Properties

Label 2352.4.a.cq.1.3
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.136768.1
Defining polynomial: \(x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 588)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.51732\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +10.6550 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +10.6550 q^{5} +9.00000 q^{9} +6.65399 q^{11} -75.9335 q^{13} +31.9651 q^{15} +104.287 q^{17} +85.4943 q^{19} +68.6733 q^{23} -11.4700 q^{25} +27.0000 q^{27} +87.7843 q^{29} +62.7683 q^{31} +19.9620 q^{33} +42.2093 q^{37} -227.800 q^{39} -313.904 q^{41} -306.591 q^{43} +95.8954 q^{45} +215.081 q^{47} +312.861 q^{51} +525.024 q^{53} +70.8986 q^{55} +256.483 q^{57} +360.491 q^{59} -800.726 q^{61} -809.075 q^{65} +40.2286 q^{67} +206.020 q^{69} +298.781 q^{71} -517.126 q^{73} -34.4099 q^{75} +1222.47 q^{79} +81.0000 q^{81} +1328.55 q^{83} +1111.18 q^{85} +263.353 q^{87} +639.938 q^{89} +188.305 q^{93} +910.946 q^{95} -1425.65 q^{97} +59.8859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} + 36q^{9} - 48q^{17} + 192q^{19} - 192q^{23} + 324q^{25} + 108q^{27} + 96q^{29} + 48q^{31} + 256q^{37} - 1008q^{41} + 112q^{43} + 864q^{47} - 144q^{51} - 648q^{53} + 2352q^{55} + 576q^{57} + 336q^{59} - 960q^{61} - 360q^{65} - 720q^{67} - 576q^{69} + 1344q^{71} - 672q^{73} + 972q^{75} + 1984q^{79} + 324q^{81} + 3120q^{83} + 680q^{85} + 288q^{87} - 2160q^{89} + 144q^{93} + 3744q^{95} - 2016q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 10.6550 0.953016 0.476508 0.879170i \(-0.341902\pi\)
0.476508 + 0.879170i \(0.341902\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 6.65399 0.182387 0.0911933 0.995833i \(-0.470932\pi\)
0.0911933 + 0.995833i \(0.470932\pi\)
\(12\) 0 0
\(13\) −75.9335 −1.62001 −0.810006 0.586422i \(-0.800536\pi\)
−0.810006 + 0.586422i \(0.800536\pi\)
\(14\) 0 0
\(15\) 31.9651 0.550224
\(16\) 0 0
\(17\) 104.287 1.48784 0.743921 0.668268i \(-0.232964\pi\)
0.743921 + 0.668268i \(0.232964\pi\)
\(18\) 0 0
\(19\) 85.4943 1.03230 0.516151 0.856498i \(-0.327364\pi\)
0.516151 + 0.856498i \(0.327364\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 68.6733 0.622581 0.311291 0.950315i \(-0.399239\pi\)
0.311291 + 0.950315i \(0.399239\pi\)
\(24\) 0 0
\(25\) −11.4700 −0.0917596
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 87.7843 0.562108 0.281054 0.959692i \(-0.409316\pi\)
0.281054 + 0.959692i \(0.409316\pi\)
\(30\) 0 0
\(31\) 62.7683 0.363662 0.181831 0.983330i \(-0.441798\pi\)
0.181831 + 0.983330i \(0.441798\pi\)
\(32\) 0 0
\(33\) 19.9620 0.105301
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 42.2093 0.187545 0.0937726 0.995594i \(-0.470107\pi\)
0.0937726 + 0.995594i \(0.470107\pi\)
\(38\) 0 0
\(39\) −227.800 −0.935314
\(40\) 0 0
\(41\) −313.904 −1.19570 −0.597848 0.801609i \(-0.703977\pi\)
−0.597848 + 0.801609i \(0.703977\pi\)
\(42\) 0 0
\(43\) −306.591 −1.08732 −0.543659 0.839306i \(-0.682962\pi\)
−0.543659 + 0.839306i \(0.682962\pi\)
\(44\) 0 0
\(45\) 95.8954 0.317672
\(46\) 0 0
\(47\) 215.081 0.667508 0.333754 0.942660i \(-0.391685\pi\)
0.333754 + 0.942660i \(0.391685\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 312.861 0.859006
\(52\) 0 0
\(53\) 525.024 1.36071 0.680354 0.732884i \(-0.261826\pi\)
0.680354 + 0.732884i \(0.261826\pi\)
\(54\) 0 0
\(55\) 70.8986 0.173818
\(56\) 0 0
\(57\) 256.483 0.596000
\(58\) 0 0
\(59\) 360.491 0.795457 0.397729 0.917503i \(-0.369799\pi\)
0.397729 + 0.917503i \(0.369799\pi\)
\(60\) 0 0
\(61\) −800.726 −1.68070 −0.840348 0.542048i \(-0.817649\pi\)
−0.840348 + 0.542048i \(0.817649\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −809.075 −1.54390
\(66\) 0 0
\(67\) 40.2286 0.0733538 0.0366769 0.999327i \(-0.488323\pi\)
0.0366769 + 0.999327i \(0.488323\pi\)
\(68\) 0 0
\(69\) 206.020 0.359447
\(70\) 0 0
\(71\) 298.781 0.499419 0.249709 0.968321i \(-0.419665\pi\)
0.249709 + 0.968321i \(0.419665\pi\)
\(72\) 0 0
\(73\) −517.126 −0.829110 −0.414555 0.910024i \(-0.636063\pi\)
−0.414555 + 0.910024i \(0.636063\pi\)
\(74\) 0 0
\(75\) −34.4099 −0.0529774
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1222.47 1.74099 0.870494 0.492178i \(-0.163799\pi\)
