Properties

Label 441.4.a.x.1.7
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 74 x^{6} + 1469 x^{4} - 8828 x^{2} + 2500\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(6.81412\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.39991 q^{2} +21.1590 q^{4} -15.5768 q^{5} +71.0573 q^{8} +O(q^{10})\) \(q+5.39991 q^{2} +21.1590 q^{4} -15.5768 q^{5} +71.0573 q^{8} -84.1131 q^{10} +31.9702 q^{11} +72.5746 q^{13} +214.431 q^{16} +29.0288 q^{17} +108.829 q^{19} -329.589 q^{20} +172.636 q^{22} -55.2869 q^{23} +117.636 q^{25} +391.896 q^{26} +17.7363 q^{29} +56.1189 q^{31} +589.448 q^{32} +156.753 q^{34} -295.816 q^{37} +587.668 q^{38} -1106.84 q^{40} -238.605 q^{41} +16.8202 q^{43} +676.457 q^{44} -298.544 q^{46} -511.909 q^{47} +635.223 q^{50} +1535.60 q^{52} +265.205 q^{53} -497.992 q^{55} +95.7742 q^{58} +254.181 q^{59} -72.8352 q^{61} +303.037 q^{62} +1467.52 q^{64} -1130.48 q^{65} -506.360 q^{67} +614.220 q^{68} -827.722 q^{71} -372.577 q^{73} -1597.38 q^{74} +2302.72 q^{76} +1028.45 q^{79} -3340.14 q^{80} -1288.44 q^{82} -453.148 q^{83} -452.175 q^{85} +90.8276 q^{86} +2271.71 q^{88} -332.065 q^{89} -1169.81 q^{92} -2764.26 q^{94} -1695.21 q^{95} -1164.54 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 68q^{4} + O(q^{10}) \) \( 8q + 68q^{4} + 804q^{16} + 976q^{22} + 536q^{25} + 64q^{37} + 2160q^{43} - 768q^{46} + 2184q^{58} + 7588q^{64} + 5392q^{79} + 2864q^{85} + 5616q^{88} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.39991 1.90916 0.954578 0.297962i \(-0.0963070\pi\)
0.954578 + 0.297962i \(0.0963070\pi\)
\(3\) 0 0
\(4\) 21.1590 2.64487
\(5\) −15.5768 −1.39323 −0.696615 0.717446i \(-0.745311\pi\)
−0.696615 + 0.717446i \(0.745311\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 71.0573 3.14032
\(9\) 0 0
\(10\) −84.1131 −2.65989
\(11\) 31.9702 0.876306 0.438153 0.898900i \(-0.355633\pi\)
0.438153 + 0.898900i \(0.355633\pi\)
\(12\) 0 0
\(13\) 72.5746 1.54835 0.774176 0.632971i \(-0.218165\pi\)
0.774176 + 0.632971i \(0.218165\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 214.431 3.35048
\(17\) 29.0288 0.414148 0.207074 0.978325i \(-0.433606\pi\)
0.207074 + 0.978325i \(0.433606\pi\)
\(18\) 0 0
\(19\) 108.829 1.31406 0.657030 0.753864i \(-0.271812\pi\)
0.657030 + 0.753864i \(0.271812\pi\)
\(20\) −329.589 −3.68492
\(21\) 0 0
\(22\) 172.636 1.67300
\(23\) −55.2869 −0.501222 −0.250611 0.968088i \(-0.580632\pi\)
−0.250611 + 0.968088i \(0.580632\pi\)
\(24\) 0 0
\(25\) 117.636 0.941088
\(26\) 391.896 2.95604
\(27\) 0 0
\(28\) 0 0
\(29\) 17.7363 0.113570 0.0567852 0.998386i \(-0.481915\pi\)
0.0567852 + 0.998386i \(0.481915\pi\)
\(30\) 0 0
\(31\) 56.1189 0.325137 0.162568 0.986697i \(-0.448022\pi\)
0.162568 + 0.986697i \(0.448022\pi\)
\(32\) 589.448 3.25627
\(33\) 0 0
\(34\) 156.753 0.790673
\(35\) 0 0
\(36\) 0 0
\(37\) −295.816 −1.31437 −0.657187 0.753728i \(-0.728254\pi\)
−0.657187 + 0.753728i \(0.728254\pi\)
\(38\) 587.668 2.50875
\(39\) 0 0
\(40\) −1106.84 −4.37518
\(41\) −238.605 −0.908873 −0.454437 0.890779i \(-0.650159\pi\)
−0.454437 + 0.890779i \(0.650159\pi\)
\(42\) 0 0
\(43\) 16.8202 0.0596526 0.0298263 0.999555i \(-0.490505\pi\)
0.0298263 + 0.999555i \(0.490505\pi\)
\(44\) 676.457 2.31772
\(45\) 0 0
\(46\) −298.544 −0.956911
\(47\) −511.909 −1.58871 −0.794357 0.607451i \(-0.792192\pi\)
−0.794357 + 0.607451i \(0.792192\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 635.223 1.79668
\(51\) 0 0
\(52\) 1535.60 4.09519
\(53\) 265.205 0.687335 0.343667 0.939091i \(-0.388331\pi\)
0.343667 + 0.939091i \(0.388331\pi\)
\(54\) 0 0
\(55\) −497.992 −1.22090
\(56\) 0 0
\(57\) 0 0
\(58\) 95.7742 0.216823
\(59\) 254.181 0.560875 0.280437 0.959872i \(-0.409520\pi\)
0.280437 + 0.959872i \(0.409520\pi\)
\(60\) 0 0
\(61\) −72.8352 −0.152878 −0.0764392 0.997074i \(-0.524355\pi\)
−0.0764392 + 0.997074i \(0.524355\pi\)
\(62\) 303.037 0.620737
\(63\) 0 0
\(64\) 1467.52 2.86625
\(65\) −1130.48 −2.15721
\(66\) 0 0
\(67\) −506.360 −0.923308 −0.461654 0.887060i \(-0.652744\pi\)
−0.461654 + 0.887060i \(0.652744\pi\)
\(68\) 614.220 1.09537
\(69\) 0 0
\(70\) 0 0
\(71\) −827.722 −1.38356 −0.691779 0.722110i \(-0.743173\pi\)
−0.691779 + 0.722110i \(0.743173\pi\)
\(72\) 0 0
\(73\) −372.577 −0.597354 −0.298677 0.954354i \(-0.596545\pi\)
−0.298677 + 0.954354i \(0.596545\pi\)
\(74\) −1597.38 −2.50934
\(75\) 0 0
\(76\) 2302.72 3.47552
\(77\) 0 0
