Properties

Label 1216.2.c.e.609.4
Level $1216$
Weight $2$
Character 1216.609
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.4
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1216.609
Dual form 1216.2.c.e.609.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} +1.73205i q^{5} -1.73205 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} +1.73205i q^{5} -1.73205 q^{7} -1.00000 q^{9} +3.00000i q^{11} -3.46410 q^{15} -3.00000 q^{17} +1.00000i q^{19} -3.46410i q^{21} -3.46410 q^{23} +2.00000 q^{25} +4.00000i q^{27} +3.46410i q^{29} -6.92820 q^{31} -6.00000 q^{33} -3.00000i q^{35} -10.3923i q^{37} +6.00000 q^{41} +1.00000i q^{43} -1.73205i q^{45} -5.19615 q^{47} -4.00000 q^{49} -6.00000i q^{51} -5.19615 q^{55} -2.00000 q^{57} +6.00000i q^{59} -5.19615i q^{61} +1.73205 q^{63} +4.00000i q^{67} -6.92820i q^{69} +3.46410 q^{71} -1.00000 q^{73} +4.00000i q^{75} -5.19615i q^{77} -11.0000 q^{81} -12.0000i q^{83} -5.19615i q^{85} -6.92820 q^{87} -13.8564i q^{93} -1.73205 q^{95} +4.00000 q^{97} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 12q^{17} + 8q^{25} - 24q^{33} + 24q^{41} - 16q^{49} - 8q^{57} - 4q^{73} - 44q^{81} + 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 1.73205i 0.774597i 0.921954 + 0.387298i \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) 0 0
\(7\) −1.73205 −0.654654 −0.327327 0.944911i \(-0.606148\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.00000i 0.904534i 0.891883 + 0.452267i \(0.149385\pi\)
−0.891883 + 0.452267i \(0.850615\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) − 3.46410i − 0.755929i
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 3.46410i 0.643268i 0.946864 + 0.321634i \(0.104232\pi\)
−0.946864 + 0.321634i \(0.895768\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) − 3.00000i − 0.507093i
\(36\) 0 0
\(37\) − 10.3923i − 1.70848i −0.519875 0.854242i \(-0.674022\pi\)
0.519875 0.854242i \(-0.325978\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 0 0
\(45\) − 1.73205i − 0.258199i
\(46\) 0 0
\(47\) −5.19615 −0.757937 −0.378968 0.925410i \(-0.623721\pi\)
−0.378968 + 0.925410i \(0.623721\pi\)
\(48\) 0 0
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) − 6.00000i − 0.840168i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −5.19615 −0.700649
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) − 5.19615i − 0.665299i −0.943051 0.332650i \(-0.892057\pi\)
0.943051 0.332650i \(-0.107943\pi\)
\(62\) 0 0
\(63\) 1.73205 0.218218
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) − 6.92820i − 0.834058i
\(70\) 0 0
\(71\) 3.46410 0.411113 0.205557 0.978645i \(-0.434100\pi\)
0.205557 + 0.978645i \(0.434100\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) 4.00000i 0.461880i
\(76\) 0 0
\(77\) − 5.19615i − 0.592157i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) − 5.19615i − 0.563602i
\(86\) 0 0
\(87\) −6.92820 −0.742781
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 13.8564i − 1.43684i
\(94\) 0 0
\(95\) −1.73205 −0.177705
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) − 3.00000i − 0.301511i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 17.3205 1.70664 0.853320 0.521387i \(-0.174585\pi\)
0.853320 + 0.521387i \(0.174585\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 20.7846 1.97279
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) − 6.00000i − 0.559503i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.19615 0.476331
