Properties

Label 177.8.a.b.1.10
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(2.41303\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.413031 q^{2} +27.0000 q^{3} -127.829 q^{4} +231.152 q^{5} +11.1518 q^{6} +302.150 q^{7} -105.665 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+0.413031 q^{2} +27.0000 q^{3} -127.829 q^{4} +231.152 q^{5} +11.1518 q^{6} +302.150 q^{7} -105.665 q^{8} +729.000 q^{9} +95.4729 q^{10} +1281.53 q^{11} -3451.39 q^{12} -14851.9 q^{13} +124.797 q^{14} +6241.10 q^{15} +16318.5 q^{16} -21166.2 q^{17} +301.100 q^{18} -1755.31 q^{19} -29548.0 q^{20} +8158.05 q^{21} +529.311 q^{22} +91962.2 q^{23} -2852.97 q^{24} -24693.8 q^{25} -6134.31 q^{26} +19683.0 q^{27} -38623.7 q^{28} +62655.4 q^{29} +2577.77 q^{30} -103275. q^{31} +20265.2 q^{32} +34601.2 q^{33} -8742.32 q^{34} +69842.6 q^{35} -93187.6 q^{36} -37921.6 q^{37} -724.998 q^{38} -401003. q^{39} -24424.8 q^{40} -45664.2 q^{41} +3369.53 q^{42} -637229. q^{43} -163817. q^{44} +168510. q^{45} +37983.3 q^{46} +229428. q^{47} +440600. q^{48} -732248. q^{49} -10199.3 q^{50} -571488. q^{51} +1.89852e6 q^{52} -1.07832e6 q^{53} +8129.69 q^{54} +296227. q^{55} -31926.8 q^{56} -47393.4 q^{57} +25878.6 q^{58} -205379. q^{59} -797796. q^{60} -2.75987e6 q^{61} -42655.6 q^{62} +220267. q^{63} -2.08040e6 q^{64} -3.43305e6 q^{65} +14291.4 q^{66} -3.11037e6 q^{67} +2.70567e6 q^{68} +2.48298e6 q^{69} +28847.1 q^{70} +4.01814e6 q^{71} -77030.1 q^{72} -5.86422e6 q^{73} -15662.8 q^{74} -666733. q^{75} +224380. q^{76} +387214. q^{77} -165627. q^{78} +4.62456e6 q^{79} +3.77206e6 q^{80} +531441. q^{81} -18860.7 q^{82} +9.51599e6 q^{83} -1.04284e6 q^{84} -4.89262e6 q^{85} -263195. q^{86} +1.69170e6 q^{87} -135413. q^{88} -6.95658e6 q^{89} +69599.7 q^{90} -4.48752e6 q^{91} -1.17555e7 q^{92} -2.78841e6 q^{93} +94760.7 q^{94} -405743. q^{95} +547161. q^{96} -9.56241e6 q^{97} -302441. q^{98} +934234. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.413031 0.0365071 0.0182536 0.999833i \(-0.494189\pi\)
0.0182536 + 0.999833i \(0.494189\pi\)
\(3\) 27.0000 0.577350
\(4\) −127.829 −0.998667
\(5\) 231.152 0.826994 0.413497 0.910505i \(-0.364307\pi\)
0.413497 + 0.910505i \(0.364307\pi\)
\(6\) 11.1518 0.0210774
\(7\) 302.150 0.332950 0.166475 0.986046i \(-0.446761\pi\)
0.166475 + 0.986046i \(0.446761\pi\)
\(8\) −105.665 −0.0729656
\(9\) 729.000 0.333333
\(10\) 95.4729 0.0301912
\(11\) 1281.53 0.290304 0.145152 0.989409i \(-0.453633\pi\)
0.145152 + 0.989409i \(0.453633\pi\)
\(12\) −3451.39 −0.576581
\(13\) −14851.9 −1.87491 −0.937457 0.348100i \(-0.886827\pi\)
−0.937457 + 0.348100i \(0.886827\pi\)
\(14\) 124.797 0.0121551
\(15\) 6241.10 0.477465
\(16\) 16318.5 0.996003
\(17\) −21166.2 −1.04489 −0.522447 0.852672i \(-0.674981\pi\)
−0.522447 + 0.852672i \(0.674981\pi\)
\(18\) 301.100 0.0121690
\(19\) −1755.31 −0.0587106 −0.0293553 0.999569i \(-0.509345\pi\)
−0.0293553 + 0.999569i \(0.509345\pi\)
\(20\) −29548.0 −0.825892
\(21\) 8158.05 0.192229
\(22\) 529.311 0.0105982
\(23\) 91962.2 1.57602 0.788011 0.615662i \(-0.211111\pi\)
0.788011 + 0.615662i \(0.211111\pi\)
\(24\) −2852.97 −0.0421267
\(25\) −24693.8 −0.316081
\(26\) −6134.31 −0.0684478
\(27\) 19683.0 0.192450
\(28\) −38623.7 −0.332507
\(29\) 62655.4 0.477052 0.238526 0.971136i \(-0.423336\pi\)
0.238526 + 0.971136i \(0.423336\pi\)
\(30\) 2577.77 0.0174309
\(31\) −103275. −0.622626 −0.311313 0.950307i \(-0.600769\pi\)
−0.311313 + 0.950307i \(0.600769\pi\)
\(32\) 20265.2 0.109327
\(33\) 34601.2 0.167607
\(34\) −8742.32 −0.0381461
\(35\) 69842.6 0.275348
\(36\) −93187.6 −0.332889
\(37\) −37921.6 −0.123078 −0.0615390 0.998105i \(-0.519601\pi\)
−0.0615390 + 0.998105i \(0.519601\pi\)
\(38\) −724.998 −0.00214335
\(39\) −401003. −1.08248
\(40\) −24424.8 −0.0603421
\(41\) −45664.2 −0.103474 −0.0517371 0.998661i \(-0.516476\pi\)
−0.0517371 + 0.998661i \(0.516476\pi\)
\(42\) 3369.53 0.00701773
\(43\) −637229. −1.22224 −0.611119 0.791539i \(-0.709280\pi\)
−0.611119 + 0.791539i \(0.709280\pi\)
\(44\) −163817. −0.289917
\(45\) 168510. 0.275665
\(46\) 37983.3 0.0575360
\(47\) 229428. 0.322332 0.161166 0.986927i \(-0.448475\pi\)
0.161166 + 0.986927i \(0.448475\pi\)
\(48\) 440600. 0.575043
\(49\) −732248. −0.889144
\(50\) −10199.3 −0.0115392
\(51\) −571488. −0.603270
\(52\) 1.89852e6 1.87242
\(53\) −1.07832e6 −0.994906 −0.497453 0.867491i \(-0.665731\pi\)
−0.497453 + 0.867491i \(0.665731\pi\)
\(54\) 8129.69 0.00702580
\(55\) 296227. 0.240080
\(56\) −31926.8 −0.0242939
\(57\) −47393.4 −0.0338966
\(58\) 25878.6 0.0174158
\(59\) −205379. −0.130189
\(60\) −797796. −0.476829
\(61\) −2.75987e6 −1.55680 −0.778402 0.627767i \(-0.783969\pi\)
−0.778402 + 0.627767i \(0.783969\pi\)
\(62\) −42655.6 −0.0227303
\(63\) 220267. 0.110983
\(64\) −2.08040e6 −0.992012
\(65\) −3.43305e6 −1.55054
\(66\) 14291.4 0.00611886
\(67\) −3.11037e6 −1.26343 −0.631715 0.775201i \(-0.717649\pi\)
−0.631715 + 0.775201i \(0.717649\pi\)
\(68\) 2.70567e6 1.04350
\(69\) 2.48298e6 0.909916
\(70\) 28847.1 0.0100522
\(71\) 4.01814e6 1.33236 0.666179 0.745792i \(-0.267929\pi\)
0.666179 + 0.745792i \(0.267929\pi\)
\(72\) −77030.1 −0.0243219
\(73\) −5.86422e6 −1.76433 −0.882165 0.470940i \(-0.843915\pi\)
−0.882165 + 0.470940i \(0.843915\pi\)
\(74\) −15662.8 −0.00449322
\(75\) −666733. −0.182489
\(76\) 224380. 0.0586323
\(77\) 387214. 0.0966570
\(78\) −165627. −0.0395183
