Properties

Label 9702.2.a.cu.1.1
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{8} +1.23607 q^{10} -1.00000 q^{11} +3.23607 q^{13} +1.00000 q^{16} +2.47214 q^{17} +7.23607 q^{19} -1.23607 q^{20} +1.00000 q^{22} -4.00000 q^{23} -3.47214 q^{25} -3.23607 q^{26} -4.47214 q^{29} -2.00000 q^{31} -1.00000 q^{32} -2.47214 q^{34} +6.94427 q^{37} -7.23607 q^{38} +1.23607 q^{40} -2.47214 q^{41} -10.4721 q^{43} -1.00000 q^{44} +4.00000 q^{46} -2.00000 q^{47} +3.47214 q^{50} +3.23607 q^{52} -8.47214 q^{53} +1.23607 q^{55} +4.47214 q^{58} +2.76393 q^{59} +0.763932 q^{61} +2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +11.4164 q^{67} +2.47214 q^{68} -6.47214 q^{71} -12.9443 q^{73} -6.94427 q^{74} +7.23607 q^{76} -1.23607 q^{80} +2.47214 q^{82} -12.1803 q^{83} -3.05573 q^{85} +10.4721 q^{86} +1.00000 q^{88} +10.0000 q^{89} -4.00000 q^{92} +2.00000 q^{94} -8.94427 q^{95} -12.4721 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 10 q^{19} + 2 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{25} - 2 q^{26} - 4 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{37} - 10 q^{38} - 2 q^{40} + 4 q^{41} - 12 q^{43} - 2 q^{44} + 8 q^{46} - 4 q^{47} - 2 q^{50} + 2 q^{52} - 8 q^{53} - 2 q^{55} + 10 q^{59} + 6 q^{61} + 4 q^{62} + 2 q^{64} - 8 q^{65} - 4 q^{67} - 4 q^{68} - 4 q^{71} - 8 q^{73} + 4 q^{74} + 10 q^{76} + 2 q^{80} - 4 q^{82} - 2 q^{83} - 24 q^{85} + 12 q^{86} + 2 q^{88} + 20 q^{89} - 8 q^{92} + 4 q^{94} - 16 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.23607 0.390879
\(11\) −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
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.47214 0.599581 0.299791 0.954005i \(-0.403083\pi\)
0.299791 + 0.954005i \(0.403083\pi\)
\(18\) 0 0
\(19\) 7.23607 1.66007 0.830034 0.557713i \(-0.188321\pi\)
0.830034 + 0.557713i \(0.188321\pi\)
\(20\) −1.23607 −0.276393
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −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
\(26\) −3.23607 −0.634645
\(27\) 0 0
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.47214 −0.423968
\(35\) 0 0
\(36\) 0 0
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) −7.23607 −1.17385
\(39\) 0 0
\(40\) 1.23607 0.195440
\(41\) −2.47214 −0.386083 −0.193041 0.981191i \(-0.561835\pi\)
−0.193041 + 0.981191i \(0.561835\pi\)
\(42\) 0 0
\(43\) −10.4721 −1.59699 −0.798493 0.602004i \(-0.794369\pi\)
−0.798493 + 0.602004i \(0.794369\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 3.47214 0.491034
\(51\) 0 0
\(52\) 3.23607 0.448762
\(53\) −8.47214 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(54\) 0 0
\(55\) 1.23607 0.166671
\(56\) 0 0
\(57\) 0 0
\(58\) 4.47214 0.587220
\(59\) 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 \(0.484427\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 11.4164 1.39474 0.697368 0.716713i \(-0.254354\pi\)
0.697368 + 0.716713i \(0.254354\pi\)
\(68\) 2.47214 0.299791
\(69\) 0 0
\(70\) 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 \(0.773582\pi\)
\(74\) −6.94427 −0.807255
\(75\) 0 0
\(76\) 7.23607 0.830034
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.23607 −0.138197
\(81\) 0 0
\(82\) 2.47214 0.273002
\(83\) −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
\(86\) 10.4721 1.12924
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) −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
\(99\) 0 0
\(100\) −3.47214 −0.347214
\(101\) 8.18034 0.813974 0.406987 0.913434i \(-0.366579\pi\)
0.406987 + 0.913434i \(0.366579\pi\)
\(102\) 0 0
\(103\) 14.9443 1.47250 0.736251 0.676708i \(-0.236594\pi\)