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) −10.3923 −0.922168 −0.461084 0.887357i \(-0.652539\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) 15.0000i 1.31056i 0.755388 + 0.655278i \(0.227449\pi\)
−0.755388 + 0.655278i \(0.772551\pi\)
\(132\) 0 0
\(133\) − 1.73205i − 0.150188i
\(134\) 0 0
\(135\) −6.92820 −0.596285
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 5.00000i 0.424094i 0.977259 + 0.212047i \(0.0680131\pi\)
−0.977259 + 0.212047i \(0.931987\pi\)
\(140\) 0 0
\(141\) − 10.3923i − 0.875190i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) − 8.00000i − 0.659829i
\(148\) 0 0
\(149\) 15.5885i 1.27706i 0.769599 + 0.638528i \(0.220456\pi\)
−0.769599 + 0.638528i \(0.779544\pi\)
\(150\) 0 0
\(151\) −17.3205 −1.40952 −0.704761 0.709444i \(-0.748946\pi\)
−0.704761 + 0.709444i \(0.748946\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) − 12.0000i − 0.963863i
\(156\) 0 0
\(157\) 13.8564i 1.10586i 0.833227 + 0.552931i \(0.186491\pi\)
−0.833227 + 0.552931i \(0.813509\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) 0 0
\(165\) − 10.3923i − 0.809040i
\(166\) 0 0
\(167\) −3.46410 −0.268060 −0.134030 0.990977i \(-0.542792\pi\)
−0.134030 + 0.990977i \(0.542792\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) − 1.00000i − 0.0764719i
\(172\) 0 0
\(173\) 10.3923i 0.790112i 0.918657 + 0.395056i \(0.129275\pi\)
−0.918657 + 0.395056i \(0.870725\pi\)
\(174\) 0 0
\(175\) −3.46410 −0.261861
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) − 6.00000i − 0.448461i −0.974536 0.224231i \(-0.928013\pi\)
0.974536 0.224231i \(-0.0719869\pi\)
\(180\) 0 0
\(181\) 24.2487i 1.80239i 0.433411 + 0.901196i \(0.357310\pi\)
−0.433411 + 0.901196i \(0.642690\pi\)
\(182\) 0 0
\(183\) 10.3923 0.768221
\(184\) 0 0
\(185\) 18.0000 1.32339
\(186\) 0 0
\(187\) − 9.00000i − 0.658145i
\(188\) 0 0
\(189\) − 6.92820i − 0.503953i
\(190\) 0 0
\(191\) 22.5167 1.62925 0.814624 0.579989i \(-0.196943\pi\)
0.814624 + 0.579989i \(0.196943\pi\)
\(192\) 0 0
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 27.7128i 1.97446i 0.159313 + 0.987228i \(0.449072\pi\)
−0.159313 + 0.987228i \(0.550928\pi\)
\(198\) 0 0
\(199\) −19.0526 −1.35060 −0.675300 0.737543i \(-0.735986\pi\)
−0.675300 + 0.737543i \(0.735986\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 10.3923i 0.725830i
\(206\) 0 0
\(207\) 3.46410 0.240772
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 22.0000i 1.51454i 0.653101 + 0.757271i \(0.273468\pi\)
−0.653101 + 0.757271i \(0.726532\pi\)
\(212\) 0 0
\(213\) 6.92820i 0.474713i
\(214\) 0 0
\(215\) −1.73205 −0.118125
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) − 2.00000i − 0.135147i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.46410 −0.231973 −0.115987 0.993251i \(-0.537003\pi\)
−0.115987 + 0.993251i \(0.537003\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.73205i 0.114457i 0.998361 + 0.0572286i \(0.0182264\pi\)
−0.998361 + 0.0572286i \(0.981774\pi\)
\(230\) 0 0
\(231\) 10.3923 0.683763
\(232\) 0 0
\(233\) 27.0000 1.76883 0.884414 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) 0 0
\(235\) − 9.00000i − 0.587095i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.73205 −0.112037 −0.0560185 0.998430i \(-0.517841\pi\)
−0.0560185 + 0.998430i \(0.517841\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) − 10.0000i − 0.641500i
\(244\) 0 0
\(245\) − 6.92820i − 0.442627i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 24.0000 1.52094
