Properties

Label 8.9.d.b
Level 8
Weight 9
Character orbit 8.d
Analytic conductor 3.259
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(3.25902888049\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -2 + \beta_{1} ) q^{2} \) \( + ( -6 - \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -99 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 1 + 3 \beta_{1} + \beta_{3} - \beta_{5} ) q^{5} \) \( + ( -216 - 7 \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{6} \) \( + ( 14 + 34 \beta_{1} + 6 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} ) q^{7} \) \( + ( -1170 - 112 \beta_{1} - 14 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} ) q^{8} \) \( + ( 2787 + 192 \beta_{1} - 24 \beta_{2} - 24 \beta_{3} - 24 \beta_{4} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 + \beta_{1} ) q^{2} \) \( + ( -6 - \beta_{1} - \beta_{2} ) q^{3} \) \( + ( -99 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} \) \( + ( 1 + 3 \beta_{1} + \beta_{3} - \beta_{5} ) q^{5} \) \( + ( -216 - 7 \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{6} \) \( + ( 14 + 34 \beta_{1} + 6 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} ) q^{7} \) \( + ( -1170 - 112 \beta_{1} - 14 \beta_{2} - 4 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} ) q^{8} \) \( + ( 2787 + 192 \beta_{1} - 24 \beta_{2} - 24 \beta_{3} - 24 \beta_{4} ) q^{9} \) \( + ( -704 + 10 \beta_{1} + 92 \beta_{2} + 6 \beta_{3} + 38 \beta_{4} + 4 \beta_{5} ) q^{10} \) \( + ( 7866 + 95 \beta_{1} - 49 \beta_{2} - 16 \beta_{3} - 16 \beta_{4} ) q^{11} \) \( + ( -4722 - 246 \beta_{1} + 206 \beta_{2} - 38 \beta_{3} + 20 \beta_{4} - 8 \beta_{5} ) q^{12} \) \( + ( -401 - 1203 \beta_{1} - 17 \beta_{3} - 128 \beta_{4} + 17 \beta_{5} ) q^{13} \) \( + ( -8512 - 68 \beta_{1} - 280 \beta_{2} + 100 \beta_{3} + 100 \beta_{4} + 24 \beta_{5} ) q^{14} \) \( + ( 1014 + 2938 \beta_{1} + 142 \beta_{3} + 152 \beta_{4} - 38 \beta_{5} ) q^{15} \) \( + ( 3092 - 1224 \beta_{1} - 452 \beta_{2} - 144 \beta_{3} - 148 \beta_{4} - 56 \beta_{5} ) q^{16} \) \( + ( -33150 + 1216 \beta_{1} - 8 \beta_{2} - 136 \beta_{3} - 136 \beta_{4} ) q^{17} \) \( + ( 45786 + 3339 \beta_{1} + 336 \beta_{2} + 408 \beta_{3} + 120 \beta_{4} - 48 \beta_{5} ) q^{18} \) \( + ( -17542 - 5249 \beta_{1} + 367 \beta_{2} + 624 \beta_{3} + 624 \beta_{4} ) q^{19} \) \( + ( -36584 - 1328 \beta_{1} - 184 \beta_{2} - 480 \beta_{3} - 920 \beta_{4} + 112 \beta_{5} ) q^{20} \) \( + ( -4620 - 12836 \beta_{1} - 908 \beta_{3} + 128 \beta_{4} - 116 \beta_{5} ) q^{21} \) \( + ( 10984 + 8201 \beta_{1} + 158 \beta_{2} + 305 \beta_{3} + 245 \beta_{4} - 98 \beta_{5} ) q^{22} \) \( + ( 4474 + 12950 \beta_{1} + 162 \beta_{3} + 808 \beta_{4} + 310 \beta_{5} ) q^{23} \) \( + ( 166452 - 3440 \beta_{1} + 1324 \beta_{2} - 840 \beta_{3} - 1452 \beta_{4} + 312 \beta_{5} ) q^{24} \) \( + ( -73247 + 6656 \beta_{1} + 1904 \beta_{2} - 