0.870494 + 0.492178i \(0.163799\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1328.55 1.75696 0.878479 0.477782i \(-0.158559\pi\)
0.878479 + 0.477782i \(0.158559\pi\)
\(84\) 0 0
\(85\) 1111.18 1.41794
\(86\) 0 0
\(87\) 263.353 0.324533
\(88\) 0 0
\(89\) 639.938 0.762172 0.381086 0.924540i \(-0.375550\pi\)
0.381086 + 0.924540i \(0.375550\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 188.305 0.209960
\(94\) 0 0
\(95\) 910.946 0.983801
\(96\) 0 0
\(97\) −1425.65 −1.49230 −0.746149 0.665779i \(-0.768099\pi\)
−0.746149 + 0.665779i \(0.768099\pi\)
\(98\) 0 0
\(99\) 59.8859 0.0607956
\(100\) 0 0
\(101\) 992.170 0.977471 0.488736 0.872432i \(-0.337458\pi\)
0.488736 + 0.872432i \(0.337458\pi\)
\(102\) 0 0
\(103\) 267.572 0.255967 0.127984 0.991776i \(-0.459149\pi\)
0.127984 + 0.991776i \(0.459149\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1536.62 1.38833 0.694164 0.719817i \(-0.255774\pi\)
0.694164 + 0.719817i \(0.255774\pi\)
\(108\) 0 0
\(109\) 998.820 0.877703 0.438852 0.898560i \(-0.355385\pi\)
0.438852 + 0.898560i \(0.355385\pi\)
\(110\) 0 0
\(111\) 126.628 0.108279
\(112\) 0 0
\(113\) 939.006 0.781719 0.390860 0.920450i \(-0.372178\pi\)
0.390860 + 0.920450i \(0.372178\pi\)
\(114\) 0 0
\(115\) 731.717 0.593330
\(116\) 0 0
\(117\) −683.401 −0.540004
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1286.72 −0.966735
\(122\) 0 0
\(123\) −941.712 −0.690336
\(124\) 0 0
\(125\) −1454.09 −1.04046
\(126\) 0 0
\(127\) 1621.27 1.13279 0.566397 0.824133i \(-0.308337\pi\)
0.566397 + 0.824133i \(0.308337\pi\)
\(128\) 0 0
\(129\) −919.773 −0.627764
\(130\) 0 0
\(131\) 1518.43 1.01271 0.506357 0.862324i \(-0.330992\pi\)
0.506357 + 0.862324i \(0.330992\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 287.686 0.183408
\(136\) 0 0
\(137\) −2484.99 −1.54969 −0.774844 0.632152i \(-0.782172\pi\)
−0.774844 + 0.632152i \(0.782172\pi\)
\(138\) 0 0
\(139\) −1655.36 −1.01011 −0.505057 0.863086i \(-0.668529\pi\)
−0.505057 + 0.863086i \(0.668529\pi\)
\(140\) 0 0
\(141\) 645.244 0.385386
\(142\) 0 0
\(143\) −505.260 −0.295469
\(144\) 0 0
\(145\) 935.346 0.535698
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 350.191 0.192542 0.0962709 0.995355i \(-0.469308\pi\)
0.0962709 + 0.995355i \(0.469308\pi\)
\(150\) 0 0
\(151\) −3338.14 −1.79903 −0.899516 0.436888i \(-0.856081\pi\)
−0.899516 + 0.436888i \(0.856081\pi\)
\(152\) 0 0
\(153\) 938.583 0.495947
\(154\) 0 0
\(155\) 668.799 0.346576
\(156\) 0 0
\(157\) 1743.67 0.886371 0.443185 0.896430i \(-0.353848\pi\)
0.443185 + 0.896430i \(0.353848\pi\)
\(158\) 0 0
\(159\) 1575.07 0.785605
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3431.56 −1.64896 −0.824480 0.565891i \(-0.808532\pi\)
−0.824480 + 0.565891i \(0.808532\pi\)
\(164\) 0 0
\(165\) 212.696 0.100354
\(166\) 0 0
\(167\) 3381.05 1.56667 0.783335 0.621600i \(-0.213517\pi\)
0.783335 + 0.621600i \(0.213517\pi\)
\(168\) 0 0
\(169\) 3568.89 1.62444
\(170\) 0 0
\(171\) 769.449 0.344101
\(172\) 0 0
\(173\) 1341.00 0.589332 0.294666 0.955600i \(-0.404792\pi\)
0.294666 + 0.955600i \(0.404792\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1081.47 0.459257
\(178\) 0 0
\(179\) −1125.15 −0.469818 −0.234909 0.972017i \(-0.575479\pi\)
−0.234909 + 0.972017i \(0.575479\pi\)
\(180\) 0 0
\(181\) 3535.04 1.45170 0.725848 0.687855i \(-0.241447\pi\)
0.725848 + 0.687855i \(0.241447\pi\)
\(182\) 0 0
\(183\) −2402.18 −0.970350
\(184\) 0 0
\(185\) 449.742 0.178734
\(186\) 0 0
\(187\) 693.925 0.271363
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2639.89 −1.00008 −0.500041 0.866001i \(-0.666682\pi\)
−0.500041 + 0.866001i \(0.666682\pi\)
\(192\) 0 0
\(193\) 1047.05 0.390510 0.195255 0.980752i \(-0.437447\pi\)
0.195255 + 0.980752i \(0.437447\pi\)
\(194\) 0 0
\(195\) −2427.22 −0.891370
\(196\) 0 0
\(197\) −4585.45 −1.65837 −0.829187 0.558972i \(-0.811196\pi\)
−0.829187 + 0.558972i \(0.811196\pi\)
\(198\) 0 0
\(199\) 830.742 0.295928 0.147964 0.988993i \(-0.452728\pi\)
0.147964 + 0.988993i \(0.452728\pi\)
\(200\) 0 0
\(201\) 120.686 0.0423508
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3344.66 −1.13952
