Properties

Label 2842.2.a.x.1.1
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.6934842544\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.1019601.1
Defining polynomial: \( x^{5} - 10x^{3} - x^{2} + 24x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 406)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.37936\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.23497 q^{3} +1.00000 q^{4} -1.14439 q^{5} -3.23497 q^{6} +1.00000 q^{8} +7.46501 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.23497 q^{3} +1.00000 q^{4} -1.14439 q^{5} -3.23497 q^{6} +1.00000 q^{8} +7.46501 q^{9} -1.14439 q^{10} -0.850683 q^{11} -3.23497 q^{12} -3.66136 q^{13} +3.70208 q^{15} +1.00000 q^{16} -0.194248 q^{17} +7.46501 q^{18} +6.99369 q^{19} -1.14439 q^{20} -0.850683 q^{22} -3.46219 q^{23} -3.23497 q^{24} -3.69036 q^{25} -3.66136 q^{26} -14.4442 q^{27} -1.00000 q^{29} +3.70208 q^{30} +3.97423 q^{31} +1.00000 q^{32} +2.75193 q^{33} -0.194248 q^{34} +7.46501 q^{36} +7.20919 q^{37} +6.99369 q^{38} +11.8444 q^{39} -1.14439 q^{40} -4.04493 q^{41} +9.77788 q^{43} -0.850683 q^{44} -8.54291 q^{45} -3.46219 q^{46} -0.0538211 q^{47} -3.23497 q^{48} -3.69036 q^{50} +0.628386 q^{51} -3.66136 q^{52} -2.67097 q^{53} -14.4442 q^{54} +0.973516 q^{55} -22.6244 q^{57} -1.00000 q^{58} +8.04354 q^{59} +3.70208 q^{60} +5.08144 q^{61} +3.97423 q^{62} +1.00000 q^{64} +4.19004 q^{65} +2.75193 q^{66} -11.6003 q^{67} -0.194248 q^{68} +11.2001 q^{69} -11.2825 q^{71} +7.46501 q^{72} +7.31077 q^{73} +7.20919 q^{74} +11.9382 q^{75} +6.99369 q^{76} +11.8444 q^{78} -13.3564 q^{79} -1.14439 q^{80} +24.3314 q^{81} -4.04493 q^{82} -9.04564 q^{83} +0.222296 q^{85} +9.77788 q^{86} +3.23497 q^{87} -0.850683 q^{88} -14.2305 q^{89} -8.54291 q^{90} -3.46219 q^{92} -12.8565 q^{93} -0.0538211 q^{94} -8.00353 q^{95} -3.23497 q^{96} -18.7779 q^{97} -6.35036 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{2} - 3 q^{3} + 5 q^{4} - 7 q^{5} - 3 q^{6} + 5 q^{8} + 8 q^{9} - 7 q^{10} - 3 q^{12} - 10 q^{13} - 10 q^{15} + 5 q^{16} - 8 q^{17} + 8 q^{18} - 2 q^{19} - 7 q^{20} + q^{23} - 3 q^{24} + 12 q^{25} - 10 q^{26} - 15 q^{27} - 5 q^{29} - 10 q^{30} - 11 q^{31} + 5 q^{32} - 9 q^{33} - 8 q^{34} + 8 q^{36} - 8 q^{37} - 2 q^{38} + 18 q^{39} - 7 q^{40} - 23 q^{41} - 3 q^{43} - 4 q^{45} + q^{46} - 16 q^{47} - 3 q^{48} + 12 q^{50} + 7 q^{51} - 10 q^{52} + 7 q^{53} - 15 q^{54} - 6 q^{55} - 34 q^{57} - 5 q^{58} + 9 q^{59} - 10 q^{60} - 15 q^{61} - 11 q^{62} + 5 q^{64} + 5 q^{65} - 9 q^{66} - 4 q^{67} - 8 q^{68} - 14 q^{69} - 22 q^{71} + 8 q^{72} - 8 q^{74} + 34 q^{75} - 2 q^{76} + 18 q^{78} - 13 q^{79} - 7 q^{80} + 17 q^{81} - 23 q^{82} - 28 q^{83} - 7 q^{85} - 3 q^{86} + 3 q^{87} - 17 q^{89} - 4 q^{90} + q^{92} + 17 q^{93} - 16 q^{94} - 9 q^{95} - 3 q^{96} - 42 q^{97} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −3.23497 −1.86771 −0.933854 0.357653i \(-0.883577\pi\)
−0.933854 + 0.357653i \(0.883577\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.14439 −0.511789 −0.255894 0.966705i \(-0.582370\pi\)
−0.255894 + 0.966705i \(0.582370\pi\)
\(6\) −3.23497 −1.32067
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 7.46501 2.48834
\(10\) −1.14439 −0.361889
\(11\) −0.850683 −0.256491 −0.128245 0.991742i \(-0.540934\pi\)
−0.128245 + 0.991742i \(0.540934\pi\)
\(12\) −3.23497 −0.933854
\(13\) −3.66136 −1.01548 −0.507739 0.861511i \(-0.669519\pi\)
−0.507739 + 0.861511i \(0.669519\pi\)
\(14\) 0 0
\(15\) 3.70208 0.955872
\(16\) 1.00000 0.250000
\(17\) −0.194248 −0.0471121 −0.0235561 0.999723i \(-0.507499\pi\)
−0.0235561 + 0.999723i \(0.507499\pi\)
\(18\) 7.46501 1.75952
\(19\) 6.99369 1.60446 0.802231 0.597014i \(-0.203646\pi\)
0.802231 + 0.597014i \(0.203646\pi\)
\(20\) −1.14439 −0.255894
\(21\) 0 0
\(22\) −0.850683 −0.181366
\(23\) −3.46219 −0.721916 −0.360958 0.932582i \(-0.617550\pi\)
−0.360958 + 0.932582i \(0.617550\pi\)
\(24\) −3.23497 −0.660335
\(25\) −3.69036 −0.738072
\(26\) −3.66136 −0.718051
\(27\) −14.4442 −2.77978
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 3.70208 0.675904
\(31\) 3.97423 0.713792 0.356896 0.934144i \(-0.383835\pi\)
0.356896 + 0.934144i \(0.383835\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.75193 0.479050
\(34\) −0.194248 −0.0333133
\(35\) 0 0
\(36\) 7.46501 1.24417
\(37\) 7.20919 1.18518 0.592592 0.805503i \(-0.298105\pi\)
0.592592 + 0.805503i \(0.298105\pi\)
\(38\) 6.99369 1.13453
\(39\) 11.8444 1.89662
\(40\) −1.14439 −0.180945
\(41\) −4.04493 −0.631712 −0.315856 0.948807i \(-0.602292\pi\)
−0.315856 + 0.948807i \(0.602292\pi\)
\(42\) 0 0
\(43\) 9.77788 1.49111 0.745556 0.666443i \(-0.232184\pi\)
0.745556 + 0.666443i \(0.232184\pi\)
\(44\) −0.850683 −0.128245
\(45\) −8.54291 −1.27350
\(46\) −3.46219 −0.510472
\(47\) −0.0538211 −0.00785061 −0.00392531 0.999992i \(-0.501249\pi\)
−0.00392531 + 0.999992i \(0.501249\pi\)
\(48\) −3.23497 −0.466927
\(49\) 0 0
\(50\) −3.69036 −0.521896
\(51\) 0.628386 0.0879917
\(52\) −3.66136 −0.507739
