta_{5} ) q^{27}$$ $$+ ( -80248 + 2648 \beta_{1} - 256 \beta_{2} + 88 \beta_{3} - 256 \beta_{4} + 256 \beta_{5} ) q^{28}$$ $$+ ( -3394 \beta_{1} + 256 \beta_{2} + 95 \beta_{3} + 2209 \beta_{4} - 95 \beta_{5} ) q^{29}$$ $$+ ( 136600 + 1768 \beta_{1} + 280 \beta_{2} + 912 \beta_{3} - 560 \beta_{4} + 112 \beta_{5} ) q^{30}$$ $$+ ( -14656 - 9024 \beta_{1} - 408 \beta_{2} - 504 \beta_{3} + 168 \beta_{4} - 168 \beta_{5} ) q^{31}$$ $$+ ( 136136 + 4648 \beta_{1} - 488 \beta_{2} + 200 \beta_{3} - 1136 \beta_{4} - 80 \beta_{5} ) q^{32}$$ $$+ ( 9156 + 10128 \beta_{1} + 636 \beta_{2} - 396 \beta_{3} + 132 \beta_{4} - 132 \beta_{5} ) q^{33}$$ $$+ ( -167886 + 658 \beta_{1} + 720 \beta_{2} - 1184 \beta_{3} - 1440 \beta_{4} + 288 \beta_{5} ) q^{34}$$ $$+ ( 10144 \beta_{1} - 800 \beta_{2} - 528 \beta_{3} + 768 \beta_{4} + 528 \beta_{5} ) q^{35}$$ $$+ ( -208899 - 12369 \beta_{1} - 768 \beta_{2} - 81 \beta_{3} + 2048 \beta_{4} ) q^{36}$$ $$+ ( -12942 \beta_{1} + 928 \beta_{2} + 25 \beta_{3} - 7097 \beta_{4} - 25 \beta_{5} ) q^{37}$$ $$+ ( 163148 - 270 \beta_{1} + 903 \beta_{2} - 1984 \beta_{3} + 4438 \beta_{4} - 398 \beta_{5} ) q^{38}$$ $$+ ( -55120 - 11728 \beta_{1} - 826 \beta_{2} + 942 \beta_{3} - 314 \beta_{4} + 314 \beta_{5} ) q^{39}$$ $$+ ( 159320 - 21800 \beta_{1} - 680 \beta_{2} - 136 \beta_{3} - 4208 \beta_{4} - 592 \beta_{5} ) q^{40}$$ $$+ ( 85690 + 13344 \beta_{1} + 248 \beta_{2} + 2664 \beta_{3} - 888 \beta_{4} + 888 \beta_{5} ) q^{41}$$ $$+ ( -182368 + 848 \beta_{1} + 536 \beta_{2} + 1440 \beta_{3} - 1872 \beta_{4} - 880 \beta_{5} ) q^{42}$$ $$+ ( 7392 \beta_{1} - 608 \beta_{2} - 560 \beta_{3} - 275 \beta_{4} + 560 \beta_{5} ) q^{43}$$ $$+ ( -181986 + 27650 \beta_{1} + 388 \beta_{2} - 526 \beta_{3} + 1912 \beta_{4} - 1688 \beta_{5} ) q^{44}$$ $$+ ( 5118 \beta_{1} - 480 \beta_{2} - 801 \beta_{3} + 14081 \beta_{4} + 801 \beta_{5} ) q^{45}$$ $$+ ( 154632 - 1160 \beta_{1} - 120 \beta_{2} + 1072 \beta_{3} + 240 \beta_{4} - 48 \beta_{5} ) q^{46}$$ $$+ ( 260064 + 18144 \beta_{1} + 588 \beta_{2} + 2268 \beta_{3} - 756 \beta_{4} + 756 \beta_{5} ) q^{47}$$ $$+ ( 142344 + 48936 \beta_{1} + 888 \beta_{2} - 1464 \beta_{3} + 8912 \beta_{4} + 1392 \beta_{5} ) q^{48}$$ $$+ ( -84279 - 18816 \beta_{1} - 672 \beta_{2} - 2016 \beta_{3} + 672 \beta_{4} - 672 \beta_{5} ) q^{49}$$ $$+ ( -24515 - 4611 \beta_{1} - 1760 \beta_{2} - 64 \beta_{3} + 3520 \beta_{4} - 704 \beta_{5} ) q^{50}$$ $$+ ( -32496 \beta_{1} + 2736 \beta_{2} + 2904 \beta_{3} - 4406 \beta_{4} - 2904 \beta_{5} ) q^{51}$$ $$+ ( 136140 - 56588 \beta_{1} + 1064 \beta_{2} + 852 \beta_{3} - 5584 \beta_{4} + 2704 \beta_{5} ) q^{52}$$ $$+ ( 38810 \beta_{1} - 2720 \beta_{2} + 365 \beta_{3} - 16589 \beta_{4} - 365 \beta_{5} ) q^{53}$$ $$+ ( -181256 - 3404 \beta_{1} - 3674 \beta_{2} + 3648 \beta_{3} - 18276 \beta_{4} + 1588 \beta_{5} ) q^{54}$$ $$+ ( -542000 + 32080 \beta_{1} + 3050 \beta_{2} - 6846 \beta_{3} + 2282 \beta_{4} - 2282 \beta_{5} ) q^{55}$$ $$+ ( -411920 - 77200 \beta_{1} + 3760 \beta_{2} + 1968 \beta_{3} + 3360 \beta_{4} - 672 \beta_{5} ) q^{56}$$ $$+ ( -311756 - 81584 \beta_{1} - 3188 \beta_{2} - 7260 \beta_{3} + 2420 \beta_{4} - 2420 \beta_{5} ) q^{57}$$ $$+ ( 521240 - 628 \beta_{1} - 3254 \beta_{2} - 3336 \beta_{3} + 8884 \beta_{4} + 4988 \beta_{5} ) q^{58}$$ $$+ ( -57088 \beta_{1} + 3584 \beta_{2} - 3456 \beta_{3} + 9141 \beta_{4} + 3456 \beta_{5} ) q^{59}$$ $$+ ( 951240 + 122904 \beta_{1} + 3840 \beta_{2} + 1560 \beta_{3} - 18688 \beta_{4} + 2304 \beta_{5} ) q^{60}$$ $$+ ( 22638 \beta_{1} - 1056 \beta_{2} + 3927 \beta_{3} + 7881 \beta_{4} - 3927 \beta_{5} ) q^{61}$$ $$+ ( -1172896 - 3424 \beta_{1} - 3360 \beta_{2} - 7872 \beta_{3} + 6720 \beta_{4} - 1344 \beta_{5} ) q^{62}$$ $$+ ( 964592 + 60272 \beta_{1} + 4094 \beta_{2} - 4026 \beta_{3} + 1342 \beta_{4} - 1342 \beta_{5} ) q^{63}$$ $$+ ( -800432 + 139792 \beta_{1} + 496 \beta_{2} + 6224 \beta_{3} - 12896 \beta_{4} - 1312 \beta_{5} ) q^{64}$$ $$+ ( 230800 - 38880 \beta_{1} - 4360 \beta_{2} + 11880 \beta_{3} - 3960 \beta_{4} + 3960 \beta_{5} ) q^{65}$$ $$+ ( 1318356 + 2676 \beta_{1} - 2640 \beta_{2} + 9120 \beta_{3} + 5280 \beta_{4} - 1056 \beta_{5} ) q^{66}$$ $$+ ( 30192 \beta_{1} - 1968 \beta_{2} + 1320 \beta_{3} + 5001 \beta_{4} - 1320 \beta_{5} ) q^{67}$$ $$+ ( 1105206 - 171342 \beta_{1} + 2816 \beta_{2} - 3406 \beta_{3} + 22528 \beta_{4} - 8192 \beta_{5} ) q^{68}$$ $$+ ( 13944 \beta_{1} - 1440 \beta_{2} - 3108 \beta_{3} + 10180 \beta_{4} + 3108 \beta_{5} ) q^{69}$$ $$+ ( -1236800 + 19616 \beta_{1} + 5040 \beta_{2} + 8576 \beta_{3} + 28896 \beta_{4} - 1632 \beta_{5} ) q^{70}$$ $$+ ( -1276272 + 12048 \beta_{1} - 3534 \beta_{2} + 22698 \beta_{3} - 7566 \beta_{4} + 7566 \beta_{5} ) q^{71}$$ $$+ ( -1891610 - 191786 \beta_{1} - 2210 \beta_{2} - 10914 \beta_{3} + 13908 \beta_{4} + 3772 \beta_{5} ) q^{72}$$ $$+ ( 347114 + 171536 \beta_{1} + 9052 \beta_{2} + 2580 \beta_{3} - 860 \beta_{4} + 860 \beta_{5} ) q^{73}$$ $$+ ( 1300264 - 29292 \beta_{1} + 6822 \beta_{2} - 14648 \beta_{3} - 43732 \beta_{4} - 14044 \beta_{5} ) q^{74}$$ $$+ ( 50592 \beta_{1} - 3360 \beta_{2} + 1776 \beta_{3} - 27731 \beta_{4} - 1776 \beta_{5} ) q^{75}$$ $$+ ( 1540326 + 175034 \beta_{1} - 13580 \beta_{2} - 2070 \beta_{3} + 55960 \beta_{4} + 1736 \beta_{5} ) q^{76}$$ $$+ ( -73592 \beta_{1} + 3488 \beta_{2} - 12380 \beta_{3} - 33604 \beta_{4} + 12380 \beta_{5} ) q^{77}$$ $$+ ( -1575800 - 50696 \beta_{1} + 6280 \beta_{2} - 10704 \beta_{3} - 12560 \beta_{4} + 2512 \beta_{5} ) q^{78}$$ $$+ ( 2669216 - 290912 \beta_{1} - 16204 \beta_{2} + 228 \beta_{3} - 76 \beta_{4} + 76 \beta_{5} ) q^{79}$$ $$+ ( -2091600 + 162416 \beta_{1} - 3760 \beta_{2} - 17616 \beta_{3} - 37920 \beta_{4} - 