\(206\) 0 0
\(207\) 618.059 0.207527
\(208\) 0 0
\(209\) 568.878 0.188278
\(210\) 0 0
\(211\) 2630.08 0.858114 0.429057 0.903277i \(-0.358846\pi\)
0.429057 + 0.903277i \(0.358846\pi\)
\(212\) 0 0
\(213\) 896.342 0.288340
\(214\) 0 0
\(215\) −3266.74 −1.03623
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1551.38 −0.478687
\(220\) 0 0
\(221\) −7918.87 −2.41032
\(222\) 0 0
\(223\) −863.988 −0.259448 −0.129724 0.991550i \(-0.541409\pi\)
−0.129724 + 0.991550i \(0.541409\pi\)
\(224\) 0 0
\(225\) −103.230 −0.0305865
\(226\) 0 0
\(227\) −4160.36 −1.21644 −0.608221 0.793767i \(-0.708117\pi\)
−0.608221 + 0.793767i \(0.708117\pi\)
\(228\) 0 0
\(229\) 181.210 0.0522914 0.0261457 0.999658i \(-0.491677\pi\)
0.0261457 + 0.999658i \(0.491677\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2150.45 0.604637 0.302319 0.953207i \(-0.402239\pi\)
0.302319 + 0.953207i \(0.402239\pi\)
\(234\) 0 0
\(235\) 2291.70 0.636146
\(236\) 0 0
\(237\) 3667.40 1.00516
\(238\) 0 0
\(239\) 6939.47 1.87815 0.939073 0.343719i \(-0.111687\pi\)
0.939073 + 0.343719i \(0.111687\pi\)
\(240\) 0 0
\(241\) 206.170 0.0551060 0.0275530 0.999620i \(-0.491228\pi\)
0.0275530 + 0.999620i \(0.491228\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6491.88 −1.67234
\(248\) 0 0
\(249\) 3985.65 1.01438
\(250\) 0 0
\(251\) 5011.93 1.26036 0.630180 0.776449i \(-0.282981\pi\)
0.630180 + 0.776449i \(0.282981\pi\)
\(252\) 0 0
\(253\) 456.951 0.113551
\(254\) 0 0
\(255\) 3333.55 0.818647
\(256\) 0 0
\(257\) 5863.08 1.42307 0.711535 0.702651i \(-0.248000\pi\)
0.711535 + 0.702651i \(0.248000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 790.059 0.187369
\(262\) 0 0
\(263\) 5639.56 1.32224 0.661122 0.750279i \(-0.270081\pi\)
0.661122 + 0.750279i \(0.270081\pi\)
\(264\) 0 0
\(265\) 5594.15 1.29678
\(266\) 0 0
\(267\) 1919.81 0.440040
\(268\) 0 0
\(269\) 2976.63 0.674679 0.337339 0.941383i \(-0.390473\pi\)
0.337339 + 0.941383i \(0.390473\pi\)
\(270\) 0 0
\(271\) −2807.33 −0.629275 −0.314637 0.949212i \(-0.601883\pi\)
−0.314637 + 0.949212i \(0.601883\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −76.3210 −0.0167357
\(276\) 0 0
\(277\) −1918.39 −0.416118 −0.208059 0.978116i \(-0.566715\pi\)
−0.208059 + 0.978116i \(0.566715\pi\)
\(278\) 0 0
\(279\) 564.915 0.121221
\(280\) 0 0
\(281\) 5209.50 1.10595 0.552977 0.833197i \(-0.313492\pi\)
0.552977 + 0.833197i \(0.313492\pi\)
\(282\) 0 0
\(283\) 7496.70 1.57467 0.787337 0.616523i \(-0.211459\pi\)
0.787337 + 0.616523i \(0.211459\pi\)
\(284\) 0 0
\(285\) 2732.84 0.567998
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5962.78 1.21367
\(290\) 0 0
\(291\) −4276.95 −0.861579
\(292\) 0 0
\(293\) 3358.94 0.669732 0.334866 0.942266i \(-0.391309\pi\)
0.334866 + 0.942266i \(0.391309\pi\)
\(294\) 0 0
\(295\) 3841.05 0.758084
\(296\) 0 0
\(297\) 179.658 0.0351003
\(298\) 0 0
\(299\) −5214.60 −1.00859
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2976.51 0.564343
\(304\) 0 0
\(305\) −8531.77 −1.60173
\(306\) 0 0
\(307\) −7330.92 −1.36286 −0.681429 0.731884i \(-0.738641\pi\)
−0.681429 + 0.731884i \(0.738641\pi\)
\(308\) 0 0
\(309\) 802.716 0.147783
\(310\) 0 0
\(311\) −2038.53 −0.371687 −0.185843 0.982579i \(-0.559502\pi\)
−0.185843 + 0.982579i \(0.559502\pi\)
\(312\) 0 0
\(313\) −4138.87 −0.747421 −0.373711 0.927545i \(-0.621915\pi\)
−0.373711 + 0.927545i \(0.621915\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7552.25 −1.33810 −0.669049 0.743219i \(-0.733298\pi\)
−0.669049 + 0.743219i \(0.733298\pi\)
\(318\) 0 0
\(319\) 584.116 0.102521
\(320\) 0 0
\(321\) 4609.87 0.801552
\(322\) 0 0
\(323\) 8915.94 1.53590
\(324\) 0 0
\(325\) 870.953 0.148652
\(326\) 0 0
\(327\) 2996.46 0.506742
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5577.70 −0.926218 −0.463109 0.886301i \(-0.653266\pi\)
−0.463109 + 0.886301i \(0.653266\pi\)
\(332\) 0 0
\(333\) 379.884 0.0625151
\(334\) 0 0
\(335\) 428.637 0.0699073
\(336\) 0 0
\(337\) 4467.16 0.722083 0.361041 0.932550i \(-0.382421\pi\)
0.361041 + 0.932550i \(0.382421\pi\)
\(338\) 0 0
\(339\) 2817.02 0.451326
\(340\) 0 0
\(341\) 417.660 0.0663271
