Properties

Label 8624.2.a.bf.1.2
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.23607 q^{3} +1.23607 q^{5} -1.47214 q^{9} +O(q^{10})\) \(q+1.23607 q^{3} +1.23607 q^{5} -1.47214 q^{9} -1.00000 q^{11} +3.23607 q^{13} +1.52786 q^{15} -2.47214 q^{17} -7.23607 q^{19} -4.00000 q^{23} -3.47214 q^{25} -5.52786 q^{27} +4.47214 q^{29} +2.00000 q^{31} -1.23607 q^{33} +6.94427 q^{37} +4.00000 q^{39} +2.47214 q^{41} +10.4721 q^{43} -1.81966 q^{45} -2.00000 q^{47} -3.05573 q^{51} +8.47214 q^{53} -1.23607 q^{55} -8.94427 q^{57} +2.76393 q^{59} +0.763932 q^{61} +4.00000 q^{65} -11.4164 q^{67} -4.94427 q^{69} -6.47214 q^{71} -12.9443 q^{73} -4.29180 q^{75} -2.41641 q^{81} -12.1803 q^{83} -3.05573 q^{85} +5.52786 q^{87} -10.0000 q^{89} +2.47214 q^{93} -8.94427 q^{95} -12.4721 q^{97} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 6 q^{9} - 2 q^{11} + 2 q^{13} + 12 q^{15} + 4 q^{17} - 10 q^{19} - 8 q^{23} + 2 q^{25} - 20 q^{27} + 4 q^{31} + 2 q^{33} - 4 q^{37} + 8 q^{39} - 4 q^{41} + 12 q^{43} - 26 q^{45} - 4 q^{47} - 24 q^{51} + 8 q^{53} + 2 q^{55} + 10 q^{59} + 6 q^{61} + 8 q^{65} + 4 q^{67} + 8 q^{69} - 4 q^{71} - 8 q^{73} - 22 q^{75} + 22 q^{81} - 2 q^{83} - 24 q^{85} + 20 q^{87} - 20 q^{89} - 4 q^{93} - 16 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.23607 0.713644 0.356822 0.934172i \(-0.383860\pi\)
0.356822 + 0.934172i \(0.383860\pi\)
\(4\) 0 0
\(5\) 1.23607 0.552786 0.276393 0.961045i \(-0.410861\pi\)
0.276393 + 0.961045i \(0.410861\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.47214 −0.490712
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 3.23607 0.897524 0.448762 0.893651i \(-0.351865\pi\)
0.448762 + 0.893651i \(0.351865\pi\)
\(14\) 0 0
\(15\) 1.52786 0.394493
\(16\) 0 0
\(17\) −2.47214 −0.599581 −0.299791 0.954005i \(-0.596917\pi\)
−0.299791 + 0.954005i \(0.596917\pi\)
\(18\) 0 0
\(19\) −7.23607 −1.66007 −0.830034 0.557713i \(-0.811679\pi\)
−0.830034 + 0.557713i \(0.811679\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) −5.52786 −1.06384
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) −1.23607 −0.215172
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 2.47214 0.386083 0.193041 0.981191i \(-0.438165\pi\)
0.193041 + 0.981191i \(0.438165\pi\)
\(42\) 0 0
\(43\) 10.4721 1.59699 0.798493 0.602004i \(-0.205631\pi\)
0.798493 + 0.602004i \(0.205631\pi\)
\(44\) 0 0
\(45\) −1.81966 −0.271259
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3.05573 −0.427888
\(52\) 0 0
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 0 0
\(55\) −1.23607 −0.166671
\(56\) 0 0
\(57\) −8.94427 −1.18470
\(58\) 0 0
\(59\) 2.76393 0.359833 0.179917 0.983682i \(-0.442417\pi\)
0.179917 + 0.983682i \(0.442417\pi\)
\(60\) 0 0
\(61\) 0.763932 0.0978115 0.0489057 0.998803i \(-0.484427\pi\)
0.0489057 + 0.998803i \(0.484427\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −11.4164 −1.39474 −0.697368 0.716713i \(-0.745646\pi\)
−0.697368 + 0.716713i \(0.745646\pi\)
\(68\) 0 0
\(69\) −4.94427 −0.595220
\(70\) 0 0
\(71\) −6.47214 −0.768101 −0.384051 0.923312i \(-0.625471\pi\)
−0.384051 + 0.923312i \(0.625471\pi\)
\(72\) 0 0
\(73\) −12.9443 −1.51501 −0.757506 0.652828i \(-0.773582\pi\)
−0.757506 + 0.652828i \(0.773582\pi\)
\(74\) 0 0
\(75\) −4.29180 −0.495574
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 0 0
\(83\) −12.1803 −1.33697 −0.668483 0.743727i \(-0.733056\pi\)
−0.668483 + 0.743727i \(0.733056\pi\)
\(84\) 0 0
\(85\) −3.05573 −0.331440
\(86\) 0 0
\(87\) 5.52786 0.592649
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.47214 0.256349
\(94\) 0 0
\(95\) −8.94427 −0.917663
\(96\) 0 0
\(97\) −12.4721 −1.26635 −0.633177 0.774007i \(-0.718249\pi\)
−0.633177 + 0.774007i \(0.718249\pi\)
\(98\) 0 0
\(99\) 1.47214 0.147955
\(100\) 0 0
\(101\) −8.18034 −0.813974 −0.406987 0.913434i \(-0.633421\pi\)
