Properties

Label 690.2.a.i.1.1
Level $690$
Weight $2$
Character 690.1
Self dual yes
Analytic conductor $5.510$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 690.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.50967773947\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 690.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} -2.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} +4.00000 q^{57} -1.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} +6.00000 q^{68} +1.00000 q^{69} -10.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} -6.00000 q^{85} -2.00000 q^{86} +2.00000 q^{88} +4.00000 q^{89} -1.00000 q^{90} +1.00000 q^{92} -8.00000 q^{93} +4.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} +16.0000 q^{97} -7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 2.00000 0.301511
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 6.00000 0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −8.00000 −0.829561
\(94\) 4.00000 0.412568
\(95\) −4.00000 −0.410391
\(96\) 1.00000 0.102062
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) −7.00000 −0.707107
\(99\) 2.00000 0.201008
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 6.00000 0.594089
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −2.00000 −0.190693
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 4.00000 0.374634
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) −2.00000 −0.180334
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) −1.00000 −0.0860663
\(136\) 6.00000 0.514496
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 1.00000 0.0851257
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) −10.0000 −0.839181
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) −7.00000 −0.577350
\(148\) −6.00000 −0.493197
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 1.00000 0.0816497
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 4.00000 0.324443
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −2.00000 −0.156174
\(165\) −2.00000 −0.155700
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −6.00000 −0.460179
\(171\) 4.00000 0.305888
\(172\) −2.00000 −0.152499
\(173\) 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
\(177\) 0 0
\(178\) 4.00000 0.299813
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 1.00000 0.0737210
\(185\) 6.00000 0.441129
\(186\) −8.00000 −0.586588
\(187\) 12.0000 0.877527
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 2.00000 0.142134
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 1.00000 0.0707107
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 2.00000 0.139686
\(206\) −16.0000 −1.11477
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −2.00000 −0.137361
\(213\) −10.0000 −0.685189
\(214\) 8.00000 0.546869
\(215\) 2.00000 0.136399
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) −10.0000 −0.675737
\(220\) −2.00000 −0.134840
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 10.0000 0.665190
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 4.00000 0.264906
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −26.0000 −1.68180 −0.840900 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 7.00000 0.447214
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −4.00000 −0.253490
\(250\) −1.00000 −0.0632456
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 6.00000 0.376473
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −4.00000 −0.247121
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 2.00000 0.123091
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) −2.00000 −0.122169
\(269\) 28.0000 1.70719 0.853595 0.520937i \(-0.174417\pi\)
0.853595 + 0.520937i \(0.174417\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 2.00000 0.120605
\(276\) 1.00000 0.0601929
\(277\) 24.0000 1.44202 0.721010 0.692925i \(-0.243678\pi\)
0.721010 + 0.692925i \(0.243678\pi\)
\(278\) 12.0000 0.719712
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 4.00000 0.238197
\(283\) 10.0000 0.594438 0.297219 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) −10.0000 −0.593391
\(285\) −4.00000 −0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) −10.0000 −0.585206
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 2.00000 0.116052
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −20.0000 −1.15087
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 2.00000 0.114520
\(306\) 6.00000 0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 8.00000 0.454369
\(311\) −2.00000 −0.113410 −0.0567048 0.998391i \(-0.518059\pi\)
−0.0567048 + 0.998391i \(0.518059\pi\)
\(312\) 0 0
