Properties

Label 4275.2.a.b.1.1
Level $4275$
Weight $2$
Character 4275.1
Self dual yes
Analytic conductor $34.136$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.1360468641\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 4275.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +5.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} +5.00000 q^{7} -1.00000 q^{11} -2.00000 q^{13} -10.0000 q^{14} -4.00000 q^{16} -1.00000 q^{17} -1.00000 q^{19} +2.00000 q^{22} -4.00000 q^{23} +4.00000 q^{26} +10.0000 q^{28} +2.00000 q^{29} -6.00000 q^{31} +8.00000 q^{32} +2.00000 q^{34} +2.00000 q^{38} +1.00000 q^{43} -2.00000 q^{44} +8.00000 q^{46} -9.00000 q^{47} +18.0000 q^{49} -4.00000 q^{52} +10.0000 q^{53} -4.00000 q^{58} +8.00000 q^{59} -1.00000 q^{61} +12.0000 q^{62} -8.00000 q^{64} -8.00000 q^{67} -2.00000 q^{68} +12.0000 q^{71} +11.0000 q^{73} -2.00000 q^{76} -5.00000 q^{77} +16.0000 q^{79} +12.0000 q^{83} -2.00000 q^{86} +6.00000 q^{89} -10.0000 q^{91} -8.00000 q^{92} +18.0000 q^{94} +10.0000 q^{97} -36.0000 q^{98} +O(q^{100})\)

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 \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 5.00000 1.88982 0.944911 0.327327i \(-0.106148\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −10.0000 −2.67261
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 10.0000 1.88982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −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
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −9.00000 −1.31278 −0.656392 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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 \(0.325650\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 12.0000 1.52400
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 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 \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) −5.00000 −0.569803
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −10.0000 −1.04828
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 18.0000 1.85656
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −36.0000 −3.63655
\(99\) 0 0
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −20.0000 −1.94257
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 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 \(0.470012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) −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
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −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
\(135\) 0 0
\(136\) 0 0
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 0 0
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −24.0000 −2.01404
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 0 0
\(146\) −22.0000 −1.82073
\(147\) 0 0
\(148\) 0 0
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 10.0000 0.805823
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −32.0000 −2.54578
\(159\) 0 0
\(160\) 0 0
\(161\) −20.0000 −1.57622
\(162\) 0 0
\(163\) 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 \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −12.0000 −0.899438
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 20.0000 1.48250
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.00000 0.0731272
\(188\) −18.0000 −1.31278
\(189\) 0 0
\(190\) 0 0
\(191\) −9.00000 −0.651217 −0.325609 0.945505i \(-0.605569\pi\)
−0.325609 + 0.945505i \(0.605569\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) −20.0000 −1.43592
\(195\) 0 0
\(196\) 36.0000 2.57143
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −21.0000 −1.48865 −0.744325 0.667817i \(-0.767229\pi\)
−0.744325 + 0.667817i \(0.767229\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 4.00000 0.281439
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 8.00000 0.554700
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 20.0000 1.37361
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) −30.0000 −2.03653
\(218\) −8.00000 −0.541828
\(219\) 0 0
\(220\) 0 0
\(221\) 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
\(225\) 0 0
\(226\) −4.00000 −0.266076
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 25.0000 1.65205 0.826023 0.563636i \(-0.190598\pi\)
0.826023 + 0.563636i \(0.190598\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 16.0000 1.04151
\(237\) 0 0
\(238\) 10.0000 0.648204
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 20.0000 1.28565
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 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
\(265\) 0 0
\(266\) 10.0000 0.613139
\(267\) 0 0
\(268\) −16.0000 −0.977356
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) 11.0000 0.660926 0.330463 0.943819i \(-0.392795\pi\)
0.330463 + 0.943819i \(0.392795\pi\)
