Properties

Label 1470.2.a.p.1.1
Level $1470$
Weight $2$
Character 1470.1
Self dual yes
Analytic conductor $11.738$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.7380090971\)
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 \(\chi\) \(=\) 1470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} +2.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} +8.00000 q^{46} -10.0000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -4.00000 q^{59} -1.00000 q^{60} +10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} +2.00000 q^{67} +4.00000 q^{68} +8.00000 q^{69} -12.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +8.00000 q^{74} +1.00000 q^{75} -2.00000 q^{78} +16.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -16.0000 q^{83} -4.00000 q^{85} -2.00000 q^{86} +2.00000 q^{88} -14.0000 q^{89} -1.00000 q^{90} +8.00000 q^{92} +2.00000 q^{93} -10.0000 q^{94} +1.00000 q^{96} -6.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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
\(8\) 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
\(13\) −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
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 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
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) 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
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 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
\(46\) 8.00000 1.17954
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) −2.00000 −0.277350
\(53\) −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
\(57\) 0 0
\(58\) 0 0
\(59\) −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 \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 2.00000 0.246183
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 4.00000 0.485071
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 8.00000 0.929981
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 2.00000 0.207390
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 1.00000 0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 4.00000 0.396059
\(103\) 20.0000 1.97066 0.985329 0.170664i \(-0.0545913\pi\)
0.985329 + 0.170664i \(0.0545913\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −2.00000 −0.190693
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −7.00000 −0.636364
\(122\) 10.0000 0.905357
\(123\) 2.00000 0.180334
\(124\) 2.00000 0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.00000 −0.176090
\(130\) 2.00000 0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) −1.00000 −0.0860663
\(136\) 4.00000 0.342997
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 8.00000 0.681005
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) −12.0000 −1.00702
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 1.00000 0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) −2.00000 −0.160128
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 16.0000 1.27289
\(159\) −2.00000 −0.158610
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 2.00000 0.156174
\(165\) −2.00000 −0.155700
\(166\) −16.0000 −1.24184
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −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
\(177\) −4.00000 −0.300658
\(178\) −14.0000 −1.04934
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 8.00000 0.589768
\(185\) −8.00000 −0.588172
\(186\) 2.00000 0.146647
\(187\) 8.00000 0.585018
\(188\) −10.0000 −0.729325
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −6.00000 −0.430775
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 2.00000 0.142134
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 1.00000 0.0707107
\(201\) 2.00000 0.141069
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) −2.00000 −0.139686
\(206\) 20.0000 1.39347
\(207\) 8.00000 0.556038
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −2.00000 −0.137361
\(213\) −12.0000 −0.822226
\(214\) 12.0000 0.820303
\(215\) 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
\(221\) −8.00000 −0.538138
\(222\) 8.00000 0.536925
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −2.00000 −0.130744
\(235\) 10.0000 0.652328
\(236\) −4.00000 −0.260378
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 0 0
\(248\) 2.00000 0.127000
\(249\) −16.0000 −1.01396
\(250\) −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
\(253\) 16.0000 1.00591
\(254\) −12.0000 −0.752947
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 2.00000 0.123091
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 2.00000 0.122169
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 2.00000 0.120605
\(276\) 8.00000 0.481543
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 4.00000 0.239904
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) −10.0000 −0.595491
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) −10.0000 −0.585206
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 8.00000 0.464991
\(297\) 2.00000 0.116052
\(298\) −16.0000 −0.926855
\(299\) −16.0000 −0.925304
