Properties

Label 690.2.a.k.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} +4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} -10.0000 q^{61} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} -6.00000 q^{68} -1.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{83} -6.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} +18.0000 q^{89} +1.00000 q^{90} -1.00000 q^{92} +4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 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
\(22\) 4.00000 0.852803
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 1.00000 0.182574
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) −6.00000 −0.840168
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 4.00000 0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −6.00000 −0.727607
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −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
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) 4.00000 0.426401
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −7.00000 −0.707107
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −6.00000 −0.594089
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 4.00000 0.381385
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 4.00000 0.374634
\(115\) −1.00000 −0.0932505
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) −10.0000 −0.905357
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 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
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 10.0000 0.827606
\(147\) −7.00000 −0.577350
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000 0.324443
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −8.00000 −0.636446
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 10.0000 0.780869
\(165\) 4.00000 0.311400
\(166\) −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
\(169\) −9.00000 −0.692308
\(170\) −6.00000 −0.460179
\(171\) 4.00000 0.305888
\(172\) −4.00000 −0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −4.00000 −0.300658
\(178\) 18.0000 1.34916
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 1.00000 0.0745356
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) −1.00000 −0.0737210
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) −7.00000 −0.500000
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 4.00000 0.284268
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 10.0000 0.698430
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 10.0000 0.675737
\(220\) 4.00000 0.269680
\(221\) 12.0000 0.807207
\(222\) −2.00000 −0.134231
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 4.00000 0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 1.00000 0.0645497
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) −7.00000 −0.447214
\(246\) 10.0000 0.637577
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) −16.0000 −1.00393
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) −2.00000 −0.123797
\(262\) 20.0000 1.23560
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 4.00000 0.246183
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) −12.0000 −0.733017
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 1.00000 0.0608581
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 4.00000 0.241209
\(276\) −1.00000 −0.0601929
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −8.00000 −0.474713
\(285\) 4.00000 0.236940
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) −2.00000 −0.117444
\(291\) 2.00000 0.117242
\(292\) 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
\(295\) −4.00000 −0.232889
\(296\) −2.00000 −0.116248
\(297\) 4.00000 0.232104
\(298\) −10.0000 −0.579284
\(299\) 2.00000 0.115663
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 6.00000 0.344691
\(304\) 4.00000 0.229416
\(305\) −10.0000 −0.572598
\(306\) −6.00000 −0.342997
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) −2.00000 −0.113228
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 6.00000 0.336463
\(319\) −8.00000 −0.447914
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 20.0000 1.10770
\(327\) −10.0000 −0.553001
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −4.00000 −0.219529
\(333\) −2.00000 −0.109599
\(334\) 8.00000 0.437741
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) −6.00000 −0.325875
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) −1.00000 −0.0538382
\(346\) 14.0000 0.752645
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −2.00000 −0.107211
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 4.00000 0.213201
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) −4.00000 −0.212598
\(355\) −8.00000 −0.424596
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 1.00000 0.0527046
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) −10.0000 −0.522708
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 10.0000 0.520579
\(370\) −2.00000 −0.103975
\(371\) 0 0
\(372\) 0 0
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) −24.0000 −1.24101
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) −4.00000 −0.203331
\(388\) 2.00000 0.101535
\(389\) 38.0000 1.92668 0.963338 0.268290i \(-0.0864585\pi\)
0.963338 + 0.268290i \(0.0864585\pi\)
\(390\) −2.00000 −0.101274
\(391\) 6.00000 0.303433
\(392\) −7.00000 −0.353553
\(393\) 20.0000 1.00887
\(394\) 6.00000 0.302276
\(395\) −8.00000 −0.402524
\(396\) 4.00000 0.201008
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −12.0000 −0.598506
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) −6.00000 −0.297044
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 10.0000 0.493865
\(411\) 2.00000 0.0986527
\(412\) 0 0
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −4.00000 −0.196352