\(278\) 26.0000 1.55938
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 13.0000 0.772770 0.386385 0.922338i \(-0.373724\pi\)
0.386385 + 0.922338i \(0.373724\pi\)
\(284\) 24.0000 1.42414
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) 22.0000 1.28745
\(293\) −28.0000 −1.63578 −0.817889 0.575376i \(-0.804856\pi\)
−0.817889 + 0.575376i \(0.804856\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −42.0000 −2.43299
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 5.00000 0.288195
\(302\) 0 0
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −10.0000 −0.569803
\(309\) 0 0
\(310\) 0 0
\(311\) 21.0000 1.19080 0.595400 0.803429i \(-0.296993\pi\)
0.595400 + 0.803429i \(0.296993\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −36.0000 −2.03160
\(315\) 0 0
\(316\) 32.0000 1.80014
\(317\) −4.00000 −0.224662 −0.112331 0.993671i \(-0.535832\pi\)
−0.112331 + 0.993671i \(0.535832\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) 0 0
\(322\) 40.0000 2.22911
\(323\) 1.00000 0.0556415
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 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
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 18.0000 0.979071
\(339\) 0 0
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 55.0000 2.96972
\(344\) 0 0
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) −25.0000 −1.34207 −0.671035 0.741426i \(-0.734150\pi\)
−0.671035 + 0.741426i \(0.734150\pi\)
\(348\) 0 0
\(349\) 9.00000 0.481759 0.240879 0.970555i \(-0.422564\pi\)
0.240879 + 0.970555i \(0.422564\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −8.00000 −0.426401
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) −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
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 16.0000 0.834058
\(369\) 0 0
\(370\) 0 0
\(371\) 50.0000 2.59587
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) 34.0000 1.74646 0.873231 0.487306i \(-0.162020\pi\)
0.873231 + 0.487306i \(0.162020\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 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
\(387\) 0 0
\(388\) 20.0000 1.01535
\(389\) 27.0000 1.36895 0.684477 0.729034i \(-0.260031\pi\)
0.684477 + 0.729034i \(0.260031\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 4.00000 0.201517
\(395\) 0 0
\(396\) 0 0
\(397\) −25.0000 −1.25471 −0.627357 0.778732i \(-0.715863\pi\)
−0.627357 + 0.778732i \(0.715863\pi\)
\(398\) 42.0000 2.10527
\(399\) 0 0
\(400\) 0 0
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) 12.0000 0.597763
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) −20.0000 −0.992583
\(407\) 0 0
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 40.0000 1.96827
\(414\) 0 0
\(415\) 0 0
\(416\) −16.0000 −0.784465
\(417\) 0 0
\(418\) −2.00000 −0.0978232
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −24.0000 −1.16830
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −5.00000 −0.241967
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 34.0000 1.63772 0.818861 0.573992i \(-0.194606\pi\)
0.818861 + 0.573992i \(0.194606\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 60.0000 2.88009
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −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
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) −40.0000 −1.88982
\(449\) 36.0000 1.69895 0.849473 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.00000 0.188144
\(453\) 0 0
\(454\) −36.0000 −1.68956
\(455\) 0 0
\(456\) 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
\(459\) 0 0
\(460\) 0 0
\(461\) −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 \(-0.629265\pi\)
−0.395029 + 0.918669i \(0.629265\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −5.00000 −0.231372 −0.115686 0.993286i \(-0.536907\pi\)
−0.115686 + 0.993286i \(0.536907\pi\)
\(468\) 0 0
\(469\) −40.0000 −1.84703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.00000 −0.0459800
\(474\) 0 0
\(475\) 0 0
\(476\) −10.0000 −0.458349
\(477\) 0 0
\(478\) −6.00000 −0.274434
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −40.0000 −1.82195
\(483\) 0 0
\(484\) −20.0000 −0.909091
\(485\) 0 0
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 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
\(498\) 0 0
\(499\) 5.00000 0.223831 0.111915 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 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
\(505\) 0 0
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 55.0000 2.43306
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 16.0000 0.705730
\(515\) 0 0
\(516\) 0 0
\(517\) 9.00000 0.395820
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 0 0
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) −14.0000 −0.611593
\(525\) 0 0