−0.406987 + 0.913434i \(0.633421\pi\)
\(102\) 0 0
\(103\) −14.9443 −1.47250 −0.736251 0.676708i \(-0.763406\pi\)
−0.736251 + 0.676708i \(0.763406\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.47214 −0.238990 −0.119495 0.992835i \(-0.538128\pi\)
−0.119495 + 0.992835i \(0.538128\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 8.58359 0.814719
\(112\) 0 0
\(113\) −0.472136 −0.0444148 −0.0222074 0.999753i \(-0.507069\pi\)
−0.0222074 + 0.999753i \(0.507069\pi\)
\(114\) 0 0
\(115\) −4.94427 −0.461056
\(116\) 0 0
\(117\) −4.76393 −0.440426
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 3.05573 0.275526
\(124\) 0 0
\(125\) −10.4721 −0.936656
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) 12.9443 1.13968
\(130\) 0 0
\(131\) 4.76393 0.416227 0.208113 0.978105i \(-0.433268\pi\)
0.208113 + 0.978105i \(0.433268\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −6.83282 −0.588075
\(136\) 0 0
\(137\) −19.8885 −1.69919 −0.849596 0.527433i \(-0.823154\pi\)
−0.849596 + 0.527433i \(0.823154\pi\)
\(138\) 0 0
\(139\) −21.7082 −1.84127 −0.920633 0.390429i \(-0.872327\pi\)
−0.920633 + 0.390429i \(0.872327\pi\)
\(140\) 0 0
\(141\) −2.47214 −0.208191
\(142\) 0 0
\(143\) −3.23607 −0.270614
\(144\) 0 0
\(145\) 5.52786 0.459064
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −22.3607 −1.83186 −0.915929 0.401340i \(-0.868545\pi\)
−0.915929 + 0.401340i \(0.868545\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 3.63932 0.294222
\(154\) 0 0
\(155\) 2.47214 0.198567
\(156\) 0 0
\(157\) 12.6525 1.00978 0.504889 0.863184i \(-0.331534\pi\)
0.504889 + 0.863184i \(0.331534\pi\)
\(158\) 0 0
\(159\) 10.4721 0.830494
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 19.4164 1.52081 0.760405 0.649449i \(-0.225000\pi\)
0.760405 + 0.649449i \(0.225000\pi\)
\(164\) 0 0
\(165\) −1.52786 −0.118944
\(166\) 0 0
\(167\) 11.4164 0.883428 0.441714 0.897156i \(-0.354371\pi\)
0.441714 + 0.897156i \(0.354371\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) 10.6525 0.814615
\(172\) 0 0
\(173\) 3.23607 0.246034 0.123017 0.992405i \(-0.460743\pi\)
0.123017 + 0.992405i \(0.460743\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.41641 0.256793
\(178\) 0 0
\(179\) 8.94427 0.668526 0.334263 0.942480i \(-0.391513\pi\)
0.334263 + 0.942480i \(0.391513\pi\)
\(180\) 0 0
\(181\) −9.23607 −0.686512 −0.343256 0.939242i \(-0.611530\pi\)
−0.343256 + 0.939242i \(0.611530\pi\)
\(182\) 0 0
\(183\) 0.944272 0.0698026
\(184\) 0 0
\(185\) 8.58359 0.631078
\(186\) 0 0
\(187\) 2.47214 0.180780
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.47214 0.178877 0.0894387 0.995992i \(-0.471493\pi\)
0.0894387 + 0.995992i \(0.471493\pi\)
\(192\) 0 0
\(193\) −14.9443 −1.07571 −0.537856 0.843037i \(-0.680766\pi\)
−0.537856 + 0.843037i \(0.680766\pi\)
\(194\) 0 0
\(195\) 4.94427 0.354067
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 18.9443 1.34292 0.671462 0.741039i \(-0.265667\pi\)
0.671462 + 0.741039i \(0.265667\pi\)
\(200\) 0 0
\(201\) −14.1115 −0.995345
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.05573 0.213421
\(206\) 0 0
\(207\) 5.88854 0.409282
\(208\) 0 0
\(209\) 7.23607 0.500529
\(210\) 0 0
\(211\) 13.5279 0.931297 0.465648 0.884970i \(-0.345821\pi\)
0.465648 + 0.884970i \(0.345821\pi\)
\(212\) 0 0
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) 12.9443 0.882792
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) −0.472136 −0.0316166 −0.0158083 0.999875i \(-0.505032\pi\)
−0.0158083 + 0.999875i \(0.505032\pi\)
\(224\) 0 0
\(225\) 5.11146 0.340764
\(226\) 0 0
\(227\) −19.2361 −1.27674 −0.638371 0.769729i \(-0.720392\pi\)
−0.638371 + 0.769729i \(0.720392\pi\)
\(228\) 0 0
\(229\) −17.2361 −1.13899 −0.569496 0.821994i \(-0.692861\pi\)
−0.569496 + 0.821994i \(0.692861\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.9443 −0.979032 −0.489516 0.871994i \(-0.662826\pi\)
