Properties

Label 6897.2.a.f.1.1
Level $6897$
Weight $2$
Character 6897.1
Self dual yes
Analytic conductor $55.073$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6897.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.0728222741\)
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 \(\chi\) \(=\) 6897.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} +5.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -3.00000 q^{5} -2.00000 q^{6} +5.00000 q^{7} +1.00000 q^{9} -6.00000 q^{10} -2.00000 q^{12} -2.00000 q^{13} +10.0000 q^{14} +3.00000 q^{15} -4.00000 q^{16} +1.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} -6.00000 q^{20} -5.00000 q^{21} -4.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +10.0000 q^{28} +2.00000 q^{29} +6.00000 q^{30} -6.00000 q^{31} -8.00000 q^{32} +2.00000 q^{34} -15.0000 q^{35} +2.00000 q^{36} +2.00000 q^{38} +2.00000 q^{39} -10.0000 q^{42} +1.00000 q^{43} -3.00000 q^{45} -8.00000 q^{46} -9.00000 q^{47} +4.00000 q^{48} +18.0000 q^{49} +8.00000 q^{50} -1.00000 q^{51} -4.00000 q^{52} +10.0000 q^{53} -2.00000 q^{54} -1.00000 q^{57} +4.00000 q^{58} -8.00000 q^{59} +6.00000 q^{60} +1.00000 q^{61} -12.0000 q^{62} +5.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} +4.00000 q^{69} -30.0000 q^{70} -12.0000 q^{71} +11.0000 q^{73} -4.00000 q^{75} +2.00000 q^{76} +4.00000 q^{78} -16.0000 q^{79} +12.0000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -10.0000 q^{84} -3.00000 q^{85} +2.00000 q^{86} -2.00000 q^{87} -6.00000 q^{89} -6.00000 q^{90} -10.0000 q^{91} -8.00000 q^{92} +6.00000 q^{93} -18.0000 q^{94} -3.00000 q^{95} +8.00000 q^{96} -10.0000 q^{97} +36.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 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 \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) −6.00000 −1.89737
\(11\) 0 0
\(12\) −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 \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 2.00000 0.471405
\(19\) 1.00000 0.229416
\(20\) −6.00000 −1.34164
\(21\) −5.00000 −1.09109
\(22\) 0 0
\(23\) −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 \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) −15.0000 −2.53546
\(36\) 2.00000 0.333333
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 2.00000 0.324443
\(39\) 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 \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) −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 \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 4.00000 0.525226
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 6.00000 0.774597
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) −12.0000 −1.52400
\(63\) 5.00000 0.629941
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000 0.242536
\(69\) 4.00000 0.481543
\(70\) −30.0000 −3.58569
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 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 \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −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 \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −6.00000 −0.632456
\(91\) −10.0000 −1.04828
\(92\) −8.00000 −0.834058
\(93\) 6.00000 0.622171
\(94\) −18.0000 −1.85656
\(95\) −3.00000 −0.307794
\(96\) 8.00000 0.816497
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 36.0000 3.63655
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −2.00000 −0.198030
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 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 \(0.561356\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −20.0000 −1.88982
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −2.00000 −0.187317
\(115\) 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
\(121\) 0 0
\(122\) 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 \(-0.598923\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(132\) 0 0
\(133\) 5.00000 0.433555
\(134\) 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 \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) −30.0000 −2.53546
\(141\) 9.00000 0.757937
\(142\) −24.0000 −2.01404
\(143\) 0 0
\(144\) −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 \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) −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
\(154\) 0 0
\(155\) 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
\(165\) 0 0
\(166\) −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
\(171\) 1.00000 0.0764719
\(172\) 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
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) −12.0000 −0.899438
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) −6.00000 −0.447214
