Properties

Label 2166.4.a.s.1.3
Level $2166$
Weight $4$
Character 2166.1
Self dual yes
Analytic conductor $127.798$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2166.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.798137072\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.140435\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +20.2020 q^{5} +6.00000 q^{6} -22.8872 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +20.2020 q^{5} +6.00000 q^{6} -22.8872 q^{7} -8.00000 q^{8} +9.00000 q^{9} -40.4040 q^{10} -57.5614 q^{11} -12.0000 q^{12} +27.3148 q^{13} +45.7744 q^{14} -60.6059 q^{15} +16.0000 q^{16} -73.4486 q^{17} -18.0000 q^{18} +80.8079 q^{20} +68.6616 q^{21} +115.123 q^{22} -188.584 q^{23} +24.0000 q^{24} +283.120 q^{25} -54.6296 q^{26} -27.0000 q^{27} -91.5488 q^{28} -7.15594 q^{29} +121.212 q^{30} -117.144 q^{31} -32.0000 q^{32} +172.684 q^{33} +146.897 q^{34} -462.367 q^{35} +36.0000 q^{36} -332.600 q^{37} -81.9443 q^{39} -161.616 q^{40} -42.7158 q^{41} -137.323 q^{42} +47.2469 q^{43} -230.245 q^{44} +181.818 q^{45} +377.168 q^{46} -507.995 q^{47} -48.0000 q^{48} +180.824 q^{49} -566.240 q^{50} +220.346 q^{51} +109.259 q^{52} +492.609 q^{53} +54.0000 q^{54} -1162.85 q^{55} +183.098 q^{56} +14.3119 q^{58} +460.041 q^{59} -242.424 q^{60} +450.745 q^{61} +234.287 q^{62} -205.985 q^{63} +64.0000 q^{64} +551.813 q^{65} -345.368 q^{66} +522.465 q^{67} -293.794 q^{68} +565.751 q^{69} +924.734 q^{70} +931.419 q^{71} -72.0000 q^{72} +350.123 q^{73} +665.199 q^{74} -849.360 q^{75} +1317.42 q^{77} +163.889 q^{78} +194.868 q^{79} +323.232 q^{80} +81.0000 q^{81} +85.4317 q^{82} -286.083 q^{83} +274.646 q^{84} -1483.81 q^{85} -94.4937 q^{86} +21.4678 q^{87} +460.491 q^{88} +839.690 q^{89} -363.636 q^{90} -625.159 q^{91} -754.335 q^{92} +351.431 q^{93} +1015.99 q^{94} +96.0000 q^{96} +605.516 q^{97} -361.648 q^{98} -518.052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 6q^{2} - 9q^{3} + 12q^{4} + 2q^{5} + 18q^{6} - 17q^{7} - 24q^{8} + 27q^{9} + O(q^{10}) \) \( 3q - 6q^{2} - 9q^{3} + 12q^{4} + 2q^{5} + 18q^{6} - 17q^{7} - 24q^{8} + 27q^{9} - 4q^{10} - 52q^{11} - 36q^{12} + 75q^{13} + 34q^{14} - 6q^{15} + 48q^{16} - 48q^{17} - 54q^{18} + 8q^{20} + 51q^{21} + 104q^{22} - 238q^{23} + 72q^{24} + 229q^{25} - 150q^{26} - 81q^{27} - 68q^{28} - 8q^{29} + 12q^{30} + 107q^{31} - 96q^{32} + 156q^{33} + 96q^{34} - 294q^{35} + 108q^{36} + 305q^{37} - 225q^{39} - 16q^{40} + 16q^{41} - 102q^{42} - 331q^{43} - 208q^{44} + 18q^{45} + 476q^{46} - 766q^{47} - 144q^{48} + 1142q^{49} - 458q^{50} + 144q^{51} + 300q^{52} - 118q^{53} + 162q^{54} - 1400q^{55} + 136q^{56} + 16q^{58} + 936q^{59} - 24q^{60} - 399q^{61} - 214q^{62} - 153q^{63} + 192q^{64} + 370q^{65} - 312q^{66} + 61q^{67} - 192q^{68} + 714q^{69} + 588q^{70} + 974q^{71} - 216q^{72} + 91q^{73} - 610q^{74} - 687q^{75} - 36q^{77} + 450q^{78} - 321q^{79} + 32q^{80} + 243q^{81} - 32q^{82} - 2148q^{83} + 204q^{84} - 1680q^{85} + 662q^{86} + 24q^{87} + 416q^{88} + 1116q^{89} - 36q^{90} + 1367q^{91} - 952q^{92} - 321q^{93} + 1532q^{94} + 288q^{96} + 1382q^{97} - 2284q^{98} - 468q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −3.00000 −0.577350
\(4\) 4.00000 0.500000
\(5\) 20.2020 1.80692 0.903460 0.428672i \(-0.141018\pi\)
0.903460 + 0.428672i \(0.141018\pi\)
\(6\) 6.00000 0.408248
\(7\) −22.8872 −1.23579 −0.617897 0.786259i \(-0.712015\pi\)
−0.617897 + 0.786259i \(0.712015\pi\)
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) −40.4040 −1.27769
\(11\) −57.5614 −1.57776 −0.788882 0.614545i \(-0.789340\pi\)
−0.788882 + 0.614545i \(0.789340\pi\)
\(12\) −12.0000 −0.288675
\(13\) 27.3148 0.582750 0.291375 0.956609i \(-0.405887\pi\)
0.291375 + 0.956609i \(0.405887\pi\)
\(14\) 45.7744 0.873838
\(15\) −60.6059 −1.04323
\(16\) 16.0000 0.250000
\(17\) −73.4486 −1.04788 −0.523938 0.851756i \(-0.675538\pi\)
−0.523938 + 0.851756i \(0.675538\pi\)
\(18\) −18.0000 −0.235702
\(19\) 0 0
\(20\) 80.8079 0.903460
\(21\) 68.6616 0.713485
\(22\) 115.123 1.11565
\(23\) −188.584 −1.70967 −0.854836 0.518899i \(-0.826342\pi\)
−0.854836 + 0.518899i \(0.826342\pi\)
\(24\) 24.0000 0.204124
\(25\) 283.120 2.26496
\(26\) −54.6296 −0.412067
\(27\) −27.0000 −0.192450
\(28\) −91.5488 −0.617897
\(29\) −7.15594 −0.0458215 −0.0229108 0.999738i \(-0.507293\pi\)
−0.0229108 + 0.999738i \(0.507293\pi\)
\(30\) 121.212 0.737672
\(31\) −117.144 −0.678698 −0.339349 0.940661i \(-0.610207\pi\)
−0.339349 + 0.940661i \(0.610207\pi\)
\(32\) −32.0000 −0.176777
\(33\) 172.684 0.910922
\(34\) 146.897 0.740960
\(35\) −462.367 −2.23298
\(36\) 36.0000 0.166667
\(37\) −332.600 −1.47781 −0.738906 0.673809i \(-0.764657\pi\)
−0.738906 + 0.673809i \(0.764657\pi\)
\(38\) 0 0
\(39\) −81.9443 −0.336451
\(40\) −161.616 −0.638843
\(41\) −42.7158 −0.162710 −0.0813548 0.996685i \(-0.525925\pi\)
−0.0813548 + 0.996685i \(0.525925\pi\)
\(42\) −137.323 −0.504510
\(43\) 47.2469 0.167560 0.0837800 0.996484i \(-0.473301\pi\)
0.0837800 + 0.996484i \(0.473301\pi\)
\(44\) −230.245 −0.788882
\(45\) 181.818 0.602307
\(46\) 377.168 1.20892
\(47\) −507.995 −1.57657 −0.788284 0.615312i \(-0.789030\pi\)
−0.788284 + 0.615312i \(0.789030\pi\)
