Properties

Label 38.6.a.c.1.1
Level $38$
Weight $6$
Character 38.1
Self dual yes
Analytic conductor $6.095$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.09458515289\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1441}) \)
Defining polynomial: \(x^{2} - x - 360\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(19.4803\) of defining polynomial
Character \(\chi\) \(=\) 38.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} -17.4803 q^{3} +16.0000 q^{4} -79.4408 q^{5} +69.9210 q^{6} +132.921 q^{7} -64.0000 q^{8} +62.5592 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -17.4803 q^{3} +16.0000 q^{4} -79.4408 q^{5} +69.9210 q^{6} +132.921 q^{7} -64.0000 q^{8} +62.5592 q^{9} +317.763 q^{10} +311.520 q^{11} -279.684 q^{12} +901.401 q^{13} -531.684 q^{14} +1388.64 q^{15} +256.000 q^{16} -157.803 q^{17} -250.237 q^{18} +361.000 q^{19} -1271.05 q^{20} -2323.49 q^{21} -1246.08 q^{22} -2522.53 q^{23} +1118.74 q^{24} +3185.83 q^{25} -3605.61 q^{26} +3154.15 q^{27} +2126.74 q^{28} +4738.28 q^{29} -5554.58 q^{30} -6587.76 q^{31} -1024.00 q^{32} -5445.44 q^{33} +631.210 q^{34} -10559.3 q^{35} +1000.95 q^{36} +8508.60 q^{37} -1444.00 q^{38} -15756.7 q^{39} +5084.21 q^{40} +19741.1 q^{41} +9293.97 q^{42} +10985.0 q^{43} +4984.32 q^{44} -4969.75 q^{45} +10090.1 q^{46} +15085.5 q^{47} -4474.94 q^{48} +860.995 q^{49} -12743.3 q^{50} +2758.43 q^{51} +14422.4 q^{52} +21699.6 q^{53} -12616.6 q^{54} -24747.4 q^{55} -8506.94 q^{56} -6310.37 q^{57} -18953.1 q^{58} -40676.1 q^{59} +22218.3 q^{60} +6151.79 q^{61} +26351.0 q^{62} +8315.44 q^{63} +4096.00 q^{64} -71608.0 q^{65} +21781.8 q^{66} +62760.3 q^{67} -2524.84 q^{68} +44094.5 q^{69} +42237.4 q^{70} -55311.0 q^{71} -4003.79 q^{72} -48528.1 q^{73} -34034.4 q^{74} -55689.2 q^{75} +5776.00 q^{76} +41407.5 q^{77} +63026.9 q^{78} +31017.6 q^{79} -20336.8 q^{80} -70337.2 q^{81} -78964.4 q^{82} +41068.7 q^{83} -37175.9 q^{84} +12536.0 q^{85} -43939.8 q^{86} -82826.3 q^{87} -19937.3 q^{88} -17065.6 q^{89} +19879.0 q^{90} +119815. q^{91} -40360.5 q^{92} +115156. q^{93} -60341.9 q^{94} -28678.1 q^{95} +17899.8 q^{96} +139045. q^{97} -3443.98 q^{98} +19488.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{2} + 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} - 128q^{8} + 239q^{9} + O(q^{10}) \) \( 2q - 8q^{2} + 3q^{3} + 32q^{4} - 45q^{5} - 12q^{6} + 114q^{7} - 128q^{8} + 239q^{9} + 180q^{10} + 661q^{11} + 48q^{12} + 1613q^{13} - 456q^{14} + 2094q^{15} + 512q^{16} + 64q^{17} - 956q^{18} + 722q^{19} - 720q^{20} - 2711q^{21} - 2644q^{22} - 3185q^{23} - 192q^{24} + 1247q^{25} - 6452q^{26} + 1791q^{27} + 1824q^{28} - 2481q^{29} - 8376q^{30} - 1180q^{31} - 2048q^{32} + 1712q^{33} - 256q^{34} - 11211q^{35} + 3824q^{36} + 10488q^{37} - 2888q^{38} - 1183q^{39} + 2880q^{40} + 16630q^{41} + 10844q^{42} + 11303q^{43} + 10576q^{44} + 1107q^{45} + 12740q^{46} - 12155q^{47} + 768q^{48} - 15588q^{49} - 4988q^{50} + 7301q^{51} + 25808q^{52} + 20585q^{53} - 7164q^{54} - 12711q^{55} - 7296q^{56} + 1083q^{57} + 9924q^{58} - 78581q^{59} + 33504q^{60} + 43621q^{61} + 4720q^{62} + 4977q^{63} + 8192q^{64} - 47100q^{65} - 6848q^{66} + 7805q^{67} + 1024q^{68} + 30527q^{69} + 44844q^{70} - 62488q^{71} - 15296q^{72} + 16218q^{73} - 41952q^{74} - 95397q^{75} + 11552q^{76} + 34795q^{77} + 4732q^{78} + 67122q^{79} - 11520q^{80} - 141130q^{81} - 66520q^{82} - 10714q^{83} - 43376q^{84} + 20175q^{85} - 45212q^{86} - 230679q^{87} - 42304q^{88} + 128188q^{89} - 4428q^{90} + 106351q^{91} - 50960q^{92} + 225908q^{93} + 48620q^{94} - 16245q^{95} - 3072q^{96} + 178558q^{97} + 62352q^{98} + 81151q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −17.4803 −1.12136 −0.560679 0.828033i \(-0.689460\pi\)
−0.560679 + 0.828033i \(0.689460\pi\)
\(4\) 16.0000 0.500000
\(5\) −79.4408 −1.42108 −0.710540 0.703657i \(-0.751549\pi\)
−0.710540 + 0.703657i \(0.751549\pi\)
\(6\) 69.9210 0.792920
\(7\) 132.921 1.02529 0.512647 0.858599i \(-0.328665\pi\)
0.512647 + 0.858599i \(0.328665\pi\)
\(8\) −64.0000 −0.353553
\(9\) 62.5592 0.257445
\(10\) 317.763 1.00485
\(11\) 311.520 0.776254 0.388127 0.921606i \(-0.373122\pi\)
0.388127 + 0.921606i \(0.373122\pi\)
\(12\) −279.684 −0.560679
\(13\) 901.401 1.47931 0.739656 0.672985i \(-0.234988\pi\)
0.739656 + 0.672985i \(0.234988\pi\)
\(14\) −531.684 −0.724993
\(15\) 1388.64 1.59354
\(16\) 256.000 0.250000
\(17\) −157.803 −0.132432 −0.0662158 0.997805i \(-0.521093\pi\)
−0.0662158 + 0.997805i \(0.521093\pi\)
\(18\) −250.237 −0.182041
\(19\) 361.000 0.229416
\(20\) −1271.05 −0.710540
\(21\) −2323.49 −1.14972
\(22\) −1246.08 −0.548894
\(23\) −2522.53 −0.994299 −0.497150 0.867665i \(-0.665620\pi\)
−0.497150 + 0.867665i \(0.665620\pi\)
\(24\) 1118.74 0.396460
\(25\) 3185.83 1.01947
\(26\) −3605.61 −1.04603
\(27\) 3154.15 0.832670
\(28\) 2126.74 0.512647
\(29\) 4738.28 1.04623 0.523113 0.852263i \(-0.324770\pi\)
0.523113 + 0.852263i \(0.324770\pi\)
\(30\) −5554.58 −1.12680
\(31\) −6587.76 −1.23121 −0.615607 0.788053i \(-0.711089\pi\)
−0.615607 + 0.788053i \(0.711089\pi\)
\(32\) −1024.00 −0.176777
\(33\) −5445.44 −0.870459
\(34\) 631.210 0.0936433
\(35\) −10559.3 −1.45702
\(36\) 1000.95 0.128723
\(37\) 8508.60 1.02177 0.510886 0.859648i \(-0.329317\pi\)
0.510886 + 0.859648i \(0.329317\pi\)
\(38\) −1444.00 −0.162221
\(39\) −15756.7 −1.65884
\(40\) 5084.21 0.502427
\(41\) 19741.1 1.83405 0.917027 0.398826i \(-0.130582\pi\)
0.917027 + 0.398826i \(0.130582\pi\)
\(42\) 9293.97 0.812977
