Properties

Label 2009.2.a.n.1.4
Level $2009$
Weight $2$
Character 2009.1
Self dual yes
Analytic conductor $16.042$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.0419457661\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.633117.1
Defining polynomial: \(x^{5} - x^{4} - 6 x^{3} + 4 x^{2} + 6 x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.08727\) of defining polynomial
Character \(\chi\) \(=\) 2009.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.08727 q^{2} -2.08727 q^{3} -0.817843 q^{4} -0.209668 q^{5} -2.26943 q^{6} -3.06376 q^{8} +1.35670 q^{9} +O(q^{10})\) \(q+1.08727 q^{2} -2.08727 q^{3} -0.817843 q^{4} -0.209668 q^{5} -2.26943 q^{6} -3.06376 q^{8} +1.35670 q^{9} -0.227965 q^{10} +6.03819 q^{11} +1.70706 q^{12} -3.67193 q^{13} +0.437633 q^{15} -1.69545 q^{16} +5.37138 q^{17} +1.47510 q^{18} +3.54285 q^{19} +0.171475 q^{20} +6.56515 q^{22} -1.30362 q^{23} +6.39489 q^{24} -4.95604 q^{25} -3.99238 q^{26} +3.43002 q^{27} -8.00307 q^{29} +0.475825 q^{30} +0.384208 q^{31} +4.28411 q^{32} -12.6033 q^{33} +5.84014 q^{34} -1.10957 q^{36} -3.68876 q^{37} +3.85204 q^{38} +7.66432 q^{39} +0.642370 q^{40} +1.00000 q^{41} +0.824527 q^{43} -4.93829 q^{44} -0.284455 q^{45} -1.41739 q^{46} -5.11625 q^{47} +3.53885 q^{48} -5.38855 q^{50} -11.2115 q^{51} +3.00307 q^{52} +1.53217 q^{53} +3.72936 q^{54} -1.26601 q^{55} -7.39489 q^{57} -8.70150 q^{58} -10.2669 q^{59} -0.357915 q^{60} -9.36070 q^{61} +0.417738 q^{62} +8.04887 q^{64} +0.769885 q^{65} -13.7032 q^{66} +11.3638 q^{67} -4.39294 q^{68} +2.72101 q^{69} -14.9494 q^{71} -4.15659 q^{72} -7.77203 q^{73} -4.01068 q^{74} +10.3446 q^{75} -2.89750 q^{76} +8.33318 q^{78} -6.04703 q^{79} +0.355480 q^{80} -11.2295 q^{81} +1.08727 q^{82} -14.1871 q^{83} -1.12620 q^{85} +0.896484 q^{86} +16.7046 q^{87} -18.4996 q^{88} -0.520905 q^{89} -0.309280 q^{90} +1.06616 q^{92} -0.801946 q^{93} -5.56275 q^{94} -0.742821 q^{95} -8.94209 q^{96} +3.65270 q^{97} +8.19200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + O(q^{10}) \) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + 2q^{11} + 2q^{12} - 5q^{13} - 5q^{15} - q^{16} - 13q^{17} + 21q^{18} + 23q^{20} + q^{22} + 2q^{23} - 2q^{24} + 22q^{25} - 10q^{27} - 5q^{29} - 33q^{30} - 17q^{31} - 12q^{32} - 3q^{33} + 8q^{34} + 15q^{36} - 7q^{37} + 3q^{38} + 5q^{39} - 7q^{40} + 5q^{41} + q^{43} - 47q^{44} + 23q^{45} - 24q^{46} - 9q^{47} + 19q^{48} + 2q^{50} + 5q^{51} - 20q^{52} + 5q^{53} - 2q^{54} - 33q^{55} - 3q^{57} - 27q^{58} - 7q^{59} - 16q^{60} - 22q^{61} + 28q^{62} - 3q^{64} - 31q^{65} + 42q^{66} - 3q^{67} - 17q^{68} + 22q^{69} - 24q^{71} - 12q^{72} - 40q^{73} - 5q^{74} - 24q^{75} + 19q^{76} + 30q^{78} - 42q^{79} - 24q^{80} + 9q^{81} - q^{82} + 12q^{83} - 23q^{85} + 16q^{86} + 32q^{87} + 26q^{88} - 8q^{89} + 59q^{90} + 12q^{92} - 11q^{93} + 23q^{94} - 17q^{95} + 17q^{96} - 16q^{97} - 45q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.08727 0.768816 0.384408 0.923163i \(-0.374406\pi\)
0.384408 + 0.923163i \(0.374406\pi\)
\(3\) −2.08727 −1.20509 −0.602543 0.798086i \(-0.705846\pi\)
−0.602543 + 0.798086i \(0.705846\pi\)
\(4\) −0.817843 −0.408922
\(5\) −0.209668 −0.0937662 −0.0468831 0.998900i \(-0.514929\pi\)
−0.0468831 + 0.998900i \(0.514929\pi\)
\(6\) −2.26943 −0.926490
\(7\) 0 0
\(8\) −3.06376 −1.08320
\(9\) 1.35670 0.452232
\(10\) −0.227965 −0.0720889
\(11\) 6.03819 1.82058 0.910292 0.413967i \(-0.135857\pi\)
0.910292 + 0.413967i \(0.135857\pi\)
\(12\) 1.70706 0.492786
\(13\) −3.67193 −1.01841 −0.509205 0.860645i \(-0.670061\pi\)
−0.509205 + 0.860645i \(0.670061\pi\)
\(14\) 0 0
\(15\) 0.437633 0.112996
\(16\) −1.69545 −0.423862
\(17\) 5.37138 1.30275 0.651375 0.758756i \(-0.274192\pi\)
0.651375 + 0.758756i \(0.274192\pi\)
\(18\) 1.47510 0.347684
\(19\) 3.54285 0.812786 0.406393 0.913698i \(-0.366786\pi\)
0.406393 + 0.913698i \(0.366786\pi\)
\(20\) 0.171475 0.0383430
\(21\) 0 0
\(22\) 6.56515 1.39969
\(23\) −1.30362 −0.271824 −0.135912 0.990721i \(-0.543396\pi\)
−0.135912 + 0.990721i \(0.543396\pi\)
\(24\) 6.39489 1.30535
\(25\) −4.95604 −0.991208
\(26\) −3.99238 −0.782971
\(27\) 3.43002 0.660107
\(28\) 0 0
\(29\) −8.00307 −1.48613 −0.743066 0.669218i \(-0.766629\pi\)
−0.743066 + 0.669218i \(0.766629\pi\)
\(30\) 0.475825 0.0868734
\(31\) 0.384208 0.0690058 0.0345029 0.999405i \(-0.489015\pi\)
0.0345029 + 0.999405i \(0.489015\pi\)
\(32\) 4.28411 0.757330
\(33\) −12.6033 −2.19396
\(34\) 5.84014 1.00158
\(35\) 0 0
\(36\) −1.10957 −0.184928
\(37\) −3.68876 −0.606429 −0.303214 0.952922i \(-0.598060\pi\)
−0.303214 + 0.952922i \(0.598060\pi\)
\(38\) 3.85204 0.624883
\(39\) 7.66432 1.22727
\(40\) 0.642370 0.101568
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) 0.824527 0.125739 0.0628696 0.998022i \(-0.479975\pi\)
0.0628696 + 0.998022i \(0.479975\pi\)
\(44\) −4.93829 −0.744476
\(45\) −0.284455 −0.0424041
\(46\) −1.41739 −0.208983
\(47\) −5.11625 −0.746282 −0.373141 0.927775i \(-0.621719\pi\)
−0.373141 + 0.927775i \(0.621719\pi\)
\(48\) 3.53885 0.510790
