Properties

Label 1850.2.a.i.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1850.1

$q$-expansion

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

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 \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 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
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 3.00000 0.639602
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) 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 \(0.589851\pi\)
\(30\) 0 0
\(31\) 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
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) 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 \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −2.00000 −0.288675
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −13.0000 −1.78569 −0.892844 0.450367i \(-0.851293\pi\)
−0.892844 + 0.450367i \(0.851293\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 12.0000 1.58944
\(58\) −3.00000 −0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −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
\(70\) 0 0
\(71\) −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
\(75\) 0 0
\(76\) −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 \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 3.00000 0.331295
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 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 \(0.903076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) −6.00000 −0.622171
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) −6.00000 −0.606092
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −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 \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −13.0000 −1.26267
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 4.00000 0.384900
\(109\) −3.00000 −0.287348 −0.143674 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −1.00000 −0.0944911
\(113\) 7.00000 0.658505 0.329252 0.944242i \(-0.393203\pi\)
0.329252 + 0.944242i \(0.393203\pi\)
\(114\) 12.0000 1.12390
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −15.0000 −1.35804
\(123\) −6.00000 −0.541002
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 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
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 4.00000 0.340503
\(139\) 3.00000 0.254457 0.127228 0.991873i \(-0.459392\pi\)
0.127228 + 0.991873i \(0.459392\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −2.00000 −0.167836
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 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 \(0.236096\pi\)
\(150\) 0 0
\(151\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) 3.00000 0.239426 0.119713 0.992809i \(-0.461803\pi\)
0.119713 + 0.992809i \(0.461803\pi\)
\(158\) −8.00000 −0.636446
\(159\) 26.0000 2.06193
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) −11.0000 −0.864242
\(163\) 5.00000 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 2.00000 0.154303
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 1.00000 0.0762493
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 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 \(0.145798\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) −9.00000 −0.658145
\(188\) −4.00000 −0.291730
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 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 \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 7.00000 0.502571
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 3.00000 0.213201
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) 3.00000 0.210559
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −18.0000 −1.24509
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) −3.00000 −0.203653
\(218\) −3.00000 −0.203186
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −23.0000 −1.54019 −0.770097 0.637927i \(-0.779792\pi\)
−0.770097 + 0.637927i \(0.779792\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 7.00000 0.465633
\(227\) −13.0000 −0.862840 −0.431420 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 12.0000 0.794719
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) −3.00000 −0.196960
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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 \(0.405985\pi\)
\(240\) 0 0
\(241\) 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
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) −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 \(-0.490185\pi\)
0.0308313 + 0.999525i \(0.490185\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 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 \(0.598608\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) 3.00000 0.179928
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) −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 \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) −3.00000 −0.175262 −0.0876309 0.996153i \(-0.527930\pi\)
−0.0876309 + 0.996153i \(0.527930\pi\)
\(294\) 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
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 16.0000 0.920697
\(303\) 20.0000 1.14897
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) −3.00000 −0.171499
\(307\) −34.0000 −1.94048 −0.970241 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) −3.00000 −0.170941
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 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 \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 3.00000 0.169300
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 26.0000 1.45801
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) 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
\(330\) 0 0
\(331\) 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 \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) −13.0000 −0.707107
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 9.00000 0.487377
\(342\) −6.00000 −0.324443
\(343\) 13.0000 0.701934
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) −9.00000 −0.483843
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 6.00000 0.321634
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.00000 0.159901
\(353\) −19.0000 −1.01127 −0.505634 0.862748i \(-0.668741\pi\)
−0.505634 + 0.862748i \(0.668741\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −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 \(0.361250\pi\)
\(360\) 0 0
