Properties

Label 441.4.a.u.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{65})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.11692\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.53113 q^{2} +12.5311 q^{4} -13.4791 q^{5} -20.5311 q^{8} +O(q^{10})\) \(q-4.53113 q^{2} +12.5311 q^{4} -13.4791 q^{5} -20.5311 q^{8} +61.0753 q^{10} -0.813227 q^{11} -34.9564 q^{13} -7.21984 q^{16} +117.732 q^{17} +93.2913 q^{19} -168.908 q^{20} +3.68484 q^{22} -120.249 q^{23} +56.6848 q^{25} +158.392 q^{26} -8.56420 q^{29} +82.1070 q^{31} +196.963 q^{32} -533.458 q^{34} +28.8132 q^{37} -422.715 q^{38} +276.740 q^{40} -70.5291 q^{41} +417.179 q^{43} -10.1906 q^{44} +544.864 q^{46} +338.261 q^{47} -256.846 q^{50} -438.043 q^{52} -149.121 q^{53} +10.9615 q^{55} +38.8055 q^{58} -94.1828 q^{59} +120.525 q^{61} -372.037 q^{62} -834.706 q^{64} +471.179 q^{65} -792.366 q^{67} +1475.31 q^{68} -449.128 q^{71} +469.420 q^{73} -130.556 q^{74} +1169.05 q^{76} -1019.85 q^{79} +97.3166 q^{80} +319.576 q^{82} -104.253 q^{83} -1586.91 q^{85} -1890.29 q^{86} +16.6965 q^{88} -1572.92 q^{89} -1506.86 q^{92} -1532.70 q^{94} -1257.48 q^{95} -550.057 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 34q^{4} - 66q^{8} + O(q^{10}) \) \( 4q - 2q^{2} + 34q^{4} - 66q^{8} - 100q^{11} - 174q^{16} - 340q^{22} - 352q^{23} - 128q^{25} - 260q^{29} + 30q^{32} + 212q^{37} + 540q^{43} - 460q^{44} + 696q^{46} - 1366q^{50} - 16q^{53} - 780q^{58} - 1678q^{64} + 756q^{65} - 1944q^{67} - 2248q^{71} + 284q^{74} - 1048q^{79} - 3284q^{85} - 4820q^{86} + 1260q^{88} - 3512q^{92} - 2192q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.53113 −1.60200 −0.800998 0.598667i \(-0.795697\pi\)
−0.800998 + 0.598667i \(0.795697\pi\)
\(3\) 0 0
\(4\) 12.5311 1.56639
\(5\) −13.4791 −1.20560 −0.602802 0.797891i \(-0.705949\pi\)
−0.602802 + 0.797891i \(0.705949\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −20.5311 −0.907356
\(9\) 0 0
\(10\) 61.0753 1.93137
\(11\) −0.813227 −0.0222906 −0.0111453 0.999938i \(-0.503548\pi\)
−0.0111453 + 0.999938i \(0.503548\pi\)
\(12\) 0 0
\(13\) −34.9564 −0.745781 −0.372891 0.927875i \(-0.621633\pi\)
−0.372891 + 0.927875i \(0.621633\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −7.21984 −0.112810
\(17\) 117.732 1.67966 0.839829 0.542851i \(-0.182655\pi\)
0.839829 + 0.542851i \(0.182655\pi\)
\(18\) 0 0
\(19\) 93.2913 1.12645 0.563224 0.826304i \(-0.309561\pi\)
0.563224 + 0.826304i \(0.309561\pi\)
\(20\) −168.908 −1.88845
\(21\) 0 0
\(22\) 3.68484 0.0357095
\(23\) −120.249 −1.09016 −0.545079 0.838384i \(-0.683501\pi\)
−0.545079 + 0.838384i \(0.683501\pi\)
\(24\) 0 0
\(25\) 56.6848 0.453479
\(26\) 158.392 1.19474
\(27\) 0 0
\(28\) 0 0
\(29\) −8.56420 −0.0548390 −0.0274195 0.999624i \(-0.508729\pi\)
−0.0274195 + 0.999624i \(0.508729\pi\)
\(30\) 0 0
\(31\) 82.1070 0.475705 0.237852 0.971301i \(-0.423557\pi\)
0.237852 + 0.971301i \(0.423557\pi\)
\(32\) 196.963 1.08808
\(33\) 0 0
\(34\) −533.458 −2.69080
\(35\) 0 0
\(36\) 0 0
\(37\) 28.8132 0.128023 0.0640117 0.997949i \(-0.479611\pi\)
0.0640117 + 0.997949i \(0.479611\pi\)
\(38\) −422.715 −1.80456
\(39\) 0 0
\(40\) 276.740 1.09391
\(41\) −70.5291 −0.268654 −0.134327 0.990937i \(-0.542887\pi\)
−0.134327 + 0.990937i \(0.542887\pi\)
\(42\) 0 0
\(43\) 417.179 1.47952 0.739758 0.672873i \(-0.234940\pi\)
0.739758 + 0.672873i \(0.234940\pi\)
\(44\) −10.1906 −0.0349159
\(45\) 0 0
\(46\) 544.864 1.74643
\(47\) 338.261 1.04980 0.524899 0.851165i \(-0.324103\pi\)
0.524899 + 0.851165i \(0.324103\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −256.846 −0.726471
\(51\) 0 0
\(52\) −438.043 −1.16819
\(53\) −149.121 −0.386477 −0.193239 0.981152i \(-0.561899\pi\)
−0.193239 + 0.981152i \(0.561899\pi\)
\(54\) 0 0
\(55\) 10.9615 0.0268737
\(56\) 0 0
\(57\) 0 0
\(58\) 38.8055 0.0878519
\(59\) −94.1828 −0.207823 −0.103911 0.994587i \(-0.533136\pi\)
−0.103911 + 0.994587i \(0.533136\pi\)
\(60\) 0 0
\(61\) 120.525 0.252977 0.126488 0.991968i \(-0.459629\pi\)
0.126488 + 0.991968i \(0.459629\pi\)
\(62\) −372.037 −0.762077
\(63\) 0 0
\(64\) −834.706 −1.63029
\(65\) 471.179 0.899116
\(66\) 0 0
\(67\) −792.366 −1.44482 −0.722410 0.691465i \(-0.756965\pi\)
−0.722410 + 0.691465i \(0.756965\pi\)
\(68\) 1475.31 2.63100
\(69\) 0 0
\(70\) 0 0
\(71\) −449.128 −0.750729 −0.375364 0.926877i \(-0.622482\pi\)
−0.375364 + 0.926877i \(0.622482\pi\)
\(72\) 0 0
\(73\) 469.420 0.752623 0.376311 0.926493i \(-0.377192\pi\)
