Properties

Label 91.2.a.d.1.3
Level $91$
Weight $2$
Character 91.1
Self dual yes
Analytic conductor $0.727$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.34292 q^{2} -1.14637 q^{3} +3.48929 q^{4} -1.34292 q^{5} -2.68585 q^{6} -1.00000 q^{7} +3.48929 q^{8} -1.68585 q^{9} +O(q^{10})\) \(q+2.34292 q^{2} -1.14637 q^{3} +3.48929 q^{4} -1.34292 q^{5} -2.68585 q^{6} -1.00000 q^{7} +3.48929 q^{8} -1.68585 q^{9} -3.14637 q^{10} +1.14637 q^{11} -4.00000 q^{12} +1.00000 q^{13} -2.34292 q^{14} +1.53948 q^{15} +1.19656 q^{16} +5.83221 q^{17} -3.94981 q^{18} -3.34292 q^{19} -4.68585 q^{20} +1.14637 q^{21} +2.68585 q^{22} -3.17513 q^{23} -4.00000 q^{24} -3.19656 q^{25} +2.34292 q^{26} +5.37169 q^{27} -3.48929 q^{28} +10.4893 q^{29} +3.60688 q^{30} +1.63565 q^{31} -4.17513 q^{32} -1.31415 q^{33} +13.6644 q^{34} +1.34292 q^{35} -5.88240 q^{36} +8.51806 q^{37} -7.83221 q^{38} -1.14637 q^{39} -4.68585 q^{40} -0.292731 q^{41} +2.68585 q^{42} -8.15371 q^{43} +4.00000 q^{44} +2.26396 q^{45} -7.43910 q^{46} -10.6142 q^{47} -1.37169 q^{48} +1.00000 q^{49} -7.48929 q^{50} -6.68585 q^{51} +3.48929 q^{52} -0.782020 q^{53} +12.5855 q^{54} -1.53948 q^{55} -3.48929 q^{56} +3.83221 q^{57} +24.5756 q^{58} +12.6430 q^{59} +5.37169 q^{60} -2.00000 q^{61} +3.83221 q^{62} +1.68585 q^{63} -12.1751 q^{64} -1.34292 q^{65} -3.07896 q^{66} -6.10038 q^{67} +20.3503 q^{68} +3.63986 q^{69} +3.14637 q^{70} +1.53948 q^{71} -5.88240 q^{72} -15.3001 q^{73} +19.9572 q^{74} +3.66442 q^{75} -11.6644 q^{76} -1.14637 q^{77} -2.68585 q^{78} +0.882404 q^{79} -1.60688 q^{80} -1.10038 q^{81} -0.685846 q^{82} -12.1292 q^{83} +4.00000 q^{84} -7.83221 q^{85} -19.1035 q^{86} -12.0246 q^{87} +4.00000 q^{88} +5.73604 q^{89} +5.30429 q^{90} -1.00000 q^{91} -11.0790 q^{92} -1.87506 q^{93} -24.8683 q^{94} +4.48929 q^{95} +4.78623 q^{96} -5.34292 q^{97} +2.34292 q^{98} -1.93260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} - 2q^{3} + 3q^{4} + 2q^{5} + 4q^{6} - 3q^{7} + 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q + q^{2} - 2q^{3} + 3q^{4} + 2q^{5} + 4q^{6} - 3q^{7} + 3q^{8} + 7q^{9} - 8q^{10} + 2q^{11} - 12q^{12} + 3q^{13} - q^{14} - 6q^{15} - q^{16} + 4q^{17} - 15q^{18} - 4q^{19} - 2q^{20} + 2q^{21} - 4q^{22} + 10q^{23} - 12q^{24} - 5q^{25} + q^{26} - 8q^{27} - 3q^{28} + 24q^{29} + 20q^{30} - 4q^{31} + 7q^{32} - 16q^{33} + 14q^{34} - 2q^{35} - q^{36} - 10q^{38} - 2q^{39} - 2q^{40} + 2q^{41} - 4q^{42} + 10q^{43} + 12q^{44} + 22q^{45} - 18q^{46} - 8q^{47} + 20q^{48} + 3q^{49} - 15q^{50} - 8q^{51} + 3q^{52} + 8q^{53} + 32q^{54} + 6q^{55} - 3q^{56} - 2q^{57} + 12q^{58} - 4q^{59} - 8q^{60} - 6q^{61} - 2q^{62} - 7q^{63} - 17q^{64} + 2q^{65} + 12q^{66} - 12q^{67} + 22q^{68} - 6q^{69} + 8q^{70} - 6q^{71} - q^{72} - 10q^{73} + 30q^{74} - 16q^{75} - 8q^{76} - 2q^{77} + 4q^{78} - 14q^{79} - 14q^{80} + 3q^{81} + 10q^{82} - 12q^{83} + 12q^{84} - 10q^{85} - 26q^{86} - 26q^{87} + 12q^{88} + 2q^{89} - 28q^{90} - 3q^{91} - 12q^{92} - 22q^{93} - 10q^{94} + 6q^{95} - 4q^{96} - 10q^{97} + q^{98} + 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 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 \(-0.607361\pi\)
