# Properties

 Label 8.10.b.a Level 8 Weight 10 Character orbit 8.b Analytic conductor 4.120 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8 = 2^{3}$$ Weight: $$k$$ = $$10$$ Character orbit: $$[\chi]$$ = 8.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.12028668931$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{28}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 - \beta_{1} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + ( -54 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( -2 + 6 \beta_{1} - \beta_{2} + \beta_{4} ) q^{5} + ( 585 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{6} + ( 597 + 17 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{7} + ( -434 + 51 \beta_{1} - 4 \beta_{2} + 15 \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( -4923 + 40 \beta_{1} + 24 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -2 - \beta_{1} ) q^{2} + ( \beta_{1} + \beta_{3} ) q^{3} + ( -54 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( -2 + 6 \beta_{1} - \beta_{2} + \beta_{4} ) q^{5} + ( 585 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{6} + ( 597 + 17 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{7} + ( -434 + 51 \beta_{1} - 4 \beta_{2} + 15 \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( -4923 + 40 \beta_{1} + 24 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{9} + ( 3330 - 28 \beta_{1} + 6 \beta_{2} + 86 \beta_{3} - 18 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 4 \beta_{7} ) q^{10} + ( 144 - 607 \beta_{1} - 56 \beta_{2} + \beta_{3} + 12 \beta_{5} - 8 \beta_{7} ) q^{11} + ( 6920 - 572 \beta_{1} + 6 \beta_{2} - 318 \beta_{3} - 32 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 12 \beta_{7} ) q^{12} + ( -294 + 1066 \beta_{1} - 115 \beta_{2} + 24 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} - 16 \beta_{7} ) q^{13} + ( -9120 - 492 \beta_{1} - 32 \beta_{2} - 692 \beta_{3} - 8 \beta_{4} - 52 \beta_{5} + 16 \beta_{6} + 24 \beta_{7} ) q^{14} + ( -21307 + 3985 \beta_{1} + 196 \beta_{2} - 14 \beta_{3} + 17 \beta_{4} + 65 \beta_{5} + 30 \beta_{6} - 17 \beta_{7} ) q^{15} + ( 23508 + 154 \beta_{1} + 16 \beta_{2} + 1194 \beta_{3} - 56 \beta_{4} + 74 \beta_{5} - 36 \beta_{6} + 12 \beta_{7} ) q^{16} + ( -11052 - 6792 \beta_{1} + 104 \beta_{2} + 186 \beta_{3} + 30 \beta_{4} - 96 \beta_{5} - 68 \beta_{6} - 30 \beta_{7} ) q^{17} + ( -3694 + 4609 \beta_{1} + 2120 \beta_{3} + 144 \beta_{4} - 184 \beta_{5} + 32 \beta_{6} + 16 \beta_{7} ) q^{18} + ( 3920 - 15595 \beta_{1} + 40 \beta_{2} - 267 \beta_{3} + 128 \beta_{4} + 220 \beta_{5} + 24 \beta_{7} ) q^{19} + ( 155968 - 3616 \beta_{1} - 164 \beta_{2} - 2580 \beta_{3} + 288 \beta_{4} + 364 \beta_{5} - 72 \beta_{6} - 56 \beta_{7} ) q^{20} + ( -7720 + 31456 \beta_{1} + 588 \beta_{2} + 904 \beta_{3} - 28 \beta_{4} - 472 \beta_{5} + 80 \beta_{7} ) q^{21} + ( -297395 + 1729 \beta_{1} + 603 \beta_{2} - 1114 \beta_{3} - 123 \beta_{4} - 400 \beta_{5} - 101 \beta_{6} - 160 \beta_{7} ) q^{22} + ( 418639 + 30099 \beta_{1} - 1460 \beta_{2} - 970 \beta_{3} - 141 \beta_{4} + 547 \beta_{5} - 166 \beta_{6} + 141 \beta_{7} ) q^{23} + ( -493716 - 4226 \beta_{1} + 200 \beta_{2} - 506 \beta_{3} + 72 \beta_{4} + 870 \beta_{5} + 276 \beta_{6} - 140 \beta_{7} ) q^{24} + ( -289463 - 57008 \beta_{1} - 1776 \beta_{2} + 164 \beta_{3} - 340 \beta_{4} - 1184 \beta_{5} + 472 \beta_{6} + 340 \beta_{7} ) q^{25} + ( 568518 - 2740 \beta_{1} - 814 \beta_{2} - 1950 \beta_{3} - 310 \beta_{4} - 818 \beta_{5} - 198 \beta_{6} - 308 \beta_{7} ) q^{26} + ( 23120 - 87850 \beta_{1} + 2472 \beta_{2} - 1354 \beta_{3} - 1408 \beta_{4} + 1052 \beta_{5} + 152 \beta_{7} ) q^{27} + ( 940656 + 4712 \beta_{1} + 872 \beta_{2} + 7472 \beta_{3} - 832 \beta_{4} + 1328 \beta_{5} + 544 \beta_{6} - 96 \beta_{7} ) q^{28} + ( -25166 + 93482 \beta_{1} + 985 \beta_{2} - 5312 \beta_{3} - 89 \beta_{4} - 1472 \beta_{5} + 128 \beta_{7} ) q^{29} + ( -1914528 + 24148 \beta_{1} - 4128 \beta_{2} + 16268 \beta_{3} + 1144 \beta_{4} - 1524 \beta_{5} + 272 \beta_{6} + 152 \beta_{7} ) q^{30} + ( 79812 + 85668 \beta_{1} + 1232 \beta_{2} - 1192 \beta_{3} + 84 \beta_{4} + 1444 \beta_{5} + 280 \beta_{6} - 84 \beta_{7} ) q^{31} + ( -1766904 - 12924 \beta_{1} - 1664 \beta_{2} - 22268 \beta_{3} + 1552 \beta_{4} + 1348 \beta_{5} - 1128 \beta_{6} + 56 \beta_{7} ) q^{32} + ( 22658 - 57816 \beta_{1} + 3928 \beta_{2} + 3094 \beta_{3} + 882 \beta_{4} - 448 \beta_{5} - 1564 \beta_{6} - 882 \beta_{7} ) q^{33} + ( 3463804 + 4230 \beta_{1} + 7168 \beta_{2} - 24264 \beta_{3} - 400 \beta_{4} - 1480 \beta_{5} + 480 \beta_{6} + 752 \beta_{7} ) q^{34} + ( 992 - 18280 \beta_{1} - 9744 \beta_{2} - 296 \beta_{3} + 6272 \beta_{4} + 1000 \beta_{5} - 496 \beta_{7} ) q^{35} + ( 7166726 - 34027 \beta_{1} - 1497 \beta_{2} + 24224 \beta_{3} + 128 \beta_{4} - 352 \beta_{5} - 2112 \beta_{6} + 704 \beta_{7} ) q^{36} + ( 14542 - 25730 \beta_{1} - 8761 \beta_{2} + 38664 \beta_{3} + 361 \beta_{4} + 1448 \beta_{5} - 1200 \beta_{7} ) q^{37} + ( -7959599 + 40405 \beta_{1} + 15351 \beta_{2} + 14366 \beta_{3} - 1943 \beta_{4} + 432 \beta_{5} + 55 \beta_{6} + 992 \beta_{7} ) q^{38} + ( -2179329 - 63053 \beta_{1} + 12300 \beta_{2} + 4998 \beta_{3} + 1283 \beta_{4} - 1149 \beta_{5} + 1018 \beta_{6} - 1283 \beta_{7} ) q^{39} + ( -11605672 - 104708 \beta_{1} + 3920 \beta_{2} - 52 \beta_{3} - 7152 \beta_{4} - 4468 \beta_{5} + 2344 \beta_{6} + 1640 \beta_{7} ) q^{40} + ( -335642 + 265968 \beta_{1} + 6960 \beta_{2} - 2836 \beta_{3} + 516 \beta_{4} + 4384 \beta_{5} + 1416 \beta_{6} - 516 \beta_{7} ) q^{41} + ( 15960328 - 85776 \beta_{1} - 29288 \beta_{2} + 7608 \beta_{3} + 2296 \beta_{4} + 4168 \beta_{5} + 504 \beta_{6} + 1488 \beta_{7} ) q^{42} + ( -87904 + 368653 \beta_{1} + 5456 \beta_{2} + 9165 \beta_{3} - 14080 \beta_{4} - 5832 \beta_{5} - 1232 \beta_{7} ) q^{43} + ( 14175400 + 241556 \beta_{1} - 1314 \beta_{2} - 46518 \beta_{3} + 3168 \beta_{4} - 8054 \beta_{5} + 3604 \beta_{6} + 316 \beta_{7} ) q^{44} + ( 159122 - 768238 \beta_{1} + 681 \beta_{2} - 148888 \beta_{3} + 1223 \beta_{4} + 8840 \beta_{5} + 272 \beta_{7} ) q^{45} + ( -16417248 - 356868 \beta_{1} - 35936 \beta_{2} - 98652 \beta_{3} - 7832 \beta_{4} + 11812 \beta_{5} - 2256 \beta_{6} - 1592 \beta_{7} ) q^{46} + ( 1112078 - 778938 \beta_{1} - 25256 \beta_{2} + 