\(526\) −46.0000 −2.00570
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) −10.0000 −0.433555
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −28.0000 −1.20717
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 3.00000 0.128980 0.0644900 0.997918i \(-0.479458\pi\)
0.0644900 + 0.997918i \(0.479458\pi\)
\(542\) −24.0000 −1.03089
\(543\) 0 0
\(544\) −8.00000 −0.342997
\(545\) 0 0
\(546\) 0 0
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) −18.0000 −0.768922
\(549\) 0 0
\(550\) 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 \(-0.834988\pi\)
−0.868613 + 0.495491i \(0.834988\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) 20.0000 0.843649
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −26.0000 −1.09286
\(567\) 0 0
\(568\) 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
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −27.0000 −1.12402 −0.562012 0.827129i \(-0.689973\pi\)
−0.562012 + 0.827129i \(0.689973\pi\)
\(578\) 32.0000 1.33102
\(579\) 0 0
\(580\) 0 0
\(581\) 60.0000 2.48922
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 0 0
\(585\) 0 0
\(586\) 56.0000 2.31334
\(587\) 7.00000 0.288921 0.144460 0.989511i \(-0.453855\pi\)
0.144460 + 0.989511i \(0.453855\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 42.0000 1.72039
\(597\) 0 0
\(598\) −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
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −26.0000 −1.05531 −0.527654 0.849460i \(-0.676928\pi\)
−0.527654 + 0.849460i \(0.676928\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 0 0
\(611\) 18.0000 0.728202
\(612\) 0 0
\(613\) −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 \(0.317103\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −42.0000 −1.68405
\(623\) 30.0000 1.20192
\(624\) 0 0
\(625\) 0 0
\(626\) −4.00000 −0.159872
\(627\) 0 0
\(628\) 36.0000 1.43656
\(629\) 0 0
\(630\) 0 0
\(631\) 15.0000 0.597141 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 8.00000 0.317721
\(635\) 0 0
\(636\) 0 0
\(637\) −36.0000 −1.42637
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 1.00000 0.0394362 0.0197181 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) −40.0000 −1.57622
\(645\) 0 0
\(646\) −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
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.00000 0.117399 0.0586995 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 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
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 20.0000 0.773823
\(669\) 0 0
\(670\) 0 0
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 0 0
\(679\) 50.0000 1.91882
\(680\) 0 0
\(681\) 0 0
\(682\) −12.0000 −0.459504
\(683\) −6.00000 −0.229584 −0.114792 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −110.000 −4.19982
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −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
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −18.0000 −0.681310
\(699\) 0 0
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 4.00000 0.150542
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) −42.0000 −1.57734 −0.788672 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 36.0000 1.34538
\(717\) 0 0
\(718\) 74.0000 2.76166
\(719\) −33.0000 −1.23069 −0.615346 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 0 0
\(721\) 10.0000 0.372419
\(722\) −2.00000 −0.0744323
\(723\) 0 0
\(724\) −28.0000 −1.04061
\(725\) 0 0
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.00000 −0.0369863
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −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 \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) 30.0000 1.09618
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 36.0000 1.31278
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) −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 \(0.587644\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 68.0000 2.45694
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) 11.0000 0.396670 0.198335 0.980134i \(-0.436447\pi\)
0.198335 + 0.980134i \(0.436447\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.00000 −0.287926
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −54.0000 −1.93599
\(779\) 0 0
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) −72.0000 −2.57143
\(785\) 0 0
\(786\) 0 0
\(787\) −40.0000 −1.42585 −0.712923 0.701242i \(-0.752629\pi\)
−0.712923 + 0.701242i \(0.752629\pi\)
\(788\) −4.00000 −0.142494
\(789\) 0 0
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 50.0000 1.77443
\(795\) 0 0
\(796\) −42.0000 −1.48865
\(797\) −44.0000 −1.55856 −0.779280 0.626676i \(-0.784415\pi\)
−0.779280 + 0.626676i \(0.784415\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 0 0
