# Properties

 Label 8.11.d.b Level 8 Weight 11 Character orbit 8.d Analytic conductor 5.083 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 8.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08285802139$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} - 51 x^{5} + 30855 x^{4} - 121569 x^{3} + 12144527 x^{2} + 279415575 x + 3348211684$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{28}\cdot 3^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 5 - \beta_{1} ) q^{2} + ( 60 + \beta_{1} + \beta_{2} ) q^{3} + ( 25 - 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( -697 - 66 \beta_{1} + 15 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{6} + ( -13 - 52 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{7} + ( -5 - 28 \beta_{1} - 40 \beta_{2} + 7 \beta_{3} - 11 \beta_{4} - \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{8} + ( -4581 - 408 \beta_{1} + 150 \beta_{2} + 18 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( 5 - \beta_{1} ) q^{2} + ( 60 + \beta_{1} + \beta_{2} ) q^{3} + ( 25 - 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( \beta_{1} - \beta_{6} ) q^{5} + ( -697 - 66 \beta_{1} + 15 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{6} + ( -13 - 52 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{7} + ( -5 - 28 \beta_{1} - 40 \beta_{2} + 7 \beta_{3} - 11 \beta_{4} - \beta_{5} - 3 \beta_{6} - 4 \beta_{7} ) q^{8} + ( -4581 - 408 \beta_{1} + 150 \beta_{2} + 18 \beta_{4} ) q^{9} + ( 1092 + 4 \beta_{1} + 162 \beta_{2} - 2 \beta_{3} + 12 \beta_{4} + 6 \beta_{5} - 20 \beta_{6} - 2 \beta_{7} ) q^{10} + ( 17940 - 71 \beta_{1} - 163 \beta_{2} + 104 \beta_{3} + 4 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} + 16 \beta_{7} ) q^{11} + ( 22712 + 830 \beta_{1} + 170 \beta_{2} + 40 \beta_{3} - 86 \beta_{4} + 6 \beta_{5} + 58 \beta_{6} - 16 \beta_{7} ) q^{12} + ( 88 + 183 \beta_{1} - 168 \beta_{2} - 280 \beta_{3} + 32 \beta_{4} - 8 \beta_{5} - 15 \beta_{6} + 48 \beta_{7} ) q^{13} + ( -49752 - 248 \beta_{1} - 1356 \beta_{2} + 44 \beta_{3} + 24 \beta_{4} + 28 \beta_{5} + 88 \beta_{6} - 20 \beta_{7} ) q^{14} + ( 2159 + 8284 \beta_{1} - 587 \beta_{2} - 575 \beta_{3} + 310 \beta_{4} - 21 \beta_{5} - 11 \beta_{6} + 86 \beta_{7} ) q^{15} + ( -248082 - 1280 \beta_{1} - 2360 \beta_{2} + 86 \beta_{3} - 326 \beta_{4} + 62 \beta_{5} - 54 \beta_{6} - 24 \beta_{7} ) q^{16} + ( -42494 + 14520 \beta_{1} - 414 \beta_{2} + 832 \beta_{3} - 426 \beta_{4} - 64 \beta_{5} - 64 \beta_{6} + 128 \beta_{7} ) q^{17} + ( 384303 + 6597 \beta_{1} + 3096 \beta_{2} + 240 \beta_{3} + 168 \beta_{4} + 168 \beta_{6} - 168 \beta_{7} ) q^{18} + ( 206388 - 45903 \beta_{1} + 3605 \beta_{2} + 104 \beta_{3} + 1604 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} + 16 \beta_{7} ) q^{19} + ( 316692 + 456 \beta_{1} + 13080 \beta_{2} - 268 \beta_{3} - 1164 \beta_{4} + 76 \beta_{5} - 364 \beta_{6} - 128 \beta_{7} ) q^{20} + ( -18136 - 72380 \beta_{1} + 2344 \beta_{2} - 104 \beta_{3} - 2336 \beta_{4} - 120 \beta_{5} + 148 \beta_{6} - 304 \beta_{7} ) q^{21} + ( 192255 - 17586 \beta_{1} - 15257 \beta_{2} + 686 \beta_{3} + 2345 \beta_{4} - 128 \beta_{5} - 727 \beta_{6} - 169 \beta_{7} ) q^{22} + ( 26369 + 107044 \beta_{1} - 2341 \beta_{2} + 1743 \beta_{3} + 3178 \beta_{4} + 69 \beta_{5} - 485 \beta_{6} - 246 \beta_{7} ) q^{23} + ( -2480526 - 36280 \beta_{1} - 6016 \beta_{2} - 870 \beta_{3} - 4290 \beta_{4} - 294 \beta_{5} + 1134 \beta_{6} - 120 \beta_{7} ) q^{24} + ( -2301255 + 196080 \beta_{1} + 10660 \beta_{2} - 4160 \beta_{3} - 6260 \beta_{4} + 320 \beta_{5} + 320 \beta_{6} - 640 \beta_{7} ) q^{25} + ( 231260 + 1436 \beta_{1} + 12974 \beta_{2} - 590 \beta_{3} + 10708 \beta_{4} - 22 \beta_{5} + 244 \beta_{6} + 1202 \beta_{7} ) q^{26} + ( 3576528 - 201582 \beta_{1} - 38394 \beta_{2} - 2808 \beta_{3} + 5076 \beta_{4} + 216 \beta_{5} + 216 \beta_{6} - 432 \beta_{7} ) q^{27} + ( 2306920 + 64656 \beta_{1} - 6992 \beta_{2} + 1448 \beta_{3} - 11608 \beta_{4} - 680 \beta_{5} - 280 \beta_{6} + 1664 \beta_{7} ) q^{28} + ( -101824 - 403261 \beta_{1} + 17216 \beta_{2} + 10176 \beta_{3} - 12800 \beta_{4} + 1344 \beta_{5} - 771 \beta_{6} + 1152 \beta_{7} ) q^{29} + ( 8382280 + 42536 \beta_{1} + 8708 \beta_{2} - 9124 \beta_{3} + 21368 \beta_{4} + 12 \beta_{5} + 1208 \beta_{6} + 2332 \beta_{7} ) q^{30} + ( 127852 + 511760 \beta_{1} - 10556 \beta_{2} + 9940 \beta_{3} + 15320 \beta_{4} + 412 \beta_{5} + 5764 \beta_{6} - 1352 \beta_{7} ) q^{31} + ( -7952140 + 215328 \beta_{1} + 32976 \beta_{2} + 3492 \beta_{3} - 20004 \beta_{4} - 204 \beta_{5} - 5572 \beta_{6} + 3120 \beta_{7} ) q^{32} + ( -9105756 + 699208 \beta_{1} - 9314 \beta_{2} - 4992 \beta_{3} - 23190 \beta_{4} + 384 \beta_{5} + 384 \beta_{6} - 768 \beta_{7} ) q^{33} + ( -14445758 - 34210 \beta_{1} - 130232 \beta_{2} - 10032 \beta_{3} + 19192 \beta_{4} - 1024 \beta_{5} - 5384 \beta_{6} - 1784 \beta_{7} ) q^{34} + ( 6470224 - 935104 \beta_{1} + 21592 \beta_{2} - 2704 \beta_{3} + 30680 \beta_{4} + 208 \beta_{5} + 208 \beta_{6} - 416 \beta_{7} ) q^{35} + ( 6247731 - 363687 \beta_{1} - 15045 \beta_{2} - 9813 \beta_{3} - 20784 \beta_{4} + 432 \beta_{5} + 8016 \beta_{6} - 4992 \beta_{7} ) q^{36} + ( -241176 - 968855 \beta_{1} + 24296 \beta_{2} - 17192 \beta_{3} - 31008 \beta_{4} - 3768 \beta_{5} + 3503 \beta_{6} - 6064 \beta_{7} ) q^{37} + ( 46867399 + 30558 \beta_{1} + 116463 \beta_{2} + 41150 \beta_{3} + 7713 \beta_{4} - 128 \beta_{5} + 4641 \beta_{6} - 5537 \beta_{7} ) q^{38} + ( 220729 + 889316 \beta_{1} - 45277 \beta_{2} - 31849 \beta_{3} + 29882 \beta_{4} - 1059 \beta_{5} - 26845 \beta_{6} + 5050 \beta_{7} ) q^{39} + ( -44924432 - 569440 \beta_{1} + 342112 \beta_{2} - 14864 \beta_{3} - 12496 \beta_{4} + 2896 \beta_{5} + 5296 \beta_{6} - 11648 \beta_{7} ) q^{40} + ( 11687250 + 533968 \beta_{1} + 133100 \beta_{2} + 54080 \beta_{3} - 9308 \beta_{4} - 4160 \beta_{5} - 4160 \beta_{6} + 8320 \beta_{7} ) q^{41} + ( -72075440 - 360880 \beta_{1} - 121816 \beta_{2} + 71768 \beta_{3} - 7696 \beta_{4} + 3576 \beta_{5} + 11120 \beta_{6} - 3752 \beta_{7} ) q^{42} + ( -1410292 - 375783 \beta_{1} + 153345 \beta_{2} + 34320 \beta_{3} + 19368 \beta_{4} - 2640 \beta_{5} - 2640 \beta_{6} + 5280 \beta_{7} ) q^{43} + ( 91117976 + 288462 \beta_{1} - 283430 \beta_{2} + 32136 \beta_{3} - 2374 \beta_{4} + 6774 \beta_{5} - 24118 \beta_{6} + 6000 \beta_{7} ) q^{44} + ( 401448 + 1563243 \beta_{1} - 88536 \beta_{2} - 63336 \beta_{3} + 56544 \beta_{4} + 648 \beta_{5} - 7059 \beta_{6} + 17616 \beta_{7} ) q^{45} + ( 106651704 + 531288 \beta_{1} - 66116 \beta_{2} - 104988 \beta_{3} - 58104 \beta_{4} + 2868 \beta_{5} - 14392 \beta_{6} - 7900 \beta_{7} ) q^{46} + ( -418630 - 1783288 \beta_{1} + 14462 \beta_{2} - 70938 \beta_{3} - 48092 \beta_{4} - 3678 \beta_{5} + 67518 \beta_{6} + 7620 \beta_{7} ) q^{47} + ( -142238972 + 1805376 \beta_{1} - 623568 \beta_{2} + 47156 \beta_{3} + 40172 \beta_{4} - 1500 \beta_{5} + 26060 \beta_{6} + 9520 \beta_{7} ) q^{48} + ( -11063055 - 4427328 \beta_{1} - 517616 \beta_{2} - 26624 \beta_{3} + 124336 \beta_{4} + 2048 \beta_{5} + 2048 \beta_{6} - 4096 \beta_{7} ) q^{49} + ( -208415155 + 1274055 \beta_{1} + 385680 \beta_{2} - 218720 \beta_{3} - 95760 \beta_{4} + 5120 \beta_{5} + 27120 \beta_{6} + 8720 \beta_{7} ) q^{50} + ( -32732736 + 6715042 \beta_{1} - 479042 \beta_{2} + 31512 \beta_{3} - 229956 \beta_{4} - 2424 \beta_{5} - 2424 \beta_{6} + 4848 \beta_{7} ) q^{51} + ( 302766860 + 1336184 \beta_{1} + 480936 \beta_{2} - 20436 \beta_{3} + 169900 \beta_{4} - 12652 \beta_{5} + 14732 \beta_{6} - 12160 \beta_{7} ) q^{52} + ( 1595416 + 6425733 \beta_{1} - 157160 \beta_{2} + 93480 \beta_{3} + 198176 \beta_{4} + 11448 \beta_{5} - 11165 \beta_{6} + 8112 \beta_{7} ) q^{53} + ( 218126574 - 2489508 \beta_{1} + 13662 \beta_{2} + 222588 \beta_{3} - 100926 \beta_{4} + 3456 \beta_{5} - 17982 \beta_{6} + 42174 \beta_{7} ) q^{54} + ( -2309785 - 8975588 \beta_{1} + 425789 \beta_{2} + 244937 \beta_{3} - 306938 \beta_{4} + 7235 \beta_{5} - 103363 \beta_{6} - 41338 \beta_{7} ) q^{55} + ( -366966176 - 4298944 \beta_{1} - 1029440 \beta_{2} - 27552 \beta_{3} + 312800 \beta_{4} - 8928 \beta_{5} - 33568 \beta_{6} + 16128 \beta_{7} ) q^{56} + ( 172905348 - 6898168 \beta_{1} + 1561454 \beta_{2} - 254592 \beta_{3} + 255834 \beta_{4} + 19584 \beta_{5} + 19584 \beta_{6} - 39168 \beta_{7} ) q^{57} + ( -406552116 - 2042612 \beta_{1} + 1706086 \beta_{2} + 422266 \beta_{3} - 284892 \beta_{4} - 31854 \beta_{5} - 106812 \beta_{6} - 5766 \beta_{7} ) q^{58} + ( -118671972 + 8652361 \beta_{1} + 771017 \beta_{2} - 247936 \beta_{3} - 270848 \beta_{4} + 19072 \beta_{5} + 19072 \beta_{6} - 38144 \beta_{7} ) q^{59} + ( 654618632 - 5169968 \beta_{1} + 1071472 \beta_{2} - 87224 \beta_{3} + 388168 \beta_{4} - 21576 \beta_{5} + 57736 \beta_{6} + 18560 \beta_{7} ) q^{60} + ( 1200152 + 5054859 \beta_{1} + 73752 \beta_{2} + 343336 \beta_{3} + 106528 \beta_{4} - 17224 \beta_{5} + 59229 \beta_{6} - 133200 \beta_{7} ) q^{61} + ( 508792096 + 2544800 \beta_{1} - 1641200 \beta_{2} - 491408 \beta_{3} - 432160 \beta_{4} - 35536 \beta_{5} + 87264 \beta_{6} - 27280 \beta_{7} ) q^{62} + ( -2432463 - 9697116 \beta_{1} + 432555 \beta_{2} + 255903 \beta_{3} - 314742 \beta_{4} + 20277 \beta_{5} + 93291 \beta_{6} - 8214 \beta_{7} ) q^{63} + ( -724858664 + 4312640 \beta_{1} + 791008 \beta_{2} - 205960 \beta_{3} + 440328 \beta_{4} + 984 \beta_{5} - 150200 \beta_{6} - 12896 \beta_{7} ) q^{64} + ( -105072832 - 6770288 \beta_{1} - 6014596 \beta_{2} + 300352 \beta_{3} + 44500 \beta_{4} - 23104 \beta_{5} - 23104 \beta_{6} + 46208 \beta_{7} ) q^{65} + ( -744715972 + 5465916 \beta_{1} - 605640 \beta_{2} - 688592 \beta_{3} - 152696 \beta_{4} + 6144 \beta_{5} - 5240 \beta_{6} + 48248 \beta_{7} ) q^{66} + ( -153950668 - 1278015 \beta_{1} + 1637061 \beta_{2} - 213720 \beta_{3} + 79716 \beta_{4} + 16440 \beta_{5} + 16440 \beta_{6} - 32880 \beta_{7} ) q^{67} + ( 662247202 + 18392918 \beta_{1} - 1669470 \beta_{2} + 110066 \beta_{3} + 