\(53\) −2.67097 −0.366886 −0.183443 0.983030i \(-0.558724\pi\)
−0.183443 + 0.983030i \(0.558724\pi\)
\(54\) −14.4442 −1.96560
\(55\) 0.973516 0.131269
\(56\) 0 0
\(57\) −22.6244 −2.99667
\(58\) −1.00000 −0.131306
\(59\) 8.04354 1.04718 0.523590 0.851970i \(-0.324592\pi\)
0.523590 + 0.851970i \(0.324592\pi\)
\(60\) 3.70208 0.477936
\(61\) 5.08144 0.650611 0.325306 0.945609i \(-0.394533\pi\)
0.325306 + 0.945609i \(0.394533\pi\)
\(62\) 3.97423 0.504727
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.19004 0.519710
\(66\) 2.75193 0.338739
\(67\) −11.6003 −1.41720 −0.708599 0.705611i \(-0.750673\pi\)
−0.708599 + 0.705611i \(0.750673\pi\)
\(68\) −0.194248 −0.0235561
\(69\) 11.2001 1.34833
\(70\) 0 0
\(71\) −11.2825 −1.33898 −0.669492 0.742819i \(-0.733488\pi\)
−0.669492 + 0.742819i \(0.733488\pi\)
\(72\) 7.46501 0.879760
\(73\) 7.31077 0.855661 0.427830 0.903859i \(-0.359278\pi\)
0.427830 + 0.903859i \(0.359278\pi\)
\(74\) 7.20919 0.838052
\(75\) 11.9382 1.37850
\(76\) 6.99369 0.802231
\(77\) 0 0
\(78\) 11.8444 1.34111
\(79\) −13.3564 −1.50271 −0.751357 0.659896i \(-0.770600\pi\)
−0.751357 + 0.659896i \(0.770600\pi\)
\(80\) −1.14439 −0.127947
\(81\) 24.3314 2.70348
\(82\) −4.04493 −0.446688
\(83\) −9.04564 −0.992888 −0.496444 0.868069i \(-0.665361\pi\)
−0.496444 + 0.868069i \(0.665361\pi\)
\(84\) 0 0
\(85\) 0.222296 0.0241114
\(86\) 9.77788 1.05438
\(87\) 3.23497 0.346825
\(88\) −0.850683 −0.0906831
\(89\) −14.2305 −1.50843 −0.754216 0.656626i \(-0.771983\pi\)
−0.754216 + 0.656626i \(0.771983\pi\)
\(90\) −8.54291 −0.900502
\(91\) 0 0
\(92\) −3.46219 −0.360958
\(93\) −12.8565 −1.33316
\(94\) −0.0538211 −0.00555122
\(95\) −8.00353 −0.821145
\(96\) −3.23497 −0.330167
\(97\) −18.7779 −1.90660 −0.953302 0.302017i \(-0.902340\pi\)
−0.953302 + 0.302017i \(0.902340\pi\)
\(98\) 0 0
\(99\) −6.35036 −0.638235
\(100\) −3.69036 −0.369036
\(101\) −13.7194 −1.36513 −0.682564 0.730826i \(-0.739135\pi\)
−0.682564 + 0.730826i \(0.739135\pi\)
\(102\) 0.628386 0.0622195
\(103\) −1.18115 −0.116382 −0.0581909 0.998305i \(-0.518533\pi\)
−0.0581909 + 0.998305i \(0.518533\pi\)
\(104\) −3.66136 −0.359026
\(105\) 0 0
\(106\) −2.67097 −0.259428
\(107\) −1.79224 −0.173263 −0.0866313 0.996240i \(-0.527610\pi\)
−0.0866313 + 0.996240i \(0.527610\pi\)
\(108\) −14.4442 −1.38989
\(109\) 3.64190 0.348830 0.174415 0.984672i \(-0.444197\pi\)
0.174415 + 0.984672i \(0.444197\pi\)
\(110\) 0.973516 0.0928211
\(111\) −23.3215 −2.21358
\(112\) 0 0
\(113\) −17.7718 −1.67183 −0.835915 0.548859i \(-0.815063\pi\)
−0.835915 + 0.548859i \(0.815063\pi\)
\(114\) −22.6244 −2.11896
\(115\) 3.96211 0.369468
\(116\) −1.00000 −0.0928477
\(117\) −27.3321 −2.52685
\(118\) 8.04354 0.740468
\(119\) 0 0
\(120\) 3.70208 0.337952
\(121\) −10.2763 −0.934213
\(122\) 5.08144 0.460052
\(123\) 13.0852 1.17985
\(124\) 3.97423 0.356896
\(125\) 9.94520 0.889526
\(126\) 0 0
\(127\) −11.5789 −1.02746 −0.513732 0.857950i \(-0.671738\pi\)
−0.513732 + 0.857950i \(0.671738\pi\)
\(128\) 1.00000 0.0883883
\(129\) −31.6311 −2.78496
\(130\) 4.19004 0.367490
\(131\) 16.1976 1.41519 0.707593 0.706620i \(-0.249781\pi\)
0.707593 + 0.706620i \(0.249781\pi\)
\(132\) 2.75193 0.239525
\(133\) 0 0
\(134\) −11.6003 −1.00211
\(135\) 16.5298 1.42266
\(136\) −0.194248 −0.0166566
\(137\) −12.7842 −1.09223 −0.546116 0.837710i \(-0.683894\pi\)
−0.546116 + 0.837710i \(0.683894\pi\)
\(138\) 11.2001 0.953412
\(139\) −0.170575 −0.0144679 −0.00723397 0.999974i \(-0.502303\pi\)
−0.00723397 + 0.999974i \(0.502303\pi\)
\(140\) 0 0
\(141\) 0.174109 0.0146627
\(142\) −11.2825 −0.946804
\(143\) 3.11465 0.260460
\(144\) 7.46501 0.622084
\(145\) 1.14439 0.0950367
\(146\) 7.31077 0.605044
\(147\) 0 0
\(148\) 7.20919 0.592592
\(149\) 14.0440 1.15053 0.575263 0.817969i \(-0.304900\pi\)
0.575263 + 0.817969i \(0.304900\pi\)
\(150\) 11.9382 0.974750
\(151\) 12.1976 0.992623 0.496311 0.868145i \(-0.334687\pi\)
0.496311 + 0.868145i \(0.334687\pi\)
\(152\) 6.99369 0.567263
\(153\) −1.45006 −0.117231
\(154\) 0 0
\(155\) −4.54808 −0.365311
\(156\) 11.8444 0.948309
\(157\) −14.3897 −1.14842 −0.574212 0.818707i \(-0.694691\pi\)
−0.574212 + 0.818707i \(0.694691\pi\)
\(158\) −13.3564 −1.06258
\(159\) 8.64051 0.685237
\(160\) −1.14439 −0.0904723
\(161\) 0 0
\(162\) 24.3314 1.91165
\(163\) 16.7674 1.31332 0.656660 0.754186i \(-0.271969\pi\)
0.656660 + 0.754186i \(0.271969\pi\)
\(164\) −4.04493 −0.315856
\(165\) −3.14929 −0.245172
\(166\) −9.04564 −0.702078
\(167\) 10.6555 0.824544 0.412272 0.911061i \(-0.364735\pi\)
0.412272 + 0.911061i \(0.364735\pi\)
\(168\) 0 0
\(169\) 0.405541 0.0311955
\(170\) 0.222296 0.0170494
\(171\) 52.2080 3.99244
\(172\) 9.77788 0.745556
\(173\) −17.9251 −1.36282 −0.681409 0.731902i \(-0.738633\pi\)
−0.681409 + 0.731902i \(0.738633\pi\)
\(174\) 3.23497 0.245242
\(175\) 0 0
\(176\) −0.850683 −0.0641226
\(177\) −26.0206 −1.95583
\(178\) −14.2305 −1.06662
\(179\) 5.28920 0.395333 0.197667 0.980269i \(-0.436664\pi\)
