Properties

Label 441.4.a.e.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -4.00000 q^{4} +7.00000 q^{5} +24.0000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} -4.00000 q^{4} +7.00000 q^{5} +24.0000 q^{8} -14.0000 q^{10} +5.00000 q^{11} +14.0000 q^{13} -16.0000 q^{16} -21.0000 q^{17} -49.0000 q^{19} -28.0000 q^{20} -10.0000 q^{22} +159.000 q^{23} -76.0000 q^{25} -28.0000 q^{26} -58.0000 q^{29} -147.000 q^{31} -160.000 q^{32} +42.0000 q^{34} +219.000 q^{37} +98.0000 q^{38} +168.000 q^{40} +350.000 q^{41} -124.000 q^{43} -20.0000 q^{44} -318.000 q^{46} +525.000 q^{47} +152.000 q^{50} -56.0000 q^{52} -303.000 q^{53} +35.0000 q^{55} +116.000 q^{58} -105.000 q^{59} +413.000 q^{61} +294.000 q^{62} +448.000 q^{64} +98.0000 q^{65} +415.000 q^{67} +84.0000 q^{68} +432.000 q^{71} +1113.00 q^{73} -438.000 q^{74} +196.000 q^{76} -103.000 q^{79} -112.000 q^{80} -700.000 q^{82} +1092.00 q^{83} -147.000 q^{85} +248.000 q^{86} +120.000 q^{88} -329.000 q^{89} -636.000 q^{92} -1050.00 q^{94} -343.000 q^{95} +882.000 q^{97} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −4.00000 −0.500000
\(5\) 7.00000 0.626099 0.313050 0.949737i \(-0.398649\pi\)
0.313050 + 0.949737i \(0.398649\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 24.0000 1.06066
\(9\) 0 0
\(10\) −14.0000 −0.442719
\(11\) 5.00000 0.137051 0.0685253 0.997649i \(-0.478171\pi\)
0.0685253 + 0.997649i \(0.478171\pi\)
\(12\) 0 0
\(13\) 14.0000 0.298685 0.149342 0.988786i \(-0.452284\pi\)
0.149342 + 0.988786i \(0.452284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −16.0000 −0.250000
\(17\) −21.0000 −0.299603 −0.149801 0.988716i \(-0.547863\pi\)
−0.149801 + 0.988716i \(0.547863\pi\)
\(18\) 0 0
\(19\) −49.0000 −0.591651 −0.295826 0.955242i \(-0.595595\pi\)
−0.295826 + 0.955242i \(0.595595\pi\)
\(20\) −28.0000 −0.313050
\(21\) 0 0
\(22\) −10.0000 −0.0969094
\(23\) 159.000 1.44147 0.720735 0.693211i \(-0.243805\pi\)
0.720735 + 0.693211i \(0.243805\pi\)
\(24\) 0 0
\(25\) −76.0000 −0.608000
\(26\) −28.0000 −0.211202
\(27\) 0 0
\(28\) 0 0
\(29\) −58.0000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −147.000 −0.851677 −0.425838 0.904799i \(-0.640021\pi\)
−0.425838 + 0.904799i \(0.640021\pi\)
\(32\) −160.000 −0.883883
\(33\) 0 0
\(34\) 42.0000 0.211851
\(35\) 0 0
\(36\) 0 0
\(37\) 219.000 0.973064 0.486532 0.873663i \(-0.338262\pi\)
0.486532 + 0.873663i \(0.338262\pi\)
\(38\) 98.0000 0.418361
\(39\) 0 0
\(40\) 168.000 0.664078
\(41\) 350.000 1.33319 0.666595 0.745420i \(-0.267751\pi\)
0.666595 + 0.745420i \(0.267751\pi\)
\(42\) 0 0
\(43\) −124.000 −0.439763 −0.219882 0.975527i \(-0.570567\pi\)
−0.219882 + 0.975527i \(0.570567\pi\)
\(44\) −20.0000 −0.0685253
\(45\) 0 0
\(46\) −318.000 −1.01927
\(47\) 525.000 1.62934 0.814671 0.579923i \(-0.196917\pi\)
0.814671 + 0.579923i \(0.196917\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 152.000 0.429921
\(51\) 0 0
\(52\) −56.0000 −0.149342
\(53\) −303.000 −0.785288 −0.392644 0.919691i \(-0.628439\pi\)
−0.392644 + 0.919691i \(0.628439\pi\)
\(54\) 0 0
\(55\) 35.0000 0.0858073
\(56\) 0 0
\(57\) 0 0
\(58\) 116.000 0.262613
\(59\) −105.000 −0.231692 −0.115846 0.993267i \(-0.536958\pi\)
−0.115846 + 0.993267i \(0.536958\pi\)
\(60\) 0 0
\(61\) 413.000 0.866873 0.433436 0.901184i \(-0.357301\pi\)
0.433436 + 0.901184i \(0.357301\pi\)
\(62\) 294.000 0.602226
\(63\) 0 0
\(64\) 448.000 0.875000
\(65\) 98.0000 0.187006
\(66\) 0 0
\(67\) 415.000 0.756721 0.378361 0.925658i \(-0.376488\pi\)
0.378361 + 0.925658i \(0.376488\pi\)
\(68\) 84.0000 0.149801
\(69\) 0 0
\(70\) 0 0
\(71\) 432.000 0.722098 0.361049 0.932547i \(-0.382419\pi\)
0.361049 + 0.932547i \(0.382419\pi\)
\(72\) 0 0
\(73\) 1113.00 1.78448 0.892238 0.451565i \(-0.149134\pi\)
0.892238 + 0.451565i \(0.149134\pi\)
\(74\) −438.000 −0.688060
\(75\) 0 0
\(76\) 196.000 0.295826
\(77\) 0 0
\(78\) 0 0
\(79\) −103.000 −0.146689 −0.0733443 0.997307i \(-0.523367\pi\)
−0.0733443 + 0.997307i \(0.523367\pi\)
\(80\) −112.000 −0.156525
\(81\) 0 0
\(82\) −700.000 −0.942708
\(83\) 1092.00 1.44413 0.722064 0.691827i \(-0.243194\pi\)
0.722064 + 0.691827i \(0.243194\pi\)
\(84\) 0 0
\(85\) −147.000 −0.187581
\(86\) 248.000 0.310960
\(87\) 0 0
\(88\) 120.000 0.145364
\(89\) −329.000 −0.391842 −0.195921 0.980620i \(-0.562770\pi\)
−0.195921 + 0.980620i \(0.562770\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −636.000 −0.720735
\(93\) 0 0
\(94\) −1050.00 −1.15212
\(95\) −343.000 −0.370432
\(96\) 0 0
