Properties

Label 5780.2.c.a.5201.2
Level $5780$
Weight $2$
Character 5780.5201
Analytic conductor $46.154$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5780 = 2^{2} \cdot 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5780.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.1535323683\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5201.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5780.5201
Dual form 5780.2.c.a.5201.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} +1.00000i q^{5} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} +1.00000i q^{5} +2.00000i q^{7} -1.00000 q^{9} +2.00000 q^{13} -2.00000 q^{15} +4.00000 q^{19} -4.00000 q^{21} +6.00000i q^{23} -1.00000 q^{25} +4.00000i q^{27} -6.00000i q^{29} +4.00000i q^{31} -2.00000 q^{35} -2.00000i q^{37} +4.00000i q^{39} +6.00000i q^{41} +10.0000 q^{43} -1.00000i q^{45} -6.00000 q^{47} +3.00000 q^{49} +6.00000 q^{53} +8.00000i q^{57} -12.0000 q^{59} +2.00000i q^{61} -2.00000i q^{63} +2.00000i q^{65} +2.00000 q^{67} -12.0000 q^{69} +12.0000i q^{71} -2.00000i q^{73} -2.00000i q^{75} +8.00000i q^{79} -11.0000 q^{81} -6.00000 q^{83} +12.0000 q^{87} -6.00000 q^{89} +4.00000i q^{91} -8.00000 q^{93} +4.00000i q^{95} -2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} + 4q^{13} - 4q^{15} + 8q^{19} - 8q^{21} - 2q^{25} - 4q^{35} + 20q^{43} - 12q^{47} + 6q^{49} + 12q^{53} - 24q^{59} + 4q^{67} - 24q^{69} - 22q^{81} - 12q^{83} + 24q^{87} - 12q^{89} - 16q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5780\mathbb{Z}\right)^\times\).

\(n\) \(581\) \(1157\) \(2891\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 6.00000i 1.25109i 0.780189 + 0.625543i \(0.215123\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 4.00000i 0.640513i
\(40\) 0 0
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) 0 0
\(65\) 2.00000i 0.248069i
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 12.0000i 1.42414i 0.702109 + 0.712069i \(0.252242\pi\)
−0.702109 + 0.712069i \(0.747758\pi\)
\(72\) 0 0
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) − 2.00000i − 0.230940i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000i 0.900070i 0.893011 + 0.450035i \(0.148589\pi\)
−0.893011 + 0.450035i \(0.851411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.00000i 0.419314i
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 4.00000i 0.410391i
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) − 4.00000i − 0.390360i
\(106\) 0 0
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 20.0000i 1.76090i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i 0.985506 + 0.169638i \(0.0542598\pi\)
−0.985506 + 0.169638i \(0.945740\pi\)
\(140\) 0 0
\(141\) − 12.0000i − 1.01058i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) 12.0000i 0.951662i
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) − 10.0000i − 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 18.0000i − 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) − 2.00000i − 0.151186i
\(176\) 0 0
\(177\) − 24.0000i − 1.80395i
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) − 10.0000i − 0.743294i −0.928374 0.371647i \(-0.878793\pi\)
0.928374 0.371647i \(-0.121207\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 26.0000i 1.87152i 0.352636 + 0.935760i \(0.385285\pi\)
−0.352636 + 0.935760i \(0.614715\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) − 8.00000i − 0.567105i −0.958957 0.283552i \(-0.908487\pi\)
0.958957 0.283552i \(-0.0915130\pi\)
\(200\) 0 0
\(201\) 4.00000i 0.282138i
\(202\) 0 0
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) − 6.00000i − 0.417029i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 16.0000i − 1.10149i −0.834675 0.550743i \(-0.814345\pi\)
