Properties

Label 882.4.a.bi.1.2
Level $882$
Weight $4$
Character 882.1
Self dual yes
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 294)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +15.8995 q^{5} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +15.8995 q^{5} +8.00000 q^{8} +31.7990 q^{10} -57.3970 q^{11} +5.69848 q^{13} +16.0000 q^{16} +51.8995 q^{17} +16.2010 q^{19} +63.5980 q^{20} -114.794 q^{22} +213.397 q^{23} +127.794 q^{25} +11.3970 q^{26} +218.191 q^{29} -251.397 q^{31} +32.0000 q^{32} +103.799 q^{34} +386.794 q^{37} +32.4020 q^{38} +127.196 q^{40} +328.503 q^{41} -37.5879 q^{43} -229.588 q^{44} +426.794 q^{46} +254.995 q^{47} +255.588 q^{50} +22.7939 q^{52} -211.588 q^{53} -912.583 q^{55} +436.382 q^{58} +412.201 q^{59} -836.693 q^{61} -502.794 q^{62} +64.0000 q^{64} +90.6030 q^{65} -165.588 q^{67} +207.598 q^{68} +465.015 q^{71} +449.658 q^{73} +773.588 q^{74} +64.8040 q^{76} -343.558 q^{79} +254.392 q^{80} +657.005 q^{82} -1502.33 q^{83} +825.176 q^{85} -75.1758 q^{86} -459.176 q^{88} +341.085 q^{89} +853.588 q^{92} +509.990 q^{94} +257.588 q^{95} -865.437 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 12q^{5} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 12q^{5} + 16q^{8} + 24q^{10} + 4q^{11} - 48q^{13} + 32q^{16} + 84q^{17} + 72q^{19} + 48q^{20} + 8q^{22} + 308q^{23} + 18q^{25} - 96q^{26} + 80q^{29} - 384q^{31} + 64q^{32} + 168q^{34} + 536q^{37} + 144q^{38} + 96q^{40} + 756q^{41} + 400q^{43} + 16q^{44} + 616q^{46} + 312q^{47} + 36q^{50} - 192q^{52} + 52q^{53} - 1152q^{55} + 160q^{58} + 864q^{59} - 1416q^{61} - 768q^{62} + 128q^{64} + 300q^{65} + 144q^{67} + 336q^{68} + 1524q^{71} - 744q^{73} + 1072q^{74} + 288q^{76} + 976q^{79} + 192q^{80} + 1512q^{82} - 312q^{83} + 700q^{85} + 800q^{86} + 32q^{88} + 108q^{89} + 1232q^{92} + 624q^{94} + 40q^{95} + 744q^{97} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 15.8995 1.42209 0.711047 0.703144i \(-0.248221\pi\)
0.711047 + 0.703144i \(0.248221\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 31.7990 1.00557
\(11\) −57.3970 −1.57326 −0.786629 0.617426i \(-0.788176\pi\)
−0.786629 + 0.617426i \(0.788176\pi\)
\(12\) 0 0
\(13\) 5.69848 0.121575 0.0607875 0.998151i \(-0.480639\pi\)
0.0607875 + 0.998151i \(0.480639\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 51.8995 0.740440 0.370220 0.928944i \(-0.379282\pi\)
0.370220 + 0.928944i \(0.379282\pi\)
\(18\) 0 0
\(19\) 16.2010 0.195619 0.0978096 0.995205i \(-0.468816\pi\)
0.0978096 + 0.995205i \(0.468816\pi\)
\(20\) 63.5980 0.711047
\(21\) 0 0
\(22\) −114.794 −1.11246
\(23\) 213.397 1.93462 0.967312 0.253590i \(-0.0816114\pi\)
0.967312 + 0.253590i \(0.0816114\pi\)
\(24\) 0 0
\(25\) 127.794 1.02235
\(26\) 11.3970 0.0859665
\(27\) 0 0
\(28\) 0 0
\(29\) 218.191 1.39714 0.698570 0.715542i \(-0.253820\pi\)
0.698570 + 0.715542i \(0.253820\pi\)
\(30\) 0 0
\(31\) −251.397 −1.45652 −0.728262 0.685299i \(-0.759671\pi\)
−0.728262 + 0.685299i \(0.759671\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 103.799 0.523570
\(35\) 0 0
\(36\) 0 0
\(37\) 386.794 1.71861 0.859304 0.511464i \(-0.170897\pi\)
0.859304 + 0.511464i \(0.170897\pi\)
\(38\) 32.4020 0.138324
\(39\) 0 0
\(40\) 127.196 0.502786
\(41\) 328.503 1.25130 0.625652 0.780102i \(-0.284833\pi\)
0.625652 + 0.780102i \(0.284833\pi\)
\(42\) 0 0
\(43\) −37.5879 −0.133305 −0.0666523 0.997776i \(-0.521232\pi\)
−0.0666523 + 0.997776i \(0.521232\pi\)
\(44\) −229.588 −0.786629
\(45\) 0 0
\(46\) 426.794 1.36799
\(47\) 254.995 0.791379 0.395690 0.918384i \(-0.370506\pi\)
0.395690 + 0.918384i \(0.370506\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 255.588 0.722912
\(51\) 0 0
\(52\) 22.7939 0.0607875
\(53\) −211.588 −0.548374 −0.274187 0.961676i \(-0.588409\pi\)
−0.274187 + 0.961676i \(0.588409\pi\)
\(54\) 0 0
\(55\) −912.583 −2.23732
\(56\) 0 0
\(57\) 0 0
\(58\) 436.382 0.987927
\(59\) 412.201 0.909559 0.454780 0.890604i \(-0.349718\pi\)
0.454780 + 0.890604i \(0.349718\pi\)
\(60\) 0 0
\(61\) −836.693 −1.75619 −0.878095 0.478486i \(-0.841186\pi\)
−0.878095 + 0.478486i \(0.841186\pi\)
\(62\) −502.794 −1.02992
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 90.6030 0.172891
\(66\) 0 0
\(67\) −165.588 −0.301937 −0.150969 0.988539i \(-0.548239\pi\)
−0.150969 + 0.988539i \(0.548239\pi\)
\(68\) 207.598 0.370220
\(69\) 0 0
\(70\) 0 0
\(71\) 465.015 0.777284 0.388642 0.921389i \(-0.372944\pi\)
0.388642 + 0.921389i \(0.372944\pi\)
