Properties

Label 4006.2.a.g.1.3
Level 4006
Weight 2
Character 4006.1
Self dual yes
Analytic conductor 31.988
Analytic rank 1
Dimension 40
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4006.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9880710497\)
Analytic rank: \(1\)
Dimension: \(40\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) = 4006.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.96542 q^{3} +1.00000 q^{4} +0.539090 q^{5} +2.96542 q^{6} -0.220082 q^{7} -1.00000 q^{8} +5.79371 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.96542 q^{3} +1.00000 q^{4} +0.539090 q^{5} +2.96542 q^{6} -0.220082 q^{7} -1.00000 q^{8} +5.79371 q^{9} -0.539090 q^{10} +5.11019 q^{11} -2.96542 q^{12} -1.83998 q^{13} +0.220082 q^{14} -1.59863 q^{15} +1.00000 q^{16} +4.75454 q^{17} -5.79371 q^{18} -4.79891 q^{19} +0.539090 q^{20} +0.652635 q^{21} -5.11019 q^{22} +2.67550 q^{23} +2.96542 q^{24} -4.70938 q^{25} +1.83998 q^{26} -8.28451 q^{27} -0.220082 q^{28} -5.41969 q^{29} +1.59863 q^{30} -6.62830 q^{31} -1.00000 q^{32} -15.1538 q^{33} -4.75454 q^{34} -0.118644 q^{35} +5.79371 q^{36} -1.59451 q^{37} +4.79891 q^{38} +5.45632 q^{39} -0.539090 q^{40} -4.76330 q^{41} -0.652635 q^{42} +10.8795 q^{43} +5.11019 q^{44} +3.12333 q^{45} -2.67550 q^{46} +2.31041 q^{47} -2.96542 q^{48} -6.95156 q^{49} +4.70938 q^{50} -14.0992 q^{51} -1.83998 q^{52} +2.96006 q^{53} +8.28451 q^{54} +2.75485 q^{55} +0.220082 q^{56} +14.2308 q^{57} +5.41969 q^{58} -4.09203 q^{59} -1.59863 q^{60} -2.30748 q^{61} +6.62830 q^{62} -1.27509 q^{63} +1.00000 q^{64} -0.991915 q^{65} +15.1538 q^{66} +4.94120 q^{67} +4.75454 q^{68} -7.93397 q^{69} +0.118644 q^{70} +5.87372 q^{71} -5.79371 q^{72} -4.62683 q^{73} +1.59451 q^{74} +13.9653 q^{75} -4.79891 q^{76} -1.12466 q^{77} -5.45632 q^{78} -9.45252 q^{79} +0.539090 q^{80} +7.18592 q^{81} +4.76330 q^{82} +7.62420 q^{83} +0.652635 q^{84} +2.56313 q^{85} -10.8795 q^{86} +16.0716 q^{87} -5.11019 q^{88} +9.82531 q^{89} -3.12333 q^{90} +0.404946 q^{91} +2.67550 q^{92} +19.6557 q^{93} -2.31041 q^{94} -2.58704 q^{95} +2.96542 q^{96} -3.10555 q^{97} +6.95156 q^{98} +29.6069 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + O(q^{10}) \) \( 40q - 40q^{2} - q^{3} + 40q^{4} - 23q^{5} + q^{6} + 12q^{7} - 40q^{8} + 29q^{9} + 23q^{10} - 28q^{11} - q^{12} - 12q^{13} - 12q^{14} - 14q^{15} + 40q^{16} - 10q^{17} - 29q^{18} + 3q^{19} - 23q^{20} - 40q^{21} + 28q^{22} - 10q^{23} + q^{24} + 43q^{25} + 12q^{26} - 7q^{27} + 12q^{28} - 37q^{29} + 14q^{30} - 16q^{31} - 40q^{32} - 11q^{33} + 10q^{34} - 24q^{35} + 29q^{36} - 17q^{37} - 3q^{38} - 10q^{39} + 23q^{40} - 58q^{41} + 40q^{42} + 37q^{43} - 28q^{44} - 66q^{45} + 10q^{46} - 34q^{47} - q^{48} + 28q^{49} - 43q^{50} - 7q^{51} - 12q^{52} - 43q^{53} + 7q^{54} + 28q^{55} - 12q^{56} - 25q^{57} + 37q^{58} - 92q^{59} - 14q^{60} - 37q^{61} + 16q^{62} + 14q^{63} + 40q^{64} - 54q^{65} + 11q^{66} - 10q^{67} - 10q^{68} - 49q^{69} + 24q^{70} - 87q^{71} - 29q^{72} + 12q^{73} + 17q^{74} - 31q^{75} + 3q^{76} - 53q^{77} + 10q^{78} + 14q^{79} - 23q^{80} - 4q^{81} + 58q^{82} - 42q^{83} - 40q^{84} - 24q^{85} - 37q^{86} + 24q^{87} + 28q^{88} - 118q^{89} + 66q^{90} - 35q^{91} - 10q^{92} - 22q^{93} + 34q^{94} - 18q^{95} + q^{96} + 2q^{97} - 28q^{98} - 48q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.96542 −1.71209 −0.856043 0.516905i \(-0.827084\pi\)
−0.856043 + 0.516905i \(0.827084\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.539090 0.241088 0.120544 0.992708i \(-0.461536\pi\)
0.120544 + 0.992708i \(0.461536\pi\)
\(6\) 2.96542 1.21063
\(7\) −0.220082 −0.0831831 −0.0415915 0.999135i \(-0.513243\pi\)
−0.0415915 + 0.999135i \(0.513243\pi\)
\(8\) −1.00000 −0.353553
\(9\) 5.79371 1.93124
\(10\) −0.539090 −0.170475
\(11\) 5.11019 1.54078 0.770390 0.637573i \(-0.220062\pi\)
0.770390 + 0.637573i \(0.220062\pi\)
\(12\) −2.96542 −0.856043
\(13\) −1.83998 −0.510319 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(14\) 0.220082 0.0588193
\(15\) −1.59863 −0.412764
\(16\) 1.00000 0.250000
\(17\) 4.75454 1.15315 0.576573 0.817046i \(-0.304390\pi\)
0.576573 + 0.817046i \(0.304390\pi\)
\(18\) −5.79371 −1.36559
\(19\) −4.79891 −1.10095 −0.550473 0.834853i \(-0.685552\pi\)
−0.550473 + 0.834853i \(0.685552\pi\)
\(20\) 0.539090 0.120544
\(21\) 0.652635 0.142417
\(22\) −5.11019 −1.08950
\(23\) 2.67550 0.557880 0.278940 0.960309i \(-0.410017\pi\)
0.278940 + 0.960309i \(0.410017\pi\)
\(24\) 2.96542 0.605314
\(25\) −4.70938 −0.941876
\(26\) 1.83998 0.360850
\(27\) −8.28451 −1.59435
\(28\) −0.220082 −0.0415915
\(29\) −5.41969 −1.00641 −0.503206 0.864167i \(-0.667846\pi\)
−0.503206 + 0.864167i \(0.667846\pi\)
\(30\) 1.59863 0.291868
\(31\) −6.62830 −1.19048 −0.595239 0.803549i \(-0.702943\pi\)
−0.595239 + 0.803549i \(0.702943\pi\)
\(32\) −1.00000 −0.176777
\(33\) −15.1538 −2.63795
\(34\) −4.75454 −0.815398
\(35\) −0.118644 −0.0200545
\(36\) 5.79371 0.965618
\(37\) −1.59451 −0.262136 −0.131068 0.991373i \(-0.541841\pi\)
−0.131068 + 0.991373i \(0.541841\pi\)
\(38\) 4.79891 0.778486
\(39\) 5.45632 0.873710
\(40\) −0.539090 −0.0852376
\(41\) −4.76330 −0.743902 −0.371951 0.928252i \(-0.621311\pi\)
−0.371951 + 0.928252i \(0.621311\pi\)
\(42\) −0.652635 −0.100704
\(43\) 10.8795 1.65911 0.829553 0.558429i \(-0.188596\pi\)
