Properties

Label 8470.2.a.ck.1.3
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.48929 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.48929 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.19656 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.48929 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.48929 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.19656 q^{9} -1.00000 q^{10} +2.48929 q^{12} +2.48929 q^{13} +1.00000 q^{14} -2.48929 q^{15} +1.00000 q^{16} +5.19656 q^{17} +3.19656 q^{18} +0.707269 q^{19} -1.00000 q^{20} +2.48929 q^{21} +5.17513 q^{23} +2.48929 q^{24} +1.00000 q^{25} +2.48929 q^{26} +0.489289 q^{27} +1.00000 q^{28} -2.39312 q^{29} -2.48929 q^{30} +1.00000 q^{32} +5.19656 q^{34} -1.00000 q^{35} +3.19656 q^{36} +7.37169 q^{37} +0.707269 q^{38} +6.19656 q^{39} -1.00000 q^{40} -4.97858 q^{41} +2.48929 q^{42} -9.15371 q^{43} -3.19656 q^{45} +5.17513 q^{46} -5.37169 q^{47} +2.48929 q^{48} +1.00000 q^{49} +1.00000 q^{50} +12.9357 q^{51} +2.48929 q^{52} -1.19656 q^{53} +0.489289 q^{54} +1.00000 q^{56} +1.76060 q^{57} -2.39312 q^{58} +3.29273 q^{59} -2.48929 q^{60} +7.78202 q^{61} +3.19656 q^{63} +1.00000 q^{64} -2.48929 q^{65} -7.58967 q^{67} +5.19656 q^{68} +12.8824 q^{69} -1.00000 q^{70} +12.1751 q^{71} +3.19656 q^{72} -0.175135 q^{73} +7.37169 q^{74} +2.48929 q^{75} +0.707269 q^{76} +6.19656 q^{78} +1.97858 q^{79} -1.00000 q^{80} -8.37169 q^{81} -4.97858 q^{82} +4.48929 q^{83} +2.48929 q^{84} -5.19656 q^{85} -9.15371 q^{86} -5.95715 q^{87} -0.978577 q^{89} -3.19656 q^{90} +2.48929 q^{91} +5.17513 q^{92} -5.37169 q^{94} -0.707269 q^{95} +2.48929 q^{96} +4.17513 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} + 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 3q^{4} - 3q^{5} + 3q^{7} + 3q^{8} + 5q^{9} - 3q^{10} + 3q^{14} + 3q^{16} + 11q^{17} + 5q^{18} + 5q^{19} - 3q^{20} - 4q^{23} + 3q^{25} - 6q^{27} + 3q^{28} + 2q^{29} + 3q^{32} + 11q^{34} - 3q^{35} + 5q^{36} - 2q^{37} + 5q^{38} + 14q^{39} - 3q^{40} + 7q^{43} - 5q^{45} - 4q^{46} + 8q^{47} + 3q^{49} + 3q^{50} - 6q^{51} + q^{53} - 6q^{54} + 3q^{56} - 20q^{57} + 2q^{58} + 7q^{59} + 13q^{61} + 5q^{63} + 3q^{64} - 9q^{67} + 11q^{68} + 22q^{69} - 3q^{70} + 17q^{71} + 5q^{72} + 19q^{73} - 2q^{74} + 5q^{76} + 14q^{78} - 9q^{79} - 3q^{80} - q^{81} + 6q^{83} - 11q^{85} + 7q^{86} + 12q^{87} + 12q^{89} - 5q^{90} - 4q^{92} + 8q^{94} - 5q^{95} - 7q^{97} + 3q^{98} + O(q^{100}) \)

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
\(3\) 2.48929 1.43719 0.718596 0.695428i \(-0.244785\pi\)
0.718596 + 0.695428i \(0.244785\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.48929 1.01625
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 3.19656 1.06552
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 2.48929 0.718596
\(13\) 2.48929 0.690404 0.345202 0.938528i \(-0.387810\pi\)
0.345202 + 0.938528i \(0.387810\pi\)
\(14\) 1.00000 0.267261
\(15\) −2.48929 −0.642732
\(16\) 1.00000 0.250000
\(17\) 5.19656 1.26035 0.630175 0.776453i \(-0.282983\pi\)
0.630175 + 0.776453i \(0.282983\pi\)
\(18\) 3.19656 0.753436
\(19\) 0.707269 0.162259 0.0811293 0.996704i \(-0.474147\pi\)
0.0811293 + 0.996704i \(0.474147\pi\)
\(20\) −1.00000 −0.223607
\(21\) 2.48929 0.543207
\(22\) 0 0
\(23\) 5.17513 1.07909 0.539545 0.841957i \(-0.318596\pi\)
0.539545 + 0.841957i \(0.318596\pi\)
\(24\) 2.48929 0.508124
\(25\) 1.00000 0.200000
\(26\) 2.48929 0.488190
\(27\) 0.489289 0.0941636
\(28\) 1.00000 0.188982
\(29\) −2.39312 −0.444390 −0.222195 0.975002i \(-0.571322\pi\)
−0.222195 + 0.975002i \(0.571322\pi\)
\(30\) −2.48929 −0.454480
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.19656 0.891202
\(35\) −1.00000 −0.169031
\(36\) 3.19656 0.532760
\(37\) 7.37169 1.21190 0.605949 0.795503i \(-0.292793\pi\)
0.605949 + 0.795503i \(0.292793\pi\)
\(38\) 0.707269 0.114734
\(39\) 6.19656 0.992243
\(40\) −1.00000 −0.158114
\(41\) −4.97858 −0.777523 −0.388762 0.921338i \(-0.627097\pi\)
−0.388762 + 0.921338i \(0.627097\pi\)
\(42\) 2.48929 0.384106
\(43\) −9.15371 −1.39593 −0.697964 0.716133i \(-0.745910\pi\)
−0.697964 + 0.716133i \(0.745910\pi\)
\(44\) 0 0
\(45\) −3.19656 −0.476515
\(46\) 5.17513 0.763032
\(47\) −5.37169 −0.783542 −0.391771 0.920063i \(-0.628137\pi\)
−0.391771 + 0.920063i \(0.628137\pi\)
\(48\) 2.48929 0.359298
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 12.9357 1.81136
\(52\) 2.48929 0.345202
\(53\) −1.19656 −0.164360 −0.0821799 0.996618i \(-0.526188\pi\)
−0.0821799 + 0.996618i \(0.526188\pi\)
\(54\) 0.489289 0.0665837
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.76060 0.233197
\(58\) −2.39312 −0.314231
\(59\) 3.29273 0.428677 0.214339 0.976759i \(-0.431240\pi\)
0.214339 + 0.976759i \(0.431240\pi\)
\(60\) −2.48929 −0.321366
\(61\) 7.78202 0.996386 0.498193 0.867066i \(-0.333997\pi\)
0.498193 + 0.867066i \(0.333997\pi\)
\(62\) 0 0
\(63\) 3.19656 0.402728
\(64\) 1.00000 0.125000