\(79\) 4.62456e6 1.05530 0.527649 0.849462i \(-0.323074\pi\)
0.527649 + 0.849462i \(0.323074\pi\)
\(80\) 3.77206e6 0.823689
\(81\) 531441. 0.111111
\(82\) −18860.7 −0.00377755
\(83\) 9.51599e6 1.82676 0.913378 0.407113i \(-0.133465\pi\)
0.913378 + 0.407113i \(0.133465\pi\)
\(84\) −1.04284e6 −0.191973
\(85\) −4.89262e6 −0.864122
\(86\) −263195. −0.0446204
\(87\) 1.69170e6 0.275426
\(88\) −135413. −0.0211822
\(89\) −6.95658e6 −1.04600 −0.522999 0.852333i \(-0.675187\pi\)
−0.522999 + 0.852333i \(0.675187\pi\)
\(90\) 69599.7 0.0100637
\(91\) −4.48752e6 −0.624254
\(92\) −1.17555e7 −1.57392
\(93\) −2.78841e6 −0.359473
\(94\) 94760.7 0.0117674
\(95\) −405743. −0.0485533
\(96\) 547161. 0.0631199
\(97\) −9.56241e6 −1.06382 −0.531908 0.846802i \(-0.678525\pi\)
−0.531908 + 0.846802i \(0.678525\pi\)
\(98\) −302441. −0.0324601
\(99\) 934234. 0.0967681
\(100\) 3.15660e6 0.315660
\(101\) −9.74084e6 −0.940744 −0.470372 0.882468i \(-0.655880\pi\)
−0.470372 + 0.882468i \(0.655880\pi\)
\(102\) −236043. −0.0220237
\(103\) −1.17131e7 −1.05619 −0.528093 0.849186i \(-0.677093\pi\)
−0.528093 + 0.849186i \(0.677093\pi\)
\(104\) 1.56934e6 0.136804
\(105\) 1.88575e6 0.158972
\(106\) −445380. −0.0363212
\(107\) −2.32996e7 −1.83867 −0.919337 0.393471i \(-0.871274\pi\)
−0.919337 + 0.393471i \(0.871274\pi\)
\(108\) −2.51607e6 −0.192194
\(109\) 1.84368e7 1.36362 0.681808 0.731531i \(-0.261194\pi\)
0.681808 + 0.731531i \(0.261194\pi\)
\(110\) 122351. 0.00876463
\(111\) −1.02388e6 −0.0710591
\(112\) 4.93064e6 0.331620
\(113\) −1.69409e7 −1.10449 −0.552247 0.833681i \(-0.686229\pi\)
−0.552247 + 0.833681i \(0.686229\pi\)
\(114\) −19574.9 −0.00123747
\(115\) 2.12572e7 1.30336
\(116\) −8.00921e6 −0.476416
\(117\) −1.08271e7 −0.624972
\(118\) −84827.9 −0.00475282
\(119\) −6.39538e6 −0.347898
\(120\) −659469. −0.0348385
\(121\) −1.78449e7 −0.915723
\(122\) −1.13991e6 −0.0568344
\(123\) −1.23293e6 −0.0597409
\(124\) 1.32015e7 0.621796
\(125\) −2.37668e7 −1.08839
\(126\) 90977.3 0.00405169
\(127\) −1.02426e7 −0.443708 −0.221854 0.975080i \(-0.571211\pi\)
−0.221854 + 0.975080i \(0.571211\pi\)
\(128\) −3.45322e6 −0.145542
\(129\) −1.72052e7 −0.705660
\(130\) −1.41796e6 −0.0566059
\(131\) 2.19669e7 0.853728 0.426864 0.904316i \(-0.359618\pi\)
0.426864 + 0.904316i \(0.359618\pi\)
\(132\) −4.42306e6 −0.167384
\(133\) −530367. −0.0195477
\(134\) −1.28468e6 −0.0461242
\(135\) 4.54976e6 0.159155
\(136\) 2.23654e6 0.0762414
\(137\) 1.76506e6 0.0586459 0.0293229 0.999570i \(-0.490665\pi\)
0.0293229 + 0.999570i \(0.490665\pi\)
\(138\) 1.02555e6 0.0332184
\(139\) −2.43007e7 −0.767481 −0.383740 0.923441i \(-0.625364\pi\)
−0.383740 + 0.923441i \(0.625364\pi\)
\(140\) −8.92793e6 −0.274981
\(141\) 6.19454e6 0.186098
\(142\) 1.65962e6 0.0486406
\(143\) −1.90332e7 −0.544296
\(144\) 1.18962e7 0.332001
\(145\) 1.44829e7 0.394519
\(146\) −2.42210e6 −0.0644107
\(147\) −1.97707e7 −0.513348
\(148\) 4.84749e6 0.122914
\(149\) 7.68149e7 1.90236 0.951181 0.308633i \(-0.0998715\pi\)
0.951181 + 0.308633i \(0.0998715\pi\)
\(150\) −275382. −0.00666217
\(151\) 4.28340e7 1.01244 0.506220 0.862404i \(-0.331042\pi\)
0.506220 + 0.862404i \(0.331042\pi\)
\(152\) 185476. 0.00428385
\(153\) −1.54302e7 −0.348298
\(154\) 159931. 0.00352867
\(155\) −2.38721e7 −0.514908
\(156\) 5.12599e7 1.08104
\(157\) 5.51222e7 1.13678 0.568392 0.822758i \(-0.307566\pi\)
0.568392 + 0.822758i \(0.307566\pi\)
\(158\) 1.91009e6 0.0385259
\(159\) −2.91146e7 −0.574409
\(160\) 4.68435e6 0.0904126
\(161\) 2.77864e7 0.524737
\(162\) 219502. 0.00405635
\(163\) 3.76340e7 0.680650 0.340325 0.940308i \(-0.389463\pi\)
0.340325 + 0.940308i \(0.389463\pi\)
\(164\) 5.83723e6 0.103336
\(165\) 7.99814e6 0.138610
\(166\) 3.93040e6 0.0666896
\(167\) 9.25680e7 1.53799 0.768995 0.639255i \(-0.220757\pi\)
0.768995 + 0.639255i \(0.220757\pi\)
\(168\) −862025. −0.0140261
\(169\) 1.57832e8 2.51531
\(170\) −2.02080e6 −0.0315466
\(171\) −1.27962e6 −0.0195702
\(172\) 8.14566e7 1.22061
\(173\) −1.09599e8 −1.60933 −0.804665 0.593729i \(-0.797655\pi\)
−0.804665 + 0.593729i \(0.797655\pi\)
\(174\) 698723. 0.0100550
\(175\) −7.46124e6 −0.105239
\(176\) 2.09126e7 0.289144
\(177\) −5.54523e6 −0.0751646
\(178\) −2.87328e6 −0.0381864
\(179\) 4.35682e7 0.567786 0.283893 0.958856i \(-0.408374\pi\)
0.283893 + 0.958856i \(0.408374\pi\)
\(180\) −2.15405e7 −0.275297
\(181\) −1.29762e8 −1.62656 −0.813282 0.581870i \(-0.802321\pi\)
−0.813282 + 0.581870i \(0.802321\pi\)
\(182\) −1.85348e6 −0.0227897
\(183\) −7.45164e7 −0.898821
\(184\) −9.71724e6 −0.114995
\(185\) −8.76564e6 −0.101785
\(186\) −1.15170e6 −0.0131233
\(187\) −2.71251e7 −0.303337
\(188\) −2.93276e7 −0.321902
\(189\) 5.94722e6 0.0640763
\(190\) −167585. −0.00177254
\(191\) 3.07331e7 0.319146 0.159573 0.987186i \(-0.448988\pi\)
0.159573 + 0.987186i \(0.448988\pi\)
\(192\) −5.61708e7 −0.572739
\(193\) 1.24448e8 1.24606 0.623029 0.782199i \(-0.285902\pi\)
0.623029 + 0.782199i \(0.285902\pi\)
\(194\) −3.94957e6 −0.0388369
\(195\) −9.26925e7 −0.895207
\(196\) 9.36029e7 0.887959
\(197\) −5.43655e7 −0.506630 −0.253315 0.967384i \(-0.581521\pi\)
−0.253315 + 0.967384i \(0.581521\pi\)
\(198\) 385868. 0.00353273
\(199\) 1.40875e8 1.26721 0.633606 0.773656i \(-0.281574\pi\)
0.633606 + 0.773656i \(0.281574\pi\)
\(200\) 2.60929e6 0.0230630
\(201\) −8.39801e7 −0.729441
\(202\) −4.02327e6 −0.0343439
\(203\) 1.89313e7 0.158835