0.736251 + 0.676708i \(0.236594\pi\)
\(104\) −3.23607 −0.317323
\(105\) 0 0
\(106\) 8.47214 0.822887
\(107\) −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 \(0.658969\pi\)
\(110\) −1.23607 −0.117854
\(111\) 0 0
\(112\) 0 0
\(113\) 0.472136 0.0444148 0.0222074 0.999753i \(-0.492931\pi\)
0.0222074 + 0.999753i \(0.492931\pi\)
\(114\) 0 0
\(115\) 4.94427 0.461056
\(116\) −4.47214 −0.415227
\(117\) 0 0
\(118\) −2.76393 −0.254441
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −0.763932 −0.0691632
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 4.76393 0.416227 0.208113 0.978105i \(-0.433268\pi\)
0.208113 + 0.978105i \(0.433268\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −11.4164 −0.986227
\(135\) 0 0
\(136\) −2.47214 −0.211984
\(137\) 19.8885 1.69919 0.849596 0.527433i \(-0.176846\pi\)
0.849596 + 0.527433i \(0.176846\pi\)
\(138\) 0 0
\(139\) 21.7082 1.84127 0.920633 0.390429i \(-0.127673\pi\)
0.920633 + 0.390429i \(0.127673\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.47214 0.543130
\(143\) −3.23607 −0.270614
\(144\) 0 0
\(145\) 5.52786 0.459064
\(146\) 12.9443 1.07128
\(147\) 0 0
\(148\) 6.94427 0.570816
\(149\) 22.3607 1.83186 0.915929 0.401340i \(-0.131455\pi\)
0.915929 + 0.401340i \(0.131455\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) −7.23607 −0.586923
\(153\) 0 0
\(154\) 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
\(159\) 0 0
\(160\) 1.23607 0.0977198
\(161\) 0 0
\(162\) 0 0
\(163\) −19.4164 −1.52081 −0.760405 0.649449i \(-0.775000\pi\)
−0.760405 + 0.649449i \(0.775000\pi\)
\(164\) −2.47214 −0.193041
\(165\) 0 0
\(166\) 12.1803 0.945378
\(167\) 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
\(170\) 3.05573 0.234364
\(171\) 0 0
\(172\) −10.4721 −0.798493
\(173\) −3.23607 −0.246034 −0.123017 0.992405i \(-0.539257\pi\)
−0.123017 + 0.992405i \(0.539257\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) 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 0
\(184\) 4.00000 0.294884
\(185\) −8.58359 −0.631078
\(186\) 0 0
\(187\) −2.47214 −0.180780
\(188\) −2.00000 −0.145865
\(189\) 0 0
\(190\) 8.94427 0.648886
\(191\) 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 \(0.680766\pi\)
\(194\) 12.4721 0.895447
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −18.9443 −1.34292 −0.671462 0.741039i \(-0.734333\pi\)
−0.671462 + 0.741039i \(0.734333\pi\)
\(200\) 3.47214 0.245517
\(201\) 0 0
\(202\) −8.18034 −0.575567
\(203\) 0 0
\(204\) 0 0
\(205\) 3.05573 0.213421
\(206\) −14.9443 −1.04122
\(207\) 0 0
\(208\) 3.23607 0.224381
\(209\) −7.23607 −0.500529
\(210\) 0 0
\(211\) −13.5279 −0.931297 −0.465648 0.884970i \(-0.654179\pi\)
−0.465648 + 0.884970i \(0.654179\pi\)
\(212\) −8.47214 −0.581869
\(213\) 0 0
\(214\) 2.47214 0.168992
\(215\) 12.9443 0.882792
\(216\) 0 0
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) 0 0
\(220\) 1.23607 0.0833357
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 0.472136 0.0316166 0.0158083 0.999875i \(-0.494968\pi\)
0.0158083 + 0.999875i \(0.494968\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −0.472136 −0.0314060
\(227\) −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 \(0.692861\pi\)
\(230\) −4.94427 −0.326016
\(231\) 0 0
\(232\) 4.47214 0.293610
\(233\) 14.9443 0.979032 0.489516 0.871994i \(-0.337174\pi\)
0.489516 + 0.871994i \(0.337174\pi\)
\(234\) 0 0
\(235\) 2.47214 0.161264
\(236\) 2.76393 0.179917
\(237\) 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 \(0.665392\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 0.763932 0.0489057
\(245\) 0 0
\(246\) 0 0
\(247\) 23.4164 1.48995
