Properties

Label 1083.2.a.e.1.1
Level $1083$
Weight $2$
Character 1083.1
Self dual yes
Analytic conductor $8.648$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1083 = 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1083.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.64779853890\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1083.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} +2.00000 q^{6} -5.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} +2.00000 q^{6} -5.00000 q^{7} +1.00000 q^{9} -6.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -10.0000 q^{14} -3.00000 q^{15} -4.00000 q^{16} -1.00000 q^{17} +2.00000 q^{18} -6.00000 q^{20} -5.00000 q^{21} +2.00000 q^{22} -4.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -10.0000 q^{28} +2.00000 q^{29} -6.00000 q^{30} +6.00000 q^{31} -8.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} +15.0000 q^{35} +2.00000 q^{36} -2.00000 q^{39} -10.0000 q^{42} -1.00000 q^{43} +2.00000 q^{44} -3.00000 q^{45} -8.00000 q^{46} -9.00000 q^{47} -4.00000 q^{48} +18.0000 q^{49} +8.00000 q^{50} -1.00000 q^{51} -4.00000 q^{52} -10.0000 q^{53} +2.00000 q^{54} -3.00000 q^{55} +4.00000 q^{58} +8.00000 q^{59} -6.00000 q^{60} -1.00000 q^{61} +12.0000 q^{62} -5.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} +2.00000 q^{66} -8.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +30.0000 q^{70} +12.0000 q^{71} -11.0000 q^{73} +4.00000 q^{75} -5.00000 q^{77} -4.00000 q^{78} -16.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} +12.0000 q^{83} -10.0000 q^{84} +3.00000 q^{85} -2.00000 q^{86} +2.00000 q^{87} +6.00000 q^{89} -6.00000 q^{90} +10.0000 q^{91} -8.00000 q^{92} +6.00000 q^{93} -18.0000 q^{94} -8.00000 q^{96} +10.0000 q^{97} +36.0000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 2.00000 0.816497
\(7\) −5.00000 −1.88982 −0.944911 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) −6.00000 −1.89737
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −10.0000 −2.67261
\(15\) −3.00000 −0.774597
\(16\) −4.00000 −1.00000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 2.00000 0.471405
\(19\) 0 0
\(20\) −6.00000 −1.34164
\(21\) −5.00000 −1.09109
\(22\) 2.00000 0.426401
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) −10.0000 −1.88982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −6.00000 −1.09545
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −8.00000 −1.41421
\(33\) 1.00000 0.174078
\(34\) −2.00000 −0.342997
\(35\) 15.0000 2.53546
\(36\) 2.00000 0.333333
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −10.0000 −1.54303
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 2.00000 0.301511
\(45\) −3.00000 −0.447214
\(46\) −8.00000 −1.17954
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −4.00000 −0.577350
\(49\) 18.0000 2.57143
\(50\) 8.00000 1.13137
\(51\) −1.00000 −0.140028
\(52\) −4.00000 −0.554700
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 2.00000 0.272166
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) −6.00000 −0.774597
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 12.0000 1.52400
\(63\) −5.00000 −0.629941
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) 2.00000 0.246183
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) 30.0000 3.58569
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) −5.00000 −0.569803
\(78\) −4.00000 −0.452911
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 12.0000 1.34164
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −10.0000 −1.09109
\(85\) 3.00000 0.325396
\(86\) −2.00000 −0.215666
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −6.00000 −0.632456
\(91\) 10.0000 1.04828
\(92\) −8.00000 −0.834058
\(93\) 6.00000 0.622171
\(94\) −18.0000 −1.85656
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 36.0000 3.63655
\(99\) 1.00000 0.100504
