Properties

Label 1694.2.a.l.1.2
Level $1694$
Weight $2$
Character 1694.1
Self dual yes
Analytic conductor $13.527$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1694.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.5266581024\)
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 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1694.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} -1.23607 q^{5} -1.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.47214 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.23607 q^{3} +1.00000 q^{4} -1.23607 q^{5} -1.23607 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.47214 q^{9} +1.23607 q^{10} +1.23607 q^{12} +3.23607 q^{13} +1.00000 q^{14} -1.52786 q^{15} +1.00000 q^{16} -2.47214 q^{17} +1.47214 q^{18} +7.23607 q^{19} -1.23607 q^{20} -1.23607 q^{21} +4.00000 q^{23} -1.23607 q^{24} -3.47214 q^{25} -3.23607 q^{26} -5.52786 q^{27} -1.00000 q^{28} -4.47214 q^{29} +1.52786 q^{30} +2.00000 q^{31} -1.00000 q^{32} +2.47214 q^{34} +1.23607 q^{35} -1.47214 q^{36} +6.94427 q^{37} -7.23607 q^{38} +4.00000 q^{39} +1.23607 q^{40} +2.47214 q^{41} +1.23607 q^{42} +10.4721 q^{43} +1.81966 q^{45} -4.00000 q^{46} -2.00000 q^{47} +1.23607 q^{48} +1.00000 q^{49} +3.47214 q^{50} -3.05573 q^{51} +3.23607 q^{52} +8.47214 q^{53} +5.52786 q^{54} +1.00000 q^{56} +8.94427 q^{57} +4.47214 q^{58} +2.76393 q^{59} -1.52786 q^{60} +0.763932 q^{61} -2.00000 q^{62} +1.47214 q^{63} +1.00000 q^{64} -4.00000 q^{65} +11.4164 q^{67} -2.47214 q^{68} +4.94427 q^{69} -1.23607 q^{70} +6.47214 q^{71} +1.47214 q^{72} -12.9443 q^{73} -6.94427 q^{74} -4.29180 q^{75} +7.23607 q^{76} -4.00000 q^{78} -1.23607 q^{80} -2.41641 q^{81} -2.47214 q^{82} +12.1803 q^{83} -1.23607 q^{84} +3.05573 q^{85} -10.4721 q^{86} -5.52786 q^{87} +10.0000 q^{89} -1.81966 q^{90} -3.23607 q^{91} +4.00000 q^{92} +2.47214 q^{93} +2.00000 q^{94} -8.94427 q^{95} -1.23607 q^{96} +12.4721 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} + 6 q^{9} - 2 q^{10} - 2 q^{12} + 2 q^{13} + 2 q^{14} - 12 q^{15} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 10 q^{19} + 2 q^{20} + 2 q^{21} + 8 q^{23} + 2 q^{24} + 2 q^{25} - 2 q^{26} - 20 q^{27} - 2 q^{28} + 12 q^{30} + 4 q^{31} - 2 q^{32} - 4 q^{34} - 2 q^{35} + 6 q^{36} - 4 q^{37} - 10 q^{38} + 8 q^{39} - 2 q^{40} - 4 q^{41} - 2 q^{42} + 12 q^{43} + 26 q^{45} - 8 q^{46} - 4 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} - 24 q^{51} + 2 q^{52} + 8 q^{53} + 20 q^{54} + 2 q^{56} + 10 q^{59} - 12 q^{60} + 6 q^{61} - 4 q^{62} - 6 q^{63} + 2 q^{64} - 8 q^{65} - 4 q^{67} + 4 q^{68} - 8 q^{69} + 2 q^{70} + 4 q^{71} - 6 q^{72} - 8 q^{73} + 4 q^{74} - 22 q^{75} + 10 q^{76} - 8 q^{78} + 2 q^{80} + 22 q^{81} + 4 q^{82} + 2 q^{83} + 2 q^{84} + 24 q^{85} - 12 q^{86} - 20 q^{87} + 20 q^{89} - 26 q^{90} - 2 q^{91} + 8 q^{92} - 4 q^{93} + 4 q^{94} + 2 q^{96} + 16 q^{97} - 2 q^{98}+O(q^{100}) \) 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
\(3\) 1.23607 0.713644 0.356822 0.934172i \(-0.383860\pi\)
0.356822 + 0.934172i \(0.383860\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) −1.23607 −0.504623
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −1.47214 −0.490712
\(10\) 1.23607 0.390879
\(11\) 0 0
\(12\) 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
\(15\) −1.52786 −0.394493
\(16\) 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 \(-0.188321\pi\)
0.830034 + 0.557713i \(0.188321\pi\)
\(20\) −1.23607 −0.276393
\(21\) −1.23607 −0.269732
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.23607 −0.252311
\(25\) −3.47214 −0.694427
\(26\) −3.23607 −0.634645
\(27\) −5.52786 −1.06384
\(28\) −1.00000 −0.188982
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 1.52786 0.278949
\(31\) 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
