Properties

Label 4560.2.a.bu.1.3
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1524.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.13264\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +4.81342 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +4.81342 q^{7} +1.00000 q^{9} +1.45186 q^{11} -2.81342 q^{13} +1.00000 q^{15} +6.26527 q^{17} -1.00000 q^{19} +4.81342 q^{21} -6.26527 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.548143 q^{29} -8.26527 q^{31} +1.45186 q^{33} +4.81342 q^{35} +6.81342 q^{37} -2.81342 q^{39} +4.54814 q^{41} +7.71713 q^{43} +1.00000 q^{45} +10.2653 q^{47} +16.1690 q^{49} +6.26527 q^{51} -0.265275 q^{53} +1.45186 q^{55} -1.00000 q^{57} -2.90371 q^{59} +11.3616 q^{61} +4.81342 q^{63} -2.81342 q^{65} -6.26527 q^{69} -6.90371 q^{71} -6.53055 q^{73} +1.00000 q^{75} +6.98840 q^{77} -10.7231 q^{79} +1.00000 q^{81} -9.16899 q^{83} +6.26527 q^{85} +0.548143 q^{87} -18.7055 q^{89} -13.5422 q^{91} -8.26527 q^{93} -1.00000 q^{95} -6.81342 q^{97} +1.45186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} + 6 q^{39} + 16 q^{41} + 4 q^{43} + 3 q^{45} + 14 q^{47} + 27 q^{49} + 2 q^{51} + 16 q^{53} + 2 q^{55} - 3 q^{57} - 4 q^{59} + 22 q^{61} + 6 q^{65} - 2 q^{69} - 16 q^{71} + 14 q^{73} + 3 q^{75} - 20 q^{77} - 8 q^{79} + 3 q^{81} - 6 q^{83} + 2 q^{85} + 4 q^{87} + 4 q^{89} - 48 q^{91} - 8 q^{93} - 3 q^{95} - 6 q^{97} + 2 q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.81342 1.81930 0.909650 0.415375i \(-0.136350\pi\)
0.909650 + 0.415375i \(0.136350\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.45186 0.437751 0.218876 0.975753i \(-0.429761\pi\)
0.218876 + 0.975753i \(0.429761\pi\)
\(12\) 0 0
\(13\) −2.81342 −0.780302 −0.390151 0.920751i \(-0.627577\pi\)
−0.390151 + 0.920751i \(0.627577\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 6.26527 1.51955 0.759776 0.650185i \(-0.225308\pi\)
0.759776 + 0.650185i \(0.225308\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 4.81342 1.05037
\(22\) 0 0
\(23\) −6.26527 −1.30640 −0.653200 0.757185i \(-0.726574\pi\)
−0.653200 + 0.757185i \(0.726574\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.548143 0.101788 0.0508938 0.998704i \(-0.483793\pi\)
0.0508938 + 0.998704i \(0.483793\pi\)
\(30\) 0 0
\(31\) −8.26527 −1.48449 −0.742244 0.670130i \(-0.766238\pi\)
−0.742244 + 0.670130i \(0.766238\pi\)
\(32\) 0 0
\(33\) 1.45186 0.252736
\(34\) 0 0
\(35\) 4.81342 0.813616
\(36\) 0 0
\(37\) 6.81342 1.12012 0.560059 0.828452i \(-0.310778\pi\)
0.560059 + 0.828452i \(0.310778\pi\)
\(38\) 0 0
\(39\) −2.81342 −0.450507
\(40\) 0 0
\(41\) 4.54814 0.710301 0.355150 0.934809i \(-0.384430\pi\)
0.355150 + 0.934809i \(0.384430\pi\)
\(42\) 0 0
\(43\) 7.71713 1.17685 0.588426 0.808551i \(-0.299748\pi\)
0.588426 + 0.808551i \(0.299748\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 10.2653 1.49734 0.748672 0.662940i \(-0.230692\pi\)
0.748672 + 0.662940i \(0.230692\pi\)
\(48\) 0 0
\(49\) 16.1690 2.30986
\(50\) 0 0
\(51\) 6.26527 0.877314
\(52\) 0 0
\(53\) −0.265275 −0.0364383 −0.0182192 0.999834i \(-0.505800\pi\)
−0.0182192 + 0.999834i \(0.505800\pi\)
\(54\) 0 0
\(55\) 1.45186 0.195768
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −2.90371 −0.378031 −0.189016 0.981974i \(-0.560530\pi\)
−0.189016 + 0.981974i \(0.560530\pi\)
\(60\) 0 0
\(61\) 11.3616 1.45470 0.727349 0.686267i \(-0.240752\pi\)
0.727349 + 0.686267i \(0.240752\pi\)
\(62\) 0 0
\(63\) 4.81342 0.606434
\(64\) 0 0
\(65\) −2.81342 −0.348962
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −6.26527 −0.754250
\(70\) 0 0
\(71\) −6.90371 −0.819320 −0.409660 0.912238i \(-0.634353\pi\)
−0.409660 + 0.912238i \(0.634353\pi\)
\(72\) 0 0
\(73\) −6.53055 −0.764343 −0.382172 0.924091i \(-0.624824\pi\)
−0.382172 + 0.924091i \(0.624824\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 6.98840 0.796402
\(78\) 0 0
\(79\) −10.7231 −1.20645 −0.603223 0.797573i \(-0.706117\pi\)
−0.603223 + 0.797573i \(0.706117\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −9.16899 −1.00643 −0.503214 0.864162i \(-0.667849\pi\)