\(48\) −48.0000 −0.144338
\(49\) 180.824 0.527184
\(50\) −566.240 −1.60157
\(51\) 220.346 0.604991
\(52\) 109.259 0.291375
\(53\) 492.609 1.27670 0.638350 0.769746i \(-0.279617\pi\)
0.638350 + 0.769746i \(0.279617\pi\)
\(54\) 54.0000 0.136083
\(55\) −1162.85 −2.85089
\(56\) 183.098 0.436919
\(57\) 0 0
\(58\) 14.3119 0.0324007
\(59\) 460.041 1.01512 0.507561 0.861616i \(-0.330547\pi\)
0.507561 + 0.861616i \(0.330547\pi\)
\(60\) −242.424 −0.521613
\(61\) 450.745 0.946097 0.473049 0.881036i \(-0.343153\pi\)
0.473049 + 0.881036i \(0.343153\pi\)
\(62\) 234.287 0.479912
\(63\) −205.985 −0.411931
\(64\) 64.0000 0.125000
\(65\) 551.813 1.05298
\(66\) −345.368 −0.644119
\(67\) 522.465 0.952675 0.476337 0.879263i \(-0.341964\pi\)
0.476337 + 0.879263i \(0.341964\pi\)
\(68\) −293.794 −0.523938
\(69\) 565.751 0.987079
\(70\) 924.734 1.57895
\(71\) 931.419 1.55689 0.778444 0.627714i \(-0.216009\pi\)
0.778444 + 0.627714i \(0.216009\pi\)
\(72\) −72.0000 −0.117851
\(73\) 350.123 0.561354 0.280677 0.959802i \(-0.409441\pi\)
0.280677 + 0.959802i \(0.409441\pi\)
\(74\) 665.199 1.04497
\(75\) −849.360 −1.30768
\(76\) 0 0
\(77\) 1317.42 1.94979
\(78\) 163.889 0.237907
\(79\) 194.868 0.277524 0.138762 0.990326i \(-0.455688\pi\)
0.138762 + 0.990326i \(0.455688\pi\)
\(80\) 323.232 0.451730
\(81\) 81.0000 0.111111
\(82\) 85.4317 0.115053
\(83\) −286.083 −0.378333 −0.189167 0.981945i \(-0.560579\pi\)
−0.189167 + 0.981945i \(0.560579\pi\)
\(84\) 274.646 0.356743
\(85\) −1483.81 −1.89343
\(86\) −94.4937 −0.118483
\(87\) 21.4678 0.0264551
\(88\) 460.491 0.557824
\(89\) 839.690 1.00008 0.500039 0.866003i \(-0.333319\pi\)
0.500039 + 0.866003i \(0.333319\pi\)
\(90\) −363.636 −0.425895
\(91\) −625.159 −0.720159
\(92\) −754.335 −0.854836
\(93\) 351.431 0.391846
\(94\) 1015.99 1.11480
\(95\) 0 0
\(96\) 96.0000 0.102062
\(97\) 605.516 0.633823 0.316911 0.948455i \(-0.397354\pi\)
0.316911 + 0.948455i \(0.397354\pi\)
\(98\) −361.648 −0.372776
\(99\) −518.052 −0.525921
\(100\) 1132.48 1.13248
\(101\) 47.2760 0.0465756 0.0232878 0.999729i \(-0.492587\pi\)
0.0232878 + 0.999729i \(0.492587\pi\)
\(102\) −440.691 −0.427794
\(103\) −430.750 −0.412069 −0.206034 0.978545i \(-0.566056\pi\)
−0.206034 + 0.978545i \(0.566056\pi\)
\(104\) −218.518 −0.206033
\(105\) 1387.10 1.28921
\(106\) −985.219 −0.902763
\(107\) 1055.78 0.953893 0.476946 0.878932i \(-0.341744\pi\)
0.476946 + 0.878932i \(0.341744\pi\)
\(108\) −108.000 −0.0962250
\(109\) −282.571 −0.248307 −0.124153 0.992263i \(-0.539621\pi\)
−0.124153 + 0.992263i \(0.539621\pi\)
\(110\) 2325.71 2.01589
\(111\) 997.799 0.853215
\(112\) −366.195 −0.308948
\(113\) −841.860 −0.700846 −0.350423 0.936592i \(-0.613962\pi\)
−0.350423 + 0.936592i \(0.613962\pi\)
\(114\) 0 0
\(115\) −3809.77 −3.08924
\(116\) −28.6238 −0.0229108
\(117\) 245.833 0.194250
\(118\) −920.081 −0.717799
\(119\) 1681.03 1.29496
\(120\) 484.848 0.368836
\(121\) 1982.31 1.48934
\(122\) −901.489 −0.668992
\(123\) 128.148 0.0939404
\(124\) −468.575 −0.339349
\(125\) 3194.34 2.28568
\(126\) 411.970 0.291279
\(127\) −955.393 −0.667539 −0.333769 0.942655i \(-0.608321\pi\)
−0.333769 + 0.942655i \(0.608321\pi\)
\(128\) −128.000 −0.0883883
\(129\) −141.741 −0.0967408
\(130\) −1103.63 −0.744572
\(131\) −1639.11 −1.09320 −0.546601 0.837393i \(-0.684079\pi\)
−0.546601 + 0.837393i \(0.684079\pi\)
\(132\) 690.736 0.455461
\(133\) 0 0
\(134\) −1044.93 −0.673643
\(135\) −545.454 −0.347742
\(136\) 587.588 0.370480
\(137\) −847.711 −0.528649 −0.264324 0.964434i \(-0.585149\pi\)
−0.264324 + 0.964434i \(0.585149\pi\)
\(138\) −1131.50 −0.697970
\(139\) 1167.60 0.712481 0.356240 0.934394i \(-0.384058\pi\)
0.356240 + 0.934394i \(0.384058\pi\)
\(140\) −1849.47 −1.11649
\(141\) 1523.98 0.910231
\(142\) −1862.84 −1.10089
\(143\) −1572.28 −0.919442
\(144\) 144.000 0.0833333
\(145\) −144.564 −0.0827959
\(146\) −700.247 −0.396937
\(147\) −542.473 −0.304370
\(148\) −1330.40 −0.738906
\(149\) −1721.57 −0.946552 −0.473276 0.880914i \(-0.656929\pi\)
−0.473276 + 0.880914i \(0.656929\pi\)
\(150\) 1698.72 0.924666
\(151\) 2327.14 1.25417 0.627085 0.778951i \(-0.284248\pi\)
0.627085 + 0.778951i \(0.284248\pi\)
\(152\) 0 0
\(153\) −661.037 −0.349292
\(154\) −2634.84 −1.37871
\(155\) −2366.53 −1.22635
\(156\) −327.777 −0.168226
\(157\) −954.679 −0.485297 −0.242649 0.970114i \(-0.578016\pi\)
−0.242649 + 0.970114i \(0.578016\pi\)
\(158\) −389.736 −0.196239
\(159\) −1477.83 −0.737103
\(160\) −646.463 −0.319421
\(161\) 4316.16 2.11280
\(162\) −162.000 −0.0785674
\(163\) 1826.52 0.877696 0.438848 0.898561i \(-0.355387\pi\)
0.438848 + 0.898561i \(0.355387\pi\)
\(164\) −170.863 −0.0813548
\(165\) 3488.56 1.64596
\(166\) 572.166 0.267522
\(167\) 1465.61 0.679116 0.339558 0.940585i \(-0.389722\pi\)
0.339558 + 0.940585i \(0.389722\pi\)
\(168\) −549.293 −0.252255
\(169\) −1450.90 −0.660402
\(170\) 2967.61 1.33886
\(171\) 0 0
\(172\) 188.987 0.0837800
\(173\) 4162.86 1.82946 0.914730 0.404065i \(-0.132403\pi\)
0.914730 + 0.404065i \(0.132403\pi\)
\(174\) −42.9356 −0.0187066
\(175\) −6479.83 −2.79902
\(176\) −920.982 −0.394441
\(177\) −1380.12 −0.586081
\(178\) −1679.38 −0.707162
