Properties

Label 370.2.a.c.1.1
Level $370$
Weight $2$
Character 370.1
Self dual yes
Analytic conductor $2.954$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
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 \(\chi\) \(=\) 370.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} +2.00000 q^{12} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -3.00000 q^{22} +2.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} -2.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} -3.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} -1.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} +3.00000 q^{41} -2.00000 q^{42} -1.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} +4.00000 q^{47} +2.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} +13.0000 q^{53} +4.00000 q^{54} +3.00000 q^{55} -1.00000 q^{56} -12.0000 q^{57} +3.00000 q^{58} +2.00000 q^{60} -15.0000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +3.00000 q^{68} +4.00000 q^{69} -1.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} +1.00000 q^{74} +2.00000 q^{75} -6.00000 q^{76} +3.00000 q^{77} -8.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -3.00000 q^{82} -4.00000 q^{83} +2.00000 q^{84} +3.00000 q^{85} +1.00000 q^{86} -6.00000 q^{87} -3.00000 q^{88} -18.0000 q^{89} -1.00000 q^{90} +2.00000 q^{92} +6.00000 q^{93} -4.00000 q^{94} -6.00000 q^{95} -2.00000 q^{96} -7.00000 q^{97} +6.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.00000 −0.816497
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 2.00000 0.577350
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) −3.00000 −0.639602
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −2.00000 −0.408248
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −2.00000 −0.365148
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) −3.00000 −0.514496
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −2.00000 −0.308607
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 3.00000 0.452267
\(45\) 1.00000 0.149071
\(46\) −2.00000 −0.294884
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 2.00000 0.288675
\(49\) −6.00000 −0.857143
\(50\) −1.00000 −0.141421
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 13.0000 1.78569 0.892844 0.450367i \(-0.148707\pi\)
0.892844 + 0.450367i \(0.148707\pi\)
\(54\) 4.00000 0.544331
\(55\) 3.00000 0.404520
\(56\) −1.00000 −0.133631
\(57\) −12.0000 −1.58944
\(58\) 3.00000 0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 2.00000 0.258199
\(61\) −15.0000 −1.92055 −0.960277 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) −3.00000 −0.381000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 3.00000 0.363803
\(69\) 4.00000 0.481543
\(70\) −1.00000 −0.119523
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 1.00000 0.116248
\(75\) 2.00000 0.230940
\(76\) −6.00000 −0.688247
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) −3.00000 −0.331295
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 2.00000 0.218218
\(85\) 3.00000 0.325396
\(86\) 1.00000 0.107833
\(87\) −6.00000 −0.643268
\(88\) −3.00000 −0.319801
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) 6.00000 0.622171
\(94\) −4.00000 −0.412568
\(95\) −6.00000 −0.615587
\(96\) −2.00000 −0.204124
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) 6.00000 0.606092
\(99\) 3.00000 0.301511
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −6.00000 −0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) −13.0000 −1.26267
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) −4.00000 −0.384900
\(109\) −3.00000 −0.287348 −0.143674 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) −3.00000 −0.286039
\(111\) −2.00000 −0.189832
\(112\) 1.00000 0.0944911
\(113\) −7.00000 −0.658505 −0.329252 0.944242i \(-0.606797\pi\)