\(313\) −20.0000 −1.13047 −0.565233 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −2.00000 −0.112154
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) 2.00000 0.110600
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) −4.00000 −0.219529
\(333\) −6.00000 −0.328798
\(334\) 8.00000 0.437741
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) −13.0000 −0.707107
\(339\) 10.0000 0.543125
\(340\) −6.00000 −0.325396
\(341\) −16.0000 −0.866449
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) −1.00000 −0.0538382
\(346\) 6.00000 0.322562
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 10.0000 0.530745
\(356\) 4.00000 0.212000
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 6.00000 0.315353
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) −2.00000 −0.104542
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 1.00000 0.0521286
\(369\) −2.00000 −0.104116
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 12.0000 0.620505
\(375\) −1.00000 −0.0516398
\(376\) 4.00000 0.206284
\(377\) 0 0
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) −4.00000 −0.205196
\(381\) 6.00000 0.307389
\(382\) −8.00000 −0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −2.00000 −0.101666
\(388\) 16.0000 0.812277
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) −7.00000 −0.353553
\(393\) −4.00000 −0.201773
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 6.00000 0.297044
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 2.00000 0.0987730
\(411\) 2.00000 0.0986527
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 8.00000 0.391293
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −20.0000 −0.973585
\(423\) 4.00000 0.194487
\(424\) −2.00000 −0.0971286
\(425\) 6.00000 0.291043
\(426\) −10.0000 −0.484502
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 2.00000 0.0964486
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) 1.00000 0.0481125
\(433\) 32.0000 1.53782 0.768911 0.639356i \(-0.220799\pi\)
0.768911 + 0.639356i \(0.220799\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 4.00000 0.191346
\(438\) −10.0000 −0.477818
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) −2.00000 −0.0953463
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −6.00000 −0.284747
\(445\) −4.00000 −0.189618
\(446\) −14.0000 −0.662919
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 1.00000 0.0471405
\(451\) −4.00000 −0.188353
\(452\) 10.0000 0.470360
\(453\) −20.0000 −0.939682
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −14.0000 −0.654177
\(459\) 6.00000 0.280056
\(460\) −1.00000 −0.0466252
\(461\) 32.0000 1.49039 0.745194 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 0 0
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 22.0000 1.01913
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.00000 −0.184506
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) −26.0000 −1.18921
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −16.0000 −0.726523
\(486\) 1.00000 0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −8.00000 −0.361773
\(490\) 7.00000 0.316228
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 0 0
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 8.00000 0.357414
\(502\) −18.0000 −0.803379
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) −13.0000 −0.577350
\(508\) 6.00000 0.266207
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) −6.00000 −0.265684
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) 6.00000 0.264649
\(515\) 16.0000 0.705044
\(516\) −2.00000 −0.0880451
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 0 0
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) −48.0000 −2.09091
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 4.00000 0.173097
\(535\) −8.00000 −0.345870
\(536\) −2.00000 −0.0863868
\(537\) −24.0000 −1.03568
\(538\) 28.0000 1.20717
\(539\) −14.0000 −0.603023
\(540\) −1.00000 −0.0430331
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −8.00000 −0.343629
\(543\) 6.00000 0.257485
\(544\) 6.00000 0.257248
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 2.00000 0.0854358
\(549\) −2.00000 −0.0853579
\(550\) 2.00000 0.0852803
\(551\) 0 0
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 24.0000 1.01966
\(555\) 6.00000 0.254686
\(556\) 12.0000 0.508913
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −8.00000 −0.338667
\(559\) 0 0
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 12.0000 0.506189
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 4.00000 0.168430
\(565\) −10.0000 −0.420703
\(566\) 10.0000 0.420331
