Properties

Label 7623.2.a.ch.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2525.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.77748\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.77748 q^{2} +1.15945 q^{4} +2.77748 q^{5} -1.00000 q^{7} +1.49406 q^{8} +O(q^{10})\) \(q-1.77748 q^{2} +1.15945 q^{4} +2.77748 q^{5} -1.00000 q^{7} +1.49406 q^{8} -4.93693 q^{10} -4.29348 q^{13} +1.77748 q^{14} -4.97458 q^{16} +2.75556 q^{17} +1.93910 q^{19} +3.22035 q^{20} -4.37009 q^{23} +2.71442 q^{25} +7.63159 q^{26} -1.15945 q^{28} -8.62809 q^{29} -0.200588 q^{31} +5.85410 q^{32} -4.89796 q^{34} -2.77748 q^{35} +1.03548 q^{37} -3.44671 q^{38} +4.14974 q^{40} -9.60616 q^{41} -4.70820 q^{43} +7.76777 q^{46} +13.0455 q^{47} +1.00000 q^{49} -4.82484 q^{50} -4.97807 q^{52} +3.90012 q^{53} -1.49406 q^{56} +15.3363 q^{58} +8.55713 q^{59} -0.988609 q^{61} +0.356542 q^{62} -0.456423 q^{64} -11.9251 q^{65} -5.41745 q^{67} +3.19493 q^{68} +4.93693 q^{70} +2.01705 q^{71} -9.97108 q^{73} -1.84055 q^{74} +2.24828 q^{76} +6.29348 q^{79} -13.8168 q^{80} +17.0748 q^{82} +1.72146 q^{83} +7.65351 q^{85} +8.36876 q^{86} +15.3035 q^{89} +4.29348 q^{91} -5.06691 q^{92} -23.1882 q^{94} +5.38581 q^{95} +11.6162 q^{97} -1.77748 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{4} + 6q^{5} - 4q^{7} - 9q^{8} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{4} + 6q^{5} - 4q^{7} - 9q^{8} - 14q^{10} + 2q^{14} - 4q^{16} + 3q^{17} + 3q^{19} + 17q^{20} + 8q^{23} + 12q^{26} - 4q^{28} - 3q^{29} - 3q^{31} + 10q^{32} - 12q^{34} - 6q^{35} - 7q^{37} + 20q^{38} - 13q^{40} - 4q^{41} + 8q^{43} - 3q^{46} + 14q^{47} + 4q^{49} - 33q^{50} - 17q^{52} + 9q^{53} + 9q^{56} + 3q^{58} + 25q^{59} - 19q^{61} - 10q^{62} + 3q^{64} - 12q^{65} - 15q^{67} + q^{68} + 14q^{70} + 7q^{71} - 11q^{73} - 8q^{74} - 26q^{76} + 8q^{79} + 4q^{80} + 3q^{82} + q^{83} + 15q^{85} - 4q^{86} + 17q^{89} + 17q^{92} - 20q^{94} - 17q^{95} - 15q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.77748 −1.25687 −0.628436 0.777862i \(-0.716304\pi\)
−0.628436 + 0.777862i \(0.716304\pi\)
\(3\) 0 0
\(4\) 1.15945 0.579725
\(5\) 2.77748 1.24213 0.621064 0.783760i \(-0.286701\pi\)
0.621064 + 0.783760i \(0.286701\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.49406 0.528231
\(9\) 0 0
\(10\) −4.93693 −1.56120
\(11\) 0 0
\(12\) 0 0
\(13\) −4.29348 −1.19080 −0.595398 0.803431i \(-0.703006\pi\)
−0.595398 + 0.803431i \(0.703006\pi\)
\(14\) 1.77748 0.475053
\(15\) 0 0
\(16\) −4.97458 −1.24364
\(17\) 2.75556 0.668321 0.334160 0.942516i \(-0.391547\pi\)
0.334160 + 0.942516i \(0.391547\pi\)
\(18\) 0 0
\(19\) 1.93910 0.444859 0.222429 0.974949i \(-0.428601\pi\)
0.222429 + 0.974949i \(0.428601\pi\)
\(20\) 3.22035 0.720093
\(21\) 0 0
\(22\) 0 0
\(23\) −4.37009 −0.911228 −0.455614 0.890178i \(-0.650580\pi\)
−0.455614 + 0.890178i \(0.650580\pi\)
\(24\) 0 0
\(25\) 2.71442 0.542884
\(26\) 7.63159 1.49668
\(27\) 0 0
\(28\) −1.15945 −0.219116
\(29\) −8.62809 −1.60220 −0.801098 0.598533i \(-0.795750\pi\)
−0.801098 + 0.598533i \(0.795750\pi\)
\(30\) 0 0
\(31\) −0.200588 −0.0360266 −0.0180133 0.999838i \(-0.505734\pi\)
−0.0180133 + 0.999838i \(0.505734\pi\)
\(32\) 5.85410 1.03487
\(33\) 0 0
\(34\) −4.89796 −0.839993
\(35\) −2.77748 −0.469481
\(36\) 0 0
\(37\) 1.03548 0.170232 0.0851159 0.996371i \(-0.472874\pi\)
0.0851159 + 0.996371i \(0.472874\pi\)
\(38\) −3.44671 −0.559130
\(39\) 0 0
\(40\) 4.14974 0.656131
\(41\) −9.60616 −1.50023 −0.750115 0.661307i \(-0.770002\pi\)
−0.750115 + 0.661307i \(0.770002\pi\)
\(42\) 0 0
\(43\) −4.70820 −0.717994 −0.358997 0.933339i \(-0.616881\pi\)
−0.358997 + 0.933339i \(0.616881\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 7.76777 1.14530
\(47\) 13.0455 1.90289 0.951443 0.307823i \(-0.0996006\pi\)
0.951443 + 0.307823i \(0.0996006\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.82484 −0.682335
\(51\) 0 0
\(52\) −4.97807 −0.690334
\(53\) 3.90012 0.535723 0.267861 0.963457i \(-0.413683\pi\)
0.267861 + 0.963457i \(0.413683\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.49406 −0.199653
\(57\) 0 0
\(58\) 15.3363 2.01375
\(59\) 8.55713 1.11404 0.557022 0.830498i \(-0.311944\pi\)
0.557022 + 0.830498i \(0.311944\pi\)
\(60\) 0 0
\(61\) −0.988609 −0.126578 −0.0632892 0.997995i \(-0.520159\pi\)
−0.0632892 + 0.997995i \(0.520159\pi\)
\(62\) 0.356542 0.0452808
\(63\) 0 0
\(64\) −0.456423 −0.0570529
\(65\) −11.9251 −1.47912
\(66\) 0 0
\(67\) −5.41745 −0.661846 −0.330923 0.943658i \(-0.607360\pi\)
−0.330923 + 0.943658i \(0.607360\pi\)
\(68\) 3.19493 0.387442
\(69\) 0 0
\(70\) 4.93693 0.590077