0.834675 0.550743i \(-0.185655\pi\)
\(212\) 0 0
\(213\) −24.0000 −1.64445
\(214\) 0 0
\(215\) 10.0000i 0.681994i
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) − 6.00000i − 0.398234i −0.979976 0.199117i \(-0.936193\pi\)
0.979976 0.199117i \(-0.0638074\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) − 6.00000i − 0.391397i
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) − 14.0000i − 0.901819i −0.892570 0.450910i \(-0.851100\pi\)
0.892570 0.450910i \(-0.148900\pi\)
\(242\) 0 0
\(243\) − 10.0000i − 0.641500i
\(244\) 0 0
\(245\) 3.00000i 0.191663i
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) − 12.0000i − 0.760469i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 6.00000i 0.371391i
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 6.00000i 0.368577i
\(266\) 0 0
\(267\) − 12.0000i − 0.734388i
\(268\) 0 0
\(269\) − 18.0000i − 1.09748i −0.835993 0.548740i \(-0.815108\pi\)
0.835993 0.548740i \(-0.184892\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) −8.00000 −0.484182
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 26.0000i − 1.56219i −0.624413 0.781094i \(-0.714662\pi\)
0.624413 0.781094i \(-0.285338\pi\)
\(278\) 0 0
\(279\) − 4.00000i − 0.239474i
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 14.0000i 0.832214i 0.909316 + 0.416107i \(0.136606\pi\)
−0.909316 + 0.416107i \(0.863394\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 0 0
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) − 12.0000i − 0.698667i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000i 0.693978i
\(300\) 0 0
\(301\) 20.0000i 1.15278i
\(302\) 0 0
\(303\) 12.0000i 0.689382i
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 2.00000 0.114146 0.0570730 0.998370i \(-0.481823\pi\)
0.0570730 + 0.998370i \(0.481823\pi\)
\(308\) 0 0
\(309\) 28.0000i 1.59286i
\(310\) 0 0
\(311\) − 12.0000i − 0.680458i −0.940343 0.340229i \(-0.889495\pi\)
0.940343 0.340229i \(-0.110505\pi\)
\(312\) 0 0
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) − 12.0000i − 0.661581i
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 2.00000i 0.109272i
\(336\) 0 0
\(337\) − 2.00000i − 0.108947i −0.998515 0.0544735i \(-0.982652\pi\)
0.998515 0.0544735i \(-0.0173480\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) − 12.0000i − 0.646058i
\(346\) 0 0
\(347\) − 30.0000i − 1.61048i −0.592946 0.805242i \(-0.702035\pi\)
0.592946 0.805242i \(-0.297965\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 8.00000i 0.427008i
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 22.0000i 1.15470i
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) − 22.0000i − 1.14839i −0.818718 0.574195i \(-0.805315\pi\)
0.818718 0.574195i \(-0.194685\pi\)
\(368\) 0 0
\(369\) − 6.00000i − 0.312348i
\(370\) 0 0
\(371\) 12.0000i 0.623009i
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 0 0
\(377\) − 12.0000i − 0.618031i
\(378\) 0 0
\(379\) 28.0000i 1.43826i 0.694874 + 0.719132i \(0.255460\pi\)
−0.694874 + 0.719132i \(0.744540\pi\)
\(380\) 0 0
\(381\) − 4.00000i − 0.204926i
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −10.0000 −0.508329
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) − 30.0000i − 1.49813i −0.662497 0.749064i \(-0.730503\pi\)
0.662497 0.749064i \(-0.269497\pi\)
\(402\) 0 0
\(403\) 8.00000i 0.398508i
\(404\) 0 0
\(405\) − 11.0000i − 0.546594i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −34.0000 −1.68119 −0.840596 0.541663i \(-0.817795\pi\)
−0.840596 + 0.541663i \(0.817795\pi\)
\(410\) 0 0
\(411\) 36.0000i 1.77575i
\(412\) 0 0
\(413\) − 24.0000i − 1.18096i
\(414\) 0 0