\(204\) 7.30530e7 0.602466
\(205\) −1.05554e7 −0.0855726
\(206\) −4.83786e6 −0.0385584
\(207\) 6.70405e7 0.525340
\(208\) −2.42362e8 −1.86742
\(209\) −2.24948e6 −0.0170439
\(210\) 778873. 0.00580362
\(211\) 2.07381e8 1.51978 0.759891 0.650051i \(-0.225252\pi\)
0.759891 + 0.650051i \(0.225252\pi\)
\(212\) 1.37841e8 0.993580
\(213\) 1.08490e8 0.769237
\(214\) −9.62345e6 −0.0671247
\(215\) −1.47297e8 −1.01078
\(216\) −2.07981e6 −0.0140422
\(217\) −3.12044e7 −0.207304
\(218\) 7.61496e6 0.0497817
\(219\) −1.58334e8 −1.01864
\(220\) −3.78666e7 −0.239760
\(221\) 3.14360e8 1.95909
\(222\) −422895. −0.00259416
\(223\) −2.57344e8 −1.55398 −0.776992 0.629511i \(-0.783255\pi\)
−0.776992 + 0.629511i \(0.783255\pi\)
\(224\) 6.12314e6 0.0364004
\(225\) −1.80018e7 −0.105360
\(226\) −6.99714e6 −0.0403219
\(227\) −1.28065e8 −0.726676 −0.363338 0.931657i \(-0.618363\pi\)
−0.363338 + 0.931657i \(0.618363\pi\)
\(228\) 6.05827e6 0.0338514
\(229\) 9.10898e7 0.501240 0.250620 0.968086i \(-0.419366\pi\)
0.250620 + 0.968086i \(0.419366\pi\)
\(230\) 8.77990e6 0.0475819
\(231\) 1.04548e7 0.0558049
\(232\) −6.62052e6 −0.0348084
\(233\) −6.28725e7 −0.325623 −0.162812 0.986657i \(-0.552056\pi\)
−0.162812 + 0.986657i \(0.552056\pi\)
\(234\) −4.47192e6 −0.0228159
\(235\) 5.30326e7 0.266566
\(236\) 2.62535e7 0.130015
\(237\) 1.24863e8 0.609277
\(238\) −2.64149e6 −0.0127008
\(239\) −2.11572e8 −1.00246 −0.501229 0.865315i \(-0.667119\pi\)
−0.501229 + 0.865315i \(0.667119\pi\)
\(240\) 1.01846e8 0.475557
\(241\) 1.41219e7 0.0649882 0.0324941 0.999472i \(-0.489655\pi\)
0.0324941 + 0.999472i \(0.489655\pi\)
\(242\) −7.37048e6 −0.0334304
\(243\) 1.43489e7 0.0641500
\(244\) 3.52792e8 1.55473
\(245\) −1.69261e8 −0.735317
\(246\) −509240. −0.00218097
\(247\) 2.60698e7 0.110077
\(248\) 1.09126e7 0.0454303
\(249\) 2.56932e8 1.05468
\(250\) −9.81641e6 −0.0397340
\(251\) 8.14251e7 0.325013 0.162506 0.986707i \(-0.448042\pi\)
0.162506 + 0.986707i \(0.448042\pi\)
\(252\) −2.81567e7 −0.110836
\(253\) 1.17852e8 0.457526
\(254\) −4.23051e6 −0.0161985
\(255\) −1.32101e8 −0.498901
\(256\) 2.64865e8 0.986699
\(257\) 2.21346e8 0.813402 0.406701 0.913561i \(-0.366679\pi\)
0.406701 + 0.913561i \(0.366679\pi\)
\(258\) −7.10628e6 −0.0257616
\(259\) −1.14580e7 −0.0409789
\(260\) 4.38845e8 1.54848
\(261\) 4.56758e7 0.159017
\(262\) 9.07301e6 0.0311672
\(263\) 4.49267e8 1.52286 0.761429 0.648249i \(-0.224498\pi\)
0.761429 + 0.648249i \(0.224498\pi\)
\(264\) −3.65616e6 −0.0122296
\(265\) −2.49256e8 −0.822781
\(266\) −219058. −0.000713631 0
\(267\) −1.87828e8 −0.603907
\(268\) 3.97597e8 1.26175
\(269\) −4.00214e8 −1.25360 −0.626801 0.779180i \(-0.715636\pi\)
−0.626801 + 0.779180i \(0.715636\pi\)
\(270\) 1.87919e6 0.00581030
\(271\) 2.23102e8 0.680944 0.340472 0.940255i \(-0.389413\pi\)
0.340472 + 0.940255i \(0.389413\pi\)
\(272\) −3.45402e8 −1.04072
\(273\) −1.21163e8 −0.360413
\(274\) 729025. 0.00214099
\(275\) −3.16458e7 −0.0917597
\(276\) −3.17398e8 −0.908704
\(277\) 2.84338e8 0.803815 0.401907 0.915680i \(-0.368347\pi\)
0.401907 + 0.915680i \(0.368347\pi\)
\(278\) −1.00370e7 −0.0280185
\(279\) −7.52871e7 −0.207542
\(280\) −7.37995e6 −0.0200909
\(281\) −3.25571e8 −0.875332 −0.437666 0.899138i \(-0.644195\pi\)
−0.437666 + 0.899138i \(0.644195\pi\)
\(282\) 2.55854e6 0.00679392
\(283\) −4.66681e7 −0.122396 −0.0611980 0.998126i \(-0.519492\pi\)
−0.0611980 + 0.998126i \(0.519492\pi\)
\(284\) −5.13636e8 −1.33058
\(285\) −1.09551e7 −0.0280323
\(286\) −7.86129e6 −0.0198707
\(287\) −1.37974e7 −0.0344518
\(288\) 1.47734e7 0.0364423
\(289\) 3.76710e7 0.0918048
\(290\) 5.98190e6 0.0144028
\(291\) −2.58185e8 −0.614194
\(292\) 7.49619e8 1.76198
\(293\) −7.70273e7 −0.178899 −0.0894495 0.995991i \(-0.528511\pi\)
−0.0894495 + 0.995991i \(0.528511\pi\)
\(294\) −8.16592e6 −0.0187408
\(295\) −4.74737e7 −0.107665
\(296\) 4.00700e6 0.00898046
\(297\) 2.52243e7 0.0558691
\(298\) 3.17269e7 0.0694498
\(299\) −1.36582e9 −2.95491
\(300\) 8.52281e7 0.182246
\(301\) −1.92539e8 −0.406945
\(302\) 1.76918e7 0.0369613
\(303\) −2.63003e8 −0.543139
\(304\) −2.86441e7 −0.0584759
\(305\) −6.37948e8 −1.28747
\(306\) −6.37315e6 −0.0127154
\(307\) −1.67909e8 −0.331200 −0.165600 0.986193i \(-0.552956\pi\)
−0.165600 + 0.986193i \(0.552956\pi\)
\(308\) −4.94973e7 −0.0965281
\(309\) −3.16253e8 −0.609790
\(310\) −9.85992e6 −0.0187978
\(311\) −2.22728e8 −0.419868 −0.209934 0.977716i \(-0.567325\pi\)
−0.209934 + 0.977716i \(0.567325\pi\)
\(312\) 4.23721e7 0.0789840
\(313\) −2.23952e8 −0.412809 −0.206404 0.978467i \(-0.566176\pi\)
−0.206404 + 0.978467i \(0.566176\pi\)
\(314\) 2.27672e7 0.0415007
\(315\) 5.09152e7 0.0917827
\(316\) −5.91154e8 −1.05389
\(317\) 7.57737e8 1.33601 0.668007 0.744155i \(-0.267147\pi\)
0.668007 + 0.744155i \(0.267147\pi\)
\(318\) −1.20252e7 −0.0209700
\(319\) 8.02947e7 0.138490
\(320\) −4.80888e8 −0.820388
\(321\) −6.29089e8 −1.06156
\(322\) 1.14766e7 0.0191566
\(323\) 3.71533e7 0.0613464
\(324\) −6.79338e7 −0.110963
\(325\) 3.66751e8 0.592625
\(326\) 1.55440e7 0.0248486
\(327\) 4.97793e8 0.787284
\(328\) 4.82513e6 0.00755006
\(329\) 6.93215e7 0.107320
\(330\) 3.30348e6 0.00506026
\(331\) 2.98249e8 0.452044 0.226022 0.974122i \(-0.427428\pi\)
0.226022 + 0.974122i \(0.427428\pi\)
\(332\) −1.21642e9 −1.82432
\(333\) −2.76448e7 −0.0410260
\(334\) 3.82335e7 0.0561476
\(335\) −7.18969e8 −1.04485
\(336\) 1.33127e8 0.191461