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 4.00000 0.228665
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 20.0000 1.13776
\(310\) −2.00000 −0.113592
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) −2.00000 −0.113228
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −2.00000 −0.112154
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −10.0000 −0.553849
\(327\) −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 \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −16.0000 −0.878114
\(333\) 8.00000 0.438397
\(334\) −18.0000 −0.984916
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −9.00000 −0.489535
\(339\) −14.0000 −0.760376
\(340\) −4.00000 −0.216930
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) −8.00000 −0.430706
\(346\) 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
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 2.00000 0.106600
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) −4.00000 −0.212598
\(355\) 12.0000 0.636894
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) 22.0000 1.15629
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 10.0000 0.522708
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 8.00000 0.417029
\(369\) 2.00000 0.104116
\(370\) −8.00000 −0.415900
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) −36.0000 −1.86401 −0.932005 0.362446i \(-0.881942\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 8.00000 0.413670
\(375\) −1.00000 −0.0516398
\(376\) −10.0000 −0.515711
\(377\) 0 0
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −18.0000 −0.916176
\(387\) −2.00000 −0.101666
\(388\) −6.00000 −0.304604
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 2.00000 0.101274
\(391\) 32.0000 1.61831
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 18.0000 0.906827
\(395\) −16.0000 −0.805047
\(396\) 2.00000 0.100504
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 2.00000 0.0997509
\(403\) −4.00000 −0.199254
\(404\) 14.0000 0.696526
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 16.0000 0.793091
\(408\) 4.00000 0.198030
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) −2.00000 −0.0987730
\(411\) −2.00000 −0.0986527
\(412\) 20.0000 0.985329
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 16.0000 0.785409
\(416\) −2.00000 −0.0980581
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) −4.00000 −0.194717
\(423\) −10.0000 −0.486217
\(424\) −2.00000 −0.0971286
\(425\) 4.00000 0.194029
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) −4.00000 −0.193122
\(430\) 2.00000 0.0964486
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 8.00000 0.379663
\(445\) 14.0000 0.663664
\(446\) −16.0000 −0.757622
\(447\) −16.0000 −0.756774
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 1.00000 0.0471405
\(451\) 4.00000 0.188353
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) −42.0000 −1.96468 −0.982339 0.187112i \(-0.940087\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) −10.0000 −0.467269
\(459\) 4.00000 0.186704
\(460\) −8.00000 −0.373002
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) −14.0000 −0.648537
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) 10.0000 0.461266
\(471\) 10.0000 0.460776
\(472\) −4.00000 −0.184115
\(473\) −4.00000 −0.183920
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) −8.00000 −0.365911
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −16.0000 −0.729537
\(482\) 20.0000 0.910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 6.00000 0.272446
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 10.0000 0.452679
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 2.00000 0.0901670
\(493\) 0 0
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) −16.0000 −0.716977
\(499\) 40.0000 1.79065 0.895323 0.445418i \(-0.146945\pi\)
0.895323 + 0.445418i \(0.146945\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −18.0000 −0.804181
\(502\) 20.0000 0.892644
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 16.0000 0.711287
\(507\) −9.00000 −0.399704
\(508\) −12.0000 −0.532414
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) −4.00000 −0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) −20.0000 −0.881305
\(516\) −2.00000 −0.0880451
\(517\) −20.0000 −0.879599
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 2.00000 0.0877058
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 2.00000 0.0870388
\(529\) 41.0000 1.78261
\(530\) 2.00000 0.0868744
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) −14.0000 −0.605839
\(535\) −12.0000 −0.518805
\(536\) 2.00000 0.0863868
\(537\) −2.00000 −0.0863064
\(538\) 10.0000 0.431131
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 14.0000 0.601351
\(543\) 22.0000 0.944110
\(544\) 4.00000 0.171499
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −14.0000 −0.598597 −0.299298 0.954160i \(-0.596753\pi\)
−0.299298 + 0.954160i \(0.596753\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 10.0000 0.426790
\(550\) 2.00000 0.0852803
\(551\) 0 0
\(552\) 8.00000 0.340503
\(553\) 0 0
\(554\) −28.0000 −1.18961