\(97\) 882.000 0.923232 0.461616 0.887080i \(-0.347270\pi\)
0.461616 + 0.887080i \(0.347270\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 304.000 0.304000
\(101\) 1379.00 1.35857 0.679285 0.733874i \(-0.262290\pi\)
0.679285 + 0.733874i \(0.262290\pi\)
\(102\) 0 0
\(103\) 679.000 0.649552 0.324776 0.945791i \(-0.394711\pi\)
0.324776 + 0.945791i \(0.394711\pi\)
\(104\) 336.000 0.316803
\(105\) 0 0
\(106\) 606.000 0.555282
\(107\) −457.000 −0.412896 −0.206448 0.978458i \(-0.566190\pi\)
−0.206448 + 0.978458i \(0.566190\pi\)
\(108\) 0 0
\(109\) −1125.00 −0.988582 −0.494291 0.869296i \(-0.664572\pi\)
−0.494291 + 0.869296i \(0.664572\pi\)
\(110\) −70.0000 −0.0606749
\(111\) 0 0
\(112\) 0 0
\(113\) 1538.00 1.28038 0.640190 0.768217i \(-0.278856\pi\)
0.640190 + 0.768217i \(0.278856\pi\)
\(114\) 0 0
\(115\) 1113.00 0.902502
\(116\) 232.000 0.185695
\(117\) 0 0
\(118\) 210.000 0.163831
\(119\) 0 0
\(120\) 0 0
\(121\) −1306.00 −0.981217
\(122\) −826.000 −0.612972
\(123\) 0 0
\(124\) 588.000 0.425838
\(125\) −1407.00 −1.00677
\(126\) 0 0
\(127\) 72.0000 0.0503068 0.0251534 0.999684i \(-0.491993\pi\)
0.0251534 + 0.999684i \(0.491993\pi\)
\(128\) 384.000 0.265165
\(129\) 0 0
\(130\) −196.000 −0.132233
\(131\) 2149.00 1.43327 0.716637 0.697446i \(-0.245680\pi\)
0.716637 + 0.697446i \(0.245680\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −830.000 −0.535083
\(135\) 0 0
\(136\) −504.000 −0.317777
\(137\) 1125.00 0.701571 0.350786 0.936456i \(-0.385915\pi\)
0.350786 + 0.936456i \(0.385915\pi\)
\(138\) 0 0
\(139\) −252.000 −0.153772 −0.0768862 0.997040i \(-0.524498\pi\)
−0.0768862 + 0.997040i \(0.524498\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −864.000 −0.510600
\(143\) 70.0000 0.0409349
\(144\) 0 0
\(145\) −406.000 −0.232527
\(146\) −2226.00 −1.26182
\(147\) 0 0
\(148\) −876.000 −0.486532
\(149\) 201.000 0.110514 0.0552569 0.998472i \(-0.482402\pi\)
0.0552569 + 0.998472i \(0.482402\pi\)
\(150\) 0 0
\(151\) 1619.00 0.872532 0.436266 0.899818i \(-0.356301\pi\)
0.436266 + 0.899818i \(0.356301\pi\)
\(152\) −1176.00 −0.627541
\(153\) 0 0
\(154\) 0 0
\(155\) −1029.00 −0.533234
\(156\) 0 0
\(157\) −679.000 −0.345160 −0.172580 0.984996i \(-0.555210\pi\)
−0.172580 + 0.984996i \(0.555210\pi\)
\(158\) 206.000 0.103725
\(159\) 0 0
\(160\) −1120.00 −0.553399
\(161\) 0 0
\(162\) 0 0
\(163\) −467.000 −0.224407 −0.112203 0.993685i \(-0.535791\pi\)
−0.112203 + 0.993685i \(0.535791\pi\)
\(164\) −1400.00 −0.666595
\(165\) 0 0
\(166\) −2184.00 −1.02115
\(167\) 1204.00 0.557894 0.278947 0.960306i \(-0.410015\pi\)
0.278947 + 0.960306i \(0.410015\pi\)
\(168\) 0 0
\(169\) −2001.00 −0.910787
\(170\) 294.000 0.132640
\(171\) 0 0
\(172\) 496.000 0.219882
\(173\) −2821.00 −1.23975 −0.619875 0.784701i \(-0.712817\pi\)
−0.619875 + 0.784701i \(0.712817\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −80.0000 −0.0342627
\(177\) 0 0
\(178\) 658.000 0.277074
\(179\) 3253.00 1.35833 0.679164 0.733987i \(-0.262343\pi\)
0.679164 + 0.733987i \(0.262343\pi\)
\(180\) 0 0
\(181\) −1582.00 −0.649664 −0.324832 0.945772i \(-0.605308\pi\)
−0.324832 + 0.945772i \(0.605308\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 3816.00 1.52891
\(185\) 1533.00 0.609235
\(186\) 0 0
\(187\) −105.000 −0.0410608
\(188\) −2100.00 −0.814671
\(189\) 0 0
\(190\) 686.000 0.261935
\(191\) −2557.00 −0.968681 −0.484340 0.874880i \(-0.660940\pi\)
−0.484340 + 0.874880i \(0.660940\pi\)
\(192\) 0 0
\(193\) −397.000 −0.148066 −0.0740329 0.997256i \(-0.523587\pi\)
−0.0740329 + 0.997256i \(0.523587\pi\)
\(194\) −1764.00 −0.652824
\(195\) 0 0
\(196\) 0 0
\(197\) −2914.00 −1.05388 −0.526939 0.849903i \(-0.676660\pi\)
−0.526939 + 0.849903i \(0.676660\pi\)
\(198\) 0 0
\(199\) −3339.00 −1.18942 −0.594712 0.803939i \(-0.702734\pi\)
−0.594712 + 0.803939i \(0.702734\pi\)
\(200\) −1824.00 −0.644881
\(201\) 0 0
\(202\) −2758.00 −0.960654
\(203\) 0 0
\(204\) 0 0
\(205\) 2450.00 0.834709
\(206\) −1358.00 −0.459303
\(207\) 0 0
\(208\) −224.000 −0.0746712
\(209\) −245.000 −0.0810861
\(210\) 0 0
\(211\) 1780.00 0.580759 0.290380 0.956911i \(-0.406218\pi\)
0.290380 + 0.956911i \(0.406218\pi\)
\(212\) 1212.00 0.392644
\(213\) 0 0
\(214\) 914.000 0.291961
\(215\) −868.000 −0.275335
\(216\) 0 0
\(217\) 0 0
\(218\) 2250.00 0.699033
\(219\) 0 0
\(220\) −140.000 −0.0429036
\(221\) −294.000 −0.0894868
\(222\) 0 0
\(223\) 1400.00 0.420408 0.210204 0.977658i \(-0.432587\pi\)
