Properties

Label 2366.2.a.n.1.1
Level 2366
Weight 2
Character 2366.1
Self dual yes
Analytic conductor 18.893
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.8926051182\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2366.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -3.00000 q^{10} +1.00000 q^{12} -1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -2.00000 q^{18} +4.00000 q^{19} -3.00000 q^{20} -1.00000 q^{21} +3.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -5.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -3.00000 q^{30} +10.0000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +3.00000 q^{35} -2.00000 q^{36} -8.00000 q^{37} +4.00000 q^{38} -3.00000 q^{40} -1.00000 q^{42} +8.00000 q^{43} +6.00000 q^{45} +3.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{50} +6.00000 q^{51} +12.0000 q^{53} -5.00000 q^{54} -1.00000 q^{56} +4.00000 q^{57} +6.00000 q^{58} -3.00000 q^{59} -3.00000 q^{60} +11.0000 q^{61} +10.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -2.00000 q^{67} +6.00000 q^{68} +3.00000 q^{69} +3.00000 q^{70} +3.00000 q^{71} -2.00000 q^{72} -2.00000 q^{73} -8.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} -4.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -1.00000 q^{84} -18.0000 q^{85} +8.00000 q^{86} +6.00000 q^{87} +6.00000 q^{89} +6.00000 q^{90} +3.00000 q^{92} +10.0000 q^{93} -6.00000 q^{94} -12.0000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) −3.00000 −0.948683
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −3.00000 −0.774597
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −2.00000 −0.471405
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −3.00000 −0.670820
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −3.00000 −0.547723
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 3.00000 0.507093
\(36\) −2.00000 −0.333333
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −1.00000 −0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 3.00000 0.442326
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 4.00000 0.565685
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) 6.00000 0.787839
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) −3.00000 −0.387298
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) 10.0000 1.27000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 6.00000 0.727607
\(69\) 3.00000 0.361158
\(70\) 3.00000 0.358569
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) −2.00000 −0.235702
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −8.00000 −0.929981
\(75\) 4.00000 0.461880
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.00000 −0.109109
\(85\) −18.0000 −1.95237
\(86\) 8.00000 0.862662
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 6.00000 0.632456
\(91\) 0 0
\(92\) 3.00000 0.312772
\(93\) 10.0000 1.03695
\(94\) −6.00000 −0.618853
\(95\) −12.0000 −1.23117
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 6.00000 0.594089
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 12.0000 1.16554
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −5.00000 −0.481125
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) −1.00000 −0.0944911
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 4.00000 0.374634
\(115\) −9.00000 −0.839254
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) −3.00000 −0.276172
\(119\) −6.00000 −0.550019
\(120\) −3.00000 −0.273861
\(121\) −11.0000 −1.00000
\(122\) 11.0000 0.995893
\(123\) 0 0
\(124\) 10.0000 0.898027
\(125\) 3.00000 0.268328
\(126\) 2.00000 0.178174
\(127\) 11.0000 0.976092 0.488046 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −2.00000 −0.172774
\(135\) 15.0000 1.29099
\(136\) 6.00000 0.514496
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 3.00000 0.255377
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 3.00000 0.253546
\(141\) −6.00000 −0.505291
\(142\) 3.00000 0.251754
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −18.0000 −1.49482