0.829553 + 0.558429i \(0.188596\pi\)
\(44\) 5.11019 0.770390
\(45\) 3.12333 0.465598
\(46\) −2.67550 −0.394481
\(47\) 2.31041 0.337008 0.168504 0.985701i \(-0.446106\pi\)
0.168504 + 0.985701i \(0.446106\pi\)
\(48\) −2.96542 −0.428021
\(49\) −6.95156 −0.993081
\(50\) 4.70938 0.666007
\(51\) −14.0992 −1.97428
\(52\) −1.83998 −0.255160
\(53\) 2.96006 0.406596 0.203298 0.979117i \(-0.434834\pi\)
0.203298 + 0.979117i \(0.434834\pi\)
\(54\) 8.28451 1.12738
\(55\) 2.75485 0.371464
\(56\) 0.220082 0.0294097
\(57\) 14.2308 1.88491
\(58\) 5.41969 0.711640
\(59\) −4.09203 −0.532737 −0.266369 0.963871i \(-0.585824\pi\)
−0.266369 + 0.963871i \(0.585824\pi\)
\(60\) −1.59863 −0.206382
\(61\) −2.30748 −0.295442 −0.147721 0.989029i \(-0.547194\pi\)
−0.147721 + 0.989029i \(0.547194\pi\)
\(62\) 6.62830 0.841795
\(63\) −1.27509 −0.160646
\(64\) 1.00000 0.125000
\(65\) −0.991915 −0.123032
\(66\) 15.1538 1.86531
\(67\) 4.94120 0.603664 0.301832 0.953361i \(-0.402402\pi\)
0.301832 + 0.953361i \(0.402402\pi\)
\(68\) 4.75454 0.576573
\(69\) −7.93397 −0.955138
\(70\) 0.118644 0.0141806
\(71\) 5.87372 0.697082 0.348541 0.937293i \(-0.386677\pi\)
0.348541 + 0.937293i \(0.386677\pi\)
\(72\) −5.79371 −0.682795
\(73\) −4.62683 −0.541529 −0.270765 0.962646i \(-0.587276\pi\)
−0.270765 + 0.962646i \(0.587276\pi\)
\(74\) 1.59451 0.185358
\(75\) 13.9653 1.61257
\(76\) −4.79891 −0.550473
\(77\) −1.12466 −0.128167
\(78\) −5.45632 −0.617806
\(79\) −9.45252 −1.06349 −0.531746 0.846904i \(-0.678464\pi\)
−0.531746 + 0.846904i \(0.678464\pi\)
\(80\) 0.539090 0.0602721
\(81\) 7.18592 0.798435
\(82\) 4.76330 0.526018
\(83\) 7.62420 0.836865 0.418433 0.908248i \(-0.362580\pi\)
0.418433 + 0.908248i \(0.362580\pi\)
\(84\) 0.652635 0.0712083
\(85\) 2.56313 0.278010
\(86\) −10.8795 −1.17316
\(87\) 16.0716 1.72306
\(88\) −5.11019 −0.544748
\(89\) 9.82531 1.04148 0.520740 0.853715i \(-0.325656\pi\)
0.520740 + 0.853715i \(0.325656\pi\)
\(90\) −3.12333 −0.329228
\(91\) 0.404946 0.0424499
\(92\) 2.67550 0.278940
\(93\) 19.6557 2.03820
\(94\) −2.31041 −0.238301
\(95\) −2.58704 −0.265425
\(96\) 2.96542 0.302657
\(97\) −3.10555 −0.315321 −0.157660 0.987493i \(-0.550395\pi\)
−0.157660 + 0.987493i \(0.550395\pi\)
\(98\) 6.95156 0.702214
\(99\) 29.6069 2.97561
\(100\) −4.70938 −0.470938
\(101\) −7.86660 −0.782756 −0.391378 0.920230i \(-0.628002\pi\)
−0.391378 + 0.920230i \(0.628002\pi\)
\(102\) 14.0992 1.39603
\(103\) 9.82927 0.968507 0.484253 0.874928i \(-0.339091\pi\)
0.484253 + 0.874928i \(0.339091\pi\)
\(104\) 1.83998 0.180425
\(105\) 0.351829 0.0343350
\(106\) −2.96006 −0.287507
\(107\) −9.15461 −0.885009 −0.442505 0.896766i \(-0.645910\pi\)
−0.442505 + 0.896766i \(0.645910\pi\)
\(108\) −8.28451 −0.797177
\(109\) −5.20154 −0.498217 −0.249108 0.968476i \(-0.580138\pi\)
−0.249108 + 0.968476i \(0.580138\pi\)
\(110\) −2.75485 −0.262665
\(111\) 4.72839 0.448799
\(112\) −0.220082 −0.0207958
\(113\) −10.1470 −0.954554 −0.477277 0.878753i \(-0.658376\pi\)
−0.477277 + 0.878753i \(0.658376\pi\)
\(114\) −14.2308 −1.33283
\(115\) 1.44233 0.134498
\(116\) −5.41969 −0.503206
\(117\) −10.6603 −0.985547
\(118\) 4.09203 0.376702
\(119\) −1.04639 −0.0959223
\(120\) 1.59863 0.145934
\(121\) 15.1140 1.37400
\(122\) 2.30748 0.208909
\(123\) 14.1252 1.27362
\(124\) −6.62830 −0.595239
\(125\) −5.23423 −0.468164
\(126\) 1.27509 0.113594
\(127\) −8.05548 −0.714808 −0.357404 0.933950i \(-0.616338\pi\)
−0.357404 + 0.933950i \(0.616338\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −32.2622 −2.84053
\(130\) 0.991915 0.0869967
\(131\) 5.25397 0.459041 0.229521 0.973304i \(-0.426284\pi\)
0.229521 + 0.973304i \(0.426284\pi\)
\(132\) −15.1538 −1.31897
\(133\) 1.05615 0.0915800
\(134\) −4.94120 −0.426855
\(135\) −4.46609 −0.384380
\(136\) −4.75454 −0.407699
\(137\) 4.67123 0.399090 0.199545 0.979889i \(-0.436054\pi\)
0.199545 + 0.979889i \(0.436054\pi\)
\(138\) 7.93397 0.675384
\(139\) −11.9008 −1.00941 −0.504706 0.863291i \(-0.668399\pi\)
−0.504706 + 0.863291i \(0.668399\pi\)
\(140\) −0.118644 −0.0100272
\(141\) −6.85134 −0.576987
\(142\) −5.87372 −0.492912
\(143\) −9.40265 −0.786289
\(144\) 5.79371 0.482809
\(145\) −2.92170 −0.242634
\(146\) 4.62683 0.382919
\(147\) 20.6143 1.70024
\(148\) −1.59451 −0.131068
\(149\) 15.6554 1.28254 0.641272 0.767314i \(-0.278407\pi\)
0.641272 + 0.767314i \(0.278407\pi\)
\(150\) −13.9653 −1.14026
\(151\) 16.0723 1.30795 0.653975 0.756517i \(-0.273100\pi\)
0.653975 + 0.756517i \(0.273100\pi\)
\(152\) 4.79891 0.389243
\(153\) 27.5464 2.22700
\(154\) 1.12466 0.0906276
\(155\) −3.57325 −0.287010
\(156\) 5.45632 0.436855
\(157\) 11.3185 0.903314 0.451657 0.892192i \(-0.350833\pi\)
0.451657 + 0.892192i \(0.350833\pi\)
\(158\) 9.45252 0.752002
\(159\) −8.77782 −0.696127
\(160\) −0.539090 −0.0426188
\(161\) −0.588828 −0.0464062
\(162\) −7.18592 −0.564579
\(163\) 10.8088 0.846612 0.423306 0.905987i \(-0.360870\pi\)
0.423306 + 0.905987i \(0.360870\pi\)
\(164\) −4.76330 −0.371951
\(165\) −8.16928 −0.635978
\(166\) −7.62420 −0.591753
\(167\) 23.4075 1.81133 0.905663 0.423999i \(-0.139374\pi\)
0.905663 + 0.423999i \(0.139374\pi\)
\(168\) −0.652635 −0.0503518
\(169\) −9.61447 −0.739574
\(170\) −2.56313 −0.196583
\(171\) −27.8035 −2.12618
\(172\) 10.8795 0.829553