0.197667 + 0.980269i \(0.436664\pi\)
\(180\) −8.54291 −0.636751
\(181\) −13.1711 −0.978997 −0.489498 0.872004i \(-0.662820\pi\)
−0.489498 + 0.872004i \(0.662820\pi\)
\(182\) 0 0
\(183\) −16.4383 −1.21515
\(184\) −3.46219 −0.255236
\(185\) −8.25016 −0.606564
\(186\) −12.8565 −0.942684
\(187\) 0.165244 0.0120838
\(188\) −0.0538211 −0.00392531
\(189\) 0 0
\(190\) −8.00353 −0.580637
\(191\) 9.90869 0.716968 0.358484 0.933536i \(-0.383294\pi\)
0.358484 + 0.933536i \(0.383294\pi\)
\(192\) −3.23497 −0.233464
\(193\) −17.3950 −1.25212 −0.626061 0.779774i \(-0.715334\pi\)
−0.626061 + 0.779774i \(0.715334\pi\)
\(194\) −18.7779 −1.34817
\(195\) −13.5546 −0.970667
\(196\) 0 0
\(197\) 17.5961 1.25367 0.626834 0.779153i \(-0.284350\pi\)
0.626834 + 0.779153i \(0.284350\pi\)
\(198\) −6.35036 −0.451300
\(199\) −11.2846 −0.799943 −0.399972 0.916528i \(-0.630980\pi\)
−0.399972 + 0.916528i \(0.630980\pi\)
\(200\) −3.69036 −0.260948
\(201\) 37.5265 2.64691
\(202\) −13.7194 −0.965291
\(203\) 0 0
\(204\) 0.628386 0.0439958
\(205\) 4.62899 0.323303
\(206\) −1.18115 −0.0822943
\(207\) −25.8453 −1.79637
\(208\) −3.66136 −0.253869
\(209\) −5.94941 −0.411529
\(210\) 0 0
\(211\) −9.87572 −0.679873 −0.339936 0.940448i \(-0.610406\pi\)
−0.339936 + 0.940448i \(0.610406\pi\)
\(212\) −2.67097 −0.183443
\(213\) 36.4984 2.50083
\(214\) −1.79224 −0.122515
\(215\) −11.1897 −0.763134
\(216\) −14.4442 −0.982801
\(217\) 0 0
\(218\) 3.64190 0.246660
\(219\) −23.6501 −1.59813
\(220\) 0.973516 0.0656345
\(221\) 0.711212 0.0478413
\(222\) −23.3215 −1.56524
\(223\) 16.5982 1.11150 0.555748 0.831351i \(-0.312432\pi\)
0.555748 + 0.831351i \(0.312432\pi\)
\(224\) 0 0
\(225\) −27.5486 −1.83657
\(226\) −17.7718 −1.18216
\(227\) −17.9930 −1.19424 −0.597120 0.802152i \(-0.703688\pi\)
−0.597120 + 0.802152i \(0.703688\pi\)
\(228\) −22.6244 −1.49833
\(229\) 2.03821 0.134689 0.0673444 0.997730i \(-0.478547\pi\)
0.0673444 + 0.997730i \(0.478547\pi\)
\(230\) 3.96211 0.261253
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) −16.4989 −1.08088 −0.540441 0.841382i \(-0.681742\pi\)
−0.540441 + 0.841382i \(0.681742\pi\)
\(234\) −27.3321 −1.78675
\(235\) 0.0615925 0.00401785
\(236\) 8.04354 0.523590
\(237\) 43.2075 2.80663
\(238\) 0 0
\(239\) 18.7414 1.21228 0.606141 0.795357i \(-0.292717\pi\)
0.606141 + 0.795357i \(0.292717\pi\)
\(240\) 3.70208 0.238968
\(241\) −17.8781 −1.15163 −0.575813 0.817581i \(-0.695315\pi\)
−0.575813 + 0.817581i \(0.695315\pi\)
\(242\) −10.2763 −0.660588
\(243\) −35.3786 −2.26954
\(244\) 5.08144 0.325306
\(245\) 0 0
\(246\) 13.0852 0.834283
\(247\) −25.6064 −1.62930
\(248\) 3.97423 0.252364
\(249\) 29.2624 1.85443
\(250\) 9.94520 0.628990
\(251\) 1.69249 0.106829 0.0534144 0.998572i \(-0.482990\pi\)
0.0534144 + 0.998572i \(0.482990\pi\)
\(252\) 0 0
\(253\) 2.94522 0.185165
\(254\) −11.5789 −0.726527
\(255\) −0.719122 −0.0450331
\(256\) 1.00000 0.0625000
\(257\) −3.55324 −0.221645 −0.110822 0.993840i \(-0.535348\pi\)
−0.110822 + 0.993840i \(0.535348\pi\)
\(258\) −31.6311 −1.96927
\(259\) 0 0
\(260\) 4.19004 0.259855
\(261\) −7.46501 −0.462073
\(262\) 16.1976 1.00069
\(263\) −18.3627 −1.13229 −0.566147 0.824304i \(-0.691567\pi\)
−0.566147 + 0.824304i \(0.691567\pi\)
\(264\) 2.75193 0.169370
\(265\) 3.05665 0.187768
\(266\) 0 0
\(267\) 46.0353 2.81731
\(268\) −11.6003 −0.708599
\(269\) −18.1820 −1.10858 −0.554289 0.832324i \(-0.687010\pi\)
−0.554289 + 0.832324i \(0.687010\pi\)
\(270\) 16.5298 1.00597
\(271\) 7.30703 0.443871 0.221935 0.975061i \(-0.428763\pi\)
0.221935 + 0.975061i \(0.428763\pi\)
\(272\) −0.194248 −0.0117780
\(273\) 0 0
\(274\) −12.7842 −0.772324
\(275\) 3.13933 0.189309
\(276\) 11.2001 0.674164
\(277\) −16.6420 −0.999918 −0.499959 0.866049i \(-0.666652\pi\)
−0.499959 + 0.866049i \(0.666652\pi\)
\(278\) −0.170575 −0.0102304
\(279\) 29.6676 1.77616
\(280\) 0 0
\(281\) 13.2562 0.790800 0.395400 0.918509i \(-0.370606\pi\)
0.395400 + 0.918509i \(0.370606\pi\)
\(282\) 0.174109 0.0103681
\(283\) 15.2786 0.908217 0.454109 0.890946i \(-0.349958\pi\)
0.454109 + 0.890946i \(0.349958\pi\)
\(284\) −11.2825 −0.669492
\(285\) 25.8912 1.53366
\(286\) 3.11465 0.184173
\(287\) 0 0
\(288\) 7.46501 0.439880
\(289\) −16.9623 −0.997780
\(290\) 1.14439 0.0672011
\(291\) 60.7458 3.56098
\(292\) 7.31077 0.427830
\(293\) 15.5191 0.906637 0.453318 0.891349i \(-0.350240\pi\)
0.453318 + 0.891349i \(0.350240\pi\)
\(294\) 0 0
\(295\) −9.20498 −0.535935
\(296\) 7.20919 0.419026
\(297\) 12.2874 0.712987
\(298\) 14.0440 0.813544
\(299\) 12.6763 0.733090
\(300\) 11.9382 0.689252
\(301\) 0 0
\(302\) 12.1976 0.701890
\(303\) 44.3817 2.54966
\(304\) 6.99369 0.401116
\(305\) −5.81517 −0.332975
\(306\) −1.45006 −0.0828947
\(307\) −3.14277 −0.179368 −0.0896838 0.995970i \(-0.528586\pi\)
−0.0896838 + 0.995970i \(0.528586\pi\)
\(308\) 0 0
\(309\) 3.82097 0.217367
\(310\) −4.54808 −0.258314
\(311\) −24.2160 −1.37316 −0.686581 0.727053i \(-0.740889\pi\)
−0.686581 + 0.727053i \(0.740889\pi\)
