Properties

Label 1386.2.a.m.1.2
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
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 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1386.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{7} -1.00000 q^{8} -1.23607 q^{10} -1.00000 q^{11} -3.23607 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.47214 q^{17} -7.23607 q^{19} +1.23607 q^{20} +1.00000 q^{22} -4.00000 q^{23} -3.47214 q^{25} +3.23607 q^{26} +1.00000 q^{28} -4.47214 q^{29} +2.00000 q^{31} -1.00000 q^{32} +2.47214 q^{34} +1.23607 q^{35} +6.94427 q^{37} +7.23607 q^{38} -1.23607 q^{40} +2.47214 q^{41} -10.4721 q^{43} -1.00000 q^{44} +4.00000 q^{46} +2.00000 q^{47} +1.00000 q^{49} +3.47214 q^{50} -3.23607 q^{52} -8.47214 q^{53} -1.23607 q^{55} -1.00000 q^{56} +4.47214 q^{58} -2.76393 q^{59} -0.763932 q^{61} -2.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +11.4164 q^{67} -2.47214 q^{68} -1.23607 q^{70} -6.47214 q^{71} +12.9443 q^{73} -6.94427 q^{74} -7.23607 q^{76} -1.00000 q^{77} +1.23607 q^{80} -2.47214 q^{82} +12.1803 q^{83} -3.05573 q^{85} +10.4721 q^{86} +1.00000 q^{88} -10.0000 q^{89} -3.23607 q^{91} -4.00000 q^{92} -2.00000 q^{94} -8.94427 q^{95} +12.4721 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 2 q^{10} - 2 q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 4 q^{17} - 10 q^{19} - 2 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{25} + 2 q^{26} + 2 q^{28} + 4 q^{31} - 2 q^{32} - 4 q^{34} - 2 q^{35} - 4 q^{37} + 10 q^{38} + 2 q^{40} - 4 q^{41} - 12 q^{43} - 2 q^{44} + 8 q^{46} + 4 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} - 8 q^{53} + 2 q^{55} - 2 q^{56} - 10 q^{59} - 6 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{65} - 4 q^{67} + 4 q^{68} + 2 q^{70} - 4 q^{71} + 8 q^{73} + 4 q^{74} - 10 q^{76} - 2 q^{77} - 2 q^{80} + 4 q^{82} + 2 q^{83} - 24 q^{85} + 12 q^{86} + 2 q^{88} - 20 q^{89} - 2 q^{91} - 8 q^{92} - 4 q^{94} + 16 q^{97} - 2 q^{98}+O(q^{100}) \) 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 \(-0.410861\pi\)
0.276393 + 0.961045i \(0.410861\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −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 \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −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 \(0.811679\pi\)
\(20\) 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
\(28\) 1.00000 0.188982
\(29\) −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 \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.47214 0.423968
\(35\) 1.23607 0.208934
\(36\) 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 \(-0.438165\pi\)
0.193041 + 0.981191i \(0.438165\pi\)
\(42\) 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 \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 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
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 4.47214 0.587220
\(59\) −2.76393 −0.359833 −0.179917 0.983682i \(-0.557583\pi\)
−0.179917 + 0.983682i \(0.557583\pi\)
\(60\) 0 0
\(61\) −0.763932 −0.0978115 −0.0489057 0.998803i \(-0.515573\pi\)
−0.0489057 + 0.998803i \(0.515573\pi\)
\(62\) −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
\(70\) −1.23607 −0.147738
\(71\) −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 \(-0.226418\pi\)
0.757506 + 0.652828i \(0.226418\pi\)
\(74\) −6.94427 −0.807255
\(75\) 0 0
\(76\) −7.23607 −0.830034
\(77\) −1.00000 −0.113961
\(78\) 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 \(-0.266944\pi\)
0.668483 + 0.743727i \(0.266944\pi\)
\(84\) 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 \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −3.23607 −0.339232
\(92\) −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 \(-0.281751\pi\)
0.633177 + 0.774007i \(0.281751\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −3.47214 −0.347214