−0.330927 + 0.943656i \(0.607361\pi\)
\(4\) 3.48929 1.74464
\(5\) −1.34292 −0.600573 −0.300287 0.953849i \(-0.597082\pi\)
−0.300287 + 0.953849i \(0.597082\pi\)
\(6\) −2.68585 −1.09649
\(7\) −1.00000 −0.377964
\(8\) 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
\(14\) −2.34292 −0.626173
\(15\) 1.53948 0.397492
\(16\) 1.19656 0.299139
\(17\) 5.83221 1.41452 0.707260 0.706954i \(-0.249931\pi\)
0.707260 + 0.706954i \(0.249931\pi\)
\(18\) −3.94981 −0.930979
\(19\) −3.34292 −0.766919 −0.383460 0.923558i \(-0.625267\pi\)
−0.383460 + 0.923558i \(0.625267\pi\)
\(20\) −4.68585 −1.04779
\(21\) 1.14637 0.250157
\(22\) 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
\(28\) −3.48929 −0.659414
\(29\) 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 \(-0.453075\pi\)
0.146886 + 0.989153i \(0.453075\pi\)
\(32\) −4.17513 −0.738067
\(33\) −1.31415 −0.228765
\(34\) 13.6644 2.34343
\(35\) 1.34292 0.226995
\(36\) −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 \(-0.507277\pi\)
−0.0228584 + 0.999739i \(0.507277\pi\)
\(42\) 2.68585 0.414435
\(43\) −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 \(-0.781809\pi\)
−0.774122 + 0.633036i \(0.781809\pi\)
\(48\) −1.37169 −0.197987
\(49\) 1.00000 0.142857
\(50\) −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
\(56\) −3.48929 −0.466276
\(57\) 3.83221 0.507589
\(58\) 24.5756 3.22693
\(59\) 12.6430 1.64598 0.822989 0.568057i \(-0.192305\pi\)
0.822989 + 0.568057i \(0.192305\pi\)
\(60\) 5.37169 0.693482
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 3.83221 0.486691
\(63\) 1.68585 0.212397
\(64\) −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
\(70\) 3.14637 0.376063
\(71\) 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 \(-0.853088\pi\)
−0.895369 + 0.445324i \(0.853088\pi\)
\(74\) 19.9572 2.31997
\(75\) 3.66442 0.423131
\(76\) −11.6644 −1.33800
\(77\) −1.14637 −0.130640
\(78\) −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 \(-0.731856\pi\)
−0.665674 + 0.746243i \(0.731856\pi\)
\(84\) 4.00000 0.436436
\(85\) −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 \(-0.401675\pi\)
0.304009 + 0.952669i \(0.401675\pi\)
\(90\) 5.30429 0.559121
\(91\) −1.00000 −0.104828
\(92\) −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 \(-0.587436\pi\)
−0.271246 + 0.962510i \(0.587436\pi\)
\(98\) 2.34292 0.236671
\(99\) −1.93260 −0.194233
\(100\) −11.1537 −1.11537
\(101\) 11.1464 1.10910 0.554552 0.832149i \(-0.312889\pi\)
0.554552 + 0.832149i \(0.312889\pi\)
\(102\) −15.6644 −1.55101
\(103\) −3.41454 −0.336444 −0.168222 0.985749i \(-0.553803\pi\)
−0.168222 + 0.985749i \(0.553803\pi\)
\(104\) 3.48929 0.342153
\(105\) −1.53948 −0.150238
\(106\) −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
\(112\) −1.19656 −0.113064
\(113\) 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
\(119\) −5.83221 −0.534638
\(120\) 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
\(126\) 3.94981 0.351877
\(127\) 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 \(-0.551176\pi\)
−0.160081 + 0.987104i \(0.551176\pi\)
\(132\) −4.58546 −0.399113
\(133\) 3.34292 0.289868