−0.329252 + 0.944242i \(0.606797\pi\)
\(114\) 12.0000 1.12390
\(115\) 2.00000 0.186501
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) −2.00000 −0.182574
\(121\) −2.00000 −0.181818
\(122\) 15.0000 1.35804
\(123\) 6.00000 0.541002
\(124\) 3.00000 0.269408
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 6.00000 0.522233
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) −3.00000 −0.257248
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) −4.00000 −0.340503
\(139\) 3.00000 0.254457 0.127228 0.991873i \(-0.459392\pi\)
0.127228 + 0.991873i \(0.459392\pi\)
\(140\) 1.00000 0.0845154
\(141\) 8.00000 0.673722
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) −12.0000 −0.989743
\(148\) −1.00000 −0.0821995
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −2.00000 −0.163299
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 6.00000 0.486664
\(153\) 3.00000 0.242536
\(154\) −3.00000 −0.241747
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) 8.00000 0.636446
\(159\) 26.0000 2.06193
\(160\) −1.00000 −0.0790569
\(161\) 2.00000 0.157622
\(162\) 11.0000 0.864242
\(163\) −5.00000 −0.391630 −0.195815 0.980641i \(-0.562735\pi\)
−0.195815 + 0.980641i \(0.562735\pi\)
\(164\) 3.00000 0.234261
\(165\) 6.00000 0.467099
\(166\) 4.00000 0.310460
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −2.00000 −0.154303
\(169\) −13.0000 −1.00000
\(170\) −3.00000 −0.230089
\(171\) −6.00000 −0.458831
\(172\) −1.00000 −0.0762493
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 6.00000 0.454859
\(175\) 1.00000 0.0755929
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) 18.0000 1.34916
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −30.0000 −2.21766
\(184\) −2.00000 −0.147442
\(185\) −1.00000 −0.0735215
\(186\) −6.00000 −0.439941
\(187\) 9.00000 0.658145
\(188\) 4.00000 0.291730
\(189\) −4.00000 −0.290957
\(190\) 6.00000 0.435286
\(191\) 21.0000 1.51951 0.759753 0.650211i \(-0.225320\pi\)
0.759753 + 0.650211i \(0.225320\pi\)
\(192\) 2.00000 0.144338
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 7.00000 0.502571
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −3.00000 −0.213201
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) −3.00000 −0.210559
\(204\) 6.00000 0.420084
\(205\) 3.00000 0.209529
\(206\) −8.00000 −0.557386
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) −18.0000 −1.24509
\(210\) −2.00000 −0.138013
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 13.0000 0.892844
\(213\) −4.00000 −0.274075
\(214\) −2.00000 −0.136717
\(215\) −1.00000 −0.0681994
\(216\) 4.00000 0.272166
\(217\) 3.00000 0.203653
\(218\) 3.00000 0.203186
\(219\) 0 0
\(220\) 3.00000 0.202260
\(221\) 0 0
\(222\) 2.00000 0.134231
\(223\) 23.0000 1.54019 0.770097 0.637927i \(-0.220208\pi\)
0.770097 + 0.637927i \(0.220208\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 7.00000 0.465633
\(227\) 13.0000 0.862840 0.431420 0.902151i \(-0.358013\pi\)
0.431420 + 0.902151i \(0.358013\pi\)
\(228\) −12.0000 −0.794719
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −2.00000 −0.131876
\(231\) 6.00000 0.394771
\(232\) 3.00000 0.196960
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) −3.00000 −0.194461
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 2.00000 0.129099
\(241\) 24.0000 1.54598 0.772988 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(242\) 2.00000 0.128565
\(243\) −10.0000 −0.641500
\(244\) −15.0000 −0.960277
\(245\) −6.00000 −0.383326
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) −3.00000 −0.190500
\(249\) −8.00000 −0.506979
\(250\) −1.00000 −0.0632456
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 1.00000 0.0629941
\(253\) 6.00000 0.377217
\(254\) 4.00000 0.250982
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 2.00000 0.124515