\(101\) −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 \(0.763406\pi\)
\(104\) 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
\(112\) 1.00000 0.0944911
\(113\) 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
\(119\) −2.47214 −0.226620
\(120\) 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 \(-0.566732\pi\)
−0.208113 + 0.978105i \(0.566732\pi\)
\(132\) 0 0
\(133\) −7.23607 −0.627447
\(134\) −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 \(-0.872327\pi\)
−0.920633 + 0.390429i \(0.872327\pi\)
\(140\) 1.23607 0.104467
\(141\) 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
\(154\) 1.00000 0.0805823
\(155\) 2.47214 0.198567
\(156\) 0 0
\(157\) −12.6525 −1.00978 −0.504889 0.863184i \(-0.668466\pi\)
−0.504889 + 0.863184i \(0.668466\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −1.23607 −0.0977198
\(161\) −4.00000 −0.315244
\(162\) 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 \(-0.645629\pi\)
−0.441714 + 0.897156i \(0.645629\pi\)
\(168\) 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 \(-0.460743\pi\)
0.123017 + 0.992405i \(0.460743\pi\)
\(174\) 0 0
\(175\) −3.47214 −0.262469
\(176\) −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 \(-0.388470\pi\)
0.343256 + 0.939242i \(0.388470\pi\)
\(182\) 3.23607 0.239873
\(183\) 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
\(196\) 1.00000 0.0714286
\(197\) −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 \(-0.265667\pi\)
0.671462 + 0.741039i \(0.265667\pi\)
\(200\) 3.47214 0.245517
\(201\) 0 0
\(202\) 8.18034 0.575567
\(203\) −4.47214 −0.313882
\(204\) 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
\(217\) 2.00000 0.135769
\(218\) 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 \(-0.505032\pi\)
−0.0158083 + 0.999875i \(0.505032\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −0.472136 −0.0314060
\(227\) 19.2361 1.27674 0.638371 0.769729i \(-0.279608\pi\)
0.638371 + 0.769729i \(0.279608\pi\)
\(228\) 0 0
\(229\) 17.2361 1.13899 0.569496 0.821994i \(-0.307139\pi\)
0.569496 + 0.821994i \(0.307139\pi\)
\(230\) 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
\(238\) 2.47214 0.160245
\(239\) −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 \(-0.334608\pi\)
0.496529 + 0.868020i \(0.334608\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −0.763932 −0.0489057
\(245\) 1.23607 0.0789695
\(246\) 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 \(-0.874008\pi\)
−0.922682 + 0.385562i \(0.874008\pi\)
\(252\) 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 \(-0.569492\pi\)
−0.216586 + 0.976264i \(0.569492\pi\)
\(258\) 0 0
\(259\) 6.94427 0.431496
\(260\) −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
\(266\) 7.23607 0.443672
\(267\) 0 0
\(268\) 11.4164 0.697368
\(269\) 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 \(0.490870\pi\)
\(272\) −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
\(280\) −1.23607 −0.0738692
\(281\) −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 \(-0.356572\pi\)
0.435500 + 0.900189i \(0.356572\pi\)
\(284\) −6.47214 −0.384051
\(285\) 0 0
\(286\) −3.23607 −0.191353
\(287\) 2.47214 0.145926
\(288\) 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 \(-0.215967\pi\)
0.778527 + 0.627611i \(0.215967\pi\)
\(294\) 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
\(301\) −10.4721 −0.603604
\(302\) −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 \(-0.767033\pi\)
−0.743915 + 0.668274i \(0.767033\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) −2.47214 −0.140408
\(311\) 21.4164 1.21441 0.607207 0.794544i \(-0.292290\pi\)
0.607207 + 0.794544i \(0.292290\pi\)
\(312\) 0 0
\(313\) 19.5279 1.10378 0.551890 0.833917i \(-0.313907\pi\)
0.551890 + 0.833917i \(0.313907\pi\)
\(314\) 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
\(322\) 4.00000 0.222911