\(415\) − 6.00000i − 0.294528i
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 36.0000i 1.75872i 0.476162 + 0.879358i \(0.342028\pi\)
−0.476162 + 0.879358i \(0.657972\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000i 1.73406i 0.498257 + 0.867029i \(0.333974\pi\)
−0.498257 + 0.867029i \(0.666026\pi\)
\(432\) 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
\(437\) 24.0000i 1.14808i
\(438\) 0 0
\(439\) − 8.00000i − 0.381819i −0.981608 0.190910i \(-0.938856\pi\)
0.981608 0.190910i \(-0.0611437\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) − 6.00000i − 0.284427i
\(446\) 0 0
\(447\) − 12.0000i − 0.567581i
\(448\) 0 0
\(449\) 6.00000i 0.283158i 0.989927 + 0.141579i \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 40.0000i − 1.87936i
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −26.0000 −1.21623 −0.608114 0.793849i \(-0.708074\pi\)
−0.608114 + 0.793849i \(0.708074\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 0 0
\(465\) − 8.00000i − 0.370991i
\(466\) 0 0
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 0 0
\(469\) 4.00000i 0.184703i
\(470\) 0 0
\(471\) − 44.0000i − 2.02741i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 24.0000i 1.09659i 0.836286 + 0.548294i \(0.184723\pi\)
−0.836286 + 0.548294i \(0.815277\pi\)
\(480\) 0 0
\(481\) − 4.00000i − 0.182384i
\(482\) 0 0
\(483\) − 24.0000i − 1.09204i
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 26.0000i 1.17817i 0.808070 + 0.589086i \(0.200512\pi\)
−0.808070 + 0.589086i \(0.799488\pi\)
\(488\) 0 0
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) − 4.00000i − 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) 36.0000 1.60836
\(502\) 0 0
\(503\) − 18.0000i − 0.802580i −0.915951 0.401290i \(-0.868562\pi\)
0.915951 0.401290i \(-0.131438\pi\)
\(504\) 0 0
\(505\) 6.00000i 0.266996i
\(506\) 0 0
\(507\) − 18.0000i − 0.799408i
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) 16.0000i 0.706417i
\(514\) 0 0
\(515\) 14.0000i 0.616914i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) − 6.00000i − 0.262865i −0.991325 0.131432i \(-0.958042\pi\)
0.991325 0.131432i \(-0.0419576\pi\)
\(522\) 0 0
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 24.0000i 1.03568i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 14.0000i − 0.601907i −0.953639 0.300954i \(-0.902695\pi\)
0.953639 0.300954i \(-0.0973049\pi\)
\(542\) 0 0
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) − 26.0000i − 1.11168i −0.831289 0.555840i \(-0.812397\pi\)
0.831289 0.555840i \(-0.187603\pi\)
\(548\) 0 0
\(549\) − 2.00000i − 0.0853579i
\(550\) 0 0
\(551\) − 24.0000i − 1.02243i
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 4.00000i 0.169791i
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) − 22.0000i − 0.923913i
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 8.00000i 0.334790i 0.985890 + 0.167395i \(0.0535355\pi\)
−0.985890 + 0.167395i \(0.946465\pi\)
\(572\) 0 0
\(573\) − 24.0000i − 1.00261i
\(574\) 0 0
\(575\) − 6.00000i − 0.250217i
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) −52.0000 −2.16105
\(580\) 0 0
\(581\) − 12.0000i − 0.497844i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) − 2.00000i − 0.0826898i
\(586\) 0 0
\(587\) 6.00000 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) 0 0
\(589\) 16.0000i 0.659269i
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) − 10.0000i − 0.407909i −0.978980 0.203954i \(-0.934621\pi\)
0.978980 0.203954i \(-0.0653794\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) 0 0
\(605\) 11.0000i 0.447214i
\(606\) 0 0
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) 0 0
\(609\) 24.0000i 0.972529i
