Properties

Label 1280.2.d.c.641.2
Level $1280$
Weight $2$
Character 1280.641
Analytic conductor $10.221$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(1\)
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 641.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.2.d.c.641.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} -1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} -1.00000i q^{5} -2.00000 q^{7} -1.00000 q^{9} -2.00000i q^{13} +2.00000 q^{15} -6.00000 q^{17} +4.00000i q^{19} -4.00000i q^{21} -6.00000 q^{23} -1.00000 q^{25} +4.00000i q^{27} -6.00000i q^{29} -4.00000 q^{31} +2.00000i q^{35} +2.00000i q^{37} +4.00000 q^{39} -6.00000 q^{41} -10.0000i q^{43} +1.00000i q^{45} -6.00000 q^{47} -3.00000 q^{49} -12.0000i q^{51} -6.00000i q^{53} -8.00000 q^{57} +12.0000i q^{59} -2.00000i q^{61} +2.00000 q^{63} -2.00000 q^{65} -2.00000i q^{67} -12.0000i q^{69} +12.0000 q^{71} -2.00000 q^{73} -2.00000i q^{75} +8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} +6.00000i q^{85} +12.0000 q^{87} +6.00000 q^{89} +4.00000i q^{91} -8.00000i q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{7} - 2q^{9} + 4q^{15} - 12q^{17} - 12q^{23} - 2q^{25} - 8q^{31} + 8q^{39} - 12q^{41} - 12q^{47} - 6q^{49} - 16q^{57} + 4q^{63} - 4q^{65} + 24q^{71} - 4q^{73} + 16q^{79} - 22q^{81} + 24q^{87} + 12q^{89} + 8q^{95} + 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) − 4.00000i − 0.872872i
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 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
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 10.0000i − 1.52499i −0.646997 0.762493i \(-0.723975\pi\)
0.646997 0.762493i \(-0.276025\pi\)
\(44\) 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
\(51\) − 12.0000i − 1.68034i
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 0 0
\(61\) − 2.00000i − 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 0 0
\(69\) − 12.0000i − 1.44463i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) − 2.00000i − 0.230940i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 6.00000i 0.650791i
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000i 0.419314i
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000i 0.597022i 0.954406 + 0.298511i \(0.0964900\pi\)
−0.954406 + 0.298511i \(0.903510\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) − 2.00000i − 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 6.00000i 0.559503i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) − 12.0000i − 1.08200i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 20.0000 1.76090
\(130\) 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 \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) − 4.00000i − 0.339276i −0.985506 0.169638i \(-0.945740\pi\)
0.985506 0.169638i \(-0.0542598\pi\)
\(140\) 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
\(149\) − 6.00000i − 0.491539i −0.969328 0.245770i \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 4.00000i 0.321288i
\(156\) 0 0
\(157\) 22.0000i 1.75579i 0.478852 + 0.877896i \(0.341053\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 0 0
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 10.0000i 0.783260i 0.920123 + 0.391630i \(0.128089\pi\)
−0.920123 + 0.391630i \(0.871911\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) −24.0000 −1.80395
\(178\) 0 0
\(179\) 12.0000i 0.896922i 0.893802 + 0.448461i \(0.148028\pi\)
−0.893802 + 0.448461i \(0.851972\pi\)
\(180\) 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
\(189\) − 8.00000i − 0.581914i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) 0 0
\(195\) − 4.00000i − 0.286446i
\(196\) 0 0
\(197\) 18.0000i 1.28245i 0.767354 + 0.641223i \(0.221573\pi\)
−0.767354 + 0.641223i \(0.778427\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 12.0000i 0.842235i
\(204\) 0 0
\(205\) 6.00000i 0.419058i
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000i 1.10149i 0.834675 + 0.550743i \(0.185655\pi\)
−0.834675 + 0.550743i \(0.814345\pi\)
\(212\) 0 0
\(213\) 24.0000i 1.64445i
\(214\) 0 0
