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

Downloads

Learn more about

Newspace parameters

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

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: \( 2^{28}\cdot 3^{2} \)
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.

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

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

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

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:

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 \(S_{10}^{\mathrm{new}}(8, [\chi])\).