\(72\) 0 0
\(73\) 449.658 0.720938 0.360469 0.932771i \(-0.382617\pi\)
0.360469 + 0.932771i \(0.382617\pi\)
\(74\) 773.588 1.21524
\(75\) 0 0
\(76\) 64.8040 0.0978096
\(77\) 0 0
\(78\) 0 0
\(79\) −343.558 −0.489282 −0.244641 0.969614i \(-0.578670\pi\)
−0.244641 + 0.969614i \(0.578670\pi\)
\(80\) 254.392 0.355524
\(81\) 0 0
\(82\) 657.005 0.884806
\(83\) −1502.33 −1.98677 −0.993387 0.114812i \(-0.963373\pi\)
−0.993387 + 0.114812i \(0.963373\pi\)
\(84\) 0 0
\(85\) 825.176 1.05298
\(86\) −75.1758 −0.0942606
\(87\) 0 0
\(88\) −459.176 −0.556231
\(89\) 341.085 0.406236 0.203118 0.979154i \(-0.434893\pi\)
0.203118 + 0.979154i \(0.434893\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 853.588 0.967312
\(93\) 0 0
\(94\) 509.990 0.559590
\(95\) 257.588 0.278189
\(96\) 0 0
\(97\) −865.437 −0.905895 −0.452947 0.891537i \(-0.649627\pi\)
−0.452947 + 0.891537i \(0.649627\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 511.176 0.511176
\(101\) 243.256 0.239652 0.119826 0.992795i \(-0.461766\pi\)
0.119826 + 0.992795i \(0.461766\pi\)
\(102\) 0 0
\(103\) −953.346 −0.912000 −0.456000 0.889980i \(-0.650718\pi\)
−0.456000 + 0.889980i \(0.650718\pi\)
\(104\) 45.5879 0.0429833
\(105\) 0 0
\(106\) −423.176 −0.387759
\(107\) 1344.95 1.21516 0.607578 0.794260i \(-0.292141\pi\)
0.607578 + 0.794260i \(0.292141\pi\)
\(108\) 0 0
\(109\) 1734.35 1.52404 0.762022 0.647551i \(-0.224207\pi\)
0.762022 + 0.647551i \(0.224207\pi\)
\(110\) −1825.17 −1.58202
\(111\) 0 0
\(112\) 0 0
\(113\) −1441.18 −1.19977 −0.599887 0.800085i \(-0.704788\pi\)
−0.599887 + 0.800085i \(0.704788\pi\)
\(114\) 0 0
\(115\) 3392.90 2.75122
\(116\) 872.764 0.698570
\(117\) 0 0
\(118\) 824.402 0.643156
\(119\) 0 0
\(120\) 0 0
\(121\) 1963.41 1.47514
\(122\) −1673.39 −1.24181
\(123\) 0 0
\(124\) −1005.59 −0.728262
\(125\) 44.4222 0.0317860
\(126\) 0 0
\(127\) 1184.70 0.827759 0.413880 0.910332i \(-0.364173\pi\)
0.413880 + 0.910332i \(0.364173\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) 181.206 0.122252
\(131\) −297.588 −0.198476 −0.0992381 0.995064i \(-0.531641\pi\)
−0.0992381 + 0.995064i \(0.531641\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −331.176 −0.213502
\(135\) 0 0
\(136\) 415.196 0.261785
\(137\) −620.985 −0.387258 −0.193629 0.981075i \(-0.562026\pi\)
−0.193629 + 0.981075i \(0.562026\pi\)
\(138\) 0 0
\(139\) −898.754 −0.548426 −0.274213 0.961669i \(-0.588417\pi\)
−0.274213 + 0.961669i \(0.588417\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 930.030 0.549623
\(143\) −327.076 −0.191269
\(144\) 0 0
\(145\) 3469.13 1.98686
\(146\) 899.316 0.509780
\(147\) 0 0
\(148\) 1547.18 0.859304
\(149\) 3054.70 1.67954 0.839769 0.542945i \(-0.182691\pi\)
0.839769 + 0.542945i \(0.182691\pi\)
\(150\) 0 0
\(151\) −65.1455 −0.0351090 −0.0175545 0.999846i \(-0.505588\pi\)
−0.0175545 + 0.999846i \(0.505588\pi\)
\(152\) 129.608 0.0691619
\(153\) 0 0
\(154\) 0 0
\(155\) −3997.08 −2.07131
\(156\) 0 0
\(157\) 1542.22 0.783966 0.391983 0.919973i \(-0.371789\pi\)
0.391983 + 0.919973i \(0.371789\pi\)
\(158\) −687.115 −0.345974
\(159\) 0 0
\(160\) 508.784 0.251393
\(161\) 0 0
\(162\) 0 0
\(163\) −2514.73 −1.20840 −0.604200 0.796833i \(-0.706507\pi\)
−0.604200 + 0.796833i \(0.706507\pi\)
\(164\) 1314.01 0.625652
\(165\) 0 0
\(166\) −3004.66 −1.40486
\(167\) −528.643 −0.244956 −0.122478 0.992471i \(-0.539084\pi\)
−0.122478 + 0.992471i \(0.539084\pi\)
\(168\) 0 0
\(169\) −2164.53 −0.985220
\(170\) 1650.35 0.744566
\(171\) 0 0
\(172\) −150.352 −0.0666523
\(173\) −96.8439 −0.0425602 −0.0212801 0.999774i \(-0.506774\pi\)
−0.0212801 + 0.999774i \(0.506774\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −918.352 −0.393314
\(177\) 0 0
\(178\) 682.171 0.287252
\(179\) 534.542 0.223204 0.111602 0.993753i \(-0.464402\pi\)
0.111602 + 0.993753i \(0.464402\pi\)
\(180\) 0 0
\(181\) 2087.00 0.857049 0.428524 0.903530i \(-0.359034\pi\)
0.428524 + 0.903530i \(0.359034\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1707.18 0.683993
\(185\) 6149.83 2.44402
\(186\) 0 0
\(187\) −2978.87 −1.16490
\(188\) 1019.98 0.395690
\(189\) 0 0
\(190\) 515.176 0.196709
\(191\) −3387.69 −1.28337 −0.641687 0.766966i \(-0.721765\pi\)
−0.641687 + 0.766966i \(0.721765\pi\)
\(192\) 0 0
\(193\) −1908.35 −0.711742 −0.355871 0.934535i \(-0.615816\pi\)
−0.355871 + 0.934535i \(0.615816\pi\)
\(194\) −1730.87 −0.640564
\(195\) 0 0
\(196\) 0 0
\(197\) 2061.88 0.745699 0.372850 0.927892i \(-0.378381\pi\)