\(65\) −2.48929 −0.308758
\(66\) 0 0
\(67\) −7.58967 −0.927226 −0.463613 0.886038i \(-0.653447\pi\)
−0.463613 + 0.886038i \(0.653447\pi\)
\(68\) 5.19656 0.630175
\(69\) 12.8824 1.55086
\(70\) −1.00000 −0.119523
\(71\) 12.1751 1.44492 0.722461 0.691411i \(-0.243011\pi\)
0.722461 + 0.691411i \(0.243011\pi\)
\(72\) 3.19656 0.376718
\(73\) −0.175135 −0.0204980 −0.0102490 0.999947i \(-0.503262\pi\)
−0.0102490 + 0.999947i \(0.503262\pi\)
\(74\) 7.37169 0.856942
\(75\) 2.48929 0.287438
\(76\) 0.707269 0.0811293
\(77\) 0 0
\(78\) 6.19656 0.701622
\(79\) 1.97858 0.222607 0.111304 0.993786i \(-0.464497\pi\)
0.111304 + 0.993786i \(0.464497\pi\)
\(80\) −1.00000 −0.111803
\(81\) −8.37169 −0.930188
\(82\) −4.97858 −0.549792
\(83\) 4.48929 0.492763 0.246382 0.969173i \(-0.420758\pi\)
0.246382 + 0.969173i \(0.420758\pi\)
\(84\) 2.48929 0.271604
\(85\) −5.19656 −0.563646
\(86\) −9.15371 −0.987070
\(87\) −5.95715 −0.638674
\(88\) 0 0
\(89\) −0.978577 −0.103729 −0.0518645 0.998654i \(-0.516516\pi\)
−0.0518645 + 0.998654i \(0.516516\pi\)
\(90\) −3.19656 −0.336947
\(91\) 2.48929 0.260948
\(92\) 5.17513 0.539545
\(93\) 0 0
\(94\) −5.37169 −0.554048
\(95\) −0.707269 −0.0725643
\(96\) 2.48929 0.254062
\(97\) 4.17513 0.423921 0.211960 0.977278i \(-0.432015\pi\)
0.211960 + 0.977278i \(0.432015\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.48929 −0.645708 −0.322854 0.946449i \(-0.604642\pi\)
−0.322854 + 0.946449i \(0.604642\pi\)
\(102\) 12.9357 1.28083
\(103\) 3.78202 0.372653 0.186327 0.982488i \(-0.440342\pi\)
0.186327 + 0.982488i \(0.440342\pi\)
\(104\) 2.48929 0.244095
\(105\) −2.48929 −0.242930
\(106\) −1.19656 −0.116220
\(107\) −4.21798 −0.407768 −0.203884 0.978995i \(-0.565356\pi\)
−0.203884 + 0.978995i \(0.565356\pi\)
\(108\) 0.489289 0.0470818
\(109\) −9.37169 −0.897645 −0.448823 0.893621i \(-0.648157\pi\)
−0.448823 + 0.893621i \(0.648157\pi\)
\(110\) 0 0
\(111\) 18.3503 1.74173
\(112\) 1.00000 0.0944911
\(113\) 10.5897 0.996193 0.498096 0.867122i \(-0.334033\pi\)
0.498096 + 0.867122i \(0.334033\pi\)
\(114\) 1.76060 0.164895
\(115\) −5.17513 −0.482584
\(116\) −2.39312 −0.222195
\(117\) 7.95715 0.735639
\(118\) 3.29273 0.303120
\(119\) 5.19656 0.476368
\(120\) −2.48929 −0.227240
\(121\) 0 0
\(122\) 7.78202 0.704551
\(123\) −12.3931 −1.11745
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 3.19656 0.284772
\(127\) 2.82487 0.250666 0.125333 0.992115i \(-0.460000\pi\)
0.125333 + 0.992115i \(0.460000\pi\)
\(128\) 1.00000 0.0883883
\(129\) −22.7862 −2.00622
\(130\) −2.48929 −0.218325
\(131\) −15.8181 −1.38204 −0.691018 0.722837i \(-0.742838\pi\)
−0.691018 + 0.722837i \(0.742838\pi\)
\(132\) 0 0
\(133\) 0.707269 0.0613280
\(134\) −7.58967 −0.655648
\(135\) −0.489289 −0.0421113
\(136\) 5.19656 0.445601
\(137\) 17.1323 1.46371 0.731855 0.681460i \(-0.238655\pi\)
0.731855 + 0.681460i \(0.238655\pi\)
\(138\) 12.8824 1.09662
\(139\) 16.4721 1.39714 0.698572 0.715540i \(-0.253819\pi\)
0.698572 + 0.715540i \(0.253819\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −13.3717 −1.12610
\(142\) 12.1751 1.02171
\(143\) 0 0
\(144\) 3.19656 0.266380
\(145\) 2.39312 0.198737
\(146\) −0.175135 −0.0144943
\(147\) 2.48929 0.205313
\(148\) 7.37169 0.605949
\(149\) −14.9357 −1.22358 −0.611791 0.791019i \(-0.709551\pi\)
−0.611791 + 0.791019i \(0.709551\pi\)
\(150\) 2.48929 0.203250
\(151\) −13.1323 −1.06869 −0.534345 0.845266i \(-0.679442\pi\)
−0.534345 + 0.845266i \(0.679442\pi\)
\(152\) 0.707269 0.0573671
\(153\) 16.6111 1.34293
\(154\) 0 0
\(155\) 0 0
\(156\) 6.19656 0.496122
\(157\) −8.88240 −0.708893 −0.354446 0.935076i \(-0.615331\pi\)
−0.354446 + 0.935076i \(0.615331\pi\)
\(158\) 1.97858 0.157407
\(159\) −2.97858 −0.236217
\(160\) −1.00000 −0.0790569
\(161\) 5.17513 0.407858
\(162\) −8.37169 −0.657742
\(163\) 10.5254 0.824413 0.412207 0.911090i \(-0.364758\pi\)
0.412207 + 0.911090i \(0.364758\pi\)
\(164\) −4.97858 −0.388762
\(165\) 0 0
\(166\) 4.48929 0.348436
\(167\) 22.5254 1.74307 0.871534 0.490335i \(-0.163126\pi\)
0.871534 + 0.490335i \(0.163126\pi\)
\(168\) 2.48929 0.192053
\(169\) −6.80344 −0.523342
\(170\) −5.19656 −0.398558
\(171\) 2.26083 0.172890
\(172\) −9.15371 −0.697964
\(173\) −13.5640 −1.03125 −0.515627 0.856813i \(-0.672441\pi\)
−0.515627 + 0.856813i \(0.672441\pi\)
\(174\) −5.95715 −0.451611
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 8.19656 0.616091
\(178\) −0.978577 −0.0733475
\(179\) 26.3503 1.96951 0.984756 0.173943i \(-0.0556507\pi\)
0.984756 + 0.173943i \(0.0556507\pi\)
\(180\) −3.19656 −0.238257
\(181\) 1.48508 0.110385 0.0551925 0.998476i \(-0.482423\pi\)
0.0551925 + 0.998476i \(0.482423\pi\)
\(182\) 2.48929 0.184518
\(183\) 19.3717 1.43200
\(184\) 5.17513 0.381516
\(185\) −7.37169 −0.541978
\(186\) 0 0
\(187\) 0 0
\(188\) −5.37169 −0.391771
\(189\) 0.489289 0.0355905
\(190\) −0.707269 −0.0513107
\(191\) 10.1966 0.737797 0.368899 0.929470i \(-0.379735\pi\)