\(43\) 10985.0 0.905997 0.452999 0.891511i \(-0.350354\pi\)
0.452999 + 0.891511i \(0.350354\pi\)
\(44\) 4984.32 0.388127
\(45\) −4969.75 −0.365850
\(46\) 10090.1 0.703076
\(47\) 15085.5 0.996127 0.498063 0.867141i \(-0.334045\pi\)
0.498063 + 0.867141i \(0.334045\pi\)
\(48\) −4474.94 −0.280340
\(49\) 860.995 0.0512284
\(50\) −12743.3 −0.720872
\(51\) 2758.43 0.148503
\(52\) 14422.4 0.739656
\(53\) 21699.6 1.06112 0.530558 0.847649i \(-0.321982\pi\)
0.530558 + 0.847649i \(0.321982\pi\)
\(54\) −12616.6 −0.588787
\(55\) −24747.4 −1.10312
\(56\) −8506.94 −0.362496
\(57\) −6310.37 −0.257257
\(58\) −18953.1 −0.739794
\(59\) −40676.1 −1.52128 −0.760639 0.649175i \(-0.775114\pi\)
−0.760639 + 0.649175i \(0.775114\pi\)
\(60\) 22218.3 0.796770
\(61\) 6151.79 0.211679 0.105839 0.994383i \(-0.466247\pi\)
0.105839 + 0.994383i \(0.466247\pi\)
\(62\) 26351.0 0.870600
\(63\) 8315.44 0.263957
\(64\) 4096.00 0.125000
\(65\) −71608.0 −2.10222
\(66\) 21781.8 0.615508
\(67\) 62760.3 1.70804 0.854019 0.520241i \(-0.174158\pi\)
0.854019 + 0.520241i \(0.174158\pi\)
\(68\) −2524.84 −0.0662158
\(69\) 44094.5 1.11497
\(70\) 42237.4 1.03027
\(71\) −55311.0 −1.30216 −0.651081 0.759008i \(-0.725684\pi\)
−0.651081 + 0.759008i \(0.725684\pi\)
\(72\) −4003.79 −0.0910207
\(73\) −48528.1 −1.06583 −0.532913 0.846170i \(-0.678903\pi\)
−0.532913 + 0.846170i \(0.678903\pi\)
\(74\) −34034.4 −0.722502
\(75\) −55689.2 −1.14319
\(76\) 5776.00 0.114708
\(77\) 41407.5 0.795889
\(78\) 63026.9 1.17298
\(79\) 31017.6 0.559166 0.279583 0.960121i \(-0.409804\pi\)
0.279583 + 0.960121i \(0.409804\pi\)
\(80\) −20336.8 −0.355270
\(81\) −70337.2 −1.19117
\(82\) −78964.4 −1.29687
\(83\) 41068.7 0.654358 0.327179 0.944962i \(-0.393902\pi\)
0.327179 + 0.944962i \(0.393902\pi\)
\(84\) −37175.9 −0.574861
\(85\) 12536.0 0.188196
\(86\) −43939.8 −0.640637
\(87\) −82826.3 −1.17320
\(88\) −19937.3 −0.274447
\(89\) −17065.6 −0.228373 −0.114187 0.993459i \(-0.536426\pi\)
−0.114187 + 0.993459i \(0.536426\pi\)
\(90\) 19879.0 0.258695
\(91\) 119815. 1.51673
\(92\) −40360.5 −0.497150
\(93\) 115156. 1.38063
\(94\) −60341.9 −0.704368
\(95\) −28678.1 −0.326018
\(96\) 17899.8 0.198230
\(97\) 139045. 1.50047 0.750234 0.661172i \(-0.229941\pi\)
0.750234 + 0.661172i \(0.229941\pi\)
\(98\) −3443.98 −0.0362239
\(99\) 19488.4 0.199843
\(100\) 50973.3 0.509733
\(101\) −122253. −1.19249 −0.596247 0.802801i \(-0.703342\pi\)
−0.596247 + 0.802801i \(0.703342\pi\)
\(102\) −11033.7 −0.105008
\(103\) −71932.4 −0.668084 −0.334042 0.942558i \(-0.608413\pi\)
−0.334042 + 0.942558i \(0.608413\pi\)
\(104\) −57689.7 −0.523016
\(105\) 184580. 1.63385
\(106\) −86798.5 −0.750322
\(107\) 14833.3 0.125250 0.0626249 0.998037i \(-0.480053\pi\)
0.0626249 + 0.998037i \(0.480053\pi\)
\(108\) 50466.4 0.416335
\(109\) 140025. 1.12886 0.564429 0.825482i \(-0.309096\pi\)
0.564429 + 0.825482i \(0.309096\pi\)
\(110\) 98989.5 0.780023
\(111\) −148733. −1.14577
\(112\) 34027.8 0.256324
\(113\) 235172. 1.73256 0.866282 0.499555i \(-0.166503\pi\)
0.866282 + 0.499555i \(0.166503\pi\)
\(114\) 25241.5 0.181908
\(115\) 200392. 1.41298
\(116\) 75812.5 0.523113
\(117\) 56391.0 0.380842
\(118\) 162704. 1.07571
\(119\) −20975.3 −0.135781
\(120\) −88873.3 −0.563401
\(121\) −64006.4 −0.397430
\(122\) −24607.2 −0.149679
\(123\) −345080. −2.05663
\(124\) −105404. −0.615607
\(125\) −4832.71 −0.0276640
\(126\) −33261.8 −0.186646
\(127\) −24783.9 −0.136351 −0.0681757 0.997673i \(-0.521718\pi\)
−0.0681757 + 0.997673i \(0.521718\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −192020. −1.01595
\(130\) 286432. 1.48649
\(131\) −152549. −0.776661 −0.388331 0.921520i \(-0.626948\pi\)
−0.388331 + 0.921520i \(0.626948\pi\)
\(132\) −87127.1 −0.435230
\(133\) 47984.5 0.235219
\(134\) −251041. −1.20777
\(135\) −250568. −1.18329
\(136\) 10099.4 0.0468216
\(137\) 192265. 0.875184 0.437592 0.899174i \(-0.355831\pi\)
0.437592 + 0.899174i \(0.355831\pi\)
\(138\) −176378. −0.788400
\(139\) −342833. −1.50503 −0.752515 0.658575i \(-0.771159\pi\)
−0.752515 + 0.658575i \(0.771159\pi\)
\(140\) −168950. −0.728512
\(141\) −263698. −1.11702
\(142\) 221244. 0.920768
\(143\) 280804. 1.14832
\(144\) 16015.2 0.0643614
\(145\) −376413. −1.48677
\(146\) 194112. 0.753652
\(147\) −15050.4 −0.0574454
\(148\) 136138. 0.510886
\(149\) 335859. 1.23934 0.619671 0.784862i \(-0.287266\pi\)
0.619671 + 0.784862i \(0.287266\pi\)
\(150\) 222757. 0.808356
\(151\) 266683. 0.951816 0.475908 0.879495i \(-0.342120\pi\)
0.475908 + 0.879495i \(0.342120\pi\)
\(152\) −23104.0 −0.0811107
\(153\) −9872.01 −0.0340939
\(154\) −165630. −0.562778
\(155\) 523337. 1.74965
\(156\) −252108. −0.829420
\(157\) −173779. −0.562664 −0.281332 0.959611i \(-0.590776\pi\)
−0.281332 + 0.959611i \(0.590776\pi\)
\(158\) −124071. −0.395390
\(159\) −379315. −1.18989
\(160\) 81347.3 0.251214
\(161\) −335298. −1.01945
\(162\) 281349. 0.842282
\(163\) −406317. −1.19783 −0.598917 0.800811i \(-0.704402\pi\)
−0.598917 + 0.800811i \(0.704402\pi\)
\(164\) 315858. 0.917027
\(165\) 432590. 1.23699
\(166\) −164275. −0.462701
\(167\) −319695. −0.887043 −0.443522 0.896264i \(-0.646271\pi\)
−0.443522 + 0.896264i \(0.646271\pi\)
\(168\) 148704. 0.406488
\(169\) 441231. 1.18836
\(170\) −50143.8 −0.133075
\(171\) 22583.9 0.0590620
\(172\) 175759. 0.452999
\(173\) −427313. −1.08550 −0.542752 0.839893i \(-0.682618\pi\)
−0.542752 + 0.839893i \(0.682618\pi\)
\(174\) 331305. 0.829574
\(175\) 423464. 1.04525
