Properties

Label 2850.2.a.y.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.7573645761\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -2.00000 q^{21} -6.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{27} -2.00000 q^{28} -8.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} -1.00000 q^{38} -4.00000 q^{41} -2.00000 q^{42} +6.00000 q^{43} -6.00000 q^{44} -4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -2.00000 q^{51} -6.00000 q^{53} +1.00000 q^{54} -2.00000 q^{56} -1.00000 q^{57} -8.00000 q^{58} -4.00000 q^{59} +2.00000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} +8.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} +1.00000 q^{72} -6.00000 q^{73} +4.00000 q^{74} -1.00000 q^{76} +12.0000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} +6.00000 q^{86} -8.00000 q^{87} -6.00000 q^{88} -4.00000 q^{89} -4.00000 q^{92} -8.00000 q^{93} +12.0000 q^{94} +1.00000 q^{96} -12.0000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) −6.00000 −1.27920
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.00000 −1.04447
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) −2.00000 −0.308607
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) −1.00000 −0.132453
\(58\) −8.00000 −1.05045
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −8.00000 −1.01600
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −4.00000 −0.441726
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) −8.00000 −0.857690
\(88\) −6.00000 −0.639602
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −8.00000 −0.829561
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) −3.00000 −0.303046
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −2.00000 −0.198030
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) −2.00000 −0.188982
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 2.00000 0.181071
\(123\) −4.00000 −0.360668
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 22.0000 1.92215 0.961074 0.276289i \(-0.0891049\pi\)
0.961074 + 0.276289i \(0.0891049\pi\)
\(132\) −6.00000 −0.522233
\(133\) 2.00000 0.173422
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −4.00000 −0.340503
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) −3.00000 −0.247436
\(148\) 4.00000 0.328798
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −2.00000 −0.161690
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 1.00000 0.0785674
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) −2.00000 −0.154303
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 6.00000 0.457496
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −8.00000 −0.606478
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) −4.00000 −0.300658
\(178\) −4.00000 −0.299813
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 12.0000 0.877527
\(188\) 12.0000 0.875190
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −22.0000 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) 1.00000 0.0721688
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) −6.00000 −0.426401
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −10.0000 −0.703598
\(203\) 16.0000 1.12298
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) −14.0000 −0.948200
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 0 0
\(222\) 4.00000 0.268462
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 28.0000 1.85843 0.929213 0.369546i \(-0.120487\pi\)
0.929213 + 0.369546i \(0.120487\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) −8.00000 −0.525226
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 8.00000 0.519656
\(238\) 4.00000 0.259281
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 25.0000 1.60706
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −4.00000 −0.255031
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) −2.00000 −0.125988
\(253\) 24.0000 1.50887
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 6.00000 0.373544
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) 22.0000 1.35916
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 2.00000 0.122628
\(267\) −4.00000 −0.244796
\(268\) 8.00000 0.488678
\(269\) 8.00000 0.487769 0.243884 0.969804i \(-0.421578\pi\)
0.243884 + 0.969804i \(0.421578\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 12.0000 0.714590
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) −6.00000 −0.351123
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) −6.00000 −0.348155
\(298\) 22.0000 1.27443
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) −10.0000 −0.574485
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 12.0000 0.683763
\(309\) 0 0
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −6.00000 −0.336463
\(319\) 48.0000 2.68748
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 8.00000 0.445823
\(323\) 2.00000 0.111283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) −14.0000 −0.774202
\(328\) −4.00000 −0.220863
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −4.00000 −0.219529
\(333\) 4.00000 0.219199
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) −13.0000 −0.707107
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 48.0000 2.59935
\(342\) −1.00000 −0.0540738
\(343\) 20.0000 1.07990
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) −8.00000 −0.428845
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −4.00000 −0.212000
\(357\) 4.00000 0.211702
\(358\) 12.0000 0.634220
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 2.00000 0.105118
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −4.00000 −0.208514
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) −8.00000 −0.414781
\(373\) 24.0000 1.24267 0.621336 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) −22.0000 −1.12562