\(71\) 2.01705 0.239380 0.119690 0.992811i \(-0.461810\pi\)
0.119690 + 0.992811i \(0.461810\pi\)
\(72\) 0 0
\(73\) −9.97108 −1.16703 −0.583513 0.812104i \(-0.698322\pi\)
−0.583513 + 0.812104i \(0.698322\pi\)
\(74\) −1.84055 −0.213960
\(75\) 0 0
\(76\) 2.24828 0.257896
\(77\) 0 0
\(78\) 0 0
\(79\) 6.29348 0.708071 0.354036 0.935232i \(-0.384809\pi\)
0.354036 + 0.935232i \(0.384809\pi\)
\(80\) −13.8168 −1.54477
\(81\) 0 0
\(82\) 17.0748 1.88560
\(83\) 1.72146 0.188955 0.0944773 0.995527i \(-0.469882\pi\)
0.0944773 + 0.995527i \(0.469882\pi\)
\(84\) 0 0
\(85\) 7.65351 0.830140
\(86\) 8.36876 0.902426
\(87\) 0 0
\(88\) 0 0
\(89\) 15.3035 1.62217 0.811086 0.584928i \(-0.198877\pi\)
0.811086 + 0.584928i \(0.198877\pi\)
\(90\) 0 0
\(91\) 4.29348 0.450079
\(92\) −5.06691 −0.528262
\(93\) 0 0
\(94\) −23.1882 −2.39168
\(95\) 5.38581 0.552572
\(96\) 0 0
\(97\) 11.6162 1.17945 0.589724 0.807605i \(-0.299236\pi\)
0.589724 + 0.807605i \(0.299236\pi\)
\(98\) −1.77748 −0.179553
\(99\) 0 0
\(100\) 3.14723 0.314723
\(101\) −3.41179 −0.339486 −0.169743 0.985488i \(-0.554294\pi\)
−0.169743 + 0.985488i \(0.554294\pi\)
\(102\) 0 0
\(103\) −18.1826 −1.79158 −0.895791 0.444475i \(-0.853390\pi\)
−0.895791 + 0.444475i \(0.853390\pi\)
\(104\) −6.41473 −0.629016
\(105\) 0 0
\(106\) −6.93240 −0.673334
\(107\) −3.24746 −0.313944 −0.156972 0.987603i \(-0.550173\pi\)
−0.156972 + 0.987603i \(0.550173\pi\)
\(108\) 0 0
\(109\) 12.6912 1.21559 0.607796 0.794093i \(-0.292054\pi\)
0.607796 + 0.794093i \(0.292054\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.97458 0.470053
\(113\) 18.4480 1.73545 0.867723 0.497048i \(-0.165583\pi\)
0.867723 + 0.497048i \(0.165583\pi\)
\(114\) 0 0
\(115\) −12.1379 −1.13186
\(116\) −10.0038 −0.928833
\(117\) 0 0
\(118\) −15.2102 −1.40021
\(119\) −2.75556 −0.252601
\(120\) 0 0
\(121\) 0 0
\(122\) 1.75724 0.159093
\(123\) 0 0
\(124\) −0.232572 −0.0208855
\(125\) −6.34817 −0.567797
\(126\) 0 0
\(127\) 19.5093 1.73117 0.865585 0.500762i \(-0.166947\pi\)
0.865585 + 0.500762i \(0.166947\pi\)
\(128\) −10.8969 −0.963161
\(129\) 0 0
\(130\) 21.1966 1.85907
\(131\) 6.89796 0.602677 0.301339 0.953517i \(-0.402567\pi\)
0.301339 + 0.953517i \(0.402567\pi\)
\(132\) 0 0
\(133\) −1.93910 −0.168141
\(134\) 9.62943 0.831856
\(135\) 0 0
\(136\) 4.11698 0.353028
\(137\) 2.61070 0.223047 0.111523 0.993762i \(-0.464427\pi\)
0.111523 + 0.993762i \(0.464427\pi\)
\(138\) 0 0
\(139\) −1.00134 −0.0849322 −0.0424661 0.999098i \(-0.513521\pi\)
−0.0424661 + 0.999098i \(0.513521\pi\)
\(140\) −3.22035 −0.272170
\(141\) 0 0
\(142\) −3.58527 −0.300869
\(143\) 0 0
\(144\) 0 0
\(145\) −23.9644 −1.99013
\(146\) 17.7234 1.46680
\(147\) 0 0
\(148\) 1.20059 0.0986877
\(149\) 14.8948 1.22023 0.610113 0.792314i \(-0.291124\pi\)
0.610113 + 0.792314i \(0.291124\pi\)
\(150\) 0 0
\(151\) 15.8046 1.28616 0.643080 0.765799i \(-0.277656\pi\)
0.643080 + 0.765799i \(0.277656\pi\)
\(152\) 2.89713 0.234988
\(153\) 0 0
\(154\) 0 0
\(155\) −0.557129 −0.0447497
\(156\) 0 0
\(157\) 4.59123 0.366420 0.183210 0.983074i \(-0.441351\pi\)
0.183210 + 0.983074i \(0.441351\pi\)
\(158\) −11.1866 −0.889955
\(159\) 0 0
\(160\) 16.2597 1.28544
\(161\) 4.37009 0.344412
\(162\) 0 0
\(163\) 8.03764 0.629557 0.314778 0.949165i \(-0.398070\pi\)
0.314778 + 0.949165i \(0.398070\pi\)
\(164\) −11.1379 −0.869721
\(165\) 0 0
\(166\) −3.05987 −0.237492
\(167\) 13.4069 1.03746 0.518729 0.854939i \(-0.326405\pi\)
0.518729 + 0.854939i \(0.326405\pi\)
\(168\) 0 0
\(169\) 5.43394 0.417995
\(170\) −13.6040 −1.04338
\(171\) 0 0
\(172\) −5.45893 −0.416239
\(173\) 20.5821 1.56483 0.782413 0.622760i \(-0.213989\pi\)
0.782413 + 0.622760i \(0.213989\pi\)
\(174\) 0 0
\(175\) −2.71442 −0.205191
\(176\) 0 0
\(177\) 0 0
\(178\) −27.2018 −2.03886
\(179\) 3.71520 0.277687 0.138843 0.990314i \(-0.455662\pi\)
0.138843 + 0.990314i \(0.455662\pi\)
\(180\) 0 0
\(181\) 4.77183 0.354687 0.177344 0.984149i \(-0.443250\pi\)
0.177344 + 0.984149i \(0.443250\pi\)
\(182\) −7.63159 −0.565691
\(183\) 0 0
\(184\) −6.52920 −0.481339
\(185\) 2.87603 0.211450
\(186\) 0 0
\(187\) 0 0
\(188\) 15.1257 1.10315
\(189\) 0 0
\(190\) −9.57319 −0.694512
\(191\) 0.829158 0.0599958 0.0299979 0.999550i \(-0.490450\pi\)
0.0299979 + 0.999550i \(0.490450\pi\)
\(192\) 0 0
\(193\) −6.73803 −0.485014 −0.242507 0.970150i \(-0.577970\pi\)
−0.242507 + 0.970150i \(0.577970\pi\)
\(194\) −20.6476 −1.48241
\(195\) 0 0
\(196\) 1.15945 0.0828179
\(197\) 10.9216 0.778129 0.389065 0.921210i \(-0.372798\pi\)
0.389065 + 0.921210i \(0.372798\pi\)
\(198\) 0 0
\(199\) 20.9746 1.48685 0.743424 0.668820i \(-0.233200\pi\)
0.743424 + 0.668820i \(0.233200\pi\)