105072 \beta_{4} + 71440 \beta_{5} - 123920 \beta_{6} + 99712 \beta_{7} ) q^{68} + ( 1304216 + 4953388 \beta_{1} - 335720 \beta_{2} - 218456 \beta_{3} + 207904 \beta_{4} + 37176 \beta_{5} - 21188 \beta_{6} + 156848 \beta_{7} ) q^{69} + ( 964562656 - 1596544 \beta_{1} + 2103840 \beta_{2} + 870976 \beta_{3} - 12832 \beta_{4} + 3328 \beta_{5} + 67040 \beta_{6} - 43744 \beta_{7} ) q^{70} + ( 3109979 + 11761740 \beta_{1} - 921511 \beta_{2} - 940395 \beta_{3} + 462382 \beta_{4} - 22137 \beta_{5} + 44825 \beta_{6} + 174222 \beta_{7} ) q^{71} + ( -544775175 - 9205620 \beta_{1} - 957624 \beta_{2} + 349245 \beta_{3} - 682281 \beta_{4} + 6885 \beta_{5} + 221103 \beta_{6} + 48276 \beta_{7} ) q^{72} + ( 352284402 + 18279720 \beta_{1} + 9466214 \beta_{2} + 346112 \beta_{3} - 261118 \beta_{4} - 26624 \beta_{5} - 26624 \beta_{6} + 53248 \beta_{7} ) q^{73} + ( -958926044 - 4789020 \beta_{1} - 3817838 \beta_{2} + 931086 \beta_{3} + 329132 \beta_{4} + 98262 \beta_{5} + 326284 \beta_{6} - 43506 \beta_{7} ) q^{74} + ( 559738860 - 8529095 \beta_{1} - 4815695 \beta_{2} + 1151280 \beta_{3} + 196920 \beta_{4} - 88560 \beta_{5} - 88560 \beta_{6} + 177120 \beta_{7} ) q^{75} + ( 458463192 - 45826338 \beta_{1} - 2077302 \beta_{2} - 53176 \beta_{3} - 1027222 \beta_{4} - 12218 \beta_{5} + 133626 \beta_{6} - 284688 \beta_{7} ) q^{76} + ( -4807288 - 20123420 \beta_{1} - 296056 \beta_{2} - 1667016 \beta_{3} - 497312 \beta_{4} - 80280 \beta_{5} - 95084 \beta_{6} + 195984 \beta_{7} ) q^{77} + ( 896565752 + 4481560 \beta_{1} + 4483228 \beta_{2} - 961532 \beta_{3} + 1507144 \beta_{4} + 154068 \beta_{5} - 464888 \beta_{6} + 80708 \beta_{7} ) q^{78} + ( 9084134 + 36757528 \beta_{1} - 1294462 \beta_{2} - 425734 \beta_{3} + 1122172 \beta_{4} - 81154 \beta_{5} - 476542 \beta_{6} - 116740 \beta_{7} ) q^{79} + ( -792193280 + 41782400 \beta_{1} + 8040064 \beta_{2} + 116224 \beta_{3} - 1546624 \beta_{4} + 67712 \beta_{5} + 574080 \beta_{6} - 321280 \beta_{7} ) q^{80} + ( -1484813835 + 9789768 \beta_{1} - 5016546 \beta_{2} - 673920 \beta_{3} - 522774 \beta_{4} + 51840 \beta_{5} + 51840 \beta_{6} - 103680 \beta_{7} ) q^{81} + ( -457683798 - 14655186 \beta_{1} - 5200976 \beta_{2} - 420640 \beta_{3} + 1425872 \beta_{4} - 66560 \beta_{5} - 171568 \beta_{6} - 294352 \beta_{7} ) q^{82} + ( 215876156 - 59776399 \beta_{1} + 1463761 \beta_{2} + 906048 \beta_{3} + 2036192 \beta_{4} - 69696 \beta_{5} - 69696 \beta_{6} + 139392 \beta_{7} ) q^{83} + ( -344936560 + 72211360 \beta_{1} - 4062752 \beta_{2} + 665872 \beta_{3} - 2370800 \beta_{4} - 140304 \beta_{5} - 245360 \beta_{6} - 36352 \beta_{7} ) q^{84} + ( -18531208 - 73175502 \beta_{1} + 2896248 \beta_{2} + 864712 \beta_{3} - 2435168 \beta_{4} - 57448 \beta_{5} - 114090 \beta_{6} - 374160 \beta_{7} ) q^{85} + ( 378520783 + 3208078 \beta_{1} - 1079529 \beta_{2} + 353550 \beta_{3} + 999033 \beta_{4} - 42240 \beta_{5} - 14727 \beta_{6} - 280953 \beta_{7} ) q^{86} + ( 11019661 + 43858196 \beta_{1} - 597793 \beta_{2} + 1297667 \beta_{3} + 1250594 \beta_{4} - 7935 \beta_{5} + 1219487 \beta_{6} - 346238 \beta_{7} ) q^{87} + ( 703173410 - 89628664 \beta_{1} + 5836672 \beta_{2} + 38666 \beta_{3} - 1249266 \beta_{4} - 21942 \beta_{5} - 1177154 \beta_{6} + 297544 \beta_{7} ) q^{88} + ( 790659474 + 92502760 \beta_{1} + 13716086 \beta_{2} + 1174784 \beta_{3} - 2462798 \beta_{4} - 90368 \beta_{5} - 90368 \beta_{6} + 180736 \beta_{7} ) q^{89} + ( 1568180652 + 7916268 \beta_{1} + 5665494 \beta_{2} - 1646070 \beta_{3} + 2672580 \beta_{4} - 77598 \beta_{5} - 185244 \beta_{6} + 359178 \beta_{7} ) q^{90} + ( 935260720 - 75034944 \beta_{1} + 4320872 \beta_{2} - 3433456 \beta_{3} + 2329832 \beta_{4} + 264112 \beta_{5} + 264112 \beta_{6} - 528224 \beta_{7} ) q^{91} + ( -1143807112 - 115192272 \beta_{1} + 8767376 \beta_{2} - 801736 \beta_{3} - 249032 \beta_{4} + 238792 \beta_{5} + 35320 \beta_{6} + 393088 \beta_{7} ) q^{92} + ( -11514944 - 44530672 \beta_{1} + 3248576 \beta_{2} + 3747392 \beta_{3} - 1554688 \beta_{4} + 359616 \beta_{5} + 112688 \beta_{6} + 52096 \beta_{7} ) q^{93} + ( -1772640720 - 8703632 \beta_{1} - 8964008 \beta_{2} + 1742824 \beta_{3} + 1718352 \beta_{4} - 369912 \beta_{5} + 1600464 \beta_{6} + 383592 \beta_{7} ) q^{94} + ( 12722927 + 51909692 \beta_{1} - 1829227 \beta_{2} + 115137 \beta_{3} + 1753942 \beta_{4} + 265707 \beta_{5} - 1795179 \beta_{6} + 701910 \beta_{7} ) q^{95} + ( 1027438360 + 151759552 \beta_{1} - 18992288 \beta_{2} - 993352 \beta_{3} - 33208 \beta_{4} - 282216 \beta_{5} - 340600 \beta_{6} + 546208 \beta_{7} ) q^{96} + ( -1185370846 + 67181496 \beta_{1} - 36922974 \beta_{2} - 1440192 \beta_{3} - 3454698 \beta_{4} + 110784 \beta_{5} + 110784 \beta_{6} - 221568 \beta_{7} ) q^{97} + ( 4353606773 + 34636815 \beta_{1} + 1407552 \beta_{2} + 4703872 \beta_{3} - 1034304 \beta_{4} + 32768 \beta_{5} - 247872 \beta_{6} + 477248 \beta_{7} ) q^{98} + ( -1687350924 + 56970915 \beta_{1} - 14709693 \beta_{2} - 2313792 \beta_{3} - 2467296 \beta_{4} + 177984 \beta_{5} + 177984 \beta_{6} - 355968 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 42q^{2} + 480q^{3} + 212q^{4} - 5412q^{6} - 72q^{8} - 35496q^{9} + O(q^{10})$$ $$8q + 42q^{2} + 480q^{3} + 212q^{4} - 5412q^{6} - 72q^{8} - 35496q^{9} + 9120q^{10} + 143328q^{11} + 180120q^{12} - 400320q^{14} - 1987312q^{16} - 370800q^{17} + 3067758q^{18} + 1753312q^{19} + 2557440q^{20} + 1549180q^{22} - 19794288q^{24} - 18792760q^{25} + 1891680q^{26} + 28949184q^{27} + 18286080q^{28} + 67026240q^{30} - 64016928q^{32} - 74308704q^{33} - 115705388q^{34} + 53736960q^{35} + 50631516q^{36} + 375128220q^{38} - 357584640q^{40} + 92669328q^{41} - 576155520q^{42} - 10190624q^{43} + 727831512q^{44} + 851947200q^{46} - 1142760480q^{48} - 80432248q^{49} - 1669361190q^{50} - 276714816q^{51} + 2420759040q^{52} + 1749768696q^{54} - 2928529920q^{56} + 1400712864q^{57} - 3245264160q^{58} - 965642016q^{59} + 5250055680q^{60} + 4060980480q^{62} - 5804705728q^{64} - 839028480q^{65} - 5970262296q^{66} - 1225582880q^{67} + 5258111400q^{68} + 7723829760q^{70} - 4343608728q^{72} + 2800072720q^{73} - 7669373280q^{74} + 4485554400q^{75} + 3753451288q^{76} + 7176312000q^{78} - 6408629760q^{80} - 11909065176q^{81} - 3638778380q^{82} + 1853422560q^{83} - 2916218880q^{84} + 3022180476q^{86} + 5815578640q^{88} + 6162596112q^{89} + 12545940960q^{90} + 7645985280q^{91} - 8903892480q^{92} - 14182177920q^{94} + 7877525568q^{96} - 9697863536q^{97} + 34759868298q^{98} - 13646747232q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 51 x^{5} + 30855 x^{4} - 121569 x^{3} + 12144527 x^{2} + 279415575 x + 3348211684$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$($$$$-39 \nu^{7} + 640 \nu^{6} - 6467 \nu^{5} - 38788 \nu^{4} + 639715 \nu^{3} + 3220592 \nu^{2} - 518764921 \nu - 5385260908$$$$)/29360128$$ $$\beta_{3}$$ $$=$$ $$($$$$39 \nu^{7} - 640 \nu^{6} + 6467 \nu^{5} + 38788 \nu^{4} - 639715 \nu^{3} + 114219920 \nu^{2} + 107723129 \nu + 5561421676$$$$)/29360128$$ $$\beta_{4}$$ $$=$$ $$($$$$73 \nu^{7} + 640 \nu^{6} - 5907 \nu^{5} - 42820 \nu^{4} + 4083379 \nu^{3} - 64144 \nu^{2} + 831567895 \nu + 25027780436$$$$)/29360128$$ $$\beta_{5}$$ $$=$$ $$($$$$-127 \nu^{7} + 2496 \nu^{6} - 17723 \nu^{5} + 547420 \nu^{4} - 1923813 \nu^{3} + 18107440 \nu^{2} - 1496543073 \nu - 3601739852$$$$)/3670016$$ $$\beta_{6}$$ $$=$$ $$($$$$-51 \nu^{7} - 2688 \nu^{6} + 81793 \nu^{5} - 1067988 \nu^{4} + 10212255 \nu^{3} - 64144848 \nu^{2} + 299736339 \nu - 56323662812$$$$)/14680064$$ $$\beta_{7}$$ $$=$$ $$($$$$147 \nu^{7} - 2560 \nu^{6} + 1247 \nu^{5} + 270164 \nu^{4} + 7125889 \nu^{3} - 75156144 \nu^{2} + 1701809933 \nu + 21257762140$$$$)/7340032$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 7 \beta_{1} + 1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{7} + 3 \beta_{6} + \beta_{5} + 11 \beta_{4} + 11 \beta_{3} + 58 \beta_{2} + 46 \beta_{1} + 131$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$36 \beta_{7} + 9 \beta_{6} + 43 \beta_{5} - 31 \beta_{4} + 67 \beta_{3} - 592 \beta_{2} - 412 \beta_{1} - 122793$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-300 \beta_{7} + 629 \beta_{6} + 303 \beta_{5} + 1773 \beta_{4} - 159 \beta_{3} - 10902 \beta_{2} - 30996 \beta_{1} + 67607$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-7556 \beta_{7} + 2137 \beta_{6} + 2883 \beta_{5} + 64497 \beta_{4} - 18502 \beta_{3} - 92393 \beta_{2} - 253747 \beta_{1} - 35764058$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-118692 \beta_{7} - 98207 \beta_{6} - 32229 \beta_{5} + 1740089 \beta_{4} - 275559 \beta_{3} - 3734114 \beta_{2} - 48934814 \beta_{1} - 2229652087$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 16.5295 + 8.54130i 16.5295 − 8.54130i 4.83019 + 15.8950i 4.83019 − 15.8950i −8.43092 + 11.1953i −8.43092 − 11.1953i −11.4287 + 6.91461i −11.4287 − 6.91461i
−27.0589 17.0826i 203.867 440.370 + 924.473i 2088.87i −5516.42 3482.58i 25367.2i 3876.45 32537.9i −17487.2 −35683.3 + 56522.5i
3.2 −27.0589 + 17.0826i 203.867 440.370 924.473i 2088.87i −5516.42 + 3482.58i 25367.2i 3876.45 + 32537.9i −17487.2 −35683.3 56522.5i
3.3 −3.66038 31.7900i −119.281 −997.203 + 232.727i 948.910i 436.614 + 3791.94i 15458.1i 11048.5 + 30849.2i −44821.0 30165.8 3473.37i
3.4 −3.66038 + 31.7900i −119.281 −997.203 232.727i 948.910i 436.614 3791.94i 15458.1i 11048.5 30849.2i −44821.0 30165.8 + 3473.37i
3.5 22.8618 22.3905i 352.099 21.3279 1023.78i 3773.58i 8049.62 7883.68i 15618.2i −22435.3 23883.0i 64924.4 84492.5 + 86271.1i
3.6 22.8618 + 22.3905i 352.099 21.3279 + 1023.78i 3773.58i 8049.62 + 7883.68i 15618.2i −22435.3 + 23883.0i 64924.4 84492.5 86271.1i
3.7 28.8575 13.8292i −196.684 641.505 798.152i 5381.00i −5675.81 + 2719.99i 6613.83i 7474.38 31904.2i −20364.2 −74415.0 155282.i
3.8 28.8575 + 13.8292i −196.684 641.505 + 798.152i 5381.00i −5675.81 2719.99i 6613.83i 7474.38 + 31904.2i −20364.2 −74415.0 + 155282.i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.8 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.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.11.d.b 8