\(36\) −1.47214 −0.245356
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) −7.23607 −1.17385
\(39\) 4.00000 0.640513
\(40\) 1.23607 0.195440
\(41\) 2.47214 0.386083 0.193041 0.981191i \(-0.438165\pi\)
0.193041 + 0.981191i \(0.438165\pi\)
\(42\) 1.23607 0.190729
\(43\) 10.4721 1.59699 0.798493 0.602004i \(-0.205631\pi\)
0.798493 + 0.602004i \(0.205631\pi\)
\(44\) 0 0
\(45\) 1.81966 0.271259
\(46\) −4.00000 −0.589768
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 1.23607 0.178411
\(49\) 1.00000 0.142857
\(50\) 3.47214 0.491034
\(51\) −3.05573 −0.427888
\(52\) 3.23607 0.448762
\(53\) 8.47214 1.16374 0.581869 0.813283i \(-0.302322\pi\)
0.581869 + 0.813283i \(0.302322\pi\)
\(54\) 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 \(0.442417\pi\)
\(60\) −1.52786 −0.197246
\(61\) 0.763932 0.0978115 0.0489057 0.998803i \(-0.484427\pi\)
0.0489057 + 0.998803i \(0.484427\pi\)
\(62\) −2.00000 −0.254000
\(63\) 1.47214 0.185472
\(64\) 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
\(69\) 4.94427 0.595220
\(70\) −1.23607 −0.147738
\(71\) 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
\(75\) −4.29180 −0.495574
\(76\) 7.23607 0.830034
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −1.23607 −0.138197
\(81\) −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
\(85\) 3.05573 0.331440
\(86\) −10.4721 −1.12924
\(87\) −5.52786 −0.592649
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −1.81966 −0.191809
\(91\) −3.23607 −0.339232
\(92\) 4.00000 0.417029
\(93\) 2.47214 0.256349
\(94\) 2.00000 0.206284
\(95\) −8.94427 −0.917663
\(96\) −1.23607 −0.126156
\(97\) 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 \(0.633421\pi\)
\(102\) 3.05573 0.302562
\(103\) −14.9443 −1.47250 −0.736251 0.676708i \(-0.763406\pi\)
−0.736251 + 0.676708i \(0.763406\pi\)
\(104\) −3.23607 −0.317323
\(105\) 1.52786 0.149104
\(106\) −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 \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 8.58359 0.814719
\(112\) −1.00000 −0.0944911
\(113\) −0.472136 −0.0444148 −0.0222074 0.999753i \(-0.507069\pi\)
−0.0222074 + 0.999753i \(0.507069\pi\)
\(114\) −8.94427 −0.837708
\(115\) −4.94427 −0.461056
\(116\) −4.47214 −0.415227
\(117\) −4.76393 −0.440426
\(118\) −2.76393 −0.254441
\(119\) 2.47214 0.226620
\(120\) 1.52786 0.139474
\(121\) 0 0
\(122\) −0.763932 −0.0691632
\(123\) 3.05573 0.275526
\(124\) 2.00000 0.179605
\(125\) 10.4721 0.936656
\(126\) −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
\(130\) 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
\(135\) 6.83282 0.588075
\(136\) 2.47214 0.211984
\(137\) −19.8885 −1.69919 −0.849596 0.527433i \(-0.823154\pi\)
−0.849596 + 0.527433i \(0.823154\pi\)
\(138\) −4.94427 −0.420884
\(139\) 21.7082 1.84127 0.920633 0.390429i \(-0.127673\pi\)
0.920633 + 0.390429i \(0.127673\pi\)
\(140\) 1.23607 0.104467
\(141\) −2.47214 −0.208191
\(142\) −6.47214 −0.543130
\(143\) 0 0
\(144\) −1.47214 −0.122678
\(145\) 5.52786 0.459064
\(146\) 12.9443 1.07128
\(147\) 1.23607 0.101949
\(148\) 6.94427 0.570816
\(149\) 22.3607 1.83186 0.915929 0.401340i \(-0.131455\pi\)
0.915929 + 0.401340i \(0.131455\pi\)
\(150\) 4.29180 0.350424
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −7.23607 −0.586923
\(153\) 3.63932 0.294222
\(154\) 0 0
\(155\) −2.47214 −0.198567
\(156\) 4.00000 0.320256
\(157\) −12.6525 −1.00978 −0.504889 0.863184i \(-0.668466\pi\)
−0.504889 + 0.863184i \(0.668466\pi\)
\(158\) 0 0
\(159\) 10.4721 0.830494
\(160\) 1.23607 0.0977198
\(161\) −4.00000 −0.315244
\(162\) 2.41641 0.189851
\(163\) −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 \(0.645629\pi\)
\(168\) 1.23607 0.0953647
\(169\) −2.52786 −0.194451
\(170\) −3.05573 −0.234364