\(134\) −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 \(-0.397013\pi\)
0.317928 + 0.948115i \(0.397013\pi\)
\(140\) 4.68585 0.396026
\(141\) 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
\(147\) −1.14637 −0.0945506
\(148\) 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
\(154\) −2.68585 −0.216432
\(155\) −2.19656 −0.176432
\(156\) −4.00000 −0.320256
\(157\) 22.8683 1.82509 0.912546 0.408975i \(-0.134114\pi\)
0.912546 + 0.408975i \(0.134114\pi\)
\(158\) 2.06740 0.164474
\(159\) 0.896480 0.0710955
\(160\) 5.60688 0.443263
\(161\) 3.17513 0.250236
\(162\) −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 \(-0.467749\pi\)
0.101148 + 0.994871i \(0.467749\pi\)
\(168\) 4.00000 0.308607
\(169\) 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 \(-0.362634\pi\)
0.418276 + 0.908320i \(0.362634\pi\)
\(174\) −28.1726 −2.13576
\(175\) 3.19656 0.241637
\(176\) 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 \(-0.421595\pi\)
0.243834 + 0.969817i \(0.421595\pi\)
\(182\) −2.34292 −0.173669
\(183\) 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
\(189\) −5.37169 −0.390733
\(190\) 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
\(196\) 3.48929 0.249235
\(197\) −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 \(-0.660065\pi\)
−0.481934 + 0.876208i \(0.660065\pi\)
\(200\) −11.1537 −0.788687
\(201\) 6.99327 0.493267
\(202\) 26.1151 1.83745
\(203\) −10.4893 −0.736204
\(204\) −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
\(210\) −3.60688 −0.248899
\(211\) 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
\(217\) −1.63565 −0.111035
\(218\) −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 \(-0.727760\pi\)
−0.656016 + 0.754747i \(0.727760\pi\)
\(224\) 4.17513 0.278963
\(225\) 5.38890 0.359260
\(226\) 38.5328 2.56316
\(227\) −19.6644 −1.30517 −0.652587 0.757714i \(-0.726316\pi\)
−0.652587 + 0.757714i \(0.726316\pi\)
\(228\) 13.3717 0.885562
\(229\) 7.76481 0.513113 0.256556 0.966529i \(-0.417412\pi\)
0.256556 + 0.966529i \(0.417412\pi\)
\(230\) 9.99013 0.658730
\(231\) 1.31415 0.0864650
\(232\) 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
\(238\) −13.6644 −0.885733
\(239\) −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 \(-0.541420\pi\)
−0.129758 + 0.991546i \(0.541420\pi\)
\(242\) −22.6932 −1.45877
\(243\) −14.8536 −0.952861
\(244\) −6.97858 −0.446758
\(245\) −1.34292 −0.0857962
\(246\) 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 \(-0.470714\pi\)
0.0918758 + 0.995770i \(0.470714\pi\)
\(252\) 5.88240 0.370557
\(253\) −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 \(-0.709319\pi\)
−0.611214 + 0.791465i \(0.709319\pi\)
\(258\) 21.8996 1.36341
\(259\) −8.51806 −0.529286
\(260\) −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
\(266\) 7.83221 0.480224
\(267\) −6.57560 −0.402420
\(268\) −21.2860 −1.30025
\(269\) −9.47208 −0.577523 −0.288761 0.957401i \(-0.593243\pi\)
−0.288761 + 0.957401i \(0.593243\pi\)
\(270\) −16.9013 −1.02858
\(271\) 29.3717 1.78420 0.892102 0.451835i \(-0.149230\pi\)
0.892102 + 0.451835i \(0.149230\pi\)
\(272\) 6.97858 0.423138
\(273\) 1.14637 0.0693812
