Properties

Label 3850.2.c.p.1849.2
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 770)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +2.00000i q^{12} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +2.00000 q^{21} +1.00000i q^{22} -4.00000i q^{23} -2.00000 q^{24} -4.00000i q^{27} -1.00000i q^{28} -2.00000 q^{29} -2.00000 q^{31} +1.00000i q^{32} -2.00000i q^{33} +1.00000 q^{36} +6.00000i q^{37} +8.00000 q^{41} +2.00000i q^{42} -12.0000i q^{43} -1.00000 q^{44} +4.00000 q^{46} +6.00000i q^{47} -2.00000i q^{48} -1.00000 q^{49} -6.00000i q^{53} +4.00000 q^{54} +1.00000 q^{56} -2.00000i q^{58} +10.0000 q^{59} -4.00000 q^{61} -2.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +8.00000i q^{67} -8.00000 q^{69} -4.00000 q^{71} +1.00000i q^{72} -4.00000i q^{73} -6.00000 q^{74} +1.00000i q^{77} +16.0000 q^{79} -11.0000 q^{81} +8.00000i q^{82} -2.00000 q^{84} +12.0000 q^{86} +4.00000i q^{87} -1.00000i q^{88} +6.00000 q^{89} +4.00000i q^{92} +4.00000i q^{93} -6.00000 q^{94} +2.00000 q^{96} -14.0000i q^{97} -1.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} + 2q^{11} - 2q^{14} + 2q^{16} + 4q^{21} - 4q^{24} - 4q^{29} - 4q^{31} + 2q^{36} + 16q^{41} - 2q^{44} + 8q^{46} - 2q^{49} + 8q^{54} + 2q^{56} + 20q^{59} - 8q^{61} - 2q^{64} + 4q^{66} - 16q^{69} - 8q^{71} - 12q^{74} + 32q^{79} - 22q^{81} - 4q^{84} + 24q^{86} + 12q^{89} - 12q^{94} + 4q^{96} - 2q^{99} + O(q^{100}) \)

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
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 1.00000i 0.377964i
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 2.00000i 0.577350i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 1.00000i 0.213201i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) − 1.00000i − 0.188982i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 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
\(33\) − 2.00000i − 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 2.00000i 0.308607i
\(43\) − 12.0000i − 1.82998i −0.403473 0.914991i \(-0.632197\pi\)
0.403473 0.914991i \(-0.367803\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) − 2.00000i − 0.262613i
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) − 2.00000i − 0.254000i
\(63\) − 1.00000i − 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 0 0
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 8.00000i 0.883452i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 4.00000i 0.428845i
\(88\) − 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
\(91\) 0 0
\(92\) 4.00000i 0.417029i
\(93\) 4.00000i 0.414781i
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) − 14.0000i − 1.42148i −0.703452 0.710742i \(-0.748359\pi\)
0.703452 0.710742i \(-0.251641\pi\)
\(98\) − 1.00000i − 0.101015i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) 1.00000i 0.0944911i
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 10.0000i 0.920575i
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 4.00000i − 0.362143i
\(123\) − 16.0000i − 1.44267i
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) − 8.00000i − 0.709885i −0.934888 0.354943i \(-0.884500\pi\)
0.934888 0.354943i \(-0.115500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 2.00000i 0.174078i
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) − 8.00000i − 0.681005i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) − 4.00000i − 0.335673i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) 2.00000i 0.164957i
\(148\) − 6.00000i − 0.493197i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) − 6.00000i − 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) 16.0000i 1.27289i
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) − 11.0000i − 0.864242i
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 0 0
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) − 2.00000i − 0.154303i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 12.0000i 0.914991i
\(173\) 24.0000i 1.82469i 0.409426 + 0.912343i \(0.365729\pi\)
−0.409426 + 0.912343i \(0.634271\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) − 20.0000i − 1.50329i
\(178\) 6.00000i 0.449719i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 8.00000i 0.591377i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) − 6.00000i − 0.437595i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 2.00000i 0.144338i
\(193\) − 22.0000i − 1.58359i −0.610784 0.791797i \(-0.709146\pi\)
0.610784 0.791797i \(-0.290854\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 14.0000i − 0.997459i −0.866758 0.498729i \(-0.833800\pi\)
0.866758 0.498729i \(-0.166200\pi\)
\(198\) − 1.00000i − 0.0710669i