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) 6.00000 0.304997
\(388\) −12.0000 −0.609208
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) −3.00000 −0.151523
\(393\) 22.0000 1.10975
\(394\) −14.0000 −0.705310
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 20.0000 1.00251
\(399\) 2.00000 0.100125
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 8.00000 0.399004
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 16.0000 0.794067
\(407\) −24.0000 −1.18964
\(408\) −2.00000 −0.0990148
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −28.0000 −1.36302
\(423\) 12.0000 0.583460
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 1.00000 0.0481125
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) 4.00000 0.191346
\(438\) −6.00000 −0.286691
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 22.0000 1.04056
\(448\) −2.00000 −0.0944911
\(449\) −4.00000 −0.188772 −0.0943858 0.995536i \(-0.530089\pi\)
−0.0943858 + 0.995536i \(0.530089\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 18.0000 0.841085
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 12.0000 0.558291
\(463\) 26.0000 1.20832 0.604161 0.796862i \(-0.293508\pi\)
0.604161 + 0.796862i \(0.293508\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) −4.00000 −0.184115
\(473\) −36.0000 −1.65528
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) −6.00000 −0.274721
\(478\) −18.0000 −0.823301
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −22.0000 −1.00207
\(483\) 8.00000 0.364013
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 2.00000 0.0905357
\(489\) −6.00000 −0.271329
\(490\) 0 0
\(491\) −34.0000 −1.53440 −0.767199 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) −4.00000 −0.180334
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) −14.0000 −0.624851
\(503\) −40.0000 −1.78351 −0.891756 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) −13.0000 −0.577350
\(508\) 12.0000 0.532414
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 6.00000 0.264135
\(517\) −72.0000 −3.16656
\(518\) −8.00000 −0.351500
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) −8.00000 −0.350150
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 22.0000 0.961074
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 16.0000 0.696971
\(528\) −6.00000 −0.261116
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 2.00000 0.0867110
\(533\) 0 0
\(534\) −4.00000 −0.173097
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 12.0000 0.517838
\(538\) 8.00000 0.344904
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 20.0000 0.859074
\(543\) 2.00000 0.0858282
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) −6.00000 −0.256307
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) −4.00000 −0.170251
\(553\) −16.0000 −0.680389
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) −8.00000 −0.338667
\(559\) 0 0
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 16.0000 0.674919
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 12.0000 0.505291
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) −22.0000 −0.919063
\(574\) 8.00000 0.333914
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) −13.0000 −0.540729
\(579\) 8.00000 0.332469
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) −12.0000 −0.497416
\(583\) 36.0000 1.49097
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) −3.00000 −0.123718
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) −14.0000 −0.575883
\(592\) 4.00000 0.164399
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −12.0000 −0.489083
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 16.0000 0.648353
\(610\) 0 0
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −6.00000 −0.240578
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 0 0
\(626\) 14.0000 0.559553
\(627\) 6.00000 0.239617
\(628\) −6.00000 −0.239426
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 8.00000 0.318223
\(633\) −28.0000 −1.11290
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 48.0000 1.90034
\(639\) 0 0
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 12.0000 0.473602
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.00000 0.0392837
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) −6.00000 −0.234978
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) −6.00000 −0.234082
\(658\) −24.0000 −0.935617
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 32.0000 1.23904
\(668\) 16.0000 0.619059
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) −2.00000 −0.0771517
\(673\) 28.0000 1.07932 0.539660 0.841883i \(-0.318553\pi\)
0.539660 + 0.841883i \(0.318553\pi\)
\(674\) 20.0000 0.770371
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) −14.0000 −0.537667
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 48.0000 1.83801
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 18.0000 0.686743
\(688\) 6.00000 0.228748
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) −18.0000 −0.684257
\(693\) 12.0000 0.455842
\(694\) −36.0000 −1.36654
\(695\) 0 0
\(696\) −8.00000 −0.303239
\(697\) 8.00000 0.303022
\(698\) −2.00000 −0.0757011
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 20.0000 0.752177
\(708\) −4.00000 −0.150329
\(709\) 50.0000 1.87779 0.938895 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −4.00000 −0.149906
\(713\) 32.0000 1.19841
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −18.0000 −0.672222
\(718\) −6.00000 −0.223918
\(719\) 14.0000 0.522112 0.261056 0.965324i \(-0.415929\pi\)
0.261056 + 0.965324i \(0.415929\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) −22.0000 −0.818189
