Properties

Label 5520.2.a.bx.1.2
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.0774219157\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.71083 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +1.71083 q^{7} +1.00000 q^{9} -2.52444 q^{11} -6.72999 q^{13} +1.00000 q^{15} +4.44082 q^{17} +3.10278 q^{19} -1.71083 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.01916 q^{29} +1.13249 q^{31} +2.52444 q^{33} -1.71083 q^{35} +2.49472 q^{37} +6.72999 q^{39} -7.39194 q^{41} +6.78389 q^{43} -1.00000 q^{45} +13.3083 q^{47} -4.07306 q^{49} -4.44082 q^{51} -12.4408 q^{53} +2.52444 q^{55} -3.10278 q^{57} -11.0192 q^{59} -2.89722 q^{61} +1.71083 q^{63} +6.72999 q^{65} +0.0836184 q^{67} +1.00000 q^{69} +13.8030 q^{71} -5.57331 q^{73} -1.00000 q^{75} -4.31889 q^{77} +1.00000 q^{81} -1.97028 q^{83} -4.44082 q^{85} -5.01916 q^{87} -7.62721 q^{89} -11.5139 q^{91} -1.13249 q^{93} -3.10278 q^{95} +9.83276 q^{97} -2.52444 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{5} + 6 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{15} - 6 q^{17} + 2 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} + 6 q^{31} + 2 q^{33} - 6 q^{35} - 8 q^{37} - 14 q^{41} + 4 q^{43} - 3 q^{45} + 18 q^{47} + 5 q^{49} + 6 q^{51} - 18 q^{53} + 2 q^{55} - 2 q^{57} - 12 q^{59} - 16 q^{61} + 6 q^{63} + 14 q^{67} + 3 q^{69} + 4 q^{71} - 3 q^{75} - 22 q^{77} + 3 q^{81} + 4 q^{83} + 6 q^{85} + 6 q^{87} - 10 q^{89} + 2 q^{91} - 6 q^{93} - 2 q^{95} + 2 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 1.71083 0.646634 0.323317 0.946291i \(-0.395202\pi\)
0.323317 + 0.946291i \(0.395202\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.52444 −0.761147 −0.380573 0.924751i \(-0.624273\pi\)
−0.380573 + 0.924751i \(0.624273\pi\)
\(12\) 0 0
\(13\) −6.72999 −1.86656 −0.933281 0.359146i \(-0.883068\pi\)
−0.933281 + 0.359146i \(0.883068\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 4.44082 1.07706 0.538528 0.842607i \(-0.318980\pi\)
0.538528 + 0.842607i \(0.318980\pi\)
\(18\) 0 0
\(19\) 3.10278 0.711825 0.355913 0.934519i \(-0.384170\pi\)
0.355913 + 0.934519i \(0.384170\pi\)
\(20\) 0 0
\(21\) −1.71083 −0.373334
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.01916 0.932034 0.466017 0.884776i \(-0.345689\pi\)
0.466017 + 0.884776i \(0.345689\pi\)
\(30\) 0 0
\(31\) 1.13249 0.203402 0.101701 0.994815i \(-0.467572\pi\)
0.101701 + 0.994815i \(0.467572\pi\)
\(32\) 0 0
\(33\) 2.52444 0.439448
\(34\) 0 0
\(35\) −1.71083 −0.289183
\(36\) 0 0
\(37\) 2.49472 0.410129 0.205065 0.978748i \(-0.434260\pi\)
0.205065 + 0.978748i \(0.434260\pi\)
\(38\) 0 0
\(39\) 6.72999 1.07766
\(40\) 0 0
\(41\) −7.39194 −1.15443 −0.577214 0.816593i \(-0.695860\pi\)
−0.577214 + 0.816593i \(0.695860\pi\)
\(42\) 0 0
\(43\) 6.78389 1.03453 0.517267 0.855824i \(-0.326950\pi\)
0.517267 + 0.855824i \(0.326950\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 13.3083 1.94122 0.970609 0.240660i \(-0.0773640\pi\)
0.970609 + 0.240660i \(0.0773640\pi\)
\(48\) 0 0
\(49\) −4.07306 −0.581865
\(50\) 0 0
\(51\) −4.44082 −0.621839
\(52\) 0 0
\(53\) −12.4408 −1.70888 −0.854439 0.519552i \(-0.826099\pi\)
−0.854439 + 0.519552i \(0.826099\pi\)
\(54\) 0 0
\(55\) 2.52444 0.340395
\(56\) 0 0
\(57\) −3.10278 −0.410973
\(58\) 0 0
\(59\) −11.0192 −1.43457 −0.717286 0.696779i \(-0.754616\pi\)
−0.717286 + 0.696779i \(0.754616\pi\)
\(60\) 0 0
\(61\) −2.89722 −0.370952 −0.185476 0.982649i \(-0.559383\pi\)
−0.185476 + 0.982649i \(0.559383\pi\)
\(62\) 0 0
\(63\) 1.71083 0.215545
\(64\) 0 0
\(65\) 6.72999 0.834752
\(66\) 0 0
\(67\) 0.0836184 0.0102156 0.00510781 0.999987i \(-0.498374\pi\)
0.00510781 + 0.999987i \(0.498374\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 13.8030 1.63812 0.819060 0.573708i \(-0.194496\pi\)
0.819060 + 0.573708i \(0.194496\pi\)
\(72\) 0 0
\(73\) −5.57331 −0.652307 −0.326154 0.945317i \(-0.605753\pi\)
−0.326154 + 0.945317i \(0.605753\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −4.31889 −0.492183
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.97028 −0.216266 −0.108133 0.994136i \(-0.534487\pi\)