0.372850 + 0.927892i \(0.378381\pi\)
\(198\) 0 0
\(199\) 3171.50 1.12976 0.564878 0.825174i \(-0.308923\pi\)
0.564878 + 0.825174i \(0.308923\pi\)
\(200\) 1022.35 0.361456
\(201\) 0 0
\(202\) 486.512 0.169460
\(203\) 0 0
\(204\) 0 0
\(205\) 5223.02 1.77947
\(206\) −1906.69 −0.644882
\(207\) 0 0
\(208\) 91.1758 0.0303938
\(209\) −929.889 −0.307760
\(210\) 0 0
\(211\) 1349.97 0.440454 0.220227 0.975449i \(-0.429320\pi\)
0.220227 + 0.975449i \(0.429320\pi\)
\(212\) −846.352 −0.274187
\(213\) 0 0
\(214\) 2689.91 0.859245
\(215\) −597.628 −0.189572
\(216\) 0 0
\(217\) 0 0
\(218\) 3468.70 1.07766
\(219\) 0 0
\(220\) −3650.33 −1.11866
\(221\) 295.748 0.0900190
\(222\) 0 0
\(223\) −1361.85 −0.408951 −0.204476 0.978872i \(-0.565549\pi\)
−0.204476 + 0.978872i \(0.565549\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2882.35 −0.848368
\(227\) −1861.81 −0.544373 −0.272186 0.962245i \(-0.587747\pi\)
−0.272186 + 0.962245i \(0.587747\pi\)
\(228\) 0 0
\(229\) −5358.78 −1.54637 −0.773184 0.634181i \(-0.781337\pi\)
−0.773184 + 0.634181i \(0.781337\pi\)
\(230\) 6785.81 1.94540
\(231\) 0 0
\(232\) 1745.53 0.493963
\(233\) −5441.12 −1.52987 −0.764935 0.644107i \(-0.777229\pi\)
−0.764935 + 0.644107i \(0.777229\pi\)
\(234\) 0 0
\(235\) 4054.29 1.12542
\(236\) 1648.80 0.454780
\(237\) 0 0
\(238\) 0 0
\(239\) 1157.28 0.313213 0.156607 0.987661i \(-0.449945\pi\)
0.156607 + 0.987661i \(0.449945\pi\)
\(240\) 0 0
\(241\) −3969.38 −1.06095 −0.530477 0.847699i \(-0.677987\pi\)
−0.530477 + 0.847699i \(0.677987\pi\)
\(242\) 3926.82 1.04308
\(243\) 0 0
\(244\) −3346.77 −0.878095
\(245\) 0 0
\(246\) 0 0
\(247\) 92.3212 0.0237824
\(248\) −2011.18 −0.514959
\(249\) 0 0
\(250\) 88.8444 0.0224761
\(251\) −5978.75 −1.50349 −0.751744 0.659455i \(-0.770787\pi\)
−0.751744 + 0.659455i \(0.770787\pi\)
\(252\) 0 0
\(253\) −12248.3 −3.04366
\(254\) 2369.41 0.585314
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −4650.15 −1.12867 −0.564335 0.825546i \(-0.690868\pi\)
−0.564335 + 0.825546i \(0.690868\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 362.412 0.0864456
\(261\) 0 0
\(262\) −595.176 −0.140344
\(263\) −3695.37 −0.866411 −0.433205 0.901295i \(-0.642618\pi\)
−0.433205 + 0.901295i \(0.642618\pi\)
\(264\) 0 0
\(265\) −3364.14 −0.779840
\(266\) 0 0
\(267\) 0 0
\(268\) −662.352 −0.150969
\(269\) 7157.69 1.62235 0.811175 0.584804i \(-0.198829\pi\)
0.811175 + 0.584804i \(0.198829\pi\)
\(270\) 0 0
\(271\) 4038.37 0.905216 0.452608 0.891710i \(-0.350494\pi\)
0.452608 + 0.891710i \(0.350494\pi\)
\(272\) 830.392 0.185110
\(273\) 0 0
\(274\) −1241.97 −0.273833
\(275\) −7334.98 −1.60842
\(276\) 0 0
\(277\) −2754.82 −0.597550 −0.298775 0.954324i \(-0.596578\pi\)
−0.298775 + 0.954324i \(0.596578\pi\)
\(278\) −1797.51 −0.387796
\(279\) 0 0
\(280\) 0 0
\(281\) 772.742 0.164050 0.0820248 0.996630i \(-0.473861\pi\)
0.0820248 + 0.996630i \(0.473861\pi\)
\(282\) 0 0
\(283\) −6745.49 −1.41688 −0.708441 0.705770i \(-0.750601\pi\)
−0.708441 + 0.705770i \(0.750601\pi\)
\(284\) 1860.06 0.388642
\(285\) 0 0
\(286\) −654.152 −0.135248
\(287\) 0 0
\(288\) 0 0
\(289\) −2219.44 −0.451749
\(290\) 6938.25 1.40492
\(291\) 0 0
\(292\) 1798.63 0.360469
\(293\) −1922.69 −0.383362 −0.191681 0.981457i \(-0.561394\pi\)
−0.191681 + 0.981457i \(0.561394\pi\)
\(294\) 0 0
\(295\) 6553.79 1.29348
\(296\) 3094.35 0.607620
\(297\) 0 0
\(298\) 6109.41 1.18761
\(299\) 1216.04 0.235202
\(300\) 0 0
\(301\) 0 0
\(302\) −130.291 −0.0248258
\(303\) 0 0
\(304\) 259.216 0.0489048
\(305\) −13303.0 −2.49747
\(306\) 0 0
\(307\) 2016.68 0.374913 0.187456 0.982273i \(-0.439976\pi\)
0.187456 + 0.982273i \(0.439976\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −7994.17 −1.46464
\(311\) −7149.99 −1.30366 −0.651831 0.758365i \(-0.725999\pi\)
−0.651831 + 0.758365i \(0.725999\pi\)
\(312\) 0 0
\(313\) 8596.49 1.55240 0.776202 0.630484i \(-0.217144\pi\)
0.776202 + 0.630484i \(0.217144\pi\)
\(314\) 3084.44 0.554347
\(315\) 0 0
\(316\) −1374.23 −0.244641
\(317\) 2853.24 0.505532 0.252766 0.967527i \(-0.418660\pi\)
0.252766 + 0.967527i \(0.418660\pi\)
\(318\) 0 0
\(319\) −12523.5 −2.19806
\(320\) 1017.57 0.177762
\(321\) 0 0
\(322\) 0 0
\(323\) 840.824 0.144844
\(324\) 0 0
\(325\) 728.232 0.124292
\(326\) −5029.47 −0.854467
\(327\) 0 0
\(328\) 2628.02 0.442403
\(329\) 0 0
\(330\) 0 0
\(331\) 1619.12 0.268866 0.134433 0.990923i \(-0.457079\pi\)
0.134433 + 0.990923i \(0.457079\pi\)
