Properties

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

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(2\)
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 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 3850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} +1.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.47214 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.23607 q^{3} +1.00000 q^{4} +1.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.47214 q^{9} +1.00000 q^{11} -1.23607 q^{12} +3.23607 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.47214 q^{17} +1.47214 q^{18} -7.23607 q^{19} +1.23607 q^{21} -1.00000 q^{22} -4.00000 q^{23} +1.23607 q^{24} -3.23607 q^{26} +5.52786 q^{27} -1.00000 q^{28} +4.47214 q^{29} +2.00000 q^{31} -1.00000 q^{32} -1.23607 q^{33} +2.47214 q^{34} -1.47214 q^{36} -6.94427 q^{37} +7.23607 q^{38} -4.00000 q^{39} -2.47214 q^{41} -1.23607 q^{42} +10.4721 q^{43} +1.00000 q^{44} +4.00000 q^{46} +2.00000 q^{47} -1.23607 q^{48} +1.00000 q^{49} +3.05573 q^{51} +3.23607 q^{52} -8.47214 q^{53} -5.52786 q^{54} +1.00000 q^{56} +8.94427 q^{57} -4.47214 q^{58} +2.76393 q^{59} -0.763932 q^{61} -2.00000 q^{62} +1.47214 q^{63} +1.00000 q^{64} +1.23607 q^{66} -11.4164 q^{67} -2.47214 q^{68} +4.94427 q^{69} +6.47214 q^{71} +1.47214 q^{72} -12.9443 q^{73} +6.94427 q^{74} -7.23607 q^{76} -1.00000 q^{77} +4.00000 q^{78} -2.41641 q^{81} +2.47214 q^{82} +12.1803 q^{83} +1.23607 q^{84} -10.4721 q^{86} -5.52786 q^{87} -1.00000 q^{88} +10.0000 q^{89} -3.23607 q^{91} -4.00000 q^{92} -2.47214 q^{93} -2.00000 q^{94} +1.23607 q^{96} -12.4721 q^{97} -1.00000 q^{98} -1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} - 10 q^{19} - 2 q^{21} - 2 q^{22} - 8 q^{23} - 2 q^{24} - 2 q^{26} + 20 q^{27} - 2 q^{28} + 4 q^{31} - 2 q^{32} + 2 q^{33} - 4 q^{34} + 6 q^{36} + 4 q^{37} + 10 q^{38} - 8 q^{39} + 4 q^{41} + 2 q^{42} + 12 q^{43} + 2 q^{44} + 8 q^{46} + 4 q^{47} + 2 q^{48} + 2 q^{49} + 24 q^{51} + 2 q^{52} - 8 q^{53} - 20 q^{54} + 2 q^{56} + 10 q^{59} - 6 q^{61} - 4 q^{62} - 6 q^{63} + 2 q^{64} - 2 q^{66} + 4 q^{67} + 4 q^{68} - 8 q^{69} + 4 q^{71} - 6 q^{72} - 8 q^{73} - 4 q^{74} - 10 q^{76} - 2 q^{77} + 8 q^{78} + 22 q^{81} - 4 q^{82} + 2 q^{83} - 2 q^{84} - 12 q^{86} - 20 q^{87} - 2 q^{88} + 20 q^{89} - 2 q^{91} - 8 q^{92} + 4 q^{93} - 4 q^{94} - 2 q^{96} - 16 q^{97} - 2 q^{98} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.23607 −0.713644 −0.356822 0.934172i \(-0.616140\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.23607 0.504623
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −1.47214 −0.490712
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −1.23607 −0.356822
\(13\) 3.23607 0.897524 0.448762 0.893651i \(-0.351865\pi\)
0.448762 + 0.893651i \(0.351865\pi\)
\(14\) 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 \(0.596917\pi\)
\(18\) 1.47214 0.346986
\(19\) −7.23607 −1.66007 −0.830034 0.557713i \(-0.811679\pi\)
−0.830034 + 0.557713i \(0.811679\pi\)
\(20\) 0 0
\(21\) 1.23607 0.269732
\(22\) −1.00000 −0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.23607 0.252311
\(25\) 0 0
\(26\) −3.23607 −0.634645
\(27\) 5.52786 1.06384
\(28\) −1.00000 −0.188982
\(29\) 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 \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.23607 −0.215172
\(34\) 2.47214 0.423968
\(35\) 0 0
\(36\) −1.47214 −0.245356
\(37\) −6.94427 −1.14163 −0.570816 0.821078i \(-0.693373\pi\)
−0.570816 + 0.821078i \(0.693373\pi\)
\(38\) 7.23607 1.17385
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −2.47214 −0.386083 −0.193041 0.981191i \(-0.561835\pi\)
−0.193041 + 0.981191i \(0.561835\pi\)
\(42\) −1.23607 −0.190729
\(43\) 10.4721 1.59699 0.798493 0.602004i \(-0.205631\pi\)
0.798493 + 0.602004i \(0.205631\pi\)
\(44\) 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 \(0.453403\pi\)
\(48\) −1.23607 −0.178411
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.05573 0.427888
\(52\) 3.23607 0.448762