−0.108133 + 0.994136i \(0.534487\pi\)
\(84\) 0 0
\(85\) −4.44082 −0.481675
\(86\) 0 0
\(87\) −5.01916 −0.538110
\(88\) 0 0
\(89\) −7.62721 −0.808483 −0.404241 0.914652i \(-0.632464\pi\)
−0.404241 + 0.914652i \(0.632464\pi\)
\(90\) 0 0
\(91\) −11.5139 −1.20698
\(92\) 0 0
\(93\) −1.13249 −0.117434
\(94\) 0 0
\(95\) −3.10278 −0.318338
\(96\) 0 0
\(97\) 9.83276 0.998366 0.499183 0.866497i \(-0.333634\pi\)
0.499183 + 0.866497i \(0.333634\pi\)
\(98\) 0 0
\(99\) −2.52444 −0.253716
\(100\) 0 0
\(101\) −10.8136 −1.07599 −0.537997 0.842947i \(-0.680819\pi\)
−0.537997 + 0.842947i \(0.680819\pi\)
\(102\) 0 0
\(103\) −15.0872 −1.48658 −0.743292 0.668967i \(-0.766737\pi\)
−0.743292 + 0.668967i \(0.766737\pi\)
\(104\) 0 0
\(105\) 1.71083 0.166960
\(106\) 0 0
\(107\) 13.8625 1.34014 0.670068 0.742299i \(-0.266265\pi\)
0.670068 + 0.742299i \(0.266265\pi\)
\(108\) 0 0
\(109\) 16.8277 1.61181 0.805903 0.592048i \(-0.201680\pi\)
0.805903 + 0.592048i \(0.201680\pi\)
\(110\) 0 0
\(111\) −2.49472 −0.236788
\(112\) 0 0
\(113\) −19.6952 −1.85277 −0.926386 0.376574i \(-0.877102\pi\)
−0.926386 + 0.376574i \(0.877102\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −6.72999 −0.622188
\(118\) 0 0
\(119\) 7.59749 0.696461
\(120\) 0 0
\(121\) −4.62721 −0.420656
\(122\) 0 0
\(123\) 7.39194 0.666509
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 18.9355 1.68026 0.840129 0.542387i \(-0.182479\pi\)
0.840129 + 0.542387i \(0.182479\pi\)
\(128\) 0 0
\(129\) −6.78389 −0.597288
\(130\) 0 0
\(131\) −12.4111 −1.08436 −0.542181 0.840261i \(-0.682401\pi\)
−0.542181 + 0.840261i \(0.682401\pi\)
\(132\) 0 0
\(133\) 5.30833 0.460290
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −13.4217 −1.14669 −0.573345 0.819314i \(-0.694355\pi\)
−0.573345 + 0.819314i \(0.694355\pi\)
\(138\) 0 0
\(139\) −15.2786 −1.29591 −0.647957 0.761677i \(-0.724376\pi\)
−0.647957 + 0.761677i \(0.724376\pi\)
\(140\) 0 0
\(141\) −13.3083 −1.12076
\(142\) 0 0
\(143\) 16.9894 1.42073
\(144\) 0 0
\(145\) −5.01916 −0.416818
\(146\) 0 0
\(147\) 4.07306 0.335940
\(148\) 0 0
\(149\) −1.68111 −0.137722 −0.0688610 0.997626i \(-0.521937\pi\)
−0.0688610 + 0.997626i \(0.521937\pi\)
\(150\) 0 0
\(151\) 2.78389 0.226550 0.113275 0.993564i \(-0.463866\pi\)
0.113275 + 0.993564i \(0.463866\pi\)
\(152\) 0 0
\(153\) 4.44082 0.359019
\(154\) 0 0
\(155\) −1.13249 −0.0909641
\(156\) 0 0
\(157\) −9.91638 −0.791413 −0.395707 0.918377i \(-0.629500\pi\)
−0.395707 + 0.918377i \(0.629500\pi\)
\(158\) 0 0
\(159\) 12.4408 0.986621
\(160\) 0 0
\(161\) −1.71083 −0.134832
\(162\) 0 0
\(163\) −7.42166 −0.581310 −0.290655 0.956828i \(-0.593873\pi\)
−0.290655 + 0.956828i \(0.593873\pi\)
\(164\) 0 0
\(165\) −2.52444 −0.196527
\(166\) 0 0
\(167\) 5.94610 0.460123 0.230062 0.973176i \(-0.426107\pi\)
0.230062 + 0.973176i \(0.426107\pi\)
\(168\) 0 0
\(169\) 32.2927 2.48406
\(170\) 0 0
\(171\) 3.10278 0.237275
\(172\) 0 0
\(173\) 13.0872 0.995001 0.497500 0.867464i \(-0.334251\pi\)
0.497500 + 0.867464i \(0.334251\pi\)
\(174\) 0 0
\(175\) 1.71083 0.129327
\(176\) 0 0
\(177\) 11.0192 0.828251
\(178\) 0 0
\(179\) −26.4494 −1.97692 −0.988461 0.151476i \(-0.951597\pi\)
−0.988461 + 0.151476i \(0.951597\pi\)
\(180\) 0 0
\(181\) −24.0383 −1.78675 −0.893377 0.449308i \(-0.851671\pi\)
−0.893377 + 0.449308i \(0.851671\pi\)
\(182\) 0 0
\(183\) 2.89722 0.214169
\(184\) 0 0
\(185\) −2.49472 −0.183415
\(186\) 0 0
\(187\) −11.2106 −0.819798
\(188\) 0 0
\(189\) −1.71083 −0.124445
\(190\) 0 0
\(191\) −25.7194 −1.86099 −0.930496 0.366302i \(-0.880624\pi\)
−0.930496 + 0.366302i \(0.880624\pi\)
\(192\) 0 0
\(193\) 26.1361 1.88132 0.940658 0.339357i \(-0.110210\pi\)
0.940658 + 0.339357i \(0.110210\pi\)
\(194\) 0 0
\(195\) −6.72999 −0.481944
\(196\) 0 0
\(197\) −14.3033 −1.01907 −0.509534 0.860451i \(-0.670182\pi\)
−0.509534 + 0.860451i \(0.670182\pi\)
\(198\) 0 0
\(199\) 25.2927 1.79295 0.896477 0.443090i \(-0.146118\pi\)
0.896477 + 0.443090i \(0.146118\pi\)
\(200\) 0 0
\(201\) −0.0836184 −0.00589799
\(202\) 0 0
\(203\) 8.58693 0.602684