−0.489516 + 0.871994i \(0.662826\pi\)
\(234\) 0 0
\(235\) −2.47214 −0.161264
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −15.4164 −0.993058 −0.496529 0.868020i \(-0.665392\pi\)
−0.496529 + 0.868020i \(0.665392\pi\)
\(242\) 0 0
\(243\) 13.5967 0.872232
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −23.4164 −1.48995
\(248\) 0 0
\(249\) −15.0557 −0.954118
\(250\) 0 0
\(251\) 29.2361 1.84536 0.922682 0.385562i \(-0.125992\pi\)
0.922682 + 0.385562i \(0.125992\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) −3.77709 −0.236530
\(256\) 0 0
\(257\) −6.94427 −0.433172 −0.216586 0.976264i \(-0.569492\pi\)
−0.216586 + 0.976264i \(0.569492\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.58359 −0.407514
\(262\) 0 0
\(263\) 4.94427 0.304877 0.152438 0.988313i \(-0.451287\pi\)
0.152438 + 0.988313i \(0.451287\pi\)
\(264\) 0 0
\(265\) 10.4721 0.643298
\(266\) 0 0
\(267\) −12.3607 −0.756461
\(268\) 0 0
\(269\) 22.7639 1.38794 0.693971 0.720003i \(-0.255860\pi\)
0.693971 + 0.720003i \(0.255860\pi\)
\(270\) 0 0
\(271\) 0.944272 0.0573604 0.0286802 0.999589i \(-0.490870\pi\)
0.0286802 + 0.999589i \(0.490870\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.47214 0.209378
\(276\) 0 0
\(277\) 3.52786 0.211969 0.105984 0.994368i \(-0.466201\pi\)
0.105984 + 0.994368i \(0.466201\pi\)
\(278\) 0 0
\(279\) −2.94427 −0.176269
\(280\) 0 0
\(281\) 28.8328 1.72002 0.860011 0.510276i \(-0.170457\pi\)
0.860011 + 0.510276i \(0.170457\pi\)
\(282\) 0 0
\(283\) 14.6525 0.870999 0.435500 0.900189i \(-0.356572\pi\)
0.435500 + 0.900189i \(0.356572\pi\)
\(284\) 0 0
\(285\) −11.0557 −0.654885
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −10.8885 −0.640503
\(290\) 0 0
\(291\) −15.4164 −0.903726
\(292\) 0 0
\(293\) 26.6525 1.55705 0.778527 0.627611i \(-0.215967\pi\)
0.778527 + 0.627611i \(0.215967\pi\)
\(294\) 0 0
\(295\) 3.41641 0.198911
\(296\) 0 0
\(297\) 5.52786 0.320759
\(298\) 0 0
\(299\) −12.9443 −0.748587
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −10.1115 −0.580888
\(304\) 0 0
\(305\) 0.944272 0.0540689
\(306\) 0 0
\(307\) −26.0689 −1.48783 −0.743915 0.668274i \(-0.767033\pi\)
−0.743915 + 0.668274i \(0.767033\pi\)
\(308\) 0 0
\(309\) −18.4721 −1.05084
\(310\) 0 0
\(311\) −21.4164 −1.21441 −0.607207 0.794544i \(-0.707710\pi\)
−0.607207 + 0.794544i \(0.707710\pi\)
\(312\) 0 0
\(313\) −19.5279 −1.10378 −0.551890 0.833917i \(-0.686093\pi\)
−0.551890 + 0.833917i \(0.686093\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −30.9443 −1.73800 −0.869002 0.494809i \(-0.835238\pi\)
−0.869002 + 0.494809i \(0.835238\pi\)
\(318\) 0 0
\(319\) −4.47214 −0.250392
\(320\) 0 0
\(321\) −3.05573 −0.170554
\(322\) 0 0
\(323\) 17.8885 0.995345
\(324\) 0 0
\(325\) −11.2361 −0.623265
\(326\) 0 0
\(327\) −12.3607 −0.683547
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 16.9443 0.931341 0.465671 0.884958i \(-0.345813\pi\)
0.465671 + 0.884958i \(0.345813\pi\)
\(332\) 0 0
\(333\) −10.2229 −0.560212
\(334\) 0 0
\(335\) −14.1115 −0.770991
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) −0.583592 −0.0316964
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −6.11146 −0.329030
\(346\) 0 0
\(347\) −2.47214 −0.132711 −0.0663556 0.997796i \(-0.521137\pi\)
−0.0663556 + 0.997796i \(0.521137\pi\)
\(348\) 0 0
\(349\) −21.7082 −1.16201 −0.581007 0.813899i \(-0.697341\pi\)
−0.581007 + 0.813899i \(0.697341\pi\)
\(350\) 0 0
\(351\) −17.8885 −0.954820
\(352\) 0 0
\(353\) 17.0557 0.907785 0.453892 0.891056i \(-0.350035\pi\)
0.453892 + 0.891056i \(0.350035\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −26.8328 −1.41618 −0.708091 0.706121i \(-0.750443\pi\)
−0.708091 + 0.706121i \(0.750443\pi\)
\(360\) 0 0
\(361\) 33.3607 1.75583
\(362\) 0 0
\(363\) 1.23607 0.0648767
\(364\) 0 0
\(365\) −16.0000 −0.837478