\(53\) −8.47214 −1.16374 −0.581869 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(54\) −5.52786 −0.752247
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 8.94427 1.18470
\(58\) −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 \(-0.515573\pi\)
−0.0489057 + 0.998803i \(0.515573\pi\)
\(62\) −2.00000 −0.254000
\(63\) 1.47214 0.185472
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.23607 0.152149
\(67\) −11.4164 −1.39474 −0.697368 0.716713i \(-0.745646\pi\)
−0.697368 + 0.716713i \(0.745646\pi\)
\(68\) −2.47214 −0.299791
\(69\) 4.94427 0.595220
\(70\) 0 0
\(71\) 6.47214 0.768101 0.384051 0.923312i \(-0.374529\pi\)
0.384051 + 0.923312i \(0.374529\pi\)
\(72\) 1.47214 0.173493
\(73\) −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
\(77\) −1.00000 −0.113961
\(78\) 4.00000 0.452911
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 2.47214 0.273002
\(83\) 12.1803 1.33697 0.668483 0.743727i \(-0.266944\pi\)
0.668483 + 0.743727i \(0.266944\pi\)
\(84\) 1.23607 0.134866
\(85\) 0 0
\(86\) −10.4721 −1.12924
\(87\) −5.52786 −0.592649
\(88\) −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
\(91\) −3.23607 −0.339232
\(92\) −4.00000 −0.417029
\(93\) −2.47214 −0.256349
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) 1.23607 0.126156
\(97\) −12.4721 −1.26635 −0.633177 0.774007i \(-0.718249\pi\)
−0.633177 + 0.774007i \(0.718249\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.47214 −0.147955
\(100\) 0 0
\(101\) 8.18034 0.813974 0.406987 0.913434i \(-0.366579\pi\)
0.406987 + 0.913434i \(0.366579\pi\)
\(102\) −3.05573 −0.302562
\(103\) 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 \(0.538128\pi\)
\(108\) 5.52786 0.531919
\(109\) −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
\(112\) −1.00000 −0.0944911
\(113\) 0.472136 0.0444148 0.0222074 0.999753i \(-0.492931\pi\)
0.0222074 + 0.999753i \(0.492931\pi\)
\(114\) −8.94427 −0.837708
\(115\) 0 0
\(116\) 4.47214 0.415227
\(117\) −4.76393 −0.440426
\(118\) −2.76393 −0.254441
\(119\) 2.47214 0.226620
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.763932 0.0691632
\(123\) 3.05573 0.275526
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) −1.47214 −0.131148
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −12.9443 −1.13968
\(130\) 0 0
\(131\) 4.76393 0.416227 0.208113 0.978105i \(-0.433268\pi\)
0.208113 + 0.978105i \(0.433268\pi\)
\(132\) −1.23607 −0.107586
\(133\) 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 \(0.176846\pi\)
\(138\) −4.94427 −0.420884
\(139\) −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
\(142\) −6.47214 −0.543130
\(143\) 3.23607 0.270614
\(144\) −1.47214 −0.122678
\(145\) 0 0
\(146\) 12.9443 1.07128
\(147\) −1.23607 −0.101949
\(148\) −6.94427 −0.570816
\(149\) −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 \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 7.23607 0.586923
\(153\) 3.63932 0.294222
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 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
\(161\) 4.00000 0.315244
\(162\) 2.41641 0.189851
\(163\) 19.4164 1.52081 0.760405 0.649449i \(-0.225000\pi\)
0.760405 + 0.649449i \(0.225000\pi\)
\(164\) −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 \(0.645629\pi\)
\(168\) −1.23607 −0.0953647
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) 10.6525 0.814615
\(172\) 10.4721 0.798493
\(173\) 3.23607 0.246034 0.123017 0.992405i \(-0.460743\pi\)
0.123017 + 0.992405i \(0.460743\pi\)
\(174\) 5.52786 0.419066
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −3.41641 −0.256793
\(178\) −10.0000 −0.749532
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 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
\(183\) 0.944272 0.0698026
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 2.47214 0.181266
\(187\) −2.47214 −0.180780
\(188\) 2.00000 0.145865
\(189\) −5.52786 −0.402093
\(190\) 0 0
\(191\) −2.47214 −0.178877 −0.0894387 0.995992i \(-0.528507\pi\)
−0.0894387 + 0.995992i \(0.528507\pi\)
\(192\) −1.23607 −0.0892055