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 12.0000 0.741362
\(263\) −1.00000 −0.0616626 −0.0308313 0.999525i \(-0.509815\pi\)
−0.0308313 + 0.999525i \(0.509815\pi\)
\(264\) −6.00000 −0.369274
\(265\) 13.0000 0.798584
\(266\) 6.00000 0.367884
\(267\) −36.0000 −2.20316
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 4.00000 0.243432
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) 3.00000 0.180907
\(276\) 4.00000 0.240772
\(277\) −20.0000 −1.20168 −0.600842 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) −3.00000 −0.179928
\(279\) 3.00000 0.179605
\(280\) −1.00000 −0.0597614
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) −8.00000 −0.476393
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −2.00000 −0.118678
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 3.00000 0.177084
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) 3.00000 0.176166
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 3.00000 0.175262 0.0876309 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) 12.0000 0.699854
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) −12.0000 −0.696311
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 2.00000 0.115470
\(301\) −1.00000 −0.0576390
\(302\) −16.0000 −0.920697
\(303\) −20.0000 −1.14897
\(304\) −6.00000 −0.344124
\(305\) −15.0000 −0.858898
\(306\) −3.00000 −0.171499
\(307\) 34.0000 1.94048 0.970241 0.242140i \(-0.0778494\pi\)
0.970241 + 0.242140i \(0.0778494\pi\)
\(308\) 3.00000 0.170941
\(309\) 16.0000 0.910208
\(310\) −3.00000 −0.170389
\(311\) 25.0000 1.41762 0.708810 0.705399i \(-0.249232\pi\)
0.708810 + 0.705399i \(0.249232\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 3.00000 0.169300
\(315\) 1.00000 0.0563436
\(316\) −8.00000 −0.450035
\(317\) 21.0000 1.17948 0.589739 0.807594i \(-0.299231\pi\)
0.589739 + 0.807594i \(0.299231\pi\)
\(318\) −26.0000 −1.45801
\(319\) −9.00000 −0.503903
\(320\) 1.00000 0.0559017
\(321\) 4.00000 0.223258
\(322\) −2.00000 −0.111456
\(323\) −18.0000 −1.00155
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 5.00000 0.276924
\(327\) −6.00000 −0.331801
\(328\) −3.00000 −0.165647
\(329\) 4.00000 0.220527
\(330\) −6.00000 −0.330289
\(331\) 30.0000 1.64895 0.824475 0.565899i \(-0.191471\pi\)
0.824475 + 0.565899i \(0.191471\pi\)
\(332\) −4.00000 −0.219529
\(333\) −1.00000 −0.0547997
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 12.0000 0.653682 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) 13.0000 0.707107
\(339\) −14.0000 −0.760376
\(340\) 3.00000 0.162698
\(341\) 9.00000 0.487377
\(342\) 6.00000 0.324443
\(343\) −13.0000 −0.701934
\(344\) 1.00000 0.0539164
\(345\) 4.00000 0.215353
\(346\) −9.00000 −0.483843
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) −6.00000 −0.321634
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 19.0000 1.01127 0.505634 0.862748i \(-0.331259\pi\)
0.505634 + 0.862748i \(0.331259\pi\)
\(354\) 0 0
\(355\) −2.00000 −0.106149
\(356\) −18.0000 −0.953998
\(357\) 6.00000 0.317554
\(358\) −24.0000 −1.26844
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.0000 0.894737
\(362\) −2.00000 −0.105118
\(363\) −4.00000 −0.209946
\(364\) 0 0
\(365\) 0 0
\(366\) 30.0000 1.56813
\(367\) 13.0000 0.678594 0.339297 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) 2.00000 0.104257
\(369\) 3.00000 0.156174
\(370\) 1.00000 0.0519875
\(371\) 13.0000 0.674926
\(372\) 6.00000 0.311086
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −9.00000 −0.465379
\(375\) 2.00000 0.103280
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −6.00000 −0.307794
\(381\) −8.00000 −0.409852
\(382\) −21.0000 −1.07445
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −2.00000 −0.102062
\(385\) 3.00000 0.152894