\(215\) −10.0000 −0.681994
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) − 4.00000i − 0.270295i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 6.00000i 0.398234i 0.979976 + 0.199117i \(0.0638074\pi\)
−0.979976 + 0.199117i \(0.936193\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 6.00000i 0.391397i
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 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
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −12.0000 −0.751469
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) − 4.00000i − 0.248548i
\(260\) 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
\(265\) −6.00000 −0.368577
\(266\) 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 \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 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
\(285\) 8.00000i 0.473879i
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 4.00000i 0.234484i
\(292\) 0 0
\(293\) − 30.0000i − 1.75262i −0.481749 0.876309i \(-0.659998\pi\)
0.481749 0.876309i \(-0.340002\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000i 0.693978i
\(300\) 0 0
\(301\) 20.0000i 1.15278i
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) − 2.00000i − 0.114146i −0.998370 0.0570730i \(-0.981823\pi\)
0.998370 0.0570730i \(-0.0181768\pi\)
\(308\) 0 0
\(309\) − 28.0000i − 1.59286i
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) − 2.00000i − 0.112687i
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) 0 0
\(325\) 2.00000i 0.110940i
\(326\) 0 0
\(327\) 4.00000 0.221201
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 8.00000i 0.439720i 0.975531 + 0.219860i \(0.0705600\pi\)
−0.975531 + 0.219860i \(0.929440\pi\)
\(332\) 0 0
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) − 12.0000i − 0.651751i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −12.0000 −0.646058
\(346\) 0 0
\(347\) − 30.0000i − 1.61048i −0.592946 0.805242i \(-0.702035\pi\)
0.592946 0.805242i \(-0.297965\pi\)
\(348\) 0 0
\(349\) 10.0000i 0.535288i 0.963518 + 0.267644i \(0.0862451\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) − 12.0000i − 0.636894i
\(356\) 0 0
\(357\) 24.0000i 1.27021i
\(358\) 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
\(365\) 2.00000i 0.104685i
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000i 0.623009i
\(372\) 0 0
\(373\) 26.0000i 1.34623i 0.739538 + 0.673114i \(0.235044\pi\)
−0.739538 + 0.673114i \(0.764956\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) − 28.0000i − 1.43826i −0.694874 0.719132i \(-0.744540\pi\)
0.694874 0.719132i \(-0.255460\pi\)
\(380\) 0 0
\(381\) 4.00000i 0.204926i
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.0000i 0.508329i
\(388\) 0 0
\(389\) − 6.00000i − 0.304212i −0.988364 0.152106i \(-0.951394\pi\)
0.988364 0.152106i \(-0.0486055\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 8.00000i − 0.402524i
\(396\) 0 0
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 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 \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 0 0
\(411\) − 36.0000i − 1.77575i
\(412\) 0 0
\(413\) − 24.0000i − 1.18096i
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) − 36.0000i − 1.75872i −0.476162 0.879358i \(-0.657972\pi\)
0.476162 0.879358i \(-0.342028\pi\)
\(420\) 0 0
\(421\) 26.0000i 1.26716i 0.773676 + 0.633581i \(0.218416\pi\)
−0.773676 + 0.633581i \(0.781584\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) − 12.0000i − 0.575356i
\(436\) 0 0
\(437\) − 24.0000i − 1.14808i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 6.00000i 0.285069i 0.989790 + 0.142534i \(0.0455251\pi\)
−0.989790 + 0.142534i \(0.954475\pi\)
\(444\) 0 0
\(445\) − 6.00000i − 0.284427i
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 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
\(459\) − 24.0000i − 1.12022i
\(460\) 0 0
\(461\) − 30.0000i − 1.39724i −0.715493 0.698620i \(-0.753798\pi\)
0.715493 0.698620i \(-0.246202\pi\)
\(462\) 0 0