\(176\) 79749.1 0.194064
\(177\) 711028. 1.70590
\(178\) 68262.2 0.161484
\(179\) 361946. 0.844329 0.422164 0.906519i \(-0.361271\pi\)
0.422164 + 0.906519i \(0.361271\pi\)
\(180\) −79516.1 −0.182925
\(181\) −416686. −0.945393 −0.472697 0.881225i \(-0.656719\pi\)
−0.472697 + 0.881225i \(0.656719\pi\)
\(182\) −479261. −1.07249
\(183\) −107535. −0.237368
\(184\) 161442. 0.351538
\(185\) −675930. −1.45202
\(186\) −460623. −0.976255
\(187\) −49158.6 −0.102801
\(188\) 241368. 0.498063
\(189\) 419253. 0.853732
\(190\) 114712. 0.230530
\(191\) 581586. 1.15353 0.576767 0.816909i \(-0.304314\pi\)
0.576767 + 0.816909i \(0.304314\pi\)
\(192\) −71599.1 −0.140170
\(193\) −182832. −0.353312 −0.176656 0.984273i \(-0.556528\pi\)
−0.176656 + 0.984273i \(0.556528\pi\)
\(194\) −556181. −1.06099
\(195\) 1.25173e6 2.35734
\(196\) 13775.9 0.0256142
\(197\) −93344.9 −0.171366 −0.0856831 0.996322i \(-0.527307\pi\)
−0.0856831 + 0.996322i \(0.527307\pi\)
\(198\) −77953.8 −0.141310
\(199\) 723166. 1.29451 0.647255 0.762274i \(-0.275917\pi\)
0.647255 + 0.762274i \(0.275917\pi\)
\(200\) −203893. −0.360436
\(201\) −1.09707e6 −1.91532
\(202\) 489012. 0.843221
\(203\) 629817. 1.07269
\(204\) 44134.8 0.0742517
\(205\) −1.56825e6 −2.60634
\(206\) 287730. 0.472407
\(207\) −157808. −0.255978
\(208\) 230759. 0.369828
\(209\) 112459. 0.178085
\(210\) −738320. −1.15530
\(211\) 85741.4 0.132582 0.0662910 0.997800i \(-0.478883\pi\)
0.0662910 + 0.997800i \(0.478883\pi\)
\(212\) 347194. 0.530558
\(213\) 966850. 1.46019
\(214\) −59333.0 −0.0885650
\(215\) −872653. −1.28749
\(216\) −201866. −0.294393
\(217\) −875652. −1.26236
\(218\) −560100. −0.798223
\(219\) 848283. 1.19517
\(220\) −395958. −0.551559
\(221\) −142243. −0.195908
\(222\) 594930. 0.810184
\(223\) 111106. 0.149615 0.0748076 0.997198i \(-0.476166\pi\)
0.0748076 + 0.997198i \(0.476166\pi\)
\(224\) −136111. −0.181248
\(225\) 199303. 0.262457
\(226\) −940688. −1.22511
\(227\) −314299. −0.404835 −0.202418 0.979299i \(-0.564880\pi\)
−0.202418 + 0.979299i \(0.564880\pi\)
\(228\) −100966. −0.128629
\(229\) 1.45815e6 1.83744 0.918718 0.394913i \(-0.129225\pi\)
0.918718 + 0.394913i \(0.129225\pi\)
\(230\) −801568. −0.999127
\(231\) −723814. −0.892477
\(232\) −303250. −0.369897
\(233\) 1.38626e6 1.67284 0.836418 0.548091i \(-0.184645\pi\)
0.836418 + 0.548091i \(0.184645\pi\)
\(234\) −225564. −0.269296
\(235\) −1.19840e6 −1.41558
\(236\) −650817. −0.760639
\(237\) −542196. −0.627026
\(238\) 83901.1 0.0960119
\(239\) 365117. 0.413464 0.206732 0.978398i \(-0.433717\pi\)
0.206732 + 0.978398i \(0.433717\pi\)
\(240\) 355493. 0.398385
\(241\) −312312. −0.346374 −0.173187 0.984889i \(-0.555407\pi\)
−0.173187 + 0.984889i \(0.555407\pi\)
\(242\) 256026. 0.281025
\(243\) 463054. 0.503056
\(244\) 98428.7 0.105839
\(245\) −68398.1 −0.0727996
\(246\) 1.38032e6 1.45426
\(247\) 325406. 0.339377
\(248\) 421617. 0.435300
\(249\) −717891. −0.733771
\(250\) 19330.8 0.0195614
\(251\) −970095. −0.971919 −0.485959 0.873981i \(-0.661530\pi\)
−0.485959 + 0.873981i \(0.661530\pi\)
\(252\) 133047. 0.131979
\(253\) −785819. −0.771829
\(254\) 99135.5 0.0964151
\(255\) −219132. −0.211035
\(256\) 65536.0 0.0625000
\(257\) 1.28988e6 1.21819 0.609096 0.793096i \(-0.291532\pi\)
0.609096 + 0.793096i \(0.291532\pi\)
\(258\) 768079. 0.718384
\(259\) 1.13097e6 1.04762
\(260\) −1.14573e6 −1.05111
\(261\) 296423. 0.269346
\(262\) 610197. 0.549183
\(263\) −501426. −0.447011 −0.223505 0.974703i \(-0.571750\pi\)
−0.223505 + 0.974703i \(0.571750\pi\)
\(264\) 348508. 0.307754
\(265\) −1.72384e6 −1.50793
\(266\) −191938. −0.166325
\(267\) 298310. 0.256089
\(268\) 1.00416e6 0.854019
\(269\) 1.42986e6 1.20479 0.602397 0.798197i \(-0.294213\pi\)
0.602397 + 0.798197i \(0.294213\pi\)
\(270\) 1.00227e6 0.836713
\(271\) −709506. −0.586857 −0.293429 0.955981i \(-0.594796\pi\)
−0.293429 + 0.955981i \(0.594796\pi\)
\(272\) −40397.4 −0.0331079
\(273\) −2.09440e6 −1.70080
\(274\) −769061. −0.618849
\(275\) 992450. 0.791365
\(276\) 705512. 0.557483
\(277\) −766740. −0.600411 −0.300205 0.953875i \(-0.597055\pi\)
−0.300205 + 0.953875i \(0.597055\pi\)
\(278\) 1.37133e6 1.06422
\(279\) −412125. −0.316970
\(280\) 675798. 0.515136
\(281\) 5975.32 0.00451435 0.00225718 0.999997i \(-0.499282\pi\)
0.00225718 + 0.999997i \(0.499282\pi\)
\(282\) 1.05479e6 0.789849
\(283\) −189246. −0.140463 −0.0702314 0.997531i \(-0.522374\pi\)
−0.0702314 + 0.997531i \(0.522374\pi\)
\(284\) −884975. −0.651081
\(285\) 501301. 0.365583
\(286\) −1.12322e6 −0.811986
\(287\) 2.62401e6 1.88044
\(288\) −64060.7 −0.0455104
\(289\) −1.39496e6 −0.982462
\(290\) 1.50565e6 1.05131
\(291\) −2.43055e6 −1.68256
\(292\) −776449. −0.532913
\(293\) 986180. 0.671100 0.335550 0.942022i \(-0.391078\pi\)
0.335550 + 0.942022i \(0.391078\pi\)
\(294\) 60201.7 0.0406200
\(295\) 3.23134e6 2.16186
\(296\) −544551. −0.361251
\(297\) 982580. 0.646363
\(298\) −1.34343e6 −0.876347
\(299\) −2.27381e6 −1.47088
\(300\) −891027. −0.571594
\(301\) 1.46013e6 0.928914
\(302\) −1.06673e6 −0.673035
\(303\) 2.13702e6 1.33721
\(304\) 92416.0 0.0573539
\(305\) −488703. −0.300812
\(306\) 39488.0 0.0241080
\(307\) −1.45722e6 −0.882428 −0.441214 0.897402i \(-0.645452\pi\)
−0.441214 + 0.897402i \(0.645452\pi\)
\(308\) 662520. 0.397944
\(309\) 1.25740e6 0.749162
\(310\) −2.09335e6 −1.23719
\(311\) 2.40526e6 1.41014 0.705069 0.709138i \(-0.250916\pi\)
0.705069 + 0.709138i \(0.250916\pi\)
\(312\) 1.00843e6 0.586488