\(332\) −6009.33 −0.993387
\(333\) 0 0
\(334\) −1057.29 −0.173210
\(335\) −2632.76 −0.429383
\(336\) 0 0
\(337\) −3278.67 −0.529972 −0.264986 0.964252i \(-0.585367\pi\)
−0.264986 + 0.964252i \(0.585367\pi\)
\(338\) −4329.05 −0.696655
\(339\) 0 0
\(340\) 3300.70 0.526488
\(341\) 14429.4 2.29149
\(342\) 0 0
\(343\) 0 0
\(344\) −300.703 −0.0471303
\(345\) 0 0
\(346\) −193.688 −0.0300946
\(347\) 2850.30 0.440957 0.220479 0.975392i \(-0.429238\pi\)
0.220479 + 0.975392i \(0.429238\pi\)
\(348\) 0 0
\(349\) −4725.32 −0.724758 −0.362379 0.932031i \(-0.618035\pi\)
−0.362379 + 0.932031i \(0.618035\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1836.70 −0.278115
\(353\) −6727.44 −1.01435 −0.507175 0.861843i \(-0.669310\pi\)
−0.507175 + 0.861843i \(0.669310\pi\)
\(354\) 0 0
\(355\) 7393.51 1.10537
\(356\) 1364.34 0.203118
\(357\) 0 0
\(358\) 1069.08 0.157829
\(359\) 7331.89 1.07789 0.538945 0.842341i \(-0.318823\pi\)
0.538945 + 0.842341i \(0.318823\pi\)
\(360\) 0 0
\(361\) −6596.53 −0.961733
\(362\) 4174.01 0.606025
\(363\) 0 0
\(364\) 0 0
\(365\) 7149.34 1.02524
\(366\) 0 0
\(367\) 2774.43 0.394616 0.197308 0.980342i \(-0.436780\pi\)
0.197308 + 0.980342i \(0.436780\pi\)
\(368\) 3414.35 0.483656
\(369\) 0 0
\(370\) 12299.7 1.72819
\(371\) 0 0
\(372\) 0 0
\(373\) 2527.35 0.350834 0.175417 0.984494i \(-0.443873\pi\)
0.175417 + 0.984494i \(0.443873\pi\)
\(374\) −5957.75 −0.823711
\(375\) 0 0
\(376\) 2039.96 0.279795
\(377\) 1243.36 0.169857
\(378\) 0 0
\(379\) 3116.40 0.422371 0.211186 0.977446i \(-0.432268\pi\)
0.211186 + 0.977446i \(0.432268\pi\)
\(380\) 1030.35 0.139095
\(381\) 0 0
\(382\) −6775.38 −0.907483
\(383\) −1518.07 −0.202532 −0.101266 0.994859i \(-0.532289\pi\)
−0.101266 + 0.994859i \(0.532289\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3816.70 −0.503277
\(387\) 0 0
\(388\) −3461.75 −0.452947
\(389\) 3246.77 0.423182 0.211591 0.977358i \(-0.432135\pi\)
0.211591 + 0.977358i \(0.432135\pi\)
\(390\) 0 0
\(391\) 11075.2 1.43247
\(392\) 0 0
\(393\) 0 0
\(394\) 4123.76 0.527289
\(395\) −5462.39 −0.695804
\(396\) 0 0
\(397\) −1830.62 −0.231427 −0.115713 0.993283i \(-0.536915\pi\)
−0.115713 + 0.993283i \(0.536915\pi\)
\(398\) 6342.99 0.798858
\(399\) 0 0
\(400\) 2044.70 0.255588
\(401\) 3385.81 0.421644 0.210822 0.977524i \(-0.432386\pi\)
0.210822 + 0.977524i \(0.432386\pi\)
\(402\) 0 0
\(403\) −1432.58 −0.177077
\(404\) 973.024 0.119826
\(405\) 0 0
\(406\) 0 0
\(407\) −22200.8 −2.70382
\(408\) 0 0
\(409\) −9253.17 −1.11868 −0.559340 0.828938i \(-0.688945\pi\)
−0.559340 + 0.828938i \(0.688945\pi\)
\(410\) 10446.0 1.25828
\(411\) 0 0
\(412\) −3813.39 −0.456000
\(413\) 0 0
\(414\) 0 0
\(415\) −23886.3 −2.82538
\(416\) 182.352 0.0214916
\(417\) 0 0
\(418\) −1859.78 −0.217619
\(419\) 3547.52 0.413622 0.206811 0.978381i \(-0.433692\pi\)
0.206811 + 0.978381i \(0.433692\pi\)
\(420\) 0 0
\(421\) 7848.87 0.908624 0.454312 0.890843i \(-0.349885\pi\)
0.454312 + 0.890843i \(0.349885\pi\)
\(422\) 2699.94 0.311448
\(423\) 0 0
\(424\) −1692.70 −0.193880
\(425\) 6632.44 0.756990
\(426\) 0 0
\(427\) 0 0
\(428\) 5379.82 0.607578
\(429\) 0 0
\(430\) −1195.26 −0.134047
\(431\) 4447.32 0.497030 0.248515 0.968628i \(-0.420057\pi\)
0.248515 + 0.968628i \(0.420057\pi\)
\(432\) 0 0
\(433\) 6994.82 0.776327 0.388164 0.921590i \(-0.373110\pi\)
0.388164 + 0.921590i \(0.373110\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6937.41 0.762022
\(437\) 3457.25 0.378450
\(438\) 0 0
\(439\) −636.182 −0.0691647 −0.0345823 0.999402i \(-0.511010\pi\)
−0.0345823 + 0.999402i \(0.511010\pi\)
\(440\) −7300.66 −0.791012
\(441\) 0 0
\(442\) 591.497 0.0636530
\(443\) 4474.24 0.479859 0.239929 0.970790i \(-0.422876\pi\)
0.239929 + 0.970790i \(0.422876\pi\)
\(444\) 0 0
\(445\) 5423.08 0.577705
\(446\) −2723.70 −0.289172
\(447\) 0 0
\(448\) 0 0
\(449\) 2389.42 0.251144 0.125572 0.992085i \(-0.459923\pi\)
0.125572 + 0.992085i \(0.459923\pi\)
\(450\) 0 0
\(451\) −18855.0 −1.96862
\(452\) −5764.70 −0.599887
\(453\) 0 0
\(454\) −3723.62 −0.384930
\(455\) 0 0
\(456\) 0 0
\(457\) −4438.34 −0.454304 −0.227152 0.973859i \(-0.572941\pi\)
−0.227152 + 0.973859i \(0.572941\pi\)
\(458\) −10717.6 −1.09345
\(459\) 0 0
\(460\) 13571.6 1.37561
\(461\) −14079.8 −1.42248 −0.711240 0.702949i \(-0.751866\pi\)
−0.711240 + 0.702949i \(0.751866\pi\)
\(462\) 0 0
\(463\) 4687.50 0.470511 0.235255 0.971934i \(-0.424407\pi\)
0.235255 + 0.971934i \(0.424407\pi\)