\(171\) −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
\(175\) 3.47214 0.262469
\(176\) 0 0
\(177\) 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 \(0.608487\pi\)
\(180\) 1.81966 0.135629
\(181\) 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
\(185\) −8.58359 −0.631078
\(186\) −2.47214 −0.181266
\(187\) 0 0
\(188\) −2.00000 −0.145865
\(189\) 5.52786 0.402093
\(190\) 8.94427 0.648886
\(191\) −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
\(195\) −4.94427 −0.354067
\(196\) 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
\(201\) 14.1115 0.995345
\(202\) 8.18034 0.575567
\(203\) 4.47214 0.313882
\(204\) −3.05573 −0.213944
\(205\) −3.05573 −0.213421
\(206\) 14.9443 1.04122
\(207\) −5.88854 −0.409282
\(208\) 3.23607 0.224381
\(209\) 0 0
\(210\) −1.52786 −0.105433
\(211\) 13.5279 0.931297 0.465648 0.884970i \(-0.345821\pi\)
0.465648 + 0.884970i \(0.345821\pi\)
\(212\) 8.47214 0.581869
\(213\) 8.00000 0.548151
\(214\) 2.47214 0.168992
\(215\) −12.9443 −0.882792
\(216\) 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 \(-0.505032\pi\)
−0.0158083 + 0.999875i \(0.505032\pi\)
\(224\) 1.00000 0.0668153
\(225\) 5.11146 0.340764
\(226\) 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 \(0.307139\pi\)
\(230\) 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 \(0.337174\pi\)
\(234\) 4.76393 0.311428
\(235\) 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 \(0.723912\pi\)
\(240\) −1.52786 −0.0986232
\(241\) −15.4164 −0.993058 −0.496529 0.868020i \(-0.665392\pi\)
−0.496529 + 0.868020i \(0.665392\pi\)
\(242\) 0 0
\(243\) 13.5967 0.872232
\(244\) 0.763932 0.0489057
\(245\) −1.23607 −0.0789695
\(246\) −3.05573 −0.194826
\(247\) 23.4164 1.48995
\(248\) −2.00000 −0.127000
\(249\) 15.0557 0.954118
\(250\) −10.4721 −0.662316
\(251\) 29.2361 1.84536 0.922682 0.385562i \(-0.125992\pi\)
0.922682 + 0.385562i \(0.125992\pi\)
\(252\) 1.47214 0.0927358
\(253\) 0 0
\(254\) −12.0000 −0.752947
\(255\) 3.77709 0.236530
\(256\) 1.00000 0.0625000
\(257\) 6.94427 0.433172 0.216586 0.976264i \(-0.430508\pi\)
0.216586 + 0.976264i \(0.430508\pi\)
\(258\) −12.9443 −0.805875
\(259\) −6.94427 −0.431496
\(260\) −4.00000 −0.248069
\(261\) 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 \(0.451287\pi\)
\(264\) 0 0
\(265\) −10.4721 −0.643298
\(266\) 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 \(0.744140\pi\)
\(270\) −6.83282 −0.415832
\(271\) −0.944272 −0.0573604 −0.0286802 0.999589i \(-0.509130\pi\)
−0.0286802 + 0.999589i \(0.509130\pi\)
\(272\) −2.47214 −0.149895
\(273\) −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
\(280\) −1.23607 −0.0738692
\(281\) −28.8328 −1.72002 −0.860011 0.510276i \(-0.829543\pi\)
−0.860011 + 0.510276i \(0.829543\pi\)
\(282\) 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
\(285\) −11.0557 −0.654885
\(286\) 0 0
\(287\) −2.47214 −0.145926
\(288\) 1.47214 0.0867464
\(289\) −10.8885 −0.640503
\(290\) −5.52786 −0.324607
\(291\) 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
\(295\) −3.41641 −0.198911
\(296\) −6.94427 −0.403628
\(297\) 0 0
\(298\) −22.3607 −1.29532
\(299\) 12.9443 0.748587
\(300\) −4.29180 −0.247787
\(301\) −10.4721 −0.603604
\(302\) 12.0000 0.690522
\(303\) −10.1115 −0.580888
\(304\) 7.23607 0.415017
\(305\) −0.944272 −0.0540689
\(306\) −3.63932 −0.208046
\(307\) 26.0689 1.48783 0.743915 0.668274i \(-0.232967\pi\)
0.743915 + 0.668274i \(0.232967\pi\)
\(308\) 0 0
\(309\) −18.4721 −1.05084
\(310\) 2.47214 0.140408
\(311\) −21.4164 −1.21441 −0.607207 0.794544i \(-0.707710\pi\)
−0.607207 + 0.794544i \(0.707710\pi\)
\(312\) −4.00000 −0.226455
\(313\) 19.5279 1.10378 0.551890 0.833917i \(-0.313907\pi\)
0.551890 + 0.833917i \(0.313907\pi\)
\(314\) 12.6525 0.714021
\(315\) −1.81966 −0.102526
\(316\) 0 0