\(204\) 0 0
\(205\) 7.39194 0.516276
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −7.83276 −0.541804
\(210\) 0 0
\(211\) 21.7875 1.49991 0.749955 0.661489i \(-0.230075\pi\)
0.749955 + 0.661489i \(0.230075\pi\)
\(212\) 0 0
\(213\) −13.8030 −0.945769
\(214\) 0 0
\(215\) −6.78389 −0.462657
\(216\) 0 0
\(217\) 1.93751 0.131527
\(218\) 0 0
\(219\) 5.57331 0.376610
\(220\) 0 0
\(221\) −29.8867 −2.01039
\(222\) 0 0
\(223\) −20.7738 −1.39112 −0.695560 0.718468i \(-0.744843\pi\)
−0.695560 + 0.718468i \(0.744843\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −9.19499 −0.607622 −0.303811 0.952732i \(-0.598259\pi\)
−0.303811 + 0.952732i \(0.598259\pi\)
\(230\) 0 0
\(231\) 4.31889 0.284162
\(232\) 0 0
\(233\) 6.98944 0.457893 0.228947 0.973439i \(-0.426472\pi\)
0.228947 + 0.973439i \(0.426472\pi\)
\(234\) 0 0
\(235\) −13.3083 −0.868139
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.2353 −1.05017 −0.525086 0.851049i \(-0.675967\pi\)
−0.525086 + 0.851049i \(0.675967\pi\)
\(240\) 0 0
\(241\) 7.77886 0.501081 0.250540 0.968106i \(-0.419392\pi\)
0.250540 + 0.968106i \(0.419392\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 4.07306 0.260218
\(246\) 0 0
\(247\) −20.8816 −1.32867
\(248\) 0 0
\(249\) 1.97028 0.124861
\(250\) 0 0
\(251\) 19.0872 1.20477 0.602386 0.798205i \(-0.294217\pi\)
0.602386 + 0.798205i \(0.294217\pi\)
\(252\) 0 0
\(253\) 2.52444 0.158710
\(254\) 0 0
\(255\) 4.44082 0.278095
\(256\) 0 0
\(257\) −27.2489 −1.69974 −0.849869 0.526993i \(-0.823319\pi\)
−0.849869 + 0.526993i \(0.823319\pi\)
\(258\) 0 0
\(259\) 4.26804 0.265203
\(260\) 0 0
\(261\) 5.01916 0.310678
\(262\) 0 0
\(263\) −15.4303 −0.951470 −0.475735 0.879589i \(-0.657818\pi\)
−0.475735 + 0.879589i \(0.657818\pi\)
\(264\) 0 0
\(265\) 12.4408 0.764233
\(266\) 0 0
\(267\) 7.62721 0.466778
\(268\) 0 0
\(269\) −12.9114 −0.787219 −0.393610 0.919278i \(-0.628774\pi\)
−0.393610 + 0.919278i \(0.628774\pi\)
\(270\) 0 0
\(271\) 3.74914 0.227744 0.113872 0.993495i \(-0.463675\pi\)
0.113872 + 0.993495i \(0.463675\pi\)
\(272\) 0 0
\(273\) 11.5139 0.696851
\(274\) 0 0
\(275\) −2.52444 −0.152229
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 1.13249 0.0678007
\(280\) 0 0
\(281\) −1.84835 −0.110263 −0.0551316 0.998479i \(-0.517558\pi\)
−0.0551316 + 0.998479i \(0.517558\pi\)
\(282\) 0 0
\(283\) −1.40753 −0.0836689 −0.0418345 0.999125i \(-0.513320\pi\)
−0.0418345 + 0.999125i \(0.513320\pi\)
\(284\) 0 0
\(285\) 3.10278 0.183793
\(286\) 0 0
\(287\) −12.6464 −0.746492
\(288\) 0 0
\(289\) 2.72088 0.160052
\(290\) 0 0
\(291\) −9.83276 −0.576407
\(292\) 0 0
\(293\) 27.5280 1.60820 0.804102 0.594492i \(-0.202647\pi\)
0.804102 + 0.594492i \(0.202647\pi\)
\(294\) 0 0
\(295\) 11.0192 0.641560
\(296\) 0 0
\(297\) 2.52444 0.146483
\(298\) 0 0
\(299\) 6.72999 0.389205
\(300\) 0 0
\(301\) 11.6061 0.668964
\(302\) 0 0
\(303\) 10.8136 0.621225
\(304\) 0 0
\(305\) 2.89722 0.165895
\(306\) 0 0
\(307\) 4.56275 0.260410 0.130205 0.991487i \(-0.458436\pi\)
0.130205 + 0.991487i \(0.458436\pi\)
\(308\) 0 0
\(309\) 15.0872 0.858280
\(310\) 0 0
\(311\) −6.84333 −0.388049 −0.194025 0.980997i \(-0.562154\pi\)
−0.194025 + 0.980997i \(0.562154\pi\)
\(312\) 0 0
\(313\) −25.7491 −1.45543 −0.727714 0.685881i \(-0.759417\pi\)
−0.727714 + 0.685881i \(0.759417\pi\)
\(314\) 0 0
\(315\) −1.71083 −0.0963944
\(316\) 0 0
\(317\) 9.23884 0.518905 0.259452 0.965756i \(-0.416458\pi\)
0.259452 + 0.965756i \(0.416458\pi\)
\(318\) 0 0
\(319\) −12.6705 −0.709415
\(320\) 0 0
\(321\) −13.8625 −0.773728
\(322\) 0 0
\(323\) 13.7789 0.766676
\(324\) 0 0
\(325\) −6.72999 −0.373313
\(326\) 0 0
\(327\) −16.8277 −0.930576
\(328\) 0 0
\(329\) 22.7683 1.25526
\(330\) 0 0
\(331\) −13.6030 −0.747690 −0.373845 0.927491i \(-0.621961\pi\)
−0.373845 + 0.927491i \(0.621961\pi\)
\(332\) 0 0
\(333\) 2.49472 0.136710
\(334\) 0 0
\(335\) −0.0836184 −0.00456856
\(336\) 0 0
\(337\) −9.25443 −0.504121 −0.252060 0.967712i \(-0.581108\pi\)
−0.252060 + 0.967712i \(0.581108\pi\)
\(338\) 0 0
\(339\) 19.6952 1.06970
\(340\) 0 0
