Properties

Label 1216.2.a.q.1.1
Level $1216$
Weight $2$
Character 1216.1
Self dual yes
Analytic conductor $9.710$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1216.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +4.00000 q^{13} +2.00000 q^{15} -3.00000 q^{17} -1.00000 q^{19} +6.00000 q^{21} -8.00000 q^{23} -4.00000 q^{25} -4.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} +10.0000 q^{33} +3.00000 q^{35} -10.0000 q^{37} +8.00000 q^{39} +10.0000 q^{41} +1.00000 q^{43} +1.00000 q^{45} +1.00000 q^{47} +2.00000 q^{49} -6.00000 q^{51} +4.00000 q^{53} +5.00000 q^{55} -2.00000 q^{57} +6.00000 q^{59} +13.0000 q^{61} +3.00000 q^{63} +4.00000 q^{65} -12.0000 q^{67} -16.0000 q^{69} -2.00000 q^{71} +9.00000 q^{73} -8.00000 q^{75} +15.0000 q^{77} -8.00000 q^{79} -11.0000 q^{81} -12.0000 q^{83} -3.00000 q^{85} +4.00000 q^{87} +12.0000 q^{89} +12.0000 q^{91} -8.00000 q^{93} -1.00000 q^{95} -8.00000 q^{97} +5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 10.0000 1.74078
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.00000 0.145865 0.0729325 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) 3.00000 0.377964
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −16.0000 −1.92617
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) −8.00000 −0.923760
\(76\) 0 0
\(77\) 15.0000 1.70941
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −20.0000 −1.89832
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 20.0000 1.80334
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −11.0000 −0.939793 −0.469897 0.882721i \(-0.655709\pi\)
−0.469897 + 0.882721i \(0.655709\pi\)
\(138\) 0 0
\(139\) −3.00000 −0.254457 −0.127228 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) 20.0000 1.67248
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 4.00000 0.329914
\(148\) 0 0
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 10.0000 0.778499
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −12.0000 −0.907115
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 26.0000 1.92198
\(184\) 0 0
\(185\) −10.0000 −0.735215
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) −12.0000 −0.872872
\(190\) 0 0
\(191\) −25.0000 −1.80894 −0.904468 0.426541i \(-0.859732\pi\)
−0.904468 + 0.426541i \(0.859732\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 0 0
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) −24.0000 −1.69283
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) 18.0000 1.23917 0.619586 0.784929i \(-0.287301\pi\)
0.619586 + 0.784929i \(0.287301\pi\)
\(212\) 0 0
\(213\) −4.00000 −0.274075
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 18.0000 1.21633
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −17.0000 −1.12339 −0.561696 0.827344i \(-0.689851\pi\)
−0.561696 + 0.827344i \(0.689851\pi\)
\(230\) 0 0
\(231\) 30.0000 1.97386
\(232\) 0 0
\(233\) 3.00000 0.196537 0.0982683 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 0 0
\(235\) 1.00000 0.0652328
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) −24.0000 −1.52094
\(250\) 0 0
\(251\) 11.0000 0.694314 0.347157 0.937807i \(-0.387147\pi\)
0.347157 + 0.937807i \(0.387147\pi\)
\(252\) 0 0
\(253\) −40.0000 −2.51478
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 0 0
\(257\) 32.0000 1.99611 0.998053 0.0623783i \(-0.0198685\pi\)
0.998053 + 0.0623783i \(0.0198685\pi\)
\(258\) 0 0
\(259\) −30.0000 −1.86411
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) 24.0000 1.46878
\(268\) 0 0
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 24.0000 1.45255
\(274\) 0 0
\(275\) −20.0000 −1.20605
\(276\) 0 0
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) −3.00000 −0.178331 −0.0891657 0.996017i \(-0.528420\pi\)