528 \beta_{3} - 528 \beta_{4} ) q^{25} \) \( + ( 296128 + 2518 \beta_{1} - 3356 \beta_{2} - 230 \beta_{3} + 762 \beta_{4} + 188 \beta_{5} ) q^{26} \) \( + ( 68724 - 4098 \beta_{1} - 210 \beta_{2} + 432 \beta_{3} + 432 \beta_{4} ) q^{27} \) \( + ( -436912 - 13024 \beta_{1} - 3472 \beta_{2} - 256 \beta_{3} + 752 \beta_{4} - 608 \beta_{5} ) q^{28} \) \( + ( -1381 - 5167 \beta_{1} + 667 \beta_{3} - 2048 \beta_{4} + 357 \beta_{5} ) q^{29} \) \( + ( -724032 - 5620 \beta_{1} + 5832 \beta_{2} + 1940 \beta_{3} + 916 \beta_{4} + 56 \beta_{5} ) q^{30} \) \( + ( 1432 + 3880 \beta_{1} + 1784 \beta_{3} - 672 \beta_{4} - 1368 \beta_{5} ) q^{31} \) \( + ( 540344 + 9904 \beta_{1} + 4968 \beta_{2} + 832 \beta_{3} + 3816 \beta_{4} - 784 \beta_{5} ) q^{32} \) \( + ( 290820 - 6976 \beta_{1} - 11512 \beta_{2} - 504 \beta_{3} - 504 \beta_{4} ) q^{33} \) \( + ( 386524 - 29894 \beta_{1} + 2160 \beta_{2} + 2184 \beta_{3} + 40 \beta_{4} - 16 \beta_{5} ) q^{34} \) \( + ( 344064 + 38528 \beta_{1} - 3808 \beta_{2} - 4704 \beta_{3} - 4704 \beta_{4} ) q^{35} \) \( + ( -1220457 + 35415 \beta_{1} + 4227 \beta_{2} + 2211 \beta_{3} + 2784 \beta_{4} + 1344 \beta_{5} ) q^{36} \) \( + ( 27129 + 77291 \beta_{1} + 4217 \beta_{3} + 2176 \beta_{4} - 121 \beta_{5} ) q^{37} \) \( + ( -1358872 - 32151 \beta_{1} - 9250 \beta_{2} - 10351 \beta_{3} - 1835 \beta_{4} + 734 \beta_{5} ) q^{38} \) \( + ( -41046 - 119578 \beta_{1} - 6766 \beta_{3} - 6680 \beta_{4} + 3206 \beta_{5} ) q^{39} \) \( + ( 2116048 - 16544 \beta_{1} - 11664 \beta_{2} + 5696 \beta_{3} + 2736 \beta_{4} + 288 \beta_{5} ) q^{40} \) \( + ( 620610 - 20992 \beta_{1} + 31568 \beta_{2} + 5840 \beta_{3} + 5840 \beta_{4} ) q^{41} \) \( + ( 3177216 + 26120 \beta_{1} + 10416 \beta_{2} - 14536 \beta_{3} - 8264 \beta_{4} - 1840 \beta_{5} ) q^{42} \) \( + ( -1645126 + 12831 \beta_{1} + 6783 \beta_{2} - 672 \beta_{3} - 672 \beta_{4} ) q^{43} \) \( + ( -1583858 - 3126 \beta_{1} + 15822 \beta_{2} + 6426 \beta_{3} + 2516 \beta_{4} + 632 \beta_{5} ) q^{44} \) \( + ( 45891 + 140745 \beta_{1} - 573 \beta_{3} + 19584 \beta_{4} - 2499 \beta_{5} ) q^{45} \) \( + ( -3208640 - 32812 \beta_{1} - 16264 \beta_{2} + 6028 \beta_{3} - 15476 \beta_{4} - 1912 \beta_{5} ) q^{46} \) \( + ( -53020 - 148036 \beta_{1} - 8972 \beta_{3} + 16 \beta_{4} - 2052 \beta_{5} ) q^{47} \) \( + ( 3511224 + 200080 \beta_{1} - 30808 \beta_{2} + 4064 \beta_{3} - 8312 \beta_{4} + 3248 \beta_{5} ) q^{48} \) \( + ( -2148671 - 189952 \beta_{1} - 56896 \beta_{2} + 14784 \beta_{3} + 14784 \beta_{4} ) q^{49} \) \( + ( 1830910 - 58671 \beta_{1} + 12256 \beta_{2} + 6544 \beta_{3} - 9520 \beta_{4} + 3808 \beta_{5} ) q^{50} \) \( + ( -731916 - 35266 \beta_{1} + 14414 \beta_{2} + 5520 \beta_{3} + 5520 \beta_{4} ) q^{51} \) \( + ( -2784920 + 273968 \beta_{1} - 13256 \beta_{2} + 2528 \beta_{3} + 792 \beta_{4} - 9072 \beta_{5} ) q^{52} \) \( + ( -2715 - 19409 \beta_{1} + 4837 \beta_{3} - 17536 \beta_{4} + 6427 \beta_{5} ) q^{53} \) \( + ( -1208304 + 58146 \beta_{1} - 7332 \beta_{2} - 6702 \beta_{3} + 1050 \beta_{4} - 420 \beta_{5} ) q^{54} \) \( + ( 70262 + 209402 \beta_{1} + 10510 \beta_{3} + 18072 \beta_{4} - 9126 \beta_{5} ) q^{55} \) \( + ( 1594208 - 419264 \beta_{1} + 51744 \beta_{2} - 11136 \beta_{3} + 22688 \beta_{4} - 7744 \beta_{5} ) q^{56} \) \( + ( 1300356 + 270016 \beta_{1} + 113416 \beta_{2} - 17400 \beta_{3} - 17400 \beta_{4} ) q^{57} \) \( + ( 1254848 + 11790 \beta_{1} - 59468 \beta_{2} + 11170 \beta_{3} + 20226 \beta_{4} + 4716 \beta_{5} ) q^{58} \) \( + ( 4246842 + 97247 \beta_{1} - 36385 \beta_{2} - 14848 \beta_{3} - 14848 \beta_{4} ) q^{59} \) \( + ( -1207536 - 757088 \beta_{1} - 24144 \beta_{2} - 22784 \beta_{3} - 19408 \beta_{4} + 13600 \beta_{5} ) q^{60} \) \( + ( -194413 - 567879 \beta_{1} - 15853 \beta_{3} - 39040 \beta_{4} + 493 \beta_{5} ) q^{61} \) \( + ( -910592 + 16560 \beta_{1} + 117280 \beta_{2} + 13776 \beta_{3} + 63952 \beta_{4} + 7648 \beta_{5} ) q^{62} \) \( + ( 190890 + 521190 \beta_{1} + 26514 \beta_{3} - 13848 \beta_{4} + 24966 \beta_{5} ) q^{63} \) \( + ( -5772080 + 491680 \beta_{1} + 103152 \beta_{2} - 28160 \beta_{3} - 36496 \beta_{4} + 5536 \beta_{5} ) q^{64} \) \( + ( 4703232 + 431104 \beta_{1} - 220784 \beta_{2} - 72432 \beta_{3} - 72432 \beta_{4} ) q^{65} \) \( + ( -2012904 + 291404 \beta_{1} - 14960 \beta_{2} + 19576 \beta_{3} + 57560 \beta_{4} - 23024 \beta_{5} ) q^{66} \) \( + ( 7867898 - 385473 \beta_{1} - 23889 \beta_{2} + 40176 \beta_{3} + 40176 \beta_{4} ) q^{67} \) \( + ( -1390118 + 381306 \beta_{1} - 52126 \beta_{2} - 32318 \beta_{3} + 13216 \beta_{4} + 8640 \beta_{5} ) q^{68} \) \( + ( -229188 - 625100 \beta_{1} - 37316 \beta_{3} + 19328 \beta_{4} - 25148 \beta_{5} ) q^{69} \) \( + ( 9582720 + 453152 \beta_{1} + 67648 \beta_{2} + 79072 \beta_{3} + 19040 \beta_{4} - 7616 \beta_{5} ) q^{70} \) \( + ( 258174 + 723506 \beta_{1} + 69686 \beta_{3} - 5192 \beta_{4} - 18670 \beta_{5} ) q^{71} \) \( + ( -10000950 - 1369296 \beta_{1} - 108714 \beta_{2} + 4212 \beta_{3} - 63726 \beta_{4} + 4620 \beta_{5} ) q^{72} \) \( + ( -9268414 + 119488 \beta_{1} + 190120 \beta_{2} + 7848 \beta_{3} + 7848 \beta_{4} ) q^{73} \) \( + ( -19088064 - 155078 \beta_{1} + 49788 \beta_{2} + 64342 \beta_{3} + 25718 \beta_{4} + 4324 \beta_{5} ) q^{74} \) \( + ( -21193542 - 552289 \beta_{1} + 58559 \beta_{2} + 67872 \beta_{3} + 67872 \beta_{4} ) q^{75} \) \( + ( 24928590 - 1153398 \beta_{1} - 498 \beta_{2} - 10790 \beta_{3} - 67244 \beta_{4} - 37000 \beta_{5} ) q^{76} \) \( + ( 61684 + 134876 \beta_{1} + 24948 \beta_{3} - 54656 \beta_{4} + 25228 \beta_{5} ) q^{77} \) \( + ( 29404224 + 204340 \beta_{1} - 395592 \beta_{2} - 79316 \beta_{3} - 87508 \beta_{4} - 6584 \beta_{5} ) q^{78} \) \( + ( -33028 - 1948 \beta_{1} - 86420 \beta_{3} + 147312 \beta_{4} - 10716 \beta_{5} ) q^{79} \) \( + ( -21812768 + 1957056 \beta_{1} - 111968 \beta_{2} - 9984 \beta_{3} + 83104 \beta_{4} - 17984 \beta_{5} ) q^{80} \) \( + ( -13548987 - 1067328 \beta_{1} + 141192 \beta_{2} + 134280 \beta_{3} + 134280 \beta_{4} ) q^{81} \) \( + ( -7872836 + 512018 \beta_{1} - 30304 \beta_{2} - 125008 \beta_{3} - 157840 \beta_{4} + 63136 \beta_{5} ) q^{82} \) \( + ( 24373626 + 956223 \beta_{1} + 58815 \beta_{2} - 99712 \beta_{3} - 99712 \beta_{4} ) q^{83} \) \( + ( 37198560 + 3612736 \beta_{1} + 355488 \beta_{2} + 60544 \beta_{3} - 71392 \beta_{4} + 11968 \beta_{5} ) q^{84} \) \( + ( 80546 + 272358 \beta_{1} - 71134 \beta_{3} + 91520 \beta_{4} + 40414 \beta_{5} ) q^{85} \) \( + ( 6428072 - 1622215 \beta_{1} + 24318 \beta_{2} + 3969 \beta_{3} - 33915 \beta_{4} + 13566 \beta_{5} ) q^{86} \) \( + ( -884238 - 2496802 \beta_{1} - 165382 \beta_{3} - 31736 \beta_{4} + 9470 \beta_{5} ) q^{87} \) \( + ( -8435404 - 1721456 \beta_{1} - 115668 \beta_{2} - 49736 \beta_{3} - 53164 \beta_{4} + 38200 \beta_{5} ) q^{88} \) \( + ( 23653314 - 546112 \beta_{1} - 403480 \beta_{2} + 15848 \beta_{3} + 15848 \beta_{4} ) q^{89} \) \( + ( -34606656 - 303330 \beta_{1} + 497940 \beta_{2} - 11502 \beta_{3} - 154254 \beta_{4} - 35316 \beta_{5} ) q^{90} \) \( + ( -45588480 + 370048 \beta_{1} + 122080 \beta_{2} - 27552 \beta_{3} - 27552 \beta_{4} ) q^{91} \) \( + ( 15314800 - 3027360 \beta_{1} - 5680 \beta_{2} + 133376 \beta_{3} + 369488 \beta_{4} + 14304 \beta_{5} ) q^{92} \) \( + ( 1204752 + 3518000 \beta_{1} + 153104 \beta_{3} + 222208 \beta_{4} - 56848 \beta_{5} ) q^{93} \) \( + ( 36662912 + 317832 \beta_{1} + 166960 \beta_{2} - 153032 \beta_{3} - 43464 \beta_{4} - 13872 \beta_{5} ) q^{94} \) \( + ( -876106 - 2734918 \beta_{1} + 107854 \beta_{3} - 470120 \beta_{4} - 1254 \beta_{5} ) q^{95} \) \( + ( -40536624 + 3203616 \beta_{1} - 249104 \beta_{2} + 325760 \beta_{3} + 202480 \beta_{4} - 43360 \beta_{5} ) q^{96} \) \( + ( 20980994 + 935616 \beta_{1} + 546168 \beta_{2} - 43272 \beta_{3} - 43272 \beta_{4} ) q^{97} \) \( + ( -43683458 - 2560383 \beta_{1} - 350336 \beta_{2} - 179648 \beta_{3} + 284480 \beta_{4} - 113792 \beta_{5} ) q^{98} \) \( + ( 50120430 + 968061 \beta_{1} - 383235 \beta_{2} - 150144 \beta_{3} - 150144 \beta_{4} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 36q^{3} \) \(\mathstrut -\mathstrut 588q^{4} \) \(\mathstrut -\mathstrut 1284q^{6} \) \(\mathstrut -\mathstrut 6824q^{8} \) \(\mathstrut +\mathstrut 16338q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 36q^{3} \) \(\mathstrut -\mathstrut 588q^{4} \) \(\mathstrut -\mathstrut 1284q^{6} \) \(\mathstrut -\mathstrut 6824q^{8} \) \(\mathstrut +\mathstrut 16338q^{9} \) \(\mathstrut -\mathstrut 4080q^{10} \) \(\mathstrut +\mathstrut 46940q^{11} \) \(\mathstrut -\mathstrut 27336q^{12} \) \(\mathstrut -\mathstrut 51744q^{14} \) \(\mathstrut +\mathstrut 