−0.503214 + 0.864162i \(0.667849\pi\)
\(84\) 0 0
\(85\) 6.26527 0.679564
\(86\) 0 0
\(87\) 0.548143 0.0587671
\(88\) 0 0
\(89\) −18.7055 −1.98278 −0.991391 0.130935i \(-0.958202\pi\)
−0.991391 + 0.130935i \(0.958202\pi\)
\(90\) 0 0
\(91\) −13.5422 −1.41960
\(92\) 0 0
\(93\) −8.26527 −0.857069
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −6.81342 −0.691798 −0.345899 0.938272i \(-0.612426\pi\)
−0.345899 + 0.938272i \(0.612426\pi\)
\(98\) 0 0
\(99\) 1.45186 0.145917
\(100\) 0 0
\(101\) 17.4343 1.73477 0.867387 0.497634i \(-0.165798\pi\)
0.867387 + 0.497634i \(0.165798\pi\)
\(102\) 0 0
\(103\) −16.5305 −1.62880 −0.814402 0.580301i \(-0.802935\pi\)
−0.814402 + 0.580301i \(0.802935\pi\)
\(104\) 0 0
\(105\) 4.81342 0.469741
\(106\) 0 0
\(107\) 16.0000 1.54678 0.773389 0.633932i \(-0.218560\pi\)
0.773389 + 0.633932i \(0.218560\pi\)
\(108\) 0 0
\(109\) 3.09629 0.296570 0.148285 0.988945i \(-0.452625\pi\)
0.148285 + 0.988945i \(0.452625\pi\)
\(110\) 0 0
\(111\) 6.81342 0.646701
\(112\) 0 0
\(113\) 13.3616 1.25695 0.628475 0.777830i \(-0.283679\pi\)
0.628475 + 0.777830i \(0.283679\pi\)
\(114\) 0 0
\(115\) −6.26527 −0.584240
\(116\) 0 0
\(117\) −2.81342 −0.260101
\(118\) 0 0
\(119\) 30.1574 2.76452
\(120\) 0 0
\(121\) −8.89211 −0.808374
\(122\) 0 0
\(123\) 4.54814 0.410092
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 0 0
\(129\) 7.71713 0.679456
\(130\) 0 0
\(131\) 4.35557 0.380548 0.190274 0.981731i \(-0.439062\pi\)
0.190274 + 0.981731i \(0.439062\pi\)
\(132\) 0 0
\(133\) −4.81342 −0.417376
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 2.90371 0.246290 0.123145 0.992389i \(-0.460702\pi\)
0.123145 + 0.992389i \(0.460702\pi\)
\(140\) 0 0
\(141\) 10.2653 0.864492
\(142\) 0 0
\(143\) −4.08468 −0.341578
\(144\) 0 0
\(145\) 0.548143 0.0455208
\(146\) 0 0
\(147\) 16.1690 1.33360
\(148\) 0 0
\(149\) 6.53055 0.535003 0.267502 0.963557i \(-0.413802\pi\)
0.267502 + 0.963557i \(0.413802\pi\)
\(150\) 0 0
\(151\) −18.9884 −1.54525 −0.772627 0.634860i \(-0.781058\pi\)
−0.772627 + 0.634860i \(0.781058\pi\)
\(152\) 0 0
\(153\) 6.26527 0.506517
\(154\) 0 0
\(155\) −8.26527 −0.663883
\(156\) 0 0
\(157\) 16.1574 1.28950 0.644750 0.764394i \(-0.276962\pi\)
0.644750 + 0.764394i \(0.276962\pi\)
\(158\) 0 0
\(159\) −0.265275 −0.0210377
\(160\) 0 0
\(161\) −30.1574 −2.37673
\(162\) 0 0
\(163\) −10.4403 −0.817744 −0.408872 0.912592i \(-0.634078\pi\)
−0.408872 + 0.912592i \(0.634078\pi\)
\(164\) 0 0
\(165\) 1.45186 0.113027
\(166\) 0 0
\(167\) 13.8921 1.07500 0.537502 0.843263i \(-0.319368\pi\)
0.537502 + 0.843263i \(0.319368\pi\)
\(168\) 0 0
\(169\) −5.08468 −0.391129
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −2.63844 −0.200597 −0.100298 0.994957i \(-0.531980\pi\)
−0.100298 + 0.994957i \(0.531980\pi\)
\(174\) 0 0
\(175\) 4.81342 0.363860
\(176\) 0 0
\(177\) −2.90371 −0.218257
\(178\) 0 0
\(179\) 2.37316 0.177379 0.0886893 0.996059i \(-0.471732\pi\)
0.0886893 + 0.996059i \(0.471732\pi\)
\(180\) 0 0
\(181\) 0.373165 0.0277371 0.0138686 0.999904i \(-0.495585\pi\)
0.0138686 + 0.999904i \(0.495585\pi\)
\(182\) 0 0
\(183\) 11.3616 0.839871
\(184\) 0 0
\(185\) 6.81342 0.500932
\(186\) 0 0
\(187\) 9.09629 0.665186
\(188\) 0 0
\(189\) 4.81342 0.350125
\(190\) 0 0
\(191\) −4.17498 −0.302091 −0.151045 0.988527i \(-0.548264\pi\)
−0.151045 + 0.988527i \(0.548264\pi\)
\(192\) 0 0
\(193\) 15.3440 1.10448 0.552241 0.833684i \(-0.313773\pi\)
0.552241 + 0.833684i \(0.313773\pi\)
\(194\) 0 0
\(195\) −2.81342 −0.201473
\(196\) 0 0
\(197\) −22.2653 −1.58634 −0.793168 0.609003i \(-0.791570\pi\)
−0.793168 + 0.609003i \(0.791570\pi\)
\(198\) 0 0
\(199\) −6.15739 −0.436485 −0.218243 0.975895i \(-0.570032\pi\)
−0.218243 + 0.975895i \(0.570032\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.63844 0.185182
\(204\) 0 0
\(205\) 4.54814 0.317656
\(206\) 0 0
\(207\) −6.26527 −0.435467
\(208\) 0 0
\(209\) −1.45186 −0.100427
\(210\) 0 0
\(211\) 23.7842 1.63737 0.818687 0.574241i \(-0.194703\pi\)
0.818687 + 0.574241i \(0.194703\pi\)
