Properties

Label 3850.2.c.j.1849.1
Level $3850$
Weight $2$
Character 3850.1849
Analytic conductor $30.742$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3850.1849
Dual form 3850.2.c.j.1849.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +3.00000 q^{9} -1.00000 q^{11} -2.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.00000i q^{17} -3.00000i q^{18} +6.00000 q^{19} +1.00000i q^{22} -4.00000i q^{23} -2.00000 q^{26} +1.00000i q^{28} +2.00000 q^{29} -2.00000 q^{31} -1.00000i q^{32} -4.00000 q^{34} -3.00000 q^{36} +10.0000i q^{37} -6.00000i q^{38} +4.00000 q^{41} +8.00000i q^{43} +1.00000 q^{44} -4.00000 q^{46} +2.00000i q^{47} -1.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +1.00000 q^{56} -2.00000i q^{58} +12.0000 q^{59} -14.0000 q^{61} +2.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} +4.00000i q^{68} -8.00000 q^{71} +3.00000i q^{72} -4.00000i q^{73} +10.0000 q^{74} -6.00000 q^{76} +1.00000i q^{77} +9.00000 q^{81} -4.00000i q^{82} +6.00000i q^{83} +8.00000 q^{86} -1.00000i q^{88} +6.00000 q^{89} -2.00000 q^{91} +4.00000i q^{92} +2.00000 q^{94} -14.0000i q^{97} +1.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 6 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} + 6 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 12 q^{19} - 4 q^{26} + 4 q^{29} - 4 q^{31} - 8 q^{34} - 6 q^{36} + 8 q^{41} + 2 q^{44} - 8 q^{46} - 2 q^{49} + 2 q^{56} + 24 q^{59} - 28 q^{61} - 2 q^{64} - 16 q^{71} + 20 q^{74} - 12 q^{76} + 18 q^{81} + 16 q^{86} + 12 q^{89} - 4 q^{91} + 4 q^{94} - 6 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times\).

\(n\) \(1751\) \(2201\) \(2927\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 0.485071i \(-0.838794\pi\)
0.874475 0.485071i \(-0.161206\pi\)
\(18\) − 3.00000i − 0.707107i
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000i 0.213201i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) 10.0000i 1.64399i 0.569495 + 0.821995i \(0.307139\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) − 6.00000i − 0.973329i
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) − 2.00000i − 0.262613i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 2.00000i 0.254000i
\(63\) − 3.00000i − 0.377964i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 4.00000i 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 3.00000i 0.353553i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) − 4.00000i − 0.441726i
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) − 1.00000i − 0.106600i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 4.00000i 0.417029i
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) 1.00000i 0.101015i
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) − 18.0000i − 1.77359i −0.462160 0.886796i \(-0.652926\pi\)
0.462160 0.886796i \(-0.347074\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 16.0000i − 1.54678i −0.633932 0.773389i \(-0.718560\pi\)
0.633932 0.773389i \(-0.281440\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) − 6.00000i − 0.554700i
\(118\) − 12.0000i − 1.10469i
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 14.0000i 1.26750i
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) −3.00000 −0.267261
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) − 6.00000i − 0.520266i
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) 2.00000i 0.167248i
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −4.00000 −0.331042
\(147\) 0 0
\(148\) − 10.0000i − 0.821995i
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 6.00000i 0.486664i
\(153\) − 12.0000i − 0.970143i
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) − 8.00000i − 0.638470i −0.947676 0.319235i \(-0.896574\pi\)
0.947676 0.319235i \(-0.103426\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) − 9.00000i − 0.707107i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 4.00000i 0.309529i 0.987951 + 0.154765i \(0.0494619\pi\)
−0.987951 + 0.154765i \(0.950538\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 18.0000 1.37649
\(172\) − 8.00000i − 0.609994i
\(173\) 14.0000i 1.06440i 0.846619 + 0.532200i \(0.178635\pi\)
−0.846619 + 0.532200i \(0.821365\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) − 6.00000i − 0.449719i
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 2.00000i 0.148250i
\(183\) 0 0
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) − 2.00000i − 0.145865i
\(189\) 0 0
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 0 0
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 3.00000i 0.213201i
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 6.00000i − 0.422159i
\(203\) − 2.00000i − 0.140372i
\(204\) 0 0
\(205\) 0 0
\(206\) −18.0000 −1.25412