\(49\) 0 0
\(50\) −5.38855 −0.762057
\(51\) −11.2115 −1.56993
\(52\) 3.00307 0.416450
\(53\) 1.53217 0.210460 0.105230 0.994448i \(-0.466442\pi\)
0.105230 + 0.994448i \(0.466442\pi\)
\(54\) 3.72936 0.507501
\(55\) −1.26601 −0.170709
\(56\) 0 0
\(57\) −7.39489 −0.979477
\(58\) −8.70150 −1.14256
\(59\) −10.2669 −1.33664 −0.668320 0.743874i \(-0.732986\pi\)
−0.668320 + 0.743874i \(0.732986\pi\)
\(60\) −0.357915 −0.0462066
\(61\) −9.36070 −1.19851 −0.599257 0.800557i \(-0.704537\pi\)
−0.599257 + 0.800557i \(0.704537\pi\)
\(62\) 0.417738 0.0530528
\(63\) 0 0
\(64\) 8.04887 1.00611
\(65\) 0.769885 0.0954925
\(66\) −13.7032 −1.68675
\(67\) 11.3638 1.38830 0.694152 0.719828i \(-0.255779\pi\)
0.694152 + 0.719828i \(0.255779\pi\)
\(68\) −4.39294 −0.532723
\(69\) 2.72101 0.327571
\(70\) 0 0
\(71\) −14.9494 −1.77416 −0.887081 0.461614i \(-0.847271\pi\)
−0.887081 + 0.461614i \(0.847271\pi\)
\(72\) −4.15659 −0.489859
\(73\) −7.77203 −0.909648 −0.454824 0.890581i \(-0.650298\pi\)
−0.454824 + 0.890581i \(0.650298\pi\)
\(74\) −4.01068 −0.466232
\(75\) 10.3446 1.19449
\(76\) −2.89750 −0.332366
\(77\) 0 0
\(78\) 8.33318 0.943547
\(79\) −6.04703 −0.680344 −0.340172 0.940363i \(-0.610485\pi\)
−0.340172 + 0.940363i \(0.610485\pi\)
\(80\) 0.355480 0.0397439
\(81\) −11.2295 −1.24772
\(82\) 1.08727 0.120069
\(83\) −14.1871 −1.55723 −0.778617 0.627500i \(-0.784078\pi\)
−0.778617 + 0.627500i \(0.784078\pi\)
\(84\) 0 0
\(85\) −1.12620 −0.122154
\(86\) 0.896484 0.0966703
\(87\) 16.7046 1.79092
\(88\) −18.4996 −1.97206
\(89\) −0.520905 −0.0552159 −0.0276079 0.999619i \(-0.508789\pi\)
−0.0276079 + 0.999619i \(0.508789\pi\)
\(90\) −0.309280 −0.0326010
\(91\) 0 0
\(92\) 1.06616 0.111155
\(93\) −0.801946 −0.0831580
\(94\) −5.56275 −0.573754
\(95\) −0.742821 −0.0762118
\(96\) −8.94209 −0.912648
\(97\) 3.65270 0.370876 0.185438 0.982656i \(-0.440630\pi\)
0.185438 + 0.982656i \(0.440630\pi\)
\(98\) 0 0
\(99\) 8.19200 0.823327
\(100\) 4.05326 0.405326
\(101\) 2.45465 0.244247 0.122123 0.992515i \(-0.461030\pi\)
0.122123 + 0.992515i \(0.461030\pi\)
\(102\) −12.1899 −1.20698
\(103\) 10.2479 1.00975 0.504876 0.863192i \(-0.331538\pi\)
0.504876 + 0.863192i \(0.331538\pi\)
\(104\) 11.2499 1.10314
\(105\) 0 0
\(106\) 1.66588 0.161805
\(107\) −5.33318 −0.515578 −0.257789 0.966201i \(-0.582994\pi\)
−0.257789 + 0.966201i \(0.582994\pi\)
\(108\) −2.80522 −0.269932
\(109\) 8.82638 0.845413 0.422707 0.906267i \(-0.361080\pi\)
0.422707 + 0.906267i \(0.361080\pi\)
\(110\) −1.37650 −0.131244
\(111\) 7.69944 0.730799
\(112\) 0 0
\(113\) −18.2045 −1.71253 −0.856267 0.516534i \(-0.827222\pi\)
−0.856267 + 0.516534i \(0.827222\pi\)
\(114\) −8.04024 −0.753038
\(115\) 0.273327 0.0254879
\(116\) 6.54525 0.607711
\(117\) −4.98170 −0.460559
\(118\) −11.1629 −1.02763
\(119\) 0 0
\(120\) −1.34080 −0.122398
\(121\) 25.4598 2.31452
\(122\) −10.1776 −0.921437
\(123\) −2.08727 −0.188203
\(124\) −0.314222 −0.0282180
\(125\) 2.08746 0.186708
\(126\) 0 0
\(127\) −13.1751 −1.16910 −0.584551 0.811357i \(-0.698729\pi\)
−0.584551 + 0.811357i \(0.698729\pi\)
\(128\) 0.183089 0.0161829
\(129\) −1.72101 −0.151527
\(130\) 0.837073 0.0734162
\(131\) −13.0506 −1.14023 −0.570117 0.821563i \(-0.693102\pi\)
−0.570117 + 0.821563i \(0.693102\pi\)
\(132\) 10.3076 0.897158
\(133\) 0 0
\(134\) 12.3555 1.06735
\(135\) −0.719163 −0.0618957
\(136\) −16.4566 −1.41114
\(137\) 12.7699 1.09100 0.545502 0.838109i \(-0.316339\pi\)
0.545502 + 0.838109i \(0.316339\pi\)
\(138\) 2.95847 0.251842
\(139\) −13.9759 −1.18542 −0.592712 0.805415i \(-0.701943\pi\)
−0.592712 + 0.805415i \(0.701943\pi\)
\(140\) 0 0
\(141\) 10.6790 0.899334
\(142\) −16.2540 −1.36400
\(143\) −22.1718 −1.85410
\(144\) −2.30021 −0.191684
\(145\) 1.67798 0.139349
\(146\) −8.45030 −0.699352
\(147\) 0 0
\(148\) 3.01683 0.247982
\(149\) −2.02136 −0.165597 −0.0827983 0.996566i \(-0.526386\pi\)
−0.0827983 + 0.996566i \(0.526386\pi\)
\(150\) 11.2474 0.918344
\(151\) 11.7648 0.957406 0.478703 0.877977i \(-0.341107\pi\)
0.478703 + 0.877977i \(0.341107\pi\)
\(152\) −10.8544 −0.880411
\(153\) 7.28733 0.589146
\(154\) 0 0
\(155\) −0.0805560 −0.00647041
\(156\) −6.26821 −0.501858
\(157\) −18.8859 −1.50726 −0.753631 0.657297i \(-0.771700\pi\)
−0.753631 + 0.657297i \(0.771700\pi\)
\(158\) −6.57475 −0.523059
\(159\) −3.19805 −0.253622
\(160\) −0.898238 −0.0710119
\(161\) 0 0
\(162\) −12.2095 −0.959266
\(163\) 16.3263 1.27877 0.639387 0.768885i \(-0.279188\pi\)
0.639387 + 0.768885i \(0.279188\pi\)
\(164\) −0.817843 −0.0638628
\(165\) 2.64251 0.205719
\(166\) −15.4252 −1.19723
\(167\) 22.5846 1.74765 0.873824 0.486242i \(-0.161633\pi\)
0.873824 + 0.486242i \(0.161633\pi\)
\(168\) 0 0
\(169\) 0.483092 0.0371609
\(170\) −1.22449 −0.0939139
\(171\) 4.80658 0.367568
\(172\) −0.674334 −0.0514175
\(173\) 22.2230 1.68958 0.844790 0.535097i \(-0.179725\pi\)
0.844790 + 0.535097i \(0.179725\pi\)
\(174\) 18.1624 1.37689
\(175\) 0 0
\(176\) −10.2374 −0.771675
\(177\) 21.4299 1.61077
\(178\) −0.566365 −0.0424508