\(274\) −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
\(280\) 4.68585 0.280033
\(281\) −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 \(-0.796386\pi\)
−0.802292 + 0.596932i \(0.796386\pi\)
\(284\) 5.37169 0.318751
\(285\) −5.14637 −0.304844
\(286\) 2.68585 0.158817
\(287\) 0.292731 0.0172794
\(288\) 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 \(-0.643406\pi\)
−0.435437 + 0.900219i \(0.643406\pi\)
\(294\) −2.68585 −0.156642
\(295\) −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
\(301\) 8.15371 0.469972
\(302\) 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 \(-0.233512\pi\)
0.742770 + 0.669546i \(0.233512\pi\)
\(308\) −4.00000 −0.227921
\(309\) 3.91431 0.222677
\(310\) −5.14637 −0.292294
\(311\) 19.4966 1.10555 0.552776 0.833330i \(-0.313568\pi\)
0.552776 + 0.833330i \(0.313568\pi\)
\(312\) −4.00000 −0.226455
\(313\) −3.48194 −0.196811 −0.0984055 0.995146i \(-0.531374\pi\)
−0.0984055 + 0.995146i \(0.531374\pi\)
\(314\) 53.5787 3.02362
\(315\) −2.26396 −0.127560
\(316\) 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
\(322\) 7.43910 0.414565
\(323\) −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
\(329\) 10.6142 0.585182
\(330\) 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
\(336\) 1.37169 0.0748320
\(337\) −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
\(343\) −1.00000 −0.0539949
\(344\) −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 \(-0.717068\pi\)
−0.630300 + 0.776351i \(0.717068\pi\)
\(350\) 7.48929 0.400319
\(351\) 5.37169 0.286720
\(352\) −4.78623 −0.255107
\(353\) −7.64973 −0.407154 −0.203577 0.979059i \(-0.565257\pi\)
−0.203577 + 0.979059i \(0.565257\pi\)
\(354\) −33.9572 −1.80480
\(355\) −2.06740 −0.109726
\(356\) 20.0147 1.06078
\(357\) 6.68585 0.353853
\(358\) 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
\(364\) −3.48929 −0.182888
\(365\) 20.5468 1.07547
\(366\) 5.37169 0.280783
\(367\) 5.33871 0.278679 0.139339 0.990245i \(-0.455502\pi\)
0.139339 + 0.990245i \(0.455502\pi\)
\(368\) −3.79923 −0.198049
\(369\) 0.493499 0.0256906
\(370\) −26.8009 −1.39331
\(371\) 0.782020 0.0406004
\(372\) −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
\(378\) −12.5855 −0.647326
\(379\) 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 \(-0.568312\pi\)
−0.212964 + 0.977060i \(0.568312\pi\)
\(384\) 23.1281 1.18025
\(385\) 1.53948 0.0784592
\(386\) −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
\(392\) 3.48929 0.176236
\(393\) 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 \(-0.488811\pi\)
0.0351436 + 0.999382i \(0.488811\pi\)
\(398\) −31.8568 −1.59684
\(399\) −3.83221 −0.191851
\(400\) −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
\(406\) −24.5756 −1.21967
\(407\) 9.76481 0.484024
\(408\) −23.3288 −1.15495
\(409\) −18.3790 −0.908785 −0.454392 0.890802i \(-0.650144\pi\)
−0.454392 + 0.890802i \(0.650144\pi\)
\(410\) 0.921039 0.0454869
\(411\) 15.0214 0.740952
\(412\) −11.9143 −0.586976
\(413\) −12.6430 −0.622121
\(414\) 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 \(-0.762235\pi\)
−0.733757 + 0.679412i \(0.762235\pi\)
\(420\) −5.37169 −0.262112