7036 \beta_{3} - 1866 \beta_{4} - 12634 \beta_{5} - 5164 \beta_{6} + 1866 \beta_{7} ) q^{47} + ( -27367064 + 601556 \beta_{1} + 1632 \beta_{2} + 117812 \beta_{3} + 5008 \beta_{4} - 10892 \beta_{5} - 648 \beta_{6} - 2216 \beta_{7} ) q^{48} + ( 2866729 + 736320 \beta_{1} - 23104 \beta_{2} - 24208 \beta_{3} - 4656 \beta_{4} + 10240 \beta_{5} + 7072 \beta_{6} + 4656 \beta_{7} ) q^{49} + ( 28986198 + 193915 \beta_{1} + 66560 \beta_{2} + 144176 \beta_{3} - 1440 \beta_{4} + 19760 \beta_{5} - 5440 \beta_{6} - 7328 \beta_{7} ) q^{50} + ( -312656 + 1229062 \beta_{1} + 16344 \beta_{2} + 12838 \beta_{3} + 14848 \beta_{4} - 17948 \beta_{5} + 4456 \beta_{7} ) q^{51} + ( 27541184 - 675040 \beta_{1} + 5012 \beta_{2} - 128316 \beta_{3} - 672 \beta_{4} - 12988 \beta_{5} + 2856 \beta_{6} - 3752 \beta_{7} ) q^{52} + ( 236934 - 526522 \beta_{1} + 53347 \beta_{2} + 346552 \beta_{3} - 2835 \beta_{4} + 9240 \beta_{5} + 7216 \beta_{7} ) q^{53} + ( -45437030 + 252498 \beta_{1} + 62070 \beta_{2} - 99812 \beta_{3} + 27626 \beta_{4} + 16048 \beta_{5} + 4790 \beta_{6} - 2592 \beta_{7} ) q^{54} + ( 1097129 - 911851 \beta_{1} - 28652 \beta_{2} + 4298 \beta_{3} - 4475 \beta_{4} - 17723 \beta_{5} + 3574 \beta_{6} + 4475 \beta_{7} ) q^{55} + ( -44602768 - 779240 \beta_{1} - 18848 \beta_{2} + 130424 \beta_{3} + 31520 \beta_{4} - 4872 \beta_{5} - 8944 \beta_{6} - 8176 \beta_{7} ) q^{56} + ( 16655850 + 355016 \beta_{1} - 19656 \beta_{2} - 2418 \beta_{3} + 3610 \beta_{4} + 13248 \beta_{5} - 24268 \beta_{6} - 3610 \beta_{7} ) q^{57} + ( 46981710 - 251204 \beta_{1} - 85526 \beta_{2} + 23322 \beta_{3} - 3006 \beta_{4} + 6934 \beta_{5} + 8370 \beta_{6} + 2204 \beta_{7} ) q^{58} + ( -100608 + 400189 \beta_{1} + 33408 \beta_{2} - 15811 \beta_{3} + 4224 \beta_{4} - 7808 \beta_{5} + 5376 \beta_{7} ) q^{59} + ( 64249456 + 1620584 \beta_{1} + 5224 \beta_{2} + 139824 \beta_{3} - 16192 \beta_{4} + 2608 \beta_{5} - 24544 \beta_{6} - 2656 \beta_{7} ) q^{60} + ( 24898 - 866190 \beta_{1} - 34623 \beta_{2} - 749160 \beta_{3} - 10289 \beta_{4} + 3448 \beta_{5} - 6416 \beta_{7} ) q^{61} + ( -43036544 + 20368 \beta_{1} - 93312 \beta_{2} + 138096 \beta_{3} + 8288 \beta_{4} - 8848 \beta_{5} + 1344 \beta_{6} + 224 \beta_{7} ) q^{62} + ( -28182429 + 1274935 \beta_{1} + 92508 \beta_{2} + 222 \beta_{3} + 5879 \beta_{4} + 17415 \beta_{5} + 22738 \beta_{6} - 5879 \beta_{7} ) q^{63} + ( -40149360 + 2008264 \beta_{1} + 11904 \beta_{2} - 284856 \beta_{3} - 63456 \beta_{4} + 3016 \beta_{5} + 21936 \beta_{6} + 15344 \beta_{7} ) q^{64} + ( -17985460 - 1296144 \beta_{1} + 25520 \beta_{2} + 24460 \beta_{3} - 924 \beta_{4} - 27232 \beta_{5} + 16456 \beta_{6} + 924 \beta_{7} ) q^{65} + ( 29839672 - 48108 \beta_{1} + 55296 \beta_{2} - 522744 \beta_{3} - 3056 \beta_{4} - 47864 \beta_{5} + 14112 \beta_{6} + 20368 \beta_{7} ) q^{66} + ( 794960 - 3156435 \beta_{1} - 99288 \beta_{2} - 30963 \beta_{3} - 55552 \beta_{4} + 49052 \beta_{5} - 22120 \beta_{7} ) q^{67} + ( 10990708 - 3591722 \beta_{1} - 3374 \beta_{2} + 337760 \beta_{3} + 6016 \beta_{4} + 34656 \beta_{5} + 35904 \beta_{6} + 12608 \beta_{7} ) q^{68} + ( -879608 + 4594816 \beta_{1} - 