\(179\) −10.0095 −0.748142 −0.374071 0.927400i \(-0.622038\pi\)
−0.374071 + 0.927400i \(0.622038\pi\)
\(180\) 0.232640 0.0173400
\(181\) −10.8101 −0.803511 −0.401755 0.915747i \(-0.631600\pi\)
−0.401755 + 0.915747i \(0.631600\pi\)
\(182\) 0 0
\(183\) 19.5383 1.44431
\(184\) 3.99398 0.294440
\(185\) 0.773414 0.0568625
\(186\) −0.871932 −0.0639332
\(187\) 32.4334 2.37177
\(188\) 4.18429 0.305171
\(189\) 0 0
\(190\) −0.807647 −0.0585929
\(191\) −19.7693 −1.43046 −0.715229 0.698890i \(-0.753678\pi\)
−0.715229 + 0.698890i \(0.753678\pi\)
\(192\) −16.8002 −1.21245
\(193\) 18.0955 1.30254 0.651270 0.758846i \(-0.274237\pi\)
0.651270 + 0.758846i \(0.274237\pi\)
\(194\) 3.97148 0.285135
\(195\) −1.60696 −0.115077
\(196\) 0 0
\(197\) 4.47538 0.318858 0.159429 0.987209i \(-0.449035\pi\)
0.159429 + 0.987209i \(0.449035\pi\)
\(198\) 8.90692 0.632987
\(199\) −10.1956 −0.722744 −0.361372 0.932422i \(-0.617692\pi\)
−0.361372 + 0.932422i \(0.617692\pi\)
\(200\) 15.1841 1.07368
\(201\) −23.7192 −1.67303
\(202\) 2.66887 0.187781
\(203\) 0 0
\(204\) 9.16926 0.641977
\(205\) −0.209668 −0.0146438
\(206\) 11.1422 0.776313
\(207\) −1.76862 −0.122928
\(208\) 6.22556 0.431665
\(209\) 21.3924 1.47974
\(210\) 0 0
\(211\) −14.0477 −0.967081 −0.483540 0.875322i \(-0.660649\pi\)
−0.483540 + 0.875322i \(0.660649\pi\)
\(212\) −1.25308 −0.0860616
\(213\) 31.2033 2.13802
\(214\) −5.79861 −0.396385
\(215\) −0.172877 −0.0117901
\(216\) −10.5087 −0.715029
\(217\) 0 0
\(218\) 9.59666 0.649968
\(219\) 16.2223 1.09620
\(220\) 1.03540 0.0698066
\(221\) −19.7233 −1.32674
\(222\) 8.37138 0.561850
\(223\) 0.701496 0.0469756 0.0234878 0.999724i \(-0.492523\pi\)
0.0234878 + 0.999724i \(0.492523\pi\)
\(224\) 0 0
\(225\) −6.72385 −0.448256
\(226\) −19.7932 −1.31662
\(227\) −9.36869 −0.621822 −0.310911 0.950439i \(-0.600634\pi\)
−0.310911 + 0.950439i \(0.600634\pi\)
\(228\) 6.04786 0.400529
\(229\) −12.4012 −0.819496 −0.409748 0.912199i \(-0.634383\pi\)
−0.409748 + 0.912199i \(0.634383\pi\)
\(230\) 0.297180 0.0195955
\(231\) 0 0
\(232\) 24.5195 1.60978
\(233\) 9.61739 0.630056 0.315028 0.949082i \(-0.397986\pi\)
0.315028 + 0.949082i \(0.397986\pi\)
\(234\) −5.41646 −0.354085
\(235\) 1.07271 0.0699760
\(236\) 8.39674 0.546581
\(237\) 12.6218 0.819873
\(238\) 0 0
\(239\) −16.5603 −1.07120 −0.535600 0.844472i \(-0.679914\pi\)
−0.535600 + 0.844472i \(0.679914\pi\)
\(240\) −0.741983 −0.0478948
\(241\) −9.29684 −0.598862 −0.299431 0.954118i \(-0.596797\pi\)
−0.299431 + 0.954118i \(0.596797\pi\)
\(242\) 27.6816 1.77944
\(243\) 13.1489 0.843501
\(244\) 7.65558 0.490098
\(245\) 0 0
\(246\) −2.26943 −0.144693
\(247\) −13.0091 −0.827750
\(248\) −1.17712 −0.0747472
\(249\) 29.6123 1.87660
\(250\) 2.26963 0.143544
\(251\) −2.89503 −0.182733 −0.0913664 0.995817i \(-0.529123\pi\)
−0.0913664 + 0.995817i \(0.529123\pi\)
\(252\) 0 0
\(253\) −7.87152 −0.494878
\(254\) −14.3249 −0.898824
\(255\) 2.35069 0.147206
\(256\) −15.8987 −0.993668
\(257\) −5.41470 −0.337760 −0.168880 0.985637i \(-0.554015\pi\)
−0.168880 + 0.985637i \(0.554015\pi\)
\(258\) −1.87120 −0.116496
\(259\) 0 0
\(260\) −0.629645 −0.0390489
\(261\) −10.8577 −0.672077
\(262\) −14.1895 −0.876631
\(263\) −26.7339 −1.64848 −0.824242 0.566238i \(-0.808398\pi\)
−0.824242 + 0.566238i \(0.808398\pi\)
\(264\) 38.6136 2.37650
\(265\) −0.321246 −0.0197340
\(266\) 0 0
\(267\) 1.08727 0.0665399
\(268\) −9.29377 −0.567708
\(269\) 14.6488 0.893151 0.446576 0.894746i \(-0.352643\pi\)
0.446576 + 0.894746i \(0.352643\pi\)
\(270\) −0.781925 −0.0475864
\(271\) −22.8241 −1.38646 −0.693232 0.720714i \(-0.743814\pi\)
−0.693232 + 0.720714i \(0.743814\pi\)
\(272\) −9.10688 −0.552186
\(273\) 0 0
\(274\) 13.8843 0.838782
\(275\) −29.9255 −1.80458
\(276\) −2.22536 −0.133951
\(277\) −8.31190 −0.499414 −0.249707 0.968321i \(-0.580334\pi\)
−0.249707 + 0.968321i \(0.580334\pi\)
\(278\) −15.1956 −0.911373
\(279\) 0.521254 0.0312067
\(280\) 0 0
\(281\) 17.3137 1.03285 0.516423 0.856333i \(-0.327263\pi\)
0.516423 + 0.856333i \(0.327263\pi\)
\(282\) 11.6110 0.691422
\(283\) 14.7818 0.878686 0.439343 0.898319i \(-0.355211\pi\)
0.439343 + 0.898319i \(0.355211\pi\)
\(284\) 12.2262 0.725493
\(285\) 1.55047 0.0918418
\(286\) −24.1068 −1.42546
\(287\) 0 0
\(288\) 5.81224 0.342489
\(289\) 11.8517 0.697158
\(290\) 1.82442 0.107134
\(291\) −7.62418 −0.446937
\(292\) 6.35631 0.371975
\(293\) −4.99638 −0.291892 −0.145946 0.989293i \(-0.546623\pi\)
−0.145946 + 0.989293i \(0.546623\pi\)
\(294\) 0 0
\(295\) 2.15264 0.125332
\(296\) 11.3015 0.656885
\(297\) 20.7111 1.20178
\(298\) −2.19777 −0.127313
\(299\) 4.78681 0.276828
\(300\) −8.46026 −0.488453
\(301\) 0 0
\(302\) 12.7915 0.736069
\(303\) −5.12352 −0.294338
\(304\) −6.00671 −0.344509
\(305\) 1.96263 0.112380
\(306\) 7.92330 0.452945
\(307\) 7.85295 0.448192 0.224096 0.974567i \(-0.428057\pi\)
0.224096 + 0.974567i \(0.428057\pi\)
\(308\) 0 0
\(309\) −21.3901 −1.21684
\(310\) −0.0875861 −0.00497456
\(311\) 27.0379 1.53318 0.766589 0.642138i \(-0.221952\pi\)