0.368899 + 0.929470i \(0.379735\pi\)
\(192\) 2.48929 0.179649
\(193\) −10.9828 −0.790558 −0.395279 0.918561i \(-0.629352\pi\)
−0.395279 + 0.918561i \(0.629352\pi\)
\(194\) 4.17513 0.299757
\(195\) −6.19656 −0.443745
\(196\) 1.00000 0.0714286
\(197\) 27.5040 1.95958 0.979789 0.200035i \(-0.0641056\pi\)
0.979789 + 0.200035i \(0.0641056\pi\)
\(198\) 0 0
\(199\) 7.56404 0.536201 0.268100 0.963391i \(-0.413604\pi\)
0.268100 + 0.963391i \(0.413604\pi\)
\(200\) 1.00000 0.0707107
\(201\) −18.8929 −1.33260
\(202\) −6.48929 −0.456585
\(203\) −2.39312 −0.167964
\(204\) 12.9357 0.905682
\(205\) 4.97858 0.347719
\(206\) 3.78202 0.263506
\(207\) 16.5426 1.14979
\(208\) 2.48929 0.172601
\(209\) 0 0
\(210\) −2.48929 −0.171777
\(211\) −20.7862 −1.43098 −0.715492 0.698621i \(-0.753797\pi\)
−0.715492 + 0.698621i \(0.753797\pi\)
\(212\) −1.19656 −0.0821799
\(213\) 30.3074 2.07663
\(214\) −4.21798 −0.288335
\(215\) 9.15371 0.624278
\(216\) 0.489289 0.0332919
\(217\) 0 0
\(218\) −9.37169 −0.634731
\(219\) −0.435961 −0.0294595
\(220\) 0 0
\(221\) 12.9357 0.870151
\(222\) 18.3503 1.23159
\(223\) 16.7606 1.12237 0.561186 0.827690i \(-0.310345\pi\)
0.561186 + 0.827690i \(0.310345\pi\)
\(224\) 1.00000 0.0668153
\(225\) 3.19656 0.213104
\(226\) 10.5897 0.704415
\(227\) −13.3717 −0.887510 −0.443755 0.896148i \(-0.646354\pi\)
−0.443755 + 0.896148i \(0.646354\pi\)
\(228\) 1.76060 0.116598
\(229\) −4.13229 −0.273069 −0.136535 0.990635i \(-0.543597\pi\)
−0.136535 + 0.990635i \(0.543597\pi\)
\(230\) −5.17513 −0.341238
\(231\) 0 0
\(232\) −2.39312 −0.157116
\(233\) 5.21798 0.341841 0.170921 0.985285i \(-0.445326\pi\)
0.170921 + 0.985285i \(0.445326\pi\)
\(234\) 7.95715 0.520175
\(235\) 5.37169 0.350411
\(236\) 3.29273 0.214339
\(237\) 4.92525 0.319929
\(238\) 5.19656 0.336843
\(239\) −30.3032 −1.96015 −0.980076 0.198623i \(-0.936353\pi\)
−0.980076 + 0.198623i \(0.936353\pi\)
\(240\) −2.48929 −0.160683
\(241\) 11.4145 0.735275 0.367638 0.929969i \(-0.380167\pi\)
0.367638 + 0.929969i \(0.380167\pi\)
\(242\) 0 0
\(243\) −22.3074 −1.43102
\(244\) 7.78202 0.498193
\(245\) −1.00000 −0.0638877
\(246\) −12.3931 −0.790156
\(247\) 1.76060 0.112024
\(248\) 0 0
\(249\) 11.1751 0.708195
\(250\) −1.00000 −0.0632456
\(251\) −12.7606 −0.805442 −0.402721 0.915323i \(-0.631935\pi\)
−0.402721 + 0.915323i \(0.631935\pi\)
\(252\) 3.19656 0.201364
\(253\) 0 0
\(254\) 2.82487 0.177248
\(255\) −12.9357 −0.810067
\(256\) 1.00000 0.0625000
\(257\) −3.58967 −0.223918 −0.111959 0.993713i \(-0.535712\pi\)
−0.111959 + 0.993713i \(0.535712\pi\)
\(258\) −22.7862 −1.41861
\(259\) 7.37169 0.458055
\(260\) −2.48929 −0.154379
\(261\) −7.64973 −0.473506
\(262\) −15.8181 −0.977248
\(263\) 2.63252 0.162328 0.0811640 0.996701i \(-0.474136\pi\)
0.0811640 + 0.996701i \(0.474136\pi\)
\(264\) 0 0
\(265\) 1.19656 0.0735039
\(266\) 0.707269 0.0433654
\(267\) −2.43596 −0.149078
\(268\) −7.58967 −0.463613
\(269\) 12.8996 0.786503 0.393252 0.919431i \(-0.371350\pi\)
0.393252 + 0.919431i \(0.371350\pi\)
\(270\) −0.489289 −0.0297772
\(271\) −8.93573 −0.542807 −0.271404 0.962466i \(-0.587488\pi\)
−0.271404 + 0.962466i \(0.587488\pi\)
\(272\) 5.19656 0.315088
\(273\) 6.19656 0.375033
\(274\) 17.1323 1.03500
\(275\) 0 0
\(276\) 12.8824 0.775430
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 16.4721 0.987929
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 20.5212 1.22419 0.612096 0.790784i \(-0.290327\pi\)
0.612096 + 0.790784i \(0.290327\pi\)
\(282\) −13.3717 −0.796273
\(283\) −0.681635 −0.0405190 −0.0202595 0.999795i \(-0.506449\pi\)
−0.0202595 + 0.999795i \(0.506449\pi\)
\(284\) 12.1751 0.722461
\(285\) −1.76060 −0.104289
\(286\) 0 0
\(287\) −4.97858 −0.293876
\(288\) 3.19656 0.188359
\(289\) 10.0042 0.588483
\(290\) 2.39312 0.140529
\(291\) 10.3931 0.609255
\(292\) −0.175135 −0.0102490
\(293\) −19.4250 −1.13482 −0.567411 0.823435i \(-0.692055\pi\)
−0.567411 + 0.823435i \(0.692055\pi\)
\(294\) 2.48929 0.145178
\(295\) −3.29273 −0.191710
\(296\) 7.37169 0.428471
\(297\) 0 0
\(298\) −14.9357 −0.865204
\(299\) 12.8824 0.745009
\(300\) 2.48929 0.143719
\(301\) −9.15371 −0.527611
\(302\) −13.1323 −0.755678
\(303\) −16.1537 −0.928007
\(304\) 0.707269 0.0405647
\(305\) −7.78202 −0.445597
\(306\) 16.6111 0.949593
\(307\) 14.1923 0.810000 0.405000 0.914317i \(-0.367272\pi\)
0.405000 + 0.914317i \(0.367272\pi\)
\(308\) 0 0
\(309\) 9.41454 0.535574
\(310\) 0 0
\(311\) −28.8929 −1.63837 −0.819183 0.573533i \(-0.805573\pi\)
−0.819183 + 0.573533i \(0.805573\pi\)
\(312\) 6.19656 0.350811
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −8.88240 −0.501263
\(315\) −3.19656 −0.180106
\(316\) 1.97858 0.111304
\(317\) 7.97437 0.447885 0.223943 0.974602i \(-0.428107\pi\)
0.223943 + 0.974602i \(0.428107\pi\)
\(318\) −2.97858 −0.167030
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −10.4998 −0.586040
\(322\) 5.17513 0.288399
\(323\) 3.67536 0.204503