0.210204 + 0.977658i \(0.432587\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −3076.00 −0.905365
\(227\) −2205.00 −0.644718 −0.322359 0.946617i \(-0.604476\pi\)
−0.322359 + 0.946617i \(0.604476\pi\)
\(228\) 0 0
\(229\) −287.000 −0.0828188 −0.0414094 0.999142i \(-0.513185\pi\)
−0.0414094 + 0.999142i \(0.513185\pi\)
\(230\) −2226.00 −0.638166
\(231\) 0 0
\(232\) −1392.00 −0.393919
\(233\) −4587.00 −1.28972 −0.644859 0.764301i \(-0.723084\pi\)
−0.644859 + 0.764301i \(0.723084\pi\)
\(234\) 0 0
\(235\) 3675.00 1.02013
\(236\) 420.000 0.115846
\(237\) 0 0
\(238\) 0 0
\(239\) −1668.00 −0.451439 −0.225720 0.974192i \(-0.572473\pi\)
−0.225720 + 0.974192i \(0.572473\pi\)
\(240\) 0 0
\(241\) 3409.00 0.911174 0.455587 0.890191i \(-0.349429\pi\)
0.455587 + 0.890191i \(0.349429\pi\)
\(242\) 2612.00 0.693825
\(243\) 0 0
\(244\) −1652.00 −0.433436
\(245\) 0 0
\(246\) 0 0
\(247\) −686.000 −0.176717
\(248\) −3528.00 −0.903340
\(249\) 0 0
\(250\) 2814.00 0.711892
\(251\) −4760.00 −1.19701 −0.598503 0.801121i \(-0.704238\pi\)
−0.598503 + 0.801121i \(0.704238\pi\)
\(252\) 0 0
\(253\) 795.000 0.197554
\(254\) −144.000 −0.0355723
\(255\) 0 0
\(256\) −4352.00 −1.06250
\(257\) −805.000 −0.195387 −0.0976936 0.995217i \(-0.531147\pi\)
−0.0976936 + 0.995217i \(0.531147\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −392.000 −0.0935031
\(261\) 0 0
\(262\) −4298.00 −1.01348
\(263\) 257.000 0.0602559 0.0301279 0.999546i \(-0.490409\pi\)
0.0301279 + 0.999546i \(0.490409\pi\)
\(264\) 0 0
\(265\) −2121.00 −0.491668
\(266\) 0 0
\(267\) 0 0
\(268\) −1660.00 −0.378361
\(269\) 3591.00 0.813930 0.406965 0.913444i \(-0.366587\pi\)
0.406965 + 0.913444i \(0.366587\pi\)
\(270\) 0 0
\(271\) −1393.00 −0.312246 −0.156123 0.987738i \(-0.549900\pi\)
−0.156123 + 0.987738i \(0.549900\pi\)
\(272\) 336.000 0.0749007
\(273\) 0 0
\(274\) −2250.00 −0.496086
\(275\) −380.000 −0.0833268
\(276\) 0 0
\(277\) 415.000 0.0900178 0.0450089 0.998987i \(-0.485668\pi\)
0.0450089 + 0.998987i \(0.485668\pi\)
\(278\) 504.000 0.108733
\(279\) 0 0
\(280\) 0 0
\(281\) 4954.00 1.05171 0.525856 0.850574i \(-0.323745\pi\)
0.525856 + 0.850574i \(0.323745\pi\)
\(282\) 0 0
\(283\) 4277.00 0.898379 0.449190 0.893437i \(-0.351713\pi\)
0.449190 + 0.893437i \(0.351713\pi\)
\(284\) −1728.00 −0.361049
\(285\) 0 0
\(286\) −140.000 −0.0289454
\(287\) 0 0
\(288\) 0 0
\(289\) −4472.00 −0.910238
\(290\) 812.000 0.164422
\(291\) 0 0
\(292\) −4452.00 −0.892238
\(293\) 7742.00 1.54366 0.771830 0.635829i \(-0.219342\pi\)
0.771830 + 0.635829i \(0.219342\pi\)
\(294\) 0 0
\(295\) −735.000 −0.145062
\(296\) 5256.00 1.03209
\(297\) 0 0
\(298\) −402.000 −0.0781451
\(299\) 2226.00 0.430545
\(300\) 0 0
\(301\) 0 0
\(302\) −3238.00 −0.616973
\(303\) 0 0
\(304\) 784.000 0.147913
\(305\) 2891.00 0.542748
\(306\) 0 0
\(307\) 7364.00 1.36901 0.684504 0.729009i \(-0.260019\pi\)
0.684504 + 0.729009i \(0.260019\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2058.00 0.377053
\(311\) 9975.00 1.81875 0.909374 0.415980i \(-0.136562\pi\)
0.909374 + 0.415980i \(0.136562\pi\)
\(312\) 0 0
\(313\) 4753.00 0.858324 0.429162 0.903228i \(-0.358809\pi\)
0.429162 + 0.903228i \(0.358809\pi\)
\(314\) 1358.00 0.244065
\(315\) 0 0
\(316\) 412.000 0.0733443
\(317\) 3477.00 0.616050 0.308025 0.951378i \(-0.400332\pi\)
0.308025 + 0.951378i \(0.400332\pi\)
\(318\) 0 0
\(319\) −290.000 −0.0508993
\(320\) 3136.00 0.547837
\(321\) 0 0
\(322\) 0 0
\(323\) 1029.00 0.177260
\(324\) 0 0
\(325\) −1064.00 −0.181600
\(326\) 934.000 0.158679
\(327\) 0 0
\(328\) 8400.00 1.41406
\(329\) 0 0
\(330\) 0 0
\(331\) 3341.00 0.554797 0.277399 0.960755i \(-0.410528\pi\)
0.277399 + 0.960755i \(0.410528\pi\)
\(332\) −4368.00 −0.722064
\(333\) 0 0
\(334\) −2408.00 −0.394491
\(335\) 2905.00 0.473782
\(336\) 0 0
\(337\) 7366.00 1.19066 0.595329 0.803482i \(-0.297022\pi\)
0.595329 + 0.803482i \(0.297022\pi\)
\(338\) 4002.00 0.644024
\(339\) 0 0
\(340\) 588.000 0.0937905
\(341\) −735.000 −0.116723
\(342\) 0 0
\(343\) 0 0
\(344\) −2976.00 −0.466439
\(345\) 0 0
\(346\) 5642.00 0.876635
\(347\) −7415.00 −1.14714 −0.573571 0.819156i \(-0.694442\pi\)
−0.573571 + 0.819156i \(0.694442\pi\)
\(348\) 0 0
\(349\) 3878.00 0.594798 0.297399 0.954753i \(-0.403881\pi\)
0.297399 + 0.954753i \(0.403881\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −800.000 −0.121137
\(353\) 1267.00 0.191036 0.0955179 0.995428i \(-0.469549\pi\)
0.0955179 + 0.995428i \(0.469549\pi\)
\(354\) 0 0
\(355\) 3024.00 0.452105
\(356\) 1316.00 0.195921