0.376311 + 0.926493i \(0.377192\pi\)
\(74\) −130.556 −0.205093
\(75\) 0 0
\(76\) 1169.05 1.76446
\(77\) 0 0
\(78\) 0 0
\(79\) −1019.85 −1.45243 −0.726217 0.687465i \(-0.758723\pi\)
−0.726217 + 0.687465i \(0.758723\pi\)
\(80\) 97.3166 0.136004
\(81\) 0 0
\(82\) 319.576 0.430382
\(83\) −104.253 −0.137870 −0.0689352 0.997621i \(-0.521960\pi\)
−0.0689352 + 0.997621i \(0.521960\pi\)
\(84\) 0 0
\(85\) −1586.91 −2.02500
\(86\) −1890.29 −2.37018
\(87\) 0 0
\(88\) 16.6965 0.0202256
\(89\) −1572.92 −1.87336 −0.936680 0.350185i \(-0.886119\pi\)
−0.936680 + 0.350185i \(0.886119\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1506.86 −1.70762
\(93\) 0 0
\(94\) −1532.70 −1.68177
\(95\) −1257.48 −1.35805
\(96\) 0 0
\(97\) −550.057 −0.575772 −0.287886 0.957665i \(-0.592952\pi\)
−0.287886 + 0.957665i \(0.592952\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 710.325 0.710325
\(101\) −65.7169 −0.0647433 −0.0323717 0.999476i \(-0.510306\pi\)
−0.0323717 + 0.999476i \(0.510306\pi\)
\(102\) 0 0
\(103\) −1829.35 −1.75001 −0.875005 0.484113i \(-0.839142\pi\)
−0.875005 + 0.484113i \(0.839142\pi\)
\(104\) 717.694 0.676689
\(105\) 0 0
\(106\) 675.685 0.619135
\(107\) −861.377 −0.778248 −0.389124 0.921185i \(-0.627222\pi\)
−0.389124 + 0.921185i \(0.627222\pi\)
\(108\) 0 0
\(109\) −1620.52 −1.42401 −0.712007 0.702173i \(-0.752213\pi\)
−0.712007 + 0.702173i \(0.752213\pi\)
\(110\) −49.6681 −0.0430515
\(111\) 0 0
\(112\) 0 0
\(113\) −380.409 −0.316689 −0.158344 0.987384i \(-0.550616\pi\)
−0.158344 + 0.987384i \(0.550616\pi\)
\(114\) 0 0
\(115\) 1620.84 1.31430
\(116\) −107.319 −0.0858993
\(117\) 0 0
\(118\) 426.754 0.332931
\(119\) 0 0
\(120\) 0 0
\(121\) −1330.34 −0.999503
\(122\) −546.112 −0.405268
\(123\) 0 0
\(124\) 1028.89 0.745140
\(125\) 920.824 0.658888
\(126\) 0 0
\(127\) 958.358 0.669610 0.334805 0.942287i \(-0.391329\pi\)
0.334805 + 0.942287i \(0.391329\pi\)
\(128\) 2206.46 1.52363
\(129\) 0 0
\(130\) −2134.97 −1.44038
\(131\) 1152.16 0.768431 0.384216 0.923243i \(-0.374472\pi\)
0.384216 + 0.923243i \(0.374472\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3590.31 2.31459
\(135\) 0 0
\(136\) −2417.17 −1.52405
\(137\) −357.377 −0.222867 −0.111434 0.993772i \(-0.535544\pi\)
−0.111434 + 0.993772i \(0.535544\pi\)
\(138\) 0 0
\(139\) 2736.29 1.66970 0.834852 0.550475i \(-0.185553\pi\)
0.834852 + 0.550475i \(0.185553\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2035.06 1.20266
\(143\) 28.4275 0.0166239
\(144\) 0 0
\(145\) 115.437 0.0661141
\(146\) −2127.00 −1.20570
\(147\) 0 0
\(148\) 361.062 0.200535
\(149\) −1409.94 −0.775212 −0.387606 0.921825i \(-0.626698\pi\)
−0.387606 + 0.921825i \(0.626698\pi\)
\(150\) 0 0
\(151\) 2352.35 1.26776 0.633879 0.773432i \(-0.281462\pi\)
0.633879 + 0.773432i \(0.281462\pi\)
\(152\) −1915.38 −1.02209
\(153\) 0 0
\(154\) 0 0
\(155\) −1106.72 −0.573511
\(156\) 0 0
\(157\) −1213.82 −0.617029 −0.308514 0.951220i \(-0.599832\pi\)
−0.308514 + 0.951220i \(0.599832\pi\)
\(158\) 4621.08 2.32679
\(159\) 0 0
\(160\) −2654.88 −1.31179
\(161\) 0 0
\(162\) 0 0
\(163\) −722.774 −0.347313 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(164\) −883.809 −0.420817
\(165\) 0 0
\(166\) 472.383 0.220868
\(167\) 753.016 0.348923 0.174462 0.984664i \(-0.444182\pi\)
0.174462 + 0.984664i \(0.444182\pi\)
\(168\) 0 0
\(169\) −975.051 −0.443810
\(170\) 7190.51 3.24404
\(171\) 0 0
\(172\) 5227.72 2.31750
\(173\) −1859.14 −0.817038 −0.408519 0.912750i \(-0.633955\pi\)
−0.408519 + 0.912750i \(0.633955\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.87137 0.00251461
\(177\) 0 0
\(178\) 7127.10 3.00112
\(179\) 522.825 0.218312 0.109156 0.994025i \(-0.465185\pi\)
0.109156 + 0.994025i \(0.465185\pi\)
\(180\) 0 0
\(181\) −2901.38 −1.19148 −0.595740 0.803177i \(-0.703141\pi\)
−0.595740 + 0.803177i \(0.703141\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 2468.85 0.989163
\(185\) −388.375 −0.154345
\(186\) 0 0
\(187\) −95.7427 −0.0374407
\(188\) 4238.79 1.64439
\(189\) 0 0
\(190\) 5697.80 2.17559
\(191\) −2604.35 −0.986619 −0.493309 0.869854i \(-0.664213\pi\)
−0.493309 + 0.869854i \(0.664213\pi\)
\(192\) 0 0
\(193\) 676.245 0.252214 0.126107 0.992017i \(-0.459752\pi\)
0.126107 + 0.992017i \(0.459752\pi\)
\(194\) 2492.38 0.922384
\(195\) 0 0
\(196\) 0 0
\(197\) 3685.99 1.33308 0.666538 0.745471i \(-0.267775\pi\)
0.666538 + 0.745471i \(0.267775\pi\)
\(198\) 0 0
\(199\) 799.801 0.284907 0.142453 0.989802i \(-0.454501\pi\)