\(366\) 0 0
\(367\) −5.41641 −0.282734 −0.141367 0.989957i \(-0.545150\pi\)
−0.141367 + 0.989957i \(0.545150\pi\)
\(368\) 0 0
\(369\) −3.63932 −0.189455
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) −12.9443 −0.668439
\(376\) 0 0
\(377\) 14.4721 0.745353
\(378\) 0 0
\(379\) −14.4721 −0.743384 −0.371692 0.928356i \(-0.621222\pi\)
−0.371692 + 0.928356i \(0.621222\pi\)
\(380\) 0 0
\(381\) 14.8328 0.759908
\(382\) 0 0
\(383\) −23.8885 −1.22065 −0.610324 0.792152i \(-0.708961\pi\)
−0.610324 + 0.792152i \(0.708961\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −15.4164 −0.783660
\(388\) 0 0
\(389\) 33.4164 1.69428 0.847140 0.531370i \(-0.178323\pi\)
0.847140 + 0.531370i \(0.178323\pi\)
\(390\) 0 0
\(391\) 9.88854 0.500085
\(392\) 0 0
\(393\) 5.88854 0.297038
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 23.7082 1.18988 0.594940 0.803770i \(-0.297176\pi\)
0.594940 + 0.803770i \(0.297176\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.3607 0.717138 0.358569 0.933503i \(-0.383265\pi\)
0.358569 + 0.933503i \(0.383265\pi\)
\(402\) 0 0
\(403\) 6.47214 0.322400
\(404\) 0 0
\(405\) −2.98684 −0.148417
\(406\) 0 0
\(407\) −6.94427 −0.344215
\(408\) 0 0
\(409\) −3.41641 −0.168930 −0.0844652 0.996426i \(-0.526918\pi\)
−0.0844652 + 0.996426i \(0.526918\pi\)
\(410\) 0 0
\(411\) −24.5836 −1.21262
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −15.0557 −0.739057
\(416\) 0 0
\(417\) −26.8328 −1.31401
\(418\) 0 0
\(419\) 17.2361 0.842037 0.421019 0.907052i \(-0.361673\pi\)
0.421019 + 0.907052i \(0.361673\pi\)
\(420\) 0 0
\(421\) 16.4721 0.802803 0.401401 0.915902i \(-0.368523\pi\)
0.401401 + 0.915902i \(0.368523\pi\)
\(422\) 0 0
\(423\) 2.94427 0.143155
\(424\) 0 0
\(425\) 8.58359 0.416365
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −23.0557 −1.11056 −0.555278 0.831665i \(-0.687388\pi\)
−0.555278 + 0.831665i \(0.687388\pi\)
\(432\) 0 0
\(433\) −28.4721 −1.36828 −0.684142 0.729349i \(-0.739823\pi\)
−0.684142 + 0.729349i \(0.739823\pi\)
\(434\) 0 0
\(435\) 6.83282 0.327608
\(436\) 0 0
\(437\) 28.9443 1.38459
\(438\) 0 0
\(439\) 8.94427 0.426887 0.213443 0.976955i \(-0.431532\pi\)
0.213443 + 0.976955i \(0.431532\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.9443 1.18514 0.592569 0.805520i \(-0.298114\pi\)
0.592569 + 0.805520i \(0.298114\pi\)
\(444\) 0 0
\(445\) −12.3607 −0.585952
\(446\) 0 0
\(447\) −27.6393 −1.30729
\(448\) 0 0
\(449\) 18.9443 0.894035 0.447018 0.894525i \(-0.352486\pi\)
0.447018 + 0.894525i \(0.352486\pi\)
\(450\) 0 0
\(451\) −2.47214 −0.116408
\(452\) 0 0
\(453\) −14.8328 −0.696906
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.9443 1.26040 0.630200 0.776433i \(-0.282973\pi\)
0.630200 + 0.776433i \(0.282973\pi\)
\(458\) 0 0
\(459\) 13.6656 0.637857
\(460\) 0 0
\(461\) −24.7639 −1.15337 −0.576686 0.816966i \(-0.695654\pi\)
−0.576686 + 0.816966i \(0.695654\pi\)
\(462\) 0 0
\(463\) 30.4721 1.41616 0.708080 0.706132i \(-0.249562\pi\)
0.708080 + 0.706132i \(0.249562\pi\)
\(464\) 0 0
\(465\) 3.05573 0.141706
\(466\) 0 0
\(467\) −27.1246 −1.25518 −0.627589 0.778545i \(-0.715958\pi\)
−0.627589 + 0.778545i \(0.715958\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 15.6393 0.720622
\(472\) 0 0
\(473\) −10.4721 −0.481509
\(474\) 0 0
\(475\) 25.1246 1.15280
\(476\) 0 0
\(477\) −12.4721 −0.571060
\(478\) 0 0
\(479\) 12.3607 0.564774 0.282387 0.959301i \(-0.408874\pi\)
0.282387 + 0.959301i \(0.408874\pi\)
\(480\) 0 0
\(481\) 22.4721 1.02464
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.4164 −0.700023
\(486\) 0 0
\(487\) −16.9443 −0.767818 −0.383909 0.923371i \(-0.625422\pi\)
−0.383909 + 0.923371i \(0.625422\pi\)
\(488\) 0 0
\(489\) 24.0000 1.08532
\(490\) 0 0
\(491\) 16.9443 0.764684 0.382342 0.924021i \(-0.375118\pi\)
0.382342 + 0.924021i \(0.375118\pi\)
\(492\) 0 0