\(207\) − 12.0000i − 0.834058i
\(208\) − 2.00000i − 0.138675i
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) −16.0000 −1.09374
\(215\) 0 0
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) − 14.0000i − 0.948200i
\(219\) 0 0
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 2.00000i 0.133930i 0.997755 + 0.0669650i \(0.0213316\pi\)
−0.997755 + 0.0669650i \(0.978668\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) − 2.00000i − 0.132745i −0.997795 0.0663723i \(-0.978857\pi\)
0.997795 0.0663723i \(-0.0211425\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) − 30.0000i − 1.96537i −0.185296 0.982683i \(-0.559325\pi\)
0.185296 0.982683i \(-0.440675\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 4.00000i 0.259281i
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) − 1.00000i − 0.0642824i
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) − 12.0000i − 0.763542i
\(248\) − 2.00000i − 0.127000i
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 3.00000i 0.188982i
\(253\) 4.00000i 0.251478i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) − 6.00000i − 0.370681i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) 12.0000i 0.733017i
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) − 4.00000i − 0.242536i
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 30.0000i − 1.80253i −0.433273 0.901263i \(-0.642641\pi\)
0.433273 0.901263i \(-0.357359\pi\)
\(278\) 14.0000i 0.839664i
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) − 6.00000i − 0.356663i −0.983970 0.178331i \(-0.942930\pi\)
0.983970 0.178331i \(-0.0570699\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) − 4.00000i − 0.236113i
\(288\) − 3.00000i − 0.176777i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 4.00000i 0.234082i
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −10.0000 −0.581238
\(297\) 0 0
\(298\) 2.00000i 0.115857i
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 24.0000i 1.38104i
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) −12.0000 −0.685994
\(307\) − 10.0000i − 0.570730i −0.958419 0.285365i \(-0.907885\pi\)
0.958419 0.285365i \(-0.0921148\pi\)
\(308\) − 1.00000i − 0.0569803i
\(309\) 0 0
\(310\) 0 0
\(311\) 14.0000 0.793867 0.396934 0.917847i \(-0.370074\pi\)
0.396934 + 0.917847i \(0.370074\pi\)
\(312\) 0 0
\(313\) 2.00000i 0.113047i 0.998401 + 0.0565233i \(0.0180015\pi\)
−0.998401 + 0.0565233i \(0.981998\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 0 0
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) 0 0
\(322\) 4.00000i 0.222911i
\(323\) − 24.0000i − 1.33540i
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 4.00000i 0.220863i
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) − 6.00000i − 0.329293i
\(333\) 30.0000i 1.64399i
\(334\) 4.00000 0.218870
\(335\) 0 0
\(336\) 0 0
\(337\) − 18.0000i − 0.980522i −0.871576 0.490261i \(-0.836901\pi\)
0.871576 0.490261i \(-0.163099\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) 0 0
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) − 18.0000i − 0.973329i
\(343\) 1.00000i 0.0539949i
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 8.00000i 0.429463i 0.976673 + 0.214731i \(0.0688876\pi\)
−0.976673 + 0.214731i \(0.931112\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000i 0.0533002i
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 4.00000i 0.211407i
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) − 20.0000i − 1.05118i
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 22.0000i 1.14839i 0.818718 + 0.574195i \(0.194685\pi\)
−0.818718 + 0.574195i \(0.805315\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) − 4.00000i − 0.206010i
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.00000i 0.204658i
\(383\) 10.0000i 0.510976i 0.966812 + 0.255488i \(0.0822362\pi\)
−0.966812 + 0.255488i \(0.917764\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 24.0000i 1.21999i
\(388\) 14.0000i 0.710742i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) − 1.00000i − 0.0505076i
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 24.0000i 1.20453i 0.798298 + 0.602263i \(0.205734\pi\)
−0.798298 + 0.602263i \(0.794266\pi\)
\(398\) − 14.0000i − 0.701757i
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) − 10.0000i − 0.495682i
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 18.0000i 0.886796i
\(413\) − 12.0000i − 0.590481i
\(414\) −12.0000 −0.589768
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 6.00000i 0.293470i
\(419\) 32.0000 1.56330 0.781651 0.623716i \(-0.214378\pi\)
0.781651 + 0.623716i \(0.214378\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 8.00000i 0.389434i