\(555\) −8.00000 −0.339581
\(556\) 4.00000 0.169638
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 2.00000 0.0846668
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 14.0000 0.590554
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) −10.0000 −0.421076
\(565\) 14.0000 0.588984
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 0.333623
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) −4.00000 −0.165663
\(584\) −10.0000 −0.413803
\(585\) 2.00000 0.0826898
\(586\) −30.0000 −1.23929
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 4.00000 0.164677
\(591\) 18.0000 0.740421
\(592\) 8.00000 0.328798
\(593\) 20.0000 0.821302 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −16.0000 −0.655386
\(597\) −10.0000 −0.409273
\(598\) −16.0000 −0.654289
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 1.00000 0.0408248
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) 7.00000 0.284590
\(606\) 14.0000 0.568711
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) 20.0000 0.809113
\(612\) 4.00000 0.161690
\(613\) −8.00000 −0.323117 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) 20.0000 0.807134
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 20.0000 0.804518
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 8.00000 0.321029
\(622\) 20.0000 0.801927
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 16.0000 0.636446
\(633\) −4.00000 −0.158986
\(634\) 6.00000 0.238290
\(635\) 12.0000 0.476205
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) −1.00000 −0.0395285
\(641\) 42.0000 1.65890 0.829450 0.558581i \(-0.188654\pi\)
0.829450 + 0.558581i \(0.188654\pi\)
\(642\) 12.0000 0.473602
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) 2.00000 0.0786281 0.0393141 0.999227i \(-0.487483\pi\)
0.0393141 + 0.999227i \(0.487483\pi\)
\(648\) 1.00000 0.0392837
\(649\) −8.00000 −0.314027
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −10.0000 −0.391630
\(653\) 2.00000 0.0782660 0.0391330 0.999234i \(-0.487540\pi\)
0.0391330 + 0.999234i \(0.487540\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 12.0000 0.468879
\(656\) 2.00000 0.0780869
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −34.0000 −1.32445 −0.662226 0.749304i \(-0.730388\pi\)
−0.662226 + 0.749304i \(0.730388\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −4.00000 −0.155464
\(663\) −8.00000 −0.310694
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 0 0
\(668\) −18.0000 −0.696441
\(669\) −16.0000 −0.618596
\(670\) −2.00000 −0.0772667
\(671\) 20.0000 0.772091
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) −14.0000 −0.537667
\(679\) 0 0
\(680\) −4.00000 −0.153393
\(681\) 12.0000 0.459841
\(682\) 4.00000 0.153168
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) −2.00000 −0.0762493
\(689\) 4.00000 0.152388
\(690\) −8.00000 −0.304555
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) −10.0000 −0.378506
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) 16.0000 0.604312 0.302156 0.953259i \(-0.402294\pi\)
0.302156 + 0.953259i \(0.402294\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 10.0000 0.376622
\(706\) −24.0000 −0.903252
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 42.0000 1.57734 0.788672 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 12.0000 0.450352
\(711\) 16.0000 0.600047
\(712\) −14.0000 −0.524672
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −2.00000 −0.0747435
\(717\) −8.00000 −0.298765
\(718\) −20.0000 −0.746393
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) 20.0000 0.743808
\(724\) 22.0000 0.817624
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) −8.00000 −0.295891
\(732\) 10.0000 0.369611
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 4.00000 0.147342
\(738\) 2.00000 0.0736210
\(739\) 48.0000 1.76571 0.882854 0.469647i \(-0.155619\pi\)
0.882854 + 0.469647i \(0.155619\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 2.00000 0.0733236
\(745\) 16.0000 0.586195
\(746\) −36.0000 −1.31805
\(747\) −16.0000 −0.585409
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −10.0000 −0.364662
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 8.00000 0.290573
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −12.0000 −0.434714
\(763\) 0 0
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) −14.0000 −0.505841
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) −18.0000 −0.647834
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 2.00000 0.0718421
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) −24.0000 −0.860442
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) −24.0000 −0.858788
\(782\) 32.0000 1.14432
\(783\) 0 0
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) −12.0000 −0.428026
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) −20.0000 −0.710221
\(794\) −14.0000 −0.496841
\(795\) 2.00000 0.0709327
\(796\) −10.0000 −0.354441
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 1.00000 0.0353553
\(801\) −14.0000 −0.494666
\(802\) −14.0000 −0.494357
\(803\) −20.0000 −0.705785
\(804\) 2.00000 0.0705346