\(337\) −1.29895e9 −1.84879 −0.924393 0.381442i \(-0.875428\pi\)
−0.924393 + 0.381442i \(0.875428\pi\)
\(338\) 6.51894e7 0.0918266
\(339\) −4.57406e8 −0.637680
\(340\) 6.25420e8 0.862970
\(341\) −1.32349e8 −0.180751
\(342\) −528523. −0.000714452 0
\(343\) −4.70083e8 −0.628991
\(344\) 6.73331e7 0.0891814
\(345\) 5.73945e8 0.752495
\(346\) −4.52678e7 −0.0587521
\(347\) 1.13535e9 1.45874 0.729369 0.684120i \(-0.239814\pi\)
0.729369 + 0.684120i \(0.239814\pi\)
\(348\) −2.16249e8 −0.275059
\(349\) 2.24671e8 0.282916 0.141458 0.989944i \(-0.454821\pi\)
0.141458 + 0.989944i \(0.454821\pi\)
\(350\) −3.08172e6 −0.00384199
\(351\) −2.92331e8 −0.360828
\(352\) 2.59705e7 0.0317381
\(353\) 1.00698e9 1.21846 0.609228 0.792995i \(-0.291480\pi\)
0.609228 + 0.792995i \(0.291480\pi\)
\(354\) −2.29035e6 −0.00274404
\(355\) 9.28800e8 1.10185
\(356\) 8.89256e8 1.04460
\(357\) −1.72675e8 −0.200859
\(358\) 1.79950e7 0.0207282
\(359\) −5.34366e7 −0.0609548 −0.0304774 0.999535i \(-0.509703\pi\)
−0.0304774 + 0.999535i \(0.509703\pi\)
\(360\) −1.78057e7 −0.0201140
\(361\) −8.90791e8 −0.996553
\(362\) −5.35956e7 −0.0593812
\(363\) −4.81811e8 −0.528693
\(364\) 5.73637e8 0.623422
\(365\) −1.35552e9 −1.45909
\(366\) −3.07776e7 −0.0328134
\(367\) 1.32145e8 0.139547 0.0697733 0.997563i \(-0.477772\pi\)
0.0697733 + 0.997563i \(0.477772\pi\)
\(368\) 1.50069e9 1.56972
\(369\) −3.32892e7 −0.0344914
\(370\) −3.62048e6 −0.00371587
\(371\) −3.25814e8 −0.331254
\(372\) 3.56441e8 0.358994
\(373\) 1.16722e8 0.116458 0.0582292 0.998303i \(-0.481455\pi\)
0.0582292 + 0.998303i \(0.481455\pi\)
\(374\) −1.12035e7 −0.0110740
\(375\) −6.41703e8 −0.628383
\(376\) −2.42426e7 −0.0235191
\(377\) −9.30555e8 −0.894432
\(378\) 2.45639e6 0.00233924
\(379\) 1.54193e9 1.45488 0.727440 0.686172i \(-0.240710\pi\)
0.727440 + 0.686172i \(0.240710\pi\)
\(380\) 5.18659e7 0.0484886
\(381\) −2.76550e8 −0.256175
\(382\) 1.26937e7 0.0116511
\(383\) −1.11000e9 −1.00955 −0.504774 0.863252i \(-0.668424\pi\)
−0.504774 + 0.863252i \(0.668424\pi\)
\(384\) −9.32370e7 −0.0840289
\(385\) 8.95052e7 0.0799347
\(386\) 5.14010e7 0.0454900
\(387\) −4.64540e8 −0.407413
\(388\) 1.22236e9 1.06240
\(389\) 3.52125e8 0.303301 0.151650 0.988434i \(-0.451541\pi\)
0.151650 + 0.988434i \(0.451541\pi\)
\(390\) −3.82849e7 −0.0326814
\(391\) −1.94649e9 −1.64678
\(392\) 7.73734e7 0.0648769
\(393\) 5.93106e8 0.492900
\(394\) −2.24546e7 −0.0184956
\(395\) 1.06897e9 0.872725
\(396\) −1.19423e8 −0.0966392
\(397\) 1.23251e9 0.988611 0.494305 0.869288i \(-0.335422\pi\)
0.494305 + 0.869288i \(0.335422\pi\)
\(398\) 5.81859e7 0.0462623
\(399\) −1.43199e7 −0.0112859
\(400\) −4.02967e8 −0.314818
\(401\) −1.95405e9 −1.51332 −0.756661 0.653808i \(-0.773171\pi\)
−0.756661 + 0.653808i \(0.773171\pi\)
\(402\) −3.46864e7 −0.0266298
\(403\) 1.53383e9 1.16737
\(404\) 1.24517e9 0.939491
\(405\) 1.22844e8 0.0918882
\(406\) 7.81924e6 0.00579860
\(407\) −4.85975e7 −0.0357301
\(408\) 6.03866e7 0.0440180
\(409\) −8.06706e8 −0.583020 −0.291510 0.956568i \(-0.594158\pi\)
−0.291510 + 0.956568i \(0.594158\pi\)
\(410\) −4.35969e6 −0.00312401
\(411\) 4.76566e7 0.0338592
\(412\) 1.49728e9 1.05478
\(413\) −6.20553e7 −0.0433465
\(414\) 2.76898e7 0.0191787
\(415\) 2.19964e9 1.51072
\(416\) −3.00978e8 −0.204979
\(417\) −6.56120e8 −0.443105
\(418\) −929105. −0.000622225 0
\(419\) −1.37468e9 −0.912960 −0.456480 0.889734i \(-0.650890\pi\)
−0.456480 + 0.889734i \(0.650890\pi\)
\(420\) −2.41054e8 −0.158760
\(421\) −8.51303e8 −0.556028 −0.278014 0.960577i \(-0.589676\pi\)
−0.278014 + 0.960577i \(0.589676\pi\)
\(422\) 8.56549e7 0.0554829
\(423\) 1.67253e8 0.107444
\(424\) 1.13941e8 0.0725939
\(425\) 5.22675e8 0.330271
\(426\) 4.48096e7 0.0280826
\(427\) −8.33894e8 −0.518338
\(428\) 2.97837e9 1.83622
\(429\) −5.13896e8 −0.314249
\(430\) −6.08381e7 −0.0369008
\(431\) 3.76775e8 0.226679 0.113339 0.993556i \(-0.463845\pi\)
0.113339 + 0.993556i \(0.463845\pi\)
\(432\) 3.21197e8 0.191681
\(433\) 1.14074e9 0.675270 0.337635 0.941277i \(-0.390373\pi\)
0.337635 + 0.941277i \(0.390373\pi\)
\(434\) −1.28884e7 −0.00756806
\(435\) 3.91039e8 0.227776
\(436\) −2.35676e9 −1.36180
\(437\) −1.61422e8 −0.0925291
\(438\) −6.53968e7 −0.0371875
\(439\) 1.15020e9 0.648854 0.324427 0.945911i \(-0.394829\pi\)
0.324427 + 0.945911i \(0.394829\pi\)
\(440\) −3.13010e7 −0.0175176
\(441\) −5.33809e8 −0.296381
\(442\) 1.29840e8 0.0715207
\(443\) 4.99658e8 0.273061 0.136531 0.990636i \(-0.456405\pi\)
0.136531 + 0.990636i \(0.456405\pi\)
\(444\) 1.30882e8 0.0709644
\(445\) −1.60803e9 −0.865034
\(446\) −1.06291e8 −0.0567315
\(447\) 2.07400e9 1.09833
\(448\) −6.28593e8 −0.330291
\(449\) 1.70526e9 0.889054 0.444527 0.895766i \(-0.353372\pi\)
0.444527 + 0.895766i \(0.353372\pi\)
\(450\) −7.43530e6 −0.00384640
\(451\) −5.85199e7 −0.0300390
\(452\) 2.16555e9 1.10302
\(453\) 1.15652e9 0.584533
\(454\) −5.28949e7 −0.0265289
\(455\) −1.03730e9 −0.516254
\(456\) 5.00785e6 0.00247328
\(457\) −6.10097e8 −0.299015 −0.149507 0.988761i \(-0.547769\pi\)
−0.149507 + 0.988761i \(0.547769\pi\)
\(458\) 3.76229e7 0.0182988
\(459\) −4.16615e8 −0.201090
\(460\) −2.71730e9 −1.30162
\(461\) 2.09113e9 0.994093 0.497046 0.867724i \(-0.334418\pi\)
0.497046 + 0.867724i \(0.334418\pi\)
\(462\) 4.31815e6 0.00203728
\(463\) −3.87258e8 −0.181329 −0.0906643 0.995882i \(-0.528899\pi\)
−0.0906643 + 0.995882i \(0.528899\pi\)
\(464\) 1.02244e9 0.475146