\(341\) −2.85891 −0.154819
\(342\) 0 0
\(343\) −18.9441 −1.02289
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −22.9511 −1.23208 −0.616040 0.787715i \(-0.711264\pi\)
−0.616040 + 0.787715i \(0.711264\pi\)
\(348\) 0 0
\(349\) −2.96526 −0.158727 −0.0793633 0.996846i \(-0.525289\pi\)
−0.0793633 + 0.996846i \(0.525289\pi\)
\(350\) 0 0
\(351\) 6.72999 0.359220
\(352\) 0 0
\(353\) −19.2489 −1.02451 −0.512257 0.858832i \(-0.671191\pi\)
−0.512257 + 0.858832i \(0.671191\pi\)
\(354\) 0 0
\(355\) −13.8030 −0.732589
\(356\) 0 0
\(357\) −7.59749 −0.402102
\(358\) 0 0
\(359\) −21.6116 −1.14062 −0.570309 0.821430i \(-0.693177\pi\)
−0.570309 + 0.821430i \(0.693177\pi\)
\(360\) 0 0
\(361\) −9.37279 −0.493305
\(362\) 0 0
\(363\) 4.62721 0.242866
\(364\) 0 0
\(365\) 5.57331 0.291721
\(366\) 0 0
\(367\) 9.71083 0.506901 0.253451 0.967348i \(-0.418434\pi\)
0.253451 + 0.967348i \(0.418434\pi\)
\(368\) 0 0
\(369\) −7.39194 −0.384809
\(370\) 0 0
\(371\) −21.2841 −1.10502
\(372\) 0 0
\(373\) −32.4494 −1.68017 −0.840083 0.542457i \(-0.817494\pi\)
−0.840083 + 0.542457i \(0.817494\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −33.7789 −1.73970
\(378\) 0 0
\(379\) −28.8222 −1.48050 −0.740248 0.672333i \(-0.765292\pi\)
−0.740248 + 0.672333i \(0.765292\pi\)
\(380\) 0 0
\(381\) −18.9355 −0.970097
\(382\) 0 0
\(383\) −2.54862 −0.130228 −0.0651141 0.997878i \(-0.520741\pi\)
−0.0651141 + 0.997878i \(0.520741\pi\)
\(384\) 0 0
\(385\) 4.31889 0.220111
\(386\) 0 0
\(387\) 6.78389 0.344844
\(388\) 0 0
\(389\) −17.5577 −0.890212 −0.445106 0.895478i \(-0.646834\pi\)
−0.445106 + 0.895478i \(0.646834\pi\)
\(390\) 0 0
\(391\) −4.44082 −0.224582
\(392\) 0 0
\(393\) 12.4111 0.626057
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 29.0872 1.45984 0.729922 0.683530i \(-0.239556\pi\)
0.729922 + 0.683530i \(0.239556\pi\)
\(398\) 0 0
\(399\) −5.30833 −0.265749
\(400\) 0 0
\(401\) −0.616650 −0.0307940 −0.0153970 0.999881i \(-0.504901\pi\)
−0.0153970 + 0.999881i \(0.504901\pi\)
\(402\) 0 0
\(403\) −7.62167 −0.379663
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −6.29776 −0.312168
\(408\) 0 0
\(409\) −8.07357 −0.399212 −0.199606 0.979876i \(-0.563966\pi\)
−0.199606 + 0.979876i \(0.563966\pi\)
\(410\) 0 0
\(411\) 13.4217 0.662042
\(412\) 0 0
\(413\) −18.8519 −0.927642
\(414\) 0 0
\(415\) 1.97028 0.0967173
\(416\) 0 0
\(417\) 15.2786 0.748197
\(418\) 0 0
\(419\) −19.9844 −0.976303 −0.488151 0.872759i \(-0.662329\pi\)
−0.488151 + 0.872759i \(0.662329\pi\)
\(420\) 0 0
\(421\) 9.04334 0.440745 0.220373 0.975416i \(-0.429273\pi\)
0.220373 + 0.975416i \(0.429273\pi\)
\(422\) 0 0
\(423\) 13.3083 0.647073
\(424\) 0 0
\(425\) 4.44082 0.215411
\(426\) 0 0
\(427\) −4.95666 −0.239870
\(428\) 0 0
\(429\) −16.9894 −0.820258
\(430\) 0 0
\(431\) −8.63778 −0.416067 −0.208034 0.978122i \(-0.566706\pi\)
−0.208034 + 0.978122i \(0.566706\pi\)
\(432\) 0 0
\(433\) −34.6902 −1.66711 −0.833553 0.552440i \(-0.813697\pi\)
−0.833553 + 0.552440i \(0.813697\pi\)
\(434\) 0 0
\(435\) 5.01916 0.240650
\(436\) 0 0
\(437\) −3.10278 −0.148426
\(438\) 0 0
\(439\) −11.4217 −0.545126 −0.272563 0.962138i \(-0.587871\pi\)
−0.272563 + 0.962138i \(0.587871\pi\)
\(440\) 0 0
\(441\) −4.07306 −0.193955
\(442\) 0 0
\(443\) 8.78943 0.417598 0.208799 0.977959i \(-0.433045\pi\)
0.208799 + 0.977959i \(0.433045\pi\)
\(444\) 0 0
\(445\) 7.62721 0.361565
\(446\) 0 0
\(447\) 1.68111 0.0795139
\(448\) 0 0
\(449\) −11.5874 −0.546845 −0.273423 0.961894i \(-0.588156\pi\)
−0.273423 + 0.961894i \(0.588156\pi\)
\(450\) 0 0
\(451\) 18.6605 0.878689
\(452\) 0 0
\(453\) −2.78389 −0.130798
\(454\) 0 0
\(455\) 11.5139 0.539779
\(456\) 0 0
\(457\) 10.6620 0.498745 0.249373 0.968408i \(-0.419776\pi\)
0.249373 + 0.968408i \(0.419776\pi\)
\(458\) 0 0
\(459\) −4.44082 −0.207280
\(460\) 0 0
\(461\) −22.3033 −1.03877 −0.519384 0.854541i \(-0.673839\pi\)
−0.519384 + 0.854541i \(0.673839\pi\)
\(462\) 0 0
\(463\) 27.9250 1.29778 0.648892 0.760881i \(-0.275233\pi\)
0.648892 + 0.760881i \(0.275233\pi\)
\(464\) 0 0
\(465\) 1.13249 0.0525182
\(466\) 0 0