\(357\) 0 0
\(358\) −6506.00 −0.960483
\(359\) −4685.00 −0.688760 −0.344380 0.938830i \(-0.611911\pi\)
−0.344380 + 0.938830i \(0.611911\pi\)
\(360\) 0 0
\(361\) −4458.00 −0.649949
\(362\) 3164.00 0.459382
\(363\) 0 0
\(364\) 0 0
\(365\) 7791.00 1.11726
\(366\) 0 0
\(367\) 4641.00 0.660104 0.330052 0.943963i \(-0.392934\pi\)
0.330052 + 0.943963i \(0.392934\pi\)
\(368\) −2544.00 −0.360367
\(369\) 0 0
\(370\) −3066.00 −0.430794
\(371\) 0 0
\(372\) 0 0
\(373\) −8797.00 −1.22116 −0.610578 0.791956i \(-0.709063\pi\)
−0.610578 + 0.791956i \(0.709063\pi\)
\(374\) 210.000 0.0290343
\(375\) 0 0
\(376\) 12600.0 1.72818
\(377\) −812.000 −0.110929
\(378\) 0 0
\(379\) 13680.0 1.85407 0.927037 0.374969i \(-0.122347\pi\)
0.927037 + 0.374969i \(0.122347\pi\)
\(380\) 1372.00 0.185216
\(381\) 0 0
\(382\) 5114.00 0.684961
\(383\) 9765.00 1.30279 0.651395 0.758739i \(-0.274184\pi\)
0.651395 + 0.758739i \(0.274184\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 794.000 0.104698
\(387\) 0 0
\(388\) −3528.00 −0.461616
\(389\) −1731.00 −0.225617 −0.112809 0.993617i \(-0.535985\pi\)
−0.112809 + 0.993617i \(0.535985\pi\)
\(390\) 0 0
\(391\) −3339.00 −0.431868
\(392\) 0 0
\(393\) 0 0
\(394\) 5828.00 0.745204
\(395\) −721.000 −0.0918416
\(396\) 0 0
\(397\) −10983.0 −1.38847 −0.694233 0.719750i \(-0.744256\pi\)
−0.694233 + 0.719750i \(0.744256\pi\)
\(398\) 6678.00 0.841050
\(399\) 0 0
\(400\) 1216.00 0.152000
\(401\) −6603.00 −0.822289 −0.411145 0.911570i \(-0.634871\pi\)
−0.411145 + 0.911570i \(0.634871\pi\)
\(402\) 0 0
\(403\) −2058.00 −0.254383
\(404\) −5516.00 −0.679285
\(405\) 0 0
\(406\) 0 0
\(407\) 1095.00 0.133359
\(408\) 0 0
\(409\) −10955.0 −1.32443 −0.662213 0.749316i \(-0.730382\pi\)
−0.662213 + 0.749316i \(0.730382\pi\)
\(410\) −4900.00 −0.590229
\(411\) 0 0
\(412\) −2716.00 −0.324776
\(413\) 0 0
\(414\) 0 0
\(415\) 7644.00 0.904167
\(416\) −2240.00 −0.264002
\(417\) 0 0
\(418\) 490.000 0.0573366
\(419\) 6636.00 0.773723 0.386861 0.922138i \(-0.373559\pi\)
0.386861 + 0.922138i \(0.373559\pi\)
\(420\) 0 0
\(421\) −16630.0 −1.92517 −0.962585 0.270980i \(-0.912652\pi\)
−0.962585 + 0.270980i \(0.912652\pi\)
\(422\) −3560.00 −0.410659
\(423\) 0 0
\(424\) −7272.00 −0.832923
\(425\) 1596.00 0.182159
\(426\) 0 0
\(427\) 0 0
\(428\) 1828.00 0.206448
\(429\) 0 0
\(430\) 1736.00 0.194692
\(431\) −4923.00 −0.550192 −0.275096 0.961417i \(-0.588710\pi\)
−0.275096 + 0.961417i \(0.588710\pi\)
\(432\) 0 0
\(433\) −8974.00 −0.995988 −0.497994 0.867180i \(-0.665930\pi\)
−0.497994 + 0.867180i \(0.665930\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4500.00 0.494291
\(437\) −7791.00 −0.852847
\(438\) 0 0
\(439\) 4179.00 0.454334 0.227167 0.973856i \(-0.427054\pi\)
0.227167 + 0.973856i \(0.427054\pi\)
\(440\) 840.000 0.0910123
\(441\) 0 0
\(442\) 588.000 0.0632767
\(443\) 12927.0 1.38641 0.693206 0.720740i \(-0.256198\pi\)
0.693206 + 0.720740i \(0.256198\pi\)
\(444\) 0 0
\(445\) −2303.00 −0.245332
\(446\) −2800.00 −0.297273
\(447\) 0 0
\(448\) 0 0
\(449\) 2826.00 0.297032 0.148516 0.988910i \(-0.452550\pi\)
0.148516 + 0.988910i \(0.452550\pi\)
\(450\) 0 0
\(451\) 1750.00 0.182715
\(452\) −6152.00 −0.640190
\(453\) 0 0
\(454\) 4410.00 0.455884
\(455\) 0 0
\(456\) 0 0
\(457\) 8479.00 0.867901 0.433951 0.900937i \(-0.357119\pi\)
0.433951 + 0.900937i \(0.357119\pi\)
\(458\) 574.000 0.0585617
\(459\) 0 0
\(460\) −4452.00 −0.451251
\(461\) 9338.00 0.943414 0.471707 0.881755i \(-0.343638\pi\)
0.471707 + 0.881755i \(0.343638\pi\)
\(462\) 0 0
\(463\) −4016.00 −0.403109 −0.201554 0.979477i \(-0.564599\pi\)
−0.201554 + 0.979477i \(0.564599\pi\)
\(464\) 928.000 0.0928477
\(465\) 0 0
\(466\) 9174.00 0.911969
\(467\) −5859.00 −0.580561 −0.290281 0.956942i \(-0.593749\pi\)
−0.290281 + 0.956942i \(0.593749\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −7350.00 −0.721341
\(471\) 0 0
\(472\) −2520.00 −0.245747
\(473\) −620.000 −0.0602698
\(474\) 0 0
\(475\) 3724.00 0.359724
\(476\) 0 0
\(477\) 0 0
\(478\) 3336.00 0.319216
\(479\) 6503.00 0.620312 0.310156 0.950686i \(-0.399619\pi\)
0.310156 + 0.950686i \(0.399619\pi\)
\(480\) 0 0
\(481\) 3066.00 0.290639
\(482\) −6818.00 −0.644297
\(483\) 0 0
\(484\) 5224.00 0.490609
\(485\) 6174.00 0.578035
\(486\) 0 0
\(487\) −16049.0 −1.49333 −0.746663 0.665203i \(-0.768345\pi\)
−0.746663 + 0.665203i \(0.768345\pi\)
\(488\) 9912.00 0.919457
\(489\) 0 0
\(490\) 0 0
\(491\) −8864.00 −0.814718 −0.407359 0.913268i \(-0.633550\pi\)