\(250\) 0 0
\(251\) − 21.0000i − 1.32551i −0.748837 0.662754i \(-0.769387\pi\)
0.748837 0.662754i \(-0.230613\pi\)
\(252\) 0 0
\(253\) − 10.3923i − 0.653359i
\(254\) 0 0
\(255\) 10.3923 0.650791
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 18.0000i 1.11847i
\(260\) 0 0
\(261\) − 3.46410i − 0.214423i
\(262\) 0 0
\(263\) 29.4449 1.81565 0.907824 0.419351i \(-0.137742\pi\)
0.907824 + 0.419351i \(0.137742\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 27.7128i 1.68968i 0.535019 + 0.844840i \(0.320304\pi\)
−0.535019 + 0.844840i \(0.679696\pi\)
\(270\) 0 0
\(271\) 3.46410 0.210429 0.105215 0.994450i \(-0.466447\pi\)
0.105215 + 0.994450i \(0.466447\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.00000i 0.361814i
\(276\) 0 0
\(277\) − 15.5885i − 0.936620i −0.883564 0.468310i \(-0.844863\pi\)
0.883564 0.468310i \(-0.155137\pi\)
\(278\) 0 0
\(279\) 6.92820 0.414781
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) − 19.0000i − 1.12943i −0.825285 0.564716i \(-0.808986\pi\)
0.825285 0.564716i \(-0.191014\pi\)
\(284\) 0 0
\(285\) − 3.46410i − 0.205196i
\(286\) 0 0
\(287\) −10.3923 −0.613438
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 8.00000i 0.468968i
\(292\) 0 0
\(293\) 13.8564i 0.809500i 0.914427 + 0.404750i \(0.132641\pi\)
−0.914427 + 0.404750i \(0.867359\pi\)
\(294\) 0 0
\(295\) −10.3923 −0.605063
\(296\) 0 0
\(297\) −12.0000 −0.696311
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 1.73205i − 0.0998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.00000 0.515339
\(306\) 0 0
\(307\) 16.0000i 0.913168i 0.889680 + 0.456584i \(0.150927\pi\)
−0.889680 + 0.456584i \(0.849073\pi\)
\(308\) 0 0
\(309\) 34.6410i 1.97066i
\(310\) 0 0
\(311\) 22.5167 1.27680 0.638401 0.769704i \(-0.279596\pi\)
0.638401 + 0.769704i \(0.279596\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) 3.00000i 0.169031i
\(316\) 0 0
\(317\) − 17.3205i − 0.972817i −0.873732 0.486408i \(-0.838307\pi\)
0.873732 0.486408i \(-0.161693\pi\)
\(318\) 0 0
\(319\) −10.3923 −0.581857
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) − 3.00000i − 0.166924i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) − 16.0000i − 0.879440i −0.898135 0.439720i \(-0.855078\pi\)
0.898135 0.439720i \(-0.144922\pi\)
\(332\) 0 0
\(333\) 10.3923i 0.569495i
\(334\) 0 0
\(335\) −6.92820 −0.378528
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) − 36.0000i − 1.95525i
\(340\) 0 0
\(341\) − 20.7846i − 1.12555i
\(342\) 0 0
\(343\) 19.0526 1.02874
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) − 3.00000i − 0.161048i −0.996753 0.0805242i \(-0.974341\pi\)
0.996753 0.0805242i \(-0.0256594\pi\)
\(348\) 0 0
\(349\) 36.3731i 1.94701i 0.228675 + 0.973503i \(0.426561\pi\)
−0.228675 + 0.973503i \(0.573439\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 6.00000i 0.318447i
\(356\) 0 0
\(357\) 10.3923i 0.550019i
\(358\) 0 0
\(359\) 29.4449 1.55404 0.777020 0.629476i \(-0.216730\pi\)
0.777020 + 0.629476i \(0.216730\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 4.00000i 0.209946i
\(364\) 0 0
\(365\) − 1.73205i − 0.0906597i
\(366\) 0 0
\(367\) 10.3923 0.542474 0.271237 0.962513i \(-0.412567\pi\)
0.271237 + 0.962513i \(0.412567\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 13.8564i − 0.717458i −0.933442 0.358729i \(-0.883210\pi\)
0.933442 0.358729i \(-0.116790\pi\)
\(374\) 0 0
\(375\) −24.2487 −1.25220
