Properties

Label 637.2.a.j.1.3
Level $637$
Weight $2$
Character 637.1
Self dual yes
Analytic conductor $5.086$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 637.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.34292 q^{2} +1.14637 q^{3} +3.48929 q^{4} +1.34292 q^{5} +2.68585 q^{6} +3.48929 q^{8} -1.68585 q^{9} +O(q^{10})\) \(q+2.34292 q^{2} +1.14637 q^{3} +3.48929 q^{4} +1.34292 q^{5} +2.68585 q^{6} +3.48929 q^{8} -1.68585 q^{9} +3.14637 q^{10} +1.14637 q^{11} +4.00000 q^{12} -1.00000 q^{13} +1.53948 q^{15} +1.19656 q^{16} -5.83221 q^{17} -3.94981 q^{18} +3.34292 q^{19} +4.68585 q^{20} +2.68585 q^{22} -3.17513 q^{23} +4.00000 q^{24} -3.19656 q^{25} -2.34292 q^{26} -5.37169 q^{27} +10.4893 q^{29} +3.60688 q^{30} -1.63565 q^{31} -4.17513 q^{32} +1.31415 q^{33} -13.6644 q^{34} -5.88240 q^{36} +8.51806 q^{37} +7.83221 q^{38} -1.14637 q^{39} +4.68585 q^{40} +0.292731 q^{41} -8.15371 q^{43} +4.00000 q^{44} -2.26396 q^{45} -7.43910 q^{46} +10.6142 q^{47} +1.37169 q^{48} -7.48929 q^{50} -6.68585 q^{51} -3.48929 q^{52} -0.782020 q^{53} -12.5855 q^{54} +1.53948 q^{55} +3.83221 q^{57} +24.5756 q^{58} -12.6430 q^{59} +5.37169 q^{60} +2.00000 q^{61} -3.83221 q^{62} -12.1751 q^{64} -1.34292 q^{65} +3.07896 q^{66} -6.10038 q^{67} -20.3503 q^{68} -3.63986 q^{69} +1.53948 q^{71} -5.88240 q^{72} +15.3001 q^{73} +19.9572 q^{74} -3.66442 q^{75} +11.6644 q^{76} -2.68585 q^{78} +0.882404 q^{79} +1.60688 q^{80} -1.10038 q^{81} +0.685846 q^{82} +12.1292 q^{83} -7.83221 q^{85} -19.1035 q^{86} +12.0246 q^{87} +4.00000 q^{88} -5.73604 q^{89} -5.30429 q^{90} -11.0790 q^{92} -1.87506 q^{93} +24.8683 q^{94} +4.48929 q^{95} -4.78623 q^{96} +5.34292 q^{97} -1.93260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9} + 8 q^{10} + 2 q^{11} + 12 q^{12} - 3 q^{13} - 6 q^{15} - q^{16} - 4 q^{17} - 15 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 10 q^{23} + 12 q^{24} - 5 q^{25} - q^{26} + 8 q^{27} + 24 q^{29} + 20 q^{30} + 4 q^{31} + 7 q^{32} + 16 q^{33} - 14 q^{34} - q^{36} + 10 q^{38} - 2 q^{39} + 2 q^{40} - 2 q^{41} + 10 q^{43} + 12 q^{44} - 22 q^{45} - 18 q^{46} + 8 q^{47} - 20 q^{48} - 15 q^{50} - 8 q^{51} - 3 q^{52} + 8 q^{53} - 32 q^{54} - 6 q^{55} - 2 q^{57} + 12 q^{58} + 4 q^{59} - 8 q^{60} + 6 q^{61} + 2 q^{62} - 17 q^{64} + 2 q^{65} - 12 q^{66} - 12 q^{67} - 22 q^{68} + 6 q^{69} - 6 q^{71} - q^{72} + 10 q^{73} + 30 q^{74} + 16 q^{75} + 8 q^{76} + 4 q^{78} - 14 q^{79} + 14 q^{80} + 3 q^{81} - 10 q^{82} + 12 q^{83} - 10 q^{85} - 26 q^{86} + 26 q^{87} + 12 q^{88} - 2 q^{89} + 28 q^{90} - 12 q^{92} - 22 q^{93} + 10 q^{94} + 6 q^{95} + 4 q^{96} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.34292 1.65670 0.828348 0.560213i \(-0.189281\pi\)
0.828348 + 0.560213i \(0.189281\pi\)
\(3\) 1.14637 0.661854 0.330927 0.943656i \(-0.392639\pi\)
0.330927 + 0.943656i \(0.392639\pi\)
\(4\) 3.48929 1.74464
\(5\) 1.34292 0.600573 0.300287 0.953849i \(-0.402918\pi\)
0.300287 + 0.953849i \(0.402918\pi\)
\(6\) 2.68585 1.09649
\(7\) 0 0
\(8\) 3.48929 1.23365
\(9\) −1.68585 −0.561949
\(10\) 3.14637 0.994968
\(11\) 1.14637 0.345642 0.172821 0.984953i \(-0.444712\pi\)
0.172821 + 0.984953i \(0.444712\pi\)
\(12\) 4.00000 1.15470
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.53948 0.397492
\(16\) 1.19656 0.299139
\(17\) −5.83221 −1.41452 −0.707260 0.706954i \(-0.750069\pi\)
−0.707260 + 0.706954i \(0.750069\pi\)
\(18\) −3.94981 −0.930979
\(19\) 3.34292 0.766919 0.383460 0.923558i \(-0.374733\pi\)
0.383460 + 0.923558i \(0.374733\pi\)
\(20\) 4.68585 1.04779
\(21\) 0 0
\(22\) 2.68585 0.572624
\(23\) −3.17513 −0.662061 −0.331031 0.943620i \(-0.607396\pi\)
−0.331031 + 0.943620i \(0.607396\pi\)
\(24\) 4.00000 0.816497
\(25\) −3.19656 −0.639312
\(26\) −2.34292 −0.459485
\(27\) −5.37169 −1.03378
\(28\) 0 0
\(29\) 10.4893 1.94781 0.973906 0.226952i \(-0.0728760\pi\)
0.973906 + 0.226952i \(0.0728760\pi\)
\(30\) 3.60688 0.658524
\(31\) −1.63565 −0.293772 −0.146886 0.989153i \(-0.546925\pi\)
−0.146886 + 0.989153i \(0.546925\pi\)
\(32\) −4.17513 −0.738067
\(33\) 1.31415 0.228765
\(34\) −13.6644 −2.34343
\(35\) 0 0
\(36\) −5.88240 −0.980401
\(37\) 8.51806 1.40036 0.700180 0.713966i \(-0.253103\pi\)
0.700180 + 0.713966i \(0.253103\pi\)
\(38\) 7.83221 1.27055
\(39\) −1.14637 −0.183565
\(40\) 4.68585 0.740897
\(41\) 0.292731 0.0457169 0.0228584 0.999739i \(-0.492723\pi\)
0.0228584 + 0.999739i \(0.492723\pi\)
\(42\) 0 0
\(43\) −8.15371 −1.24343 −0.621715 0.783244i \(-0.713564\pi\)
−0.621715 + 0.783244i \(0.713564\pi\)
\(44\) 4.00000 0.603023
\(45\) −2.26396 −0.337491
\(46\) −7.43910 −1.09683
\(47\) 10.6142 1.54824 0.774122 0.633036i \(-0.218191\pi\)
0.774122 + 0.633036i \(0.218191\pi\)
\(48\) 1.37169 0.197987
\(49\) 0 0
\(50\) −7.48929 −1.05915
\(51\) −6.68585 −0.936206
\(52\) −3.48929 −0.483877
\(53\) −0.782020 −0.107419 −0.0537093 0.998557i \(-0.517104\pi\)
−0.0537093 + 0.998557i \(0.517104\pi\)
\(54\) −12.5855 −1.71266
\(55\) 1.53948 0.207584
\(56\) 0 0
\(57\) 3.83221 0.507589
\(58\) 24.5756 3.22693