155484 \beta_{2} + 1321336 \beta_{3} + 15148 \beta_{4} - 37160 \beta_{5} - 20048 \beta_{7} ) q^{69} + ( -6885744 - 18960 \beta_{1} + 91504 \beta_{2} + 470528 \beta_{3} - 120624 \beta_{4} - 62432 \beta_{5} - 18704 \beta_{6} + 15168 \beta_{7} ) q^{70} + ( 69175533 + 3457401 \beta_{1} - 7452 \beta_{2} - 42654 \beta_{3} + 10713 \beta_{4} + 74793 \beta_{5} - 46578 \beta_{6} - 10713 \beta_{7} ) q^{71} + ( 15626338 - 7135995 \beta_{1} + 18276 \beta_{2} - 897143 \beta_{3} - 16996 \beta_{4} + 57625 \beta_{5} - 20530 \beta_{6} + 18894 \beta_{7} ) q^{72} + ( -64782312 - 2639768 \beta_{1} + 205272 \beta_{2} + 96374 \beta_{3} + 13010 \beta_{4} - 57344 \beta_{5} + 50596 \beta_{6} - 13010 \beta_{7} ) q^{73} + ( -8331502 + 67172 \beta_{1} + 31926 \beta_{2} - 212378 \beta_{3} + 13214 \beta_{4} - 62166 \beta_{5} - 54034 \beta_{6} - 22556 \beta_{7} ) q^{74} + ( 1324768 - 5458883 \beta_{1} - 233616 \beta_{2} + 26749 \beta_{3} + 145024 \beta_{4} + 96040 \beta_{5} - 12656 \beta_{7} ) q^{75} + ( -12951096 + 8041252 \beta_{1} - 82634 \beta_{2} + 57618 \beta_{3} + 93152 \beta_{4} + 54354 \beta_{5} + 12708 \beta_{6} + 24556 \beta_{7} ) q^{76} + ( -1828424 + 5582464 \beta_{1} + 118076 \beta_{2} - 1574968 \beta_{3} + 58996 \beta_{4} - 107672 \beta_{5} + 25296 \beta_{7} ) q^{77} + ( 39406880 + 1953340 \beta_{1} + 85920 \beta_{2} + 681956 \beta_{3} + 61416 \beta_{4} - 102556 \beta_{5} + 20528 \beta_{6} + 16456 \beta_{7} ) q^{78} + ( -32064430 + 3888362 \beta_{1} + 31784 \beta_{2} - 42924 \beta_{3} + 11914 \beta_{4} + 78666 \beta_{5} - 31764 \beta_{6} - 11914 \beta_{7} ) q^{79} + ( 108621136 + 11276776 \beta_{1} + 69568 \beta_{2} + 352680 \beta_{3} + 121376 \beta_{4} + 120872 \beta_{5} - 1936 \beta_{6} - 43984 \beta_{7} ) q^{80} + ( 30295035 - 5368216 \beta_{1} - 43368 \beta_{2} + 122214 \beta_{3} + 19042 \beta_{4} - 65088 \beta_{5} - 97852 \beta_{6} - 19042 \beta_{7} ) q^{81} + ( -131573780 + 596838 \beta_{1} - 283648 \beta_{2} + 715152 \beta_{3} + 44832 \beta_{4} - 51312 \beta_{5} + 8256 \beta_{6} + 2592 \beta_{7} ) q^{82} + ( 1352576 - 4564535 \beta_{1} + 628416 \beta_{2} - 26167 \beta_{3} - 198784 \beta_{4} + 22368 \beta_{5} + 61376 \beta_{7} ) q^{83} + ( -155523584 - 15319040 \beta_{1} + 111216 \beta_{2} + 20784 \beta_{3} - 147840 \beta_{4} + 94000 \beta_{5} - 76832 \beta_{6} - 39904 \beta_{7} ) q^{84} + ( -228740 + 2678212 \beta_{1} + 146270 \beta_{2} + 1609368 \beta_{3} - 58574 \beta_{4} - 25480 \beta_{5} + 12528 \beta_{7} ) q^{85} + ( 186293725 - 818047 \beta_{1} - 484469 \beta_{2} - 1385098 \beta_{3} + 235829 \beta_{4} + 22880 \beta_{5} + 11275 \beta_{6} - 80960 \beta_{7} ) q^{86} + ( 67071491 + 1948311 \beta_{1} - 74276 \beta_{2} - 131234 \beta_{3} - 49257 \beta_{4} - 16537 \beta_{5} + 159890 \beta_{6} + 49257 \beta_{7} ) q^{87} + ( 197447580 - 15596618 \beta_{1} - 117528 \beta_{2} + 1889630 \beta_{3} - 64408 \beta_{4} - 31682 \beta_{5} + 36580 \beta_{6} - 10556 \beta_{7} ) q^{88} + ( 93645000 - 3402648 \beta_{1} - 927656 \beta_{2} - 274794 \beta_{3} - 119886 \beta_{4} - 84864 \beta_{5} + 15716 \beta_{6} + 119886 \beta_{7} ) q^{89} + ( -402485042 + 2101404 \beta_{1} + 718314 \beta_{2} + 