\(323\) 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
\(329\) 2.00000 0.110264
\(330\) 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
\(343\) 1.00000 0.0539949
\(344\) 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 \(-0.302659\pi\)
0.581007 + 0.813899i \(0.302659\pi\)
\(350\) 3.47214 0.185593
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 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
\(356\) −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
\(364\) −3.23607 −0.169616
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) −5.41641 −0.282734 −0.141367 0.989957i \(-0.545150\pi\)
−0.141367 + 0.989957i \(0.545150\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −8.58359 −0.446240
\(371\) −8.47214 −0.439851
\(372\) 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 \(-0.291039\pi\)
0.610324 + 0.792152i \(0.291039\pi\)
\(384\) 0 0
\(385\) −1.23607 −0.0629959
\(386\) 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
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −23.7082 −1.18988 −0.594940 0.803770i \(-0.702824\pi\)
−0.594940 + 0.803770i \(0.702824\pi\)
\(398\) −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
\(406\) 4.47214 0.221948
\(407\) −6.94427 −0.344215
\(408\) 0 0
\(409\) 3.41641 0.168930 0.0844652 0.996426i \(-0.473082\pi\)
0.0844652 + 0.996426i \(0.473082\pi\)
\(410\) −3.05573 −0.150912
\(411\) 0 0
\(412\) −14.9443 −0.736251
\(413\) −2.76393 −0.136004
\(414\) 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 \(-0.638327\pi\)
−0.421019 + 0.907052i \(0.638327\pi\)
\(420\) 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
\(427\) −0.763932 −0.0369693
\(428\) −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 \(-0.260177\pi\)
0.684142 + 0.729349i \(0.260177\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 28.9443 1.38459
\(438\) 0 0
\(439\) 8.94427 0.426887 0.213443 0.976955i \(-0.431532\pi\)
0.213443 + 0.976955i \(0.431532\pi\)
\(440\) 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
\(448\) 1.00000 0.0472456
\(449\) −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
\(455\) −4.00000 −0.187523
\(456\) 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 \(-0.695654\pi\)
−0.576686 + 0.816966i \(0.695654\pi\)
\(462\) 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 \(-0.284042\pi\)
0.627589 + 0.778545i \(0.284042\pi\)
\(468\) 0 0
\(469\) 11.4164 0.527161
\(470\) −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
\(476\) −2.47214 −0.113310
\(477\) 0 0
\(478\) 20.0000 0.914779
\(479\) −12.3607 −0.564774 −0.282387 0.959301i \(-0.591126\pi\)
−0.282387 + 0.959301i \(0.591126\pi\)
\(480\) 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
\(490\) −1.23607 −0.0558399
\(491\) 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
\(497\) −6.47214 −0.290315
\(498\) 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 \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 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 \(-0.679092\pi\)
−0.533417 + 0.845852i \(0.679092\pi\)
\(510\) 0 0
\(511\) 12.9443 0.572621
\(512\) −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
\(518\) −6.94427 −0.305114
\(519\) 0 0
\(520\) 4.00000 0.175412
\(521\) 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 \(0.601156\pi\)
\(524\) −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
\(532\) −7.23607 −0.313723
\(533\) −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
\(539\) −1.00000 −0.0430730
\(540\) 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
\(560\) 1.23607 0.0522334
\(561\) 0 0
\(562\) 28.8328 1.21624
\(563\) −31.2361 −1.31644 −0.658222 0.752824i \(-0.728691\pi\)
−0.658222 + 0.752824i \(0.728691\pi\)
\(564\) 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
\(574\) −2.47214 −0.103185
\(575\) 13.8885 0.579192
\(576\) 0 0
\(577\) 26.9443 1.12170 0.560852 0.827916i \(-0.310474\pi\)
