Properties

Label 2850.2.a.u.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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} -4.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} -5.00000 q^{31} +1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -1.00000 q^{38} +4.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} +1.00000 q^{44} -5.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +4.00000 q^{51} -4.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} +1.00000 q^{57} +3.00000 q^{58} -11.0000 q^{61} -5.00000 q^{62} +1.00000 q^{64} -1.00000 q^{66} +1.00000 q^{67} -4.00000 q^{68} +5.00000 q^{69} +2.00000 q^{71} +1.00000 q^{72} +3.00000 q^{73} +6.00000 q^{74} -1.00000 q^{76} +4.00000 q^{78} +17.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} -3.00000 q^{83} -4.00000 q^{86} -3.00000 q^{87} +1.00000 q^{88} +7.00000 q^{89} -5.00000 q^{92} +5.00000 q^{93} -1.00000 q^{96} -10.0000 q^{97} -7.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −1.00000 −0.162221
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) −4.00000 −0.554700
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 3.00000 0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) 1.00000 0.122169 0.0610847 0.998133i \(-0.480544\pi\)
0.0610847 + 0.998133i \(0.480544\pi\)
\(68\) −4.00000 −0.485071
\(69\) 5.00000 0.601929
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 1.00000 0.117851
\(73\) 3.00000 0.351123 0.175562 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 17.0000 1.91265 0.956325 0.292306i \(-0.0944227\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −3.00000 −0.321634
\(88\) 1.00000 0.106600
\(89\) 7.00000 0.741999 0.370999 0.928633i \(-0.379015\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.00000 −0.521286
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −7.00000 −0.707107
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 4.00000 0.396059
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −10.0000 −0.966736 −0.483368 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) −5.00000 −0.470360 −0.235180 0.971952i \(-0.575568\pi\)
−0.235180 + 0.971952i \(0.575568\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −11.0000 −0.995893
\(123\) 2.00000 0.180334
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) 1.00000 0.0863868
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 4.00000 0.341743 0.170872 0.985293i \(-0.445342\pi\)
0.170872 + 0.985293i \(0.445342\pi\)
\(138\) 5.00000 0.425628
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 3.00000 0.248282
\(147\) 7.00000 0.577350
\(148\) 6.00000 0.493197
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 17.0000 1.35245
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −4.00000 −0.304997
\(173\) −7.00000 −0.532200 −0.266100 0.963945i \(-0.585735\pi\)
−0.266100 + 0.963945i \(0.585735\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 7.00000 0.524672
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 11.0000 0.813143
\(184\) −5.00000 −0.368605
\(185\) 0 0
\(186\) 5.00000 0.366618
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.0000 0.795932 0.397966 0.917400i \(-0.369716\pi\)
0.397966 + 0.917400i \(0.369716\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 20.0000 1.42494 0.712470 0.701702i \(-0.247576\pi\)
0.712470 + 0.701702i \(0.247576\pi\)
\(198\) 1.00000 0.0710669
\(199\) 6.00000 0.425329 0.212664 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(200\) 0 0
\(201\) −1.00000 −0.0705346
\(202\) −8.00000 −0.562878
\(203\) 0 0
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) −13.0000 −0.905753
\(207\) −5.00000 −0.347524
\(208\) −4.00000 −0.277350
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) −9.00000 −0.618123
\(213\) −2.00000 −0.137038
\(214\) −10.0000 −0.683586
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) −6.00000 −0.402694
\(223\) 15.0000 1.00447 0.502237 0.864730i \(-0.332510\pi\)
0.502237 + 0.864730i \(0.332510\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −5.00000 −0.332595
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 1.00000 0.0662266
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) −17.0000 −1.10427
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) −11.0000 −0.704203
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 4.00000 0.254514
\(248\) −5.00000 −0.317500
\(249\) 3.00000 0.190117