\(567\) 0 0
\(568\) −10.0000 −0.419591
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) −4.00000 −0.167542
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 19.0000 0.790296
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) 16.0000 0.663221
\(583\) −4.00000 −0.165663
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) −7.00000 −0.288675
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) −6.00000 −0.246598
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) 26.0000 1.06233 0.531166 0.847268i \(-0.321754\pi\)
0.531166 + 0.847268i \(0.321754\pi\)
\(600\) 1.00000 0.0408248
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −2.00000 −0.0814463
\(604\) −20.0000 −0.813788
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 12.0000 0.484281
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −16.0000 −0.643614
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 8.00000 0.321288
\(621\) 1.00000 0.0401286
\(622\) −2.00000 −0.0801927
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −20.0000 −0.799361
\(627\) 8.00000 0.319489
\(628\) −6.00000 −0.239426
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) −6.00000 −0.238290
\(635\) −6.00000 −0.238103
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) 0 0
\(639\) −10.0000 −0.395594
\(640\) −1.00000 −0.0395285
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 8.00000 0.315735
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 24.0000 0.944267
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 2.00000 0.0782062
\(655\) 4.00000 0.156293
\(656\) −2.00000 −0.0780869
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 18.0000 0.701180 0.350590 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) −14.0000 −0.541271
\(670\) 2.00000 0.0772667
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 28.0000 1.07852
\(675\) 1.00000 0.0384900
\(676\) −13.0000 −0.500000
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 10.0000 0.384048
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) 8.00000 0.306561
\(682\) −16.0000 −0.612672
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 4.00000 0.152944
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) −2.00000 −0.0762493
\(689\) 0 0
\(690\) −1.00000 −0.0380693
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 26.0000 0.984115
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 2.00000 0.0753778
\(705\) −4.00000 −0.150649
\(706\) −6.00000 −0.225813
\(707\) 0 0
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 10.0000 0.375293
\(711\) 0 0
\(712\) 4.00000 0.149906
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) −26.0000 −0.970988
\(718\) −32.0000 −1.19423
\(719\) −22.0000 −0.820462 −0.410231 0.911982i \(-0.634552\pi\)
−0.410231 + 0.911982i \(0.634552\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 2.00000 0.0743808
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) −12.0000 −0.443836
\(732\) −2.00000 −0.0739221
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 16.0000 0.590571
\(735\) 7.00000 0.258199
\(736\) 1.00000 0.0368605
\(737\) −4.00000 −0.147342
\(738\) −2.00000 −0.0736210
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −8.00000 −0.293294
\(745\) −10.0000 −0.366372
\(746\) −6.00000 −0.219676
\(747\) −4.00000 −0.146352
\(748\) 12.0000 0.438763
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 4.00000 0.145865
\(753\) −18.0000 −0.655956
\(754\) 0 0
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) −8.00000 −0.290573
\(759\) 2.00000 0.0725954
\(760\) −4.00000 −0.145095
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 6.00000 0.217357
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) −6.00000 −0.216930
\(766\) 16.0000 0.578103
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −14.0000 −0.503871
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) −2.00000 −0.0718885
\(775\) −8.00000 −0.287368
\(776\) 16.0000 0.574367
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) 6.00000 0.214560
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 6.00000 0.214149
\(786\) −4.00000 −0.142675
\(787\) 18.0000 0.641631 0.320815 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) 14.0000 0.498729
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) 0 0
\(794\) 8.00000 0.283909
\(795\) 2.00000 0.0709327
\(796\) 24.0000 0.850657
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) −20.0000 −0.705785
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 0 0
\(807\) 28.0000 0.985647
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) −12.0000 −0.420600
\(815\) 8.00000 0.280228