\(464\) 3491.05 0.349285
\(465\) 0 0
\(466\) −10882.2 −1.08178
\(467\) 8447.26 0.837029 0.418514 0.908210i \(-0.362551\pi\)
0.418514 + 0.908210i \(0.362551\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 8108.58 0.795789
\(471\) 0 0
\(472\) 3297.61 0.321578
\(473\) 2157.43 0.209723
\(474\) 0 0
\(475\) 2070.39 0.199992
\(476\) 0 0
\(477\) 0 0
\(478\) 2314.55 0.221475
\(479\) −4369.41 −0.416792 −0.208396 0.978045i \(-0.566824\pi\)
−0.208396 + 0.978045i \(0.566824\pi\)
\(480\) 0 0
\(481\) 2204.14 0.208940
\(482\) −7938.75 −0.750208
\(483\) 0 0
\(484\) 7853.65 0.737570
\(485\) −13760.0 −1.28827
\(486\) 0 0
\(487\) −14477.7 −1.34712 −0.673561 0.739132i \(-0.735236\pi\)
−0.673561 + 0.739132i \(0.735236\pi\)
\(488\) −6693.55 −0.620907
\(489\) 0 0
\(490\) 0 0
\(491\) −9306.12 −0.855355 −0.427677 0.903931i \(-0.640668\pi\)
−0.427677 + 0.903931i \(0.640668\pi\)
\(492\) 0 0
\(493\) 11324.0 1.03450
\(494\) 184.642 0.0168167
\(495\) 0 0
\(496\) −4022.35 −0.364131
\(497\) 0 0
\(498\) 0 0
\(499\) −12237.5 −1.09785 −0.548926 0.835871i \(-0.684963\pi\)
−0.548926 + 0.835871i \(0.684963\pi\)
\(500\) 177.689 0.0158930
\(501\) 0 0
\(502\) −11957.5 −1.06313
\(503\) 5524.30 0.489694 0.244847 0.969562i \(-0.421262\pi\)
0.244847 + 0.969562i \(0.421262\pi\)
\(504\) 0 0
\(505\) 3867.65 0.340808
\(506\) −24496.7 −2.15219
\(507\) 0 0
\(508\) 4738.81 0.413880
\(509\) −10079.6 −0.877743 −0.438871 0.898550i \(-0.644622\pi\)
−0.438871 + 0.898550i \(0.644622\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) −9300.30 −0.798091
\(515\) −15157.7 −1.29695
\(516\) 0 0
\(517\) −14635.9 −1.24504
\(518\) 0 0
\(519\) 0 0
\(520\) 724.824 0.0611262
\(521\) 5706.61 0.479868 0.239934 0.970789i \(-0.422874\pi\)
0.239934 + 0.970789i \(0.422874\pi\)
\(522\) 0 0
\(523\) 10657.3 0.891032 0.445516 0.895274i \(-0.353020\pi\)
0.445516 + 0.895274i \(0.353020\pi\)
\(524\) −1190.35 −0.0992381
\(525\) 0 0
\(526\) −7390.73 −0.612645
\(527\) −13047.4 −1.07847
\(528\) 0 0
\(529\) 33371.3 2.74277
\(530\) −6728.28 −0.551430
\(531\) 0 0
\(532\) 0 0
\(533\) 1871.97 0.152127
\(534\) 0 0
\(535\) 21384.1 1.72807
\(536\) −1324.70 −0.106751
\(537\) 0 0
\(538\) 14315.4 1.14717
\(539\) 0 0
\(540\) 0 0
\(541\) −4010.48 −0.318714 −0.159357 0.987221i \(-0.550942\pi\)
−0.159357 + 0.987221i \(0.550942\pi\)
\(542\) 8076.74 0.640084
\(543\) 0 0
\(544\) 1660.78 0.130892
\(545\) 27575.3 2.16733
\(546\) 0 0
\(547\) −17619.8 −1.37728 −0.688638 0.725105i \(-0.741791\pi\)
−0.688638 + 0.725105i \(0.741791\pi\)
\(548\) −2483.94 −0.193629
\(549\) 0 0
\(550\) −14670.0 −1.13733
\(551\) 3534.91 0.273307
\(552\) 0 0
\(553\) 0 0
\(554\) −5509.65 −0.422532
\(555\) 0 0
\(556\) −3595.01 −0.274213
\(557\) 10337.7 0.786395 0.393198 0.919454i \(-0.371369\pi\)
0.393198 + 0.919454i \(0.371369\pi\)
\(558\) 0 0
\(559\) −214.194 −0.0162065
\(560\) 0 0
\(561\) 0 0
\(562\) 1545.48 0.116001
\(563\) −24023.7 −1.79836 −0.899180 0.437580i \(-0.855836\pi\)
−0.899180 + 0.437580i \(0.855836\pi\)
\(564\) 0 0
\(565\) −22914.0 −1.70619
\(566\) −13491.0 −1.00189
\(567\) 0 0
\(568\) 3720.12 0.274811
\(569\) 13179.2 0.971001 0.485501 0.874236i \(-0.338637\pi\)
0.485501 + 0.874236i \(0.338637\pi\)
\(570\) 0 0
\(571\) 7776.26 0.569924 0.284962 0.958539i \(-0.408019\pi\)
0.284962 + 0.958539i \(0.408019\pi\)
\(572\) −1308.30 −0.0956344
\(573\) 0 0
\(574\) 0 0
\(575\) 27270.8 1.97787
\(576\) 0 0
\(577\) −20167.0 −1.45505 −0.727525 0.686081i \(-0.759330\pi\)
−0.727525 + 0.686081i \(0.759330\pi\)
\(578\) −4438.88 −0.319435
\(579\) 0 0
\(580\) 13876.5 0.993432
\(581\) 0 0
\(582\) 0 0
\(583\) 12144.5 0.862734
\(584\) 3597.26 0.254890
\(585\) 0 0
\(586\) −3845.39 −0.271078
\(587\) −8365.08 −0.588184 −0.294092 0.955777i \(-0.595017\pi\)
−0.294092 + 0.955777i \(0.595017\pi\)
\(588\) 0 0
\(589\) −4072.88 −0.284924
\(590\) 13107.6 0.914628
\(591\) 0 0
\(592\) 6188.70 0.429652
\(593\) −27621.9 −1.91281 −0.956403 0.292050i \(-0.905663\pi\)
−0.956403 + 0.292050i \(0.905663\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12218.8 0.839769
\(597\) 0 0
\(598\) 2432.08 0.166313
\(599\) −538.318 −0.0367197 −0.0183598 0.999831i \(-0.505844\pi\)
−0.0183598 + 0.999831i \(0.505844\pi\)
\(600\) 0 0
\(601\) −6958.64 −0.472294 −0.236147 0.971717i \(-0.575885\pi\)
−0.236147 + 0.971717i \(0.575885\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −260.582 −0.0175545
\(605\) 31217.3 2.09779
\(606\) 0 0
\(607\) 17297.6 1.15665 0.578326 0.815806i \(-0.303706\pi\)