\(312\) 11.8444 0.670555
\(313\) 10.3528 0.585175 0.292588 0.956239i \(-0.405484\pi\)
0.292588 + 0.956239i \(0.405484\pi\)
\(314\) −14.3897 −0.812058
\(315\) 0 0
\(316\) −13.3564 −0.751357
\(317\) −34.5329 −1.93956 −0.969780 0.243983i \(-0.921546\pi\)
−0.969780 + 0.243983i \(0.921546\pi\)
\(318\) 8.64051 0.484536
\(319\) 0.850683 0.0476291
\(320\) −1.14439 −0.0639736
\(321\) 5.79784 0.323604
\(322\) 0 0
\(323\) −1.35851 −0.0755896
\(324\) 24.3314 1.35174
\(325\) 13.5117 0.749496
\(326\) 16.7674 0.928658
\(327\) −11.7814 −0.651514
\(328\) −4.04493 −0.223344
\(329\) 0 0
\(330\) −3.14929 −0.173363
\(331\) 5.10043 0.280345 0.140173 0.990127i \(-0.455234\pi\)
0.140173 + 0.990127i \(0.455234\pi\)
\(332\) −9.04564 −0.496444
\(333\) 53.8167 2.94914
\(334\) 10.6555 0.583041
\(335\) 13.2753 0.725306
\(336\) 0 0
\(337\) 24.7613 1.34884 0.674418 0.738350i \(-0.264394\pi\)
0.674418 + 0.738350i \(0.264394\pi\)
\(338\) 0.405541 0.0220585
\(339\) 57.4912 3.12249
\(340\) 0.222296 0.0120557
\(341\) −3.38081 −0.183081
\(342\) 52.2080 2.82308
\(343\) 0 0
\(344\) 9.77788 0.527188
\(345\) −12.8173 −0.690059
\(346\) −17.9251 −0.963658
\(347\) 6.42032 0.344661 0.172331 0.985039i \(-0.444870\pi\)
0.172331 + 0.985039i \(0.444870\pi\)
\(348\) 3.23497 0.173412
\(349\) 2.54039 0.135984 0.0679920 0.997686i \(-0.478341\pi\)
0.0679920 + 0.997686i \(0.478341\pi\)
\(350\) 0 0
\(351\) 52.8852 2.82281
\(352\) −0.850683 −0.0453415
\(353\) 1.20628 0.0642038 0.0321019 0.999485i \(-0.489780\pi\)
0.0321019 + 0.999485i \(0.489780\pi\)
\(354\) −26.0206 −1.38298
\(355\) 12.9116 0.685276
\(356\) −14.2305 −0.754216
\(357\) 0 0
\(358\) 5.28920 0.279543
\(359\) 0.315996 0.0166777 0.00833883 0.999965i \(-0.497346\pi\)
0.00833883 + 0.999965i \(0.497346\pi\)
\(360\) −8.54291 −0.450251
\(361\) 29.9117 1.57430
\(362\) −13.1711 −0.692255
\(363\) 33.2436 1.74484
\(364\) 0 0
\(365\) −8.36640 −0.437917
\(366\) −16.4383 −0.859243
\(367\) −24.2997 −1.26843 −0.634217 0.773155i \(-0.718678\pi\)
−0.634217 + 0.773155i \(0.718678\pi\)
\(368\) −3.46219 −0.180479
\(369\) −30.1955 −1.57191
\(370\) −8.25016 −0.428905
\(371\) 0 0
\(372\) −12.8565 −0.666578
\(373\) −9.98510 −0.517009 −0.258504 0.966010i \(-0.583230\pi\)
−0.258504 + 0.966010i \(0.583230\pi\)
\(374\) 0.165244 0.00854454
\(375\) −32.1724 −1.66137
\(376\) −0.0538211 −0.00277561
\(377\) 3.66136 0.188570
\(378\) 0 0
\(379\) −13.0891 −0.672343 −0.336172 0.941801i \(-0.609132\pi\)
−0.336172 + 0.941801i \(0.609132\pi\)
\(380\) −8.00353 −0.410573
\(381\) 37.4575 1.91901
\(382\) 9.90869 0.506973
\(383\) 2.96390 0.151448 0.0757241 0.997129i \(-0.475873\pi\)
0.0757241 + 0.997129i \(0.475873\pi\)
\(384\) −3.23497 −0.165084
\(385\) 0 0
\(386\) −17.3950 −0.885384
\(387\) 72.9920 3.71039
\(388\) −18.7779 −0.953302
\(389\) −34.9114 −1.77008 −0.885040 0.465514i \(-0.845869\pi\)
−0.885040 + 0.465514i \(0.845869\pi\)
\(390\) −13.5546 −0.686365
\(391\) 0.672523 0.0340110
\(392\) 0 0
\(393\) −52.3985 −2.64316
\(394\) 17.5961 0.886477
\(395\) 15.2850 0.769072
\(396\) −6.35036 −0.319117
\(397\) −15.4156 −0.773685 −0.386842 0.922146i \(-0.626434\pi\)
−0.386842 + 0.922146i \(0.626434\pi\)
\(398\) −11.2846 −0.565645
\(399\) 0 0
\(400\) −3.69036 −0.184518
\(401\) 38.0366 1.89946 0.949728 0.313077i \(-0.101360\pi\)
0.949728 + 0.313077i \(0.101360\pi\)
\(402\) 37.5265 1.87165
\(403\) −14.5511 −0.724840
\(404\) −13.7194 −0.682564
\(405\) −27.8447 −1.38361
\(406\) 0 0
\(407\) −6.13274 −0.303989
\(408\) 0.628386 0.0311098
\(409\) 29.2856 1.44808 0.724041 0.689757i \(-0.242283\pi\)
0.724041 + 0.689757i \(0.242283\pi\)
\(410\) 4.62899 0.228610
\(411\) 41.3566 2.03997
\(412\) −1.18115 −0.0581909
\(413\) 0 0
\(414\) −25.8453 −1.27023
\(415\) 10.3518 0.508149
\(416\) −3.66136 −0.179513
\(417\) 0.551803 0.0270219
\(418\) −5.94941 −0.290995
\(419\) 6.29412 0.307488 0.153744 0.988111i \(-0.450867\pi\)
0.153744 + 0.988111i \(0.450867\pi\)
\(420\) 0 0
\(421\) 2.94479 0.143520 0.0717602 0.997422i \(-0.477138\pi\)
0.0717602 + 0.997422i \(0.477138\pi\)
\(422\) −9.87572 −0.480743
\(423\) −0.401775 −0.0195350
\(424\) −2.67097 −0.129714
\(425\) 0.716846 0.0347721
\(426\) 36.4984 1.76836
\(427\) 0 0
\(428\) −1.79224 −0.0866313
\(429\) −10.0758 −0.486464
\(430\) −11.1897 −0.539617
\(431\) 5.45942 0.262971 0.131486 0.991318i \(-0.458025\pi\)
0.131486 + 0.991318i \(0.458025\pi\)
\(432\) −14.4442 −0.694945
\(433\) 16.9899 0.816482 0.408241 0.912874i \(-0.366142\pi\)
0.408241 + 0.912874i \(0.366142\pi\)
\(434\) 0 0
\(435\) −3.70208 −0.177501
\(436\) 3.64190 0.174415
\(437\) −24.2135 −1.15829
\(438\) −23.6501 −1.13005
\(439\) 22.6419 1.08064 0.540321 0.841459i \(-0.318303\pi\)
0.540321 + 0.841459i \(0.318303\pi\)
\(440\) 0.973516 0.0464106
\(441\) 0 0
\(442\) 0.711212 0.0338289
\(443\) −17.1387 −0.814285 −0.407143 0.913365i \(-0.633475\pi\)
−0.407143 + 0.913365i \(0.633475\pi\)
\(444\) −23.3215 −1.10679
\(445\) 16.2853 0.771998
\(446\) 16.5982 0.785946
\(447\) −45.4317 −2.14885
\(448\) 0 0