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2195.15 0.342559
\(346\) 0 0
\(347\) −9788.41 −1.51432 −0.757161 0.653229i \(-0.773414\pi\)
−0.757161 + 0.653229i \(0.773414\pi\)
\(348\) 0 0
\(349\) 4746.95 0.728075 0.364038 0.931384i \(-0.381398\pi\)
0.364038 + 0.931384i \(0.381398\pi\)
\(350\) 0 0
\(351\) −2050.20 −0.311771
\(352\) 0 0
\(353\) 9434.66 1.42254 0.711269 0.702920i \(-0.248121\pi\)
0.711269 + 0.702920i \(0.248121\pi\)
\(354\) 0 0
\(355\) 3183.52 0.475954
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11307.5 −1.66237 −0.831183 0.555999i \(-0.812336\pi\)
−0.831183 + 0.555999i \(0.812336\pi\)
\(360\) 0 0
\(361\) 450.275 0.0656474
\(362\) 0 0
\(363\) −3860.17 −0.558145
\(364\) 0 0
\(365\) −5510.00 −0.790155
\(366\) 0 0
\(367\) 6089.61 0.866144 0.433072 0.901359i \(-0.357430\pi\)
0.433072 + 0.901359i \(0.357430\pi\)
\(368\) 0 0
\(369\) −2825.14 −0.398565
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2601.84 −0.361175 −0.180588 0.983559i \(-0.557800\pi\)
−0.180588 + 0.983559i \(0.557800\pi\)
\(374\) 0 0
\(375\) −4362.28 −0.600713
\(376\) 0 0
\(377\) −6665.77 −0.910622
\(378\) 0 0
\(379\) −10416.2 −1.41173 −0.705865 0.708347i \(-0.749441\pi\)
−0.705865 + 0.708347i \(0.749441\pi\)
\(380\) 0 0
\(381\) 4863.82 0.654018
\(382\) 0 0
\(383\) −1664.46 −0.222063 −0.111031 0.993817i \(-0.535415\pi\)
−0.111031 + 0.993817i \(0.535415\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2759.32 −0.362440
\(388\) 0 0
\(389\) −1334.16 −0.173893 −0.0869465 0.996213i \(-0.527711\pi\)
−0.0869465 + 0.996213i \(0.527711\pi\)
\(390\) 0 0
\(391\) 7161.73 0.926302
\(392\) 0 0
\(393\) 4555.28 0.584691
\(394\) 0 0
\(395\) 13025.4 1.65919
\(396\) 0 0
\(397\) −4444.88 −0.561920 −0.280960 0.959720i \(-0.590653\pi\)
−0.280960 + 0.959720i \(0.590653\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2649.82 −0.329989 −0.164995 0.986294i \(-0.552761\pi\)
−0.164995 + 0.986294i \(0.552761\pi\)
\(402\) 0 0
\(403\) −4766.21 −0.589137
\(404\) 0 0
\(405\) 863.059 0.105891
\(406\) 0 0
\(407\) 280.860 0.0342057
\(408\) 0 0
\(409\) 9592.66 1.15972 0.579862 0.814715i \(-0.303107\pi\)
0.579862 + 0.814715i \(0.303107\pi\)
\(410\) 0 0
\(411\) −7454.98 −0.894713
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 14155.8 1.67441
\(416\) 0 0
\(417\) −4966.09 −0.583190
\(418\) 0 0
\(419\) 16850.1 1.96463 0.982317 0.187224i \(-0.0599491\pi\)
0.982317 + 0.187224i \(0.0599491\pi\)
\(420\) 0 0
\(421\) 1691.10 0.195770 0.0978849 0.995198i \(-0.468792\pi\)
0.0978849 + 0.995198i \(0.468792\pi\)
\(422\) 0 0
\(423\) 1935.73 0.222503
\(424\) 0 0
\(425\) −1196.17 −0.136524
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1515.78 −0.170589
\(430\) 0 0
\(431\) 6789.34 0.758773 0.379386 0.925238i \(-0.376135\pi\)
0.379386 + 0.925238i \(0.376135\pi\)
\(432\) 0 0
\(433\) −10386.6 −1.15276 −0.576382 0.817180i \(-0.695536\pi\)
−0.576382 + 0.817180i \(0.695536\pi\)
\(434\) 0 0
\(435\) 2806.04 0.309286
\(436\) 0 0
\(437\) 5871.17 0.642692
\(438\) 0 0
\(439\) −4212.26 −0.457950 −0.228975 0.973432i \(-0.573537\pi\)
−0.228975 + 0.973432i \(0.573537\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1809.96 0.194117 0.0970585 0.995279i \(-0.469057\pi\)
0.0970585 + 0.995279i \(0.469057\pi\)
\(444\) 0 0
\(445\) 6818.57 0.726362
\(446\) 0 0
\(447\) 1050.57 0.111164
\(448\) 0 0
\(449\) 1768.29 0.185859 0.0929297 0.995673i \(-0.470377\pi\)
0.0929297 + 0.995673i \(0.470377\pi\)
\(450\) 0 0
\(451\) −2088.71 −0.218079
\(452\) 0 0
\(453\) −10014.4 −1.03867
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4372.61 0.447575 0.223788 0.974638i \(-0.428158\pi\)
0.223788 + 0.974638i \(0.428158\pi\)
\(458\) 0 0
\(459\) 2815.75 0.286335
\(460\) 0 0
\(461\) 1164.34 0.117633 0.0588165 0.998269i \(-0.481267\pi\)
0.0588165 + 0.998269i \(0.481267\pi\)
\(462\) 0 0
\(463\) 14893.9 1.49498 0.747491 0.664272i \(-0.231258\pi\)
0.747491 + 0.664272i \(0.231258\pi\)
\(464\) 0 0
\(465\) 2006.40 0.200096
\(466\) 0 0
\(467\) 19160.3 1.89857 0.949285 0.314418i \(-0.101809\pi\)
0.949285 + 0.314418i \(0.101809\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5231.02 0.511746
\(472\) 0 0