\(386\) 10.0000 0.508987
\(387\) −1.00000 −0.0508329
\(388\) −7.00000 −0.355371
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 6.00000 0.303046
\(393\) −24.0000 −1.21064
\(394\) 18.0000 0.906827
\(395\) −8.00000 −0.402524
\(396\) 3.00000 0.150756
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −8.00000 −0.401004
\(399\) −12.0000 −0.600751
\(400\) 1.00000 0.0500000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) −11.0000 −0.546594
\(406\) 3.00000 0.148888
\(407\) −3.00000 −0.148704
\(408\) −6.00000 −0.297044
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) −3.00000 −0.148159
\(411\) 16.0000 0.789222
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −2.00000 −0.0982946
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) 18.0000 0.880409
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 2.00000 0.0975900
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 3.00000 0.146038
\(423\) 4.00000 0.194487
\(424\) −13.0000 −0.631336
\(425\) 3.00000 0.145521
\(426\) 4.00000 0.193801
\(427\) −15.0000 −0.725901
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) 1.00000 0.0482243
\(431\) −31.0000 −1.49322 −0.746609 0.665263i \(-0.768319\pi\)
−0.746609 + 0.665263i \(0.768319\pi\)
\(432\) −4.00000 −0.192450
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) −3.00000 −0.144005
\(435\) −6.00000 −0.287678
\(436\) −3.00000 −0.143674
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) 37.0000 1.76591 0.882957 0.469454i \(-0.155549\pi\)
0.882957 + 0.469454i \(0.155549\pi\)
\(440\) −3.00000 −0.143019
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −34.0000 −1.61539 −0.807694 0.589601i \(-0.799285\pi\)
−0.807694 + 0.589601i \(0.799285\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −18.0000 −0.853282
\(446\) −23.0000 −1.08908
\(447\) 36.0000 1.70274
\(448\) 1.00000 0.0472456
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 9.00000 0.423793
\(452\) −7.00000 −0.329252
\(453\) 32.0000 1.50349
\(454\) −13.0000 −0.610120
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 9.00000 0.421002 0.210501 0.977594i \(-0.432490\pi\)
0.210501 + 0.977594i \(0.432490\pi\)
\(458\) 6.00000 0.280362
\(459\) −12.0000 −0.560112
\(460\) 2.00000 0.0932505
\(461\) −1.00000 −0.0465746 −0.0232873 0.999729i \(-0.507413\pi\)
−0.0232873 + 0.999729i \(0.507413\pi\)
\(462\) −6.00000 −0.279145
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) −3.00000 −0.139272
\(465\) 6.00000 0.278243
\(466\) −4.00000 −0.185296
\(467\) −37.0000 −1.71216 −0.856078 0.516847i \(-0.827106\pi\)
−0.856078 + 0.516847i \(0.827106\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.00000 −0.184506
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 16.0000 0.734904
\(475\) −6.00000 −0.275299
\(476\) 3.00000 0.137505
\(477\) 13.0000 0.595229
\(478\) −9.00000 −0.411650
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) −24.0000 −1.09317
\(483\) 4.00000 0.182006
\(484\) −2.00000 −0.0909091
\(485\) −7.00000 −0.317854
\(486\) 10.0000 0.453609
\(487\) 34.0000 1.54069 0.770344 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) 15.0000 0.679018
\(489\) −10.0000 −0.452216
\(490\) 6.00000 0.271052
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 6.00000 0.270501
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 3.00000 0.134704
\(497\) −2.00000 −0.0897123
\(498\) 8.00000 0.358489
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) 1.00000 0.0447214
\(501\) 24.0000 1.07224
\(502\) −6.00000 −0.267793
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −10.0000 −0.444994
\(506\) −6.00000 −0.266733
\(507\) −26.0000 −1.15470
\(508\) −4.00000 −0.177471
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) −6.00000 −0.265684
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 24.0000 1.05963