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 10.0000i 0.513665i 0.966456 + 0.256833i \(0.0826790\pi\)
−0.966456 + 0.256833i \(0.917321\pi\)
\(380\) 0 0
\(381\) − 20.7846i − 1.06483i
\(382\) 0 0
\(383\) 27.7128 1.41606 0.708029 0.706183i \(-0.249584\pi\)
0.708029 + 0.706183i \(0.249584\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) 0 0
\(387\) − 1.00000i − 0.0508329i
\(388\) 0 0
\(389\) 22.5167i 1.14164i 0.821075 + 0.570820i \(0.193375\pi\)
−0.821075 + 0.570820i \(0.806625\pi\)
\(390\) 0 0
\(391\) 10.3923 0.525561
\(392\) 0 0
\(393\) −30.0000 −1.51330
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 22.5167i − 1.13008i −0.825064 0.565039i \(-0.808861\pi\)
0.825064 0.565039i \(-0.191139\pi\)
\(398\) 0 0
\(399\) 3.46410 0.173422
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 19.0526i − 0.946729i
\(406\) 0 0
\(407\) 31.1769 1.54538
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) − 6.00000i − 0.295958i
\(412\) 0 0
\(413\) − 10.3923i − 0.511372i
\(414\) 0 0
\(415\) 20.7846 1.02028
\(416\) 0 0
\(417\) −10.0000 −0.489702
\(418\) 0 0
\(419\) 12.0000i 0.586238i 0.956076 + 0.293119i \(0.0946933\pi\)
−0.956076 + 0.293119i \(0.905307\pi\)
\(420\) 0 0
\(421\) − 13.8564i − 0.675320i −0.941268 0.337660i \(-0.890365\pi\)
0.941268 0.337660i \(-0.109635\pi\)
\(422\) 0 0
\(423\) 5.19615 0.252646
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 9.00000i 0.435541i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.7128 −1.33488 −0.667440 0.744664i \(-0.732610\pi\)
−0.667440 + 0.744664i \(0.732610\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) − 12.0000i − 0.575356i
\(436\) 0 0
\(437\) − 3.46410i − 0.165710i
\(438\) 0 0
\(439\) −24.2487 −1.15733 −0.578664 0.815566i \(-0.696426\pi\)
−0.578664 + 0.815566i \(0.696426\pi\)
\(440\) 0 0
\(441\) 4.00000 0.190476
\(442\) 0 0
\(443\) − 27.0000i − 1.28281i −0.767203 0.641404i \(-0.778352\pi\)
0.767203 0.641404i \(-0.221648\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −31.1769 −1.47462
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 18.0000i 0.847587i
\(452\) 0 0
\(453\) − 34.6410i − 1.62758i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −37.0000 −1.73079 −0.865393 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 0 0
\(459\) − 12.0000i − 0.560112i
\(460\) 0 0
\(461\) 22.5167i 1.04871i 0.851501 + 0.524353i \(0.175693\pi\)
−0.851501 + 0.524353i \(0.824307\pi\)
\(462\) 0 0
\(463\) 25.9808 1.20743 0.603714 0.797201i \(-0.293687\pi\)
0.603714 + 0.797201i \(0.293687\pi\)
\(464\) 0 0
\(465\) 24.0000 1.11297
\(466\) 0 0
\(467\) − 3.00000i − 0.138823i −0.997588 0.0694117i \(-0.977888\pi\)
0.997588 0.0694117i \(-0.0221122\pi\)
\(468\) 0 0
\(469\) − 6.92820i − 0.319915i
\(470\) 0 0
\(471\) −27.7128 −1.27694
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) 2.00000i 0.0917663i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.3923 0.474837 0.237418 0.971408i \(-0.423699\pi\)
0.237418 + 0.971408i \(0.423699\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 12.0000i 0.546019i
\(484\) 0 0
\(485\) 6.92820i 0.314594i
\(486\) 0 0
\(487\) −24.2487 −1.09881 −0.549407 0.835555i \(-0.685146\pi\)
−0.549407 + 0.835555i \(0.685146\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) − 12.0000i − 0.541552i −0.962642 0.270776i \(-0.912720\pi\)
0.962642 0.270776i \(-0.0872803\pi\)
\(492\) 0 0
\(493\) − 10.3923i − 0.468046i
\(494\) 0 0
\(495\) 5.19615 0.233550
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) − 29.0000i − 1.29822i −0.760695 0.649109i \(-0.775142\pi\)