−0.0891657 + 0.996017i \(0.528420\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 30.0000 1.77084
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) −20.0000 −1.16052
\(298\) 0 0
\(299\) −32.0000 −1.85061
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 0 0
\(303\) 20.0000 1.14897
\(304\) 0 0
\(305\) 13.0000 0.744378
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) −7.00000 −0.396934 −0.198467 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) 0 0
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) 10.0000 0.559893
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.00000 0.165395
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) −20.0000 −1.08625
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) −16.0000 −0.861411
\(346\) 0 0
\(347\) 19.0000 1.01997 0.509987 0.860182i \(-0.329650\pi\)
0.509987 + 0.860182i \(0.329650\pi\)
\(348\) 0 0
\(349\) 11.0000 0.588817 0.294408 0.955680i \(-0.404877\pi\)
0.294408 + 0.955680i \(0.404877\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −2.00000 −0.106149
\(356\) 0 0
\(357\) −18.0000 −0.952661
\(358\) 0 0
\(359\) −21.0000 −1.10834 −0.554169 0.832404i \(-0.686964\pi\)
−0.554169 + 0.832404i \(0.686964\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 28.0000 1.46962
\(364\) 0 0
\(365\) 9.00000 0.471082
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) 0 0
\(375\) −18.0000 −0.929516
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) 0 0
\(385\) 15.0000 0.764471
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) 0 0
\(389\) 21.0000 1.06474 0.532371 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −5.00000 −0.250943 −0.125471 0.992097i \(-0.540044\pi\)
−0.125471 + 0.992097i \(0.540044\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 28.0000 1.39825 0.699127 0.714998i \(-0.253572\pi\)
0.699127 + 0.714998i \(0.253572\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) −50.0000 −2.47841
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) −22.0000 −1.08518
\(412\) 0 0
\(413\) 18.0000 0.885722
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −6.00000 −0.293821
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 40.0000 1.94948 0.974740 0.223341i \(-0.0716964\pi\)
0.974740 + 0.223341i \(0.0716964\pi\)
\(422\) 0 0
\(423\) 1.00000 0.0486217
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 39.0000 1.88734
\(428\) 0 0
\(429\) 40.0000 1.93122
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 0 0
\(437\) 8.00000 0.382692
\(438\) 0 0
\(439\) −2.00000 −0.0954548 −0.0477274 0.998860i \(-0.515198\pi\)
−0.0477274 + 0.998860i \(0.515198\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −5.00000 −0.237557 −0.118779 0.992921i \(-0.537898\pi\)
−0.118779 + 0.992921i \(0.537898\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) 30.0000 1.41895
\(448\) 0 0
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 0 0
\(451\) 50.0000 2.35441
\(452\) 0 0
\(453\) −4.00000 −0.187936
\(454\) 0 0
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −13.0000 −0.608114 −0.304057 0.952654i \(-0.598341\pi\)
−0.304057 + 0.952654i \(0.598341\pi\)
\(458\) 0 0
\(459\) 12.0000 0.560112
\(460\) 0 0
\(461\) 19.0000 0.884918 0.442459 0.896789i \(-0.354106\pi\)
0.442459 + 0.896789i \(0.354106\pi\)
\(462\) 0 0
\(463\) −19.0000 −0.883005 −0.441502 0.897260i \(-0.645554\pi\)
−0.441502 + 0.897260i \(0.645554\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) −5.00000 −0.231372 −0.115686 0.993286i \(-0.536907\pi\)
−0.115686 + 0.993286i \(0.536907\pi\)
\(468\) 0 0
\(469\) −36.0000 −1.66233
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 0 0