\(324\) −8.37169 −0.465094
\(325\) 2.48929 0.138081
\(326\) 10.5254 0.582948
\(327\) −23.3288 −1.29009
\(328\) −4.97858 −0.274896
\(329\) −5.37169 −0.296151
\(330\) 0 0
\(331\) 8.62831 0.474255 0.237127 0.971479i \(-0.423794\pi\)
0.237127 + 0.971479i \(0.423794\pi\)
\(332\) 4.48929 0.246382
\(333\) 23.5640 1.29130
\(334\) 22.5254 1.23254
\(335\) 7.58967 0.414668
\(336\) 2.48929 0.135802
\(337\) −7.80344 −0.425081 −0.212540 0.977152i \(-0.568174\pi\)
−0.212540 + 0.977152i \(0.568174\pi\)
\(338\) −6.80344 −0.370058
\(339\) 26.3608 1.43172
\(340\) −5.19656 −0.281823
\(341\) 0 0
\(342\) 2.26083 0.122251
\(343\) 1.00000 0.0539949
\(344\) −9.15371 −0.493535
\(345\) −12.8824 −0.693565
\(346\) −13.5640 −0.729207
\(347\) 27.1966 1.45999 0.729994 0.683454i \(-0.239523\pi\)
0.729994 + 0.683454i \(0.239523\pi\)
\(348\) −5.95715 −0.319337
\(349\) −28.0533 −1.50166 −0.750830 0.660495i \(-0.770346\pi\)
−0.750830 + 0.660495i \(0.770346\pi\)
\(350\) 1.00000 0.0534522
\(351\) 1.21798 0.0650110
\(352\) 0 0
\(353\) 12.5254 0.666660 0.333330 0.942810i \(-0.391828\pi\)
0.333330 + 0.942810i \(0.391828\pi\)
\(354\) 8.19656 0.435642
\(355\) −12.1751 −0.646189
\(356\) −0.978577 −0.0518645
\(357\) 12.9357 0.684632
\(358\) 26.3503 1.39266
\(359\) −21.3461 −1.12660 −0.563301 0.826252i \(-0.690469\pi\)
−0.563301 + 0.826252i \(0.690469\pi\)
\(360\) −3.19656 −0.168473
\(361\) −18.4998 −0.973672
\(362\) 1.48508 0.0780540
\(363\) 0 0
\(364\) 2.48929 0.130474
\(365\) 0.175135 0.00916697
\(366\) 19.3717 1.01257
\(367\) 24.7178 1.29026 0.645128 0.764074i \(-0.276804\pi\)
0.645128 + 0.764074i \(0.276804\pi\)
\(368\) 5.17513 0.269773
\(369\) −15.9143 −0.828466
\(370\) −7.37169 −0.383236
\(371\) −1.19656 −0.0621222
\(372\) 0 0
\(373\) −12.9185 −0.668896 −0.334448 0.942414i \(-0.608550\pi\)
−0.334448 + 0.942414i \(0.608550\pi\)
\(374\) 0 0
\(375\) −2.48929 −0.128546
\(376\) −5.37169 −0.277024
\(377\) −5.95715 −0.306809
\(378\) 0.489289 0.0251663
\(379\) 35.9227 1.84523 0.922613 0.385726i \(-0.126049\pi\)
0.922613 + 0.385726i \(0.126049\pi\)
\(380\) −0.707269 −0.0362821
\(381\) 7.03190 0.360255
\(382\) 10.1966 0.521701
\(383\) 11.6325 0.594394 0.297197 0.954816i \(-0.403948\pi\)
0.297197 + 0.954816i \(0.403948\pi\)
\(384\) 2.48929 0.127031
\(385\) 0 0
\(386\) −10.9828 −0.559009
\(387\) −29.2604 −1.48739
\(388\) 4.17513 0.211960
\(389\) −25.2860 −1.28205 −0.641026 0.767520i \(-0.721491\pi\)
−0.641026 + 0.767520i \(0.721491\pi\)
\(390\) −6.19656 −0.313775
\(391\) 26.8929 1.36003
\(392\) 1.00000 0.0505076
\(393\) −39.3759 −1.98625
\(394\) 27.5040 1.38563
\(395\) −1.97858 −0.0995530
\(396\) 0 0
\(397\) 5.17935 0.259944 0.129972 0.991518i \(-0.458511\pi\)
0.129972 + 0.991518i \(0.458511\pi\)
\(398\) 7.56404 0.379151
\(399\) 1.76060 0.0881401
\(400\) 1.00000 0.0500000
\(401\) 32.7434 1.63513 0.817563 0.575839i \(-0.195324\pi\)
0.817563 + 0.575839i \(0.195324\pi\)
\(402\) −18.8929 −0.942291
\(403\) 0 0
\(404\) −6.48929 −0.322854
\(405\) 8.37169 0.415993
\(406\) −2.39312 −0.118768
\(407\) 0 0
\(408\) 12.9357 0.640414
\(409\) −17.3717 −0.858975 −0.429487 0.903073i \(-0.641306\pi\)
−0.429487 + 0.903073i \(0.641306\pi\)
\(410\) 4.97858 0.245874
\(411\) 42.6472 2.10363
\(412\) 3.78202 0.186327
\(413\) 3.29273 0.162025
\(414\) 16.5426 0.813025
\(415\) −4.48929 −0.220371
\(416\) 2.48929 0.122047
\(417\) 41.0038 2.00796
\(418\) 0 0
\(419\) 4.31415 0.210760 0.105380 0.994432i \(-0.466394\pi\)
0.105380 + 0.994432i \(0.466394\pi\)
\(420\) −2.48929 −0.121465
\(421\) −40.7005 −1.98362 −0.991811 0.127711i \(-0.959237\pi\)
−0.991811 + 0.127711i \(0.959237\pi\)
\(422\) −20.7862 −1.01186
\(423\) −17.1709 −0.834879
\(424\) −1.19656 −0.0581100
\(425\) 5.19656 0.252070
\(426\) 30.3074 1.46840
\(427\) 7.78202 0.376598
\(428\) −4.21798 −0.203884
\(429\) 0 0
\(430\) 9.15371 0.441431
\(431\) −31.7862 −1.53109 −0.765544 0.643383i \(-0.777530\pi\)
−0.765544 + 0.643383i \(0.777530\pi\)
\(432\) 0.489289 0.0235409
\(433\) −28.0722 −1.34907 −0.674533 0.738245i \(-0.735655\pi\)
−0.674533 + 0.738245i \(0.735655\pi\)
\(434\) 0 0
\(435\) 5.95715 0.285624
\(436\) −9.37169 −0.448823
\(437\) 3.66021 0.175092
\(438\) −0.435961 −0.0208310
\(439\) 28.9442 1.38143 0.690715 0.723128i \(-0.257296\pi\)
0.690715 + 0.723128i \(0.257296\pi\)
\(440\) 0 0
\(441\) 3.19656 0.152217
\(442\) 12.9357 0.615290
\(443\) 7.79923 0.370553 0.185276 0.982686i \(-0.440682\pi\)
0.185276 + 0.982686i \(0.440682\pi\)
\(444\) 18.3503 0.870865
\(445\) 0.978577 0.0463890
\(446\) 16.7606 0.793637
\(447\) −37.1793 −1.75852
\(448\) 1.00000 0.0472456
\(449\) −11.7862 −0.556227 −0.278113 0.960548i \(-0.589709\pi\)
−0.278113 + 0.960548i \(0.589709\pi\)
\(450\) 3.19656 0.150687
\(451\) 0 0
\(452\) 10.5897 0.498096
\(453\) −32.6901 −1.53591
\(454\) −13.3717 −0.627565
\(455\) −2.48929 −0.116700
\(456\) 1.76060 0.0824475
\(457\) −30.1109 −1.40853 −0.704263 0.709939i \(-0.748723\pi\)