\(199\) 6.00000 0.425329 0.212664 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) − 2.00000i − 0.140372i
\(204\) 0 0
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 8.00000i 0.548151i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) − 2.00000i − 0.135769i
\(218\) 6.00000i 0.406371i
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 12.0000i 0.805387i
\(223\) − 6.00000i − 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 2.00000i 0.131306i
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) − 32.0000i − 2.07862i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) 10.0000i 0.641500i
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 16.0000 1.02012
\(247\) 0 0
\(248\) 2.00000i 0.127000i
\(249\) 0 0
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 1.00000i 0.0629941i
\(253\) − 4.00000i − 0.251478i
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.0000i 0.873296i 0.899632 + 0.436648i \(0.143834\pi\)
−0.899632 + 0.436648i \(0.856166\pi\)
\(258\) − 24.0000i − 1.49417i
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 4.00000i 0.247121i
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) − 12.0000i − 0.734388i
\(268\) − 8.00000i − 0.488678i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 8.00000i − 0.479808i
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 12.0000i 0.714590i
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000i 0.472225i
\(288\) − 1.00000i − 0.0589256i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −28.0000 −1.64139
\(292\) 4.00000i 0.234082i
\(293\) − 4.00000i − 0.233682i −0.993151 0.116841i \(-0.962723\pi\)
0.993151 0.116841i \(-0.0372769\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) − 4.00000i − 0.232104i
\(298\) − 6.00000i − 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) − 8.00000i − 0.460348i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) − 1.00000i − 0.0569803i
\(309\) −28.0000 −1.59286
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 2.00000i 0.113047i 0.998401 + 0.0565233i \(0.0180015\pi\)
−0.998401 + 0.0565233i \(0.981998\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) − 12.0000i − 0.672927i
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 4.00000i 0.222911i
\(323\) 0 0
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) − 12.0000i − 0.663602i
\(328\) − 8.00000i − 0.441726i
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) − 6.00000i − 0.328798i
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) − 14.0000i − 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) 13.0000i 0.707107i
\(339\) 4.00000 0.217250
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) − 28.0000i − 1.50312i −0.659665 0.751559i \(-0.729302\pi\)
0.659665 0.751559i \(-0.270698\pi\)
\(348\) − 4.00000i − 0.214423i
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000i 0.0533002i
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 20.0000 1.06299
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) − 12.0000i − 0.634220i
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) − 6.00000i − 0.315353i
\(363\) − 2.00000i − 0.104973i
\(364\) 0 0
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) 22.0000i 1.14839i 0.818718 + 0.574195i \(0.194685\pi\)
−0.818718 + 0.574195i \(0.805315\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) − 4.00000i − 0.207390i
\(373\) 14.0000i 0.724893i 0.932005 + 0.362446i \(0.118058\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) 4.00000i 0.205738i
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) − 8.00000i − 0.409316i
\(383\) − 2.00000i − 0.102195i −0.998694 0.0510976i \(-0.983728\pi\)
0.998694 0.0510976i \(-0.0162720\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 12.0000i 0.609994i
\(388\) 14.0000i 0.710742i
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000i 0.0505076i
\(393\) − 8.00000i − 0.403547i
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) 34.0000i 1.70641i 0.521575 + 0.853206i \(0.325345\pi\)
−0.521575 + 0.853206i \(0.674655\pi\)
\(398\) 6.00000i 0.300753i
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 16.0000i 0.798007i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 6.00000i 0.297409i
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 14.0000i 0.689730i
\(413\) 10.0000i 0.492068i
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) 16.0000i 0.783523i
\(418\) 0 0
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) − 12.0000i − 0.584151i
\(423\) − 6.00000i − 0.291730i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) − 4.00000i − 0.193574i
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) 14.0000i 0.672797i 0.941720 + 0.336399i \(0.109209\pi\)