\(467\) −24.4308 −1.13052 −0.565261 0.824912i \(-0.691224\pi\)
−0.565261 + 0.824912i \(0.691224\pi\)
\(468\) 0 0
\(469\) 0.143057 0.00660576
\(470\) 0 0
\(471\) 9.91638 0.456923
\(472\) 0 0
\(473\) −17.1255 −0.787431
\(474\) 0 0
\(475\) 3.10278 0.142365
\(476\) 0 0
\(477\) −12.4408 −0.569626
\(478\) 0 0
\(479\) −19.2106 −0.877753 −0.438877 0.898547i \(-0.644624\pi\)
−0.438877 + 0.898547i \(0.644624\pi\)
\(480\) 0 0
\(481\) −16.7894 −0.765532
\(482\) 0 0
\(483\) 1.71083 0.0778455
\(484\) 0 0
\(485\) −9.83276 −0.446483
\(486\) 0 0
\(487\) −22.4933 −1.01927 −0.509634 0.860392i \(-0.670219\pi\)
−0.509634 + 0.860392i \(0.670219\pi\)
\(488\) 0 0
\(489\) 7.42166 0.335619
\(490\) 0 0
\(491\) 8.53857 0.385340 0.192670 0.981264i \(-0.438285\pi\)
0.192670 + 0.981264i \(0.438285\pi\)
\(492\) 0 0
\(493\) 22.2892 1.00385
\(494\) 0 0
\(495\) 2.52444 0.113465
\(496\) 0 0
\(497\) 23.6147 1.05926
\(498\) 0 0
\(499\) −17.0247 −0.762130 −0.381065 0.924548i \(-0.624443\pi\)
−0.381065 + 0.924548i \(0.624443\pi\)
\(500\) 0 0
\(501\) −5.94610 −0.265652
\(502\) 0 0
\(503\) 33.8625 1.50985 0.754927 0.655809i \(-0.227672\pi\)
0.754927 + 0.655809i \(0.227672\pi\)
\(504\) 0 0
\(505\) 10.8136 0.481199
\(506\) 0 0
\(507\) −32.2927 −1.43417
\(508\) 0 0
\(509\) 34.7738 1.54132 0.770662 0.637244i \(-0.219926\pi\)
0.770662 + 0.637244i \(0.219926\pi\)
\(510\) 0 0
\(511\) −9.53500 −0.421804
\(512\) 0 0
\(513\) −3.10278 −0.136991
\(514\) 0 0
\(515\) 15.0872 0.664821
\(516\) 0 0
\(517\) −33.5960 −1.47755
\(518\) 0 0
\(519\) −13.0872 −0.574464
\(520\) 0 0
\(521\) −12.7995 −0.560755 −0.280378 0.959890i \(-0.590460\pi\)
−0.280378 + 0.959890i \(0.590460\pi\)
\(522\) 0 0
\(523\) 24.4111 1.06742 0.533711 0.845667i \(-0.320797\pi\)
0.533711 + 0.845667i \(0.320797\pi\)
\(524\) 0 0
\(525\) −1.71083 −0.0746668
\(526\) 0 0
\(527\) 5.02920 0.219076
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −11.0192 −0.478191
\(532\) 0 0
\(533\) 49.7477 2.15481
\(534\) 0 0
\(535\) −13.8625 −0.599327
\(536\) 0 0
\(537\) 26.4494 1.14138
\(538\) 0 0
\(539\) 10.2822 0.442885
\(540\) 0 0
\(541\) 27.1849 1.16877 0.584386 0.811476i \(-0.301335\pi\)
0.584386 + 0.811476i \(0.301335\pi\)
\(542\) 0 0
\(543\) 24.0383 1.03158
\(544\) 0 0
\(545\) −16.8277 −0.720821
\(546\) 0 0
\(547\) −26.8433 −1.14774 −0.573869 0.818947i \(-0.694558\pi\)
−0.573869 + 0.818947i \(0.694558\pi\)
\(548\) 0 0
\(549\) −2.89722 −0.123651
\(550\) 0 0
\(551\) 15.5733 0.663445
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 2.49472 0.105895
\(556\) 0 0
\(557\) −32.8519 −1.39198 −0.695990 0.718051i \(-0.745034\pi\)
−0.695990 + 0.718051i \(0.745034\pi\)
\(558\) 0 0
\(559\) −45.6555 −1.93102
\(560\) 0 0
\(561\) 11.2106 0.473311
\(562\) 0 0
\(563\) 26.7441 1.12713 0.563565 0.826072i \(-0.309429\pi\)
0.563565 + 0.826072i \(0.309429\pi\)
\(564\) 0 0
\(565\) 19.6952 0.828585
\(566\) 0 0
\(567\) 1.71083 0.0718482
\(568\) 0 0
\(569\) −34.3799 −1.44128 −0.720641 0.693309i \(-0.756152\pi\)
−0.720641 + 0.693309i \(0.756152\pi\)
\(570\) 0 0
\(571\) −39.9532 −1.67199 −0.835996 0.548736i \(-0.815109\pi\)
−0.835996 + 0.548736i \(0.815109\pi\)
\(572\) 0 0
\(573\) 25.7194 1.07444
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 23.5194 0.979126 0.489563 0.871968i \(-0.337156\pi\)
0.489563 + 0.871968i \(0.337156\pi\)
\(578\) 0 0
\(579\) −26.1361 −1.08618
\(580\) 0 0
\(581\) −3.37082 −0.139845
\(582\) 0 0
\(583\) 31.4061 1.30071
\(584\) 0 0
\(585\) 6.72999 0.278251
\(586\) 0 0
\(587\) 8.36274 0.345167 0.172584 0.984995i \(-0.444788\pi\)
0.172584 + 0.984995i \(0.444788\pi\)
\(588\) 0 0
\(589\) 3.51388 0.144787
\(590\) 0 0
\(591\) 14.3033 0.588359
\(592\) 0 0
\(593\) 5.87662 0.241324 0.120662 0.992694i \(-0.461498\pi\)
0.120662 + 0.992694i \(0.461498\pi\)
\(594\) 0 0
\(595\) −7.59749 −0.311467
\(596\) 0 0
\(597\) −25.2927 −1.03516
\(598\) 0 0
\(599\) 9.12550 0.372858 0.186429 0.982468i \(-0.440309\pi\)
0.186429 + 0.982468i \(0.440309\pi\)
\(600\) 0 0
\(601\) 15.3764 0.627215 0.313607 0.949553i \(-0.398462\pi\)
0.313607 + 0.949553i \(0.398462\pi\)