\(313\) −3.20380e6 −1.84843 −0.924217 0.381867i \(-0.875281\pi\)
−0.924217 + 0.381867i \(0.875281\pi\)
\(314\) 695117. 0.397863
\(315\) −660585. −0.375104
\(316\) 496282. 0.279583
\(317\) −603046. −0.337056 −0.168528 0.985697i \(-0.553901\pi\)
−0.168528 + 0.985697i \(0.553901\pi\)
\(318\) 1.51726e6 0.841380
\(319\) 1.47607e6 0.812137
\(320\) −325389. −0.177635
\(321\) −259289. −0.140450
\(322\) 1.34119e6 0.720860
\(323\) −56966.7 −0.0303819
\(324\) −1.12540e6 −0.595584
\(325\) 2.87171e6 1.50811
\(326\) 1.62527e6 0.846996
\(327\) −2.44767e6 −1.26585
\(328\) −1.26343e6 −0.648436
\(329\) 2.00518e6 1.02132
\(330\) −1.73036e6 −0.874685
\(331\) −3.77416e6 −1.89344 −0.946719 0.322062i \(-0.895624\pi\)
−0.946719 + 0.322062i \(0.895624\pi\)
\(332\) 657099. 0.327179
\(333\) 532292. 0.263051
\(334\) 1.27878e6 0.627234
\(335\) −4.98572e6 −2.42726
\(336\) −594814. −0.287431
\(337\) −2.12850e6 −1.02094 −0.510470 0.859896i \(-0.670528\pi\)
−0.510470 + 0.859896i \(0.670528\pi\)
\(338\) −1.76492e6 −0.840300
\(339\) −4.11087e6 −1.94283
\(340\) 200575. 0.0940979
\(341\) −2.05222e6 −0.955735
\(342\) −90335.5 −0.0417632
\(343\) −2.11956e6 −0.972770
\(344\) −703037. −0.320318
\(345\) −3.50290e6 −1.58446
\(346\) 1.70925e6 0.767567
\(347\) 2.40305e6 1.07137 0.535685 0.844418i \(-0.320053\pi\)
0.535685 + 0.844418i \(0.320053\pi\)
\(348\) −1.32522e6 −0.586598
\(349\) 3.71870e6 1.63428 0.817141 0.576438i \(-0.195558\pi\)
0.817141 + 0.576438i \(0.195558\pi\)
\(350\) −1.69386e6 −0.739106
\(351\) 2.84315e6 1.23178
\(352\) −318996. −0.137224
\(353\) −2.04192e6 −0.872174 −0.436087 0.899905i \(-0.643636\pi\)
−0.436087 + 0.899905i \(0.643636\pi\)
\(354\) −2.84411e6 −1.20625
\(355\) 4.39394e6 1.85048
\(356\) −273049. −0.114187
\(357\) 366653. 0.152260
\(358\) −1.44778e6 −0.597030
\(359\) −949493. −0.388826 −0.194413 0.980920i \(-0.562280\pi\)
−0.194413 + 0.980920i \(0.562280\pi\)
\(360\) 318064. 0.129348
\(361\) 130321. 0.0526316
\(362\) 1.66674e6 0.668494
\(363\) 1.11885e6 0.445661
\(364\) 1.91704e6 0.758365
\(365\) 3.85511e6 1.51462
\(366\) 430139. 0.167844
\(367\) 1.74418e6 0.675967 0.337984 0.941152i \(-0.390255\pi\)
0.337984 + 0.941152i \(0.390255\pi\)
\(368\) −645768. −0.248575
\(369\) 1.23499e6 0.472169
\(370\) 2.70372e6 1.02673
\(371\) 2.88434e6 1.08796
\(372\) 1.84249e6 0.690316
\(373\) 4.59987e6 1.71188 0.855940 0.517076i \(-0.172980\pi\)
0.855940 + 0.517076i \(0.172980\pi\)
\(374\) 196634. 0.0726910
\(375\) 84477.0 0.0310213
\(376\) −965471. −0.352184
\(377\) 4.27109e6 1.54770
\(378\) −1.67701e6 −0.603680
\(379\) −6993.69 −0.00250097 −0.00125048 0.999999i \(-0.500398\pi\)
−0.00125048 + 0.999999i \(0.500398\pi\)
\(380\) −458850. −0.163009
\(381\) 433228. 0.152899
\(382\) −2.32634e6 −0.815672
\(383\) 1.37680e6 0.479594 0.239797 0.970823i \(-0.422919\pi\)
0.239797 + 0.970823i \(0.422919\pi\)
\(384\) 286396. 0.0991150
\(385\) −3.28944e6 −1.13102
\(386\) 731327. 0.249829
\(387\) 687210. 0.233245
\(388\) 2.22472e6 0.750234
\(389\) 1.40115e6 0.469473 0.234736 0.972059i \(-0.424577\pi\)
0.234736 + 0.972059i \(0.424577\pi\)
\(390\) −5.00690e6 −1.66689
\(391\) 398062. 0.131677
\(392\) −55103.7 −0.0181120
\(393\) 2.66660e6 0.870916
\(394\) 373380. 0.121174
\(395\) −2.46407e6 −0.794620
\(396\) 311815. 0.0999215
\(397\) 3.33402e6 1.06168 0.530839 0.847473i \(-0.321877\pi\)
0.530839 + 0.847473i \(0.321877\pi\)
\(398\) −2.89266e6 −0.915357
\(399\) −838781. −0.263764
\(400\) 815574. 0.254867
\(401\) −5.32612e6 −1.65406 −0.827028 0.562161i \(-0.809970\pi\)
−0.827028 + 0.562161i \(0.809970\pi\)
\(402\) 4.38826e6 1.35434
\(403\) −5.93822e6 −1.82135
\(404\) −1.95605e6 −0.596247
\(405\) 5.58764e6 1.69274
\(406\) −2.51927e6 −0.758506
\(407\) 2.65060e6 0.793155
\(408\) −176539. −0.0525039
\(409\) −3.00861e6 −0.889319 −0.444659 0.895700i \(-0.646675\pi\)
−0.444659 + 0.895700i \(0.646675\pi\)
\(410\) 6.27300e6 1.84296
\(411\) −3.36085e6 −0.981395
\(412\) −1.15092e6 −0.334042
\(413\) −5.40670e6 −1.55976
\(414\) 631231. 0.181004
\(415\) −3.26253e6 −0.929895
\(416\) −923035. −0.261508
\(417\) 5.99280e6 1.68768
\(418\) −449835. −0.125925
\(419\) 2.47470e6 0.688633 0.344317 0.938854i \(-0.388111\pi\)
0.344317 + 0.938854i \(0.388111\pi\)
\(420\) 2.95328e6 0.816924
\(421\) −3.25587e6 −0.895286 −0.447643 0.894212i \(-0.647737\pi\)
−0.447643 + 0.894212i \(0.647737\pi\)
\(422\) −342966. −0.0937496
\(423\) 943736. 0.256448
\(424\) −1.38878e6 −0.375161
\(425\) −502733. −0.135010
\(426\) −3.86740e6 −1.03251
\(427\) 817702. 0.217033
\(428\) 237332. 0.0626249
\(429\) −4.90853e6 −1.28768
\(430\) 3.49061e6 0.910396
\(431\) −3.54756e6 −0.919891 −0.459945 0.887947i \(-0.652131\pi\)
−0.459945 + 0.887947i \(0.652131\pi\)
\(432\) 807462. 0.208168
\(433\) 2.58960e6 0.663763 0.331881 0.943321i \(-0.392317\pi\)
0.331881 + 0.943321i \(0.392317\pi\)
\(434\) 3.50261e6 0.892621
\(435\) 6.57979e6 1.66720
\(436\) 2.24040e6 0.564429
\(437\) −910634. −0.228108
\(438\) −3.39313e6 −0.845114
\(439\) −3.47682e6 −0.861036 −0.430518 0.902582i \(-0.641669\pi\)
−0.430518 + 0.902582i \(0.641669\pi\)
\(440\) 1.58383e6 0.390011
\(441\) 53863.2 0.0131885
\(442\) 568974. 0.138528
\(443\) −3.22496e6 −0.780756 −0.390378 0.920655i \(-0.627656\pi\)
−0.390378 + 0.920655i \(0.627656\pi\)
\(444\) −2.37972e6 −0.572886
\(445\) 1.35570e6 0.324537
\(446\) −444424. −0.105794
\(447\) −5.87089e6 −1.38975
\(448\) 544444. 0.128162
\(449\) −4.47721e6 −1.04807 −0.524036 0.851696i \(-0.675574\pi\)
−0.524036 + 0.851696i \(0.675574\pi\)