\(449\) −12.1802 −0.574818 −0.287409 0.957808i \(-0.592794\pi\)
−0.287409 + 0.957808i \(0.592794\pi\)
\(450\) −27.5486 −1.29865
\(451\) 3.44095 0.162028
\(452\) −17.7718 −0.835915
\(453\) −39.4587 −1.85393
\(454\) −17.9930 −0.844455
\(455\) 0 0
\(456\) −22.6244 −1.05948
\(457\) 31.5102 1.47398 0.736991 0.675902i \(-0.236246\pi\)
0.736991 + 0.675902i \(0.236246\pi\)
\(458\) 2.03821 0.0952394
\(459\) 2.80575 0.130961
\(460\) 3.96211 0.184734
\(461\) 6.40796 0.298448 0.149224 0.988803i \(-0.452322\pi\)
0.149224 + 0.988803i \(0.452322\pi\)
\(462\) 0 0
\(463\) −0.902939 −0.0419632 −0.0209816 0.999780i \(-0.506679\pi\)
−0.0209816 + 0.999780i \(0.506679\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 14.7129 0.682294
\(466\) −16.4989 −0.764299
\(467\) 36.4112 1.68491 0.842454 0.538768i \(-0.181110\pi\)
0.842454 + 0.538768i \(0.181110\pi\)
\(468\) −27.3321 −1.26343
\(469\) 0 0
\(470\) 0.0615925 0.00284105
\(471\) 46.5502 2.14492
\(472\) 8.04354 0.370234
\(473\) −8.31787 −0.382456
\(474\) 43.2075 1.98459
\(475\) −25.8092 −1.18421
\(476\) 0 0
\(477\) −19.9388 −0.912937
\(478\) 18.7414 0.857213
\(479\) −31.4283 −1.43600 −0.717998 0.696045i \(-0.754941\pi\)
−0.717998 + 0.696045i \(0.754941\pi\)
\(480\) 3.70208 0.168976
\(481\) −26.3954 −1.20353
\(482\) −17.8781 −0.814323
\(483\) 0 0
\(484\) −10.2763 −0.467106
\(485\) 21.4893 0.975778
\(486\) −35.3786 −1.60481
\(487\) 14.4460 0.654609 0.327304 0.944919i \(-0.393860\pi\)
0.327304 + 0.944919i \(0.393860\pi\)
\(488\) 5.08144 0.230026
\(489\) −54.2418 −2.45290
\(490\) 0 0
\(491\) 29.2405 1.31961 0.659803 0.751439i \(-0.270640\pi\)
0.659803 + 0.751439i \(0.270640\pi\)
\(492\) 13.0852 0.589927
\(493\) 0.194248 0.00874850
\(494\) −25.6064 −1.15209
\(495\) 7.26731 0.326641
\(496\) 3.97423 0.178448
\(497\) 0 0
\(498\) 29.2624 1.31128
\(499\) 28.4905 1.27541 0.637704 0.770282i \(-0.279884\pi\)
0.637704 + 0.770282i \(0.279884\pi\)
\(500\) 9.94520 0.444763
\(501\) −34.4700 −1.54001
\(502\) 1.69249 0.0755393
\(503\) −19.9091 −0.887701 −0.443850 0.896101i \(-0.646388\pi\)
−0.443850 + 0.896101i \(0.646388\pi\)
\(504\) 0 0
\(505\) 15.7004 0.698657
\(506\) 2.94522 0.130931
\(507\) −1.31191 −0.0582641
\(508\) −11.5789 −0.513732
\(509\) −3.57199 −0.158326 −0.0791628 0.996862i \(-0.525225\pi\)
−0.0791628 + 0.996862i \(0.525225\pi\)
\(510\) −0.719122 −0.0318432
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −101.018 −4.46005
\(514\) −3.55324 −0.156727
\(515\) 1.35170 0.0595628
\(516\) −31.6311 −1.39248
\(517\) 0.0457847 0.00201361
\(518\) 0 0
\(519\) 57.9871 2.54535
\(520\) 4.19004 0.183745
\(521\) 9.36877 0.410453 0.205227 0.978714i \(-0.434207\pi\)
0.205227 + 0.978714i \(0.434207\pi\)
\(522\) −7.46501 −0.326735
\(523\) 13.6159 0.595381 0.297690 0.954662i \(-0.403784\pi\)
0.297690 + 0.954662i \(0.403784\pi\)
\(524\) 16.1976 0.707593
\(525\) 0 0
\(526\) −18.3627 −0.800653
\(527\) −0.771986 −0.0336283
\(528\) 2.75193 0.119762
\(529\) −11.0133 −0.478838
\(530\) 3.05665 0.132772
\(531\) 60.0451 2.60574
\(532\) 0 0
\(533\) 14.8099 0.641490
\(534\) 46.0353 1.99214
\(535\) 2.05103 0.0886738
\(536\) −11.6003 −0.501055
\(537\) −17.1104 −0.738367
\(538\) −18.1820 −0.783883
\(539\) 0 0
\(540\) 16.5298 0.711330
\(541\) 10.6970 0.459901 0.229951 0.973202i \(-0.426144\pi\)
0.229951 + 0.973202i \(0.426144\pi\)
\(542\) 7.30703 0.313864
\(543\) 42.6079 1.82848
\(544\) −0.194248 −0.00832832
\(545\) −4.16776 −0.178527
\(546\) 0 0
\(547\) −23.6925 −1.01302 −0.506509 0.862235i \(-0.669064\pi\)
−0.506509 + 0.862235i \(0.669064\pi\)
\(548\) −12.7842 −0.546116
\(549\) 37.9330 1.61894
\(550\) 3.13933 0.133861
\(551\) −6.99369 −0.297941
\(552\) 11.2001 0.476706
\(553\) 0 0
\(554\) −16.6420 −0.707049
\(555\) 26.6890 1.13288
\(556\) −0.170575 −0.00723397
\(557\) 12.2781 0.520242 0.260121 0.965576i \(-0.416238\pi\)
0.260121 + 0.965576i \(0.416238\pi\)
\(558\) 29.6676 1.25593
\(559\) −35.8003 −1.51419
\(560\) 0 0
\(561\) −0.534558 −0.0225690
\(562\) 13.2562 0.559180
\(563\) −37.3831 −1.57551 −0.787755 0.615989i \(-0.788756\pi\)
−0.787755 + 0.615989i \(0.788756\pi\)
\(564\) 0.174109 0.00733133
\(565\) 20.3379 0.855624
\(566\) 15.2786 0.642207
\(567\) 0 0
\(568\) −11.2825 −0.473402
\(569\) −14.2478 −0.597300 −0.298650 0.954363i \(-0.596536\pi\)
−0.298650 + 0.954363i \(0.596536\pi\)
\(570\) 25.8912 1.08446
\(571\) −6.59865 −0.276145 −0.138072 0.990422i \(-0.544091\pi\)
−0.138072 + 0.990422i \(0.544091\pi\)
\(572\) 3.11465 0.130230
\(573\) −32.0543 −1.33909
\(574\) 0 0
\(575\) 12.7767 0.532826
\(576\) 7.46501 0.311042
\(577\) 21.0911 0.878032 0.439016 0.898479i \(-0.355327\pi\)
0.439016 + 0.898479i \(0.355327\pi\)
\(578\) −16.9623 −0.705537
\(579\) 56.2723 2.33860
\(580\) 1.14439 0.0475184
\(581\) 0 0
\(582\) 60.7458 2.51800
\(583\) 2.27215 0.0941029
\(584\) 7.31077 0.302522
\(585\) 31.2787 1.29321
\(586\) 15.5191 0.641089
\(587\) −23.5851 −0.973459 −0.486730 0.873553i \(-0.661810\pi\)
−0.486730 + 0.873553i \(0.661810\pi\)