\(473\) 5.00000 0.229900
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 0 0
\(483\) −48.0000 −2.18408
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) −6.00000 −0.270226
\(494\) 0 0
\(495\) 5.00000 0.224733
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 19.0000 0.850557 0.425278 0.905063i \(-0.360176\pi\)
0.425278 + 0.905063i \(0.360176\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 6.00000 0.266469
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 27.0000 1.19441
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) 6.00000 0.264392
\(516\) 0 0
\(517\) 5.00000 0.219900
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 32.0000 1.40195 0.700973 0.713188i \(-0.252749\pi\)
0.700973 + 0.713188i \(0.252749\pi\)
\(522\) 0 0
\(523\) 26.0000 1.13690 0.568450 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) 0 0
\(525\) −24.0000 −1.04745
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 40.0000 1.73259
\(534\) 0 0
\(535\) 2.00000 0.0864675
\(536\) 0 0
\(537\) 36.0000 1.55351
\(538\) 0 0
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) −11.0000 −0.472927 −0.236463 0.971640i \(-0.575988\pi\)
−0.236463 + 0.971640i \(0.575988\pi\)
\(542\) 0 0
\(543\) −20.0000 −0.858282
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 13.0000 0.554826
\(550\) 0 0
\(551\) −2.00000 −0.0852029
\(552\) 0 0
\(553\) −24.0000 −1.02058
\(554\) 0 0
\(555\) −20.0000 −0.848953
\(556\) 0 0
\(557\) −1.00000 −0.0423714 −0.0211857 0.999776i \(-0.506744\pi\)
−0.0211857 + 0.999776i \(0.506744\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −30.0000 −1.26660
\(562\) 0 0
\(563\) 42.0000 1.77009 0.885044 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) 0 0
\(565\) −10.0000 −0.420703
\(566\) 0 0
\(567\) −33.0000 −1.38587
\(568\) 0 0
\(569\) −8.00000 −0.335377 −0.167689 0.985840i \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) −50.0000 −2.08878
\(574\) 0 0
\(575\) 32.0000 1.33449
\(576\) 0 0
\(577\) −13.0000 −0.541197 −0.270599 0.962692i \(-0.587222\pi\)
−0.270599 + 0.962692i \(0.587222\pi\)
\(578\) 0 0
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 0 0
\(583\) 20.0000 0.828315
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −4.00000 −0.164538
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) −9.00000 −0.368964
\(596\) 0 0
\(597\) 14.0000 0.572982
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) 47.0000 1.89831 0.949156 0.314806i \(-0.101939\pi\)
0.949156 + 0.314806i \(0.101939\pi\)
\(614\) 0 0
\(615\) 20.0000 0.806478
\(616\) 0 0
\(617\) 9.00000 0.362326 0.181163 0.983453i \(-0.442014\pi\)
0.181163 + 0.983453i \(0.442014\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 32.0000 1.28412
\(622\) 0 0
\(623\) 36.0000 1.44231
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −10.0000 −0.399362
\(628\) 0 0
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 36.0000 1.43087
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) 8.00000 0.316972
\(638\) 0 0
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −11.0000 −0.433798 −0.216899 0.976194i \(-0.569594\pi\)
−0.216899 + 0.976194i \(0.569594\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) 7.00000 0.275198 0.137599 0.990488i \(-0.456061\pi\)
0.137599 + 0.990488i \(0.456061\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) 0 0
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) 0 0
\(655\) −9.00000 −0.351659
\(656\) 0 0
\(657\) 9.00000 0.351123
\(658\) 0 0
\(659\) −34.0000 −1.32445 −0.662226 0.749304i \(-0.730388\pi\)
−0.662226 + 0.749304i \(0.730388\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) 0 0
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) −3.00000 −0.116335