\(450\) −797213. −0.185585
\(451\) 6.14975e6 1.42369
\(452\) 3.76275e6 0.866282
\(453\) −4.66169e6 −1.06733
\(454\) 1.25720e6 0.286262
\(455\) −9.51821e6 −2.15539
\(456\) 403864. 0.0909542
\(457\) 563938. 0.126311 0.0631555 0.998004i \(-0.479884\pi\)
0.0631555 + 0.998004i \(0.479884\pi\)
\(458\) −5.83259e6 −1.29926
\(459\) −497733. −0.110272
\(460\) 3.20627e6 0.706489
\(461\) −3.05325e6 −0.669130 −0.334565 0.942373i \(-0.608589\pi\)
−0.334565 + 0.942373i \(0.608589\pi\)
\(462\) 2.89526e6 0.631076
\(463\) 7.31911e6 1.58674 0.793370 0.608740i \(-0.208325\pi\)
0.793370 + 0.608740i \(0.208325\pi\)
\(464\) 1.21300e6 0.261557
\(465\) −9.14806e6 −1.96199
\(466\) −5.54502e6 −1.18287
\(467\) −3.83007e6 −0.812671 −0.406336 0.913724i \(-0.633194\pi\)
−0.406336 + 0.913724i \(0.633194\pi\)
\(468\) 902256. 0.190421
\(469\) 8.34216e6 1.75124
\(470\) 4.79361e6 1.00096
\(471\) 3.03771e6 0.630948
\(472\) 2.60327e6 0.537853
\(473\) 3.42203e6 0.703284
\(474\) 2.16879e6 0.443374
\(475\) 1.15009e6 0.233882
\(476\) −335604. −0.0678907
\(477\) 1.35751e6 0.273179
\(478\) −1.46047e6 −0.292363
\(479\) −852140. −0.169696 −0.0848481 0.996394i \(-0.527040\pi\)
−0.0848481 + 0.996394i \(0.527040\pi\)
\(480\) −1.42197e6 −0.281701
\(481\) 7.66967e6 1.51152
\(482\) 1.24925e6 0.244924
\(483\) 5.86109e6 1.14317
\(484\) −1.02410e6 −0.198715
\(485\) −1.10459e7 −2.13228
\(486\) −1.85222e6 −0.355714
\(487\) −1.76953e6 −0.338093 −0.169046 0.985608i \(-0.554069\pi\)
−0.169046 + 0.985608i \(0.554069\pi\)
\(488\) −393715. −0.0748397
\(489\) 7.10253e6 1.34320
\(490\) 273592. 0.0514771
\(491\) −1.12702e6 −0.210973 −0.105487 0.994421i \(-0.533640\pi\)
−0.105487 + 0.994421i \(0.533640\pi\)
\(492\) −5.52127e6 −1.02832
\(493\) −747713. −0.138553
\(494\) −1.30162e6 −0.239976
\(495\) −1.54818e6 −0.283993
\(496\) −1.68647e6 −0.307803
\(497\) −7.35199e6 −1.33510
\(498\) 2.87156e6 0.518854
\(499\) −5.96123e6 −1.07173 −0.535864 0.844304i \(-0.680014\pi\)
−0.535864 + 0.844304i \(0.680014\pi\)
\(500\) −77323.4 −0.0138320
\(501\) 5.58835e6 0.994694
\(502\) 3.88038e6 0.687250
\(503\) −5.66845e6 −0.998951 −0.499476 0.866328i \(-0.666474\pi\)
−0.499476 + 0.866328i \(0.666474\pi\)
\(504\) −532188. −0.0933230
\(505\) 9.71188e6 1.69463
\(506\) 3.14327e6 0.545765
\(507\) −7.71283e6 −1.33258
\(508\) −396542. −0.0681757
\(509\) 8.42220e6 1.44089 0.720446 0.693512i \(-0.243937\pi\)
0.720446 + 0.693512i \(0.243937\pi\)
\(510\) 876526. 0.149224
\(511\) −6.45040e6 −1.09278
\(512\) −262144. −0.0441942
\(513\) 1.13865e6 0.191028
\(514\) −5.15951e6 −0.861392
\(515\) 5.71436e6 0.949401
\(516\) −3.07232e6 −0.507974
\(517\) 4.69943e6 0.773247
\(518\) −4.52389e6 −0.740777
\(519\) 7.46955e6 1.21724
\(520\) 4.58291e6 0.743247
\(521\) 5.81861e6 0.939128 0.469564 0.882899i \(-0.344411\pi\)
0.469564 + 0.882899i \(0.344411\pi\)
\(522\) −1.18569e6 −0.190457
\(523\) 4.81037e6 0.768996 0.384498 0.923126i \(-0.374375\pi\)
0.384498 + 0.923126i \(0.374375\pi\)
\(524\) −2.44079e6 −0.388331
\(525\) −7.40226e6 −1.17210
\(526\) 2.00571e6 0.316084
\(527\) 1.03957e6 0.163052
\(528\) −1.39403e6 −0.217615
\(529\) −73173.3 −0.0113688
\(530\) 6.89534e6 1.06627
\(531\) −2.54466e6 −0.391646
\(532\) 767752. 0.117609
\(533\) 1.77947e7 2.71314
\(534\) −1.19324e6 −0.181082
\(535\) −1.17836e6 −0.177990
\(536\) −4.01666e6 −0.603883
\(537\) −6.32691e6 −0.946795
\(538\) −5.71944e6 −0.851918
\(539\) 268217. 0.0397662
\(540\) −4.00909e6 −0.591645
\(541\) 3.11994e6 0.458304 0.229152 0.973391i \(-0.426405\pi\)
0.229152 + 0.973391i \(0.426405\pi\)
\(542\) 2.83802e6 0.414971
\(543\) 7.28378e6 1.06012
\(544\) 161590. 0.0234108
\(545\) −1.11237e7 −1.60420
\(546\) 8.37760e6 1.20265
\(547\) 2.61616e6 0.373848 0.186924 0.982374i \(-0.440148\pi\)
0.186924 + 0.982374i \(0.440148\pi\)
\(548\) 3.07624e6 0.437592
\(549\) 384851. 0.0544957
\(550\) −3.96980e6 −0.559580
\(551\) 1.71052e6 0.240021
\(552\) −2.82205e6 −0.394200
\(553\) 4.12290e6 0.573310
\(554\) 3.06696e6 0.424555
\(555\) 1.18154e7 1.62823
\(556\) −5.48532e6 −0.752515
\(557\) 1.91062e6 0.260937 0.130469 0.991452i \(-0.458352\pi\)
0.130469 + 0.991452i \(0.458352\pi\)
\(558\) 1.64850e6 0.224132
\(559\) 9.90185e6 1.34025
\(560\) −2.70319e6 −0.364256
\(561\) 859305. 0.115276
\(562\) −23901.3 −0.00319213
\(563\) 84059.8 0.0111768 0.00558840 0.999984i \(-0.498221\pi\)
0.00558840 + 0.999984i \(0.498221\pi\)
\(564\) −4.21917e6 −0.558508
\(565\) −1.86822e7 −2.46211
\(566\) 756986. 0.0993222
\(567\) −9.34930e6 −1.22130
\(568\) 3.53990e6 0.460384
\(569\) −2.22862e6 −0.288572 −0.144286 0.989536i \(-0.546089\pi\)
−0.144286 + 0.989536i \(0.546089\pi\)
\(570\) −2.00520e6 −0.258506
\(571\) 8.28284e6 1.06314 0.531568 0.847015i \(-0.321603\pi\)
0.531568 + 0.847015i \(0.321603\pi\)
\(572\) 4.49287e6 0.574161
\(573\) −1.01663e7 −1.29353
\(574\) −1.04960e7 −1.32968
\(575\) −8.03637e6 −1.01366
\(576\) 256243. 0.0321807
\(577\) −1.50122e7 −1.87717 −0.938585 0.345047i \(-0.887863\pi\)
−0.938585 + 0.345047i \(0.887863\pi\)
\(578\) 5.57982e6 0.694705
\(579\) 3.19595e6 0.396190
\(580\) −6.02260e6 −0.743385
\(581\) 5.45889e6 0.670910
\(582\) 9.72218e6 1.18975
\(583\) 6.75986e6 0.823695
\(584\) 3.10580e6 0.376826
\(585\) −4.47974e6 −0.541207
\(586\) −3.94472e6 −0.474539
\(587\) −4.07944e6 −0.488658 −0.244329 0.969692i \(-0.578568\pi\)
−0.244329 + 0.969692i \(0.578568\pi\)
\(588\) −240807. −0.0287227
\(589\) −2.37818e6 −0.282460
\(590\) −1.29253e7 −1.52866
\(591\) 1.63169e6 0.192163
\(592\) 2.17820e6 0.255443