\(179\) −2637.74 −1.10142 −0.550710 0.834697i \(-0.685643\pi\)
−0.550710 + 0.834697i \(0.685643\pi\)
\(180\) 727.271 0.301153
\(181\) −779.832 −0.320246 −0.160123 0.987097i \(-0.551189\pi\)
−0.160123 + 0.987097i \(0.551189\pi\)
\(182\) 1250.32 0.509229
\(183\) −1352.23 −0.546230
\(184\) 1508.67 0.604460
\(185\) −6719.17 −2.67029
\(186\) −702.862 −0.277077
\(187\) 4227.80 1.65330
\(188\) −2031.98 −0.788284
\(189\) 617.955 0.237828
\(190\) 0 0
\(191\) 3961.70 1.50083 0.750415 0.660967i \(-0.229854\pi\)
0.750415 + 0.660967i \(0.229854\pi\)
\(192\) −192.000 −0.0721688
\(193\) −364.405 −0.135909 −0.0679545 0.997688i \(-0.521647\pi\)
−0.0679545 + 0.997688i \(0.521647\pi\)
\(194\) −1211.03 −0.448180
\(195\) −1655.44 −0.607940
\(196\) 723.297 0.263592
\(197\) 2309.32 0.835190 0.417595 0.908633i \(-0.362873\pi\)
0.417595 + 0.908633i \(0.362873\pi\)
\(198\) 1036.10 0.371882
\(199\) 3676.55 1.30967 0.654833 0.755774i \(-0.272739\pi\)
0.654833 + 0.755774i \(0.272739\pi\)
\(200\) −2264.96 −0.800785
\(201\) −1567.39 −0.550027
\(202\) −94.5519 −0.0329339
\(203\) 163.780 0.0566259
\(204\) 881.383 0.302496
\(205\) −862.945 −0.294003
\(206\) 861.501 0.291377
\(207\) −1697.25 −0.569890
\(208\) 437.036 0.145688
\(209\) 0 0
\(210\) −2774.20 −0.911610
\(211\) 1342.60 0.438051 0.219025 0.975719i \(-0.429712\pi\)
0.219025 + 0.975719i \(0.429712\pi\)
\(212\) 1970.44 0.638350
\(213\) −2794.26 −0.898870
\(214\) −2111.57 −0.674504
\(215\) 954.480 0.302767
\(216\) 216.000 0.0680414
\(217\) 2681.09 0.838730
\(218\) 565.143 0.175579
\(219\) −1050.37 −0.324098
\(220\) −4651.41 −1.42545
\(221\) −2006.23 −0.610650
\(222\) −1995.60 −0.603314
\(223\) −2239.58 −0.672528 −0.336264 0.941768i \(-0.609163\pi\)
−0.336264 + 0.941768i \(0.609163\pi\)
\(224\) 732.391 0.218459
\(225\) 2548.08 0.754987
\(226\) 1683.72 0.495573
\(227\) −6233.41 −1.82258 −0.911291 0.411763i \(-0.864913\pi\)
−0.911291 + 0.411763i \(0.864913\pi\)
\(228\) 0 0
\(229\) −2291.74 −0.661320 −0.330660 0.943750i \(-0.607271\pi\)
−0.330660 + 0.943750i \(0.607271\pi\)
\(230\) 7619.53 2.18442
\(231\) −3952.26 −1.12571
\(232\) 57.2475 0.0162004
\(233\) 1599.07 0.449609 0.224804 0.974404i \(-0.427826\pi\)
0.224804 + 0.974404i \(0.427826\pi\)
\(234\) −491.666 −0.137356
\(235\) −10262.5 −2.84873
\(236\) 1840.16 0.507561
\(237\) −584.604 −0.160228
\(238\) −3362.06 −0.915673
\(239\) −2443.37 −0.661291 −0.330646 0.943755i \(-0.607266\pi\)
−0.330646 + 0.943755i \(0.607266\pi\)
\(240\) −969.695 −0.260806
\(241\) −996.491 −0.266347 −0.133173 0.991093i \(-0.542517\pi\)
−0.133173 + 0.991093i \(0.542517\pi\)
\(242\) −3964.62 −1.05312
\(243\) −243.000 −0.0641500
\(244\) 1802.98 0.473049
\(245\) 3653.01 0.952580
\(246\) −256.295 −0.0664259
\(247\) 0 0
\(248\) 937.150 0.239956
\(249\) 858.248 0.218431
\(250\) −6388.68 −1.61622
\(251\) 759.890 0.191091 0.0955455 0.995425i \(-0.469540\pi\)
0.0955455 + 0.995425i \(0.469540\pi\)
\(252\) −823.939 −0.205966
\(253\) 10855.1 2.69746
\(254\) 1910.79 0.472021
\(255\) 4451.42 1.09317
\(256\) 256.000 0.0625000
\(257\) −4773.20 −1.15854 −0.579268 0.815137i \(-0.696662\pi\)
−0.579268 + 0.815137i \(0.696662\pi\)
\(258\) 283.481 0.0684061
\(259\) 7612.27 1.82627
\(260\) 2207.25 0.526492
\(261\) −64.4035 −0.0152738
\(262\) 3278.22 0.773011
\(263\) 4356.43 1.02140 0.510702 0.859758i \(-0.329386\pi\)
0.510702 + 0.859758i \(0.329386\pi\)
\(264\) −1381.47 −0.322060
\(265\) 9951.69 2.30690
\(266\) 0 0
\(267\) −2519.07 −0.577396
\(268\) 2089.86 0.476337
\(269\) −140.715 −0.0318941 −0.0159471 0.999873i \(-0.505076\pi\)
−0.0159471 + 0.999873i \(0.505076\pi\)
\(270\) 1090.91 0.245891
\(271\) 1523.06 0.341399 0.170700 0.985323i \(-0.445397\pi\)
0.170700 + 0.985323i \(0.445397\pi\)
\(272\) −1175.18 −0.261969
\(273\) 1875.48 0.415784
\(274\) 1695.42 0.373811
\(275\) −16296.8 −3.57357
\(276\) 2263.01 0.493540
\(277\) 2217.54 0.481008 0.240504 0.970648i \(-0.422687\pi\)
0.240504 + 0.970648i \(0.422687\pi\)
\(278\) −2335.21 −0.503800
\(279\) −1054.29 −0.226233
\(280\) 3698.94 0.789477
\(281\) 170.520 0.0362005 0.0181003 0.999836i \(-0.494238\pi\)
0.0181003 + 0.999836i \(0.494238\pi\)
\(282\) −3047.97 −0.643631
\(283\) 2176.30 0.457130 0.228565 0.973529i \(-0.426597\pi\)
0.228565 + 0.973529i \(0.426597\pi\)
\(284\) 3725.68 0.778444
\(285\) 0 0
\(286\) 3144.55 0.650144
\(287\) 977.646 0.201075
\(288\) −288.000 −0.0589256
\(289\) 481.691 0.0980441
\(290\) 289.128 0.0585455
\(291\) −1816.55 −0.365938
\(292\) 1400.49 0.280677
\(293\) −9879.00 −1.96975 −0.984876 0.173261i \(-0.944569\pi\)
−0.984876 + 0.173261i \(0.944569\pi\)
\(294\) 1084.95 0.215222
\(295\) 9293.73 1.83424
\(296\) 2660.80 0.522485
\(297\) 1554.16 0.303641
\(298\) 3443.13 0.669313
\(299\) −5151.12 −0.996312
\(300\) −3397.44 −0.653838
\(301\) −1081.35 −0.207069
\(302\) −4654.27 −0.886832
\(303\) −141.828 −0.0268904
\(304\) 0 0
\(305\) 9105.94 1.70952
\(306\) 1322.07 0.246987
\(307\) −1343.71 −0.249803 −0.124902 0.992169i \(-0.539862\pi\)
−0.124902 + 0.992169i \(0.539862\pi\)
\(308\) 5269.67 0.974895
\(309\) 1292.25 0.237908
\(310\) 4733.07 0.867162
\(311\) 88.3559 0.0161100 0.00805499 0.999968i \(-0.497436\pi\)