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) − 12.0000i − 0.483887i
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 0 0
\(623\) − 12.0000i − 0.480770i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 0 0
\(633\) 32.0000 1.27189
\(634\) 0 0
\(635\) − 2.00000i − 0.0793676i
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) − 12.0000i − 0.474713i
\(640\) 0 0
\(641\) 18.0000i 0.710957i 0.934684 + 0.355479i \(0.115682\pi\)
−0.934684 + 0.355479i \(0.884318\pi\)
\(642\) 0 0
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 0 0
\(645\) −20.0000 −0.787499
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) − 16.0000i − 0.627089i
\(652\) 0 0
\(653\) 42.0000i 1.64359i 0.569785 + 0.821794i \(0.307026\pi\)
−0.569785 + 0.821794i \(0.692974\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 0 0
\(669\) 20.0000i 0.773245i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) − 46.0000i − 1.77317i −0.462566 0.886585i \(-0.653071\pi\)
0.462566 0.886585i \(-0.346929\pi\)
\(674\) 0 0
\(675\) − 4.00000i − 0.153960i
\(676\) 0 0
\(677\) − 18.0000i − 0.691796i −0.938272 0.345898i \(-0.887574\pi\)
0.938272 0.345898i \(-0.112426\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 42.0000i 1.60709i 0.595247 + 0.803543i \(0.297054\pi\)
−0.595247 + 0.803543i \(0.702946\pi\)
\(684\) 0 0
\(685\) 18.0000i 0.687745i
\(686\) 0 0
\(687\) − 28.0000i − 1.06827i
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 8.00000i 0.304334i 0.988355 + 0.152167i \(0.0486252\pi\)
−0.988355 + 0.152167i \(0.951375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) − 8.00000i − 0.301726i
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 12.0000i 0.451306i
\(708\) 0 0
\(709\) 34.0000i 1.27690i 0.769665 + 0.638448i \(0.220423\pi\)
−0.769665 + 0.638448i \(0.779577\pi\)
\(710\) 0 0
\(711\) − 8.00000i − 0.300023i
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 48.0000i − 1.79259i
\(718\) 0 0
\(719\) 24.0000i 0.895049i 0.894272 + 0.447524i \(0.147694\pi\)
−0.894272 + 0.447524i \(0.852306\pi\)
\(720\) 0 0
\(721\) 28.0000i 1.04277i
\(722\) 0 0
\(723\) 28.0000 1.04133
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) −46.0000 −1.70605 −0.853023 0.521874i \(-0.825233\pi\)
−0.853023 + 0.521874i \(0.825233\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 16.0000i 0.587775i
\(742\) 0 0
\(743\) − 6.00000i − 0.220119i −0.993925 0.110059i \(-0.964896\pi\)
0.993925 0.110059i \(-0.0351041\pi\)
\(744\) 0 0
\(745\) − 6.00000i − 0.219823i
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 4.00000i 0.145962i 0.997333 + 0.0729810i \(0.0232513\pi\)
−0.997333 + 0.0729810i \(0.976749\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 20.0000i − 0.727875i
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 12.0000i 0.432169i
\(772\) 0 0
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) − 4.00000i − 0.143684i
\(776\) 0 0
\(777\) 8.00000i 0.286998i
\(778\) 0 0
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) − 22.0000i − 0.785214i
\(786\) 0 0
\(787\) − 26.0000i − 0.926800i −0.886149 0.463400i \(-0.846629\pi\)
0.886149 0.463400i \(-0.153371\pi\)
\(788\) 0 0
\(789\) 36.0000i 1.28163i
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 4.00000i 0.142044i
\(794\) 0 0
\(795\) −12.0000 −0.425596
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) − 12.0000i − 0.422944i
\(806\) 0 0
\(807\) 36.0000 1.26726
\(808\) 0 0
\(809\) − 6.00000i − 0.210949i −0.994422 0.105474i \(-0.966364\pi\)
0.994422 0.105474i \(-0.0336361\pi\)
\(810\) 0 0
\(811\) 16.0000i 0.561836i 0.959732 + 0.280918i \(0.0906389\pi\)
−0.959732 + 0.280918i \(0.909361\pi\)
\(812\) 0 0
\(813\) 40.0000i 1.40286i
\(814\) 0 0