−0.704263 + 0.709939i \(0.748723\pi\)
\(458\) −4.13229 −0.193089
\(459\) 2.54262 0.118679
\(460\) −5.17513 −0.241292
\(461\) 18.9101 0.880731 0.440366 0.897819i \(-0.354849\pi\)
0.440366 + 0.897819i \(0.354849\pi\)
\(462\) 0 0
\(463\) −26.8543 −1.24802 −0.624011 0.781415i \(-0.714498\pi\)
−0.624011 + 0.781415i \(0.714498\pi\)
\(464\) −2.39312 −0.111098
\(465\) 0 0
\(466\) 5.21798 0.241718
\(467\) −39.9332 −1.84789 −0.923944 0.382527i \(-0.875054\pi\)
−0.923944 + 0.382527i \(0.875054\pi\)
\(468\) 7.95715 0.367820
\(469\) −7.58967 −0.350459
\(470\) 5.37169 0.247778
\(471\) −22.1109 −1.01881
\(472\) 3.29273 0.151560
\(473\) 0 0
\(474\) 4.92525 0.226224
\(475\) 0.707269 0.0324517
\(476\) 5.19656 0.238184
\(477\) −3.82487 −0.175129
\(478\) −30.3032 −1.38604
\(479\) −18.2646 −0.834530 −0.417265 0.908785i \(-0.637011\pi\)
−0.417265 + 0.908785i \(0.637011\pi\)
\(480\) −2.48929 −0.113620
\(481\) 18.3503 0.836700
\(482\) 11.4145 0.519918
\(483\) 12.8824 0.586170
\(484\) 0 0
\(485\) −4.17513 −0.189583
\(486\) −22.3074 −1.01189
\(487\) 13.1323 0.595081 0.297540 0.954709i \(-0.403834\pi\)
0.297540 + 0.954709i \(0.403834\pi\)
\(488\) 7.78202 0.352275
\(489\) 26.2008 1.18484
\(490\) −1.00000 −0.0451754
\(491\) −19.4145 −0.876166 −0.438083 0.898935i \(-0.644342\pi\)
−0.438083 + 0.898935i \(0.644342\pi\)
\(492\) −12.3931 −0.558725
\(493\) −12.4360 −0.560088
\(494\) 1.76060 0.0792130
\(495\) 0 0
\(496\) 0 0
\(497\) 12.1751 0.546129
\(498\) 11.1751 0.500770
\(499\) −10.3503 −0.463342 −0.231671 0.972794i \(-0.574419\pi\)
−0.231671 + 0.972794i \(0.574419\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 56.0722 2.50512
\(502\) −12.7606 −0.569533
\(503\) −22.0894 −0.984920 −0.492460 0.870335i \(-0.663902\pi\)
−0.492460 + 0.870335i \(0.663902\pi\)
\(504\) 3.19656 0.142386
\(505\) 6.48929 0.288770
\(506\) 0 0
\(507\) −16.9357 −0.752142
\(508\) 2.82487 0.125333
\(509\) −30.8652 −1.36808 −0.684038 0.729447i \(-0.739778\pi\)
−0.684038 + 0.729447i \(0.739778\pi\)
\(510\) −12.9357 −0.572804
\(511\) −0.175135 −0.00774750
\(512\) 1.00000 0.0441942
\(513\) 0.346059 0.0152789
\(514\) −3.58967 −0.158334
\(515\) −3.78202 −0.166656
\(516\) −22.7862 −1.00311
\(517\) 0 0
\(518\) 7.37169 0.323894
\(519\) −33.7648 −1.48211
\(520\) −2.48929 −0.109163
\(521\) −0.350269 −0.0153456 −0.00767279 0.999971i \(-0.502442\pi\)
−0.00767279 + 0.999971i \(0.502442\pi\)
\(522\) −7.64973 −0.334820
\(523\) 25.1898 1.10147 0.550737 0.834679i \(-0.314347\pi\)
0.550737 + 0.834679i \(0.314347\pi\)
\(524\) −15.8181 −0.691018
\(525\) 2.48929 0.108641
\(526\) 2.63252 0.114783
\(527\) 0 0
\(528\) 0 0
\(529\) 3.78202 0.164436
\(530\) 1.19656 0.0519751
\(531\) 10.5254 0.456764
\(532\) 0.707269 0.0306640
\(533\) −12.3931 −0.536805
\(534\) −2.43596 −0.105414
\(535\) 4.21798 0.182359
\(536\) −7.58967 −0.327824
\(537\) 65.5934 2.83057
\(538\) 12.8996 0.556142
\(539\) 0 0
\(540\) −0.489289 −0.0210556
\(541\) 0.243614 0.0104738 0.00523689 0.999986i \(-0.498333\pi\)
0.00523689 + 0.999986i \(0.498333\pi\)
\(542\) −8.93573 −0.383823
\(543\) 3.69679 0.158644
\(544\) 5.19656 0.222801
\(545\) 9.37169 0.401439
\(546\) 6.19656 0.265188
\(547\) 28.4826 1.21783 0.608913 0.793237i \(-0.291606\pi\)
0.608913 + 0.793237i \(0.291606\pi\)
\(548\) 17.1323 0.731855
\(549\) 24.8757 1.06167
\(550\) 0 0
\(551\) −1.69258 −0.0721062
\(552\) 12.8824 0.548312
\(553\) 1.97858 0.0841377
\(554\) −10.0000 −0.424859
\(555\) −18.3503 −0.778926
\(556\) 16.4721 0.698572
\(557\) −18.3931 −0.779341 −0.389671 0.920954i \(-0.627411\pi\)
−0.389671 + 0.920954i \(0.627411\pi\)
\(558\) 0 0
\(559\) −22.7862 −0.963755
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 20.5212 0.865634
\(563\) −20.4464 −0.861715 −0.430857 0.902420i \(-0.641789\pi\)
−0.430857 + 0.902420i \(0.641789\pi\)
\(564\) −13.3717 −0.563050
\(565\) −10.5897 −0.445511
\(566\) −0.681635 −0.0286513
\(567\) −8.37169 −0.351578
\(568\) 12.1751 0.510857
\(569\) −7.67115 −0.321591 −0.160796 0.986988i \(-0.551406\pi\)
−0.160796 + 0.986988i \(0.551406\pi\)
\(570\) −1.76060 −0.0737433
\(571\) −0.149501 −0.00625642 −0.00312821 0.999995i \(-0.500996\pi\)
−0.00312821 + 0.999995i \(0.500996\pi\)
\(572\) 0 0
\(573\) 25.3822 1.06036
\(574\) −4.97858 −0.207802
\(575\) 5.17513 0.215818
\(576\) 3.19656 0.133190
\(577\) 14.0256 0.583895 0.291947 0.956434i \(-0.405697\pi\)
0.291947 + 0.956434i \(0.405697\pi\)
\(578\) 10.0042 0.416120
\(579\) −27.3393 −1.13618
\(580\) 2.39312 0.0993687
\(581\) 4.48929 0.186247
\(582\) 10.3931 0.430808
\(583\) 0 0
\(584\) −0.175135 −0.00724713
\(585\) −7.95715 −0.328988
\(586\) −19.4250 −0.802440
\(587\) 28.8396 1.19034 0.595168 0.803601i \(-0.297086\pi\)
0.595168 + 0.803601i \(0.297086\pi\)
\(588\) 2.48929 0.102657
\(589\) 0 0
\(590\) −3.29273 −0.135560
\(591\) 68.4653 2.81629
\(592\) 7.37169 0.302975
\(593\) 22.9529 0.942564 0.471282 0.881982i \(-0.343791\pi\)
0.471282 + 0.881982i \(0.343791\pi\)