−0.407359 + 0.913268i \(0.633550\pi\)
\(492\) 0 0
\(493\) 1218.00 0.111270
\(494\) 1372.00 0.124958
\(495\) 0 0
\(496\) 2352.00 0.212919
\(497\) 0 0
\(498\) 0 0
\(499\) −10211.0 −0.916046 −0.458023 0.888940i \(-0.651442\pi\)
−0.458023 + 0.888940i \(0.651442\pi\)
\(500\) 5628.00 0.503384
\(501\) 0 0
\(502\) 9520.00 0.846411
\(503\) −1680.00 −0.148921 −0.0744607 0.997224i \(-0.523724\pi\)
−0.0744607 + 0.997224i \(0.523724\pi\)
\(504\) 0 0
\(505\) 9653.00 0.850600
\(506\) −1590.00 −0.139692
\(507\) 0 0
\(508\) −288.000 −0.0251534
\(509\) −9457.00 −0.823525 −0.411762 0.911291i \(-0.635087\pi\)
−0.411762 + 0.911291i \(0.635087\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 5632.00 0.486136
\(513\) 0 0
\(514\) 1610.00 0.138160
\(515\) 4753.00 0.406684
\(516\) 0 0
\(517\) 2625.00 0.223302
\(518\) 0 0
\(519\) 0 0
\(520\) 2352.00 0.198350
\(521\) −18081.0 −1.52043 −0.760214 0.649673i \(-0.774906\pi\)
−0.760214 + 0.649673i \(0.774906\pi\)
\(522\) 0 0
\(523\) −20377.0 −1.70368 −0.851839 0.523803i \(-0.824513\pi\)
−0.851839 + 0.523803i \(0.824513\pi\)
\(524\) −8596.00 −0.716637
\(525\) 0 0
\(526\) −514.000 −0.0426073
\(527\) 3087.00 0.255165
\(528\) 0 0
\(529\) 13114.0 1.07783
\(530\) 4242.00 0.347662
\(531\) 0 0
\(532\) 0 0
\(533\) 4900.00 0.398204
\(534\) 0 0
\(535\) −3199.00 −0.258514
\(536\) 9960.00 0.802624
\(537\) 0 0
\(538\) −7182.00 −0.575535
\(539\) 0 0
\(540\) 0 0
\(541\) −6193.00 −0.492159 −0.246079 0.969250i \(-0.579142\pi\)
−0.246079 + 0.969250i \(0.579142\pi\)
\(542\) 2786.00 0.220791
\(543\) 0 0
\(544\) 3360.00 0.264814
\(545\) −7875.00 −0.618950
\(546\) 0 0
\(547\) −18464.0 −1.44326 −0.721630 0.692279i \(-0.756607\pi\)
−0.721630 + 0.692279i \(0.756607\pi\)
\(548\) −4500.00 −0.350786
\(549\) 0 0
\(550\) 760.000 0.0589209
\(551\) 2842.00 0.219734
\(552\) 0 0
\(553\) 0 0
\(554\) −830.000 −0.0636522
\(555\) 0 0
\(556\) 1008.00 0.0768862
\(557\) 9413.00 0.716053 0.358027 0.933711i \(-0.383450\pi\)
0.358027 + 0.933711i \(0.383450\pi\)
\(558\) 0 0
\(559\) −1736.00 −0.131351
\(560\) 0 0
\(561\) 0 0
\(562\) −9908.00 −0.743672
\(563\) 3199.00 0.239470 0.119735 0.992806i \(-0.461795\pi\)
0.119735 + 0.992806i \(0.461795\pi\)
\(564\) 0 0
\(565\) 10766.0 0.801644
\(566\) −8554.00 −0.635250
\(567\) 0 0
\(568\) 10368.0 0.765901
\(569\) −21583.0 −1.59017 −0.795085 0.606498i \(-0.792574\pi\)
−0.795085 + 0.606498i \(0.792574\pi\)
\(570\) 0 0
\(571\) 20267.0 1.48537 0.742686 0.669640i \(-0.233551\pi\)
0.742686 + 0.669640i \(0.233551\pi\)
\(572\) −280.000 −0.0204675
\(573\) 0 0
\(574\) 0 0
\(575\) −12084.0 −0.876413
\(576\) 0 0
\(577\) −13951.0 −1.00656 −0.503282 0.864122i \(-0.667874\pi\)
−0.503282 + 0.864122i \(0.667874\pi\)
\(578\) 8944.00 0.643636
\(579\) 0 0
\(580\) 1624.00 0.116264
\(581\) 0 0
\(582\) 0 0
\(583\) −1515.00 −0.107624
\(584\) 26712.0 1.89272
\(585\) 0 0
\(586\) −15484.0 −1.09153
\(587\) −20972.0 −1.47463 −0.737314 0.675550i \(-0.763906\pi\)
−0.737314 + 0.675550i \(0.763906\pi\)
\(588\) 0 0
\(589\) 7203.00 0.503895
\(590\) 1470.00 0.102574
\(591\) 0 0
\(592\) −3504.00 −0.243266
\(593\) −189.000 −0.0130882 −0.00654410 0.999979i \(-0.502083\pi\)
−0.00654410 + 0.999979i \(0.502083\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −804.000 −0.0552569
\(597\) 0 0
\(598\) −4452.00 −0.304441
\(599\) 10281.0 0.701286 0.350643 0.936509i \(-0.385963\pi\)
0.350643 + 0.936509i \(0.385963\pi\)
\(600\) 0 0
\(601\) 6090.00 0.413338 0.206669 0.978411i \(-0.433738\pi\)
0.206669 + 0.978411i \(0.433738\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −6476.00 −0.436266
\(605\) −9142.00 −0.614339
\(606\) 0 0
\(607\) −4949.00 −0.330929 −0.165464 0.986216i \(-0.552912\pi\)
−0.165464 + 0.986216i \(0.552912\pi\)
\(608\) 7840.00 0.522951
\(609\) 0 0
\(610\) −5782.00 −0.383781
\(611\) 7350.00 0.486660
\(612\) 0 0
\(613\) −15797.0 −1.04084 −0.520420 0.853910i \(-0.674225\pi\)
−0.520420 + 0.853910i \(0.674225\pi\)
\(614\) −14728.0 −0.968035
\(615\) 0 0
\(616\) 0 0
\(617\) 9378.00 0.611903 0.305951 0.952047i \(-0.401025\pi\)
0.305951 + 0.952047i \(0.401025\pi\)
\(618\) 0 0
\(619\) 24353.0 1.58131 0.790654 0.612263i \(-0.209741\pi\)
0.790654 + 0.612263i \(0.209741\pi\)
\(620\) 4116.00 0.266617
\(621\) 0 0
\(622\) −19950.0 −1.28605
\(623\) 0 0
\(624\) 0 0
\(625\) −349.000 −0.0223360
\(626\) −9506.00 −0.606927
\(627\) 0 0
\(628\) 2716.00 0.172580
\(629\) −4599.00 −0.291533
\(630\) 0 0
\(631\) −12640.0 −0.797449 −0.398725 0.917071i \(-0.630547\pi\)
