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

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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:

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 \(S_{8}^{\mathrm{new}}(8, [\chi])\).