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 $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.49908020387$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 3 x^{5} - 10 x^{4} - 24 x^{3} - 320 x^{2} - 3072 x + 32768$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{15}$$ Twist minimal: yes 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{2} + 116q^{4} + 268q^{6} - 688q^{7} + 1512q^{8} - 2918q^{9} + O(q^{10})$$ $$6q + 6q^{2} + 116q^{4} + 268q^{6} - 688q^{7} + 1512q^{8} - 2918q^{9} - 1656q^{10} - 4088q^{12} + 12048q^{14} + 17872q^{15} + 35344q^{16} + 1452q^{17} - 89062q^{18} - 114768q^{20} + 152860q^{22} - 1296q^{23} + 282512q^{24} - 39314q^{25} - 316968q^{26} - 480800q^{28} + 821648q^{30} - 89280q^{31} + 817056q^{32} + 53880q^{33} - 1009108q^{34} - 1253556q^{36} + 974124q^{38} - 328208q^{39} + 954464q^{40} + 521244q^{41} - 1093088q^{42} - 1096344q^{44} + 929840q^{46} + 1566432q^{47} + 853920q^{48} - 511050q^{49} - 148626q^{50} + 823952q^{52} - 1077064q^{54} - 3270256q^{55} - 2468928q^{56} - 1889896q^{57} + 3130744q^{58} + 5715168q^{60} - 7055808q^{62} + 5776816q^{63} - 4792768q^{64} + 1416480q^{65} + 7926264q^{66} + 6608040q^{68} - 7406912q^{70} - 7597104q^{71} - 11363944q^{72} + 2089564q^{73} + 7744200q^{74} + 9241288q^{76} - 9471184q^{78} + 16015904q^{79} - 12600384q^{80} - 723058q^{81} + 10715932q^{82} + 4220608q^{84} - 5639076q^{86} - 37453776q^{87} + 1541200q^{88} + 2169084q^{89} - 121864q^{90} + 669600q^{92} + 15503712q^{94} + 48537936q^{95} + 21402176q^{96} - 1088308q^{97} - 14983242q^{98} + O(q^{100})$$

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.8.b.a 6
3.b odd 2 1 72.8.d.b 6
4.b odd 2 1 32.8.b.a 6
8.b even 2 1 inner 8.8.b.a 6
8.d odd 2 1 32.8.b.a 6
12.b even 2 1 288.8.d.b 6
16.e even 4 2 256.8.a.r 6
16.f odd 4 2 256.8.a.q 6
24.f even 2 1 288.8.d.b 6
24.h odd 2 1 72.8.d.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.b.a 6 1.a even 1 1 trivial
8.8.b.a 6 8.b even 2 1 inner
32.8.b.a 6 4.b odd 2 1
32.8.b.a 6 8.d odd 2 1
72.8.d.b 6 3.b odd 2 1
72.8.d.b 6 24.h odd 2 1
256.8.a.q 6 16.f odd 4 2
256.8.a.r 6 16.e even 4 2
288.8.d.b 6 12.b even 2 1
288.8.d.b 6 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(8, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 6 T - 40 T^{2} - 192 T^{3} - 5120 T^{4} - 98304 T^{5} + 2097152 T^{6}$$
$3$ $$1 - 5102 T^{2} + 14791383 T^{4} - 32202570276 T^{6} + 70746726356127 T^{8} - 116717395105211022 T^{10} +$$$$10\!\cdots\!09$$$$T^{12}$$
$5$ $$1 - 214718 T^{2} + 31745660775 T^{4} - 2880459231122500 T^{6} +$$$$19\!\cdots\!75$$$$T^{8} -$$$$79\!\cdots\!50$$$$T^{10} +$$$$22\!\cdots\!25$$$$T^{12}$$
$7$ $$( 1 + 344 T + 1422245 T^{2} + 124668368 T^{3} + 1171279914035 T^{4} + 233308737060056 T^{5} + 558545864083284007 T^{6} )^{2}$$
$11$ $$1 - 64629022 T^{2} + 2363172524483783 T^{4} -$$$$55\!\cdots\!04$$$$T^{6} +$$$$89\!\cdots\!03$$$$T^{8} -$$$$93\!\cdots\!82$$$$T^{10} +$$$$54\!\cdots\!21$$$$T^{12}$$
$13$ $$1 - 205410958 T^{2} + 21942186159527543 T^{4} -$$$$16\!\cdots\!64$$$$T^{6} +$$$$86\!\cdots\!27$$$$T^{8} -$$$$31\!\cdots\!18$$$$T^{10} +$$$$61\!\cdots\!69$$$$T^{12}$$
$17$ $$( 1 - 726 T + 323301023 T^{2} - 9708009717300 T^{3} + 132662912757362479 T^{4} -$$$$12\!\cdots\!54$$$$T^{5} +$$$$69\!\cdots\!17$$$$T^{6} )^{2}$$