\(59\) −12.6430 −1.64598 −0.822989 0.568057i \(-0.807695\pi\)
−0.822989 + 0.568057i \(0.807695\pi\)
\(60\) 5.37169 0.693482
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −3.83221 −0.486691
\(63\) 0 0
\(64\) −12.1751 −1.52189
\(65\) −1.34292 −0.166569
\(66\) 3.07896 0.378994
\(67\) −6.10038 −0.745281 −0.372640 0.927976i \(-0.621547\pi\)
−0.372640 + 0.927976i \(0.621547\pi\)
\(68\) −20.3503 −2.46783
\(69\) −3.63986 −0.438188
\(70\) 0 0
\(71\) 1.53948 0.182703 0.0913514 0.995819i \(-0.470881\pi\)
0.0913514 + 0.995819i \(0.470881\pi\)
\(72\) −5.88240 −0.693248
\(73\) 15.3001 1.79074 0.895369 0.445324i \(-0.146912\pi\)
0.895369 + 0.445324i \(0.146912\pi\)
\(74\) 19.9572 2.31997
\(75\) −3.66442 −0.423131
\(76\) 11.6644 1.33800
\(77\) 0 0
\(78\) −2.68585 −0.304112
\(79\) 0.882404 0.0992782 0.0496391 0.998767i \(-0.484193\pi\)
0.0496391 + 0.998767i \(0.484193\pi\)
\(80\) 1.60688 0.179655
\(81\) −1.10038 −0.122265
\(82\) 0.685846 0.0757390
\(83\) 12.1292 1.33135 0.665674 0.746243i \(-0.268144\pi\)
0.665674 + 0.746243i \(0.268144\pi\)
\(84\) 0 0
\(85\) −7.83221 −0.849523
\(86\) −19.1035 −2.05999
\(87\) 12.0246 1.28917
\(88\) 4.00000 0.426401
\(89\) −5.73604 −0.608019 −0.304009 0.952669i \(-0.598325\pi\)
−0.304009 + 0.952669i \(0.598325\pi\)
\(90\) −5.30429 −0.559121
\(91\) 0 0
\(92\) −11.0790 −1.15506
\(93\) −1.87506 −0.194434
\(94\) 24.8683 2.56497
\(95\) 4.48929 0.460591
\(96\) −4.78623 −0.488493
\(97\) 5.34292 0.542492 0.271246 0.962510i \(-0.412564\pi\)
0.271246 + 0.962510i \(0.412564\pi\)
\(98\) 0 0
\(99\) −1.93260 −0.194233
\(100\) −11.1537 −1.11537
\(101\) −11.1464 −1.10910 −0.554552 0.832149i \(-0.687111\pi\)
−0.554552 + 0.832149i \(0.687111\pi\)
\(102\) −15.6644 −1.55101
\(103\) 3.41454 0.336444 0.168222 0.985749i \(-0.446197\pi\)
0.168222 + 0.985749i \(0.446197\pi\)
\(104\) −3.48929 −0.342153
\(105\) 0 0
\(106\) −1.83221 −0.177960
\(107\) 4.97858 0.481297 0.240649 0.970612i \(-0.422640\pi\)
0.240649 + 0.970612i \(0.422640\pi\)
\(108\) −18.7434 −1.80358
\(109\) −13.4966 −1.29274 −0.646372 0.763023i \(-0.723714\pi\)
−0.646372 + 0.763023i \(0.723714\pi\)
\(110\) 3.60688 0.343903
\(111\) 9.76481 0.926835
\(112\) 0 0
\(113\) 16.4464 1.54715 0.773576 0.633704i \(-0.218466\pi\)
0.773576 + 0.633704i \(0.218466\pi\)
\(114\) 8.97858 0.840921
\(115\) −4.26396 −0.397616
\(116\) 36.6002 3.39824
\(117\) 1.68585 0.155857
\(118\) −29.6216 −2.72689
\(119\) 0 0
\(120\) 5.37169 0.490366
\(121\) −9.68585 −0.880531
\(122\) 4.68585 0.424237
\(123\) 0.335577 0.0302579
\(124\) −5.70727 −0.512528
\(125\) −11.0073 −0.984527
\(126\) 0 0
\(127\) 12.0575 1.06993 0.534967 0.844873i \(-0.320324\pi\)
0.534967 + 0.844873i \(0.320324\pi\)
\(128\) −20.1751 −1.78325
\(129\) −9.34713 −0.822969
\(130\) −3.14637 −0.275955
\(131\) 3.66442 0.320162 0.160081 0.987104i \(-0.448824\pi\)
0.160081 + 0.987104i \(0.448824\pi\)
\(132\) 4.58546 0.399113
\(133\) 0 0
\(134\) −14.2927 −1.23470
\(135\) −7.21377 −0.620862
\(136\) −20.3503 −1.74502
\(137\) −13.1035 −1.11951 −0.559755 0.828658i \(-0.689105\pi\)
−0.559755 + 0.828658i \(0.689105\pi\)
\(138\) −8.52792 −0.725945
\(139\) −7.49663 −0.635856 −0.317928 0.948115i \(-0.602987\pi\)
−0.317928 + 0.948115i \(0.602987\pi\)
\(140\) 0 0
\(141\) 12.1678 1.02471
\(142\) 3.60688 0.302683
\(143\) −1.14637 −0.0958639
\(144\) −2.01721 −0.168101
\(145\) 14.0863 1.16980
\(146\) 35.8469 2.96671
\(147\) 0 0
\(148\) 29.7220 2.44313
\(149\) 2.16779 0.177592 0.0887961 0.996050i \(-0.471698\pi\)
0.0887961 + 0.996050i \(0.471698\pi\)
\(150\) −8.58546 −0.701000
\(151\) 14.9112 1.21345 0.606727 0.794910i \(-0.292482\pi\)
0.606727 + 0.794910i \(0.292482\pi\)
\(152\) 11.6644 0.946110
\(153\) 9.83221 0.794887
\(154\) 0 0
\(155\) −2.19656 −0.176432
\(156\) −4.00000 −0.320256
\(157\) −22.8683 −1.82509 −0.912546 0.408975i \(-0.865886\pi\)
−0.912546 + 0.408975i \(0.865886\pi\)
\(158\) 2.06740 0.164474
\(159\) −0.896480 −0.0710955
\(160\) −5.60688 −0.443263
\(161\) 0 0
\(162\) −2.57812 −0.202556
\(163\) 7.07896 0.554467 0.277234 0.960803i \(-0.410582\pi\)
0.277234 + 0.960803i \(0.410582\pi\)
\(164\) 1.02142 0.0797597
\(165\) 1.76481 0.137390
\(166\) 28.4177 2.20564
\(167\) −2.61423 −0.202295 −0.101148 0.994871i \(-0.532251\pi\)
−0.101148 + 0.994871i \(0.532251\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −18.3503 −1.40740
\(171\) −5.63565 −0.430969
\(172\) −28.4507 −2.16934
\(173\) −11.0031 −0.836553 −0.418276 0.908320i \(-0.637366\pi\)
−0.418276 + 0.908320i \(0.637366\pi\)
\(174\) 28.1726 2.13576
\(175\) 0 0
\(176\) 1.37169 0.103395
\(177\) −14.4935 −1.08940
\(178\) −13.4391 −1.00730
\(179\) 23.9614 1.79096 0.895478 0.445105i \(-0.146834\pi\)
0.895478 + 0.445105i \(0.146834\pi\)
\(180\) −7.89962 −0.588803
\(181\) −6.56090 −0.487668 −0.243834 0.969817i \(-0.578405\pi\)
−0.243834 + 0.969817i \(0.578405\pi\)
\(182\) 0 0
\(183\) 2.29273 0.169484
\(184\) −11.0790 −0.816752
\(185\) 11.4391 0.841019
\(186\) −4.39312 −0.322119
\(187\) −6.68585 −0.488917
\(188\) 37.0361 2.70114