\(594\) 0 0
\(595\) −5.19656 −0.213038
\(596\) −14.9357 −0.611791
\(597\) 18.8291 0.770623
\(598\) 12.8824 0.526801
\(599\) −0.807653 −0.0329998 −0.0164999 0.999864i \(-0.505252\pi\)
−0.0164999 + 0.999864i \(0.505252\pi\)
\(600\) 2.48929 0.101625
\(601\) −7.21377 −0.294256 −0.147128 0.989117i \(-0.547003\pi\)
−0.147128 + 0.989117i \(0.547003\pi\)
\(602\) −9.15371 −0.373077
\(603\) −24.2608 −0.987977
\(604\) −13.1323 −0.534345
\(605\) 0 0
\(606\) −16.1537 −0.656200
\(607\) 2.52540 0.102503 0.0512515 0.998686i \(-0.483679\pi\)
0.0512515 + 0.998686i \(0.483679\pi\)
\(608\) 0.707269 0.0286835
\(609\) −5.95715 −0.241396
\(610\) −7.78202 −0.315085
\(611\) −13.3717 −0.540961
\(612\) 16.6111 0.671464
\(613\) 3.19656 0.129108 0.0645539 0.997914i \(-0.479438\pi\)
0.0645539 + 0.997914i \(0.479438\pi\)
\(614\) 14.1923 0.572757
\(615\) 12.3931 0.499739
\(616\) 0 0
\(617\) −36.6577 −1.47578 −0.737892 0.674919i \(-0.764178\pi\)
−0.737892 + 0.674919i \(0.764178\pi\)
\(618\) 9.41454 0.378708
\(619\) −35.6174 −1.43158 −0.715791 0.698314i \(-0.753934\pi\)
−0.715791 + 0.698314i \(0.753934\pi\)
\(620\) 0 0
\(621\) 2.53213 0.101611
\(622\) −28.8929 −1.15850
\(623\) −0.978577 −0.0392059
\(624\) 6.19656 0.248061
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −8.88240 −0.354446
\(629\) 38.3074 1.52742
\(630\) −3.19656 −0.127354
\(631\) −35.6535 −1.41934 −0.709671 0.704533i \(-0.751157\pi\)
−0.709671 + 0.704533i \(0.751157\pi\)
\(632\) 1.97858 0.0787036
\(633\) −51.7429 −2.05660
\(634\) 7.97437 0.316703
\(635\) −2.82487 −0.112101
\(636\) −2.97858 −0.118108
\(637\) 2.48929 0.0986292
\(638\) 0 0
\(639\) 38.9185 1.53959
\(640\) −1.00000 −0.0395285
\(641\) 5.97858 0.236140 0.118070 0.993005i \(-0.462329\pi\)
0.118070 + 0.993005i \(0.462329\pi\)
\(642\) −10.4998 −0.414393
\(643\) −41.3717 −1.63154 −0.815770 0.578376i \(-0.803687\pi\)
−0.815770 + 0.578376i \(0.803687\pi\)
\(644\) 5.17513 0.203929
\(645\) 22.7862 0.897207
\(646\) 3.67536 0.144605
\(647\) 42.2902 1.66260 0.831300 0.555824i \(-0.187597\pi\)
0.831300 + 0.555824i \(0.187597\pi\)
\(648\) −8.37169 −0.328871
\(649\) 0 0
\(650\) 2.48929 0.0976379
\(651\) 0 0
\(652\) 10.5254 0.412207
\(653\) 18.7262 0.732812 0.366406 0.930455i \(-0.380588\pi\)
0.366406 + 0.930455i \(0.380588\pi\)
\(654\) −23.3288 −0.912230
\(655\) 15.8181 0.618066
\(656\) −4.97858 −0.194381
\(657\) −0.559828 −0.0218410
\(658\) −5.37169 −0.209410
\(659\) −23.9143 −0.931569 −0.465785 0.884898i \(-0.654228\pi\)
−0.465785 + 0.884898i \(0.654228\pi\)
\(660\) 0 0
\(661\) 41.6174 1.61873 0.809364 0.587307i \(-0.199812\pi\)
0.809364 + 0.587307i \(0.199812\pi\)
\(662\) 8.62831 0.335349
\(663\) 32.2008 1.25057
\(664\) 4.48929 0.174218
\(665\) −0.707269 −0.0274267
\(666\) 23.5640 0.913088
\(667\) −12.3847 −0.479537
\(668\) 22.5254 0.871534
\(669\) 41.7220 1.61306
\(670\) 7.58967 0.293215
\(671\) 0 0
\(672\) 2.48929 0.0960264
\(673\) 25.5254 0.983932 0.491966 0.870614i \(-0.336278\pi\)
0.491966 + 0.870614i \(0.336278\pi\)
\(674\) −7.80344 −0.300577
\(675\) 0.489289 0.0188327
\(676\) −6.80344 −0.261671
\(677\) −21.8610 −0.840186 −0.420093 0.907481i \(-0.638003\pi\)
−0.420093 + 0.907481i \(0.638003\pi\)
\(678\) 26.3608 1.01238
\(679\) 4.17513 0.160227
\(680\) −5.19656 −0.199279
\(681\) −33.2860 −1.27552
\(682\) 0 0
\(683\) −11.9744 −0.458186 −0.229093 0.973404i \(-0.573576\pi\)
−0.229093 + 0.973404i \(0.573576\pi\)
\(684\) 2.26083 0.0864448
\(685\) −17.1323 −0.654591
\(686\) 1.00000 0.0381802
\(687\) −10.2865 −0.392453
\(688\) −9.15371 −0.348982
\(689\) −2.97858 −0.113475
\(690\) −12.8824 −0.490425
\(691\) −29.4611 −1.12075 −0.560377 0.828238i \(-0.689344\pi\)
−0.560377 + 0.828238i \(0.689344\pi\)
\(692\) −13.5640 −0.515627
\(693\) 0 0
\(694\) 27.1966 1.03237
\(695\) −16.4721 −0.624821
\(696\) −5.95715 −0.225805
\(697\) −25.8715 −0.979952
\(698\) −28.0533 −1.06183
\(699\) 12.9891 0.491291
\(700\) 1.00000 0.0377964
\(701\) 8.58546 0.324268 0.162134 0.986769i \(-0.448162\pi\)
0.162134 + 0.986769i \(0.448162\pi\)
\(702\) 1.21798 0.0459697
\(703\) 5.21377 0.196641
\(704\) 0 0
\(705\) 13.3717 0.503607
\(706\) 12.5254 0.471400
\(707\) −6.48929 −0.244055
\(708\) 8.19656 0.308045
\(709\) −18.7005 −0.702313 −0.351157 0.936317i \(-0.614212\pi\)
−0.351157 + 0.936317i \(0.614212\pi\)
\(710\) −12.1751 −0.456925
\(711\) 6.32464 0.237192
\(712\) −0.978577 −0.0366737
\(713\) 0 0
\(714\) 12.9357 0.484108
\(715\) 0 0
\(716\) 26.3503 0.984756
\(717\) −75.4334 −2.81711
\(718\) −21.3461 −0.796628
\(719\) −17.1365 −0.639084 −0.319542 0.947572i \(-0.603529\pi\)
−0.319542 + 0.947572i \(0.603529\pi\)
\(720\) −3.19656 −0.119129
\(721\) 3.78202 0.140850
\(722\) −18.4998 −0.688490
\(723\) 28.4141 1.05673
\(724\) 1.48508 0.0551925
\(725\) −2.39312 −0.0888781
\(726\) 0 0
\(727\) 7.41454 0.274990 0.137495 0.990502i \(-0.456095\pi\)
0.137495 + 0.990502i \(0.456095\pi\)
\(728\) 2.48929 0.0922592
\(729\) −30.4145 −1.12646