\(193\) 14.9443 1.07571 0.537856 0.843037i \(-0.319234\pi\)
0.537856 + 0.843037i \(0.319234\pi\)
\(194\) 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 \(0.721573\pi\)
\(198\) 1.47214 0.104620
\(199\) 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
\(202\) −8.18034 −0.575567
\(203\) −4.47214 −0.313882
\(204\) 3.05573 0.213944
\(205\) 0 0
\(206\) −14.9443 −1.04122
\(207\) 5.88854 0.409282
\(208\) 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
\(213\) −8.00000 −0.548151
\(214\) 2.47214 0.168992
\(215\) 0 0
\(216\) −5.52786 −0.376124
\(217\) −2.00000 −0.135769
\(218\) 10.0000 0.677285
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) −8.58359 −0.576093
\(223\) 0.472136 0.0316166 0.0158083 0.999875i \(-0.494968\pi\)
0.0158083 + 0.999875i \(0.494968\pi\)
\(224\) 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 \(0.279608\pi\)
\(228\) 8.94427 0.592349
\(229\) 17.2361 1.13899 0.569496 0.821994i \(-0.307139\pi\)
0.569496 + 0.821994i \(0.307139\pi\)
\(230\) 0 0
\(231\) 1.23607 0.0813273
\(232\) −4.47214 −0.293610
\(233\) 14.9443 0.979032 0.489516 0.871994i \(-0.337174\pi\)
0.489516 + 0.871994i \(0.337174\pi\)
\(234\) 4.76393 0.311428
\(235\) 0 0
\(236\) 2.76393 0.179917
\(237\) 0 0
\(238\) −2.47214 −0.160245
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 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
\(243\) −13.5967 −0.872232
\(244\) −0.763932 −0.0489057
\(245\) 0 0
\(246\) −3.05573 −0.194826
\(247\) −23.4164 −1.48995
\(248\) −2.00000 −0.127000
\(249\) −15.0557 −0.954118
\(250\) 0 0
\(251\) 29.2361 1.84536 0.922682 0.385562i \(-0.125992\pi\)
0.922682 + 0.385562i \(0.125992\pi\)
\(252\) 1.47214 0.0927358
\(253\) −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 \(0.569492\pi\)
\(258\) 12.9443 0.805875
\(259\) 6.94427 0.431496
\(260\) 0 0
\(261\) −6.58359 −0.407514
\(262\) −4.76393 −0.294317
\(263\) 4.94427 0.304877 0.152438 0.988313i \(-0.451287\pi\)
0.152438 + 0.988313i \(0.451287\pi\)
\(264\) 1.23607 0.0760747
\(265\) 0 0
\(266\) −7.23607 −0.443672
\(267\) −12.3607 −0.756461
\(268\) −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 \(-0.490870\pi\)
0.0286802 + 0.999589i \(0.490870\pi\)
\(272\) −2.47214 −0.149895
\(273\) 4.00000 0.242091
\(274\) −19.8885 −1.20151
\(275\) 0 0
\(276\) 4.94427 0.297610
\(277\) −3.52786 −0.211969 −0.105984 0.994368i \(-0.533799\pi\)
−0.105984 + 0.994368i \(0.533799\pi\)
\(278\) 21.7082 1.30197
\(279\) −2.94427 −0.176269
\(280\) 0 0
\(281\) 28.8328 1.72002 0.860011 0.510276i \(-0.170457\pi\)
0.860011 + 0.510276i \(0.170457\pi\)
\(282\) 2.47214 0.147214
\(283\) −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
\(287\) 2.47214 0.145926
\(288\) 1.47214 0.0867464
\(289\) −10.8885 −0.640503
\(290\) 0 0
\(291\) 15.4164 0.903726
\(292\) −12.9443 −0.757506
\(293\) 26.6525 1.55705 0.778527 0.627611i \(-0.215967\pi\)
0.778527 + 0.627611i \(0.215967\pi\)
\(294\) 1.23607 0.0720889
\(295\) 0 0
\(296\) 6.94427 0.403628
\(297\) 5.52786 0.320759
\(298\) 22.3607 1.29532
\(299\) −12.9443 −0.748587
\(300\) 0 0
\(301\) −10.4721 −0.603604
\(302\) −12.0000 −0.690522
\(303\) −10.1115 −0.580888
\(304\) −7.23607 −0.415017
\(305\) 0 0
\(306\) −3.63932 −0.208046
\(307\) 26.0689 1.48783 0.743915 0.668274i \(-0.232967\pi\)
0.743915 + 0.668274i \(0.232967\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −18.4721 −1.05084
\(310\) 0 0
\(311\) −21.4164 −1.21441 −0.607207 0.794544i \(-0.707710\pi\)
−0.607207 + 0.794544i \(0.707710\pi\)
\(312\) 4.00000 0.226455
\(313\) −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 \(0.164762\pi\)
\(318\) −10.4721 −0.587248
\(319\) 4.47214 0.250392
\(320\) 0 0
\(321\) 3.05573 0.170554
\(322\) −4.00000 −0.222911
\(323\) 17.8885 0.995345
\(324\) −2.41641 −0.134245
\(325\) 0 0
\(326\) −19.4164 −1.07538
\(327\) 12.3607 0.683547
\(328\) 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
\(333\) 10.2229 0.560212
\(334\) 11.4164 0.624678
\(335\) 0 0
\(336\) 1.23607 0.0674330
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 2.52786 0.137498
\(339\) −0.583592 −0.0316964