\(78\) 0 0
\(79\) 1028.45 1.46468 0.732341 0.680938i \(-0.238428\pi\)
0.732341 + 0.680938i \(0.238428\pi\)
\(80\) −3340.14 −4.66799
\(81\) 0 0
\(82\) −1288.44 −1.73518
\(83\) −453.148 −0.599270 −0.299635 0.954054i \(-0.596865\pi\)
−0.299635 + 0.954054i \(0.596865\pi\)
\(84\) 0 0
\(85\) −452.175 −0.577003
\(86\) 90.8276 0.113886
\(87\) 0 0
\(88\) 2271.71 2.75188
\(89\) −332.065 −0.395493 −0.197746 0.980253i \(-0.563362\pi\)
−0.197746 + 0.980253i \(0.563362\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1169.81 −1.32567
\(93\) 0 0
\(94\) −2764.26 −3.03310
\(95\) −1695.21 −1.83079
\(96\) 0 0
\(97\) −1164.54 −1.21898 −0.609489 0.792795i \(-0.708625\pi\)
−0.609489 + 0.792795i \(0.708625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2489.06 2.48906
\(101\) 1863.54 1.83593 0.917967 0.396656i \(-0.129829\pi\)
0.917967 + 0.396656i \(0.129829\pi\)
\(102\) 0 0
\(103\) 986.280 0.943506 0.471753 0.881731i \(-0.343621\pi\)
0.471753 + 0.881731i \(0.343621\pi\)
\(104\) 5156.95 4.86232
\(105\) 0 0
\(106\) 1432.08 1.31223
\(107\) −695.685 −0.628546 −0.314273 0.949333i \(-0.601761\pi\)
−0.314273 + 0.949333i \(0.601761\pi\)
\(108\) 0 0
\(109\) 1715.00 1.50703 0.753517 0.657428i \(-0.228356\pi\)
0.753517 + 0.657428i \(0.228356\pi\)
\(110\) −2689.11 −2.33088
\(111\) 0 0
\(112\) 0 0
\(113\) 877.721 0.730700 0.365350 0.930870i \(-0.380949\pi\)
0.365350 + 0.930870i \(0.380949\pi\)
\(114\) 0 0
\(115\) 861.191 0.698317
\(116\) 375.281 0.300379
\(117\) 0 0
\(118\) 1372.56 1.07080
\(119\) 0 0
\(120\) 0 0
\(121\) −308.908 −0.232087
\(122\) −393.303 −0.291869
\(123\) 0 0
\(124\) 1187.42 0.859946
\(125\) 114.708 0.0820784
\(126\) 0 0
\(127\) 781.088 0.545751 0.272875 0.962049i \(-0.412025\pi\)
0.272875 + 0.962049i \(0.412025\pi\)
\(128\) 3208.88 2.21584
\(129\) 0 0
\(130\) −6104.48 −4.11845
\(131\) −1961.24 −1.30805 −0.654024 0.756474i \(-0.726920\pi\)
−0.654024 + 0.756474i \(0.726920\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2734.29 −1.76274
\(135\) 0 0
\(136\) 2062.71 1.30056
\(137\) −1220.31 −0.761009 −0.380504 0.924779i \(-0.624250\pi\)
−0.380504 + 0.924779i \(0.624250\pi\)
\(138\) 0 0
\(139\) 1068.10 0.651765 0.325882 0.945410i \(-0.394339\pi\)
0.325882 + 0.945410i \(0.394339\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4469.62 −2.64143
\(143\) 2320.22 1.35683
\(144\) 0 0
\(145\) −276.274 −0.158230
\(146\) −2011.88 −1.14044
\(147\) 0 0
\(148\) −6259.16 −3.47635
\(149\) −2590.82 −1.42449 −0.712243 0.701933i \(-0.752321\pi\)
−0.712243 + 0.701933i \(0.752321\pi\)
\(150\) 0 0
\(151\) 1929.54 1.03989 0.519946 0.854199i \(-0.325952\pi\)
0.519946 + 0.854199i \(0.325952\pi\)
\(152\) 7733.12 4.12657
\(153\) 0 0
\(154\) 0 0
\(155\) −874.151 −0.452990
\(156\) 0 0
\(157\) 2625.73 1.33475 0.667377 0.744720i \(-0.267417\pi\)
0.667377 + 0.744720i \(0.267417\pi\)
\(158\) 5553.54 2.79630
\(159\) 0 0
\(160\) −9181.70 −4.53673
\(161\) 0 0
\(162\) 0 0
\(163\) −3100.17 −1.48972 −0.744858 0.667223i \(-0.767483\pi\)
−0.744858 + 0.667223i \(0.767483\pi\)
\(164\) −5048.63 −2.40385
\(165\) 0 0
\(166\) −2446.95 −1.14410
\(167\) −3264.73 −1.51277 −0.756386 0.654126i \(-0.773037\pi\)
−0.756386 + 0.654126i \(0.773037\pi\)
\(168\) 0 0
\(169\) 3070.07 1.39739
\(170\) −2441.70 −1.10159
\(171\) 0 0
\(172\) 355.899 0.157773
\(173\) 2036.31 0.894900 0.447450 0.894309i \(-0.352332\pi\)
0.447450 + 0.894309i \(0.352332\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6855.39 2.93605
\(177\) 0 0
\(178\) −1793.12 −0.755057
\(179\) −3582.35 −1.49585 −0.747925 0.663783i \(-0.768950\pi\)
−0.747925 + 0.663783i \(0.768950\pi\)
\(180\) 0 0
\(181\) −1637.35 −0.672392 −0.336196 0.941792i \(-0.609140\pi\)
−0.336196 + 0.941792i \(0.609140\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3928.53 −1.57400
\(185\) 4607.86 1.83122
\(186\) 0 0
\(187\) 928.056 0.362921
\(188\) −10831.5 −4.20195
\(189\) 0 0
\(190\) −9153.97 −3.49526
\(191\) 2824.78 1.07013 0.535063 0.844812i \(-0.320288\pi\)
0.535063 + 0.844812i \(0.320288\pi\)
\(192\) 0 0
\(193\) −1657.53 −0.618195 −0.309098 0.951030i \(-0.600027\pi\)
−0.309098 + 0.951030i \(0.600027\pi\)
\(194\) −6288.39 −2.32722
\(195\) 0 0
\(196\) 0 0
\(197\) 1890.78 0.683819 0.341909 0.939733i \(-0.388926\pi\)
0.341909 + 0.939733i \(0.388926\pi\)
\(198\) 0 0
\(199\) −1392.75 −0.496126 −0.248063 0.968744i \(-0.579794\pi\)
−0.248063 + 0.968744i \(0.579794\pi\)