\(200\) 4.05552 0.286768
\(201\) 0 0
\(202\) 6.06440 0.426690
\(203\) 8.62809 0.605573
\(204\) 0 0
\(205\) −26.6810 −1.86348
\(206\) 32.3192 2.25179
\(207\) 0 0
\(208\) 21.3582 1.48093
\(209\) 0 0
\(210\) 0 0
\(211\) −6.02361 −0.414682 −0.207341 0.978269i \(-0.566481\pi\)
−0.207341 + 0.978269i \(0.566481\pi\)
\(212\) 4.52199 0.310572
\(213\) 0 0
\(214\) 5.77231 0.394587
\(215\) −13.0770 −0.891841
\(216\) 0 0
\(217\) 0.200588 0.0136168
\(218\) −22.5583 −1.52784
\(219\) 0 0
\(220\) 0 0
\(221\) −11.8309 −0.795833
\(222\) 0 0
\(223\) −4.62507 −0.309718 −0.154859 0.987937i \(-0.549492\pi\)
−0.154859 + 0.987937i \(0.549492\pi\)
\(224\) −5.85410 −0.391144
\(225\) 0 0
\(226\) −32.7911 −2.18123
\(227\) −17.3003 −1.14826 −0.574132 0.818763i \(-0.694660\pi\)
−0.574132 + 0.818763i \(0.694660\pi\)
\(228\) 0 0
\(229\) −4.42612 −0.292486 −0.146243 0.989249i \(-0.546718\pi\)
−0.146243 + 0.989249i \(0.546718\pi\)
\(230\) 21.5749 1.42260
\(231\) 0 0
\(232\) −12.8909 −0.846330
\(233\) 10.5330 0.690042 0.345021 0.938595i \(-0.387872\pi\)
0.345021 + 0.938595i \(0.387872\pi\)
\(234\) 0 0
\(235\) 36.2338 2.36363
\(236\) 9.92157 0.645839
\(237\) 0 0
\(238\) 4.89796 0.317487
\(239\) 9.75646 0.631093 0.315546 0.948910i \(-0.397812\pi\)
0.315546 + 0.948910i \(0.397812\pi\)
\(240\) 0 0
\(241\) 12.5501 0.808422 0.404211 0.914666i \(-0.367546\pi\)
0.404211 + 0.914666i \(0.367546\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.14624 −0.0733807
\(245\) 2.77748 0.177447
\(246\) 0 0
\(247\) −8.32546 −0.529736
\(248\) −0.299691 −0.0190304
\(249\) 0 0
\(250\) 11.2838 0.713648
\(251\) 10.9912 0.693758 0.346879 0.937910i \(-0.387241\pi\)
0.346879 + 0.937910i \(0.387241\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −34.6775 −2.17586
\(255\) 0 0
\(256\) 20.2819 1.26762
\(257\) −5.33280 −0.332651 −0.166325 0.986071i \(-0.553190\pi\)
−0.166325 + 0.986071i \(0.553190\pi\)
\(258\) 0 0
\(259\) −1.03548 −0.0643416
\(260\) −13.8265 −0.857484
\(261\) 0 0
\(262\) −12.2610 −0.757488
\(263\) 8.18034 0.504421 0.252211 0.967672i \(-0.418842\pi\)
0.252211 + 0.967672i \(0.418842\pi\)
\(264\) 0 0
\(265\) 10.8325 0.665436
\(266\) 3.44671 0.211331
\(267\) 0 0
\(268\) −6.28126 −0.383689
\(269\) −12.8439 −0.783107 −0.391554 0.920155i \(-0.628062\pi\)
−0.391554 + 0.920155i \(0.628062\pi\)
\(270\) 0 0
\(271\) −22.0472 −1.33927 −0.669636 0.742689i \(-0.733550\pi\)
−0.669636 + 0.742689i \(0.733550\pi\)
\(272\) −13.7077 −0.831153
\(273\) 0 0
\(274\) −4.64047 −0.280341
\(275\) 0 0
\(276\) 0 0
\(277\) 24.0770 1.44664 0.723322 0.690511i \(-0.242614\pi\)
0.723322 + 0.690511i \(0.242614\pi\)
\(278\) 1.77986 0.106749
\(279\) 0 0
\(280\) −4.14974 −0.247994
\(281\) −15.4418 −0.921182 −0.460591 0.887612i \(-0.652363\pi\)
−0.460591 + 0.887612i \(0.652363\pi\)
\(282\) 0 0
\(283\) −3.35732 −0.199572 −0.0997860 0.995009i \(-0.531816\pi\)
−0.0997860 + 0.995009i \(0.531816\pi\)
\(284\) 2.33867 0.138774
\(285\) 0 0
\(286\) 0 0
\(287\) 9.60616 0.567034
\(288\) 0 0
\(289\) −9.40691 −0.553348
\(290\) 42.5963 2.50134
\(291\) 0 0
\(292\) −11.5610 −0.676555
\(293\) −6.83705 −0.399425 −0.199712 0.979855i \(-0.564001\pi\)
−0.199712 + 0.979855i \(0.564001\pi\)
\(294\) 0 0
\(295\) 23.7673 1.38379
\(296\) 1.54707 0.0899218
\(297\) 0 0
\(298\) −26.4752 −1.53367
\(299\) 18.7629 1.08509
\(300\) 0 0
\(301\) 4.70820 0.271376
\(302\) −28.0924 −1.61654
\(303\) 0 0
\(304\) −9.64618 −0.553246
\(305\) −2.74585 −0.157227
\(306\) 0 0
\(307\) −11.7970 −0.673293 −0.336646 0.941631i \(-0.609293\pi\)
−0.336646 + 0.941631i \(0.609293\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0.990289 0.0562446
\(311\) −25.3139 −1.43542 −0.717709 0.696343i \(-0.754809\pi\)
−0.717709 + 0.696343i \(0.754809\pi\)
\(312\) 0 0
\(313\) −19.6543 −1.11093 −0.555464 0.831540i \(-0.687460\pi\)
−0.555464 + 0.831540i \(0.687460\pi\)
\(314\) −8.16083 −0.460542
\(315\) 0 0
\(316\) 7.29697 0.410487
\(317\) −3.61056 −0.202789 −0.101395 0.994846i \(-0.532330\pi\)
−0.101395 + 0.994846i \(0.532330\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.26771 −0.0708670
\(321\) 0 0
\(322\) −7.76777 −0.432881
\(323\) 5.34329 0.297308
\(324\) 0 0
\(325\) −11.6543 −0.646464
\(326\) −14.2868 −0.791272
\(327\) 0 0
\(328\) −14.3522 −0.792469
\(329\) −13.0455 −0.719224
\(330\) 0 0
\(331\) −26.5335 −1.45841 −0.729205 0.684295i \(-0.760110\pi\)
−0.729205 + 0.684295i \(0.760110\pi\)
\(332\) 1.99595 0.109542
\(333\) 0 0
\(334\) −23.8306 −1.30395
\(335\) −15.0469 −0.822098
\(336\) 0 0
\(337\) −0.685979 −0.0373676 −0.0186838 0.999825i \(-0.505948\pi\)
−0.0186838 + 0.999825i \(0.505948\pi\)
