Properties

Label 3850.2.a.m.1.1
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} +2.00000 q^{21} -1.00000 q^{22} -6.00000 q^{23} -2.00000 q^{24} -2.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +4.00000 q^{29} +1.00000 q^{32} +2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} +6.00000 q^{38} +4.00000 q^{39} +2.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -6.00000 q^{46} +4.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +4.00000 q^{51} -2.00000 q^{52} +12.0000 q^{53} +4.00000 q^{54} -1.00000 q^{56} -12.0000 q^{57} +4.00000 q^{58} +2.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +8.00000 q^{67} -2.00000 q^{68} +12.0000 q^{69} -12.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -8.00000 q^{74} +6.00000 q^{76} +1.00000 q^{77} +4.00000 q^{78} +10.0000 q^{79} -11.0000 q^{81} +12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{86} -8.00000 q^{87} -1.00000 q^{88} +14.0000 q^{89} +2.00000 q^{91} -6.00000 q^{92} +4.00000 q^{94} -2.00000 q^{96} -4.00000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) −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
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −2.00000 −0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 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
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −1.00000 −0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) −2.00000 −0.408248
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 4.00000 0.769800
\(28\) −1.00000 −0.188982
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 6.00000 0.973329
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 2.00000 0.308607
\(43\) −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
\(46\) −6.00000 −0.884652
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) −2.00000 −0.277350
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −12.0000 −1.58944
\(58\) 4.00000 0.525226
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.00000 −0.242536
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 1.00000 0.113961
\(78\) 4.00000 0.452911
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −8.00000 −0.857690
\(88\) −1.00000 −0.106600
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 4.00000 0.396059
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 4.00000 0.384900
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 16.0000 1.51865
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −12.0000 −1.12390
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 2.00000 0.174078
\(133\) −6.00000 −0.520266
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 12.0000 1.02151
\(139\) −6.00000 −0.508913 −0.254457 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) −12.0000 −1.00702
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) −2.00000 −0.164957
\(148\) −8.00000 −0.657596
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 6.00000 0.486664
\(153\) −2.00000 −0.161690
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 10.0000 0.795557
\(159\) −24.0000 −1.90332
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) −11.0000 −0.864242
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) −4.00000 −0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −8.00000 −0.606478
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) −4.00000 −0.295689
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 4.00000 0.291730
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −2.00000 −0.144338
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 10.0000 0.703598
\(203\) −4.00000 −0.280745
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) −6.00000 −0.417029
\(208\) −2.00000 −0.138675
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 12.0000 0.824163
\(213\) 24.0000 1.64445
\(214\) −20.0000 −1.36717
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 16.0000 1.07385
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −12.0000 −0.794719
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 4.00000 0.262613
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) −20.0000 −1.29914
\(238\) 2.00000 0.129641
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 1.00000 0.0642824
\(243\) 10.0000 0.641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) −24.0000 −1.52094
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 6.00000 0.377217
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 28.0000 1.74659 0.873296 0.487190i \(-0.161978\pi\)
0.873296 + 0.487190i \(0.161978\pi\)
\(258\) 8.00000 0.498058
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) −6.00000 −0.370681
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) −28.0000 −1.71357
\(268\) 8.00000 0.488678
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) −2.00000 −0.121268
\(273\) −4.00000 −0.242091
\(274\) 22.0000 1.32907
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −6.00000 −0.359856
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) −8.00000 −0.476393
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 6.00000 0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −2.00000 −0.116642
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) −4.00000 −0.232104
\(298\) 4.00000 0.231714
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −6.00000 −0.345261
\(303\) −20.0000 −1.14897
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 1.00000 0.0569803
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 4.00000 0.226455
\(313\) 12.0000 0.678280 0.339140 0.940736i \(-0.389864\pi\)
0.339140 + 0.940736i \(0.389864\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) −24.0000 −1.34585
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 40.0000 2.23258
\(322\) 6.00000 0.334367
\(323\) −12.0000 −0.667698
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) −8.00000 −0.442401
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000 0.658586
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) −8.00000 −0.428845
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) −1.00000 −0.0533002
\(353\) −20.0000 −1.06449 −0.532246 0.846590i \(-0.678652\pi\)
−0.532246 + 0.846590i \(0.678652\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) −4.00000 −0.211702
\(358\) −4.00000 −0.211407
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 2.00000 0.105118
\(363\) −2.00000 −0.104973
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) −8.00000 −0.412021
