Properties

Label 2842.2.a.z.1.5
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
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: \(0\)
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 1.5
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 \(-0.116423\pi\)
0.933854 + 0.357653i \(0.116423\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.14439 0.511789 0.255894 0.966705i \(-0.417630\pi\)
0.255894 + 0.966705i \(0.417630\pi\)
\(6\) 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 \(-0.330481\pi\)
0.507739 + 0.861511i \(0.330481\pi\)
\(14\) 0 0
\(15\) 3.70208 0.955872
\(16\) 1.00000 0.250000
\(17\) 0.194248 0.0471121 0.0235561 0.999723i \(-0.492501\pi\)
0.0235561 + 0.999723i \(0.492501\pi\)
\(18\) 7.46501 1.75952
\(19\) −6.99369 −1.60446 −0.802231 0.597014i \(-0.796354\pi\)
−0.802231 + 0.597014i \(0.796354\pi\)
\(20\) 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 \(-0.616165\pi\)
−0.356896 + 0.934144i \(0.616165\pi\)
\(32\) 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 \(-0.397708\pi\)
0.315856 + 0.948807i \(0.397708\pi\)
\(42\) 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 \(-0.498751\pi\)
0.00392531 + 0.999992i \(0.498751\pi\)
\(48\) 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 \(-0.675408\pi\)
−0.523590 + 0.851970i \(0.675408\pi\)
\(60\) 3.70208 0.477936
\(61\) −5.08144 −0.650611 −0.325306 0.945609i \(-0.605467\pi\)
−0.325306 + 0.945609i \(0.605467\pi\)
\(62\) −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 \(-0.640722\pi\)
−0.427830 + 0.903859i \(0.640722\pi\)
\(74\) 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 \(-0.334639\pi\)
0.496444 + 0.868069i \(0.334639\pi\)
\(84\) 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 \(-0.228017\pi\)
0.754216 + 0.656626i \(0.228017\pi\)
\(90\) 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 \(-0.0976601\pi\)
0.953302 + 0.302017i \(0.0976601\pi\)
\(98\) 0 0
\(99\) −6.35036 −0.638235
\(100\) −3.69036 −0.369036
\(101\) 13.7194 1.36513 0.682564 0.730826i \(-0.260865\pi\)
0.682564 + 0.730826i \(0.260865\pi\)
\(102\) 0.628386 0.0622195
\(103\) 1.18115 0.116382 0.0581909 0.998305i \(-0.481467\pi\)
0.0581909 + 0.998305i \(0.481467\pi\)
\(104\) 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 \(-0.750219\pi\)
−0.707593 + 0.706620i \(0.750219\pi\)
\(132\) −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 \(-0.497697\pi\)
0.00723397 + 0.999974i \(0.497697\pi\)
\(140\) 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 \(-0.305309\pi\)
0.574212 + 0.818707i \(0.305309\pi\)
\(158\) −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 \(-0.635265\pi\)
−0.412272 + 0.911061i \(0.635265\pi\)
\(168\) 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 \(-0.261367\pi\)
0.681409 + 0.731902i \(0.261367\pi\)
\(174\) −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 \(-0.337180\pi\)
0.489498 + 0.872004i \(0.337180\pi\)
\(182\) 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 \(-0.369020\pi\)
0.399972 + 0.916528i \(0.369020\pi\)
\(200\) −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 \(-0.687568\pi\)
−0.555748 + 0.831351i \(0.687568\pi\)
\(224\) 0 0
\(225\) −27.5486 −1.83657
\(226\) −17.7718 −1.18216
\(227\) 17.9930 1.19424 0.597120 0.802152i \(-0.296312\pi\)
0.597120 + 0.802152i \(0.296312\pi\)
\(228\) −22.6244 −1.49833
\(229\) −2.03821 −0.134689 −0.0673444 0.997730i \(-0.521453\pi\)
−0.0673444 + 0.997730i \(0.521453\pi\)
\(230\) −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 \(-0.304685\pi\)
0.575813 + 0.817581i \(0.304685\pi\)
\(242\) −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 \(-0.517010\pi\)
−0.0534144 + 0.998572i \(0.517010\pi\)
\(252\) 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 \(-0.464652\pi\)