\(173\) −7.35810 −0.559426 −0.279713 0.960084i \(-0.590239\pi\)
−0.279713 + 0.960084i \(0.590239\pi\)
\(174\) −16.0716 −1.21839
\(175\) 1.03645 0.0783482
\(176\) 5.11019 0.385195
\(177\) 12.1346 0.912092
\(178\) −9.82531 −0.736438
\(179\) −13.1350 −0.981759 −0.490879 0.871228i \(-0.663324\pi\)
−0.490879 + 0.871228i \(0.663324\pi\)
\(180\) 3.12333 0.232799
\(181\) −12.5443 −0.932413 −0.466206 0.884676i \(-0.654380\pi\)
−0.466206 + 0.884676i \(0.654380\pi\)
\(182\) −0.404946 −0.0300166
\(183\) 6.84264 0.505822
\(184\) −2.67550 −0.197240
\(185\) −0.859583 −0.0631978
\(186\) −19.6557 −1.44123
\(187\) 24.2966 1.77674
\(188\) 2.31041 0.168504
\(189\) 1.82327 0.132623
\(190\) 2.58704 0.187684
\(191\) 6.85138 0.495749 0.247874 0.968792i \(-0.420268\pi\)
0.247874 + 0.968792i \(0.420268\pi\)
\(192\) −2.96542 −0.214011
\(193\) 7.46745 0.537519 0.268760 0.963207i \(-0.413386\pi\)
0.268760 + 0.963207i \(0.413386\pi\)
\(194\) 3.10555 0.222965
\(195\) 2.94144 0.210641
\(196\) −6.95156 −0.496540
\(197\) −11.3676 −0.809905 −0.404953 0.914338i \(-0.632712\pi\)
−0.404953 + 0.914338i \(0.632712\pi\)
\(198\) −29.6069 −2.10407
\(199\) 6.57036 0.465760 0.232880 0.972505i \(-0.425185\pi\)
0.232880 + 0.972505i \(0.425185\pi\)
\(200\) 4.70938 0.333004
\(201\) −14.6527 −1.03352
\(202\) 7.86660 0.553492
\(203\) 1.19277 0.0837164
\(204\) −14.0992 −0.987142
\(205\) −2.56785 −0.179346
\(206\) −9.82927 −0.684838
\(207\) 15.5010 1.07740
\(208\) −1.83998 −0.127580
\(209\) −24.5233 −1.69631
\(210\) −0.351829 −0.0242785
\(211\) −12.4627 −0.857970 −0.428985 0.903312i \(-0.641129\pi\)
−0.428985 + 0.903312i \(0.641129\pi\)
\(212\) 2.96006 0.203298
\(213\) −17.4180 −1.19346
\(214\) 9.15461 0.625796
\(215\) 5.86502 0.399991
\(216\) 8.28451 0.563690
\(217\) 1.45877 0.0990277
\(218\) 5.20154 0.352293
\(219\) 13.7205 0.927144
\(220\) 2.75485 0.185732
\(221\) −8.74828 −0.588473
\(222\) −4.72839 −0.317348
\(223\) −13.6367 −0.913179 −0.456589 0.889678i \(-0.650929\pi\)
−0.456589 + 0.889678i \(0.650929\pi\)
\(224\) 0.220082 0.0147048
\(225\) −27.2848 −1.81899
\(226\) 10.1470 0.674971
\(227\) −28.8214 −1.91294 −0.956471 0.291829i \(-0.905736\pi\)
−0.956471 + 0.291829i \(0.905736\pi\)
\(228\) 14.2308 0.942456
\(229\) −17.4261 −1.15155 −0.575775 0.817608i \(-0.695299\pi\)
−0.575775 + 0.817608i \(0.695299\pi\)
\(230\) −1.44233 −0.0951046
\(231\) 3.33509 0.219433
\(232\) 5.41969 0.355820
\(233\) 2.26966 0.148690 0.0743452 0.997233i \(-0.476313\pi\)
0.0743452 + 0.997233i \(0.476313\pi\)
\(234\) 10.6603 0.696887
\(235\) 1.24552 0.0812487
\(236\) −4.09203 −0.266369
\(237\) 28.0307 1.82079
\(238\) 1.04639 0.0678273
\(239\) 23.3995 1.51359 0.756793 0.653654i \(-0.226765\pi\)
0.756793 + 0.653654i \(0.226765\pi\)
\(240\) −1.59863 −0.103191
\(241\) 13.4798 0.868308 0.434154 0.900839i \(-0.357047\pi\)
0.434154 + 0.900839i \(0.357047\pi\)
\(242\) −15.1140 −0.971566
\(243\) 3.54427 0.227365
\(244\) −2.30748 −0.147721
\(245\) −3.74752 −0.239420
\(246\) −14.1252 −0.900588
\(247\) 8.82990 0.561833
\(248\) 6.62830 0.420898
\(249\) −22.6090 −1.43278
\(250\) 5.23423 0.331042
\(251\) −31.0300 −1.95860 −0.979299 0.202421i \(-0.935119\pi\)
−0.979299 + 0.202421i \(0.935119\pi\)
\(252\) −1.27509 −0.0803231
\(253\) 13.6723 0.859570
\(254\) 8.05548 0.505445
\(255\) −7.60074 −0.475977
\(256\) 1.00000 0.0625000
\(257\) −28.0792 −1.75153 −0.875765 0.482737i \(-0.839643\pi\)
−0.875765 + 0.482737i \(0.839643\pi\)
\(258\) 32.2622 2.00856
\(259\) 0.350922 0.0218053
\(260\) −0.991915 −0.0615160
\(261\) −31.4001 −1.94362
\(262\) −5.25397 −0.324591
\(263\) −0.786856 −0.0485197 −0.0242598 0.999706i \(-0.507723\pi\)
−0.0242598 + 0.999706i \(0.507723\pi\)
\(264\) 15.1538 0.932655
\(265\) 1.59574 0.0980255
\(266\) −1.05615 −0.0647568
\(267\) −29.1361 −1.78310
\(268\) 4.94120 0.301832
\(269\) −17.0961 −1.04237 −0.521184 0.853444i \(-0.674510\pi\)
−0.521184 + 0.853444i \(0.674510\pi\)
\(270\) 4.46609 0.271798
\(271\) −4.65637 −0.282855 −0.141427 0.989949i \(-0.545169\pi\)
−0.141427 + 0.989949i \(0.545169\pi\)
\(272\) 4.75454 0.288287
\(273\) −1.20084 −0.0726779
\(274\) −4.67123 −0.282199
\(275\) −24.0658 −1.45122
\(276\) −7.93397 −0.477569
\(277\) −20.2844 −1.21877 −0.609385 0.792874i \(-0.708584\pi\)
−0.609385 + 0.792874i \(0.708584\pi\)
\(278\) 11.9008 0.713762
\(279\) −38.4024 −2.29909
\(280\) 0.118644 0.00709032
\(281\) 1.65735 0.0988694 0.0494347 0.998777i \(-0.484258\pi\)
0.0494347 + 0.998777i \(0.484258\pi\)
\(282\) 6.85134 0.407991
\(283\) −27.8194 −1.65369 −0.826846 0.562428i \(-0.809867\pi\)
−0.826846 + 0.562428i \(0.809867\pi\)
\(284\) 5.87372 0.348541
\(285\) 7.67166 0.454430
\(286\) 9.40265 0.555991
\(287\) 1.04832 0.0618801
\(288\) −5.79371 −0.341397
\(289\) 5.60569 0.329746
\(290\) 2.92170 0.171568
\(291\) 9.20925 0.539856
\(292\) −4.62683 −0.270765
\(293\) −0.258205 −0.0150845 −0.00754226 0.999972i \(-0.502401\pi\)
−0.00754226 + 0.999972i \(0.502401\pi\)
\(294\) −20.6143 −1.20225
\(295\) −2.20597 −0.128437
\(296\) 1.59451 0.0926789
\(297\) −42.3354 −2.45655
\(298\) −15.6554 −0.906895
\(299\) −4.92287 −0.284697
\(300\) 13.9653 0.806286
\(301\) −2.39438 −0.138010
\(302\) −16.0723 −0.924860
\(303\) 23.3278 1.34015
\(304\) −4.79891 −0.275236
\(305\) −1.24394 −0.0712277