\(473\) −2040.05 −0.198312
\(474\) 0 0
\(475\) −980.616 −0.0947237
\(476\) 0 0
\(477\) 4725.21 0.453569
\(478\) 0 0
\(479\) −9406.28 −0.897253 −0.448626 0.893719i \(-0.648087\pi\)
−0.448626 + 0.893719i \(0.648087\pi\)
\(480\) 0 0
\(481\) −3205.10 −0.303825
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15190.4 −1.42218
\(486\) 0 0
\(487\) −12460.8 −1.15945 −0.579725 0.814812i \(-0.696840\pi\)
−0.579725 + 0.814812i \(0.696840\pi\)
\(488\) 0 0
\(489\) −10294.7 −0.952028
\(490\) 0 0
\(491\) −12868.0 −1.18274 −0.591369 0.806401i \(-0.701412\pi\)
−0.591369 + 0.806401i \(0.701412\pi\)
\(492\) 0 0
\(493\) 9154.76 0.836328
\(494\) 0 0
\(495\) 638.087 0.0579392
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 13705.0 1.22950 0.614750 0.788722i \(-0.289257\pi\)
0.614750 + 0.788722i \(0.289257\pi\)
\(500\) 0 0
\(501\) 10143.2 0.904517
\(502\) 0 0
\(503\) 10126.1 0.897616 0.448808 0.893628i \(-0.351849\pi\)
0.448808 + 0.893628i \(0.351849\pi\)
\(504\) 0 0
\(505\) 10571.6 0.931546
\(506\) 0 0
\(507\) 10706.7 0.937870
\(508\) 0 0
\(509\) 6236.06 0.543042 0.271521 0.962432i \(-0.412473\pi\)
0.271521 + 0.962432i \(0.412473\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2308.35 0.198667
\(514\) 0 0
\(515\) 2850.99 0.243941
\(516\) 0 0
\(517\) 1431.15 0.121744
\(518\) 0 0
\(519\) 4023.00 0.340251
\(520\) 0 0
\(521\) −17016.8 −1.43094 −0.715468 0.698645i \(-0.753787\pi\)
−0.715468 + 0.698645i \(0.753787\pi\)
\(522\) 0 0
\(523\) −5814.62 −0.486148 −0.243074 0.970008i \(-0.578156\pi\)
−0.243074 + 0.970008i \(0.578156\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6545.92 0.541071
\(528\) 0 0
\(529\) −7450.98 −0.612393
\(530\) 0 0
\(531\) 3244.42 0.265152
\(532\) 0 0
\(533\) 23835.8 1.93704
\(534\) 0 0
\(535\) 16372.8 1.32310
\(536\) 0 0
\(537\) −3375.44 −0.271250
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −10069.2 −0.800199 −0.400100 0.916472i \(-0.631025\pi\)
−0.400100 + 0.916472i \(0.631025\pi\)
\(542\) 0 0
\(543\) 10605.1 0.838137
\(544\) 0 0
\(545\) 10642.5 0.836466
\(546\) 0 0
\(547\) 7437.51 0.581362 0.290681 0.956820i \(-0.406118\pi\)
0.290681 + 0.956820i \(0.406118\pi\)
\(548\) 0 0
\(549\) −7206.53 −0.560232
\(550\) 0 0
\(551\) 7505.06 0.580265
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1349.23 0.103192
\(556\) 0 0
\(557\) 21787.1 1.65736 0.828680 0.559723i \(-0.189092\pi\)
0.828680 + 0.559723i \(0.189092\pi\)
\(558\) 0 0
\(559\) 23280.5 1.76147
\(560\) 0 0
\(561\) 2081.77 0.156671
\(562\) 0 0
\(563\) 2572.48 0.192570 0.0962851 0.995354i \(-0.469304\pi\)
0.0962851 + 0.995354i \(0.469304\pi\)
\(564\) 0 0
\(565\) 10005.2 0.744991
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17580.6 −1.29528 −0.647642 0.761945i \(-0.724245\pi\)
−0.647642 + 0.761945i \(0.724245\pi\)
\(570\) 0 0
\(571\) −7220.75 −0.529210 −0.264605 0.964357i \(-0.585242\pi\)
−0.264605 + 0.964357i \(0.585242\pi\)
\(572\) 0 0
\(573\) −7919.67 −0.577398
\(574\) 0 0
\(575\) −787.679 −0.0571278
\(576\) 0 0
\(577\) −11155.1 −0.804839 −0.402419 0.915455i \(-0.631831\pi\)
−0.402419 + 0.915455i \(0.631831\pi\)
\(578\) 0 0
\(579\) 3141.16 0.225461
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3493.50 0.248175
\(584\) 0 0
\(585\) −7281.67 −0.514633
\(586\) 0 0
\(587\) −15216.2 −1.06992 −0.534958 0.844879i \(-0.679673\pi\)
−0.534958 + 0.844879i \(0.679673\pi\)
\(588\) 0 0
\(589\) 5366.33 0.375409
\(590\) 0 0
\(591\) −13756.3 −0.957462
\(592\) 0 0
\(593\) −3843.89 −0.266188 −0.133094 0.991103i \(-0.542491\pi\)
−0.133094 + 0.991103i \(0.542491\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2492.23 0.170854
\(598\) 0 0
\(599\) −18747.0 −1.27876 −0.639382 0.768889i \(-0.720810\pi\)
−0.639382 + 0.768889i \(0.720810\pi\)
\(600\) 0 0
\(601\) 9864.63 0.669529 0.334764 0.942302i \(-0.391343\pi\)
0.334764 + 0.942302i \(0.391343\pi\)
\(602\) 0 0
\(603\) 362.057 0.0244513
\(604\) 0 0
\(605\) −13710.1 −0.921314
\(606\) 0 0
\(607\) 22292.8 1.49067 0.745336 0.666689i \(-0.232289\pi\)
0.745336 + 0.666689i \(0.232289\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16331.9 −1.08137
\(612\) 0 0
\(613\) 6046.78 0.398413 0.199206 0.979958i \(-0.436164\pi\)