\(361\) 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 \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) −2.00000 −0.104257
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) 13.0000 0.674926
\(372\) −6.00000 −0.311086
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) −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 \(0.365203\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 21.0000 1.07445
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 8.00000 0.401004
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) 3.00000 0.148704
\(408\) 6.00000 0.297044
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 16.0000 0.789222
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) −2.00000 −0.0982946
\(415\) 0 0
\(416\) 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 \(0.405307\pi\)
\(420\) 0 0
\(421\) 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
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) 15.0000 0.725901
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) 0 0
\(431\) −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 \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) −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 \(0.155549\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 34.0000 1.61539 0.807694 0.589601i \(-0.200715\pi\)
0.807694 + 0.589601i \(0.200715\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −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 \(0.575825\pi\)
\(450\) 0 0
\(451\) 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 \(-0.567510\pi\)
−0.210501 + 0.977594i \(0.567510\pi\)
\(458\) −6.00000 −0.280362
\(459\) −12.0000 −0.560112
\(460\) 0 0
\(461\) −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 \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −4.00000 −0.185296
\(467\) 37.0000 1.71216 0.856078 0.516847i \(-0.172894\pi\)
0.856078 + 0.516847i \(0.172894\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 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 \(0.558505\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 24.0000 1.09317
\(483\) −4.00000 −0.182006
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) −15.0000 −0.679018
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) −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
\(495\) 0 0
\(496\) 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 \(0.663890\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 6.00000 0.267793
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −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 \(0.205978\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −24.0000 −1.05963
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) −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 \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 1.00000 0.0436021
\(527\) −9.00000 −0.392046
\(528\) −6.00000 −0.261116
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) 0 0
\(534\) 36.0000 1.55787
\(535\) 0 0
\(536\) 0 0
\(537\) −48.0000 −2.07135
\(538\) −10.0000 −0.431131
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −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
\(545\) 0 0
\(546\) 0 0
\(547\) −23.0000 −0.983409 −0.491704 0.870762i \(-0.663626\pi\)
−0.491704 + 0.870762i \(0.663626\pi\)
\(548\) −8.00000 −0.341743
\(549\) −15.0000 −0.640184
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 4.00000 0.170251
\(553\) 8.00000 0.340195
\(554\) 20.0000 0.849719
\(555\) 0 0
\(556\) 3.00000 0.127228
\(557\) 40.0000 1.69485 0.847427 0.530912i \(-0.178150\pi\)
0.847427 + 0.530912i \(0.178150\pi\)
\(558\) 3.00000 0.127000
\(559\) 0 0
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) −24.0000 −1.01238
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 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 \(0.432780\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −8.00000 −0.332756
\(579\) −20.0000 −0.831172
\(580\) 0 0
\(581\) −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 \(-0.756916\pi\)
−0.722302 + 0.691577i \(0.756916\pi\)
\(588\) 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 \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 12.0000 0.492366
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 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
\(605\) 0 0
\(606\) 20.0000 0.812444
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −6.00000 −0.243332
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) −11.0000 −0.444286 −0.222143 0.975014i \(-0.571305\pi\)
−0.222143 + 0.975014i \(0.571305\pi\)
\(614\) −34.0000 −1.37213
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 16.0000 0.643614
\(619\) −37.0000 −1.48716 −0.743578 0.668649i \(-0.766873\pi\)
−0.743578 + 0.668649i \(0.766873\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 25.0000 1.00241
\(623\) 18.0000 0.721155
\(624\) 0 0
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) 36.0000 1.43770
\(628\) 3.00000 0.119713
\(629\) −3.00000 −0.119618
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) 26.0000 1.03097
\(637\) 0 0
\(638\) −9.00000 −0.356313
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −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 \(-0.569594\pi\)
−0.216899 + 0.976194i \(0.569594\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) 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 \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) 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 \(0.252657\pi\)
\(660\) 0 0
\(661\) −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
\(665\) 0 0
\(666\) 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 \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −12.0000 −0.462223
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) −14.0000 −0.537667
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 26.0000 0.996322
\(682\) 9.00000 0.344628
\(683\) 31.0000 1.18618 0.593091 0.805135i \(-0.297907\pi\)
0.593091 + 0.805135i \(0.297907\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 12.0000 0.457829
\(688\) 1.00000 0.0381246
\(689\) 0 0
\(690\) 0 0
\(691\) 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
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) −9.00000 −0.340899
\(698\) −22.0000 −0.832712
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) −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 \(0.616125\pi\)
\(710\) 0 0
\(711\) −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 \(0.608954\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 17.0000 0.632674