\(514\) 22.0000 0.970378
\(515\) 8.00000 0.352522
\(516\) −2.00000 −0.0880451
\(517\) 12.0000 0.527759
\(518\) 1.00000 0.0439375
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −23.0000 −1.00765 −0.503824 0.863806i \(-0.668074\pi\)
−0.503824 + 0.863806i \(0.668074\pi\)
\(522\) 3.00000 0.131306
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) −12.0000 −0.524222
\(525\) 2.00000 0.0872872
\(526\) 1.00000 0.0436021
\(527\) 9.00000 0.392046
\(528\) 6.00000 0.261116
\(529\) −19.0000 −0.826087
\(530\) −13.0000 −0.564684
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) 0 0
\(534\) 36.0000 1.55787
\(535\) 2.00000 0.0864675
\(536\) 0 0
\(537\) 48.0000 2.07135
\(538\) 10.0000 0.431131
\(539\) −18.0000 −0.775315
\(540\) −4.00000 −0.172133
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 10.0000 0.429537
\(543\) 4.00000 0.171656
\(544\) −3.00000 −0.128624
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 23.0000 0.983409 0.491704 0.870762i \(-0.336374\pi\)
0.491704 + 0.870762i \(0.336374\pi\)
\(548\) 8.00000 0.341743
\(549\) −15.0000 −0.640184
\(550\) −3.00000 −0.127920
\(551\) 18.0000 0.766826
\(552\) −4.00000 −0.170251
\(553\) −8.00000 −0.340195
\(554\) 20.0000 0.849719
\(555\) −2.00000 −0.0848953
\(556\) 3.00000 0.127228
\(557\) −40.0000 −1.69485 −0.847427 0.530912i \(-0.821850\pi\)
−0.847427 + 0.530912i \(0.821850\pi\)
\(558\) −3.00000 −0.127000
\(559\) 0 0
\(560\) 1.00000 0.0422577
\(561\) 18.0000 0.759961
\(562\) 24.0000 1.01238
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) 8.00000 0.336861
\(565\) −7.00000 −0.294492
\(566\) 20.0000 0.840663
\(567\) −11.0000 −0.461957
\(568\) 2.00000 0.0839181
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 12.0000 0.502625
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 0 0
\(573\) 42.0000 1.75458
\(574\) −3.00000 −0.125218
\(575\) 2.00000 0.0834058
\(576\) 1.00000 0.0416667
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 8.00000 0.332756
\(579\) −20.0000 −0.831172
\(580\) −3.00000 −0.124568
\(581\) −4.00000 −0.165948
\(582\) 14.0000 0.580319
\(583\) 39.0000 1.61521
\(584\) 0 0
\(585\) 0 0
\(586\) −3.00000 −0.123929
\(587\) 35.0000 1.44460 0.722302 0.691577i \(-0.243084\pi\)
0.722302 + 0.691577i \(0.243084\pi\)
\(588\) −12.0000 −0.494872
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) −1.00000 −0.0410997
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 12.0000 0.492366
\(595\) 3.00000 0.122988
\(596\) 18.0000 0.737309
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −2.00000 −0.0816497
\(601\) 21.0000 0.856608 0.428304 0.903635i \(-0.359111\pi\)
0.428304 + 0.903635i \(0.359111\pi\)
\(602\) 1.00000 0.0407570
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) −2.00000 −0.0813116
\(606\) 20.0000 0.812444
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 6.00000 0.243332
\(609\) −6.00000 −0.243132
\(610\) 15.0000 0.607332
\(611\) 0 0
\(612\) 3.00000 0.121268
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) −34.0000 −1.37213
\(615\) 6.00000 0.241943
\(616\) −3.00000 −0.120873
\(617\) 36.0000 1.44931 0.724653 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) −16.0000 −0.643614
\(619\) −37.0000 −1.48716 −0.743578 0.668649i \(-0.766873\pi\)
−0.743578 + 0.668649i \(0.766873\pi\)
\(620\) 3.00000 0.120483
\(621\) −8.00000 −0.321029
\(622\) −25.0000 −1.00241
\(623\) −18.0000 −0.721155
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −26.0000 −1.03917
\(627\) −36.0000 −1.43770
\(628\) −3.00000 −0.119713
\(629\) −3.00000 −0.119618
\(630\) −1.00000 −0.0398410
\(631\) −41.0000 −1.63218 −0.816092 0.577922i \(-0.803864\pi\)
−0.816092 + 0.577922i \(0.803864\pi\)
\(632\) 8.00000 0.318223
\(633\) −6.00000 −0.238479
\(634\) −21.0000 −0.834017
\(635\) −4.00000 −0.158735
