Properties

Label 2850.2.a.ba.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{21} -2.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{27} +2.00000 q^{28} +1.00000 q^{32} -2.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} -8.00000 q^{41} +2.00000 q^{42} +6.00000 q^{43} -2.00000 q^{44} +8.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +2.00000 q^{51} +10.0000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +1.00000 q^{57} -8.00000 q^{59} +2.00000 q^{61} +2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +2.00000 q^{68} +8.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -4.00000 q^{74} +1.00000 q^{76} -4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -8.00000 q^{82} +16.0000 q^{83} +2.00000 q^{84} +6.00000 q^{86} -2.00000 q^{88} +16.0000 q^{89} +8.00000 q^{92} +8.00000 q^{94} +1.00000 q^{96} -8.00000 q^{97} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −2.00000 −0.426401
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 2.00000 0.308607
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 2.00000 0.242536
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −8.00000 −0.883452
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 16.0000 1.69600 0.847998 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −3.00000 −0.303046
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 2.00000 0.198030
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 2.00000 0.188982
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −2.00000 −0.174078
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 8.00000 0.681005
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) −3.00000 −0.247436
\(148\) −4.00000 −0.328798
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 1.00000 0.0811107
\(153\) 2.00000 0.161690
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −8.00000 −0.636446
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 1.00000 0.0785674
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 2.00000 0.154303
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 6.00000 0.457496
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −8.00000 −0.601317
\(178\) 16.0000 1.19925
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 8.00000 0.589768
\(185\) 0 0
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 8.00000 0.583460
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −2.00000 −0.142134
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −12.0000 −0.836080
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 10.0000 0.686803
\(213\) 8.00000 0.548151
\(214\) −20.0000 −1.36717
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 1.00000 0.0662266
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) −8.00000 −0.519656
\(238\) 4.00000 0.259281
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −8.00000 −0.510061
\(247\) 0 0
\(248\) 0 0
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 2.00000 0.125988
\(253\) −16.0000 −1.00591
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 6.00000 0.373544
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) 16.0000 0.979184
\(268\) 0 0
\(269\) 8.00000 0.487769 0.243884 0.969804i \(-0.421578\pi\)
0.243884 + 0.969804i \(0.421578\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 8.00000 0.481543
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 8.00000 0.479808
\(279\) 0 0
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 8.00000 0.476393
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −16.0000 −0.944450
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 2.00000 0.117041
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) −2.00000 −0.116052
\(298\) 2.00000 0.115857
\(299\) 0 0
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) −8.00000 −0.460348
\(303\) 2.00000 0.114897
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 24.0000 1.36975 0.684876 0.728659i \(-0.259856\pi\)
0.684876 + 0.728659i \(0.259856\pi\)
\(308\) −4.00000 −0.227921
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) 0 0
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 10.0000 0.560772
\(319\) 0 0
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 16.0000 0.891645
\(323\) 2.00000 0.111283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 10.0000 0.553849
\(327\) 2.00000 0.110600
\(328\) −8.00000 −0.441726
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 16.0000 0.878114
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) −13.0000 −0.707107
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) −20.0000 −1.07990
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) 16.0000 0.847998
\(357\) 4.00000 0.211702
\(358\) −8.00000 −0.422813
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −6.00000 −0.315353
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −26.0000 −1.35719 −0.678594 0.734513i \(-0.737411\pi\)
−0.678594 + 0.734513i \(0.737411\pi\)
\(368\) 8.00000 0.417029
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) 0 0
\(378\) 2.00000 0.102869
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 10.0000 0.511645
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 6.00000 0.304997
\(388\) −8.00000 −0.406138
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) −3.00000 −0.151523
\(393\) −6.00000 −0.302660
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −4.00000 −0.200502
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 2.00000 0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −12.0000 −0.591198
\(413\) −16.0000 −0.787309
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) −2.00000 −0.0978232
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 4.00000 0.194717