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 5.00000 0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 15.0000 0.926703
\(263\) 11.0000 0.678289 0.339145 0.940734i \(-0.389862\pi\)
0.339145 + 0.940734i \(0.389862\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 0 0
\(266\) 0 0
\(267\) −7.00000 −0.428393
\(268\) 1.00000 0.0610847
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −4.00000 −0.242536
\(273\) 0 0
\(274\) 4.00000 0.241649
\(275\) 0 0
\(276\) 5.00000 0.300965
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) −2.00000 −0.119952
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 3.00000 0.175562
\(293\) −11.0000 −0.642627 −0.321313 0.946973i \(-0.604124\pi\)
−0.321313 + 0.946973i \(0.604124\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 20.0000 1.15663
\(300\) 0 0
\(301\) 0 0
\(302\) −24.0000 −1.38104
\(303\) 8.00000 0.459588
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) 25.0000 1.42683 0.713413 0.700744i \(-0.247149\pi\)
0.713413 + 0.700744i \(0.247149\pi\)
\(308\) 0 0
\(309\) 13.0000 0.739544
\(310\) 0 0
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) 4.00000 0.226455
\(313\) −21.0000 −1.18699 −0.593495 0.804838i \(-0.702252\pi\)
−0.593495 + 0.804838i \(0.702252\pi\)
\(314\) −14.0000 −0.790066
\(315\) 0 0
\(316\) 17.0000 0.956325
\(317\) 13.0000 0.730153 0.365076 0.930978i \(-0.381043\pi\)
0.365076 + 0.930978i \(0.381043\pi\)
\(318\) 9.00000 0.504695
\(319\) 3.00000 0.167968
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) −6.00000 −0.331801
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) 13.0000 0.714545 0.357272 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) −3.00000 −0.164646
\(333\) 6.00000 0.328798
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 3.00000 0.163178
\(339\) 5.00000 0.271563
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −7.00000 −0.376322
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −3.00000 −0.160817
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 1.00000 0.0533002
\(353\) −28.0000 −1.49029 −0.745145 0.666903i \(-0.767620\pi\)
−0.745145 + 0.666903i \(0.767620\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.00000 0.370999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 0 0
\(366\) 11.0000 0.574979
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) −5.00000 −0.260643
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 5.00000 0.259238
\(373\) −28.0000 −1.44979 −0.724893 0.688862i \(-0.758111\pi\)
−0.724893 + 0.688862i \(0.758111\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −5.00000 −0.256158
\(382\) 11.0000 0.562809
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −4.00000 −0.203331
\(388\) −10.0000 −0.507673
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) −7.00000 −0.353553
\(393\) −15.0000 −0.756650
\(394\) 20.0000 1.00759
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) 3.00000 0.150566 0.0752828 0.997162i \(-0.476014\pi\)
0.0752828 + 0.997162i \(0.476014\pi\)
\(398\) 6.00000 0.300753
\(399\) 0 0
\(400\) 0 0
\(401\) −17.0000 −0.848939 −0.424470 0.905442i \(-0.639539\pi\)
−0.424470 + 0.905442i \(0.639539\pi\)
\(402\) −1.00000 −0.0498755
\(403\) 20.0000 0.996271
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 4.00000 0.198030
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) −13.0000 −0.640464
\(413\) 0 0
\(414\) −5.00000 −0.245737
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) 2.00000 0.0979404
\(418\) −1.00000 −0.0489116
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) −13.0000 −0.632830
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) 0 0
\(428\) −10.0000 −0.483368
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 38.0000 1.83040 0.915198 0.403005i \(-0.132034\pi\)
0.915198 + 0.403005i \(0.132034\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 22.0000 1.05725 0.528626 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) 5.00000 0.239182
\(438\) −3.00000 −0.143346
\(439\) 21.0000 1.00228 0.501138 0.865368i \(-0.332915\pi\)
0.501138 + 0.865368i \(0.332915\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 16.0000 0.761042
\(443\) 13.0000 0.617649 0.308824 0.951119i \(-0.400064\pi\)
0.308824 + 0.951119i \(0.400064\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) 15.0000 0.710271
\(447\) 0 0
\(448\) 0 0
\(449\) −27.0000 −1.27421 −0.637104 0.770778i \(-0.719868\pi\)