\(636\) 26.0000 1.03097
\(637\) 0 0
\(638\) 9.00000 0.356313
\(639\) −2.00000 −0.0791188
\(640\) −1.00000 −0.0395285
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) −4.00000 −0.157867
\(643\) 11.0000 0.433798 0.216899 0.976194i \(-0.430406\pi\)
0.216899 + 0.976194i \(0.430406\pi\)
\(644\) 2.00000 0.0788110
\(645\) −2.00000 −0.0787499
\(646\) 18.0000 0.708201
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) −5.00000 −0.195815
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 6.00000 0.234619
\(655\) −12.0000 −0.468879
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) −4.00000 −0.155936
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 6.00000 0.233550
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) −30.0000 −1.16598
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) −6.00000 −0.232670
\(666\) 1.00000 0.0387492
\(667\) −6.00000 −0.232321
\(668\) 12.0000 0.464294
\(669\) 46.0000 1.77846
\(670\) 0 0
\(671\) −45.0000 −1.73721
\(672\) −2.00000 −0.0771517
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −12.0000 −0.462223
\(675\) −4.00000 −0.153960
\(676\) −13.0000 −0.500000
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) 14.0000 0.537667
\(679\) −7.00000 −0.268635
\(680\) −3.00000 −0.115045
\(681\) 26.0000 0.996322
\(682\) −9.00000 −0.344628
\(683\) −31.0000 −1.18618 −0.593091 0.805135i \(-0.702093\pi\)
−0.593091 + 0.805135i \(0.702093\pi\)
\(684\) −6.00000 −0.229416
\(685\) 8.00000 0.305664
\(686\) 13.0000 0.496342
\(687\) −12.0000 −0.457829
\(688\) −1.00000 −0.0381246
\(689\) 0 0
\(690\) −4.00000 −0.152277
\(691\) 5.00000 0.190209 0.0951045 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) 9.00000 0.342129
\(693\) 3.00000 0.113961
\(694\) 28.0000 1.06287
\(695\) 3.00000 0.113796
\(696\) 6.00000 0.227429
\(697\) 9.00000 0.340899
\(698\) 22.0000 0.832712
\(699\) 8.00000 0.302588
\(700\) 1.00000 0.0377964
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 6.00000 0.226294
\(704\) 3.00000 0.113067
\(705\) 8.00000 0.301297
\(706\) −19.0000 −0.715074
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) 2.00000 0.0750587
\(711\) −8.00000 −0.300023
\(712\) 18.0000 0.674579
\(713\) 6.00000 0.224702
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 18.0000 0.672222
\(718\) −16.0000 −0.597115
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 1.00000 0.0372678
\(721\) 8.00000 0.297936
\(722\) −17.0000 −0.632674
\(723\) 48.0000 1.78514
\(724\) 2.00000 0.0743294
\(725\) −3.00000 −0.111417
\(726\) 4.00000 0.148454
\(727\) 52.0000 1.92857 0.964287 0.264861i \(-0.0853260\pi\)
0.964287 + 0.264861i \(0.0853260\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) −30.0000 −1.10883
\(733\) −41.0000 −1.51437 −0.757185 0.653201i \(-0.773426\pi\)
−0.757185 + 0.653201i \(0.773426\pi\)
\(734\) −13.0000 −0.479839
\(735\) −12.0000 −0.442627
\(736\) −2.00000 −0.0737210
\(737\) 0 0
\(738\) −3.00000 −0.110432
\(739\) 27.0000 0.993211 0.496606 0.867976i \(-0.334580\pi\)
0.496606 + 0.867976i \(0.334580\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) −13.0000 −0.477245
\(743\) 3.00000 0.110059 0.0550297 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) −6.00000 −0.219971
\(745\) 18.0000 0.659469
\(746\) −22.0000 −0.805477
\(747\) −4.00000 −0.146352
\(748\) 9.00000 0.329073
\(749\) 2.00000 0.0730784
\(750\) −2.00000 −0.0730297
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 4.00000 0.145865
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) −4.00000 −0.145479
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) −16.0000 −0.581146
\(759\) 12.0000 0.435572
\(760\) 6.00000 0.217643
\(761\) −21.0000 −0.761249 −0.380625 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) 8.00000 0.289809
\(763\) −3.00000 −0.108607