\(588\) 0 0
\(589\) 27.7945 1.14525
\(590\) −9.20498 −0.378963
\(591\) −56.9227 −2.34149
\(592\) 7.20919 0.296296
\(593\) 24.8808 1.02173 0.510865 0.859661i \(-0.329325\pi\)
0.510865 + 0.859661i \(0.329325\pi\)
\(594\) 12.2874 0.504158
\(595\) 0 0
\(596\) 14.0440 0.575263
\(597\) 36.5053 1.49406
\(598\) 12.6763 0.518373
\(599\) 4.77907 0.195268 0.0976338 0.995222i \(-0.468873\pi\)
0.0976338 + 0.995222i \(0.468873\pi\)
\(600\) 11.9382 0.487375
\(601\) 44.3292 1.80823 0.904113 0.427292i \(-0.140533\pi\)
0.904113 + 0.427292i \(0.140533\pi\)
\(602\) 0 0
\(603\) −86.5961 −3.52647
\(604\) 12.1976 0.496311
\(605\) 11.7602 0.478119
\(606\) 44.3817 1.80288
\(607\) −42.8992 −1.74123 −0.870613 0.491968i \(-0.836278\pi\)
−0.870613 + 0.491968i \(0.836278\pi\)
\(608\) 6.99369 0.283632
\(609\) 0 0
\(610\) −5.81517 −0.235449
\(611\) 0.197058 0.00797212
\(612\) −1.45006 −0.0586154
\(613\) 26.0405 1.05177 0.525884 0.850557i \(-0.323735\pi\)
0.525884 + 0.850557i \(0.323735\pi\)
\(614\) −3.14277 −0.126832
\(615\) −14.9746 −0.603836
\(616\) 0 0
\(617\) −38.2301 −1.53909 −0.769543 0.638595i \(-0.779516\pi\)
−0.769543 + 0.638595i \(0.779516\pi\)
\(618\) 3.82097 0.153702
\(619\) −33.8413 −1.36020 −0.680098 0.733122i \(-0.738063\pi\)
−0.680098 + 0.733122i \(0.738063\pi\)
\(620\) −4.54808 −0.182655
\(621\) 50.0084 2.00677
\(622\) −24.2160 −0.970973
\(623\) 0 0
\(624\) 11.8444 0.474154
\(625\) 7.07059 0.282823
\(626\) 10.3528 0.413781
\(627\) 19.2461 0.768617
\(628\) −14.3897 −0.574212
\(629\) −1.40037 −0.0558365
\(630\) 0 0
\(631\) −36.3237 −1.44602 −0.723012 0.690835i \(-0.757243\pi\)
−0.723012 + 0.690835i \(0.757243\pi\)
\(632\) −13.3564 −0.531290
\(633\) 31.9476 1.26980
\(634\) −34.5329 −1.37148
\(635\) 13.2509 0.525845
\(636\) 8.64051 0.342618
\(637\) 0 0
\(638\) 0.850683 0.0336789
\(639\) −84.2238 −3.33184
\(640\) −1.14439 −0.0452361
\(641\) −5.01376 −0.198031 −0.0990157 0.995086i \(-0.531569\pi\)
−0.0990157 + 0.995086i \(0.531569\pi\)
\(642\) 5.79784 0.228823
\(643\) 21.6758 0.854811 0.427406 0.904060i \(-0.359428\pi\)
0.427406 + 0.904060i \(0.359428\pi\)
\(644\) 0 0
\(645\) 36.1985 1.42531
\(646\) −1.35851 −0.0534499
\(647\) 23.0945 0.907940 0.453970 0.891017i \(-0.350007\pi\)
0.453970 + 0.891017i \(0.350007\pi\)
\(648\) 24.3314 0.955826
\(649\) −6.84250 −0.268592
\(650\) 13.5117 0.529974
\(651\) 0 0
\(652\) 16.7674 0.656660
\(653\) −33.9240 −1.32755 −0.663774 0.747933i \(-0.731046\pi\)
−0.663774 + 0.747933i \(0.731046\pi\)
\(654\) −11.7814 −0.460690
\(655\) −18.5364 −0.724276
\(656\) −4.04493 −0.157928
\(657\) 54.5750 2.12917
\(658\) 0 0
\(659\) −5.52520 −0.215231 −0.107616 0.994193i \(-0.534322\pi\)
−0.107616 + 0.994193i \(0.534322\pi\)
\(660\) −3.14929 −0.122586
\(661\) 1.62365 0.0631527 0.0315763 0.999501i \(-0.489947\pi\)
0.0315763 + 0.999501i \(0.489947\pi\)
\(662\) 5.10043 0.198234
\(663\) −2.30075 −0.0893536
\(664\) −9.04564 −0.351039
\(665\) 0 0
\(666\) 53.8167 2.08536
\(667\) 3.46219 0.134056
\(668\) 10.6555 0.412272
\(669\) −53.6945 −2.07595
\(670\) 13.2753 0.512869
\(671\) −4.32269 −0.166876
\(672\) 0 0
\(673\) −1.69886 −0.0654862 −0.0327431 0.999464i \(-0.510424\pi\)
−0.0327431 + 0.999464i \(0.510424\pi\)
\(674\) 24.7613 0.953771
\(675\) 53.3042 2.05168
\(676\) 0.405541 0.0155977
\(677\) −20.7277 −0.796628 −0.398314 0.917249i \(-0.630405\pi\)
−0.398314 + 0.917249i \(0.630405\pi\)
\(678\) 57.4912 2.20794
\(679\) 0 0
\(680\) 0.222296 0.00852468
\(681\) 58.2068 2.23049
\(682\) −3.38081 −0.129458
\(683\) −30.3817 −1.16252 −0.581261 0.813717i \(-0.697441\pi\)
−0.581261 + 0.813717i \(0.697441\pi\)
\(684\) 52.2080 1.99622
\(685\) 14.6302 0.558991
\(686\) 0 0
\(687\) −6.59355 −0.251560
\(688\) 9.77788 0.372778
\(689\) 9.77939 0.372565
\(690\) −12.8173 −0.487945
\(691\) 4.62046 0.175771 0.0878853 0.996131i \(-0.471989\pi\)
0.0878853 + 0.996131i \(0.471989\pi\)
\(692\) −17.9251 −0.681409
\(693\) 0 0
\(694\) 6.42032 0.243712
\(695\) 0.195204 0.00740453
\(696\) 3.23497 0.122621
\(697\) 0.785720 0.0297613
\(698\) 2.54039 0.0961553
\(699\) 53.3735 2.01877
\(700\) 0 0
\(701\) 36.9179 1.39437 0.697186 0.716890i \(-0.254435\pi\)
0.697186 + 0.716890i \(0.254435\pi\)
\(702\) 52.8852 1.99602
\(703\) 50.4189 1.90158
\(704\) −0.850683 −0.0320613
\(705\) −0.199250 −0.00750418
\(706\) 1.20628 0.0453989
\(707\) 0 0
\(708\) −26.0206 −0.977914
\(709\) −32.0742 −1.20457 −0.602286 0.798281i \(-0.705743\pi\)
−0.602286 + 0.798281i \(0.705743\pi\)
\(710\) 12.9116 0.484564
\(711\) −99.7058 −3.73926
\(712\) −14.2305 −0.533311
\(713\) −13.7595 −0.515298
\(714\) 0 0
\(715\) −3.56439 −0.133301
\(716\) 5.28920 0.197667
\(717\) −60.6279 −2.26419
\(718\) 0.315996 0.0117929
\(719\) −35.8354 −1.33644 −0.668218 0.743966i \(-0.732943\pi\)
−0.668218 + 0.743966i \(0.732943\pi\)
\(720\) −8.54291 −0.318376
\(721\) 0 0
\(722\) 29.9117 1.11320
\(723\) 57.8349 2.15090
\(724\) −13.1711 −0.489498
\(725\) 3.69036 0.137057
\(726\) 33.2436 1.23379
\(727\) 12.8401 0.476215 0.238107 0.971239i \(-0.423473\pi\)