\(378\) −4.00000 −0.205738
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 8.00000 0.409316
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −2.00000 −0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) −4.00000 −0.203069
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 16.0000 0.802008
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) −16.0000 −0.798007
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 8.00000 0.396545
\(408\) 4.00000 0.198030
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) −44.0000 −2.17036
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 12.0000 0.587643
\(418\) −6.00000 −0.293470
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −20.0000 −0.973585
\(423\) 4.00000 0.194487
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) 24.0000 1.16280
\(427\) −2.00000 −0.0967868
\(428\) −20.0000 −0.966736
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −22.0000 −1.05970 −0.529851 0.848091i \(-0.677752\pi\)
−0.529851 + 0.848091i \(0.677752\pi\)
\(432\) 4.00000 0.192450
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) −36.0000 −1.72211
\(438\) −12.0000 −0.573382
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 4.00000 0.190261
\(443\) −40.0000 −1.90046 −0.950229 0.311553i \(-0.899151\pi\)
−0.950229 + 0.311553i \(0.899151\pi\)
\(444\) 16.0000 0.759326
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) −8.00000 −0.378387
\(448\) −1.00000 −0.0472456
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 12.0000 0.563809
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) −12.0000 −0.561951
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 6.00000 0.280362
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −26.0000 −1.21094 −0.605470 0.795868i \(-0.707015\pi\)
−0.605470 + 0.795868i \(0.707015\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 10.0000 0.464739 0.232370 0.972628i \(-0.425352\pi\)
0.232370 + 0.972628i \(0.425352\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) −38.0000 −1.75843 −0.879215 0.476425i \(-0.841932\pi\)
−0.879215 + 0.476425i \(0.841932\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) −20.0000 −0.918630
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) 12.0000 0.549442
\(478\) 6.00000 0.274434
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) −8.00000 −0.364390
\(483\) −12.0000 −0.546019
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) 18.0000 0.815658 0.407829 0.913058i \(-0.366286\pi\)
0.407829 + 0.913058i \(0.366286\pi\)
\(488\) 2.00000 0.0905357
\(489\) −48.0000 −2.17064
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) −24.0000 −1.07547
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) 18.0000 0.799408
\(508\) −8.00000 −0.354943
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) 24.0000 1.05963
\(514\) 28.0000 1.23503
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) −4.00000 −0.175920
\(518\) 8.00000 0.351500
\(519\) −36.0000 −1.58022
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 4.00000 0.175075
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) 0 0
\(534\) −28.0000 −1.21168
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 8.00000 0.345225
\(538\) 14.0000 0.603583
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 28.0000 1.20270
\(543\) −4.00000 −0.171656
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 22.0000 0.939793
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 12.0000 0.510754
\(553\) −10.0000 −0.425243
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) −6.00000 −0.254457
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −2.00000 −0.0843649
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) −32.0000 −1.34506
\(567\) 11.0000 0.461957
\(568\) −12.0000 −0.503509
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 2.00000 0.0836242
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) −13.0000 −0.540729
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 8.00000 0.331611
\(583\) −12.0000 −0.496989
\(584\) 6.00000 0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 0 0
\(590\) 0 0
\(591\) −4.00000 −0.164538
\(592\) −8.00000 −0.328798
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) −32.0000 −1.30967
\(598\) 12.0000 0.490716
\(599\) 44.0000 1.79779 0.898896 0.438163i \(-0.144371\pi\)
0.898896 + 0.438163i \(0.144371\pi\)
\(600\) 0 0
\(601\) 44.0000 1.79480 0.897399 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 4.00000 0.163028
\(603\) 8.00000 0.325785
\(604\) −6.00000 −0.244137
\(605\) 0 0
\(606\) −20.0000 −0.812444
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 6.00000 0.243332
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) −2.00000 −0.0808452
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) −16.0000 −0.643614
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 16.0000 0.641542
\(623\) −14.0000 −0.560898
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) 12.0000 0.479616
\(627\) 12.0000 0.479234
\(628\) 10.0000 0.399043
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 10.0000 0.397779
\(633\) 40.0000 1.58986
\(634\) 0 0
\(635\) 0 0
\(636\) −24.0000 −0.951662
\(637\) −2.00000 −0.0792429
\(638\) −4.00000 −0.158362
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 40.0000 1.57867
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 24.0000 0.939913
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) −8.00000 −0.312825
\(655\) 0 0
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) −4.00000 −0.155936
\(659\) 8.00000 0.311636 0.155818 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 12.0000 0.466393
\(663\) −8.00000 −0.310694
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 32.0000 1.23719
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 2.00000 0.0771517
\(673\) 42.0000 1.61898 0.809491 0.587133i \(-0.199743\pi\)
0.809491 + 0.587133i \(0.199743\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) −12.0000 −0.460857
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) −48.0000 −1.83936
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −12.0000 −0.457829
\(688\) −4.00000 −0.152499
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 18.0000 0.684257
\(693\) 1.00000 0.0379869
\(694\) −20.0000 −0.759190
\(695\) 0 0
\(696\) −8.00000 −0.303239
\(697\) 0 0
\(698\) 6.00000 0.227103
\(699\) −52.0000 −1.96682
\(700\) 0 0