0.00805499 + 0.999968i \(0.497436\pi\)
\(312\) 655.555 0.118953
\(313\) −1892.46 −0.341751 −0.170876 0.985293i \(-0.554660\pi\)
−0.170876 + 0.985293i \(0.554660\pi\)
\(314\) 1909.36 0.343157
\(315\) −4161.30 −0.744326
\(316\) 779.472 0.138762
\(317\) 8378.86 1.48455 0.742277 0.670093i \(-0.233746\pi\)
0.742277 + 0.670093i \(0.233746\pi\)
\(318\) 2955.66 0.521211
\(319\) 411.906 0.0722956
\(320\) 1292.93 0.225865
\(321\) −3167.35 −0.550730
\(322\) −8632.31 −1.49398
\(323\) 0 0
\(324\) 324.000 0.0555556
\(325\) 7733.36 1.31991
\(326\) −3653.05 −0.620625
\(327\) 847.714 0.143360
\(328\) 341.727 0.0575265
\(329\) 11626.6 1.94831
\(330\) −6977.12 −1.16387
\(331\) 4833.50 0.802639 0.401319 0.915938i \(-0.368552\pi\)
0.401319 + 0.915938i \(0.368552\pi\)
\(332\) −1144.33 −0.189167
\(333\) −2993.40 −0.492604
\(334\) −2931.22 −0.480208
\(335\) 10554.8 1.72141
\(336\) 1098.59 0.178371
\(337\) −2625.77 −0.424436 −0.212218 0.977222i \(-0.568069\pi\)
−0.212218 + 0.977222i \(0.568069\pi\)
\(338\) 2901.81 0.466975
\(339\) 2525.58 0.404633
\(340\) −5935.23 −0.946714
\(341\) 6742.95 1.07082
\(342\) 0 0
\(343\) 3711.75 0.584302
\(344\) −377.975 −0.0592414
\(345\) 11429.3 1.78357
\(346\) −8325.73 −1.29362
\(347\) 8041.49 1.24406 0.622031 0.782992i \(-0.286308\pi\)
0.622031 + 0.782992i \(0.286308\pi\)
\(348\) 85.8713 0.0132275
\(349\) 8603.95 1.31965 0.659826 0.751418i \(-0.270630\pi\)
0.659826 + 0.751418i \(0.270630\pi\)
\(350\) 12959.7 1.97921
\(351\) −737.499 −0.112150
\(352\) 1841.96 0.278912
\(353\) −10359.4 −1.56198 −0.780988 0.624547i \(-0.785284\pi\)
−0.780988 + 0.624547i \(0.785284\pi\)
\(354\) 2760.24 0.414422
\(355\) 18816.5 2.81317
\(356\) 3358.76 0.500039
\(357\) −5043.10 −0.747644
\(358\) 5275.49 0.778821
\(359\) −3544.68 −0.521117 −0.260558 0.965458i \(-0.583907\pi\)
−0.260558 + 0.965458i \(0.583907\pi\)
\(360\) −1454.54 −0.212948
\(361\) 0 0
\(362\) 1559.66 0.226448
\(363\) −5946.93 −0.859870
\(364\) −2500.64 −0.360079
\(365\) 7073.18 1.01432
\(366\) 2704.47 0.386243
\(367\) 330.253 0.0469729 0.0234864 0.999724i \(-0.492523\pi\)
0.0234864 + 0.999724i \(0.492523\pi\)
\(368\) −3017.34 −0.427418
\(369\) −384.443 −0.0542365
\(370\) 13438.3 1.88818
\(371\) −11274.5 −1.57774
\(372\) 1405.72 0.195923
\(373\) −12016.2 −1.66802 −0.834012 0.551747i \(-0.813961\pi\)
−0.834012 + 0.551747i \(0.813961\pi\)
\(374\) −8455.60 −1.16906
\(375\) −9583.02 −1.31964
\(376\) 4063.96 0.557401
\(377\) −195.463 −0.0267025
\(378\) −1235.91 −0.168170
\(379\) 5463.94 0.740538 0.370269 0.928925i \(-0.379266\pi\)
0.370269 + 0.928925i \(0.379266\pi\)
\(380\) 0 0
\(381\) 2866.18 0.385404
\(382\) −7923.40 −1.06125
\(383\) 7380.58 0.984673 0.492337 0.870405i \(-0.336143\pi\)
0.492337 + 0.870405i \(0.336143\pi\)
\(384\) 384.000 0.0510310
\(385\) 26614.5 3.52311
\(386\) 728.809 0.0961021
\(387\) 425.222 0.0558533
\(388\) 2422.06 0.316911
\(389\) 2035.83 0.265349 0.132675 0.991160i \(-0.457643\pi\)
0.132675 + 0.991160i \(0.457643\pi\)
\(390\) 3310.88 0.429879
\(391\) 13851.2 1.79152
\(392\) −1446.59 −0.186388
\(393\) 4917.32 0.631161
\(394\) −4618.64 −0.590568
\(395\) 3936.72 0.501463
\(396\) −2072.21 −0.262961
\(397\) 13314.3 1.68318 0.841592 0.540114i \(-0.181619\pi\)
0.841592 + 0.540114i \(0.181619\pi\)
\(398\) −7353.09 −0.926074
\(399\) 0 0
\(400\) 4529.92 0.566240
\(401\) 13204.3 1.64437 0.822186 0.569219i \(-0.192754\pi\)
0.822186 + 0.569219i \(0.192754\pi\)
\(402\) 3134.79 0.388928
\(403\) −3199.75 −0.395511
\(404\) 189.104 0.0232878
\(405\) 1636.36 0.200769
\(406\) −327.559 −0.0400406
\(407\) 19144.9 2.33164
\(408\) −1762.77 −0.213897
\(409\) 2761.86 0.333900 0.166950 0.985965i \(-0.446608\pi\)
0.166950 + 0.985965i \(0.446608\pi\)
\(410\) 1725.89 0.207892
\(411\) 2543.13 0.305215
\(412\) −1723.00 −0.206034
\(413\) −10529.0 −1.25448
\(414\) 3394.51 0.402973
\(415\) −5779.44 −0.683618
\(416\) −874.073 −0.103017
\(417\) −3502.81 −0.411351
\(418\) 0 0
\(419\) −15182.0 −1.77015 −0.885073 0.465453i \(-0.845891\pi\)
−0.885073 + 0.465453i \(0.845891\pi\)
\(420\) 5548.40 0.644606
\(421\) 15980.4 1.84996 0.924982 0.380011i \(-0.124080\pi\)
0.924982 + 0.380011i \(0.124080\pi\)
\(422\) −2685.21 −0.309749
\(423\) −4571.95 −0.525522
\(424\) −3940.88 −0.451382
\(425\) −20794.8 −2.37340
\(426\) 5588.51 0.635597
\(427\) −10316.3 −1.16918
\(428\) 4223.14 0.476946
\(429\) 4716.83 0.530840
\(430\) −1908.96 −0.214089
\(431\) 4762.00 0.532198 0.266099 0.963946i \(-0.414265\pi\)
0.266099 + 0.963946i \(0.414265\pi\)
\(432\) −432.000 −0.0481125
\(433\) 9974.90 1.10707 0.553537 0.832825i \(-0.313278\pi\)
0.553537 + 0.832825i \(0.313278\pi\)
\(434\) −5362.18 −0.593072
\(435\) 433.693 0.0478022
\(436\) −1130.29 −0.124153
\(437\) 0 0
\(438\) 2100.74 0.229172
\(439\) −8956.89 −0.973779 −0.486890 0.873464i \(-0.661869\pi\)
−0.486890 + 0.873464i \(0.661869\pi\)
\(440\) 9302.83 1.00794
\(441\) 1627.42 0.175728
\(442\) 4012.46 0.431795
\(443\) 11368.1 1.21922 0.609612 0.792700i \(-0.291325\pi\)
0.609612 + 0.792700i \(0.291325\pi\)
\(444\) 3991.19 0.426607
\(445\) 16963.4 1.80706
\(446\) 4479.17 0.475549
\(447\) 5164.70 0.546492
\(448\) −1464.78 −0.154474