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) −10.4721 −0.662316
\(251\) 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
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.94427 0.433172 0.216586 0.976264i \(-0.430508\pi\)
0.216586 + 0.976264i \(0.430508\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) 0 0
\(262\) −4.76393 −0.294317
\(263\) 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
\(267\) 0 0
\(268\) 11.4164 0.697368
\(269\) −22.7639 −1.38794 −0.693971 0.720003i \(-0.744140\pi\)
−0.693971 + 0.720003i \(0.744140\pi\)
\(270\) 0 0
\(271\) −0.944272 −0.0573604 −0.0286802 0.999589i \(-0.509130\pi\)
−0.0286802 + 0.999589i \(0.509130\pi\)
\(272\) 2.47214 0.149895
\(273\) 0 0
\(274\) −19.8885 −1.20151
\(275\) 3.47214 0.209378
\(276\) 0 0
\(277\) 3.52786 0.211969 0.105984 0.994368i \(-0.466201\pi\)
0.105984 + 0.994368i \(0.466201\pi\)
\(278\) −21.7082 −1.30197
\(279\) 0 0
\(280\) 0 0
\(281\) −28.8328 −1.72002 −0.860011 0.510276i \(-0.829543\pi\)
−0.860011 + 0.510276i \(0.829543\pi\)
\(282\) 0 0
\(283\) −14.6525 −0.870999 −0.435500 0.900189i \(-0.643428\pi\)
−0.435500 + 0.900189i \(0.643428\pi\)
\(284\) −6.47214 −0.384051
\(285\) 0 0
\(286\) 3.23607 0.191353
\(287\) 0 0
\(288\) 0 0
\(289\) −10.8885 −0.640503
\(290\) −5.52786 −0.324607
\(291\) 0 0
\(292\) −12.9443 −0.757506
\(293\) −26.6525 −1.55705 −0.778527 0.627611i \(-0.784033\pi\)
−0.778527 + 0.627611i \(0.784033\pi\)
\(294\) 0 0
\(295\) −3.41641 −0.198911
\(296\) −6.94427 −0.403628
\(297\) 0 0
\(298\) −22.3607 −1.29532
\(299\) −12.9443 −0.748587
\(300\) 0 0
\(301\) 0 0
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) 7.23607 0.415017
\(305\) −0.944272 −0.0540689
\(306\) 0 0
\(307\) 26.0689 1.48783 0.743915 0.668274i \(-0.232967\pi\)
0.743915 + 0.668274i \(0.232967\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.47214 −0.140408
\(311\) −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 \(0.686093\pi\)
\(314\) −12.6525 −0.714021
\(315\) 0 0
\(316\) 0 0
\(317\) 30.9443 1.73800 0.869002 0.494809i \(-0.164762\pi\)
0.869002 + 0.494809i \(0.164762\pi\)
\(318\) 0 0
\(319\) 4.47214 0.250392
\(320\) −1.23607 −0.0690983
\(321\) 0 0
\(322\) 0 0
\(323\) 17.8885 0.995345
\(324\) 0 0
\(325\) −11.2361 −0.623265
\(326\) 19.4164 1.07538
\(327\) 0 0
\(328\) 2.47214 0.136501
\(329\) 0 0
\(330\) 0 0
\(331\) −16.9443 −0.931341 −0.465671 0.884958i \(-0.654187\pi\)
−0.465671 + 0.884958i \(0.654187\pi\)
\(332\) −12.1803 −0.668483
\(333\) 0 0
\(334\) −11.4164 −0.624678
\(335\) −14.1115 −0.770991
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 2.52786 0.137498
\(339\) 0 0
\(340\) −3.05573 −0.165720
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) 0 0
\(344\) 10.4721 0.564620
\(345\) 0 0
\(346\) 3.23607 0.173972
\(347\) −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
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −17.0557 −0.907785 −0.453892 0.891056i \(-0.649965\pi\)
−0.453892 + 0.891056i \(0.649965\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −8.94427 −0.472719
\(359\) −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
\(362\) 9.23607 0.485437
\(363\) 0 0
\(364\) 0 0
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 5.41641 0.282734 0.141367 0.989957i \(-0.454850\pi\)
0.141367 + 0.989957i \(0.454850\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 8.58359 0.446240
\(371\) 0 0
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 2.47214 0.127831
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) −14.4721 −0.745353
\(378\) 0 0
\(379\) 14.4721 0.743384 0.371692 0.928356i \(-0.378778\pi\)