\(338\) −9.65874 −0.525366
\(339\) 0 0
\(340\) 8.87387 0.481253
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −7.03436 −0.379267
\(345\) 0 0
\(346\) −36.5843 −1.96678
\(347\) −21.5055 −1.15447 −0.577237 0.816577i \(-0.695869\pi\)
−0.577237 + 0.816577i \(0.695869\pi\)
\(348\) 0 0
\(349\) 19.4429 1.04075 0.520377 0.853937i \(-0.325792\pi\)
0.520377 + 0.853937i \(0.325792\pi\)
\(350\) 4.82484 0.257898
\(351\) 0 0
\(352\) 0 0
\(353\) 20.9307 1.11403 0.557015 0.830502i \(-0.311947\pi\)
0.557015 + 0.830502i \(0.311947\pi\)
\(354\) 0 0
\(355\) 5.60232 0.297340
\(356\) 17.7437 0.940413
\(357\) 0 0
\(358\) −6.60370 −0.349017
\(359\) −9.77127 −0.515708 −0.257854 0.966184i \(-0.583015\pi\)
−0.257854 + 0.966184i \(0.583015\pi\)
\(360\) 0 0
\(361\) −15.2399 −0.802100
\(362\) −8.48185 −0.445796
\(363\) 0 0
\(364\) 4.97807 0.260922
\(365\) −27.6945 −1.44960
\(366\) 0 0
\(367\) 9.89969 0.516759 0.258380 0.966043i \(-0.416811\pi\)
0.258380 + 0.966043i \(0.416811\pi\)
\(368\) 21.7394 1.13324
\(369\) 0 0
\(370\) −5.11210 −0.265765
\(371\) −3.90012 −0.202484
\(372\) 0 0
\(373\) −4.27475 −0.221338 −0.110669 0.993857i \(-0.535299\pi\)
−0.110669 + 0.993857i \(0.535299\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 19.4909 1.00516
\(377\) 37.0445 1.90789
\(378\) 0 0
\(379\) 4.32594 0.222209 0.111104 0.993809i \(-0.464561\pi\)
0.111104 + 0.993809i \(0.464561\pi\)
\(380\) 6.24458 0.320340
\(381\) 0 0
\(382\) −1.47382 −0.0754070
\(383\) 1.32210 0.0675561 0.0337781 0.999429i \(-0.489246\pi\)
0.0337781 + 0.999429i \(0.489246\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 11.9767 0.609600
\(387\) 0 0
\(388\) 13.4684 0.683756
\(389\) 38.5092 1.95249 0.976246 0.216665i \(-0.0695180\pi\)
0.976246 + 0.216665i \(0.0695180\pi\)
\(390\) 0 0
\(391\) −12.0420 −0.608992
\(392\) 1.49406 0.0754616
\(393\) 0 0
\(394\) −19.4129 −0.978008
\(395\) 17.4800 0.879516
\(396\) 0 0
\(397\) −0.410109 −0.0205828 −0.0102914 0.999947i \(-0.503276\pi\)
−0.0102914 + 0.999947i \(0.503276\pi\)
\(398\) −37.2820 −1.86878
\(399\) 0 0
\(400\) −13.5031 −0.675154
\(401\) 1.56684 0.0782443 0.0391221 0.999234i \(-0.487544\pi\)
0.0391221 + 0.999234i \(0.487544\pi\)
\(402\) 0 0
\(403\) 0.861219 0.0429004
\(404\) −3.95580 −0.196808
\(405\) 0 0
\(406\) −15.3363 −0.761127
\(407\) 0 0
\(408\) 0 0
\(409\) −6.77852 −0.335176 −0.167588 0.985857i \(-0.553598\pi\)
−0.167588 + 0.985857i \(0.553598\pi\)
\(410\) 47.4250 2.34215
\(411\) 0 0
\(412\) −21.0818 −1.03863
\(413\) −8.55713 −0.421069
\(414\) 0 0
\(415\) 4.78133 0.234706
\(416\) −25.1344 −1.23232
\(417\) 0 0
\(418\) 0 0
\(419\) −28.7218 −1.40315 −0.701577 0.712594i \(-0.747520\pi\)
−0.701577 + 0.712594i \(0.747520\pi\)
\(420\) 0 0
\(421\) 12.2256 0.595838 0.297919 0.954591i \(-0.403707\pi\)
0.297919 + 0.954591i \(0.403707\pi\)
\(422\) 10.7069 0.521202
\(423\) 0 0
\(424\) 5.82703 0.282985
\(425\) 7.47973 0.362820
\(426\) 0 0
\(427\) 0.988609 0.0478421
\(428\) −3.76527 −0.182001
\(429\) 0 0
\(430\) 23.2441 1.12093
\(431\) −3.69129 −0.177803 −0.0889016 0.996040i \(-0.528336\pi\)
−0.0889016 + 0.996040i \(0.528336\pi\)
\(432\) 0 0
\(433\) 29.0749 1.39725 0.698625 0.715488i \(-0.253796\pi\)
0.698625 + 0.715488i \(0.253796\pi\)
\(434\) −0.356542 −0.0171145
\(435\) 0 0
\(436\) 14.7148 0.704709
\(437\) −8.47403 −0.405368
\(438\) 0 0
\(439\) −14.2017 −0.677811 −0.338905 0.940820i \(-0.610057\pi\)
−0.338905 + 0.940820i \(0.610057\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 21.0293 1.00026
\(443\) −27.2356 −1.29400 −0.647001 0.762489i \(-0.723977\pi\)
−0.647001 + 0.762489i \(0.723977\pi\)
\(444\) 0 0
\(445\) 42.5053 2.01495
\(446\) 8.22100 0.389275
\(447\) 0 0
\(448\) 0.456423 0.0215640
\(449\) 41.9159 1.97813 0.989067 0.147467i \(-0.0471119\pi\)
0.989067 + 0.147467i \(0.0471119\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 21.3896 1.00608
\(453\) 0 0
\(454\) 30.7511 1.44322
\(455\) 11.9251 0.559056
\(456\) 0 0
\(457\) 19.4977 0.912063 0.456032 0.889964i \(-0.349270\pi\)
0.456032 + 0.889964i \(0.349270\pi\)
\(458\) 7.86736 0.367617
\(459\) 0 0
\(460\) −14.0733 −0.656169
\(461\) −12.2251 −0.569380 −0.284690 0.958620i \(-0.591891\pi\)
−0.284690 + 0.958620i \(0.591891\pi\)
\(462\) 0 0
\(463\) 13.8550 0.643894 0.321947 0.946758i \(-0.395663\pi\)
0.321947 + 0.946758i \(0.395663\pi\)
\(464\) 42.9211 1.99256
\(465\) 0 0
\(466\) −18.7223 −0.867294
\(467\) 16.4207 0.759861 0.379930 0.925015i \(-0.375948\pi\)
0.379930 + 0.925015i \(0.375948\pi\)
\(468\) 0 0
\(469\) 5.41745 0.250154
\(470\) −64.4050 −2.97078
\(471\) 0 0
\(472\) 12.7849 0.588473
\(473\) 0 0
\(474\) 0 0
\(475\) 5.26352 0.241507