0.578326 + 0.815806i \(0.303706\pi\)
\(608\) 518.432 0.0345809
\(609\) 0 0
\(610\) −26606.0 −1.76598
\(611\) 1453.08 0.0962120
\(612\) 0 0
\(613\) −839.158 −0.0552908 −0.0276454 0.999618i \(-0.508801\pi\)
−0.0276454 + 0.999618i \(0.508801\pi\)
\(614\) 4033.37 0.265103
\(615\) 0 0
\(616\) 0 0
\(617\) 16040.0 1.04659 0.523295 0.852152i \(-0.324703\pi\)
0.523295 + 0.852152i \(0.324703\pi\)
\(618\) 0 0
\(619\) 5429.28 0.352538 0.176269 0.984342i \(-0.443597\pi\)
0.176269 + 0.984342i \(0.443597\pi\)
\(620\) −15988.3 −1.03566
\(621\) 0 0
\(622\) −14300.0 −0.921828
\(623\) 0 0
\(624\) 0 0
\(625\) −15268.0 −0.977149
\(626\) 17193.0 1.09772
\(627\) 0 0
\(628\) 6168.88 0.391983
\(629\) 20074.4 1.27253
\(630\) 0 0
\(631\) −1807.86 −0.114057 −0.0570284 0.998373i \(-0.518163\pi\)
−0.0570284 + 0.998373i \(0.518163\pi\)
\(632\) −2748.46 −0.172987
\(633\) 0 0
\(634\) 5706.47 0.357465
\(635\) 18836.2 1.17715
\(636\) 0 0
\(637\) 0 0
\(638\) −25047.0 −1.55426
\(639\) 0 0
\(640\) 2035.14 0.125697
\(641\) 5904.56 0.363832 0.181916 0.983314i \(-0.441770\pi\)
0.181916 + 0.983314i \(0.441770\pi\)
\(642\) 0 0
\(643\) −8092.42 −0.496320 −0.248160 0.968719i \(-0.579826\pi\)
−0.248160 + 0.968719i \(0.579826\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1681.65 0.102420
\(647\) 20192.7 1.22698 0.613490 0.789702i \(-0.289765\pi\)
0.613490 + 0.789702i \(0.289765\pi\)
\(648\) 0 0
\(649\) −23659.1 −1.43097
\(650\) 1456.46 0.0878880
\(651\) 0 0
\(652\) −10058.9 −0.604200
\(653\) 21180.8 1.26933 0.634664 0.772789i \(-0.281139\pi\)
0.634664 + 0.772789i \(0.281139\pi\)
\(654\) 0 0
\(655\) −4731.50 −0.282252
\(656\) 5256.04 0.312826
\(657\) 0 0
\(658\) 0 0
\(659\) −28411.3 −1.67944 −0.839718 0.543023i \(-0.817280\pi\)
−0.839718 + 0.543023i \(0.817280\pi\)
\(660\) 0 0
\(661\) −16704.9 −0.982975 −0.491488 0.870885i \(-0.663547\pi\)
−0.491488 + 0.870885i \(0.663547\pi\)
\(662\) 3238.23 0.190117
\(663\) 0 0
\(664\) −12018.7 −0.702431
\(665\) 0 0
\(666\) 0 0
\(667\) 46561.3 2.70294
\(668\) −2114.57 −0.122478
\(669\) 0 0
\(670\) −5265.53 −0.303620
\(671\) 48023.7 2.76294
\(672\) 0 0
\(673\) −9047.09 −0.518187 −0.259093 0.965852i \(-0.583424\pi\)
−0.259093 + 0.965852i \(0.583424\pi\)
\(674\) −6557.35 −0.374747
\(675\) 0 0
\(676\) −8658.11 −0.492610
\(677\) −7844.26 −0.445316 −0.222658 0.974897i \(-0.571473\pi\)
−0.222658 + 0.974897i \(0.571473\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 6601.41 0.372283
\(681\) 0 0
\(682\) 28858.8 1.62033
\(683\) −25766.1 −1.44350 −0.721751 0.692153i \(-0.756662\pi\)
−0.721751 + 0.692153i \(0.756662\pi\)
\(684\) 0 0
\(685\) −9873.35 −0.550717
\(686\) 0 0
\(687\) 0 0
\(688\) −601.406 −0.0333261
\(689\) −1205.73 −0.0666686
\(690\) 0 0
\(691\) −24674.1 −1.35839 −0.679195 0.733958i \(-0.737671\pi\)
−0.679195 + 0.733958i \(0.737671\pi\)
\(692\) −387.376 −0.0212801
\(693\) 0 0
\(694\) 5700.60 0.311804
\(695\) −14289.7 −0.779914
\(696\) 0 0
\(697\) 17049.1 0.926515
\(698\) −9450.63 −0.512481
\(699\) 0 0
\(700\) 0 0
\(701\) −29377.9 −1.58286 −0.791431 0.611258i \(-0.790664\pi\)
−0.791431 + 0.611258i \(0.790664\pi\)
\(702\) 0 0
\(703\) 6266.45 0.336193
\(704\) −3673.41 −0.196657
\(705\) 0 0
\(706\) −13454.9 −0.717253
\(707\) 0 0
\(708\) 0 0
\(709\) 30594.3 1.62058 0.810291 0.586028i \(-0.199309\pi\)
0.810291 + 0.586028i \(0.199309\pi\)
\(710\) 14787.0 0.781615
\(711\) 0 0
\(712\) 2728.68 0.143626
\(713\) −53647.4 −2.81782
\(714\) 0 0
\(715\) −5200.34 −0.272002
\(716\) 2138.17 0.111602
\(717\) 0 0
\(718\) 14663.8 0.762183
\(719\) 1946.94 0.100985 0.0504927 0.998724i \(-0.483921\pi\)
0.0504927 + 0.998724i \(0.483921\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −13193.1 −0.680048
\(723\) 0 0
\(724\) 8348.02 0.428524
\(725\) 27883.5 1.42837
\(726\) 0 0
\(727\) 15750.6 0.803518 0.401759 0.915745i \(-0.368399\pi\)
0.401759 + 0.915745i \(0.368399\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 14298.7 0.724956
\(731\) −1950.79 −0.0987040
\(732\) 0 0
\(733\) 14349.6 0.723074 0.361537 0.932358i \(-0.382252\pi\)
0.361537 + 0.932358i \(0.382252\pi\)
\(734\) 5548.86 0.279036
\(735\) 0 0
\(736\) 6828.70 0.341996
\(737\) 9504.24 0.475025
\(738\) 0 0
\(739\) −17758.1 −0.883953 −0.441976 0.897027i \(-0.645722\pi\)
−0.441976 + 0.897027i \(0.645722\pi\)
\(740\) 24599.3 1.22201
\(741\) 0 0
\(742\) 0 0
\(743\) 29187.6 1.44117 0.720586 0.693366i \(-0.243873\pi\)
0.720586 + 0.693366i \(0.243873\pi\)