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) 30.0000i 1.38823i 0.719862 + 0.694117i \(0.244205\pi\)
−0.719862 + 0.694117i \(0.755795\pi\)
\(468\) 0 0
\(469\) 4.00000i 0.184703i
\(470\) 0 0
\(471\) −44.0000 −2.02741
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 4.00000i − 0.183533i
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 24.0000i 1.09204i
\(484\) 0 0
\(485\) − 2.00000i − 0.0908153i
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 36.0000i 1.62136i
\(494\) 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 \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) − 36.0000i − 1.60836i
\(502\) 0 0
\(503\) 18.0000 0.802580 0.401290 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 18.0000i 0.799408i
\(508\) 0 0
\(509\) − 6.00000i − 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) −16.0000 −0.706417
\(514\) 0 0
\(515\) 14.0000i 0.616914i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 14.0000i 0.612177i 0.952003 + 0.306089i \(0.0990204\pi\)
−0.952003 + 0.306089i \(0.900980\pi\)
\(524\) 0 0
\(525\) 4.00000i 0.174574i
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) − 12.0000i − 0.520756i
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) −24.0000 −1.03568
\(538\) 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
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 4.00000i 0.169791i
\(556\) 0 0
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.0000i 0.758610i 0.925272 + 0.379305i \(0.123837\pi\)
−0.925272 + 0.379305i \(0.876163\pi\)
\(564\) 0 0
\(565\) 6.00000i 0.252422i
\(566\) 0 0
\(567\) 22.0000 0.923913
\(568\) 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
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 0 0
\(579\) 52.0000i 2.16105i
\(580\) 0 0
\(581\) 12.0000i 0.497844i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) − 6.00000i − 0.247647i −0.992304 0.123823i \(-0.960484\pi\)
0.992304 0.123823i \(-0.0395156\pi\)
\(588\) 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 \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) − 12.0000i − 0.491952i
\(596\) 0 0
\(597\) − 16.0000i − 0.654836i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 2.00000i 0.0814463i
\(604\) 0 0
\(605\) − 11.0000i − 0.447214i
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 12.0000i 0.485468i
\(612\) 0 0
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 0 0
\(621\) − 24.0000i − 0.963087i
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 12.0000i − 0.478471i
\(630\) 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
\(637\) 6.00000i 0.237729i
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 0 0
\(645\) − 20.0000i − 0.787499i
\(646\) 0 0
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 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 \(-0.692974\pi\)
0.569785 0.821794i \(-0.307026\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) − 36.0000i − 1.40236i −0.712984 0.701180i \(-0.752657\pi\)
0.712984 0.701180i \(-0.247343\pi\)
\(660\) 0 0
\(661\) − 22.0000i − 0.855701i −0.903850 0.427850i \(-0.859271\pi\)
0.903850 0.427850i \(-0.140729\pi\)
\(662\) 0 0
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 36.0000i 1.39393i
\(668\) 0 0
\(669\) − 20.0000i − 0.773245i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 0 0
\(675\) − 4.00000i − 0.153960i
\(676\) 0 0
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 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 \(-0.702946\pi\)
0.595247 0.803543i \(-0.297054\pi\)
\(684\) 0 0
\(685\) 18.0000i 0.687745i
\(686\) 0 0
\(687\) −28.0000 −1.06827
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) 12.0000i 0.453882i
\(700\) 0 0
\(701\) 30.0000i 1.13308i 0.824033 + 0.566542i \(0.191719\pi\)
−0.824033 + 0.566542i \(0.808281\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 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 \(-0.779577\pi\)
0.769665 0.638448i \(-0.220423\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 48.0000i − 1.79259i
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 28.0000i 1.04133i