\(189\) 0 0
\(190\) 10.5181 0.763060
\(191\) −4.39312 −0.317875 −0.158937 0.987289i \(-0.550807\pi\)
−0.158937 + 0.987289i \(0.550807\pi\)
\(192\) −13.9572 −1.00727
\(193\) −8.29273 −0.596924 −0.298462 0.954422i \(-0.596474\pi\)
−0.298462 + 0.954422i \(0.596474\pi\)
\(194\) 12.5181 0.898744
\(195\) −1.53948 −0.110245
\(196\) 0 0
\(197\) −3.17092 −0.225919 −0.112959 0.993600i \(-0.536033\pi\)
−0.112959 + 0.993600i \(0.536033\pi\)
\(198\) −4.52792 −0.321786
\(199\) 13.5970 0.963867 0.481934 0.876208i \(-0.339935\pi\)
0.481934 + 0.876208i \(0.339935\pi\)
\(200\) −11.1537 −0.788687
\(201\) −6.99327 −0.493267
\(202\) −26.1151 −1.83745
\(203\) 0 0
\(204\) −23.3288 −1.63335
\(205\) 0.393115 0.0274564
\(206\) 8.00000 0.557386
\(207\) 5.35279 0.372045
\(208\) −1.19656 −0.0829663
\(209\) 3.83221 0.265080
\(210\) 0 0
\(211\) 9.27552 0.638553 0.319277 0.947662i \(-0.396560\pi\)
0.319277 + 0.947662i \(0.396560\pi\)
\(212\) −2.72869 −0.187407
\(213\) 1.76481 0.120923
\(214\) 11.6644 0.797364
\(215\) −10.9498 −0.746771
\(216\) −18.7434 −1.27533
\(217\) 0 0
\(218\) −31.6216 −2.14168
\(219\) 17.5395 1.18521
\(220\) 5.37169 0.362159
\(221\) 5.83221 0.392317
\(222\) 22.8782 1.53548
\(223\) 19.5928 1.31203 0.656016 0.754747i \(-0.272240\pi\)
0.656016 + 0.754747i \(0.272240\pi\)
\(224\) 0 0
\(225\) 5.38890 0.359260
\(226\) 38.5328 2.56316
\(227\) 19.6644 1.30517 0.652587 0.757714i \(-0.273684\pi\)
0.652587 + 0.757714i \(0.273684\pi\)
\(228\) 13.3717 0.885562
\(229\) −7.76481 −0.513113 −0.256556 0.966529i \(-0.582588\pi\)
−0.256556 + 0.966529i \(0.582588\pi\)
\(230\) −9.99013 −0.658730
\(231\) 0 0
\(232\) 36.6002 2.40292
\(233\) 12.1966 0.799023 0.399512 0.916728i \(-0.369180\pi\)
0.399512 + 0.916728i \(0.369180\pi\)
\(234\) 3.94981 0.258207
\(235\) 14.2541 0.929835
\(236\) −44.1151 −2.87165
\(237\) 1.01156 0.0657077
\(238\) 0 0
\(239\) −10.2927 −0.665781 −0.332891 0.942965i \(-0.608024\pi\)
−0.332891 + 0.942965i \(0.608024\pi\)
\(240\) 1.84208 0.118906
\(241\) 4.02877 0.259516 0.129758 0.991546i \(-0.458580\pi\)
0.129758 + 0.991546i \(0.458580\pi\)
\(242\) −22.6932 −1.45877
\(243\) 14.8536 0.952861
\(244\) 6.97858 0.446758
\(245\) 0 0
\(246\) 0.786230 0.0501282
\(247\) −3.34292 −0.212705
\(248\) −5.70727 −0.362412
\(249\) 13.9044 0.881158
\(250\) −25.7894 −1.63106
\(251\) −2.91117 −0.183752 −0.0918758 0.995770i \(-0.529286\pi\)
−0.0918758 + 0.995770i \(0.529286\pi\)
\(252\) 0 0
\(253\) −3.63986 −0.228836
\(254\) 28.2499 1.77256
\(255\) −8.97858 −0.562260
\(256\) −22.9185 −1.43241
\(257\) 19.5970 1.22243 0.611214 0.791465i \(-0.290681\pi\)
0.611214 + 0.791465i \(0.290681\pi\)
\(258\) −21.8996 −1.36341
\(259\) 0 0
\(260\) −4.68585 −0.290604
\(261\) −17.6833 −1.09457
\(262\) 8.58546 0.530412
\(263\) 7.56825 0.466678 0.233339 0.972395i \(-0.425035\pi\)
0.233339 + 0.972395i \(0.425035\pi\)
\(264\) 4.58546 0.282216
\(265\) −1.05019 −0.0645128
\(266\) 0 0
\(267\) −6.57560 −0.402420
\(268\) −21.2860 −1.30025
\(269\) 9.47208 0.577523 0.288761 0.957401i \(-0.406757\pi\)
0.288761 + 0.957401i \(0.406757\pi\)
\(270\) −16.9013 −1.02858
\(271\) −29.3717 −1.78420 −0.892102 0.451835i \(-0.850770\pi\)
−0.892102 + 0.451835i \(0.850770\pi\)
\(272\) −6.97858 −0.423138
\(273\) 0 0
\(274\) −30.7005 −1.85469
\(275\) −3.66442 −0.220973
\(276\) −12.7005 −0.764483
\(277\) −1.90383 −0.114390 −0.0571949 0.998363i \(-0.518216\pi\)
−0.0571949 + 0.998363i \(0.518216\pi\)
\(278\) −17.5640 −1.05342
\(279\) 2.75746 0.165085
\(280\) 0 0
\(281\) −20.5756 −1.22744 −0.613719 0.789525i \(-0.710327\pi\)
−0.613719 + 0.789525i \(0.710327\pi\)
\(282\) 28.5082 1.69764
\(283\) 26.9933 1.60458 0.802292 0.596932i \(-0.203614\pi\)
0.802292 + 0.596932i \(0.203614\pi\)
\(284\) 5.37169 0.318751
\(285\) 5.14637 0.304844
\(286\) −2.68585 −0.158817
\(287\) 0 0
\(288\) 7.03863 0.414756
\(289\) 17.0147 1.00086
\(290\) 33.0031 1.93801
\(291\) 6.12494 0.359050
\(292\) 53.3864 3.12420
\(293\) 14.9070 0.870874 0.435437 0.900219i \(-0.356594\pi\)
0.435437 + 0.900219i \(0.356594\pi\)
\(294\) 0 0
\(295\) −16.9786 −0.988531
\(296\) 29.7220 1.72755
\(297\) −6.15792 −0.357319
\(298\) 5.07896 0.294216
\(299\) 3.17513 0.183623
\(300\) −12.7862 −0.738213
\(301\) 0 0
\(302\) 34.9357 2.01033
\(303\) −12.7778 −0.734066
\(304\) 4.00000 0.229416
\(305\) 2.68585 0.153791
\(306\) 23.0361 1.31689
\(307\) −26.0288 −1.48554 −0.742770 0.669546i \(-0.766488\pi\)
−0.742770 + 0.669546i \(0.766488\pi\)
\(308\) 0 0
\(309\) 3.91431 0.222677
\(310\) −5.14637 −0.292294
\(311\) −19.4966 −1.10555 −0.552776 0.833330i \(-0.686432\pi\)
−0.552776 + 0.833330i \(0.686432\pi\)
\(312\) −4.00000 −0.226455
\(313\) 3.48194 0.196811 0.0984055 0.995146i \(-0.468626\pi\)
0.0984055 + 0.995146i \(0.468626\pi\)
\(314\) −53.5787 −3.02362
\(315\) 0 0
\(316\) 3.07896 0.173205
\(317\) 5.02142 0.282031 0.141016 0.990007i \(-0.454963\pi\)
0.141016 + 0.990007i \(0.454963\pi\)
\(318\) −2.10038 −0.117784
\(319\) 12.0246 0.673246
\(320\) −16.3503 −0.914008
\(321\) 5.70727 0.318549
\(322\) 0 0