−0.637104 + 0.770778i \(0.719868\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) −5.00000 −0.235180
\(453\) 24.0000 1.12762
\(454\) −2.00000 −0.0938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 5.00000 0.233635
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 4.00000 0.186299 0.0931493 0.995652i \(-0.470307\pi\)
0.0931493 + 0.995652i \(0.470307\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) 11.0000 0.509019 0.254510 0.967070i \(-0.418086\pi\)
0.254510 + 0.967070i \(0.418086\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) −17.0000 −0.780836
\(475\) 0 0
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) −37.0000 −1.69057 −0.845287 0.534313i \(-0.820570\pi\)
−0.845287 + 0.534313i \(0.820570\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) −4.00000 −0.182195
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −11.0000 −0.497947
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 2.00000 0.0901670
\(493\) −12.0000 −0.540453
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −5.00000 −0.224507
\(497\) 0 0
\(498\) 3.00000 0.134433
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 16.0000 0.714115
\(503\) 36.0000 1.60516 0.802580 0.596544i \(-0.203460\pi\)
0.802580 + 0.596544i \(0.203460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −5.00000 −0.222277
\(507\) −3.00000 −0.133235
\(508\) 5.00000 0.221839
\(509\) −27.0000 −1.19675 −0.598377 0.801215i \(-0.704187\pi\)
−0.598377 + 0.801215i \(0.704187\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −7.00000 −0.308757
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) 7.00000 0.307266
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 3.00000 0.131306
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 15.0000 0.655278
\(525\) 0 0
\(526\) 11.0000 0.479623
\(527\) 20.0000 0.871214
\(528\) −1.00000 −0.0435194
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) −7.00000 −0.302920
\(535\) 0 0
\(536\) 1.00000 0.0431934
\(537\) 4.00000 0.172613
\(538\) −2.00000 −0.0862261
\(539\) −7.00000 −0.301511
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 4.00000 0.170872
\(549\) −11.0000 −0.469469
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) 5.00000 0.212814
\(553\) 0 0
\(554\) 1.00000 0.0424859
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −5.00000 −0.211667
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) −3.00000 −0.126547
\(563\) −26.0000 −1.09577 −0.547885 0.836554i \(-0.684567\pi\)
−0.547885 + 0.836554i \(0.684567\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.00000 0.336265
\(567\) 0 0
\(568\) 2.00000 0.0839181
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) −4.00000 −0.167248
\(573\) −11.0000 −0.459532
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 27.0000 1.12402 0.562012 0.827129i \(-0.310027\pi\)
0.562012 + 0.827129i \(0.310027\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) −9.00000 −0.372742
\(584\) 3.00000 0.124141
\(585\) 0 0
\(586\) −11.0000 −0.454406
\(587\) 17.0000 0.701665 0.350833 0.936438i \(-0.385899\pi\)
0.350833 + 0.936438i \(0.385899\pi\)
\(588\) 7.00000 0.288675
\(589\) 5.00000 0.206021
\(590\) 0 0
\(591\) −20.0000 −0.822690
\(592\) 6.00000 0.246598
\(593\) 28.0000 1.14982 0.574911 0.818216i \(-0.305037\pi\)
0.574911 + 0.818216i \(0.305037\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 0 0
\(597\) −6.00000 −0.245564
\(598\) 20.0000 0.817861
\(599\) 28.0000 1.14405 0.572024 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) 1.00000 0.0407231
\(604\) −24.0000 −0.976546
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) 25.0000 1.01472 0.507359 0.861735i \(-0.330622\pi\)
0.507359 + 0.861735i \(0.330622\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 25.0000 1.00892
\(615\) 0 0
\(616\) 0 0
\(617\) 46.0000 1.85189 0.925945 0.377658i \(-0.123271\pi\)
0.925945 + 0.377658i \(0.123271\pi\)
\(618\) 13.0000 0.522937
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) 32.0000 1.28308
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −21.0000 −0.839329
\(627\) 1.00000 0.0399362
\(628\) −14.0000 −0.558661
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) 17.0000 0.676224
\(633\) 13.0000 0.516704
\(634\) 13.0000 0.516296
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) 28.0000 1.10940
\(638\) 3.00000 0.118771
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 10.0000 0.394669