\(100\) 8.00000 0.800000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −2.00000 −0.198030
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 15.0000 1.46385
\(106\) −20.0000 −1.94257
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 2.00000 0.192450
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) −6.00000 −0.572078
\(111\) 0 0
\(112\) 20.0000 1.88982
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 12.0000 1.11901
\(116\) 4.00000 0.371391
\(117\) −2.00000 −0.184900
\(118\) 16.0000 1.47292
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 12.0000 1.07763
\(125\) 3.00000 0.268328
\(126\) −10.0000 −0.890871
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 12.0000 1.05247
\(131\) 7.00000 0.611593 0.305796 0.952097i \(-0.401077\pi\)
0.305796 + 0.952097i \(0.401077\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −16.0000 −1.38219
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) −8.00000 −0.681005
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 30.0000 2.53546
\(141\) −9.00000 −0.757937
\(142\) 24.0000 2.01404
\(143\) −2.00000 −0.167248
\(144\) −4.00000 −0.333333
\(145\) −6.00000 −0.498273
\(146\) −22.0000 −1.82073
\(147\) 18.0000 1.48461
\(148\) 0 0
\(149\) −21.0000 −1.72039 −0.860194 0.509968i \(-0.829657\pi\)
−0.860194 + 0.509968i \(0.829657\pi\)
\(150\) 8.00000 0.653197
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) −10.0000 −0.805823
\(155\) −18.0000 −1.44579
\(156\) −4.00000 −0.320256
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −32.0000 −2.54578
\(159\) −10.0000 −0.793052
\(160\) 24.0000 1.89737
\(161\) 20.0000 1.57622
\(162\) 2.00000 0.157135
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 24.0000 1.86276
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 4.00000 0.303239
\(175\) −20.0000 −1.51186
\(176\) −4.00000 −0.301511
\(177\) 8.00000 0.601317
\(178\) 12.0000 0.899438
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) −6.00000 −0.447214
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 20.0000 1.48250
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 0 0
\(186\) 12.0000 0.879883
\(187\) −1.00000 −0.0731272
\(188\) −18.0000 −1.31278
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) −8.00000 −0.577350
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 20.0000 1.43592
\(195\) 6.00000 0.429669
\(196\) 36.0000 2.57143
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 2.00000 0.142134
\(199\) −21.0000 −1.48865 −0.744325 0.667817i \(-0.767229\pi\)
−0.744325 + 0.667817i \(0.767229\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 4.00000 0.281439
\(203\) −10.0000 −0.701862
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) −4.00000 −0.278019
\(208\) 8.00000 0.554700
\(209\) 0 0
\(210\) 30.0000 2.07020
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −20.0000 −1.37361
\(213\) 12.0000 0.822226
\(214\) −12.0000 −0.820303
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) −30.0000 −2.03653
\(218\) −8.00000 −0.541828
\(219\) −11.0000 −0.743311
\(220\) −6.00000 −0.404520
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 40.0000 2.67261
\(225\) 4.00000 0.266667
\(226\) −4.00000 −0.266076
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) 25.0000 1.65205 0.826023 0.563636i \(-0.190598\pi\)
0.826023 + 0.563636i \(0.190598\pi\)
\(230\) 24.0000 1.58251
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) −4.00000 −0.261488
\(235\) 27.0000 1.76129
\(236\) 16.0000 1.04151
\(237\) −16.0000 −1.03931
\(238\) 10.0000 0.648204
\(239\) −3.00000 −0.194054 −0.0970269 0.995282i \(-0.530933\pi\)
−0.0970269 + 0.995282i \(0.530933\pi\)
\(240\) 12.0000 0.774597
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) −20.0000 −1.28565
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) −54.0000 −3.44993