0.371692 + 0.928356i \(0.378778\pi\)
\(380\) −8.94427 −0.458831
\(381\) 0 0
\(382\) −2.47214 −0.126485
\(383\) −23.8885 −1.22065 −0.610324 0.792152i \(-0.708961\pi\)
−0.610324 + 0.792152i \(0.708961\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.9443 0.760643
\(387\) 0 0
\(388\) −12.4721 −0.633177
\(389\) −33.4164 −1.69428 −0.847140 0.531370i \(-0.821677\pi\)
−0.847140 + 0.531370i \(0.821677\pi\)
\(390\) 0 0
\(391\) −9.88854 −0.500085
\(392\) 0 0
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) 23.7082 1.18988 0.594940 0.803770i \(-0.297176\pi\)
0.594940 + 0.803770i \(0.297176\pi\)
\(398\) 18.9443 0.949591
\(399\) 0 0
\(400\) −3.47214 −0.173607
\(401\) −14.3607 −0.717138 −0.358569 0.933503i \(-0.616735\pi\)
−0.358569 + 0.933503i \(0.616735\pi\)
\(402\) 0 0
\(403\) −6.47214 −0.322400
\(404\) 8.18034 0.406987
\(405\) 0 0
\(406\) 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 \(0.526918\pi\)
\(410\) −3.05573 −0.150912
\(411\) 0 0
\(412\) 14.9443 0.736251
\(413\) 0 0
\(414\) 0 0
\(415\) 15.0557 0.739057
\(416\) −3.23607 −0.158661
\(417\) 0 0
\(418\) 7.23607 0.353928
\(419\) 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 \(0.368523\pi\)
\(422\) 13.5279 0.658526
\(423\) 0 0
\(424\) 8.47214 0.411443
\(425\) −8.58359 −0.416365
\(426\) 0 0
\(427\) 0 0
\(428\) −2.47214 −0.119495
\(429\) 0 0
\(430\) −12.9443 −0.624228
\(431\) −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
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −28.9443 −1.38459
\(438\) 0 0
\(439\) −8.94427 −0.426887 −0.213443 0.976955i \(-0.568468\pi\)
−0.213443 + 0.976955i \(0.568468\pi\)
\(440\) −1.23607 −0.0589272
\(441\) 0 0
\(442\) −8.00000 −0.380521
\(443\) 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
\(446\) −0.472136 −0.0223563
\(447\) 0 0
\(448\) 0 0
\(449\) −18.9443 −0.894035 −0.447018 0.894525i \(-0.647514\pi\)
−0.447018 + 0.894525i \(0.647514\pi\)
\(450\) 0 0
\(451\) 2.47214 0.116408
\(452\) 0.472136 0.0222074
\(453\) 0 0
\(454\) 19.2361 0.902793
\(455\) 0 0
\(456\) 0 0
\(457\) 26.9443 1.26040 0.630200 0.776433i \(-0.282973\pi\)
0.630200 + 0.776433i \(0.282973\pi\)
\(458\) 17.2361 0.805389
\(459\) 0 0
\(460\) 4.94427 0.230528
\(461\) 24.7639 1.15337 0.576686 0.816966i \(-0.304346\pi\)
0.576686 + 0.816966i \(0.304346\pi\)
\(462\) 0 0
\(463\) −30.4721 −1.41616 −0.708080 0.706132i \(-0.750438\pi\)
−0.708080 + 0.706132i \(0.750438\pi\)
\(464\) −4.47214 −0.207614
\(465\) 0 0
\(466\) −14.9443 −0.692280
\(467\) −27.1246 −1.25518 −0.627589 0.778545i \(-0.715958\pi\)
−0.627589 + 0.778545i \(0.715958\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −2.47214 −0.114031
\(471\) 0 0
\(472\) −2.76393 −0.127220
\(473\) 10.4721 0.481509
\(474\) 0 0
\(475\) −25.1246 −1.15280
\(476\) 0 0
\(477\) 0 0
\(478\) 20.0000 0.914779
\(479\) 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
\(482\) 15.4164 0.702198
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 15.4164 0.700023
\(486\) 0 0
\(487\) 16.9443 0.767818 0.383909 0.923371i \(-0.374578\pi\)
0.383909 + 0.923371i \(0.374578\pi\)
\(488\) −0.763932 −0.0345816
\(489\) 0 0
\(490\) 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
\(494\) −23.4164 −1.05355
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) −32.3607 −1.44866 −0.724331 0.689452i \(-0.757851\pi\)
−0.724331 + 0.689452i \(0.757851\pi\)
\(500\) 10.4721 0.468328
\(501\) 0 0
\(502\) −29.2361 −1.30487
\(503\) 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
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) 24.0689 1.06683 0.533417 0.845852i \(-0.320908\pi\)