0.560852 + 0.827916i \(0.310474\pi\)
\(578\) 10.8885 0.452904
\(579\) 0 0
\(580\) −5.52786 −0.229532
\(581\) 12.1803 0.505326
\(582\) 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 \(-0.461678\pi\)
0.120102 + 0.992762i \(0.461678\pi\)
\(588\) 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 \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 0 0
\(595\) −3.05573 −0.125273
\(596\) 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 \(-0.751500\pi\)
−0.710430 + 0.703768i \(0.751500\pi\)
\(602\) 10.4721 0.426812
\(603\) 0 0
\(604\) 12.0000 0.488273
\(605\) 1.23607 0.0502533
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 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
\(616\) 1.00000 0.0402911
\(617\) −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 \(-0.378462\pi\)
0.372613 + 0.927987i \(0.378462\pi\)
\(620\) 2.47214 0.0992834
\(621\) 0 0
\(622\) −21.4164 −0.858720
\(623\) −10.0000 −0.400642
\(624\) 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
\(637\) −3.23607 −0.128218
\(638\) −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 \(-0.620638\pi\)
−0.369989 + 0.929036i \(0.620638\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −17.8885 −0.703815
\(647\) 28.8328 1.13353 0.566767 0.823878i \(-0.308194\pi\)
0.566767 + 0.823878i \(0.308194\pi\)
\(648\) 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
\(658\) −2.00000 −0.0779681
\(659\) −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 \(-0.519355\pi\)
−0.0607667 + 0.998152i \(0.519355\pi\)
\(662\) 16.9443 0.658558
\(663\) 0 0
\(664\) −12.1803 −0.472689
\(665\) −8.94427 −0.346844
\(666\) 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 \(-0.332982\pi\)
0.500954 + 0.865474i \(0.332982\pi\)
\(678\) 0 0
\(679\) 12.4721 0.478637
\(680\) 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
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −10.4721 −0.399246
\(689\) 27.4164 1.04448
\(690\) 0 0
\(691\) 12.6525 0.481323 0.240661 0.970609i \(-0.422636\pi\)
0.240661 + 0.970609i \(0.422636\pi\)
\(692\) 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
\(700\) −3.47214 −0.131234
\(701\) 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
\(707\) −8.18034 −0.307653
\(708\) 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 \(-0.601629\pi\)
−0.313879 + 0.949463i \(0.601629\pi\)
\(720\) 0 0
\(721\) −14.9443 −0.556554
\(722\) −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 \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 3.23607 0.119937
\(729\) 0 0
\(730\) −16.0000 −0.592187
\(731\) 25.8885 0.957522
\(732\) 0 0
\(733\) 49.1246 1.81446 0.907229 0.420636i \(-0.138193\pi\)
0.907229 + 0.420636i \(0.138193\pi\)
\(734\) 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
\(742\) 8.47214 0.311022
\(743\) −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
\(749\) −2.47214 −0.0903299
\(750\) 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 \(-0.433655\pi\)
0.206922 + 0.978357i \(0.433655\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 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 \(-0.786218\pi\)
−0.782818 + 0.622251i \(0.786218\pi\)
\(770\) 1.23607 0.0445448
\(771\) 0 0
\(772\) −14.9443 −0.537856
\(773\) −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
\(776\) −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
\(784\) 1.00000 0.0357143
\(785\) −15.6393 −0.558191
\(786\) 0 0
\(787\) −28.1803 −1.00452 −0.502260 0.864716i \(-0.667498\pi\)
−0.502260 + 0.864716i \(0.667498\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) 0.472136 0.0167872
\(792\) 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 \(-0.236375\pi\)
0.736716 + 0.676202i \(0.236375\pi\)
\(798\) 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
\(805\) −4.94427 −0.174263
\(806\) 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 \(-0.473345\pi\)