0.766589 + 0.642138i \(0.221952\pi\)
\(312\) −23.4816 −1.32938
\(313\) 0.168393 0.00951815 0.00475907 0.999989i \(-0.498485\pi\)
0.00475907 + 0.999989i \(0.498485\pi\)
\(314\) −20.5341 −1.15881
\(315\) 0 0
\(316\) 4.94552 0.278207
\(317\) 5.33690 0.299750 0.149875 0.988705i \(-0.452113\pi\)
0.149875 + 0.988705i \(0.452113\pi\)
\(318\) −3.47715 −0.194989
\(319\) −48.3240 −2.70563
\(320\) −1.68759 −0.0943390
\(321\) 11.1318 0.621316
\(322\) 0 0
\(323\) 19.0300 1.05886
\(324\) 9.18394 0.510219
\(325\) 18.1982 1.00946
\(326\) 17.7511 0.983142
\(327\) −18.4230 −1.01880
\(328\) −3.06376 −0.169168
\(329\) 0 0
\(330\) 2.87312 0.158160
\(331\) −8.09405 −0.444889 −0.222445 0.974945i \(-0.571404\pi\)
−0.222445 + 0.974945i \(0.571404\pi\)
\(332\) 11.6028 0.636786
\(333\) −5.00453 −0.274247
\(334\) 24.5556 1.34362
\(335\) −2.38261 −0.130176
\(336\) 0 0
\(337\) 20.1540 1.09786 0.548929 0.835869i \(-0.315036\pi\)
0.548929 + 0.835869i \(0.315036\pi\)
\(338\) 0.525252 0.0285699
\(339\) 37.9977 2.06375
\(340\) 0.921058 0.0499514
\(341\) 2.31992 0.125631
\(342\) 5.22605 0.282592
\(343\) 0 0
\(344\) −2.52615 −0.136201
\(345\) −0.570508 −0.0307151
\(346\) 24.1624 1.29898
\(347\) 18.2807 0.981359 0.490679 0.871340i \(-0.336749\pi\)
0.490679 + 0.871340i \(0.336749\pi\)
\(348\) −13.6617 −0.732345
\(349\) −9.74567 −0.521674 −0.260837 0.965383i \(-0.583999\pi\)
−0.260837 + 0.965383i \(0.583999\pi\)
\(350\) 0 0
\(351\) −12.5948 −0.672260
\(352\) 25.8683 1.37878
\(353\) −26.2143 −1.39525 −0.697624 0.716464i \(-0.745759\pi\)
−0.697624 + 0.716464i \(0.745759\pi\)
\(354\) 23.3000 1.23838
\(355\) 3.13439 0.166356
\(356\) 0.426019 0.0225790
\(357\) 0 0
\(358\) −10.8830 −0.575184
\(359\) 0.473108 0.0249697 0.0124849 0.999922i \(-0.496026\pi\)
0.0124849 + 0.999922i \(0.496026\pi\)
\(360\) 0.871502 0.0459322
\(361\) −6.44820 −0.339379
\(362\) −11.7535 −0.617752
\(363\) −53.1414 −2.78920
\(364\) 0 0
\(365\) 1.62954 0.0852942
\(366\) 21.2434 1.11041
\(367\) −10.2670 −0.535932 −0.267966 0.963428i \(-0.586352\pi\)
−0.267966 + 0.963428i \(0.586352\pi\)
\(368\) 2.21022 0.115216
\(369\) 1.35670 0.0706268
\(370\) 0.840910 0.0437168
\(371\) 0 0
\(372\) 0.655866 0.0340051
\(373\) 20.1653 1.04412 0.522061 0.852908i \(-0.325163\pi\)
0.522061 + 0.852908i \(0.325163\pi\)
\(374\) 35.2639 1.82345
\(375\) −4.35709 −0.224999
\(376\) 15.6749 0.808374
\(377\) 29.3867 1.51349
\(378\) 0 0
\(379\) −26.8293 −1.37813 −0.689063 0.724701i \(-0.741978\pi\)
−0.689063 + 0.724701i \(0.741978\pi\)
\(380\) 0.607511 0.0311647
\(381\) 27.5000 1.40887
\(382\) −21.4946 −1.09976
\(383\) 24.1679 1.23492 0.617461 0.786602i \(-0.288161\pi\)
0.617461 + 0.786602i \(0.288161\pi\)
\(384\) −0.382156 −0.0195018
\(385\) 0 0
\(386\) 19.6746 1.00141
\(387\) 1.11863 0.0568634
\(388\) −2.98734 −0.151659
\(389\) 15.9595 0.809180 0.404590 0.914498i \(-0.367414\pi\)
0.404590 + 0.914498i \(0.367414\pi\)
\(390\) −1.74720 −0.0884728
\(391\) −7.00224 −0.354119
\(392\) 0 0
\(393\) 27.2401 1.37408
\(394\) 4.86595 0.245143
\(395\) 1.26786 0.0637932
\(396\) −6.69977 −0.336676
\(397\) 8.73399 0.438346 0.219173 0.975686i \(-0.429664\pi\)
0.219173 + 0.975686i \(0.429664\pi\)
\(398\) −11.0853 −0.555657
\(399\) 0 0
\(400\) 8.40270 0.420135
\(401\) 1.77238 0.0885086 0.0442543 0.999020i \(-0.485909\pi\)
0.0442543 + 0.999020i \(0.485909\pi\)
\(402\) −25.7892 −1.28625
\(403\) −1.41079 −0.0702763
\(404\) −2.00752 −0.0998778
\(405\) 2.35445 0.116994
\(406\) 0 0
\(407\) −22.2735 −1.10405
\(408\) 34.3494 1.70055
\(409\) 27.0895 1.33949 0.669746 0.742590i \(-0.266403\pi\)
0.669746 + 0.742590i \(0.266403\pi\)
\(410\) −0.227965 −0.0112584
\(411\) −26.6542 −1.31475
\(412\) −8.38114 −0.412909
\(413\) 0 0
\(414\) −1.92297 −0.0945087
\(415\) 2.97457 0.146016
\(416\) −15.7310 −0.771273
\(417\) 29.1716 1.42854
\(418\) 23.2593 1.13765
\(419\) 18.3386 0.895898 0.447949 0.894059i \(-0.352155\pi\)
0.447949 + 0.894059i \(0.352155\pi\)
\(420\) 0 0
\(421\) 16.7202 0.814894 0.407447 0.913229i \(-0.366419\pi\)
0.407447 + 0.913229i \(0.366419\pi\)
\(422\) −15.2736 −0.743507
\(423\) −6.94120 −0.337493
\(424\) −4.69420 −0.227970
\(425\) −26.6208 −1.29130
\(426\) 33.9265 1.64374
\(427\) 0 0
\(428\) 4.36171 0.210831
\(429\) 46.2786 2.23435
\(430\) −0.187964 −0.00906441
\(431\) 0.204738 0.00986186 0.00493093 0.999988i \(-0.498430\pi\)
0.00493093 + 0.999988i \(0.498430\pi\)
\(432\) −5.81541 −0.279794
\(433\) −31.9997 −1.53781 −0.768903 0.639366i \(-0.779197\pi\)
−0.768903 + 0.639366i \(0.779197\pi\)
\(434\) 0 0
\(435\) −3.50240 −0.167927
\(436\) −7.21859 −0.345708
\(437\) −4.61854 −0.220935
\(438\) 17.6381 0.842779
\(439\) −24.9052 −1.18866 −0.594330 0.804221i \(-0.702583\pi\)
−0.594330 + 0.804221i \(0.702583\pi\)
\(440\) 3.87876 0.184912
\(441\) 0 0
\(442\) −21.4446 −1.02002
\(443\) 19.0300 0.904142 0.452071 0.891982i \(-0.350685\pi\)
0.452071 + 0.891982i \(0.350685\pi\)
\(444\) −6.29694 −0.298839
\(445\) 0.109217 0.00517738
\(446\) 0.762716 0.0361156
\(447\) 4.21913 0.199558
\(448\) 0 0