\(493\) −11.0557 −0.497925
\(494\) 0 0
\(495\) 1.81966 0.0817876
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 32.3607 1.44866 0.724331 0.689452i \(-0.242149\pi\)
0.724331 + 0.689452i \(0.242149\pi\)
\(500\) 0 0
\(501\) 14.1115 0.630453
\(502\) 0 0
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 0 0
\(505\) −10.1115 −0.449954
\(506\) 0 0
\(507\) −3.12461 −0.138769
\(508\) 0 0
\(509\) −24.0689 −1.06683 −0.533417 0.845852i \(-0.679092\pi\)
−0.533417 + 0.845852i \(0.679092\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 40.0000 1.76604
\(514\) 0 0
\(515\) −18.4721 −0.813980
\(516\) 0 0
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 10.3607 0.453910 0.226955 0.973905i \(-0.427123\pi\)
0.226955 + 0.973905i \(0.427123\pi\)
\(522\) 0 0
\(523\) −14.2918 −0.624937 −0.312468 0.949928i \(-0.601156\pi\)
−0.312468 + 0.949928i \(0.601156\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.94427 −0.215376
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.06888 −0.176575
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −3.05573 −0.132111
\(536\) 0 0
\(537\) 11.0557 0.477090
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −26.9443 −1.15842 −0.579212 0.815177i \(-0.696640\pi\)
−0.579212 + 0.815177i \(0.696640\pi\)
\(542\) 0 0
\(543\) −11.4164 −0.489925
\(544\) 0 0
\(545\) −12.3607 −0.529473
\(546\) 0 0
\(547\) 0.944272 0.0403742 0.0201871 0.999796i \(-0.493574\pi\)
0.0201871 + 0.999796i \(0.493574\pi\)
\(548\) 0 0
\(549\) −1.12461 −0.0479973
\(550\) 0 0
\(551\) −32.3607 −1.37861
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 10.6099 0.450365
\(556\) 0 0
\(557\) 24.8328 1.05220 0.526100 0.850423i \(-0.323654\pi\)
0.526100 + 0.850423i \(0.323654\pi\)
\(558\) 0 0
\(559\) 33.8885 1.43333
\(560\) 0 0
\(561\) 3.05573 0.129013
\(562\) 0 0
\(563\) 31.2361 1.31644 0.658222 0.752824i \(-0.271309\pi\)
0.658222 + 0.752824i \(0.271309\pi\)
\(564\) 0 0
\(565\) −0.583592 −0.0245519
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −36.8328 −1.54411 −0.772056 0.635555i \(-0.780772\pi\)
−0.772056 + 0.635555i \(0.780772\pi\)
\(570\) 0 0
\(571\) 10.1115 0.423151 0.211576 0.977362i \(-0.432141\pi\)
0.211576 + 0.977362i \(0.432141\pi\)
\(572\) 0 0
\(573\) 3.05573 0.127655
\(574\) 0 0
\(575\) 13.8885 0.579192
\(576\) 0 0
\(577\) −26.9443 −1.12170 −0.560852 0.827916i \(-0.689526\pi\)
−0.560852 + 0.827916i \(0.689526\pi\)
\(578\) 0 0
\(579\) −18.4721 −0.767676
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −8.47214 −0.350880
\(584\) 0 0
\(585\) −5.88854 −0.243461
\(586\) 0 0
\(587\) −5.81966 −0.240203 −0.120102 0.992762i \(-0.538322\pi\)
−0.120102 + 0.992762i \(0.538322\pi\)
\(588\) 0 0
\(589\) −14.4721 −0.596314
\(590\) 0 0
\(591\) 22.2492 0.915211
\(592\) 0 0
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 23.4164 0.958370
\(598\) 0 0
\(599\) 32.3607 1.32222 0.661111 0.750288i \(-0.270085\pi\)
0.661111 + 0.750288i \(0.270085\pi\)
\(600\) 0 0
\(601\) 34.8328 1.42086 0.710430 0.703768i \(-0.248500\pi\)
0.710430 + 0.703768i \(0.248500\pi\)
\(602\) 0 0
\(603\) 16.8065 0.684414
\(604\) 0 0
\(605\) 1.23607 0.0502533
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6.47214 −0.261835
\(612\) 0 0
\(613\) 28.4721 1.14998 0.574989 0.818161i \(-0.305006\pi\)
0.574989 + 0.818161i \(0.305006\pi\)
\(614\) 0 0
\(615\) 3.77709 0.152307
\(616\) 0 0
\(617\) 21.4164 0.862192 0.431096 0.902306i \(-0.358127\pi\)
0.431096 + 0.902306i \(0.358127\pi\)
\(618\) 0 0
\(619\) 18.5410 0.745227 0.372613 0.927987i \(-0.378462\pi\)
0.372613 + 0.927987i \(0.378462\pi\)
\(620\) 0 0
\(621\) 22.1115 0.887302
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 0 0
\(627\) 8.94427 0.357200
\(628\) 0 0
\(629\) −17.1672 −0.684500
\(630\) 0 0
\(631\) 31.4164 1.25067 0.625334 0.780357i \(-0.284963\pi\)
0.625334 + 0.780357i \(0.284963\pi\)