0.760695 0.649109i \(-0.224858\pi\)
\(500\) 0 0
\(501\) − 6.92820i − 0.309529i
\(502\) 0 0
\(503\) −31.1769 −1.39011 −0.695055 0.718957i \(-0.744620\pi\)
−0.695055 + 0.718957i \(0.744620\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 26.0000i 1.15470i
\(508\) 0 0
\(509\) 27.7128i 1.22835i 0.789170 + 0.614174i \(0.210511\pi\)
−0.789170 + 0.614174i \(0.789489\pi\)
\(510\) 0 0
\(511\) 1.73205 0.0766214
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 30.0000i 1.32196i
\(516\) 0 0
\(517\) − 15.5885i − 0.685580i
\(518\) 0 0
\(519\) −20.7846 −0.912343
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) 34.0000i 1.48672i 0.668894 + 0.743358i \(0.266768\pi\)
−0.668894 + 0.743358i \(0.733232\pi\)
\(524\) 0 0
\(525\) − 6.92820i − 0.302372i
\(526\) 0 0
\(527\) 20.7846 0.905392
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) − 6.00000i − 0.260378i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −10.3923 −0.449299
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) − 12.0000i − 0.516877i
\(540\) 0 0
\(541\) − 5.19615i − 0.223400i −0.993742 0.111700i \(-0.964370\pi\)
0.993742 0.111700i \(-0.0356296\pi\)
\(542\) 0 0
\(543\) −48.4974 −2.08122
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 0 0
\(549\) 5.19615i 0.221766i
\(550\) 0 0
\(551\) −3.46410 −0.147576
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 36.0000i 1.52811i
\(556\) 0 0
\(557\) − 29.4449i − 1.24762i −0.781576 0.623809i \(-0.785584\pi\)
0.781576 0.623809i \(-0.214416\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 0 0
\(563\) − 18.0000i − 0.758610i −0.925272 0.379305i \(-0.876163\pi\)
0.925272 0.379305i \(-0.123837\pi\)
\(564\) 0 0
\(565\) − 31.1769i − 1.31162i
\(566\) 0 0
\(567\) 19.0526 0.800132
\(568\) 0 0
\(569\) 12.0000 0.503066 0.251533 0.967849i \(-0.419065\pi\)
0.251533 + 0.967849i \(0.419065\pi\)
\(570\) 0 0
\(571\) − 28.0000i − 1.17176i −0.810397 0.585882i \(-0.800748\pi\)
0.810397 0.585882i \(-0.199252\pi\)
\(572\) 0 0
\(573\) 45.0333i 1.88129i
\(574\) 0 0
\(575\) −6.92820 −0.288926
\(576\) 0 0
\(577\) 37.0000 1.54033 0.770165 0.637845i \(-0.220174\pi\)
0.770165 + 0.637845i \(0.220174\pi\)
\(578\) 0 0
\(579\) 40.0000i 1.66234i
\(580\) 0 0
\(581\) 20.7846i 0.862291i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.00000i 0.123823i 0.998082 + 0.0619116i \(0.0197197\pi\)
−0.998082 + 0.0619116i \(0.980280\pi\)
\(588\) 0 0
\(589\) − 6.92820i − 0.285472i
\(590\) 0 0
\(591\) −55.4256 −2.27991
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 9.00000i 0.368964i
\(596\) 0 0
\(597\) − 38.1051i − 1.55954i
\(598\) 0 0
\(599\) −27.7128 −1.13231 −0.566157 0.824297i \(-0.691571\pi\)
−0.566157 + 0.824297i \(0.691571\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) 3.46410i 0.140836i
\(606\) 0 0
\(607\) −27.7128 −1.12483 −0.562414 0.826856i \(-0.690127\pi\)
−0.562414 + 0.826856i \(0.690127\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 1.73205i − 0.0699569i −0.999388 0.0349784i \(-0.988864\pi\)
0.999388 0.0349784i \(-0.0111363\pi\)
\(614\) 0 0
\(615\) −20.7846 −0.838116
\(616\) 0 0
\(617\) 15.0000 0.603877 0.301939 0.953327i \(-0.402366\pi\)
0.301939 + 0.953327i \(0.402366\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 0 0
\(621\) − 13.8564i − 0.556038i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) − 6.00000i − 0.239617i
\(628\) 0 0
\(629\) 31.1769i 1.24310i
\(630\) 0 0