\(146\) −2.00000 −0.165521
\(147\) 1.00000 0.0824786
\(148\) −8.00000 −0.657596
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 4.00000 0.326599
\(151\) 1.00000 0.0813788 0.0406894 0.999172i \(-0.487045\pi\)
0.0406894 + 0.999172i \(0.487045\pi\)
\(152\) 4.00000 0.324443
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) −30.0000 −2.40966
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −4.00000 −0.318223
\(159\) 12.0000 0.951662
\(160\) −3.00000 −0.237171
\(161\) −3.00000 −0.236433
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) −18.0000 −1.38054
\(171\) −8.00000 −0.611775
\(172\) 8.00000 0.609994
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 6.00000 0.454859
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) −3.00000 −0.225494
\(178\) 6.00000 0.449719
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 6.00000 0.447214
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) 11.0000 0.813143
\(184\) 3.00000 0.221163
\(185\) 24.0000 1.76452
\(186\) 10.0000 0.733236
\(187\) 0 0
\(188\) −6.00000 −0.437595
\(189\) 5.00000 0.363696
\(190\) −12.0000 −0.870572
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 1.00000 0.0721688
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 4.00000 0.282843
\(201\) −2.00000 −0.141069
\(202\) −6.00000 −0.422159
\(203\) −6.00000 −0.421117
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 3.00000 0.207020
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 12.0000 0.824163
\(213\) 3.00000 0.205557
\(214\) −18.0000 −1.23045
\(215\) −24.0000 −1.63679
\(216\) −5.00000 −0.340207
\(217\) −10.0000 −0.678844
\(218\) 16.0000 1.08366
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −8.00000 −0.533333
\(226\) −18.0000 −1.19734
\(227\) 15.0000 0.995585 0.497792 0.867296i \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) 4.00000 0.264906
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −3.00000 −0.196537 −0.0982683 0.995160i \(-0.531330\pi\)
−0.0982683 + 0.995160i \(0.531330\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) −3.00000 −0.195283
\(237\) −4.00000 −0.259828
\(238\) −6.00000 −0.388922
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) −3.00000 −0.193649
\(241\) 28.0000 1.80364 0.901819 0.432113i \(-0.142232\pi\)
0.901819 + 0.432113i \(0.142232\pi\)
\(242\) −11.0000 −0.707107
\(243\) 16.0000 1.02640
\(244\) 11.0000 0.704203
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 10.0000 0.635001
\(249\) 0 0
\(250\) 3.00000 0.189737
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 11.0000 0.690201
\(255\) −18.0000 −1.12720
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 8.00000 0.498058
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −12.0000 −0.742781
\(262\) 15.0000 0.926703
\(263\) −27.0000 −1.66489 −0.832446 0.554107i \(-0.813060\pi\)
−0.832446 + 0.554107i \(0.813060\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) −4.00000 −0.245256
\(267\) 6.00000 0.367194
\(268\) −2.00000 −0.122169
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 15.0000 0.912871
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) −3.00000 −0.181237
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −16.0000 −0.959616
\(279\) −20.0000 −1.19737
\(280\) 3.00000 0.179284
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −6.00000 −0.357295
\(283\) −31.0000 −1.84276 −0.921379 0.388664i \(-0.872937\pi\)
−0.921379 + 0.388664i \(0.872937\pi\)
\(284\) 3.00000 0.178017
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 0 0
\(288\) −2.00000 −0.117851
\(289\) 19.0000 1.11765
\(290\) −18.0000 −1.05700
\(291\) −2.00000 −0.117242
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 1.00000 0.0583212
\(295\) 9.00000 0.524000
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 4.00000 0.230940
\(301\) −8.00000 −0.461112
\(302\) 1.00000 0.0575435
\(303\) −6.00000 −0.344691
\(304\) 4.00000 0.229416
\(305\) −33.0000 −1.88957