\(476\) −3.19493 −0.146439
\(477\) 0 0
\(478\) −17.3420 −0.793202
\(479\) 24.7914 1.13275 0.566374 0.824149i \(-0.308346\pi\)
0.566374 + 0.824149i \(0.308346\pi\)
\(480\) 0 0
\(481\) −4.44581 −0.202711
\(482\) −22.3076 −1.01608
\(483\) 0 0
\(484\) 0 0
\(485\) 32.2639 1.46503
\(486\) 0 0
\(487\) −14.3342 −0.649544 −0.324772 0.945792i \(-0.605288\pi\)
−0.324772 + 0.945792i \(0.605288\pi\)
\(488\) −1.47704 −0.0668627
\(489\) 0 0
\(490\) −4.93693 −0.223028
\(491\) 22.0193 0.993719 0.496859 0.867831i \(-0.334486\pi\)
0.496859 + 0.867831i \(0.334486\pi\)
\(492\) 0 0
\(493\) −23.7752 −1.07078
\(494\) 14.7984 0.665810
\(495\) 0 0
\(496\) 0.997839 0.0448043
\(497\) −2.01705 −0.0904770
\(498\) 0 0
\(499\) −25.3117 −1.13311 −0.566554 0.824024i \(-0.691724\pi\)
−0.566554 + 0.824024i \(0.691724\pi\)
\(500\) −7.36038 −0.329166
\(501\) 0 0
\(502\) −19.5367 −0.871964
\(503\) 26.0214 1.16024 0.580119 0.814531i \(-0.303006\pi\)
0.580119 + 0.814531i \(0.303006\pi\)
\(504\) 0 0
\(505\) −9.47619 −0.421685
\(506\) 0 0
\(507\) 0 0
\(508\) 22.6201 1.00360
\(509\) −38.1269 −1.68994 −0.844972 0.534810i \(-0.820383\pi\)
−0.844972 + 0.534810i \(0.820383\pi\)
\(510\) 0 0
\(511\) 9.97108 0.441095
\(512\) −14.2570 −0.630077
\(513\) 0 0
\(514\) 9.47896 0.418099
\(515\) −50.5018 −2.22538
\(516\) 0 0
\(517\) 0 0
\(518\) 1.84055 0.0808691
\(519\) 0 0
\(520\) −17.8168 −0.781319
\(521\) 26.0420 1.14092 0.570460 0.821325i \(-0.306765\pi\)
0.570460 + 0.821325i \(0.306765\pi\)
\(522\) 0 0
\(523\) 19.1230 0.836189 0.418095 0.908403i \(-0.362698\pi\)
0.418095 + 0.908403i \(0.362698\pi\)
\(524\) 7.99784 0.349387
\(525\) 0 0
\(526\) −14.5404 −0.633993
\(527\) −0.552731 −0.0240773
\(528\) 0 0
\(529\) −3.90228 −0.169664
\(530\) −19.2546 −0.836368
\(531\) 0 0
\(532\) −2.24828 −0.0974755
\(533\) 41.2438 1.78647
\(534\) 0 0
\(535\) −9.01977 −0.389959
\(536\) −8.09401 −0.349608
\(537\) 0 0
\(538\) 22.8298 0.984265
\(539\) 0 0
\(540\) 0 0
\(541\) 10.8860 0.468027 0.234014 0.972233i \(-0.424814\pi\)
0.234014 + 0.972233i \(0.424814\pi\)
\(542\) 39.1886 1.68329
\(543\) 0 0
\(544\) 16.1313 0.691624
\(545\) 35.2495 1.50992
\(546\) 0 0
\(547\) −12.9091 −0.551951 −0.275976 0.961165i \(-0.589001\pi\)
−0.275976 + 0.961165i \(0.589001\pi\)
\(548\) 3.02697 0.129306
\(549\) 0 0
\(550\) 0 0
\(551\) −16.7307 −0.712751
\(552\) 0 0
\(553\) −6.29348 −0.267626
\(554\) −42.7964 −1.81825
\(555\) 0 0
\(556\) −1.16100 −0.0492373
\(557\) 0.762626 0.0323135 0.0161567 0.999869i \(-0.494857\pi\)
0.0161567 + 0.999869i \(0.494857\pi\)
\(558\) 0 0
\(559\) 20.2146 0.854985
\(560\) 13.8168 0.583867
\(561\) 0 0
\(562\) 27.4476 1.15781
\(563\) 22.2281 0.936803 0.468402 0.883516i \(-0.344830\pi\)
0.468402 + 0.883516i \(0.344830\pi\)
\(564\) 0 0
\(565\) 51.2392 2.15565
\(566\) 5.96758 0.250836
\(567\) 0 0
\(568\) 3.01360 0.126448
\(569\) 20.5131 0.859955 0.429978 0.902839i \(-0.358521\pi\)
0.429978 + 0.902839i \(0.358521\pi\)
\(570\) 0 0
\(571\) −19.5654 −0.818785 −0.409393 0.912358i \(-0.634259\pi\)
−0.409393 + 0.912358i \(0.634259\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −17.0748 −0.712688
\(575\) −11.8623 −0.494691
\(576\) 0 0
\(577\) −19.5716 −0.814776 −0.407388 0.913255i \(-0.633560\pi\)
−0.407388 + 0.913255i \(0.633560\pi\)
\(578\) 16.7206 0.695487
\(579\) 0 0
\(580\) −27.7855 −1.15373
\(581\) −1.72146 −0.0714182
\(582\) 0 0
\(583\) 0 0
\(584\) −14.8974 −0.616460
\(585\) 0 0
\(586\) 12.1528 0.502026
\(587\) −1.57068 −0.0648290 −0.0324145 0.999475i \(-0.510320\pi\)
−0.0324145 + 0.999475i \(0.510320\pi\)
\(588\) 0 0
\(589\) −0.388959 −0.0160268
\(590\) −42.2460 −1.73924
\(591\) 0 0
\(592\) −5.15107 −0.211708
\(593\) 30.1230 1.23700 0.618502 0.785783i \(-0.287740\pi\)
0.618502 + 0.785783i \(0.287740\pi\)
\(594\) 0 0
\(595\) −7.65351 −0.313763
\(596\) 17.2697 0.707396
\(597\) 0 0
\(598\) −33.3507 −1.36381
\(599\) −5.77792 −0.236079 −0.118040 0.993009i \(-0.537661\pi\)
−0.118040 + 0.993009i \(0.537661\pi\)
\(600\) 0 0
\(601\) 45.5645 1.85862 0.929308 0.369305i \(-0.120404\pi\)
0.929308 + 0.369305i \(0.120404\pi\)
\(602\) −8.36876 −0.341085
\(603\) 0 0
\(604\) 18.3246 0.745619
\(605\) 0 0
\(606\) 0 0
\(607\) 34.7211 1.40928 0.704642 0.709563i \(-0.251108\pi\)
0.704642 + 0.709563i \(0.251108\pi\)
\(608\) 11.3517 0.460371
\(609\) 0 0
\(610\) 4.88070 0.197614
\(611\) −56.0107 −2.26595
\(612\) 0 0
\(613\) −24.1423 −0.975097 −0.487548 0.873096i \(-0.662109\pi\)
−0.487548 + 0.873096i \(0.662109\pi\)
\(614\) 20.9690 0.846242
\(615\) 0 0
\(616\) 0 0
\(617\) 13.4967 0.543358 0.271679 0.962388i \(-0.412421\pi\)
0.271679 + 0.962388i \(0.412421\pi\)