\(200\) 8358.89 2.95532
\(201\) 0 0
\(202\) 10063.0 3.50508
\(203\) 0 0
\(204\) 0 0
\(205\) 3716.69 1.26627
\(206\) 5325.82 1.80130
\(207\) 0 0
\(208\) 15562.2 5.18772
\(209\) 3479.29 1.15152
\(210\) 0 0
\(211\) 3314.53 1.08143 0.540714 0.841206i \(-0.318154\pi\)
0.540714 + 0.841206i \(0.318154\pi\)
\(212\) 5611.47 1.81791
\(213\) 0 0
\(214\) −3756.64 −1.19999
\(215\) −262.005 −0.0831097
\(216\) 0 0
\(217\) 0 0
\(218\) 9260.82 2.87716
\(219\) 0 0
\(220\) −10537.0 −3.22911
\(221\) 2106.75 0.641247
\(222\) 0 0
\(223\) −5576.50 −1.67457 −0.837287 0.546764i \(-0.815860\pi\)
−0.837287 + 0.546764i \(0.815860\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 4739.61 1.39502
\(227\) −1447.92 −0.423357 −0.211678 0.977339i \(-0.567893\pi\)
−0.211678 + 0.977339i \(0.567893\pi\)
\(228\) 0 0
\(229\) 1617.55 0.466771 0.233385 0.972384i \(-0.425020\pi\)
0.233385 + 0.972384i \(0.425020\pi\)
\(230\) 4650.35 1.33320
\(231\) 0 0
\(232\) 1260.29 0.356647
\(233\) −6092.11 −1.71291 −0.856454 0.516224i \(-0.827337\pi\)
−0.856454 + 0.516224i \(0.827337\pi\)
\(234\) 0 0
\(235\) 7973.89 2.21344
\(236\) 5378.22 1.48344
\(237\) 0 0
\(238\) 0 0
\(239\) −1595.90 −0.431927 −0.215963 0.976401i \(-0.569289\pi\)
−0.215963 + 0.976401i \(0.569289\pi\)
\(240\) 0 0
\(241\) 2188.83 0.585041 0.292521 0.956259i \(-0.405506\pi\)
0.292521 + 0.956259i \(0.405506\pi\)
\(242\) −1668.07 −0.443090
\(243\) 0 0
\(244\) −1541.12 −0.404344
\(245\) 0 0
\(246\) 0 0
\(247\) 7898.24 2.03463
\(248\) 3987.65 1.02103
\(249\) 0 0
\(250\) 619.413 0.156700
\(251\) −6203.07 −1.55990 −0.779949 0.625843i \(-0.784755\pi\)
−0.779949 + 0.625843i \(0.784755\pi\)
\(252\) 0 0
\(253\) −1767.53 −0.439224
\(254\) 4217.80 1.04192
\(255\) 0 0
\(256\) 5587.48 1.36413
\(257\) −268.323 −0.0651266 −0.0325633 0.999470i \(-0.510367\pi\)
−0.0325633 + 0.999470i \(0.510367\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −23919.8 −5.70554
\(261\) 0 0
\(262\) −10590.5 −2.49727
\(263\) 3724.96 0.873349 0.436675 0.899619i \(-0.356156\pi\)
0.436675 + 0.899619i \(0.356156\pi\)
\(264\) 0 0
\(265\) −4131.04 −0.957615
\(266\) 0 0
\(267\) 0 0
\(268\) −10714.1 −2.44203
\(269\) −8557.25 −1.93957 −0.969786 0.243958i \(-0.921554\pi\)
−0.969786 + 0.243958i \(0.921554\pi\)
\(270\) 0 0
\(271\) −5279.66 −1.18346 −0.591728 0.806137i \(-0.701554\pi\)
−0.591728 + 0.806137i \(0.701554\pi\)
\(272\) 6224.67 1.38760
\(273\) 0 0
\(274\) −6589.56 −1.45288
\(275\) 3760.84 0.824681
\(276\) 0 0
\(277\) −441.548 −0.0957764 −0.0478882 0.998853i \(-0.515249\pi\)
−0.0478882 + 0.998853i \(0.515249\pi\)
\(278\) 5767.66 1.24432
\(279\) 0 0
\(280\) 0 0
\(281\) −3766.49 −0.799609 −0.399804 0.916601i \(-0.630922\pi\)
−0.399804 + 0.916601i \(0.630922\pi\)
\(282\) 0 0
\(283\) 1811.02 0.380403 0.190201 0.981745i \(-0.439086\pi\)
0.190201 + 0.981745i \(0.439086\pi\)
\(284\) −17513.8 −3.65933
\(285\) 0 0
\(286\) 12529.0 2.59040
\(287\) 0 0
\(288\) 0 0
\(289\) −4070.33 −0.828481
\(290\) −1491.85 −0.302085
\(291\) 0 0
\(292\) −7883.35 −1.57993
\(293\) 5815.74 1.15959 0.579794 0.814763i \(-0.303133\pi\)
0.579794 + 0.814763i \(0.303133\pi\)
\(294\) 0 0
\(295\) −3959.33 −0.781427
\(296\) −21019.9 −4.12755
\(297\) 0 0
\(298\) −13990.2 −2.71957
\(299\) −4012.42 −0.776068
\(300\) 0 0
\(301\) 0 0
\(302\) 10419.3 1.98532
\(303\) 0 0
\(304\) 23336.4 4.40274
\(305\) 1134.54 0.212995
\(306\) 0 0
\(307\) −1974.93 −0.367150 −0.183575 0.983006i \(-0.558767\pi\)
−0.183575 + 0.983006i \(0.558767\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4720.33 −0.864829
\(311\) −1159.07 −0.211334 −0.105667 0.994402i \(-0.533698\pi\)
−0.105667 + 0.994402i \(0.533698\pi\)
\(312\) 0 0
\(313\) 4493.67 0.811494 0.405747 0.913986i \(-0.367011\pi\)
0.405747 + 0.913986i \(0.367011\pi\)
\(314\) 14178.7 2.54825
\(315\) 0 0
\(316\) 21761.0 3.87390
\(317\) 4677.50 0.828752 0.414376 0.910106i \(-0.364000\pi\)
0.414376 + 0.910106i \(0.364000\pi\)
\(318\) 0 0
\(319\) 567.031 0.0995225
\(320\) −22859.2 −3.99334
\(321\) 0 0
\(322\) 0 0
\(323\) 3159.19 0.544216
\(324\) 0 0
\(325\) 8537.38 1.45713
\(326\) −16740.6 −2.84410
\(327\) 0 0
\(328\) −16954.6 −2.85415
\(329\) 0 0
\(330\) 0 0
\(331\) −2982.71 −0.495301 −0.247650 0.968849i \(-0.579658\pi\)
−0.247650 + 0.968849i \(0.579658\pi\)
\(332\) −9588.15 −1.58499
\(333\) 0 0
\(334\) −17629.3 −2.88811
\(335\) 7887.45 1.28638
\(336\) 0 0
\(337\) 7328.53 1.18460 0.592301 0.805717i \(-0.298220\pi\)