\(306\) −12.0000 −0.685994
\(307\) 13.0000 0.741949 0.370975 0.928643i \(-0.379024\pi\)
0.370975 + 0.928643i \(0.379024\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) −30.0000 −1.70389
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 2.00000 0.112867
\(315\) −6.00000 −0.338062
\(316\) −4.00000 −0.225018
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 12.0000 0.672927
\(319\) 0 0
\(320\) −3.00000 −0.167705
\(321\) −18.0000 −1.00466
\(322\) −3.00000 −0.167183
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 16.0000 0.884802
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 0 0
\(333\) 16.0000 0.876795
\(334\) 12.0000 0.656611
\(335\) 6.00000 0.327815
\(336\) −1.00000 −0.0545545
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) −18.0000 −0.976187
\(341\) 0 0
\(342\) −8.00000 −0.432590
\(343\) −1.00000 −0.0539949
\(344\) 8.00000 0.431331
\(345\) −9.00000 −0.484544
\(346\) 9.00000 0.483843
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 6.00000 0.321634
\(349\) −29.0000 −1.55233 −0.776167 0.630527i \(-0.782839\pi\)
−0.776167 + 0.630527i \(0.782839\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) −3.00000 −0.159448
\(355\) −9.00000 −0.477670
\(356\) 6.00000 0.317999
\(357\) −6.00000 −0.317554
\(358\) 6.00000 0.317110
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) 6.00000 0.316228
\(361\) −3.00000 −0.157895
\(362\) 5.00000 0.262794
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 11.0000 0.574979
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) 24.0000 1.24770
\(371\) −12.0000 −0.623009
\(372\) 10.0000 0.518476
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) −6.00000 −0.309426
\(377\) 0 0
\(378\) 5.00000 0.257172
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) −12.0000 −0.615587
\(381\) 11.0000 0.563547
\(382\) −24.0000 −1.22795
\(383\) 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
\(386\) −5.00000 −0.254493
\(387\) −16.0000 −0.813326
\(388\) −2.00000 −0.101535
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 1.00000 0.0505076
\(393\) 15.0000 0.756650
\(394\) −12.0000 −0.604551
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 7.00000 0.351320 0.175660 0.984451i \(-0.443794\pi\)
0.175660 + 0.984451i \(0.443794\pi\)
\(398\) 14.0000 0.701757
\(399\) −4.00000 −0.200250
\(400\) 4.00000 0.200000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) −3.00000 −0.149071
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) 14.0000 0.689730
\(413\) 3.00000 0.147620
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 0 0
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 3.00000 0.146385
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −4.00000 −0.194717
\(423\) 12.0000 0.583460
\(424\) 12.0000 0.582772
\(425\) 24.0000 1.16417
\(426\) 3.00000 0.145350
\(427\) −11.0000 −0.532327
\(428\) −18.0000 −0.870063
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) −5.00000 −0.240563
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) −10.0000 −0.480015
\(435\) −18.0000 −0.863034
\(436\) 16.0000 0.766261
\(437\) 12.0000 0.574038
\(438\) −2.00000 −0.0955637
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) −8.00000 −0.379663
\(445\) −18.0000 −0.853282
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) −8.00000 −0.377124
\(451\) 0 0
\(452\) −18.0000 −0.846649
\(453\) 1.00000 0.0469841
\(454\) 15.0000 0.703985
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 22.0000 1.02799
\(459\) −30.0000 −1.40028
\(460\) −9.00000 −0.419627
\(461\) −21.0000 −0.978068 −0.489034 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) 31.0000 1.44069 0.720346 0.693615i \(-0.243983\pi\)
0.720346 + 0.693615i \(0.243983\pi\)
\(464\) 6.00000 0.278543
\(465\) −30.0000 −1.39122
\(466\) −3.00000 −0.138972
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) 2.00000 0.0923514
\(470\) 18.0000 0.830278
\(471\) 2.00000 0.0921551
\(472\) −3.00000 −0.138086