\(323\) −19.4966 −1.08482
\(324\) −3.83956 −0.213309
\(325\) 3.19656 0.177313
\(326\) 16.5855 0.918584
\(327\) −15.4721 −0.855608
\(328\) 1.02142 0.0563986
\(329\) 0 0
\(330\) 4.13481 0.227614
\(331\) −6.14950 −0.338007 −0.169004 0.985615i \(-0.554055\pi\)
−0.169004 + 0.985615i \(0.554055\pi\)
\(332\) 42.3221 2.32273
\(333\) −14.3601 −0.786931
\(334\) −6.12494 −0.335142
\(335\) −8.19235 −0.447596
\(336\) 0 0
\(337\) −25.6258 −1.39593 −0.697963 0.716134i \(-0.745910\pi\)
−0.697963 + 0.716134i \(0.745910\pi\)
\(338\) 2.34292 0.127438
\(339\) 18.8536 1.02399
\(340\) −27.3288 −1.48211
\(341\) −1.87506 −0.101540
\(342\) −13.2039 −0.713985
\(343\) 0 0
\(344\) −28.4507 −1.53396
\(345\) −4.88806 −0.263164
\(346\) −25.7795 −1.38591
\(347\) −16.7005 −0.896532 −0.448266 0.893900i \(-0.647958\pi\)
−0.448266 + 0.893900i \(0.647958\pi\)
\(348\) 41.9572 2.24914
\(349\) 23.5500 1.26060 0.630300 0.776351i \(-0.282932\pi\)
0.630300 + 0.776351i \(0.282932\pi\)
\(350\) 0 0
\(351\) 5.37169 0.286720
\(352\) −4.78623 −0.255107
\(353\) 7.64973 0.407154 0.203577 0.979059i \(-0.434743\pi\)
0.203577 + 0.979059i \(0.434743\pi\)
\(354\) −33.9572 −1.80480
\(355\) 2.06740 0.109726
\(356\) −20.0147 −1.06078
\(357\) 0 0
\(358\) 56.1396 2.96707
\(359\) 18.3748 0.969786 0.484893 0.874573i \(-0.338858\pi\)
0.484893 + 0.874573i \(0.338858\pi\)
\(360\) −7.89962 −0.416346
\(361\) −7.82487 −0.411835
\(362\) −15.3717 −0.807918
\(363\) −11.1035 −0.582784
\(364\) 0 0
\(365\) 20.5468 1.07547
\(366\) 5.37169 0.280783
\(367\) −5.33871 −0.278679 −0.139339 0.990245i \(-0.544498\pi\)
−0.139339 + 0.990245i \(0.544498\pi\)
\(368\) −3.79923 −0.198049
\(369\) −0.493499 −0.0256906
\(370\) 26.8009 1.39331
\(371\) 0 0
\(372\) −6.54262 −0.339219
\(373\) 21.5212 1.11433 0.557163 0.830403i \(-0.311890\pi\)
0.557163 + 0.830403i \(0.311890\pi\)
\(374\) −15.6644 −0.809988
\(375\) −12.6184 −0.651614
\(376\) 37.0361 1.90999
\(377\) −10.4893 −0.540226
\(378\) 0 0
\(379\) 4.61002 0.236801 0.118400 0.992966i \(-0.462223\pi\)
0.118400 + 0.992966i \(0.462223\pi\)
\(380\) 15.6644 0.803568
\(381\) 13.8223 0.708140
\(382\) −10.2927 −0.526622
\(383\) 8.33558 0.425928 0.212964 0.977060i \(-0.431688\pi\)
0.212964 + 0.977060i \(0.431688\pi\)
\(384\) −23.1281 −1.18025
\(385\) 0 0
\(386\) −19.4292 −0.988922
\(387\) 13.7459 0.698744
\(388\) 18.6430 0.946455
\(389\) −6.44223 −0.326634 −0.163317 0.986574i \(-0.552219\pi\)
−0.163317 + 0.986574i \(0.552219\pi\)
\(390\) −3.60688 −0.182642
\(391\) 18.5181 0.936498
\(392\) 0 0
\(393\) 4.20077 0.211901
\(394\) −7.42923 −0.374279
\(395\) 1.18500 0.0596238
\(396\) −6.74338 −0.338868
\(397\) −1.40046 −0.0702872 −0.0351436 0.999382i \(-0.511189\pi\)
−0.0351436 + 0.999382i \(0.511189\pi\)
\(398\) 31.8568 1.59684
\(399\) 0 0
\(400\) −3.82487 −0.191243
\(401\) −6.97858 −0.348494 −0.174247 0.984702i \(-0.555749\pi\)
−0.174247 + 0.984702i \(0.555749\pi\)
\(402\) −16.3847 −0.817194
\(403\) 1.63565 0.0814777
\(404\) −38.8929 −1.93499
\(405\) −1.47773 −0.0734291
\(406\) 0 0
\(407\) 9.76481 0.484024
\(408\) −23.3288 −1.15495
\(409\) 18.3790 0.908785 0.454392 0.890802i \(-0.349856\pi\)
0.454392 + 0.890802i \(0.349856\pi\)
\(410\) 0.921039 0.0454869
\(411\) −15.0214 −0.740952
\(412\) 11.9143 0.586976
\(413\) 0 0
\(414\) 12.5412 0.616365
\(415\) 16.2885 0.799572
\(416\) 4.17513 0.204703
\(417\) −8.59388 −0.420844
\(418\) 8.97858 0.439157
\(419\) 30.0393 1.46751 0.733757 0.679412i \(-0.237765\pi\)
0.733757 + 0.679412i \(0.237765\pi\)
\(420\) 0 0
\(421\) −8.31729 −0.405360 −0.202680 0.979245i \(-0.564965\pi\)
−0.202680 + 0.979245i \(0.564965\pi\)
\(422\) 21.7318 1.05789
\(423\) −17.8940 −0.870034
\(424\) −2.72869 −0.132517
\(425\) 18.6430 0.904318
\(426\) 4.13481 0.200332
\(427\) 0 0
\(428\) 17.3717 0.839692
\(429\) −1.31415 −0.0634479
\(430\) −25.6546 −1.23717
\(431\) 9.64973 0.464811 0.232406 0.972619i \(-0.425340\pi\)
0.232406 + 0.972619i \(0.425340\pi\)
\(432\) −6.42754 −0.309245
\(433\) −26.3074 −1.26425 −0.632127 0.774865i \(-0.717818\pi\)
−0.632127 + 0.774865i \(0.717818\pi\)
\(434\) 0 0
\(435\) 16.1481 0.774240
\(436\) −47.0937 −2.25538
\(437\) −10.6142 −0.507748
\(438\) 41.0937 1.96353
\(439\) −33.8139 −1.61385 −0.806925 0.590654i \(-0.798870\pi\)
−0.806925 + 0.590654i \(0.798870\pi\)
\(440\) 5.37169 0.256085
\(441\) 0 0
\(442\) 13.6644 0.649950
\(443\) 26.4464 1.25651 0.628254 0.778008i \(-0.283770\pi\)
0.628254 + 0.778008i \(0.283770\pi\)
\(444\) 34.0722 1.61700
\(445\) −7.70306 −0.365160
\(446\) 45.9044 2.17364
\(447\) 2.48508 0.117540
\(448\) 0 0
\(449\) 2.64300 0.124731 0.0623655 0.998053i \(-0.480136\pi\)
0.0623655 + 0.998053i \(0.480136\pi\)
\(450\) 12.6258 0.595185
\(451\) 0.335577 0.0158017
\(452\) 57.3864 2.69923
\(453\) 17.0937 0.803130
\(454\) 46.0722 2.16228
\(455\) 0 0
\(456\) 13.3717 0.626187
\(457\) −33.6890 −1.57590 −0.787952 0.615737i \(-0.788859\pi\)
−0.787952 + 0.615737i \(0.788859\pi\)
\(458\) −18.1923 −0.850073
\(459\) 31.3288 1.46231
\(460\) −14.8782 −0.693699