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 0 0
\(669\) −4.00000 −0.154649
\(670\) 0 0
\(671\) 65.0000 2.50930
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 16.0000 0.615840
\(676\) 0 0
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) −11.0000 −0.420288
\(686\) 0 0
\(687\) −34.0000 −1.29718
\(688\) 0 0
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) −33.0000 −1.25538 −0.627690 0.778464i \(-0.715999\pi\)
−0.627690 + 0.778464i \(0.715999\pi\)
\(692\) 0 0
\(693\) 15.0000 0.569803
\(694\) 0 0
\(695\) −3.00000 −0.113796
\(696\) 0 0
\(697\) −30.0000 −1.13633
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) 0 0
\(705\) 2.00000 0.0753244
\(706\) 0 0
\(707\) 30.0000 1.12827
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 20.0000 0.747958
\(716\) 0 0
\(717\) −42.0000 −1.56852
\(718\) 0 0
\(719\) −13.0000 −0.484818 −0.242409 0.970174i \(-0.577938\pi\)
−0.242409 + 0.970174i \(0.577938\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) −52.0000 −1.93390
\(724\) 0 0
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) −7.00000 −0.259616 −0.129808 0.991539i \(-0.541436\pi\)
−0.129808 + 0.991539i \(0.541436\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 0 0
\(735\) 4.00000 0.147542
\(736\) 0 0
\(737\) −60.0000 −2.21013
\(738\) 0 0
\(739\) −43.0000 −1.58178 −0.790890 0.611958i \(-0.790382\pi\)
−0.790890 + 0.611958i \(0.790382\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) 15.0000 0.549557
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 22.0000 0.801725
\(754\) 0 0
\(755\) −2.00000 −0.0727875
\(756\) 0 0
\(757\) 5.00000 0.181728 0.0908640 0.995863i \(-0.471037\pi\)
0.0908640 + 0.995863i \(0.471037\pi\)
\(758\) 0 0
\(759\) −80.0000 −2.90382
\(760\) 0 0
\(761\) 9.00000 0.326250 0.163125 0.986605i \(-0.447843\pi\)
0.163125 + 0.986605i \(0.447843\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 31.0000 1.11789 0.558944 0.829205i \(-0.311207\pi\)
0.558944 + 0.829205i \(0.311207\pi\)
\(770\) 0 0
\(771\) 64.0000 2.30490
\(772\) 0 0
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) 0 0
\(777\) −60.0000 −2.15249
\(778\) 0 0
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) −10.0000 −0.357828
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) 42.0000 1.49524
\(790\) 0 0
\(791\) −30.0000 −1.06668
\(792\) 0 0
\(793\) 52.0000 1.84657
\(794\) 0 0
\(795\) 8.00000 0.283731
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) −3.00000 −0.106132
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) 45.0000 1.58802
\(804\) 0 0
\(805\) −24.0000 −0.845889
\(806\) 0 0
\(807\) 48.0000 1.68968
\(808\) 0 0
\(809\) 39.0000 1.37117 0.685583 0.727994i \(-0.259547\pi\)
0.685583 + 0.727994i \(0.259547\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) 0 0
\(823\) −53.0000 −1.84746 −0.923732 0.383040i \(-0.874877\pi\)
−0.923732 + 0.383040i \(0.874877\pi\)
\(824\) 0 0
\(825\) −40.0000 −1.39262
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 48.0000 1.66711 0.833554 0.552437i \(-0.186302\pi\)
0.833554 + 0.552437i \(0.186302\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 6.00000 0.207639
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 0 0
\(839\) 10.0000 0.345238 0.172619 0.984989i \(-0.444777\pi\)
0.172619 + 0.984989i \(0.444777\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 44.0000 1.51544
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 42.0000 1.44314
\(848\) 0 0
\(849\) −6.00000 −0.205919
\(850\) 0 0
\(851\) 80.0000 2.74236
\(852\) 0 0
\(853\) −42.0000 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 1.00000 0.0341196 0.0170598 0.999854i \(-0.494569\pi\)