\(465\) −6.44547e8 −0.297282
\(466\) −2.59683e7 −0.0118876
\(467\) 2.46901e8 0.112180 0.0560898 0.998426i \(-0.482137\pi\)
0.0560898 + 0.998426i \(0.482137\pi\)
\(468\) 1.38402e9 0.624139
\(469\) −9.39800e8 −0.420659
\(470\) 2.19041e7 0.00973157
\(471\) 1.48830e9 0.656322
\(472\) 2.17015e7 0.00949931
\(473\) −8.16627e8 −0.354821
\(474\) 5.15723e7 0.0222429
\(475\) 4.33453e7 0.0185573
\(476\) 8.17518e8 0.347434
\(477\) −7.86095e8 −0.331635
\(478\) −8.73860e7 −0.0365969
\(479\) −3.29359e9 −1.36929 −0.684645 0.728877i \(-0.740043\pi\)
−0.684645 + 0.728877i \(0.740043\pi\)
\(480\) 1.26477e8 0.0521998
\(481\) 5.63209e8 0.230761
\(482\) 5.83280e6 0.00237253
\(483\) 7.50233e8 0.302957
\(484\) 2.28110e9 0.914503
\(485\) −2.21037e9 −0.879769
\(486\) 5.92654e6 0.00234193
\(487\) 1.64421e9 0.645070 0.322535 0.946558i \(-0.395465\pi\)
0.322535 + 0.946558i \(0.395465\pi\)
\(488\) 2.91623e8 0.113593
\(489\) 1.01612e9 0.392974
\(490\) −6.99099e7 −0.0268443
\(491\) 2.85157e9 1.08717 0.543587 0.839353i \(-0.317066\pi\)
0.543587 + 0.839353i \(0.317066\pi\)
\(492\) 1.57605e8 0.0596613
\(493\) −1.32618e9 −0.498469
\(494\) 1.07676e7 0.00401861
\(495\) 2.15950e8 0.0800267
\(496\) −1.68529e9 −0.620138
\(497\) 1.21408e9 0.443609
\(498\) 1.06121e8 0.0385033
\(499\) 3.73412e9 1.34535 0.672676 0.739937i \(-0.265145\pi\)
0.672676 + 0.739937i \(0.265145\pi\)
\(500\) 3.03809e9 1.08694
\(501\) 2.49934e9 0.887958
\(502\) 3.36311e7 0.0118653
\(503\) −4.65904e9 −1.63233 −0.816166 0.577817i \(-0.803905\pi\)
−0.816166 + 0.577817i \(0.803905\pi\)
\(504\) −2.32747e7 −0.00809798
\(505\) −2.25161e9 −0.777990
\(506\) 4.86766e7 0.0167030
\(507\) 4.26146e9 1.45221
\(508\) 1.30931e9 0.443117
\(509\) 4.17061e9 1.40180 0.700902 0.713258i \(-0.252781\pi\)
0.700902 + 0.713258i \(0.252781\pi\)
\(510\) −5.45617e7 −0.0182134
\(511\) −1.77187e9 −0.587435
\(512\) 5.51410e8 0.181564
\(513\) −3.45498e7 −0.0112989
\(514\) 9.14226e7 0.0296950
\(515\) −2.70750e9 −0.873460
\(516\) 2.19933e9 0.704719
\(517\) 2.94018e8 0.0935743
\(518\) −4.73251e6 −0.00149602
\(519\) −2.95917e9 −0.929148
\(520\) 3.62755e8 0.113136
\(521\) −1.08528e9 −0.336209 −0.168104 0.985769i \(-0.553765\pi\)
−0.168104 + 0.985769i \(0.553765\pi\)
\(522\) 1.88655e7 0.00580527
\(523\) −4.07601e9 −1.24589 −0.622945 0.782266i \(-0.714064\pi\)
−0.622945 + 0.782266i \(0.714064\pi\)
\(524\) −2.80802e9 −0.852590
\(525\) −2.01454e8 −0.0607599
\(526\) 1.85561e8 0.0555952
\(527\) 2.18593e9 0.650579
\(528\) 5.64641e8 0.166937
\(529\) 5.05223e9 1.48384
\(530\) −1.02950e8 −0.0300374
\(531\) −1.49721e8 −0.0433963
\(532\) 6.77965e7 0.0195217
\(533\) 6.78202e8 0.194005
\(534\) −7.75787e7 −0.0220469
\(535\) −5.38574e9 −1.52057
\(536\) 3.28659e8 0.0921869
\(537\) 1.17634e9 0.327811
\(538\) −1.65301e8 −0.0457654
\(539\) −9.38396e8 −0.258122
\(540\) −5.81593e8 −0.158943
\(541\) −4.61079e9 −1.25194 −0.625972 0.779846i \(-0.715297\pi\)
−0.625972 + 0.779846i \(0.715297\pi\)
\(542\) 9.21481e7 0.0248593
\(543\) −3.50356e9 −0.939097
\(544\) −4.28939e8 −0.114235
\(545\) 4.26169e9 1.12770
\(546\) −5.00441e7 −0.0131577
\(547\) −3.70849e9 −0.968817 −0.484409 0.874842i \(-0.660965\pi\)
−0.484409 + 0.874842i \(0.660965\pi\)
\(548\) −2.25627e8 −0.0585677
\(549\) −2.01194e9 −0.518934
\(550\) −1.30707e7 −0.00334988
\(551\) −1.09980e8 −0.0280080
\(552\) −2.62365e8 −0.0663926
\(553\) 1.39731e9 0.351362
\(554\) 1.17441e8 0.0293450
\(555\) −2.36672e8 −0.0587654
\(556\) 3.10635e9 0.766458
\(557\) −4.17975e9 −1.02484 −0.512421 0.858734i \(-0.671251\pi\)
−0.512421 + 0.858734i \(0.671251\pi\)
\(558\) −3.10959e7 −0.00757676
\(559\) 9.46409e9 2.29159
\(560\) 1.13973e9 0.274248
\(561\) −7.32378e8 −0.175132
\(562\) −1.34471e8 −0.0319559
\(563\) 5.00396e9 1.18178 0.590888 0.806754i \(-0.298778\pi\)
0.590888 + 0.806754i \(0.298778\pi\)
\(564\) −7.91845e8 −0.185850
\(565\) −3.91593e9 −0.913410
\(566\) −1.92754e7 −0.00446833
\(567\) 1.60575e8 0.0369945
\(568\) −4.24579e8 −0.0972163
\(569\) 5.07854e8 0.115570 0.0577851 0.998329i \(-0.481596\pi\)
0.0577851 + 0.998329i \(0.481596\pi\)
\(570\) −4.52478e6 −0.00102338
\(571\) −5.27683e9 −1.18617 −0.593085 0.805140i \(-0.702090\pi\)
−0.593085 + 0.805140i \(0.702090\pi\)
\(572\) 2.43300e9 0.543571
\(573\) 8.29792e8 0.184259
\(574\) −5.69877e6 −0.00125774
\(575\) −2.27090e9 −0.498150
\(576\) −1.51661e9 −0.330671
\(577\) −1.54218e9 −0.334209 −0.167105 0.985939i \(-0.553442\pi\)
−0.167105 + 0.985939i \(0.553442\pi\)
\(578\) 1.55593e7 0.00335153
\(579\) 3.36010e9 0.719411
\(580\) −1.85134e9 −0.393993
\(581\) 2.87526e9 0.608219
\(582\) −1.06638e8 −0.0224225
\(583\) −1.38190e9 −0.288826
\(584\) 6.19645e8 0.128735
\(585\) −2.50270e9 −0.516848
\(586\) −3.18147e7 −0.00653109
\(587\) 9.40246e8 0.191870 0.0959352 0.995388i \(-0.469416\pi\)
0.0959352 + 0.995388i \(0.469416\pi\)
\(588\) 2.52728e9 0.512663
\(589\) 1.81279e8 0.0365547
\(590\) −1.96081e7 −0.00393056
\(591\) −1.46787e9 −0.292503
\(592\) −6.18824e8 −0.122586
\(593\) 3.28316e9 0.646547 0.323274 0.946306i \(-0.395217\pi\)
0.323274 + 0.946306i \(0.395217\pi\)
\(594\) 1.04184e7 0.00203962
\(595\) −1.47830e9 −0.287710
\(596\) −9.81920e9 −1.89983
\(597\) 3.80363e9 0.731625
\(598\) −5.64125e8 −0.107875
\(599\) 7.40433e8 0.140764 0.0703820 0.997520i \(-0.477578\pi\)
0.0703820 + 0.997520i \(0.477578\pi\)
\(600\) 7.04507e7 0.0133155
\(601\) −2.01096e9 −0.377870 −0.188935 0.981990i \(-0.560504\pi\)
