Properties

Label 2009.2.a.n.1.5
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 1.5
Root \(-2.03121\) of defining polynomial
Character \(\chi\) \(=\) 2009.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.03121 q^{2} -3.03121 q^{3} +2.12582 q^{4} +3.82713 q^{5} -6.15703 q^{6} +0.255573 q^{8} +6.18825 q^{9} +O(q^{10})\) \(q+2.03121 q^{2} -3.03121 q^{3} +2.12582 q^{4} +3.82713 q^{5} -6.15703 q^{6} +0.255573 q^{8} +6.18825 q^{9} +7.77372 q^{10} -5.96294 q^{11} -6.44382 q^{12} -1.44574 q^{13} -11.6009 q^{15} -3.73252 q^{16} -6.06148 q^{17} +12.5696 q^{18} +0.0743284 q^{19} +8.13581 q^{20} -12.1120 q^{22} -4.43383 q^{23} -0.774695 q^{24} +9.64695 q^{25} -2.93660 q^{26} -9.66425 q^{27} -1.92662 q^{29} -23.5638 q^{30} -1.76471 q^{31} -8.09269 q^{32} +18.0749 q^{33} -12.3122 q^{34} +13.1551 q^{36} +0.497233 q^{37} +0.150977 q^{38} +4.38234 q^{39} +0.978111 q^{40} +1.00000 q^{41} +4.10393 q^{43} -12.6762 q^{44} +23.6832 q^{45} -9.00606 q^{46} +2.92536 q^{47} +11.3141 q^{48} +19.5950 q^{50} +18.3736 q^{51} -3.07338 q^{52} +3.08431 q^{53} -19.6301 q^{54} -22.8210 q^{55} -0.225305 q^{57} -3.91337 q^{58} -11.4408 q^{59} -24.6614 q^{60} -2.94851 q^{61} -3.58450 q^{62} -8.97293 q^{64} -5.53303 q^{65} +36.7140 q^{66} -1.12488 q^{67} -12.8856 q^{68} +13.4399 q^{69} +5.87671 q^{71} +1.58155 q^{72} -15.7737 q^{73} +1.00999 q^{74} -29.2420 q^{75} +0.158009 q^{76} +8.90146 q^{78} -14.5736 q^{79} -14.2849 q^{80} +10.7297 q^{81} +2.03121 q^{82} +14.4941 q^{83} -23.1981 q^{85} +8.33596 q^{86} +5.83998 q^{87} -1.52396 q^{88} -0.670099 q^{89} +48.1057 q^{90} -9.42555 q^{92} +5.34921 q^{93} +5.94203 q^{94} +0.284465 q^{95} +24.5307 q^{96} +10.5587 q^{97} -36.9002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + O(q^{10}) \) \( 5q - q^{2} - 4q^{3} + 3q^{4} + 5q^{5} - 12q^{6} - 3q^{8} + q^{9} + 2q^{11} + 2q^{12} - 5q^{13} - 5q^{15} - q^{16} - 13q^{17} + 21q^{18} + 23q^{20} + q^{22} + 2q^{23} - 2q^{24} + 22q^{25} - 10q^{27} - 5q^{29} - 33q^{30} - 17q^{31} - 12q^{32} - 3q^{33} + 8q^{34} + 15q^{36} - 7q^{37} + 3q^{38} + 5q^{39} - 7q^{40} + 5q^{41} + q^{43} - 47q^{44} + 23q^{45} - 24q^{46} - 9q^{47} + 19q^{48} + 2q^{50} + 5q^{51} - 20q^{52} + 5q^{53} - 2q^{54} - 33q^{55} - 3q^{57} - 27q^{58} - 7q^{59} - 16q^{60} - 22q^{61} + 28q^{62} - 3q^{64} - 31q^{65} + 42q^{66} - 3q^{67} - 17q^{68} + 22q^{69} - 24q^{71} - 12q^{72} - 40q^{73} - 5q^{74} - 24q^{75} + 19q^{76} + 30q^{78} - 42q^{79} - 24q^{80} + 9q^{81} - q^{82} + 12q^{83} - 23q^{85} + 16q^{86} + 32q^{87} + 26q^{88} - 8q^{89} + 59q^{90} + 12q^{92} - 11q^{93} + 23q^{94} - 17q^{95} + 17q^{96} - 16q^{97} - 45q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.03121 1.43628 0.718142 0.695897i \(-0.244993\pi\)
0.718142 + 0.695897i \(0.244993\pi\)