\(631\) 36.3731 1.44799 0.723994 0.689806i \(-0.242304\pi\)
0.723994 + 0.689806i \(0.242304\pi\)
\(632\) 0 0
\(633\) −44.0000 −1.74884
\(634\) 0 0
\(635\) − 18.0000i − 0.714308i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3.46410 −0.137038
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 5.00000i 0.197181i 0.995128 + 0.0985904i \(0.0314334\pi\)
−0.995128 + 0.0985904i \(0.968567\pi\)
\(644\) 0 0
\(645\) − 3.46410i − 0.136399i
\(646\) 0 0
\(647\) −19.0526 −0.749033 −0.374517 0.927220i \(-0.622191\pi\)
−0.374517 + 0.927220i \(0.622191\pi\)
\(648\) 0 0
\(649\) −18.0000 −0.706562
\(650\) 0 0
\(651\) 24.0000i 0.940634i
\(652\) 0 0
\(653\) − 22.5167i − 0.881145i −0.897717 0.440573i \(-0.854775\pi\)
0.897717 0.440573i \(-0.145225\pi\)
\(654\) 0 0
\(655\) −25.9808 −1.01515
\(656\) 0 0
\(657\) 1.00000 0.0390137
\(658\) 0 0
\(659\) − 6.00000i − 0.233727i −0.993148 0.116863i \(-0.962716\pi\)
0.993148 0.116863i \(-0.0372840\pi\)
\(660\) 0 0
\(661\) 34.6410i 1.34738i 0.739014 + 0.673690i \(0.235292\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) − 12.0000i − 0.464642i
\(668\) 0 0
\(669\) − 6.92820i − 0.267860i
\(670\) 0 0
\(671\) 15.5885 0.601786
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 8.00000i 0.307920i
\(676\) 0 0
\(677\) − 10.3923i − 0.399409i −0.979856 0.199704i \(-0.936002\pi\)
0.979856 0.199704i \(-0.0639982\pi\)
\(678\) 0 0
\(679\) −6.92820 −0.265880
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) − 5.19615i − 0.198535i
\(686\) 0 0
\(687\) −3.46410 −0.132164
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 7.00000i − 0.266293i −0.991096 0.133146i \(-0.957492\pi\)
0.991096 0.133146i \(-0.0425080\pi\)
\(692\) 0 0
\(693\) 5.19615i 0.197386i
\(694\) 0 0
\(695\) −8.66025 −0.328502
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) 0 0
\(699\) 54.0000i 2.04247i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 10.3923 0.391953
\(704\) 0 0
\(705\) 18.0000 0.677919
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 6.92820i − 0.260194i −0.991501 0.130097i \(-0.958471\pi\)
0.991501 0.130097i \(-0.0415289\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3.46410i − 0.129369i
\(718\) 0 0
\(719\) 29.4449 1.09811 0.549054 0.835787i \(-0.314988\pi\)
0.549054 + 0.835787i \(0.314988\pi\)
\(720\) 0 0
\(721\) −30.0000 −1.11726
\(722\) 0 0
\(723\) − 20.0000i − 0.743808i
\(724\) 0 0
\(725\) 6.92820i 0.257307i
\(726\) 0 0
\(727\) −19.0526 −0.706620 −0.353310 0.935506i \(-0.614944\pi\)
−0.353310 + 0.935506i \(0.614944\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) − 3.00000i − 0.110959i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 13.8564 0.511101
\(736\) 0 0
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) 19.0000i 0.698926i 0.936950 + 0.349463i \(0.113636\pi\)
−0.936950 + 0.349463i \(0.886364\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.92820 0.254171 0.127086 0.991892i \(-0.459438\pi\)
0.127086 + 0.991892i \(0.459438\pi\)
\(744\) 0 0
\(745\) −27.0000 −0.989203
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) − 10.3923i − 0.379727i
\(750\) 0 0
\(751\) −20.7846 −0.758441 −0.379221 0.925306i \(-0.623808\pi\)
−0.379221 + 0.925306i \(0.623808\pi\)
\(752\) 0 0
\(753\) 42.0000 1.53057
\(754\) 0 0
\(755\) − 30.0000i − 1.09181i
\(756\) 0 0
\(757\) − 36.3731i − 1.32200i −0.750385 0.661001i \(-0.770132\pi\)
0.750385 0.661001i \(-0.229868\pi\)
\(758\) 0 0
\(759\) 20.7846 0.754434
\(760\) 0 0
\(761\) −39.0000 −1.41375 −0.706874 0.707339i \(-0.749895\pi\)