0.0836421 + 0.996496i \(0.473345\pi\)
\(812\) −4.47214 −0.156941
\(813\) 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
\(826\) 2.76393 0.0961695
\(827\) −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 \(-0.434826\pi\)
0.203321 + 0.979112i \(0.434826\pi\)
\(830\) −15.0557 −0.522592
\(831\) 0 0
\(832\) −3.23607 −0.112190
\(833\) −2.47214 −0.0856544
\(834\) 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 \(-0.593844\pi\)
−0.290567 + 0.956855i \(0.593844\pi\)
\(840\) 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
\(847\) 1.00000 0.0343604
\(848\) −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 \(-0.311917\pi\)
0.557092 + 0.830451i \(0.311917\pi\)
\(854\) 0.763932 0.0261412
\(855\) 0 0
\(856\) 2.47214 0.0844959
\(857\) 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 \(0.416929\pi\)
\(860\) −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
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) −36.9443 −1.25181
\(872\) 10.0000 0.338643
\(873\) 0 0
\(874\) −28.9443 −0.979055
\(875\) −10.4721 −0.354023
\(876\) 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 \(-0.661436\pi\)
−0.485701 + 0.874125i \(0.661436\pi\)
\(882\) 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 \(-0.232572\pi\)
0.744743 + 0.667351i \(0.232572\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 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
\(896\) −1.00000 −0.0334077
\(897\) 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
\(910\) 4.00000 0.132599
\(911\) 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
\(917\) −4.76393 −0.157319
\(918\) 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 \(-0.729557\pi\)
−0.660267 + 0.751031i \(0.729557\pi\)
\(930\) 0 0
\(931\) −7.23607 −0.237153
\(932\) 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 \(-0.515894\pi\)
−0.0499131 + 0.998754i \(0.515894\pi\)
\(938\) −11.4164 −0.372759
\(939\) 0 0
\(940\) 2.47214 0.0806322
\(941\) 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
\(944\) −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
\(952\) 2.47214 0.0801224
\(953\) −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
\(959\) 19.8885 0.642235
\(960\) 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 \(-0.801060\pi\)
−0.810969 + 0.585089i \(0.801060\pi\)
\(972\) 0 0
\(973\) −21.7082 −0.695933
\(974\) −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
\(980\) 1.23607 0.0394847
\(981\) 0 0
\(982\) −16.9443 −0.540713
\(983\) −14.0000 −0.446531 −0.223265 0.974758i \(-0.571672\pi\)
−0.223265 + 0.974758i \(0.571672\pi\)
\(984\) 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
\(994\) 6.47214 0.205284
\(995\) 23.4164 0.742350
\(996\) 0 0
\(997\) 24.1803 0.765799 0.382900 0.923790i \(-0.374926\pi\)
0.382900 + 0.923790i \(0.374926\pi\)
\(998\) 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 1386.2.a.m.1.2 2
3.2 odd 2 154.2.a.d.1.2 2
7.6 odd 2 9702.2.a.cu.1.1 2
12.11 even 2 1232.2.a.p.1.1 2
15.2 even 4 3850.2.c.q.1849.3 4
15.8 even 4 3850.2.c.q.1849.2 4
15.14 odd 2 3850.2.a.bj.1.1 2
21.2 odd 6 1078.2.e.q.67.1 4
21.5 even 6 1078.2.e.n.67.2 4
21.11 odd 6 1078.2.e.q.177.1 4
21.17 even 6 1078.2.e.n.177.2 4
21.20 even 2 1078.2.a.w.1.1 2
24.5 odd 2 4928.2.a.bt.1.1 2
24.11 even 2 4928.2.a.bk.1.2 2
33.32 even 2 1694.2.a.l.1.2 2
84.83 odd 2 8624.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.d.1.2 2 3.2 odd 2
1078.2.a.w.1.1 2 21.20 even 2
1078.2.e.n.67.2 4 21.5 even 6
1078.2.e.n.177.2 4 21.17 even 6
1078.2.e.q.67.1 4 21.2 odd 6
1078.2.e.q.177.1 4 21.11 odd 6
1232.2.a.p.1.1 2 12.11 even 2
1386.2.a.m.1.2 2 1.1 even 1 trivial
1694.2.a.l.1.2 2 33.32 even 2
3850.2.a.bj.1.1 2 15.14 odd 2
3850.2.c.q.1849.2 4 15.8 even 4
3850.2.c.q.1849.3 4 15.2 even 4
4928.2.a.bk.1.2 2 24.11 even 2
4928.2.a.bt.1.1 2 24.5 odd 2
8624.2.a.bf.1.2 2 84.83 odd 2
9702.2.a.cu.1.1 2 7.6 odd 2