\(306\) −27.5464 −1.57472
\(307\) 31.9661 1.82440 0.912201 0.409744i \(-0.134382\pi\)
0.912201 + 0.409744i \(0.134382\pi\)
\(308\) −1.12466 −0.0640834
\(309\) −29.1479 −1.65817
\(310\) 3.57325 0.202947
\(311\) 2.69476 0.152806 0.0764029 0.997077i \(-0.475656\pi\)
0.0764029 + 0.997077i \(0.475656\pi\)
\(312\) −5.45632 −0.308903
\(313\) −11.7999 −0.666970 −0.333485 0.942755i \(-0.608225\pi\)
−0.333485 + 0.942755i \(0.608225\pi\)
\(314\) −11.3185 −0.638739
\(315\) −0.687388 −0.0387299
\(316\) −9.45252 −0.531746
\(317\) −20.0170 −1.12427 −0.562133 0.827047i \(-0.690019\pi\)
−0.562133 + 0.827047i \(0.690019\pi\)
\(318\) 8.77782 0.492236
\(319\) −27.6956 −1.55066
\(320\) 0.539090 0.0301360
\(321\) 27.1472 1.51521
\(322\) 0.588828 0.0328141
\(323\) −22.8166 −1.26955
\(324\) 7.18592 0.399218
\(325\) 8.66518 0.480658
\(326\) −10.8088 −0.598645
\(327\) 15.4247 0.852990
\(328\) 4.76330 0.263009
\(329\) −0.508480 −0.0280334
\(330\) 8.16928 0.449704
\(331\) 6.57648 0.361476 0.180738 0.983531i \(-0.442151\pi\)
0.180738 + 0.983531i \(0.442151\pi\)
\(332\) 7.62420 0.418433
\(333\) −9.23812 −0.506246
\(334\) −23.4075 −1.28080
\(335\) 2.66375 0.145536
\(336\) 0.652635 0.0356041
\(337\) −8.95217 −0.487656 −0.243828 0.969818i \(-0.578403\pi\)
−0.243828 + 0.969818i \(0.578403\pi\)
\(338\) 9.61447 0.522958
\(339\) 30.0902 1.63428
\(340\) 2.56313 0.139005
\(341\) −33.8719 −1.83426
\(342\) 27.8035 1.50344
\(343\) 3.07048 0.165791
\(344\) −10.8795 −0.586582
\(345\) −4.27712 −0.230272
\(346\) 7.35810 0.395574
\(347\) 15.9803 0.857867 0.428934 0.903336i \(-0.358889\pi\)
0.428934 + 0.903336i \(0.358889\pi\)
\(348\) 16.0716 0.861531
\(349\) −28.3421 −1.51712 −0.758560 0.651603i \(-0.774097\pi\)
−0.758560 + 0.651603i \(0.774097\pi\)
\(350\) −1.03645 −0.0554005
\(351\) 15.2433 0.813630
\(352\) −5.11019 −0.272374
\(353\) 2.23439 0.118924 0.0594622 0.998231i \(-0.481061\pi\)
0.0594622 + 0.998231i \(0.481061\pi\)
\(354\) −12.1346 −0.644946
\(355\) 3.16646 0.168058
\(356\) 9.82531 0.520740
\(357\) 3.10298 0.164227
\(358\) 13.1350 0.694208
\(359\) 4.83027 0.254932 0.127466 0.991843i \(-0.459316\pi\)
0.127466 + 0.991843i \(0.459316\pi\)
\(360\) −3.12333 −0.164614
\(361\) 4.02952 0.212080
\(362\) 12.5443 0.659316
\(363\) −44.8194 −2.35241
\(364\) 0.404946 0.0212250
\(365\) −2.49427 −0.130556
\(366\) −6.84264 −0.357670
\(367\) 30.9066 1.61331 0.806655 0.591023i \(-0.201276\pi\)
0.806655 + 0.591023i \(0.201276\pi\)
\(368\) 2.67550 0.139470
\(369\) −27.5972 −1.43665
\(370\) 0.859583 0.0446876
\(371\) −0.651456 −0.0338219
\(372\) 19.6557 1.01910
\(373\) 0.175729 0.00909892 0.00454946 0.999990i \(-0.498552\pi\)
0.00454946 + 0.999990i \(0.498552\pi\)
\(374\) −24.2966 −1.25635
\(375\) 15.5217 0.801536
\(376\) −2.31041 −0.119150
\(377\) 9.97213 0.513591
\(378\) −1.82327 −0.0937789
\(379\) −20.4684 −1.05139 −0.525696 0.850673i \(-0.676195\pi\)
−0.525696 + 0.850673i \(0.676195\pi\)
\(380\) −2.58704 −0.132712
\(381\) 23.8879 1.22381
\(382\) −6.85138 −0.350547
\(383\) −23.6401 −1.20795 −0.603976 0.797002i \(-0.706418\pi\)
−0.603976 + 0.797002i \(0.706418\pi\)
\(384\) 2.96542 0.151328
\(385\) −0.606292 −0.0308995
\(386\) −7.46745 −0.380083
\(387\) 63.0325 3.20412
\(388\) −3.10555 −0.157660
\(389\) −11.5581 −0.586019 −0.293010 0.956109i \(-0.594657\pi\)
−0.293010 + 0.956109i \(0.594657\pi\)
\(390\) −2.94144 −0.148946
\(391\) 12.7208 0.643317
\(392\) 6.95156 0.351107
\(393\) −15.5802 −0.785918
\(394\) 11.3676 0.572689
\(395\) −5.09576 −0.256395
\(396\) 29.6069 1.48780
\(397\) 8.09403 0.406227 0.203114 0.979155i \(-0.434894\pi\)
0.203114 + 0.979155i \(0.434894\pi\)
\(398\) −6.57036 −0.329342
\(399\) −3.13193 −0.156793
\(400\) −4.70938 −0.235469
\(401\) 11.8819 0.593355 0.296677 0.954978i \(-0.404121\pi\)
0.296677 + 0.954978i \(0.404121\pi\)
\(402\) 14.6527 0.730812
\(403\) 12.1960 0.607524
\(404\) −7.86660 −0.391378
\(405\) 3.87386 0.192493
\(406\) −1.19277 −0.0591964
\(407\) −8.14824 −0.403893
\(408\) 14.0992 0.698015
\(409\) 39.2750 1.94203 0.971013 0.239027i \(-0.0768286\pi\)
0.971013 + 0.239027i \(0.0768286\pi\)
\(410\) 2.56785 0.126817
\(411\) −13.8521 −0.683276
\(412\) 9.82927 0.484253
\(413\) 0.900582 0.0443147
\(414\) −15.5010 −0.761835
\(415\) 4.11013 0.201758
\(416\) 1.83998 0.0902125
\(417\) 35.2908 1.72820
\(418\) 24.5233 1.19948
\(419\) −27.0626 −1.32210 −0.661048 0.750344i \(-0.729888\pi\)
−0.661048 + 0.750344i \(0.729888\pi\)
\(420\) 0.351829 0.0171675
\(421\) 21.1878 1.03263 0.516316 0.856398i \(-0.327303\pi\)
0.516316 + 0.856398i \(0.327303\pi\)
\(422\) 12.4627 0.606676
\(423\) 13.3859 0.650842
\(424\) −2.96006 −0.143753
\(425\) −22.3910 −1.08612
\(426\) 17.4180 0.843907
\(427\) 0.507834 0.0245758
\(428\) −9.15461 −0.442505
\(429\) 27.8828 1.34619
\(430\) −5.86502 −0.282836
\(431\) −19.4929 −0.938939 −0.469470 0.882949i \(-0.655555\pi\)
−0.469470 + 0.882949i \(0.655555\pi\)
\(432\) −8.28451 −0.398589
\(433\) 16.8948 0.811913 0.405957 0.913892i \(-0.366938\pi\)
0.405957 + 0.913892i \(0.366938\pi\)
\(434\) −1.45877 −0.0700231
\(435\) 8.66406 0.415410
\(436\) −5.20154 −0.249108
\(437\) −12.8395 −0.614195
\(438\) −13.7205 −0.655590
\(439\) −33.8285 −1.61455 −0.807273 0.590178i \(-0.799058\pi\)
−0.807273 + 0.590178i \(0.799058\pi\)
\(440\) −2.75485 −0.131332