−0.398725 + 0.917071i \(0.630547\pi\)
\(632\) −2472.00 −0.155587
\(633\) 0 0
\(634\) −6954.00 −0.435613
\(635\) 504.000 0.0314971
\(636\) 0 0
\(637\) 0 0
\(638\) 580.000 0.0359913
\(639\) 0 0
\(640\) 2688.00 0.166020
\(641\) 1041.00 0.0641451 0.0320726 0.999486i \(-0.489789\pi\)
0.0320726 + 0.999486i \(0.489789\pi\)
\(642\) 0 0
\(643\) −9548.00 −0.585593 −0.292797 0.956175i \(-0.594586\pi\)
−0.292797 + 0.956175i \(0.594586\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −2058.00 −0.125342
\(647\) −3241.00 −0.196935 −0.0984674 0.995140i \(-0.531394\pi\)
−0.0984674 + 0.995140i \(0.531394\pi\)
\(648\) 0 0
\(649\) −525.000 −0.0317535
\(650\) 2128.00 0.128411
\(651\) 0 0
\(652\) 1868.00 0.112203
\(653\) 8853.00 0.530543 0.265272 0.964174i \(-0.414538\pi\)
0.265272 + 0.964174i \(0.414538\pi\)
\(654\) 0 0
\(655\) 15043.0 0.897372
\(656\) −5600.00 −0.333298
\(657\) 0 0
\(658\) 0 0
\(659\) −7044.00 −0.416381 −0.208191 0.978088i \(-0.566757\pi\)
−0.208191 + 0.978088i \(0.566757\pi\)
\(660\) 0 0
\(661\) 12089.0 0.711358 0.355679 0.934608i \(-0.384250\pi\)
0.355679 + 0.934608i \(0.384250\pi\)
\(662\) −6682.00 −0.392301
\(663\) 0 0
\(664\) 26208.0 1.53173
\(665\) 0 0
\(666\) 0 0
\(667\) −9222.00 −0.535348
\(668\) −4816.00 −0.278947
\(669\) 0 0
\(670\) −5810.00 −0.335015
\(671\) 2065.00 0.118805
\(672\) 0 0
\(673\) 982.000 0.0562456 0.0281228 0.999604i \(-0.491047\pi\)
0.0281228 + 0.999604i \(0.491047\pi\)
\(674\) −14732.0 −0.841922
\(675\) 0 0
\(676\) 8004.00 0.455394
\(677\) −30513.0 −1.73222 −0.866108 0.499857i \(-0.833386\pi\)
−0.866108 + 0.499857i \(0.833386\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −3528.00 −0.198960
\(681\) 0 0
\(682\) 1470.00 0.0825355
\(683\) −11475.0 −0.642868 −0.321434 0.946932i \(-0.604165\pi\)
−0.321434 + 0.946932i \(0.604165\pi\)
\(684\) 0 0
\(685\) 7875.00 0.439253
\(686\) 0 0
\(687\) 0 0
\(688\) 1984.00 0.109941
\(689\) −4242.00 −0.234553
\(690\) 0 0
\(691\) 28315.0 1.55883 0.779416 0.626506i \(-0.215516\pi\)
0.779416 + 0.626506i \(0.215516\pi\)
\(692\) 11284.0 0.619875
\(693\) 0 0
\(694\) 14830.0 0.811151
\(695\) −1764.00 −0.0962767
\(696\) 0 0
\(697\) −7350.00 −0.399428
\(698\) −7756.00 −0.420586
\(699\) 0 0
\(700\) 0 0
\(701\) −10614.0 −0.571876 −0.285938 0.958248i \(-0.592305\pi\)
−0.285938 + 0.958248i \(0.592305\pi\)
\(702\) 0 0
\(703\) −10731.0 −0.575715
\(704\) 2240.00 0.119919
\(705\) 0 0
\(706\) −2534.00 −0.135083
\(707\) 0 0
\(708\) 0 0
\(709\) 10299.0 0.545539 0.272769 0.962079i \(-0.412060\pi\)
0.272769 + 0.962079i \(0.412060\pi\)
\(710\) −6048.00 −0.319686
\(711\) 0 0
\(712\) −7896.00 −0.415611
\(713\) −23373.0 −1.22767
\(714\) 0 0
\(715\) 490.000 0.0256293
\(716\) −13012.0 −0.679164
\(717\) 0 0
\(718\) 9370.00 0.487027
\(719\) 32529.0 1.68724 0.843621 0.536939i \(-0.180420\pi\)
0.843621 + 0.536939i \(0.180420\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 8916.00 0.459583
\(723\) 0 0
\(724\) 6328.00 0.324832
\(725\) 4408.00 0.225806
\(726\) 0 0
\(727\) −29456.0 −1.50270 −0.751350 0.659904i \(-0.770597\pi\)
−0.751350 + 0.659904i \(0.770597\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −15582.0 −0.790021
\(731\) 2604.00 0.131754
\(732\) 0 0
\(733\) −27867.0 −1.40422 −0.702109 0.712070i \(-0.747758\pi\)
−0.702109 + 0.712070i \(0.747758\pi\)
\(734\) −9282.00 −0.466764
\(735\) 0 0
\(736\) −25440.0 −1.27409
\(737\) 2075.00 0.103709
\(738\) 0 0
\(739\) 19539.0 0.972603 0.486302 0.873791i \(-0.338346\pi\)
0.486302 + 0.873791i \(0.338346\pi\)
\(740\) −6132.00 −0.304617
\(741\) 0 0
\(742\) 0 0
\(743\) −1248.00 −0.0616214 −0.0308107 0.999525i \(-0.509809\pi\)
−0.0308107 + 0.999525i \(0.509809\pi\)
\(744\) 0 0
\(745\) 1407.00 0.0691926
\(746\) 17594.0 0.863488
\(747\) 0 0
\(748\) 420.000 0.0205304
\(749\) 0 0
\(750\) 0 0
\(751\) 28093.0 1.36502 0.682509 0.730877i \(-0.260889\pi\)
0.682509 + 0.730877i \(0.260889\pi\)
\(752\) −8400.00 −0.407336
\(753\) 0 0
\(754\) 1624.00 0.0784385
\(755\) 11333.0 0.546292
\(756\) 0 0
\(757\) 35954.0 1.72625 0.863124 0.504991i \(-0.168504\pi\)
0.863124 + 0.504991i \(0.168504\pi\)
\(758\) −27360.0 −1.31103
\(759\) 0 0
\(760\) −8232.00 −0.392903
\(761\) −861.000 −0.0410134 −0.0205067 0.999790i \(-0.506528\pi\)
−0.0205067 + 0.999790i \(0.506528\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 10228.0 0.484340
\(765\) 0 0
\(766\) −19530.0 −0.921211
\(767\) −1470.00 −0.0692029
\(768\) 0 0
\(769\) −24710.0 −1.15873 −0.579366 0.815067i \(-0.696700\pi\)