\(317\) −30.9443 −1.73800 −0.869002 0.494809i \(-0.835238\pi\)
−0.869002 + 0.494809i \(0.835238\pi\)
\(318\) −10.4721 −0.587248
\(319\) 0 0
\(320\) −1.23607 −0.0690983
\(321\) −3.05573 −0.170554
\(322\) 4.00000 0.222911
\(323\) −17.8885 −0.995345
\(324\) −2.41641 −0.134245
\(325\) −11.2361 −0.623265
\(326\) 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
\(335\) −14.1115 −0.770991
\(336\) −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
\(340\) 3.05573 0.165720
\(341\) 0 0
\(342\) 10.6525 0.576020
\(343\) −1.00000 −0.0539949
\(344\) −10.4721 −0.564620
\(345\) −6.11146 −0.329030
\(346\) −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 \(-0.697341\pi\)
−0.581007 + 0.813899i \(0.697341\pi\)
\(350\) −3.47214 −0.185593
\(351\) −17.8885 −0.954820
\(352\) 0 0
\(353\) −17.0557 −0.907785 −0.453892 0.891056i \(-0.649965\pi\)
−0.453892 + 0.891056i \(0.649965\pi\)
\(354\) −3.41641 −0.181580
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) 3.05573 0.161726
\(358\) 8.94427 0.472719
\(359\) −26.8328 −1.41618 −0.708091 0.706121i \(-0.750443\pi\)
−0.708091 + 0.706121i \(0.750443\pi\)
\(360\) −1.81966 −0.0959045
\(361\) 33.3607 1.75583
\(362\) −9.23607 −0.485437
\(363\) 0 0
\(364\) −3.23607 −0.169616
\(365\) 16.0000 0.837478
\(366\) −0.944272 −0.0493579
\(367\) −5.41641 −0.282734 −0.141367 0.989957i \(-0.545150\pi\)
−0.141367 + 0.989957i \(0.545150\pi\)
\(368\) 4.00000 0.208514
\(369\) −3.63932 −0.189455
\(370\) 8.58359 0.446240
\(371\) −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 \(0.450355\pi\)
\(374\) 0 0
\(375\) 12.9443 0.668439
\(376\) 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 \(0.378778\pi\)
\(380\) −8.94427 −0.458831
\(381\) 14.8328 0.759908
\(382\) 2.47214 0.126485
\(383\) −23.8885 −1.22065 −0.610324 0.792152i \(-0.708961\pi\)
−0.610324 + 0.792152i \(0.708961\pi\)
\(384\) −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 \(0.178323\pi\)
\(390\) 4.94427 0.250363
\(391\) −9.88854 −0.500085
\(392\) −1.00000 −0.0505076
\(393\) −5.88854 −0.297038
\(394\) 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
\(399\) −8.94427 −0.447774
\(400\) −3.47214 −0.173607
\(401\) 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
\(405\) 2.98684 0.148417
\(406\) −4.47214 −0.221948
\(407\) 0 0
\(408\) 3.05573 0.151281
\(409\) −3.41641 −0.168930 −0.0844652 0.996426i \(-0.526918\pi\)
−0.0844652 + 0.996426i \(0.526918\pi\)
\(410\) 3.05573 0.150912
\(411\) −24.5836 −1.21262
\(412\) −14.9443 −0.736251
\(413\) −2.76393 −0.136004
\(414\) 5.88854 0.289406
\(415\) −15.0557 −0.739057
\(416\) −3.23607 −0.158661
\(417\) 26.8328 1.31401
\(418\) 0 0
\(419\) 17.2361 0.842037 0.421019 0.907052i \(-0.361673\pi\)
0.421019 + 0.907052i \(0.361673\pi\)
\(420\) 1.52786 0.0745521
\(421\) 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
\(425\) 8.58359 0.416365
\(426\) −8.00000 −0.387601
\(427\) −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 \(0.687388\pi\)
\(432\) −5.52786 −0.265959
\(433\) 28.4721 1.36828 0.684142 0.729349i \(-0.260177\pi\)
0.684142 + 0.729349i \(0.260177\pi\)
\(434\) 2.00000 0.0960031
\(435\) 6.83282 0.327608
\(436\) 10.0000 0.478913
\(437\) 28.9443 1.38459
\(438\) 16.0000 0.764510
\(439\) −8.94427 −0.426887 −0.213443 0.976955i \(-0.568468\pi\)
−0.213443 + 0.976955i \(0.568468\pi\)
\(440\) 0 0
\(441\) −1.47214 −0.0701017
\(442\) 8.00000 0.380521
\(443\) −24.9443 −1.18514 −0.592569 0.805520i \(-0.701886\pi\)
−0.592569 + 0.805520i \(0.701886\pi\)
\(444\) 8.58359 0.407359
\(445\) −12.3607 −0.585952
\(446\) 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 \(0.352486\pi\)
\(450\) −5.11146 −0.240956
\(451\) 0 0
\(452\) −0.472136 −0.0222074
\(453\) −14.8328 −0.696906
\(454\) −19.2361 −0.902793
\(455\) 4.00000 0.187523
\(456\) −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