\(421\) −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
\(427\) 2.00000 0.0967868
\(428\) 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 \(-0.282182\pi\)
0.632127 + 0.774865i \(0.282182\pi\)
\(434\) −3.83221 −0.183952
\(435\) 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 \(-0.201130\pi\)
0.806925 + 0.590654i \(0.201130\pi\)
\(440\) −5.37169 −0.256085
\(441\) −1.68585 −0.0802784
\(442\) 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
\(448\) 12.1751 0.575221
\(449\) 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
\(455\) 1.34292 0.0629572
\(456\) 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 \(-0.220096\pi\)
0.770320 + 0.637657i \(0.220096\pi\)
\(462\) 3.07896 0.143246
\(463\) −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 \(-0.481020\pi\)
0.0595922 + 0.998223i \(0.481020\pi\)
\(468\) −5.88240 −0.271914
\(469\) 6.10038 0.281690
\(470\) 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
\(476\) −20.3503 −0.932753
\(477\) 1.31836 0.0603638
\(478\) −24.1151 −1.10300
\(479\) 0.513847 0.0234783 0.0117391 0.999931i \(-0.496263\pi\)
0.0117391 + 0.999931i \(0.496263\pi\)
\(480\) −6.42754 −0.293376
\(481\) 8.51806 0.388390
\(482\) −9.43910 −0.429939
\(483\) −3.63986 −0.165620
\(484\) −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
\(490\) −3.14637 −0.142138
\(491\) −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
\(497\) −1.53948 −0.0690551
\(498\) 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 \(-0.736559\pi\)
−0.676626 + 0.736327i \(0.736559\pi\)
\(504\) 5.88240 0.262023
\(505\) −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 \(-0.424521\pi\)
0.234908 + 0.972018i \(0.424521\pi\)
\(510\) 21.0361 0.931495
\(511\) 15.3001 0.676836
\(512\) −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
\(518\) −19.9572 −0.876867
\(519\) −12.6136 −0.553676
\(520\) −4.68585 −0.205488
\(521\) −16.2646 −0.712564 −0.356282 0.934378i \(-0.615956\pi\)
−0.356282 + 0.934378i \(0.615956\pi\)
\(522\) −41.4307 −1.81337
\(523\) 7.22219 0.315804 0.157902 0.987455i \(-0.449527\pi\)
0.157902 + 0.987455i \(0.449527\pi\)
\(524\) −12.7862 −0.558569
\(525\) −3.66442 −0.159929
\(526\) 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
\(532\) 11.6644 0.505717
\(533\) −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
\(539\) 1.14637 0.0493775
\(540\) −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
\(546\) 2.68585 0.114944
\(547\) −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
\(553\) −0.882404 −0.0375236
\(554\) −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
\(560\) 1.60688 0.0679033
\(561\) −7.66442 −0.323592
\(562\) −48.2070 −2.03349
\(563\) −41.3717 −1.74361 −0.871804 0.489854i \(-0.837050\pi\)
−0.871804 + 0.489854i \(0.837050\pi\)
\(564\) 42.4569 1.78776
\(565\) −22.0863 −0.929178
\(566\) −63.2432 −2.65831
\(567\) 1.10038 0.0462118
\(568\) 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
\(574\) 0.685846 0.0286267
\(575\) 10.1495 0.423263
\(576\) 20.5254 0.855225
\(577\) 37.5296 1.56238 0.781189 0.624294i \(-0.214613\pi\)
0.781189 + 0.624294i \(0.214613\pi\)
\(578\) 39.8641 1.65813