0.592301 + 0.805717i \(0.298220\pi\)
\(338\) 16578.1 2.66784
\(339\) 0 0
\(340\) −9567.57 −1.52610
\(341\) 1794.13 0.284920
\(342\) 0 0
\(343\) 0 0
\(344\) 1195.20 0.187328
\(345\) 0 0
\(346\) 10995.9 1.70850
\(347\) 8309.33 1.28550 0.642749 0.766076i \(-0.277794\pi\)
0.642749 + 0.766076i \(0.277794\pi\)
\(348\) 0 0
\(349\) 334.303 0.0512745 0.0256373 0.999671i \(-0.491839\pi\)
0.0256373 + 0.999671i \(0.491839\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 18844.8 2.85349
\(353\) −5458.19 −0.822975 −0.411487 0.911415i \(-0.634991\pi\)
−0.411487 + 0.911415i \(0.634991\pi\)
\(354\) 0 0
\(355\) 12893.2 1.92761
\(356\) −7026.17 −1.04603
\(357\) 0 0
\(358\) −19344.3 −2.85581
\(359\) 7196.90 1.05804 0.529022 0.848608i \(-0.322559\pi\)
0.529022 + 0.848608i \(0.322559\pi\)
\(360\) 0 0
\(361\) 4984.82 0.726755
\(362\) −8841.52 −1.28370
\(363\) 0 0
\(364\) 0 0
\(365\) 5803.55 0.832251
\(366\) 0 0
\(367\) −6324.43 −0.899544 −0.449772 0.893143i \(-0.648495\pi\)
−0.449772 + 0.893143i \(0.648495\pi\)
\(368\) −11855.2 −1.67934
\(369\) 0 0
\(370\) 24882.0 3.49609
\(371\) 0 0
\(372\) 0 0
\(373\) −10931.8 −1.51749 −0.758746 0.651386i \(-0.774188\pi\)
−0.758746 + 0.651386i \(0.774188\pi\)
\(374\) 5011.42 0.692872
\(375\) 0 0
\(376\) −36374.9 −4.98907
\(377\) 1287.20 0.175847
\(378\) 0 0
\(379\) 6024.02 0.816446 0.408223 0.912882i \(-0.366149\pi\)
0.408223 + 0.912882i \(0.366149\pi\)
\(380\) −35868.9 −4.84220
\(381\) 0 0
\(382\) 15253.6 2.04304
\(383\) 6848.10 0.913633 0.456816 0.889561i \(-0.348990\pi\)
0.456816 + 0.889561i \(0.348990\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8950.51 −1.18023
\(387\) 0 0
\(388\) −24640.4 −3.22404
\(389\) 5161.88 0.672796 0.336398 0.941720i \(-0.390791\pi\)
0.336398 + 0.941720i \(0.390791\pi\)
\(390\) 0 0
\(391\) −1604.91 −0.207580
\(392\) 0 0
\(393\) 0 0
\(394\) 10210.0 1.30552
\(395\) −16020.0 −2.04064
\(396\) 0 0
\(397\) −344.768 −0.0435854 −0.0217927 0.999763i \(-0.506937\pi\)
−0.0217927 + 0.999763i \(0.506937\pi\)
\(398\) −7520.70 −0.947182
\(399\) 0 0
\(400\) 25224.8 3.15310
\(401\) 5098.84 0.634972 0.317486 0.948263i \(-0.397161\pi\)
0.317486 + 0.948263i \(0.397161\pi\)
\(402\) 0 0
\(403\) 4072.80 0.503426
\(404\) 39430.7 4.85582
\(405\) 0 0
\(406\) 0 0
\(407\) −9457.28 −1.15179
\(408\) 0 0
\(409\) 323.124 0.0390647 0.0195323 0.999809i \(-0.493782\pi\)
0.0195323 + 0.999809i \(0.493782\pi\)
\(410\) 20069.8 2.41750
\(411\) 0 0
\(412\) 20868.7 2.49545
\(413\) 0 0
\(414\) 0 0
\(415\) 7058.58 0.834921
\(416\) 42779.0 5.04185
\(417\) 0 0
\(418\) 18787.8 2.19843
\(419\) −4415.98 −0.514880 −0.257440 0.966294i \(-0.582879\pi\)
−0.257440 + 0.966294i \(0.582879\pi\)
\(420\) 0 0
\(421\) 1379.37 0.159683 0.0798415 0.996808i \(-0.474559\pi\)
0.0798415 + 0.996808i \(0.474559\pi\)
\(422\) 17898.1 2.06461
\(423\) 0 0
\(424\) 18844.8 2.15845
\(425\) 3414.83 0.389750
\(426\) 0 0
\(427\) 0 0
\(428\) −14720.0 −1.66243
\(429\) 0 0
\(430\) −1414.80 −0.158669
\(431\) 1655.79 0.185050 0.0925248 0.995710i \(-0.470506\pi\)
0.0925248 + 0.995710i \(0.470506\pi\)
\(432\) 0 0
\(433\) −8612.65 −0.955883 −0.477942 0.878392i \(-0.658617\pi\)
−0.477942 + 0.878392i \(0.658617\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 36287.6 3.98592
\(437\) −6016.83 −0.658636
\(438\) 0 0
\(439\) 5975.39 0.649635 0.324817 0.945777i \(-0.394697\pi\)
0.324817 + 0.945777i \(0.394697\pi\)
\(440\) −35386.0 −3.83400
\(441\) 0 0
\(442\) 11376.3 1.22424
\(443\) 806.532 0.0865000 0.0432500 0.999064i \(-0.486229\pi\)
0.0432500 + 0.999064i \(0.486229\pi\)
\(444\) 0 0
\(445\) 5172.51 0.551012
\(446\) −30112.6 −3.19702
\(447\) 0 0
\(448\) 0 0
\(449\) 6253.04 0.657237 0.328618 0.944463i \(-0.393417\pi\)
0.328618 + 0.944463i \(0.393417\pi\)
\(450\) 0 0
\(451\) −7628.23 −0.796451
\(452\) 18571.7 1.93261
\(453\) 0 0
\(454\) −7818.65 −0.808254
\(455\) 0 0
\(456\) 0 0
\(457\) 160.288 0.0164069 0.00820344 0.999966i \(-0.497389\pi\)
0.00820344 + 0.999966i \(0.497389\pi\)
\(458\) 8734.60 0.891138
\(459\) 0 0
\(460\) 18221.9 1.84696
\(461\) 2408.80 0.243360 0.121680 0.992569i \(-0.461172\pi\)
0.121680 + 0.992569i \(0.461172\pi\)
\(462\) 0 0
\(463\) −1092.89 −0.109699 −0.0548496 0.998495i \(-0.517468\pi\)
−0.0548496 + 0.998495i \(0.517468\pi\)
\(464\) 3803.20 0.380516
\(465\) 0 0
\(466\) −32896.8 −3.27021
\(467\) 15054.8 1.49177 0.745884 0.666076i \(-0.232027\pi\)