\(416\) −2.00000 −0.0980581
\(417\) −20.0000 −0.979404
\(418\) 16.0000 0.782586
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −6.00000 −0.291043
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −8.00000 −0.386244
\(430\) −4.00000 −0.192897
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) −10.0000 −0.478913
\(437\) −4.00000 −0.191346
\(438\) 10.0000 0.477818
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 4.00000 0.190693
\(441\) −7.00000 −0.333333
\(442\) 12.0000 0.570782
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 18.0000 0.853282
\(446\) 16.0000 0.757622
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 40.0000 1.88353
\(452\) −6.00000 −0.282216
\(453\) 8.00000 0.375873
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 14.0000 0.654177
\(459\) −6.00000 −0.280056
\(460\) −1.00000 −0.0466252
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) −4.00000 −0.184115
\(473\) −16.0000 −0.735681
\(474\) −8.00000 −0.367452
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 16.0000 0.731823
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 1.00000 0.0456435
\(481\) 4.00000 0.182384
\(482\) 18.0000 0.819878
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 2.00000 0.0908153
\(486\) 1.00000 0.0453609
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) −10.0000 −0.452679
\(489\) 20.0000 0.904431
\(490\) −7.00000 −0.316228
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 10.0000 0.450835
\(493\) 12.0000 0.540453
\(494\) −8.00000 −0.359937
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.00000 0.357414
\(502\) 20.0000 0.892644
\(503\) −40.0000 −1.78351 −0.891756 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) −4.00000 −0.177822
\(507\) −9.00000 −0.399704
\(508\) −16.0000 −0.709885
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) −6.00000 −0.265684
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) −2.00000 −0.0877058
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 0 0
\(528\) 4.00000 0.174078
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) 18.0000 0.778936
\(535\) −12.0000 −0.518805
\(536\) −12.0000 −0.518321
\(537\) 4.00000 0.172613
\(538\) 14.0000 0.603583
\(539\) −28.0000 −1.20605
\(540\) 1.00000 0.0430331
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) −2.00000 −0.0858282
\(544\) −6.00000 −0.257248
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 2.00000 0.0854358
\(549\) −10.0000 −0.426790
\(550\) 4.00000 0.170561
\(551\) −8.00000 −0.340811
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) −2.00000 −0.0848953
\(556\) −20.0000 −0.848189
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 18.0000 0.759284
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 4.00000 0.167542
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −8.00000 −0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 19.0000 0.790296
\(579\) 2.00000 0.0831172
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) 24.0000 0.993978
\(584\) 10.0000 0.413803
\(585\) −2.00000 −0.0826898
\(586\) 22.0000 0.908812
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −7.00000 −0.288675
\(589\) 0 0
\(590\) −4.00000 −0.164677
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −16.0000 −0.654836
\(598\) 2.00000 0.0817861
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 1.00000 0.0408248
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 8.00000 0.325515
\(605\) 5.00000 0.203279
\(606\) 6.00000 0.243733
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 4.00000 0.161427
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 8.00000 0.320771
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) 16.0000 0.638978
\(628\) −10.0000 −0.399043
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −8.00000 −0.318223
\(633\) 4.00000 0.158986
\(634\) 30.0000 1.19145
\(635\) −16.0000 −0.634941
\(636\) 6.00000 0.237915
\(637\) 14.0000 0.554700
\(638\) −8.00000 −0.316723
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −12.0000 −0.473602
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) −24.0000 −0.944267
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −10.0000 −0.391031
\(655\) 20.0000 0.781465
\(656\) 10.0000 0.390434
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 4.00000 0.155700
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 28.0000 1.08825
\(663\) 12.0000 0.466041
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 2.00000 0.0774403
\(668\) 8.00000 0.309529
\(669\) 16.0000 0.618596
\(670\) −12.0000 −0.463600
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 2.00000 0.0770371
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 4.00000 0.152944
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) −1.00000 −0.0380693
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −20.0000 −0.758643
\(696\) −2.00000 −0.0758098
\(697\) −60.0000 −2.27266
\(698\) 14.0000 0.529908
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −50.0000 −1.88847 −0.944237 0.329267i \(-0.893198\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) −2.00000 −0.0754851
\(703\) −8.00000 −0.301726
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 34.0000 1.27961
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) −8.00000 −0.300235
\(711\) −8.00000 −0.300023
\(712\) 18.0000 0.674579
\(713\) 0 0
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 4.00000 0.149487
\(717\) 16.0000 0.597531
\(718\) 8.00000 0.298557
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 18.0000 0.669427
\(724\) −2.00000 −0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 5.00000 0.185567
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) 24.0000 0.887672
\(732\) −10.0000 −0.369611
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) −8.00000 −0.295285