\(593\) −545602. −0.0637147 −0.0318573 0.999492i \(-0.510142\pi\)
−0.0318573 + 0.999492i \(0.510142\pi\)
\(594\) −3.93032e6 −0.457048
\(595\) 1.66629e6 0.192956
\(596\) 5.37374e6 0.619671
\(597\) −1.26411e7 −1.45161
\(598\) 9.09526e6 1.04007
\(599\) −8.45396e6 −0.962704 −0.481352 0.876527i \(-0.659854\pi\)
−0.481352 + 0.876527i \(0.659854\pi\)
\(600\) 3.56411e6 0.404178
\(601\) −5.39862e6 −0.609673 −0.304836 0.952405i \(-0.598602\pi\)
−0.304836 + 0.952405i \(0.598602\pi\)
\(602\) −5.84052e6 −0.656841
\(603\) 3.92623e6 0.439727
\(604\) 4.26693e6 0.475908
\(605\) 5.08472e6 0.564779
\(606\) −8.54806e6 −0.945553
\(607\) −1.14275e7 −1.25886 −0.629432 0.777055i \(-0.716712\pi\)
−0.629432 + 0.777055i \(0.716712\pi\)
\(608\) −369664. −0.0405554
\(609\) −1.10094e7 −1.20287
\(610\) 1.95481e6 0.212706
\(611\) 1.35981e7 1.47358
\(612\) −157952. −0.0170470
\(613\) −894639. −0.0961605 −0.0480802 0.998843i \(-0.515310\pi\)
−0.0480802 + 0.998843i \(0.515310\pi\)
\(614\) 5.82888e6 0.623971
\(615\) 2.74134e7 2.92264
\(616\) −2.65008e6 −0.281389
\(617\) 1.18925e7 1.25765 0.628826 0.777546i \(-0.283536\pi\)
0.628826 + 0.777546i \(0.283536\pi\)
\(618\) −5.02958e6 −0.529738
\(619\) 1.03513e7 1.08585 0.542924 0.839782i \(-0.317317\pi\)
0.542924 + 0.839782i \(0.317317\pi\)
\(620\) 8.37339e6 0.874826
\(621\) −7.95645e6 −0.827923
\(622\) −9.62106e6 −0.997118
\(623\) −2.26837e6 −0.234150
\(624\) −4.03372e6 −0.414710
\(625\) −9.57182e6 −0.980154
\(626\) 1.28152e7 1.30704
\(627\) −1.96581e6 −0.199697
\(628\) −2.78047e6 −0.281332
\(629\) −1.34268e6 −0.135315
\(630\) 2.64234e6 0.265239
\(631\) −1.17088e6 −0.117068 −0.0585342 0.998285i \(-0.518643\pi\)
−0.0585342 + 0.998285i \(0.518643\pi\)
\(632\) −1.98513e6 −0.197695
\(633\) −1.49878e6 −0.148672
\(634\) 2.41219e6 0.238335
\(635\) 1.96885e6 0.193766
\(636\) −6.06904e6 −0.594946
\(637\) 776102. 0.0757828
\(638\) −5.90427e6 −0.574268
\(639\) −3.46021e6 −0.335236
\(640\) 1.30156e6 0.125607
\(641\) −4.72990e6 −0.454681 −0.227341 0.973815i \(-0.573003\pi\)
−0.227341 + 0.973815i \(0.573003\pi\)
\(642\) 1.03716e6 0.0993131
\(643\) −1.49075e7 −1.42193 −0.710963 0.703229i \(-0.751741\pi\)
−0.710963 + 0.703229i \(0.751741\pi\)
\(644\) −5.36476e6 −0.509725
\(645\) 1.52542e7 1.44374
\(646\) 227867. 0.0214832
\(647\) −4.27732e6 −0.401708 −0.200854 0.979621i \(-0.564372\pi\)
−0.200854 + 0.979621i \(0.564372\pi\)
\(648\) 4.50158e6 0.421141
\(649\) −1.26714e7 −1.18090
\(650\) −1.14869e7 −1.06639
\(651\) 1.53066e7 1.41555
\(652\) −6.50108e6 −0.598917
\(653\) 2.25370e6 0.206829 0.103415 0.994638i \(-0.467023\pi\)
0.103415 + 0.994638i \(0.467023\pi\)
\(654\) 9.79069e6 0.895094
\(655\) 1.21186e7 1.10370
\(656\) 5.05372e6 0.458513
\(657\) −3.03588e6 −0.274392
\(658\) −8.02071e6 −0.722184
\(659\) −2.93133e6 −0.262937 −0.131468 0.991320i \(-0.541969\pi\)
−0.131468 + 0.991320i \(0.541969\pi\)
\(660\) 6.92144e6 0.618496
\(661\) 6.56037e6 0.584016 0.292008 0.956416i \(-0.405677\pi\)
0.292008 + 0.956416i \(0.405677\pi\)
\(662\) 1.50967e7 1.33886
\(663\) 2.48645e6 0.219683
\(664\) −2.62840e6 −0.231351
\(665\) −3.81192e6 −0.334264
\(666\) −2.12917e6 −0.186005
\(667\) −1.19525e7 −1.04026
\(668\) −5.11512e6 −0.443522
\(669\) −1.94216e6 −0.167772
\(670\) 1.99429e7 1.71633
\(671\) 1.91640e6 0.164316
\(672\) 2.37926e6 0.203244
\(673\) −3.19596e6 −0.271996 −0.135998 0.990709i \(-0.543424\pi\)
−0.135998 + 0.990709i \(0.543424\pi\)
\(674\) 8.51402e6 0.721913
\(675\) 1.00486e7 0.848880
\(676\) 7.05970e6 0.594182
\(677\) −2.06961e6 −0.173547 −0.0867734 0.996228i \(-0.527656\pi\)
−0.0867734 + 0.996228i \(0.527656\pi\)
\(678\) 1.64435e7 1.37379
\(679\) 1.84820e7 1.53842
\(680\) −802301. −0.0665373
\(681\) 5.49402e6 0.453965
\(682\) 8.20887e6 0.675807
\(683\) −2.61594e6 −0.214574 −0.107287 0.994228i \(-0.534216\pi\)
−0.107287 + 0.994228i \(0.534216\pi\)
\(684\) 361342. 0.0295310
\(685\) −1.52737e7 −1.24371
\(686\) 8.47824e6 0.687852
\(687\) −2.54888e7 −2.06043
\(688\) 2.81215e6 0.226499
\(689\) 1.95601e7 1.56972
\(690\) 1.40116e7 1.12038
\(691\) −1.40203e7 −1.11703 −0.558513 0.829496i \(-0.688628\pi\)
−0.558513 + 0.829496i \(0.688628\pi\)
\(692\) −6.83701e6 −0.542752
\(693\) 2.59042e6 0.204898
\(694\) −9.61221e6 −0.757573
\(695\) 2.72349e7 2.13877
\(696\) 5.30088e6 0.414787
\(697\) −3.11520e6 −0.242887
\(698\) −1.48748e7 −1.15561
\(699\) −2.42321e7 −1.87585
\(700\) 6.77543e6 0.522627
\(701\) 3.42664e6 0.263375 0.131687 0.991291i \(-0.457961\pi\)
0.131687 + 0.991291i \(0.457961\pi\)
\(702\) −1.13726e7 −0.870999
\(703\) 3.07161e6 0.234411
\(704\) 1.27598e6 0.0970318
\(705\) 2.09484e7 1.58737
\(706\) 8.16770e6 0.616720
\(707\) −1.62500e7 −1.22266
\(708\) 1.13764e7 0.852950
\(709\) 8.04969e6 0.601400 0.300700 0.953719i \(-0.402780\pi\)
0.300700 + 0.953719i \(0.402780\pi\)
\(710\) −1.75758e7 −1.30848
\(711\) 1.94044e6 0.143955
\(712\) 1.09220e6 0.0807422
\(713\) 1.66178e7 1.22420
\(714\) −1.46661e6 −0.107664
\(715\) −2.23073e7 −1.63186
\(716\) 5.79114e6 0.422164
\(717\) −6.38234e6 −0.463641
\(718\) 3.79797e6 0.274942
\(719\) −2.66646e7 −1.92359 −0.961795 0.273771i \(-0.911729\pi\)
−0.961795 + 0.273771i \(0.911729\pi\)
\(720\) −1.27226e6 −0.0914626
\(721\) −9.56132e6 −0.684983
\(722\) −521284. −0.0372161
\(723\) 5.45929e6 0.388410
\(724\) −6.66698e6 −0.472697
\(725\) 1.50954e7 1.06659
\(726\) −4.47540e6 −0.315130
\(727\) 1.25706e7 0.882102 0.441051 0.897482i \(-0.354606\pi\)
0.441051 + 0.897482i \(0.354606\pi\)