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 6.00000 0.379473
\(251\) 7.00000 0.441836 0.220918 0.975292i \(-0.429095\pi\)
0.220918 + 0.975292i \(0.429095\pi\)
\(252\) −10.0000 −0.629941
\(253\) −4.00000 −0.251478
\(254\) 4.00000 0.250982
\(255\) 3.00000 0.187867
\(256\) 16.0000 1.00000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) 12.0000 0.744208
\(261\) 2.00000 0.123797
\(262\) 14.0000 0.864923
\(263\) 23.0000 1.41824 0.709120 0.705087i \(-0.249092\pi\)
0.709120 + 0.705087i \(0.249092\pi\)
\(264\) 0 0
\(265\) 30.0000 1.84289
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) −16.0000 −0.977356
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −6.00000 −0.365148
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 4.00000 0.242536
\(273\) 10.0000 0.605228
\(274\) −18.0000 −1.08742
\(275\) 4.00000 0.241209
\(276\) −8.00000 −0.481543
\(277\) −11.0000 −0.660926 −0.330463 0.943819i \(-0.607205\pi\)
−0.330463 + 0.943819i \(0.607205\pi\)
\(278\) −26.0000 −1.55938
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) −18.0000 −1.07188
\(283\) −13.0000 −0.772770 −0.386385 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 24.0000 1.42414
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) −8.00000 −0.471405
\(289\) −16.0000 −0.941176
\(290\) −12.0000 −0.704664
\(291\) 10.0000 0.586210
\(292\) −22.0000 −1.28745
\(293\) 28.0000 1.63578 0.817889 0.575376i \(-0.195144\pi\)
0.817889 + 0.575376i \(0.195144\pi\)
\(294\) 36.0000 2.09956
\(295\) −24.0000 −1.39733
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) −42.0000 −2.43299
\(299\) 8.00000 0.462652
\(300\) 8.00000 0.461880
\(301\) 5.00000 0.288195
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) 3.00000 0.171780
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −10.0000 −0.569803
\(309\) 2.00000 0.113776
\(310\) −36.0000 −2.04466
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) 0 0
\(313\) −2.00000 −0.113047 −0.0565233 0.998401i \(-0.518002\pi\)
−0.0565233 + 0.998401i \(0.518002\pi\)
\(314\) −36.0000 −2.03160
\(315\) 15.0000 0.845154
\(316\) −32.0000 −1.80014
\(317\) 4.00000 0.224662 0.112331 0.993671i \(-0.464168\pi\)
0.112331 + 0.993671i \(0.464168\pi\)
\(318\) −20.0000 −1.12154
\(319\) 2.00000 0.111979
\(320\) 24.0000 1.34164
\(321\) −6.00000 −0.334887
\(322\) 40.0000 2.22911
\(323\) 0 0
\(324\) 2.00000 0.111111
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) 45.0000 2.48093
\(330\) −6.00000 −0.330289
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 24.0000 1.31717
\(333\) 0 0
\(334\) −20.0000 −1.09435
\(335\) 24.0000 1.31126
\(336\) 20.0000 1.09109
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −18.0000 −0.979071
\(339\) −2.00000 −0.108625
\(340\) 6.00000 0.325396
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) −55.0000 −2.96972
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) −12.0000 −0.645124
\(347\) −25.0000 −1.34207 −0.671035 0.741426i \(-0.734150\pi\)
−0.671035 + 0.741426i \(0.734150\pi\)
\(348\) 4.00000 0.214423
\(349\) 9.00000 0.481759 0.240879 0.970555i \(-0.422564\pi\)
0.240879 + 0.970555i \(0.422564\pi\)
\(350\) −40.0000 −2.13809
\(351\) −2.00000 −0.106752
\(352\) −8.00000 −0.426401
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 16.0000 0.850390
\(355\) −36.0000 −1.91068
\(356\) 12.0000 0.635999
\(357\) 5.00000 0.264628
\(358\) 36.0000 1.90266
\(359\) 37.0000 1.95279 0.976393 0.216003i \(-0.0693022\pi\)
0.976393 + 0.216003i \(0.0693022\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 28.0000 1.47165
\(363\) −10.0000 −0.524864
\(364\) 20.0000 1.04828
\(365\) 33.0000 1.72730
\(366\) −2.00000 −0.104542
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 16.0000 0.834058
\(369\) 0 0
\(370\) 0 0
\(371\) 50.0000 2.59587
\(372\) 12.0000 0.622171
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) −2.00000 −0.103418