0.199206 + 0.979958i \(0.436164\pi\)
\(614\) 0 0
\(615\) −10034.0 −0.657901
\(616\) 0 0
\(617\) −9383.68 −0.612273 −0.306137 0.951988i \(-0.599036\pi\)
−0.306137 + 0.951988i \(0.599036\pi\)
\(618\) 0 0
\(619\) −3989.50 −0.259049 −0.129525 0.991576i \(-0.541345\pi\)
−0.129525 + 0.991576i \(0.541345\pi\)
\(620\) 0 0
\(621\) 1854.18 0.119816
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −14059.7 −0.899821
\(626\) 0 0
\(627\) 1706.63 0.108702
\(628\) 0 0
\(629\) 4401.88 0.279038
\(630\) 0 0
\(631\) −11434.0 −0.721363 −0.360681 0.932689i \(-0.617456\pi\)
−0.360681 + 0.932689i \(0.617456\pi\)
\(632\) 0 0
\(633\) 7890.24 0.495432
\(634\) 0 0
\(635\) 17274.7 1.07957
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2689.03 0.166473
\(640\) 0 0
\(641\) −15097.3 −0.930274 −0.465137 0.885239i \(-0.653995\pi\)
−0.465137 + 0.885239i \(0.653995\pi\)
\(642\) 0 0
\(643\) 4170.19 0.255764 0.127882 0.991789i \(-0.459182\pi\)
0.127882 + 0.991789i \(0.459182\pi\)
\(644\) 0 0
\(645\) −9800.23 −0.598269
\(646\) 0 0
\(647\) −4563.88 −0.277318 −0.138659 0.990340i \(-0.544279\pi\)
−0.138659 + 0.990340i \(0.544279\pi\)
\(648\) 0 0
\(649\) 2398.71 0.145081
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3819.04 0.228868 0.114434 0.993431i \(-0.463495\pi\)
0.114434 + 0.993431i \(0.463495\pi\)
\(654\) 0 0
\(655\) 16178.9 0.965134
\(656\) 0 0
\(657\) −4654.13 −0.276370
\(658\) 0 0
\(659\) 4326.02 0.255717 0.127859 0.991792i \(-0.459190\pi\)
0.127859 + 0.991792i \(0.459190\pi\)
\(660\) 0 0
\(661\) 29317.8 1.72516 0.862579 0.505923i \(-0.168848\pi\)
0.862579 + 0.505923i \(0.168848\pi\)
\(662\) 0 0
\(663\) −23756.6 −1.39160
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6028.43 0.349958
\(668\) 0 0
\(669\) −2591.96 −0.149792
\(670\) 0 0
\(671\) −5328.02 −0.306536
\(672\) 0 0
\(673\) −5483.56 −0.314080 −0.157040 0.987592i \(-0.550195\pi\)
−0.157040 + 0.987592i \(0.550195\pi\)
\(674\) 0 0
\(675\) −309.689 −0.0176591
\(676\) 0 0
\(677\) 12069.3 0.685174 0.342587 0.939486i \(-0.388697\pi\)
0.342587 + 0.939486i \(0.388697\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −12481.1 −0.702314
\(682\) 0 0
\(683\) −30796.7 −1.72533 −0.862667 0.505772i \(-0.831208\pi\)
−0.862667 + 0.505772i \(0.831208\pi\)
\(684\) 0 0
\(685\) −26477.7 −1.47688
\(686\) 0 0
\(687\) 543.631 0.0301904
\(688\) 0 0
\(689\) −39866.9 −2.20436
\(690\) 0 0
\(691\) −6501.96 −0.357954 −0.178977 0.983853i \(-0.557279\pi\)
−0.178977 + 0.983853i \(0.557279\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −17638.0 −0.962656
\(696\) 0 0
\(697\) −32736.1 −1.77901
\(698\) 0 0
\(699\) 6451.34 0.349088
\(700\) 0 0
\(701\) −2235.98 −0.120473 −0.0602367 0.998184i \(-0.519186\pi\)
−0.0602367 + 0.998184i \(0.519186\pi\)
\(702\) 0 0
\(703\) 3608.66 0.193603
\(704\) 0 0
\(705\) 6875.11 0.367279
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 35564.5 1.88385 0.941926 0.335820i \(-0.109014\pi\)
0.941926 + 0.335820i \(0.109014\pi\)
\(710\) 0 0
\(711\) 11002.2 0.580330
\(712\) 0 0
\(713\) 4310.50 0.226409
\(714\) 0 0
\(715\) −5383.57 −0.281586
\(716\) 0 0
\(717\) 20818.4 1.08435
\(718\) 0 0
\(719\) −9924.15 −0.514754 −0.257377 0.966311i \(-0.582858\pi\)
−0.257377 + 0.966311i \(0.582858\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 618.509 0.0318155
\(724\) 0 0
\(725\) −1006.88 −0.0515788
\(726\) 0 0
\(727\) 880.081 0.0448974 0.0224487 0.999748i \(-0.492854\pi\)
0.0224487 + 0.999748i \(0.492854\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −31973.5 −1.61776
\(732\) 0 0
\(733\) 17336.2 0.873568 0.436784 0.899566i \(-0.356117\pi\)
0.436784 + 0.899566i \(0.356117\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 267.681 0.0133787
\(738\) 0 0
\(739\) 24586.4 1.22385 0.611924 0.790917i \(-0.290396\pi\)
0.611924 + 0.790917i \(0.290396\pi\)
\(740\) 0 0
\(741\) −19475.6 −0.965527
\(742\) 0 0
\(743\) −14579.3 −0.719870 −0.359935 0.932977i \(-0.617201\pi\)
−0.359935 + 0.932977i \(0.617201\pi\)
\(744\) 0 0
\(745\) 3731.30 0.183496
\(746\) 0 0
\(747\) 11957.0 0.585652
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16086.0 −0.781604 −0.390802 0.920475i \(-0.627802\pi\)