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 2.00000 0.0739221
\(733\) −18.0000 −0.664845 −0.332423 0.943131i \(-0.607866\pi\)
−0.332423 + 0.943131i \(0.607866\pi\)
\(734\) 10.0000 0.369107
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −48.0000 −1.76810
\(738\) −4.00000 −0.147242
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 24.0000 0.878702
\(747\) −4.00000 −0.146352
\(748\) 12.0000 0.438763
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 12.0000 0.437595
\(753\) −14.0000 −0.510188
\(754\) 0 0
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 20.0000 0.726433
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 12.0000 0.434714
\(763\) 28.0000 1.01367
\(764\) −22.0000 −0.795932
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 8.00000 0.287926
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) −12.0000 −0.430775
\(777\) −8.00000 −0.286998
\(778\) −30.0000 −1.07555
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) −8.00000 −0.285897
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 22.0000 0.784714
\(787\) −44.0000 −1.56843 −0.784215 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) −14.0000 −0.498729
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 28.0000 0.995565
\(792\) −6.00000 −0.213201
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 2.00000 0.0707992
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) −4.00000 −0.141333
\(802\) 0 0
\(803\) 36.0000 1.27041
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 0 0
\(807\) 8.00000 0.281613
\(808\) −10.0000 −0.351799
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 16.0000 0.561490
\(813\) 20.0000 0.701431
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) −6.00000 −0.209913
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) 0 0
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) −6.00000 −0.209274
\(823\) −14.0000 −0.488009 −0.244005 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) −4.00000 −0.139010
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) −8.00000 −0.276520
\(838\) −14.0000 −0.483622
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −2.00000 −0.0689246
\(843\) 16.0000 0.551069
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) −50.0000 −1.71802
\(848\) −6.00000 −0.206041
\(849\) −22.0000 −0.755038
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 36.0000 1.22616
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 16.0000 0.543075
\(869\) −48.0000 −1.62829
\(870\) 0 0
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) −12.0000 −0.406138
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −8.00000 −0.269987
\(879\) −10.0000 −0.337292
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) −3.00000 −0.101015
\(883\) −14.0000 −0.471138 −0.235569 0.971858i \(-0.575695\pi\)
−0.235569 + 0.971858i \(0.575695\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 4.00000 0.134231
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 16.0000 0.535720
\(893\) −12.0000 −0.401565
\(894\) 22.0000 0.735790
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −4.00000 −0.133482
\(899\) 64.0000 2.13452
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 24.0000 0.799113
\(903\) −12.0000 −0.399335
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 28.0000 0.929213
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 24.0000 0.794284
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) 18.0000 0.594737
\(917\) −44.0000 −1.45301
\(918\) −2.00000 −0.0660098
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) −34.0000 −1.11973
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) 0 0
\(926\) 26.0000 0.854413
\(927\) 0 0
\(928\) −8.00000 −0.262613
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) −14.0000 −0.458585
\(933\) −6.00000 −0.196431
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) 34.0000 1.11073 0.555366 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) −16.0000 −0.522419
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) −40.0000 −1.30396 −0.651981 0.758235i \(-0.726062\pi\)
−0.651981 + 0.758235i \(0.726062\pi\)
\(942\) −6.00000 −0.195491
\(943\) 16.0000 0.521032
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) −36.0000 −1.17046
\(947\) −44.0000 −1.42981 −0.714904 0.699223i \(-0.753530\pi\)
−0.714904 + 0.699223i \(0.753530\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 4.00000 0.129641
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −18.0000 −0.582162
\(957\) 48.0000 1.55162
\(958\) 18.0000 0.581554
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) 18.0000 0.578841 0.289420 0.957202i \(-0.406537\pi\)
0.289420 + 0.957202i \(0.406537\pi\)
\(968\) 25.0000 0.803530
\(969\) 2.00000 0.0642493
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −6.00000 −0.191859
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) −34.0000 −1.08498
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −4.00000 −0.127515
\(985\) 0 0
\(986\) 16.0000 0.509544
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −8.00000 −0.254000
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) 58.0000 1.83688 0.918439 0.395562i \(-0.129450\pi\)
0.918439 + 0.395562i \(0.129450\pi\)
\(998\) 24.0000 0.759707
\(999\) 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 2850.2.a.y.1.1 1
3.2 odd 2 8550.2.a.g.1.1 1
5.2 odd 4 2850.2.d.j.799.2 2
5.3 odd 4 2850.2.d.j.799.1 2
5.4 even 2 570.2.a.b.1.1 1
15.14 odd 2 1710.2.a.s.1.1 1
20.19 odd 2 4560.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.b.1.1 1 5.4 even 2
1710.2.a.s.1.1 1 15.14 odd 2
2850.2.a.y.1.1 1 1.1 even 1 trivial
2850.2.d.j.799.1 2 5.3 odd 4
2850.2.d.j.799.2 2 5.2 odd 4
4560.2.a.r.1.1 1 20.19 odd 2
8550.2.a.g.1.1 1 3.2 odd 2