422266 \beta_{3} - 157438 \beta_{4} + 6262 \beta_{5} + 140594 \beta_{6} + 10332 \beta_{7} ) q^{90} + ( -2123104 + 7739656 \beta_{1} - 28336 \beta_{2} - 392760 \beta_{3} + 108416 \beta_{4} - 111624 \beta_{5} + 11440 \beta_{7} ) q^{91} + ( -374999728 + 18770488 \beta_{1} + 208440 \beta_{2} - 2126192 \beta_{3} - 130496 \beta_{4} + 18064 \beta_{5} + 28768 \beta_{6} - 110368 \beta_{7} ) q^{92} + ( 1999584 - 8255584 \beta_{1} + 121968 \beta_{2} - 807616 \beta_{3} - 242928 \beta_{4} + 92736 \beta_{5} - 17280 \beta_{7} ) q^{93} + ( 383696704 - 1790664 \beta_{1} + 820032 \beta_{2} - 2605176 \beta_{3} - 162992 \beta_{4} + 185992 \beta_{5} - 29856 \beta_{6} - 9200 \beta_{7} ) q^{94} + ( -180852003 - 10500807 \beta_{1} - 863836 \beta_{2} - 113646 \beta_{3} - 103959 \beta_{4} - 198231 \beta_{5} - 16082 \beta_{6} + 103959 \beta_{7} ) q^{95} + ( 530595344 + 24832776 \beta_{1} - 471040 \beta_{2} - 16632 \beta_{3} + 59936 \beta_{4} - 157816 \beta_{5} - 94416 \beta_{6} + 25200 \beta_{7} ) q^{96} + ( -5335308 + 17344952 \beta_{1} + 517288 \beta_{2} - 139702 \beta_{3} + 54478 \beta_{4} + 303136 \beta_{5} + 40732 \beta_{6} - 54478 \beta_{7} ) q^{97} + ( -381838066 - 2033337 \beta_{1} - 786432 \beta_{2} + 2240832 \beta_{3} - 7552 \beta_{4} + 264512 \beta_{5} - 74496 \beta_{6} - 102784 \beta_{7} ) q^{98} + ( -3435648 + 12684675 \beta_{1} - 551232 \beta_{2} - 125565 \beta_{3} + 84864 \beta_{4} - 148128 \beta_{5} - 66624 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 18q^{2} - 428q^{4} + 4684q^{6} + 4800q^{7} - 3384q^{8} - 39368q^{9} + O(q^{10})$$ $$8q - 18q^{2} - 428q^{4} + 4684q^{6} + 4800q^{7} - 3384q^{8} - 39368q^{9} + 26392q^{10} + 54760q^{12} - 72336q^{14} - 163136q^{15} + 185616q^{16} - 102000q^{17} - 23614q^{18} + 1245264q^{20} - 2373124q^{22} + 3412032q^{23} - 3961456q^{24} - 2423384q^{25} + 4551240q^{26} + 7509920q^{28} - 15284368q^{30} + 803584q^{31} - 14113248q^{32} + 58272q^{33} + 27757244q^{34} + 57226188q^{36} - 63661140q^{38} - 17590208q^{39} - 93063648q^{40} - 2180784q^{41} + 127541344q^{42} + 114013320q^{44} - 131840944q^{46} + 7432320q^{47} - 217917408q^{48} + 24436680q^{49} + 231784902q^{50} + 219270896q^{52} - 362934280q^{54} + 7056832q^{55} - 358503360q^{56} + 134003744q^{57} + 375425192q^{58} + 516952992q^{60} - 344291904q^{62} - 223198400q^{63} - 316815296q^{64} - 146501760q^{65} + 239713176q^{66} + 79875048q^{68} - 56202048q^{70} + 560234688q^{71} + 112273016q^{72} - 523987120q^{73} - 65773608q^{74} - 87532760q^{76} + 318117968q^{78} - 248943744q^{79} + 890441280q^{80} + 231960296q^{81} - 1051981172q^{82} - 1275608768q^{84} + 1492810428q^{86} + 540527424q^{87} + 1544767952q^{88} + 744827856q^{89} - 3218579800q^{90} - 2959012128q^{92} + 3068552352q^{94} - 1465245504q^{95} + 4296343616q^{96} - 9932784q^{97} - 3062604162q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 59 x^{6} - 313 x^{5} - 315 x^{4} - 92091 x^{3} + 1261649 x^{2} - 16074123 x + 251007534$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$4 \nu^{2} + 2 \nu + 58$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 318 \nu^{6} + 2947 \nu^{5} + 70006 \nu^{4} + 49989 \nu^{3} - 742730 \nu^{2} + 