\(602\) 0 0
\(603\) 0.0836184 0.00340521
\(604\) 0 0
\(605\) 4.62721 0.188123
\(606\) 0 0
\(607\) −11.1511 −0.452611 −0.226305 0.974056i \(-0.572665\pi\)
−0.226305 + 0.974056i \(0.572665\pi\)
\(608\) 0 0
\(609\) −8.58693 −0.347960
\(610\) 0 0
\(611\) −89.5649 −3.62341
\(612\) 0 0
\(613\) 5.47002 0.220932 0.110466 0.993880i \(-0.464766\pi\)
0.110466 + 0.993880i \(0.464766\pi\)
\(614\) 0 0
\(615\) −7.39194 −0.298072
\(616\) 0 0
\(617\) 11.1169 0.447550 0.223775 0.974641i \(-0.428162\pi\)
0.223775 + 0.974641i \(0.428162\pi\)
\(618\) 0 0
\(619\) 18.8433 0.757377 0.378689 0.925524i \(-0.376375\pi\)
0.378689 + 0.925524i \(0.376375\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −13.0489 −0.522792
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.83276 0.312810
\(628\) 0 0
\(629\) 11.0786 0.441733
\(630\) 0 0
\(631\) 4.10329 0.163349 0.0816747 0.996659i \(-0.473973\pi\)
0.0816747 + 0.996659i \(0.473973\pi\)
\(632\) 0 0
\(633\) −21.7875 −0.865974
\(634\) 0 0
\(635\) −18.9355 −0.751434
\(636\) 0 0
\(637\) 27.4116 1.08609
\(638\) 0 0
\(639\) 13.8030 0.546040
\(640\) 0 0
\(641\) 15.6116 0.616622 0.308311 0.951286i \(-0.400236\pi\)
0.308311 + 0.951286i \(0.400236\pi\)
\(642\) 0 0
\(643\) 0.602517 0.0237609 0.0118805 0.999929i \(-0.496218\pi\)
0.0118805 + 0.999929i \(0.496218\pi\)
\(644\) 0 0
\(645\) 6.78389 0.267115
\(646\) 0 0
\(647\) 27.3466 1.07511 0.537554 0.843230i \(-0.319349\pi\)
0.537554 + 0.843230i \(0.319349\pi\)
\(648\) 0 0
\(649\) 27.8172 1.09192
\(650\) 0 0
\(651\) −1.93751 −0.0759369
\(652\) 0 0
\(653\) 45.8172 1.79296 0.896482 0.443079i \(-0.146114\pi\)
0.896482 + 0.443079i \(0.146114\pi\)
\(654\) 0 0
\(655\) 12.4111 0.484942
\(656\) 0 0
\(657\) −5.57331 −0.217436
\(658\) 0 0
\(659\) −4.83779 −0.188453 −0.0942267 0.995551i \(-0.530038\pi\)
−0.0942267 + 0.995551i \(0.530038\pi\)
\(660\) 0 0
\(661\) 36.6933 1.42720 0.713602 0.700552i \(-0.247063\pi\)
0.713602 + 0.700552i \(0.247063\pi\)
\(662\) 0 0
\(663\) 29.8867 1.16070
\(664\) 0 0
\(665\) −5.30833 −0.205848
\(666\) 0 0
\(667\) −5.01916 −0.194343
\(668\) 0 0
\(669\) 20.7738 0.803163
\(670\) 0 0
\(671\) 7.31386 0.282349
\(672\) 0 0
\(673\) 37.2983 1.43774 0.718871 0.695143i \(-0.244659\pi\)
0.718871 + 0.695143i \(0.244659\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 30.1275 1.15789 0.578946 0.815366i \(-0.303464\pi\)
0.578946 + 0.815366i \(0.303464\pi\)
\(678\) 0 0
\(679\) 16.8222 0.645577
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 20.8972 0.799610 0.399805 0.916600i \(-0.369078\pi\)
0.399805 + 0.916600i \(0.369078\pi\)
\(684\) 0 0
\(685\) 13.4217 0.512815
\(686\) 0 0
\(687\) 9.19499 0.350811
\(688\) 0 0
\(689\) 83.7266 3.18973
\(690\) 0 0
\(691\) −1.21611 −0.0462631 −0.0231316 0.999732i \(-0.507364\pi\)
−0.0231316 + 0.999732i \(0.507364\pi\)
\(692\) 0 0
\(693\) −4.31889 −0.164061
\(694\) 0 0
\(695\) 15.2786 0.579551
\(696\) 0 0
\(697\) −32.8263 −1.24338
\(698\) 0 0
\(699\) −6.98944 −0.264365
\(700\) 0 0
\(701\) 2.42669 0.0916547 0.0458273 0.998949i \(-0.485408\pi\)
0.0458273 + 0.998949i \(0.485408\pi\)
\(702\) 0 0
\(703\) 7.74055 0.291940
\(704\) 0 0
\(705\) 13.3083 0.501221
\(706\) 0 0
\(707\) −18.5003 −0.695774
\(708\) 0 0
\(709\) 3.00502 0.112856 0.0564280 0.998407i \(-0.482029\pi\)
0.0564280 + 0.998407i \(0.482029\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.13249 −0.0424122
\(714\) 0 0
\(715\) −16.9894 −0.635369
\(716\) 0 0
\(717\) 16.2353 0.606317
\(718\) 0 0
\(719\) −18.3814 −0.685510 −0.342755 0.939425i \(-0.611360\pi\)
−0.342755 + 0.939425i \(0.611360\pi\)
\(720\) 0 0
\(721\) −25.8116 −0.961276
\(722\) 0 0
\(723\) −7.77886 −0.289299
\(724\) 0 0
\(725\) 5.01916 0.186407
\(726\) 0 0
\(727\) 23.5335 0.872811 0.436405 0.899750i \(-0.356251\pi\)
0.436405 + 0.899750i \(0.356251\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.1260 1.11425
\(732\) 0 0
\(733\) −22.6308 −0.835887 −0.417944 0.908473i \(-0.637249\pi\)
−0.417944 + 0.908473i \(0.637249\pi\)
\(734\) 0 0
\(735\) −4.07306 −0.150237
\(736\) 0 0
\(737\) −0.211090 −0.00777558