3.b odd 2 1 72.11.b.b 8
4.b odd 2 1 32.11.d.b 8
8.b even 2 1 32.11.d.b 8
8.d odd 2 1 inner 8.11.d.b 8
12.b even 2 1 288.11.b.b 8
16.e even 4 2 256.11.c.m 16
16.f odd 4 2 256.11.c.m 16
24.f even 2 1 72.11.b.b 8
24.h odd 2 1 288.11.b.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.11.d.b 8 1.a even 1 1 trivial
8.11.d.b 8 8.d odd 2 1 inner
32.11.d.b 8 4.b odd 2 1
32.11.d.b 8 8.b even 2 1
72.11.b.b 8 3.b odd 2 1
72.11.b.b 8 24.f even 2 1
256.11.c.m 16 16.e even 4 2
256.11.c.m 16 16.f odd 4 2
288.11.b.b 8 12.b even 2 1
288.11.b.b 8 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 240 T_{3}^{3} - 80424 T_{3}^{2} + 9637056 T_{3} + 1684044432$$ acting on $$S_{11}^{\mathrm{new}}(8, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 42 T + 776 T^{2} - 7872 T^{3} + 537600 T^{4} - 8060928 T^{5} + 813694976 T^{6} - 45097156608 T^{7} + 1099511627776 T^{8}$$
$3$ $$( 1 - 240 T + 155772 T^{2} - 32878224 T^{3} + 13106837286 T^{4} - 1941426248976 T^{5} + 543143379712572 T^{6} - 49413871702715760 T^{7} + 12157665459056928801 T^{8} )^{2}$$
$5$ $$1 - 29666120 T^{2} + 474519111223900 T^{4} -$$$$56\!\cdots\!00$$$$T^{6} +$$$$58\!\cdots\!50$$$$T^{8} -$$$$53\!\cdots\!00$$$$T^{10} +$$$$43\!\cdots\!00$$$$T^{12} -$$$$25\!\cdots\!00$$$$T^{14} +$$$$82\!\cdots\!25$$$$T^{16}$$
$7$ $$1 - 1089684872 T^{2} + 669297395540350108 T^{4} -$$$$28\!\cdots\!84$$$$T^{6} +$$$$89\!\cdots\!70$$$$T^{8} -$$$$22\!\cdots\!84$$$$T^{10} +$$$$42\!\cdots\!08$$$$T^{12} -$$$$55\!\cdots\!72$$$$T^{14} +$$$$40\!\cdots\!01$$$$T^{16}$$
$11$ $$( 1 - 71664 T + 52584995516 T^{2} - 4110626988784272 T^{3} +$$$$19\!\cdots\!78$$$$T^{4} -$$$$10\!\cdots\!72$$$$T^{5} +$$$$35\!\cdots\!16$$$$T^{6} -$$$$12\!\cdots\!64$$$$T^{7} +$$$$45\!\cdots\!01$$$$T^{8} )^{2}$$
$13$ $$1 - 459250928072 T^{2} +$$$$14\!\cdots\!28$$$$T^{4} -$$$$28\!\cdots\!44$$$$T^{6} +$$$$45\!\cdots\!10$$$$T^{8} -$$$$54\!\cdots\!44$$$$T^{10} +$$$$51\!\cdots\!28$$$$T^{12} -$$$$31\!\cdots\!72$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16}$$
$17$ $$( 1 + 185400 T + 3991264862492 T^{2} - 74312252090724216 T^{3} +$$$$10\!\cdots\!46$$$$T^{4} -$$$$14\!\cdots\!84$$$$T^{5} +$$$$16\!\cdots\!92$$$$T^{6} +$$$$15\!\cdots\!00$$$$T^{7} +$$$$16\!\cdots\!01$$$$T^{8} )^{2}$$
$19$ $$( 1 - 876656 T + 13744021629436 T^{2} - 15629661673389367568 T^{3} +$$$$11\!\cdots\!18$$$$T^{4} -$$$$95\!\cdots\!68$$$$T^{5} +$$$$51\!\cdots\!36$$$$T^{6} -$$$$20\!\cdots\!56$$$$T^{7} +$$$$14\!\cdots\!01$$$$T^{8} )^{2}$$
$23$ $$1 - 205026617368712 T^{2} +$$$$20\!\cdots\!08$$$$T^{4} -$$$$13\!\cdots\!44$$$$T^{6} +$$$$65\!\cdots\!70$$$$T^{8} -$$$$23\!\cdots\!44$$$$T^{10} +$$$$61\!\cdots\!08$$$$T^{12} -$$$$10\!\cdots\!12$$$$T^{14} +$$$$86\!\cdots\!01$$$$T^{16}$$
$29$ $$1 - 356194589173448 T^{2} +$$$$30\!\cdots\!68$$$$T^{4} -$$$$29\!\cdots\!36$$$$T^{6} +$$$$43\!\cdots\!30$$$$T^{8} -$$$$52\!\cdots\!36$$$$T^{10} +$$$$94\!\cdots\!68$$$$T^{12} -$$$$19\!\cdots\!48$$$$T^{14} +$$$$98\!\cdots\!01$$$$T^{16}$$
$31$ $$1 - 2237050600650248 T^{2} +$$$$38\!\cdots\!68$$$$T^{4} -$$$$44\!\cdots\!56$$$$T^{6} +$$$$41\!\cdots\!70$$$$T^{8} -$$$$29\!\cdots\!56$$$$T^{10} +$$$$17\!\cdots\!68$$$$T^{12} -$$$$67\!\cdots\!48$$$$T^{14} +$$$$20\!\cdots\!01$$$$T^{16}$$