\(579\) 9.50650 0.395077
\(580\) −49.1512 −2.04089
\(581\) 12.1292 0.503202
\(582\) 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 \(-0.342088\pi\)
0.475995 + 0.879448i \(0.342088\pi\)
\(588\) −4.00000 −0.164957
\(589\) −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 \(-0.572855\pi\)
−0.226889 + 0.973921i \(0.572855\pi\)
\(594\) 14.4275 0.591969
\(595\) 7.83221 0.321089
\(596\) 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 \(-0.0693520\pi\)
0.976359 + 0.216156i \(0.0693520\pi\)
\(602\) 19.1035 0.778601
\(603\) 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 \(-0.132467\pi\)
0.914649 + 0.404249i \(0.132467\pi\)
\(608\) 13.9572 0.566037
\(609\) 12.0246 0.487260
\(610\) 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
\(616\) −4.00000 −0.161165
\(617\) 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 \(-0.469335\pi\)
0.0961874 + 0.995363i \(0.469335\pi\)
\(620\) −7.66442 −0.307811
\(621\) −17.0558 −0.684428
\(622\) 45.6791 1.83157
\(623\) −5.73604 −0.229810
\(624\) −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
\(630\) −5.30429 −0.211328
\(631\) −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
\(637\) 1.00000 0.0396214
\(638\) 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 \(-0.289121\pi\)
0.615086 + 0.788460i \(0.289121\pi\)
\(644\) 11.0790 0.436572
\(645\) −12.5525 −0.494253
\(646\) −45.6791 −1.79722
\(647\) 14.9112 0.586219 0.293109 0.956079i \(-0.405310\pi\)
0.293109 + 0.956079i \(0.405310\pi\)
\(648\) −3.83956 −0.150832
\(649\) 14.4935 0.568920
\(650\) −7.48929 −0.293754
\(651\) 1.87506 0.0734893
\(652\) 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
\(658\) 24.8683 0.969468
\(659\) −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 \(-0.413553\pi\)
0.268254 + 0.963348i \(0.413553\pi\)
\(662\) −14.4078 −0.559975
\(663\) −6.68585 −0.259657
\(664\) −42.3221 −1.64242
\(665\) −4.48929 −0.174087
\(666\) −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
\(672\) −4.78623 −0.184633
\(673\) 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 \(-0.263019\pi\)
0.677604 + 0.735427i \(0.263019\pi\)
\(678\) −44.1726 −1.69644
\(679\) 5.34292 0.205043
\(680\) −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
\(686\) −2.34292 −0.0894532
\(687\) −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 \(-0.477710\pi\)
0.0699684 + 0.997549i \(0.477710\pi\)
\(692\) 38.3931 1.45949
\(693\) 1.93260 0.0734132
\(694\) −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
\(700\) 11.1537 0.421571
\(701\) −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
\(707\) −11.1464 −0.419202
\(708\) −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
\(714\) 15.6644 0.586226
\(715\) −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 \(-0.393220\pi\)
0.329202 + 0.944260i \(0.393220\pi\)
\(720\) 2.70896 0.100957
\(721\) 3.41454 0.127164
\(722\) −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 \(-0.645549\pi\)
−0.441488 + 0.897267i \(0.645549\pi\)
\(728\) −3.48929 −0.129322
\(729\) 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 \(-0.303461\pi\)
0.578954 + 0.815360i \(0.303461\pi\)