\(423\) 8.00000 0.388973
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 4.00000 0.193574
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000 0.963366 0.481683 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) 1.00000 0.0481125
\(433\) 12.0000 0.576683 0.288342 0.957528i \(-0.406896\pi\)
0.288342 + 0.957528i \(0.406896\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 8.00000 0.382692
\(438\) 2.00000 0.0955637
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −12.0000 −0.568216
\(447\) 2.00000 0.0945968
\(448\) 2.00000 0.0944911
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 2.00000 0.0940721
\(453\) −8.00000 −0.375873
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 10.0000 0.467269
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) −4.00000 −0.186097
\(463\) −18.0000 −0.836531 −0.418265 0.908325i \(-0.637362\pi\)
−0.418265 + 0.908325i \(0.637362\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) −8.00000 −0.368230
\(473\) −12.0000 −0.551761
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 10.0000 0.457869
\(478\) 6.00000 0.274434
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 10.0000 0.455488
\(483\) 16.0000 0.728025
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 2.00000 0.0905357
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) −8.00000 −0.360668
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 16.0000 0.716977
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 6.00000 0.267793
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 2.00000 0.0890871
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) −13.0000 −0.577350
\(508\) −16.0000 −0.709885
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −30.0000 −1.32324
\(515\) 0 0
\(516\) 6.00000 0.264135
\(517\) −16.0000 −0.703679
\(518\) −8.00000 −0.351500
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −20.0000 −0.876216 −0.438108 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 2.00000 0.0867110
\(533\) 0 0
\(534\) 16.0000 0.692388
\(535\) 0 0
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 8.00000 0.344904
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) −4.00000 −0.171815
\(543\) −6.00000 −0.257485
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) −18.0000 −0.768922
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 8.00000 0.340503
\(553\) −16.0000 −0.680389
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −20.0000 −0.843649
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 2.00000 0.0840663
\(567\) 2.00000 0.0839921
\(568\) 8.00000 0.335673
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 10.0000 0.417756
\(574\) −16.0000 −0.667827
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −13.0000 −0.540729
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 32.0000 1.32758
\(582\) −8.00000 −0.331611
\(583\) −20.0000 −0.828315
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) 44.0000 1.81607 0.908037 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) −3.00000 −0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) −4.00000 −0.164399
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 12.0000 0.489083
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) −44.0000 −1.78590 −0.892952 0.450151i \(-0.851370\pi\)
−0.892952 + 0.450151i \(0.851370\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 46.0000 1.85792 0.928961 0.370177i \(-0.120703\pi\)
0.928961 + 0.370177i \(0.120703\pi\)
\(614\) 24.0000 0.968561
\(615\) 0 0
\(616\) −4.00000 −0.161165
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −12.0000 −0.482711
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) −14.0000 −0.561349
\(623\) 32.0000 1.28205
\(624\) 0 0
\(625\) 0 0
\(626\) −18.0000 −0.719425
\(627\) −2.00000 −0.0798723
\(628\) 6.00000 0.239426
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −8.00000 −0.318223
\(633\) 4.00000 0.158986
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) 0 0
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) −20.0000 −0.789337
\(643\) −46.0000 −1.81406 −0.907031 0.421063i \(-0.861657\pi\)
−0.907031 + 0.421063i \(0.861657\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) −8.00000 −0.312348
\(657\) 2.00000 0.0780274
\(658\) 16.0000 0.623745
\(659\) −44.0000 −1.71400 −0.856998 0.515319i \(-0.827673\pi\)
−0.856998 + 0.515319i \(0.827673\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −12.0000 −0.466393
\(663\) 0 0
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 0 0
\(668\) 0 0
\(669\) −12.0000 −0.463947
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 2.00000 0.0771517
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 16.0000 0.616297
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 2.00000 0.0768095
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 10.0000 0.381524
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) 0 0
\(691\) 40.0000 1.52167 0.760836 0.648944i \(-0.224789\pi\)
0.760836 + 0.648944i \(0.224789\pi\)
\(692\) 6.00000 0.228086
\(693\) −4.00000 −0.151947
\(694\) 8.00000 0.303676
\(695\) 0 0
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) −26.0000 −0.984115
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 4.00000 0.150435
\(708\) −8.00000 −0.300658
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 16.0000 0.599625
\(713\) 0 0
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) −8.00000 −0.298974
\(717\) 6.00000 0.224074
\(718\) −30.0000 −1.11959
\(719\) 22.0000 0.820462 0.410231 0.911982i \(-0.365448\pi\)
0.410231 + 0.911982i \(0.365448\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 1.00000 0.0372161
\(723\) 10.0000 0.371904