\(423\) 6.00000i 0.291730i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 14.0000i 0.677507i
\(428\) 16.0000i 0.773389i
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) − 24.0000i − 1.14808i
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 8.00000i 0.380521i
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.00000 0.0947027
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 14.0000i 0.658505i
\(453\) 0 0
\(454\) −2.00000 −0.0938647
\(455\) 0 0
\(456\) 0 0
\(457\) 2.00000i 0.0935561i 0.998905 + 0.0467780i \(0.0148953\pi\)
−0.998905 + 0.0467780i \(0.985105\pi\)
\(458\) 20.0000i 0.934539i
\(459\) 0 0
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 32.0000i 1.48717i 0.668644 + 0.743583i \(0.266875\pi\)
−0.668644 + 0.743583i \(0.733125\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −30.0000 −1.38972
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 6.00000i 0.277350i
\(469\) −12.0000 −0.554109
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000i 0.552345i
\(473\) − 8.00000i − 0.367840i
\(474\) 0 0
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) − 18.0000i − 0.824163i
\(478\) 16.0000i 0.731823i
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) − 12.0000i − 0.546585i
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) − 28.0000i − 1.26880i −0.773004 0.634401i \(-0.781247\pi\)
0.773004 0.634401i \(-0.218753\pi\)
\(488\) − 14.0000i − 0.633750i
\(489\) 0 0
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) − 8.00000i − 0.360302i
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 8.00000i 0.358849i
\(498\) 0 0
\(499\) −44.0000 −1.96971 −0.984855 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 12.0000i − 0.535586i
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) − 8.00000i − 0.354943i
\(509\) 28.0000 1.24108 0.620539 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) − 1.00000i − 0.0441942i
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.00000i − 0.0879599i
\(518\) − 10.0000i − 0.439375i
\(519\) 0 0
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) 34.0000i 1.48672i 0.668894 + 0.743358i \(0.266768\pi\)
−0.668894 + 0.743358i \(0.733232\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 36.0000 1.56227
\(532\) 6.00000i 0.260133i
\(533\) − 8.00000i − 0.346518i
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 12.0000i 0.517357i
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 20.0000i 0.859074i
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) − 12.0000i − 0.513083i −0.966533 0.256541i \(-0.917417\pi\)
0.966533 0.256541i \(-0.0825830\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −42.0000 −1.79252
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) −30.0000 −1.27458
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) 6.00000i 0.254000i
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000i 0.421825i
\(563\) 34.0000i 1.43293i 0.697623 + 0.716465i \(0.254241\pi\)
−0.697623 + 0.716465i \(0.745759\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.00000 −0.252199
\(567\) − 9.00000i − 0.377964i
\(568\) − 8.00000i − 0.335673i
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) − 2.00000i − 0.0836242i
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) −3.00000 −0.125000
\(577\) 14.0000i 0.582828i 0.956597 + 0.291414i \(0.0941257\pi\)
−0.956597 + 0.291414i \(0.905874\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) 0 0
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 6.00000i 0.248495i
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) 10.0000i 0.410997i
\(593\) 12.0000i 0.492781i 0.969171 + 0.246390i \(0.0792446\pi\)
−0.969171 + 0.246390i \(0.920755\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) 0 0
\(598\) 8.00000i 0.327144i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) − 8.00000i − 0.326056i
\(603\) − 36.0000i − 1.46603i
\(604\) 24.0000 0.976546
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) − 6.00000i − 0.243332i
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000 0.161823
\(612\) 12.0000i 0.485071i
\(613\) − 46.0000i − 1.85792i −0.370177 0.928961i \(-0.620703\pi\)
0.370177 0.928961i \(-0.379297\pi\)
\(614\) −10.0000 −0.403567
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 14.0000i − 0.561349i
\(623\) − 6.00000i − 0.240385i
\(624\) 0 0
\(625\) 0 0
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) 8.00000i 0.319235i
\(629\) 40.0000 1.59490
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 2.00000i 0.0791808i
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) − 4.00000i − 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) 9.00000i 0.353553i
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) − 10.0000i − 0.391330i −0.980671 0.195665i \(-0.937313\pi\)