0.745884 + 0.666076i \(0.232027\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 43058.3 4.22581
\(471\) 0 0
\(472\) 18061.5 1.76133
\(473\) 537.745 0.0522739
\(474\) 0 0
\(475\) 12802.2 1.23665
\(476\) 0 0
\(477\) 0 0
\(478\) −8617.73 −0.824615
\(479\) 10455.6 0.997346 0.498673 0.866790i \(-0.333821\pi\)
0.498673 + 0.866790i \(0.333821\pi\)
\(480\) 0 0
\(481\) −21468.7 −2.03511
\(482\) 11819.5 1.11693
\(483\) 0 0
\(484\) −6536.18 −0.613841
\(485\) 18139.7 1.69831
\(486\) 0 0
\(487\) −12717.7 −1.18335 −0.591675 0.806176i \(-0.701533\pi\)
−0.591675 + 0.806176i \(0.701533\pi\)
\(488\) −5175.47 −0.480087
\(489\) 0 0
\(490\) 0 0
\(491\) 20983.1 1.92863 0.964313 0.264763i \(-0.0852937\pi\)
0.964313 + 0.264763i \(0.0852937\pi\)
\(492\) 0 0
\(493\) 514.863 0.0470350
\(494\) 42649.8 3.88442
\(495\) 0 0
\(496\) 12033.6 1.08937
\(497\) 0 0
\(498\) 0 0
\(499\) 12718.1 1.14096 0.570480 0.821311i \(-0.306757\pi\)
0.570480 + 0.821311i \(0.306757\pi\)
\(500\) 2427.11 0.217087
\(501\) 0 0
\(502\) −33496.0 −2.97809
\(503\) 15675.9 1.38957 0.694785 0.719218i \(-0.255500\pi\)
0.694785 + 0.719218i \(0.255500\pi\)
\(504\) 0 0
\(505\) −29028.0 −2.55788
\(506\) −9544.50 −0.838547
\(507\) 0 0
\(508\) 16527.0 1.44344
\(509\) 10218.3 0.889819 0.444910 0.895575i \(-0.353236\pi\)
0.444910 + 0.895575i \(0.353236\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 4500.87 0.388501
\(513\) 0 0
\(514\) −1448.92 −0.124337
\(515\) −15363.1 −1.31452
\(516\) 0 0
\(517\) −16365.8 −1.39220
\(518\) 0 0
\(519\) 0 0
\(520\) −80328.7 −6.77432
\(521\) 4808.13 0.404315 0.202157 0.979353i \(-0.435205\pi\)
0.202157 + 0.979353i \(0.435205\pi\)
\(522\) 0 0
\(523\) 9936.18 0.830744 0.415372 0.909652i \(-0.363651\pi\)
0.415372 + 0.909652i \(0.363651\pi\)
\(524\) −41497.8 −3.45962
\(525\) 0 0
\(526\) 20114.4 1.66736
\(527\) 1629.06 0.134655
\(528\) 0 0
\(529\) −9110.36 −0.748777
\(530\) −22307.2 −1.82823
\(531\) 0 0
\(532\) 0 0
\(533\) −17316.6 −1.40725
\(534\) 0 0
\(535\) 10836.5 0.875709
\(536\) −35980.5 −2.89948
\(537\) 0 0
\(538\) −46208.4 −3.70294
\(539\) 0 0
\(540\) 0 0
\(541\) −2048.66 −0.162808 −0.0814038 0.996681i \(-0.525940\pi\)
−0.0814038 + 0.996681i \(0.525940\pi\)
\(542\) −28509.7 −2.25940
\(543\) 0 0
\(544\) 17111.0 1.34858
\(545\) −26714.1 −2.09964
\(546\) 0 0
\(547\) 6154.72 0.481091 0.240546 0.970638i \(-0.422674\pi\)
0.240546 + 0.970638i \(0.422674\pi\)
\(548\) −25820.5 −2.01277
\(549\) 0 0
\(550\) 20308.2 1.57444
\(551\) 1930.22 0.149238
\(552\) 0 0
\(553\) 0 0
\(554\) −2384.32 −0.182852
\(555\) 0 0
\(556\) 22600.0 1.72384
\(557\) −4446.75 −0.338267 −0.169134 0.985593i \(-0.554097\pi\)
−0.169134 + 0.985593i \(0.554097\pi\)
\(558\) 0 0
\(559\) 1220.72 0.0923631
\(560\) 0 0
\(561\) 0 0
\(562\) −20338.7 −1.52658
\(563\) −9686.59 −0.725117 −0.362559 0.931961i \(-0.618097\pi\)
−0.362559 + 0.931961i \(0.618097\pi\)
\(564\) 0 0
\(565\) −13672.1 −1.01803
\(566\) 9779.35 0.726248
\(567\) 0 0
\(568\) −58815.7 −4.34481
\(569\) 7306.92 0.538351 0.269176 0.963091i \(-0.413249\pi\)
0.269176 + 0.963091i \(0.413249\pi\)
\(570\) 0 0
\(571\) 9109.34 0.667625 0.333813 0.942639i \(-0.391665\pi\)
0.333813 + 0.942639i \(0.391665\pi\)
\(572\) 49093.6 3.58864
\(573\) 0 0
\(574\) 0 0
\(575\) −6503.72 −0.471694
\(576\) 0 0
\(577\) 18707.7 1.34976 0.674880 0.737927i \(-0.264195\pi\)
0.674880 + 0.737927i \(0.264195\pi\)
\(578\) −21979.4 −1.58170
\(579\) 0 0
\(580\) −5845.67 −0.418497
\(581\) 0 0
\(582\) 0 0
\(583\) 8478.66 0.602316
\(584\) −26474.3 −1.87588
\(585\) 0 0
\(586\) 31404.4 2.21383
\(587\) 24610.4 1.73046 0.865230 0.501375i \(-0.167172\pi\)
0.865230 + 0.501375i \(0.167172\pi\)
\(588\) 0 0
\(589\) 6107.38 0.427250
\(590\) −21380.0 −1.49187
\(591\) 0 0
\(592\) −63432.0 −4.40378
\(593\) −18840.0 −1.30466 −0.652332 0.757933i \(-0.726209\pi\)
−0.652332 + 0.757933i \(0.726209\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −54819.2 −3.76759
\(597\) 0 0
\(598\) −21666.7 −1.48163
\(599\) 20647.6 1.40841 0.704206 0.709996i \(-0.251303\pi\)
0.704206 + 0.709996i \(0.251303\pi\)
\(600\) 0 0
\(601\) 15772.8 1.07053 0.535264 0.844685i \(-0.320212\pi\)
0.535264 + 0.844685i \(0.320212\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 40827.1 2.75038
\(605\) 4811.79 0.323350
\(606\) 0 0
\(607\) −5182.64 −0.346552 −0.173276 0.984873i \(-0.555435\pi\)
−0.173276 + 0.984873i \(0.555435\pi\)