0.533417 + 0.845852i \(0.320908\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.94427 −0.306299
\(515\) −18.4721 −0.813980
\(516\) 0 0
\(517\) 2.00000 0.0879599
\(518\) 0 0
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) −10.3607 −0.453910 −0.226955 0.973905i \(-0.572877\pi\)
−0.226955 + 0.973905i \(0.572877\pi\)
\(522\) 0 0
\(523\) 14.2918 0.624937 0.312468 0.949928i \(-0.398844\pi\)
0.312468 + 0.949928i \(0.398844\pi\)
\(524\) 4.76393 0.208113
\(525\) 0 0
\(526\) −4.94427 −0.215580
\(527\) −4.94427 −0.215376
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −10.4721 −0.454881
\(531\) 0 0
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) 3.05573 0.132111
\(536\) −11.4164 −0.493114
\(537\) 0 0
\(538\) 22.7639 0.981423
\(539\) 0 0
\(540\) 0 0
\(541\) −26.9443 −1.15842 −0.579212 0.815177i \(-0.696640\pi\)
−0.579212 + 0.815177i \(0.696640\pi\)
\(542\) 0.944272 0.0405600
\(543\) 0 0
\(544\) −2.47214 −0.105992
\(545\) 12.3607 0.529473
\(546\) 0 0
\(547\) −0.944272 −0.0403742 −0.0201871 0.999796i \(-0.506426\pi\)
−0.0201871 + 0.999796i \(0.506426\pi\)
\(548\) 19.8885 0.849596
\(549\) 0 0
\(550\) −3.47214 −0.148052
\(551\) −32.3607 −1.37861
\(552\) 0 0
\(553\) 0 0
\(554\) −3.52786 −0.149885
\(555\) 0 0
\(556\) 21.7082 0.920633
\(557\) −24.8328 −1.05220 −0.526100 0.850423i \(-0.676346\pi\)
−0.526100 + 0.850423i \(0.676346\pi\)
\(558\) 0 0
\(559\) −33.8885 −1.43333
\(560\) 0 0
\(561\) 0 0
\(562\) 28.8328 1.21624
\(563\) 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
\(566\) 14.6525 0.615889
\(567\) 0 0
\(568\) 6.47214 0.271565
\(569\) 36.8328 1.54411 0.772056 0.635555i \(-0.219228\pi\)
0.772056 + 0.635555i \(0.219228\pi\)
\(570\) 0 0
\(571\) −10.1115 −0.423151 −0.211576 0.977362i \(-0.567859\pi\)
−0.211576 + 0.977362i \(0.567859\pi\)
\(572\) −3.23607 −0.135307
\(573\) 0 0
\(574\) 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 \(0.689526\pi\)
\(578\) 10.8885 0.452904
\(579\) 0 0
\(580\) 5.52786 0.229532
\(581\) 0 0
\(582\) 0 0
\(583\) 8.47214 0.350880
\(584\) 12.9443 0.535638
\(585\) 0 0
\(586\) 26.6525 1.10100
\(587\) −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
\(590\) 3.41641 0.140651
\(591\) 0 0
\(592\) 6.94427 0.285408
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22.3607 0.915929
\(597\) 0 0
\(598\) 12.9443 0.529331
\(599\) 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
\(603\) 0 0
\(604\) 12.0000 0.488273
\(605\) −1.23607 −0.0502533
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −7.23607 −0.293461
\(609\) 0 0
\(610\) 0.944272 0.0382325
\(611\) −6.47214 −0.261835
\(612\) 0 0
\(613\) 28.4721 1.14998 0.574989 0.818161i \(-0.305006\pi\)
0.574989 + 0.818161i \(0.305006\pi\)
\(614\) −26.0689 −1.05205
\(615\) 0 0
\(616\) 0 0
\(617\) −21.4164 −0.862192 −0.431096 0.902306i \(-0.641873\pi\)
−0.431096 + 0.902306i \(0.641873\pi\)
\(618\) 0 0
\(619\) −18.5410 −0.745227 −0.372613 0.927987i \(-0.621538\pi\)
−0.372613 + 0.927987i \(0.621538\pi\)
\(620\) 2.47214 0.0992834
\(621\) 0 0
\(622\) 21.4164 0.858720
\(623\) 0 0
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 19.5279 0.780490
\(627\) 0 0
\(628\) 12.6525 0.504889
\(629\) 17.1672 0.684500
\(630\) 0 0
\(631\) −31.4164 −1.25067 −0.625334 0.780357i \(-0.715037\pi\)
−0.625334 + 0.780357i \(0.715037\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −30.9443 −1.22895
\(635\) 14.8328 0.588622
\(636\) 0 0
\(637\) 0 0
\(638\) −4.47214 −0.177054
\(639\) 0 0
\(640\) 1.23607 0.0488599
\(641\) −27.5279 −1.08729 −0.543643 0.839317i \(-0.682955\pi\)
−0.543643 + 0.839317i \(0.682955\pi\)
\(642\) 0 0