\(764\) 21.0000 0.759753
\(765\) 3.00000 0.108465
\(766\) 12.0000 0.433578
\(767\) 0 0
\(768\) 2.00000 0.0721688
\(769\) 8.00000 0.288487 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) −3.00000 −0.108112
\(771\) −44.0000 −1.58462
\(772\) −10.0000 −0.359908
\(773\) −41.0000 −1.47467 −0.737334 0.675529i \(-0.763915\pi\)
−0.737334 + 0.675529i \(0.763915\pi\)
\(774\) 1.00000 0.0359443
\(775\) 3.00000 0.107763
\(776\) 7.00000 0.251285
\(777\) −2.00000 −0.0717496
\(778\) −15.0000 −0.537776
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) −6.00000 −0.214560
\(783\) 12.0000 0.428845
\(784\) −6.00000 −0.214286
\(785\) −3.00000 −0.107075
\(786\) 24.0000 0.856052
\(787\) −10.0000 −0.356462 −0.178231 0.983989i \(-0.557037\pi\)
−0.178231 + 0.983989i \(0.557037\pi\)
\(788\) −18.0000 −0.641223
\(789\) −2.00000 −0.0712019
\(790\) 8.00000 0.284627
\(791\) −7.00000 −0.248891
\(792\) −3.00000 −0.106600
\(793\) 0 0
\(794\) −18.0000 −0.638796
\(795\) 26.0000 0.922125
\(796\) 8.00000 0.283552
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 12.0000 0.424795
\(799\) 12.0000 0.424529
\(800\) −1.00000 −0.0353553
\(801\) −18.0000 −0.635999
\(802\) −34.0000 −1.20058
\(803\) 0 0
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) −20.0000 −0.704033
\(808\) 10.0000 0.351799
\(809\) 34.0000 1.19538 0.597688 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) 11.0000 0.386501
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −3.00000 −0.105279
\(813\) −20.0000 −0.701431
\(814\) 3.00000 0.105150
\(815\) −5.00000 −0.175142
\(816\) 6.00000 0.210042
\(817\) 6.00000 0.209913
\(818\) 0 0
\(819\) 0 0
\(820\) 3.00000 0.104765
\(821\) 8.00000 0.279202 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) −16.0000 −0.558064
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) −8.00000 −0.278693
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −11.0000 −0.382507 −0.191254 0.981541i \(-0.561255\pi\)
−0.191254 + 0.981541i \(0.561255\pi\)
\(828\) 2.00000 0.0695048
\(829\) −39.0000 −1.35453 −0.677263 0.735741i \(-0.736834\pi\)
−0.677263 + 0.735741i \(0.736834\pi\)
\(830\) 4.00000 0.138842
\(831\) −40.0000 −1.38758
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) −6.00000 −0.207763
\(835\) 12.0000 0.415277
\(836\) −18.0000 −0.622543
\(837\) −12.0000 −0.414781
\(838\) −12.0000 −0.414533
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −20.0000 −0.689655
\(842\) −34.0000 −1.17172
\(843\) −48.0000 −1.65321
\(844\) −3.00000 −0.103264
\(845\) −13.0000 −0.447214
\(846\) −4.00000 −0.137523
\(847\) −2.00000 −0.0687208
\(848\) 13.0000 0.446422
\(849\) −40.0000 −1.37280
\(850\) −3.00000 −0.102899
\(851\) −2.00000 −0.0685591
\(852\) −4.00000 −0.137038
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 15.0000 0.513289
\(855\) −6.00000 −0.205196
\(856\) −2.00000 −0.0683586
\(857\) 33.0000 1.12726 0.563629 0.826028i \(-0.309405\pi\)
0.563629 + 0.826028i \(0.309405\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 6.00000 0.204479
\(862\) 31.0000 1.05586
\(863\) 33.0000 1.12333 0.561667 0.827364i \(-0.310160\pi\)
0.561667 + 0.827364i \(0.310160\pi\)
\(864\) 4.00000 0.136083
\(865\) 9.00000 0.306009
\(866\) 22.0000 0.747590
\(867\) −16.0000 −0.543388
\(868\) 3.00000 0.101827
\(869\) −24.0000 −0.814144
\(870\) 6.00000 0.203419
\(871\) 0 0
\(872\) 3.00000 0.101593
\(873\) −7.00000 −0.236914
\(874\) 12.0000 0.405906
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −43.0000 −1.45201 −0.726003 0.687691i \(-0.758624\pi\)
−0.726003 + 0.687691i \(0.758624\pi\)
\(878\) −37.0000 −1.24869
\(879\) 6.00000 0.202375
\(880\) 3.00000 0.101130
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 6.00000 0.202031
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 34.0000 1.14225