−0.188935 + 0.981990i \(0.560504\pi\)
\(602\) −7.95245e7 −0.0148564
\(603\) −2.26746e9 −0.421143
\(604\) −5.47545e9 −1.01109
\(605\) −4.12487e9 −0.757298
\(606\) −1.08628e8 −0.0198285
\(607\) −4.88190e9 −0.885990 −0.442995 0.896524i \(-0.646084\pi\)
−0.442995 + 0.896524i \(0.646084\pi\)
\(608\) −3.55718e7 −0.00641864
\(609\) 5.11146e8 0.0917033
\(610\) −2.63493e8 −0.0470017
\(611\) −3.40744e9 −0.604344
\(612\) 1.97243e9 0.347834
\(613\) −6.53926e9 −1.14661 −0.573306 0.819341i \(-0.694339\pi\)
−0.573306 + 0.819341i \(0.694339\pi\)
\(614\) −6.93517e7 −0.0120912
\(615\) −2.84995e8 −0.0494054
\(616\) −4.09151e7 −0.00705264
\(617\) 1.08583e9 0.186108 0.0930541 0.995661i \(-0.470337\pi\)
0.0930541 + 0.995661i \(0.470337\pi\)
\(618\) −1.30622e8 −0.0222617
\(619\) −1.46677e9 −0.248568 −0.124284 0.992247i \(-0.539663\pi\)
−0.124284 + 0.992247i \(0.539663\pi\)
\(620\) 3.05156e9 0.514222
\(621\) 1.81009e9 0.303305
\(622\) −9.19935e7 −0.0153282
\(623\) −2.10193e9 −0.348266
\(624\) −6.54377e9 −1.07816
\(625\) −3.56453e9 −0.584012
\(626\) −9.24990e7 −0.0150705
\(627\) −6.07359e7 −0.00984032
\(628\) −7.04624e9 −1.13527
\(629\) 8.02657e8 0.128603
\(630\) 2.10296e7 0.00335072
\(631\) 5.56162e9 0.881248 0.440624 0.897692i \(-0.354757\pi\)
0.440624 + 0.897692i \(0.354757\pi\)
\(632\) −4.88656e8 −0.0770005
\(633\) 5.59930e9 0.877446
\(634\) 3.12969e8 0.0487741
\(635\) −2.36760e9 −0.366944
\(636\) 3.72171e9 0.573644
\(637\) 1.08753e10 1.66707
\(638\) 3.31642e7 0.00505588
\(639\) 2.92922e9 0.444119
\(640\) −7.98218e8 −0.120363
\(641\) −8.98365e8 −0.134726 −0.0673628 0.997729i \(-0.521458\pi\)
−0.0673628 + 0.997729i \(0.521458\pi\)
\(642\) −2.59833e8 −0.0387545
\(643\) 6.66060e9 0.988042 0.494021 0.869450i \(-0.335527\pi\)
0.494021 + 0.869450i \(0.335527\pi\)
\(644\) −3.55192e9 −0.524038
\(645\) −3.97701e9 −0.583576
\(646\) 1.53455e7 0.00223958
\(647\) 2.89308e9 0.419948 0.209974 0.977707i \(-0.432662\pi\)
0.209974 + 0.977707i \(0.432662\pi\)
\(648\) −5.61550e7 −0.00810729
\(649\) −2.63199e8 −0.0377944
\(650\) 1.51480e8 0.0216350
\(651\) −8.42519e8 −0.119687
\(652\) −4.81074e9 −0.679743
\(653\) 5.97319e9 0.839480 0.419740 0.907644i \(-0.362121\pi\)
0.419740 + 0.907644i \(0.362121\pi\)
\(654\) 2.05604e8 0.0287415
\(655\) 5.07769e9 0.706028
\(656\) −7.45172e8 −0.103061
\(657\) −4.27501e9 −0.588110
\(658\) 2.86320e7 0.00391796
\(659\) 8.14255e9 1.10831 0.554155 0.832414i \(-0.313041\pi\)
0.554155 + 0.832414i \(0.313041\pi\)
\(660\) −1.02240e9 −0.138425
\(661\) 3.91035e9 0.526635 0.263318 0.964709i \(-0.415183\pi\)
0.263318 + 0.964709i \(0.415183\pi\)
\(662\) 1.23186e8 0.0165028
\(663\) 8.48772e9 1.13108
\(664\) −1.00551e9 −0.133290
\(665\) −1.22595e8 −0.0161658
\(666\) −1.14182e7 −0.00149774
\(667\) 5.76193e9 0.751844
\(668\) −1.18329e10 −1.53594
\(669\) −6.94828e9 −0.897193
\(670\) −2.96956e8 −0.0381444
\(671\) −3.53685e9 −0.451947
\(672\) 1.65325e8 0.0210158
\(673\) −8.56370e9 −1.08295 −0.541475 0.840717i \(-0.682134\pi\)
−0.541475 + 0.840717i \(0.682134\pi\)
\(674\) −5.36505e8 −0.0674939
\(675\) −4.86049e8 −0.0608298
\(676\) −2.01755e10 −2.51195
\(677\) −1.32503e10 −1.64122 −0.820608 0.571492i \(-0.806365\pi\)
−0.820608 + 0.571492i \(0.806365\pi\)
\(678\) −1.88923e8 −0.0232799
\(679\) −2.88928e9 −0.354198
\(680\) 5.16981e8 0.0630512
\(681\) −3.45776e9 −0.419546
\(682\) −5.46643e7 −0.00659870
\(683\) 1.52557e10 1.83214 0.916070 0.401018i \(-0.131344\pi\)
0.916070 + 0.401018i \(0.131344\pi\)
\(684\) 1.63573e8 0.0195441
\(685\) 4.07997e8 0.0484998
\(686\) −1.94159e8 −0.0229627
\(687\) 2.45942e9 0.289391
\(688\) −1.03986e10 −1.21735
\(689\) 1.60151e10 1.86536
\(690\) 2.37057e8 0.0274714
\(691\) 2.37047e9 0.273314 0.136657 0.990618i \(-0.456364\pi\)
0.136657 + 0.990618i \(0.456364\pi\)
\(692\) 1.40100e10 1.60719
\(693\) 2.82279e8 0.0322190
\(694\) 4.68936e8 0.0532544
\(695\) −5.61716e9 −0.634702
\(696\) −1.78754e8 −0.0200966
\(697\) 9.66540e8 0.108120
\(698\) 9.27960e7 0.0103285
\(699\) −1.69756e9 −0.187999
\(700\) 9.53766e8 0.105099
\(701\) −8.71012e9 −0.955017 −0.477508 0.878627i \(-0.658460\pi\)
−0.477508 + 0.878627i \(0.658460\pi\)
\(702\) −1.20742e8 −0.0131728
\(703\) 6.65641e7 0.00722598
\(704\) −2.66609e9 −0.287985
\(705\) 1.43188e9 0.153902
\(706\) 4.15914e8 0.0444823
\(707\) −2.94320e9 −0.313221
\(708\) 7.08844e8 0.0750644
\(709\) −8.22562e9 −0.866775 −0.433388 0.901208i \(-0.642682\pi\)
−0.433388 + 0.901208i \(0.642682\pi\)
\(710\) 3.83623e8 0.0402254
\(711\) 3.37130e9 0.351766
\(712\) 7.35071e8 0.0763219
\(713\) −9.49736e9 −0.981272
\(714\) −7.13203e7 −0.00733279
\(715\) −4.39955e9 −0.450130
\(716\) −5.56930e9 −0.567029
\(717\) −5.71245e9 −0.578770
\(718\) −2.20710e7 −0.00222529
\(719\) −5.78538e9 −0.580471 −0.290236 0.956955i \(-0.593734\pi\)
−0.290236 + 0.956955i \(0.593734\pi\)
\(720\) 2.74983e9 0.274563
\(721\) −3.53911e9 −0.351658
\(722\) −3.67924e8 −0.0363813
\(723\) 3.81292e8 0.0375210
\(724\) 1.65873e10 1.62440
\(725\) −1.54720e9 −0.150787
\(726\) −1.99003e8 −0.0193011
\(727\) −1.00474e10 −0.969798 −0.484899 0.874570i \(-0.661144\pi\)
−0.484899 + 0.874570i \(0.661144\pi\)
\(728\) 4.74176e8 0.0455491
\(729\) 3.87420e8 0.0370370
\(730\) −5.59874e8 −0.0532672
\(731\) 1.34877e10 1.27711
\(732\) 9.52539e9 0.897623
\(733\) 7.89455e9 0.740395 0.370198 0.928953i \(-0.379290\pi\)
0.370198 + 0.928953i \(0.379290\pi\)