−0.579366 + 0.815067i \(0.696700\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1588.00 0.0740329
\(773\) 16499.0 0.767694 0.383847 0.923397i \(-0.374599\pi\)
0.383847 + 0.923397i \(0.374599\pi\)
\(774\) 0 0
\(775\) 11172.0 0.517819
\(776\) 21168.0 0.979236
\(777\) 0 0
\(778\) 3462.00 0.159536
\(779\) −17150.0 −0.788784
\(780\) 0 0
\(781\) 2160.00 0.0989640
\(782\) 6678.00 0.305377
\(783\) 0 0
\(784\) 0 0
\(785\) −4753.00 −0.216104
\(786\) 0 0
\(787\) −16471.0 −0.746033 −0.373016 0.927825i \(-0.621676\pi\)
−0.373016 + 0.927825i \(0.621676\pi\)
\(788\) 11656.0 0.526939
\(789\) 0 0
\(790\) 1442.00 0.0649418
\(791\) 0 0
\(792\) 0 0
\(793\) 5782.00 0.258922
\(794\) 21966.0 0.981794
\(795\) 0 0
\(796\) 13356.0 0.594712
\(797\) −36470.0 −1.62087 −0.810435 0.585828i \(-0.800769\pi\)
−0.810435 + 0.585828i \(0.800769\pi\)
\(798\) 0 0
\(799\) −11025.0 −0.488156
\(800\) 12160.0 0.537401
\(801\) 0 0
\(802\) 13206.0 0.581446
\(803\) 5565.00 0.244564
\(804\) 0 0
\(805\) 0 0
\(806\) 4116.00 0.179876
\(807\) 0 0
\(808\) 33096.0 1.44098
\(809\) −35751.0 −1.55369 −0.776847 0.629690i \(-0.783182\pi\)
−0.776847 + 0.629690i \(0.783182\pi\)
\(810\) 0 0
\(811\) 16492.0 0.714072 0.357036 0.934091i \(-0.383787\pi\)
0.357036 + 0.934091i \(0.383787\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −2190.00 −0.0942991
\(815\) −3269.00 −0.140501
\(816\) 0 0
\(817\) 6076.00 0.260186
\(818\) 21910.0 0.936510
\(819\) 0 0
\(820\) −9800.00 −0.417355
\(821\) 41473.0 1.76299 0.881497 0.472190i \(-0.156536\pi\)
0.881497 + 0.472190i \(0.156536\pi\)
\(822\) 0 0
\(823\) −25065.0 −1.06162 −0.530809 0.847492i \(-0.678112\pi\)
−0.530809 + 0.847492i \(0.678112\pi\)
\(824\) 16296.0 0.688954
\(825\) 0 0
\(826\) 0 0
\(827\) −9732.00 −0.409208 −0.204604 0.978845i \(-0.565591\pi\)
−0.204604 + 0.978845i \(0.565591\pi\)
\(828\) 0 0
\(829\) −27755.0 −1.16281 −0.581406 0.813614i \(-0.697497\pi\)
−0.581406 + 0.813614i \(0.697497\pi\)
\(830\) −15288.0 −0.639342
\(831\) 0 0
\(832\) 6272.00 0.261349
\(833\) 0 0
\(834\) 0 0
\(835\) 8428.00 0.349297
\(836\) 980.000 0.0405431
\(837\) 0 0
\(838\) −13272.0 −0.547105
\(839\) 21112.0 0.868733 0.434367 0.900736i \(-0.356972\pi\)
0.434367 + 0.900736i \(0.356972\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) 33260.0 1.36130
\(843\) 0 0
\(844\) −7120.00 −0.290380
\(845\) −14007.0 −0.570243
\(846\) 0 0
\(847\) 0 0
\(848\) 4848.00 0.196322
\(849\) 0 0
\(850\) −3192.00 −0.128806
\(851\) 34821.0 1.40264
\(852\) 0 0
\(853\) 21238.0 0.852492 0.426246 0.904607i \(-0.359836\pi\)
0.426246 + 0.904607i \(0.359836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −10968.0 −0.437942
\(857\) −35609.0 −1.41935 −0.709673 0.704531i \(-0.751158\pi\)
−0.709673 + 0.704531i \(0.751158\pi\)
\(858\) 0 0
\(859\) −2177.00 −0.0864706 −0.0432353 0.999065i \(-0.513767\pi\)
−0.0432353 + 0.999065i \(0.513767\pi\)
\(860\) 3472.00 0.137668
\(861\) 0 0
\(862\) 9846.00 0.389044
\(863\) 32247.0 1.27196 0.635980 0.771706i \(-0.280596\pi\)
0.635980 + 0.771706i \(0.280596\pi\)
\(864\) 0 0
\(865\) −19747.0 −0.776206
\(866\) 17948.0 0.704270
\(867\) 0 0
\(868\) 0 0
\(869\) −515.000 −0.0201038
\(870\) 0 0
\(871\) 5810.00 0.226021
\(872\) −27000.0 −1.04855
\(873\) 0 0
\(874\) 15582.0 0.603054
\(875\) 0 0
\(876\) 0 0
\(877\) 27631.0 1.06389 0.531946 0.846779i \(-0.321461\pi\)
0.531946 + 0.846779i \(0.321461\pi\)
\(878\) −8358.00 −0.321263
\(879\) 0 0
\(880\) −560.000 −0.0214518
\(881\) 24402.0 0.933172 0.466586 0.884476i \(-0.345484\pi\)
0.466586 + 0.884476i \(0.345484\pi\)
\(882\) 0 0
\(883\) −19612.0 −0.747448 −0.373724 0.927540i \(-0.621919\pi\)
−0.373724 + 0.927540i \(0.621919\pi\)
\(884\) 1176.00 0.0447434
\(885\) 0 0
\(886\) −25854.0 −0.980341
\(887\) 2261.00 0.0855884 0.0427942 0.999084i \(-0.486374\pi\)
0.0427942 + 0.999084i \(0.486374\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 4606.00 0.173476
\(891\) 0 0
\(892\) −5600.00 −0.210204
\(893\) −25725.0 −0.964003
\(894\) 0 0
\(895\) 22771.0 0.850448
\(896\) 0 0
\(897\) 0 0
\(898\) −5652.00 −0.210033
\(899\) 8526.00 0.316305
\(900\) 0 0
\(901\) 6363.00 0.235274
\(902\) −3500.00 −0.129199
\(903\) 0 0
\(904\) 36912.0 1.35805
\(905\) −11074.0 −0.406754
\(906\) 0 0
\(907\) −23833.0 −0.872505 −0.436252 0.899824i \(-0.643695\pi\)
−0.436252 + 0.899824i \(0.643695\pi\)
\(908\) 8820.00 0.322359
\(909\) 0 0
\(910\) 0 0
\(911\) −31824.0 −1.15738 −0.578692 0.815546i \(-0.696437\pi\)
−0.578692 + 0.815546i \(0.696437\pi\)
\(912\) 0 0