\(212\) 0 0
\(213\) −6.90371 −0.473035
\(214\) 0 0
\(215\) 7.71713 0.526304
\(216\) 0 0
\(217\) −39.7842 −2.70073
\(218\) 0 0
\(219\) −6.53055 −0.441294
\(220\) 0 0
\(221\) −17.6268 −1.18571
\(222\) 0 0
\(223\) −16.5305 −1.10697 −0.553484 0.832860i \(-0.686702\pi\)
−0.553484 + 0.832860i \(0.686702\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 20.0847 1.33307 0.666534 0.745475i \(-0.267777\pi\)
0.666534 + 0.745475i \(0.267777\pi\)
\(228\) 0 0
\(229\) −27.8921 −1.84316 −0.921581 0.388185i \(-0.873102\pi\)
−0.921581 + 0.388185i \(0.873102\pi\)
\(230\) 0 0
\(231\) 6.98840 0.459803
\(232\) 0 0
\(233\) 20.3380 1.33239 0.666193 0.745780i \(-0.267923\pi\)
0.666193 + 0.745780i \(0.267923\pi\)
\(234\) 0 0
\(235\) 10.2653 0.669633
\(236\) 0 0
\(237\) −10.7231 −0.696542
\(238\) 0 0
\(239\) 0.921307 0.0595944 0.0297972 0.999556i \(-0.490514\pi\)
0.0297972 + 0.999556i \(0.490514\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 16.1690 1.03300
\(246\) 0 0
\(247\) 2.81342 0.179013
\(248\) 0 0
\(249\) −9.16899 −0.581061
\(250\) 0 0
\(251\) −15.6092 −0.985247 −0.492623 0.870243i \(-0.663962\pi\)
−0.492623 + 0.870243i \(0.663962\pi\)
\(252\) 0 0
\(253\) −9.09629 −0.571879
\(254\) 0 0
\(255\) 6.26527 0.392347
\(256\) 0 0
\(257\) 7.16899 0.447189 0.223595 0.974682i \(-0.428221\pi\)
0.223595 + 0.974682i \(0.428221\pi\)
\(258\) 0 0
\(259\) 32.7958 2.03783
\(260\) 0 0
\(261\) 0.548143 0.0339292
\(262\) 0 0
\(263\) 24.4227 1.50597 0.752983 0.658040i \(-0.228614\pi\)
0.752983 + 0.658040i \(0.228614\pi\)
\(264\) 0 0
\(265\) −0.265275 −0.0162957
\(266\) 0 0
\(267\) −18.7055 −1.14476
\(268\) 0 0
\(269\) −10.1750 −0.620379 −0.310190 0.950675i \(-0.600393\pi\)
−0.310190 + 0.950675i \(0.600393\pi\)
\(270\) 0 0
\(271\) −2.37316 −0.144159 −0.0720797 0.997399i \(-0.522964\pi\)
−0.0720797 + 0.997399i \(0.522964\pi\)
\(272\) 0 0
\(273\) −13.5422 −0.819608
\(274\) 0 0
\(275\) 1.45186 0.0875503
\(276\) 0 0
\(277\) −16.3380 −0.981654 −0.490827 0.871257i \(-0.663305\pi\)
−0.490827 + 0.871257i \(0.663305\pi\)
\(278\) 0 0
\(279\) −8.26527 −0.494829
\(280\) 0 0
\(281\) 5.64443 0.336718 0.168359 0.985726i \(-0.446153\pi\)
0.168359 + 0.985726i \(0.446153\pi\)
\(282\) 0 0
\(283\) 21.3440 1.26877 0.634384 0.773018i \(-0.281254\pi\)
0.634384 + 0.773018i \(0.281254\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) 21.8921 1.29225
\(288\) 0 0
\(289\) 22.2537 1.30904
\(290\) 0 0
\(291\) −6.81342 −0.399410
\(292\) 0 0
\(293\) 10.6384 0.621504 0.310752 0.950491i \(-0.399419\pi\)
0.310752 + 0.950491i \(0.399419\pi\)
\(294\) 0 0
\(295\) −2.90371 −0.169061
\(296\) 0 0
\(297\) 1.45186 0.0842453
\(298\) 0 0
\(299\) 17.6268 1.01939
\(300\) 0 0
\(301\) 37.1458 2.14105
\(302\) 0 0
\(303\) 17.4343 1.00157
\(304\) 0 0
\(305\) 11.3616 0.650561
\(306\) 0 0
\(307\) −22.1574 −1.26459 −0.632294 0.774728i \(-0.717887\pi\)
−0.632294 + 0.774728i \(0.717887\pi\)
\(308\) 0 0
\(309\) −16.5305 −0.940390
\(310\) 0 0
\(311\) −4.17498 −0.236741 −0.118371 0.992969i \(-0.537767\pi\)
−0.118371 + 0.992969i \(0.537767\pi\)
\(312\) 0 0
\(313\) −12.3732 −0.699373 −0.349686 0.936867i \(-0.613712\pi\)
−0.349686 + 0.936867i \(0.613712\pi\)
\(314\) 0 0
\(315\) 4.81342 0.271205
\(316\) 0 0
\(317\) −3.73473 −0.209763 −0.104882 0.994485i \(-0.533446\pi\)
−0.104882 + 0.994485i \(0.533446\pi\)
\(318\) 0 0
\(319\) 0.795825 0.0445576
\(320\) 0 0
\(321\) 16.0000 0.893033
\(322\) 0 0
\(323\) −6.26527 −0.348609
\(324\) 0 0
\(325\) −2.81342 −0.156060
\(326\) 0 0
\(327\) 3.09629 0.171225
\(328\) 0 0
\(329\) 49.4111 2.72412
\(330\) 0 0
\(331\) −14.9884 −0.823837 −0.411918 0.911221i \(-0.635141\pi\)
−0.411918 + 0.911221i \(0.635141\pi\)
\(332\) 0 0
\(333\) 6.81342 0.373373
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 32.9708 1.79603 0.898017 0.439961i \(-0.145008\pi\)
0.898017 + 0.439961i \(0.145008\pi\)
\(338\) 0 0
\(339\) 13.3616 0.725700
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 44.1342 2.38302
\(344\) 0 0
\(345\) −6.26527 −0.337311
\(346\) 0 0
\(347\) −35.1458 −1.88672 −0.943362 0.331765i \(-0.892356\pi\)