\(734\) 5.45800e7 0.00509445
\(735\) −4.57003e9 −0.424535
\(736\) 1.86364e9 0.172301
\(737\) −3.98603e9 −0.366779
\(738\) −1.37495e7 −0.00125918
\(739\) 8.45416e9 0.770575 0.385288 0.922797i \(-0.374102\pi\)
0.385288 + 0.922797i \(0.374102\pi\)
\(740\) 1.12051e9 0.101649
\(741\) 7.03884e8 0.0635532
\(742\) −1.34571e8 −0.0120931
\(743\) 5.05511e9 0.452137 0.226068 0.974111i \(-0.427413\pi\)
0.226068 + 0.974111i \(0.427413\pi\)
\(744\) 2.94639e8 0.0262292
\(745\) 1.77559e10 1.57324
\(746\) 4.82097e7 0.00425156
\(747\) 6.93715e9 0.608919
\(748\) 3.46739e9 0.302933
\(749\) −7.03997e9 −0.612187
\(750\) −2.65043e8 −0.0229405
\(751\) −1.42778e10 −1.23005 −0.615023 0.788509i \(-0.710853\pi\)
−0.615023 + 0.788509i \(0.710853\pi\)
\(752\) 3.74392e9 0.321043
\(753\) 2.19848e9 0.187646
\(754\) −3.84348e8 −0.0326532
\(755\) 9.90116e9 0.837282
\(756\) −7.60230e8 −0.0639909
\(757\) 1.01019e10 0.846388 0.423194 0.906039i \(-0.360909\pi\)
0.423194 + 0.906039i \(0.360909\pi\)
\(758\) 6.36864e8 0.0531135
\(759\) 3.18201e9 0.264153
\(760\) 4.28731e7 0.00354272
\(761\) −3.03041e9 −0.249261 −0.124631 0.992203i \(-0.539775\pi\)
−0.124631 + 0.992203i \(0.539775\pi\)
\(762\) −1.14224e8 −0.00935222
\(763\) 5.57067e9 0.454016
\(764\) −3.92859e9 −0.318720
\(765\) −3.56672e9 −0.288041
\(766\) −4.58464e8 −0.0368557
\(767\) 3.05028e9 0.244093
\(768\) 7.15135e9 0.569671
\(769\) −1.80433e10 −1.43078 −0.715392 0.698723i \(-0.753752\pi\)
−0.715392 + 0.698723i \(0.753752\pi\)
\(770\) 3.69684e7 0.00291819
\(771\) 5.97633e9 0.469618
\(772\) −1.59081e10 −1.24440
\(773\) −1.03732e10 −0.807766 −0.403883 0.914811i \(-0.632340\pi\)
−0.403883 + 0.914811i \(0.632340\pi\)
\(774\) −1.91869e8 −0.0148735
\(775\) 2.55024e9 0.196800
\(776\) 1.01042e9 0.0776220
\(777\) −3.09366e8 −0.0236592
\(778\) 1.45439e8 0.0110726
\(779\) 8.01549e7 0.00607503
\(780\) 1.18488e10 0.894014
\(781\) 5.14936e9 0.386789
\(782\) −8.03963e8 −0.0601191
\(783\) 1.23325e9 0.0918087
\(784\) −1.19492e10 −0.885590
\(785\) 1.27416e10 0.940113
\(786\) 2.44971e8 0.0179944
\(787\) −1.10014e10 −0.804518 −0.402259 0.915526i \(-0.631775\pi\)
−0.402259 + 0.915526i \(0.631775\pi\)
\(788\) 6.94950e9 0.505955
\(789\) 1.21302e10 0.879222
\(790\) 4.41520e8 0.0318607
\(791\) −5.11871e9 −0.367742
\(792\) −9.87163e7 −0.00706075
\(793\) 4.09894e10 2.91887
\(794\) 5.09067e8 0.0360913
\(795\) −6.72990e9 −0.475033
\(796\) −1.80080e10 −1.26552
\(797\) 1.83467e10 1.28367 0.641834 0.766843i \(-0.278174\pi\)
0.641834 + 0.766843i \(0.278174\pi\)
\(798\) −5.91457e6 −0.000412015 0
\(799\) −4.85612e9 −0.336803
\(800\) −5.00426e8 −0.0345561
\(801\) −5.07135e9 −0.348666
\(802\) −8.07085e8 −0.0552470
\(803\) −7.51515e9 −0.512193
\(804\) 1.07351e10 0.728469
\(805\) 6.42288e9 0.433954
\(806\) 6.33518e8 0.0426174
\(807\) −1.08058e10 −0.723767
\(808\) 1.02927e9 0.0686420
\(809\) 1.83197e10 1.21646 0.608231 0.793760i \(-0.291879\pi\)
0.608231 + 0.793760i \(0.291879\pi\)
\(810\) 5.07382e7 0.00335458
\(811\) −8.15379e9 −0.536768 −0.268384 0.963312i \(-0.586490\pi\)
−0.268384 + 0.963312i \(0.586490\pi\)
\(812\) −2.41998e9 −0.158623
\(813\) 6.02376e9 0.393143
\(814\) −2.00723e7 −0.00130440
\(815\) 8.69918e9 0.562894
\(816\) −9.32585e9 −0.600859
\(817\) 1.11853e9 0.0717583
\(818\) −3.33195e8 −0.0212844
\(819\) −3.27140e9 −0.208085
\(820\) 1.34929e9 0.0854586
\(821\) −1.71557e10 −1.08195 −0.540977 0.841038i \(-0.681945\pi\)
−0.540977 + 0.841038i \(0.681945\pi\)
\(822\) 1.96837e7 0.00123610
\(823\) 3.13687e10 1.96154 0.980771 0.195160i \(-0.0625227\pi\)
0.980771 + 0.195160i \(0.0625227\pi\)
\(824\) 1.23767e9 0.0770653
\(825\) −8.54437e8 −0.0529775
\(826\) −2.56308e7 −0.00158246
\(827\) −3.08831e10 −1.89868 −0.949338 0.314256i \(-0.898245\pi\)
−0.949338 + 0.314256i \(0.898245\pi\)
\(828\) −8.56974e9 −0.524640
\(829\) 2.08621e10 1.27180 0.635899 0.771773i \(-0.280630\pi\)
0.635899 + 0.771773i \(0.280630\pi\)
\(830\) 9.08519e8 0.0551519
\(831\) 7.67713e9 0.464083
\(832\) 3.08980e10 1.85994
\(833\) 1.54989e10 0.929062
\(834\) −2.70998e8 −0.0161765
\(835\) 2.13973e10 1.27191
\(836\) 2.87550e8 0.0170212
\(837\) −2.03275e9 −0.119824
\(838\) −5.67784e8 −0.0333295
\(839\) 3.33343e10 1.94861 0.974303 0.225241i \(-0.0723169\pi\)
0.974303 + 0.225241i \(0.0723169\pi\)
\(840\) −1.99259e8 −0.0115995
\(841\) −1.33242e10 −0.772421
\(842\) −3.51615e8 −0.0202990
\(843\) −8.79040e9 −0.505373
\(844\) −2.65094e10 −1.51776
\(845\) 3.64831e10 2.08014
\(846\) 6.90805e7 0.00392247
\(847\) −5.39183e9 −0.304891
\(848\) −1.75966e10 −0.990930
\(849\) −1.26004e9 −0.0706654
\(850\) 2.15881e8 0.0120573
\(851\) −3.48735e9 −0.193973
\(852\) −1.38682e10 −0.768212
\(853\) −5.30610e9 −0.292721 −0.146360 0.989231i \(-0.546756\pi\)
−0.146360 + 0.989231i \(0.546756\pi\)
\(854\) −3.44424e8 −0.0189231
\(855\) −2.95787e8 −0.0161844
\(856\) 2.46196e9 0.134160
\(857\) −3.22916e9 −0.175250 −0.0876248 0.996154i \(-0.527928\pi\)
−0.0876248 + 0.996154i \(0.527928\pi\)
\(858\) −2.12255e8 −0.0114723
\(859\) −1.09723e10 −0.590637 −0.295318 0.955399i \(-0.595426\pi\)
−0.295318 + 0.955399i \(0.595426\pi\)
\(860\) 1.88288e10 1.00944
\(861\) −3.72531e8 −0.0198908
\(862\) 1.55620e8 0.00827540
\(863\) 1.83908e10 0.974009 0.487005 0.873399i \(-0.338090\pi\)
0.487005 + 0.873399i \(0.338090\pi\)
\(864\) 3.98881e8 0.0210400
\(865\) −2.53340e10 −1.33091
\(866\) 4.71159e8 0.0246522
\(867\) 1.01712e9 0.0530035