\(473\) 0 0
\(474\) −4.00000 −0.183726
\(475\) 16.0000 0.734130
\(476\) −6.00000 −0.275010
\(477\) −24.0000 −1.09888
\(478\) −21.0000 −0.960518
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) −3.00000 −0.136931
\(481\) 0 0
\(482\) 28.0000 1.27537
\(483\) −3.00000 −0.136505
\(484\) −11.0000 −0.500000
\(485\) 6.00000 0.272446
\(486\) 16.0000 0.725775
\(487\) 13.0000 0.589086 0.294543 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) 11.0000 0.497947
\(489\) −20.0000 −0.904431
\(490\) −3.00000 −0.135526
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) −3.00000 −0.134568
\(498\) 0 0
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 3.00000 0.134164
\(501\) 12.0000 0.536120
\(502\) 3.00000 0.133897
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 2.00000 0.0890871
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 11.0000 0.488046
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) −18.0000 −0.797053
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) −20.0000 −0.883022
\(514\) −6.00000 −0.264649
\(515\) −42.0000 −1.85074
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) 9.00000 0.395056
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) −12.0000 −0.525226
\(523\) −1.00000 −0.0437269 −0.0218635 0.999761i \(-0.506960\pi\)
−0.0218635 + 0.999761i \(0.506960\pi\)
\(524\) 15.0000 0.655278
\(525\) −4.00000 −0.174574
\(526\) −27.0000 −1.17726
\(527\) 60.0000 2.61364
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) −36.0000 −1.56374
\(531\) 6.00000 0.260378
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 6.00000 0.259645
\(535\) 54.0000 2.33462
\(536\) −2.00000 −0.0863868
\(537\) 6.00000 0.258919
\(538\) 9.00000 0.388018
\(539\) 0 0
\(540\) 15.0000 0.645497
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −2.00000 −0.0859074
\(543\) 5.00000 0.214571
\(544\) 6.00000 0.257248
\(545\) −48.0000 −2.05609
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −3.00000 −0.128154
\(549\) −22.0000 −0.938937
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 3.00000 0.127688
\(553\) 4.00000 0.170097
\(554\) 26.0000 1.10463
\(555\) 24.0000 1.01874
\(556\) −16.0000 −0.678551
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) −20.0000 −0.846668
\(559\) 0 0
\(560\) 3.00000 0.126773
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) −6.00000 −0.252646
\(565\) 54.0000 2.27180
\(566\) −31.0000 −1.30303
\(567\) −1.00000 −0.0419961
\(568\) 3.00000 0.125877
\(569\) −45.0000 −1.88650 −0.943249 0.332086i \(-0.892248\pi\)
−0.943249 + 0.332086i \(0.892248\pi\)
\(570\) −12.0000 −0.502625
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) −2.00000 −0.0833333
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) 19.0000 0.790296
\(579\) −5.00000 −0.207793
\(580\) −18.0000 −0.747409
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 1.00000 0.0412393
\(589\) 40.0000 1.64817
\(590\) 9.00000 0.370524
\(591\) −12.0000 −0.493614
\(592\) −8.00000 −0.328798
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 18.0000 0.737928
\(596\) 0 0
\(597\) 14.0000 0.572982
\(598\) 0 0
\(599\) 3.00000 0.122577 0.0612883 0.998120i \(-0.480479\pi\)
0.0612883 + 0.998120i \(0.480479\pi\)
\(600\) 4.00000 0.163299
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −8.00000 −0.326056
\(603\) 4.00000 0.162893
\(604\) 1.00000 0.0406894
\(605\) 33.0000 1.34164
\(606\) −6.00000 −0.243733
\(607\) −10.0000 −0.405887 −0.202944 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 4.00000 0.162221
\(609\) −6.00000 −0.243132
\(610\) −33.0000 −1.33613
\(611\) 0 0
\(612\) −12.0000 −0.485071
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 13.0000 0.524637
\(615\) 0 0
\(616\) 0 0
\(617\) 3.00000 0.120775 0.0603877 0.998175i \(-0.480766\pi\)
0.0603877 + 0.998175i \(0.480766\pi\)
\(618\) 14.0000 0.563163
\(619\) 37.0000 1.48716 0.743578 0.668649i \(-0.233127\pi\)
0.743578 + 0.668649i \(0.233127\pi\)
\(620\) −30.0000 −1.20483