0.980671 0.195665i \(-0.0626866\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.00000 0.156174
\(657\) − 12.0000i − 0.468165i
\(658\) − 2.00000i − 0.0779681i
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) 20.0000 0.777910 0.388955 0.921257i \(-0.372836\pi\)
0.388955 + 0.921257i \(0.372836\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) 30.0000 1.16248
\(667\) − 8.00000i − 0.309761i
\(668\) − 4.00000i − 0.154765i
\(669\) 0 0
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) − 22.0000i − 0.848038i −0.905653 0.424019i \(-0.860619\pi\)
0.905653 0.424019i \(-0.139381\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 26.0000i 0.999261i 0.866239 + 0.499631i \(0.166531\pi\)
−0.866239 + 0.499631i \(0.833469\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) − 2.00000i − 0.0765840i
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) −18.0000 −0.688247
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 8.00000i 0.304997i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) 3.00000i 0.113961i
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) 0 0
\(697\) − 16.0000i − 0.606043i
\(698\) − 10.0000i − 0.378506i
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 60.0000i 2.26294i
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) −18.0000 −0.676004 −0.338002 0.941145i \(-0.609751\pi\)
−0.338002 + 0.941145i \(0.609751\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000i 0.224860i
\(713\) 8.00000i 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) − 16.0000i − 0.597115i
\(719\) 26.0000 0.969636 0.484818 0.874615i \(-0.338886\pi\)
0.484818 + 0.874615i \(0.338886\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) − 17.0000i − 0.632674i
\(723\) 0 0
\(724\) −20.0000 −0.743294
\(725\) 0 0
\(726\) 0 0
\(727\) − 10.0000i − 0.370879i −0.982656 0.185440i \(-0.940629\pi\)
0.982656 0.185440i \(-0.0593710\pi\)
\(728\) − 2.00000i − 0.0741249i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 12.0000i 0.442026i
\(738\) − 12.0000i − 0.441726i
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.00000i 0.220267i
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 10.0000 0.366126
\(747\) 18.0000i 0.658586i
\(748\) − 4.00000i − 0.146254i
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) 2.00000i 0.0729325i
\(753\) 0 0
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 0 0
\(757\) 26.0000i 0.944986i 0.881334 + 0.472493i \(0.156646\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) 4.00000i 0.145287i
\(759\) 0 0
\(760\) 0 0
\(761\) −48.0000 −1.74000 −0.869999 0.493053i \(-0.835881\pi\)
−0.869999 + 0.493053i \(0.835881\pi\)
\(762\) 0 0
\(763\) − 14.0000i − 0.506834i
\(764\) 4.00000 0.144715
\(765\) 0 0
\(766\) 10.0000 0.361315
\(767\) − 24.0000i − 0.866590i
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.00000i 0.0719816i
\(773\) 48.0000i 1.72644i 0.504828 + 0.863220i \(0.331556\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(774\) 24.0000 0.862662
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) − 30.0000i − 1.07555i
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 16.0000i 0.572159i
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 0 0
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) − 3.00000i − 0.106600i
\(793\) 28.0000i 0.994309i
\(794\) 24.0000 0.851728
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) − 16.0000i − 0.566749i −0.959009 0.283375i \(-0.908546\pi\)
0.959009 0.283375i \(-0.0914540\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 18.0000i 0.635602i
\(803\) 4.00000i 0.141157i
\(804\) 0 0
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) 6.00000i 0.211079i
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 2.00000i 0.0701862i
\(813\) 0 0
\(814\) −10.0000 −0.350500
\(815\) 0 0
\(816\) 0 0
\(817\) 48.0000i 1.67931i
\(818\) 16.0000i 0.559427i
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 18.0000 0.627060
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 20.0000i 0.695468i 0.937593 + 0.347734i \(0.113049\pi\)
−0.937593 + 0.347734i \(0.886951\pi\)
\(828\) 12.0000i 0.417029i
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.00000i 0.0693375i
\(833\) 4.00000i 0.138592i
\(834\) 0 0
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) − 32.0000i − 1.10542i
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 2.00000i 0.0689246i
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) − 1.00000i − 0.0343604i
\(848\) − 6.00000i − 0.206041i
\(849\) 0 0
\(850\) 0 0
\(851\) 40.0000 1.37118
\(852\) 0 0
\(853\) − 2.00000i − 0.0684787i −0.999414 0.0342393i \(-0.989099\pi\)
0.999414 0.0342393i \(-0.0109009\pi\)
\(854\) 14.0000 0.479070