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) −10.6525 −0.576020
\(343\) −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 \(0.521137\pi\)
\(348\) −5.52786 −0.296325
\(349\) 21.7082 1.16201 0.581007 0.813899i \(-0.302659\pi\)
0.581007 + 0.813899i \(0.302659\pi\)
\(350\) 0 0
\(351\) 17.8885 0.954820
\(352\) −1.00000 −0.0533002
\(353\) 17.0557 0.907785 0.453892 0.891056i \(-0.350035\pi\)
0.453892 + 0.891056i \(0.350035\pi\)
\(354\) 3.41641 0.181580
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −3.05573 −0.161726
\(358\) 8.94427 0.472719
\(359\) 26.8328 1.41618 0.708091 0.706121i \(-0.249557\pi\)
0.708091 + 0.706121i \(0.249557\pi\)
\(360\) 0 0
\(361\) 33.3607 1.75583
\(362\) −9.23607 −0.485437
\(363\) −1.23607 −0.0648767
\(364\) −3.23607 −0.169616
\(365\) 0 0
\(366\) −0.944272 −0.0493579
\(367\) 5.41641 0.282734 0.141367 0.989957i \(-0.454850\pi\)
0.141367 + 0.989957i \(0.454850\pi\)
\(368\) −4.00000 −0.208514
\(369\) 3.63932 0.189455
\(370\) 0 0
\(371\) 8.47214 0.439851
\(372\) −2.47214 −0.128174
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 2.47214 0.127831
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) 14.4721 0.745353
\(378\) 5.52786 0.284323
\(379\) 14.4721 0.743384 0.371692 0.928356i \(-0.378778\pi\)
0.371692 + 0.928356i \(0.378778\pi\)
\(380\) 0 0
\(381\) −14.8328 −0.759908
\(382\) 2.47214 0.126485
\(383\) 23.8885 1.22065 0.610324 0.792152i \(-0.291039\pi\)
0.610324 + 0.792152i \(0.291039\pi\)
\(384\) 1.23607 0.0630778
\(385\) 0 0
\(386\) −14.9443 −0.760643
\(387\) −15.4164 −0.783660
\(388\) −12.4721 −0.633177
\(389\) 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
\(392\) −1.00000 −0.0505076
\(393\) −5.88854 −0.297038
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −1.47214 −0.0739776
\(397\) 23.7082 1.18988 0.594940 0.803770i \(-0.297176\pi\)
0.594940 + 0.803770i \(0.297176\pi\)
\(398\) −18.9443 −0.949591
\(399\) −8.94427 −0.447774
\(400\) 0 0
\(401\) 14.3607 0.717138 0.358569 0.933503i \(-0.383265\pi\)
0.358569 + 0.933503i \(0.383265\pi\)
\(402\) −14.1115 −0.703815
\(403\) 6.47214 0.322400
\(404\) 8.18034 0.406987
\(405\) 0 0
\(406\) 4.47214 0.221948
\(407\) −6.94427 −0.344215
\(408\) −3.05573 −0.151281
\(409\) 3.41641 0.168930 0.0844652 0.996426i \(-0.473082\pi\)
0.0844652 + 0.996426i \(0.473082\pi\)
\(410\) 0 0
\(411\) −24.5836 −1.21262
\(412\) 14.9443 0.736251
\(413\) −2.76393 −0.136004
\(414\) −5.88854 −0.289406
\(415\) 0 0
\(416\) −3.23607 −0.158661
\(417\) 26.8328 1.31401
\(418\) 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
\(423\) −2.94427 −0.143155
\(424\) 8.47214 0.411443
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 0.763932 0.0369693
\(428\) −2.47214 −0.119495
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 23.0557 1.11056 0.555278 0.831665i \(-0.312612\pi\)
0.555278 + 0.831665i \(0.312612\pi\)
\(432\) 5.52786 0.265959
\(433\) −28.4721 −1.36828 −0.684142 0.729349i \(-0.739823\pi\)
−0.684142 + 0.729349i \(0.739823\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 28.9443 1.38459
\(438\) −16.0000 −0.764510
\(439\) 8.94427 0.426887 0.213443 0.976955i \(-0.431532\pi\)
0.213443 + 0.976955i \(0.431532\pi\)
\(440\) 0 0
\(441\) −1.47214 −0.0701017
\(442\) 8.00000 0.380521
\(443\) 24.9443 1.18514 0.592569 0.805520i \(-0.298114\pi\)
0.592569 + 0.805520i \(0.298114\pi\)
\(444\) 8.58359 0.407359
\(445\) 0 0
\(446\) −0.472136 −0.0223563
\(447\) 27.6393 1.30729
\(448\) −1.00000 −0.0472456
\(449\) 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
\(452\) 0.472136 0.0222074
\(453\) −14.8328 −0.696906
\(454\) −19.2361 −0.902793
\(455\) 0 0
\(456\) −8.94427 −0.418854
\(457\) −26.9443 −1.26040 −0.630200 0.776433i \(-0.717027\pi\)
−0.630200 + 0.776433i \(0.717027\pi\)
\(458\) −17.2361 −0.805389
\(459\) −13.6656 −0.637857
\(460\) 0 0
\(461\) 24.7639 1.15337 0.576686 0.816966i \(-0.304346\pi\)
0.576686 + 0.816966i \(0.304346\pi\)
\(462\) −1.23607 −0.0575071
\(463\) 30.4721 1.41616 0.708080 0.706132i \(-0.249562\pi\)
0.708080 + 0.706132i \(0.249562\pi\)