\(816\) 6.00000 0.210042
\(817\) −8.00000 −0.279885
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 44.0000 1.53561 0.767805 0.640683i \(-0.221349\pi\)
0.767805 + 0.640683i \(0.221349\pi\)
\(822\) 2.00000 0.0697580
\(823\) −10.0000 −0.348578 −0.174289 0.984695i \(-0.555763\pi\)
−0.174289 + 0.984695i \(0.555763\pi\)
\(824\) −16.0000 −0.557386
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 1.00000 0.0347524
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 4.00000 0.138842
\(831\) 24.0000 0.832551
\(832\) 0 0
\(833\) −42.0000 −1.45521
\(834\) 12.0000 0.415526
\(835\) −8.00000 −0.276851
\(836\) 8.00000 0.276686
\(837\) −8.00000 −0.276520
\(838\) −6.00000 −0.207267
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 22.0000 0.758170
\(843\) 12.0000 0.413302
\(844\) −20.0000 −0.688428
\(845\) 13.0000 0.447214
\(846\) 4.00000 0.137523
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) 10.0000 0.343199
\(850\) 6.00000 0.205798
\(851\) −6.00000 −0.205677
\(852\) −10.0000 −0.342594
\(853\) −16.0000 −0.547830 −0.273915 0.961754i \(-0.588319\pi\)
−0.273915 + 0.961754i \(0.588319\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 8.00000 0.273434
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) −40.0000 −1.36241
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) 32.0000 1.08740
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 2.00000 0.0677285
\(873\) 16.0000 0.541518
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −12.0000 −0.404980
\(879\) 22.0000 0.742042
\(880\) −2.00000 −0.0674200
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) −7.00000 −0.235702
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) −28.0000 −0.940148 −0.470074 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) −4.00000 −0.134080
\(891\) 2.00000 0.0670025
\(892\) −14.0000 −0.468755
\(893\) 16.0000 0.535420
\(894\) 10.0000 0.334450
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −12.0000 −0.399778
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) −6.00000 −0.199447
\(906\) −20.0000 −0.664455
\(907\) 38.0000 1.26177 0.630885 0.775877i \(-0.282692\pi\)
0.630885 + 0.775877i \(0.282692\pi\)
\(908\) 8.00000 0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 4.00000 0.132453
\(913\) −8.00000 −0.264761
\(914\) −8.00000 −0.264616
\(915\) 2.00000 0.0661180
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 48.0000 1.58337 0.791687 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 12.0000 0.395413
\(922\) 32.0000 1.05386
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 34.0000 1.11731
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 8.00000 0.262330
\(931\) −28.0000 −0.917663
\(932\) 22.0000 0.720634
\(933\) −2.00000 −0.0654771
\(934\) −4.00000 −0.130884
\(935\) −12.0000 −0.392442
\(936\) 0 0
\(937\) 40.0000 1.30674 0.653372 0.757037i \(-0.273354\pi\)
0.653372 + 0.757037i \(0.273354\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) −4.00000 −0.130466
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) −6.00000 −0.195491
\(943\) −2.00000 −0.0651290
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 4.00000 0.129777
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 8.00000 0.258874
\(956\) −26.0000 −0.840900
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 2.00000 0.0644157
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −38.0000 −1.22200 −0.610999 0.791632i \(-0.709232\pi\)
−0.610999 + 0.791632i \(0.709232\pi\)
\(968\) −7.00000 −0.224989
\(969\) 24.0000 0.770991
\(970\) −16.0000 −0.513729
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) −8.00000 −0.255812
\(979\) 8.00000 0.255681
\(980\) 7.00000 0.223607
\(981\) 2.00000 0.0638551
\(982\) −12.0000 −0.382935
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −14.0000 −0.446077
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) −2.00000 −0.0635642
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) −8.00000 −0.254000
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) −4.00000 −0.126745
\(997\) −16.0000 −0.506725 −0.253363 0.967371i \(-0.581537\pi\)
−0.253363 + 0.967371i \(0.581537\pi\)
\(998\) −36.0000 −1.13956
\(999\) −6.00000 −0.189832
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 690.2.a.i.1.1 1
3.2 odd 2 2070.2.a.g.1.1 1
4.3 odd 2 5520.2.a.d.1.1 1
5.2 odd 4 3450.2.d.r.2899.2 2
5.3 odd 4 3450.2.d.r.2899.1 2
5.4 even 2 3450.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.i.1.1 1 1.1 even 1 trivial
2070.2.a.g.1.1 1 3.2 odd 2
3450.2.a.c.1.1 1 5.4 even 2
3450.2.d.r.2899.1 2 5.3 odd 4
3450.2.d.r.2899.2 2 5.2 odd 4
5520.2.a.d.1.1 1 4.3 odd 2