0.142453 + 0.989802i \(0.454501\pi\)
\(200\) −1163.80 −0.411467
\(201\) 0 0
\(202\) 297.772 0.103719
\(203\) 0 0
\(204\) 0 0
\(205\) 950.665 0.323890
\(206\) 8289.01 2.80351
\(207\) 0 0
\(208\) 252.380 0.0841316
\(209\) −75.8670 −0.0251092
\(210\) 0 0
\(211\) −667.385 −0.217747 −0.108874 0.994056i \(-0.534724\pi\)
−0.108874 + 0.994056i \(0.534724\pi\)
\(212\) −1868.65 −0.605375
\(213\) 0 0
\(214\) 3903.01 1.24675
\(215\) −5623.18 −1.78371
\(216\) 0 0
\(217\) 0 0
\(218\) 7342.77 2.28126
\(219\) 0 0
\(220\) 137.360 0.0420947
\(221\) −4115.48 −1.25266
\(222\) 0 0
\(223\) −2646.82 −0.794818 −0.397409 0.917642i \(-0.630091\pi\)
−0.397409 + 0.917642i \(0.630091\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1723.68 0.507334
\(227\) 4121.11 1.20497 0.602485 0.798131i \(-0.294178\pi\)
0.602485 + 0.798131i \(0.294178\pi\)
\(228\) 0 0
\(229\) −4066.92 −1.17358 −0.586790 0.809739i \(-0.699609\pi\)
−0.586790 + 0.809739i \(0.699609\pi\)
\(230\) −7344.25 −2.10550
\(231\) 0 0
\(232\) 175.833 0.0497585
\(233\) 3904.67 1.09787 0.548934 0.835865i \(-0.315034\pi\)
0.548934 + 0.835865i \(0.315034\pi\)
\(234\) 0 0
\(235\) −4559.44 −1.26564
\(236\) −1180.22 −0.325532
\(237\) 0 0
\(238\) 0 0
\(239\) −5425.12 −1.46829 −0.734146 0.678991i \(-0.762417\pi\)
−0.734146 + 0.678991i \(0.762417\pi\)
\(240\) 0 0
\(241\) 1602.89 0.428429 0.214215 0.976787i \(-0.431281\pi\)
0.214215 + 0.976787i \(0.431281\pi\)
\(242\) 6027.94 1.60120
\(243\) 0 0
\(244\) 1510.31 0.396261
\(245\) 0 0
\(246\) 0 0
\(247\) −3261.13 −0.840084
\(248\) −1685.75 −0.431634
\(249\) 0 0
\(250\) −4172.37 −1.05554
\(251\) 3805.93 0.957085 0.478542 0.878064i \(-0.341165\pi\)
0.478542 + 0.878064i \(0.341165\pi\)
\(252\) 0 0
\(253\) 97.7897 0.0243003
\(254\) −4342.44 −1.07271
\(255\) 0 0
\(256\) −3320.09 −0.810569
\(257\) 4589.34 1.11391 0.556956 0.830542i \(-0.311969\pi\)
0.556956 + 0.830542i \(0.311969\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 5904.41 1.40837
\(261\) 0 0
\(262\) −5220.58 −1.23102
\(263\) −877.175 −0.205661 −0.102831 0.994699i \(-0.532790\pi\)
−0.102831 + 0.994699i \(0.532790\pi\)
\(264\) 0 0
\(265\) 2010.00 0.465938
\(266\) 0 0
\(267\) 0 0
\(268\) −9929.24 −2.26315
\(269\) −6123.55 −1.38795 −0.693977 0.719997i \(-0.744143\pi\)
−0.693977 + 0.719997i \(0.744143\pi\)
\(270\) 0 0
\(271\) 3489.76 0.782243 0.391122 0.920339i \(-0.372087\pi\)
0.391122 + 0.920339i \(0.372087\pi\)
\(272\) −850.006 −0.189482
\(273\) 0 0
\(274\) 1619.32 0.357032
\(275\) −46.0976 −0.0101083
\(276\) 0 0
\(277\) −4891.70 −1.06106 −0.530530 0.847666i \(-0.678007\pi\)
−0.530530 + 0.847666i \(0.678007\pi\)
\(278\) −12398.5 −2.67486
\(279\) 0 0
\(280\) 0 0
\(281\) −6914.46 −1.46791 −0.733954 0.679199i \(-0.762327\pi\)
−0.733954 + 0.679199i \(0.762327\pi\)
\(282\) 0 0
\(283\) −3559.85 −0.747742 −0.373871 0.927481i \(-0.621970\pi\)
−0.373871 + 0.927481i \(0.621970\pi\)
\(284\) −5628.09 −1.17593
\(285\) 0 0
\(286\) −128.809 −0.0266315
\(287\) 0 0
\(288\) 0 0
\(289\) 8947.80 1.82125
\(290\) −523.061 −0.105914
\(291\) 0 0
\(292\) 5882.36 1.17890
\(293\) 3285.11 0.655011 0.327505 0.944849i \(-0.393792\pi\)
0.327505 + 0.944849i \(0.393792\pi\)
\(294\) 0 0
\(295\) 1269.49 0.250552
\(296\) −591.568 −0.116163
\(297\) 0 0
\(298\) 6388.61 1.24189
\(299\) 4203.47 0.813020
\(300\) 0 0
\(301\) 0 0
\(302\) −10658.8 −2.03094
\(303\) 0 0
\(304\) −673.548 −0.127075
\(305\) −1624.56 −0.304990
\(306\) 0 0
\(307\) 9094.65 1.69075 0.845373 0.534176i \(-0.179378\pi\)
0.845373 + 0.534176i \(0.179378\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 5014.71 0.918762
\(311\) −8163.06 −1.48838 −0.744188 0.667971i \(-0.767163\pi\)
−0.744188 + 0.667971i \(0.767163\pi\)
\(312\) 0 0
\(313\) −2979.62 −0.538076 −0.269038 0.963130i \(-0.586706\pi\)
−0.269038 + 0.963130i \(0.586706\pi\)
\(314\) 5499.98 0.988478
\(315\) 0 0
\(316\) −12779.9 −2.27508
\(317\) 3888.11 0.688889 0.344445 0.938807i \(-0.388067\pi\)
0.344445 + 0.938807i \(0.388067\pi\)
\(318\) 0 0
\(319\) 6.96463 0.00122240
\(320\) 11251.0 1.96548
\(321\) 0 0
\(322\) 0 0
\(323\) 10983.4 1.89205
\(324\) 0 0
\(325\) −1981.50 −0.338196
\(326\) 3274.98 0.556394
\(327\) 0 0
\(328\) 1448.04 0.243764
\(329\) 0 0
\(330\) 0 0
\(331\) −4893.03 −0.812524 −0.406262 0.913757i \(-0.633168\pi\)
−0.406262 + 0.913757i \(0.633168\pi\)
\(332\) −1306.41 −0.215959
\(333\) 0 0
\(334\) −3412.01 −0.558973
\(335\) 10680.3 1.74188
\(336\) 0 0