\(441\) −40.2753 −1.91787
\(442\) 8.74828 0.416113
\(443\) −21.2006 −1.00727 −0.503635 0.863916i \(-0.668004\pi\)
−0.503635 + 0.863916i \(0.668004\pi\)
\(444\) 4.72839 0.224399
\(445\) 5.29672 0.251089
\(446\) 13.6367 0.645715
\(447\) −46.4249 −2.19582
\(448\) −0.220082 −0.0103979
\(449\) −11.5835 −0.546660 −0.273330 0.961920i \(-0.588125\pi\)
−0.273330 + 0.961920i \(0.588125\pi\)
\(450\) 27.2848 1.28622
\(451\) −24.3414 −1.14619
\(452\) −10.1470 −0.477277
\(453\) −47.6612 −2.23932
\(454\) 28.8214 1.35265
\(455\) 0.218303 0.0102342
\(456\) −14.2308 −0.666417
\(457\) 12.3740 0.578830 0.289415 0.957204i \(-0.406539\pi\)
0.289415 + 0.957204i \(0.406539\pi\)
\(458\) 17.4261 0.814268
\(459\) −39.3891 −1.83852
\(460\) 1.44233 0.0672491
\(461\) −26.7956 −1.24799 −0.623997 0.781426i \(-0.714492\pi\)
−0.623997 + 0.781426i \(0.714492\pi\)
\(462\) −3.33509 −0.155162
\(463\) −1.70450 −0.0792147 −0.0396073 0.999215i \(-0.512611\pi\)
−0.0396073 + 0.999215i \(0.512611\pi\)
\(464\) −5.41969 −0.251603
\(465\) 10.5962 0.491386
\(466\) −2.26966 −0.105140
\(467\) −22.2192 −1.02818 −0.514091 0.857736i \(-0.671871\pi\)
−0.514091 + 0.857736i \(0.671871\pi\)
\(468\) −10.6603 −0.492773
\(469\) −1.08747 −0.0502146
\(470\) −1.24552 −0.0574515
\(471\) −33.5641 −1.54655
\(472\) 4.09203 0.188351
\(473\) 55.5962 2.55632
\(474\) −28.0307 −1.28749
\(475\) 22.5999 1.03695
\(476\) −1.04639 −0.0479611
\(477\) 17.1497 0.785232
\(478\) −23.3995 −1.07027
\(479\) −31.3616 −1.43295 −0.716473 0.697615i \(-0.754245\pi\)
−0.716473 + 0.697615i \(0.754245\pi\)
\(480\) 1.59863 0.0729670
\(481\) 2.93387 0.133773
\(482\) −13.4798 −0.613987
\(483\) 1.74612 0.0794513
\(484\) 15.1140 0.687001
\(485\) −1.67417 −0.0760201
\(486\) −3.54427 −0.160771
\(487\) 7.48526 0.339190 0.169595 0.985514i \(-0.445754\pi\)
0.169595 + 0.985514i \(0.445754\pi\)
\(488\) 2.30748 0.104455
\(489\) −32.0527 −1.44947
\(490\) 3.74752 0.169296
\(491\) 37.2492 1.68103 0.840516 0.541786i \(-0.182252\pi\)
0.840516 + 0.541786i \(0.182252\pi\)
\(492\) 14.1252 0.636812
\(493\) −25.7682 −1.16054
\(494\) −8.82990 −0.397276
\(495\) 15.9608 0.717384
\(496\) −6.62830 −0.297620
\(497\) −1.29270 −0.0579855
\(498\) 22.6090 1.01313
\(499\) 21.4802 0.961585 0.480792 0.876835i \(-0.340349\pi\)
0.480792 + 0.876835i \(0.340349\pi\)
\(500\) −5.23423 −0.234082
\(501\) −69.4130 −3.10114
\(502\) 31.0300 1.38494
\(503\) −42.8529 −1.91072 −0.955359 0.295447i \(-0.904532\pi\)
−0.955359 + 0.295447i \(0.904532\pi\)
\(504\) 1.27509 0.0567970
\(505\) −4.24081 −0.188713
\(506\) −13.6723 −0.607808
\(507\) 28.5109 1.26621
\(508\) −8.05548 −0.357404
\(509\) 0.0331688 0.00147018 0.000735090 1.00000i \(-0.499766\pi\)
0.000735090 1.00000i \(0.499766\pi\)
\(510\) 7.60074 0.336566
\(511\) 1.01828 0.0450461
\(512\) −1.00000 −0.0441942
\(513\) 39.7566 1.75530
\(514\) 28.0792 1.23852
\(515\) 5.29886 0.233496
\(516\) −32.2622 −1.42026
\(517\) 11.8066 0.519255
\(518\) −0.350922 −0.0154186
\(519\) 21.8198 0.957785
\(520\) 0.991915 0.0434984
\(521\) 20.8175 0.912029 0.456015 0.889972i \(-0.349276\pi\)
0.456015 + 0.889972i \(0.349276\pi\)
\(522\) 31.4001 1.37434
\(523\) 8.83260 0.386222 0.193111 0.981177i \(-0.438142\pi\)
0.193111 + 0.981177i \(0.438142\pi\)
\(524\) 5.25397 0.229521
\(525\) −3.07351 −0.134139
\(526\) 0.786856 0.0343086
\(527\) −31.5146 −1.37280
\(528\) −15.1538 −0.659487
\(529\) −15.8417 −0.688770
\(530\) −1.59574 −0.0693145
\(531\) −23.7080 −1.02884
\(532\) 1.05615 0.0457900
\(533\) 8.76438 0.379628
\(534\) 29.1361 1.26084
\(535\) −4.93515 −0.213365
\(536\) −4.94120 −0.213427
\(537\) 38.9509 1.68085
\(538\) 17.0961 0.737066
\(539\) −35.5238 −1.53012
\(540\) −4.46609 −0.192190
\(541\) −9.64488 −0.414666 −0.207333 0.978270i \(-0.566478\pi\)
−0.207333 + 0.978270i \(0.566478\pi\)
\(542\) 4.65637 0.200008
\(543\) 37.1992 1.59637
\(544\) −4.75454 −0.203849
\(545\) −2.80410 −0.120114
\(546\) 1.20084 0.0513910
\(547\) 4.51428 0.193017 0.0965083 0.995332i \(-0.469233\pi\)
0.0965083 + 0.995332i \(0.469233\pi\)
\(548\) 4.67123 0.199545
\(549\) −13.3689 −0.570569
\(550\) 24.0658 1.02617
\(551\) 26.0086 1.10800
\(552\) 7.93397 0.337692
\(553\) 2.08033 0.0884645
\(554\) 20.2844 0.861801
\(555\) 2.54902 0.108200
\(556\) −11.9008 −0.504706
\(557\) −12.2984 −0.521101 −0.260551 0.965460i \(-0.583904\pi\)
−0.260551 + 0.965460i \(0.583904\pi\)
\(558\) 38.4024 1.62570
\(559\) −20.0180 −0.846673
\(560\) −0.118644 −0.00501362
\(561\) −72.0496 −3.04194
\(562\) −1.65735 −0.0699112
\(563\) 8.74121 0.368398 0.184199 0.982889i \(-0.441031\pi\)
0.184199 + 0.982889i \(0.441031\pi\)
\(564\) −6.85134 −0.288493
\(565\) −5.47017 −0.230132
\(566\) 27.8194 1.16934
\(567\) −1.58149 −0.0664163
\(568\) −5.87372 −0.246456
\(569\) −11.2391 −0.471169 −0.235585 0.971854i \(-0.575701\pi\)
−0.235585 + 0.971854i \(0.575701\pi\)
\(570\) −7.67166 −0.321331
\(571\) 9.65898 0.404216 0.202108 0.979363i \(-0.435221\pi\)
0.202108 + 0.979363i \(0.435221\pi\)
\(572\) −9.40265 −0.393145
\(573\) −20.3172 −0.848764
\(574\) −1.04832 −0.0437558
\(575\) −12.5999 −0.525454
\(576\) 5.79371 0.241404
\(577\) 0.979135 0.0407619 0.0203810 0.999792i \(-0.493512\pi\)
0.0203810 + 0.999792i \(0.493512\pi\)
\(578\) −5.60569 −0.233166
\(579\) −22.1441 −0.920279
\(580\) −2.92170 −0.121317