\(868\) 3.98884e9 0.207027
\(869\) 5.92650e9 0.306358
\(870\) 1.61511e8 0.00831544
\(871\) 4.61951e10 2.36882
\(872\) −1.94813e9 −0.0994970
\(873\) −6.97100e9 −0.354605
\(874\) −6.66724e7 −0.00337797
\(875\) −7.18113e9 −0.362380
\(876\) 2.02397e10 1.01728
\(877\) 2.93823e10 1.47091 0.735456 0.677572i \(-0.236968\pi\)
0.735456 + 0.677572i \(0.236968\pi\)
\(878\) 4.75068e8 0.0236878
\(879\) −2.07974e9 −0.103287
\(880\) 4.83399e9 0.239120
\(881\) 7.96639e9 0.392506 0.196253 0.980553i \(-0.437123\pi\)
0.196253 + 0.980553i \(0.437123\pi\)
\(882\) −2.20480e8 −0.0108200
\(883\) −2.11047e10 −1.03161 −0.515806 0.856706i \(-0.672507\pi\)
−0.515806 + 0.856706i \(0.672507\pi\)
\(884\) −4.01844e10 −1.95648
\(885\) −1.28179e9 −0.0621607
\(886\) 2.06374e8 0.00996868
\(887\) −3.91929e10 −1.88571 −0.942854 0.333206i \(-0.891870\pi\)
−0.942854 + 0.333206i \(0.891870\pi\)
\(888\) 1.08189e8 0.00518487
\(889\) −3.09480e9 −0.147733
\(890\) −6.64165e8 −0.0315799
\(891\) 6.81056e8 0.0322560
\(892\) 3.28961e10 1.55191
\(893\) −4.02717e8 −0.0189243
\(894\) 8.56627e8 0.0400969
\(895\) 1.00709e10 0.469555
\(896\) −1.04339e9 −0.0484584
\(897\) −3.68771e10 −1.70602
\(898\) 7.04325e8 0.0324568
\(899\) −6.47071e9 −0.297025
\(900\) 2.30116e9 0.105220
\(901\) 2.28240e10 1.03957
\(902\) −2.41706e7 −0.00109664
\(903\) −5.19855e9 −0.234950
\(904\) 1.79007e9 0.0805901
\(905\) −2.99946e10 −1.34516
\(906\) 4.77678e8 0.0213396
\(907\) 7.45412e9 0.331719 0.165860 0.986149i \(-0.446960\pi\)
0.165860 + 0.986149i \(0.446960\pi\)
\(908\) 1.63705e10 0.725707
\(909\) −7.10107e9 −0.313581
\(910\) −4.28436e8 −0.0188470
\(911\) 3.46618e9 0.151893 0.0759463 0.997112i \(-0.475802\pi\)
0.0759463 + 0.997112i \(0.475802\pi\)
\(912\) −7.73390e8 −0.0337611
\(913\) 1.21950e10 0.530315
\(914\) −2.51989e8 −0.0109162
\(915\) −1.72246e10 −0.743319
\(916\) −1.16440e10 −0.500572
\(917\) 6.63730e9 0.284249
\(918\) −1.72075e8 −0.00734122
\(919\) −1.94364e9 −0.0826060 −0.0413030 0.999147i \(-0.513151\pi\)
−0.0413030 + 0.999147i \(0.513151\pi\)
\(920\) −2.24616e9 −0.0951005
\(921\) −4.53355e9 −0.191218
\(922\) 8.63700e8 0.0362915
\(923\) −5.96772e10 −2.49806
\(924\) −1.33643e9 −0.0557306
\(925\) 9.36428e8 0.0389026
\(926\) −1.59949e8 −0.00661979
\(927\) −8.53883e9 −0.352062
\(928\) 1.26973e9 0.0521546
\(929\) 3.65536e10 1.49581 0.747904 0.663807i \(-0.231061\pi\)
0.747904 + 0.663807i \(0.231061\pi\)
\(930\) −2.66218e8 −0.0108529
\(931\) 1.28532e9 0.0522022
\(932\) 8.03696e9 0.325189
\(933\) −6.01365e9 −0.242411
\(934\) 1.01978e8 0.00409536
\(935\) −6.27002e9 −0.250858
\(936\) 1.14405e9 0.0456014
\(937\) −2.70613e10 −1.07463 −0.537317 0.843380i \(-0.680562\pi\)
−0.537317 + 0.843380i \(0.680562\pi\)
\(938\) −3.88167e8 −0.0153571
\(939\) −6.04669e9 −0.238335
\(940\) −6.77912e9 −0.266211
\(941\) −3.52735e10 −1.38002 −0.690010 0.723800i \(-0.742394\pi\)
−0.690010 + 0.723800i \(0.742394\pi\)
\(942\) 6.14714e8 0.0239604
\(943\) −4.19938e9 −0.163078
\(944\) −3.35148e9 −0.129669
\(945\) 1.37471e9 0.0529908
\(946\) −3.37292e8 −0.0129535
\(947\) −1.60900e10 −0.615648 −0.307824 0.951443i \(-0.599601\pi\)
−0.307824 + 0.951443i \(0.599601\pi\)
\(948\) −1.59612e10 −0.608465
\(949\) 8.70950e10 3.30797
\(950\) 1.79030e7 0.000677474 0
\(951\) 2.04589e10 0.771349
\(952\) 6.75771e8 0.0253846
\(953\) 2.34617e10 0.878081 0.439040 0.898467i \(-0.355319\pi\)
0.439040 + 0.898467i \(0.355319\pi\)
\(954\) −3.24682e8 −0.0121071
\(955\) 7.10400e9 0.263932
\(956\) 2.70452e10 1.00112
\(957\) 2.16796e9 0.0799574
\(958\) −1.36035e9 −0.0499888
\(959\) 5.33313e8 0.0195262
\(960\) −1.29840e10 −0.473651
\(961\) −1.68470e10 −0.612337
\(962\) 2.32623e8 0.00842441
\(963\) −1.69854e10 −0.612891
\(964\) −1.80520e9 −0.0649016
\(965\) 2.87664e10 1.03048
\(966\) 3.09869e8 0.0110601
\(967\) 2.75570e10 0.980029 0.490014 0.871714i \(-0.336992\pi\)
0.490014 + 0.871714i \(0.336992\pi\)
\(968\) 1.88559e9 0.0668163
\(969\) 1.00314e9 0.0354183
\(970\) −9.12951e8 −0.0321179
\(971\) −3.36601e10 −1.17991 −0.589954 0.807437i \(-0.700854\pi\)
−0.589954 + 0.807437i \(0.700854\pi\)
\(972\) −1.83421e9 −0.0640645
\(973\) −7.34247e9 −0.255533
\(974\) 6.79111e8 0.0235496
\(975\) 9.90228e9 0.342152
\(976\) −4.50370e10 −1.55058
\(977\) −3.08878e9 −0.105963 −0.0529817 0.998595i \(-0.516872\pi\)
−0.0529817 + 0.998595i \(0.516872\pi\)
\(978\) 4.19689e8 0.0143463
\(979\) −8.91505e9 −0.303658
\(980\) 2.16365e10 0.734337
\(981\) 1.34404e10 0.454538
\(982\) 1.17779e9 0.0396896
\(983\) −4.00750e10 −1.34566 −0.672832 0.739796i \(-0.734922\pi\)
−0.672832 + 0.739796i \(0.734922\pi\)
\(984\) 1.30279e8 0.00435903
\(985\) −1.25667e10 −0.418980
\(986\) −5.47754e8 −0.0181977
\(987\) 1.87168e9 0.0619615
\(988\) −3.33248e9 −0.109931
\(989\) −5.86010e10 −1.92627
\(990\) 8.91940e7 0.00292154
\(991\) −3.18674e9 −0.104013 −0.0520066 0.998647i \(-0.516562\pi\)
−0.0520066 + 0.998647i \(0.516562\pi\)
\(992\) −2.09288e9 −0.0680697
\(993\) 8.05272e9 0.260988
\(994\) 5.01453e8 0.0161949
\(995\) 3.25636e10 1.04798
\(996\) −3.28434e10 −1.05327
\(997\) −4.16475e8 −0.0133093 −0.00665465 0.999978i \(-0.502118\pi\)
−0.00665465 + 0.999978i \(0.502118\pi\)
\(998\) 1.54231e9 0.0491150
\(999\) −7.46410e8 −0.0236864
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.8.a.b.1.10 17
3.2 odd 2 531.8.a.d.1.8 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.10 17 1.1 even 1 trivial
531.8.a.d.1.8 17 3.2 odd 2