20496q^{16} \) \(\mathstrut -\mathstrut 201076q^{17} \) \(\mathstrut +\mathstrut 267990q^{18} \) \(\mathstrut -\mathstrut 95268q^{19} \) \(\mathstrut -\mathstrut 216480q^{20} \) \(\mathstrut +\mathstrut 49404q^{22} \) \(\mathstrut +\mathstrut 1009296q^{24} \) \(\mathstrut -\mathstrut 447930q^{25} \) \(\mathstrut +\mathstrut 1765104q^{26} \) \(\mathstrut +\mathstrut 419256q^{27} \) \(\mathstrut -\mathstrut 2600640q^{28} \) \(\mathstrut -\mathstrut 4325280q^{30} \) \(\mathstrut +\mathstrut 3232096q^{32} \) \(\mathstrut +\mathstrut 1736856q^{33} \) \(\mathstrut +\mathstrut 2378916q^{34} \) \(\mathstrut +\mathstrut 1989120q^{35} \) \(\mathstrut -\mathstrut 7392228q^{36} \) \(\mathstrut -\mathstrut 8088196q^{38} \) \(\mathstrut +\mathstrut 12694080q^{40} \) \(\mathstrut +\mathstrut 3817100q^{41} \) \(\mathstrut +\mathstrut 19064640q^{42} \) \(\mathstrut -\mathstrut 9881508q^{43} \) \(\mathstrut -\mathstrut 9479368q^{44} \) \(\mathstrut -\mathstrut 19226976q^{46} \) \(\mathstrut +\mathstrut 20590944q^{48} \) \(\mathstrut -\mathstrut 12655482q^{49} \) \(\mathstrut +\mathstrut 11106610q^{50} \) \(\mathstrut -\mathstrut 4303176q^{51} \) \(\mathstrut -\mathstrut 17270880q^{52} \) \(\mathstrut -\mathstrut 7366536q^{54} \) \(\mathstrut +\mathstrut 10545024q^{56} \) \(\mathstrut +\mathstrut 7523736q^{57} \) \(\mathstrut +\mathstrut 7354800q^{58} \) \(\mathstrut +\mathstrut 25243484q^{59} \) \(\mathstrut -\mathstrut 5760960q^{60} \) \(\mathstrut -\mathstrut 5304960q^{62} \) \(\mathstrut -\mathstrut 35364288q^{64} \) \(\mathstrut +\mathstrut 27060480q^{65} \) \(\mathstrut -\mathstrut 12683256q^{66} \) \(\mathstrut +\mathstrut 47850204q^{67} \) \(\mathstrut -\mathstrut 9160216q^{68} \) \(\mathstrut +\mathstrut 56582400q^{70} \) \(\mathstrut -\mathstrut 57502200q^{72} \) \(\mathstrut -\mathstrut 55484916q^{73} \) \(\mathstrut -\mathstrut 114255984q^{74} \) \(\mathstrut -\mathstrut 126075300q^{75} \) \(\mathstrut +\mathstrut 151972920q^{76} \) \(\mathstrut +\mathstrut 175397280q^{78} \) \(\mathstrut -\mathstrut 134958720q^{80} \) \(\mathstrut -\mathstrut 79145442q^{81} \) \(\mathstrut -\mathstrut 48197916q^{82} \) \(\mathstrut +\mathstrut 144646364q^{83} \) \(\mathstrut +\mathstrut 216531840q^{84} \) \(\mathstrut +\mathstrut 41826428q^{86} \) \(\mathstrut -\mathstrut 47377776q^{88} \) \(\mathstrut +\mathstrut 142173452q^{89} \) \(\mathstrut -\mathstrut 205943760q^{90} \) \(\mathstrut -\mathstrut 273971712q^{91} \) \(\mathstrut +\mathstrut 97636800q^{92} \) \(\mathstrut +\mathstrut 220009536q^{94} \) \(\mathstrut -\mathstrut 250689984q^{96} \) \(\mathstrut +\mathstrut 125193612q^{97} \) \(\mathstrut -\mathstrut 257093774q^{98} \) \(\mathstrut +\mathstrut 298320276q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(x^{5}\mathstrut +\mathstrut \) \(78\) \(x^{4}\mathstrut -\mathstrut \) \(514\) \(x^{3}\mathstrut +\mathstrut \) \(4237\) \(x^{2}\mathstrut -\mathstrut \) \(18333\) \(x\mathstrut +\mathstrut \) \(238980\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} + 80 \nu^{3} - 594 \nu^{2} + 4831 \nu - 19068 \)\()/2048\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 66 \nu^{4} + 272 \nu^{3} - 1810 \nu^{2} + 19487 \nu - 125820 \)\()/2048\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{5} - 10 \nu^{4} - 192 \nu^{3} - 570 \nu^{2} + 3507 \nu - 27340 \)\()/512\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{5} + 54 \nu^{4} + 208 \nu^{3} + 6438 \nu^{2} + 17691 \nu + 226452 \)\()/2048\)
\(\beta_{5}\)\(=\)\((\)\( -37 \nu^{5} + 138 \nu^{4} + 944 \nu^{3} + 15002 \nu^{2} + 7301 \nu - 350996 \)\()/2048\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(7\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut -\mathstrut \) \(55\) \(\beta_{1}\mathstrut -\mathstrut \) \(848\)\()/32\)
\(\nu^{3}\)\(=\)\((\)\(16\) \(\beta_{5}\mathstrut -\mathstrut \) \(35\) \(\beta_{4}\mathstrut -\mathstrut \) \(51\) \(\beta_{3}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(123\) \(\beta_{1}\mathstrut +\mathstrut \) \(7000\)\()/32\)
\(\nu^{4}\)\(=\)\((\)\(48\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(95\) \(\beta_{3}\mathstrut -\mathstrut \) \(1080\) \(\beta_{2}\mathstrut +\mathstrut \) \(4041\) \(\beta_{1}\mathstrut -\mathstrut \) \(14432\)\()/32\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(1184\) \(\beta_{5}\mathstrut +\mathstrut \) \(2109\) \(\beta_{4}\mathstrut -\mathstrut \) \(1155\) \(\beta_{3}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut -\mathstrut \) \(2709\) \(\beta_{1}\mathstrut -\mathstrut \) \(521048\)\()/32\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
5.83239 4.16154i
5.83239 + 4.16154i
−0.793226 7.99733i
−0.793226 + 7.99733i
−4.53916 7.17456i
−4.53916 + 7.17456i
−13.6648 8.32309i −17.4864 117.452 + 227.466i 796.785i 238.948 + 145.541i 1779.55i 288.260 4085.84i −6255.23 6631.71 10887.9i
3.2 −13.6648 + 8.32309i −17.4864 117.452 227.466i 796.785i 238.948 145.541i 1779.55i 288.260 + 4085.84i −6255.23 6631.71 + 10887.9i
3.3 −0.413549 15.9947i 117.102 −255.658 + 13.2291i 816.841i −48.4273 1873.00i 1333.52i 317.322 + 4083.69i 7151.84 −13065.1 + 337.803i
3.4 −0.413549 + 15.9947i 117.102 −255.658 13.2291i 816.841i −48.4273 + 1873.00i 1333.52i 317.322 4083.69i 7151.84 −13065.1 337.803i
3.5 7.07833 14.3491i −117.615 −155.795 203.136i 306.178i −832.521 + 1687.68i 4321.70i −4017.58 + 797.653i 7272.39 4393.38 + 2167.23i
3.6 7.07833 + 14.3491i −117.615 −155.795 + 203.136i 306.178i −832.521 1687.68i 4321.70i −4017.58 797.653i 7272.39 4393.38 2167.23i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.6
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{3} \) \(\mathstrut +\mathstrut 18 T_{3}^{2} \) \(\mathstrut -\mathstrut 13764 T_{3} \) \(\mathstrut -\mathstrut 240840 \) acting on \(S_{9}^{\mathrm{new}}(8, [\chi])\).