\(581\) −1.67795 −0.0696130
\(582\) −9.20925 −0.381736
\(583\) 15.1265 0.626474
\(584\) 4.62683 0.191459
\(585\) −5.74687 −0.237604
\(586\) 0.258205 0.0106664
\(587\) 26.5042 1.09395 0.546973 0.837150i \(-0.315780\pi\)
0.546973 + 0.837150i \(0.315780\pi\)
\(588\) 20.6143 0.850119
\(589\) 31.8086 1.31065
\(590\) 2.20597 0.0908185
\(591\) 33.7096 1.38663
\(592\) −1.59451 −0.0655339
\(593\) −2.78421 −0.114334 −0.0571669 0.998365i \(-0.518207\pi\)
−0.0571669 + 0.998365i \(0.518207\pi\)
\(594\) 42.3354 1.73704
\(595\) −0.564097 −0.0231257
\(596\) 15.6554 0.641272
\(597\) −19.4839 −0.797421
\(598\) 4.92287 0.201311
\(599\) −8.26439 −0.337674 −0.168837 0.985644i \(-0.554001\pi\)
−0.168837 + 0.985644i \(0.554001\pi\)
\(600\) −13.9653 −0.570131
\(601\) −11.5506 −0.471160 −0.235580 0.971855i \(-0.575699\pi\)
−0.235580 + 0.971855i \(0.575699\pi\)
\(602\) 2.39438 0.0975875
\(603\) 28.6279 1.16582
\(604\) 16.0723 0.653975
\(605\) 8.14782 0.331256
\(606\) −23.3278 −0.947626
\(607\) −1.08893 −0.0441983 −0.0220992 0.999756i \(-0.507035\pi\)
−0.0220992 + 0.999756i \(0.507035\pi\)
\(608\) 4.79891 0.194621
\(609\) −3.53708 −0.143330
\(610\) 1.24394 0.0503656
\(611\) −4.25112 −0.171982
\(612\) 27.5464 1.11350
\(613\) −33.0129 −1.33338 −0.666690 0.745335i \(-0.732289\pi\)
−0.666690 + 0.745335i \(0.732289\pi\)
\(614\) −31.9661 −1.29005
\(615\) 7.61474 0.307056
\(616\) 1.12466 0.0453138
\(617\) −16.1373 −0.649663 −0.324831 0.945772i \(-0.605308\pi\)
−0.324831 + 0.945772i \(0.605308\pi\)
\(618\) 29.1479 1.17250
\(619\) 17.3721 0.698245 0.349122 0.937077i \(-0.386480\pi\)
0.349122 + 0.937077i \(0.386480\pi\)
\(620\) −3.57325 −0.143505
\(621\) −22.1652 −0.889458
\(622\) −2.69476 −0.108050
\(623\) −2.16237 −0.0866336
\(624\) 5.45632 0.218427
\(625\) 20.7252 0.829008
\(626\) 11.7999 0.471619
\(627\) 72.7219 2.90423
\(628\) 11.3185 0.451657
\(629\) −7.58116 −0.302281
\(630\) 0.687388 0.0273862
\(631\) −6.58014 −0.261951 −0.130976 0.991386i \(-0.541811\pi\)
−0.130976 + 0.991386i \(0.541811\pi\)
\(632\) 9.45252 0.376001
\(633\) 36.9572 1.46892
\(634\) 20.0170 0.794976
\(635\) −4.34262 −0.172332
\(636\) −8.77782 −0.348063
\(637\) 12.7908 0.506788
\(638\) 27.6956 1.09648
\(639\) 34.0306 1.34623
\(640\) −0.539090 −0.0213094
\(641\) 37.3463 1.47509 0.737545 0.675298i \(-0.235985\pi\)
0.737545 + 0.675298i \(0.235985\pi\)
\(642\) −27.1472 −1.07142
\(643\) 34.0081 1.34115 0.670573 0.741843i \(-0.266048\pi\)
0.670573 + 0.741843i \(0.266048\pi\)
\(644\) −0.588828 −0.0232031
\(645\) −17.3922 −0.684818
\(646\) 22.8166 0.897708
\(647\) 16.1058 0.633183 0.316592 0.948562i \(-0.397462\pi\)
0.316592 + 0.948562i \(0.397462\pi\)
\(648\) −7.18592 −0.282290
\(649\) −20.9111 −0.820831
\(650\) −8.66518 −0.339876
\(651\) −4.32586 −0.169544
\(652\) 10.8088 0.423306
\(653\) 28.2006 1.10357 0.551787 0.833985i \(-0.313946\pi\)
0.551787 + 0.833985i \(0.313946\pi\)
\(654\) −15.4247 −0.603155
\(655\) 2.83236 0.110669
\(656\) −4.76330 −0.185976
\(657\) −26.8065 −1.04582
\(658\) 0.508480 0.0198226
\(659\) 5.64841 0.220031 0.110015 0.993930i \(-0.464910\pi\)
0.110015 + 0.993930i \(0.464910\pi\)
\(660\) −8.16928 −0.317989
\(661\) 10.7607 0.418543 0.209272 0.977858i \(-0.432891\pi\)
0.209272 + 0.977858i \(0.432891\pi\)
\(662\) −6.57648 −0.255602
\(663\) 25.9423 1.00752
\(664\) −7.62420 −0.295876
\(665\) 0.569361 0.0220789
\(666\) 9.23812 0.357970
\(667\) −14.5004 −0.561456
\(668\) 23.4075 0.905663
\(669\) 40.4384 1.56344
\(670\) −2.66375 −0.102910
\(671\) −11.7916 −0.455212
\(672\) −0.652635 −0.0251759
\(673\) 9.75137 0.375888 0.187944 0.982180i \(-0.439818\pi\)
0.187944 + 0.982180i \(0.439818\pi\)
\(674\) 8.95217 0.344825
\(675\) 39.0149 1.50169
\(676\) −9.61447 −0.369787
\(677\) −31.3089 −1.20330 −0.601648 0.798761i \(-0.705489\pi\)
−0.601648 + 0.798761i \(0.705489\pi\)
\(678\) −30.0902 −1.15561
\(679\) 0.683474 0.0262293
\(680\) −2.56313 −0.0982914
\(681\) 85.4674 3.27512
\(682\) 33.8719 1.29702
\(683\) −34.6757 −1.32683 −0.663414 0.748253i \(-0.730893\pi\)
−0.663414 + 0.748253i \(0.730893\pi\)
\(684\) −27.8035 −1.06309
\(685\) 2.51821 0.0962159
\(686\) −3.07048 −0.117232
\(687\) 51.6757 1.97155
\(688\) 10.8795 0.414776
\(689\) −5.44646 −0.207494
\(690\) 4.27712 0.162827
\(691\) −21.4456 −0.815827 −0.407914 0.913020i \(-0.633744\pi\)
−0.407914 + 0.913020i \(0.633744\pi\)
\(692\) −7.35810 −0.279713
\(693\) −6.51595 −0.247520
\(694\) −15.9803 −0.606604
\(695\) −6.41559 −0.243357
\(696\) −16.0716 −0.609194
\(697\) −22.6473 −0.857828
\(698\) 28.3421 1.07277
\(699\) −6.73049 −0.254570
\(700\) 1.03645 0.0391741
\(701\) 17.7909 0.671952 0.335976 0.941871i \(-0.390934\pi\)
0.335976 + 0.941871i \(0.390934\pi\)
\(702\) −15.2433 −0.575323
\(703\) 7.65190 0.288597
\(704\) 5.11019 0.192597
\(705\) −3.69349 −0.139105
\(706\) −2.23439 −0.0840922
\(707\) 1.73130 0.0651121
\(708\) 12.1346 0.456046
\(709\) 13.7638 0.516909 0.258455 0.966023i \(-0.416787\pi\)
0.258455 + 0.966023i \(0.416787\pi\)
\(710\) −3.16646 −0.118835
\(711\) −54.7651 −2.05385
\(712\) −9.82531 −0.368219
\(713\) −17.7340 −0.664144
\(714\) −3.10298 −0.116126
\(715\) −5.06887 −0.189565
\(716\) −13.1350 −0.490879
\(717\) −69.3893 −2.59139
\(718\) −4.83027 −0.180264
\(719\) −4.58682 −0.171060 −0.0855298 0.996336i \(-0.527258\pi\)