\(815\) 10.0000 0.350285
\(816\) 0 0
\(817\) 40.0000 1.39942
\(818\) 0 0
\(819\) − 4.00000i − 0.139771i
\(820\) 0 0
\(821\) 54.0000i 1.88461i 0.334751 + 0.942306i \(0.391348\pi\)
−0.334751 + 0.942306i \(0.608652\pi\)
\(822\) 0 0
\(823\) 38.0000i 1.32460i 0.749240 + 0.662298i \(0.230419\pi\)
−0.749240 + 0.662298i \(0.769581\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 30.0000i − 1.04320i −0.853189 0.521601i \(-0.825335\pi\)
0.853189 0.521601i \(-0.174665\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 52.0000 1.80386
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) − 48.0000i − 1.65714i −0.559883 0.828572i \(-0.689154\pi\)
0.559883 0.828572i \(-0.310846\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) − 12.0000i − 0.413302i
\(844\) 0 0
\(845\) − 9.00000i − 0.309609i
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) − 50.0000i − 1.71197i −0.517003 0.855984i \(-0.672952\pi\)
0.517003 0.855984i \(-0.327048\pi\)
\(854\) 0 0
\(855\) − 4.00000i − 0.136797i
\(856\) 0 0
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) − 24.0000i − 0.817918i
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 26.0000i 0.877958i 0.898497 + 0.438979i \(0.144660\pi\)
−0.898497 + 0.438979i \(0.855340\pi\)
\(878\) 0 0
\(879\) − 60.0000i − 2.02375i
\(880\) 0 0
\(881\) 18.0000i 0.606435i 0.952921 + 0.303218i \(0.0980609\pi\)
−0.952921 + 0.303218i \(0.901939\pi\)
\(882\) 0 0
\(883\) 14.0000 0.471138 0.235569 0.971858i \(-0.424305\pi\)
0.235569 + 0.971858i \(0.424305\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 0 0
\(887\) − 18.0000i − 0.604381i −0.953248 0.302190i \(-0.902282\pi\)
0.953248 0.302190i \(-0.0977178\pi\)
\(888\) 0 0
\(889\) − 4.00000i − 0.134156i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 12.0000i 0.401116i
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −40.0000 −1.33112
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) − 46.0000i − 1.52740i −0.645568 0.763702i \(-0.723379\pi\)
0.645568 0.763702i \(-0.276621\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 12.0000i 0.397578i 0.980042 + 0.198789i \(0.0637008\pi\)
−0.980042 + 0.198789i \(0.936299\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) − 4.00000i − 0.132236i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 4.00000i 0.131804i
\(922\) 0 0
\(923\) 24.0000i 0.789970i
\(924\) 0 0
\(925\) 2.00000i 0.0657596i
\(926\) 0 0
\(927\) −14.0000 −0.459820
\(928\) 0 0
\(929\) − 42.0000i − 1.37798i −0.724773 0.688988i \(-0.758055\pi\)
0.724773 0.688988i \(-0.241945\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 44.0000 1.43589
\(940\) 0 0
\(941\) − 18.0000i − 0.586783i −0.955992 0.293392i \(-0.905216\pi\)
0.955992 0.293392i \(-0.0947840\pi\)
\(942\) 0 0
\(943\) −36.0000 −1.17232
\(944\) 0 0
\(945\) − 8.00000i − 0.260240i
\(946\) 0 0
\(947\) − 18.0000i − 0.584921i −0.956278 0.292461i \(-0.905526\pi\)
0.956278 0.292461i \(-0.0944741\pi\)
\(948\) 0 0
\(949\) − 4.00000i − 0.129845i
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) − 12.0000i − 0.388311i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 36.0000i 1.16250i
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) − 6.00000i − 0.193347i
\(964\) 0 0
\(965\) −26.0000 −0.836970
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) − 4.00000i − 0.128103i
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 2.00000i − 0.0638551i
\(982\) 0 0
\(983\) 18.0000i 0.574111i 0.957914 + 0.287055i \(0.0926764\pi\)
−0.957914 + 0.287055i \(0.907324\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) 60.0000i 1.90789i
\(990\) 0 0
\(991\) 4.00000i 0.127064i 0.997980 + 0.0635321i \(0.0202365\pi\)