\(632\) 0 0
\(633\) 16.7214 0.664614
\(634\) 0 0
\(635\) 14.8328 0.588622
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 9.52786 0.376916
\(640\) 0 0
\(641\) 27.5279 1.08729 0.543643 0.839317i \(-0.317045\pi\)
0.543643 + 0.839317i \(0.317045\pi\)
\(642\) 0 0
\(643\) −18.7639 −0.739977 −0.369989 0.929036i \(-0.620638\pi\)
−0.369989 + 0.929036i \(0.620638\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) −28.8328 −1.13353 −0.566767 0.823878i \(-0.691806\pi\)
−0.566767 + 0.823878i \(0.691806\pi\)
\(648\) 0 0
\(649\) −2.76393 −0.108494
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 46.3607 1.81423 0.907117 0.420879i \(-0.138278\pi\)
0.907117 + 0.420879i \(0.138278\pi\)
\(654\) 0 0
\(655\) 5.88854 0.230084
\(656\) 0 0
\(657\) 19.0557 0.743435
\(658\) 0 0
\(659\) −16.5836 −0.646005 −0.323003 0.946398i \(-0.604692\pi\)
−0.323003 + 0.946398i \(0.604692\pi\)
\(660\) 0 0
\(661\) 3.12461 0.121533 0.0607667 0.998152i \(-0.480645\pi\)
0.0607667 + 0.998152i \(0.480645\pi\)
\(662\) 0 0
\(663\) −9.88854 −0.384039
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −17.8885 −0.692647
\(668\) 0 0
\(669\) −0.583592 −0.0225630
\(670\) 0 0
\(671\) −0.763932 −0.0294913
\(672\) 0 0
\(673\) −3.88854 −0.149892 −0.0749462 0.997188i \(-0.523878\pi\)
−0.0749462 + 0.997188i \(0.523878\pi\)
\(674\) 0 0
\(675\) 19.1935 0.738758
\(676\) 0 0
\(677\) 26.0689 1.00191 0.500954 0.865474i \(-0.332982\pi\)
0.500954 + 0.865474i \(0.332982\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −23.7771 −0.911140
\(682\) 0 0
\(683\) −32.9443 −1.26058 −0.630289 0.776361i \(-0.717064\pi\)
−0.630289 + 0.776361i \(0.717064\pi\)
\(684\) 0 0
\(685\) −24.5836 −0.939291
\(686\) 0 0
\(687\) −21.3050 −0.812835
\(688\) 0 0
\(689\) 27.4164 1.04448
\(690\) 0 0
\(691\) 12.6525 0.481323 0.240661 0.970609i \(-0.422636\pi\)
0.240661 + 0.970609i \(0.422636\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −26.8328 −1.01783
\(696\) 0 0
\(697\) −6.11146 −0.231488
\(698\) 0 0
\(699\) −18.4721 −0.698680
\(700\) 0 0
\(701\) −42.7214 −1.61356 −0.806782 0.590850i \(-0.798793\pi\)
−0.806782 + 0.590850i \(0.798793\pi\)
\(702\) 0 0
\(703\) −50.2492 −1.89519
\(704\) 0 0
\(705\) −3.05573 −0.115085
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.47214 0.167955 0.0839773 0.996468i \(-0.473238\pi\)
0.0839773 + 0.996468i \(0.473238\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) −24.7214 −0.923236
\(718\) 0 0
\(719\) 16.8328 0.627758 0.313879 0.949463i \(-0.398371\pi\)
0.313879 + 0.949463i \(0.398371\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −19.0557 −0.708690
\(724\) 0 0
\(725\) −15.5279 −0.576690
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) 24.0557 0.890953
\(730\) 0 0
\(731\) −25.8885 −0.957522
\(732\) 0 0
\(733\) −49.1246 −1.81446 −0.907229 0.420636i \(-0.861807\pi\)
−0.907229 + 0.420636i \(0.861807\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.4164 0.420529
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −28.9443 −1.06329
\(742\) 0 0
\(743\) −21.8885 −0.803013 −0.401506 0.915856i \(-0.631513\pi\)
−0.401506 + 0.915856i \(0.631513\pi\)
\(744\) 0 0
\(745\) −27.6393 −1.01263
\(746\) 0 0
\(747\) 17.9311 0.656065
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.9443 0.618305 0.309153 0.951012i \(-0.399955\pi\)
0.309153 + 0.951012i \(0.399955\pi\)
\(752\) 0 0
\(753\) 36.1378 1.31693
\(754\) 0 0
\(755\) −14.8328 −0.539821
\(756\) 0 0
\(757\) −23.3050 −0.847033 −0.423516 0.905888i \(-0.639204\pi\)
−0.423516 + 0.905888i \(0.639204\pi\)
\(758\) 0 0
\(759\) 4.94427 0.179466
\(760\) 0 0
\(761\) 11.4164 0.413844 0.206922 0.978357i \(-0.433655\pi\)
0.206922 + 0.978357i \(0.433655\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 4.49845 0.162642
\(766\) 0 0
\(767\) 8.94427 0.322959
\(768\) 0 0
\(769\) 43.4164 1.56564 0.782818 0.622251i \(-0.213782\pi\)