−0.943362 + 0.331765i \(0.892356\pi\)
\(348\) 0 0
\(349\) −4.19257 −0.224423 −0.112212 0.993684i \(-0.535793\pi\)
−0.112212 + 0.993684i \(0.535793\pi\)
\(350\) 0 0
\(351\) −2.81342 −0.150169
\(352\) 0 0
\(353\) −3.80743 −0.202649 −0.101325 0.994853i \(-0.532308\pi\)
−0.101325 + 0.994853i \(0.532308\pi\)
\(354\) 0 0
\(355\) −6.90371 −0.366411
\(356\) 0 0
\(357\) 30.1574 1.59610
\(358\) 0 0
\(359\) 1.27126 0.0670947 0.0335474 0.999437i \(-0.489320\pi\)
0.0335474 + 0.999437i \(0.489320\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −8.89211 −0.466715
\(364\) 0 0
\(365\) −6.53055 −0.341825
\(366\) 0 0
\(367\) −10.0903 −0.526709 −0.263355 0.964699i \(-0.584829\pi\)
−0.263355 + 0.964699i \(0.584829\pi\)
\(368\) 0 0
\(369\) 4.54814 0.236767
\(370\) 0 0
\(371\) −1.27688 −0.0662923
\(372\) 0 0
\(373\) −14.2829 −0.739539 −0.369769 0.929124i \(-0.620563\pi\)
−0.369769 + 0.929124i \(0.620563\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −1.54215 −0.0794250
\(378\) 0 0
\(379\) −20.7958 −1.06821 −0.534105 0.845418i \(-0.679351\pi\)
−0.534105 + 0.845418i \(0.679351\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) −3.46945 −0.177281 −0.0886403 0.996064i \(-0.528252\pi\)
−0.0886403 + 0.996064i \(0.528252\pi\)
\(384\) 0 0
\(385\) 6.98840 0.356162
\(386\) 0 0
\(387\) 7.71713 0.392284
\(388\) 0 0
\(389\) 8.72312 0.442280 0.221140 0.975242i \(-0.429022\pi\)
0.221140 + 0.975242i \(0.429022\pi\)
\(390\) 0 0
\(391\) −39.2537 −1.98514
\(392\) 0 0
\(393\) 4.35557 0.219710
\(394\) 0 0
\(395\) −10.7231 −0.539539
\(396\) 0 0
\(397\) 19.4462 0.975979 0.487989 0.872850i \(-0.337730\pi\)
0.487989 + 0.872850i \(0.337730\pi\)
\(398\) 0 0
\(399\) −4.81342 −0.240972
\(400\) 0 0
\(401\) 27.9824 1.39737 0.698687 0.715427i \(-0.253768\pi\)
0.698687 + 0.715427i \(0.253768\pi\)
\(402\) 0 0
\(403\) 23.2537 1.15835
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 9.89211 0.490334
\(408\) 0 0
\(409\) −25.9648 −1.28388 −0.641939 0.766756i \(-0.721870\pi\)
−0.641939 + 0.766756i \(0.721870\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) −13.9768 −0.687753
\(414\) 0 0
\(415\) −9.16899 −0.450088
\(416\) 0 0
\(417\) 2.90371 0.142196
\(418\) 0 0
\(419\) 28.7055 1.40236 0.701178 0.712986i \(-0.252658\pi\)
0.701178 + 0.712986i \(0.252658\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 10.2653 0.499115
\(424\) 0 0
\(425\) 6.26527 0.303910
\(426\) 0 0
\(427\) 54.6879 2.64653
\(428\) 0 0
\(429\) −4.08468 −0.197210
\(430\) 0 0
\(431\) −22.6879 −1.09284 −0.546420 0.837512i \(-0.684010\pi\)
−0.546420 + 0.837512i \(0.684010\pi\)
\(432\) 0 0
\(433\) 31.3440 1.50629 0.753147 0.657852i \(-0.228535\pi\)
0.753147 + 0.657852i \(0.228535\pi\)
\(434\) 0 0
\(435\) 0.548143 0.0262814
\(436\) 0 0
\(437\) 6.26527 0.299709
\(438\) 0 0
\(439\) 10.1926 0.486465 0.243232 0.969968i \(-0.421792\pi\)
0.243232 + 0.969968i \(0.421792\pi\)
\(440\) 0 0
\(441\) 16.1690 0.769952
\(442\) 0 0
\(443\) 13.5189 0.642304 0.321152 0.947028i \(-0.395930\pi\)
0.321152 + 0.947028i \(0.395930\pi\)
\(444\) 0 0
\(445\) −18.7055 −0.886727
\(446\) 0 0
\(447\) 6.53055 0.308884
\(448\) 0 0
\(449\) −15.9824 −0.754256 −0.377128 0.926161i \(-0.623088\pi\)
−0.377128 + 0.926161i \(0.623088\pi\)
\(450\) 0 0
\(451\) 6.60325 0.310935
\(452\) 0 0
\(453\) −18.9884 −0.892153
\(454\) 0 0
\(455\) −13.5422 −0.634866
\(456\) 0 0
\(457\) −23.0963 −1.08040 −0.540199 0.841537i \(-0.681651\pi\)
−0.540199 + 0.841537i \(0.681651\pi\)
\(458\) 0 0
\(459\) 6.26527 0.292438
\(460\) 0 0
\(461\) −29.2537 −1.36248 −0.681240 0.732060i \(-0.738559\pi\)
−0.681240 + 0.732060i \(0.738559\pi\)
\(462\) 0 0
\(463\) −15.5365 −0.722044 −0.361022 0.932557i \(-0.617572\pi\)
−0.361022 + 0.932557i \(0.617572\pi\)
\(464\) 0 0
\(465\) −8.26527 −0.383293
\(466\) 0 0
\(467\) 11.5422 0.534107 0.267054 0.963682i \(-0.413950\pi\)
0.267054 + 0.963682i \(0.413950\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 16.1574 0.744493
\(472\) 0 0
\(473\) 11.2042 0.515169
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −0.265275 −0.0121461