−0.390802 + 0.920475i \(0.627802\pi\)
\(752\) 0 0
\(753\) 15035.8 0.727669
\(754\) 0 0
\(755\) −35568.0 −1.71451
\(756\) 0 0
\(757\) −37017.3 −1.77730 −0.888651 0.458584i \(-0.848357\pi\)
−0.888651 + 0.458584i \(0.848357\pi\)
\(758\) 0 0
\(759\) 1370.85 0.0655584
\(760\) 0 0
\(761\) −24606.5 −1.17212 −0.586060 0.810267i \(-0.699322\pi\)
−0.586060 + 0.810267i \(0.699322\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 10000.6 0.472646
\(766\) 0 0
\(767\) −27373.4 −1.28865
\(768\) 0 0
\(769\) 12715.6 0.596275 0.298137 0.954523i \(-0.403635\pi\)
0.298137 + 0.954523i \(0.403635\pi\)
\(770\) 0 0
\(771\) 17589.2 0.821610
\(772\) 0 0
\(773\) −3174.23 −0.147696 −0.0738480 0.997270i \(-0.523528\pi\)
−0.0738480 + 0.997270i \(0.523528\pi\)
\(774\) 0 0
\(775\) −719.949 −0.0333695
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −26837.0 −1.23432
\(780\) 0 0
\(781\) 1988.08 0.0910874
\(782\) 0 0
\(783\) 2370.18 0.108178
\(784\) 0 0
\(785\) 18578.9 0.844726
\(786\) 0 0
\(787\) −15569.9 −0.705218 −0.352609 0.935771i \(-0.614705\pi\)
−0.352609 + 0.935771i \(0.614705\pi\)
\(788\) 0 0
\(789\) 16918.7 0.763398
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 60801.9 2.72275
\(794\) 0 0
\(795\) 16782.5 0.748695
\(796\) 0 0
\(797\) 31514.5 1.40063 0.700314 0.713835i \(-0.253043\pi\)
0.700314 + 0.713835i \(0.253043\pi\)
\(798\) 0 0
\(799\) 22430.2 0.993146
\(800\) 0 0
\(801\) 5759.44 0.254057
\(802\) 0 0
\(803\) −3440.95 −0.151219
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8929.90 0.389526
\(808\) 0 0
\(809\) 11029.1 0.479310 0.239655 0.970858i \(-0.422966\pi\)
0.239655 + 0.970858i \(0.422966\pi\)
\(810\) 0 0
\(811\) −20830.5 −0.901920 −0.450960 0.892544i \(-0.648918\pi\)
−0.450960 + 0.892544i \(0.648918\pi\)
\(812\) 0 0
\(813\) −8422.00 −0.363312
\(814\) 0 0
\(815\) −36563.4 −1.57149
\(816\) 0 0
\(817\) −26211.8 −1.12244
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9349.06 −0.397423 −0.198712 0.980058i \(-0.563676\pi\)
−0.198712 + 0.980058i \(0.563676\pi\)
\(822\) 0 0
\(823\) 32890.6 1.39307 0.696533 0.717525i \(-0.254725\pi\)
0.696533 + 0.717525i \(0.254725\pi\)
\(824\) 0 0
\(825\) −228.963 −0.00966238
\(826\) 0 0
\(827\) 13081.2 0.550033 0.275016 0.961440i \(-0.411317\pi\)
0.275016 + 0.961440i \(0.411317\pi\)
\(828\) 0 0
\(829\) −28791.2 −1.20622 −0.603112 0.797657i \(-0.706073\pi\)
−0.603112 + 0.797657i \(0.706073\pi\)
\(830\) 0 0
\(831\) −5755.17 −0.240246
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 36025.3 1.49306
\(836\) 0 0
\(837\) 1694.74 0.0699868
\(838\) 0 0
\(839\) −36766.8 −1.51291 −0.756455 0.654046i \(-0.773070\pi\)
−0.756455 + 0.654046i \(0.773070\pi\)
\(840\) 0 0
\(841\) −16682.9 −0.684034
\(842\) 0 0
\(843\) 15628.5 0.638523
\(844\) 0 0
\(845\) 38026.7 1.54812
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 22490.1 0.909138
\(850\) 0 0
\(851\) 2898.65 0.116762
\(852\) 0 0
\(853\) −5899.55 −0.236808 −0.118404 0.992966i \(-0.537778\pi\)
−0.118404 + 0.992966i \(0.537778\pi\)
\(854\) 0 0
\(855\) 8198.51 0.327934
\(856\) 0 0
\(857\) −42277.0 −1.68513 −0.842564 0.538596i \(-0.818955\pi\)
−0.842564 + 0.538596i \(0.818955\pi\)
\(858\) 0 0
\(859\) −6343.46 −0.251963 −0.125981 0.992033i \(-0.540208\pi\)
−0.125981 + 0.992033i \(0.540208\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6929.78 −0.273340 −0.136670 0.990617i \(-0.543640\pi\)
−0.136670 + 0.990617i \(0.543640\pi\)
\(864\) 0 0
\(865\) 14288.4 0.561643
\(866\) 0 0
\(867\) 17888.3 0.700715
\(868\) 0 0
\(869\) 8134.27 0.317533
\(870\) 0 0
\(871\) −3054.69 −0.118834
\(872\) 0 0
\(873\) −12830.9 −0.497433
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3287.88 0.126595 0.0632976 0.997995i \(-0.479838\pi\)
0.0632976 + 0.997995i \(0.479838\pi\)
\(878\) 0 0
\(879\) 10076.8 0.386670
\(880\) 0 0
\(881\) 46875.9 1.79261 0.896304 0.443439i \(-0.146242\pi\)
0.896304 + 0.443439i \(0.146242\pi\)
\(882\) 0 0
\(883\) −42479.4 −1.61897 −0.809483 0.587144i \(-0.800252\pi\)
−0.809483 + 0.587144i \(0.800252\pi\)
\(884\) 0 0
\(885\) 11523.2 0.437680
\(886\) 0 0
\(887\) 3680.15 0.139309 0.0696547 0.997571i \(-0.477810\pi\)