0.782818 + 0.622251i \(0.213782\pi\)
\(770\) 0 0
\(771\) −8.58359 −0.309131
\(772\) 0 0
\(773\) −15.7082 −0.564985 −0.282492 0.959270i \(-0.591161\pi\)
−0.282492 + 0.959270i \(0.591161\pi\)
\(774\) 0 0
\(775\) −6.94427 −0.249446
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17.8885 −0.640924
\(780\) 0 0
\(781\) 6.47214 0.231591
\(782\) 0 0
\(783\) −24.7214 −0.883469
\(784\) 0 0
\(785\) 15.6393 0.558191
\(786\) 0 0
\(787\) −28.1803 −1.00452 −0.502260 0.864716i \(-0.667498\pi\)
−0.502260 + 0.864716i \(0.667498\pi\)
\(788\) 0 0
\(789\) 6.11146 0.217574
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.47214 0.0877881
\(794\) 0 0
\(795\) 12.9443 0.459086
\(796\) 0 0
\(797\) 41.5967 1.47343 0.736716 0.676202i \(-0.236375\pi\)
0.736716 + 0.676202i \(0.236375\pi\)
\(798\) 0 0
\(799\) 4.94427 0.174916
\(800\) 0 0
\(801\) 14.7214 0.520154
\(802\) 0 0
\(803\) 12.9443 0.456793
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 28.1378 0.990496
\(808\) 0 0
\(809\) −21.0557 −0.740280 −0.370140 0.928976i \(-0.620690\pi\)
−0.370140 + 0.928976i \(0.620690\pi\)
\(810\) 0 0
\(811\) 4.76393 0.167284 0.0836421 0.996496i \(-0.473345\pi\)
0.0836421 + 0.996496i \(0.473345\pi\)
\(812\) 0 0
\(813\) 1.16718 0.0409349
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) −75.7771 −2.65110
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.41641 −0.0494330 −0.0247165 0.999695i \(-0.507868\pi\)
−0.0247165 + 0.999695i \(0.507868\pi\)
\(822\) 0 0
\(823\) 46.2492 1.61215 0.806073 0.591816i \(-0.201589\pi\)
0.806073 + 0.591816i \(0.201589\pi\)
\(824\) 0 0
\(825\) 4.29180 0.149421
\(826\) 0 0
\(827\) −16.9443 −0.589210 −0.294605 0.955619i \(-0.595188\pi\)
−0.294605 + 0.955619i \(0.595188\pi\)
\(828\) 0 0
\(829\) −11.7082 −0.406643 −0.203321 0.979112i \(-0.565174\pi\)
−0.203321 + 0.979112i \(0.565174\pi\)
\(830\) 0 0
\(831\) 4.36068 0.151270
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 14.1115 0.488347
\(836\) 0 0
\(837\) −11.0557 −0.382142
\(838\) 0 0
\(839\) 16.8328 0.581133 0.290567 0.956855i \(-0.406156\pi\)
0.290567 + 0.956855i \(0.406156\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) 35.6393 1.22748
\(844\) 0 0
\(845\) −3.12461 −0.107490
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 18.1115 0.621584
\(850\) 0 0
\(851\) −27.7771 −0.952186
\(852\) 0 0
\(853\) −32.5410 −1.11418 −0.557092 0.830451i \(-0.688083\pi\)
−0.557092 + 0.830451i \(0.688083\pi\)
\(854\) 0 0
\(855\) 13.1672 0.450308
\(856\) 0 0
\(857\) 46.4721 1.58746 0.793729 0.608272i \(-0.208137\pi\)
0.793729 + 0.608272i \(0.208137\pi\)
\(858\) 0 0
\(859\) 15.1246 0.516045 0.258023 0.966139i \(-0.416929\pi\)
0.258023 + 0.966139i \(0.416929\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −0.583592 −0.0198657 −0.00993285 0.999951i \(-0.503162\pi\)
−0.00993285 + 0.999951i \(0.503162\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) 0 0
\(867\) −13.4590 −0.457091
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −36.9443 −1.25181
\(872\) 0 0
\(873\) 18.3607 0.621415
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.05573 0.305790 0.152895 0.988242i \(-0.451140\pi\)
0.152895 + 0.988242i \(0.451140\pi\)
\(878\) 0 0
\(879\) 32.9443 1.11118
\(880\) 0 0
\(881\) −28.8328 −0.971402 −0.485701 0.874125i \(-0.661436\pi\)
−0.485701 + 0.874125i \(0.661436\pi\)
\(882\) 0 0
\(883\) 2.83282 0.0953318 0.0476659 0.998863i \(-0.484822\pi\)
0.0476659 + 0.998863i \(0.484822\pi\)
\(884\) 0 0
\(885\) 4.22291 0.141952
\(886\) 0 0
\(887\) −44.3607 −1.48949 −0.744743 0.667351i \(-0.767428\pi\)
−0.744743 + 0.667351i \(0.767428\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.41641 0.0809527
\(892\) 0 0
\(893\) 14.4721 0.484292
\(894\) 0 0
\(895\) 11.0557 0.369552
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) 8.94427 0.298308
\(900\) 0 0