\(734\) 12.5082 0.461686
\(735\) 1.53948 0.0567846
\(736\) 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
\(742\) 1.83221 0.0672626
\(743\) 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
\(749\) −4.97858 −0.181913
\(750\) −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
\(756\) −18.7434 −0.681690
\(757\) −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 \(-0.464614\pi\)
0.110938 + 0.993827i \(0.464614\pi\)
\(762\) −32.3847 −1.17317
\(763\) 13.4966 0.488611
\(764\) −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 \(-0.478013\pi\)
0.0690206 + 0.997615i \(0.478013\pi\)
\(770\) 3.60688 0.129983
\(771\) 22.4653 0.809070
\(772\) −28.9357 −1.04142
\(773\) −6.53635 −0.235096 −0.117548 0.993067i \(-0.537503\pi\)
−0.117548 + 0.993067i \(0.537503\pi\)
\(774\) 32.2056 1.15761
\(775\) −5.22846 −0.187812
\(776\) −18.6430 −0.669245
\(777\) 9.76481 0.350311
\(778\) −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
\(784\) 1.19656 0.0427342
\(785\) −30.7104 −1.09610
\(786\) 9.84208 0.351055
\(787\) 30.8066 1.09814 0.549068 0.835778i \(-0.314983\pi\)
0.549068 + 0.835778i \(0.314983\pi\)
\(788\) −11.0643 −0.394148
\(789\) −8.67598 −0.308873
\(790\) −2.77636 −0.0987786
\(791\) −16.4464 −0.584768
\(792\) −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 \(-0.741274\pi\)
−0.687460 + 0.726222i \(0.741274\pi\)
\(798\) −8.97858 −0.317838
\(799\) −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
\(805\) −4.26396 −0.150285
\(806\) 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 \(-0.428587\pi\)
0.222472 + 0.974939i \(0.428587\pi\)
\(812\) −36.6002 −1.28441
\(813\) −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
\(819\) 1.68585 0.0589082
\(820\) 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
\(826\) −29.6216 −1.03067
\(827\) 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 \(-0.765695\pi\)
−0.741099 + 0.671396i \(0.765695\pi\)
\(830\) 38.1627 1.32465
\(831\) 2.18248 0.0757094
\(832\) −12.1751 −0.422097
\(833\) 5.83221 0.202074
\(834\) −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 \(-0.743849\pi\)
−0.693311 + 0.720639i \(0.743849\pi\)
\(840\) −5.37169 −0.185341
\(841\) 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
\(847\) 9.68585 0.332810
\(848\) −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 \(-0.390540\pi\)
0.337141 + 0.941454i \(0.390540\pi\)
\(854\) 4.68585 0.160346
\(855\) −7.56825 −0.258829
\(856\) 17.3717 0.593752
\(857\) 1.66442 0.0568556 0.0284278 0.999596i \(-0.490950\pi\)
0.0284278 + 0.999596i \(0.490950\pi\)
\(858\) −3.07896 −0.105114
\(859\) −41.2944 −1.40895 −0.704474 0.709730i \(-0.748817\pi\)
−0.704474 + 0.709730i \(0.748817\pi\)
\(860\) 38.2070 1.30285
\(861\) −0.335577 −0.0114364
\(862\) 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
\(868\) −5.70727 −0.193717
\(869\) 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
\(875\) −11.0073 −0.372116
\(876\) 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 \(-0.456475\pi\)
0.136313 + 0.990666i \(0.456475\pi\)
\(882\) −3.94981 −0.132997
\(883\) −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 \(-0.723661\pi\)