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −10.0000 −0.391630
\(653\) 48.0000 1.87839 0.939193 0.343391i \(-0.111576\pi\)
0.939193 + 0.343391i \(0.111576\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 3.00000 0.117041
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 13.0000 0.505259
\(663\) −16.0000 −0.621389
\(664\) −3.00000 −0.116423
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) −15.0000 −0.580802
\(668\) −18.0000 −0.696441
\(669\) −15.0000 −0.579934
\(670\) 0 0
\(671\) −11.0000 −0.424650
\(672\) 0 0
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 15.0000 0.576497 0.288248 0.957556i \(-0.406927\pi\)
0.288248 + 0.957556i \(0.406927\pi\)
\(678\) 5.00000 0.192024
\(679\) 0 0
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) −5.00000 −0.191460
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) −5.00000 −0.190762
\(688\) −4.00000 −0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 26.0000 0.989087 0.494543 0.869153i \(-0.335335\pi\)
0.494543 + 0.869153i \(0.335335\pi\)
\(692\) −7.00000 −0.266100
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 8.00000 0.303022
\(698\) −1.00000 −0.0378506
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 4.00000 0.150970
\(703\) −6.00000 −0.226294
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −28.0000 −1.05379
\(707\) 0 0
\(708\) 0 0
\(709\) −43.0000 −1.61490 −0.807449 0.589937i \(-0.799153\pi\)
−0.807449 + 0.589937i \(0.799153\pi\)
\(710\) 0 0
\(711\) 17.0000 0.637550
\(712\) 7.00000 0.262336
\(713\) 25.0000 0.936257
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) −29.0000 −1.08152 −0.540759 0.841178i \(-0.681863\pi\)
−0.540759 + 0.841178i \(0.681863\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) 4.00000 0.148762
\(724\) 0 0
\(725\) 0 0
\(726\) 10.0000 0.371135
\(727\) 10.0000 0.370879 0.185440 0.982656i \(-0.440629\pi\)
0.185440 + 0.982656i \(0.440629\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 11.0000 0.406572
\(733\) −41.0000 −1.51437 −0.757185 0.653201i \(-0.773426\pi\)
−0.757185 + 0.653201i \(0.773426\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) 1.00000 0.0368355
\(738\) −2.00000 −0.0736210
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 5.00000 0.183309
\(745\) 0 0
\(746\) −28.0000 −1.02515
\(747\) −3.00000 −0.109764
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) −16.0000 −0.583072
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) −8.00000 −0.290573
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) −5.00000 −0.181131
\(763\) 0 0
\(764\) 11.0000 0.397966
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) 7.00000 0.252099
\(772\) −10.0000 −0.359908
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) 2.00000 0.0715656
\(782\) 20.0000 0.715199
\(783\) −3.00000 −0.107211
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) −15.0000 −0.535032
\(787\) −37.0000 −1.31891 −0.659454 0.751745i \(-0.729212\pi\)
−0.659454 + 0.751745i \(0.729212\pi\)
\(788\) 20.0000 0.712470
\(789\) −11.0000 −0.391610
\(790\) 0 0
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 44.0000 1.56249
\(794\) 3.00000 0.106466
\(795\) 0 0
\(796\) 6.00000 0.212664
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 7.00000 0.247333
\(802\) −17.0000 −0.600291
\(803\) 3.00000 0.105868
\(804\) −1.00000 −0.0352673
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) 2.00000 0.0704033
\(808\) −8.00000 −0.281439
\(809\) −44.0000 −1.54696 −0.773479 0.633822i \(-0.781485\pi\)
−0.773479 + 0.633822i \(0.781485\pi\)
\(810\) 0 0
\(811\) −35.0000 −1.22902 −0.614508 0.788911i \(-0.710645\pi\)
−0.614508 + 0.788911i \(0.710645\pi\)
\(812\) 0 0
\(813\) −2.00000 −0.0701431
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 4.00000 0.139942
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) 26.0000 0.907406 0.453703 0.891153i \(-0.350103\pi\)
0.453703 + 0.891153i \(0.350103\pi\)
\(822\) −4.00000 −0.139516
\(823\) 26.0000 0.906303 0.453152 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) −13.0000 −0.452876
\(825\) 0 0
\(826\) 0 0
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) −5.00000 −0.173762
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) −1.00000 −0.0346896
\(832\) −4.00000 −0.138675
\(833\) 28.0000 0.970143
\(834\) 2.00000 0.0692543
\(835\) 0 0
\(836\) −1.00000 −0.0345857
\(837\) 5.00000 0.172825
\(838\) −12.0000 −0.414533