\(744\) 0 0
\(745\) 48568.2 2.38846
\(746\) 5054.69 0.248077
\(747\) 0 0
\(748\) −11915.5 −0.582451
\(749\) 0 0
\(750\) 0 0
\(751\) 13781.9 0.669654 0.334827 0.942280i \(-0.391322\pi\)
0.334827 + 0.942280i \(0.391322\pi\)
\(752\) 4079.92 0.197845
\(753\) 0 0
\(754\) 2486.72 0.120107
\(755\) −1035.78 −0.0499283
\(756\) 0 0
\(757\) 36952.7 1.77420 0.887099 0.461579i \(-0.152717\pi\)
0.887099 + 0.461579i \(0.152717\pi\)
\(758\) 6232.80 0.298662
\(759\) 0 0
\(760\) 2060.70 0.0983547
\(761\) −28816.5 −1.37266 −0.686332 0.727288i \(-0.740780\pi\)
−0.686332 + 0.727288i \(0.740780\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −13550.8 −0.641687
\(765\) 0 0
\(766\) −3036.14 −0.143212
\(767\) 2348.92 0.110580
\(768\) 0 0
\(769\) 25285.2 1.18571 0.592854 0.805310i \(-0.298001\pi\)
0.592854 + 0.805310i \(0.298001\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7633.41 −0.355871
\(773\) 18418.2 0.856995 0.428497 0.903543i \(-0.359043\pi\)
0.428497 + 0.903543i \(0.359043\pi\)
\(774\) 0 0
\(775\) −32127.0 −1.48908
\(776\) −6923.49 −0.320282
\(777\) 0 0
\(778\) 6493.55 0.299235
\(779\) 5322.07 0.244779
\(780\) 0 0
\(781\) −26690.5 −1.22287
\(782\) 22150.4 1.01291
\(783\) 0 0
\(784\) 0 0
\(785\) 24520.5 1.11487
\(786\) 0 0
\(787\) −11075.8 −0.501664 −0.250832 0.968031i \(-0.580704\pi\)
−0.250832 + 0.968031i \(0.580704\pi\)
\(788\) 8247.52 0.372850
\(789\) 0 0
\(790\) −10924.8 −0.492008
\(791\) 0 0
\(792\) 0 0
\(793\) −4767.88 −0.213509
\(794\) −3661.25 −0.163643
\(795\) 0 0
\(796\) 12686.0 0.564878
\(797\) 4838.83 0.215057 0.107528 0.994202i \(-0.465706\pi\)
0.107528 + 0.994202i \(0.465706\pi\)
\(798\) 0 0
\(799\) 13234.1 0.585969
\(800\) 4089.41 0.180728
\(801\) 0 0
\(802\) 6771.62 0.298147
\(803\) −25809.0 −1.13422
\(804\) 0 0
\(805\) 0 0
\(806\) −2865.16 −0.125212
\(807\) 0 0
\(808\) 1946.05 0.0847299
\(809\) 31509.9 1.36938 0.684690 0.728834i \(-0.259938\pi\)
0.684690 + 0.728834i \(0.259938\pi\)
\(810\) 0 0
\(811\) 29463.3 1.27570 0.637851 0.770160i \(-0.279823\pi\)
0.637851 + 0.770160i \(0.279823\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −44401.6 −1.91189
\(815\) −39983.0 −1.71846
\(816\) 0 0
\(817\) −608.962 −0.0260770
\(818\) −18506.3 −0.791026
\(819\) 0 0
\(820\) 20892.1 0.889736
\(821\) −3502.22 −0.148877 −0.0744386 0.997226i \(-0.523716\pi\)
−0.0744386 + 0.997226i \(0.523716\pi\)
\(822\) 0 0
\(823\) 39993.0 1.69389 0.846943 0.531684i \(-0.178440\pi\)
0.846943 + 0.531684i \(0.178440\pi\)
\(824\) −7626.77 −0.322441
\(825\) 0 0
\(826\) 0 0
\(827\) 10733.6 0.451322 0.225661 0.974206i \(-0.427546\pi\)
0.225661 + 0.974206i \(0.427546\pi\)
\(828\) 0 0
\(829\) 14537.5 0.609056 0.304528 0.952503i \(-0.401501\pi\)
0.304528 + 0.952503i \(0.401501\pi\)
\(830\) −47772.6 −1.99785
\(831\) 0 0
\(832\) 364.703 0.0151969
\(833\) 0 0
\(834\) 0 0
\(835\) −8405.16 −0.348351
\(836\) −3719.56 −0.153880
\(837\) 0 0
\(838\) 7095.03 0.292475
\(839\) 7353.57 0.302591 0.151295 0.988489i \(-0.451656\pi\)
0.151295 + 0.988489i \(0.451656\pi\)
\(840\) 0 0
\(841\) 23218.3 0.951998
\(842\) 15697.7 0.642494
\(843\) 0 0
\(844\) 5399.88 0.220227
\(845\) −34414.9 −1.40107
\(846\) 0 0
\(847\) 0 0
\(848\) −3385.41 −0.137094
\(849\) 0 0
\(850\) 13264.9 0.535273
\(851\) 82540.7 3.32486
\(852\) 0 0
\(853\) −19293.6 −0.774442 −0.387221 0.921987i \(-0.626565\pi\)
−0.387221 + 0.921987i \(0.626565\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10759.6 0.429622
\(857\) −14161.6 −0.564468 −0.282234 0.959346i \(-0.591076\pi\)
−0.282234 + 0.959346i \(0.591076\pi\)
\(858\) 0 0
\(859\) −8219.13 −0.326465 −0.163232 0.986588i \(-0.552192\pi\)
−0.163232 + 0.986588i \(0.552192\pi\)
\(860\) −2390.51 −0.0947858
\(861\) 0 0
\(862\) 8894.65 0.351454
\(863\) −2574.95 −0.101567 −0.0507835 0.998710i \(-0.516172\pi\)
−0.0507835 + 0.998710i \(0.516172\pi\)
\(864\) 0 0
\(865\) −1539.77 −0.0605246
\(866\) 13989.6 0.548946
\(867\) 0 0
\(868\) 0 0
\(869\) 19719.2 0.769766
\(870\) 0 0
\(871\) −943.600 −0.0367080
\(872\) 13874.8 0.538831
\(873\) 0 0
\(874\) 6914.49 0.267604
\(875\) 0 0
\(876\) 0 0
\(877\) 30981.1 1.19288 0.596442 0.802656i \(-0.296581\pi\)
0.596442 + 0.802656i \(0.296581\pi\)
\(878\) −1272.36 −0.0489068
\(879\) 0 0
\(880\) −14601.3 −0.559330
\(881\) −41781.8 −1.59780 −0.798902 0.601461i \(-0.794585\pi\)
−0.798902 + 0.601461i \(0.794585\pi\)
\(882\) 0 0
\(883\) −39289.6 −1.49740 −0.748699 0.662911i \(-0.769321\pi\)
−0.748699 + 0.662911i \(0.769321\pi\)