10915961 \nu - 529423278$$$$)/8912896$$ $$\beta_{4}$$ $$=$$ $$($$$$-55 \nu^{7} - 2258 \nu^{6} - 22875 \nu^{5} - 188326 \nu^{4} - 3365165 \nu^{3} + 4878490 \nu^{2} + 55540095 \nu + 120340126$$$$)/4456448$$ $$\beta_{5}$$ $$=$$ $$($$$$-545 \nu^{7} + 770 \nu^{6} - 30237 \nu^{5} + 273462 \nu^{4} + 17893 \nu^{3} + 49386870 \nu^{2} - 600770151 \nu + 7650083474$$$$)/8912896$$ $$\beta_{6}$$ $$=$$ $$($$$$-11 \nu^{7} + 22 \nu^{6} + 1377 \nu^{5} - 4078 \nu^{4} - 123529 \nu^{3} + 501650 \nu^{2} - 14934797 \nu + 79996934$$$$)/131072$$ $$\beta_{7}$$ $$=$$ $$($$$$703 \nu^{7} + 10306 \nu^{6} - 39357 \nu^{5} + 642294 \nu^{4} - 14156923 \nu^{3} - 59013834 \nu^{2} - 114709831 \nu + 4806907090$$$$)/8912896$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} - \beta_{1} - 58$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} - 2 \beta_{6} + \beta_{5} - 4 \beta_{4} - 15 \beta_{3} - 2 \beta_{2} - 57 \beta_{1} + 774$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$14 \beta_{7} - 10 \beta_{6} + 33 \beta_{5} - 12 \beta_{4} + 657 \beta_{3} + 4 \beta_{2} + 301 \beta_{1} + 9346$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-16 \beta_{7} + 88 \beta_{6} - 128 \beta_{5} - 72 \beta_{4} + 608 \beta_{3} + 99 \beta_{2} + 574 \beta_{1} + 97102$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$1066 \beta_{7} + 410 \beta_{6} + 1467 \beta_{5} - 2972 \beta_{4} - 29877 \beta_{3} - 469 \beta_{2} + 116944 \beta_{1} - 3751528$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$13522 \beta_{7} - 23454 \beta_{6} - 81689 \beta_{5} + 1428 \beta_{4} + 109815 \beta_{3} + 159882 \beta_{2} - 4238335 \beta_{1} + 74348202$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 9.73909 + 3.55976i 9.73909 − 3.55976i 3.68032 + 10.3002i 3.68032 − 10.3002i −2.43481 + 11.2224i −2.43481 − 11.2224i −10.4846 + 6.16784i −10.4846 − 6.16784i
−21.4782 7.11952i 100.481i 410.625 + 305.829i 2583.09i 715.380 2158.16i 6967.65 −6642.12 9492.10i 9586.48 −18390.4 + 55480.1i
5.2 −21.4782 + 7.11952i 100.481i 410.625 305.829i 2583.09i 715.380 + 2158.16i 6967.65 −6642.12 + 9492.10i 9586.48 −18390.4 55480.1i
5.3 −9.36065 20.6004i 150.106i −336.757 + 385.667i 292.339i −3092.24 + 1405.08i −9955.46 11097.2 + 3327.24i −2848.66 6022.31 2736.48i
5.4 −9.36065 + 20.6004i 150.106i −336.757 385.667i 292.339i −3092.24 1405.08i −9955.46 11097.2 3327.24i −2848.66 6022.31 + 2736.48i
5.5 2.86961 22.4447i 247.414i −495.531 128.815i 1417.55i 5553.14 + 709.983i 5087.57 −4313.20 + 10752.4i −41530.8 31816.6 + 4067.83i
5.6 2.86961 + 22.4447i 247.414i −495.531 + 128.815i 1417.55i 5553.14 709.983i 5087.57 −4313.20 10752.4i −41530.8 31816.6 4067.83i
5.7 18.9692 12.3357i 67.6316i 207.662 467.996i 506.862i −834.282 1282.92i 300.249 −1833.85 11439.2i 15109.0 −6252.48 9614.77i
5.8 18.9692 + 12.3357i 67.6316i 207.662 + 467.996i 506.862i −834.282 + 1282.92i 300.249 −1833.85 + 11439.2i 15109.0 −6252.48 + 9614.77i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.8 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 8.10.b.a 8
3.b odd 2 1 72.10.d.b 8
4.b odd 2 1 32.10.b.a 8
8.b even 2 1 inner 8.10.b.a 8
8.d odd 2 1 32.10.b.a 8
12.b even 2 1 288.10.d.b 8