\(618\) 0 0
\(619\) −43.4856 −1.74783 −0.873917 0.486074i \(-0.838428\pi\)
−0.873917 + 0.486074i \(0.838428\pi\)
\(620\) −0.645964 −0.0259425
\(621\) 0 0
\(622\) 44.9950 1.80414
\(623\) −15.3035 −0.613123
\(624\) 0 0
\(625\) −31.2040 −1.24816
\(626\) 34.9353 1.39629
\(627\) 0 0
\(628\) 5.32330 0.212423
\(629\) 2.85332 0.113769
\(630\) 0 0
\(631\) 6.42012 0.255581 0.127790 0.991801i \(-0.459212\pi\)
0.127790 + 0.991801i \(0.459212\pi\)
\(632\) 9.40286 0.374026
\(633\) 0 0
\(634\) 6.41771 0.254880
\(635\) 54.1868 2.15034
\(636\) 0 0
\(637\) −4.29348 −0.170114
\(638\) 0 0
\(639\) 0 0
\(640\) −30.2660 −1.19637
\(641\) 6.39600 0.252627 0.126313 0.991990i \(-0.459686\pi\)
0.126313 + 0.991990i \(0.459686\pi\)
\(642\) 0 0
\(643\) 0.652660 0.0257384 0.0128692 0.999917i \(-0.495903\pi\)
0.0128692 + 0.999917i \(0.495903\pi\)
\(644\) 5.06691 0.199664
\(645\) 0 0
\(646\) −9.49761 −0.373678
\(647\) −17.9621 −0.706165 −0.353082 0.935592i \(-0.614866\pi\)
−0.353082 + 0.935592i \(0.614866\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 20.7153 0.812522
\(651\) 0 0
\(652\) 9.31925 0.364970
\(653\) 16.9181 0.662055 0.331028 0.943621i \(-0.392605\pi\)
0.331028 + 0.943621i \(0.392605\pi\)
\(654\) 0 0
\(655\) 19.1590 0.748603
\(656\) 47.7866 1.86575
\(657\) 0 0
\(658\) 23.1882 0.903972
\(659\) −23.6249 −0.920297 −0.460148 0.887842i \(-0.652204\pi\)
−0.460148 + 0.887842i \(0.652204\pi\)
\(660\) 0 0
\(661\) −20.9819 −0.816103 −0.408051 0.912959i \(-0.633792\pi\)
−0.408051 + 0.912959i \(0.633792\pi\)
\(662\) 47.1628 1.83303
\(663\) 0 0
\(664\) 2.57197 0.0998118
\(665\) −5.38581 −0.208853
\(666\) 0 0
\(667\) 37.7056 1.45997
\(668\) 15.5446 0.601440
\(669\) 0 0
\(670\) 26.7456 1.03327
\(671\) 0 0
\(672\) 0 0
\(673\) 12.0986 0.466368 0.233184 0.972433i \(-0.425086\pi\)
0.233184 + 0.972433i \(0.425086\pi\)
\(674\) 1.21932 0.0469663
\(675\) 0 0
\(676\) 6.30038 0.242322
\(677\) 9.42988 0.362420 0.181210 0.983444i \(-0.441999\pi\)
0.181210 + 0.983444i \(0.441999\pi\)
\(678\) 0 0
\(679\) −11.6162 −0.445790
\(680\) 11.4348 0.438506
\(681\) 0 0
\(682\) 0 0
\(683\) 15.2986 0.585385 0.292692 0.956207i \(-0.405449\pi\)
0.292692 + 0.956207i \(0.405449\pi\)
\(684\) 0 0
\(685\) 7.25117 0.277053
\(686\) 1.77748 0.0678647
\(687\) 0 0
\(688\) 23.4213 0.892929
\(689\) −16.7451 −0.637936
\(690\) 0 0
\(691\) 22.8193 0.868086 0.434043 0.900892i \(-0.357087\pi\)
0.434043 + 0.900892i \(0.357087\pi\)
\(692\) 23.8639 0.907169
\(693\) 0 0
\(694\) 38.2256 1.45102
\(695\) −2.78119 −0.105497
\(696\) 0 0
\(697\) −26.4703 −1.00263
\(698\) −34.5594 −1.30809
\(699\) 0 0
\(700\) −3.14723 −0.118954
\(701\) 32.3242 1.22087 0.610433 0.792068i \(-0.290995\pi\)
0.610433 + 0.792068i \(0.290995\pi\)
\(702\) 0 0
\(703\) 2.00789 0.0757292
\(704\) 0 0
\(705\) 0 0
\(706\) −37.2040 −1.40019
\(707\) 3.41179 0.128314
\(708\) 0 0
\(709\) 14.5598 0.546804 0.273402 0.961900i \(-0.411851\pi\)
0.273402 + 0.961900i \(0.411851\pi\)
\(710\) −9.95804 −0.373718
\(711\) 0 0
\(712\) 22.8645 0.856882
\(713\) 0.876587 0.0328285
\(714\) 0 0
\(715\) 0 0
\(716\) 4.30759 0.160982
\(717\) 0 0
\(718\) 17.3683 0.648178
\(719\) −44.8602 −1.67300 −0.836501 0.547965i \(-0.815403\pi\)
−0.836501 + 0.547965i \(0.815403\pi\)
\(720\) 0 0
\(721\) 18.1826 0.677155
\(722\) 27.0887 1.00814
\(723\) 0 0
\(724\) 5.53270 0.205621
\(725\) −23.4202 −0.869806
\(726\) 0 0
\(727\) 28.3582 1.05175 0.525874 0.850562i \(-0.323738\pi\)
0.525874 + 0.850562i \(0.323738\pi\)
\(728\) 6.41473 0.237746
\(729\) 0 0
\(730\) 49.2266 1.82196
\(731\) −12.9737 −0.479850
\(732\) 0 0
\(733\) 6.25696 0.231106 0.115553 0.993301i \(-0.463136\pi\)
0.115553 + 0.993301i \(0.463136\pi\)
\(734\) −17.5965 −0.649500
\(735\) 0 0
\(736\) −25.5830 −0.943001
\(737\) 0 0
\(738\) 0 0
\(739\) 9.35201 0.344019 0.172010 0.985095i \(-0.444974\pi\)
0.172010 + 0.985095i \(0.444974\pi\)
\(740\) 3.33461 0.122583
\(741\) 0 0
\(742\) 6.93240 0.254496
\(743\) −25.2066 −0.924740 −0.462370 0.886687i \(-0.653001\pi\)
−0.462370 + 0.886687i \(0.653001\pi\)
\(744\) 0 0
\(745\) 41.3700 1.51568
\(746\) 7.59830 0.278193
\(747\) 0 0
\(748\) 0 0
\(749\) 3.24746 0.118660
\(750\) 0 0
\(751\) −34.7493 −1.26802 −0.634010 0.773325i \(-0.718592\pi\)
−0.634010 + 0.773325i \(0.718592\pi\)
\(752\) −64.8960 −2.36651
\(753\) 0 0
\(754\) −65.8460 −2.39797
\(755\) 43.8970 1.59758
\(756\) 0 0
\(757\) 34.7960 1.26468 0.632341 0.774690i \(-0.282094\pi\)
0.632341 + 0.774690i \(0.282094\pi\)
\(758\) −7.68929 −0.279288
\(759\) 0 0
\(760\) 8.04674 0.291886
\(761\) 12.2255 0.443172 0.221586 0.975141i \(-0.428877\pi\)
0.221586 + 0.975141i \(0.428877\pi\)