\(449\) −5.95010 −0.280802 −0.140401 0.990095i \(-0.544839\pi\)
−0.140401 + 0.990095i \(0.544839\pi\)
\(450\) −7.31064 −0.344627
\(451\) 6.03819 0.284327
\(452\) 14.8884 0.700292
\(453\) −24.5563 −1.15376
\(454\) −10.1863 −0.478067
\(455\) 0 0
\(456\) 22.6561 1.06097
\(457\) −31.8410 −1.48946 −0.744729 0.667367i \(-0.767421\pi\)
−0.744729 + 0.667367i \(0.767421\pi\)
\(458\) −13.4835 −0.630042
\(459\) 18.4239 0.859955
\(460\) −0.223539 −0.0104225
\(461\) −4.99638 −0.232705 −0.116352 0.993208i \(-0.537120\pi\)
−0.116352 + 0.993208i \(0.537120\pi\)
\(462\) 0 0
\(463\) 17.6038 0.818119 0.409060 0.912508i \(-0.365857\pi\)
0.409060 + 0.912508i \(0.365857\pi\)
\(464\) 13.5688 0.629914
\(465\) 0.168142 0.00779740
\(466\) 10.4567 0.484397
\(467\) 42.9444 1.98723 0.993614 0.112833i \(-0.0359925\pi\)
0.993614 + 0.112833i \(0.0359925\pi\)
\(468\) 4.07425 0.188332
\(469\) 0 0
\(470\) 1.16633 0.0537987
\(471\) 39.4201 1.81638
\(472\) 31.4554 1.44785
\(473\) 4.97865 0.228919
\(474\) 13.7233 0.630331
\(475\) −17.5585 −0.805640
\(476\) 0 0
\(477\) 2.07869 0.0951767
\(478\) −18.0056 −0.823556
\(479\) −2.85249 −0.130334 −0.0651669 0.997874i \(-0.520758\pi\)
−0.0651669 + 0.997874i \(0.520758\pi\)
\(480\) 1.87487 0.0855755
\(481\) 13.5449 0.617594
\(482\) −10.1082 −0.460415
\(483\) 0 0
\(484\) −20.8221 −0.946459
\(485\) −0.765853 −0.0347756
\(486\) 14.2964 0.648497
\(487\) −6.23225 −0.282410 −0.141205 0.989980i \(-0.545098\pi\)
−0.141205 + 0.989980i \(0.545098\pi\)
\(488\) 28.6789 1.29823
\(489\) −34.0774 −1.54103
\(490\) 0 0
\(491\) −2.13846 −0.0965071 −0.0482536 0.998835i \(-0.515366\pi\)
−0.0482536 + 0.998835i \(0.515366\pi\)
\(492\) 1.70706 0.0769602
\(493\) −42.9875 −1.93606
\(494\) −14.1444 −0.636388
\(495\) −1.71760 −0.0772002
\(496\) −0.651404 −0.0292489
\(497\) 0 0
\(498\) 32.1965 1.44276
\(499\) 25.1110 1.12412 0.562060 0.827096i \(-0.310009\pi\)
0.562060 + 0.827096i \(0.310009\pi\)
\(500\) −1.70721 −0.0763489
\(501\) −47.1402 −2.10607
\(502\) −3.14768 −0.140488
\(503\) −37.1641 −1.65707 −0.828534 0.559939i \(-0.810824\pi\)
−0.828534 + 0.559939i \(0.810824\pi\)
\(504\) 0 0
\(505\) −0.514660 −0.0229021
\(506\) −8.55847 −0.380470
\(507\) −1.00834 −0.0447821
\(508\) 10.7752 0.478071
\(509\) 2.36319 0.104747 0.0523734 0.998628i \(-0.483321\pi\)
0.0523734 + 0.998628i \(0.483321\pi\)
\(510\) 2.55584 0.113174
\(511\) 0 0
\(512\) −17.6523 −0.780131
\(513\) 12.1520 0.536526
\(514\) −5.88725 −0.259675
\(515\) −2.14864 −0.0946805
\(516\) 1.40752 0.0619625
\(517\) −30.8929 −1.35867
\(518\) 0 0
\(519\) −46.3853 −2.03609
\(520\) −2.35874 −0.103438
\(521\) 2.27368 0.0996116 0.0498058 0.998759i \(-0.484140\pi\)
0.0498058 + 0.998759i \(0.484140\pi\)
\(522\) −11.8053 −0.516704
\(523\) 35.7432 1.56294 0.781471 0.623941i \(-0.214469\pi\)
0.781471 + 0.623941i \(0.214469\pi\)
\(524\) 10.6733 0.466266
\(525\) 0 0
\(526\) −29.0670 −1.26738
\(527\) 2.06373 0.0898974
\(528\) 21.3683 0.929935
\(529\) −21.3006 −0.926112
\(530\) −0.349282 −0.0151718
\(531\) −13.9291 −0.604472
\(532\) 0 0
\(533\) −3.67193 −0.159049
\(534\) 1.18216 0.0511569
\(535\) 1.11820 0.0483438
\(536\) −34.8158 −1.50381
\(537\) 20.8925 0.901576
\(538\) 15.9272 0.686669
\(539\) 0 0
\(540\) 0.588163 0.0253105
\(541\) −22.5224 −0.968314 −0.484157 0.874981i \(-0.660874\pi\)
−0.484157 + 0.874981i \(0.660874\pi\)
\(542\) −24.8159 −1.06594
\(543\) 22.5637 0.968299
\(544\) 23.0116 0.986612
\(545\) −1.85060 −0.0792712
\(546\) 0 0
\(547\) 3.63915 0.155599 0.0777994 0.996969i \(-0.475211\pi\)
0.0777994 + 0.996969i \(0.475211\pi\)
\(548\) −10.4438 −0.446135
\(549\) −12.6996 −0.542007
\(550\) −32.5371 −1.38739
\(551\) −28.3537 −1.20791
\(552\) −8.33652 −0.354826
\(553\) 0 0
\(554\) −9.03729 −0.383957
\(555\) −1.61432 −0.0685242
\(556\) 11.4301 0.484745
\(557\) 4.80008 0.203386 0.101693 0.994816i \(-0.467574\pi\)
0.101693 + 0.994816i \(0.467574\pi\)
\(558\) 0.566744 0.0239922
\(559\) −3.02761 −0.128054
\(560\) 0 0
\(561\) −67.6973 −2.85818
\(562\) 18.8246 0.794069
\(563\) −31.7115 −1.33648 −0.668241 0.743945i \(-0.732952\pi\)
−0.668241 + 0.743945i \(0.732952\pi\)
\(564\) −8.73374 −0.367757
\(565\) 3.81689 0.160578
\(566\) 16.0718 0.675548
\(567\) 0 0
\(568\) 45.8012 1.92178
\(569\) −35.4970 −1.48811 −0.744055 0.668118i \(-0.767100\pi\)
−0.744055 + 0.668118i \(0.767100\pi\)
\(570\) 1.68578 0.0706095
\(571\) −38.7912 −1.62336 −0.811681 0.584100i \(-0.801447\pi\)
−0.811681 + 0.584100i \(0.801447\pi\)
\(572\) 18.1331 0.758182
\(573\) 41.2639 1.72383
\(574\) 0 0
\(575\) 6.46080 0.269434
\(576\) 10.9199 0.454995
\(577\) 0.335828 0.0139807 0.00699035 0.999976i \(-0.497775\pi\)
0.00699035 + 0.999976i \(0.497775\pi\)
\(578\) 12.8860 0.535987
\(579\) −37.7701 −1.56967
\(580\) −1.37233 −0.0569828
\(581\) 0 0
\(582\) −8.28954 −0.343613
\(583\) 9.25154 0.383160
\(584\) 23.8116 0.985332
\(585\) 1.04450 0.0431848
\(586\) −5.43242 −0.224411
\(587\) −8.19386 −0.338197 −0.169098 0.985599i \(-0.554086\pi\)
−0.169098 + 0.985599i \(0.554086\pi\)
\(588\) 0 0