\(464\) 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 \(0.284042\pi\)
\(468\) −4.76393 −0.220213
\(469\) 11.4164 0.527161
\(470\) 0 0
\(471\) −15.6393 −0.720622
\(472\) −2.76393 −0.127220
\(473\) 10.4721 0.481509
\(474\) 0 0
\(475\) 0 0
\(476\) 2.47214 0.113310
\(477\) 12.4721 0.571060
\(478\) −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
\(483\) −4.94427 −0.224972
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 13.5967 0.616761
\(487\) −16.9443 −0.767818 −0.383909 0.923371i \(-0.625422\pi\)
−0.383909 + 0.923371i \(0.625422\pi\)
\(488\) 0.763932 0.0345816
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) −16.9443 −0.764684 −0.382342 0.924021i \(-0.624882\pi\)
−0.382342 + 0.924021i \(0.624882\pi\)
\(492\) 3.05573 0.137763
\(493\) −11.0557 −0.497925
\(494\) 23.4164 1.05355
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) −6.47214 −0.290315
\(498\) 15.0557 0.674663
\(499\) −32.3607 −1.44866 −0.724331 0.689452i \(-0.757851\pi\)
−0.724331 + 0.689452i \(0.757851\pi\)
\(500\) 0 0
\(501\) 14.1115 0.630453
\(502\) −29.2361 −1.30487
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) −1.47214 −0.0655741
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) 3.12461 0.138769
\(508\) 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
\(511\) 12.9443 0.572621
\(512\) −1.00000 −0.0441942
\(513\) −40.0000 −1.76604
\(514\) 6.94427 0.306299
\(515\) 0 0
\(516\) −12.9443 −0.569840
\(517\) 2.00000 0.0879599
\(518\) −6.94427 −0.305114
\(519\) −4.00000 −0.175581
\(520\) 0 0
\(521\) −10.3607 −0.453910 −0.226955 0.973905i \(-0.572877\pi\)
−0.226955 + 0.973905i \(0.572877\pi\)
\(522\) 6.58359 0.288156
\(523\) 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
\(528\) −1.23607 −0.0537930
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.06888 −0.176575
\(532\) 7.23607 0.313723
\(533\) −8.00000 −0.346518
\(534\) 12.3607 0.534899
\(535\) 0 0
\(536\) 11.4164 0.493114
\(537\) 11.0557 0.477090
\(538\) 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
\(543\) −11.4164 −0.489925
\(544\) 2.47214 0.105992
\(545\) 0 0
\(546\) −4.00000 −0.171184
\(547\) 0.944272 0.0403742 0.0201871 0.999796i \(-0.493574\pi\)
0.0201871 + 0.999796i \(0.493574\pi\)
\(548\) 19.8885 0.849596
\(549\) 1.12461 0.0479973
\(550\) 0 0
\(551\) −32.3607 −1.37861
\(552\) −4.94427 −0.210442
\(553\) 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 \(0.676346\pi\)
\(558\) 2.94427 0.124641
\(559\) 33.8885 1.43333
\(560\) 0 0
\(561\) 3.05573 0.129013
\(562\) −28.8328 −1.21624
\(563\) −31.2361 −1.31644 −0.658222 0.752824i \(-0.728691\pi\)
−0.658222 + 0.752824i \(0.728691\pi\)
\(564\) −2.47214 −0.104096
\(565\) 0 0
\(566\) 14.6525 0.615889
\(567\) 2.41641 0.101480
\(568\) −6.47214 −0.271565
\(569\) −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 \(-0.567859\pi\)
−0.211576 + 0.977362i \(0.567859\pi\)
\(572\) 3.23607 0.135307
\(573\) 3.05573 0.127655
\(574\) −2.47214 −0.103185
\(575\) 0 0
\(576\) −1.47214 −0.0613390
\(577\) −26.9443 −1.12170 −0.560852 0.827916i \(-0.689526\pi\)
−0.560852 + 0.827916i \(0.689526\pi\)
\(578\) 10.8885 0.452904
\(579\) −18.4721 −0.767676
\(580\) 0 0
\(581\) −12.1803 −0.505326
\(582\) −15.4164 −0.639031
\(583\) −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 \(0.461678\pi\)
\(588\) −1.23607 −0.0509746
\(589\) −14.4721 −0.596314
\(590\) 0 0
\(591\) 22.2492 0.915211
\(592\) −6.94427 −0.285408
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) −5.52786 −0.226811
\(595\) 0 0
\(596\) −22.3607 −0.915929
\(597\) −23.4164 −0.958370
\(598\) 12.9443 0.529331
\(599\) −32.3607 −1.32222 −0.661111 0.750288i \(-0.729915\pi\)
−0.661111 + 0.750288i \(0.729915\pi\)
\(600\) 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
\(603\) 16.8065 0.684414
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) 10.1115 0.410750
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 7.23607 0.293461
\(609\) 5.52786 0.224000
\(610\) 0 0
\(611\) 6.47214 0.261835
\(612\) 3.63932 0.147111