\(762\) 0 0
\(763\) −12.6912 −0.459451
\(764\) 0.961368 0.0347811
\(765\) 0 0
\(766\) −2.35001 −0.0849094
\(767\) −36.7398 −1.32660
\(768\) 0 0
\(769\) −2.61946 −0.0944603 −0.0472301 0.998884i \(-0.515039\pi\)
−0.0472301 + 0.998884i \(0.515039\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −7.81241 −0.281175
\(773\) 0.212804 0.00765405 0.00382702 0.999993i \(-0.498782\pi\)
0.00382702 + 0.999993i \(0.498782\pi\)
\(774\) 0 0
\(775\) −0.544479 −0.0195583
\(776\) 17.3554 0.623022
\(777\) 0 0
\(778\) −68.4494 −2.45403
\(779\) −18.6273 −0.667391
\(780\) 0 0
\(781\) 0 0
\(782\) 21.4045 0.765425
\(783\) 0 0
\(784\) −4.97458 −0.177663
\(785\) 12.7521 0.455141
\(786\) 0 0
\(787\) 29.1375 1.03864 0.519320 0.854580i \(-0.326185\pi\)
0.519320 + 0.854580i \(0.326185\pi\)
\(788\) 12.6630 0.451101
\(789\) 0 0
\(790\) −31.0705 −1.10544
\(791\) −18.4480 −0.655937
\(792\) 0 0
\(793\) 4.24457 0.150729
\(794\) 0.728962 0.0258699
\(795\) 0 0
\(796\) 24.3190 0.861963
\(797\) −32.2284 −1.14159 −0.570794 0.821093i \(-0.693365\pi\)
−0.570794 + 0.821093i \(0.693365\pi\)
\(798\) 0 0
\(799\) 35.9477 1.27174
\(800\) 15.8905 0.561813
\(801\) 0 0
\(802\) −2.78503 −0.0983430
\(803\) 0 0
\(804\) 0 0
\(805\) 12.1379 0.427804
\(806\) −1.53080 −0.0539202
\(807\) 0 0
\(808\) −5.09743 −0.179327
\(809\) 29.3461 1.03175 0.515877 0.856663i \(-0.327466\pi\)
0.515877 + 0.856663i \(0.327466\pi\)
\(810\) 0 0
\(811\) 1.03940 0.0364982 0.0182491 0.999833i \(-0.494191\pi\)
0.0182491 + 0.999833i \(0.494191\pi\)
\(812\) 10.0038 0.351066
\(813\) 0 0
\(814\) 0 0
\(815\) 22.3244 0.781990
\(816\) 0 0
\(817\) −9.12966 −0.319406
\(818\) 12.0487 0.421274
\(819\) 0 0
\(820\) −30.9352 −1.08031
\(821\) −28.5323 −0.995786 −0.497893 0.867239i \(-0.665893\pi\)
−0.497893 + 0.867239i \(0.665893\pi\)
\(822\) 0 0
\(823\) 26.6577 0.929227 0.464614 0.885513i \(-0.346193\pi\)
0.464614 + 0.885513i \(0.346193\pi\)
\(824\) −27.1659 −0.946370
\(825\) 0 0
\(826\) 15.2102 0.529229
\(827\) −1.71964 −0.0597978 −0.0298989 0.999553i \(-0.509519\pi\)
−0.0298989 + 0.999553i \(0.509519\pi\)
\(828\) 0 0
\(829\) 27.9605 0.971107 0.485554 0.874207i \(-0.338618\pi\)
0.485554 + 0.874207i \(0.338618\pi\)
\(830\) −8.49873 −0.294995
\(831\) 0 0
\(832\) 1.95964 0.0679383
\(833\) 2.75556 0.0954744
\(834\) 0 0
\(835\) 37.2375 1.28866
\(836\) 0 0
\(837\) 0 0
\(838\) 51.0526 1.76358
\(839\) −35.8624 −1.23811 −0.619054 0.785348i \(-0.712484\pi\)
−0.619054 + 0.785348i \(0.712484\pi\)
\(840\) 0 0
\(841\) 45.4439 1.56703
\(842\) −21.7308 −0.748892
\(843\) 0 0
\(844\) −6.98407 −0.240402
\(845\) 15.0927 0.519204
\(846\) 0 0
\(847\) 0 0
\(848\) −19.4014 −0.666248
\(849\) 0 0
\(850\) −13.2951 −0.456018
\(851\) −4.52515 −0.155120
\(852\) 0 0
\(853\) −15.1427 −0.518478 −0.259239 0.965813i \(-0.583472\pi\)
−0.259239 + 0.965813i \(0.583472\pi\)
\(854\) −1.75724 −0.0601314
\(855\) 0 0
\(856\) −4.85191 −0.165835
\(857\) −25.3267 −0.865142 −0.432571 0.901600i \(-0.642394\pi\)
−0.432571 + 0.901600i \(0.642394\pi\)
\(858\) 0 0
\(859\) 41.5291 1.41696 0.708478 0.705733i \(-0.249382\pi\)
0.708478 + 0.705733i \(0.249382\pi\)
\(860\) −15.1621 −0.517023
\(861\) 0 0
\(862\) 6.56121 0.223476
\(863\) 24.0504 0.818684 0.409342 0.912381i \(-0.365758\pi\)
0.409342 + 0.912381i \(0.365758\pi\)
\(864\) 0 0
\(865\) 57.1664 1.94372
\(866\) −51.6801 −1.75616
\(867\) 0 0
\(868\) 0.232572 0.00789399
\(869\) 0 0
\(870\) 0 0
\(871\) 23.2597 0.788124
\(872\) 18.9614 0.642114
\(873\) 0 0
\(874\) 15.0625 0.509495
\(875\) 6.34817 0.214607
\(876\) 0 0
\(877\) 33.8052 1.14152 0.570760 0.821117i \(-0.306649\pi\)
0.570760 + 0.821117i \(0.306649\pi\)
\(878\) 25.2433 0.851921
\(879\) 0 0
\(880\) 0 0
\(881\) 36.8296 1.24082 0.620410 0.784278i \(-0.286966\pi\)
0.620410 + 0.784278i \(0.286966\pi\)
\(882\) 0 0
\(883\) 53.4103 1.79740 0.898701 0.438563i \(-0.144512\pi\)
0.898701 + 0.438563i \(0.144512\pi\)
\(884\) −13.7174 −0.461365
\(885\) 0 0
\(886\) 48.4108 1.62639
\(887\) −11.4616 −0.384843 −0.192421 0.981312i \(-0.561634\pi\)
−0.192421 + 0.981312i \(0.561634\pi\)
\(888\) 0 0
\(889\) −19.5093 −0.654321
\(890\) −75.5525 −2.53253
\(891\) 0 0
\(892\) −5.36254 −0.179551
\(893\) 25.2965 0.846516
\(894\) 0 0
\(895\) 10.3189 0.344923
\(896\) 10.8969 0.364041
\(897\) 0 0
\(898\) −74.5049 −2.48626
\(899\) 1.73069 0.0577217
\(900\) 0 0
\(901\) 10.7470 0.358034
\(902\) 0 0
\(903\) 0 0
\(904\) 27.5626 0.916717
\(905\) 13.2537 0.440567
\(906\) 0 0
\(907\) 57.0582 1.89459 0.947293 0.320369i \(-0.103807\pi\)
0.947293 + 0.320369i \(0.103807\pi\)
\(908\) −20.0589 −0.665677
\(909\) 0 0
\(910\) −21.1966 −0.702661