\(460\) −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
\(465\) −3.05573 −0.141706
\(466\) −14.9443 −0.692280
\(467\) −27.1246 −1.25518 −0.627589 0.778545i \(-0.715958\pi\)
−0.627589 + 0.778545i \(0.715958\pi\)
\(468\) −4.76393 −0.220213
\(469\) −11.4164 −0.527161
\(470\) −2.47214 −0.114031
\(471\) −15.6393 −0.720622
\(472\) −2.76393 −0.127220
\(473\) 0 0
\(474\) 0 0
\(475\) −25.1246 −1.15280
\(476\) 2.47214 0.113310
\(477\) −12.4721 −0.571060
\(478\) 20.0000 0.914779
\(479\) −12.3607 −0.564774 −0.282387 0.959301i \(-0.591126\pi\)
−0.282387 + 0.959301i \(0.591126\pi\)
\(480\) 1.52786 0.0697371
\(481\) 22.4721 1.02464
\(482\) 15.4164 0.702198
\(483\) −4.94427 −0.224972
\(484\) 0 0
\(485\) −15.4164 −0.700023
\(486\) −13.5967 −0.616761
\(487\) 16.9443 0.767818 0.383909 0.923371i \(-0.374578\pi\)
0.383909 + 0.923371i \(0.374578\pi\)
\(488\) −0.763932 −0.0345816
\(489\) −24.0000 −1.08532
\(490\) 1.23607 0.0558399
\(491\) 16.9443 0.764684 0.382342 0.924021i \(-0.375118\pi\)
0.382342 + 0.924021i \(0.375118\pi\)
\(492\) 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 \(0.757851\pi\)
\(500\) 10.4721 0.468328
\(501\) −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
\(505\) 10.1115 0.449954
\(506\) 0 0
\(507\) −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 \(0.320908\pi\)
\(510\) −3.77709 −0.167252
\(511\) 12.9443 0.572621
\(512\) −1.00000 −0.0441942
\(513\) −40.0000 −1.76604
\(514\) −6.94427 −0.306299
\(515\) 18.4721 0.813980
\(516\) 12.9443 0.569840
\(517\) 0 0
\(518\) 6.94427 0.305114
\(519\) 4.00000 0.175581
\(520\) 4.00000 0.175412
\(521\) −10.3607 −0.453910 −0.226955 0.973905i \(-0.572877\pi\)
−0.226955 + 0.973905i \(0.572877\pi\)
\(522\) −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
\(525\) 4.29180 0.187309
\(526\) −4.94427 −0.215580
\(527\) −4.94427 −0.215376
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 10.4721 0.454881
\(531\) −4.06888 −0.176575
\(532\) −7.23607 −0.313723
\(533\) 8.00000 0.346518
\(534\) −12.3607 −0.534899
\(535\) 3.05573 0.132111
\(536\) −11.4164 −0.493114
\(537\) −11.0557 −0.477090
\(538\) 22.7639 0.981423
\(539\) 0 0
\(540\) 6.83282 0.294038
\(541\) 26.9443 1.15842 0.579212 0.815177i \(-0.303360\pi\)
0.579212 + 0.815177i \(0.303360\pi\)
\(542\) 0.944272 0.0405600
\(543\) 11.4164 0.489925
\(544\) 2.47214 0.105992
\(545\) −12.3607 −0.529473
\(546\) 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
\(555\) −10.6099 −0.450365
\(556\) 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
\(560\) 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 \(0.728691\pi\)
\(564\) −2.47214 −0.104096
\(565\) 0.583592 0.0245519
\(566\) 14.6525 0.615889
\(567\) 2.41641 0.101480
\(568\) −6.47214 −0.271565
\(569\) 36.8328 1.54411 0.772056 0.635555i \(-0.219228\pi\)
0.772056 + 0.635555i \(0.219228\pi\)
\(570\) 11.0557 0.463073
\(571\) 10.1115 0.423151 0.211576 0.977362i \(-0.432141\pi\)
0.211576 + 0.977362i \(0.432141\pi\)
\(572\) 0 0
\(573\) −3.05573 −0.127655
\(574\) 2.47214 0.103185
\(575\) −13.8885 −0.579192
\(576\) −1.47214 −0.0613390
\(577\) 26.9443 1.12170 0.560852 0.827916i \(-0.310474\pi\)
0.560852 + 0.827916i \(0.310474\pi\)
\(578\) 10.8885 0.452904
\(579\) 18.4721 0.767676
\(580\) 5.52786 0.229532
\(581\) −12.1803 −0.505326
\(582\) −15.4164 −0.639031
\(583\) 0 0
\(584\) 12.9443 0.535638
\(585\) 5.88854 0.243461
\(586\) −26.6525 −1.10100
\(587\) −5.81966 −0.240203 −0.120102 0.992762i \(-0.538322\pi\)
−0.120102 + 0.992762i \(0.538322\pi\)
\(588\) 1.23607 0.0509746
\(589\) 14.4721 0.596314
\(590\) 3.41641 0.140651
\(591\) −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 \(0.664020\pi\)
\(594\) 0 0
\(595\) −3.05573 −0.125273
\(596\) 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 \(0.729915\pi\)