\(449\) 7630.20 0.801985 0.400993 0.916081i \(-0.368665\pi\)
0.400993 + 0.916081i \(0.368665\pi\)
\(450\) −5096.16 −0.533856
\(451\) 2458.78 0.256717
\(452\) −3367.44 −0.350423
\(453\) −6981.41 −0.724095
\(454\) 12466.8 1.28876
\(455\) −12629.4 −1.30127
\(456\) 0 0
\(457\) −1500.40 −0.153579 −0.0767896 0.997047i \(-0.524467\pi\)
−0.0767896 + 0.997047i \(0.524467\pi\)
\(458\) 4583.47 0.467624
\(459\) 1983.11 0.201664
\(460\) −15239.1 −1.54462
\(461\) −14495.8 −1.46450 −0.732252 0.681034i \(-0.761531\pi\)
−0.732252 + 0.681034i \(0.761531\pi\)
\(462\) 7904.51 0.795998
\(463\) −12265.2 −1.23113 −0.615566 0.788086i \(-0.711072\pi\)
−0.615566 + 0.788086i \(0.711072\pi\)
\(464\) −114.495 −0.0114554
\(465\) 7099.60 0.708035
\(466\) −3198.15 −0.317921
\(467\) −1890.82 −0.187360 −0.0936798 0.995602i \(-0.529863\pi\)
−0.0936798 + 0.995602i \(0.529863\pi\)
\(468\) 983.332 0.0971251
\(469\) −11957.8 −1.17731
\(470\) 20525.0 2.01436
\(471\) 2864.04 0.280187
\(472\) −3680.32 −0.358900
\(473\) −2719.59 −0.264370
\(474\) 1169.21 0.113299
\(475\) 0 0
\(476\) 6724.13 0.647479
\(477\) 4433.49 0.425567
\(478\) 4886.74 0.467603
\(479\) −11662.9 −1.11251 −0.556254 0.831013i \(-0.687762\pi\)
−0.556254 + 0.831013i \(0.687762\pi\)
\(480\) 1939.39 0.184418
\(481\) −9084.88 −0.861195
\(482\) 1992.98 0.188336
\(483\) −12948.5 −1.21983
\(484\) 7929.24 0.744669
\(485\) 12232.6 1.14527
\(486\) 486.000 0.0453609
\(487\) 13286.6 1.23629 0.618147 0.786063i \(-0.287884\pi\)
0.618147 + 0.786063i \(0.287884\pi\)
\(488\) −3605.96 −0.334496
\(489\) −5479.57 −0.506738
\(490\) −7306.02 −0.673576
\(491\) 4037.71 0.371119 0.185559 0.982633i \(-0.440590\pi\)
0.185559 + 0.982633i \(0.440590\pi\)
\(492\) 512.590 0.0469702
\(493\) 525.594 0.0480153
\(494\) 0 0
\(495\) −10465.7 −0.950298
\(496\) −1874.30 −0.169674
\(497\) −21317.6 −1.92399
\(498\) −1716.50 −0.154454
\(499\) 3544.83 0.318013 0.159006 0.987278i \(-0.449171\pi\)
0.159006 + 0.987278i \(0.449171\pi\)
\(500\) 12777.4 1.14284
\(501\) −4396.83 −0.392088
\(502\) −1519.78 −0.135122
\(503\) −12958.4 −1.14868 −0.574339 0.818617i \(-0.694741\pi\)
−0.574339 + 0.818617i \(0.694741\pi\)
\(504\) 1647.88 0.145640
\(505\) 955.068 0.0841584
\(506\) −21710.3 −1.90739
\(507\) 4352.71 0.381283
\(508\) −3821.57 −0.333769
\(509\) −8236.28 −0.717224 −0.358612 0.933487i \(-0.616750\pi\)
−0.358612 + 0.933487i \(0.616750\pi\)
\(510\) −8902.84 −0.772989
\(511\) −8013.34 −0.693717
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 9546.39 0.819209
\(515\) −8702.01 −0.744575
\(516\) −566.962 −0.0483704
\(517\) 29240.9 2.48745
\(518\) −15224.5 −1.29137
\(519\) −12488.6 −1.05624
\(520\) −4414.50 −0.372286
\(521\) 7427.58 0.624584 0.312292 0.949986i \(-0.398903\pi\)
0.312292 + 0.949986i \(0.398903\pi\)
\(522\) 128.807 0.0108002
\(523\) −4604.88 −0.385005 −0.192502 0.981297i \(-0.561660\pi\)
−0.192502 + 0.981297i \(0.561660\pi\)
\(524\) −6556.43 −0.546601
\(525\) 19439.5 1.61602
\(526\) −8712.87 −0.722241
\(527\) 8604.04 0.711191
\(528\) 2762.94 0.227731
\(529\) 23396.8 1.92298
\(530\) −19903.4 −1.63122
\(531\) 4140.36 0.338374
\(532\) 0 0
\(533\) −1166.77 −0.0948191
\(534\) 5038.14 0.408280
\(535\) 21328.9 1.72361
\(536\) −4179.72 −0.336821
\(537\) 7913.23 0.635905
\(538\) 281.429 0.0225525
\(539\) −10408.5 −0.831772
\(540\) −2181.81 −0.173871
\(541\) 9030.87 0.717685 0.358842 0.933398i \(-0.383172\pi\)
0.358842 + 0.933398i \(0.383172\pi\)
\(542\) −3046.12 −0.241406
\(543\) 2339.50 0.184894
\(544\) 2350.35 0.185240
\(545\) −5708.50 −0.448670
\(546\) −3750.95 −0.294004
\(547\) 12875.6 1.00644 0.503218 0.864159i \(-0.332149\pi\)
0.503218 + 0.864159i \(0.332149\pi\)
\(548\) −3390.85 −0.264324
\(549\) 4056.70 0.315366
\(550\) 32593.6 2.52690
\(551\) 0 0
\(552\) −4526.01 −0.348985
\(553\) −4459.98 −0.342962
\(554\) −4435.09 −0.340124
\(555\) 20157.5 1.54169
\(556\) 4670.41 0.356240
\(557\) −4891.20 −0.372077 −0.186038 0.982542i \(-0.559565\pi\)
−0.186038 + 0.982542i \(0.559565\pi\)
\(558\) 2108.59 0.159971
\(559\) 1290.54 0.0976456
\(560\) −7397.87 −0.558245
\(561\) −12683.4 −0.954534
\(562\) −341.039 −0.0255976
\(563\) 2367.05 0.177192 0.0885961 0.996068i \(-0.471762\pi\)
0.0885961 + 0.996068i \(0.471762\pi\)
\(564\) 6095.94 0.455116
\(565\) −17007.2 −1.26637
\(566\) −4352.61 −0.323240
\(567\) −1853.86 −0.137310
\(568\) −7451.35 −0.550443
\(569\) −675.013 −0.0497329 −0.0248665 0.999691i \(-0.507916\pi\)
−0.0248665 + 0.999691i \(0.507916\pi\)
\(570\) 0 0
\(571\) 17245.3 1.26391 0.631955 0.775005i \(-0.282253\pi\)
0.631955 + 0.775005i \(0.282253\pi\)
\(572\) −6289.10 −0.459721
\(573\) −11885.1 −0.866504
\(574\) −1955.29 −0.142182
\(575\) −53391.9 −3.87234
\(576\) 576.000 0.0416667
\(577\) 24732.5 1.78445 0.892226 0.451589i \(-0.149143\pi\)
0.892226 + 0.451589i \(0.149143\pi\)
\(578\) −963.382 −0.0693277
\(579\) 1093.21 0.0784671
\(580\) −578.257 −0.0413979
\(581\) 6547.64 0.467542
\(582\) 3633.09 0.258757
\(583\) −28355.3 −2.01433
\(584\) −2800.99 −0.198469
\(585\) 4966.31 0.350995
\(586\) 19758.0 1.39282
\(587\) 13718.5 0.964607 0.482304 0.876004i \(-0.339800\pi\)
0.482304 + 0.876004i \(0.339800\pi\)
\(588\) −2169.89 −0.152185
\(589\) 0 0