\(461\) −33.0790 −1.54064 −0.770320 0.637657i \(-0.779904\pi\)
−0.770320 + 0.637657i \(0.779904\pi\)
\(462\) 0 0
\(463\) −2.51806 −0.117024 −0.0585120 0.998287i \(-0.518636\pi\)
−0.0585120 + 0.998287i \(0.518636\pi\)
\(464\) 12.5510 0.582667
\(465\) −2.51806 −0.116772
\(466\) 28.5756 1.32374
\(467\) −2.57560 −0.119184 −0.0595922 0.998223i \(-0.518980\pi\)
−0.0595922 + 0.998223i \(0.518980\pi\)
\(468\) 5.88240 0.271914
\(469\) 0 0
\(470\) 33.3963 1.54045
\(471\) −26.2155 −1.20794
\(472\) −44.1151 −2.03056
\(473\) −9.34713 −0.429782
\(474\) 2.37000 0.108858
\(475\) −10.6858 −0.490300
\(476\) 0 0
\(477\) 1.31836 0.0603638
\(478\) −24.1151 −1.10300
\(479\) −0.513847 −0.0234783 −0.0117391 0.999931i \(-0.503737\pi\)
−0.0117391 + 0.999931i \(0.503737\pi\)
\(480\) −6.42754 −0.293376
\(481\) −8.51806 −0.388390
\(482\) 9.43910 0.429939
\(483\) 0 0
\(484\) −33.7967 −1.53621
\(485\) 7.17513 0.325806
\(486\) 34.8009 1.57860
\(487\) 36.0575 1.63392 0.816962 0.576692i \(-0.195657\pi\)
0.816962 + 0.576692i \(0.195657\pi\)
\(488\) 6.97858 0.315905
\(489\) 8.11508 0.366976
\(490\) 0 0
\(491\) −9.22846 −0.416475 −0.208237 0.978078i \(-0.566773\pi\)
−0.208237 + 0.978078i \(0.566773\pi\)
\(492\) 1.17092 0.0527893
\(493\) −61.1758 −2.75522
\(494\) −7.83221 −0.352388
\(495\) −2.59533 −0.116651
\(496\) −1.95715 −0.0878788
\(497\) 0 0
\(498\) 32.5770 1.45981
\(499\) 1.00314 0.0449065 0.0224533 0.999748i \(-0.492852\pi\)
0.0224533 + 0.999748i \(0.492852\pi\)
\(500\) −38.4078 −1.71765
\(501\) −2.99686 −0.133890
\(502\) −6.82065 −0.304421
\(503\) 30.3503 1.35325 0.676626 0.736327i \(-0.263441\pi\)
0.676626 + 0.736327i \(0.263441\pi\)
\(504\) 0 0
\(505\) −14.9687 −0.666099
\(506\) −8.52792 −0.379112
\(507\) 1.14637 0.0509119
\(508\) 42.0722 1.86665
\(509\) −10.5995 −0.469816 −0.234908 0.972018i \(-0.575479\pi\)
−0.234908 + 0.972018i \(0.575479\pi\)
\(510\) −21.0361 −0.931495
\(511\) 0 0
\(512\) −13.3461 −0.589818
\(513\) −17.9572 −0.792828
\(514\) 45.9143 2.02519
\(515\) 4.58546 0.202060
\(516\) −32.6148 −1.43579
\(517\) 12.1678 0.535139
\(518\) 0 0
\(519\) −12.6136 −0.553676
\(520\) −4.68585 −0.205488
\(521\) 16.2646 0.712564 0.356282 0.934378i \(-0.384044\pi\)
0.356282 + 0.934378i \(0.384044\pi\)
\(522\) −41.4307 −1.81337
\(523\) −7.22219 −0.315804 −0.157902 0.987455i \(-0.550473\pi\)
−0.157902 + 0.987455i \(0.550473\pi\)
\(524\) 12.7862 0.558569
\(525\) 0 0
\(526\) 17.7318 0.773144
\(527\) 9.53948 0.415546
\(528\) 1.57246 0.0684326
\(529\) −12.9185 −0.561675
\(530\) −2.46052 −0.106878
\(531\) 21.3142 0.924955
\(532\) 0 0
\(533\) −0.292731 −0.0126796
\(534\) −15.4061 −0.666688
\(535\) 6.68585 0.289054
\(536\) −21.2860 −0.919415
\(537\) 27.4685 1.18535
\(538\) 22.1923 0.956780
\(539\) 0 0
\(540\) −25.1709 −1.08318
\(541\) 17.3534 0.746081 0.373041 0.927815i \(-0.378315\pi\)
0.373041 + 0.927815i \(0.378315\pi\)
\(542\) −68.8156 −2.95588
\(543\) −7.52119 −0.322765
\(544\) 24.3503 1.04401
\(545\) −18.1249 −0.776387
\(546\) 0 0
\(547\) −34.1109 −1.45848 −0.729238 0.684261i \(-0.760125\pi\)
−0.729238 + 0.684261i \(0.760125\pi\)
\(548\) −45.7220 −1.95315
\(549\) −3.37169 −0.143900
\(550\) −8.58546 −0.366085
\(551\) 35.0649 1.49381
\(552\) −12.7005 −0.540571
\(553\) 0 0
\(554\) −4.46052 −0.189509
\(555\) 13.1134 0.556632
\(556\) −26.1579 −1.10934
\(557\) −13.2222 −0.560242 −0.280121 0.959965i \(-0.590375\pi\)
−0.280121 + 0.959965i \(0.590375\pi\)
\(558\) 6.46052 0.273496
\(559\) 8.15371 0.344865
\(560\) 0 0
\(561\) −7.66442 −0.323592
\(562\) −48.2070 −2.03349
\(563\) 41.3717 1.74361 0.871804 0.489854i \(-0.162950\pi\)
0.871804 + 0.489854i \(0.162950\pi\)
\(564\) 42.4569 1.78776
\(565\) 22.0863 0.929178
\(566\) 63.2432 2.65831
\(567\) 0 0
\(568\) 5.37169 0.225391
\(569\) −2.68164 −0.112420 −0.0562100 0.998419i \(-0.517902\pi\)
−0.0562100 + 0.998419i \(0.517902\pi\)
\(570\) 12.0575 0.505035
\(571\) −28.9315 −1.21075 −0.605373 0.795942i \(-0.706976\pi\)
−0.605373 + 0.795942i \(0.706976\pi\)
\(572\) −4.00000 −0.167248
\(573\) −5.03612 −0.210387
\(574\) 0 0
\(575\) 10.1495 0.423263
\(576\) 20.5254 0.855225
\(577\) −37.5296 −1.56238 −0.781189 0.624294i \(-0.785387\pi\)
−0.781189 + 0.624294i \(0.785387\pi\)
\(578\) 39.8641 1.65813
\(579\) −9.50650 −0.395077
\(580\) 49.1512 2.04089
\(581\) 0 0
\(582\) 14.3503 0.594838
\(583\) −0.896480 −0.0371284
\(584\) 53.3864 2.20914
\(585\) 2.26396 0.0936033
\(586\) 34.9259 1.44277
\(587\) −23.0649 −0.951990 −0.475995 0.879448i \(-0.657912\pi\)
−0.475995 + 0.879448i \(0.657912\pi\)
\(588\) 0 0
\(589\) −5.46787 −0.225299
\(590\) −39.7795 −1.63770
\(591\) −3.63504 −0.149525
\(592\) 10.1923 0.418903
\(593\) 11.0502 0.453777 0.226889 0.973921i \(-0.427145\pi\)
0.226889 + 0.973921i \(0.427145\pi\)
\(594\) −14.4275 −0.591969
\(595\) 0 0
\(596\) 7.56404 0.309835
\(597\) 15.5872 0.637940
\(598\) 7.43910 0.304207
\(599\) −44.6044 −1.82248 −0.911242 0.411870i \(-0.864876\pi\)
−0.911242 + 0.411870i \(0.864876\pi\)
\(600\) −12.7862 −0.521996
\(601\) −47.8715 −1.95272 −0.976359 0.216156i \(-0.930648\pi\)