\(730\) 0.175135 0.00648203
\(731\) −47.5678 −1.75936
\(732\) 19.3717 0.715998
\(733\) −38.9546 −1.43882 −0.719411 0.694584i \(-0.755588\pi\)
−0.719411 + 0.694584i \(0.755588\pi\)
\(734\) 24.7178 0.912349
\(735\) −2.48929 −0.0918188
\(736\) 5.17513 0.190758
\(737\) 0 0
\(738\) −15.9143 −0.585814
\(739\) −1.84208 −0.0677619 −0.0338810 0.999426i \(-0.510787\pi\)
−0.0338810 + 0.999426i \(0.510787\pi\)
\(740\) −7.37169 −0.270989
\(741\) 4.38263 0.161000
\(742\) −1.19656 −0.0439270
\(743\) 7.56404 0.277498 0.138749 0.990328i \(-0.455692\pi\)
0.138749 + 0.990328i \(0.455692\pi\)
\(744\) 0 0
\(745\) 14.9357 0.547203
\(746\) −12.9185 −0.472981
\(747\) 14.3503 0.525049
\(748\) 0 0
\(749\) −4.21798 −0.154122
\(750\) −2.48929 −0.0908960
\(751\) −11.9614 −0.436476 −0.218238 0.975896i \(-0.570031\pi\)
−0.218238 + 0.975896i \(0.570031\pi\)
\(752\) −5.37169 −0.195885
\(753\) −31.7648 −1.15757
\(754\) −5.95715 −0.216947
\(755\) 13.1323 0.477933
\(756\) 0.489289 0.0177953
\(757\) −35.1966 −1.27924 −0.639620 0.768691i \(-0.720908\pi\)
−0.639620 + 0.768691i \(0.720908\pi\)
\(758\) 35.9227 1.30477
\(759\) 0 0
\(760\) −0.707269 −0.0256553
\(761\) 27.3288 0.990670 0.495335 0.868702i \(-0.335045\pi\)
0.495335 + 0.868702i \(0.335045\pi\)
\(762\) 7.03190 0.254739
\(763\) −9.37169 −0.339278
\(764\) 10.1966 0.368899
\(765\) −16.6111 −0.600575
\(766\) 11.6325 0.420300
\(767\) 8.19656 0.295961
\(768\) 2.48929 0.0898245
\(769\) 43.2860 1.56093 0.780467 0.625198i \(-0.214982\pi\)
0.780467 + 0.625198i \(0.214982\pi\)
\(770\) 0 0
\(771\) −8.93573 −0.321813
\(772\) −10.9828 −0.395279
\(773\) −39.7324 −1.42908 −0.714538 0.699596i \(-0.753363\pi\)
−0.714538 + 0.699596i \(0.753363\pi\)
\(774\) −29.2604 −1.05174
\(775\) 0 0
\(776\) 4.17513 0.149879
\(777\) 18.3503 0.658312
\(778\) −25.2860 −0.906547
\(779\) −3.52119 −0.126160
\(780\) −6.19656 −0.221872
\(781\) 0 0
\(782\) 26.8929 0.961688
\(783\) −1.17092 −0.0418454
\(784\) 1.00000 0.0357143
\(785\) 8.88240 0.317027
\(786\) −39.3759 −1.40449
\(787\) −2.97858 −0.106175 −0.0530874 0.998590i \(-0.516906\pi\)
−0.0530874 + 0.998590i \(0.516906\pi\)
\(788\) 27.5040 0.979789
\(789\) 6.55310 0.233296
\(790\) −1.97858 −0.0703946
\(791\) 10.5897 0.376525
\(792\) 0 0
\(793\) 19.3717 0.687909
\(794\) 5.17935 0.183808
\(795\) 2.97858 0.105639
\(796\) 7.56404 0.268100
\(797\) 33.6258 1.19109 0.595543 0.803323i \(-0.296937\pi\)
0.595543 + 0.803323i \(0.296937\pi\)
\(798\) 1.76060 0.0623244
\(799\) −27.9143 −0.987537
\(800\) 1.00000 0.0353553
\(801\) −3.12808 −0.110525
\(802\) 32.7434 1.15621
\(803\) 0 0
\(804\) −18.8929 −0.666301
\(805\) −5.17513 −0.182400
\(806\) 0 0
\(807\) 32.1109 1.13036
\(808\) −6.48929 −0.228292
\(809\) 29.3545 1.03205 0.516024 0.856574i \(-0.327411\pi\)
0.516024 + 0.856574i \(0.327411\pi\)
\(810\) 8.37169 0.294151
\(811\) −28.9013 −1.01486 −0.507431 0.861692i \(-0.669405\pi\)
−0.507431 + 0.861692i \(0.669405\pi\)
\(812\) −2.39312 −0.0839819
\(813\) −22.2436 −0.780118
\(814\) 0 0
\(815\) −10.5254 −0.368689
\(816\) 12.9357 0.452841
\(817\) −6.47414 −0.226501
\(818\) −17.3717 −0.607387
\(819\) 7.95715 0.278045
\(820\) 4.97858 0.173859
\(821\) −42.1495 −1.47103 −0.735514 0.677510i \(-0.763059\pi\)
−0.735514 + 0.677510i \(0.763059\pi\)
\(822\) 42.6472 1.48749
\(823\) −42.3931 −1.47773 −0.738866 0.673853i \(-0.764638\pi\)
−0.738866 + 0.673853i \(0.764638\pi\)
\(824\) 3.78202 0.131753
\(825\) 0 0
\(826\) 3.29273 0.114569
\(827\) 54.3969 1.89156 0.945782 0.324802i \(-0.105298\pi\)
0.945782 + 0.324802i \(0.105298\pi\)
\(828\) 16.5426 0.574896
\(829\) 29.6174 1.02865 0.514327 0.857594i \(-0.328042\pi\)
0.514327 + 0.857594i \(0.328042\pi\)
\(830\) −4.48929 −0.155825
\(831\) −24.8929 −0.863525
\(832\) 2.48929 0.0863006
\(833\) 5.19656 0.180050
\(834\) 41.0038 1.41984
\(835\) −22.5254 −0.779524
\(836\) 0 0
\(837\) 0 0
\(838\) 4.31415 0.149030
\(839\) 46.4998 1.60535 0.802675 0.596416i \(-0.203409\pi\)
0.802675 + 0.596416i \(0.203409\pi\)
\(840\) −2.48929 −0.0858886
\(841\) −23.2730 −0.802517
\(842\) −40.7005 −1.40263
\(843\) 51.0832 1.75940
\(844\) −20.7862 −0.715492
\(845\) 6.80344 0.234046
\(846\) −17.1709 −0.590349
\(847\) 0 0
\(848\) −1.19656 −0.0410900
\(849\) −1.69679 −0.0582336
\(850\) 5.19656 0.178240
\(851\) 38.1495 1.30775
\(852\) 30.3074 1.03832
\(853\) −23.5536 −0.806458 −0.403229 0.915099i \(-0.632112\pi\)
−0.403229 + 0.915099i \(0.632112\pi\)
\(854\) 7.78202 0.266295
\(855\) −2.26083 −0.0773186
\(856\) −4.21798 −0.144168
\(857\) 18.9786 0.648296 0.324148 0.946006i \(-0.394922\pi\)
0.324148 + 0.946006i \(0.394922\pi\)
\(858\) 0 0
\(859\) −16.3246 −0.556989 −0.278495 0.960438i \(-0.589835\pi\)
−0.278495 + 0.960438i \(0.589835\pi\)
\(860\) 9.15371 0.312139
\(861\) −12.3931 −0.422356
\(862\) −31.7862 −1.08264
\(863\) −38.3074 −1.30400 −0.652000 0.758219i \(-0.726070\pi\)
−0.652000 + 0.758219i \(0.726070\pi\)
\(864\) 0.489289 0.0166459