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) −8.00000 −0.301941
\(703\) −48.0000 −1.81035
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −20.0000 −0.752710
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) −12.0000 −0.448148
\(718\) −6.00000 −0.223918
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 17.0000 0.632674
\(723\) 16.0000 0.595046
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) −4.00000 −0.147844
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) −12.0000 −0.440534
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 26.0000 0.951928
\(747\) 12.0000 0.439057
\(748\) 2.00000 0.0731272
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −52.0000 −1.88997 −0.944986 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 4.00000 0.145287
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) 16.0000 0.579619
\(763\) −4.00000 −0.144810
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) −2.00000 −0.0721688
\(769\) −28.0000 −1.00971 −0.504853 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 0 0
\(771\) −56.0000 −2.01679
\(772\) −2.00000 −0.0719816
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −4.00000 −0.143592
\(777\) −16.0000 −0.573997
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 12.0000 0.429119
\(783\) 16.0000 0.571793
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 48.0000 1.71102 0.855508 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) 2.00000 0.0712470
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) −1.00000 −0.0355335
\(793\) −4.00000 −0.142044
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 12.0000 0.424795
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) −34.0000 −1.20058
\(803\) −6.00000 −0.211735
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 0 0
\(807\) −28.0000 −0.985647
\(808\) 10.0000 0.351799
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) −30.0000 −1.05344 −0.526721 0.850038i \(-0.676579\pi\)
−0.526721 + 0.850038i \(0.676579\pi\)
\(812\) −4.00000 −0.140372
\(813\) −56.0000 −1.96401
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) −24.0000 −0.839654
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) −44.0000 −1.53468
\(823\) −14.0000 −0.488009 −0.244005 0.969774i \(-0.578461\pi\)
−0.244005 + 0.969774i \(0.578461\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) −6.00000 −0.208514
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) −44.0000 −1.52634
\(832\) −2.00000 −0.0693375
\(833\) −2.00000 −0.0692959
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) −16.0000 −0.552711
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −26.0000 −0.896019
\(843\) 4.00000 0.137767
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) −1.00000 −0.0343604
\(848\) 12.0000 0.412082
\(849\) 64.0000 2.19647
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 24.0000 0.822226
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) −4.00000 −0.136558
\(859\) 48.0000 1.63774 0.818869 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −22.0000 −0.749323
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) 26.0000 0.883006
\(868\) 0 0
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 4.00000 0.135457
\(873\) −4.00000 −0.135379
\(874\) −36.0000 −1.21772
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 12.0000 0.404980
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 1.00000 0.0336718
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −40.0000 −1.34383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 16.0000 0.536925
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) −16.0000 −0.535720
\(893\) 24.0000 0.803129
\(894\) −8.00000 −0.267560
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −24.0000 −0.801337
\(898\) 14.0000 0.467186
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 12.0000 0.398673
\(907\) 4.00000 0.132818 0.0664089 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 24.0000 0.796468
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −12.0000 −0.397360
\(913\) −12.0000 −0.397142
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 6.00000 0.198137
\(918\) −8.00000 −0.264039
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) −26.0000 −0.856264
\(923\) 24.0000 0.789970
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 10.0000 0.328620
\(927\) 8.00000 0.262754
\(928\) 4.00000 0.131306
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 26.0000 0.851658
\(933\) −32.0000 −1.04763
\(934\) −38.0000 −1.24340
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −6.00000 −0.196011 −0.0980057 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) −8.00000 −0.261209
\(939\) −24.0000 −0.783210
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −20.0000 −0.651635
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) −20.0000 −0.649570
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 12.0000 0.388514
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 8.00000 0.258603
\(958\) 8.00000 0.258468
\(959\) −22.0000 −0.710417
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 16.0000 0.515861
\(963\) −20.0000 −0.644491
\(964\) −8.00000 −0.257663
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) 1.00000 0.0321412
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 10.0000 0.320750
\(973\) 6.00000 0.192351
\(974\) 18.0000 0.576757
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) −48.0000 −1.53487
\(979\) −14.0000 −0.447442
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 12.0000 0.382935
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8.00000 −0.254772
\(987\) 8.00000 0.254643
\(988\) −12.0000 −0.381771
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) −24.0000 −0.761617
\(994\) 12.0000 0.380617
\(995\) 0 0
\(996\) −24.0000 −0.760469
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −28.0000 −0.886325
\(999\) −32.0000 −1.01244
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 3850.2.a.m.1.1 1
5.2 odd 4 3850.2.c.c.1849.2 2
5.3 odd 4 3850.2.c.c.1849.1 2
5.4 even 2 770.2.a.e.1.1 1
15.14 odd 2 6930.2.a.bk.1.1 1
20.19 odd 2 6160.2.a.a.1.1 1
35.34 odd 2 5390.2.a.c.1.1 1
55.54 odd 2 8470.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.e.1.1 1 5.4 even 2
3850.2.a.m.1.1 1 1.1 even 1 trivial
3850.2.c.c.1849.1 2 5.3 odd 4
3850.2.c.c.1849.2 2 5.2 odd 4
5390.2.a.c.1.1 1 35.34 odd 2
6160.2.a.a.1.1 1 20.19 odd 2
6930.2.a.bk.1.1 1 15.14 odd 2
8470.2.a.bg.1.1 1 55.54 odd 2