\(600\) 4.29180 0.175212
\(601\) 34.8328 1.42086 0.710430 0.703768i \(-0.248500\pi\)
0.710430 + 0.703768i \(0.248500\pi\)
\(602\) 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
\(610\) 0.944272 0.0382325
\(611\) −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
\(615\) −3.77709 −0.152307
\(616\) 0 0
\(617\) 21.4164 0.862192 0.431096 0.902306i \(-0.358127\pi\)
0.431096 + 0.902306i \(0.358127\pi\)
\(618\) 18.4721 0.743058
\(619\) 18.5410 0.745227 0.372613 0.927987i \(-0.378462\pi\)
0.372613 + 0.927987i \(0.378462\pi\)
\(620\) −2.47214 −0.0992834
\(621\) −22.1115 −0.887302
\(622\) 21.4164 0.858720
\(623\) −10.0000 −0.400642
\(624\) 4.00000 0.160128
\(625\) 4.41641 0.176656
\(626\) −19.5279 −0.780490
\(627\) 0 0
\(628\) −12.6525 −0.504889
\(629\) −17.1672 −0.684500
\(630\) 1.81966 0.0724970
\(631\) −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
\(635\) −14.8328 −0.588622
\(636\) 10.4721 0.415247
\(637\) 3.23607 0.128218
\(638\) 0 0
\(639\) −9.52786 −0.376916
\(640\) 1.23607 0.0488599
\(641\) 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 \(-0.620638\pi\)
−0.369989 + 0.929036i \(0.620638\pi\)
\(644\) −4.00000 −0.157622
\(645\) −16.0000 −0.629999
\(646\) 17.8885 0.703815
\(647\) −28.8328 −1.13353 −0.566767 0.823878i \(-0.691806\pi\)
−0.566767 + 0.823878i \(0.691806\pi\)
\(648\) 2.41641 0.0949255
\(649\) 0 0
\(650\) 11.2361 0.440715
\(651\) −2.47214 −0.0968906
\(652\) −19.4164 −0.760405
\(653\) 46.3607 1.81423 0.907117 0.420879i \(-0.138278\pi\)
0.907117 + 0.420879i \(0.138278\pi\)
\(654\) −12.3607 −0.483341
\(655\) 5.88854 0.230084
\(656\) 2.47214 0.0965207
\(657\) 19.0557 0.743435
\(658\) −2.00000 −0.0779681
\(659\) −16.5836 −0.646005 −0.323003 0.946398i \(-0.604692\pi\)
−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
\(663\) −9.88854 −0.384039
\(664\) −12.1803 −0.472689
\(665\) 8.94427 0.346844
\(666\) 10.2229 0.396130
\(667\) −17.8885 −0.692647
\(668\) −11.4164 −0.441714
\(669\) −0.583592 −0.0225630
\(670\) 14.1115 0.545173
\(671\) 0 0
\(672\) 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
\(675\) 19.1935 0.738758
\(676\) −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
\(680\) −3.05573 −0.117182
\(681\) 23.7771 0.911140
\(682\) 0 0
\(683\) 32.9443 1.26058 0.630289 0.776361i \(-0.282936\pi\)
0.630289 + 0.776361i \(0.282936\pi\)
\(684\) −10.6525 −0.407308
\(685\) 24.5836 0.939291
\(686\) 1.00000 0.0381802
\(687\) 21.3050 0.812835
\(688\) 10.4721 0.399246
\(689\) 27.4164 1.04448
\(690\) 6.11146 0.232659
\(691\) 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
\(696\) 5.52786 0.209533
\(697\) −6.11146 −0.231488
\(698\) 21.7082 0.821668
\(699\) 18.4721 0.698680
\(700\) 3.47214 0.131234
\(701\) 42.7214 1.61356 0.806782 0.590850i \(-0.201207\pi\)
0.806782 + 0.590850i \(0.201207\pi\)
\(702\) 17.8885 0.675160
\(703\) 50.2492 1.89519
\(704\) 0 0
\(705\) 3.05573 0.115085
\(706\) 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 \(0.473238\pi\)
\(710\) 8.00000 0.300235
\(711\) 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 \(0.398371\pi\)
\(720\) 1.81966 0.0678147
\(721\) 14.9443 0.556554
\(722\) −33.3607 −1.24156
\(723\) −19.0557 −0.708690
\(724\) 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
\(729\) 24.0557 0.890953
\(730\) −16.0000 −0.592187
\(731\) −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
\(735\) −1.52786 −0.0563561
\(736\) −4.00000 −0.147442
\(737\) 0 0
\(738\) 3.63932 0.133965
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −8.58359 −0.315539
\(741\) 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
\(745\) −27.6393 −1.01263
\(746\) −6.00000 −0.219676
\(747\) −17.9311 −0.656065
\(748\) 0 0
\(749\) 2.47214 0.0903299
\(750\) −12.9443 −0.472658
\(751\) −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