\(643\) 18.7639 0.739977 0.369989 0.929036i \(-0.379362\pi\)
0.369989 + 0.929036i \(0.379362\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −17.8885 −0.703815
\(647\) −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
\(650\) 11.2361 0.440715
\(651\) 0 0
\(652\) −19.4164 −0.760405
\(653\) −46.3607 −1.81423 −0.907117 0.420879i \(-0.861722\pi\)
−0.907117 + 0.420879i \(0.861722\pi\)
\(654\) 0 0
\(655\) −5.88854 −0.230084
\(656\) −2.47214 −0.0965207
\(657\) 0 0
\(658\) 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 \(0.480645\pi\)
\(662\) 16.9443 0.658558
\(663\) 0 0
\(664\) 12.1803 0.472689
\(665\) 0 0
\(666\) 0 0
\(667\) 17.8885 0.692647
\(668\) 11.4164 0.441714
\(669\) 0 0
\(670\) 14.1115 0.545173
\(671\) −0.763932 −0.0294913
\(672\) 0 0
\(673\) −3.88854 −0.149892 −0.0749462 0.997188i \(-0.523878\pi\)
−0.0749462 + 0.997188i \(0.523878\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) −2.52786 −0.0972255
\(677\) −26.0689 −1.00191 −0.500954 0.865474i \(-0.667018\pi\)
−0.500954 + 0.865474i \(0.667018\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 3.05573 0.117182
\(681\) 0 0
\(682\) −2.00000 −0.0765840
\(683\) −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
\(687\) 0 0
\(688\) −10.4721 −0.399246
\(689\) −27.4164 −1.04448
\(690\) 0 0
\(691\) −12.6525 −0.481323 −0.240661 0.970609i \(-0.577364\pi\)
−0.240661 + 0.970609i \(0.577364\pi\)
\(692\) −3.23607 −0.123017
\(693\) 0 0
\(694\) 2.47214 0.0938410
\(695\) −26.8328 −1.01783
\(696\) 0 0
\(697\) −6.11146 −0.231488
\(698\) 21.7082 0.821668
\(699\) 0 0
\(700\) 0 0
\(701\) 42.7214 1.61356 0.806782 0.590850i \(-0.201207\pi\)
0.806782 + 0.590850i \(0.201207\pi\)
\(702\) 0 0
\(703\) 50.2492 1.89519
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 17.0557 0.641901
\(707\) 0 0
\(708\) 0 0
\(709\) 4.47214 0.167955 0.0839773 0.996468i \(-0.473238\pi\)
0.0839773 + 0.996468i \(0.473238\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 8.94427 0.334263
\(717\) 0 0
\(718\) 26.8328 1.00139
\(719\) 16.8328 0.627758 0.313879 0.949463i \(-0.398371\pi\)
0.313879 + 0.949463i \(0.398371\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −33.3607 −1.24156
\(723\) 0 0
\(724\) −9.23607 −0.343256
\(725\) 15.5279 0.576690
\(726\) 0 0
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −16.0000 −0.592187
\(731\) −25.8885 −0.957522
\(732\) 0 0
\(733\) −49.1246 −1.81446 −0.907229 0.420636i \(-0.861807\pi\)
−0.907229 + 0.420636i \(0.861807\pi\)
\(734\) −5.41641 −0.199923
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) −11.4164 −0.420529
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −8.58359 −0.315539
\(741\) 0 0
\(742\) 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
\(746\) 6.00000 0.219676
\(747\) 0 0
\(748\) −2.47214 −0.0903902
\(749\) 0 0
\(750\) 0 0
\(751\) −16.9443 −0.618305 −0.309153 0.951012i \(-0.600045\pi\)
−0.309153 + 0.951012i \(0.600045\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 0 0
\(754\) 14.4721 0.527044
\(755\) −14.8328 −0.539821
\(756\) 0 0
\(757\) −23.3050 −0.847033 −0.423516 0.905888i \(-0.639204\pi\)
−0.423516 + 0.905888i \(0.639204\pi\)
\(758\) −14.4721 −0.525652
\(759\) 0 0
\(760\) 8.94427 0.324443
\(761\) −11.4164 −0.413844 −0.206922 0.978357i \(-0.566345\pi\)
−0.206922 + 0.978357i \(0.566345\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.47214 0.0894387
\(765\) 0 0
\(766\) 23.8885 0.863128
\(767\) 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
\(771\) 0 0
\(772\) −14.9443 −0.537856
\(773\) 15.7082 0.564985 0.282492 0.959270i \(-0.408839\pi\)
0.282492 + 0.959270i \(0.408839\pi\)