−0.0855298 + 0.996336i \(0.527258\pi\)
\(720\) 3.12333 0.116400
\(721\) −2.16324 −0.0805634
\(722\) −4.02952 −0.149963
\(723\) −39.9732 −1.48662
\(724\) −12.5443 −0.466206
\(725\) 25.5234 0.947915
\(726\) 44.8194 1.66340
\(727\) −11.2167 −0.416004 −0.208002 0.978128i \(-0.566696\pi\)
−0.208002 + 0.978128i \(0.566696\pi\)
\(728\) −0.404946 −0.0150083
\(729\) −32.0680 −1.18770
\(730\) 2.49427 0.0923173
\(731\) 51.7270 1.91319
\(732\) 6.84264 0.252911
\(733\) −36.0414 −1.33122 −0.665609 0.746300i \(-0.731828\pi\)
−0.665609 + 0.746300i \(0.731828\pi\)
\(734\) −30.9066 −1.14078
\(735\) 11.1130 0.409908
\(736\) −2.67550 −0.0986201
\(737\) 25.2505 0.930113
\(738\) 27.5972 1.01587
\(739\) −26.3421 −0.969010 −0.484505 0.874789i \(-0.661000\pi\)
−0.484505 + 0.874789i \(0.661000\pi\)
\(740\) −0.859583 −0.0315989
\(741\) −26.1844 −0.961907
\(742\) 0.651456 0.0239157
\(743\) −11.1231 −0.408068 −0.204034 0.978964i \(-0.565405\pi\)
−0.204034 + 0.978964i \(0.565405\pi\)
\(744\) −19.6557 −0.720613
\(745\) 8.43969 0.309206
\(746\) −0.175729 −0.00643391
\(747\) 44.1724 1.61618
\(748\) 24.2966 0.888372
\(749\) 2.01476 0.0736178
\(750\) −15.5217 −0.566772
\(751\) −5.39268 −0.196782 −0.0983909 0.995148i \(-0.531370\pi\)
−0.0983909 + 0.995148i \(0.531370\pi\)
\(752\) 2.31041 0.0842521
\(753\) 92.0170 3.35329
\(754\) −9.97213 −0.363164
\(755\) 8.66444 0.315331
\(756\) 1.82327 0.0663117
\(757\) −28.1986 −1.02489 −0.512447 0.858719i \(-0.671261\pi\)
−0.512447 + 0.858719i \(0.671261\pi\)
\(758\) 20.4684 0.743446
\(759\) −40.5441 −1.47166
\(760\) 2.58704 0.0938419
\(761\) −42.0164 −1.52309 −0.761546 0.648111i \(-0.775559\pi\)
−0.761546 + 0.648111i \(0.775559\pi\)
\(762\) −23.8879 −0.865366
\(763\) 1.14476 0.0414432
\(764\) 6.85138 0.247874
\(765\) 14.8500 0.536903
\(766\) 23.6401 0.854151
\(767\) 7.52927 0.271866
\(768\) −2.96542 −0.107005
\(769\) 9.93916 0.358415 0.179208 0.983811i \(-0.442647\pi\)
0.179208 + 0.983811i \(0.442647\pi\)
\(770\) 0.606292 0.0218493
\(771\) 83.2665 2.99877
\(772\) 7.46745 0.268760
\(773\) −6.95910 −0.250301 −0.125151 0.992138i \(-0.539941\pi\)
−0.125151 + 0.992138i \(0.539941\pi\)
\(774\) −63.0325 −2.26566
\(775\) 31.2152 1.12128
\(776\) 3.10555 0.111483
\(777\) −1.04063 −0.0373325
\(778\) 11.5581 0.414378
\(779\) 22.8586 0.818995
\(780\) 2.94144 0.105321
\(781\) 30.0158 1.07405
\(782\) −12.7208 −0.454894
\(783\) 44.8995 1.60458
\(784\) −6.95156 −0.248270
\(785\) 6.10168 0.217778
\(786\) 15.5802 0.555728
\(787\) 6.10260 0.217534 0.108767 0.994067i \(-0.465310\pi\)
0.108767 + 0.994067i \(0.465310\pi\)
\(788\) −11.3676 −0.404953
\(789\) 2.33336 0.0830698
\(790\) 5.09576 0.181299
\(791\) 2.23318 0.0794027
\(792\) −29.6069 −1.05204
\(793\) 4.24572 0.150770
\(794\) −8.09403 −0.287246
\(795\) −4.73203 −0.167828
\(796\) 6.57036 0.232880
\(797\) −11.7296 −0.415485 −0.207742 0.978184i \(-0.566612\pi\)
−0.207742 + 0.978184i \(0.566612\pi\)
\(798\) 3.13193 0.110869
\(799\) 10.9850 0.388620
\(800\) 4.70938 0.166502
\(801\) 56.9250 2.01134
\(802\) −11.8819 −0.419565
\(803\) −23.6440 −0.834377
\(804\) −14.6527 −0.516762
\(805\) −0.317431 −0.0111880
\(806\) −12.1960 −0.429584
\(807\) 50.6971 1.78462
\(808\) 7.86660 0.276746
\(809\) 20.6454 0.725853 0.362926 0.931818i \(-0.381778\pi\)
0.362926 + 0.931818i \(0.381778\pi\)
\(810\) −3.87386 −0.136113
\(811\) 41.4740 1.45635 0.728175 0.685391i \(-0.240369\pi\)
0.728175 + 0.685391i \(0.240369\pi\)
\(812\) 1.19277 0.0418582
\(813\) 13.8081 0.484271
\(814\) 8.14824 0.285596
\(815\) 5.82692 0.204108
\(816\) −14.0992 −0.493571
\(817\) −52.2096 −1.82658
\(818\) −39.2750 −1.37322
\(819\) 2.34614 0.0819808
\(820\) −2.56785 −0.0896730
\(821\) 13.8087 0.481927 0.240964 0.970534i \(-0.422537\pi\)
0.240964 + 0.970534i \(0.422537\pi\)
\(822\) 13.8521 0.483149
\(823\) 12.6286 0.440206 0.220103 0.975477i \(-0.429361\pi\)
0.220103 + 0.975477i \(0.429361\pi\)
\(824\) −9.82927 −0.342419
\(825\) 71.3653 2.48462
\(826\) −0.900582 −0.0313353
\(827\) 49.3348 1.71554 0.857770 0.514034i \(-0.171849\pi\)
0.857770 + 0.514034i \(0.171849\pi\)
\(828\) 15.5010 0.538699
\(829\) −28.0521 −0.974289 −0.487145 0.873321i \(-0.661962\pi\)
−0.487145 + 0.873321i \(0.661962\pi\)
\(830\) −4.11013 −0.142665
\(831\) 60.1517 2.08664
\(832\) −1.83998 −0.0637899
\(833\) −33.0515 −1.14517
\(834\) −35.2908 −1.22202
\(835\) 12.6187 0.436689
\(836\) −24.5233 −0.848157
\(837\) 54.9122 1.89804
\(838\) 27.0626 0.934863
\(839\) 17.5700 0.606584 0.303292 0.952898i \(-0.401914\pi\)
0.303292 + 0.952898i \(0.401914\pi\)
\(840\) −0.351829 −0.0121392
\(841\) 0.373036 0.0128633
\(842\) −21.1878 −0.730181
\(843\) −4.91474 −0.169273
\(844\) −12.4627 −0.428985
\(845\) −5.18306 −0.178303
\(846\) −13.3859 −0.460215
\(847\) −3.32632 −0.114294
\(848\) 2.96006 0.101649
\(849\) 82.4962 2.83126
\(850\) 22.3910 0.768004
\(851\) −4.26610 −0.146240
\(852\) −17.4180 −0.596732
\(853\) −11.0901 −0.379717 −0.189858 0.981811i \(-0.560803\pi\)
−0.189858 + 0.981811i \(0.560803\pi\)
\(854\) −0.507834 −0.0173777
\(855\) −14.9886 −0.512598
\(856\) 9.15461 0.312898
\(857\) −9.82809 −0.335721 −0.167861 0.985811i \(-0.553686\pi\)
−0.167861 + 0.985811i \(0.553686\pi\)
\(858\) −27.8828 −0.951903
\(859\) 0.914828 0.0312135 0.0156068 0.999878i \(-0.495032\pi\)