\(589\) 1.36119 0.0560870
\(590\) 2.34050 0.0963570
\(591\) −9.34132 −0.384251
\(592\) 6.25410 0.257042
\(593\) −30.1222 −1.23697 −0.618486 0.785796i \(-0.712254\pi\)
−0.618486 + 0.785796i \(0.712254\pi\)
\(594\) 22.5186 0.923948
\(595\) 0 0
\(596\) 1.65316 0.0677160
\(597\) 21.2809 0.870968
\(598\) 5.20456 0.212830
\(599\) −39.6072 −1.61831 −0.809153 0.587599i \(-0.800073\pi\)
−0.809153 + 0.587599i \(0.800073\pi\)
\(600\) −31.6933 −1.29387
\(601\) 35.0329 1.42902 0.714511 0.699624i \(-0.246649\pi\)
0.714511 + 0.699624i \(0.246649\pi\)
\(602\) 0 0
\(603\) 15.4172 0.627836
\(604\) −9.62176 −0.391504
\(605\) −5.33809 −0.217024
\(606\) −5.57065 −0.226292
\(607\) −13.4657 −0.546555 −0.273278 0.961935i \(-0.588108\pi\)
−0.273278 + 0.961935i \(0.588108\pi\)
\(608\) 15.1780 0.615547
\(609\) 0 0
\(610\) 2.13391 0.0863996
\(611\) 18.7865 0.760022
\(612\) −5.95990 −0.240915
\(613\) −6.49820 −0.262460 −0.131230 0.991352i \(-0.541893\pi\)
−0.131230 + 0.991352i \(0.541893\pi\)
\(614\) 8.53828 0.344577
\(615\) 0.437633 0.0176471
\(616\) 0 0
\(617\) 6.63045 0.266932 0.133466 0.991053i \(-0.457389\pi\)
0.133466 + 0.991053i \(0.457389\pi\)
\(618\) −23.2568 −0.935525
\(619\) 4.50756 0.181174 0.0905871 0.995889i \(-0.471126\pi\)
0.0905871 + 0.995889i \(0.471126\pi\)
\(620\) 0.0658822 0.00264589
\(621\) −4.47144 −0.179433
\(622\) 29.3975 1.17873
\(623\) 0 0
\(624\) −12.9944 −0.520194
\(625\) 24.3425 0.973701
\(626\) 0.183089 0.00731771
\(627\) −44.6518 −1.78322
\(628\) 15.4457 0.616352
\(629\) −19.8137 −0.790025
\(630\) 0 0
\(631\) −24.2677 −0.966081 −0.483040 0.875598i \(-0.660468\pi\)
−0.483040 + 0.875598i \(0.660468\pi\)
\(632\) 18.5266 0.736949
\(633\) 29.3213 1.16542
\(634\) 5.80265 0.230453
\(635\) 2.76239 0.109622
\(636\) 2.61551 0.103712
\(637\) 0 0
\(638\) −52.5413 −2.08013
\(639\) −20.2818 −0.802334
\(640\) −0.0383878 −0.00151741
\(641\) 22.2964 0.880653 0.440327 0.897838i \(-0.354863\pi\)
0.440327 + 0.897838i \(0.354863\pi\)
\(642\) 12.1033 0.477678
\(643\) 16.2405 0.640464 0.320232 0.947339i \(-0.396239\pi\)
0.320232 + 0.947339i \(0.396239\pi\)
\(644\) 0 0
\(645\) 0.360840 0.0142081
\(646\) 20.6907 0.814067
\(647\) −13.5814 −0.533939 −0.266969 0.963705i \(-0.586022\pi\)
−0.266969 + 0.963705i \(0.586022\pi\)
\(648\) 34.4044 1.35153
\(649\) −61.9937 −2.43347
\(650\) 19.7864 0.776087
\(651\) 0 0
\(652\) −13.3524 −0.522918
\(653\) −47.1222 −1.84403 −0.922016 0.387151i \(-0.873459\pi\)
−0.922016 + 0.387151i \(0.873459\pi\)
\(654\) −20.0308 −0.783267
\(655\) 2.73628 0.106915
\(656\) −1.69545 −0.0661960
\(657\) −10.5443 −0.411372
\(658\) 0 0
\(659\) 5.27483 0.205478 0.102739 0.994708i \(-0.467239\pi\)
0.102739 + 0.994708i \(0.467239\pi\)
\(660\) −2.16116 −0.0841230
\(661\) 47.8044 1.85938 0.929689 0.368346i \(-0.120076\pi\)
0.929689 + 0.368346i \(0.120076\pi\)
\(662\) −8.80042 −0.342038
\(663\) 41.1679 1.59883
\(664\) 43.4657 1.68680
\(665\) 0 0
\(666\) −5.44128 −0.210845
\(667\) 10.4330 0.403966
\(668\) −18.4707 −0.714651
\(669\) −1.46421 −0.0566097
\(670\) −2.59054 −0.100081
\(671\) −56.5217 −2.18200
\(672\) 0 0
\(673\) −5.11960 −0.197346 −0.0986730 0.995120i \(-0.531460\pi\)
−0.0986730 + 0.995120i \(0.531460\pi\)
\(674\) 21.9128 0.844051
\(675\) −16.9993 −0.654303
\(676\) −0.395094 −0.0151959
\(677\) −30.7985 −1.18368 −0.591842 0.806054i \(-0.701599\pi\)
−0.591842 + 0.806054i \(0.701599\pi\)
\(678\) 41.3137 1.58664
\(679\) 0 0
\(680\) 3.45041 0.132317
\(681\) 19.5550 0.749349
\(682\) 2.52238 0.0965870
\(683\) −18.8534 −0.721407 −0.360703 0.932681i \(-0.617463\pi\)
−0.360703 + 0.932681i \(0.617463\pi\)
\(684\) −3.93103 −0.150307
\(685\) −2.67743 −0.102299
\(686\) 0 0
\(687\) 25.8847 0.987563
\(688\) −1.39794 −0.0532960
\(689\) −5.62603 −0.214335
\(690\) −0.620296 −0.0236143
\(691\) 6.22638 0.236863 0.118431 0.992962i \(-0.462213\pi\)
0.118431 + 0.992962i \(0.462213\pi\)
\(692\) −18.1749 −0.690906
\(693\) 0 0
\(694\) 19.8761 0.754485
\(695\) 2.93030 0.111153
\(696\) −51.1787 −1.93992
\(697\) 5.37138 0.203455
\(698\) −10.5962 −0.401071
\(699\) −20.0741 −0.759272
\(700\) 0 0
\(701\) 40.6743 1.53625 0.768124 0.640301i \(-0.221190\pi\)
0.768124 + 0.640301i \(0.221190\pi\)
\(702\) −13.6939 −0.516845
\(703\) −13.0687 −0.492897
\(704\) 48.6006 1.83171
\(705\) −2.23904 −0.0843271
\(706\) −28.5021 −1.07269
\(707\) 0 0
\(708\) −17.5263 −0.658677
\(709\) −47.5660 −1.78638 −0.893189 0.449681i \(-0.851538\pi\)
−0.893189 + 0.449681i \(0.851538\pi\)
\(710\) 3.40793 0.127897
\(711\) −8.20398 −0.307673
\(712\) 1.59593 0.0598099
\(713\) −0.500862 −0.0187574
\(714\) 0 0
\(715\) 4.64871 0.173852
\(716\) 8.18617 0.305932
\(717\) 34.5659 1.29089
\(718\) 0.514397 0.0191971
\(719\) 0.700457 0.0261226 0.0130613 0.999915i \(-0.495842\pi\)
0.0130613 + 0.999915i \(0.495842\pi\)
\(720\) 0.482279 0.0179735
\(721\) 0 0
\(722\) −7.01094 −0.260920
\(723\) 19.4050 0.721680
\(724\) 8.84099 0.328573
\(725\) 39.6635 1.47307
\(726\) −57.7791 −2.14438
\(727\) −38.6447 −1.43325 −0.716625 0.697458i \(-0.754314\pi\)