\(478\) 0 0
\(479\) 16.3556 0.747305 0.373653 0.927569i \(-0.378105\pi\)
0.373653 + 0.927569i \(0.378105\pi\)
\(480\) 0 0
\(481\) −19.1690 −0.874031
\(482\) 0 0
\(483\) −30.1574 −1.37221
\(484\) 0 0
\(485\) −6.81342 −0.309381
\(486\) 0 0
\(487\) −41.4111 −1.87651 −0.938257 0.345939i \(-0.887560\pi\)
−0.938257 + 0.345939i \(0.887560\pi\)
\(488\) 0 0
\(489\) −10.4403 −0.472125
\(490\) 0 0
\(491\) 11.0787 0.499974 0.249987 0.968249i \(-0.419574\pi\)
0.249987 + 0.968249i \(0.419574\pi\)
\(492\) 0 0
\(493\) 3.43426 0.154671
\(494\) 0 0
\(495\) 1.45186 0.0652561
\(496\) 0 0
\(497\) −33.2305 −1.49059
\(498\) 0 0
\(499\) 16.3500 0.731925 0.365962 0.930630i \(-0.380740\pi\)
0.365962 + 0.930630i \(0.380740\pi\)
\(500\) 0 0
\(501\) 13.8921 0.620654
\(502\) 0 0
\(503\) −11.8921 −0.530243 −0.265121 0.964215i \(-0.585412\pi\)
−0.265121 + 0.964215i \(0.585412\pi\)
\(504\) 0 0
\(505\) 17.4343 0.775815
\(506\) 0 0
\(507\) −5.08468 −0.225819
\(508\) 0 0
\(509\) 32.8981 1.45818 0.729091 0.684416i \(-0.239943\pi\)
0.729091 + 0.684416i \(0.239943\pi\)
\(510\) 0 0
\(511\) −31.4343 −1.39057
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −16.5305 −0.728423
\(516\) 0 0
\(517\) 14.9037 0.655465
\(518\) 0 0
\(519\) −2.63844 −0.115815
\(520\) 0 0
\(521\) −25.0787 −1.09872 −0.549359 0.835587i \(-0.685128\pi\)
−0.549359 + 0.835587i \(0.685128\pi\)
\(522\) 0 0
\(523\) −23.9648 −1.04791 −0.523954 0.851747i \(-0.675544\pi\)
−0.523954 + 0.851747i \(0.675544\pi\)
\(524\) 0 0
\(525\) 4.81342 0.210075
\(526\) 0 0
\(527\) −51.7842 −2.25576
\(528\) 0 0
\(529\) 16.2537 0.706681
\(530\) 0 0
\(531\) −2.90371 −0.126010
\(532\) 0 0
\(533\) −12.7958 −0.554249
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) 0 0
\(537\) 2.37316 0.102410
\(538\) 0 0
\(539\) 23.4751 1.01114
\(540\) 0 0
\(541\) 11.0611 0.475554 0.237777 0.971320i \(-0.423581\pi\)
0.237777 + 0.971320i \(0.423581\pi\)
\(542\) 0 0
\(543\) 0.373165 0.0160140
\(544\) 0 0
\(545\) 3.09629 0.132630
\(546\) 0 0
\(547\) −21.0963 −0.902012 −0.451006 0.892521i \(-0.648935\pi\)
−0.451006 + 0.892521i \(0.648935\pi\)
\(548\) 0 0
\(549\) 11.3616 0.484900
\(550\) 0 0
\(551\) −0.548143 −0.0233517
\(552\) 0 0
\(553\) −51.6149 −2.19489
\(554\) 0 0
\(555\) 6.81342 0.289213
\(556\) 0 0
\(557\) −39.5916 −1.67755 −0.838776 0.544477i \(-0.816728\pi\)
−0.838776 + 0.544477i \(0.816728\pi\)
\(558\) 0 0
\(559\) −21.7115 −0.918299
\(560\) 0 0
\(561\) 9.09629 0.384045
\(562\) 0 0
\(563\) 28.0847 1.18363 0.591814 0.806074i \(-0.298412\pi\)
0.591814 + 0.806074i \(0.298412\pi\)
\(564\) 0 0
\(565\) 13.3616 0.562125
\(566\) 0 0
\(567\) 4.81342 0.202145
\(568\) 0 0
\(569\) 41.0787 1.72211 0.861054 0.508513i \(-0.169805\pi\)
0.861054 + 0.508513i \(0.169805\pi\)
\(570\) 0 0
\(571\) 11.8194 0.494627 0.247313 0.968936i \(-0.420452\pi\)
0.247313 + 0.968936i \(0.420452\pi\)
\(572\) 0 0
\(573\) −4.17498 −0.174412
\(574\) 0 0
\(575\) −6.26527 −0.261280
\(576\) 0 0
\(577\) −29.4343 −1.22536 −0.612682 0.790329i \(-0.709909\pi\)
−0.612682 + 0.790329i \(0.709909\pi\)
\(578\) 0 0
\(579\) 15.3440 0.637674
\(580\) 0 0
\(581\) −44.1342 −1.83099
\(582\) 0 0
\(583\) −0.385141 −0.0159509
\(584\) 0 0
\(585\) −2.81342 −0.116321
\(586\) 0 0
\(587\) −31.7115 −1.30887 −0.654437 0.756116i \(-0.727094\pi\)
−0.654437 + 0.756116i \(0.727094\pi\)
\(588\) 0 0
\(589\) 8.26527 0.340565
\(590\) 0 0
\(591\) −22.2653 −0.915871
\(592\) 0 0
\(593\) 14.5305 0.596698 0.298349 0.954457i \(-0.403564\pi\)
0.298349 + 0.954457i \(0.403564\pi\)
\(594\) 0 0
\(595\) 30.1574 1.23633
\(596\) 0 0
\(597\) −6.15739 −0.252005
\(598\) 0 0
\(599\) −17.8074 −0.727592 −0.363796 0.931479i \(-0.618519\pi\)
−0.363796 + 0.931479i \(0.618519\pi\)
\(600\) 0 0
\(601\) 20.6879 0.843878 0.421939 0.906624i \(-0.361350\pi\)
0.421939 + 0.906624i \(0.361350\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.89211 −0.361516
\(606\) 0 0
\(607\) 17.2417 0.699819 0.349909 0.936784i \(-0.386212\pi\)
0.349909 + 0.936784i \(0.386212\pi\)
\(608\) 0 0
\(609\) 2.63844 0.106915
\(610\) 0 0