−0.706874 + 0.707339i \(0.749895\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 5.19615i 0.187867i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 41.0000 1.47850 0.739249 0.673432i \(-0.235181\pi\)
0.739249 + 0.673432i \(0.235181\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 24.2487i − 0.872166i −0.899907 0.436083i \(-0.856365\pi\)
0.899907 0.436083i \(-0.143635\pi\)
\(774\) 0 0
\(775\) −13.8564 −0.497737
\(776\) 0 0
\(777\) −36.0000 −1.29149
\(778\) 0 0
\(779\) 6.00000i 0.214972i
\(780\) 0 0
\(781\) 10.3923i 0.371866i
\(782\) 0 0
\(783\) −13.8564 −0.495188
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 0 0
\(789\) 58.8897i 2.09653i
\(790\) 0 0
\(791\) 31.1769 1.10852
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 34.6410i − 1.22705i −0.789676 0.613524i \(-0.789751\pi\)
0.789676 0.613524i \(-0.210249\pi\)
\(798\) 0 0
\(799\) 15.5885 0.551480
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 3.00000i − 0.105868i
\(804\) 0 0
\(805\) 10.3923i 0.366281i
\(806\) 0 0
\(807\) −55.4256 −1.95107
\(808\) 0 0
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) 0 0
\(811\) 4.00000i 0.140459i 0.997531 + 0.0702295i \(0.0223732\pi\)
−0.997531 + 0.0702295i \(0.977627\pi\)
\(812\) 0 0
\(813\) 6.92820i 0.242983i
\(814\) 0 0
\(815\) 6.92820 0.242684
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 5.19615i − 0.181347i −0.995881 0.0906735i \(-0.971098\pi\)
0.995881 0.0906735i \(-0.0289020\pi\)
\(822\) 0 0
\(823\) 29.4449 1.02638 0.513192 0.858274i \(-0.328463\pi\)
0.513192 + 0.858274i \(0.328463\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) − 13.8564i − 0.481253i −0.970618 0.240626i \(-0.922647\pi\)
0.970618 0.240626i \(-0.0773529\pi\)
\(830\) 0 0
\(831\) 31.1769 1.08152
\(832\) 0 0
\(833\) 12.0000 0.415775
\(834\) 0 0
\(835\) − 6.00000i − 0.207639i
\(836\) 0 0
\(837\) − 27.7128i − 0.957895i
\(838\) 0 0
\(839\) −24.2487 −0.837158 −0.418579 0.908180i \(-0.637472\pi\)
−0.418579 + 0.908180i \(0.637472\pi\)
\(840\) 0 0
\(841\) 17.0000 0.586207
\(842\) 0 0
\(843\) 60.0000i 2.06651i
\(844\) 0 0
\(845\) 22.5167i 0.774597i
\(846\) 0 0
\(847\) −3.46410 −0.119028
\(848\) 0 0
\(849\) 38.0000 1.30416
\(850\) 0 0
\(851\) 36.0000i 1.23406i
\(852\) 0 0
\(853\) − 27.7128i − 0.948869i −0.880291 0.474434i \(-0.842653\pi\)
0.880291 0.474434i \(-0.157347\pi\)
\(854\) 0 0
\(855\) 1.73205 0.0592349
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 1.00000i 0.0341196i 0.999854 + 0.0170598i \(0.00543056\pi\)
−0.999854 + 0.0170598i \(0.994569\pi\)
\(860\) 0 0
\(861\) − 20.7846i − 0.708338i
\(862\) 0 0
\(863\) 24.2487 0.825436 0.412718 0.910859i \(-0.364580\pi\)
0.412718 + 0.910859i \(0.364580\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) − 16.0000i − 0.543388i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) − 21.0000i − 0.709930i
\(876\) 0 0
\(877\) − 31.1769i − 1.05277i −0.850246 0.526385i \(-0.823547\pi\)
0.850246 0.526385i \(-0.176453\pi\)
\(878\) 0 0
\(879\) −27.7128 −0.934730
\(880\) 0 0
\(881\) 51.0000 1.71823 0.859117 0.511780i \(-0.171014\pi\)
0.859117 + 0.511780i \(0.171014\pi\)
\(882\) 0 0
\(883\) − 49.0000i − 1.64898i −0.565876 0.824491i \(-0.691462\pi\)
0.565876 0.824491i \(-0.308538\pi\)
\(884\) 0 0
\(885\) − 20.7846i − 0.698667i
\(886\) 0 0
\(887\) 31.1769 1.04682 0.523409 0.852081i \(-0.324660\pi\)
0.523409 + 0.852081i \(0.324660\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) − 33.0000i − 1.10554i