−0.646243 + 0.763132i \(0.723661\pi\)
\(888\) −34.0722 −1.14339
\(889\) −12.0575 −0.404397
\(890\) −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
\(896\) 20.1751 0.674004
\(897\) 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
\(903\) −9.34713 −0.311053
\(904\) 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
\(910\) 3.14637 0.104301
\(911\) 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
\(917\) 3.66442 0.121010
\(918\) 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
\(924\) 4.58546 0.150851
\(925\) −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 \(-0.590735\pi\)
−0.281208 + 0.959647i \(0.590735\pi\)
\(930\) 5.89962 0.193456
\(931\) −3.34292 −0.109560
\(932\) 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 \(-0.818350\pi\)
−0.841538 + 0.540197i \(0.818350\pi\)
\(938\) 14.2927 0.466674
\(939\) 3.99158 0.130260
\(940\) 49.7367 1.62223
\(941\) 32.7575 1.06786 0.533931 0.845528i \(-0.320714\pi\)
0.533931 + 0.845528i \(0.320714\pi\)
\(942\) −61.4208 −2.00120
\(943\) 0.929460 0.0302674
\(944\) 15.1281 0.492377
\(945\) 7.21377 0.234664
\(946\) −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
\(952\) −20.3503 −0.659556
\(953\) 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
\(959\) 13.1035 0.423135
\(960\) −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
\(966\) −8.52792 −0.274381
\(967\) −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 \(-0.572647\pi\)
−0.226251 + 0.974069i \(0.572647\pi\)
\(972\) −51.8286 −1.66240
\(973\) −7.49663 −0.240331
\(974\) 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
\(980\) −4.68585 −0.149684
\(981\) 22.7533 0.726455
\(982\) −21.6216 −0.689972
\(983\) −35.0367 −1.11750 −0.558749 0.829337i \(-0.688719\pi\)
−0.558749 + 0.829337i \(0.688719\pi\)
\(984\) 1.17092 0.0373277
\(985\) 4.25831 0.135681
\(986\) 143.330 4.56456
\(987\) −12.1678 −0.387305
\(988\) −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
\(994\) −3.60688 −0.114403
\(995\) 18.2598 0.578873
\(996\) 48.5166 1.53731
\(997\) −38.4422 −1.21748 −0.608739 0.793371i \(-0.708324\pi\)
−0.608739 + 0.793371i \(0.708324\pi\)
\(998\) 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 91.2.a.d.1.3 3
3.2 odd 2 819.2.a.i.1.1 3
4.3 odd 2 1456.2.a.t.1.2 3
5.4 even 2 2275.2.a.m.1.1 3
7.2 even 3 637.2.e.j.508.1 6
7.3 odd 6 637.2.e.i.79.1 6
7.4 even 3 637.2.e.j.79.1 6
7.5 odd 6 637.2.e.i.508.1 6
7.6 odd 2 637.2.a.j.1.3 3
8.3 odd 2 5824.2.a.bs.1.2 3
8.5 even 2 5824.2.a.by.1.2 3
13.5 odd 4 1183.2.c.f.337.1 6
13.8 odd 4 1183.2.c.f.337.6 6
13.12 even 2 1183.2.a.i.1.1 3
21.20 even 2 5733.2.a.x.1.1 3
91.90 odd 2 8281.2.a.bg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.3 3 1.1 even 1 trivial
637.2.a.j.1.3 3 7.6 odd 2
637.2.e.i.79.1 6 7.3 odd 6
637.2.e.i.508.1 6 7.5 odd 6
637.2.e.j.79.1 6 7.4 even 3
637.2.e.j.508.1 6 7.2 even 3
819.2.a.i.1.1 3 3.2 odd 2
1183.2.a.i.1.1 3 13.12 even 2
1183.2.c.f.337.1 6 13.5 odd 4
1183.2.c.f.337.6 6 13.8 odd 4
1456.2.a.t.1.2 3 4.3 odd 2
2275.2.a.m.1.1 3 5.4 even 2
5733.2.a.x.1.1 3 21.20 even 2
5824.2.a.bs.1.2 3 8.3 odd 2
5824.2.a.by.1.2 3 8.5 even 2
8281.2.a.bg.1.1 3 91.90 odd 2