16.e even 4 2 256.10.a.p 8
16.f odd 4 2 256.10.a.s 8
24.f even 2 1 288.10.d.b 8
24.h odd 2 1 72.10.d.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.10.b.a 8 1.a even 1 1 trivial
8.10.b.a 8 8.b even 2 1 inner
32.10.b.a 8 4.b odd 2 1
32.10.b.a 8 8.d odd 2 1
72.10.d.b 8 3.b odd 2 1
72.10.d.b 8 24.h odd 2 1
256.10.a.p 8 16.e even 4 2
256.10.a.s 8 16.f odd 4 2
288.10.d.b 8 12.b even 2 1
288.10.d.b 8 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(8, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 18 T + 376 T^{2} + 5952 T^{3} + 35840 T^{4} + 3047424 T^{5} + 98566144 T^{6} + 2415919104 T^{7} + 68719476736 T^{8}$$
$3$ $$1 - 59048 T^{2} + 1879063164 T^{4} - 39962459092824 T^{6} + 781246552565911590 T^{8} -$$$$15\!\cdots\!36$$$$T^{10} +$$$$28\!\cdots\!44$$$$T^{12} -$$$$34\!\cdots\!12$$$$T^{14} +$$$$22\!\cdots\!41$$$$T^{16}$$
$5$ $$1 - 6600808 T^{2} + 17461449664316 T^{4} - 24225043271070456600 T^{6} +$$$$30\!\cdots\!50$$$$T^{8} -$$$$92\!\cdots\!00$$$$T^{10} +$$$$25\!\cdots\!00$$$$T^{12} -$$$$36\!\cdots\!00$$$$T^{14} +$$$$21\!\cdots\!25$$$$T^{16}$$
$7$ $$( 1 - 2400 T + 77478044 T^{2} + 87749860128 T^{3} + 2890250727437094 T^{4} + 3541023369910281696 T^{5} +$$$$12\!\cdots\!56$$$$T^{6} -$$$$15\!\cdots\!00$$$$T^{7} +$$$$26\!\cdots\!01$$$$T^{8} )^{2}$$
$11$ $$1 - 10626901608 T^{2} + 58781092281363321020 T^{4} -$$$$22\!\cdots\!52$$$$T^{6} +$$$$60\!\cdots\!38$$$$T^{8} -$$$$12\!\cdots\!12$$$$T^{10} +$$$$18\!\cdots\!20$$$$T^{12} -$$$$18\!\cdots\!28$$$$T^{14} +$$$$95\!\cdots\!21$$$$T^{16}$$
$13$ $$1 - 52177437864 T^{2} +$$$$12\!\cdots\!60$$$$T^{4} -$$$$19\!\cdots\!44$$$$T^{6} +$$$$22\!\cdots\!38$$$$T^{8} -$$$$21\!\cdots\!76$$$$T^{10} +$$$$15\!\cdots\!60$$$$T^{12} -$$$$74\!\cdots\!96$$$$T^{14} +$$$$15\!\cdots\!81$$$$T^{16}$$
$17$ $$( 1 + 51000 T + 277845292252 T^{2} + 28596206694990600 T^{3} +$$$$42\!\cdots\!18$$$$T^{4} +$$$$33\!\cdots\!00$$$$T^{5} +$$$$39\!\cdots\!68$$$$T^{6} +$$$$85\!\cdots\!00$$$$T^{7} +$$$$19\!\cdots\!81$$$$T^{8} )^{2}$$
$19$ $$1 - 1485325196328 T^{2} +$$$$12\!\cdots\!88$$$$T^{4} -$$$$65\!\cdots\!60$$$$T^{6} +$$$$25\!\cdots\!58$$$$T^{8} -$$$$68\!\cdots\!60$$$$T^{10} +$$$$13\!\cdots\!28$$$$T^{12} -$$$$16\!\cdots\!88$$$$T^{14} +$$$$11\!\cdots\!61$$$$T^{16}$$
$23$ $$( 1 - 1706016 T + 3178422376156 T^{2} - 135236563190841504 T^{3} +$$$$67\!\cdots\!42$$$$T^{4} -$$$$24\!\cdots\!52$$$$T^{5} +$$$$10\!\cdots\!64$$$$T^{6} -$$$$99\!\cdots\!52$$$$T^{7} +$$$$10\!\cdots\!61$$$$T^{8} )^{2}$$
$29$ $$1 - 78392306727720 T^{2} +$$$$28\!\cdots\!40$$$$T^{4} -$$$$67\!\cdots\!00$$$$T^{6} +$$$$11\!\cdots\!38$$$$T^{8} -$$$$14\!\cdots\!00$$$$T^{10} +$$$$12\!\cdots\!40$$$$T^{12} -$$$$73\!\cdots\!20$$$$T^{14} +$$$$19\!\cdots\!41$$$$T^{16}$$
$31$ $$( 1 - 401792 T + 86180537957500 T^{2} - 37247601810974485376 T^{3} +$$$$32\!\cdots\!74$$$$T^{4} -$$$$98\!\cdots\!96$$$$T^{5} +$$$$60\!\cdots\!00$$$$T^{6} -$$$$74\!\cdots\!12$$$$T^{7} +$$$$48\!\cdots\!81$$$$T^{8} )^{2}$$
$37$ $$1 - 713593321977192 T^{2} +$$$$24\!\cdots\!96$$$$T^{4} -$$$$53\!\cdots\!04$$$$T^{6} +$$$$82\!\cdots\!86$$$$T^{8} -$$$$90\!\cdots\!16$$$$T^{10} +$$$$70\!\cdots\!36$$$$T^{12} -$$$$34\!\cdots\!88$$$$T^{14} +$$$$81\!\cdots\!81$$$$T^{16}$$