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −34.0000 −1.17172
\(843\) 3.00000 0.103325
\(844\) −13.0000 −0.447478
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) −30.0000 −1.02839
\(852\) −2.00000 −0.0685189
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −10.0000 −0.341793
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 4.00000 0.136558
\(859\) −18.0000 −0.614152 −0.307076 0.951685i \(-0.599351\pi\)
−0.307076 + 0.951685i \(0.599351\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 38.0000 1.29429
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 22.0000 0.747590
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 17.0000 0.576686
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 6.00000 0.203186
\(873\) −10.0000 −0.338449
\(874\) 5.00000 0.169128
\(875\) 0 0
\(876\) −3.00000 −0.101361
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 21.0000 0.708716
\(879\) 11.0000 0.371021
\(880\) 0 0
\(881\) −48.0000 −1.61716 −0.808581 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) −7.00000 −0.235702
\(883\) 38.0000 1.27880 0.639401 0.768874i \(-0.279182\pi\)
0.639401 + 0.768874i \(0.279182\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) 13.0000 0.436744
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 15.0000 0.502237
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −20.0000 −0.667781
\(898\) −27.0000 −0.901002
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) −5.00000 −0.166298
\(905\) 0 0
\(906\) 24.0000 0.797347
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) −2.00000 −0.0663723
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 1.00000 0.0331133
\(913\) −3.00000 −0.0992855
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 5.00000 0.165205
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) 14.0000 0.461817 0.230909 0.972975i \(-0.425830\pi\)
0.230909 + 0.972975i \(0.425830\pi\)
\(920\) 0 0
\(921\) −25.0000 −0.823778
\(922\) 4.00000 0.131733
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) −13.0000 −0.426976
\(928\) 3.00000 0.0984798
\(929\) −12.0000 −0.393707 −0.196854 0.980433i \(-0.563072\pi\)
−0.196854 + 0.980433i \(0.563072\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 24.0000 0.786146
\(933\) −32.0000 −1.04763
\(934\) 11.0000 0.359931
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 21.0000 0.685309
\(940\) 0 0
\(941\) 33.0000 1.07577 0.537885 0.843018i \(-0.319224\pi\)
0.537885 + 0.843018i \(0.319224\pi\)
\(942\) 14.0000 0.456145
\(943\) 10.0000 0.325645
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) −17.0000 −0.552134
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) −13.0000 −0.421554
\(952\) 0 0
\(953\) −3.00000 −0.0971795 −0.0485898 0.998819i \(-0.515473\pi\)
−0.0485898 + 0.998819i \(0.515473\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) 0 0
\(957\) −3.00000 −0.0969762
\(958\) −37.0000 −1.19542
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −24.0000 −0.773791
\(963\) −10.0000 −0.322245
\(964\) −4.00000 −0.128831
\(965\) 0 0
\(966\) 0 0
\(967\) −14.0000 −0.450210 −0.225105 0.974335i \(-0.572272\pi\)
−0.225105 + 0.974335i \(0.572272\pi\)
\(968\) −10.0000 −0.321412
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) −11.0000 −0.352101
\(977\) −46.0000 −1.47167 −0.735835 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 10.0000 0.319765
\(979\) 7.00000 0.223721
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 36.0000 1.14881
\(983\) −34.0000 −1.08443 −0.542216 0.840239i \(-0.682414\pi\)
−0.542216 + 0.840239i \(0.682414\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 20.0000 0.635963
\(990\) 0 0
\(991\) 13.0000 0.412959 0.206479 0.978451i \(-0.433799\pi\)
0.206479 + 0.978451i \(0.433799\pi\)
\(992\) −5.00000 −0.158750
\(993\) −13.0000 −0.412543
\(994\) 0 0
\(995\) 0 0
\(996\) 3.00000 0.0950586
\(997\) −37.0000 −1.17180 −0.585901 0.810383i \(-0.699259\pi\)
−0.585901 + 0.810383i \(0.699259\pi\)
\(998\) −24.0000 −0.759707
\(999\) −6.00000 −0.189832
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.u.1.1 yes 1
3.2 odd 2 8550.2.a.h.1.1 1
5.2 odd 4 2850.2.d.g.799.2 2
5.3 odd 4 2850.2.d.g.799.1 2
5.4 even 2 2850.2.a.l.1.1 1
15.14 odd 2 8550.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.l.1.1 1 5.4 even 2
2850.2.a.u.1.1 yes 1 1.1 even 1 trivial
2850.2.d.g.799.1 2 5.3 odd 4
2850.2.d.g.799.2 2 5.2 odd 4
8550.2.a.h.1.1 1 3.2 odd 2
8550.2.a.z.1.1 1 15.14 odd 2