\(611\) −28.8805 −1.16838
\(612\) 0 0
\(613\) −3.62684 −0.146486 −0.0732432 0.997314i \(-0.523335\pi\)
−0.0732432 + 0.997314i \(0.523335\pi\)
\(614\) 0 0
\(615\) 4.54814 0.183399
\(616\) 0 0
\(617\) 23.3264 0.939084 0.469542 0.882910i \(-0.344419\pi\)
0.469542 + 0.882910i \(0.344419\pi\)
\(618\) 0 0
\(619\) 24.8805 1.00003 0.500016 0.866016i \(-0.333327\pi\)
0.500016 + 0.866016i \(0.333327\pi\)
\(620\) 0 0
\(621\) −6.26527 −0.251417
\(622\) 0 0
\(623\) −90.0375 −3.60728
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.45186 −0.0579816
\(628\) 0 0
\(629\) 42.6879 1.70208
\(630\) 0 0
\(631\) 36.3148 1.44567 0.722834 0.691022i \(-0.242839\pi\)
0.722834 + 0.691022i \(0.242839\pi\)
\(632\) 0 0
\(633\) 23.7842 0.945338
\(634\) 0 0
\(635\) −4.00000 −0.158735
\(636\) 0 0
\(637\) −45.4901 −1.80238
\(638\) 0 0
\(639\) −6.90371 −0.273107
\(640\) 0 0
\(641\) 14.3556 0.567011 0.283506 0.958971i \(-0.408503\pi\)
0.283506 + 0.958971i \(0.408503\pi\)
\(642\) 0 0
\(643\) −37.1634 −1.46558 −0.732790 0.680455i \(-0.761782\pi\)
−0.732790 + 0.680455i \(0.761782\pi\)
\(644\) 0 0
\(645\) 7.71713 0.303862
\(646\) 0 0
\(647\) −40.9532 −1.61004 −0.805018 0.593250i \(-0.797845\pi\)
−0.805018 + 0.593250i \(0.797845\pi\)
\(648\) 0 0
\(649\) −4.21578 −0.165484
\(650\) 0 0
\(651\) −39.7842 −1.55927
\(652\) 0 0
\(653\) −18.7958 −0.735537 −0.367769 0.929917i \(-0.619878\pi\)
−0.367769 + 0.929917i \(0.619878\pi\)
\(654\) 0 0
\(655\) 4.35557 0.170186
\(656\) 0 0
\(657\) −6.53055 −0.254781
\(658\) 0 0
\(659\) −8.53055 −0.332303 −0.166152 0.986100i \(-0.553134\pi\)
−0.166152 + 0.986100i \(0.553134\pi\)
\(660\) 0 0
\(661\) 13.4343 0.522532 0.261266 0.965267i \(-0.415860\pi\)
0.261266 + 0.965267i \(0.415860\pi\)
\(662\) 0 0
\(663\) −17.6268 −0.684570
\(664\) 0 0
\(665\) −4.81342 −0.186656
\(666\) 0 0
\(667\) −3.43426 −0.132975
\(668\) 0 0
\(669\) −16.5305 −0.639108
\(670\) 0 0
\(671\) 16.4954 0.636796
\(672\) 0 0
\(673\) −6.81342 −0.262638 −0.131319 0.991340i \(-0.541921\pi\)
−0.131319 + 0.991340i \(0.541921\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 47.3032 1.81801 0.909004 0.416787i \(-0.136844\pi\)
0.909004 + 0.416787i \(0.136844\pi\)
\(678\) 0 0
\(679\) −32.7958 −1.25859
\(680\) 0 0
\(681\) 20.0847 0.769647
\(682\) 0 0
\(683\) 11.2537 0.430610 0.215305 0.976547i \(-0.430925\pi\)
0.215305 + 0.976547i \(0.430925\pi\)
\(684\) 0 0
\(685\) −6.00000 −0.229248
\(686\) 0 0
\(687\) −27.8921 −1.06415
\(688\) 0 0
\(689\) 0.746329 0.0284329
\(690\) 0 0
\(691\) −9.62684 −0.366222 −0.183111 0.983092i \(-0.558617\pi\)
−0.183111 + 0.983092i \(0.558617\pi\)
\(692\) 0 0
\(693\) 6.98840 0.265467
\(694\) 0 0
\(695\) 2.90371 0.110144
\(696\) 0 0
\(697\) 28.4954 1.07934
\(698\) 0 0
\(699\) 20.3380 0.769253
\(700\) 0 0
\(701\) 16.7231 0.631624 0.315812 0.948822i \(-0.397723\pi\)
0.315812 + 0.948822i \(0.397723\pi\)
\(702\) 0 0
\(703\) −6.81342 −0.256973
\(704\) 0 0
\(705\) 10.2653 0.386613
\(706\) 0 0
\(707\) 83.9184 3.15608
\(708\) 0 0
\(709\) −32.9532 −1.23758 −0.618792 0.785555i \(-0.712378\pi\)
−0.618792 + 0.785555i \(0.712378\pi\)
\(710\) 0 0
\(711\) −10.7231 −0.402148
\(712\) 0 0
\(713\) 51.7842 1.93933
\(714\) 0 0
\(715\) −4.08468 −0.152758
\(716\) 0 0
\(717\) 0.921307 0.0344069
\(718\) 0 0
\(719\) 15.6444 0.583439 0.291719 0.956504i \(-0.405773\pi\)
0.291719 + 0.956504i \(0.405773\pi\)
\(720\) 0 0
\(721\) −79.5684 −2.96328
\(722\) 0 0
\(723\) −2.00000 −0.0743808
\(724\) 0 0
\(725\) 0.548143 0.0203575
\(726\) 0 0
\(727\) −24.2829 −0.900602 −0.450301 0.892877i \(-0.648683\pi\)
−0.450301 + 0.892877i \(0.648683\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 48.3500 1.78829
\(732\) 0 0
\(733\) −18.5305 −0.684441 −0.342221 0.939620i \(-0.611179\pi\)
−0.342221 + 0.939620i \(0.611179\pi\)
\(734\) 0 0
\(735\) 16.1690 0.596402
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 9.44624 0.347486 0.173743 0.984791i \(-0.444414\pi\)
0.173743 + 0.984791i \(0.444414\pi\)
\(740\) 0 0
\(741\) 2.81342 0.103353
\(742\) 0 0
\(743\) −41.9768 −1.53998 −0.769990 0.638056i \(-0.779739\pi\)