0.110822 + 0.993840i \(0.464652\pi\)
\(258\) 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 \(-0.312990\pi\)
0.554289 + 0.832324i \(0.312990\pi\)
\(270\) 16.5298 1.00597
\(271\) −7.30703 −0.443871 −0.221935 0.975061i \(-0.571237\pi\)
−0.221935 + 0.975061i \(0.571237\pi\)
\(272\) 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 \(-0.650042\pi\)
−0.454109 + 0.890946i \(0.650042\pi\)
\(284\) −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 \(-0.649760\pi\)
−0.453318 + 0.891349i \(0.649760\pi\)
\(294\) 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 \(-0.471414\pi\)
0.0896838 + 0.995970i \(0.471414\pi\)
\(308\) 0 0
\(309\) 3.82097 0.217367
\(310\) −4.54808 −0.258314
\(311\) 24.2160 1.37316 0.686581 0.727053i \(-0.259111\pi\)
0.686581 + 0.727053i \(0.259111\pi\)
\(312\) 11.8444 0.670555
\(313\) −10.3528 −0.585175 −0.292588 0.956239i \(-0.594516\pi\)
−0.292588 + 0.956239i \(0.594516\pi\)
\(314\) 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 \(-0.521659\pi\)
−0.0679920 + 0.997686i \(0.521659\pi\)
\(350\) 0 0
\(351\) 52.8852 2.82281
\(352\) −0.850683 −0.0453415
\(353\) −1.20628 −0.0642038 −0.0321019 0.999485i \(-0.510220\pi\)
−0.0321019 + 0.999485i \(0.510220\pi\)
\(354\) −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 \(-0.281322\pi\)
0.634217 + 0.773155i \(0.281322\pi\)
\(368\) −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 \(-0.524127\pi\)
−0.0757241 + 0.997129i \(0.524127\pi\)
\(384\) 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 \(-0.373566\pi\)
0.386842 + 0.922146i \(0.373566\pi\)
\(398\) 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 \(-0.757717\pi\)
−0.724041 + 0.689757i \(0.757717\pi\)
\(410\) 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 \(-0.549133\pi\)
−0.153744 + 0.988111i \(0.549133\pi\)
\(420\) 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 \(-0.633858\pi\)
−0.408241 + 0.912874i \(0.633858\pi\)
\(434\) 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 \(-0.681697\pi\)
−0.540321 + 0.841459i \(0.681697\pi\)
\(440\) −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 \(-0.547678\pi\)
−0.149224 + 0.988803i \(0.547678\pi\)
\(462\) 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 \(-0.818890\pi\)
−0.842454 + 0.538768i \(0.818890\pi\)
\(468\) 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 \(-0.245059\pi\)
0.717998 + 0.696045i \(0.245059\pi\)
\(480\) 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 \(-0.353612\pi\)
0.443850 + 0.896101i \(0.353612\pi\)
\(504\) 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 \(-0.474775\pi\)
0.0791628 + 0.996862i \(0.474775\pi\)
\(510\) 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 \(-0.565793\pi\)
−0.205227 + 0.978714i \(0.565793\pi\)
\(522\) −7.46501 −0.326735
\(523\) −13.6159 −0.595381 −0.297690 0.954662i \(-0.596216\pi\)
−0.297690 + 0.954662i \(0.596216\pi\)
\(524\) −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 \(-0.211244\pi\)
0.787755 + 0.615989i \(0.211244\pi\)
\(564\) 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 \(-0.644673\pi\)
−0.439016 + 0.898479i \(0.644673\pi\)
\(578\) −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 \(-0.338190\pi\)
0.486730 + 0.873553i \(0.338190\pi\)
\(588\) 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 \(-0.670675\pi\)
−0.510865 + 0.859661i \(0.670675\pi\)
\(594\) −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 \(-0.859467\pi\)
−0.904113 + 0.427292i \(0.859467\pi\)
\(602\) 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 \(-0.163722\pi\)
0.870613 + 0.491968i \(0.163722\pi\)
\(608\) −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 \(-0.261937\pi\)
0.680098 + 0.733122i \(0.261937\pi\)