\(735\) −7.00000 −0.258199
\(736\) −1.00000 −0.0368605
\(737\) −48.0000 −1.76810
\(738\) 10.0000 0.368105
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) −18.0000 −0.659027
\(747\) −4.00000 −0.146352
\(748\) −24.0000 −0.877527
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 4.00000 0.145671
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 12.0000 0.435860
\(759\) −4.00000 −0.145191
\(760\) 4.00000 0.145095
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) −16.0000 −0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) −32.0000 −1.15621
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 2.00000 0.0719816
\(773\) 54.0000 1.94225 0.971123 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) 38.0000 1.36237
\(779\) 40.0000 1.43315
\(780\) −2.00000 −0.0716115
\(781\) −32.0000 −1.14505
\(782\) 6.00000 0.214560
\(783\) −2.00000 −0.0714742
\(784\) −7.00000 −0.250000
\(785\) −10.0000 −0.356915
\(786\) 20.0000 0.713376
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 6.00000 0.213741
\(789\) −24.0000 −0.854423
\(790\) −8.00000 −0.284627
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) 20.0000 0.710221
\(794\) −2.00000 −0.0709773
\(795\) 6.00000 0.212798
\(796\) −16.0000 −0.567105
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 18.0000 0.635999
\(802\) 10.0000 0.353112
\(803\) 40.0000 1.41157
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) 14.0000 0.492823
\(808\) 6.00000 0.211079
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 1.00000 0.0351364
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 20.0000 0.700569
\(816\) −6.00000 −0.210042
\(817\) −16.0000 −0.559769
\(818\) 26.0000 0.909069
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 2.00000 0.0697580
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) −4.00000 −0.138842
\(831\) −10.0000 −0.346896
\(832\) −2.00000 −0.0693375
\(833\) 42.0000 1.45521
\(834\) −20.0000 −0.692543
\(835\) 8.00000 0.276851
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −2.00000 −0.0689246
\(843\) 18.0000 0.619953
\(844\) 4.00000 0.137686
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −4.00000 −0.137280
\(850\) −6.00000 −0.205798
\(851\) 2.00000 0.0685591
\(852\) −8.00000 −0.274075
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) −12.0000 −0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −8.00000 −0.273115
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −32.0000 −1.08992
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) 34.0000 1.15537
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) −2.00000 −0.0678064
\(871\) 24.0000 0.813209
\(872\) −10.0000 −0.338643
\(873\) 2.00000 0.0676897
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) −50.0000 −1.68838 −0.844190 0.536044i \(-0.819918\pi\)
−0.844190 + 0.536044i \(0.819918\pi\)
\(878\) −8.00000 −0.269987
\(879\) 22.0000 0.742042
\(880\) 4.00000 0.134840
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) −7.00000 −0.235702
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 12.0000 0.403604
\(885\) −4.00000 −0.134459
\(886\) −4.00000 −0.134383
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 18.0000 0.603361
\(891\) 4.00000 0.134005
\(892\) 16.0000 0.535720
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) 2.00000 0.0667781
\(898\) −30.0000 −1.00111
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −2.00000 −0.0664822
\(906\) 8.00000 0.265782
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −20.0000 −0.663723
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 4.00000 0.132453
\(913\) −16.0000 −0.529523
\(914\) 10.0000 0.330771
\(915\) −10.0000 −0.330590
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 4.00000 0.131804
\(922\) −18.0000 −0.592798
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 16.0000 0.525793
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −28.0000 −0.917663
\(932\) −6.00000 −0.196537
\(933\) 8.00000 0.261908
\(934\) 12.0000 0.392652
\(935\) −24.0000 −0.784884
\(936\) −2.00000 −0.0653720
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) −10.0000 −0.325818
\(943\) −10.0000 −0.325645
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) −8.00000 −0.259828
\(949\) −20.0000 −0.649227
\(950\) 4.00000 0.129777
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) −8.00000 −0.258603
\(958\) 32.0000 1.03387
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) −31.0000 −1.00000
\(962\) 4.00000 0.128965
\(963\) −12.0000 −0.386695
\(964\) 18.0000 0.579741
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 5.00000 0.160706
\(969\) −24.0000 −0.770991
\(970\) 2.00000 0.0642161
\(971\) 52.0000 1.66876 0.834380 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) −2.00000 −0.0640513
\(976\) −10.0000 −0.320092
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 20.0000 0.639529
\(979\) 72.0000 2.30113
\(980\) −7.00000 −0.223607
\(981\) −10.0000 −0.319275
\(982\) −4.00000 −0.127645
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 10.0000 0.318788
\(985\) 6.00000 0.191176
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 4.00000 0.127193
\(990\) 4.00000 0.127128
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) −4.00000 −0.126745
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 20.0000 0.633089
\(999\) −2.00000 −0.0632772
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.k.1.1 1
3.2 odd 2 2070.2.a.b.1.1 1
4.3 odd 2 5520.2.a.i.1.1 1
5.2 odd 4 3450.2.d.t.2899.2 2
5.3 odd 4 3450.2.d.t.2899.1 2
5.4 even 2 3450.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.k.1.1 1 1.1 even 1 trivial
2070.2.a.b.1.1 1 3.2 odd 2
3450.2.a.d.1.1 1 5.4 even 2
3450.2.d.t.2899.1 2 5.3 odd 4
3450.2.d.t.2899.2 2 5.2 odd 4
5520.2.a.i.1.1 1 4.3 odd 2