\(887\) −33.0000 −1.10803 −0.554016 0.832506i \(-0.686905\pi\)
−0.554016 + 0.832506i \(0.686905\pi\)
\(888\) 2.00000 0.0671156
\(889\) −4.00000 −0.134156
\(890\) 18.0000 0.603361
\(891\) −33.0000 −1.10554
\(892\) 23.0000 0.770097
\(893\) −24.0000 −0.803129
\(894\) −36.0000 −1.20402
\(895\) 24.0000 0.802232
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) −9.00000 −0.300167
\(900\) 1.00000 0.0333333
\(901\) 39.0000 1.29928
\(902\) −9.00000 −0.299667
\(903\) −2.00000 −0.0665558
\(904\) 7.00000 0.232817
\(905\) 2.00000 0.0664822
\(906\) −32.0000 −1.06313
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 13.0000 0.431420
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −12.0000 −0.397360
\(913\) −12.0000 −0.397142
\(914\) −9.00000 −0.297694
\(915\) −30.0000 −0.991769
\(916\) −6.00000 −0.198246
\(917\) −12.0000 −0.396275
\(918\) 12.0000 0.396059
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 68.0000 2.24068
\(922\) 1.00000 0.0329332
\(923\) 0 0
\(924\) 6.00000 0.197386
\(925\) −1.00000 −0.0328798
\(926\) −12.0000 −0.394344
\(927\) 8.00000 0.262754
\(928\) 3.00000 0.0984798
\(929\) −3.00000 −0.0984268 −0.0492134 0.998788i \(-0.515671\pi\)
−0.0492134 + 0.998788i \(0.515671\pi\)
\(930\) −6.00000 −0.196748
\(931\) 36.0000 1.17985
\(932\) 4.00000 0.131024
\(933\) 50.0000 1.63693
\(934\) 37.0000 1.21068
\(935\) 9.00000 0.294331
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 52.0000 1.69696
\(940\) 4.00000 0.130466
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 6.00000 0.195491
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 3.00000 0.0975384
\(947\) 9.00000 0.292461 0.146230 0.989251i \(-0.453286\pi\)
0.146230 + 0.989251i \(0.453286\pi\)
\(948\) −16.0000 −0.519656
\(949\) 0 0
\(950\) 6.00000 0.194666
\(951\) 42.0000 1.36194
\(952\) −3.00000 −0.0972306
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) −13.0000 −0.420891
\(955\) 21.0000 0.679544
\(956\) 9.00000 0.291081
\(957\) −18.0000 −0.581857
\(958\) 8.00000 0.258468
\(959\) 8.00000 0.258333
\(960\) 2.00000 0.0645497
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) 24.0000 0.772988
\(965\) −10.0000 −0.321911
\(966\) −4.00000 −0.128698
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 2.00000 0.0642824
\(969\) −36.0000 −1.15649
\(970\) 7.00000 0.224756
\(971\) 9.00000 0.288824 0.144412 0.989518i \(-0.453871\pi\)
0.144412 + 0.989518i \(0.453871\pi\)
\(972\) −10.0000 −0.320750
\(973\) 3.00000 0.0961756
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) −15.0000 −0.480138
\(977\) 35.0000 1.11975 0.559875 0.828577i \(-0.310849\pi\)
0.559875 + 0.828577i \(0.310849\pi\)
\(978\) 10.0000 0.319765
\(979\) −54.0000 −1.72585
\(980\) −6.00000 −0.191663
\(981\) −3.00000 −0.0957826
\(982\) 20.0000 0.638226
\(983\) −27.0000 −0.861166 −0.430583 0.902551i \(-0.641692\pi\)
−0.430583 + 0.902551i \(0.641692\pi\)
\(984\) −6.00000 −0.191273
\(985\) −18.0000 −0.573528
\(986\) 9.00000 0.286618
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) −3.00000 −0.0953463
\(991\) 33.0000 1.04828 0.524140 0.851632i \(-0.324387\pi\)
0.524140 + 0.851632i \(0.324387\pi\)
\(992\) −3.00000 −0.0952501
\(993\) 60.0000 1.90404
\(994\) 2.00000 0.0634361
\(995\) 8.00000 0.253617
\(996\) −8.00000 −0.253490
\(997\) −12.0000 −0.380044 −0.190022 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 22.0000 0.696398
\(999\) 4.00000 0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 370.2.a.c.1.1 1
3.2 odd 2 3330.2.a.p.1.1 1
4.3 odd 2 2960.2.a.c.1.1 1
5.2 odd 4 1850.2.b.c.149.1 2
5.3 odd 4 1850.2.b.c.149.2 2
5.4 even 2 1850.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.c.1.1 1 1.1 even 1 trivial
1850.2.a.i.1.1 1 5.4 even 2
1850.2.b.c.149.1 2 5.2 odd 4
1850.2.b.c.149.2 2 5.3 odd 4
2960.2.a.c.1.1 1 4.3 odd 2
3330.2.a.p.1.1 1 3.2 odd 2