7776 \beta_{5} ) q^{80}$$ $$+ ( -116195 + 116752 \beta_{1} + 8284 \beta_{2} - 9708 \beta_{3} + 3236 \beta_{4} - 3236 \beta_{5} ) q^{81}$$ $$+ ( 1779930 + 56858 \beta_{1} + 17760 \beta_{2} + 11072 \beta_{3} - 35520 \beta_{4} + 7104 \beta_{5} ) q^{82}$$ $$+ ( 106112 \beta_{1} - 9088 \beta_{2} - 10560 \beta_{3} - 8775 \beta_{4} + 10560 \beta_{5} ) q^{83}$$ $$+ ( 700304 - 208528 \beta_{1} - 6688 \beta_{2} + 5616 \beta_{3} - 62912 \beta_{4} + 3776 \beta_{5} ) q^{84}$$ $$+ ( -288828 \beta_{1} + 22240 \beta_{2} + 11266 \beta_{3} + 74014 \beta_{4} - 11266 \beta_{5} ) q^{85}$$ $$+ ( -940292 + 14778 \beta_{1} + 6435 \beta_{2} + 5248 \beta_{3} + 24110 \beta_{4} - 3910 \beta_{5} ) q^{86}$$ $$+ ( -6226320 - 135696 \beta_{1} - 882 \beta_{2} - 35946 \beta_{3} + 11982 \beta_{4} - 11982 \beta_{5} ) q^{87}$$ $$+ ( 243572 - 181324 \beta_{1} - 24908 \beta_{2} + 34788 \beta_{3} - 21896 \beta_{4} + 5096 \beta_{5} ) q^{88}$$ $$+ ( 357466 + 38928 \beta_{1} + 476 \beta_{2} + 9108 \beta_{3} - 3036 \beta_{4} + 3036 \beta_{5} ) q^{89}$$ $$+ ( -28520 + 43852 \beta_{1} - 5270 \beta_{2} + 1272 \beta_{3} + 121332 \beta_{4} + 23356 \beta_{5} ) q^{90}$$ $$+ ( 293088 \beta_{1} - 18272 \beta_{2} + 18640 \beta_{3} + 65152 \beta_{4} - 18640 \beta_{5} ) q^{91}$$ $$+ ( 109848 + 147336 \beta_{1} + 1280 \beta_{2} + 392 \beta_{3} - 21248 \beta_{4} + 4864 \beta_{5} ) q^{92}$$ $$+ ( 21024 \beta_{1} + 2304 \beta_{2} + 26640 \beta_{3} - 140560 \beta_{4} - 26640 \beta_{5} ) q^{93}$$ $$+ ( 2576784 + 229488 \beta_{1} + 15120 \beta_{2} + 15456 \beta_{3} - 30240 \beta_{4} + 6048 \beta_{5} ) q^{94}$$ $$+ ( 8090000 + 172560 \beta_{1} + 9730 \beta_{2} - 774 \beta_{3} + 258 \beta_{4} - 258 \beta_{5} ) q^{95}$$ $$+ ( 3551056 + 160528 \beta_{1} + 6256 \beta_{2} + 38736 \beta_{3} + 118944 \beta_{4} + 9184 \beta_{5} ) q^{96}$$ $$+ ( -164494 + 223600 \beta_{1} + 19460 \beta_{2} - 38004 \beta_{3} + 12668 \beta_{4} - 12668 \beta_{5} ) q^{97}$$ $$+ ( -2490039 - 54711 \beta_{1} - 13440 \beta_{2} - 16128 \beta_{3} + 26880 \beta_{4} - 5376 \beta_{5} ) q^{98}$$ $$+ ( -289152 \beta_{1} + 19584 \beta_{2} - 7488 \beta_{3} + 54531 \beta_{4} + 7488 \beta_{5} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut +\mathstrut 6q^{2}$$ $$\mathstrut +\mathstrut 116q^{4}$$ $$\mathstrut +\mathstrut 268q^{6}$$ $$\mathstrut -\mathstrut 688q^{7}$$ $$\mathstrut +\mathstrut 1512q^{8}$$ $$\mathstrut -\mathstrut 2918q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut +\mathstrut 6q^{2}$$ $$\mathstrut +\mathstrut 116q^{4}$$ $$\mathstrut +\mathstrut 268q^{6}$$ $$\mathstrut -\mathstrut 688q^{7}$$ $$\mathstrut +\mathstrut 1512q^{8}$$ $$\mathstrut -\mathstrut 2918q^{9}$$ $$\mathstrut -\mathstrut 1656q^{10}$$ $$\mathstrut -\mathstrut 4088q^{12}$$ $$\mathstrut +\mathstrut 12048q^{14}$$ $$\mathstrut +\mathstrut 17872q^{15}$$ $$\mathstrut +\mathstrut 