\(590\) −18587.5 −1.29701
\(591\) −6927.97 −0.482197
\(592\) −5321.59 −0.369453
\(593\) 7164.95 0.496170 0.248085 0.968738i \(-0.420199\pi\)
0.248085 + 0.968738i \(0.420199\pi\)
\(594\) −3108.31 −0.214706
\(595\) 33960.2 2.33989
\(596\) −6886.26 −0.473276
\(597\) −11029.6 −0.756136
\(598\) 10302.2 0.704499
\(599\) −1507.84 −0.102852 −0.0514262 0.998677i \(-0.516377\pi\)
−0.0514262 + 0.998677i \(0.516377\pi\)
\(600\) 6794.88 0.462333
\(601\) 8574.64 0.581975 0.290987 0.956727i \(-0.406016\pi\)
0.290987 + 0.956727i \(0.406016\pi\)
\(602\) 2162.70 0.146420
\(603\) 4702.18 0.317558
\(604\) 9308.54 0.627085
\(605\) 40046.6 2.69112
\(606\) 283.656 0.0190144
\(607\) 504.688 0.0337474 0.0168737 0.999858i \(-0.494629\pi\)
0.0168737 + 0.999858i \(0.494629\pi\)
\(608\) 0 0
\(609\) −491.339 −0.0326930
\(610\) −18211.9 −1.20882
\(611\) −13875.8 −0.918745
\(612\) −2644.15 −0.174646
\(613\) 14944.4 0.984666 0.492333 0.870407i \(-0.336144\pi\)
0.492333 + 0.870407i \(0.336144\pi\)
\(614\) 2687.42 0.176637
\(615\) 2588.83 0.169743
\(616\) −10539.3 −0.689355
\(617\) −3482.10 −0.227203 −0.113601 0.993526i \(-0.536239\pi\)
−0.113601 + 0.993526i \(0.536239\pi\)
\(618\) −2584.50 −0.168226
\(619\) 3577.83 0.232319 0.116159 0.993231i \(-0.462942\pi\)
0.116159 + 0.993231i \(0.462942\pi\)
\(620\) −9466.14 −0.613176
\(621\) 5091.76 0.329026
\(622\) −176.712 −0.0113915
\(623\) −19218.2 −1.23589
\(624\) −1311.11 −0.0841128
\(625\) 29142.0 1.86509
\(626\) 3784.92 0.241655
\(627\) 0 0
\(628\) −3818.72 −0.242649
\(629\) 24429.0 1.54856
\(630\) 8322.61 0.526318
\(631\) 3649.43 0.230240 0.115120 0.993352i \(-0.463275\pi\)
0.115120 + 0.993352i \(0.463275\pi\)
\(632\) −1558.94 −0.0981194
\(633\) −4027.81 −0.252909
\(634\) −16757.7 −1.04974
\(635\) −19300.8 −1.20619
\(636\) −5911.31 −0.368552
\(637\) 4939.17 0.307217
\(638\) −823.811 −0.0511207
\(639\) 8382.77 0.518963
\(640\) −2585.85 −0.159711
\(641\) −14669.2 −0.903898 −0.451949 0.892044i \(-0.649271\pi\)
−0.451949 + 0.892044i \(0.649271\pi\)
\(642\) 6334.71 0.389425
\(643\) 7008.11 0.429818 0.214909 0.976634i \(-0.431055\pi\)
0.214909 + 0.976634i \(0.431055\pi\)
\(644\) 17264.6 1.05640
\(645\) −2863.44 −0.174803
\(646\) 0 0
\(647\) 9659.76 0.586962 0.293481 0.955965i \(-0.405186\pi\)
0.293481 + 0.955965i \(0.405186\pi\)
\(648\) −648.000 −0.0392837
\(649\) −26480.6 −1.60162
\(650\) −15466.7 −0.933315
\(651\) −8043.28 −0.484241
\(652\) 7306.10 0.438848
\(653\) 6799.09 0.407456 0.203728 0.979027i \(-0.434694\pi\)
0.203728 + 0.979027i \(0.434694\pi\)
\(654\) −1695.43 −0.101371
\(655\) −33113.2 −1.97533
\(656\) −683.454 −0.0406774
\(657\) 3151.11 0.187118
\(658\) −23253.2 −1.37766
\(659\) −19348.2 −1.14370 −0.571851 0.820357i \(-0.693775\pi\)
−0.571851 + 0.820357i \(0.693775\pi\)
\(660\) 13954.2 0.822982
\(661\) 5128.00 0.301749 0.150875 0.988553i \(-0.451791\pi\)
0.150875 + 0.988553i \(0.451791\pi\)
\(662\) −9667.01 −0.567551
\(663\) 6018.69 0.352559
\(664\) 2288.66 0.133761
\(665\) 0 0
\(666\) 5986.79 0.348324
\(667\) 1349.49 0.0783398
\(668\) 5862.45 0.339558
\(669\) 6718.75 0.388284
\(670\) −21109.6 −1.21722
\(671\) −25945.5 −1.49272
\(672\) −2197.17 −0.126128
\(673\) −23921.1 −1.37012 −0.685061 0.728486i \(-0.740224\pi\)
−0.685061 + 0.728486i \(0.740224\pi\)
\(674\) 5251.55 0.300122
\(675\) −7644.24 −0.435892
\(676\) −5803.61 −0.330201
\(677\) −766.891 −0.0435362 −0.0217681 0.999763i \(-0.506930\pi\)
−0.0217681 + 0.999763i \(0.506930\pi\)
\(678\) −5051.16 −0.286119
\(679\) −13858.6 −0.783274
\(680\) 11870.5 0.669428
\(681\) 18700.2 1.05227
\(682\) −13485.9 −0.757187
\(683\) −6573.45 −0.368267 −0.184133 0.982901i \(-0.558948\pi\)
−0.184133 + 0.982901i \(0.558948\pi\)
\(684\) 0 0
\(685\) −17125.5 −0.955226
\(686\) −7423.50 −0.413164
\(687\) 6875.21 0.381813
\(688\) 755.950 0.0418900
\(689\) 13455.5 0.743998
\(690\) −22858.6 −1.26118
\(691\) −12816.9 −0.705613 −0.352806 0.935696i \(-0.614773\pi\)
−0.352806 + 0.935696i \(0.614773\pi\)
\(692\) 16651.5 0.914730
\(693\) 11856.8 0.649930
\(694\) −16083.0 −0.879685
\(695\) 23587.9 1.28740
\(696\) −171.743 −0.00935328
\(697\) 3137.42 0.170500
\(698\) −17207.9 −0.933135
\(699\) −4797.22 −0.259582
\(700\) −25919.3 −1.39951
\(701\) −31041.6 −1.67250 −0.836251 0.548347i \(-0.815257\pi\)
−0.836251 + 0.548347i \(0.815257\pi\)
\(702\) 1475.00 0.0793023
\(703\) 0 0
\(704\) −3683.93 −0.197220
\(705\) 30787.5 1.64472
\(706\) 20718.9 1.10448
\(707\) −1082.01 −0.0575578
\(708\) −5520.49 −0.293040
\(709\) −13641.0 −0.722564 −0.361282 0.932457i \(-0.617661\pi\)
−0.361282 + 0.932457i \(0.617661\pi\)
\(710\) −37633.0 −1.98921
\(711\) 1753.81 0.0925079
\(712\) −6717.52 −0.353581
\(713\) 22091.4 1.16035
\(714\) 10086.2 0.528664
\(715\) −31763.1 −1.66136
\(716\) −10551.0 −0.550710
\(717\) 7330.11 0.381797
\(718\) 7089.35 0.368485
\(719\) 27533.8 1.42815 0.714073 0.700071i \(-0.246848\pi\)
0.714073 + 0.700071i \(0.246848\pi\)
\(720\) 2909.09 0.150577
\(721\) 9858.67 0.509232
\(722\) 0 0
\(723\) 2989.47 0.153775
\(724\) −3119.33 −0.160123
\(725\) −2025.99 −0.103784
\(726\) 11893.9 0.608020
\(727\) 20867.3 1.06455 0.532273 0.846573i \(-0.321338\pi\)
0.532273 + 0.846573i \(0.321338\pi\)