−0.976359 + 0.216156i \(0.930648\pi\)
\(602\) 0 0
\(603\) 10.2843 0.418809
\(604\) 52.0294 2.11705
\(605\) −13.0073 −0.528824
\(606\) −29.9374 −1.21612
\(607\) −45.0691 −1.82930 −0.914649 0.404249i \(-0.867533\pi\)
−0.914649 + 0.404249i \(0.867533\pi\)
\(608\) −13.9572 −0.566037
\(609\) 0 0
\(610\) 6.29273 0.254785
\(611\) −10.6142 −0.429406
\(612\) 34.3074 1.38680
\(613\) −25.5212 −1.03079 −0.515396 0.856952i \(-0.672355\pi\)
−0.515396 + 0.856952i \(0.672355\pi\)
\(614\) −60.9834 −2.46109
\(615\) 0.450654 0.0181721
\(616\) 0 0
\(617\) 29.2432 1.17729 0.588643 0.808393i \(-0.299663\pi\)
0.588643 + 0.808393i \(0.299663\pi\)
\(618\) 9.17092 0.368909
\(619\) −4.78623 −0.192375 −0.0961874 0.995363i \(-0.530665\pi\)
−0.0961874 + 0.995363i \(0.530665\pi\)
\(620\) −7.66442 −0.307811
\(621\) 17.0558 0.684428
\(622\) −45.6791 −1.83157
\(623\) 0 0
\(624\) −1.37169 −0.0549116
\(625\) 1.20077 0.0480307
\(626\) 8.15792 0.326056
\(627\) 4.39312 0.175444
\(628\) −79.7942 −3.18413
\(629\) −49.6791 −1.98084
\(630\) 0 0
\(631\) −28.3931 −1.13031 −0.565156 0.824984i \(-0.691184\pi\)
−0.565156 + 0.824984i \(0.691184\pi\)
\(632\) 3.07896 0.122475
\(633\) 10.6331 0.422629
\(634\) 11.7648 0.467240
\(635\) 16.1923 0.642574
\(636\) −3.12808 −0.124036
\(637\) 0 0
\(638\) 28.1726 1.11536
\(639\) −2.59533 −0.102670
\(640\) −27.0937 −1.07097
\(641\) 5.96137 0.235460 0.117730 0.993046i \(-0.462438\pi\)
0.117730 + 0.993046i \(0.462438\pi\)
\(642\) 13.3717 0.527739
\(643\) −31.1940 −1.23017 −0.615086 0.788460i \(-0.710879\pi\)
−0.615086 + 0.788460i \(0.710879\pi\)
\(644\) 0 0
\(645\) −12.5525 −0.494253
\(646\) −45.6791 −1.79722
\(647\) −14.9112 −0.586219 −0.293109 0.956079i \(-0.594690\pi\)
−0.293109 + 0.956079i \(0.594690\pi\)
\(648\) −3.83956 −0.150832
\(649\) −14.4935 −0.568920
\(650\) 7.48929 0.293754
\(651\) 0 0
\(652\) 24.7005 0.967348
\(653\) −3.57246 −0.139801 −0.0699006 0.997554i \(-0.522268\pi\)
−0.0699006 + 0.997554i \(0.522268\pi\)
\(654\) −36.2499 −1.41748
\(655\) 4.92104 0.192281
\(656\) 0.350269 0.0136757
\(657\) −25.7936 −1.00630
\(658\) 0 0
\(659\) −3.90383 −0.152071 −0.0760357 0.997105i \(-0.524226\pi\)
−0.0760357 + 0.997105i \(0.524226\pi\)
\(660\) 6.15792 0.239697
\(661\) −13.7936 −0.536508 −0.268254 0.963348i \(-0.586447\pi\)
−0.268254 + 0.963348i \(0.586447\pi\)
\(662\) −14.4078 −0.559975
\(663\) 6.68585 0.259657
\(664\) 42.3221 1.64242
\(665\) 0 0
\(666\) −33.6447 −1.30371
\(667\) −33.3049 −1.28957
\(668\) −9.12181 −0.352933
\(669\) 22.4605 0.868374
\(670\) −19.1940 −0.741530
\(671\) 2.29273 0.0885099
\(672\) 0 0
\(673\) 5.70306 0.219837 0.109918 0.993941i \(-0.464941\pi\)
0.109918 + 0.993941i \(0.464941\pi\)
\(674\) −60.0393 −2.31263
\(675\) 17.1709 0.660909
\(676\) 3.48929 0.134203
\(677\) −35.2614 −1.35521 −0.677604 0.735427i \(-0.736981\pi\)
−0.677604 + 0.735427i \(0.736981\pi\)
\(678\) 44.1726 1.69644
\(679\) 0 0
\(680\) −27.3288 −1.04801
\(681\) 22.5426 0.863835
\(682\) −4.39312 −0.168221
\(683\) 1.03612 0.0396459 0.0198229 0.999804i \(-0.493690\pi\)
0.0198229 + 0.999804i \(0.493690\pi\)
\(684\) −19.6644 −0.751888
\(685\) −17.5970 −0.672348
\(686\) 0 0
\(687\) −8.90131 −0.339606
\(688\) −9.75639 −0.371959
\(689\) 0.782020 0.0297926
\(690\) −11.4523 −0.435983
\(691\) −3.67850 −0.139937 −0.0699684 0.997549i \(-0.522290\pi\)
−0.0699684 + 0.997549i \(0.522290\pi\)
\(692\) −38.3931 −1.45949
\(693\) 0 0
\(694\) −39.1281 −1.48528
\(695\) −10.0674 −0.381878
\(696\) 41.9572 1.59038
\(697\) −1.70727 −0.0646674
\(698\) 55.1758 2.08843
\(699\) 13.9817 0.528837
\(700\) 0 0
\(701\) −0.0617493 −0.00233224 −0.00116612 0.999999i \(-0.500371\pi\)
−0.00116612 + 0.999999i \(0.500371\pi\)
\(702\) 12.5855 0.475008
\(703\) 28.4752 1.07396
\(704\) −13.9572 −0.526030
\(705\) 16.3404 0.615415
\(706\) 17.9227 0.674531
\(707\) 0 0
\(708\) −50.5720 −1.90061
\(709\) −42.9834 −1.61428 −0.807138 0.590363i \(-0.798985\pi\)
−0.807138 + 0.590363i \(0.798985\pi\)
\(710\) 4.84377 0.181783
\(711\) −1.48760 −0.0557892
\(712\) −20.0147 −0.750082
\(713\) 5.19342 0.194495
\(714\) 0 0
\(715\) −1.53948 −0.0575733
\(716\) 83.6081 3.12458
\(717\) −11.7992 −0.440650
\(718\) 43.0508 1.60664
\(719\) −17.6546 −0.658404 −0.329202 0.944260i \(-0.606780\pi\)
−0.329202 + 0.944260i \(0.606780\pi\)
\(720\) −2.70896 −0.100957
\(721\) 0 0
\(722\) −18.3331 −0.682286
\(723\) 4.61844 0.171762
\(724\) −22.8929 −0.850807
\(725\) −33.5296 −1.24526
\(726\) −26.0147 −0.965496
\(727\) 23.8077 0.882977 0.441488 0.897267i \(-0.354451\pi\)
0.441488 + 0.897267i \(0.354451\pi\)
\(728\) 0 0
\(729\) 20.3288 0.752920
\(730\) 48.1396 1.78173
\(731\) 47.5542 1.75885
\(732\) 8.00000 0.295689
\(733\) −31.3492 −1.15791 −0.578954 0.815360i \(-0.696539\pi\)
−0.578954 + 0.815360i \(0.696539\pi\)
\(734\) −12.5082 −0.461686
\(735\) 0 0
\(736\) 13.2566 0.488645
\(737\) −6.99327 −0.257600
\(738\) −1.15623 −0.0425615
\(739\) −24.8108 −0.912680 −0.456340 0.889806i \(-0.650840\pi\)
−0.456340 + 0.889806i \(0.650840\pi\)
\(740\) 39.9143 1.46728
\(741\) −3.83221 −0.140780