−0.941720 + 0.336399i \(0.890791\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) − 8.00000i − 0.382255i
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) − 20.0000i − 0.950229i −0.879924 0.475114i \(-0.842407\pi\)
0.879924 0.475114i \(-0.157593\pi\)
\(444\) −12.0000 −0.569495
\(445\) 0 0
\(446\) 6.00000 0.284108
\(447\) 12.0000i 0.567581i
\(448\) − 1.00000i − 0.0472456i
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) − 2.00000i − 0.0940721i
\(453\) 16.0000i 0.751746i
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) − 26.0000i − 1.21623i −0.793849 0.608114i \(-0.791926\pi\)
0.793849 0.608114i \(-0.208074\pi\)
\(458\) − 2.00000i − 0.0934539i
\(459\) 0 0
\(460\) 0 0
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 2.00000i 0.0930484i
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) 30.0000i 1.38823i 0.719862 + 0.694117i \(0.244205\pi\)
−0.719862 + 0.694117i \(0.755795\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) − 10.0000i − 0.460287i
\(473\) − 12.0000i − 0.551761i
\(474\) 32.0000 1.46981
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 16.0000i − 0.728780i
\(483\) − 8.00000i − 0.364013i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) − 20.0000i − 0.906287i −0.891438 0.453143i \(-0.850303\pi\)
0.891438 0.453143i \(-0.149697\pi\)
\(488\) 4.00000i 0.181071i
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 16.0000i 0.721336i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) − 4.00000i − 0.179425i
\(498\) 0 0
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) 0 0
\(501\) −32.0000 −1.42965
\(502\) 10.0000i 0.446322i
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) − 26.0000i − 1.15470i
\(508\) 8.00000i 0.354943i
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) 24.0000 1.05654
\(517\) 6.00000i 0.263880i
\(518\) − 6.00000i − 0.263625i
\(519\) 48.0000 2.10697
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) − 2.00000i − 0.0870388i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 0 0
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 24.0000i 1.03568i
\(538\) − 10.0000i − 0.431131i
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 20.0000i 0.859074i
\(543\) 12.0000i 0.514969i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 44.0000i 1.88130i 0.339372 + 0.940652i \(0.389785\pi\)
−0.339372 + 0.940652i \(0.610215\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 0 0
\(552\) 8.00000i 0.340503i
\(553\) 16.0000i 0.680389i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 2.00000i 0.0846668i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) − 2.00000i − 0.0843649i
\(563\) 32.0000i 1.34864i 0.738440 + 0.674320i \(0.235563\pi\)
−0.738440 + 0.674320i \(0.764437\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) − 11.0000i − 0.461957i
\(568\) 4.00000i 0.167836i
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 16.0000i 0.668410i
\(574\) −8.00000 −0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 26.0000i 1.08239i 0.840896 + 0.541197i \(0.182029\pi\)
−0.840896 + 0.541197i \(0.817971\pi\)
\(578\) 17.0000i 0.707107i
\(579\) −44.0000 −1.82858
\(580\) 0 0
\(581\) 0 0
\(582\) − 28.0000i − 1.16064i
\(583\) − 6.00000i − 0.248495i
\(584\) −4.00000 −0.165521
\(585\) 0 0
\(586\) 4.00000 0.165238
\(587\) − 2.00000i − 0.0825488i −0.999148 0.0412744i \(-0.986858\pi\)
0.999148 0.0412744i \(-0.0131418\pi\)
\(588\) − 2.00000i − 0.0824786i
\(589\) 0 0
\(590\) 0 0
\(591\) −28.0000 −1.15177
\(592\) 6.00000i 0.246598i
\(593\) − 12.0000i − 0.492781i −0.969171 0.246390i \(-0.920755\pi\)
0.969171 0.246390i \(-0.0792446\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) − 12.0000i − 0.491127i
\(598\) 0 0
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 12.0000i 0.489083i
\(603\) − 8.00000i − 0.325785i
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) − 24.0000i − 0.974130i −0.873366 0.487065i \(-0.838067\pi\)
0.873366 0.487065i \(-0.161933\pi\)
\(608\) 0 0
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) − 28.0000i − 1.12633i
\(619\) 22.0000 0.884255 0.442127 0.896952i \(-0.354224\pi\)
0.442127 + 0.896952i \(0.354224\pi\)
\(620\) 0 0
\(621\) −16.0000 −0.642058
\(622\) − 6.00000i − 0.240578i
\(623\) 6.00000i 0.240385i
\(624\) 0 0
\(625\) 0 0
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) 6.00000i 0.239426i
\(629\) 0 0
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) − 16.0000i − 0.636446i
\(633\) 24.0000i 0.953914i
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) − 2.00000i − 0.0791808i
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) − 24.0000i − 0.947204i
\(643\) 34.0000i 1.34083i 0.741987 + 0.670415i \(0.233884\pi\)