\(3\) −3.03121 −1.75007 −0.875036 0.484059i \(-0.839162\pi\)
−0.875036 + 0.484059i \(0.839162\pi\)
\(4\) 2.12582 1.06291
\(5\) 3.82713 1.71155 0.855773 0.517351i \(-0.173082\pi\)
0.855773 + 0.517351i \(0.173082\pi\)
\(6\) −6.15703 −2.51360
\(7\) 0 0
\(8\) 0.255573 0.0903586
\(9\) 6.18825 2.06275
\(10\) 7.77372 2.45827
\(11\) −5.96294 −1.79789 −0.898947 0.438057i \(-0.855667\pi\)
−0.898947 + 0.438057i \(0.855667\pi\)
\(12\) −6.44382 −1.86017
\(13\) −1.44574 −0.400976 −0.200488 0.979696i \(-0.564253\pi\)
−0.200488 + 0.979696i \(0.564253\pi\)
\(14\) 0 0
\(15\) −11.6009 −2.99533
\(16\) −3.73252 −0.933131
\(17\) −6.06148 −1.47012 −0.735062 0.677999i \(-0.762847\pi\)
−0.735062 + 0.677999i \(0.762847\pi\)
\(18\) 12.5696 2.96269
\(19\) 0.0743284 0.0170521 0.00852605 0.999964i \(-0.497286\pi\)
0.00852605 + 0.999964i \(0.497286\pi\)
\(20\) 8.13581 1.81922
\(21\) 0 0
\(22\) −12.1120 −2.58229
\(23\) −4.43383 −0.924518 −0.462259 0.886745i \(-0.652961\pi\)
−0.462259 + 0.886745i \(0.652961\pi\)
\(24\) −0.774695 −0.158134
\(25\) 9.64695 1.92939
\(26\) −2.93660 −0.575915
\(27\) −9.66425 −1.85989
\(28\) 0 0
\(29\) −1.92662 −0.357764 −0.178882 0.983871i \(-0.557248\pi\)
−0.178882 + 0.983871i \(0.557248\pi\)
\(30\) −23.5638 −4.30214
\(31\) −1.76471 −0.316951 −0.158476 0.987363i \(-0.550658\pi\)
−0.158476 + 0.987363i \(0.550658\pi\)
\(32\) −8.09269 −1.43060
\(33\) 18.0749 3.14644
\(34\) −12.3122 −2.11152
\(35\) 0 0
\(36\) 13.1551 2.19252
\(37\) 0.497233 0.0817446 0.0408723 0.999164i \(-0.486986\pi\)
0.0408723 + 0.999164i \(0.486986\pi\)
\(38\) 0.150977 0.0244917
\(39\) 4.38234 0.701736
\(40\) 0.978111 0.154653
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) 4.10393 0.625844 0.312922 0.949779i \(-0.398692\pi\)
0.312922 + 0.949779i \(0.398692\pi\)
\(44\) −12.6762 −1.91100
\(45\) 23.6832 3.53049
\(46\) −9.00606 −1.32787
\(47\) 2.92536 0.426708 0.213354 0.976975i \(-0.431561\pi\)
0.213354 + 0.976975i \(0.431561\pi\)
\(48\) 11.3141 1.63305
\(49\) 0 0
\(50\) 19.5950 2.77115
\(51\) 18.3736 2.57282
\(52\) −3.07338 −0.426202
\(53\) 3.08431 0.423663 0.211832 0.977306i \(-0.432057\pi\)
0.211832 + 0.977306i \(0.432057\pi\)
\(54\) −19.6301 −2.67132
\(55\) −22.8210 −3.07718
\(56\) 0 0
\(57\) −0.225305 −0.0298424
\(58\) −3.91337 −0.513850
\(59\) −11.4408 −1.48947 −0.744735 0.667360i \(-0.767424\pi\)
−0.744735 + 0.667360i \(0.767424\pi\)
\(60\) −24.6614 −3.18377
\(61\) −2.94851 −0.377517 −0.188759 0.982023i \(-0.560446\pi\)
−0.188759 + 0.982023i \(0.560446\pi\)
\(62\) −3.58450 −0.455232
\(63\) 0 0
\(64\) −8.97293 −1.12162
\(65\) −5.53303 −0.686288
\(66\) 36.7140 4.51919
\(67\) −1.12488 −0.137426 −0.0687129 0.997636i \(-0.521889\pi\)
−0.0687129 + 0.997636i \(0.521889\pi\)
\(68\) −12.8856 −1.56261
\(69\) 13.4399 1.61797
\(70\) 0 0
\(71\) 5.87671 0.697437 0.348719 0.937227i \(-0.386617\pi\)