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) −10.0000 −0.514344
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 18.0000 0.920960
\(383\) 34.0000 1.73732 0.868659 0.495410i \(-0.164982\pi\)
0.868659 + 0.495410i \(0.164982\pi\)
\(384\) 0 0
\(385\) 15.0000 0.764471
\(386\) −8.00000 −0.407189
\(387\) −1.00000 −0.0508329
\(388\) 20.0000 1.01535
\(389\) −27.0000 −1.36895 −0.684477 0.729034i \(-0.739969\pi\)
−0.684477 + 0.729034i \(0.739969\pi\)
\(390\) 12.0000 0.607644
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 7.00000 0.353103
\(394\) −4.00000 −0.201517
\(395\) 48.0000 2.41514
\(396\) 2.00000 0.100504
\(397\) 25.0000 1.25471 0.627357 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) −42.0000 −2.10527
\(399\) 0 0
\(400\) −16.0000 −0.800000
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) −16.0000 −0.798007
\(403\) −12.0000 −0.597763
\(404\) 4.00000 0.199007
\(405\) −3.00000 −0.149071
\(406\) −20.0000 −0.992583
\(407\) 0 0
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 4.00000 0.197066
\(413\) −40.0000 −1.96827
\(414\) −8.00000 −0.393179
\(415\) −36.0000 −1.76717
\(416\) 16.0000 0.784465
\(417\) −13.0000 −0.636613
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 30.0000 1.46385
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −24.0000 −1.16830
\(423\) −9.00000 −0.437595
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 24.0000 1.16280
\(427\) 5.00000 0.241967
\(428\) −12.0000 −0.580042
\(429\) −2.00000 −0.0965609
\(430\) 6.00000 0.289346
\(431\) 34.0000 1.63772 0.818861 0.573992i \(-0.194606\pi\)
0.818861 + 0.573992i \(0.194606\pi\)
\(432\) −4.00000 −0.192450
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) −60.0000 −2.88009
\(435\) −6.00000 −0.287678
\(436\) −8.00000 −0.383131
\(437\) 0 0
\(438\) −22.0000 −1.05120
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 4.00000 0.190261
\(443\) −5.00000 −0.237557 −0.118779 0.992921i \(-0.537898\pi\)
−0.118779 + 0.992921i \(0.537898\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) −24.0000 −1.13643
\(447\) −21.0000 −0.993266
\(448\) 40.0000 1.88982
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 8.00000 0.377124
\(451\) 0 0
\(452\) −4.00000 −0.188144
\(453\) 0 0
\(454\) −36.0000 −1.68956
\(455\) −30.0000 −1.40642
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 50.0000 2.33635
\(459\) −1.00000 −0.0466760
\(460\) 24.0000 1.11901
\(461\) 27.0000 1.25752 0.628758 0.777601i \(-0.283564\pi\)
0.628758 + 0.777601i \(0.283564\pi\)
\(462\) −10.0000 −0.465242
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) −8.00000 −0.371391
\(465\) −18.0000 −0.834730
\(466\) 18.0000 0.833834
\(467\) −5.00000 −0.231372 −0.115686 0.993286i \(-0.536907\pi\)
−0.115686 + 0.993286i \(0.536907\pi\)
\(468\) −4.00000 −0.184900
\(469\) 40.0000 1.84703
\(470\) 54.0000 2.49083
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) −1.00000 −0.0459800
\(474\) −32.0000 −1.46981
\(475\) 0 0
\(476\) 10.0000 0.458349
\(477\) −10.0000 −0.457869
\(478\) −6.00000 −0.274434
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 24.0000 1.09545
\(481\) 0 0
\(482\) −40.0000 −1.82195
\(483\) 20.0000 0.910032
\(484\) −20.0000 −0.909091
\(485\) −30.0000 −1.36223
\(486\) 2.00000 0.0907218
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −108.000 −4.87894
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −2.00000 −0.0900755
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) −24.0000 −1.07763
\(497\) −60.0000 −2.69137
\(498\) 24.0000 1.07547
\(499\) 5.00000 0.223831 0.111915 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 6.00000 0.268328
\(501\) −10.0000 −0.446767
\(502\) 14.0000 0.624851
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) −8.00000 −0.355643
\(507\) −9.00000 −0.399704
\(508\) 4.00000 0.177471