\(801\) 0 0
\(802\) 72.0000 2.54241
\(803\) −11.0000 −0.388182
\(804\) 0 0
\(805\) 0 0
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) 0 0
\(809\) 55.0000 1.93370 0.966849 0.255351i \(-0.0821909\pi\)
0.966849 + 0.255351i \(0.0821909\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 20.0000 0.701862
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 28.0000 0.978997
\(819\) 0 0
\(820\) 0 0
\(821\) 45.0000 1.57051 0.785255 0.619172i \(-0.212532\pi\)
0.785255 + 0.619172i \(0.212532\pi\)
\(822\) 0 0
\(823\) −43.0000 −1.49889 −0.749443 0.662069i \(-0.769679\pi\)
−0.749443 + 0.662069i \(0.769679\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −80.0000 −2.78356
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 52.0000 1.80603 0.903017 0.429604i \(-0.141347\pi\)
0.903017 + 0.429604i \(0.141347\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 16.0000 0.554700
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 2.00000 0.0691714
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 24.0000 0.826114
\(845\) 0 0
\(846\) 0 0
\(847\) −50.0000 −1.71802
\(848\) −40.0000 −1.37361
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 0 0
\(857\) −8.00000 −0.273275 −0.136637 0.990621i \(-0.543630\pi\)
−0.136637 + 0.990621i \(0.543630\pi\)
\(858\) 0 0
\(859\) 27.0000 0.921228 0.460614 0.887601i \(-0.347629\pi\)
0.460614 + 0.887601i \(0.347629\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −68.0000 −2.31609
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 12.0000 0.407777
\(867\) 0 0
\(868\) −60.0000 −2.03653
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 0 0
\(877\) 6.00000 0.202606 0.101303 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) −52.0000 −1.75491
\(879\) 0 0
\(880\) 0 0
\(881\) 37.0000 1.24656 0.623281 0.781998i \(-0.285799\pi\)
0.623281 + 0.781998i \(0.285799\pi\)
\(882\) 0 0
\(883\) −35.0000 −1.17784 −0.588922 0.808190i \(-0.700447\pi\)
−0.588922 + 0.808190i \(0.700447\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 10.0000 0.335957
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 10.0000 0.335389
\(890\) 0 0
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) 9.00000 0.301174
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −72.0000 −2.40267
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −10.0000 −0.333148
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 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
\(909\) 0 0
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) 50.0000 1.65205
\(917\) −35.0000 −1.15580
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 54.0000 1.77840
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 0 0
\(926\) 34.0000 1.11731
\(927\) 0 0
\(928\) 16.0000 0.525226
\(929\) 2.00000 0.0656179 0.0328089 0.999462i \(-0.489555\pi\)
0.0328089 + 0.999462i \(0.489555\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −32.0000 −1.04151
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 0 0
\(949\) −22.0000 −0.714150
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −32.0000 −1.03658 −0.518291 0.855204i \(-0.673432\pi\)
−0.518291 + 0.855204i \(0.673432\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) −32.0000 −1.03387
\(959\) −45.0000 −1.45313
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 40.0000 1.28831
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) −65.0000 −2.08380
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 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
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −44.0000 −1.40338 −0.701691 0.712481i \(-0.747571\pi\)
−0.701691 + 0.712481i \(0.747571\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) −120.000 −3.80617
\(995\) 0 0
\(996\) 0 0
\(997\) 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 4275.2.a.b.1.1 1
3.2 odd 2 1425.2.a.j.1.1 1
5.4 even 2 171.2.a.d.1.1 1
15.2 even 4 1425.2.c.b.799.2 2
15.8 even 4 1425.2.c.b.799.1 2
15.14 odd 2 57.2.a.a.1.1 1
20.19 odd 2 2736.2.a.v.1.1 1
35.34 odd 2 8379.2.a.p.1.1 1
60.59 even 2 912.2.a.g.1.1 1
95.94 odd 2 3249.2.a.b.1.1 1
105.104 even 2 2793.2.a.b.1.1 1
120.29 odd 2 3648.2.a.bh.1.1 1
120.59 even 2 3648.2.a.r.1.1 1
165.164 even 2 6897.2.a.f.1.1 1
195.194 odd 2 9633.2.a.o.1.1 1
285.284 even 2 1083.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.a.1.1 1 15.14 odd 2
171.2.a.d.1.1 1 5.4 even 2
912.2.a.g.1.1 1 60.59 even 2
1083.2.a.e.1.1 1 285.284 even 2
1425.2.a.j.1.1 1 3.2 odd 2
1425.2.c.b.799.1 2 15.8 even 4
1425.2.c.b.799.2 2 15.2 even 4
2736.2.a.v.1.1 1 20.19 odd 2
2793.2.a.b.1.1 1 105.104 even 2
3249.2.a.b.1.1 1 95.94 odd 2
3648.2.a.r.1.1 1 120.59 even 2
3648.2.a.bh.1.1 1 120.29 odd 2
4275.2.a.b.1.1 1 1.1 even 1 trivial
6897.2.a.f.1.1 1 165.164 even 2
8379.2.a.p.1.1 1 35.34 odd 2
9633.2.a.o.1.1 1 195.194 odd 2