0.0156068 + 0.999878i \(0.495032\pi\)
\(860\) 5.86502 0.199995
\(861\) −3.10869 −0.105944
\(862\) 19.4929 0.663930
\(863\) 49.6226 1.68917 0.844586 0.535420i \(-0.179847\pi\)
0.844586 + 0.535420i \(0.179847\pi\)
\(864\) 8.28451 0.281845
\(865\) −3.96667 −0.134871
\(866\) −16.8948 −0.574109
\(867\) −16.6232 −0.564554
\(868\) 1.45877 0.0495138
\(869\) −48.3042 −1.63861
\(870\) −8.66406 −0.293739
\(871\) −9.09172 −0.308061
\(872\) 5.20154 0.176146
\(873\) −17.9926 −0.608958
\(874\) 12.8395 0.434301
\(875\) 1.15196 0.0389433
\(876\) 13.7205 0.463572
\(877\) −43.6707 −1.47465 −0.737327 0.675536i \(-0.763912\pi\)
−0.737327 + 0.675536i \(0.763912\pi\)
\(878\) 33.8285 1.14166
\(879\) 0.765687 0.0258260
\(880\) 2.75485 0.0928660
\(881\) 33.0418 1.11321 0.556604 0.830778i \(-0.312104\pi\)
0.556604 + 0.830778i \(0.312104\pi\)
\(882\) 40.2753 1.35614
\(883\) −9.37760 −0.315581 −0.157791 0.987473i \(-0.550437\pi\)
−0.157791 + 0.987473i \(0.550437\pi\)
\(884\) −8.74828 −0.294236
\(885\) 6.54163 0.219895
\(886\) 21.2006 0.712248
\(887\) −19.9083 −0.668454 −0.334227 0.942493i \(-0.608475\pi\)
−0.334227 + 0.942493i \(0.608475\pi\)
\(888\) −4.72839 −0.158674
\(889\) 1.77286 0.0594599
\(890\) −5.29672 −0.177547
\(891\) 36.7214 1.23021
\(892\) −13.6367 −0.456589
\(893\) −11.0875 −0.371028
\(894\) 46.4249 1.55268
\(895\) −7.08096 −0.236691
\(896\) 0.220082 0.00735242
\(897\) 14.5984 0.487425
\(898\) 11.5835 0.386547
\(899\) 35.9233 1.19811
\(900\) −27.2848 −0.909493
\(901\) 14.0737 0.468864
\(902\) 24.3414 0.810478
\(903\) 7.10033 0.236284
\(904\) 10.1470 0.337486
\(905\) −6.76252 −0.224794
\(906\) 47.6612 1.58344
\(907\) −29.9600 −0.994805 −0.497402 0.867520i \(-0.665713\pi\)
−0.497402 + 0.867520i \(0.665713\pi\)
\(908\) −28.8214 −0.956471
\(909\) −45.5768 −1.51169
\(910\) −0.218303 −0.00723666
\(911\) 35.2589 1.16818 0.584089 0.811690i \(-0.301452\pi\)
0.584089 + 0.811690i \(0.301452\pi\)
\(912\) 14.2308 0.471228
\(913\) 38.9611 1.28942
\(914\) −12.3740 −0.409295
\(915\) 3.68880 0.121948
\(916\) −17.4261 −0.575775
\(917\) −1.15630 −0.0381845
\(918\) 39.3891 1.30003
\(919\) −30.6263 −1.01027 −0.505135 0.863041i \(-0.668557\pi\)
−0.505135 + 0.863041i \(0.668557\pi\)
\(920\) −1.44233 −0.0475523
\(921\) −94.7929 −3.12353
\(922\) 26.7956 0.882466
\(923\) −10.8075 −0.355735
\(924\) 3.33509 0.109716
\(925\) 7.50915 0.246899
\(926\) 1.70450 0.0560132
\(927\) 56.9479 1.87041
\(928\) 5.41969 0.177910
\(929\) −3.02603 −0.0992809 −0.0496405 0.998767i \(-0.515808\pi\)
−0.0496405 + 0.998767i \(0.515808\pi\)
\(930\) −10.5962 −0.347462
\(931\) 33.3599 1.09333
\(932\) 2.26966 0.0743452
\(933\) −7.99109 −0.261616
\(934\) 22.2192 0.727034
\(935\) 13.0981 0.428352
\(936\) 10.6603 0.348443
\(937\) −40.6266 −1.32721 −0.663607 0.748081i \(-0.730975\pi\)
−0.663607 + 0.748081i \(0.730975\pi\)
\(938\) 1.08747 0.0355071
\(939\) 34.9917 1.14191
\(940\) 1.24552 0.0406244
\(941\) −0.630592 −0.0205567 −0.0102784 0.999947i \(-0.503272\pi\)
−0.0102784 + 0.999947i \(0.503272\pi\)
\(942\) 33.5641 1.09358
\(943\) −12.7442 −0.415008
\(944\) −4.09203 −0.133184
\(945\) 0.982906 0.0319739
\(946\) −55.5962 −1.80759
\(947\) −47.9657 −1.55868 −0.779338 0.626604i \(-0.784444\pi\)
−0.779338 + 0.626604i \(0.784444\pi\)
\(948\) 28.0307 0.910394
\(949\) 8.51328 0.276353
\(950\) −22.5999 −0.733237
\(951\) 59.3587 1.92484
\(952\) 1.04639 0.0339136
\(953\) 28.3270 0.917602 0.458801 0.888539i \(-0.348279\pi\)
0.458801 + 0.888539i \(0.348279\pi\)
\(954\) −17.1497 −0.555243
\(955\) 3.69351 0.119519
\(956\) 23.3995 0.756793
\(957\) 82.1291 2.65486
\(958\) 31.3616 1.01325
\(959\) −1.02805 −0.0331975
\(960\) −1.59863 −0.0515955
\(961\) 12.9344 0.417238
\(962\) −2.93387 −0.0945917
\(963\) −53.0391 −1.70916
\(964\) 13.4798 0.434154
\(965\) 4.02563 0.129590
\(966\) −1.74612 −0.0561806
\(967\) 7.77713 0.250096 0.125048 0.992151i \(-0.460092\pi\)
0.125048 + 0.992151i \(0.460092\pi\)
\(968\) −15.1140 −0.485783
\(969\) 67.6608 2.17358
\(970\) 1.67417 0.0537543
\(971\) −10.9041 −0.349930 −0.174965 0.984575i \(-0.555981\pi\)
−0.174965 + 0.984575i \(0.555981\pi\)
\(972\) 3.54427 0.113683
\(973\) 2.61915 0.0839660
\(974\) −7.48526 −0.239843
\(975\) −25.6959 −0.822927
\(976\) −2.30748 −0.0738606
\(977\) −51.5261 −1.64847 −0.824233 0.566251i \(-0.808393\pi\)
−0.824233 + 0.566251i \(0.808393\pi\)
\(978\) 32.0527 1.02493
\(979\) 50.2092 1.60469
\(980\) −3.74752 −0.119710
\(981\) −30.1362 −0.962174
\(982\) −37.2492 −1.18867
\(983\) −49.3988 −1.57558 −0.787789 0.615945i \(-0.788774\pi\)
−0.787789 + 0.615945i \(0.788774\pi\)
\(984\) −14.1252 −0.450294
\(985\) −6.12814 −0.195259
\(986\) 25.7682 0.820625
\(987\) 1.50785 0.0479955
\(988\) 8.82990 0.280917
\(989\) 29.1080 0.925581
\(990\) −15.9608 −0.507267
\(991\) 43.5919 1.38474 0.692371 0.721542i \(-0.256566\pi\)
0.692371 + 0.721542i \(0.256566\pi\)
\(992\) 6.62830 0.210449
\(993\) −19.5020 −0.618878
\(994\) 1.29270 0.0410019
\(995\) 3.54201 0.112289
\(996\) −22.6090 −0.716392
\(997\) 19.2513 0.609693 0.304847 0.952401i \(-0.401395\pi\)
0.304847 + 0.952401i \(0.401395\pi\)
\(998\) −21.4802 −0.679943
\(999\) 13.2097 0.417937
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 4006.2.a.g.1.3 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.g.1.3 40 1.1 even 1 trivial