\(755\) 14.8328 0.539821
\(756\) 5.52786 0.201046
\(757\) −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 \(0.433655\pi\)
\(762\) −14.8328 −0.537336
\(763\) −10.0000 −0.362024
\(764\) −2.47214 −0.0894387
\(765\) −4.49845 −0.162642
\(766\) 23.8885 0.863128
\(767\) 8.94427 0.322959
\(768\) 1.23607 0.0446028
\(769\) 43.4164 1.56564 0.782818 0.622251i \(-0.213782\pi\)
0.782818 + 0.622251i \(0.213782\pi\)
\(770\) 0 0
\(771\) 8.58359 0.309131
\(772\) 14.9443 0.537856
\(773\) 15.7082 0.564985 0.282492 0.959270i \(-0.408839\pi\)
0.282492 + 0.959270i \(0.408839\pi\)
\(774\) 15.4164 0.554131
\(775\) −6.94427 −0.249446
\(776\) −12.4721 −0.447724
\(777\) −8.58359 −0.307935
\(778\) −33.4164 −1.19804
\(779\) 17.8885 0.640924
\(780\) −4.94427 −0.177033
\(781\) 0 0
\(782\) 9.88854 0.353614
\(783\) 24.7214 0.883469
\(784\) 1.00000 0.0357143
\(785\) 15.6393 0.558191
\(786\) 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
\(792\) 0 0
\(793\) 2.47214 0.0877881
\(794\) 23.7082 0.841373
\(795\) −12.9443 −0.459086
\(796\) 18.9443 0.671462
\(797\) −41.5967 −1.47343 −0.736716 0.676202i \(-0.763625\pi\)
−0.736716 + 0.676202i \(0.763625\pi\)
\(798\) 8.94427 0.316624
\(799\) 4.94427 0.174916
\(800\) 3.47214 0.122759
\(801\) −14.7214 −0.520154
\(802\) −14.3607 −0.507093
\(803\) 0 0
\(804\) 14.1115 0.497673
\(805\) 4.94427 0.174263
\(806\) −6.47214 −0.227971
\(807\) −28.1378 −0.990496
\(808\) 8.18034 0.287783
\(809\) 21.0557 0.740280 0.370140 0.928976i \(-0.379310\pi\)
0.370140 + 0.928976i \(0.379310\pi\)
\(810\) −2.98684 −0.104947
\(811\) −4.76393 −0.167284 −0.0836421 0.996496i \(-0.526655\pi\)
−0.0836421 + 0.996496i \(0.526655\pi\)
\(812\) 4.47214 0.156941
\(813\) −1.16718 −0.0409349
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) −3.05573 −0.106972
\(817\) 75.7771 2.65110
\(818\) 3.41641 0.119452
\(819\) 4.76393 0.166465
\(820\) −3.05573 −0.106711
\(821\) 1.41641 0.0494330 0.0247165 0.999695i \(-0.492132\pi\)
0.0247165 + 0.999695i \(0.492132\pi\)
\(822\) 24.5836 0.857451
\(823\) −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 \(0.595188\pi\)
\(828\) −5.88854 −0.204641
\(829\) 11.7082 0.406643 0.203321 0.979112i \(-0.434826\pi\)
0.203321 + 0.979112i \(0.434826\pi\)
\(830\) 15.0557 0.522592
\(831\) −4.36068 −0.151270
\(832\) 3.23607 0.112190
\(833\) −2.47214 −0.0856544
\(834\) −26.8328 −0.929144
\(835\) 14.1115 0.488347
\(836\) 0 0
\(837\) −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 \(0.406156\pi\)
\(840\) −1.52786 −0.0527163
\(841\) −9.00000 −0.310345
\(842\) −16.4721 −0.567667
\(843\) −35.6393 −1.22748
\(844\) 13.5279 0.465648
\(845\) 3.12461 0.107490
\(846\) −2.94427 −0.101226
\(847\) 0 0
\(848\) 8.47214 0.290934
\(849\) −18.1115 −0.621584
\(850\) −8.58359 −0.294415
\(851\) 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
\(855\) 13.1672 0.450308
\(856\) 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
\(861\) −3.05573 −0.104139
\(862\) 23.0557 0.785281
\(863\) 0.583592 0.0198657 0.00993285 0.999951i \(-0.496838\pi\)
0.00993285 + 0.999951i \(0.496838\pi\)
\(864\) 5.52786 0.188062
\(865\) −4.00000 −0.136004
\(866\) −28.4721 −0.967523
\(867\) −13.4590 −0.457091
\(868\) −2.00000 −0.0678844
\(869\) 0 0
\(870\) −6.83282 −0.231654
\(871\) 36.9443 1.25181
\(872\) −10.0000 −0.338643
\(873\) −18.3607 −0.621415
\(874\) −28.9443 −0.979055
\(875\) −10.4721 −0.354023
\(876\) −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 \(-0.515178\pi\)
−0.0476659 + 0.998863i \(0.515178\pi\)
\(884\) −8.00000 −0.269069
\(885\) −4.22291 −0.141952
\(886\) 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
\(890\) 12.3607 0.414331
\(891\) 0 0
\(892\) −0.472136 −0.0158083
\(893\) −14.4721 −0.484292
\(894\) −27.6393 −0.924397
\(895\) 11.0557 0.369552