\(774\) 0 0
\(775\) 6.94427 0.249446
\(776\) 12.4721 0.447724
\(777\) 0 0
\(778\) 33.4164 1.19804
\(779\) −17.8885 −0.640924
\(780\) 0 0
\(781\) 6.47214 0.231591
\(782\) 9.88854 0.353614
\(783\) 0 0
\(784\) 0 0
\(785\) −15.6393 −0.558191
\(786\) 0 0
\(787\) 28.1803 1.00452 0.502260 0.864716i \(-0.332502\pi\)
0.502260 + 0.864716i \(0.332502\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.47214 0.0877881
\(794\) −23.7082 −0.841373
\(795\) 0 0
\(796\) −18.9443 −0.671462
\(797\) −41.5967 −1.47343 −0.736716 0.676202i \(-0.763625\pi\)
−0.736716 + 0.676202i \(0.763625\pi\)
\(798\) 0 0
\(799\) −4.94427 −0.174916
\(800\) 3.47214 0.122759
\(801\) 0 0
\(802\) 14.3607 0.507093
\(803\) 12.9443 0.456793
\(804\) 0 0
\(805\) 0 0
\(806\) 6.47214 0.227971
\(807\) 0 0
\(808\) −8.18034 −0.287783
\(809\) 21.0557 0.740280 0.370140 0.928976i \(-0.379310\pi\)
0.370140 + 0.928976i \(0.379310\pi\)
\(810\) 0 0
\(811\) −4.76393 −0.167284 −0.0836421 0.996496i \(-0.526655\pi\)
−0.0836421 + 0.996496i \(0.526655\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.94427 0.243397
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) −75.7771 −2.65110
\(818\) 3.41641 0.119452
\(819\) 0 0
\(820\) 3.05573 0.106711
\(821\) 1.41641 0.0494330 0.0247165 0.999695i \(-0.492132\pi\)
0.0247165 + 0.999695i \(0.492132\pi\)
\(822\) 0 0
\(823\) −46.2492 −1.61215 −0.806073 0.591816i \(-0.798411\pi\)
−0.806073 + 0.591816i \(0.798411\pi\)
\(824\) −14.9443 −0.520608
\(825\) 0 0
\(826\) 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 \(0.565174\pi\)
\(830\) −15.0557 −0.522592
\(831\) 0 0
\(832\) 3.23607 0.112190
\(833\) 0 0
\(834\) 0 0
\(835\) −14.1115 −0.488347
\(836\) −7.23607 −0.250265
\(837\) 0 0
\(838\) −17.2361 −0.595410
\(839\) 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
\(842\) −16.4721 −0.567667
\(843\) 0 0
\(844\) −13.5279 −0.465648
\(845\) 3.12461 0.107490
\(846\) 0 0
\(847\) 0 0
\(848\) −8.47214 −0.290934
\(849\) 0 0
\(850\) 8.58359 0.294415
\(851\) −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
\(855\) 0 0
\(856\) 2.47214 0.0844959
\(857\) −46.4721 −1.58746 −0.793729 0.608272i \(-0.791863\pi\)
−0.793729 + 0.608272i \(0.791863\pi\)
\(858\) 0 0
\(859\) −15.1246 −0.516045 −0.258023 0.966139i \(-0.583071\pi\)
−0.258023 + 0.966139i \(0.583071\pi\)
\(860\) 12.9443 0.441396
\(861\) 0 0
\(862\) 23.0557 0.785281
\(863\) −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
\(866\) 28.4721 0.967523
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 36.9443 1.25181
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) 28.9443 0.979055
\(875\) 0 0
\(876\) 0 0
\(877\) 9.05573 0.305790 0.152895 0.988242i \(-0.451140\pi\)
0.152895 + 0.988242i \(0.451140\pi\)
\(878\) 8.94427 0.301855
\(879\) 0 0
\(880\) 1.23607 0.0416678
\(881\) 28.8328 0.971402 0.485701 0.874125i \(-0.338564\pi\)
0.485701 + 0.874125i \(0.338564\pi\)
\(882\) 0 0
\(883\) −2.83282 −0.0953318 −0.0476659 0.998863i \(-0.515178\pi\)
−0.0476659 + 0.998863i \(0.515178\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −24.9443 −0.838019
\(887\) −44.3607 −1.48949 −0.744743 0.667351i \(-0.767428\pi\)
−0.744743 + 0.667351i \(0.767428\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 12.3607 0.414331
\(891\) 0 0
\(892\) 0.472136 0.0158083
\(893\) −14.4721 −0.484292
\(894\) 0 0
\(895\) −11.0557 −0.369552
\(896\) 0 0
\(897\) 0 0
\(898\) 18.9443 0.632179
\(899\) 8.94427 0.298308
\(900\) 0 0
\(901\) −20.9443 −0.697755
\(902\) −2.47214 −0.0823131
\(903\) 0 0
\(904\) −0.472136 −0.0157030
\(905\) 11.4164 0.379494
\(906\) 0 0