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −20.0000 −1.48250
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 0 0
\(186\) 12.0000 0.879883
\(187\) 0 0
\(188\) −18.0000 −1.31278
\(189\) −5.00000 −0.363696
\(190\) −6.00000 −0.435286
\(191\) 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 \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −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
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) −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 \(0.703780\pi\)
\(228\) −2.00000 −0.132453
\(229\) 25.0000 1.65205 0.826023 0.563636i \(-0.190598\pi\)
0.826023 + 0.563636i \(0.190598\pi\)
\(230\) 24.0000 1.58251
\(231\) 0 0
\(232\) 0 0
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) −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 \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) −12.0000 −0.774597
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) −54.0000 −3.44993
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 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
\(253\) 0 0
\(254\) 4.00000 0.250982
\(255\) 3.00000 0.187867
\(256\) 16.0000 1.00000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) −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 \(-0.750908\pi\)
−0.709120 + 0.705087i \(0.750908\pi\)
\(264\) 0 0
\(265\) −30.0000 −1.84289
\(266\) 10.0000 0.613139
\(267\) 6.00000 0.367194
\(268\) 16.0000 0.977356
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 6.00000 0.365148
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) −4.00000 −0.242536
\(273\) 10.0000 0.605228
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 11.0000 0.660926 0.330463 0.943819i \(-0.392795\pi\)
0.330463 + 0.943819i \(0.392795\pi\)
\(278\) 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 \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) −24.0000 −1.42414
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) 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
\(297\) 0 0
\(298\) 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
\(304\) −4.00000 −0.229416
\(305\) −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 \(0.388747\pi\)
\(308\) 0 0
\(309\) 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 \(-0.535832\pi\)
−0.112331 + 0.993671i \(0.535832\pi\)
\(318\) −20.0000 −1.12154
\(319\) 0 0
\(320\) 24.0000 1.34164
\(321\) 6.00000 0.334887
\(322\) −40.0000 −2.22911
\(323\) 1.00000 0.0556415
\(324\) 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
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −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
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 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 \(-0.265850\pi\)
0.671035 + 0.741426i \(0.265850\pi\)
\(348\) −4.00000 −0.214423
\(349\) −9.00000 −0.481759 −0.240879 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(350\) 40.0000 2.13809
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −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 \(-0.930698\pi\)
−0.976393 + 0.216003i \(0.930698\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −28.0000 −1.47165
\(363\) 0 0
\(364\) −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 \(0.635947\pi\)
\(374\) 0 0
\(375\) −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 \(-0.162020\pi\)
0.873231 + 0.487306i \(0.162020\pi\)
\(380\) −6.00000 −0.307794
\(381\) −2.00000 −0.102463
\(382\) 18.0000 0.920960
\(383\) −34.0000 −1.73732 −0.868659 0.495410i \(-0.835018\pi\)
−0.868659 + 0.495410i \(0.835018\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −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
\(396\) 0 0
\(397\) 25.0000 1.25471 0.627357 0.778732i \(-0.284137\pi\)
0.627357 + 0.778732i \(0.284137\pi\)
\(398\) −42.0000 −2.10527
\(399\) −5.00000 −0.250313
\(400\) −16.0000 −0.800000
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) −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 \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −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
\(429\) 0 0
\(430\) −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 \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) −60.0000 −2.88009
\(435\) 6.00000 0.287678
\(436\) −8.00000 −0.383131
\(437\) −4.00000 −0.191346
\(438\) −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 \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 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 \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 50.0000 2.33635
\(459\) −1.00000 −0.0466760
\(460\) 24.0000 1.11901
\(461\) −27.0000 −1.25752 −0.628758 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) 0 0