−0.997980 + 0.0635321i \(0.979763\pi\)
\(992\) 0 0
\(993\) − 16.0000i − 0.507745i
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) 0 0
\(999\) 8.00000 0.253109
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 5780.2.c.a.5201.2 2
17.4 even 4 5780.2.a.f.1.1 1
17.13 even 4 20.2.a.a.1.1 1
17.16 even 2 inner 5780.2.c.a.5201.1 2
51.47 odd 4 180.2.a.a.1.1 1
68.47 odd 4 80.2.a.b.1.1 1
85.13 odd 4 100.2.c.a.49.1 2
85.47 odd 4 100.2.c.a.49.2 2
85.64 even 4 100.2.a.a.1.1 1
119.13 odd 4 980.2.a.h.1.1 1
119.30 even 12 980.2.i.i.361.1 2
119.47 odd 12 980.2.i.c.361.1 2
119.81 even 12 980.2.i.i.961.1 2
119.115 odd 12 980.2.i.c.961.1 2
136.13 even 4 320.2.a.f.1.1 1
136.115 odd 4 320.2.a.a.1.1 1
153.13 even 12 1620.2.i.h.541.1 2
153.47 odd 12 1620.2.i.b.1081.1 2
153.115 even 12 1620.2.i.h.1081.1 2
153.149 odd 12 1620.2.i.b.541.1 2
187.98 odd 4 2420.2.a.a.1.1 1
204.47 even 4 720.2.a.h.1.1 1
221.47 odd 4 3380.2.f.b.3041.1 2
221.64 even 4 3380.2.a.c.1.1 1
221.200 odd 4 3380.2.f.b.3041.2 2
255.47 even 4 900.2.d.c.649.2 2
255.98 even 4 900.2.d.c.649.1 2
255.149 odd 4 900.2.a.b.1.1 1
272.13 even 4 1280.2.d.c.641.1 2
272.115 odd 4 1280.2.d.g.641.2 2
272.149 even 4 1280.2.d.c.641.2 2
272.251 odd 4 1280.2.d.g.641.1 2
323.132 odd 4 7220.2.a.f.1.1 1
340.47 even 4 400.2.c.b.49.1 2
340.183 even 4 400.2.c.b.49.2 2
340.319 odd 4 400.2.a.c.1.1 1
357.251 even 4 8820.2.a.g.1.1 1
408.149 odd 4 2880.2.a.m.1.1 1
408.251 even 4 2880.2.a.f.1.1 1
476.251 even 4 3920.2.a.h.1.1 1
595.13 even 4 4900.2.e.f.2549.2 2
595.132 even 4 4900.2.e.f.2549.1 2
595.489 odd 4 4900.2.a.e.1.1 1
680.13 odd 4 1600.2.c.d.449.2 2
680.149 even 4 1600.2.a.c.1.1 1
680.387 even 4 1600.2.c.e.449.2 2
680.523 even 4 1600.2.c.e.449.1 2
680.557 odd 4 1600.2.c.d.449.1 2
680.659 odd 4 1600.2.a.w.1.1 1
748.659 even 4 9680.2.a.ba.1.1 1
1020.47 odd 4 3600.2.f.j.2449.1 2
1020.659 even 4 3600.2.a.be.1.1 1
1020.863 odd 4 3600.2.f.j.2449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.2.a.a.1.1 1 17.13 even 4
80.2.a.b.1.1 1 68.47 odd 4
100.2.a.a.1.1 1 85.64 even 4
100.2.c.a.49.1 2 85.13 odd 4
100.2.c.a.49.2 2 85.47 odd 4
180.2.a.a.1.1 1 51.47 odd 4
320.2.a.a.1.1 1 136.115 odd 4
320.2.a.f.1.1 1 136.13 even 4
400.2.a.c.1.1 1 340.319 odd 4
400.2.c.b.49.1 2 340.47 even 4
400.2.c.b.49.2 2 340.183 even 4
720.2.a.h.1.1 1 204.47 even 4
900.2.a.b.1.1 1 255.149 odd 4
900.2.d.c.649.1 2 255.98 even 4
900.2.d.c.649.2 2 255.47 even 4
980.2.a.h.1.1 1 119.13 odd 4
980.2.i.c.361.1 2 119.47 odd 12
980.2.i.c.961.1 2 119.115 odd 12
980.2.i.i.361.1 2 119.30 even 12
980.2.i.i.961.1 2 119.81 even 12
1280.2.d.c.641.1 2 272.13 even 4
1280.2.d.c.641.2 2 272.149 even 4
1280.2.d.g.641.1 2 272.251 odd 4
1280.2.d.g.641.2 2 272.115 odd 4
1600.2.a.c.1.1 1 680.149 even 4
1600.2.a.w.1.1 1 680.659 odd 4
1600.2.c.d.449.1 2 680.557 odd 4
1600.2.c.d.449.2 2 680.13 odd 4
1600.2.c.e.449.1 2 680.523 even 4
1600.2.c.e.449.2 2 680.387 even 4
1620.2.i.b.541.1 2 153.149 odd 12
1620.2.i.b.1081.1 2 153.47 odd 12
1620.2.i.h.541.1 2 153.13 even 12
1620.2.i.h.1081.1 2 153.115 even 12
2420.2.a.a.1.1 1 187.98 odd 4
2880.2.a.f.1.1 1 408.251 even 4
2880.2.a.m.1.1 1 408.149 odd 4
3380.2.a.c.1.1 1 221.64 even 4
3380.2.f.b.3041.1 2 221.47 odd 4
3380.2.f.b.3041.2 2 221.200 odd 4
3600.2.a.be.1.1 1 1020.659 even 4
3600.2.f.j.2449.1 2 1020.47 odd 4
3600.2.f.j.2449.2 2 1020.863 odd 4
3920.2.a.h.1.1 1 476.251 even 4
4900.2.a.e.1.1 1 595.489 odd 4
4900.2.e.f.2549.1 2 595.132 even 4
4900.2.e.f.2549.2 2 595.13 even 4
5780.2.a.f.1.1 1 17.4 even 4
5780.2.c.a.5201.1 2 17.16 even 2 inner
5780.2.c.a.5201.2 2 1.1 even 1 trivial
7220.2.a.f.1.1 1 323.132 odd 4
8820.2.a.g.1.1 1 357.251 even 4
9680.2.a.ba.1.1 1 748.659 even 4