\(608\) 64149.2 4.27894
\(609\) 0 0
\(610\) 6126.39 0.406640
\(611\) −37151.6 −2.45989
\(612\) 0 0
\(613\) −28839.1 −1.90016 −0.950081 0.312004i \(-0.899000\pi\)
−0.950081 + 0.312004i \(0.899000\pi\)
\(614\) −10664.4 −0.700946
\(615\) 0 0
\(616\) 0 0
\(617\) 5114.80 0.333734 0.166867 0.985979i \(-0.446635\pi\)
0.166867 + 0.985979i \(0.446635\pi\)
\(618\) 0 0
\(619\) −29213.9 −1.89694 −0.948471 0.316864i \(-0.897370\pi\)
−0.948471 + 0.316864i \(0.897370\pi\)
\(620\) −18496.1 −1.19810
\(621\) 0 0
\(622\) −6258.87 −0.403469
\(623\) 0 0
\(624\) 0 0
\(625\) −16491.3 −1.05544
\(626\) 24265.4 1.54927
\(627\) 0 0
\(628\) 55557.8 3.53025
\(629\) −8587.18 −0.544345
\(630\) 0 0
\(631\) 19557.5 1.23387 0.616934 0.787015i \(-0.288374\pi\)
0.616934 + 0.787015i \(0.288374\pi\)
\(632\) 73079.0 4.59957
\(633\) 0 0
\(634\) 25258.1 1.58222
\(635\) −12166.8 −0.760356
\(636\) 0 0
\(637\) 0 0
\(638\) 3061.92 0.190004
\(639\) 0 0
\(640\) −49983.9 −3.08717
\(641\) −14632.3 −0.901624 −0.450812 0.892619i \(-0.648865\pi\)
−0.450812 + 0.892619i \(0.648865\pi\)
\(642\) 0 0
\(643\) 23808.1 1.46019 0.730094 0.683347i \(-0.239476\pi\)
0.730094 + 0.683347i \(0.239476\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 17059.3 1.03899
\(647\) 17322.2 1.05256 0.526279 0.850312i \(-0.323587\pi\)
0.526279 + 0.850312i \(0.323587\pi\)
\(648\) 0 0
\(649\) 8126.23 0.491498
\(650\) 46101.1 2.78190
\(651\) 0 0
\(652\) −65596.4 −3.94011
\(653\) 3241.68 0.194268 0.0971338 0.995271i \(-0.469033\pi\)
0.0971338 + 0.995271i \(0.469033\pi\)
\(654\) 0 0
\(655\) 30549.8 1.82241
\(656\) −51164.2 −3.04516
\(657\) 0 0
\(658\) 0 0
\(659\) 16358.2 0.966958 0.483479 0.875356i \(-0.339373\pi\)
0.483479 + 0.875356i \(0.339373\pi\)
\(660\) 0 0
\(661\) −12572.6 −0.739814 −0.369907 0.929069i \(-0.620611\pi\)
−0.369907 + 0.929069i \(0.620611\pi\)
\(662\) −16106.4 −0.945606
\(663\) 0 0
\(664\) −32199.5 −1.88190
\(665\) 0 0
\(666\) 0 0
\(667\) −980.582 −0.0569240
\(668\) −69078.5 −4.00109
\(669\) 0 0
\(670\) 42591.5 2.45590
\(671\) −2328.55 −0.133968
\(672\) 0 0
\(673\) 13130.7 0.752082 0.376041 0.926603i \(-0.377285\pi\)
0.376041 + 0.926603i \(0.377285\pi\)
\(674\) 39573.4 2.26159
\(675\) 0 0
\(676\) 64959.6 3.69592
\(677\) −18624.2 −1.05729 −0.528644 0.848843i \(-0.677299\pi\)
−0.528644 + 0.848843i \(0.677299\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −32130.4 −1.81198
\(681\) 0 0
\(682\) 9688.13 0.543956
\(683\) −25377.0 −1.42170 −0.710852 0.703341i \(-0.751691\pi\)
−0.710852 + 0.703341i \(0.751691\pi\)
\(684\) 0 0
\(685\) 19008.5 1.06026
\(686\) 0 0
\(687\) 0 0
\(688\) 3606.78 0.199865
\(689\) 19247.2 1.06424
\(690\) 0 0
\(691\) −2135.42 −0.117562 −0.0587810 0.998271i \(-0.518721\pi\)
−0.0587810 + 0.998271i \(0.518721\pi\)
\(692\) 43086.2 2.36690
\(693\) 0 0
\(694\) 44869.6 2.45422
\(695\) −16637.6 −0.908058
\(696\) 0 0
\(697\) −6926.41 −0.376408
\(698\) 1805.20 0.0978910
\(699\) 0 0
\(700\) 0 0
\(701\) −9679.27 −0.521513 −0.260757 0.965405i \(-0.583972\pi\)
−0.260757 + 0.965405i \(0.583972\pi\)
\(702\) 0 0
\(703\) −32193.4 −1.72717
\(704\) 46916.8 2.51171
\(705\) 0 0
\(706\) −29473.7 −1.57119
\(707\) 0 0
\(708\) 0 0
\(709\) −25743.2 −1.36362 −0.681809 0.731530i \(-0.738807\pi\)
−0.681809 + 0.731530i \(0.738807\pi\)
\(710\) 69622.3 3.68011
\(711\) 0 0
\(712\) −23595.7 −1.24197
\(713\) −3102.63 −0.162966
\(714\) 0 0
\(715\) −36141.6 −1.89038
\(716\) −75798.9 −3.95634
\(717\) 0 0
\(718\) 38862.6 2.01997
\(719\) −10508.5 −0.545065 −0.272532 0.962147i \(-0.587861\pi\)
−0.272532 + 0.962147i \(0.587861\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 26917.5 1.38749
\(723\) 0 0
\(724\) −34644.6 −1.77839
\(725\) 2086.42 0.106880
\(726\) 0 0
\(727\) 24259.4 1.23759 0.618797 0.785551i \(-0.287620\pi\)
0.618797 + 0.785551i \(0.287620\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 31338.6 1.58890
\(731\) 488.271 0.0247050
\(732\) 0 0
\(733\) −19632.2 −0.989263 −0.494632 0.869103i \(-0.664697\pi\)
−0.494632 + 0.869103i \(0.664697\pi\)
\(734\) −34151.3 −1.71737
\(735\) 0 0
\(736\) −32588.7 −1.63212
\(737\) −16188.4 −0.809101
\(738\) 0 0
\(739\) −26353.6 −1.31182 −0.655909 0.754840i \(-0.727714\pi\)
−0.655909 + 0.754840i \(0.727714\pi\)
\(740\) 97497.6 4.84335
\(741\) 0 0
\(742\) 0 0
\(743\) −31464.6 −1.55360 −0.776799 0.629749i \(-0.783158\pi\)
−0.776799 + 0.629749i \(0.783158\pi\)
\(744\) 0 0