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) 46.0000 1.70605 0.853023 0.521874i \(-0.174767\pi\)
0.853023 + 0.521874i \(0.174767\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 60.0000i 2.21918i
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 20.0000i − 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) 0 0
\(741\) 16.0000i 0.587775i
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 20.0000i 0.727875i
\(756\) 0 0
\(757\) − 22.0000i − 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 4.00000i 0.144810i
\(764\) 0 0
\(765\) − 6.00000i − 0.216930i
\(766\) 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
\(773\) − 30.0000i − 1.07903i −0.841978 0.539513i \(-0.818609\pi\)
0.841978 0.539513i \(-0.181391\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 0 0
\(779\) − 24.0000i − 0.859889i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) 22.0000 0.785214
\(786\) 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
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) − 12.0000i − 0.425596i
\(796\) 0 0
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 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
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) − 16.0000i − 0.561836i −0.959732 0.280918i \(-0.909361\pi\)
0.959732 0.280918i \(-0.0906389\pi\)
\(812\) 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 \(-0.608652\pi\)
0.334751 0.942306i \(-0.391348\pi\)
\(822\) 0 0
\(823\) −38.0000 −1.32460 −0.662298 0.749240i \(-0.730419\pi\)
−0.662298 + 0.749240i \(0.730419\pi\)
\(824\) 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
\(829\) − 2.00000i − 0.0694629i −0.999397 0.0347314i \(-0.988942\pi\)
0.999397 0.0347314i \(-0.0110576\pi\)
\(830\) 0 0
\(831\) −52.0000 −1.80386
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 18.0000i 0.622916i
\(836\) 0 0
\(837\) − 16.0000i − 0.553041i
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 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
\(847\) −22.0000 −0.755929
\(848\) 0 0
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) − 12.0000i − 0.411355i
\(852\) 0 0
\(853\) 50.0000i 1.71197i 0.517003 + 0.855984i \(0.327048\pi\)
−0.517003 + 0.855984i \(0.672952\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) − 4.00000i − 0.136478i −0.997669 0.0682391i \(-0.978262\pi\)
0.997669 0.0682391i \(-0.0217381\pi\)
\(860\) 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
\(867\) 38.0000i 1.29055i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) − 2.00000i − 0.0676123i
\(876\) 0 0
\(877\) − 26.0000i − 0.877958i −0.898497 0.438979i \(-0.855340\pi\)
0.898497 0.438979i \(-0.144660\pi\)
\(878\) 0 0
\(879\) 60.0000 2.02375
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) − 14.0000i − 0.471138i −0.971858 0.235569i \(-0.924305\pi\)
0.971858 0.235569i \(-0.0756953\pi\)
\(884\) 0 0
\(885\) 24.0000i 0.806751i
\(886\) 0 0
\(887\) −18.0000 −0.604381 −0.302190 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 24.0000i − 0.803129i
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) 24.0000i 0.800445i
\(900\) 0 0
\(901\) 36.0000i 1.19933i
\(902\) 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
\(909\) − 6.00000i − 0.199007i
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 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 \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 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
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) − 12.0000i − 0.393284i
\(932\) 0 0
\(933\) − 24.0000i − 0.785725i
\(934\) 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
\(939\) 44.0000i 1.43589i
\(940\) 0 0
\(941\) 18.0000i 0.586783i 0.955992 + 0.293392i \(0.0947840\pi\)
−0.955992 + 0.293392i \(0.905216\pi\)
\(942\) 0 0
\(943\) 36.0000 1.17232
\(944\) 0 0
\(945\) −8.00000 −0.260240
\(946\) 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 \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 12.0000i 0.388311i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 6.00000i 0.193347i
\(964\) 0 0
\(965\) − 26.0000i − 0.836970i