35344q^{16}$$ $$\mathstrut +\mathstrut 1452q^{17}$$ $$\mathstrut -\mathstrut 89062q^{18}$$ $$\mathstrut -\mathstrut 114768q^{20}$$ $$\mathstrut +\mathstrut 152860q^{22}$$ $$\mathstrut -\mathstrut 1296q^{23}$$ $$\mathstrut +\mathstrut 282512q^{24}$$ $$\mathstrut -\mathstrut 39314q^{25}$$ $$\mathstrut -\mathstrut 316968q^{26}$$ $$\mathstrut -\mathstrut 480800q^{28}$$ $$\mathstrut +\mathstrut 821648q^{30}$$ $$\mathstrut -\mathstrut 89280q^{31}$$ $$\mathstrut +\mathstrut 817056q^{32}$$ $$\mathstrut +\mathstrut 53880q^{33}$$ $$\mathstrut -\mathstrut 1009108q^{34}$$ $$\mathstrut -\mathstrut 1253556q^{36}$$ $$\mathstrut +\mathstrut 974124q^{38}$$ $$\mathstrut -\mathstrut 328208q^{39}$$ $$\mathstrut +\mathstrut 954464q^{40}$$ $$\mathstrut +\mathstrut 521244q^{41}$$ $$\mathstrut -\mathstrut 1093088q^{42}$$ $$\mathstrut -\mathstrut 1096344q^{44}$$ $$\mathstrut +\mathstrut 929840q^{46}$$ $$\mathstrut +\mathstrut 1566432q^{47}$$ $$\mathstrut +\mathstrut 853920q^{48}$$ $$\mathstrut -\mathstrut 511050q^{49}$$ $$\mathstrut -\mathstrut 148626q^{50}$$ $$\mathstrut +\mathstrut 823952q^{52}$$ $$\mathstrut -\mathstrut 1077064q^{54}$$ $$\mathstrut -\mathstrut 3270256q^{55}$$ $$\mathstrut -\mathstrut 2468928q^{56}$$ $$\mathstrut -\mathstrut 1889896q^{57}$$ $$\mathstrut +\mathstrut 3130744q^{58}$$ $$\mathstrut +\mathstrut 5715168q^{60}$$ $$\mathstrut -\mathstrut 7055808q^{62}$$ $$\mathstrut +\mathstrut 5776816q^{63}$$ $$\mathstrut -\mathstrut 4792768q^{64}$$ $$\mathstrut +\mathstrut 1416480q^{65}$$ $$\mathstrut +\mathstrut 7926264q^{66}$$ $$\mathstrut +\mathstrut 6608040q^{68}$$ $$\mathstrut -\mathstrut 7406912q^{70}$$ $$\mathstrut -\mathstrut 7597104q^{71}$$ $$\mathstrut -\mathstrut 11363944q^{72}$$ $$\mathstrut +\mathstrut 2089564q^{73}$$ $$\mathstrut +\mathstrut 7744200q^{74}$$ $$\mathstrut +\mathstrut 9241288q^{76}$$ $$\mathstrut -\mathstrut 9471184q^{78}$$ $$\mathstrut +\mathstrut 16015904q^{79}$$ $$\mathstrut -\mathstrut 12600384q^{80}$$ $$\mathstrut -\mathstrut 723058q^{81}$$ $$\mathstrut +\mathstrut 10715932q^{82}$$ $$\mathstrut +\mathstrut 4220608q^{84}$$ $$\mathstrut -\mathstrut 5639076q^{86}$$ $$\mathstrut -\mathstrut 37453776q^{87}$$ $$\mathstrut +\mathstrut 1541200q^{88}$$ $$\mathstrut +\mathstrut 2169084q^{89}$$ $$\mathstrut -\mathstrut 121864q^{90}$$ $$\mathstrut +\mathstrut 669600q^{92}$$ $$\mathstrut +\mathstrut 15503712q^{94}$$ $$\mathstrut +\mathstrut 48537936q^{95}$$ $$\mathstrut +\mathstrut 21402176q^{96}$$ $$\mathstrut -\mathstrut 1088308q^{97}$$ $$\mathstrut -\mathstrut 14983242q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut -\mathstrut$$ $$3$$ $$x^{5}\mathstrut -\mathstrut$$ $$10$$ $$x^{4}\mathstrut -\mathstrut$$ $$24$$ $$x^{3}\mathstrut -\mathstrut$$ $$320$$ $$x^{2}\mathstrut -\mathstrut$$ $$3072$$ $$x\mathstrut +\mathstrut$$ $$32768$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} + 10 \nu^{3} + 24 \nu^{2} + 320 \nu + 2560$$$$)/512$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 3 \nu^{4} - 10 \nu^{3} - 24 \nu^{2} + 7872 \nu - 6656$$$$)/256$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{5} - 49 \nu^{4} + 242 \nu^{3} + 760 \nu^{2} + 3136 \nu + 26624$$$$)/512$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{5} + 7 \nu^{4} + 50 \nu^{3} - 104 \nu^{2} - 1088 \nu - 15872$$$$)/256$$ $$\beta_{5}$$ $$=$$ $$($$$$15 \nu^{5} + 51 \nu^{4} - 182 \nu^{3} + 3032 \nu^{2} - 2496 \nu - 90112$$$$)/512$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$16$$$$)/32$$ $$\nu^{2}$$ $$=$$ $$($$$$4$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$152$$$$)/32$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$4$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$68$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$28$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$206$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1000$$$$)/32$$ $$\nu^{4}$$ $$=$$ $$($$$$28$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$164$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$132$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$11$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2086$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$11816$$$$)/32$$ $$\nu^{5}$$ $$=$$ $$($$$$140$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$1076$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$20$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$319$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$7090$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$136136$$$$)/32$$

## 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
 −4.85268 + 2.90715i −4.85268 − 2.90715i 0.776001 + 5.60338i 0.776001 − 5.60338i 5.57668 + 0.949035i 5.57668 − 0.949035i
−9.70536 5.81430i 40.2163i 60.3879 + 112.860i 324.492i 233.829 390.313i −956.960 70.1132 1446.46i 569.651 1886.69 3149.31i
5.2 −9.70536 + 5.81430i 40.2163i 60.3879 112.860i 324.492i 233.829 + 390.313i −956.960 70.1132 + 1446.46i 569.651 1886.69 + 3149.31i
5.3 1.55200 11.2068i 21.9408i −123.183 34.7858i 184.916i −245.885 34.0522i 1051.96 −581.015 + 1326.49i 1705.60 −2072.30 286.989i
5.4 1.55200 + 11.2068i 21.9408i −123.183 + 34.7858i 184.916i −245.885 + 34.0522i 1051.96 −581.015 1326.49i 1705.60 −2072.30 + 286.989i
5.5 11.1534 1.89807i 76.9497i 120.795 42.3397i 338.443i 146.056 + 858.247i −438.996 1266.90 701.506i −3734.25 −642.387 3774.77i
5.6 11.1534 + 1.89807i 76.9497i 120.795 + 42.3397i 338.443i 146.056 858.247i −438.996 1266.90 + 701.506i −3734.25 −642.387 + 3774.77i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.6 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

There are no other newforms in $$S_{8}^{\mathrm{new}}(8, [\chi])$$.