\(745\) 40356.7 1.98464
\(746\) −59030.4 −2.89713
\(747\) 0 0
\(748\) 19636.7 0.959880
\(749\) 0 0
\(750\) 0 0
\(751\) 5411.52 0.262942 0.131471 0.991320i \(-0.458030\pi\)
0.131471 + 0.991320i \(0.458030\pi\)
\(752\) −109769. −5.32296
\(753\) 0 0
\(754\) 6950.77 0.335719
\(755\) −30056.0 −1.44881
\(756\) 0 0
\(757\) 3607.94 0.173227 0.0866135 0.996242i \(-0.472395\pi\)
0.0866135 + 0.996242i \(0.472395\pi\)
\(758\) 32529.1 1.55872
\(759\) 0 0
\(760\) −120457. −5.74926
\(761\) 4663.70 0.222154 0.111077 0.993812i \(-0.464570\pi\)
0.111077 + 0.993812i \(0.464570\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 59769.5 2.83035
\(765\) 0 0
\(766\) 36979.1 1.74427
\(767\) 18447.1 0.868431
\(768\) 0 0
\(769\) 9725.21 0.456047 0.228023 0.973656i \(-0.426774\pi\)
0.228023 + 0.973656i \(0.426774\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −35071.7 −1.63505
\(773\) −3092.80 −0.143907 −0.0719536 0.997408i \(-0.522923\pi\)
−0.0719536 + 0.997408i \(0.522923\pi\)
\(774\) 0 0
\(775\) 6601.60 0.305982
\(776\) −82748.8 −3.82798
\(777\) 0 0
\(778\) 27873.7 1.28447
\(779\) −25967.2 −1.19431
\(780\) 0 0
\(781\) −26462.4 −1.21242
\(782\) −8666.37 −0.396303
\(783\) 0 0
\(784\) 0 0
\(785\) −40900.4 −1.85962
\(786\) 0 0
\(787\) −22812.9 −1.03328 −0.516640 0.856203i \(-0.672817\pi\)
−0.516640 + 0.856203i \(0.672817\pi\)
\(788\) 40006.9 1.80861
\(789\) 0 0
\(790\) −86506.3 −3.89589
\(791\) 0 0
\(792\) 0 0
\(793\) −5285.98 −0.236710
\(794\) −1861.71 −0.0832113
\(795\) 0 0
\(796\) −29469.1 −1.31219
\(797\) −34305.4 −1.52467 −0.762334 0.647184i \(-0.775947\pi\)
−0.762334 + 0.647184i \(0.775947\pi\)
\(798\) 0 0
\(799\) −14860.1 −0.657963
\(800\) 69340.3 3.06444
\(801\) 0 0
\(802\) 27533.3 1.21226
\(803\) −11911.4 −0.523465
\(804\) 0 0
\(805\) 0 0
\(806\) 21992.8 0.961119
\(807\) 0 0
\(808\) 132418. 5.76542
\(809\) 3264.80 0.141884 0.0709421 0.997480i \(-0.477399\pi\)
0.0709421 + 0.997480i \(0.477399\pi\)
\(810\) 0 0
\(811\) −27264.0 −1.18048 −0.590239 0.807228i \(-0.700967\pi\)
−0.590239 + 0.807228i \(0.700967\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −51068.4 −2.19895
\(815\) 48290.6 2.07552
\(816\) 0 0
\(817\) 1830.53 0.0783871
\(818\) 1744.84 0.0745805
\(819\) 0 0
\(820\) 78641.4 3.34912
\(821\) −36334.2 −1.54454 −0.772272 0.635292i \(-0.780880\pi\)
−0.772272 + 0.635292i \(0.780880\pi\)
\(822\) 0 0
\(823\) 7791.80 0.330019 0.165009 0.986292i \(-0.447235\pi\)
0.165009 + 0.986292i \(0.447235\pi\)
\(824\) 70082.4 2.96291
\(825\) 0 0
\(826\) 0 0
\(827\) 36082.7 1.51719 0.758596 0.651561i \(-0.225886\pi\)
0.758596 + 0.651561i \(0.225886\pi\)
\(828\) 0 0
\(829\) 42995.0 1.80130 0.900650 0.434545i \(-0.143091\pi\)
0.900650 + 0.434545i \(0.143091\pi\)
\(830\) 38115.7 1.59399
\(831\) 0 0
\(832\) 106505. 4.43796
\(833\) 0 0
\(834\) 0 0
\(835\) 50854.0 2.10764
\(836\) 73618.3 3.04562
\(837\) 0 0
\(838\) −23845.9 −0.982986
\(839\) 28252.8 1.16257 0.581283 0.813701i \(-0.302551\pi\)
0.581283 + 0.813701i \(0.302551\pi\)
\(840\) 0 0
\(841\) −24074.4 −0.987102
\(842\) 7448.49 0.304860
\(843\) 0 0
\(844\) 70132.0 2.86024
\(845\) −47821.8 −1.94689
\(846\) 0 0
\(847\) 0 0
\(848\) 56868.2 2.30290
\(849\) 0 0
\(850\) 18439.8 0.744093
\(851\) 16354.7 0.658793
\(852\) 0 0
\(853\) −28994.8 −1.16385 −0.581924 0.813243i \(-0.697700\pi\)
−0.581924 + 0.813243i \(0.697700\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −49433.5 −1.97384
\(857\) −8632.87 −0.344099 −0.172050 0.985088i \(-0.555039\pi\)
−0.172050 + 0.985088i \(0.555039\pi\)
\(858\) 0 0
\(859\) −33947.1 −1.34838 −0.674191 0.738557i \(-0.735507\pi\)
−0.674191 + 0.738557i \(0.735507\pi\)
\(860\) −5543.76 −0.219815
\(861\) 0 0
\(862\) 8941.09 0.353288
\(863\) −70.6612 −0.00278718 −0.00139359 0.999999i \(-0.500444\pi\)
−0.00139359 + 0.999999i \(0.500444\pi\)
\(864\) 0 0
\(865\) −31719.1 −1.24680
\(866\) −46507.5 −1.82493
\(867\) 0 0
\(868\) 0 0
\(869\) 32879.8 1.28351
\(870\) 0 0
\(871\) −36748.8 −1.42961
\(872\) 121863. 4.73257
\(873\) 0 0
\(874\) −32490.3 −1.25744
\(875\) 0 0
\(876\) 0 0
\(877\) −11175.3 −0.430289 −0.215144 0.976582i \(-0.569022\pi\)
−0.215144 + 0.976582i \(0.569022\pi\)
\(878\) 32266.5 1.24025
\(879\) 0 0
\(880\) −106785. −4.09059
\(881\) 14341.3 0.548433 0.274216 0.961668i \(-0.411582\pi\)
0.274216 + 0.961668i \(0.411582\pi\)
\(882\) 0 0
\(883\) −23559.2 −0.897884 −0.448942 0.893561i \(-0.648199\pi\)