\(896\) 1.00000 0.0334077
\(897\) 16.0000 0.534224
\(898\) −18.9443 −0.632179
\(899\) −8.94427 −0.298308
\(900\) 5.11146 0.170382
\(901\) −20.9443 −0.697755
\(902\) 0 0
\(903\) −12.9443 −0.430758
\(904\) 0.472136 0.0157030
\(905\) −11.4164 −0.379494
\(906\) 14.8328 0.492787
\(907\) −24.3607 −0.808883 −0.404442 0.914564i \(-0.632534\pi\)
−0.404442 + 0.914564i \(0.632534\pi\)
\(908\) 19.2361 0.638371
\(909\) 12.0426 0.399427
\(910\) −4.00000 −0.132599
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 8.94427 0.296174
\(913\) 0 0
\(914\) 26.9443 0.891237
\(915\) −1.16718 −0.0385859
\(916\) 17.2361 0.569496
\(917\) 4.76393 0.157319
\(918\) −13.6656 −0.451033
\(919\) 22.1115 0.729390 0.364695 0.931127i \(-0.381173\pi\)
0.364695 + 0.931127i \(0.381173\pi\)
\(920\) 4.94427 0.163008
\(921\) 32.2229 1.06178
\(922\) 24.7639 0.815557
\(923\) 20.9443 0.689389
\(924\) 0 0
\(925\) −24.1115 −0.792780
\(926\) 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 \(0.270443\pi\)
\(930\) 3.05573 0.100201
\(931\) 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
\(940\) 2.47214 0.0806322
\(941\) 11.8197 0.385310 0.192655 0.981267i \(-0.438290\pi\)
0.192655 + 0.981267i \(0.438290\pi\)
\(942\) 15.6393 0.509557
\(943\) 9.88854 0.322015
\(944\) 2.76393 0.0899583
\(945\) −6.83282 −0.222272
\(946\) 0 0
\(947\) 16.9443 0.550615 0.275307 0.961356i \(-0.411220\pi\)
0.275307 + 0.961356i \(0.411220\pi\)
\(948\) 0 0
\(949\) −41.8885 −1.35976
\(950\) 25.1246 0.815150
\(951\) −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
\(955\) 3.05573 0.0988810
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) 12.3607 0.399355
\(959\) 19.8885 0.642235
\(960\) −1.52786 −0.0493116
\(961\) −27.0000 −0.870968
\(962\) −22.4721 −0.724531
\(963\) 3.63932 0.117275
\(964\) −15.4164 −0.496529
\(965\) −18.4721 −0.594639
\(966\) 4.94427 0.159079
\(967\) −45.8885 −1.47568 −0.737838 0.674978i \(-0.764153\pi\)
−0.737838 + 0.674978i \(0.764153\pi\)
\(968\) 0 0
\(969\) −22.1115 −0.710322
\(970\) 15.4164 0.494991
\(971\) 50.5410 1.62194 0.810969 0.585089i \(-0.198940\pi\)
0.810969 + 0.585089i \(0.198940\pi\)
\(972\) 13.5967 0.436116
\(973\) −21.7082 −0.695933
\(974\) −16.9443 −0.542929
\(975\) −13.8885 −0.444789
\(976\) 0.763932 0.0244529
\(977\) −28.8328 −0.922443 −0.461222 0.887285i \(-0.652589\pi\)
−0.461222 + 0.887285i \(0.652589\pi\)
\(978\) 24.0000 0.767435
\(979\) 0 0
\(980\) −1.23607 −0.0394847
\(981\) −14.7214 −0.470017
\(982\) −16.9443 −0.540713
\(983\) 14.0000 0.446531 0.223265 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) −3.05573 −0.0974131
\(985\) 22.2492 0.708919
\(986\) −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
\(995\) −23.4164 −0.742350
\(996\) 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 1694.2.a.l.1.2 2
11.10 odd 2 154.2.a.d.1.2 2
33.32 even 2 1386.2.a.m.1.2 2
44.43 even 2 1232.2.a.p.1.1 2
55.32 even 4 3850.2.c.q.1849.3 4
55.43 even 4 3850.2.c.q.1849.2 4
55.54 odd 2 3850.2.a.bj.1.1 2
77.10 even 6 1078.2.e.n.177.2 4
77.32 odd 6 1078.2.e.q.177.1 4
77.54 even 6 1078.2.e.n.67.2 4
77.65 odd 6 1078.2.e.q.67.1 4
77.76 even 2 1078.2.a.w.1.1 2
88.21 odd 2 4928.2.a.bt.1.1 2
88.43 even 2 4928.2.a.bk.1.2 2
231.230 odd 2 9702.2.a.cu.1.1 2
308.307 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 11.10 odd 2
1078.2.a.w.1.1 2 77.76 even 2
1078.2.e.n.67.2 4 77.54 even 6
1078.2.e.n.177.2 4 77.10 even 6
1078.2.e.q.67.1 4 77.65 odd 6
1078.2.e.q.177.1 4 77.32 odd 6
1232.2.a.p.1.1 2 44.43 even 2
1386.2.a.m.1.2 2 33.32 even 2
1694.2.a.l.1.2 2 1.1 even 1 trivial
3850.2.a.bj.1.1 2 55.54 odd 2
3850.2.c.q.1849.2 4 55.43 even 4
3850.2.c.q.1849.3 4 55.32 even 4
4928.2.a.bk.1.2 2 88.43 even 2
4928.2.a.bt.1.1 2 88.21 odd 2
8624.2.a.bf.1.2 2 308.307 odd 2
9702.2.a.cu.1.1 2 231.230 odd 2