\(892\) 0 0
\(893\) − 5.19615i − 0.173883i
\(894\) 0 0
\(895\) 10.3923 0.347376
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 24.0000i − 0.800445i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 3.46410 0.115278
\(904\) 0 0
\(905\) −42.0000 −1.39613
\(906\) 0 0
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 51.9615 1.72156 0.860781 0.508975i \(-0.169976\pi\)
0.860781 + 0.508975i \(0.169976\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 18.0000i 0.595062i
\(916\) 0 0
\(917\) − 25.9808i − 0.857960i
\(918\) 0 0
\(919\) 3.46410 0.114270 0.0571351 0.998366i \(-0.481803\pi\)
0.0571351 + 0.998366i \(0.481803\pi\)
\(920\) 0 0
\(921\) −32.0000 −1.05444
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 20.7846i − 0.683394i
\(926\) 0 0
\(927\) −17.3205 −0.568880
\(928\) 0 0
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 0 0
\(931\) − 4.00000i − 0.131095i
\(932\) 0 0
\(933\) 45.0333i 1.47432i
\(934\) 0 0
\(935\) 15.5885 0.509797
\(936\) 0 0
\(937\) 49.0000 1.60076 0.800380 0.599493i \(-0.204631\pi\)
0.800380 + 0.599493i \(0.204631\pi\)
\(938\) 0 0
\(939\) − 52.0000i − 1.69696i
\(940\) 0 0
\(941\) 38.1051i 1.24219i 0.783735 + 0.621096i \(0.213312\pi\)
−0.783735 + 0.621096i \(0.786688\pi\)
\(942\) 0 0
\(943\) −20.7846 −0.676840
\(944\) 0 0
\(945\) 12.0000 0.390360
\(946\) 0 0
\(947\) − 36.0000i − 1.16984i −0.811090 0.584921i \(-0.801125\pi\)
0.811090 0.584921i \(-0.198875\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 34.6410 1.12331
\(952\) 0 0
\(953\) −12.0000 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(954\) 0 0
\(955\) 39.0000i 1.26201i
\(956\) 0 0
\(957\) − 20.7846i − 0.671871i
\(958\) 0 0
\(959\) 5.19615 0.167793
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) − 6.00000i − 0.193347i
\(964\) 0 0
\(965\) 34.6410i 1.11513i
\(966\) 0 0
\(967\) 24.2487 0.779786 0.389893 0.920860i \(-0.372512\pi\)
0.389893 + 0.920860i \(0.372512\pi\)
\(968\) 0 0
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) − 24.0000i − 0.770197i −0.922876 0.385098i \(-0.874168\pi\)
0.922876 0.385098i \(-0.125832\pi\)
\(972\) 0 0
\(973\) − 8.66025i − 0.277635i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −36.0000 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 34.6410 1.10488 0.552438 0.833554i \(-0.313697\pi\)
0.552438 + 0.833554i \(0.313697\pi\)
\(984\) 0 0
\(985\) −48.0000 −1.52941
\(986\) 0 0
\(987\) 18.0000i 0.572946i
\(988\) 0 0
\(989\) − 3.46410i − 0.110152i
\(990\) 0 0
\(991\) −31.1769 −0.990367 −0.495184 0.868788i \(-0.664899\pi\)
−0.495184 + 0.868788i \(0.664899\pi\)
\(992\) 0 0
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) − 33.0000i − 1.04617i
\(996\) 0 0
\(997\) 19.0526i 0.603401i 0.953403 + 0.301700i \(0.0975542\pi\)
−0.953403 + 0.301700i \(0.902446\pi\)
\(998\) 0 0
\(999\) 41.5692 1.31519
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.c.e.609.4 yes 4
4.3 odd 2 inner 1216.2.c.e.609.2 yes 4
8.3 odd 2 inner 1216.2.c.e.609.3 yes 4
8.5 even 2 inner 1216.2.c.e.609.1 4
16.3 odd 4 4864.2.a.z.1.2 2
16.5 even 4 4864.2.a.z.1.1 2
16.11 odd 4 4864.2.a.q.1.1 2
16.13 even 4 4864.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.e.609.1 4 8.5 even 2 inner
1216.2.c.e.609.2 yes 4 4.3 odd 2 inner
1216.2.c.e.609.3 yes 4 8.3 odd 2 inner
1216.2.c.e.609.4 yes 4 1.1 even 1 trivial
4864.2.a.q.1.1 2 16.11 odd 4
4864.2.a.q.1.2 2 16.13 even 4
4864.2.a.z.1.1 2 16.5 even 4
4864.2.a.z.1.2 2 16.3 odd 4