\(613\) −28.4721 −1.14998 −0.574989 0.818161i \(-0.694994\pi\)
−0.574989 + 0.818161i \(0.694994\pi\)
\(614\) −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 \(0.641873\pi\)
\(618\) 18.4721 0.743058
\(619\) 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
\(622\) 21.4164 0.858720
\(623\) −10.0000 −0.400642
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 19.5279 0.780490
\(627\) 8.94427 0.357200
\(628\) 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
\(633\) 16.7214 0.664614
\(634\) −30.9443 −1.22895
\(635\) 0 0
\(636\) 10.4721 0.415247
\(637\) 3.23607 0.128218
\(638\) −4.47214 −0.177054
\(639\) −9.52786 −0.376916
\(640\) 0 0
\(641\) 27.5279 1.08729 0.543643 0.839317i \(-0.317045\pi\)
0.543643 + 0.839317i \(0.317045\pi\)
\(642\) −3.05573 −0.120600
\(643\) 18.7639 0.739977 0.369989 0.929036i \(-0.379362\pi\)
0.369989 + 0.929036i \(0.379362\pi\)
\(644\) 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 \(0.308194\pi\)
\(648\) 2.41641 0.0949255
\(649\) 2.76393 0.108494
\(650\) 0 0
\(651\) 2.47214 0.0968906
\(652\) 19.4164 0.760405
\(653\) −46.3607 −1.81423 −0.907117 0.420879i \(-0.861722\pi\)
−0.907117 + 0.420879i \(0.861722\pi\)
\(654\) −12.3607 −0.483341
\(655\) 0 0
\(656\) −2.47214 −0.0965207
\(657\) 19.0557 0.743435
\(658\) 2.00000 0.0779681
\(659\) 16.5836 0.646005 0.323003 0.946398i \(-0.395308\pi\)
0.323003 + 0.946398i \(0.395308\pi\)
\(660\) 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
\(663\) 9.88854 0.384039
\(664\) −12.1803 −0.472689
\(665\) 0 0
\(666\) −10.2229 −0.396130
\(667\) −17.8885 −0.692647
\(668\) −11.4164 −0.441714
\(669\) −0.583592 −0.0225630
\(670\) 0 0
\(671\) −0.763932 −0.0294913
\(672\) −1.23607 −0.0476824
\(673\) 3.88854 0.149892 0.0749462 0.997188i \(-0.476122\pi\)
0.0749462 + 0.997188i \(0.476122\pi\)
\(674\) 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 \(0.332982\pi\)
\(678\) 0.583592 0.0224127
\(679\) 12.4721 0.478637
\(680\) 0 0
\(681\) −23.7771 −0.911140
\(682\) −2.00000 −0.0765840
\(683\) −32.9443 −1.26058 −0.630289 0.776361i \(-0.717064\pi\)
−0.630289 + 0.776361i \(0.717064\pi\)
\(684\) 10.6525 0.407308
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) −21.3050 −0.812835
\(688\) 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
\(693\) 1.47214 0.0559218
\(694\) 2.47214 0.0938410
\(695\) 0 0
\(696\) 5.52786 0.209533
\(697\) 6.11146 0.231488
\(698\) −21.7082 −0.821668
\(699\) −18.4721 −0.698680
\(700\) 0 0
\(701\) −42.7214 −1.61356 −0.806782 0.590850i \(-0.798793\pi\)
−0.806782 + 0.590850i \(0.798793\pi\)
\(702\) −17.8885 −0.675160
\(703\) 50.2492 1.89519
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −17.0557 −0.641901
\(707\) −8.18034 −0.307653
\(708\) −3.41641 −0.128396
\(709\) 4.47214 0.167955 0.0839773 0.996468i \(-0.473238\pi\)
0.0839773 + 0.996468i \(0.473238\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) −8.00000 −0.299602
\(714\) 3.05573 0.114358
\(715\) 0 0
\(716\) −8.94427 −0.334263
\(717\) −24.7214 −0.923236
\(718\) −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
\(721\) −14.9443 −0.556554
\(722\) −33.3607 −1.24156
\(723\) −19.0557 −0.708690
\(724\) 9.23607 0.343256
\(725\) 0 0
\(726\) 1.23607 0.0458748
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 3.23607 0.119937
\(729\) 24.0557 0.890953
\(730\) 0 0
\(731\) −25.8885 −0.957522
\(732\) 0.944272 0.0349013
\(733\) −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
\(738\) −3.63932 −0.133965
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 28.9443 1.06329
\(742\) −8.47214 −0.311022
\(743\) −21.8885 −0.803013 −0.401506 0.915856i \(-0.631513\pi\)
−0.401506 + 0.915856i \(0.631513\pi\)
\(744\) 2.47214 0.0906329
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) −17.9311 −0.656065
\(748\) −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
\(753\) −36.1378 −1.31693
\(754\) −14.4721 −0.527044
\(755\) 0 0
\(756\) −5.52786 −0.201046
\(757\) 23.3050 0.847033 0.423516 0.905888i \(-0.360796\pi\)
0.423516 + 0.905888i \(0.360796\pi\)
\(758\) −14.4721 −0.525652
\(759\) 4.94427 0.179466