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 6.00000 0.265684
\(511\) 55.0000 2.43306
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) 16.0000 0.705730
\(515\) −6.00000 −0.264392
\(516\) −2.00000 −0.0880451
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 4.00000 0.175075
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 14.0000 0.611593
\(525\) −20.0000 −0.872872
\(526\) 46.0000 2.00570
\(527\) −6.00000 −0.261364
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 60.0000 2.60623
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 12.0000 0.519291
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) 18.0000 0.776757
\(538\) 28.0000 1.20717
\(539\) 18.0000 0.775315
\(540\) −6.00000 −0.258199
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) 24.0000 1.03089
\(543\) 14.0000 0.600798
\(544\) 8.00000 0.342997
\(545\) 12.0000 0.514024
\(546\) 20.0000 0.855921
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) −18.0000 −0.768922
\(549\) −1.00000 −0.0426790
\(550\) 8.00000 0.341121
\(551\) 0 0
\(552\) 0 0
\(553\) 80.0000 3.40195
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −26.0000 −1.10265
\(557\) −41.0000 −1.73723 −0.868613 0.495491i \(-0.834988\pi\)
−0.868613 + 0.495491i \(0.834988\pi\)
\(558\) 12.0000 0.508001
\(559\) 2.00000 0.0845910
\(560\) −60.0000 −2.53546
\(561\) −1.00000 −0.0422200
\(562\) −20.0000 −0.843649
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −18.0000 −0.757937
\(565\) 6.00000 0.252422
\(566\) −26.0000 −1.09286
\(567\) −5.00000 −0.209980
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) −4.00000 −0.167248
\(573\) 9.00000 0.375980
\(574\) 0 0
\(575\) −16.0000 −0.667246
\(576\) −8.00000 −0.333333
\(577\) 27.0000 1.12402 0.562012 0.827129i \(-0.310027\pi\)
0.562012 + 0.827129i \(0.310027\pi\)
\(578\) −32.0000 −1.33102
\(579\) −4.00000 −0.166234
\(580\) −12.0000 −0.498273
\(581\) −60.0000 −2.48922
\(582\) 20.0000 0.829027
\(583\) −10.0000 −0.414158
\(584\) 0 0
\(585\) 6.00000 0.248069
\(586\) 56.0000 2.31334
\(587\) 7.00000 0.288921 0.144460 0.989511i \(-0.453855\pi\)
0.144460 + 0.989511i \(0.453855\pi\)
\(588\) 36.0000 1.48461
\(589\) 0 0
\(590\) −48.0000 −1.97613
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 2.00000 0.0820610
\(595\) −15.0000 −0.614940
\(596\) −42.0000 −1.72039
\(597\) −21.0000 −0.859473
\(598\) 16.0000 0.654289
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 10.0000 0.407570
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) 30.0000 1.21967
\(606\) 4.00000 0.162489
\(607\) −26.0000 −1.05531 −0.527654 0.849460i \(-0.676928\pi\)
−0.527654 + 0.849460i \(0.676928\pi\)
\(608\) 0 0
\(609\) −10.0000 −0.405220
\(610\) 6.00000 0.242933
\(611\) 18.0000 0.728202
\(612\) −2.00000 −0.0808452
\(613\) 33.0000 1.33286 0.666429 0.745569i \(-0.267822\pi\)
0.666429 + 0.745569i \(0.267822\pi\)
\(614\) 24.0000 0.968561
\(615\) 0 0
\(616\) 0 0
\(617\) 27.0000 1.08698 0.543490 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) 4.00000 0.160904
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −36.0000 −1.44579
\(621\) −4.00000 −0.160514
\(622\) −42.0000 −1.68405
\(623\) −30.0000 −1.20192
\(624\) 8.00000 0.320256
\(625\) −29.0000 −1.16000
\(626\) −4.00000 −0.159872
\(627\) 0 0
\(628\) −36.0000 −1.43656
\(629\) 0 0
\(630\) 30.0000 1.19523
\(631\) 15.0000 0.597141 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 8.00000 0.317721
\(635\) −6.00000 −0.238103
\(636\) −20.0000 −0.793052
\(637\) −36.0000 −1.42637
\(638\) 4.00000 0.158362
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) −12.0000 −0.473602
\(643\) −1.00000 −0.0394362 −0.0197181 0.999806i \(-0.506277\pi\)
−0.0197181 + 0.999806i \(0.506277\pi\)
\(644\) 40.0000 1.57622
\(645\) 3.00000 0.118125
\(646\) 0 0
\(647\) −39.0000 −1.53325 −0.766624 0.642096i \(-0.778065\pi\)