\(884\) 1182.99 0.0450095
\(885\) 0 0
\(886\) 8948.48 0.339312
\(887\) 5832.44 0.220783 0.110391 0.993888i \(-0.464790\pi\)
0.110391 + 0.993888i \(0.464790\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 10846.2 0.408499
\(891\) 0 0
\(892\) −5447.39 −0.204476
\(893\) 4131.18 0.154809
\(894\) 0 0
\(895\) 8498.95 0.317418
\(896\) 0 0
\(897\) 0 0
\(898\) 4778.84 0.177586
\(899\) −54852.5 −2.03497
\(900\) 0 0
\(901\) −10981.3 −0.406038
\(902\) −37710.1 −1.39203
\(903\) 0 0
\(904\) −11529.4 −0.424184
\(905\) 33182.3 1.21880
\(906\) 0 0
\(907\) 42387.9 1.55178 0.775892 0.630866i \(-0.217300\pi\)
0.775892 + 0.630866i \(0.217300\pi\)
\(908\) −7447.24 −0.272186
\(909\) 0 0
\(910\) 0 0
\(911\) −2275.12 −0.0827423 −0.0413711 0.999144i \(-0.513173\pi\)
−0.0413711 + 0.999144i \(0.513173\pi\)
\(912\) 0 0
\(913\) 86229.3 3.12571
\(914\) −8876.68 −0.321241
\(915\) 0 0
\(916\) −21435.1 −0.773184
\(917\) 0 0
\(918\) 0 0
\(919\) −31284.3 −1.12293 −0.561465 0.827500i \(-0.689762\pi\)
−0.561465 + 0.827500i \(0.689762\pi\)
\(920\) 27143.2 0.972702
\(921\) 0 0
\(922\) −28159.7 −1.00585
\(923\) 2649.88 0.0944983
\(924\) 0 0
\(925\) 49429.9 1.75702
\(926\) 9374.99 0.332701
\(927\) 0 0
\(928\) 6982.11 0.246982
\(929\) 32196.6 1.13707 0.568535 0.822659i \(-0.307511\pi\)
0.568535 + 0.822659i \(0.307511\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −21764.5 −0.764935
\(933\) 0 0
\(934\) 16894.5 0.591869
\(935\) −47362.6 −1.65660
\(936\) 0 0
\(937\) −22293.6 −0.777269 −0.388635 0.921392i \(-0.627053\pi\)
−0.388635 + 0.921392i \(0.627053\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 16217.2 0.562708
\(941\) −31809.7 −1.10198 −0.550991 0.834511i \(-0.685750\pi\)
−0.550991 + 0.834511i \(0.685750\pi\)
\(942\) 0 0
\(943\) 70101.4 2.42080
\(944\) 6595.22 0.227390
\(945\) 0 0
\(946\) 4314.86 0.148296
\(947\) −18982.6 −0.651376 −0.325688 0.945477i \(-0.605596\pi\)
−0.325688 + 0.945477i \(0.605596\pi\)
\(948\) 0 0
\(949\) 2562.37 0.0876481
\(950\) 4140.78 0.141415
\(951\) 0 0
\(952\) 0 0
\(953\) −9254.58 −0.314570 −0.157285 0.987553i \(-0.550274\pi\)
−0.157285 + 0.987553i \(0.550274\pi\)
\(954\) 0 0
\(955\) −53862.5 −1.82508
\(956\) 4629.10 0.156607
\(957\) 0 0
\(958\) −8738.81 −0.294716
\(959\) 0 0
\(960\) 0 0
\(961\) 33409.4 1.12146
\(962\) 4408.28 0.147743
\(963\) 0 0
\(964\) −15877.5 −0.530477
\(965\) −30341.8 −1.01216
\(966\) 0 0
\(967\) 15317.5 0.509387 0.254694 0.967022i \(-0.418025\pi\)
0.254694 + 0.967022i \(0.418025\pi\)
\(968\) 15707.3 0.521541
\(969\) 0 0
\(970\) −27520.0 −0.910943
\(971\) 22432.6 0.741398 0.370699 0.928753i \(-0.379118\pi\)
0.370699 + 0.928753i \(0.379118\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −28955.5 −0.952559
\(975\) 0 0
\(976\) −13387.1 −0.439048
\(977\) 627.864 0.0205600 0.0102800 0.999947i \(-0.496728\pi\)
0.0102800 + 0.999947i \(0.496728\pi\)
\(978\) 0 0
\(979\) −19577.3 −0.639114
\(980\) 0 0
\(981\) 0 0
\(982\) −18612.2 −0.604827
\(983\) 45032.6 1.46116 0.730579 0.682828i \(-0.239250\pi\)
0.730579 + 0.682828i \(0.239250\pi\)
\(984\) 0 0
\(985\) 32782.8 1.06045
\(986\) 22648.0 0.731500
\(987\) 0 0
\(988\) 369.285 0.0118912
\(989\) −8021.14 −0.257894
\(990\) 0 0
\(991\) −31981.0 −1.02514 −0.512569 0.858646i \(-0.671306\pi\)
−0.512569 + 0.858646i \(0.671306\pi\)
\(992\) −8044.70 −0.257479
\(993\) 0 0
\(994\) 0 0
\(995\) 50425.2 1.60662
\(996\) 0 0
\(997\) −12378.8 −0.393221 −0.196611 0.980482i \(-0.562993\pi\)
−0.196611 + 0.980482i \(0.562993\pi\)
\(998\) −24475.1 −0.776298
\(999\) 0 0
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 882.4.a.bi.1.2 2
3.2 odd 2 294.4.a.k.1.1 yes 2
7.2 even 3 882.4.g.y.361.1 4
7.3 odd 6 882.4.g.bd.667.2 4
7.4 even 3 882.4.g.y.667.1 4
7.5 odd 6 882.4.g.bd.361.2 4
7.6 odd 2 882.4.a.bc.1.1 2
12.11 even 2 2352.4.a.bn.1.1 2
21.2 odd 6 294.4.e.n.67.2 4
21.5 even 6 294.4.e.o.67.1 4
21.11 odd 6 294.4.e.n.79.2 4
21.17 even 6 294.4.e.o.79.1 4
21.20 even 2 294.4.a.j.1.2 2
84.83 odd 2 2352.4.a.cd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.j.1.2 2 21.20 even 2
294.4.a.k.1.1 yes 2 3.2 odd 2
294.4.e.n.67.2 4 21.2 odd 6
294.4.e.n.79.2 4 21.11 odd 6
294.4.e.o.67.1 4 21.5 even 6
294.4.e.o.79.1 4 21.17 even 6
882.4.a.bc.1.1 2 7.6 odd 2
882.4.a.bi.1.2 2 1.1 even 1 trivial
882.4.g.y.361.1 4 7.2 even 3
882.4.g.y.667.1 4 7.4 even 3
882.4.g.bd.361.2 4 7.5 odd 6
882.4.g.bd.667.2 4 7.3 odd 6
2352.4.a.bn.1.1 2 12.11 even 2
2352.4.a.cd.1.2 2 84.83 odd 2