\(865\) 13.5640 0.461191
\(866\) −28.0722 −0.953933
\(867\) 24.9034 0.845763
\(868\) 0 0
\(869\) 0 0
\(870\) 5.95715 0.201966
\(871\) −18.8929 −0.640161
\(872\) −9.37169 −0.317366
\(873\) 13.3461 0.451696
\(874\) 3.66021 0.123809
\(875\) −1.00000 −0.0338062
\(876\) −0.435961 −0.0147298
\(877\) −20.4826 −0.691647 −0.345823 0.938300i \(-0.612400\pi\)
−0.345823 + 0.938300i \(0.612400\pi\)
\(878\) 28.9442 0.976818
\(879\) −48.3545 −1.63096
\(880\) 0 0
\(881\) −13.0937 −0.441136 −0.220568 0.975372i \(-0.570791\pi\)
−0.220568 + 0.975372i \(0.570791\pi\)
\(882\) 3.19656 0.107634
\(883\) 21.1537 0.711879 0.355940 0.934509i \(-0.384161\pi\)
0.355940 + 0.934509i \(0.384161\pi\)
\(884\) 12.9357 0.435076
\(885\) −8.19656 −0.275524
\(886\) 7.79923 0.262020
\(887\) −14.7350 −0.494752 −0.247376 0.968920i \(-0.579568\pi\)
−0.247376 + 0.968920i \(0.579568\pi\)
\(888\) 18.3503 0.615795
\(889\) 2.82487 0.0947429
\(890\) 0.978577 0.0328020
\(891\) 0 0
\(892\) 16.7606 0.561186
\(893\) −3.79923 −0.127136
\(894\) −37.1793 −1.24346
\(895\) −26.3503 −0.880792
\(896\) 1.00000 0.0334077
\(897\) 32.0680 1.07072
\(898\) −11.7862 −0.393312
\(899\) 0 0
\(900\) 3.19656 0.106552
\(901\) −6.21798 −0.207151
\(902\) 0 0
\(903\) −22.7862 −0.758278
\(904\) 10.5897 0.352207
\(905\) −1.48508 −0.0493657
\(906\) −32.6901 −1.08605
\(907\) −27.3032 −0.906588 −0.453294 0.891361i \(-0.649751\pi\)
−0.453294 + 0.891361i \(0.649751\pi\)
\(908\) −13.3717 −0.443755
\(909\) −20.7434 −0.688015
\(910\) −2.48929 −0.0825191
\(911\) −26.1709 −0.867081 −0.433541 0.901134i \(-0.642736\pi\)
−0.433541 + 0.901134i \(0.642736\pi\)
\(912\) 1.76060 0.0582992
\(913\) 0 0
\(914\) −30.1109 −0.995979
\(915\) −19.3717 −0.640408
\(916\) −4.13229 −0.136535
\(917\) −15.8181 −0.522361
\(918\) 2.54262 0.0839188
\(919\) −39.1281 −1.29072 −0.645358 0.763880i \(-0.723292\pi\)
−0.645358 + 0.763880i \(0.723292\pi\)
\(920\) −5.17513 −0.170619
\(921\) 35.3288 1.16413
\(922\) 18.9101 0.622771
\(923\) 30.3074 0.997581
\(924\) 0 0
\(925\) 7.37169 0.242380
\(926\) −26.8543 −0.882485
\(927\) 12.0894 0.397069
\(928\) −2.39312 −0.0785579
\(929\) 60.2302 1.97609 0.988044 0.154175i \(-0.0492718\pi\)
0.988044 + 0.154175i \(0.0492718\pi\)
\(930\) 0 0
\(931\) 0.707269 0.0231798
\(932\) 5.21798 0.170921
\(933\) −71.9227 −2.35464
\(934\) −39.9332 −1.30665
\(935\) 0 0
\(936\) 7.95715 0.260088
\(937\) 3.36327 0.109873 0.0549366 0.998490i \(-0.482504\pi\)
0.0549366 + 0.998490i \(0.482504\pi\)
\(938\) −7.58967 −0.247812
\(939\) −24.8929 −0.812349
\(940\) 5.37169 0.175205
\(941\) −6.22640 −0.202975 −0.101487 0.994837i \(-0.532360\pi\)
−0.101487 + 0.994837i \(0.532360\pi\)
\(942\) −22.1109 −0.720411
\(943\) −25.7648 −0.839018
\(944\) 3.29273 0.107169
\(945\) −0.489289 −0.0159166
\(946\) 0 0
\(947\) −23.2432 −0.755301 −0.377651 0.925948i \(-0.623268\pi\)
−0.377651 + 0.925948i \(0.623268\pi\)
\(948\) 4.92525 0.159965
\(949\) −0.435961 −0.0141519
\(950\) 0.707269 0.0229468
\(951\) 19.8505 0.643697
\(952\) 5.19656 0.168421
\(953\) −9.87567 −0.319904 −0.159952 0.987125i \(-0.551134\pi\)
−0.159952 + 0.987125i \(0.551134\pi\)
\(954\) −3.82487 −0.123835
\(955\) −10.1966 −0.329953
\(956\) −30.3032 −0.980076
\(957\) 0 0
\(958\) −18.2646 −0.590102
\(959\) 17.1323 0.553230
\(960\) −2.48929 −0.0803414
\(961\) −31.0000 −1.00000
\(962\) 18.3503 0.591636
\(963\) −13.4830 −0.434484
\(964\) 11.4145 0.367638
\(965\) 10.9828 0.353548
\(966\) 12.8824 0.414485
\(967\) −37.1365 −1.19423 −0.597115 0.802156i \(-0.703686\pi\)
−0.597115 + 0.802156i \(0.703686\pi\)
\(968\) 0 0
\(969\) 9.14904 0.293910
\(970\) −4.17513 −0.134055
\(971\) 28.8223 0.924953 0.462477 0.886632i \(-0.346961\pi\)
0.462477 + 0.886632i \(0.346961\pi\)
\(972\) −22.3074 −0.715511
\(973\) 16.4721 0.528071
\(974\) 13.1323 0.420786
\(975\) 6.19656 0.198449
\(976\) 7.78202 0.249096
\(977\) 43.6111 1.39524 0.697621 0.716467i \(-0.254242\pi\)
0.697621 + 0.716467i \(0.254242\pi\)
\(978\) 26.2008 0.837808
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −29.9572 −0.956458
\(982\) −19.4145 −0.619543
\(983\) 12.1923 0.388875 0.194438 0.980915i \(-0.437712\pi\)
0.194438 + 0.980915i \(0.437712\pi\)
\(984\) −12.3931 −0.395078
\(985\) −27.5040 −0.876350
\(986\) −12.4360 −0.396042
\(987\) −13.3717 −0.425626
\(988\) 1.76060 0.0560120
\(989\) −47.3717 −1.50633
\(990\) 0 0
\(991\) 22.3717 0.710660 0.355330 0.934741i \(-0.384368\pi\)
0.355330 + 0.934741i \(0.384368\pi\)
\(992\) 0 0
\(993\) 21.4783 0.681595
\(994\) 12.1751 0.386172
\(995\) −7.56404 −0.239796
\(996\) 11.1751 0.354098
\(997\) 32.4036 1.02623 0.513116 0.858319i \(-0.328491\pi\)
0.513116 + 0.858319i \(0.328491\pi\)
\(998\) −10.3503 −0.327632
\(999\) 3.60688 0.114117
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 8470.2.a.ck.1.3 yes 3
11.10 odd 2 8470.2.a.cg.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cg.1.3 3 11.10 odd 2
8470.2.a.ck.1.3 yes 3 1.1 even 1 trivial