\(907\) −24.3607 −0.808883 −0.404442 0.914564i \(-0.632534\pi\)
−0.404442 + 0.914564i \(0.632534\pi\)
\(908\) −19.2361 −0.638371
\(909\) 0 0
\(910\) 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
\(914\) −26.9443 −0.891237
\(915\) 0 0
\(916\) −17.2361 −0.569496
\(917\) 0 0
\(918\) 0 0
\(919\) −22.1115 −0.729390 −0.364695 0.931127i \(-0.618827\pi\)
−0.364695 + 0.931127i \(0.618827\pi\)
\(920\) −4.94427 −0.163008
\(921\) 0 0
\(922\) −24.7639 −0.815557
\(923\) −20.9443 −0.689389
\(924\) 0 0
\(925\) −24.1115 −0.792780
\(926\) 30.4721 1.00138
\(927\) 0 0
\(928\) 4.47214 0.146805
\(929\) 40.2492 1.32053 0.660267 0.751031i \(-0.270443\pi\)
0.660267 + 0.751031i \(0.270443\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 14.9443 0.489516
\(933\) 0 0
\(934\) 27.1246 0.887544
\(935\) 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
\(939\) 0 0
\(940\) 2.47214 0.0806322
\(941\) −11.8197 −0.385310 −0.192655 0.981267i \(-0.561710\pi\)
−0.192655 + 0.981267i \(0.561710\pi\)
\(942\) 0 0
\(943\) 9.88854 0.322015
\(944\) 2.76393 0.0899583
\(945\) 0 0
\(946\) −10.4721 −0.340479
\(947\) −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
\(950\) 25.1246 0.815150
\(951\) 0 0
\(952\) 0 0
\(953\) −22.9443 −0.743238 −0.371619 0.928385i \(-0.621197\pi\)
−0.371619 + 0.928385i \(0.621197\pi\)
\(954\) 0 0
\(955\) −3.05573 −0.0988810
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) −12.3607 −0.399355
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −22.4721 −0.724531
\(963\) 0 0
\(964\) −15.4164 −0.496529
\(965\) 18.4721 0.594639
\(966\) 0 0
\(967\) 45.8885 1.47568 0.737838 0.674978i \(-0.235847\pi\)
0.737838 + 0.674978i \(0.235847\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −15.4164 −0.494991
\(971\) 50.5410 1.62194 0.810969 0.585089i \(-0.198940\pi\)
0.810969 + 0.585089i \(0.198940\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.9443 −0.542929
\(975\) 0 0
\(976\) 0.763932 0.0244529
\(977\) 28.8328 0.922443 0.461222 0.887285i \(-0.347411\pi\)
0.461222 + 0.887285i \(0.347411\pi\)
\(978\) 0 0
\(979\) −10.0000 −0.319601
\(980\) 0 0
\(981\) 0 0
\(982\) −16.9443 −0.540713
\(983\) 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
\(986\) 11.0557 0.352086
\(987\) 0 0
\(988\) 23.4164 0.744975
\(989\) 41.8885 1.33198
\(990\) 0 0
\(991\) −0.360680 −0.0114574 −0.00572869 0.999984i \(-0.501824\pi\)
−0.00572869 + 0.999984i \(0.501824\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 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 \(0.625074\pi\)
\(998\) 32.3607 1.02436
\(999\) 0 0
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 9702.2.a.cu.1.1 2
3.2 odd 2 1078.2.a.w.1.1 2
7.6 odd 2 1386.2.a.m.1.2 2
12.11 even 2 8624.2.a.bf.1.2 2
21.2 odd 6 1078.2.e.n.67.2 4
21.5 even 6 1078.2.e.q.67.1 4
21.11 odd 6 1078.2.e.n.177.2 4
21.17 even 6 1078.2.e.q.177.1 4
21.20 even 2 154.2.a.d.1.2 2
84.83 odd 2 1232.2.a.p.1.1 2
105.62 odd 4 3850.2.c.q.1849.3 4
105.83 odd 4 3850.2.c.q.1849.2 4
105.104 even 2 3850.2.a.bj.1.1 2
168.83 odd 2 4928.2.a.bk.1.2 2
168.125 even 2 4928.2.a.bt.1.1 2
231.230 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 21.20 even 2
1078.2.a.w.1.1 2 3.2 odd 2
1078.2.e.n.67.2 4 21.2 odd 6
1078.2.e.n.177.2 4 21.11 odd 6
1078.2.e.q.67.1 4 21.5 even 6
1078.2.e.q.177.1 4 21.17 even 6
1232.2.a.p.1.1 2 84.83 odd 2
1386.2.a.m.1.2 2 7.6 odd 2
1694.2.a.l.1.2 2 231.230 odd 2
3850.2.a.bj.1.1 2 105.104 even 2
3850.2.c.q.1849.2 4 105.83 odd 4
3850.2.c.q.1849.3 4 105.62 odd 4
4928.2.a.bk.1.2 2 168.83 odd 2
4928.2.a.bt.1.1 2 168.125 even 2
8624.2.a.bf.1.2 2 12.11 even 2
9702.2.a.cu.1.1 2 1.1 even 1 trivial