−0.769990 + 0.638056i \(0.779739\pi\)
\(744\) 0 0
\(745\) 6.53055 0.239261
\(746\) 0 0
\(747\) −9.16899 −0.335476
\(748\) 0 0
\(749\) 77.0147 2.81406
\(750\) 0 0
\(751\) −26.4578 −0.965461 −0.482730 0.875769i \(-0.660355\pi\)
−0.482730 + 0.875769i \(0.660355\pi\)
\(752\) 0 0
\(753\) −15.6092 −0.568832
\(754\) 0 0
\(755\) −18.9884 −0.691058
\(756\) 0 0
\(757\) 12.9037 0.468993 0.234497 0.972117i \(-0.424656\pi\)
0.234497 + 0.972117i \(0.424656\pi\)
\(758\) 0 0
\(759\) −9.09629 −0.330174
\(760\) 0 0
\(761\) −40.7231 −1.47621 −0.738106 0.674685i \(-0.764280\pi\)
−0.738106 + 0.674685i \(0.764280\pi\)
\(762\) 0 0
\(763\) 14.9037 0.539551
\(764\) 0 0
\(765\) 6.26527 0.226521
\(766\) 0 0
\(767\) 8.16936 0.294979
\(768\) 0 0
\(769\) −5.25367 −0.189452 −0.0947261 0.995503i \(-0.530198\pi\)
−0.0947261 + 0.995503i \(0.530198\pi\)
\(770\) 0 0
\(771\) 7.16899 0.258185
\(772\) 0 0
\(773\) −38.9884 −1.40232 −0.701158 0.713006i \(-0.747333\pi\)
−0.701158 + 0.713006i \(0.747333\pi\)
\(774\) 0 0
\(775\) −8.26527 −0.296897
\(776\) 0 0
\(777\) 32.7958 1.17654
\(778\) 0 0
\(779\) −4.54814 −0.162954
\(780\) 0 0
\(781\) −10.0232 −0.358659
\(782\) 0 0
\(783\) 0.548143 0.0195890
\(784\) 0 0
\(785\) 16.1574 0.576682
\(786\) 0 0
\(787\) 11.2889 0.402404 0.201202 0.979550i \(-0.435515\pi\)
0.201202 + 0.979550i \(0.435515\pi\)
\(788\) 0 0
\(789\) 24.4227 0.869470
\(790\) 0 0
\(791\) 64.3148 2.28677
\(792\) 0 0
\(793\) −31.9648 −1.13510
\(794\) 0 0
\(795\) −0.265275 −0.00940833
\(796\) 0 0
\(797\) −27.6995 −0.981168 −0.490584 0.871394i \(-0.663217\pi\)
−0.490584 + 0.871394i \(0.663217\pi\)
\(798\) 0 0
\(799\) 64.3148 2.27529
\(800\) 0 0
\(801\) −18.7055 −0.660927
\(802\) 0 0
\(803\) −9.48143 −0.334592
\(804\) 0 0
\(805\) −30.1574 −1.06291
\(806\) 0 0
\(807\) −10.1750 −0.358176
\(808\) 0 0
\(809\) −0.723121 −0.0254236 −0.0127118 0.999919i \(-0.504046\pi\)
−0.0127118 + 0.999919i \(0.504046\pi\)
\(810\) 0 0
\(811\) 42.5073 1.49263 0.746317 0.665590i \(-0.231820\pi\)
0.746317 + 0.665590i \(0.231820\pi\)
\(812\) 0 0
\(813\) −2.37316 −0.0832305
\(814\) 0 0
\(815\) −10.4403 −0.365706
\(816\) 0 0
\(817\) −7.71713 −0.269988
\(818\) 0 0
\(819\) −13.5422 −0.473201
\(820\) 0 0
\(821\) −4.90371 −0.171141 −0.0855704 0.996332i \(-0.527271\pi\)
−0.0855704 + 0.996332i \(0.527271\pi\)
\(822\) 0 0
\(823\) 1.37915 0.0480743 0.0240371 0.999711i \(-0.492348\pi\)
0.0240371 + 0.999711i \(0.492348\pi\)
\(824\) 0 0
\(825\) 1.45186 0.0505472
\(826\) 0 0
\(827\) 16.9764 0.590328 0.295164 0.955447i \(-0.404626\pi\)
0.295164 + 0.955447i \(0.404626\pi\)
\(828\) 0 0
\(829\) −13.4343 −0.466591 −0.233296 0.972406i \(-0.574951\pi\)
−0.233296 + 0.972406i \(0.574951\pi\)
\(830\) 0 0
\(831\) −16.3380 −0.566758
\(832\) 0 0
\(833\) 101.303 3.50995
\(834\) 0 0
\(835\) 13.8921 0.480756
\(836\) 0 0
\(837\) −8.26527 −0.285690
\(838\) 0 0
\(839\) 43.2185 1.49207 0.746034 0.665908i \(-0.231956\pi\)
0.746034 + 0.665908i \(0.231956\pi\)
\(840\) 0 0
\(841\) −28.6995 −0.989639
\(842\) 0 0
\(843\) 5.64443 0.194404
\(844\) 0 0
\(845\) −5.08468 −0.174918
\(846\) 0 0
\(847\) −42.8014 −1.47067
\(848\) 0 0
\(849\) 21.3440 0.732523
\(850\) 0 0
\(851\) −42.6879 −1.46332
\(852\) 0 0
\(853\) −19.0963 −0.653844 −0.326922 0.945051i \(-0.606012\pi\)
−0.326922 + 0.945051i \(0.606012\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) −25.5422 −0.872503 −0.436252 0.899825i \(-0.643694\pi\)
−0.436252 + 0.899825i \(0.643694\pi\)
\(858\) 0 0
\(859\) −51.3991 −1.75371 −0.876857 0.480751i \(-0.840364\pi\)
−0.876857 + 0.480751i \(0.840364\pi\)
\(860\) 0 0
\(861\) 21.8921 0.746081
\(862\) 0 0
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) −2.63844 −0.0897096
\(866\) 0 0
\(867\) 22.2537 0.755774
\(868\) 0 0
\(869\) −15.5684 −0.528123
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −6.81342 −0.230599
\(874\) 0 0
\(875\) 4.81342 0.162723
\(876\) 0 0
\(877\) −49.6819 −1.67764 −0.838820 0.544409i \(-0.816754\pi\)
−0.838820 + 0.544409i \(0.816754\pi\)
\(878\) 0 0
\(879\) 10.6384 0.358826
\(880\) 0 0