\(621\) −15.0000 −0.601929
\(622\) −6.00000 −0.240578
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) −48.0000 −1.91389
\(630\) −6.00000 −0.239046
\(631\) −5.00000 −0.199047 −0.0995234 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) −4.00000 −0.159111
\(633\) −4.00000 −0.158986
\(634\) 12.0000 0.476581
\(635\) −33.0000 −1.30957
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) −3.00000 −0.118585
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) −18.0000 −0.710403
\(643\) −41.0000 −1.61688 −0.808441 0.588577i \(-0.799688\pi\)
−0.808441 + 0.588577i \(0.799688\pi\)
\(644\) −3.00000 −0.118217
\(645\) −24.0000 −0.944999
\(646\) 24.0000 0.944267
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) −20.0000 −0.783260
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 16.0000 0.625650
\(655\) −45.0000 −1.75830
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 6.00000 0.233904
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −23.0000 −0.894596 −0.447298 0.894385i \(-0.647614\pi\)
−0.447298 + 0.894385i \(0.647614\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 16.0000 0.619987
\(667\) 18.0000 0.696963
\(668\) 12.0000 0.464294
\(669\) −8.00000 −0.309298
\(670\) 6.00000 0.231800
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 14.0000 0.539260
\(675\) −20.0000 −0.769800
\(676\) 0 0
\(677\) −51.0000 −1.96009 −0.980045 0.198778i \(-0.936303\pi\)
−0.980045 + 0.198778i \(0.936303\pi\)
\(678\) −18.0000 −0.691286
\(679\) 2.00000 0.0767530
\(680\) −18.0000 −0.690268
\(681\) 15.0000 0.574801
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) −8.00000 −0.305888
\(685\) 9.00000 0.343872
\(686\) −1.00000 −0.0381802
\(687\) 22.0000 0.839352
\(688\) 8.00000 0.304997
\(689\) 0 0
\(690\) −9.00000 −0.342624
\(691\) 1.00000 0.0380418 0.0190209 0.999819i \(-0.493945\pi\)
0.0190209 + 0.999819i \(0.493945\pi\)
\(692\) 9.00000 0.342129
\(693\) 0 0
\(694\) −24.0000 −0.911028
\(695\) 48.0000 1.82074
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) −29.0000 −1.09767
\(699\) −3.00000 −0.113470
\(700\) −4.00000 −0.151186
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) 18.0000 0.677919
\(706\) 6.00000 0.225813
\(707\) 6.00000 0.225653
\(708\) −3.00000 −0.112747
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −9.00000 −0.337764
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 30.0000 1.12351
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) −21.0000 −0.784259
\(718\) 9.00000 0.335877
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 6.00000 0.223607
\(721\) −14.0000 −0.521387
\(722\) −3.00000 −0.111648
\(723\) 28.0000 1.04133
\(724\) 5.00000 0.185824
\(725\) 24.0000 0.891338
\(726\) −11.0000 −0.408248
\(727\) −52.0000 −1.92857 −0.964287 0.264861i \(-0.914674\pi\)
−0.964287 + 0.264861i \(0.914674\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 6.00000 0.222070
\(731\) 48.0000 1.77534
\(732\) 11.0000 0.406572
\(733\) −35.0000 −1.29275 −0.646377 0.763018i \(-0.723717\pi\)
−0.646377 + 0.763018i \(0.723717\pi\)
\(734\) −10.0000 −0.369107
\(735\) −3.00000 −0.110657
\(736\) 3.00000 0.110581
\(737\) 0 0
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 24.0000 0.882258
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 10.0000 0.366618
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) 0 0
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 3.00000 0.109545
\(751\) 17.0000 0.620339 0.310169 0.950681i \(-0.399614\pi\)
0.310169 + 0.950681i \(0.399614\pi\)
\(752\) −6.00000 −0.218797
\(753\) 3.00000 0.109326
\(754\) 0 0
\(755\) −3.00000 −0.109181
\(756\) 5.00000 0.181848
\(757\) −52.0000 −1.88997 −0.944986 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 11.0000 0.398488
\(763\) −16.0000 −0.579239
\(764\) −24.0000 −0.868290
\(765\) 36.0000 1.30158
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 10.0000 0.360609 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −5.00000 −0.179954