\(724\) −6.00000 −0.222988
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 22.0000 0.815935 0.407967 0.912996i \(-0.366238\pi\)
0.407967 + 0.912996i \(0.366238\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 2.00000 0.0739221
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) −26.0000 −0.959678
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) −8.00000 −0.294484
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 20.0000 0.734223
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −32.0000 −1.17160
\(747\) 16.0000 0.585409
\(748\) −4.00000 −0.146254
\(749\) −40.0000 −1.46157
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 8.00000 0.291730
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) −28.0000 −1.01701
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) −16.0000 −0.579619
\(763\) 4.00000 0.144810
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 4.00000 0.143963
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) −8.00000 −0.287183
\(777\) −8.00000 −0.286998
\(778\) 14.0000 0.501924
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −6.00000 −0.214013
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) −10.0000 −0.356235
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 4.00000 0.142224
\(792\) −2.00000 −0.0710669
\(793\) 0 0
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 22.0000 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 2.00000 0.0707992
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 16.0000 0.565332
\(802\) −12.0000 −0.423735
\(803\) −4.00000 −0.141157
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.00000 0.281613
\(808\) 2.00000 0.0703598
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) −4.00000 −0.140286
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) 6.00000 0.209913
\(818\) −18.0000 −0.629355
\(819\) 0 0
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −18.0000 −0.627822
\(823\) −34.0000 −1.18517 −0.592583 0.805510i \(-0.701892\pi\)
−0.592583 + 0.805510i \(0.701892\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 8.00000 0.278019
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) 30.0000 1.03633
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −26.0000 −0.896019
\(843\) −20.0000 −0.688837
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) −14.0000 −0.481046
\(848\) 10.0000 0.343401
\(849\) 2.00000 0.0686398
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 8.00000 0.274075
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 20.0000 0.681203
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 12.0000 0.407777
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 2.00000 0.0677285
\(873\) −8.00000 −0.270759
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) 16.0000 0.539974
\(879\) 22.0000 0.742042
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −3.00000 −0.101015
\(883\) 18.0000 0.605748 0.302874 0.953031i \(-0.402054\pi\)
0.302874 + 0.953031i \(0.402054\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −4.00000 −0.134231
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −12.0000 −0.401790
\(893\) 8.00000 0.267710
\(894\) 2.00000 0.0668900
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 24.0000 0.800890
\(899\) 0 0
\(900\) 0 0
\(901\) 20.0000 0.666297
\(902\) 16.0000 0.532742
\(903\) 12.0000 0.399335
\(904\) 2.00000 0.0665190
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 4.00000 0.132745
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 1.00000 0.0331133
\(913\) −32.0000 −1.05905
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) −12.0000 −0.396275
\(918\) 2.00000 0.0660098
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 24.0000 0.790827
\(922\) −6.00000 −0.197599
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) 0 0
\(926\) −18.0000 −0.591517
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) 6.00000 0.196537
\(933\) −14.0000 −0.458339
\(934\) −20.0000 −0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) 58.0000 1.89478 0.947389 0.320085i \(-0.103712\pi\)
0.947389 + 0.320085i \(0.103712\pi\)
\(938\) 0 0
\(939\) −18.0000 −0.587408
\(940\) 0 0
\(941\) 16.0000 0.521585 0.260793 0.965395i \(-0.416016\pi\)
0.260793 + 0.965395i \(0.416016\pi\)
\(942\) 6.00000 0.195491
\(943\) −64.0000 −2.08413
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 4.00000 0.129641
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) −30.0000 −0.969256
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −20.0000 −0.644491
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 16.0000 0.514792
\(967\) 6.00000 0.192947 0.0964735 0.995336i \(-0.469244\pi\)
0.0964735 + 0.995336i \(0.469244\pi\)
\(968\) −7.00000 −0.224989
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000 0.512936
\(974\) −20.0000 −0.640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 10.0000 0.319765
\(979\) −32.0000 −1.02272
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −30.0000 −0.957338
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −8.00000 −0.255031
\(985\) 0 0
\(986\) 0 0
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) −16.0000 −0.506471
\(999\) −4.00000 −0.126554
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 2850.2.a.ba.1.1 1
3.2 odd 2 8550.2.a.o.1.1 1
5.2 odd 4 2850.2.d.n.799.2 2
5.3 odd 4 2850.2.d.n.799.1 2
5.4 even 2 570.2.a.c.1.1 1
15.14 odd 2 1710.2.a.n.1.1 1
20.19 odd 2 4560.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.c.1.1 1 5.4 even 2
1710.2.a.n.1.1 1 15.14 odd 2
2850.2.a.ba.1.1 1 1.1 even 1 trivial
2850.2.d.n.799.1 2 5.3 odd 4
2850.2.d.n.799.2 2 5.2 odd 4
4560.2.a.bd.1.1 1 20.19 odd 2
8550.2.a.o.1.1 1 3.2 odd 2