\(463\) 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
\(473\) 0 0
\(474\) 32.0000 1.46981
\(475\) 4.00000 0.183533
\(476\) 10.0000 0.458349
\(477\) 10.0000 0.457869
\(478\) 6.00000 0.274434
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) −24.0000 −1.09545
\(481\) 0 0
\(482\) −40.0000 −1.82195
\(483\) 20.0000 0.910032
\(484\) 0 0
\(485\) 30.0000 1.36223
\(486\) −2.00000 −0.0907218
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 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
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 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 \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 4.00000 0.177471
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 6.00000 0.265684
\(511\) 55.0000 2.43306
\(512\) 32.0000 1.41421
\(513\) −1.00000 −0.0441511
\(514\) −16.0000 −0.705730
\(515\) 6.00000 0.264392
\(516\) −2.00000 −0.0880451
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 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
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −60.0000 −2.60623
\(531\) −8.00000 −0.347170
\(532\) 10.0000 0.433555
\(533\) 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
\(539\) 0 0
\(540\) 6.00000 0.258199
\(541\) −3.00000 −0.128980 −0.0644900 0.997918i \(-0.520542\pi\)
−0.0644900 + 0.997918i \(0.520542\pi\)
\(542\) −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
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 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 \(-0.165012\pi\)
0.868613 + 0.495491i \(0.165012\pi\)
\(558\) −12.0000 −0.508001
\(559\) −2.00000 −0.0845910
\(560\) 60.0000 2.53546
\(561\) 0 0
\(562\) −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 \(0.376853\pi\)
\(570\) 6.00000 0.251312
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) −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
\(583\) 0 0
\(584\) 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
\(589\) −6.00000 −0.247226
\(590\) 48.0000 1.97613
\(591\) −2.00000 −0.0822690
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) −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
\(605\) 0 0
\(606\) 4.00000 0.162489
\(607\) −26.0000 −1.05531 −0.527654 0.849460i \(-0.676928\pi\)
−0.527654 + 0.849460i \(0.676928\pi\)
\(608\) −8.00000 −0.324443
\(609\) −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 \(-0.732178\pi\)
−0.666429 + 0.745569i \(0.732178\pi\)
\(614\) 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
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 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
\(646\) 2.00000 0.0786889
\(647\) −39.0000 −1.53325 −0.766624 0.642096i \(-0.778065\pi\)
−0.766624 + 0.642096i \(0.778065\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −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 \(0.412090\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) −8.00000 −0.310929
\(663\) 2.00000 0.0776736
\(664\) 0 0
\(665\) −15.0000 −0.581675
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) −20.0000 −0.773823
\(669\) −12.0000 −0.463947
\(670\) −48.0000 −1.85440
\(671\) 0 0
\(672\) 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
\(682\) 0 0
\(683\) −6.00000 −0.229584 −0.114792 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 2.00000 0.0764719
\(685\) 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
\(693\) 0 0
\(694\) 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 \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 4.00000 0.150970
\(703\) 0 0
\(704\) 0 0
\(705\) −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
\(715\) 0 0
\(716\) −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
\(722\) 2.00000 0.0744323
\(723\) 20.0000 0.743808
\(724\) −28.0000 −1.04061
\(725\) 8.00000 0.297113
\(726\) 0 0
\(727\) −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 \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −16.0000 −0.590571
\(735\) 54.0000 1.99182
\(736\) 32.0000 1.17954
\(737\) 0 0
\(738\) 0 0
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 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
\(748\) 0 0
\(749\) −30.0000 −1.09618
\(750\) −6.00000 −0.219089
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 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
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) −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 \(-0.563553\pi\)
−0.198335 + 0.980134i \(0.563553\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) −8.00000 −0.287926
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) 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