\(760\) 0 0
\(761\) −11.4164 −0.413844 −0.206922 0.978357i \(-0.566345\pi\)
−0.206922 + 0.978357i \(0.566345\pi\)
\(762\) 14.8328 0.537336
\(763\) 10.0000 0.362024
\(764\) −2.47214 −0.0894387
\(765\) 0 0
\(766\) −23.8885 −0.863128
\(767\) 8.94427 0.322959
\(768\) −1.23607 −0.0446028
\(769\) −43.4164 −1.56564 −0.782818 0.622251i \(-0.786218\pi\)
−0.782818 + 0.622251i \(0.786218\pi\)
\(770\) 0 0
\(771\) 8.58359 0.309131
\(772\) 14.9443 0.537856
\(773\) −15.7082 −0.564985 −0.282492 0.959270i \(-0.591161\pi\)
−0.282492 + 0.959270i \(0.591161\pi\)
\(774\) 15.4164 0.554131
\(775\) 0 0
\(776\) 12.4721 0.447724
\(777\) −8.58359 −0.307935
\(778\) −33.4164 −1.19804
\(779\) 17.8885 0.640924
\(780\) 0 0
\(781\) 6.47214 0.231591
\(782\) −9.88854 −0.353614
\(783\) 24.7214 0.883469
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 5.88854 0.210037
\(787\) 28.1803 1.00452 0.502260 0.864716i \(-0.332502\pi\)
0.502260 + 0.864716i \(0.332502\pi\)
\(788\) −18.0000 −0.641223
\(789\) −6.11146 −0.217574
\(790\) 0 0
\(791\) −0.472136 −0.0167872
\(792\) 1.47214 0.0523101
\(793\) −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 \(0.236375\pi\)
\(798\) 8.94427 0.316624
\(799\) −4.94427 −0.174916
\(800\) 0 0
\(801\) −14.7214 −0.520154
\(802\) −14.3607 −0.507093
\(803\) −12.9443 −0.456793
\(804\) 14.1115 0.497673
\(805\) 0 0
\(806\) −6.47214 −0.227971
\(807\) 28.1378 0.990496
\(808\) −8.18034 −0.287783
\(809\) −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 \(0.473345\pi\)
\(812\) −4.47214 −0.156941
\(813\) −1.16718 −0.0409349
\(814\) 6.94427 0.243397
\(815\) 0 0
\(816\) 3.05573 0.106972
\(817\) −75.7771 −2.65110
\(818\) −3.41641 −0.119452
\(819\) 4.76393 0.166465
\(820\) 0 0
\(821\) −1.41641 −0.0494330 −0.0247165 0.999695i \(-0.507868\pi\)
−0.0247165 + 0.999695i \(0.507868\pi\)
\(822\) 24.5836 0.857451
\(823\) 46.2492 1.61215 0.806073 0.591816i \(-0.201589\pi\)
0.806073 + 0.591816i \(0.201589\pi\)
\(824\) −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 \(0.595188\pi\)
\(828\) 5.88854 0.204641
\(829\) 11.7082 0.406643 0.203321 0.979112i \(-0.434826\pi\)
0.203321 + 0.979112i \(0.434826\pi\)
\(830\) 0 0
\(831\) 4.36068 0.151270
\(832\) 3.23607 0.112190
\(833\) −2.47214 −0.0856544
\(834\) −26.8328 −0.929144
\(835\) 0 0
\(836\) −7.23607 −0.250265
\(837\) 11.0557 0.382142
\(838\) −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
\(843\) −35.6393 −1.22748
\(844\) −13.5279 −0.465648
\(845\) 0 0
\(846\) 2.94427 0.101226
\(847\) −1.00000 −0.0343604
\(848\) −8.47214 −0.290934
\(849\) 18.1115 0.621584
\(850\) 0 0
\(851\) 27.7771 0.952186
\(852\) −8.00000 −0.274075
\(853\) −32.5410 −1.11418 −0.557092 0.830451i \(-0.688083\pi\)
−0.557092 + 0.830451i \(0.688083\pi\)
\(854\) −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 \(0.208137\pi\)
\(858\) 4.00000 0.136558
\(859\) 15.1246 0.516045 0.258023 0.966139i \(-0.416929\pi\)
0.258023 + 0.966139i \(0.416929\pi\)
\(860\) 0 0
\(861\) −3.05573 −0.104139
\(862\) −23.0557 −0.785281
\(863\) −0.583592 −0.0198657 −0.00993285 0.999951i \(-0.503162\pi\)
−0.00993285 + 0.999951i \(0.503162\pi\)
\(864\) −5.52786 −0.188062
\(865\) 0 0
\(866\) 28.4721 0.967523
\(867\) 13.4590 0.457091
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) −36.9443 −1.25181
\(872\) 10.0000 0.338643
\(873\) 18.3607 0.621415
\(874\) −28.9443 −0.979055
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) −9.05573 −0.305790 −0.152895 0.988242i \(-0.548860\pi\)
−0.152895 + 0.988242i \(0.548860\pi\)
\(878\) −8.94427 −0.301855
\(879\) −32.9443 −1.11118
\(880\) 0 0
\(881\) 28.8328 0.971402 0.485701 0.874125i \(-0.338564\pi\)
0.485701 + 0.874125i \(0.338564\pi\)
\(882\) 1.47214 0.0495694
\(883\) 2.83282 0.0953318 0.0476659 0.998863i \(-0.484822\pi\)
0.0476659 + 0.998863i \(0.484822\pi\)
\(884\) −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 \(0.232572\pi\)
\(888\) −8.58359 −0.288046
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) −2.41641 −0.0809527
\(892\) 0.472136 0.0158083
\(893\) −14.4721 −0.484292