0.348719 + 0.937227i \(0.386617\pi\)
\(72\) 1.58155 0.186387
\(73\) −15.7737 −1.84617 −0.923087 0.384591i \(-0.874343\pi\)
−0.923087 + 0.384591i \(0.874343\pi\)
\(74\) 1.00999 0.117408
\(75\) −29.2420 −3.37657
\(76\) 0.158009 0.0181249
\(77\) 0 0
\(78\) 8.90146 1.00789
\(79\) −14.5736 −1.63965 −0.819827 0.572611i \(-0.805931\pi\)
−0.819827 + 0.572611i \(0.805931\pi\)
\(80\) −14.2849 −1.59710
\(81\) 10.7297 1.19218
\(82\) 2.03121 0.224310
\(83\) 14.4941 1.59093 0.795465 0.606000i \(-0.207227\pi\)
0.795465 + 0.606000i \(0.207227\pi\)
\(84\) 0 0
\(85\) −23.1981 −2.51619
\(86\) 8.33596 0.898890
\(87\) 5.83998 0.626112
\(88\) −1.52396 −0.162455
\(89\) −0.670099 −0.0710303 −0.0355152 0.999369i \(-0.511307\pi\)
−0.0355152 + 0.999369i \(0.511307\pi\)
\(90\) 48.1057 5.07079
\(91\) 0 0
\(92\) −9.42555 −0.982681
\(93\) 5.34921 0.554687
\(94\) 5.94203 0.612873
\(95\) 0.284465 0.0291855
\(96\) 24.5307 2.50365
\(97\) 10.5587 1.07207 0.536036 0.844195i \(-0.319921\pi\)
0.536036 + 0.844195i \(0.319921\pi\)
\(98\) 0 0
\(99\) −36.9002 −3.70861
\(100\) 20.5077 2.05077
\(101\) 3.20947 0.319355 0.159677 0.987169i \(-0.448955\pi\)
0.159677 + 0.987169i \(0.448955\pi\)
\(102\) 37.3207 3.69530
\(103\) −5.79008 −0.570513 −0.285257 0.958451i \(-0.592079\pi\)
−0.285257 + 0.958451i \(0.592079\pi\)
\(104\) −0.369491 −0.0362316
\(105\) 0 0
\(106\) 6.26490 0.608500
\(107\) −5.90146 −0.570516 −0.285258 0.958451i \(-0.592079\pi\)
−0.285258 + 0.958451i \(0.592079\pi\)
\(108\) −20.5445 −1.97689
\(109\) −13.4426 −1.28756 −0.643782 0.765209i \(-0.722636\pi\)
−0.643782 + 0.765209i \(0.722636\pi\)
\(110\) −46.3542 −4.41970
\(111\) −1.50722 −0.143059
\(112\) 0 0
\(113\) 6.01283 0.565639 0.282820 0.959173i \(-0.408730\pi\)
0.282820 + 0.959173i \(0.408730\pi\)
\(114\) −0.457643 −0.0428622
\(115\) −16.9689 −1.58236
\(116\) −4.09564 −0.380271
\(117\) −8.94659 −0.827112
\(118\) −23.2388 −2.13930
\(119\) 0 0
\(120\) −2.96486 −0.270654
\(121\) 24.5567 2.23242
\(122\) −5.98904 −0.542222
\(123\) −3.03121 −0.273315
\(124\) −3.75146 −0.336891
\(125\) 17.7845 1.59070
\(126\) 0 0
\(127\) −6.27284 −0.556625 −0.278312 0.960491i \(-0.589775\pi\)
−0.278312 + 0.960491i \(0.589775\pi\)
\(128\) −2.04053 −0.180360
\(129\) −12.4399 −1.09527
\(130\) −11.2388 −0.985705
\(131\) −12.0382 −1.05178 −0.525892 0.850551i \(-0.676268\pi\)
−0.525892 + 0.850551i \(0.676268\pi\)
\(132\) 38.4241 3.34439
\(133\) 0 0
\(134\) −2.28487 −0.197382
\(135\) −36.9864 −3.18328
\(136\) −1.54915 −0.132838
\(137\) 6.46697 0.552510 0.276255 0.961084i \(-0.410907\pi\)
0.276255 + 0.961084i \(0.410907\pi\)
\(138\) 27.2993 2.32387
\(139\) 8.66331 0.734812 0.367406 0.930061i \(-0.380246\pi\)
0.367406 + 0.930061i \(0.380246\pi\)
\(140\) 0 0
\(141\) −8.86739 −0.746769