\(738\) 0 0
\(739\) 2.18137 0.0802430 0.0401215 0.999195i \(-0.487226\pi\)
0.0401215 + 0.999195i \(0.487226\pi\)
\(740\) 0 0
\(741\) 20.8816 0.767106
\(742\) 0 0
\(743\) 25.7633 0.945163 0.472582 0.881287i \(-0.343322\pi\)
0.472582 + 0.881287i \(0.343322\pi\)
\(744\) 0 0
\(745\) 1.68111 0.0615912
\(746\) 0 0
\(747\) −1.97028 −0.0720888
\(748\) 0 0
\(749\) 23.7164 0.866577
\(750\) 0 0
\(751\) −33.3850 −1.21823 −0.609117 0.793080i \(-0.708476\pi\)
−0.609117 + 0.793080i \(0.708476\pi\)
\(752\) 0 0
\(753\) −19.0872 −0.695576
\(754\) 0 0
\(755\) −2.78389 −0.101316
\(756\) 0 0
\(757\) 30.9935 1.12648 0.563239 0.826294i \(-0.309555\pi\)
0.563239 + 0.826294i \(0.309555\pi\)
\(758\) 0 0
\(759\) −2.52444 −0.0916313
\(760\) 0 0
\(761\) 8.44082 0.305979 0.152990 0.988228i \(-0.451110\pi\)
0.152990 + 0.988228i \(0.451110\pi\)
\(762\) 0 0
\(763\) 28.7894 1.04225
\(764\) 0 0
\(765\) −4.44082 −0.160558
\(766\) 0 0
\(767\) 74.1588 2.67772
\(768\) 0 0
\(769\) 25.4372 0.917291 0.458645 0.888619i \(-0.348335\pi\)
0.458645 + 0.888619i \(0.348335\pi\)
\(770\) 0 0
\(771\) 27.2489 0.981345
\(772\) 0 0
\(773\) −35.8711 −1.29019 −0.645096 0.764101i \(-0.723183\pi\)
−0.645096 + 0.764101i \(0.723183\pi\)
\(774\) 0 0
\(775\) 1.13249 0.0406804
\(776\) 0 0
\(777\) −4.26804 −0.153115
\(778\) 0 0
\(779\) −22.9355 −0.821751
\(780\) 0 0
\(781\) −34.8449 −1.24685
\(782\) 0 0
\(783\) −5.01916 −0.179370
\(784\) 0 0
\(785\) 9.91638 0.353931
\(786\) 0 0
\(787\) 24.0524 0.857377 0.428689 0.903452i \(-0.358976\pi\)
0.428689 + 0.903452i \(0.358976\pi\)
\(788\) 0 0
\(789\) 15.4303 0.549332
\(790\) 0 0
\(791\) −33.6952 −1.19807
\(792\) 0 0
\(793\) 19.4983 0.692405
\(794\) 0 0
\(795\) −12.4408 −0.441230
\(796\) 0 0
\(797\) −49.9291 −1.76858 −0.884289 0.466940i \(-0.845356\pi\)
−0.884289 + 0.466940i \(0.845356\pi\)
\(798\) 0 0
\(799\) 59.0999 2.09080
\(800\) 0 0
\(801\) −7.62721 −0.269494
\(802\) 0 0
\(803\) 14.0695 0.496501
\(804\) 0 0
\(805\) 1.71083 0.0602989
\(806\) 0 0
\(807\) 12.9114 0.454501
\(808\) 0 0
\(809\) 20.8207 0.732019 0.366009 0.930611i \(-0.380724\pi\)
0.366009 + 0.930611i \(0.380724\pi\)
\(810\) 0 0
\(811\) −11.0347 −0.387482 −0.193741 0.981053i \(-0.562062\pi\)
−0.193741 + 0.981053i \(0.562062\pi\)
\(812\) 0 0
\(813\) −3.74914 −0.131488
\(814\) 0 0
\(815\) 7.42166 0.259970
\(816\) 0 0
\(817\) 21.0489 0.736407
\(818\) 0 0
\(819\) −11.5139 −0.402327
\(820\) 0 0
\(821\) −26.3627 −0.920066 −0.460033 0.887902i \(-0.652162\pi\)
−0.460033 + 0.887902i \(0.652162\pi\)
\(822\) 0 0
\(823\) 17.0177 0.593200 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(824\) 0 0
\(825\) 2.52444 0.0878896
\(826\) 0 0
\(827\) 26.4408 0.919437 0.459719 0.888065i \(-0.347950\pi\)
0.459719 + 0.888065i \(0.347950\pi\)
\(828\) 0 0
\(829\) 14.1602 0.491806 0.245903 0.969294i \(-0.420916\pi\)
0.245903 + 0.969294i \(0.420916\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) −18.0877 −0.626702
\(834\) 0 0
\(835\) −5.94610 −0.205773
\(836\) 0 0
\(837\) −1.13249 −0.0391447
\(838\) 0 0
\(839\) 3.08719 0.106582 0.0532908 0.998579i \(-0.483029\pi\)
0.0532908 + 0.998579i \(0.483029\pi\)
\(840\) 0 0
\(841\) −3.80807 −0.131313
\(842\) 0 0
\(843\) 1.84835 0.0636605
\(844\) 0 0
\(845\) −32.2927 −1.11090
\(846\) 0 0
\(847\) −7.91638 −0.272010
\(848\) 0 0
\(849\) 1.40753 0.0483063
\(850\) 0 0
\(851\) −2.49472 −0.0855179
\(852\) 0 0
\(853\) −16.2822 −0.557491 −0.278746 0.960365i \(-0.589919\pi\)
−0.278746 + 0.960365i \(0.589919\pi\)
\(854\) 0 0
\(855\) −3.10278 −0.106113
\(856\) 0 0
\(857\) −29.5889 −1.01074 −0.505369 0.862903i \(-0.668644\pi\)
−0.505369 + 0.862903i \(0.668644\pi\)
\(858\) 0 0
\(859\) −30.8958 −1.05415 −0.527075 0.849819i \(-0.676711\pi\)
−0.527075 + 0.849819i \(0.676711\pi\)
\(860\) 0 0
\(861\) 12.6464 0.430987
\(862\) 0 0
\(863\) −28.2127 −0.960371 −0.480186 0.877167i \(-0.659431\pi\)
−0.480186 + 0.877167i \(0.659431\pi\)
\(864\) 0 0
\(865\) −13.0872 −0.444978
\(866\) 0 0
\(867\) −2.72088 −0.0924059
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −0.562751 −0.0190681
\(872\) 0 0