$41$ $$( 1 + 1090392 T + 1059071398836988 T^{2} +$$$$88\!\cdots\!32$$$$T^{3} +$$$$48\!\cdots\!74$$$$T^{4} +$$$$28\!\cdots\!52$$$$T^{5} +$$$$11\!\cdots\!48$$$$T^{6} +$$$$38\!\cdots\!52$$$$T^{7} +$$$$11\!\cdots\!41$$$$T^{8} )^{2}$$
$43$ $$1 - 1649449028392296 T^{2} +$$$$12\!\cdots\!08$$$$T^{4} -$$$$66\!\cdots\!36$$$$T^{6} +$$$$34\!\cdots\!62$$$$T^{8} -$$$$16\!\cdots\!64$$$$T^{10} +$$$$76\!\cdots\!08$$$$T^{12} -$$$$26\!\cdots\!04$$$$T^{14} +$$$$40\!\cdots\!01$$$$T^{16}$$
$47$ $$( 1 - 3716160 T + 1896060860372156 T^{2} -$$$$18\!\cdots\!60$$$$T^{3} +$$$$22\!\cdots\!38$$$$T^{4} -$$$$20\!\cdots\!20$$$$T^{5} +$$$$23\!\cdots\!84$$$$T^{6} -$$$$52\!\cdots\!80$$$$T^{7} +$$$$15\!\cdots\!21$$$$T^{8} )^{2}$$
$53$ $$1 - 8766485591758824 T^{2} +$$$$43\!\cdots\!44$$$$T^{4} -$$$$11\!\cdots\!80$$$$T^{6} +$$$$33\!\cdots\!70$$$$T^{8} -$$$$12\!\cdots\!20$$$$T^{10} +$$$$51\!\cdots\!24$$$$T^{12} -$$$$11\!\cdots\!56$$$$T^{14} +$$$$14\!\cdots\!41$$$$T^{16}$$
$59$ $$1 - 65444821078817512 T^{2} +$$$$19\!\cdots\!36$$$$T^{4} -$$$$32\!\cdots\!40$$$$T^{6} +$$$$34\!\cdots\!90$$$$T^{8} -$$$$24\!\cdots\!40$$$$T^{10} +$$$$10\!\cdots\!76$$$$T^{12} -$$$$27\!\cdots\!32$$$$T^{14} +$$$$31\!\cdots\!81$$$$T^{16}$$
$61$ $$1 - 33480681785208872 T^{2} +$$$$58\!\cdots\!36$$$$T^{4} -$$$$62\!\cdots\!40$$$$T^{6} +$$$$64\!\cdots\!50$$$$T^{8} -$$$$84\!\cdots\!40$$$$T^{10} +$$$$10\!\cdots\!96$$$$T^{12} -$$$$85\!\cdots\!52$$$$T^{14} +$$$$34\!\cdots\!21$$$$T^{16}$$
$67$ $$1 - 104404589351487656 T^{2} +$$$$63\!\cdots\!84$$$$T^{4} -$$$$26\!\cdots\!20$$$$T^{6} +$$$$82\!\cdots\!50$$$$T^{8} -$$$$19\!\cdots\!80$$$$T^{10} +$$$$34\!\cdots\!04$$$$T^{12} -$$$$42\!\cdots\!24$$$$T^{14} +$$$$30\!\cdots\!61$$$$T^{16}$$
$71$ $$( 1 - 280117344 T + 145831167184353692 T^{2} -$$$$30\!\cdots\!96$$$$T^{3} +$$$$97\!\cdots\!34$$$$T^{4} -$$$$13\!\cdots\!76$$$$T^{5} +$$$$30\!\cdots\!12$$$$T^{6} -$$$$26\!\cdots\!04$$$$T^{7} +$$$$44\!\cdots\!21$$$$T^{8} )^{2}$$
$73$ $$( 1 + 261993560 T + 145417231313639804 T^{2} +$$$$33\!\cdots\!76$$$$T^{3} +$$$$99\!\cdots\!54$$$$T^{4} +$$$$19\!\cdots\!88$$$$T^{5} +$$$$50\!\cdots\!76$$$$T^{6} +$$$$53\!\cdots\!20$$$$T^{7} +$$$$12\!\cdots\!61$$$$T^{8} )^{2}$$
$79$ $$( 1 + 124471872 T + 432348011474041148 T^{2} +$$$$40\!\cdots\!20$$$$T^{3} +$$$$75\!\cdots\!42$$$$T^{4} +$$$$48\!\cdots\!80$$$$T^{5} +$$$$62\!\cdots\!28$$$$T^{6} +$$$$21\!\cdots\!48$$$$T^{7} +$$$$20\!\cdots\!21$$$$T^{8} )^{2}$$
$83$ $$1 - 620814363205633576 T^{2} +$$$$25\!\cdots\!08$$$$T^{4} -$$$$71\!\cdots\!76$$$$T^{6} +$$$$15\!\cdots\!22$$$$T^{8} -$$$$24\!\cdots\!84$$$$T^{10} +$$$$30\!\cdots\!48$$$$T^{12} -$$$$26\!\cdots\!04$$$$T^{14} +$$$$14\!\cdots\!61$$$$T^{16}$$
$89$ $$( 1 - 372413928 T + 500734440582085948 T^{2} -$$$$17\!\cdots\!80$$$$T^{3} +$$$$28\!\cdots\!22$$$$T^{4} -$$$$61\!\cdots\!20$$$$T^{5} +$$$$61\!\cdots\!88$$$$T^{6} -$$$$16\!\cdots\!12$$$$T^{7} +$$$$15\!\cdots\!61$$$$T^{8} )^{2}$$
$97$ $$( 1 + 4966392 T + 2046417225380869532 T^{2} -$$$$22\!\cdots\!04$$$$T^{3} +$$$$19\!\cdots\!82$$$$T^{4} -$$$$17\!\cdots\!68$$$$T^{5} +$$$$11\!\cdots\!48$$$$T^{6} +$$$$21\!\cdots\!96$$$$T^{7} +$$$$33\!\cdots\!21$$$$T^{8} )^{2}$$