−0.766624 + 0.642096i \(0.778065\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) −16.0000 −0.627572
\(651\) −30.0000 −1.17579
\(652\) 0 0
\(653\) 3.00000 0.117399 0.0586995 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) −8.00000 −0.312825
\(655\) −21.0000 −0.820538
\(656\) 0 0
\(657\) −11.0000 −0.429151
\(658\) 90.0000 3.50857
\(659\) 14.0000 0.545363 0.272681 0.962104i \(-0.412090\pi\)
0.272681 + 0.962104i \(0.412090\pi\)
\(660\) −6.00000 −0.233550
\(661\) −12.0000 −0.466746 −0.233373 0.972387i \(-0.574976\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) 8.00000 0.310929
\(663\) 2.00000 0.0776736
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) −20.0000 −0.773823
\(669\) −12.0000 −0.463947
\(670\) 48.0000 1.85440
\(671\) −1.00000 −0.0386046
\(672\) 40.0000 1.54303
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) 28.0000 1.07852
\(675\) 4.00000 0.153960
\(676\) −18.0000 −0.692308
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) −4.00000 −0.153619
\(679\) −50.0000 −1.91882
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 12.0000 0.459504
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 0 0
\(685\) 27.0000 1.03162
\(686\) −110.000 −4.19982
\(687\) 25.0000 0.953809
\(688\) 4.00000 0.152499
\(689\) 20.0000 0.761939
\(690\) 24.0000 0.913664
\(691\) −31.0000 −1.17930 −0.589648 0.807661i \(-0.700733\pi\)
−0.589648 + 0.807661i \(0.700733\pi\)
\(692\) −12.0000 −0.456172
\(693\) −5.00000 −0.189934
\(694\) −50.0000 −1.89797
\(695\) 39.0000 1.47935
\(696\) 0 0
\(697\) 0 0
\(698\) 18.0000 0.681310
\(699\) 9.00000 0.340411
\(700\) −40.0000 −1.51186
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) −4.00000 −0.150970
\(703\) 0 0
\(704\) −8.00000 −0.301511
\(705\) 27.0000 1.01688
\(706\) −4.00000 −0.150542
\(707\) −10.0000 −0.376089
\(708\) 16.0000 0.601317
\(709\) −42.0000 −1.57734 −0.788672 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) −72.0000 −2.70211
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 10.0000 0.374241
\(715\) 6.00000 0.224387
\(716\) 36.0000 1.34538
\(717\) −3.00000 −0.112037
\(718\) 74.0000 2.76166
\(719\) 33.0000 1.23069 0.615346 0.788257i \(-0.289016\pi\)
0.615346 + 0.788257i \(0.289016\pi\)
\(720\) 12.0000 0.447214
\(721\) −10.0000 −0.372419
\(722\) 0 0
\(723\) −20.0000 −0.743808
\(724\) 28.0000 1.04061
\(725\) 8.00000 0.297113
\(726\) −20.0000 −0.742270
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 66.0000 2.44277
\(731\) 1.00000 0.0369863
\(732\) −2.00000 −0.0739221
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −16.0000 −0.590571
\(735\) −54.0000 −1.99182
\(736\) 32.0000 1.17954
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −5.00000 −0.183928 −0.0919640 0.995762i \(-0.529314\pi\)
−0.0919640 + 0.995762i \(0.529314\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 100.000 3.67112
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 63.0000 2.30814
\(746\) −32.0000 −1.17160
\(747\) 12.0000 0.439057
\(748\) −2.00000 −0.0731272
\(749\) 30.0000 1.09618
\(750\) 6.00000 0.219089
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 36.0000 1.31278
\(753\) 7.00000 0.255094
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) −10.0000 −0.363696
\(757\) −17.0000 −0.617876 −0.308938 0.951082i \(-0.599973\pi\)
−0.308938 + 0.951082i \(0.599973\pi\)
\(758\) −68.0000 −2.46987
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 15.0000 0.543750 0.271875 0.962333i \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) 4.00000 0.144905
\(763\) 20.0000 0.724049
\(764\) 18.0000 0.651217
\(765\) 3.00000 0.108465
\(766\) 68.0000 2.45694
\(767\) −16.0000 −0.577727
\(768\) 16.0000 0.577350
\(769\) 11.0000 0.396670 0.198335 0.980134i \(-0.436447\pi\)
0.198335 + 0.980134i \(0.436447\pi\)
\(770\) 30.0000 1.08112
\(771\) 8.00000 0.288113
\(772\) −8.00000 −0.287926