−0.716625 + 0.697458i \(0.754314\pi\)
\(728\) 0 0
\(729\) 6.24313 0.231227
\(730\) 1.77175 0.0655756
\(731\) 4.42885 0.163807
\(732\) −15.9793 −0.590611
\(733\) 31.3572 1.15820 0.579101 0.815255i \(-0.303404\pi\)
0.579101 + 0.815255i \(0.303404\pi\)
\(734\) −11.1630 −0.412033
\(735\) 0 0
\(736\) −5.58485 −0.205860
\(737\) 68.6166 2.52752
\(738\) 1.47510 0.0542991
\(739\) 51.5448 1.89611 0.948054 0.318110i \(-0.103048\pi\)
0.948054 + 0.318110i \(0.103048\pi\)
\(740\) −0.632531 −0.0232523
\(741\) 27.1535 0.997510
\(742\) 0 0
\(743\) −15.7347 −0.577251 −0.288626 0.957442i \(-0.593198\pi\)
−0.288626 + 0.957442i \(0.593198\pi\)
\(744\) 2.45697 0.0900769
\(745\) 0.423814 0.0155274
\(746\) 21.9252 0.802737
\(747\) −19.2476 −0.704231
\(748\) −26.5254 −0.969866
\(749\) 0 0
\(750\) −4.73733 −0.172983
\(751\) −12.1375 −0.442904 −0.221452 0.975171i \(-0.571080\pi\)
−0.221452 + 0.975171i \(0.571080\pi\)
\(752\) 8.67433 0.316320
\(753\) 6.04271 0.220209
\(754\) 31.9513 1.16360
\(755\) −2.46670 −0.0897723
\(756\) 0 0
\(757\) −17.9846 −0.653660 −0.326830 0.945083i \(-0.605981\pi\)
−0.326830 + 0.945083i \(0.605981\pi\)
\(758\) −29.1707 −1.05953
\(759\) 16.4300 0.596371
\(760\) 2.27582 0.0825528
\(761\) −23.2298 −0.842079 −0.421039 0.907042i \(-0.638335\pi\)
−0.421039 + 0.907042i \(0.638335\pi\)
\(762\) 29.8999 1.08316
\(763\) 0 0
\(764\) 16.1682 0.584945
\(765\) −1.52792 −0.0552420
\(766\) 26.2770 0.949428
\(767\) 37.6995 1.36125
\(768\) 33.1848 1.19745
\(769\) 34.9703 1.26106 0.630530 0.776165i \(-0.282838\pi\)
0.630530 + 0.776165i \(0.282838\pi\)
\(770\) 0 0
\(771\) 11.3019 0.407030
\(772\) −14.7992 −0.532636
\(773\) 26.7805 0.963228 0.481614 0.876384i \(-0.340051\pi\)
0.481614 + 0.876384i \(0.340051\pi\)
\(774\) 1.21626 0.0437175
\(775\) −1.90415 −0.0683991
\(776\) −11.1910 −0.401733
\(777\) 0 0
\(778\) 17.3523 0.622111
\(779\) 3.54285 0.126936
\(780\) 1.31424 0.0470573
\(781\) −90.2671 −3.23001
\(782\) −7.61333 −0.272252
\(783\) −27.4506 −0.981006
\(784\) 0 0
\(785\) 3.95977 0.141330
\(786\) 29.6173 1.05642
\(787\) −0.177103 −0.00631303 −0.00315652 0.999995i \(-0.501005\pi\)
−0.00315652 + 0.999995i \(0.501005\pi\)
\(788\) −3.66016 −0.130388
\(789\) 55.8009 1.98657
\(790\) 1.37851 0.0490452
\(791\) 0 0
\(792\) −25.0983 −0.891829
\(793\) 34.3718 1.22058
\(794\) 9.49621 0.337008
\(795\) 0.670528 0.0237812
\(796\) 8.33837 0.295546
\(797\) 15.4863 0.548555 0.274277 0.961651i \(-0.411561\pi\)
0.274277 + 0.961651i \(0.411561\pi\)
\(798\) 0 0
\(799\) −27.4813 −0.972219
\(800\) −21.2322 −0.750672
\(801\) −0.706711 −0.0249704
\(802\) 1.92706 0.0680469
\(803\) −46.9290 −1.65609
\(804\) 19.3986 0.684137
\(805\) 0 0
\(806\) −1.53391 −0.0540296
\(807\) −30.5759 −1.07632
\(808\) −7.52045 −0.264569
\(809\) −12.8695 −0.452467 −0.226234 0.974073i \(-0.572641\pi\)
−0.226234 + 0.974073i \(0.572641\pi\)
\(810\) 2.55993 0.0899467
\(811\) −24.8951 −0.874185 −0.437092 0.899417i \(-0.643992\pi\)
−0.437092 + 0.899417i \(0.643992\pi\)
\(812\) 0 0
\(813\) 47.6400 1.67081
\(814\) −24.2173 −0.848815
\(815\) −3.42309 −0.119906
\(816\) 19.0085 0.665431
\(817\) 2.92118 0.102199
\(818\) 29.4537 1.02982
\(819\) 0 0
\(820\) 0.171475 0.00598817
\(821\) −21.8002 −0.760834 −0.380417 0.924815i \(-0.624220\pi\)
−0.380417 + 0.924815i \(0.624220\pi\)
\(822\) −28.9803 −1.01080
\(823\) −1.76071 −0.0613744 −0.0306872 0.999529i \(-0.509770\pi\)
−0.0306872 + 0.999529i \(0.509770\pi\)
\(824\) −31.3970 −1.09376
\(825\) 62.4627 2.17467
\(826\) 0 0
\(827\) 9.93134 0.345347 0.172673 0.984979i \(-0.444760\pi\)
0.172673 + 0.984979i \(0.444760\pi\)
\(828\) 1.44645 0.0502678
\(829\) −4.85768 −0.168714 −0.0843571 0.996436i \(-0.526884\pi\)
−0.0843571 + 0.996436i \(0.526884\pi\)
\(830\) 3.23416 0.112259
\(831\) 17.3492 0.601837
\(832\) −29.5549 −1.02463
\(833\) 0 0
\(834\) 31.7174 1.09828
\(835\) −4.73526 −0.163870
\(836\) −17.4956 −0.605100
\(837\) 1.31784 0.0455512
\(838\) 19.9390 0.688781
\(839\) −15.9065 −0.549152 −0.274576 0.961565i \(-0.588537\pi\)
−0.274576 + 0.961565i \(0.588537\pi\)
\(840\) 0 0
\(841\) 35.0491 1.20859
\(842\) 18.1794 0.626504
\(843\) −36.1383 −1.24467
\(844\) 11.4888 0.395460
\(845\) −0.101289 −0.00348444
\(846\) −7.54696 −0.259470
\(847\) 0 0
\(848\) −2.59771 −0.0892058
\(849\) −30.8536 −1.05889
\(850\) −28.9440 −0.992770
\(851\) 4.80875 0.164842
\(852\) −25.5194 −0.874282
\(853\) −1.08814 −0.0372572 −0.0186286 0.999826i \(-0.505930\pi\)
−0.0186286 + 0.999826i \(0.505930\pi\)
\(854\) 0 0
\(855\) −1.00778 −0.0344655
\(856\) 16.3396 0.558475
\(857\) 1.53174 0.0523232 0.0261616 0.999658i \(-0.491672\pi\)
0.0261616 + 0.999658i \(0.491672\pi\)
\(858\) 50.3174 1.71781
\(859\) −10.9042 −0.372047 −0.186024 0.982545i \(-0.559560\pi\)
−0.186024 + 0.982545i \(0.559560\pi\)
\(860\) 0.141386 0.00482122
\(861\) 0 0
\(862\) 0.222605 0.00758196
\(863\) −54.7487 −1.86367 −0.931834 0.362884i \(-0.881792\pi\)
−0.931834 + 0.362884i \(0.881792\pi\)
\(864\) 14.6946 0.499919
\(865\) −4.65943 −0.158426