\(337\) −1722.10 −0.278364 −0.139182 0.990267i \(-0.544447\pi\)
−0.139182 + 0.990267i \(0.544447\pi\)
\(338\) 4418.08 0.710982
\(339\) 0 0
\(340\) −19885.8 −3.17194
\(341\) −66.7716 −0.0106038
\(342\) 0 0
\(343\) 0 0
\(344\) −8565.16 −1.34245
\(345\) 0 0
\(346\) 8423.99 1.30889
\(347\) 238.058 0.0368289 0.0184145 0.999830i \(-0.494138\pi\)
0.0184145 + 0.999830i \(0.494138\pi\)
\(348\) 0 0
\(349\) 10053.1 1.54192 0.770959 0.636884i \(-0.219777\pi\)
0.770959 + 0.636884i \(0.219777\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −160.176 −0.0242539
\(353\) −3470.16 −0.523224 −0.261612 0.965173i \(-0.584254\pi\)
−0.261612 + 0.965173i \(0.584254\pi\)
\(354\) 0 0
\(355\) 6053.82 0.905081
\(356\) −19710.5 −2.93442
\(357\) 0 0
\(358\) −2368.99 −0.349734
\(359\) −1407.54 −0.206928 −0.103464 0.994633i \(-0.532993\pi\)
−0.103464 + 0.994633i \(0.532993\pi\)
\(360\) 0 0
\(361\) 1844.27 0.268884
\(362\) 13146.5 1.90875
\(363\) 0 0
\(364\) 0 0
\(365\) −6327.34 −0.907364
\(366\) 0 0
\(367\) −11133.0 −1.58348 −0.791742 0.610855i \(-0.790826\pi\)
−0.791742 + 0.610855i \(0.790826\pi\)
\(368\) 868.179 0.122981
\(369\) 0 0
\(370\) 1759.78 0.247261
\(371\) 0 0
\(372\) 0 0
\(373\) 9025.94 1.25294 0.626468 0.779447i \(-0.284500\pi\)
0.626468 + 0.779447i \(0.284500\pi\)
\(374\) 433.823 0.0599798
\(375\) 0 0
\(376\) −6944.88 −0.952540
\(377\) 299.373 0.0408979
\(378\) 0 0
\(379\) −5855.75 −0.793640 −0.396820 0.917896i \(-0.629886\pi\)
−0.396820 + 0.917896i \(0.629886\pi\)
\(380\) −15757.6 −2.12723
\(381\) 0 0
\(382\) 11800.6 1.58056
\(383\) −7788.03 −1.03903 −0.519517 0.854460i \(-0.673888\pi\)
−0.519517 + 0.854460i \(0.673888\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3064.16 −0.404045
\(387\) 0 0
\(388\) −6892.84 −0.901884
\(389\) 3814.05 0.497121 0.248560 0.968616i \(-0.420043\pi\)
0.248560 + 0.968616i \(0.420043\pi\)
\(390\) 0 0
\(391\) −14157.1 −1.83109
\(392\) 0 0
\(393\) 0 0
\(394\) −16701.7 −2.13558
\(395\) 13746.6 1.75106
\(396\) 0 0
\(397\) 10165.3 1.28509 0.642545 0.766248i \(-0.277879\pi\)
0.642545 + 0.766248i \(0.277879\pi\)
\(398\) −3624.00 −0.456419
\(399\) 0 0
\(400\) −409.255 −0.0511569
\(401\) −11502.5 −1.43244 −0.716222 0.697873i \(-0.754130\pi\)
−0.716222 + 0.697873i \(0.754130\pi\)
\(402\) 0 0
\(403\) −2870.16 −0.354772
\(404\) −823.507 −0.101413
\(405\) 0 0
\(406\) 0 0
\(407\) −23.4317 −0.00285372
\(408\) 0 0
\(409\) −3266.27 −0.394882 −0.197441 0.980315i \(-0.563263\pi\)
−0.197441 + 0.980315i \(0.563263\pi\)
\(410\) −4307.59 −0.518870
\(411\) 0 0
\(412\) −22923.8 −2.74120
\(413\) 0 0
\(414\) 0 0
\(415\) 1405.23 0.166217
\(416\) −6885.12 −0.811468
\(417\) 0 0
\(418\) 343.763 0.0402249
\(419\) −6822.93 −0.795518 −0.397759 0.917490i \(-0.630212\pi\)
−0.397759 + 0.917490i \(0.630212\pi\)
\(420\) 0 0
\(421\) 1431.63 0.165733 0.0828665 0.996561i \(-0.473592\pi\)
0.0828665 + 0.996561i \(0.473592\pi\)
\(422\) 3024.01 0.348830
\(423\) 0 0
\(424\) 3061.62 0.350673
\(425\) 6673.61 0.761689
\(426\) 0 0
\(427\) 0 0
\(428\) −10794.0 −1.21904
\(429\) 0 0
\(430\) 25479.3 2.85750
\(431\) −15142.2 −1.69228 −0.846141 0.532959i \(-0.821080\pi\)
−0.846141 + 0.532959i \(0.821080\pi\)
\(432\) 0 0
\(433\) 5475.65 0.607721 0.303860 0.952717i \(-0.401724\pi\)
0.303860 + 0.952717i \(0.401724\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −20306.9 −2.23056
\(437\) −11218.2 −1.22801
\(438\) 0 0
\(439\) −1780.54 −0.193578 −0.0967890 0.995305i \(-0.530857\pi\)
−0.0967890 + 0.995305i \(0.530857\pi\)
\(440\) −225.052 −0.0243840
\(441\) 0 0
\(442\) 18647.8 2.00675
\(443\) 3259.64 0.349594 0.174797 0.984605i \(-0.444073\pi\)
0.174797 + 0.984605i \(0.444073\pi\)
\(444\) 0 0
\(445\) 21201.5 2.25853
\(446\) 11993.1 1.27330
\(447\) 0 0
\(448\) 0 0
\(449\) 6826.19 0.717478 0.358739 0.933438i \(-0.383207\pi\)
0.358739 + 0.933438i \(0.383207\pi\)
\(450\) 0 0
\(451\) 57.3562 0.00598846
\(452\) −4766.95 −0.496059
\(453\) 0 0
\(454\) −18673.3 −1.93036
\(455\) 0 0
\(456\) 0 0
\(457\) 3700.03 0.378731 0.189365 0.981907i \(-0.439357\pi\)
0.189365 + 0.981907i \(0.439357\pi\)
\(458\) 18427.8 1.88007
\(459\) 0 0
\(460\) 20311.0 2.05871
\(461\) −9400.80 −0.949759 −0.474880 0.880051i \(-0.657508\pi\)
−0.474880 + 0.880051i \(0.657508\pi\)
\(462\) 0 0
\(463\) 15483.9 1.55420 0.777102 0.629374i \(-0.216689\pi\)
0.777102 + 0.629374i \(0.216689\pi\)
\(464\) 61.8321 0.00618639
\(465\) 0 0
\(466\) −17692.6 −1.75878
\(467\) −2205.62 −0.218552 −0.109276 0.994011i \(-0.534853\pi\)