−0.741987 + 0.670415i \(0.766116\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) 42.0000i 1.65119i 0.564263 + 0.825595i \(0.309160\pi\)
−0.564263 + 0.825595i \(0.690840\pi\)
\(648\) 11.0000i 0.432121i
\(649\) 10.0000 0.392534
\(650\) 0 0
\(651\) −4.00000 −0.156772
\(652\) 8.00000i 0.313304i
\(653\) 22.0000i 0.860927i 0.902608 + 0.430463i \(0.141650\pi\)
−0.902608 + 0.430463i \(0.858350\pi\)
\(654\) 12.0000 0.469237
\(655\) 0 0
\(656\) 8.00000 0.312348
\(657\) 4.00000i 0.156055i
\(658\) − 6.00000i − 0.233904i
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 28.0000i 1.08825i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 8.00000i 0.309761i
\(668\) 16.0000i 0.619059i
\(669\) −12.0000 −0.463947
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 2.00000i 0.0771517i
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 4.00000i 0.153619i
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) − 2.00000i − 0.0765840i
\(683\) 20.0000i 0.765279i 0.923898 + 0.382639i \(0.124985\pi\)
−0.923898 + 0.382639i \(0.875015\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 4.00000i 0.152610i
\(688\) − 12.0000i − 0.457496i
\(689\) 0 0
\(690\) 0 0
\(691\) 6.00000 0.228251 0.114125 0.993466i \(-0.463593\pi\)
0.114125 + 0.993466i \(0.463593\pi\)
\(692\) − 24.0000i − 0.912343i
\(693\) − 1.00000i − 0.0379869i
\(694\) 28.0000 1.06287
\(695\) 0 0
\(696\) 4.00000 0.151620
\(697\) 0 0
\(698\) 20.0000i 0.757011i
\(699\) −36.0000 −1.36165
\(700\) 0 0
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 20.0000i 0.751646i
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) − 6.00000i − 0.224860i
\(713\) 8.00000i 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 8.00000i 0.298557i
\(719\) 34.0000 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) − 19.0000i − 0.707107i
\(723\) 32.0000i 1.19009i
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) − 14.0000i − 0.519231i −0.965712 0.259616i \(-0.916404\pi\)
0.965712 0.259616i \(-0.0835959\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) − 8.00000i − 0.295689i
\(733\) 4.00000i 0.147743i 0.997268 + 0.0738717i \(0.0235355\pi\)
−0.997268 + 0.0738717i \(0.976464\pi\)
\(734\) −22.0000 −0.812035
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 8.00000i 0.294684i
\(738\) − 8.00000i − 0.294484i
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 6.00000i 0.220267i
\(743\) − 8.00000i − 0.293492i −0.989174 0.146746i \(-0.953120\pi\)
0.989174 0.146746i \(-0.0468799\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 52.0000 1.89751 0.948753 0.316017i \(-0.102346\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) 6.00000i 0.218797i
\(753\) − 20.0000i − 0.728841i
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 26.0000i 0.944986i 0.881334 + 0.472493i \(0.156646\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) 16.0000i 0.581146i
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) − 16.0000i − 0.579619i
\(763\) 6.00000i 0.217215i
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 2.00000 0.0722629
\(767\) 0 0
\(768\) − 2.00000i − 0.0721688i
\(769\) −12.0000 −0.432731 −0.216366 0.976312i \(-0.569420\pi\)
−0.216366 + 0.976312i \(0.569420\pi\)
\(770\) 0 0
\(771\) 28.0000 1.00840
\(772\) 22.0000i 0.791797i
\(773\) 18.0000i 0.647415i 0.946157 + 0.323708i \(0.104929\pi\)
−0.946157 + 0.323708i \(0.895071\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) 12.0000i 0.430498i
\(778\) − 22.0000i − 0.788738i
\(779\) 0 0
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 8.00000i 0.285897i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) − 48.0000i − 1.71102i −0.517790 0.855508i \(-0.673245\pi\)
0.517790 0.855508i \(-0.326755\pi\)
\(788\) 14.0000i 0.498729i
\(789\) −48.0000 −1.70885
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 1.00000i 0.0355335i
\(793\) 0 0
\(794\) −34.0000 −1.20661
\(795\) 0 0
\(796\) −6.00000 −0.212664
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) − 6.00000i − 0.211867i
\(803\) − 4.00000i − 0.141157i
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 0 0
\(807\) 20.0000i 0.704033i
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 2.00000i 0.0701862i
\(813\) − 40.0000i − 1.40286i
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) − 4.00000i − 0.139857i
\(819\) 0 0
\(820\) 0 0
\(821\) −34.0000 −1.18661 −0.593304 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) − 12.0000i − 0.418548i
\(823\) 32.0000i 1.11545i 0.830026 + 0.557725i \(0.188326\pi\)
−0.830026 + 0.557725i \(0.811674\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −10.0000 −0.347945
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) 0 0
\(833\) 0 0
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 26.0000i 0.898155i