$37$ $$1 - 16551679761900872 T^{2} +$$$$11\!\cdots\!28$$$$T^{4} -$$$$55\!\cdots\!04$$$$T^{6} +$$$$25\!\cdots\!90$$$$T^{8} -$$$$12\!\cdots\!04$$$$T^{10} +$$$$62\!\cdots\!28$$$$T^{12} -$$$$20\!\cdots\!72$$$$T^{14} +$$$$28\!\cdots\!01$$$$T^{16}$$
$41$ $$( 1 - 46334664 T + 39266837717699036 T^{2} -$$$$13\!\cdots\!72$$$$T^{3} +$$$$70\!\cdots\!58$$$$T^{4} -$$$$17\!\cdots\!72$$$$T^{5} +$$$$70\!\cdots\!36$$$$T^{6} -$$$$11\!\cdots\!64$$$$T^{7} +$$$$32\!\cdots\!01$$$$T^{8} )^{2}$$
$43$ $$( 1 + 5095312 T + 78132909730807996 T^{2} +$$$$71\!\cdots\!72$$$$T^{3} +$$$$24\!\cdots\!90$$$$T^{4} +$$$$15\!\cdots\!28$$$$T^{5} +$$$$36\!\cdots\!96$$$$T^{6} +$$$$51\!\cdots\!88$$$$T^{7} +$$$$21\!\cdots\!01$$$$T^{8} )^{2}$$
$47$ $$1 - 134538168632987912 T^{2} +$$$$92\!\cdots\!08$$$$T^{4} -$$$$39\!\cdots\!24$$$$T^{6} +$$$$15\!\cdots\!10$$$$T^{8} -$$$$11\!\cdots\!24$$$$T^{10} +$$$$70\!\cdots\!08$$$$T^{12} -$$$$28\!\cdots\!12$$$$T^{14} +$$$$58\!\cdots\!01$$$$T^{16}$$
$53$ $$1 - 946032264727585352 T^{2} +$$$$44\!\cdots\!68$$$$T^{4} -$$$$13\!\cdots\!24$$$$T^{6} +$$$$27\!\cdots\!10$$$$T^{8} -$$$$40\!\cdots\!24$$$$T^{10} +$$$$41\!\cdots\!68$$$$T^{12} -$$$$27\!\cdots\!52$$$$T^{14} +$$$$87\!\cdots\!01$$$$T^{16}$$
$59$ $$( 1 + 482821008 T + 1514368752837832508 T^{2} +$$$$78\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!70$$$$T^{4} +$$$$39\!\cdots\!36$$$$T^{5} +$$$$39\!\cdots\!08$$$$T^{6} +$$$$64\!\cdots\!08$$$$T^{7} +$$$$68\!\cdots\!01$$$$T^{8} )^{2}$$
$61$ $$1 - 2607734110174077128 T^{2} +$$$$31\!\cdots\!88$$$$T^{4} -$$$$28\!\cdots\!36$$$$T^{6} +$$$$22\!\cdots\!50$$$$T^{8} -$$$$14\!\cdots\!36$$$$T^{10} +$$$$82\!\cdots\!88$$$$T^{12} -$$$$34\!\cdots\!28$$$$T^{14} +$$$$67\!\cdots\!01$$$$T^{16}$$
$67$ $$( 1 + 612791440 T + 6896527760373520252 T^{2} +$$$$31\!\cdots\!84$$$$T^{3} +$$$$18\!\cdots\!06$$$$T^{4} +$$$$57\!\cdots\!16$$$$T^{5} +$$$$22\!\cdots\!52$$$$T^{6} +$$$$37\!\cdots\!60$$$$T^{7} +$$$$11\!\cdots\!01$$$$T^{8} )^{2}$$
$71$ $$1 - 17068544268164346248 T^{2} +$$$$14\!\cdots\!28$$$$T^{4} -$$$$78\!\cdots\!16$$$$T^{6} +$$$$30\!\cdots\!70$$$$T^{8} -$$$$83\!\cdots\!16$$$$T^{10} +$$$$16\!\cdots\!28$$$$T^{12} -$$$$20\!\cdots\!48$$$$T^{14} +$$$$12\!\cdots\!01$$$$T^{16}$$
$73$ $$( 1 - 1400036360 T + 8051296934129630812 T^{2} -$$$$11\!\cdots\!04$$$$T^{3} +$$$$55\!\cdots\!86$$$$T^{4} -$$$$48\!\cdots\!96$$$$T^{5} +$$$$14\!\cdots\!12$$$$T^{6} -$$$$11\!\cdots\!40$$$$T^{7} +$$$$34\!\cdots\!01$$$$T^{8} )^{2}$$
$79$ $$1 - 45980157940147603208 T^{2} +$$$$10\!\cdots\!08$$$$T^{4} -$$$$16\!\cdots\!56$$$$T^{6} +$$$$18\!\cdots\!10$$$$T^{8} -$$$$15\!\cdots\!56$$$$T^{10} +$$$$87\!\cdots\!08$$$$T^{12} -$$$$33\!\cdots\!08$$$$T^{14} +$$$$64\!\cdots\!01$$$$T^{16}$$
$83$ $$( 1 - 926711280 T + 43393122038825346812 T^{2} -$$$$33\!\cdots\!84$$$$T^{3} +$$$$94\!\cdots\!46$$$$T^{4} -$$$$52\!\cdots\!16$$$$T^{5} +$$$$10\!\cdots\!12$$$$T^{6} -$$$$34\!\cdots\!20$$$$T^{7} +$$$$57\!\cdots\!01$$$$T^{8} )^{2}$$
$89$ $$( 1 - 3081298056 T + 77020954462551268316 T^{2} -$$$$35\!\cdots\!08$$$$T^{3} +$$$$28\!\cdots\!58$$$$T^{4} -$$$$11\!\cdots\!08$$$$T^{5} +$$$$74\!\cdots\!16$$$$T^{6} -$$$$93\!\cdots\!56$$$$T^{7} +$$$$94\!\cdots\!01$$$$T^{8} )^{2}$$
$97$ $$( 1 + 4848931768 T +$$$$12\!\cdots\!96$$$$T^{2} +$$$$10\!\cdots\!08$$$$T^{3} +$$$$76\!\cdots\!90$$$$T^{4} +$$$$76\!\cdots\!92$$$$T^{5} +$$$$66\!\cdots\!96$$$$T^{6} +$$$$19\!\cdots\!32$$$$T^{7} +$$$$29\!\cdots\!01$$$$T^{8} )^{2}$$