\(901\) −20.9443 −0.697755
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.4164 −0.379494
\(906\) 0 0
\(907\) 24.3607 0.808883 0.404442 0.914564i \(-0.367466\pi\)
0.404442 + 0.914564i \(0.367466\pi\)
\(908\) 0 0
\(909\) 12.0426 0.399427
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) 12.1803 0.403110
\(914\) 0 0
\(915\) 1.16718 0.0385859
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 22.1115 0.729390 0.364695 0.931127i \(-0.381173\pi\)
0.364695 + 0.931127i \(0.381173\pi\)
\(920\) 0 0
\(921\) −32.2229 −1.06178
\(922\) 0 0
\(923\) −20.9443 −0.689389
\(924\) 0 0
\(925\) −24.1115 −0.792780
\(926\) 0 0
\(927\) 22.0000 0.722575
\(928\) 0 0
\(929\) −40.2492 −1.32053 −0.660267 0.751031i \(-0.729557\pi\)
−0.660267 + 0.751031i \(0.729557\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −26.4721 −0.866659
\(934\) 0 0
\(935\) 3.05573 0.0999330
\(936\) 0 0
\(937\) 3.05573 0.0998263 0.0499131 0.998754i \(-0.484106\pi\)
0.0499131 + 0.998754i \(0.484106\pi\)
\(938\) 0 0
\(939\) −24.1378 −0.787706
\(940\) 0 0
\(941\) 11.8197 0.385310 0.192655 0.981267i \(-0.438290\pi\)
0.192655 + 0.981267i \(0.438290\pi\)
\(942\) 0 0
\(943\) −9.88854 −0.322015
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16.9443 −0.550615 −0.275307 0.961356i \(-0.588780\pi\)
−0.275307 + 0.961356i \(0.588780\pi\)
\(948\) 0 0
\(949\) −41.8885 −1.35976
\(950\) 0 0
\(951\) −38.2492 −1.24032
\(952\) 0 0
\(953\) 22.9443 0.743238 0.371619 0.928385i \(-0.378803\pi\)
0.371619 + 0.928385i \(0.378803\pi\)
\(954\) 0 0
\(955\) 3.05573 0.0988810
\(956\) 0 0
\(957\) −5.52786 −0.178690
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 3.63932 0.117275
\(964\) 0 0
\(965\) −18.4721 −0.594639
\(966\) 0 0
\(967\) −45.8885 −1.47568 −0.737838 0.674978i \(-0.764153\pi\)
−0.737838 + 0.674978i \(0.764153\pi\)
\(968\) 0 0
\(969\) 22.1115 0.710322
\(970\) 0 0
\(971\) 50.5410 1.62194 0.810969 0.585089i \(-0.198940\pi\)
0.810969 + 0.585089i \(0.198940\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −13.8885 −0.444789
\(976\) 0 0
\(977\) −28.8328 −0.922443 −0.461222 0.887285i \(-0.652589\pi\)
−0.461222 + 0.887285i \(0.652589\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) 14.7214 0.470017
\(982\) 0 0
\(983\) 14.0000 0.446531 0.223265 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) 0 0
\(985\) 22.2492 0.708919
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −41.8885 −1.33198
\(990\) 0 0
\(991\) 0.360680 0.0114574 0.00572869 0.999984i \(-0.498176\pi\)
0.00572869 + 0.999984i \(0.498176\pi\)
\(992\) 0 0
\(993\) 20.9443 0.664646
\(994\) 0 0
\(995\) 23.4164 0.742350
\(996\) 0 0
\(997\) −24.1803 −0.765799 −0.382900 0.923790i \(-0.625074\pi\)
−0.382900 + 0.923790i \(0.625074\pi\)
\(998\) 0 0
\(999\) −38.3870 −1.21451
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 8624.2.a.bf.1.2 2
4.3 odd 2 1078.2.a.w.1.1 2
7.6 odd 2 1232.2.a.p.1.1 2
12.11 even 2 9702.2.a.cu.1.1 2
28.3 even 6 1078.2.e.q.177.1 4
28.11 odd 6 1078.2.e.n.177.2 4
28.19 even 6 1078.2.e.q.67.1 4
28.23 odd 6 1078.2.e.n.67.2 4
28.27 even 2 154.2.a.d.1.2 2
56.13 odd 2 4928.2.a.bk.1.2 2
56.27 even 2 4928.2.a.bt.1.1 2
84.83 odd 2 1386.2.a.m.1.2 2
140.27 odd 4 3850.2.c.q.1849.3 4
140.83 odd 4 3850.2.c.q.1849.2 4
140.139 even 2 3850.2.a.bj.1.1 2
308.307 odd 2 1694.2.a.l.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.2 2 28.27 even 2
1078.2.a.w.1.1 2 4.3 odd 2
1078.2.e.n.67.2 4 28.23 odd 6
1078.2.e.n.177.2 4 28.11 odd 6
1078.2.e.q.67.1 4 28.19 even 6
1078.2.e.q.177.1 4 28.3 even 6
1232.2.a.p.1.1 2 7.6 odd 2
1386.2.a.m.1.2 2 84.83 odd 2
1694.2.a.l.1.2 2 308.307 odd 2
3850.2.a.bj.1.1 2 140.139 even 2
3850.2.c.q.1849.2 4 140.83 odd 4
3850.2.c.q.1849.3 4 140.27 odd 4
4928.2.a.bk.1.2 2 56.13 odd 2
4928.2.a.bt.1.1 2 56.27 even 2
8624.2.a.bf.1.2 2 1.1 even 1 trivial
9702.2.a.cu.1.1 2 12.11 even 2