\(873\) 9.83276 0.332789
\(874\) 0 0
\(875\) −1.71083 −0.0578367
\(876\) 0 0
\(877\) 14.1078 0.476386 0.238193 0.971218i \(-0.423445\pi\)
0.238193 + 0.971218i \(0.423445\pi\)
\(878\) 0 0
\(879\) −27.5280 −0.928497
\(880\) 0 0
\(881\) −18.6222 −0.627398 −0.313699 0.949523i \(-0.601568\pi\)
−0.313699 + 0.949523i \(0.601568\pi\)
\(882\) 0 0
\(883\) 26.1133 0.878784 0.439392 0.898295i \(-0.355194\pi\)
0.439392 + 0.898295i \(0.355194\pi\)
\(884\) 0 0
\(885\) −11.0192 −0.370405
\(886\) 0 0
\(887\) −14.0383 −0.471360 −0.235680 0.971831i \(-0.575732\pi\)
−0.235680 + 0.971831i \(0.575732\pi\)
\(888\) 0 0
\(889\) 32.3955 1.08651
\(890\) 0 0
\(891\) −2.52444 −0.0845719
\(892\) 0 0
\(893\) 41.2927 1.38181
\(894\) 0 0
\(895\) 26.4494 0.884106
\(896\) 0 0
\(897\) −6.72999 −0.224708
\(898\) 0 0
\(899\) 5.68417 0.189578
\(900\) 0 0
\(901\) −55.2474 −1.84056
\(902\) 0 0
\(903\) −11.6061 −0.386226
\(904\) 0 0
\(905\) 24.0383 0.799061
\(906\) 0 0
\(907\) −33.8953 −1.12547 −0.562737 0.826636i \(-0.690252\pi\)
−0.562737 + 0.826636i \(0.690252\pi\)
\(908\) 0 0
\(909\) −10.8136 −0.358665
\(910\) 0 0
\(911\) −14.6277 −0.484638 −0.242319 0.970197i \(-0.577908\pi\)
−0.242319 + 0.970197i \(0.577908\pi\)
\(912\) 0 0
\(913\) 4.97385 0.164610
\(914\) 0 0
\(915\) −2.89722 −0.0957793
\(916\) 0 0
\(917\) −21.2333 −0.701185
\(918\) 0 0
\(919\) −5.68665 −0.187585 −0.0937927 0.995592i \(-0.529899\pi\)
−0.0937927 + 0.995592i \(0.529899\pi\)
\(920\) 0 0
\(921\) −4.56275 −0.150348
\(922\) 0 0
\(923\) −92.8943 −3.05765
\(924\) 0 0
\(925\) 2.49472 0.0820258
\(926\) 0 0
\(927\) −15.0872 −0.495528
\(928\) 0 0
\(929\) −27.7719 −0.911166 −0.455583 0.890193i \(-0.650569\pi\)
−0.455583 + 0.890193i \(0.650569\pi\)
\(930\) 0 0
\(931\) −12.6378 −0.414186
\(932\) 0 0
\(933\) 6.84333 0.224040
\(934\) 0 0
\(935\) 11.2106 0.366625
\(936\) 0 0
\(937\) −7.32391 −0.239262 −0.119631 0.992818i \(-0.538171\pi\)
−0.119631 + 0.992818i \(0.538171\pi\)
\(938\) 0 0
\(939\) 25.7491 0.840292
\(940\) 0 0
\(941\) 44.0711 1.43668 0.718338 0.695694i \(-0.244903\pi\)
0.718338 + 0.695694i \(0.244903\pi\)
\(942\) 0 0
\(943\) 7.39194 0.240715
\(944\) 0 0
\(945\) 1.71083 0.0556534
\(946\) 0 0
\(947\) 4.97225 0.161576 0.0807882 0.996731i \(-0.474256\pi\)
0.0807882 + 0.996731i \(0.474256\pi\)
\(948\) 0 0
\(949\) 37.5083 1.21757
\(950\) 0 0
\(951\) −9.23884 −0.299590
\(952\) 0 0
\(953\) −15.0177 −0.486471 −0.243236 0.969967i \(-0.578209\pi\)
−0.243236 + 0.969967i \(0.578209\pi\)
\(954\) 0 0
\(955\) 25.7194 0.832261
\(956\) 0 0
\(957\) 12.6705 0.409581
\(958\) 0 0
\(959\) −22.9622 −0.741488
\(960\) 0 0
\(961\) −29.7175 −0.958628
\(962\) 0 0
\(963\) 13.8625 0.446712
\(964\) 0 0
\(965\) −26.1361 −0.841350
\(966\) 0 0
\(967\) 29.5834 0.951337 0.475668 0.879625i \(-0.342206\pi\)
0.475668 + 0.879625i \(0.342206\pi\)
\(968\) 0 0
\(969\) −13.7789 −0.442641
\(970\) 0 0
\(971\) 33.2444 1.06686 0.533431 0.845843i \(-0.320902\pi\)
0.533431 + 0.845843i \(0.320902\pi\)
\(972\) 0 0
\(973\) −26.1391 −0.837982
\(974\) 0 0
\(975\) 6.72999 0.215532
\(976\) 0 0
\(977\) 4.91136 0.157128 0.0785641 0.996909i \(-0.474966\pi\)
0.0785641 + 0.996909i \(0.474966\pi\)
\(978\) 0 0
\(979\) 19.2544 0.615374
\(980\) 0 0
\(981\) 16.8277 0.537268
\(982\) 0 0
\(983\) 5.83422 0.186083 0.0930413 0.995662i \(-0.470341\pi\)
0.0930413 + 0.995662i \(0.470341\pi\)
\(984\) 0 0
\(985\) 14.3033 0.455741
\(986\) 0 0
\(987\) −22.7683 −0.724723
\(988\) 0 0
\(989\) −6.78389 −0.215715
\(990\) 0 0
\(991\) 12.4353 0.395020 0.197510 0.980301i \(-0.436715\pi\)
0.197510 + 0.980301i \(0.436715\pi\)
\(992\) 0 0
\(993\) 13.6030 0.431679
\(994\) 0 0
\(995\) −25.2927 −0.801834
\(996\) 0 0
\(997\) 14.8816 0.471306 0.235653 0.971837i \(-0.424277\pi\)
0.235653 + 0.971837i \(0.424277\pi\)
\(998\) 0 0
\(999\) −2.49472 −0.0789294
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 5520.2.a.bx.1.2 3
4.3 odd 2 2760.2.a.r.1.2 3
12.11 even 2 8280.2.a.bm.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.r.1.2 3 4.3 odd 2
5520.2.a.bx.1.2 3 1.1 even 1 trivial
8280.2.a.bm.1.2 3 12.11 even 2