0.238107 + 0.971239i \(0.423473\pi\)
\(728\) 0 0
\(729\) 41.4546 1.53536
\(730\) −8.36640 −0.309654
\(731\) −1.89934 −0.0702495
\(732\) −16.4383 −0.607576
\(733\) 9.13293 0.337332 0.168666 0.985673i \(-0.446054\pi\)
0.168666 + 0.985673i \(0.446054\pi\)
\(734\) −24.2997 −0.896919
\(735\) 0 0
\(736\) −3.46219 −0.127618
\(737\) 9.86815 0.363498
\(738\) −30.1955 −1.11151
\(739\) 6.32454 0.232652 0.116326 0.993211i \(-0.462888\pi\)
0.116326 + 0.993211i \(0.462888\pi\)
\(740\) −8.25016 −0.303282
\(741\) 82.8358 3.04305
\(742\) 0 0
\(743\) 32.2973 1.18487 0.592436 0.805618i \(-0.298166\pi\)
0.592436 + 0.805618i \(0.298166\pi\)
\(744\) −12.8565 −0.471342
\(745\) −16.0718 −0.588826
\(746\) −9.98510 −0.365581
\(747\) −67.5258 −2.47064
\(748\) 0.165244 0.00604190
\(749\) 0 0
\(750\) −32.1724 −1.17477
\(751\) 19.8619 0.724771 0.362385 0.932028i \(-0.381962\pi\)
0.362385 + 0.932028i \(0.381962\pi\)
\(752\) −0.0538211 −0.00196265
\(753\) −5.47514 −0.199525
\(754\) 3.66136 0.133339
\(755\) −13.9588 −0.508013
\(756\) 0 0
\(757\) 24.1136 0.876425 0.438213 0.898871i \(-0.355612\pi\)
0.438213 + 0.898871i \(0.355612\pi\)
\(758\) −13.0891 −0.475419
\(759\) −9.52770 −0.345834
\(760\) −8.00353 −0.290319
\(761\) −39.6245 −1.43639 −0.718194 0.695843i \(-0.755031\pi\)
−0.718194 + 0.695843i \(0.755031\pi\)
\(762\) 37.4575 1.35694
\(763\) 0 0
\(764\) 9.90869 0.358484
\(765\) 1.65945 0.0599974
\(766\) 2.96390 0.107090
\(767\) −29.4503 −1.06339
\(768\) −3.23497 −0.116732
\(769\) 11.8143 0.426034 0.213017 0.977049i \(-0.431671\pi\)
0.213017 + 0.977049i \(0.431671\pi\)
\(770\) 0 0
\(771\) 11.4946 0.413968
\(772\) −17.3950 −0.626061
\(773\) −18.0208 −0.648164 −0.324082 0.946029i \(-0.605055\pi\)
−0.324082 + 0.946029i \(0.605055\pi\)
\(774\) 72.9920 2.62364
\(775\) −14.6663 −0.526830
\(776\) −18.7779 −0.674087
\(777\) 0 0
\(778\) −34.9114 −1.25164
\(779\) −28.2890 −1.01356
\(780\) −13.5546 −0.485334
\(781\) 9.59781 0.343437
\(782\) 0.672523 0.0240494
\(783\) 14.4442 0.516192
\(784\) 0 0
\(785\) 16.4675 0.587750
\(786\) −52.3985 −1.86899
\(787\) 7.31843 0.260874 0.130437 0.991457i \(-0.458362\pi\)
0.130437 + 0.991457i \(0.458362\pi\)
\(788\) 17.5961 0.626834
\(789\) 59.4028 2.11480
\(790\) 15.2850 0.543816
\(791\) 0 0
\(792\) −6.35036 −0.225650
\(793\) −18.6050 −0.660682
\(794\) −15.4156 −0.547078
\(795\) −9.88815 −0.350696
\(796\) −11.2846 −0.399972
\(797\) −53.5394 −1.89646 −0.948232 0.317578i \(-0.897131\pi\)
−0.948232 + 0.317578i \(0.897131\pi\)
\(798\) 0 0
\(799\) 0.0104546 0.000369859 0
\(800\) −3.69036 −0.130474
\(801\) −106.231 −3.75349
\(802\) 38.0366 1.34312
\(803\) −6.21915 −0.219469
\(804\) 37.5265 1.32346
\(805\) 0 0
\(806\) −14.5511 −0.512540
\(807\) 58.8183 2.07050
\(808\) −13.7194 −0.482646
\(809\) 8.89326 0.312670 0.156335 0.987704i \(-0.450032\pi\)
0.156335 + 0.987704i \(0.450032\pi\)
\(810\) −27.8447 −0.978361
\(811\) −4.11053 −0.144340 −0.0721701 0.997392i \(-0.522992\pi\)
−0.0721701 + 0.997392i \(0.522992\pi\)
\(812\) 0 0
\(813\) −23.6380 −0.829021
\(814\) −6.13274 −0.214952
\(815\) −19.1885 −0.672142
\(816\) 0.628386 0.0219979
\(817\) 68.3834 2.39243
\(818\) 29.2856 1.02395
\(819\) 0 0
\(820\) 4.62899 0.161652
\(821\) 14.2064 0.495806 0.247903 0.968785i \(-0.420259\pi\)
0.247903 + 0.968785i \(0.420259\pi\)
\(822\) 41.3566 1.44248
\(823\) 18.8402 0.656728 0.328364 0.944551i \(-0.393503\pi\)
0.328364 + 0.944551i \(0.393503\pi\)
\(824\) −1.18115 −0.0411472
\(825\) −10.1556 −0.353573
\(826\) 0 0
\(827\) 7.13405 0.248075 0.124038 0.992278i \(-0.460416\pi\)
0.124038 + 0.992278i \(0.460416\pi\)
\(828\) −25.8453 −0.898185
\(829\) −1.21203 −0.0420956 −0.0210478 0.999778i \(-0.506700\pi\)
−0.0210478 + 0.999778i \(0.506700\pi\)
\(830\) 10.3518 0.359315
\(831\) 53.8362 1.86756
\(832\) −3.66136 −0.126935
\(833\) 0 0
\(834\) 0.551803 0.0191074
\(835\) −12.1940 −0.421992
\(836\) −5.94941 −0.205765
\(837\) −57.4044 −1.98419
\(838\) 6.29412 0.217427
\(839\) −13.0004 −0.448823 −0.224412 0.974494i \(-0.572046\pi\)
−0.224412 + 0.974494i \(0.572046\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 2.94479 0.101484
\(843\) −42.8834 −1.47698
\(844\) −9.87572 −0.339936
\(845\) −0.464099 −0.0159655
\(846\) −0.401775 −0.0138133
\(847\) 0 0
\(848\) −2.67097 −0.0917216
\(849\) −49.4257 −1.69629
\(850\) 0.716846 0.0245876
\(851\) −24.9596 −0.855603
\(852\) 36.4984 1.25042
\(853\) −23.8201 −0.815584 −0.407792 0.913075i \(-0.633701\pi\)
−0.407792 + 0.913075i \(0.633701\pi\)
\(854\) 0 0
\(855\) −59.7465 −2.04329
\(856\) −1.79224 −0.0612576
\(857\) 7.27140 0.248386 0.124193 0.992258i \(-0.460366\pi\)
0.124193 + 0.992258i \(0.460366\pi\)
\(858\) −10.0758 −0.343982
\(859\) −9.83263 −0.335485 −0.167742 0.985831i \(-0.553648\pi\)
−0.167742 + 0.985831i \(0.553648\pi\)
\(860\) −11.1897 −0.381567
\(861\) 0 0
\(862\) 5.45942 0.185949
\(863\) 21.9413 0.746890 0.373445 0.927652i \(-0.378176\pi\)
0.373445 + 0.927652i \(0.378176\pi\)
\(864\) −14.4442 −0.491400