\(894\) −27.6393 −0.924397
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 16.0000 0.534224
\(898\) −18.9443 −0.632179
\(899\) 8.94427 0.298308
\(900\) 0 0
\(901\) 20.9443 0.697755
\(902\) 2.47214 0.0823131
\(903\) 12.9443 0.430758
\(904\) −0.472136 −0.0157030
\(905\) 0 0
\(906\) 14.8328 0.492787
\(907\) 24.3607 0.808883 0.404442 0.914564i \(-0.367466\pi\)
0.404442 + 0.914564i \(0.367466\pi\)
\(908\) 19.2361 0.638371
\(909\) −12.0426 −0.399427
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 8.94427 0.296174
\(913\) 12.1803 0.403110
\(914\) 26.9443 0.891237
\(915\) 0 0
\(916\) 17.2361 0.569496
\(917\) −4.76393 −0.157319
\(918\) 13.6656 0.451033
\(919\) −22.1115 −0.729390 −0.364695 0.931127i \(-0.618827\pi\)
−0.364695 + 0.931127i \(0.618827\pi\)
\(920\) 0 0
\(921\) −32.2229 −1.06178
\(922\) −24.7639 −0.815557
\(923\) 20.9443 0.689389
\(924\) 1.23607 0.0406637
\(925\) 0 0
\(926\) −30.4721 −1.00138
\(927\) −22.0000 −0.722575
\(928\) −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
\(931\) −7.23607 −0.237153
\(932\) 14.9443 0.489516
\(933\) 26.4721 0.866659
\(934\) −27.1246 −0.887544
\(935\) 0 0
\(936\) 4.76393 0.155714
\(937\) 3.05573 0.0998263 0.0499131 0.998754i \(-0.484106\pi\)
0.0499131 + 0.998754i \(0.484106\pi\)
\(938\) −11.4164 −0.372759
\(939\) 24.1378 0.787706
\(940\) 0 0
\(941\) −11.8197 −0.385310 −0.192655 0.981267i \(-0.561710\pi\)
−0.192655 + 0.981267i \(0.561710\pi\)
\(942\) 15.6393 0.509557
\(943\) 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
\(950\) 0 0
\(951\) −38.2492 −1.24032
\(952\) −2.47214 −0.0801224
\(953\) −22.9443 −0.743238 −0.371619 0.928385i \(-0.621197\pi\)
−0.371619 + 0.928385i \(0.621197\pi\)
\(954\) −12.4721 −0.403800
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) −5.52786 −0.178690
\(958\) −12.3607 −0.399355
\(959\) −19.8885 −0.642235
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 22.4721 0.724531
\(963\) 3.63932 0.117275
\(964\) 15.4164 0.496529
\(965\) 0 0
\(966\) 4.94427 0.159079
\(967\) −45.8885 −1.47568 −0.737838 0.674978i \(-0.764153\pi\)
−0.737838 + 0.674978i \(0.764153\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −22.1115 −0.710322
\(970\) 0 0
\(971\) 50.5410 1.62194 0.810969 0.585089i \(-0.198940\pi\)
0.810969 + 0.585089i \(0.198940\pi\)
\(972\) −13.5967 −0.436116
\(973\) 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 \(0.347411\pi\)
\(978\) 24.0000 0.767435
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) 14.7214 0.470017
\(982\) 16.9443 0.540713
\(983\) −14.0000 −0.446531 −0.223265 0.974758i \(-0.571672\pi\)
−0.223265 + 0.974758i \(0.571672\pi\)
\(984\) −3.05573 −0.0974131
\(985\) 0 0
\(986\) 11.0557 0.352086
\(987\) 2.47214 0.0786890
\(988\) −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
\(993\) 20.9443 0.664646
\(994\) 6.47214 0.205284
\(995\) 0 0
\(996\) −15.0557 −0.477059
\(997\) −24.1803 −0.765799 −0.382900 0.923790i \(-0.625074\pi\)
−0.382900 + 0.923790i \(0.625074\pi\)
\(998\) 32.3607 1.02436
\(999\) −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 3850.2.a.bj.1.1 2
5.2 odd 4 3850.2.c.q.1849.2 4
5.3 odd 4 3850.2.c.q.1849.3 4
5.4 even 2 154.2.a.d.1.2 2
15.14 odd 2 1386.2.a.m.1.2 2
20.19 odd 2 1232.2.a.p.1.1 2
35.4 even 6 1078.2.e.q.177.1 4
35.9 even 6 1078.2.e.q.67.1 4
35.19 odd 6 1078.2.e.n.67.2 4
35.24 odd 6 1078.2.e.n.177.2 4
35.34 odd 2 1078.2.a.w.1.1 2
40.19 odd 2 4928.2.a.bk.1.2 2
40.29 even 2 4928.2.a.bt.1.1 2
55.54 odd 2 1694.2.a.l.1.2 2
105.104 even 2 9702.2.a.cu.1.1 2
140.139 even 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 5.4 even 2
1078.2.a.w.1.1 2 35.34 odd 2
1078.2.e.n.67.2 4 35.19 odd 6
1078.2.e.n.177.2 4 35.24 odd 6
1078.2.e.q.67.1 4 35.9 even 6
1078.2.e.q.177.1 4 35.4 even 6
1232.2.a.p.1.1 2 20.19 odd 2
1386.2.a.m.1.2 2 15.14 odd 2
1694.2.a.l.1.2 2 55.54 odd 2
3850.2.a.bj.1.1 2 1.1 even 1 trivial
3850.2.c.q.1849.2 4 5.2 odd 4
3850.2.c.q.1849.3 4 5.3 odd 4
4928.2.a.bk.1.2 2 40.19 odd 2
4928.2.a.bt.1.1 2 40.29 even 2
8624.2.a.bf.1.2 2 140.139 even 2
9702.2.a.cu.1.1 2 105.104 even 2