$19$ $$1 - 2002416334 T^{2} + 2872909317854295863 T^{4} -$$$$28\!\cdots\!20$$$$T^{6} +$$$$22\!\cdots\!23$$$$T^{8} -$$$$12\!\cdots\!94$$$$T^{10} +$$$$51\!\cdots\!61$$$$T^{12}$$
$23$ $$( 1 + 648 T + 9708521717 T^{2} + 6547475963760 T^{3} + 33055821794793732499 T^{4} +$$$$75\!\cdots\!32$$$$T^{5} +$$$$39\!\cdots\!23$$$$T^{6} )^{2}$$
$29$ $$1 - 47836636078 T^{2} +$$$$14\!\cdots\!79$$$$T^{4} -$$$$30\!\cdots\!44$$$$T^{6} +$$$$41\!\cdots\!99$$$$T^{8} -$$$$42\!\cdots\!58$$$$T^{10} +$$$$26\!\cdots\!41$$$$T^{12}$$
$31$ $$( 1 + 44640 T + 48354349725 T^{2} + 4324137771289408 T^{3} +$$$$13\!\cdots\!75$$$$T^{4} +$$$$33\!\cdots\!40$$$$T^{5} +$$$$20\!\cdots\!31$$$$T^{6} )^{2}$$
$37$ $$1 - 78937168126 T^{2} +$$$$11\!\cdots\!51$$$$T^{4} -$$$$18\!\cdots\!48$$$$T^{6} +$$$$10\!\cdots\!39$$$$T^{8} -$$$$64\!\cdots\!46$$$$T^{10} +$$$$73\!\cdots\!69$$$$T^{12}$$
$41$ $$( 1 - 260622 T + 351195126263 T^{2} - 83834535574878564 T^{3} +$$$$68\!\cdots\!03$$$$T^{4} -$$$$98\!\cdots\!42$$$$T^{5} +$$$$73\!\cdots\!41$$$$T^{6} )^{2}$$
$43$ $$1 - 1505999929054 T^{2} +$$$$97\!\cdots\!71$$$$T^{4} -$$$$34\!\cdots\!32$$$$T^{6} +$$$$71\!\cdots\!79$$$$T^{8} -$$$$82\!\cdots\!54$$$$T^{10} +$$$$40\!\cdots\!49$$$$T^{12}$$
$47$ $$( 1 - 783216 T + 1470820452333 T^{2} - 778339576797138720 T^{3} +$$$$74\!\cdots\!79$$$$T^{4} -$$$$20\!\cdots\!04$$$$T^{5} +$$$$13\!\cdots\!47$$$$T^{6} )^{2}$$
$53$ $$1 - 3131635055902 T^{2} +$$$$64\!\cdots\!63$$$$T^{4} -$$$$86\!\cdots\!56$$$$T^{6} +$$$$89\!\cdots\!47$$$$T^{8} -$$$$59\!\cdots\!22$$$$T^{10} +$$$$26\!\cdots\!09$$$$T^{12}$$
$59$ $$1 - 8312323804862 T^{2} +$$$$38\!\cdots\!03$$$$T^{4} -$$$$11\!\cdots\!64$$$$T^{6} +$$$$23\!\cdots\!83$$$$T^{8} -$$$$31\!\cdots\!02$$$$T^{10} +$$$$23\!\cdots\!81$$$$T^{12}$$
$61$ $$1 - 14350504934382 T^{2} +$$$$96\!\cdots\!03$$$$T^{4} -$$$$38\!\cdots\!24$$$$T^{6} +$$$$95\!\cdots\!23$$$$T^{8} -$$$$13\!\cdots\!42$$$$T^{10} +$$$$96\!\cdots\!21$$$$T^{12}$$
$67$ $$1 - 35072892237678 T^{2} +$$$$51\!\cdots\!43$$$$T^{4} -$$$$41\!\cdots\!64$$$$T^{6} +$$$$19\!\cdots\!47$$$$T^{8} -$$$$47\!\cdots\!98$$$$T^{10} +$$$$49\!\cdots\!89$$$$T^{12}$$
$71$ $$( 1 + 3798552 T + 17820666583269 T^{2} + 72937737977373055056 T^{3} +$$$$16\!\cdots\!79$$$$T^{4} +$$$$31\!\cdots\!12$$$$T^{5} +$$$$75\!\cdots\!71$$$$T^{6} )^{2}$$
$73$ $$( 1 - 1044782 T + 23207418111255 T^{2} - 22868192285089705636 T^{3} +$$$$25\!\cdots\!35$$$$T^{4} -$$$$12\!\cdots\!38$$$$T^{5} +$$$$13\!\cdots\!73$$$$T^{6} )^{2}$$
$79$ $$( 1 - 8007952 T + 49828330384013 T^{2} -$$$$25\!\cdots\!96$$$$T^{3} +$$$$95\!\cdots\!67$$$$T^{4} -$$$$29\!\cdots\!12$$$$T^{5} +$$$$70\!\cdots\!79$$$$T^{6} )^{2}$$
$83$ $$1 - 124932236904014 T^{2} +$$$$70\!\cdots\!51$$$$T^{4} -$$$$23\!\cdots\!92$$$$T^{6} +$$$$51\!\cdots\!79$$$$T^{8} -$$$$67\!\cdots\!74$$$$T^{10} +$$$$39\!\cdots\!89$$$$T^{12}$$
$89$ $$( 1 - 1084542 T + 130289056617383 T^{2} - 94766843887733093316 T^{3} +$$$$57\!\cdots\!07$$$$T^{4} -$$$$21\!\cdots\!22$$$$T^{5} +$$$$86\!\cdots\!89$$$$T^{6} )^{2}$$
$97$ $$( 1 + 544154 T + 184934786492783 T^{2} + 86271378317799707180 T^{3} +$$$$14\!\cdots\!79$$$$T^{4} +$$$$35\!\cdots\!26$$$$T^{5} +$$$$52\!\cdots\!97$$$$T^{6} )^{2}$$