\(742\) 0 0
\(743\) 21.3717 0.784051 0.392026 0.919954i \(-0.371774\pi\)
0.392026 + 0.919954i \(0.371774\pi\)
\(744\) −6.54262 −0.239864
\(745\) 2.91117 0.106657
\(746\) 50.4225 1.84610
\(747\) −20.4479 −0.748149
\(748\) −23.3288 −0.852987
\(749\) 0 0
\(750\) −29.5640 −1.07953
\(751\) 19.2243 0.701503 0.350751 0.936469i \(-0.385926\pi\)
0.350751 + 0.936469i \(0.385926\pi\)
\(752\) 12.7005 0.463141
\(753\) −3.33727 −0.121617
\(754\) −24.5756 −0.894990
\(755\) 20.0246 0.728768
\(756\) 0 0
\(757\) −19.8610 −0.721860 −0.360930 0.932593i \(-0.617541\pi\)
−0.360930 + 0.932593i \(0.617541\pi\)
\(758\) 10.8009 0.392307
\(759\) −4.17262 −0.151456
\(760\) 15.6644 0.568208
\(761\) −6.12073 −0.221876 −0.110938 0.993827i \(-0.535386\pi\)
−0.110938 + 0.993827i \(0.535386\pi\)
\(762\) 32.3847 1.17317
\(763\) 0 0
\(764\) −15.3288 −0.554578
\(765\) 13.2039 0.477388
\(766\) 19.5296 0.705634
\(767\) 12.6430 0.456512
\(768\) −26.2730 −0.948045
\(769\) −3.82800 −0.138041 −0.0690206 0.997615i \(-0.521987\pi\)
−0.0690206 + 0.997615i \(0.521987\pi\)
\(770\) 0 0
\(771\) 22.4653 0.809070
\(772\) −28.9357 −1.04142
\(773\) 6.53635 0.235096 0.117548 0.993067i \(-0.462497\pi\)
0.117548 + 0.993067i \(0.462497\pi\)
\(774\) 32.2056 1.15761
\(775\) 5.22846 0.187812
\(776\) 18.6430 0.669245
\(777\) 0 0
\(778\) −15.0937 −0.541134
\(779\) 0.978577 0.0350612
\(780\) −5.37169 −0.192337
\(781\) 1.76481 0.0631498
\(782\) 43.3864 1.55149
\(783\) −56.3452 −2.01361
\(784\) 0 0
\(785\) −30.7104 −1.09610
\(786\) 9.84208 0.351055
\(787\) −30.8066 −1.09814 −0.549068 0.835778i \(-0.685017\pi\)
−0.549068 + 0.835778i \(0.685017\pi\)
\(788\) −11.0643 −0.394148
\(789\) 8.67598 0.308873
\(790\) 2.77636 0.0987786
\(791\) 0 0
\(792\) −6.74338 −0.239616
\(793\) −2.00000 −0.0710221
\(794\) −3.28117 −0.116444
\(795\) −1.20390 −0.0426981
\(796\) 47.4439 1.68161
\(797\) 38.8156 1.37492 0.687460 0.726222i \(-0.258726\pi\)
0.687460 + 0.726222i \(0.258726\pi\)
\(798\) 0 0
\(799\) −61.9044 −2.19002
\(800\) 13.3461 0.471854
\(801\) 9.67008 0.341675
\(802\) −16.3503 −0.577348
\(803\) 17.5395 0.618955
\(804\) −24.4015 −0.860576
\(805\) 0 0
\(806\) 3.83221 0.134984
\(807\) 10.8585 0.382236
\(808\) −38.8929 −1.36825
\(809\) 1.04033 0.0365759 0.0182880 0.999833i \(-0.494178\pi\)
0.0182880 + 0.999833i \(0.494178\pi\)
\(810\) −3.46221 −0.121650
\(811\) −12.6712 −0.444944 −0.222472 0.974939i \(-0.571413\pi\)
−0.222472 + 0.974939i \(0.571413\pi\)
\(812\) 0 0
\(813\) −33.6707 −1.18088
\(814\) 22.8782 0.801880
\(815\) 9.50650 0.332998
\(816\) −8.00000 −0.280056
\(817\) −27.2572 −0.953610
\(818\) 43.0607 1.50558
\(819\) 0 0
\(820\) 1.37169 0.0479016
\(821\) 27.0361 0.943567 0.471783 0.881714i \(-0.343610\pi\)
0.471783 + 0.881714i \(0.343610\pi\)
\(822\) −35.1940 −1.22753
\(823\) −31.6363 −1.10277 −0.551386 0.834251i \(-0.685901\pi\)
−0.551386 + 0.834251i \(0.685901\pi\)
\(824\) 11.9143 0.415055
\(825\) −4.20077 −0.146252
\(826\) 0 0
\(827\) 56.4800 1.96400 0.982002 0.188872i \(-0.0604832\pi\)
0.982002 + 0.188872i \(0.0604832\pi\)
\(828\) 18.6774 0.649085
\(829\) 42.6760 1.48220 0.741099 0.671396i \(-0.234305\pi\)
0.741099 + 0.671396i \(0.234305\pi\)
\(830\) 38.1627 1.32465
\(831\) −2.18248 −0.0757094
\(832\) 12.1751 0.422097
\(833\) 0 0
\(834\) −20.1348 −0.697211
\(835\) −3.51071 −0.121493
\(836\) 13.3717 0.462470
\(837\) 8.78623 0.303697
\(838\) 70.3797 2.43122
\(839\) 40.1642 1.38662 0.693311 0.720639i \(-0.256151\pi\)
0.693311 + 0.720639i \(0.256151\pi\)
\(840\) 0 0
\(841\) 81.0252 2.79397
\(842\) −19.4868 −0.671558
\(843\) −23.5872 −0.812385
\(844\) 32.3650 1.11405
\(845\) 1.34292 0.0461980
\(846\) −41.9242 −1.44138
\(847\) 0 0
\(848\) −0.935731 −0.0321331
\(849\) 30.9442 1.06200
\(850\) 43.6791 1.49818
\(851\) −27.0460 −0.927124
\(852\) 6.15792 0.210967
\(853\) −19.6932 −0.674282 −0.337141 0.941454i \(-0.609460\pi\)
−0.337141 + 0.941454i \(0.609460\pi\)
\(854\) 0 0
\(855\) −7.56825 −0.258829
\(856\) 17.3717 0.593752
\(857\) −1.66442 −0.0568556 −0.0284278 0.999596i \(-0.509050\pi\)
−0.0284278 + 0.999596i \(0.509050\pi\)
\(858\) −3.07896 −0.105114
\(859\) 41.2944 1.40895 0.704474 0.709730i \(-0.251183\pi\)
0.704474 + 0.709730i \(0.251183\pi\)
\(860\) −38.2070 −1.30285
\(861\) 0 0
\(862\) 22.6086 0.770051
\(863\) −31.3288 −1.06645 −0.533223 0.845975i \(-0.679019\pi\)
−0.533223 + 0.845975i \(0.679019\pi\)
\(864\) 22.4275 0.763000
\(865\) −14.7764 −0.502411
\(866\) −61.6363 −2.09449
\(867\) 19.5051 0.662426
\(868\) 0 0
\(869\) 1.01156 0.0343147
\(870\) 37.8337 1.28268
\(871\) 6.10038 0.206704
\(872\) −47.0937 −1.59479
\(873\) −9.00735 −0.304852
\(874\) −24.8683 −0.841184
\(875\) 0 0
\(876\) 61.2003 2.06777
\(877\) −53.8041 −1.81683 −0.908417 0.418065i \(-0.862708\pi\)
−0.908417 + 0.418065i \(0.862708\pi\)
\(878\) −79.2234 −2.67366
\(879\) 17.0888 0.576392
\(880\) 1.84208 0.0620964
\(881\) −8.09196 −0.272625 −0.136313 0.990666i \(-0.543525\pi\)
−0.136313 + 0.990666i \(0.543525\pi\)
\(882\) 0 0
\(883\) −27.0705 −0.910996 −0.455498 0.890237i \(-0.650539\pi\)