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) −16.0000 −0.575108
\(775\) 40.0000 1.43684
\(776\) −2.00000 −0.0717958
\(777\) 8.00000 0.286998
\(778\) −30.0000 −1.07555
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 18.0000 0.643679
\(783\) −30.0000 −1.07211
\(784\) 1.00000 0.0357143
\(785\) −6.00000 −0.214149
\(786\) 15.0000 0.535032
\(787\) −53.0000 −1.88925 −0.944623 0.328158i \(-0.893572\pi\)
−0.944623 + 0.328158i \(0.893572\pi\)
\(788\) −12.0000 −0.427482
\(789\) −27.0000 −0.961225
\(790\) 12.0000 0.426941
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 0 0
\(794\) 7.00000 0.248421
\(795\) −36.0000 −1.27679
\(796\) 14.0000 0.496217
\(797\) −15.0000 −0.531327 −0.265664 0.964066i \(-0.585591\pi\)
−0.265664 + 0.964066i \(0.585591\pi\)
\(798\) −4.00000 −0.141598
\(799\) −36.0000 −1.27359
\(800\) 4.00000 0.141421
\(801\) −12.0000 −0.423999
\(802\) −6.00000 −0.211867
\(803\) 0 0
\(804\) −2.00000 −0.0705346
\(805\) 9.00000 0.317208
\(806\) 0 0
\(807\) 9.00000 0.316815
\(808\) −6.00000 −0.211079
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) −3.00000 −0.105409
\(811\) 19.0000 0.667180 0.333590 0.942718i \(-0.391740\pi\)
0.333590 + 0.942718i \(0.391740\pi\)
\(812\) −6.00000 −0.210559
\(813\) −2.00000 −0.0701431
\(814\) 0 0
\(815\) 60.0000 2.10171
\(816\) 6.00000 0.210042
\(817\) 32.0000 1.11954
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) −3.00000 −0.104637
\(823\) −13.0000 −0.453152 −0.226576 0.973994i \(-0.572753\pi\)
−0.226576 + 0.973994i \(0.572753\pi\)
\(824\) 14.0000 0.487713
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) −24.0000 −0.834562 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(828\) −6.00000 −0.208514
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) −16.0000 −0.554035
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) −50.0000 −1.72825
\(838\) 0 0
\(839\) 54.0000 1.86429 0.932144 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 3.00000 0.103510
\(841\) 7.00000 0.241379
\(842\) −26.0000 −0.896019
\(843\) −18.0000 −0.619953
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 11.0000 0.377964
\(848\) 12.0000 0.412082
\(849\) −31.0000 −1.06392
\(850\) 24.0000 0.823193
\(851\) −24.0000 −0.822709
\(852\) 3.00000 0.102778
\(853\) −17.0000 −0.582069 −0.291034 0.956713i \(-0.593999\pi\)
−0.291034 + 0.956713i \(0.593999\pi\)
\(854\) −11.0000 −0.376412
\(855\) 24.0000 0.820783
\(856\) −18.0000 −0.615227
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −24.0000 −0.818393
\(861\) 0 0
\(862\) 33.0000 1.12398
\(863\) 57.0000 1.94030 0.970151 0.242500i \(-0.0779676\pi\)
0.970151 + 0.242500i \(0.0779676\pi\)
\(864\) −5.00000 −0.170103
\(865\) −27.0000 −0.918028
\(866\) −4.00000 −0.135926
\(867\) 19.0000 0.645274
\(868\) −10.0000 −0.339422
\(869\) 0 0
\(870\) −18.0000 −0.610257
\(871\) 0 0
\(872\) 16.0000 0.541828
\(873\) 4.00000 0.135379
\(874\) 12.0000 0.405906
\(875\) −3.00000 −0.101419
\(876\) −2.00000 −0.0675737
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 14.0000 0.472477
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 9.00000 0.302532
\(886\) 18.0000 0.604722
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) −8.00000 −0.268462
\(889\) −11.0000 −0.368928
\(890\) −18.0000 −0.603361
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −9.00000 −0.300334
\(899\) 60.0000 2.00111
\(900\) −8.00000 −0.266667
\(901\) 72.0000 2.39867
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) −18.0000 −0.598671
\(905\) −15.0000 −0.498617
\(906\) 1.00000 0.0332228
\(907\) 14.0000 0.464862 0.232431 0.972613i \(-0.425332\pi\)
0.232431 + 0.972613i \(0.425332\pi\)
\(908\) 15.0000 0.497792
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 21.0000 0.695761 0.347881 0.937539i \(-0.386901\pi\)
0.347881 + 0.937539i \(0.386901\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) −29.0000 −0.959235
\(915\) −33.0000 −1.09095
\(916\) 22.0000 0.726900