−0.109276 + 0.994011i \(0.534853\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 20659.4 2.02755
\(471\) 0 0
\(472\) 1933.68 0.188569
\(473\) −339.261 −0.0329794
\(474\) 0 0
\(475\) 5288.20 0.510820
\(476\) 0 0
\(477\) 0 0
\(478\) 24581.9 2.35220
\(479\) 2349.32 0.224098 0.112049 0.993703i \(-0.464259\pi\)
0.112049 + 0.993703i \(0.464259\pi\)
\(480\) 0 0
\(481\) −1007.21 −0.0954775
\(482\) −7262.91 −0.686342
\(483\) 0 0
\(484\) −16670.6 −1.56561
\(485\) 7414.25 0.694152
\(486\) 0 0
\(487\) 10394.3 0.967167 0.483583 0.875298i \(-0.339335\pi\)
0.483583 + 0.875298i \(0.339335\pi\)
\(488\) −2474.50 −0.229540
\(489\) 0 0
\(490\) 0 0
\(491\) −12586.7 −1.15689 −0.578444 0.815722i \(-0.696340\pi\)
−0.578444 + 0.815722i \(0.696340\pi\)
\(492\) 0 0
\(493\) −1008.28 −0.0921108
\(494\) 14776.6 1.34581
\(495\) 0 0
\(496\) −592.799 −0.0536642
\(497\) 0 0
\(498\) 0 0
\(499\) 10627.9 0.953450 0.476725 0.879053i \(-0.341824\pi\)
0.476725 + 0.879053i \(0.341824\pi\)
\(500\) 11539.0 1.03208
\(501\) 0 0
\(502\) −17245.2 −1.53325
\(503\) −6719.02 −0.595599 −0.297800 0.954628i \(-0.596253\pi\)
−0.297800 + 0.954628i \(0.596253\pi\)
\(504\) 0 0
\(505\) 885.802 0.0780548
\(506\) −443.098 −0.0389291
\(507\) 0 0
\(508\) 12009.3 1.04887
\(509\) 3904.33 0.339993 0.169997 0.985445i \(-0.445624\pi\)
0.169997 + 0.985445i \(0.445624\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −2607.89 −0.225105
\(513\) 0 0
\(514\) −20794.9 −1.78448
\(515\) 24657.9 2.10982
\(516\) 0 0
\(517\) −275.083 −0.0234007
\(518\) 0 0
\(519\) 0 0
\(520\) −9673.84 −0.815819
\(521\) −15699.7 −1.32018 −0.660092 0.751184i \(-0.729483\pi\)
−0.660092 + 0.751184i \(0.729483\pi\)
\(522\) 0 0
\(523\) −10152.1 −0.848794 −0.424397 0.905476i \(-0.639514\pi\)
−0.424397 + 0.905476i \(0.639514\pi\)
\(524\) 14437.8 1.20366
\(525\) 0 0
\(526\) 3974.59 0.329469
\(527\) 9666.61 0.799021
\(528\) 0 0
\(529\) 2292.83 0.188447
\(530\) −9107.59 −0.746431
\(531\) 0 0
\(532\) 0 0
\(533\) 2465.44 0.200357
\(534\) 0 0
\(535\) 11610.6 0.938258
\(536\) 16268.2 1.31097
\(537\) 0 0
\(538\) 27746.6 2.22350
\(539\) 0 0
\(540\) 0 0
\(541\) −19846.6 −1.57722 −0.788608 0.614896i \(-0.789198\pi\)
−0.788608 + 0.614896i \(0.789198\pi\)
\(542\) −15812.6 −1.25315
\(543\) 0 0
\(544\) 23188.8 1.82760
\(545\) 21843.0 1.71679
\(546\) 0 0
\(547\) −22798.9 −1.78210 −0.891052 0.453901i \(-0.850032\pi\)
−0.891052 + 0.453901i \(0.850032\pi\)
\(548\) −4478.34 −0.349097
\(549\) 0 0
\(550\) 208.874 0.0161935
\(551\) −798.965 −0.0617733
\(552\) 0 0
\(553\) 0 0
\(554\) 22164.9 1.69981
\(555\) 0 0
\(556\) 34288.8 2.61541
\(557\) 17998.3 1.36914 0.684570 0.728947i \(-0.259990\pi\)
0.684570 + 0.728947i \(0.259990\pi\)
\(558\) 0 0
\(559\) −14583.1 −1.10340
\(560\) 0 0
\(561\) 0 0
\(562\) 31330.3 2.35158
\(563\) 195.636 0.0146449 0.00732246 0.999973i \(-0.497669\pi\)
0.00732246 + 0.999973i \(0.497669\pi\)
\(564\) 0 0
\(565\) 5127.55 0.381801
\(566\) 16130.1 1.19788
\(567\) 0 0
\(568\) 9221.11 0.681178
\(569\) 19660.4 1.44852 0.724260 0.689527i \(-0.242182\pi\)
0.724260 + 0.689527i \(0.242182\pi\)
\(570\) 0 0
\(571\) −15764.5 −1.15538 −0.577691 0.816255i \(-0.696046\pi\)
−0.577691 + 0.816255i \(0.696046\pi\)
\(572\) 356.228 0.0260396
\(573\) 0 0
\(574\) 0 0
\(575\) −6816.30 −0.494364
\(576\) 0 0
\(577\) −22306.4 −1.60941 −0.804704 0.593676i \(-0.797676\pi\)
−0.804704 + 0.593676i \(0.797676\pi\)
\(578\) −40543.6 −2.91764
\(579\) 0 0
\(580\) 1446.56 0.103560
\(581\) 0 0
\(582\) 0 0
\(583\) 121.269 0.00861483
\(584\) −9637.72 −0.682897
\(585\) 0 0
\(586\) −14885.3 −1.04932
\(587\) −15953.2 −1.12173 −0.560866 0.827906i \(-0.689532\pi\)
−0.560866 + 0.827906i \(0.689532\pi\)
\(588\) 0 0
\(589\) 7659.87 0.535856
\(590\) −5752.24 −0.401383
\(591\) 0 0
\(592\) −208.027 −0.0144423
\(593\) −3155.68 −0.218530 −0.109265 0.994013i \(-0.534850\pi\)
−0.109265 + 0.994013i \(0.534850\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −17668.1 −1.21429
\(597\) 0 0
\(598\) −19046.5 −1.30246
\(599\) −25456.3 −1.73642 −0.868212 0.496194i \(-0.834730\pi\)
−0.868212 + 0.496194i \(0.834730\pi\)
\(600\) 0 0
\(601\) −5580.96 −0.378789 −0.189395 0.981901i \(-0.560653\pi\)
−0.189395 + 0.981901i \(0.560653\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 29477.6 1.98581
\(605\) 17931.7 1.20500
\(606\) 0 0
\(607\) −381.133 −0.0254855 −0.0127427 0.999919i \(-0.504056\pi\)
−0.0127427 + 0.999919i \(0.504056\pi\)
\(608\) 18374.9 1.22566
\(609\) 0 0