0.0170598 + 0.999854i \(0.494569\pi\)
\(860\) 0 0
\(861\) 60.0000 2.04479
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) −40.0000 −1.35691
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) −8.00000 −0.270759
\(874\) 0 0
\(875\) −27.0000 −0.912767
\(876\) 0 0
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −51.0000 −1.71823 −0.859117 0.511780i \(-0.828986\pi\)
−0.859117 + 0.511780i \(0.828986\pi\)
\(882\) 0 0
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) 0 0
\(887\) −22.0000 −0.738688 −0.369344 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) −55.0000 −1.84257
\(892\) 0 0
\(893\) −1.00000 −0.0334637
\(894\) 0 0
\(895\) 18.0000 0.601674
\(896\) 0 0
\(897\) −64.0000 −2.13690
\(898\) 0 0
\(899\) −8.00000 −0.266815
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) −60.0000 −1.98571
\(914\) 0 0
\(915\) 26.0000 0.859533
\(916\) 0 0
\(917\) −27.0000 −0.891619
\(918\) 0 0
\(919\) 28.0000 0.923635 0.461817 0.886975i \(-0.347198\pi\)
0.461817 + 0.886975i \(0.347198\pi\)
\(920\) 0 0
\(921\) 24.0000 0.790827
\(922\) 0 0
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 40.0000 1.31519
\(926\) 0 0
\(927\) 6.00000 0.197066
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) −14.0000 −0.458339
\(934\) 0 0
\(935\) −15.0000 −0.490552
\(936\) 0 0
\(937\) −31.0000 −1.01273 −0.506363 0.862320i \(-0.669010\pi\)
−0.506363 + 0.862320i \(0.669010\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) 0 0
\(943\) −80.0000 −2.60516
\(944\) 0 0
\(945\) −12.0000 −0.390360
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) −60.0000 −1.94563
\(952\) 0 0
\(953\) −16.0000 −0.518291 −0.259145 0.965838i \(-0.583441\pi\)
−0.259145 + 0.965838i \(0.583441\pi\)
\(954\) 0 0
\(955\) −25.0000 −0.808981
\(956\) 0 0
\(957\) 20.0000 0.646508
\(958\) 0 0
\(959\) −33.0000 −1.06563
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) 0 0
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) −9.00000 −0.288527
\(974\) 0 0
\(975\) −32.0000 −1.02482
\(976\) 0 0
\(977\) 56.0000 1.79160 0.895799 0.444459i \(-0.146604\pi\)
0.895799 + 0.444459i \(0.146604\pi\)
\(978\) 0 0
\(979\) 60.0000 1.91761
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 38.0000 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) 0 0
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 7.00000 0.221915
\(996\) 0 0
\(997\) −37.0000 −1.17180 −0.585901 0.810383i \(-0.699259\pi\)
−0.585901 + 0.810383i \(0.699259\pi\)
\(998\) 0 0
\(999\) 40.0000 1.26554
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 1216.2.a.q.1.1 1
4.3 odd 2 1216.2.a.c.1.1 1
8.3 odd 2 76.2.a.a.1.1 1
8.5 even 2 304.2.a.a.1.1 1
24.5 odd 2 2736.2.a.q.1.1 1
24.11 even 2 684.2.a.b.1.1 1
40.3 even 4 1900.2.c.b.1749.2 2
40.19 odd 2 1900.2.a.b.1.1 1
40.27 even 4 1900.2.c.b.1749.1 2
40.29 even 2 7600.2.a.p.1.1 1
56.27 even 2 3724.2.a.a.1.1 1
88.43 even 2 9196.2.a.f.1.1 1
152.11 odd 6 1444.2.e.a.653.1 2
152.27 even 6 1444.2.e.c.653.1 2
152.37 odd 2 5776.2.a.p.1.1 1
152.75 even 2 1444.2.a.a.1.1 1
152.83 odd 6 1444.2.e.a.429.1 2
152.107 even 6 1444.2.e.c.429.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.2.a.a.1.1 1 8.3 odd 2
304.2.a.a.1.1 1 8.5 even 2
684.2.a.b.1.1 1 24.11 even 2
1216.2.a.c.1.1 1 4.3 odd 2
1216.2.a.q.1.1 1 1.1 even 1 trivial
1444.2.a.a.1.1 1 152.75 even 2
1444.2.e.a.429.1 2 152.83 odd 6
1444.2.e.a.653.1 2 152.11 odd 6
1444.2.e.c.429.1 2 152.107 even 6
1444.2.e.c.653.1 2 152.27 even 6
1900.2.a.b.1.1 1 40.19 odd 2
1900.2.c.b.1749.1 2 40.27 even 4
1900.2.c.b.1749.2 2 40.3 even 4
2736.2.a.q.1.1 1 24.5 odd 2
3724.2.a.a.1.1 1 56.27 even 2
5776.2.a.p.1.1 1 152.37 odd 2
7600.2.a.p.1.1 1 40.29 even 2
9196.2.a.f.1.1 1 88.43 even 2