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 10.0000 0.352017
\(808\) 14.0000 0.492518
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) −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
\(813\) 14.0000 0.491001
\(814\) 16.0000 0.560800
\(815\) 10.0000 0.350285
\(816\) 4.00000 0.140028
\(817\) 0 0
\(818\) 32.0000 1.11885
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 20.0000 0.696733
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 8.00000 0.278187 0.139094 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) 8.00000 0.278019
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 16.0000 0.555368
\(831\) −28.0000 −0.971309
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 4.00000 0.138509
\(835\) 18.0000 0.622916
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) −36.0000 −1.24360
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 38.0000 1.30957
\(843\) 14.0000 0.482186
\(844\) −4.00000 −0.137686
\(845\) 9.00000 0.309609
\(846\) −10.0000 −0.343807
\(847\) 0 0
\(848\) −2.00000 −0.0686803
\(849\) −4.00000 −0.137280
\(850\) 4.00000 0.137199
\(851\) 64.0000 2.19389
\(852\) −12.0000 −0.411113
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) −4.00000 −0.136558
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) −38.0000 −1.29129
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) −2.00000 −0.0677285
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 12.0000 0.405211 0.202606 0.979260i \(-0.435059\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) 26.0000 0.877457
\(879\) −30.0000 −1.01187
\(880\) −2.00000 −0.0674200
\(881\) 46.0000 1.54978 0.774890 0.632096i \(-0.217805\pi\)
0.774890 + 0.632096i \(0.217805\pi\)
\(882\) 0 0
\(883\) 34.0000 1.14419 0.572096 0.820187i \(-0.306131\pi\)
0.572096 + 0.820187i \(0.306131\pi\)
\(884\) −8.00000 −0.269069
\(885\) 4.00000 0.134459
\(886\) 28.0000 0.940678
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 8.00000 0.268462
\(889\) 0 0
\(890\) 14.0000 0.469281
\(891\) 2.00000 0.0670025
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −16.0000 −0.535120
\(895\) 2.00000 0.0668526
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 6.00000 0.200223
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) −8.00000 −0.266519
\(902\) 4.00000 0.133185
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 12.0000 0.398234
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) −32.0000 −1.05905
\(914\) −42.0000 −1.38924
\(915\) −10.0000 −0.330590
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) −8.00000 −0.263752
\(921\) 20.0000 0.659022
\(922\) 18.0000 0.592798
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 16.0000 0.525793
\(927\) 20.0000 0.656886
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) −2.00000 −0.0655826
\(931\) 0 0
\(932\) −14.0000 −0.458585
\(933\) 20.0000 0.654771
\(934\) −8.00000 −0.261768
\(935\) −8.00000 −0.261628
\(936\) −2.00000 −0.0653720
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 10.0000 0.326164
\(941\) −50.0000 −1.62995 −0.814977 0.579494i \(-0.803250\pi\)
−0.814977 + 0.579494i \(0.803250\pi\)
\(942\) 10.0000 0.325818
\(943\) 16.0000 0.521032
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 16.0000 0.519656
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −58.0000 −1.87880 −0.939402 0.342817i \(-0.888619\pi\)
−0.939402 + 0.342817i \(0.888619\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) −4.00000 −0.129234
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −27.0000 −0.870968
\(962\) −16.0000 −0.515861
\(963\) 12.0000 0.386695
\(964\) 20.0000 0.644157
\(965\) 18.0000 0.579441
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 28.0000 0.897178
\(975\) −2.00000 −0.0640513
\(976\) 10.0000 0.320092
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) −10.0000 −0.319765
\(979\) −28.0000 −0.894884
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 6.00000 0.191468
\(983\) 46.0000 1.46717 0.733586 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 2.00000 0.0637577
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) −2.00000 −0.0635642
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 2.00000 0.0635001
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 10.0000 0.317021
\(996\) −16.0000 −0.506979
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 40.0000 1.26618
\(999\) 8.00000 0.253109
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 1470.2.a.p.1.1 yes 1
3.2 odd 2 4410.2.a.n.1.1 1
5.4 even 2 7350.2.a.o.1.1 1
7.2 even 3 1470.2.i.c.361.1 2
7.3 odd 6 1470.2.i.g.961.1 2
7.4 even 3 1470.2.i.c.961.1 2
7.5 odd 6 1470.2.i.g.361.1 2
7.6 odd 2 1470.2.a.n.1.1 1
21.20 even 2 4410.2.a.e.1.1 1
35.34 odd 2 7350.2.a.bh.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1470.2.a.n.1.1 1 7.6 odd 2
1470.2.a.p.1.1 yes 1 1.1 even 1 trivial
1470.2.i.c.361.1 2 7.2 even 3
1470.2.i.c.961.1 2 7.4 even 3
1470.2.i.g.361.1 2 7.5 odd 6
1470.2.i.g.961.1 2 7.3 odd 6
4410.2.a.e.1.1 1 21.20 even 2
4410.2.a.n.1.1 1 3.2 odd 2
7350.2.a.o.1.1 1 5.4 even 2
7350.2.a.bh.1.1 1 35.34 odd 2