\(610\) 7361.07 0.488592
\(611\) −11824.4 −0.782919
\(612\) 0 0
\(613\) −8235.98 −0.542656 −0.271328 0.962487i \(-0.587463\pi\)
−0.271328 + 0.962487i \(0.587463\pi\)
\(614\) −41209.0 −2.70857
\(615\) 0 0
\(616\) 0 0
\(617\) 27419.8 1.78911 0.894555 0.446958i \(-0.147493\pi\)
0.894555 + 0.446958i \(0.147493\pi\)
\(618\) 0 0
\(619\) 16373.4 1.06317 0.531585 0.847005i \(-0.321597\pi\)
0.531585 + 0.847005i \(0.321597\pi\)
\(620\) −13868.5 −0.898342
\(621\) 0 0
\(622\) 36987.9 2.38437
\(623\) 0 0
\(624\) 0 0
\(625\) −19497.4 −1.24784
\(626\) 13501.0 0.861996
\(627\) 0 0
\(628\) −15210.6 −0.966508
\(629\) 3392.24 0.215035
\(630\) 0 0
\(631\) 4059.60 0.256118 0.128059 0.991767i \(-0.459125\pi\)
0.128059 + 0.991767i \(0.459125\pi\)
\(632\) 20938.7 1.31788
\(633\) 0 0
\(634\) −17617.5 −1.10360
\(635\) −12917.8 −0.807284
\(636\) 0 0
\(637\) 0 0
\(638\) −31.5577 −0.00195827
\(639\) 0 0
\(640\) −29741.0 −1.83690
\(641\) 6388.63 0.393660 0.196830 0.980438i \(-0.436935\pi\)
0.196830 + 0.980438i \(0.436935\pi\)
\(642\) 0 0
\(643\) 18308.0 1.12286 0.561428 0.827525i \(-0.310252\pi\)
0.561428 + 0.827525i \(0.310252\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −49767.0 −3.03105
\(647\) 3303.90 0.200757 0.100379 0.994949i \(-0.467995\pi\)
0.100379 + 0.994949i \(0.467995\pi\)
\(648\) 0 0
\(649\) 76.5919 0.00463251
\(650\) 8978.42 0.541789
\(651\) 0 0
\(652\) −9057.18 −0.544028
\(653\) −4371.27 −0.261961 −0.130981 0.991385i \(-0.541813\pi\)
−0.130981 + 0.991385i \(0.541813\pi\)
\(654\) 0 0
\(655\) −15530.0 −0.926423
\(656\) 509.209 0.0303068
\(657\) 0 0
\(658\) 0 0
\(659\) −6259.75 −0.370023 −0.185012 0.982736i \(-0.559232\pi\)
−0.185012 + 0.982736i \(0.559232\pi\)
\(660\) 0 0
\(661\) 14845.7 0.873574 0.436787 0.899565i \(-0.356116\pi\)
0.436787 + 0.899565i \(0.356116\pi\)
\(662\) 22171.0 1.30166
\(663\) 0 0
\(664\) 2140.43 0.125098
\(665\) 0 0
\(666\) 0 0
\(667\) 1029.84 0.0597832
\(668\) 9436.14 0.546550
\(669\) 0 0
\(670\) −48394.0 −2.79048
\(671\) −98.0138 −0.00563902
\(672\) 0 0
\(673\) 9409.13 0.538923 0.269462 0.963011i \(-0.413154\pi\)
0.269462 + 0.963011i \(0.413154\pi\)
\(674\) 7803.05 0.445938
\(675\) 0 0
\(676\) −12218.5 −0.695180
\(677\) 2950.63 0.167507 0.0837533 0.996487i \(-0.473309\pi\)
0.0837533 + 0.996487i \(0.473309\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 32581.1 1.83740
\(681\) 0 0
\(682\) 302.551 0.0169872
\(683\) −6280.85 −0.351874 −0.175937 0.984401i \(-0.556296\pi\)
−0.175937 + 0.984401i \(0.556296\pi\)
\(684\) 0 0
\(685\) 4817.11 0.268689
\(686\) 0 0
\(687\) 0 0
\(688\) −3011.97 −0.166904
\(689\) 5212.72 0.288228
\(690\) 0 0
\(691\) −32763.2 −1.80372 −0.901861 0.432027i \(-0.857798\pi\)
−0.901861 + 0.432027i \(0.857798\pi\)
\(692\) −23297.1 −1.27980
\(693\) 0 0
\(694\) −1078.67 −0.0589998
\(695\) −36882.5 −2.01300
\(696\) 0 0
\(697\) −8303.53 −0.451246
\(698\) −45551.9 −2.47015
\(699\) 0 0
\(700\) 0 0
\(701\) 1775.97 0.0956883 0.0478442 0.998855i \(-0.484765\pi\)
0.0478442 + 0.998855i \(0.484765\pi\)
\(702\) 0 0
\(703\) 2688.02 0.144212
\(704\) 678.805 0.0363401
\(705\) 0 0
\(706\) 15723.7 0.838203
\(707\) 0 0
\(708\) 0 0
\(709\) 8862.43 0.469444 0.234722 0.972063i \(-0.424582\pi\)
0.234722 + 0.972063i \(0.424582\pi\)
\(710\) −27430.7 −1.44994
\(711\) 0 0
\(712\) 32293.8 1.69981
\(713\) −9873.28 −0.518594
\(714\) 0 0
\(715\) −383.175 −0.0200419
\(716\) 6551.59 0.341961
\(717\) 0 0
\(718\) 6377.75 0.331498
\(719\) 27499.2 1.42635 0.713177 0.700984i \(-0.247256\pi\)
0.713177 + 0.700984i \(0.247256\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −8356.64 −0.430750
\(723\) 0 0
\(724\) −36357.6 −1.86632
\(725\) −485.460 −0.0248683
\(726\) 0 0
\(727\) −25434.9 −1.29756 −0.648781 0.760975i \(-0.724721\pi\)
−0.648781 + 0.760975i \(0.724721\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 28670.0 1.45359
\(731\) 49115.3 2.48508
\(732\) 0 0
\(733\) 24155.6 1.21720 0.608600 0.793477i \(-0.291731\pi\)
0.608600 + 0.793477i \(0.291731\pi\)
\(734\) 50445.2 2.53674
\(735\) 0 0
\(736\) −23684.6 −1.18618
\(737\) 644.373 0.0322060
\(738\) 0 0
\(739\) 27512.9 1.36952 0.684762 0.728767i \(-0.259906\pi\)
0.684762 + 0.728767i \(0.259906\pi\)
\(740\) −4866.78 −0.241765
\(741\) 0 0
\(742\) 0 0
\(743\) −5995.09 −0.296014 −0.148007 0.988986i \(-0.547286\pi\)
−0.148007 + 0.988986i \(0.547286\pi\)
\(744\) 0 0
\(745\) 19004.6 0.934598
\(746\) −40897.7 −2.00720
\(747\) 0 0
\(748\) −1199.76 −0.0586467
\(749\) 0 0