0.0696547 + 0.997571i \(0.477810\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 538.973 0.0202652
\(892\) 0 0
\(893\) 18388.2 0.689069
\(894\) 0 0
\(895\) −11988.5 −0.447745
\(896\) 0 0
\(897\) −15643.8 −0.582309
\(898\) 0 0
\(899\) 5510.07 0.204417
\(900\) 0 0
\(901\) 54753.1 2.02452
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 37666.0 1.38349
\(906\) 0 0
\(907\) 7102.18 0.260005 0.130002 0.991514i \(-0.458502\pi\)
0.130002 + 0.991514i \(0.458502\pi\)
\(908\) 0 0
\(909\) 8929.53 0.325824
\(910\) 0 0
\(911\) −38119.0 −1.38632 −0.693161 0.720783i \(-0.743782\pi\)
−0.693161 + 0.720783i \(0.743782\pi\)
\(912\) 0 0
\(913\) 8840.17 0.320446
\(914\) 0 0
\(915\) −25595.3 −0.924760
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −17190.6 −0.617045 −0.308522 0.951217i \(-0.599834\pi\)
−0.308522 + 0.951217i \(0.599834\pi\)
\(920\) 0 0
\(921\) −21992.8 −0.786847
\(922\) 0 0
\(923\) −22687.5 −0.809065
\(924\) 0 0
\(925\) −484.139 −0.0172091
\(926\) 0 0
\(927\) 2408.15 0.0853225
\(928\) 0 0
\(929\) −32256.5 −1.13918 −0.569591 0.821928i \(-0.692899\pi\)
−0.569591 + 0.821928i \(0.692899\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −6115.59 −0.214593
\(934\) 0 0
\(935\) 7393.80 0.258613
\(936\) 0 0
\(937\) −26213.6 −0.913940 −0.456970 0.889482i \(-0.651065\pi\)
−0.456970 + 0.889482i \(0.651065\pi\)
\(938\) 0 0
\(939\) −12416.6 −0.431524
\(940\) 0 0
\(941\) −34240.1 −1.18618 −0.593089 0.805137i \(-0.702092\pi\)
−0.593089 + 0.805137i \(0.702092\pi\)
\(942\) 0 0
\(943\) −21556.8 −0.744418
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36004.5 −1.23547 −0.617735 0.786386i \(-0.711950\pi\)
−0.617735 + 0.786386i \(0.711950\pi\)
\(948\) 0 0
\(949\) 39267.2 1.34317
\(950\) 0 0
\(951\) −22656.8 −0.772551
\(952\) 0 0
\(953\) −29488.1 −1.00232 −0.501161 0.865354i \(-0.667093\pi\)
−0.501161 + 0.865354i \(0.667093\pi\)
\(954\) 0 0
\(955\) −28128.2 −0.953095
\(956\) 0 0
\(957\) 1752.35 0.0591905
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −25851.1 −0.867750
\(962\) 0 0
\(963\) 13829.6 0.462776
\(964\) 0 0
\(965\) 11156.4 0.372163
\(966\) 0 0
\(967\) 56009.0 1.86260 0.931298 0.364259i \(-0.118678\pi\)
0.931298 + 0.364259i \(0.118678\pi\)
\(968\) 0 0
\(969\) 26747.8 0.886753
\(970\) 0 0
\(971\) 13576.9 0.448715 0.224358 0.974507i \(-0.427972\pi\)
0.224358 + 0.974507i \(0.427972\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 2612.86 0.0858241
\(976\) 0 0
\(977\) −31775.1 −1.04051 −0.520254 0.854011i \(-0.674163\pi\)
−0.520254 + 0.854011i \(0.674163\pi\)
\(978\) 0 0
\(979\) 4258.14 0.139010
\(980\) 0 0
\(981\) 8989.38 0.292568
\(982\) 0 0
\(983\) 32755.4 1.06280 0.531401 0.847120i \(-0.321666\pi\)
0.531401 + 0.847120i \(0.321666\pi\)
\(984\) 0 0
\(985\) −48858.2 −1.58046
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −21054.6 −0.676944
\(990\) 0 0
\(991\) 40797.0 1.30773 0.653865 0.756611i \(-0.273146\pi\)
0.653865 + 0.756611i \(0.273146\pi\)
\(992\) 0 0
\(993\) −16733.1 −0.534752
\(994\) 0 0
\(995\) 8851.60 0.282025
\(996\) 0 0
\(997\) 17370.5 0.551784 0.275892 0.961189i \(-0.411027\pi\)
0.275892 + 0.961189i \(0.411027\pi\)
\(998\) 0 0
\(999\) 1139.65 0.0360931
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 2352.4.a.cq.1.3 4
4.3 odd 2 588.4.a.j.1.3 4
7.6 odd 2 2352.4.a.cl.1.2 4
12.11 even 2 1764.4.a.bc.1.2 4
28.3 even 6 588.4.i.k.373.3 8
28.11 odd 6 588.4.i.l.373.2 8
28.19 even 6 588.4.i.k.361.3 8
28.23 odd 6 588.4.i.l.361.2 8
28.27 even 2 588.4.a.k.1.2 yes 4
84.11 even 6 1764.4.k.bb.1549.3 8
84.23 even 6 1764.4.k.bb.361.3 8
84.47 odd 6 1764.4.k.bd.361.2 8
84.59 odd 6 1764.4.k.bd.1549.2 8
84.83 odd 2 1764.4.a.ba.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.a.j.1.3 4 4.3 odd 2
588.4.a.k.1.2 yes 4 28.27 even 2
588.4.i.k.361.3 8 28.19 even 6
588.4.i.k.373.3 8 28.3 even 6
588.4.i.l.361.2 8 28.23 odd 6
588.4.i.l.373.2 8 28.11 odd 6
1764.4.a.ba.1.3 4 84.83 odd 2
1764.4.a.bc.1.2 4 12.11 even 2
1764.4.k.bb.361.3 8 84.23 even 6
1764.4.k.bb.1549.3 8 84.11 even 6
1764.4.k.bd.361.2 8 84.47 odd 6
1764.4.k.bd.1549.2 8 84.59 odd 6
2352.4.a.cl.1.2 4 7.6 odd 2
2352.4.a.cq.1.3 4 1.1 even 1 trivial