\(728\) 5001.27 0.254615
\(729\) 729.000 0.0370370
\(730\) −14146.4 −0.717234
\(731\) −3470.21 −0.175582
\(732\) −5408.94 −0.273115
\(733\) −18422.0 −0.928282 −0.464141 0.885761i \(-0.653637\pi\)
−0.464141 + 0.885761i \(0.653637\pi\)
\(734\) −660.505 −0.0332148
\(735\) −10959.0 −0.549972
\(736\) 6034.68 0.302230
\(737\) −30073.8 −1.50310
\(738\) 768.885 0.0383510
\(739\) −15508.0 −0.771949 −0.385975 0.922509i \(-0.626135\pi\)
−0.385975 + 0.922509i \(0.626135\pi\)
\(740\) −26876.7 −1.33514
\(741\) 0 0
\(742\) 22548.9 1.11563
\(743\) −11877.0 −0.586441 −0.293221 0.956045i \(-0.594727\pi\)
−0.293221 + 0.956045i \(0.594727\pi\)
\(744\) −2811.45 −0.138539
\(745\) −34779.0 −1.71034
\(746\) 24032.3 1.17947
\(747\) −2574.75 −0.126111
\(748\) 16911.2 0.826650
\(749\) −24164.0 −1.17881
\(750\) 19166.0 0.933126
\(751\) 26834.2 1.30385 0.651927 0.758281i \(-0.273961\pi\)
0.651927 + 0.758281i \(0.273961\pi\)
\(752\) −8127.92 −0.394142
\(753\) −2279.67 −0.110326
\(754\) 390.926 0.0188815
\(755\) 47012.8 2.26618
\(756\) 2471.82 0.118914
\(757\) −39104.9 −1.87753 −0.938767 0.344552i \(-0.888031\pi\)
−0.938767 + 0.344552i \(0.888031\pi\)
\(758\) −10927.9 −0.523639
\(759\) −32565.4 −1.55738
\(760\) 0 0
\(761\) 13627.0 0.649117 0.324559 0.945866i \(-0.394784\pi\)
0.324559 + 0.945866i \(0.394784\pi\)
\(762\) −5732.36 −0.272522
\(763\) 6467.27 0.306856
\(764\) 15846.8 0.750415
\(765\) −13354.3 −0.631143
\(766\) −14761.2 −0.696269
\(767\) 12565.9 0.591563
\(768\) −768.000 −0.0360844
\(769\) 23166.4 1.08635 0.543173 0.839621i \(-0.317223\pi\)
0.543173 + 0.839621i \(0.317223\pi\)
\(770\) −53228.9 −2.49122
\(771\) 14319.6 0.668881
\(772\) −1457.62 −0.0679545
\(773\) 32201.6 1.49833 0.749167 0.662381i \(-0.230454\pi\)
0.749167 + 0.662381i \(0.230454\pi\)
\(774\) −850.443 −0.0394943
\(775\) −33165.7 −1.53722
\(776\) −4844.12 −0.224090
\(777\) −22836.8 −1.05440
\(778\) −4071.67 −0.187630
\(779\) 0 0
\(780\) −6621.75 −0.303970
\(781\) −53613.7 −2.45640
\(782\) −27702.4 −1.26680
\(783\) 193.210 0.00881836
\(784\) 2893.19 0.131796
\(785\) −19286.4 −0.876893
\(786\) −9834.65 −0.446298
\(787\) −36081.7 −1.63427 −0.817137 0.576443i \(-0.804440\pi\)
−0.817137 + 0.576443i \(0.804440\pi\)
\(788\) 9237.29 0.417595
\(789\) −13069.3 −0.589708
\(790\) −7873.44 −0.354588
\(791\) 19267.8 0.866100
\(792\) 4144.42 0.185941
\(793\) 12312.0 0.551339
\(794\) −26628.5 −1.19019
\(795\) −29855.1 −1.33189
\(796\) 14706.2 0.654833
\(797\) −39094.9 −1.73753 −0.868765 0.495224i \(-0.835086\pi\)
−0.868765 + 0.495224i \(0.835086\pi\)
\(798\) 0 0
\(799\) 37311.5 1.65205
\(800\) −9059.84 −0.400392
\(801\) 7557.21 0.333359
\(802\) −26408.7 −1.16275
\(803\) −20153.6 −0.885684
\(804\) −6269.58 −0.275014
\(805\) 87194.9 3.81766
\(806\) 6399.51 0.279669
\(807\) 422.144 0.0184141
\(808\) −378.208 −0.0164670
\(809\) 7261.33 0.315568 0.157784 0.987474i \(-0.449565\pi\)
0.157784 + 0.987474i \(0.449565\pi\)
\(810\) −3272.72 −0.141965
\(811\) 34133.3 1.47791 0.738953 0.673757i \(-0.235321\pi\)
0.738953 + 0.673757i \(0.235321\pi\)
\(812\) 655.118 0.0283130
\(813\) −4569.18 −0.197107
\(814\) −38289.8 −1.64872
\(815\) 36899.4 1.58593
\(816\) 3525.53 0.151248
\(817\) 0 0
\(818\) −5523.72 −0.236103
\(819\) −5626.43 −0.240053
\(820\) −3451.78 −0.147002
\(821\) 13880.3 0.590043 0.295022 0.955491i \(-0.404673\pi\)
0.295022 + 0.955491i \(0.404673\pi\)
\(822\) −5086.27 −0.215820
\(823\) 13016.1 0.551291 0.275646 0.961259i \(-0.411108\pi\)
0.275646 + 0.961259i \(0.411108\pi\)
\(824\) 3446.00 0.145688
\(825\) 48890.3 2.06320
\(826\) 21058.1 0.887051
\(827\) −19864.1 −0.835238 −0.417619 0.908622i \(-0.637135\pi\)
−0.417619 + 0.908622i \(0.637135\pi\)
\(828\) −6789.02 −0.284945
\(829\) 11325.6 0.474494 0.237247 0.971449i \(-0.423755\pi\)
0.237247 + 0.971449i \(0.423755\pi\)
\(830\) 11558.9 0.483391
\(831\) −6652.63 −0.277710
\(832\) 1748.15 0.0728438
\(833\) −13281.3 −0.552424
\(834\) 7005.62 0.290869
\(835\) 29608.3 1.22711
\(836\) 0 0
\(837\) 3162.88 0.130615
\(838\) 30364.1 1.25168
\(839\) 4507.99 0.185498 0.0927491 0.995690i \(-0.470435\pi\)
0.0927491 + 0.995690i \(0.470435\pi\)
\(840\) −11096.8 −0.455805
\(841\) −24337.8 −0.997900
\(842\) −31960.7 −1.30812
\(843\) −511.559 −0.0209004
\(844\) 5370.42 0.219025
\(845\) −29311.1 −1.19329
\(846\) 9143.91 0.371600
\(847\) −45369.5 −1.84051
\(848\) 7881.75 0.319175
\(849\) −6528.91 −0.263924
\(850\) 41589.5 1.67825
\(851\) 62722.9 2.52657
\(852\) −11177.0 −0.449435
\(853\) −31512.5 −1.26491 −0.632454 0.774598i \(-0.717952\pi\)
−0.632454 + 0.774598i \(0.717952\pi\)
\(854\) 20632.6 0.826736
\(855\) 0 0
\(856\) −8446.28 −0.337252
\(857\) −6340.51 −0.252728 −0.126364 0.991984i \(-0.540331\pi\)
−0.126364 + 0.991984i \(0.540331\pi\)
\(858\) −9433.65 −0.375361
\(859\) −20206.0 −0.802582 −0.401291 0.915951i \(-0.631438\pi\)
−0.401291 + 0.915951i \(0.631438\pi\)
\(860\) 3817.92 0.151384
\(861\) −2932.94 −0.116091
\(862\) −9524.00 −0.376321
\(863\) −34553.8 −1.36295 −0.681474 0.731842i \(-0.738661\pi\)
−0.681474 + 0.731842i \(0.738661\pi\)
\(864\) 864.000 0.0340207
\(865\) 84098.1 3.30569
\(866\) −19949.8 −0.782820
\(867\) −1445.07 −0.0566058