\(839\) 10.0000 0.345238 0.172619 0.984989i \(-0.444777\pi\)
0.172619 + 0.984989i \(0.444777\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 18.0000i 0.620321i
\(843\) 4.00000i 0.137767i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 1.00000i 0.0343604i
\(848\) − 6.00000i − 0.206041i
\(849\) 56.0000 1.92192
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) − 8.00000i − 0.274075i
\(853\) − 8.00000i − 0.273915i −0.990577 0.136957i \(-0.956268\pi\)
0.990577 0.136957i \(-0.0437323\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) − 12.0000i − 0.409912i −0.978771 0.204956i \(-0.934295\pi\)
0.978771 0.204956i \(-0.0657052\pi\)
\(858\) 0 0
\(859\) −2.00000 −0.0682391 −0.0341196 0.999418i \(-0.510863\pi\)
−0.0341196 + 0.999418i \(0.510863\pi\)
\(860\) 0 0
\(861\) 16.0000 0.545279
\(862\) 8.00000i 0.272481i
\(863\) − 32.0000i − 1.08929i −0.838666 0.544646i \(-0.816664\pi\)
0.838666 0.544646i \(-0.183336\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) − 34.0000i − 1.15470i
\(868\) 2.00000i 0.0678844i
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) − 6.00000i − 0.203186i
\(873\) 14.0000i 0.473828i
\(874\) 0 0
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) − 10.0000i − 0.337676i −0.985644 0.168838i \(-0.945999\pi\)
0.985644 0.168838i \(-0.0540015\pi\)
\(878\) − 4.00000i − 0.134993i
\(879\) −8.00000 −0.269833
\(880\) 0 0
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 1.00000i 0.0336718i
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 52.0000i 1.74599i 0.487730 + 0.872995i \(0.337825\pi\)
−0.487730 + 0.872995i \(0.662175\pi\)
\(888\) − 12.0000i − 0.402694i
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) 6.00000i 0.200895i
\(893\) 0 0
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 10.0000i 0.333704i
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) 0 0
\(902\) 8.00000i 0.266371i
\(903\) − 24.0000i − 0.798670i
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) − 48.0000i − 1.59381i −0.604102 0.796907i \(-0.706468\pi\)
0.604102 0.796907i \(-0.293532\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 0 0
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 4.00000i 0.132092i
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) −56.0000 −1.84526
\(922\) − 20.0000i − 0.658665i
\(923\) 0 0
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 14.0000i 0.459820i
\(928\) − 2.00000i − 0.0656532i
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18.0000i 0.589610i
\(933\) 12.0000i 0.392862i
\(934\) −30.0000 −0.981630
\(935\) 0 0
\(936\) 0 0
\(937\) − 36.0000i − 1.17607i −0.808836 0.588034i \(-0.799902\pi\)
0.808836 0.588034i \(-0.200098\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) − 12.0000i − 0.390981i
\(943\) − 32.0000i − 1.04206i
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 32.0000i 1.03931i
\(949\) 0 0
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 0 0
\(957\) 4.00000i 0.129302i
\(958\) − 20.0000i − 0.646171i
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 16.0000 0.515325
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 48.0000i 1.54358i 0.635880 + 0.771788i \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(968\) − 1.00000i − 0.0321412i
\(969\) 0 0
\(970\) 0 0
\(971\) 26.0000 0.834380 0.417190 0.908819i \(-0.363015\pi\)
0.417190 + 0.908819i \(0.363015\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) − 8.00000i − 0.256468i
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) − 16.0000i − 0.511624i
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 36.0000i 1.14881i
\(983\) − 22.0000i − 0.701691i −0.936433 0.350846i \(-0.885894\pi\)
0.936433 0.350846i \(-0.114106\pi\)
\(984\) −16.0000 −0.510061
\(985\) 0 0
\(986\) 0 0
\(987\) 12.0000i 0.381964i
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) − 2.00000i − 0.0635001i
\(993\) − 56.0000i − 1.77711i
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) 0 0
\(997\) 48.0000i 1.52018i 0.649821 + 0.760088i \(0.274844\pi\)
−0.649821 + 0.760088i \(0.725156\pi\)
\(998\) 40.0000i 1.26618i
\(999\) 24.0000 0.759326
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.p.1849.2 2
5.2 odd 4 770.2.a.b.1.1 1
5.3 odd 4 3850.2.a.bb.1.1 1
5.4 even 2 inner 3850.2.c.p.1849.1 2
15.2 even 4 6930.2.a.s.1.1 1
20.7 even 4 6160.2.a.p.1.1 1
35.27 even 4 5390.2.a.q.1.1 1
55.32 even 4 8470.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.b.1.1 1 5.2 odd 4
3850.2.a.bb.1.1 1 5.3 odd 4
3850.2.c.p.1849.1 2 5.4 even 2 inner
3850.2.c.p.1849.2 2 1.1 even 1 trivial
5390.2.a.q.1.1 1 35.27 even 4
6160.2.a.p.1.1 1 20.7 even 4
6930.2.a.s.1.1 1 15.2 even 4
8470.2.a.v.1.1 1 55.32 even 4