−0.455498 + 0.890237i \(0.650539\pi\)
\(884\) 20.3503 0.684454
\(885\) −19.4637 −0.654264
\(886\) 61.9620 2.08165
\(887\) 38.4935 1.29249 0.646243 0.763132i \(-0.276339\pi\)
0.646243 + 0.763132i \(0.276339\pi\)
\(888\) 34.0722 1.14339
\(889\) 0 0
\(890\) −18.0477 −0.604959
\(891\) −1.26144 −0.0422599
\(892\) 68.3650 2.28903
\(893\) 35.4826 1.18738
\(894\) 5.82235 0.194728
\(895\) 32.1783 1.07560
\(896\) 0 0
\(897\) 3.63986 0.121532
\(898\) 6.19235 0.206641
\(899\) −17.1568 −0.572213
\(900\) 18.8034 0.626781
\(901\) 4.56090 0.151946
\(902\) 0.786230 0.0261786
\(903\) 0 0
\(904\) 57.3864 1.90864
\(905\) −8.81079 −0.292881
\(906\) 40.0491 1.33054
\(907\) 15.7031 0.521411 0.260706 0.965418i \(-0.416045\pi\)
0.260706 + 0.965418i \(0.416045\pi\)
\(908\) 68.6148 2.27706
\(909\) 18.7911 0.623260
\(910\) 0 0
\(911\) 44.9399 1.48893 0.744463 0.667663i \(-0.232705\pi\)
0.744463 + 0.667663i \(0.232705\pi\)
\(912\) 4.58546 0.151840
\(913\) 13.9044 0.460170
\(914\) −78.9307 −2.61080
\(915\) 3.07896 0.101787
\(916\) −27.0937 −0.895200
\(917\) 0 0
\(918\) 73.4011 2.42260
\(919\) −27.2432 −0.898669 −0.449334 0.893364i \(-0.648339\pi\)
−0.449334 + 0.893364i \(0.648339\pi\)
\(920\) −14.8782 −0.490519
\(921\) −29.8385 −0.983211
\(922\) −77.5015 −2.55237
\(923\) −1.53948 −0.0506726
\(924\) 0 0
\(925\) −27.2285 −0.895266
\(926\) −5.89962 −0.193873
\(927\) −5.75639 −0.189065
\(928\) −43.7942 −1.43761
\(929\) 17.1422 0.562416 0.281208 0.959647i \(-0.409265\pi\)
0.281208 + 0.959647i \(0.409265\pi\)
\(930\) −5.89962 −0.193456
\(931\) 0 0
\(932\) 42.5573 1.39401
\(933\) −22.3503 −0.731715
\(934\) −6.03442 −0.197452
\(935\) −8.97858 −0.293631
\(936\) 5.88240 0.192272
\(937\) 51.5197 1.68308 0.841538 0.540197i \(-0.181650\pi\)
0.841538 + 0.540197i \(0.181650\pi\)
\(938\) 0 0
\(939\) 3.99158 0.130260
\(940\) 49.7367 1.62223
\(941\) −32.7575 −1.06786 −0.533931 0.845528i \(-0.679286\pi\)
−0.533931 + 0.845528i \(0.679286\pi\)
\(942\) −61.4208 −2.00120
\(943\) −0.929460 −0.0302674
\(944\) −15.1281 −0.492377
\(945\) 0 0
\(946\) −21.8996 −0.712018
\(947\) 20.9295 0.680116 0.340058 0.940404i \(-0.389553\pi\)
0.340058 + 0.940404i \(0.389553\pi\)
\(948\) 3.52962 0.114637
\(949\) −15.3001 −0.496662
\(950\) −25.0361 −0.812279
\(951\) 5.75639 0.186664
\(952\) 0 0
\(953\) 46.4120 1.50343 0.751716 0.659487i \(-0.229226\pi\)
0.751716 + 0.659487i \(0.229226\pi\)
\(954\) 3.08883 0.100004
\(955\) −5.89962 −0.190907
\(956\) −35.9143 −1.16155
\(957\) 13.7845 0.445591
\(958\) −1.20390 −0.0388964
\(959\) 0 0
\(960\) −18.7434 −0.604940
\(961\) −28.3246 −0.913698
\(962\) −19.9572 −0.643444
\(963\) −8.39312 −0.270464
\(964\) 14.0575 0.452763
\(965\) −11.1365 −0.358497
\(966\) 0 0
\(967\) −23.2186 −0.746660 −0.373330 0.927699i \(-0.621784\pi\)
−0.373330 + 0.927699i \(0.621784\pi\)
\(968\) −33.7967 −1.08627
\(969\) −22.3503 −0.717994
\(970\) 16.8108 0.539762
\(971\) 14.1004 0.452503 0.226251 0.974069i \(-0.427353\pi\)
0.226251 + 0.974069i \(0.427353\pi\)
\(972\) 51.8286 1.66240
\(973\) 0 0
\(974\) 84.4800 2.70692
\(975\) 3.66442 0.117355
\(976\) 2.39312 0.0766018
\(977\) −2.95402 −0.0945074 −0.0472537 0.998883i \(-0.515047\pi\)
−0.0472537 + 0.998883i \(0.515047\pi\)
\(978\) 19.0130 0.607969
\(979\) −6.57560 −0.210157
\(980\) 0 0
\(981\) 22.7533 0.726455
\(982\) −21.6216 −0.689972
\(983\) 35.0367 1.11750 0.558749 0.829337i \(-0.311281\pi\)
0.558749 + 0.829337i \(0.311281\pi\)
\(984\) 1.17092 0.0373277
\(985\) −4.25831 −0.135681
\(986\) −143.330 −4.56456
\(987\) 0 0
\(988\) −11.6644 −0.371095
\(989\) 25.8891 0.823227
\(990\) −6.08065 −0.193256
\(991\) 47.9718 1.52388 0.761938 0.647650i \(-0.224248\pi\)
0.761938 + 0.647650i \(0.224248\pi\)
\(992\) 6.82908 0.216823
\(993\) −7.04958 −0.223712
\(994\) 0 0
\(995\) 18.2598 0.578873
\(996\) 48.5166 1.53731
\(997\) 38.4422 1.21748 0.608739 0.793371i \(-0.291676\pi\)
0.608739 + 0.793371i \(0.291676\pi\)
\(998\) 2.35027 0.0743965
\(999\) −45.7564 −1.44767
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 637.2.a.j.1.3 3
3.2 odd 2 5733.2.a.x.1.1 3
7.2 even 3 637.2.e.i.508.1 6
7.3 odd 6 637.2.e.j.79.1 6
7.4 even 3 637.2.e.i.79.1 6
7.5 odd 6 637.2.e.j.508.1 6
7.6 odd 2 91.2.a.d.1.3 3
13.12 even 2 8281.2.a.bg.1.1 3
21.20 even 2 819.2.a.i.1.1 3
28.27 even 2 1456.2.a.t.1.2 3
35.34 odd 2 2275.2.a.m.1.1 3
56.13 odd 2 5824.2.a.by.1.2 3
56.27 even 2 5824.2.a.bs.1.2 3
91.34 even 4 1183.2.c.f.337.6 6
91.83 even 4 1183.2.c.f.337.1 6
91.90 odd 2 1183.2.a.i.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.3 3 7.6 odd 2
637.2.a.j.1.3 3 1.1 even 1 trivial
637.2.e.i.79.1 6 7.4 even 3
637.2.e.i.508.1 6 7.2 even 3
637.2.e.j.79.1 6 7.3 odd 6
637.2.e.j.508.1 6 7.5 odd 6
819.2.a.i.1.1 3 21.20 even 2
1183.2.a.i.1.1 3 91.90 odd 2
1183.2.c.f.337.1 6 91.83 even 4
1183.2.c.f.337.6 6 91.34 even 4
1456.2.a.t.1.2 3 28.27 even 2
2275.2.a.m.1.1 3 35.34 odd 2
5733.2.a.x.1.1 3 3.2 odd 2
5824.2.a.bs.1.2 3 56.27 even 2
5824.2.a.by.1.2 3 56.13 odd 2
8281.2.a.bg.1.1 3 13.12 even 2