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 48.0000 1.54198
\(970\) 0 0
\(971\) − 24.0000i − 0.770197i −0.922876 0.385098i \(-0.874168\pi\)
0.922876 0.385098i \(-0.125832\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000i 0.0638551i
\(982\) 0 0
\(983\) 18.0000 0.574111 0.287055 0.957914i \(-0.407324\pi\)
0.287055 + 0.957914i \(0.407324\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 24.0000i 0.763928i
\(988\) 0 0
\(989\) 60.0000i 1.90789i
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) 0 0
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) 8.00000i 0.253617i
\(996\) 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 1280.2.d.c.641.2 2
4.3 odd 2 1280.2.d.g.641.1 2
8.3 odd 2 1280.2.d.g.641.2 2
8.5 even 2 inner 1280.2.d.c.641.1 2
16.3 odd 4 80.2.a.b.1.1 1
16.5 even 4 320.2.a.f.1.1 1
16.11 odd 4 320.2.a.a.1.1 1
16.13 even 4 20.2.a.a.1.1 1
48.5 odd 4 2880.2.a.m.1.1 1
48.11 even 4 2880.2.a.f.1.1 1
48.29 odd 4 180.2.a.a.1.1 1
48.35 even 4 720.2.a.h.1.1 1
80.3 even 4 400.2.c.b.49.2 2
80.13 odd 4 100.2.c.a.49.1 2
80.19 odd 4 400.2.a.c.1.1 1
80.27 even 4 1600.2.c.e.449.2 2
80.29 even 4 100.2.a.a.1.1 1
80.37 odd 4 1600.2.c.d.449.1 2
80.43 even 4 1600.2.c.e.449.1 2
80.53 odd 4 1600.2.c.d.449.2 2
80.59 odd 4 1600.2.a.w.1.1 1
80.67 even 4 400.2.c.b.49.1 2
80.69 even 4 1600.2.a.c.1.1 1
80.77 odd 4 100.2.c.a.49.2 2
112.13 odd 4 980.2.a.h.1.1 1
112.45 odd 12 980.2.i.c.961.1 2
112.61 odd 12 980.2.i.c.361.1 2
112.83 even 4 3920.2.a.h.1.1 1
112.93 even 12 980.2.i.i.361.1 2
112.109 even 12 980.2.i.i.961.1 2
144.13 even 12 1620.2.i.h.541.1 2
144.29 odd 12 1620.2.i.b.1081.1 2
144.61 even 12 1620.2.i.h.1081.1 2
144.77 odd 12 1620.2.i.b.541.1 2
176.109 odd 4 2420.2.a.a.1.1 1
176.131 even 4 9680.2.a.ba.1.1 1
208.77 even 4 3380.2.a.c.1.1 1
208.109 odd 4 3380.2.f.b.3041.2 2
208.125 odd 4 3380.2.f.b.3041.1 2
240.29 odd 4 900.2.a.b.1.1 1
240.77 even 4 900.2.d.c.649.2 2
240.83 odd 4 3600.2.f.j.2449.2 2
240.173 even 4 900.2.d.c.649.1 2
240.179 even 4 3600.2.a.be.1.1 1
240.227 odd 4 3600.2.f.j.2449.1 2
272.13 even 4 5780.2.c.a.5201.1 2
272.157 even 4 5780.2.c.a.5201.2 2
272.237 even 4 5780.2.a.f.1.1 1
304.189 odd 4 7220.2.a.f.1.1 1
336.125 even 4 8820.2.a.g.1.1 1
560.13 even 4 4900.2.e.f.2549.2 2
560.237 even 4 4900.2.e.f.2549.1 2
560.349 odd 4 4900.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.2.a.a.1.1 1 16.13 even 4
80.2.a.b.1.1 1 16.3 odd 4
100.2.a.a.1.1 1 80.29 even 4
100.2.c.a.49.1 2 80.13 odd 4
100.2.c.a.49.2 2 80.77 odd 4
180.2.a.a.1.1 1 48.29 odd 4
320.2.a.a.1.1 1 16.11 odd 4
320.2.a.f.1.1 1 16.5 even 4
400.2.a.c.1.1 1 80.19 odd 4
400.2.c.b.49.1 2 80.67 even 4
400.2.c.b.49.2 2 80.3 even 4
720.2.a.h.1.1 1 48.35 even 4
900.2.a.b.1.1 1 240.29 odd 4
900.2.d.c.649.1 2 240.173 even 4
900.2.d.c.649.2 2 240.77 even 4
980.2.a.h.1.1 1 112.13 odd 4
980.2.i.c.361.1 2 112.61 odd 12
980.2.i.c.961.1 2 112.45 odd 12
980.2.i.i.361.1 2 112.93 even 12
980.2.i.i.961.1 2 112.109 even 12
1280.2.d.c.641.1 2 8.5 even 2 inner
1280.2.d.c.641.2 2 1.1 even 1 trivial
1280.2.d.g.641.1 2 4.3 odd 2
1280.2.d.g.641.2 2 8.3 odd 2
1600.2.a.c.1.1 1 80.69 even 4
1600.2.a.w.1.1 1 80.59 odd 4
1600.2.c.d.449.1 2 80.37 odd 4
1600.2.c.d.449.2 2 80.53 odd 4
1600.2.c.e.449.1 2 80.43 even 4
1600.2.c.e.449.2 2 80.27 even 4
1620.2.i.b.541.1 2 144.77 odd 12
1620.2.i.b.1081.1 2 144.29 odd 12
1620.2.i.h.541.1 2 144.13 even 12
1620.2.i.h.1081.1 2 144.61 even 12
2420.2.a.a.1.1 1 176.109 odd 4
2880.2.a.f.1.1 1 48.11 even 4
2880.2.a.m.1.1 1 48.5 odd 4
3380.2.a.c.1.1 1 208.77 even 4
3380.2.f.b.3041.1 2 208.125 odd 4
3380.2.f.b.3041.2 2 208.109 odd 4
3600.2.a.be.1.1 1 240.179 even 4
3600.2.f.j.2449.1 2 240.227 odd 4
3600.2.f.j.2449.2 2 240.83 odd 4
3920.2.a.h.1.1 1 112.83 even 4
4900.2.a.e.1.1 1 560.349 odd 4
4900.2.e.f.2549.1 2 560.237 even 4
4900.2.e.f.2549.2 2 560.13 even 4
5780.2.a.f.1.1 1 272.237 even 4
5780.2.c.a.5201.1 2 272.13 even 4
5780.2.c.a.5201.2 2 272.157 even 4
7220.2.a.f.1.1 1 304.189 odd 4
8820.2.a.g.1.1 1 336.125 even 4
9680.2.a.ba.1.1 1 176.131 even 4