\(723\) −48.0000 −1.78514
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 4.00000 0.148454
\(727\) −52.0000 −1.92857 −0.964287 0.264861i \(-0.914674\pi\)
−0.964287 + 0.264861i \(0.914674\pi\)
\(728\) 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 \(-0.226574\pi\)
0.757185 + 0.653201i \(0.226574\pi\)
\(734\) −13.0000 −0.479839
\(735\) 0 0
\(736\) −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 \(0.334580\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 13.0000 0.477245
\(743\) −3.00000 −0.110059 −0.0550297 0.998485i \(-0.517525\pi\)
−0.0550297 + 0.998485i \(0.517525\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 4.00000 0.146352
\(748\) −9.00000 −0.329073
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 50.0000 1.81728 0.908640 0.417579i \(-0.137121\pi\)
0.908640 + 0.417579i \(0.137121\pi\)
\(758\) 16.0000 0.581146
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −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
\(765\) 0 0
\(766\) 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 \(0.453925\pi\)
\(770\) 0 0
\(771\) −44.0000 −1.58462
\(772\) 10.0000 0.359908
\(773\) 41.0000 1.47467 0.737334 0.675529i \(-0.236085\pi\)
0.737334 + 0.675529i \(0.236085\pi\)
\(774\) 1.00000 0.0359443
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 24.0000 0.856052
\(787\) 10.0000 0.356462 0.178231 0.983989i \(-0.442963\pi\)
0.178231 + 0.983989i \(0.442963\pi\)
\(788\) 18.0000 0.641223
\(789\) −2.00000 −0.0712019
\(790\) 0 0
\(791\) −7.00000 −0.248891
\(792\) 3.00000 0.106600
\(793\) 0 0
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) −12.0000 −0.424795
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) 34.0000 1.20058
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 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 \(0.296086\pi\)
\(810\) 0 0
\(811\) −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
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) −6.00000 −0.209913
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 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 \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) 11.0000 0.382507 0.191254 0.981541i \(-0.438745\pi\)
0.191254 + 0.981541i \(0.438745\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −39.0000 −1.35453 −0.677263 0.735741i \(-0.736834\pi\)
−0.677263 + 0.735741i \(0.736834\pi\)
\(830\) 0 0
\(831\) −40.0000 −1.38758
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) −6.00000 −0.207763
\(835\) 0 0
\(836\) −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 \(0.286552\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 34.0000 1.17172
\(843\) 48.0000 1.65321
\(844\) −3.00000 −0.103264
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) 2.00000 0.0687208
\(848\) −13.0000 −0.446422
\(849\) −40.0000 −1.37280
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) 4.00000 0.137038
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 15.0000 0.513289
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) −31.0000 −1.05586
\(863\) −33.0000 −1.12333 −0.561667 0.827364i \(-0.689840\pi\)
−0.561667 + 0.827364i \(0.689840\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 22.0000 0.747590
\(867\) 16.0000 0.543388
\(868\) −3.00000 −0.101827
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) −3.00000 −0.101593
\(873\) 7.00000 0.236914
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) 0 0
\(877\) 43.0000 1.45201 0.726003 0.687691i \(-0.241376\pi\)
0.726003 + 0.687691i \(0.241376\pi\)
\(878\) 37.0000 1.24869
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −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 \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 34.0000 1.14225
\(887\) 33.0000 1.10803 0.554016 0.832506i \(-0.313095\pi\)
0.554016 + 0.832506i \(0.313095\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) −33.0000 −1.10554
\(892\) −23.0000 −0.770097
\(893\) 24.0000 0.803129
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) −9.00000 −0.300167
\(900\) 0 0
\(901\) 39.0000 1.29928
\(902\) 9.00000 0.299667
\(903\) 2.00000 0.0665558
\(904\) 7.00000 0.232817
\(905\) 0 0
\(906\) −32.0000 −1.06313
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) −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
\(915\) 0 0
\(916\) −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 \(0.323020\pi\)
\(920\) 0 0
\(921\) 68.0000 2.24068
\(922\) −1.00000 −0.0329332
\(923\) 0 0
\(924\) 6.00000 0.197386
\(925\) 0 0
\(926\) −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 \(0.515671\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) −4.00000 −0.131024
\(933\) −50.0000 −1.63693
\(934\) 37.0000 1.21068
\(935\) 0 0
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 52.0000 1.69696
\(940\) 0 0
\(941\) −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
\(945\) 0 0
\(946\) 3.00000 0.0975384
\(947\) −9.00000 −0.292461 −0.146230 0.989251i \(-0.546714\pi\)
−0.146230 + 0.989251i \(0.546714\pi\)
\(948\) 16.0000 0.519656
\(949\) 0 0
\(950\) 0 0
\(951\) 42.0000 1.36194
\(952\) 3.00000 0.0972306
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) −13.0000 −0.420891
\(955\) 0 0
\(956\) 9.00000 0.291081
\(957\) 18.0000 0.581857
\(958\) −8.00000 −0.258468
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −2.00000 −0.0644491
\(964\) 24.0000 0.772988
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) −2.00000 −0.0642824
\(969\) −36.0000 −1.15649
\(970\) 0 0
\(971\) 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 \(-0.689151\pi\)
−0.559875 + 0.828577i \(0.689151\pi\)
\(978\) −10.0000 −0.319765
\(979\) −54.0000 −1.72585
\(980\) 0 0
\(981\) −3.00000 −0.0957826
\(982\) −20.0000 −0.638226
\(983\) 27.0000 0.861166 0.430583 0.902551i \(-0.358308\pi\)
0.430583 + 0.902551i \(0.358308\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) −2.00000 −0.0635963
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) 12.0000 0.380044 0.190022 0.981780i \(-0.439144\pi\)
0.190022 + 0.981780i \(0.439144\pi\)
\(998\) −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 1850.2.a.i.1.1 1
5.2 odd 4 1850.2.b.c.149.2 2
5.3 odd 4 1850.2.b.c.149.1 2
5.4 even 2 370.2.a.c.1.1 1
15.14 odd 2 3330.2.a.p.1.1 1
20.19 odd 2 2960.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.c.1.1 1 5.4 even 2
1850.2.a.i.1.1 1 1.1 even 1 trivial
1850.2.b.c.149.1 2 5.3 odd 4
1850.2.b.c.149.2 2 5.2 odd 4
2960.2.a.c.1.1 1 20.19 odd 2
3330.2.a.p.1.1 1 15.14 odd 2