\(865\) 20.5134 0.697475
\(866\) 16.9899 0.577340
\(867\) 54.8724 1.86356
\(868\) 0 0
\(869\) 11.3621 0.385432
\(870\) −3.70208 −0.125512
\(871\) 42.4727 1.43913
\(872\) 3.64190 0.123330
\(873\) −140.177 −4.74427
\(874\) −24.2135 −0.819032
\(875\) 0 0
\(876\) −23.6501 −0.799063
\(877\) 36.5659 1.23474 0.617372 0.786672i \(-0.288197\pi\)
0.617372 + 0.786672i \(0.288197\pi\)
\(878\) 22.6419 0.764129
\(879\) −50.2039 −1.69333
\(880\) 0.973516 0.0328172
\(881\) 33.4505 1.12698 0.563489 0.826124i \(-0.309459\pi\)
0.563489 + 0.826124i \(0.309459\pi\)
\(882\) 0 0
\(883\) −23.9555 −0.806166 −0.403083 0.915163i \(-0.632061\pi\)
−0.403083 + 0.915163i \(0.632061\pi\)
\(884\) 0.711212 0.0239207
\(885\) 29.7778 1.00097
\(886\) −17.1387 −0.575787
\(887\) −7.39554 −0.248318 −0.124159 0.992262i \(-0.539623\pi\)
−0.124159 + 0.992262i \(0.539623\pi\)
\(888\) −23.3215 −0.782618
\(889\) 0 0
\(890\) 16.2853 0.545885
\(891\) −20.6983 −0.693418
\(892\) 16.5982 0.555748
\(893\) −0.376408 −0.0125960
\(894\) −45.4317 −1.51946
\(895\) −6.05292 −0.202327
\(896\) 0 0
\(897\) −41.0074 −1.36920
\(898\) −12.1802 −0.406457
\(899\) −3.97423 −0.132548
\(900\) −27.5486 −0.918286
\(901\) 0.518832 0.0172848
\(902\) 3.44095 0.114571
\(903\) 0 0
\(904\) −17.7718 −0.591081
\(905\) 15.0729 0.501039
\(906\) −39.4587 −1.31093
\(907\) 28.5892 0.949289 0.474644 0.880178i \(-0.342577\pi\)
0.474644 + 0.880178i \(0.342577\pi\)
\(908\) −17.9930 −0.597120
\(909\) −102.415 −3.39690
\(910\) 0 0
\(911\) −15.2600 −0.505587 −0.252793 0.967520i \(-0.581349\pi\)
−0.252793 + 0.967520i \(0.581349\pi\)
\(912\) −22.6244 −0.749167
\(913\) 7.69497 0.254666
\(914\) 31.5102 1.04226
\(915\) 18.8119 0.621901
\(916\) 2.03821 0.0673444
\(917\) 0 0
\(918\) 2.80575 0.0926036
\(919\) −22.3019 −0.735673 −0.367836 0.929891i \(-0.619901\pi\)
−0.367836 + 0.929891i \(0.619901\pi\)
\(920\) 3.96211 0.130627
\(921\) 10.1668 0.335006
\(922\) 6.40796 0.211035
\(923\) 41.3092 1.35971
\(924\) 0 0
\(925\) −26.6045 −0.874752
\(926\) −0.902939 −0.0296724
\(927\) −8.81727 −0.289597
\(928\) −1.00000 −0.0328266
\(929\) 44.9700 1.47542 0.737710 0.675118i \(-0.235907\pi\)
0.737710 + 0.675118i \(0.235907\pi\)
\(930\) 14.7129 0.482455
\(931\) 0 0
\(932\) −16.4989 −0.540441
\(933\) 78.3379 2.56467
\(934\) 36.4112 1.19141
\(935\) −0.189104 −0.00618435
\(936\) −27.3321 −0.893377
\(937\) −20.6347 −0.674105 −0.337053 0.941486i \(-0.609430\pi\)
−0.337053 + 0.941486i \(0.609430\pi\)
\(938\) 0 0
\(939\) −33.4910 −1.09294
\(940\) 0.0615925 0.00200893
\(941\) −33.2588 −1.08421 −0.542103 0.840312i \(-0.682372\pi\)
−0.542103 + 0.840312i \(0.682372\pi\)
\(942\) 46.5502 1.51669
\(943\) 14.0043 0.456043
\(944\) 8.04354 0.261795
\(945\) 0 0
\(946\) −8.31787 −0.270437
\(947\) 0.0959893 0.00311923 0.00155962 0.999999i \(-0.499504\pi\)
0.00155962 + 0.999999i \(0.499504\pi\)
\(948\) 43.2075 1.40332
\(949\) −26.7673 −0.868905
\(950\) −25.8092 −0.837362
\(951\) 111.713 3.62253
\(952\) 0 0
\(953\) −11.4213 −0.369974 −0.184987 0.982741i \(-0.559224\pi\)
−0.184987 + 0.982741i \(0.559224\pi\)
\(954\) −19.9388 −0.645544
\(955\) −11.3394 −0.366936
\(956\) 18.7414 0.606141
\(957\) −2.75193 −0.0889573
\(958\) −31.4283 −1.01540
\(959\) 0 0
\(960\) 3.70208 0.119484
\(961\) −15.2055 −0.490501
\(962\) −26.3954 −0.851023
\(963\) −13.3791 −0.431136
\(964\) −17.8781 −0.575813
\(965\) 19.9068 0.640822
\(966\) 0 0
\(967\) −9.43755 −0.303491 −0.151746 0.988420i \(-0.548489\pi\)
−0.151746 + 0.988420i \(0.548489\pi\)
\(968\) −10.2763 −0.330294
\(969\) 4.39474 0.141179
\(970\) 21.4893 0.689980
\(971\) 18.5280 0.594592 0.297296 0.954785i \(-0.403915\pi\)
0.297296 + 0.954785i \(0.403915\pi\)
\(972\) −35.3786 −1.13477
\(973\) 0 0
\(974\) 14.4460 0.462878
\(975\) −43.7100 −1.39984
\(976\) 5.08144 0.162653
\(977\) −40.6766 −1.30136 −0.650680 0.759352i \(-0.725516\pi\)
−0.650680 + 0.759352i \(0.725516\pi\)
\(978\) −54.2418 −1.73446
\(979\) 12.1057 0.386899
\(980\) 0 0
\(981\) 27.1868 0.868008
\(982\) 29.2405 0.933102
\(983\) −52.6149 −1.67816 −0.839078 0.544012i \(-0.816905\pi\)
−0.839078 + 0.544012i \(0.816905\pi\)
\(984\) 13.0852 0.417142
\(985\) −20.1368 −0.641613
\(986\) 0.194248 0.00618612
\(987\) 0 0
\(988\) −25.6064 −0.814648
\(989\) −33.8528 −1.07646
\(990\) 7.26731 0.230970
\(991\) −24.7835 −0.787273 −0.393637 0.919266i \(-0.628783\pi\)
−0.393637 + 0.919266i \(0.628783\pi\)
\(992\) 3.97423 0.126182
\(993\) −16.4997 −0.523603
\(994\) 0 0
\(995\) 12.9140 0.409402
\(996\) 29.2624 0.927213
\(997\) −16.1891 −0.512713 −0.256356 0.966582i \(-0.582522\pi\)
−0.256356 + 0.966582i \(0.582522\pi\)
\(998\) 28.4905 0.901850
\(999\) −104.131 −3.29455
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 2842.2.a.x.1.1 5
7.3 odd 6 406.2.e.a.233.1 10
7.5 odd 6 406.2.e.a.291.1 yes 10
7.6 odd 2 2842.2.a.z.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
406.2.e.a.233.1 10 7.3 odd 6
406.2.e.a.291.1 yes 10 7.5 odd 6
2842.2.a.x.1.1 5 1.1 even 1 trivial
2842.2.a.z.1.5 5 7.6 odd 2