\(773\) −20.0000 −0.719350 −0.359675 0.933078i \(-0.617112\pi\)
−0.359675 + 0.933078i \(0.617112\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 24.0000 0.862105
\(776\) 0 0
\(777\) 0 0
\(778\) −54.0000 −1.93599
\(779\) 0 0
\(780\) 12.0000 0.429669
\(781\) 12.0000 0.429394
\(782\) 8.00000 0.286079
\(783\) 2.00000 0.0714742
\(784\) −72.0000 −2.57143
\(785\) 54.0000 1.92734
\(786\) 14.0000 0.499363
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) −4.00000 −0.142494
\(789\) 23.0000 0.818822
\(790\) 96.0000 3.41553
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 50.0000 1.77443
\(795\) 30.0000 1.06399
\(796\) −42.0000 −1.48865
\(797\) 44.0000 1.55856 0.779280 0.626676i \(-0.215585\pi\)
0.779280 + 0.626676i \(0.215585\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) −32.0000 −1.13137
\(801\) 6.00000 0.212000
\(802\) −72.0000 −2.54241
\(803\) −11.0000 −0.388182
\(804\) −16.0000 −0.564276
\(805\) −60.0000 −2.11472
\(806\) −24.0000 −0.845364
\(807\) 14.0000 0.492823
\(808\) 0 0
\(809\) −55.0000 −1.93370 −0.966849 0.255351i \(-0.917809\pi\)
−0.966849 + 0.255351i \(0.917809\pi\)
\(810\) −6.00000 −0.210819
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) −20.0000 −0.701862
\(813\) 12.0000 0.420858
\(814\) 0 0
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) 28.0000 0.978997
\(819\) 10.0000 0.349428
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) −18.0000 −0.627822
\(823\) 43.0000 1.49889 0.749443 0.662069i \(-0.230321\pi\)
0.749443 + 0.662069i \(0.230321\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) −80.0000 −2.78356
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −8.00000 −0.278019
\(829\) −52.0000 −1.80603 −0.903017 0.429604i \(-0.858653\pi\)
−0.903017 + 0.429604i \(0.858653\pi\)
\(830\) −72.0000 −2.49916
\(831\) −11.0000 −0.381586
\(832\) 16.0000 0.554700
\(833\) −18.0000 −0.623663
\(834\) −26.0000 −0.900306
\(835\) 30.0000 1.03819
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 56.0000 1.93449
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −52.0000 −1.79204
\(843\) −10.0000 −0.344418
\(844\) −24.0000 −0.826114
\(845\) 27.0000 0.928828
\(846\) −18.0000 −0.618853
\(847\) 50.0000 1.71802
\(848\) 40.0000 1.37361
\(849\) −13.0000 −0.446159
\(850\) −8.00000 −0.274398
\(851\) 0 0
\(852\) 24.0000 0.822226
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 0 0
\(857\) 8.00000 0.273275 0.136637 0.990621i \(-0.456370\pi\)
0.136637 + 0.990621i \(0.456370\pi\)
\(858\) −4.00000 −0.136558
\(859\) 27.0000 0.921228 0.460614 0.887601i \(-0.347629\pi\)
0.460614 + 0.887601i \(0.347629\pi\)
\(860\) 6.00000 0.204598
\(861\) 0 0
\(862\) 68.0000 2.31609
\(863\) 44.0000 1.49778 0.748889 0.662696i \(-0.230588\pi\)
0.748889 + 0.662696i \(0.230588\pi\)
\(864\) −8.00000 −0.272166
\(865\) 18.0000 0.612018
\(866\) −12.0000 −0.407777
\(867\) −16.0000 −0.543388
\(868\) −60.0000 −2.03653
\(869\) −16.0000 −0.542763
\(870\) −12.0000 −0.406838
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) −15.0000 −0.507093
\(876\) −22.0000 −0.743311
\(877\) 6.00000 0.202606 0.101303 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) −52.0000 −1.75491
\(879\) 28.0000 0.944417
\(880\) 12.0000 0.404520
\(881\) −37.0000 −1.24656 −0.623281 0.781998i \(-0.714201\pi\)
−0.623281 + 0.781998i \(0.714201\pi\)
\(882\) 36.0000 1.21218
\(883\) 35.0000 1.17784 0.588922 0.808190i \(-0.299553\pi\)
0.588922 + 0.808190i \(0.299553\pi\)
\(884\) 4.00000 0.134535
\(885\) −24.0000 −0.806751
\(886\) −10.0000 −0.335957
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) −10.0000 −0.335389
\(890\) −36.0000 −1.20672
\(891\) 1.00000 0.0335013
\(892\) −24.0000 −0.803579
\(893\) 0 0
\(894\) −42.0000 −1.40469
\(895\) −54.0000 −1.80502
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 72.0000 2.40267
\(899\) 12.0000 0.400222
\(900\) 8.00000 0.266667