\(750\) 0 0
\(751\) 1545.09 0.0750747 0.0375373 0.999295i \(-0.488049\pi\)
0.0375373 + 0.999295i \(0.488049\pi\)
\(752\) −2442.19 −0.118428
\(753\) 0 0
\(754\) −1356.50 −0.0655183
\(755\) −31707.5 −1.52841
\(756\) 0 0
\(757\) −5157.82 −0.247641 −0.123820 0.992305i \(-0.539515\pi\)
−0.123820 + 0.992305i \(0.539515\pi\)
\(758\) 26533.1 1.27141
\(759\) 0 0
\(760\) 25817.5 1.23223
\(761\) −3289.96 −0.156716 −0.0783581 0.996925i \(-0.524968\pi\)
−0.0783581 + 0.996925i \(0.524968\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −32635.4 −1.54543
\(765\) 0 0
\(766\) 35288.6 1.66453
\(767\) 3292.29 0.154990
\(768\) 0 0
\(769\) 11146.5 0.522697 0.261348 0.965245i \(-0.415833\pi\)
0.261348 + 0.965245i \(0.415833\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8474.12 0.395065
\(773\) −15830.2 −0.736576 −0.368288 0.929712i \(-0.620056\pi\)
−0.368288 + 0.929712i \(0.620056\pi\)
\(774\) 0 0
\(775\) 4654.22 0.215722
\(776\) 11293.3 0.522430
\(777\) 0 0
\(778\) −17282.0 −0.796386
\(779\) −6579.75 −0.302624
\(780\) 0 0
\(781\) 365.243 0.0167342
\(782\) 64147.9 2.93341
\(783\) 0 0
\(784\) 0 0
\(785\) 16361.2 0.743892
\(786\) 0 0
\(787\) 15163.4 0.686809 0.343404 0.939188i \(-0.388420\pi\)
0.343404 + 0.939188i \(0.388420\pi\)
\(788\) 46189.6 2.08812
\(789\) 0 0
\(790\) −62287.8 −2.80519
\(791\) 0 0
\(792\) 0 0
\(793\) −4213.10 −0.188665
\(794\) −46060.2 −2.05871
\(795\) 0 0
\(796\) 10022.4 0.446275
\(797\) 29398.3 1.30658 0.653289 0.757109i \(-0.273389\pi\)
0.653289 + 0.757109i \(0.273389\pi\)
\(798\) 0 0
\(799\) 39824.1 1.76330
\(800\) 11164.8 0.493420
\(801\) 0 0
\(802\) 52119.5 2.29477
\(803\) −381.745 −0.0167764
\(804\) 0 0
\(805\) 0 0
\(806\) 13005.1 0.568343
\(807\) 0 0
\(808\) 1349.24 0.0587453
\(809\) 20712.9 0.900155 0.450078 0.892989i \(-0.351396\pi\)
0.450078 + 0.892989i \(0.351396\pi\)
\(810\) 0 0
\(811\) −27369.9 −1.18506 −0.592532 0.805547i \(-0.701872\pi\)
−0.592532 + 0.805547i \(0.701872\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 106.172 0.00457165
\(815\) 9742.31 0.418722
\(816\) 0 0
\(817\) 38919.2 1.66660
\(818\) 14799.9 0.632600
\(819\) 0 0
\(820\) 11912.9 0.507338
\(821\) 35362.7 1.50325 0.751623 0.659592i \(-0.229271\pi\)
0.751623 + 0.659592i \(0.229271\pi\)
\(822\) 0 0
\(823\) −29190.4 −1.23635 −0.618174 0.786042i \(-0.712127\pi\)
−0.618174 + 0.786042i \(0.712127\pi\)
\(824\) 37558.6 1.58788
\(825\) 0 0
\(826\) 0 0
\(827\) 7302.08 0.307035 0.153518 0.988146i \(-0.450940\pi\)
0.153518 + 0.988146i \(0.450940\pi\)
\(828\) 0 0
\(829\) −4250.77 −0.178088 −0.0890442 0.996028i \(-0.528381\pi\)
−0.0890442 + 0.996028i \(0.528381\pi\)
\(830\) −6367.28 −0.266279
\(831\) 0 0
\(832\) 29178.3 1.21584
\(833\) 0 0
\(834\) 0 0
\(835\) −10149.9 −0.420663
\(836\) −950.699 −0.0393309
\(837\) 0 0
\(838\) 30915.6 1.27442
\(839\) −39527.7 −1.62652 −0.813258 0.581903i \(-0.802308\pi\)
−0.813258 + 0.581903i \(0.802308\pi\)
\(840\) 0 0
\(841\) −24315.7 −0.996993
\(842\) −6486.92 −0.265504
\(843\) 0 0
\(844\) −8363.09 −0.341078
\(845\) 13142.8 0.535059
\(846\) 0 0
\(847\) 0 0
\(848\) 1076.63 0.0435985
\(849\) 0 0
\(850\) −30239.0 −1.22022
\(851\) −3464.76 −0.139566
\(852\) 0 0
\(853\) −31656.1 −1.27067 −0.635337 0.772235i \(-0.719139\pi\)
−0.635337 + 0.772235i \(0.719139\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 17685.1 0.706148
\(857\) 1193.16 0.0475583 0.0237792 0.999717i \(-0.492430\pi\)
0.0237792 + 0.999717i \(0.492430\pi\)
\(858\) 0 0
\(859\) −29060.0 −1.15427 −0.577134 0.816650i \(-0.695829\pi\)
−0.577134 + 0.816650i \(0.695829\pi\)
\(860\) −70464.8 −2.79399
\(861\) 0 0
\(862\) 68611.2 2.71103
\(863\) −23063.0 −0.909702 −0.454851 0.890567i \(-0.650308\pi\)
−0.454851 + 0.890567i \(0.650308\pi\)
\(864\) 0 0
\(865\) 25059.4 0.985024
\(866\) −24810.9 −0.973566
\(867\) 0 0
\(868\) 0 0
\(869\) 829.371 0.0323757
\(870\) 0 0
\(871\) 27698.2 1.07752
\(872\) 33271.1 1.29209
\(873\) 0 0
\(874\) 50831.1 1.96726
\(875\) 0 0
\(876\) 0 0
\(877\) 33871.0 1.30415 0.652077 0.758153i \(-0.273898\pi\)
0.652077 + 0.758153i \(0.273898\pi\)
\(878\) 8067.88 0.310111
\(879\) 0 0
\(880\) −79.1405 −0.00303162
\(881\) −43331.1 −1.65705 −0.828525 0.559953i \(-0.810819\pi\)
−0.828525 + 0.559953i \(0.810819\pi\)
\(882\) 0 0
\(883\) −40897.3 −1.55867 −0.779334 0.626609i \(-0.784442\pi\)
−0.779334 + 0.626609i \(0.784442\pi\)
\(884\) −51571.6 −1.96215
\(885\) 0 0
\(886\) −14769.8 −0.560047
\(887\) −45065.8 −1.70593 −0.852965 0.521968i \(-0.825198\pi\)