−0.448942 + 0.893561i \(0.648199\pi\)
\(884\) 44576.8 1.69602
\(885\) 0 0
\(886\) 4355.20 0.165142
\(887\) 31744.4 1.20166 0.600831 0.799376i \(-0.294837\pi\)
0.600831 + 0.799376i \(0.294837\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 27931.1 1.05197
\(891\) 0 0
\(892\) −117993. −4.42904
\(893\) −55710.7 −2.08767
\(894\) 0 0
\(895\) 55801.4 2.08406
\(896\) 0 0
\(897\) 0 0
\(898\) 33765.8 1.25477
\(899\) 995.339 0.0369259
\(900\) 0 0
\(901\) 7698.59 0.284658
\(902\) −41191.8 −1.52055
\(903\) 0 0
\(904\) 62368.5 2.29463
\(905\) 25504.6 0.936796
\(906\) 0 0
\(907\) 6062.67 0.221949 0.110974 0.993823i \(-0.464603\pi\)
0.110974 + 0.993823i \(0.464603\pi\)
\(908\) −30636.6 −1.11973
\(909\) 0 0
\(910\) 0 0
\(911\) −25862.9 −0.940589 −0.470295 0.882509i \(-0.655852\pi\)
−0.470295 + 0.882509i \(0.655852\pi\)
\(912\) 0 0
\(913\) −14487.2 −0.525144
\(914\) 865.539 0.0313233
\(915\) 0 0
\(916\) 34225.7 1.23455
\(917\) 0 0
\(918\) 0 0
\(919\) 1455.14 0.0522314 0.0261157 0.999659i \(-0.491686\pi\)
0.0261157 + 0.999659i \(0.491686\pi\)
\(920\) 61193.9 2.19294
\(921\) 0 0
\(922\) 13007.3 0.464612
\(923\) −60071.6 −2.14223
\(924\) 0 0
\(925\) −34798.6 −1.23694
\(926\) −5901.48 −0.209433
\(927\) 0 0
\(928\) 10454.6 0.369816
\(929\) −3077.74 −0.108695 −0.0543474 0.998522i \(-0.517308\pi\)
−0.0543474 + 0.998522i \(0.517308\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −128903. −4.53042
\(933\) 0 0
\(934\) 81294.8 2.84802
\(935\) −14456.1 −0.505632
\(936\) 0 0
\(937\) −5354.80 −0.186695 −0.0933477 0.995634i \(-0.529757\pi\)
−0.0933477 + 0.995634i \(0.529757\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 168719. 5.85428
\(941\) 47797.8 1.65586 0.827930 0.560832i \(-0.189518\pi\)
0.827930 + 0.560832i \(0.189518\pi\)
\(942\) 0 0
\(943\) 13191.7 0.455547
\(944\) 54504.4 1.87920
\(945\) 0 0
\(946\) 2903.78 0.0997990
\(947\) 2491.69 0.0855006 0.0427503 0.999086i \(-0.486388\pi\)
0.0427503 + 0.999086i \(0.486388\pi\)
\(948\) 0 0
\(949\) −27039.6 −0.924914
\(950\) 69130.9 2.36095
\(951\) 0 0
\(952\) 0 0
\(953\) 13130.4 0.446313 0.223156 0.974783i \(-0.428364\pi\)
0.223156 + 0.974783i \(0.428364\pi\)
\(954\) 0 0
\(955\) −44001.0 −1.49093
\(956\) −33767.7 −1.14239
\(957\) 0 0
\(958\) 56459.3 1.90409
\(959\) 0 0
\(960\) 0 0
\(961\) −26641.7 −0.894286
\(962\) −115929. −3.88534
\(963\) 0 0
\(964\) 46313.4 1.54736
\(965\) 25819.0 0.861287
\(966\) 0 0
\(967\) 43314.8 1.44044 0.720222 0.693743i \(-0.244040\pi\)
0.720222 + 0.693743i \(0.244040\pi\)
\(968\) −21950.2 −0.728827
\(969\) 0 0
\(970\) 97952.8 3.24235
\(971\) 19755.8 0.652929 0.326464 0.945210i \(-0.394143\pi\)
0.326464 + 0.945210i \(0.394143\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −68674.1 −2.25920
\(975\) 0 0
\(976\) −15618.1 −0.512217
\(977\) 11140.1 0.364794 0.182397 0.983225i \(-0.441614\pi\)
0.182397 + 0.983225i \(0.441614\pi\)
\(978\) 0 0
\(979\) −10616.2 −0.346573
\(980\) 0 0
\(981\) 0 0
\(982\) 113307. 3.68205
\(983\) 3575.76 0.116021 0.0580107 0.998316i \(-0.481524\pi\)
0.0580107 + 0.998316i \(0.481524\pi\)
\(984\) 0 0
\(985\) −29452.2 −0.952716
\(986\) 2780.21 0.0897971
\(987\) 0 0
\(988\) 167119. 5.38133
\(989\) −929.937 −0.0298992
\(990\) 0 0
\(991\) 22391.8 0.717760 0.358880 0.933384i \(-0.383159\pi\)
0.358880 + 0.933384i \(0.383159\pi\)
\(992\) 33079.2 1.05873
\(993\) 0 0
\(994\) 0 0
\(995\) 21694.5 0.691218
\(996\) 0 0
\(997\) −17466.5 −0.554834 −0.277417 0.960750i \(-0.589478\pi\)
−0.277417 + 0.960750i \(0.589478\pi\)
\(998\) 68676.4 2.17827
\(999\) 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.x.1.7 yes 8
3.2 odd 2 inner 441.4.a.x.1.2 yes 8
7.2 even 3 441.4.e.z.361.2 16
7.3 odd 6 441.4.e.z.226.1 16
7.4 even 3 441.4.e.z.226.2 16
7.5 odd 6 441.4.e.z.361.1 16
7.6 odd 2 inner 441.4.a.x.1.8 yes 8
21.2 odd 6 441.4.e.z.361.7 16
21.5 even 6 441.4.e.z.361.8 16
21.11 odd 6 441.4.e.z.226.7 16
21.17 even 6 441.4.e.z.226.8 16
21.20 even 2 inner 441.4.a.x.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
441.4.a.x.1.1 8 21.20 even 2 inner
441.4.a.x.1.2 yes 8 3.2 odd 2 inner
441.4.a.x.1.7 yes 8 1.1 even 1 trivial
441.4.a.x.1.8 yes 8 7.6 odd 2 inner
441.4.e.z.226.1 16 7.3 odd 6
441.4.e.z.226.2 16 7.4 even 3
441.4.e.z.226.7 16 21.11 odd 6
441.4.e.z.226.8 16 21.17 even 6
441.4.e.z.361.1 16 7.5 odd 6
441.4.e.z.361.2 16 7.2 even 3
441.4.e.z.361.7 16 21.2 odd 6
441.4.e.z.361.8 16 21.5 even 6