\(620\) −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 \(-0.640572\pi\)
−0.427406 + 0.904060i \(0.640572\pi\)
\(644\) 0 0
\(645\) 36.1985 1.42531
\(646\) −1.35851 −0.0534499
\(647\) −23.0945 −0.907940 −0.453970 0.891017i \(-0.649993\pi\)
−0.453970 + 0.891017i \(0.649993\pi\)
\(648\) 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 \(-0.510053\pi\)
−0.0315763 + 0.999501i \(0.510053\pi\)
\(662\) 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 \(-0.369595\pi\)
0.398314 + 0.917249i \(0.369595\pi\)
\(678\) −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 \(-0.528011\pi\)
−0.0878853 + 0.996131i \(0.528011\pi\)
\(692\) 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 \(-0.267057\pi\)
0.668218 + 0.743966i \(0.267057\pi\)
\(720\) 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 \(-0.576527\pi\)
−0.238107 + 0.971239i \(0.576527\pi\)
\(728\) 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 \(-0.553946\pi\)
−0.168666 + 0.985673i \(0.553946\pi\)
\(734\) 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 \(-0.244969\pi\)
0.718194 + 0.695843i \(0.244969\pi\)
\(762\) −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 \(-0.568329\pi\)
−0.213017 + 0.977049i \(0.568329\pi\)
\(770\) 0 0
\(771\) 11.4946 0.413968
\(772\) −17.3950 −0.626061
\(773\) 18.0208 0.648164 0.324082 0.946029i \(-0.394945\pi\)
0.324082 + 0.946029i \(0.394945\pi\)
\(774\) 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 \(-0.541638\pi\)
−0.130437 + 0.991457i \(0.541638\pi\)
\(788\) 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 \(-0.102869\pi\)
0.948232 + 0.317578i \(0.102869\pi\)
\(798\) 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 \(-0.477008\pi\)
0.0721701 + 0.997392i \(0.477008\pi\)
\(812\) 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 \(-0.493300\pi\)
0.0210478 + 0.999778i \(0.493300\pi\)
\(830\) 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 \(-0.427954\pi\)
0.224412 + 0.974494i \(0.427954\pi\)
\(840\) 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 \(-0.366299\pi\)
0.407792 + 0.913075i \(0.366299\pi\)
\(854\) 0 0
\(855\) −59.7465 −2.04329
\(856\) −1.79224 −0.0612576
\(857\) −7.27140 −0.248386 −0.124193 0.992258i \(-0.539634\pi\)
−0.124193 + 0.992258i \(0.539634\pi\)
\(858\) −10.0758 −0.343982
\(859\) 9.83263 0.335485 0.167742 0.985831i \(-0.446352\pi\)
0.167742 + 0.985831i \(0.446352\pi\)
\(860\) 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 \(-0.690541\pi\)
−0.563489 + 0.826124i \(0.690541\pi\)
\(882\) 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 \(-0.460377\pi\)
0.124159 + 0.992262i \(0.460377\pi\)
\(888\) 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 \(-0.764093\pi\)
−0.737710 + 0.675118i \(0.764093\pi\)
\(930\) −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 \(-0.390570\pi\)
0.337053 + 0.941486i \(0.390570\pi\)
\(938\) 0 0
\(939\) −33.4910 −1.09294
\(940\) 0.0615925 0.00200893
\(941\) 33.2588 1.08421 0.542103 0.840312i \(-0.317628\pi\)
0.542103 + 0.840312i \(0.317628\pi\)
\(942\) 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 \(-0.596085\pi\)
−0.297296 + 0.954785i \(0.596085\pi\)
\(972\) 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 \(-0.183095\pi\)
0.839078 + 0.544012i \(0.183095\pi\)
\(984\) 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 \(-0.417478\pi\)
0.256356 + 0.966582i \(0.417478\pi\)
\(998\) 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.z.1.5 5
7.2 even 3 406.2.e.a.291.1 yes 10
7.4 even 3 406.2.e.a.233.1 10
7.6 odd 2 2842.2.a.x.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
406.2.e.a.233.1 10 7.4 even 3
406.2.e.a.291.1 yes 10 7.2 even 3
2842.2.a.x.1.1 5 7.6 odd 2
2842.2.a.z.1.5 5 1.1 even 1 trivial