\(901\) 10.0000 0.333148
\(902\) 0 0
\(903\) 5.00000 0.166390
\(904\) 0 0
\(905\) −42.0000 −1.39613
\(906\) 0 0
\(907\) 26.0000 0.863316 0.431658 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) −36.0000 −1.19470
\(909\) 2.00000 0.0663358
\(910\) −60.0000 −1.98898
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) −58.0000 −1.91847
\(915\) 3.00000 0.0991769
\(916\) 50.0000 1.65205
\(917\) −35.0000 −1.15580
\(918\) −2.00000 −0.0660098
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 54.0000 1.77840
\(923\) −24.0000 −0.789970
\(924\) −10.0000 −0.328976
\(925\) 0 0
\(926\) 34.0000 1.11731
\(927\) 2.00000 0.0656886
\(928\) −16.0000 −0.525226
\(929\) −2.00000 −0.0656179 −0.0328089 0.999462i \(-0.510445\pi\)
−0.0328089 + 0.999462i \(0.510445\pi\)
\(930\) −36.0000 −1.18049
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) −21.0000 −0.687509
\(934\) −10.0000 −0.327210
\(935\) 3.00000 0.0981105
\(936\) 0 0
\(937\) 21.0000 0.686040 0.343020 0.939328i \(-0.388550\pi\)
0.343020 + 0.939328i \(0.388550\pi\)
\(938\) 80.0000 2.61209
\(939\) −2.00000 −0.0652675
\(940\) 54.0000 1.76129
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) −36.0000 −1.17294
\(943\) 0 0
\(944\) −32.0000 −1.04151
\(945\) 15.0000 0.487950
\(946\) −2.00000 −0.0650256
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) −32.0000 −1.03931
\(949\) 22.0000 0.714150
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) 32.0000 1.03658 0.518291 0.855204i \(-0.326568\pi\)
0.518291 + 0.855204i \(0.326568\pi\)
\(954\) −20.0000 −0.647524
\(955\) −27.0000 −0.873699
\(956\) −6.00000 −0.194054
\(957\) 2.00000 0.0646508
\(958\) −32.0000 −1.03387
\(959\) 45.0000 1.45313
\(960\) 24.0000 0.774597
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) −40.0000 −1.28831
\(965\) 12.0000 0.386294
\(966\) 40.0000 1.28698
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −60.0000 −1.92648
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 2.00000 0.0641500
\(973\) 65.0000 2.08380
\(974\) 32.0000 1.02535
\(975\) −8.00000 −0.256205
\(976\) 4.00000 0.128037
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) −108.000 −3.44993
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 44.0000 1.40338 0.701691 0.712481i \(-0.252429\pi\)
0.701691 + 0.712481i \(0.252429\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) −4.00000 −0.127386
\(987\) 45.0000 1.43237
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) −6.00000 −0.190693
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −48.0000 −1.52400
\(993\) 4.00000 0.126936
\(994\) −120.000 −3.80617
\(995\) 63.0000 1.99723
\(996\) 24.0000 0.760469
\(997\) −47.0000 −1.48850 −0.744252 0.667898i \(-0.767194\pi\)
−0.744252 + 0.667898i \(0.767194\pi\)
\(998\) 10.0000 0.316544
\(999\) 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 1083.2.a.e.1.1 1
3.2 odd 2 3249.2.a.b.1.1 1
19.18 odd 2 57.2.a.a.1.1 1
57.56 even 2 171.2.a.d.1.1 1
76.75 even 2 912.2.a.g.1.1 1
95.18 even 4 1425.2.c.b.799.2 2
95.37 even 4 1425.2.c.b.799.1 2
95.94 odd 2 1425.2.a.j.1.1 1
133.132 even 2 2793.2.a.b.1.1 1
152.37 odd 2 3648.2.a.bh.1.1 1
152.75 even 2 3648.2.a.r.1.1 1
209.208 even 2 6897.2.a.f.1.1 1
228.227 odd 2 2736.2.a.v.1.1 1
247.246 odd 2 9633.2.a.o.1.1 1
285.284 even 2 4275.2.a.b.1.1 1
399.398 odd 2 8379.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.a.1.1 1 19.18 odd 2
171.2.a.d.1.1 1 57.56 even 2
912.2.a.g.1.1 1 76.75 even 2
1083.2.a.e.1.1 1 1.1 even 1 trivial
1425.2.a.j.1.1 1 95.94 odd 2
1425.2.c.b.799.1 2 95.37 even 4
1425.2.c.b.799.2 2 95.18 even 4
2736.2.a.v.1.1 1 228.227 odd 2
2793.2.a.b.1.1 1 133.132 even 2
3249.2.a.b.1.1 1 3.2 odd 2
3648.2.a.r.1.1 1 152.75 even 2
3648.2.a.bh.1.1 1 152.37 odd 2
4275.2.a.b.1.1 1 285.284 even 2
6897.2.a.f.1.1 1 209.208 even 2
8379.2.a.p.1.1 1 399.398 odd 2
9633.2.a.o.1.1 1 247.246 odd 2