−0.852965 + 0.521968i \(0.825198\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −96066.5 −3.61816
\(891\) 0 0
\(892\) −33167.7 −1.24500
\(893\) 31556.8 1.18254
\(894\) 0 0
\(895\) −7047.19 −0.263197
\(896\) 0 0
\(897\) 0 0
\(898\) −30930.3 −1.14940
\(899\) −703.180 −0.0260872
\(900\) 0 0
\(901\) −17556.3 −0.649150
\(902\) −259.888 −0.00959349
\(903\) 0 0
\(904\) 7810.22 0.287350
\(905\) 39107.9 1.43645
\(906\) 0 0
\(907\) 25282.5 0.925570 0.462785 0.886471i \(-0.346850\pi\)
0.462785 + 0.886471i \(0.346850\pi\)
\(908\) 51642.2 1.88745
\(909\) 0 0
\(910\) 0 0
\(911\) 41646.1 1.51460 0.757298 0.653070i \(-0.226519\pi\)
0.757298 + 0.653070i \(0.226519\pi\)
\(912\) 0 0
\(913\) 84.7812 0.00307322
\(914\) −16765.3 −0.606725
\(915\) 0 0
\(916\) −50963.1 −1.83829
\(917\) 0 0
\(918\) 0 0
\(919\) −26112.5 −0.937292 −0.468646 0.883386i \(-0.655258\pi\)
−0.468646 + 0.883386i \(0.655258\pi\)
\(920\) −33277.7 −1.19254
\(921\) 0 0
\(922\) 42596.3 1.52151
\(923\) 15699.9 0.559879
\(924\) 0 0
\(925\) 1633.27 0.0580559
\(926\) −70159.4 −2.48983
\(927\) 0 0
\(928\) −1686.83 −0.0596691
\(929\) 32357.0 1.14273 0.571366 0.820695i \(-0.306414\pi\)
0.571366 + 0.820695i \(0.306414\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 48929.9 1.71969
\(933\) 0 0
\(934\) 9993.95 0.350120
\(935\) 1290.52 0.0451386
\(936\) 0 0
\(937\) −32947.0 −1.14870 −0.574350 0.818610i \(-0.694745\pi\)
−0.574350 + 0.818610i \(0.694745\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −57134.9 −1.98249
\(941\) −501.602 −0.0173770 −0.00868849 0.999962i \(-0.502766\pi\)
−0.00868849 + 0.999962i \(0.502766\pi\)
\(942\) 0 0
\(943\) 8481.06 0.292875
\(944\) 679.984 0.0234445
\(945\) 0 0
\(946\) 1537.24 0.0528328
\(947\) 6436.90 0.220878 0.110439 0.993883i \(-0.464774\pi\)
0.110439 + 0.993883i \(0.464774\pi\)
\(948\) 0 0
\(949\) −16409.2 −0.561292
\(950\) −23961.5 −0.818331
\(951\) 0 0
\(952\) 0 0
\(953\) −47511.2 −1.61494 −0.807470 0.589908i \(-0.799164\pi\)
−0.807470 + 0.589908i \(0.799164\pi\)
\(954\) 0 0
\(955\) 35104.2 1.18947
\(956\) −67982.9 −2.29992
\(957\) 0 0
\(958\) −10645.1 −0.359004
\(959\) 0 0
\(960\) 0 0
\(961\) −23049.4 −0.773705
\(962\) 4563.78 0.152955
\(963\) 0 0
\(964\) 20086.1 0.671087
\(965\) −9115.15 −0.304069
\(966\) 0 0
\(967\) 7817.32 0.259967 0.129984 0.991516i \(-0.458508\pi\)
0.129984 + 0.991516i \(0.458508\pi\)
\(968\) 27313.4 0.906905
\(969\) 0 0
\(970\) −33594.9 −1.11203
\(971\) 1503.50 0.0496905 0.0248453 0.999691i \(-0.492091\pi\)
0.0248453 + 0.999691i \(0.492091\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −47097.9 −1.54940
\(975\) 0 0
\(976\) −870.168 −0.0285383
\(977\) −33389.1 −1.09336 −0.546680 0.837342i \(-0.684108\pi\)
−0.546680 + 0.837342i \(0.684108\pi\)
\(978\) 0 0
\(979\) 1279.14 0.0417584
\(980\) 0 0
\(981\) 0 0
\(982\) 57032.2 1.85333
\(983\) −5451.02 −0.176867 −0.0884337 0.996082i \(-0.528186\pi\)
−0.0884337 + 0.996082i \(0.528186\pi\)
\(984\) 0 0
\(985\) −49683.7 −1.60716
\(986\) 4568.64 0.147561
\(987\) 0 0
\(988\) −40865.6 −1.31590
\(989\) −50165.4 −1.61291
\(990\) 0 0
\(991\) −46530.0 −1.49150 −0.745750 0.666226i \(-0.767908\pi\)
−0.745750 + 0.666226i \(0.767908\pi\)
\(992\) 16172.0 0.517603
\(993\) 0 0
\(994\) 0 0
\(995\) −10780.6 −0.343484
\(996\) 0 0
\(997\) −11410.0 −0.362444 −0.181222 0.983442i \(-0.558005\pi\)
−0.181222 + 0.983442i \(0.558005\pi\)
\(998\) −48156.5 −1.52742
\(999\) 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 441.4.a.u.1.1 4
3.2 odd 2 49.4.a.e.1.3 4
7.2 even 3 441.4.e.y.361.4 8
7.3 odd 6 441.4.e.y.226.3 8
7.4 even 3 441.4.e.y.226.4 8
7.5 odd 6 441.4.e.y.361.3 8
7.6 odd 2 inner 441.4.a.u.1.2 4
12.11 even 2 784.4.a.bf.1.3 4
15.14 odd 2 1225.4.a.bb.1.2 4
21.2 odd 6 49.4.c.e.18.2 8
21.5 even 6 49.4.c.e.18.1 8
21.11 odd 6 49.4.c.e.30.2 8
21.17 even 6 49.4.c.e.30.1 8
21.20 even 2 49.4.a.e.1.4 yes 4
84.83 odd 2 784.4.a.bf.1.2 4
105.104 even 2 1225.4.a.bb.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.a.e.1.3 4 3.2 odd 2
49.4.a.e.1.4 yes 4 21.20 even 2
49.4.c.e.18.1 8 21.5 even 6
49.4.c.e.18.2 8 21.2 odd 6
49.4.c.e.30.1 8 21.17 even 6
49.4.c.e.30.2 8 21.11 odd 6
441.4.a.u.1.1 4 1.1 even 1 trivial
441.4.a.u.1.2 4 7.6 odd 2 inner
441.4.e.y.226.3 8 7.3 odd 6
441.4.e.y.226.4 8 7.4 even 3
441.4.e.y.361.3 8 7.5 odd 6
441.4.e.y.361.4 8 7.2 even 3
784.4.a.bf.1.2 4 84.83 odd 2
784.4.a.bf.1.3 4 12.11 even 2
1225.4.a.bb.1.1 4 105.104 even 2
1225.4.a.bb.1.2 4 15.14 odd 2