\(855\) 0 0
\(856\) 16.0000 0.546869
\(857\) 32.0000i 1.09310i 0.837427 + 0.546550i \(0.184059\pi\)
−0.837427 + 0.546550i \(0.815941\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 16.0000i − 0.544962i
\(863\) 44.0000i 1.49778i 0.662696 + 0.748889i \(0.269412\pi\)
−0.662696 + 0.748889i \(0.730588\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) − 2.00000i − 0.0678844i
\(869\) 0 0
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 14.0000i 0.474100i
\(873\) − 42.0000i − 1.42148i
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0000i 1.14810i 0.818821 + 0.574049i \(0.194628\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) 28.0000i 0.944954i
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 44.0000i 1.48072i 0.672212 + 0.740359i \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) 16.0000i 0.537227i 0.963248 + 0.268614i \(0.0865655\pi\)
−0.963248 + 0.268614i \(0.913434\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) − 2.00000i − 0.0669650i
\(893\) 12.0000i 0.401565i
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) − 18.0000i − 0.600668i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 4.00000i 0.133185i
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) − 52.0000i − 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) 2.00000i 0.0663723i
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) − 6.00000i − 0.198571i
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) 20.0000 0.660819
\(917\) − 6.00000i − 0.198137i
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 30.0000i − 0.987997i
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) 32.0000 1.05159
\(927\) − 54.0000i − 1.77359i
\(928\) − 2.00000i − 0.0656532i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 30.0000i 0.982683i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 12.0000i 0.392023i 0.980602 + 0.196011i \(0.0627990\pi\)
−0.980602 + 0.196011i \(0.937201\pi\)
\(938\) 12.0000i 0.391814i
\(939\) 0 0
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 0 0
\(943\) − 16.0000i − 0.521032i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) − 4.00000i − 0.129983i −0.997886 0.0649913i \(-0.979298\pi\)
0.997886 0.0649913i \(-0.0207020\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 0 0
\(952\) − 4.00000i − 0.129641i
\(953\) 22.0000i 0.712650i 0.934362 + 0.356325i \(0.115970\pi\)
−0.934362 + 0.356325i \(0.884030\pi\)
\(954\) −18.0000 −0.582772
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) − 16.0000i − 0.516937i
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) − 20.0000i − 0.644826i
\(963\) − 48.0000i − 1.54678i
\(964\) −12.0000 −0.386494
\(965\) 0 0
\(966\) 0 0
\(967\) 32.0000i 1.02905i 0.857475 + 0.514525i \(0.172032\pi\)
−0.857475 + 0.514525i \(0.827968\pi\)
\(968\) 1.00000i 0.0321412i
\(969\) 0 0
\(970\) 0 0
\(971\) 56.0000 1.79713 0.898563 0.438845i \(-0.144612\pi\)
0.898563 + 0.438845i \(0.144612\pi\)
\(972\) 0 0
\(973\) 14.0000i 0.448819i
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 2.00000i 0.0639857i 0.999488 + 0.0319928i \(0.0101854\pi\)
−0.999488 + 0.0319928i \(0.989815\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 42.0000 1.34096
\(982\) 36.0000i 1.14881i
\(983\) − 18.0000i − 0.574111i −0.957914 0.287055i \(-0.907324\pi\)
0.957914 0.287055i \(-0.0926764\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 12.0000i 0.381771i
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 2.00000i 0.0635001i
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) − 42.0000i − 1.33015i −0.746775 0.665077i \(-0.768399\pi\)
0.746775 0.665077i \(-0.231601\pi\)
\(998\) 44.0000i 1.39280i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3850.2.c.j.1849.1 2
5.2 odd 4 3850.2.a.u.1.1 1
5.3 odd 4 154.2.a.a.1.1 1
5.4 even 2 inner 3850.2.c.j.1849.2 2
15.8 even 4 1386.2.a.l.1.1 1
20.3 even 4 1232.2.a.e.1.1 1
35.3 even 12 1078.2.e.i.177.1 2
35.13 even 4 1078.2.a.d.1.1 1
35.18 odd 12 1078.2.e.j.177.1 2
35.23 odd 12 1078.2.e.j.67.1 2
35.33 even 12 1078.2.e.i.67.1 2
40.3 even 4 4928.2.a.w.1.1 1
40.13 odd 4 4928.2.a.v.1.1 1
55.43 even 4 1694.2.a.g.1.1 1
105.83 odd 4 9702.2.a.ba.1.1 1
140.83 odd 4 8624.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.a.1.1 1 5.3 odd 4
1078.2.a.d.1.1 1 35.13 even 4
1078.2.e.i.67.1 2 35.33 even 12
1078.2.e.i.177.1 2 35.3 even 12
1078.2.e.j.67.1 2 35.23 odd 12
1078.2.e.j.177.1 2 35.18 odd 12
1232.2.a.e.1.1 1 20.3 even 4
1386.2.a.l.1.1 1 15.8 even 4
1694.2.a.g.1.1 1 55.43 even 4
3850.2.a.u.1.1 1 5.2 odd 4
3850.2.c.j.1849.1 2 1.1 even 1 trivial
3850.2.c.j.1849.2 2 5.4 even 2 inner
4928.2.a.v.1.1 1 40.13 odd 4
4928.2.a.w.1.1 1 40.3 even 4
8624.2.a.r.1.1 1 140.83 odd 4
9702.2.a.ba.1.1 1 105.83 odd 4