\(917\) −15.0000 −0.495344
\(918\) −30.0000 −0.990148
\(919\) −7.00000 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) −9.00000 −0.296721
\(921\) 13.0000 0.428365
\(922\) −21.0000 −0.691598
\(923\) 0 0
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 31.0000 1.01872
\(927\) −28.0000 −0.919641
\(928\) 6.00000 0.196960
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) −30.0000 −0.983739
\(931\) 4.00000 0.131095
\(932\) −3.00000 −0.0982683
\(933\) −6.00000 −0.196431
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) 0 0
\(937\) 8.00000 0.261349 0.130674 0.991425i \(-0.458286\pi\)
0.130674 + 0.991425i \(0.458286\pi\)
\(938\) 2.00000 0.0653023
\(939\) 8.00000 0.261070
\(940\) 18.0000 0.587095
\(941\) −15.0000 −0.488986 −0.244493 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) 2.00000 0.0651635
\(943\) 0 0
\(944\) −3.00000 −0.0976417
\(945\) −15.0000 −0.487950
\(946\) 0 0
\(947\) 42.0000 1.36482 0.682408 0.730971i \(-0.260933\pi\)
0.682408 + 0.730971i \(0.260933\pi\)
\(948\) −4.00000 −0.129914
\(949\) 0 0
\(950\) 16.0000 0.519109
\(951\) 12.0000 0.389127
\(952\) −6.00000 −0.194461
\(953\) −15.0000 −0.485898 −0.242949 0.970039i \(-0.578115\pi\)
−0.242949 + 0.970039i \(0.578115\pi\)
\(954\) −24.0000 −0.777029
\(955\) 72.0000 2.32987
\(956\) −21.0000 −0.679189
\(957\) 0 0
\(958\) −30.0000 −0.969256
\(959\) 3.00000 0.0968751
\(960\) −3.00000 −0.0968246
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 36.0000 1.16008
\(964\) 28.0000 0.901819
\(965\) 15.0000 0.482867
\(966\) −3.00000 −0.0965234
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) −11.0000 −0.353553
\(969\) 24.0000 0.770991
\(970\) 6.00000 0.192648
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) 16.0000 0.513200
\(973\) 16.0000 0.512936
\(974\) 13.0000 0.416547
\(975\) 0 0
\(976\) 11.0000 0.352101
\(977\) −39.0000 −1.24772 −0.623860 0.781536i \(-0.714437\pi\)
−0.623860 + 0.781536i \(0.714437\pi\)
\(978\) −20.0000 −0.639529
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) −32.0000 −1.02168
\(982\) 12.0000 0.382935
\(983\) 30.0000 0.956851 0.478426 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 36.0000 1.14647
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 17.0000 0.540023 0.270011 0.962857i \(-0.412973\pi\)
0.270011 + 0.962857i \(0.412973\pi\)
\(992\) 10.0000 0.317500
\(993\) 10.0000 0.317340
\(994\) −3.00000 −0.0951542
\(995\) −42.0000 −1.33149
\(996\) 0 0
\(997\) 35.0000 1.10846 0.554231 0.832363i \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(998\) 22.0000 0.696398
\(999\) 40.0000 1.26554
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 2366.2.a.n.1.1 1
13.4 even 6 182.2.g.b.29.1 2
13.5 odd 4 2366.2.d.f.337.1 2
13.8 odd 4 2366.2.d.f.337.2 2
13.10 even 6 182.2.g.b.113.1 yes 2
13.12 even 2 2366.2.a.f.1.1 1
39.17 odd 6 1638.2.r.c.757.1 2
39.23 odd 6 1638.2.r.c.1387.1 2
52.23 odd 6 1456.2.s.e.113.1 2
52.43 odd 6 1456.2.s.e.1121.1 2
91.4 even 6 1274.2.e.d.471.1 2
91.10 odd 6 1274.2.h.i.373.1 2
91.17 odd 6 1274.2.e.i.471.1 2
91.23 even 6 1274.2.e.d.165.1 2
91.30 even 6 1274.2.h.j.263.1 2
91.62 odd 6 1274.2.g.g.295.1 2
91.69 odd 6 1274.2.g.g.393.1 2
91.75 odd 6 1274.2.e.i.165.1 2
91.82 odd 6 1274.2.h.i.263.1 2
91.88 even 6 1274.2.h.j.373.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.g.b.29.1 2 13.4 even 6
182.2.g.b.113.1 yes 2 13.10 even 6
1274.2.e.d.165.1 2 91.23 even 6
1274.2.e.d.471.1 2 91.4 even 6
1274.2.e.i.165.1 2 91.75 odd 6
1274.2.e.i.471.1 2 91.17 odd 6
1274.2.g.g.295.1 2 91.62 odd 6
1274.2.g.g.393.1 2 91.69 odd 6
1274.2.h.i.263.1 2 91.82 odd 6
1274.2.h.i.373.1 2 91.10 odd 6
1274.2.h.j.263.1 2 91.30 even 6
1274.2.h.j.373.1 2 91.88 even 6
1456.2.s.e.113.1 2 52.23 odd 6
1456.2.s.e.1121.1 2 52.43 odd 6
1638.2.r.c.757.1 2 39.17 odd 6
1638.2.r.c.1387.1 2 39.23 odd 6
2366.2.a.f.1.1 1 13.12 even 2
2366.2.a.n.1.1 1 1.1 even 1 trivial
2366.2.d.f.337.1 2 13.5 odd 4
2366.2.d.f.337.2 2 13.8 odd 4