\(913\) 5460.00 0.197919
\(914\) −16958.0 −0.613699
\(915\) 0 0
\(916\) 1148.00 0.0414094
\(917\) 0 0
\(918\) 0 0
\(919\) −16819.0 −0.603708 −0.301854 0.953354i \(-0.597606\pi\)
−0.301854 + 0.953354i \(0.597606\pi\)
\(920\) 26712.0 0.957248
\(921\) 0 0
\(922\) −18676.0 −0.667095
\(923\) 6048.00 0.215680
\(924\) 0 0
\(925\) −16644.0 −0.591623
\(926\) 8032.00 0.285041
\(927\) 0 0
\(928\) 9280.00 0.328266
\(929\) 1799.00 0.0635342 0.0317671 0.999495i \(-0.489887\pi\)
0.0317671 + 0.999495i \(0.489887\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18348.0 0.644859
\(933\) 0 0
\(934\) 11718.0 0.410519
\(935\) −735.000 −0.0257081
\(936\) 0 0
\(937\) −14154.0 −0.493480 −0.246740 0.969082i \(-0.579359\pi\)
−0.246740 + 0.969082i \(0.579359\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −14700.0 −0.510065
\(941\) 12047.0 0.417344 0.208672 0.977986i \(-0.433086\pi\)
0.208672 + 0.977986i \(0.433086\pi\)
\(942\) 0 0
\(943\) 55650.0 1.92175
\(944\) 1680.00 0.0579230
\(945\) 0 0
\(946\) 1240.00 0.0426172
\(947\) 24379.0 0.836548 0.418274 0.908321i \(-0.362635\pi\)
0.418274 + 0.908321i \(0.362635\pi\)
\(948\) 0 0
\(949\) 15582.0 0.532996
\(950\) −7448.00 −0.254363
\(951\) 0 0
\(952\) 0 0
\(953\) 52330.0 1.77874 0.889368 0.457192i \(-0.151145\pi\)
0.889368 + 0.457192i \(0.151145\pi\)
\(954\) 0 0
\(955\) −17899.0 −0.606490
\(956\) 6672.00 0.225720
\(957\) 0 0
\(958\) −13006.0 −0.438627
\(959\) 0 0
\(960\) 0 0
\(961\) −8182.00 −0.274647
\(962\) −6132.00 −0.205513
\(963\) 0 0
\(964\) −13636.0 −0.455587
\(965\) −2779.00 −0.0927038
\(966\) 0 0
\(967\) −12416.0 −0.412897 −0.206449 0.978457i \(-0.566191\pi\)
−0.206449 + 0.978457i \(0.566191\pi\)
\(968\) −31344.0 −1.04074
\(969\) 0 0
\(970\) −12348.0 −0.408732
\(971\) 36813.0 1.21667 0.608334 0.793681i \(-0.291838\pi\)
0.608334 + 0.793681i \(0.291838\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 32098.0 1.05594
\(975\) 0 0
\(976\) −6608.00 −0.216718
\(977\) −34995.0 −1.14595 −0.572973 0.819574i \(-0.694210\pi\)
−0.572973 + 0.819574i \(0.694210\pi\)
\(978\) 0 0
\(979\) −1645.00 −0.0537022
\(980\) 0 0
\(981\) 0 0
\(982\) 17728.0 0.576093
\(983\) −14301.0 −0.464019 −0.232010 0.972713i \(-0.574530\pi\)
−0.232010 + 0.972713i \(0.574530\pi\)
\(984\) 0 0
\(985\) −20398.0 −0.659832
\(986\) −2436.00 −0.0786796
\(987\) 0 0
\(988\) 2744.00 0.0883586
\(989\) −19716.0 −0.633905
\(990\) 0 0
\(991\) −2665.00 −0.0854253 −0.0427127 0.999087i \(-0.513600\pi\)
−0.0427127 + 0.999087i \(0.513600\pi\)
\(992\) 23520.0 0.752783
\(993\) 0 0
\(994\) 0 0
\(995\) −23373.0 −0.744697
\(996\) 0 0
\(997\) −24871.0 −0.790043 −0.395021 0.918672i \(-0.629263\pi\)
−0.395021 + 0.918672i \(0.629263\pi\)
\(998\) 20422.0 0.647743
\(999\) 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.e.1.1 1
3.2 odd 2 49.4.a.c.1.1 1
7.2 even 3 441.4.e.k.361.1 2
7.3 odd 6 63.4.e.b.37.1 2
7.4 even 3 441.4.e.k.226.1 2
7.5 odd 6 63.4.e.b.46.1 2
7.6 odd 2 441.4.a.d.1.1 1
12.11 even 2 784.4.a.r.1.1 1
15.14 odd 2 1225.4.a.d.1.1 1
21.2 odd 6 49.4.c.a.18.1 2
21.5 even 6 7.4.c.a.4.1 yes 2
21.11 odd 6 49.4.c.a.30.1 2
21.17 even 6 7.4.c.a.2.1 2
21.20 even 2 49.4.a.d.1.1 1
84.47 odd 6 112.4.i.c.81.1 2
84.59 odd 6 112.4.i.c.65.1 2
84.83 odd 2 784.4.a.b.1.1 1
105.17 odd 12 175.4.k.a.149.1 4
105.38 odd 12 175.4.k.a.149.2 4
105.47 odd 12 175.4.k.a.74.2 4
105.59 even 6 175.4.e.a.51.1 2
105.68 odd 12 175.4.k.a.74.1 4
105.89 even 6 175.4.e.a.151.1 2
105.104 even 2 1225.4.a.c.1.1 1
168.5 even 6 448.4.i.f.193.1 2
168.59 odd 6 448.4.i.a.65.1 2
168.101 even 6 448.4.i.f.65.1 2
168.131 odd 6 448.4.i.a.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.c.a.2.1 2 21.17 even 6
7.4.c.a.4.1 yes 2 21.5 even 6
49.4.a.c.1.1 1 3.2 odd 2
49.4.a.d.1.1 1 21.20 even 2
49.4.c.a.18.1 2 21.2 odd 6
49.4.c.a.30.1 2 21.11 odd 6
63.4.e.b.37.1 2 7.3 odd 6
63.4.e.b.46.1 2 7.5 odd 6
112.4.i.c.65.1 2 84.59 odd 6
112.4.i.c.81.1 2 84.47 odd 6
175.4.e.a.51.1 2 105.59 even 6
175.4.e.a.151.1 2 105.89 even 6
175.4.k.a.74.1 4 105.68 odd 12
175.4.k.a.74.2 4 105.47 odd 12
175.4.k.a.149.1 4 105.17 odd 12
175.4.k.a.149.2 4 105.38 odd 12
441.4.a.d.1.1 1 7.6 odd 2
441.4.a.e.1.1 1 1.1 even 1 trivial
441.4.e.k.226.1 2 7.4 even 3
441.4.e.k.361.1 2 7.2 even 3
448.4.i.a.65.1 2 168.59 odd 6
448.4.i.a.193.1 2 168.131 odd 6
448.4.i.f.65.1 2 168.101 even 6
448.4.i.f.193.1 2 168.5 even 6
784.4.a.b.1.1 1 84.83 odd 2
784.4.a.r.1.1 1 12.11 even 2
1225.4.a.c.1.1 1 105.104 even 2
1225.4.a.d.1.1 1 15.14 odd 2