\(866\) −34.7923 −1.18229
\(867\) −24.7377 −0.840136
\(868\) 0 0
\(869\) −36.5131 −1.23862
\(870\) −3.80806 −0.129105
\(871\) −41.7270 −1.41386
\(872\) −27.0419 −0.915753
\(873\) 4.95561 0.167722
\(874\) −5.02160 −0.169858
\(875\) 0 0
\(876\) −13.2673 −0.448261
\(877\) −0.0143469 −0.000484459 0 −0.000242229 1.00000i \(-0.500077\pi\)
−0.000242229 1.00000i \(0.500077\pi\)
\(878\) −27.0787 −0.913862
\(879\) 10.4288 0.351755
\(880\) 2.14646 0.0723570
\(881\) 56.3960 1.90003 0.950014 0.312206i \(-0.101068\pi\)
0.950014 + 0.312206i \(0.101068\pi\)
\(882\) 0 0
\(883\) −26.5606 −0.893835 −0.446917 0.894575i \(-0.647478\pi\)
−0.446917 + 0.894575i \(0.647478\pi\)
\(884\) 16.1306 0.542531
\(885\) −4.49314 −0.151035
\(886\) 20.6907 0.695119
\(887\) 29.3990 0.987121 0.493560 0.869712i \(-0.335695\pi\)
0.493560 + 0.869712i \(0.335695\pi\)
\(888\) −23.5892 −0.791603
\(889\) 0 0
\(890\) 0.118748 0.00398045
\(891\) −67.8057 −2.27158
\(892\) −0.573714 −0.0192094
\(893\) −18.1261 −0.606567
\(894\) 4.58734 0.153423
\(895\) 2.09866 0.0701504
\(896\) 0 0
\(897\) −9.99137 −0.333602
\(898\) −6.46936 −0.215885
\(899\) −3.07484 −0.102552
\(900\) 5.49905 0.183302
\(901\) 8.22986 0.274177
\(902\) 6.56515 0.218595
\(903\) 0 0
\(904\) 55.7741 1.85502
\(905\) 2.26653 0.0753421
\(906\) −26.6994 −0.887027
\(907\) −12.9258 −0.429196 −0.214598 0.976702i \(-0.568844\pi\)
−0.214598 + 0.976702i \(0.568844\pi\)
\(908\) 7.66212 0.254276
\(909\) 3.33022 0.110456
\(910\) 0 0
\(911\) 33.8152 1.12035 0.560174 0.828375i \(-0.310734\pi\)
0.560174 + 0.828375i \(0.310734\pi\)
\(912\) 12.5376 0.415163
\(913\) −85.6643 −2.83507
\(914\) −34.6198 −1.14512
\(915\) −4.09655 −0.135428
\(916\) 10.1423 0.335110
\(917\) 0 0
\(918\) 20.0318 0.661147
\(919\) −2.96555 −0.0978246 −0.0489123 0.998803i \(-0.515575\pi\)
−0.0489123 + 0.998803i \(0.515575\pi\)
\(920\) −0.837408 −0.0276085
\(921\) −16.3912 −0.540110
\(922\) −5.43242 −0.178907
\(923\) 54.8930 1.80683
\(924\) 0 0
\(925\) 18.2817 0.601097
\(926\) 19.1401 0.628983
\(927\) 13.9032 0.456642
\(928\) −34.2860 −1.12549
\(929\) 24.4362 0.801726 0.400863 0.916138i \(-0.368710\pi\)
0.400863 + 0.916138i \(0.368710\pi\)
\(930\) 0.182816 0.00599477
\(931\) 0 0
\(932\) −7.86552 −0.257644
\(933\) −56.4354 −1.84761
\(934\) 46.6921 1.52781
\(935\) −6.80023 −0.222391
\(936\) 15.2627 0.498878
\(937\) −35.9240 −1.17359 −0.586793 0.809737i \(-0.699610\pi\)
−0.586793 + 0.809737i \(0.699610\pi\)
\(938\) 0 0
\(939\) −0.351482 −0.0114702
\(940\) −0.877310 −0.0286147
\(941\) 42.9108 1.39885 0.699426 0.714705i \(-0.253439\pi\)
0.699426 + 0.714705i \(0.253439\pi\)
\(942\) 42.8603 1.39646
\(943\) −1.30362 −0.0424518
\(944\) 17.4070 0.566550
\(945\) 0 0
\(946\) 5.41314 0.175996
\(947\) 44.6039 1.44943 0.724716 0.689048i \(-0.241971\pi\)
0.724716 + 0.689048i \(0.241971\pi\)
\(948\) −10.3226 −0.335264
\(949\) 28.5384 0.926395
\(950\) −19.0909 −0.619389
\(951\) −11.1395 −0.361225
\(952\) 0 0
\(953\) −12.8800 −0.417224 −0.208612 0.977999i \(-0.566895\pi\)
−0.208612 + 0.977999i \(0.566895\pi\)
\(954\) 2.26010 0.0731734
\(955\) 4.14499 0.134129
\(956\) 13.5438 0.438037
\(957\) 100.865 3.26051
\(958\) −3.10143 −0.100203
\(959\) 0 0
\(960\) 3.52245 0.113687
\(961\) −30.8524 −0.995238
\(962\) 14.7270 0.474816
\(963\) −7.23552 −0.233161
\(964\) 7.60336 0.244888
\(965\) −3.79403 −0.122134
\(966\) 0 0
\(967\) 49.0909 1.57866 0.789329 0.613970i \(-0.210429\pi\)
0.789329 + 0.613970i \(0.210429\pi\)
\(968\) −78.0025 −2.50710
\(969\) −39.7207 −1.27601
\(970\) −0.832689 −0.0267360
\(971\) 5.24060 0.168179 0.0840895 0.996458i \(-0.473202\pi\)
0.0840895 + 0.996458i \(0.473202\pi\)
\(972\) −10.7537 −0.344926
\(973\) 0 0
\(974\) −6.77614 −0.217121
\(975\) −37.9847 −1.21648
\(976\) 15.8706 0.508004
\(977\) 44.0772 1.41015 0.705077 0.709130i \(-0.250912\pi\)
0.705077 + 0.709130i \(0.250912\pi\)
\(978\) −37.0513 −1.18477
\(979\) −3.14533 −0.100525
\(980\) 0 0
\(981\) 11.9747 0.382323
\(982\) −2.32508 −0.0741963
\(983\) 54.1938 1.72851 0.864256 0.503052i \(-0.167789\pi\)
0.864256 + 0.503052i \(0.167789\pi\)
\(984\) 6.39489 0.203862
\(985\) −0.938341 −0.0298980
\(986\) −46.7390 −1.48847
\(987\) 0 0
\(988\) 10.6394 0.338485
\(989\) −1.07487 −0.0341789
\(990\) −1.86749 −0.0593528
\(991\) −0.908400 −0.0288563 −0.0144281 0.999896i \(-0.504593\pi\)
−0.0144281 + 0.999896i \(0.504593\pi\)
\(992\) 1.64599 0.0522602
\(993\) 16.8945 0.536130
\(994\) 0 0
\(995\) 2.13768 0.0677689
\(996\) −24.2182 −0.767382
\(997\) −36.8530 −1.16715 −0.583573 0.812061i \(-0.698346\pi\)
−0.583573 + 0.812061i \(0.698346\pi\)
\(998\) 27.3024 0.864242
\(999\) −12.6525 −0.400308
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 2009.2.a.n.1.4 5
7.6 odd 2 287.2.a.e.1.4 5
21.20 even 2 2583.2.a.r.1.2 5
28.27 even 2 4592.2.a.bb.1.2 5
35.34 odd 2 7175.2.a.n.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
287.2.a.e.1.4 5 7.6 odd 2
2009.2.a.n.1.4 5 1.1 even 1 trivial
2583.2.a.r.1.2 5 21.20 even 2
4592.2.a.bb.1.2 5 28.27 even 2
7175.2.a.n.1.2 5 35.34 odd 2