\(728\) −7.66817e6 −0.536245
\(729\) 8.99764e6 0.627061
\(730\) −1.54204e7 −1.07100
\(731\) −1.73345e6 −0.119983
\(732\) −1.72056e6 −0.118684
\(733\) −1.12272e7 −0.771814 −0.385907 0.922538i \(-0.626111\pi\)
−0.385907 + 0.922538i \(0.626111\pi\)
\(734\) −6.97671e6 −0.477981
\(735\) 1.19562e6 0.0816345
\(736\) 2.58307e6 0.175769
\(737\) 1.95511e7 1.32587
\(738\) −4.93996e6 −0.333874
\(739\) 1.05677e7 0.711820 0.355910 0.934520i \(-0.384171\pi\)
0.355910 + 0.934520i \(0.384171\pi\)
\(740\) −1.08149e7 −0.726010
\(741\) −5.68818e6 −0.380564
\(742\) −1.15373e7 −0.769301
\(743\) −5.83579e6 −0.387817 −0.193909 0.981020i \(-0.562117\pi\)
−0.193909 + 0.981020i \(0.562117\pi\)
\(744\) −7.36997e6 −0.488127
\(745\) −2.66809e7 −1.76120
\(746\) −1.83995e7 −1.21048
\(747\) 2.56923e6 0.168462
\(748\) −786538. −0.0514003
\(749\) 1.97165e6 0.128418
\(750\) −337908. −0.0219354
\(751\) 4.48767e6 0.290349 0.145175 0.989406i \(-0.453626\pi\)
0.145175 + 0.989406i \(0.453626\pi\)
\(752\) 3.86188e6 0.249032
\(753\) 1.69575e7 1.08987
\(754\) −1.70844e7 −1.09439
\(755\) −2.11855e7 −1.35261
\(756\) 6.70805e6 0.426866
\(757\) 7.81327e6 0.495557 0.247778 0.968817i \(-0.420300\pi\)
0.247778 + 0.968817i \(0.420300\pi\)
\(758\) 27974.8 0.00176845
\(759\) 1.37363e7 0.865497
\(760\) 1.83540e6 0.115265
\(761\) 1.81639e7 1.13697 0.568483 0.822695i \(-0.307531\pi\)
0.568483 + 0.822695i \(0.307531\pi\)
\(762\) −1.73291e6 −0.108116
\(763\) 1.86123e7 1.15741
\(764\) 9.30537e6 0.576767
\(765\) 784240. 0.0484502
\(766\) −5.50720e6 −0.339124
\(767\) −3.66655e7 −2.25045
\(768\) −1.14559e6 −0.0700849
\(769\) −4.47189e6 −0.272694 −0.136347 0.990661i \(-0.543536\pi\)
−0.136347 + 0.990661i \(0.543536\pi\)
\(770\) 1.31578e7 0.799753
\(771\) −2.25474e7 −1.36603
\(772\) −2.92531e6 −0.176656
\(773\) −2.53207e7 −1.52415 −0.762075 0.647489i \(-0.775819\pi\)
−0.762075 + 0.647489i \(0.775819\pi\)
\(774\) −2.74884e6 −0.164929
\(775\) −2.09875e7 −1.25518
\(776\) −8.89889e6 −0.530496
\(777\) −1.97697e7 −1.17475
\(778\) −5.60460e6 −0.331967
\(779\) 7.12654e6 0.420761
\(780\) 2.00276e7 1.17867
\(781\) −1.72305e7 −1.01081
\(782\) −1.59225e6 −0.0931095
\(783\) 1.49452e7 0.871161
\(784\) 220415. 0.0128071
\(785\) 1.38052e7 0.799590
\(786\) −1.06664e7 −0.615831
\(787\) 4.73524e6 0.272524 0.136262 0.990673i \(-0.456491\pi\)
0.136262 + 0.990673i \(0.456491\pi\)
\(788\) −1.49352e6 −0.0856831
\(789\) 8.76506e6 0.501259
\(790\) 9.85626e6 0.561881
\(791\) 3.12593e7 1.77639
\(792\) −1.24726e6 −0.0706552
\(793\) 5.54523e6 0.313139
\(794\) −1.33361e7 −0.750719
\(795\) 3.01331e7 1.69093
\(796\) 1.15707e7 0.647255
\(797\) 1.54780e7 0.863116 0.431558 0.902085i \(-0.357964\pi\)
0.431558 + 0.902085i \(0.357964\pi\)
\(798\) 3.35512e6 0.186510
\(799\) −2.38053e6 −0.131919
\(800\) −3.26229e6 −0.180218
\(801\) −1.06761e6 −0.0587937
\(802\) 2.13045e7 1.16959
\(803\) −1.51175e7 −0.827351
\(804\) −1.75530e7 −0.957662
\(805\) 2.66363e7 1.44872
\(806\) 2.37529e7 1.28789
\(807\) −2.49943e7 −1.35101
\(808\) 7.82420e6 0.421611
\(809\) 5.54794e6 0.298031 0.149015 0.988835i \(-0.452390\pi\)
0.149015 + 0.988835i \(0.452390\pi\)
\(810\) −2.23506e7 −1.19695
\(811\) 9.82955e6 0.524785 0.262392 0.964961i \(-0.415489\pi\)
0.262392 + 0.964961i \(0.415489\pi\)
\(812\) 1.00771e7 0.536345
\(813\) 1.24023e7 0.658078
\(814\) −1.06024e7 −0.560845
\(815\) 3.22782e7 1.70222
\(816\) 706158. 0.0371258
\(817\) 3.96557e6 0.207850
\(818\) 1.20344e7 0.628843
\(819\) 7.49555e6 0.390475
\(820\) −2.50920e7 −1.30317
\(821\) 1.78229e7 0.922830 0.461415 0.887185i \(-0.347342\pi\)
0.461415 + 0.887185i \(0.347342\pi\)
\(822\) 1.34434e7 0.693951
\(823\) −1.29509e7 −0.666502 −0.333251 0.942838i \(-0.608146\pi\)
−0.333251 + 0.942838i \(0.608146\pi\)
\(824\) 4.60367e6 0.236204
\(825\) −1.73483e7 −0.887404
\(826\) 2.16268e7 1.10292
\(827\) −2.57276e7 −1.30808 −0.654042 0.756458i \(-0.726928\pi\)
−0.654042 + 0.756458i \(0.726928\pi\)
\(828\) −2.52492e6 −0.127989
\(829\) 2.26984e7 1.14712 0.573559 0.819164i \(-0.305562\pi\)
0.573559 + 0.819164i \(0.305562\pi\)
\(830\) 1.30501e7 0.657535
\(831\) 1.34028e7 0.673276
\(832\) 3.69214e6 0.184914
\(833\) −135867. −0.00678426
\(834\) −2.39712e7 −1.19337
\(835\) 2.53968e7 1.26056
\(836\) 1.79934e6 0.0890424
\(837\) −2.07788e7 −1.02519
\(838\) −9.89881e6 −0.486937
\(839\) −9.22885e6 −0.452629 −0.226315 0.974054i \(-0.572668\pi\)
−0.226315 + 0.974054i \(0.572668\pi\)
\(840\) −1.18131e7 −0.577652
\(841\) 1.94015e6 0.0945898
\(842\) 1.30235e7 0.633063
\(843\) −104450. −0.00506221
\(844\) 1.37186e6 0.0662910
\(845\) −3.50517e7 −1.68876
\(846\) −3.77495e6 −0.181336
\(847\) −8.50780e6 −0.407482
\(848\) 5.55511e6 0.265279
\(849\) 3.30807e6 0.157509
\(850\) 2.01093e6 0.0954662
\(851\) −2.14632e7 −1.01595
\(852\) 1.54696e7 0.730096
\(853\) −3.17055e7 −1.49198 −0.745989 0.665959i \(-0.768023\pi\)
−0.745989 + 0.665959i \(0.768023\pi\)
\(854\) −3.27081e6 −0.153465
\(855\) −1.79408e6 −0.0839318
\(856\) −949328. −0.0442825
\(857\) 3.01353e7 1.40160 0.700799 0.713359i \(-0.252827\pi\)
0.700799 + 0.713359i \(0.252827\pi\)
\(858\) 1.96341e7 0.910528
\(859\) −1.58417e7 −0.732520 −0.366260 0.930513i \(-0.619362\pi\)
−0.366260 + 0.930513i \(0.619362\pi\)
\(860\) −1.39624e7 −0.643747
\(861\) −4.58683e7 −2.10865
\(862\) 1.41902e7 0.650461
\(863\) −2.32831e7 −1.06418 −0.532089 0.846689i \(-0.678593\pi\)
−0.532089 + 0.846689i \(0.678593\pi\)
\(864\) −3.22985e6 −0.147197
\(865\) 3.39461e7 1.54259
\(866\) −1.03584e7 −0.469351