\(781\) 0 0
\(782\) −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 \(-0.784415\pi\)
−0.779280 + 0.626676i \(0.784415\pi\)
\(798\) −10.0000 −0.353996
\(799\) −9.00000 −0.318397
\(800\) −32.0000 −1.13137
\(801\) −6.00000 −0.212000
\(802\) 72.0000 2.54241
\(803\) 0 0
\(804\) −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 \(-0.0821909\pi\)
0.966849 + 0.255351i \(0.0821909\pi\)
\(810\) −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
\(817\) 1.00000 0.0349856
\(818\) 28.0000 0.978997
\(819\) −10.0000 −0.349428
\(820\) 0 0
\(821\) 45.0000 1.57051 0.785255 0.619172i \(-0.212532\pi\)
0.785255 + 0.619172i \(0.212532\pi\)
\(822\) 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
\(825\) 0 0
\(826\) −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 \(-0.141347\pi\)
0.903017 + 0.429604i \(0.141347\pi\)
\(830\) 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 \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 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
\(847\) 0 0
\(848\) −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 \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 10.0000 0.342193
\(855\) −3.00000 −0.102598
\(856\) 0 0
\(857\) 8.00000 0.273275 0.136637 0.990621i \(-0.456370\pi\)
0.136637 + 0.990621i \(0.456370\pi\)
\(858\) 0 0
\(859\) 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 \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 8.00000 0.272166
\(865\) 18.0000 0.612018
\(866\) 12.0000 0.407777
\(867\) 16.0000 0.543388
\(868\) −60.0000 −2.03653
\(869\) 0 0
\(870\) 12.0000 0.406838
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) −8.00000 −0.270604
\(875\) 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
\(880\) 0 0
\(881\) −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
\(891\) 0 0
\(892\) 24.0000 0.803579
\(893\) −9.00000 −0.301174
\(894\) −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 \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) −36.0000 −1.19470
\(909\) −2.00000 −0.0663358
\(910\) 60.0000 1.98898
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 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 \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) −54.0000 −1.77840
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 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
\(931\) 18.0000 0.589926
\(932\) −18.0000 −0.589610
\(933\) 21.0000 0.687509
\(934\) −10.0000 −0.327210
\(935\) 0 0
\(936\) 0 0
\(937\) −21.0000 −0.686040 −0.343020 0.939328i \(-0.611450\pi\)
−0.343020 + 0.939328i \(0.611450\pi\)
\(938\) 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
\(946\) 0 0
\(947\) −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
\(950\) 8.00000 0.259554
\(951\) 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
\(957\) 0 0
\(958\) 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 \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) −1.00000 −0.0321246
\(970\) 60.0000 1.92648
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −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 \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −108.000 −3.44993
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) −44.0000 −1.40338 −0.701691 0.712481i \(-0.747571\pi\)
−0.701691 + 0.712481i \(0.747571\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 4.00000 0.127386
\(987\) 45.0000 1.43237
\(988\) −4.00000 −0.127257
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 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 \(-0.232806\pi\)
0.744252 + 0.667898i \(0.232806\pi\)
\(998\) 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 6897.2.a.f.1.1 1
11.10 odd 2 57.2.a.a.1.1 1
33.32 even 2 171.2.a.d.1.1 1
44.43 even 2 912.2.a.g.1.1 1
55.32 even 4 1425.2.c.b.799.1 2
55.43 even 4 1425.2.c.b.799.2 2
55.54 odd 2 1425.2.a.j.1.1 1
77.76 even 2 2793.2.a.b.1.1 1
88.21 odd 2 3648.2.a.bh.1.1 1
88.43 even 2 3648.2.a.r.1.1 1
132.131 odd 2 2736.2.a.v.1.1 1
143.142 odd 2 9633.2.a.o.1.1 1
165.164 even 2 4275.2.a.b.1.1 1
209.208 even 2 1083.2.a.e.1.1 1
231.230 odd 2 8379.2.a.p.1.1 1
627.626 odd 2 3249.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.a.1.1 1 11.10 odd 2
171.2.a.d.1.1 1 33.32 even 2
912.2.a.g.1.1 1 44.43 even 2
1083.2.a.e.1.1 1 209.208 even 2
1425.2.a.j.1.1 1 55.54 odd 2
1425.2.c.b.799.1 2 55.32 even 4
1425.2.c.b.799.2 2 55.43 even 4
2736.2.a.v.1.1 1 132.131 odd 2
2793.2.a.b.1.1 1 77.76 even 2
3249.2.a.b.1.1 1 627.626 odd 2
3648.2.a.r.1.1 1 88.43 even 2
3648.2.a.bh.1.1 1 88.21 odd 2
4275.2.a.b.1.1 1 165.164 even 2
6897.2.a.f.1.1 1 1.1 even 1 trivial
8379.2.a.p.1.1 1 231.230 odd 2
9633.2.a.o.1.1 1 143.142 odd 2