\(911\) −6.67566 −0.221175 −0.110587 0.993866i \(-0.535273\pi\)
−0.110587 + 0.993866i \(0.535273\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −34.6568 −1.14635
\(915\) 0 0
\(916\) −5.13186 −0.169562
\(917\) −6.89796 −0.227791
\(918\) 0 0
\(919\) −29.1339 −0.961038 −0.480519 0.876984i \(-0.659552\pi\)
−0.480519 + 0.876984i \(0.659552\pi\)
\(920\) −18.1347 −0.597885
\(921\) 0 0
\(922\) 21.7299 0.715637
\(923\) −8.66015 −0.285052
\(924\) 0 0
\(925\) 2.81073 0.0924161
\(926\) −24.6270 −0.809292
\(927\) 0 0
\(928\) −50.5097 −1.65806
\(929\) −2.34472 −0.0769277 −0.0384638 0.999260i \(-0.512246\pi\)
−0.0384638 + 0.999260i \(0.512246\pi\)
\(930\) 0 0
\(931\) 1.93910 0.0635513
\(932\) 12.2125 0.400035
\(933\) 0 0
\(934\) −29.1876 −0.955047
\(935\) 0 0
\(936\) 0 0
\(937\) 8.78780 0.287085 0.143542 0.989644i \(-0.454151\pi\)
0.143542 + 0.989644i \(0.454151\pi\)
\(938\) −9.62943 −0.314412
\(939\) 0 0
\(940\) 42.0113 1.37026
\(941\) −8.88871 −0.289764 −0.144882 0.989449i \(-0.546280\pi\)
−0.144882 + 0.989449i \(0.546280\pi\)
\(942\) 0 0
\(943\) 41.9798 1.36705
\(944\) −42.5681 −1.38547
\(945\) 0 0
\(946\) 0 0
\(947\) 7.86275 0.255505 0.127752 0.991806i \(-0.459224\pi\)
0.127752 + 0.991806i \(0.459224\pi\)
\(948\) 0 0
\(949\) 42.8106 1.38969
\(950\) −9.35582 −0.303543
\(951\) 0 0
\(952\) −4.11698 −0.133432
\(953\) 17.4644 0.565727 0.282864 0.959160i \(-0.408716\pi\)
0.282864 + 0.959160i \(0.408716\pi\)
\(954\) 0 0
\(955\) 2.30297 0.0745225
\(956\) 11.3121 0.365860
\(957\) 0 0
\(958\) −44.0663 −1.42372
\(959\) −2.61070 −0.0843038
\(960\) 0 0
\(961\) −30.9598 −0.998702
\(962\) 7.90236 0.254782
\(963\) 0 0
\(964\) 14.5512 0.468663
\(965\) −18.7148 −0.602450
\(966\) 0 0
\(967\) −45.6122 −1.46679 −0.733395 0.679802i \(-0.762066\pi\)
−0.733395 + 0.679802i \(0.762066\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −57.3485 −1.84135
\(971\) −2.38378 −0.0764992 −0.0382496 0.999268i \(-0.512178\pi\)
−0.0382496 + 0.999268i \(0.512178\pi\)
\(972\) 0 0
\(973\) 1.00134 0.0321013
\(974\) 25.4788 0.816393
\(975\) 0 0
\(976\) 4.91791 0.157418
\(977\) 19.2662 0.616380 0.308190 0.951325i \(-0.400277\pi\)
0.308190 + 0.951325i \(0.400277\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.22035 0.102870
\(981\) 0 0
\(982\) −39.1390 −1.24898
\(983\) 16.2256 0.517517 0.258759 0.965942i \(-0.416687\pi\)
0.258759 + 0.965942i \(0.416687\pi\)
\(984\) 0 0
\(985\) 30.3345 0.966537
\(986\) 42.2600 1.34583
\(987\) 0 0
\(988\) −9.65296 −0.307101
\(989\) 20.5753 0.654256
\(990\) 0 0
\(991\) −50.5214 −1.60487 −0.802433 0.596743i \(-0.796461\pi\)
−0.802433 + 0.596743i \(0.796461\pi\)
\(992\) −1.17426 −0.0372828
\(993\) 0 0
\(994\) 3.58527 0.113718
\(995\) 58.2566 1.84686
\(996\) 0 0
\(997\) −45.0206 −1.42582 −0.712909 0.701257i \(-0.752623\pi\)
−0.712909 + 0.701257i \(0.752623\pi\)
\(998\) 44.9912 1.42417
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7623.2.a.ch.1.2 4
3.2 odd 2 847.2.a.l.1.3 4
11.3 even 5 693.2.m.g.64.1 8
11.4 even 5 693.2.m.g.379.1 8
11.10 odd 2 7623.2.a.co.1.3 4
21.20 even 2 5929.2.a.bi.1.3 4
33.2 even 10 847.2.f.s.323.2 8
33.5 odd 10 847.2.f.p.729.1 8
33.8 even 10 847.2.f.q.372.1 8
33.14 odd 10 77.2.f.a.64.2 8
33.17 even 10 847.2.f.s.729.2 8
33.20 odd 10 847.2.f.p.323.1 8
33.26 odd 10 77.2.f.a.71.2 yes 8
33.29 even 10 847.2.f.q.148.1 8
33.32 even 2 847.2.a.k.1.2 4
231.26 even 30 539.2.q.b.214.2 16
231.47 even 30 539.2.q.b.361.1 16
231.59 even 30 539.2.q.b.324.1 16
231.80 even 30 539.2.q.b.471.2 16
231.125 even 10 539.2.f.d.148.2 8
231.146 even 10 539.2.f.d.295.2 8
231.158 odd 30 539.2.q.c.324.1 16
231.179 odd 30 539.2.q.c.471.2 16
231.191 odd 30 539.2.q.c.214.2 16
231.212 odd 30 539.2.q.c.361.1 16
231.230 odd 2 5929.2.a.bb.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.a.64.2 8 33.14 odd 10
77.2.f.a.71.2 yes 8 33.26 odd 10
539.2.f.d.148.2 8 231.125 even 10
539.2.f.d.295.2 8 231.146 even 10
539.2.q.b.214.2 16 231.26 even 30
539.2.q.b.324.1 16 231.59 even 30
539.2.q.b.361.1 16 231.47 even 30
539.2.q.b.471.2 16 231.80 even 30
539.2.q.c.214.2 16 231.191 odd 30
539.2.q.c.324.1 16 231.158 odd 30
539.2.q.c.361.1 16 231.212 odd 30
539.2.q.c.471.2 16 231.179 odd 30
693.2.m.g.64.1 8 11.3 even 5
693.2.m.g.379.1 8 11.4 even 5
847.2.a.k.1.2 4 33.32 even 2
847.2.a.l.1.3 4 3.2 odd 2
847.2.f.p.323.1 8 33.20 odd 10
847.2.f.p.729.1 8 33.5 odd 10
847.2.f.q.148.1 8 33.29 even 10
847.2.f.q.372.1 8 33.8 even 10
847.2.f.s.323.2 8 33.2 even 10
847.2.f.s.729.2 8 33.17 even 10
5929.2.a.bb.1.2 4 231.230 odd 2
5929.2.a.bi.1.3 4 21.20 even 2
7623.2.a.ch.1.2 4 1.1 even 1 trivial
7623.2.a.co.1.3 4 11.10 odd 2