\(867\) 2.43842e7 1.10169
\(868\) −1.40104e7 −0.631178
\(869\) 9.66261e6 0.434055
\(870\) −2.63191e7 −1.17889
\(871\) 5.65722e7 2.52672
\(872\) −8.96160e6 −0.399111
\(873\) 8.69856e6 0.386289
\(874\) 3.64254e6 0.161297
\(875\) −642369. −0.0283638
\(876\) 1.35725e7 0.597586
\(877\) 2.89258e7 1.26995 0.634975 0.772532i \(-0.281010\pi\)
0.634975 + 0.772532i \(0.281010\pi\)
\(878\) 1.39073e7 0.608844
\(879\) −1.72387e7 −0.752544
\(880\) −6.33533e6 −0.275780
\(881\) −2.35636e7 −1.02283 −0.511413 0.859335i \(-0.670878\pi\)
−0.511413 + 0.859335i \(0.670878\pi\)
\(882\) −215453. −0.00932569
\(883\) −1.91319e6 −0.0825764 −0.0412882 0.999147i \(-0.513146\pi\)
−0.0412882 + 0.999147i \(0.513146\pi\)
\(884\) −2.27589e6 −0.0979538
\(885\) −5.64846e7 −2.42422
\(886\) 1.28998e7 0.552078
\(887\) 3.48791e7 1.48852 0.744262 0.667888i \(-0.232801\pi\)
0.744262 + 0.667888i \(0.232801\pi\)
\(888\) 9.51888e6 0.405092
\(889\) −3.29430e6 −0.139800
\(890\) −5.42280e6 −0.229482
\(891\) −2.19114e7 −0.924648
\(892\) 1.77770e6 0.0748076
\(893\) 5.44586e6 0.228527
\(894\) 2.34836e7 0.982699
\(895\) −2.87533e7 −1.19986
\(896\) −2.17778e6 −0.0906241
\(897\) 3.97468e7 1.64938
\(898\) 1.79088e7 0.741099
\(899\) −3.12146e7 −1.28813
\(900\) 3.18885e6 0.131229
\(901\) −3.42426e6 −0.140525
\(902\) −2.45990e7 −1.00670
\(903\) −2.55235e7 −1.04165
\(904\) −1.50510e7 −0.612554
\(905\) 3.31018e7 1.34348
\(906\) 1.86467e7 0.754714
\(907\) 1.34601e7 0.543287 0.271644 0.962398i \(-0.412433\pi\)
0.271644 + 0.962398i \(0.412433\pi\)
\(908\) −5.02878e6 −0.202418
\(909\) −7.64806e6 −0.307002
\(910\) 3.80728e7 1.52409
\(911\) −1.42316e7 −0.568144 −0.284072 0.958803i \(-0.591685\pi\)
−0.284072 + 0.958803i \(0.591685\pi\)
\(912\) −1.61546e6 −0.0643143
\(913\) 1.27937e7 0.507948
\(914\) −2.25575e6 −0.0893154
\(915\) 8.54265e6 0.337318
\(916\) 2.33303e7 0.918718
\(917\) −2.02770e7 −0.796307
\(918\) 1.99093e6 0.0779740
\(919\) 1.48289e7 0.579188 0.289594 0.957150i \(-0.406480\pi\)
0.289594 + 0.957150i \(0.406480\pi\)
\(920\) −1.28251e7 −0.499563
\(921\) 2.54726e7 0.989518
\(922\) 1.22130e7 0.473146
\(923\) −4.98574e7 −1.92631
\(924\) −1.15810e7 −0.446238
\(925\) 2.71070e7 1.04166
\(926\) −2.92764e7 −1.12199
\(927\) −4.50003e6 −0.171995
\(928\) −4.85200e6 −0.184948
\(929\) 1.48557e7 0.564748 0.282374 0.959304i \(-0.408878\pi\)
0.282374 + 0.959304i \(0.408878\pi\)
\(930\) 3.65922e7 1.38734
\(931\) 310819. 0.0117526
\(932\) 2.21801e7 0.836418
\(933\) −4.20446e7 −1.58127
\(934\) 1.53203e7 0.574645
\(935\) 3.90520e6 0.146088
\(936\) −3.60902e6 −0.134648
\(937\) 9.74272e6 0.362519 0.181260 0.983435i \(-0.441983\pi\)
0.181260 + 0.983435i \(0.441983\pi\)
\(938\) −3.33686e7 −1.23832
\(939\) 5.60032e7 2.07276
\(940\) −1.91744e7 −0.707788
\(941\) −2.16418e6 −0.0796745 −0.0398373 0.999206i \(-0.512684\pi\)
−0.0398373 + 0.999206i \(0.512684\pi\)
\(942\) −1.21508e7 −0.446148
\(943\) −4.97976e7 −1.82360
\(944\) −1.04131e7 −0.380320
\(945\) −3.33058e7 −1.21322
\(946\) −1.36881e7 −0.497297
\(947\) 1.60138e7 0.580254 0.290127 0.956988i \(-0.406302\pi\)
0.290127 + 0.956988i \(0.406302\pi\)
\(948\) −8.67514e6 −0.313513
\(949\) −4.37433e7 −1.57669
\(950\) −4.60034e6 −0.165379
\(951\) 1.05414e7 0.377961
\(952\) 1.34242e6 0.0480060
\(953\) −2.67154e6 −0.0952860 −0.0476430 0.998864i \(-0.515171\pi\)
−0.0476430 + 0.998864i \(0.515171\pi\)
\(954\) −5.43005e6 −0.193167
\(955\) −4.62016e7 −1.63926
\(956\) 5.84188e6 0.206732
\(957\) −2.58020e7 −0.910697
\(958\) 3.40856e6 0.119993
\(959\) 2.55561e7 0.897321
\(960\) 5.68789e6 0.199192
\(961\) 1.47694e7 0.515888
\(962\) −3.06787e7 −1.06881
\(963\) 927957. 0.0322450
\(964\) −4.99699e6 −0.173187
\(965\) 1.45243e7 0.502085
\(966\) −2.34443e7 −0.808342
\(967\) −7.08291e6 −0.243582 −0.121791 0.992556i \(-0.538864\pi\)
−0.121791 + 0.992556i \(0.538864\pi\)
\(968\) 4.09641e6 0.140513
\(969\) 995793. 0.0340690
\(970\) 4.41834e7 1.50775
\(971\) 2.85104e7 0.970411 0.485205 0.874400i \(-0.338745\pi\)
0.485205 + 0.874400i \(0.338745\pi\)
\(972\) 7.40887e6 0.251528
\(973\) −4.55697e7 −1.54310
\(974\) 7.07813e6 0.239068
\(975\) −5.01983e7 −1.69113
\(976\) 1.57486e6 0.0529197
\(977\) −2.48752e7 −0.833739 −0.416869 0.908966i \(-0.636873\pi\)
−0.416869 + 0.908966i \(0.636873\pi\)
\(978\) −2.84101e7 −0.949786
\(979\) −5.31626e6 −0.177276
\(980\) −1.09437e6 −0.0363998
\(981\) 8.75985e6 0.290619
\(982\) 4.50808e6 0.149181
\(983\) −2.66082e6 −0.0878277 −0.0439138 0.999035i \(-0.513983\pi\)
−0.0439138 + 0.999035i \(0.513983\pi\)
\(984\) 2.20851e7 0.727129
\(985\) 7.41539e6 0.243525
\(986\) 2.99085e6 0.0979721
\(987\) −3.50510e7 −1.14527
\(988\) 5.20649e6 0.169689
\(989\) −2.77099e7 −0.900833
\(990\) 6.19271e6 0.200813
\(991\) 5.19709e7 1.68103 0.840516 0.541786i \(-0.182252\pi\)
0.840516 + 0.541786i \(0.182252\pi\)
\(992\) 6.74587e6 0.217650
\(993\) 6.59734e7 2.12322
\(994\) 2.94080e7 0.944058
\(995\) −5.74489e7 −1.83960
\(996\) −1.14863e7 −0.366885
\(997\) −1.86644e7 −0.594672 −0.297336 0.954773i \(-0.596098\pi\)
−0.297336 + 0.954773i \(0.596098\pi\)
\(998\) 2.38449e7 0.757826
\(999\) 2.68374e7 0.850799
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 38.6.a.c.1.1 2
3.2 odd 2 342.6.a.i.1.2 2
4.3 odd 2 304.6.a.f.1.2 2
5.4 even 2 950.6.a.d.1.2 2
19.18 odd 2 722.6.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.6.a.c.1.1 2 1.1 even 1 trivial
304.6.a.f.1.2 2 4.3 odd 2
342.6.a.i.1.2 2 3.2 odd 2
722.6.a.c.1.2 2 19.18 odd 2
950.6.a.d.1.2 2 5.4 even 2