\(868\) 10724.4 0.419365
\(869\) −11216.9 −0.437867
\(870\) −867.385 −0.0338013
\(871\) 14271.0 0.555172
\(872\) 2260.57 0.0877896
\(873\) 5449.64 0.211274
\(874\) 0 0
\(875\) −73109.5 −2.82463
\(876\) −4201.48 −0.162049
\(877\) 30588.2 1.17775 0.588877 0.808223i \(-0.299570\pi\)
0.588877 + 0.808223i \(0.299570\pi\)
\(878\) 17913.8 0.688566
\(879\) 29637.0 1.13724
\(880\) −18605.7 −0.712723
\(881\) 48445.7 1.85264 0.926321 0.376735i \(-0.122953\pi\)
0.926321 + 0.376735i \(0.122953\pi\)
\(882\) −3254.84 −0.124259
\(883\) 10270.4 0.391422 0.195711 0.980662i \(-0.437299\pi\)
0.195711 + 0.980662i \(0.437299\pi\)
\(884\) −8024.92 −0.305325
\(885\) −27881.2 −1.05900
\(886\) −22736.2 −0.862121
\(887\) 27630.3 1.04592 0.522962 0.852356i \(-0.324827\pi\)
0.522962 + 0.852356i \(0.324827\pi\)
\(888\) −7982.39 −0.301657
\(889\) 21866.3 0.824940
\(890\) −33926.8 −1.27779
\(891\) −4662.47 −0.175307
\(892\) −8958.34 −0.336264
\(893\) 0 0
\(894\) −10329.4 −0.386428
\(895\) −53287.6 −1.99018
\(896\) 2929.56 0.109230
\(897\) 15453.4 0.575221
\(898\) −15260.4 −0.567089
\(899\) 838.273 0.0310990
\(900\) 10192.3 0.377493
\(901\) −36181.5 −1.33782
\(902\) −4917.56 −0.181527
\(903\) 3244.05 0.119552
\(904\) 6734.88 0.247786
\(905\) −15754.2 −0.578658
\(906\) 13962.8 0.512013
\(907\) 40746.5 1.49169 0.745847 0.666118i \(-0.232045\pi\)
0.745847 + 0.666118i \(0.232045\pi\)
\(908\) −24933.6 −0.911291
\(909\) 425.484 0.0155252
\(910\) 25258.9 0.920137
\(911\) 13335.5 0.484989 0.242494 0.970153i \(-0.422034\pi\)
0.242494 + 0.970153i \(0.422034\pi\)
\(912\) 0 0
\(913\) 16467.3 0.596921
\(914\) 3000.80 0.108597
\(915\) −27317.8 −0.986993
\(916\) −9166.94 −0.330660
\(917\) 37514.6 1.35097
\(918\) −3966.22 −0.142598
\(919\) 16604.5 0.596009 0.298005 0.954564i \(-0.403679\pi\)
0.298005 + 0.954564i \(0.403679\pi\)
\(920\) 30478.1 1.09221
\(921\) 4031.13 0.144224
\(922\) 28991.6 1.03556
\(923\) 25441.5 0.907277
\(924\) −15809.0 −0.562856
\(925\) −94165.6 −3.34719
\(926\) 24530.5 0.870541
\(927\) −3876.75 −0.137356
\(928\) 228.990 0.00810018
\(929\) −16679.4 −0.589056 −0.294528 0.955643i \(-0.595162\pi\)
−0.294528 + 0.955643i \(0.595162\pi\)
\(930\) −14199.2 −0.500656
\(931\) 0 0
\(932\) 6396.29 0.224804
\(933\) −265.068 −0.00930110
\(934\) 3781.65 0.132483
\(935\) 85409.9 2.98738
\(936\) −1966.66 −0.0686778
\(937\) 6867.24 0.239427 0.119713 0.992808i \(-0.461802\pi\)
0.119713 + 0.992808i \(0.461802\pi\)
\(938\) 23915.5 0.832483
\(939\) 5677.38 0.197310
\(940\) −41050.0 −1.42437
\(941\) 48163.9 1.66854 0.834272 0.551353i \(-0.185888\pi\)
0.834272 + 0.551353i \(0.185888\pi\)
\(942\) −5728.07 −0.198122
\(943\) 8055.52 0.278180
\(944\) 7360.65 0.253780
\(945\) 12483.9 0.429737
\(946\) 5439.19 0.186938
\(947\) −45311.2 −1.55482 −0.777412 0.628992i \(-0.783468\pi\)
−0.777412 + 0.628992i \(0.783468\pi\)
\(948\) −2338.42 −0.0801142
\(949\) 9563.54 0.327129
\(950\) 0 0
\(951\) −25136.6 −0.857108
\(952\) −13448.3 −0.457837
\(953\) 6964.61 0.236732 0.118366 0.992970i \(-0.462234\pi\)
0.118366 + 0.992970i \(0.462234\pi\)
\(954\) −8866.97 −0.300921
\(955\) 80034.1 2.71188
\(956\) −9773.49 −0.330646
\(957\) −1235.72 −0.0417399
\(958\) 23325.8 0.786661
\(959\) 19401.7 0.653300
\(960\) −3878.78 −0.130403
\(961\) −16068.4 −0.539369
\(962\) 18169.8 0.608957
\(963\) 9502.06 0.317964
\(964\) −3985.96 −0.133173
\(965\) −7361.70 −0.245577
\(966\) 25896.9 0.862547
\(967\) 46299.5 1.53970 0.769851 0.638224i \(-0.220331\pi\)
0.769851 + 0.638224i \(0.220331\pi\)
\(968\) −15858.5 −0.526561
\(969\) 0 0
\(970\) −24465.2 −0.809826
\(971\) 44402.3 1.46750 0.733748 0.679422i \(-0.237769\pi\)
0.733748 + 0.679422i \(0.237769\pi\)
\(972\) −972.000 −0.0320750
\(973\) −26723.2 −0.880478
\(974\) −26573.3 −0.874192
\(975\) −23200.1 −0.762049
\(976\) 7211.92 0.236524
\(977\) 36365.1 1.19081 0.595405 0.803426i \(-0.296992\pi\)
0.595405 + 0.803426i \(0.296992\pi\)
\(978\) 10959.1 0.358318
\(979\) −48333.7 −1.57789
\(980\) 14612.0 0.476290
\(981\) −2543.14 −0.0827689
\(982\) −8075.42 −0.262421
\(983\) 3682.23 0.119476 0.0597381 0.998214i \(-0.480973\pi\)
0.0597381 + 0.998214i \(0.480973\pi\)
\(984\) −1025.18 −0.0332130
\(985\) 46652.9 1.50912
\(986\) −1051.19 −0.0339519
\(987\) −34879.7 −1.12486
\(988\) 0 0
\(989\) −8909.99 −0.286472
\(990\) 20931.4 0.671962
\(991\) −4412.76 −0.141449 −0.0707246 0.997496i \(-0.522531\pi\)
−0.0707246 + 0.997496i \(0.522531\pi\)
\(992\) 3748.60 0.119978
\(993\) −14500.5 −0.463404
\(994\) 42635.2 1.36047
\(995\) 74273.5 2.36646
\(996\) 3432.99 0.109215
\(997\) −24825.9 −0.788610 −0.394305 0.918980i \(-0.629015\pi\)
−0.394305 + 0.918980i \(0.629015\pi\)
\(998\) −7089.66 −0.224869
\(999\) 8980.19 0.284405
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 2166.4.a.s.1.3 3
19.7 even 3 114.4.e.e.49.1 yes 6
19.11 even 3 114.4.e.e.7.1 6
19.18 odd 2 2166.4.a.w.1.3 3
57.11 odd 6 342.4.g.g.235.3 6
57.26 odd 6 342.4.g.g.163.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.4.e.e.7.1 6 19.11 even 3
114.4.e.e.49.1 yes 6 19.7 even 3
342.4.g.g.163.3 6 57.26 odd 6
342.4.g.g.235.3 6 57.11 odd 6
2166.4.a.s.1.3 3 1.1 even 1 trivial
2166.4.a.w.1.3 3 19.18 odd 2