\(881\) −21.2889 −0.717240 −0.358620 0.933484i \(-0.616753\pi\)
−0.358620 + 0.933484i \(0.616753\pi\)
\(882\) 0 0
\(883\) 30.6208 1.03047 0.515237 0.857048i \(-0.327704\pi\)
0.515237 + 0.857048i \(0.327704\pi\)
\(884\) 0 0
\(885\) −2.90371 −0.0976073
\(886\) 0 0
\(887\) 33.0611 1.11008 0.555042 0.831823i \(-0.312702\pi\)
0.555042 + 0.831823i \(0.312702\pi\)
\(888\) 0 0
\(889\) −19.2537 −0.645747
\(890\) 0 0
\(891\) 1.45186 0.0486391
\(892\) 0 0
\(893\) −10.2653 −0.343514
\(894\) 0 0
\(895\) 2.37316 0.0793261
\(896\) 0 0
\(897\) 17.6268 0.588543
\(898\) 0 0
\(899\) −4.53055 −0.151102
\(900\) 0 0
\(901\) −1.66202 −0.0553699
\(902\) 0 0
\(903\) 37.1458 1.23613
\(904\) 0 0
\(905\) 0.373165 0.0124044
\(906\) 0 0
\(907\) −51.7490 −1.71830 −0.859149 0.511725i \(-0.829007\pi\)
−0.859149 + 0.511725i \(0.829007\pi\)
\(908\) 0 0
\(909\) 17.4343 0.578258
\(910\) 0 0
\(911\) −3.78422 −0.125377 −0.0626884 0.998033i \(-0.519967\pi\)
−0.0626884 + 0.998033i \(0.519967\pi\)
\(912\) 0 0
\(913\) −13.3121 −0.440565
\(914\) 0 0
\(915\) 11.3616 0.375602
\(916\) 0 0
\(917\) 20.9652 0.692331
\(918\) 0 0
\(919\) 2.72312 0.0898275 0.0449137 0.998991i \(-0.485699\pi\)
0.0449137 + 0.998991i \(0.485699\pi\)
\(920\) 0 0
\(921\) −22.1574 −0.730111
\(922\) 0 0
\(923\) 19.4230 0.639317
\(924\) 0 0
\(925\) 6.81342 0.224024
\(926\) 0 0
\(927\) −16.5305 −0.542934
\(928\) 0 0
\(929\) 19.8426 0.651015 0.325508 0.945539i \(-0.394465\pi\)
0.325508 + 0.945539i \(0.394465\pi\)
\(930\) 0 0
\(931\) −16.1690 −0.529917
\(932\) 0 0
\(933\) −4.17498 −0.136683
\(934\) 0 0
\(935\) 9.09629 0.297480
\(936\) 0 0
\(937\) −1.28886 −0.0421051 −0.0210525 0.999778i \(-0.506702\pi\)
−0.0210525 + 0.999778i \(0.506702\pi\)
\(938\) 0 0
\(939\) −12.3732 −0.403783
\(940\) 0 0
\(941\) −21.7898 −0.710328 −0.355164 0.934804i \(-0.615575\pi\)
−0.355164 + 0.934804i \(0.615575\pi\)
\(942\) 0 0
\(943\) −28.4954 −0.927937
\(944\) 0 0
\(945\) 4.81342 0.156580
\(946\) 0 0
\(947\) −12.6384 −0.410694 −0.205347 0.978689i \(-0.565832\pi\)
−0.205347 + 0.978689i \(0.565832\pi\)
\(948\) 0 0
\(949\) 18.3732 0.596418
\(950\) 0 0
\(951\) −3.73473 −0.121107
\(952\) 0 0
\(953\) 29.6763 0.961311 0.480655 0.876910i \(-0.340399\pi\)
0.480655 + 0.876910i \(0.340399\pi\)
\(954\) 0 0
\(955\) −4.17498 −0.135099
\(956\) 0 0
\(957\) 0.795825 0.0257254
\(958\) 0 0
\(959\) −28.8805 −0.932600
\(960\) 0 0
\(961\) 37.3148 1.20370
\(962\) 0 0
\(963\) 16.0000 0.515593
\(964\) 0 0
\(965\) 15.3440 0.493940
\(966\) 0 0
\(967\) −10.2597 −0.329928 −0.164964 0.986300i \(-0.552751\pi\)
−0.164964 + 0.986300i \(0.552751\pi\)
\(968\) 0 0
\(969\) −6.26527 −0.201270
\(970\) 0 0
\(971\) 49.5916 1.59147 0.795736 0.605644i \(-0.207084\pi\)
0.795736 + 0.605644i \(0.207084\pi\)
\(972\) 0 0
\(973\) 13.9768 0.448075
\(974\) 0 0
\(975\) −2.81342 −0.0901015
\(976\) 0 0
\(977\) −9.71152 −0.310699 −0.155349 0.987860i \(-0.549650\pi\)
−0.155349 + 0.987860i \(0.549650\pi\)
\(978\) 0 0
\(979\) −27.1578 −0.867966
\(980\) 0 0
\(981\) 3.09629 0.0988568
\(982\) 0 0
\(983\) −4.83101 −0.154085 −0.0770427 0.997028i \(-0.524548\pi\)
−0.0770427 + 0.997028i \(0.524548\pi\)
\(984\) 0 0
\(985\) −22.2653 −0.709431
\(986\) 0 0
\(987\) 49.4111 1.57277
\(988\) 0 0
\(989\) −48.3500 −1.53744
\(990\) 0 0
\(991\) 10.1926 0.323778 0.161889 0.986809i \(-0.448241\pi\)
0.161889 + 0.986809i \(0.448241\pi\)
\(992\) 0 0
\(993\) −14.9884 −0.475642
\(994\) 0 0
\(995\) −6.15739 −0.195202
\(996\) 0 0
\(997\) −17.9648 −0.568951 −0.284476 0.958683i \(-0.591820\pi\)
−0.284476 + 0.958683i \(0.591820\pi\)
\(998\) 0 0
\(999\) 6.81342 0.215567
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 4560.2.a.bu.1.3 3
4.3 odd 2 1140.2.a.f.1.1 3
12.11 even 2 3420.2.a.k.1.1 3
20.3 even 4 5700.2.f.p.3649.3 6
20.7 even 4 5700.2.f.p.3649.4 6
20.19 odd 2 5700.2.a.z.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1140.2.a.f.1.1 3 4.3 odd 2
3420.2.a.k.1.1 3 12.11 even 2
4560.2.a.bu.1.3 3 1.1 even 1 trivial
5700.2.a.z.1.3 3 20.19 odd 2
5700.2.f.p.3649.3 6 20.3 even 4
5700.2.f.p.3649.4 6 20.7 even 4