\(142\) 11.9368 1.00172
\(143\) 8.62085 0.720912
\(144\) −23.0978 −1.92481
\(145\) −7.37342 −0.612329
\(146\) −32.0398 −2.65163
\(147\) 0 0
\(148\) 1.05703 0.0868872
\(149\) 8.01997 0.657022 0.328511 0.944500i \(-0.393453\pi\)
0.328511 + 0.944500i \(0.393453\pi\)
\(150\) −59.3966 −4.84971
\(151\) −4.83256 −0.393268 −0.196634 0.980477i \(-0.563001\pi\)
−0.196634 + 0.980477i \(0.563001\pi\)
\(152\) 0.0189963 0.00154080
\(153\) −37.5099 −3.03250
\(154\) 0 0
\(155\) −6.75378 −0.542477
\(156\) 9.31608 0.745883
\(157\) 10.2117 0.814982 0.407491 0.913209i \(-0.366404\pi\)
0.407491 + 0.913209i \(0.366404\pi\)
\(158\) −29.6020 −2.35501
\(159\) −9.34921 −0.741441
\(160\) −30.9718 −2.44854
\(161\) 0 0
\(162\) 21.7942 1.71232
\(163\) −19.2954 −1.51133 −0.755665 0.654958i \(-0.772686\pi\)
−0.755665 + 0.654958i \(0.772686\pi\)
\(164\) 2.12582 0.165999
\(165\) 69.1752 5.38528
\(166\) 29.4405 2.28503
\(167\) −2.42397 −0.187572 −0.0937861 0.995592i \(-0.529897\pi\)
−0.0937861 + 0.995592i \(0.529897\pi\)
\(168\) 0 0
\(169\) −10.9098 −0.839218
\(170\) −47.1202 −3.61396
\(171\) 0.459963 0.0351742
\(172\) 8.72423 0.665217
\(173\) 8.79388 0.668587 0.334293 0.942469i \(-0.391502\pi\)
0.334293 + 0.942469i \(0.391502\pi\)
\(174\) 11.8622 0.899274
\(175\) 0 0
\(176\) 22.2568 1.67767
\(177\) 34.6796 2.60668
\(178\) −1.36111 −0.102020
\(179\) 14.4831 1.08252 0.541259 0.840856i \(-0.317948\pi\)
0.541259 + 0.840856i \(0.317948\pi\)
\(180\) 50.3464 3.75260
\(181\) 3.07539 0.228592 0.114296 0.993447i \(-0.463539\pi\)
0.114296 + 0.993447i \(0.463539\pi\)
\(182\) 0 0
\(183\) 8.93755 0.660682
\(184\) −1.13317 −0.0835382
\(185\) 1.90298 0.139910
\(186\) 10.8654 0.796688
\(187\) 36.1442 2.64313
\(188\) 6.21880 0.453552
\(189\) 0 0
\(190\) 0.577808 0.0419186
\(191\) 4.90956 0.355243 0.177622 0.984099i \(-0.443160\pi\)
0.177622 + 0.984099i \(0.443160\pi\)
\(192\) 27.1988 1.96291
\(193\) 11.3050 0.813754 0.406877 0.913483i \(-0.366618\pi\)
0.406877 + 0.913483i \(0.366618\pi\)
\(194\) 21.4469 1.53980
\(195\) 16.7718 1.20105
\(196\) 0 0
\(197\) −17.8594 −1.27243 −0.636215 0.771512i \(-0.719501\pi\)
−0.636215 + 0.771512i \(0.719501\pi\)
\(198\) −74.9520 −5.32661
\(199\) −13.6330 −0.966419 −0.483209 0.875505i \(-0.660529\pi\)
−0.483209 + 0.875505i \(0.660529\pi\)
\(200\) 2.46550 0.174337
\(201\) 3.40974 0.240505
\(202\) 6.51912 0.458684
\(203\) 0 0
\(204\) 39.0591 2.73468
\(205\) 3.82713 0.267299
\(206\) −11.7609 −0.819419
\(207\) −27.4377 −1.90705
\(208\) 5.39625 0.374163
\(209\) −0.443216 −0.0306579
\(210\) 0 0
\(211\) 22.4460 1.54525 0.772625 0.634863i \(-0.218944\pi\)
0.772625 + 0.634863i \(0.218944\pi\)
\(212\) 6.55670 0.450316
\(213\) −17.8136 −1.22056
\(214\) −11.9871 −0.819423
\(215\) 15.7063 1.07116
\(216\) −2.46992 −0.168057
\(217\) 0 0
\(218\) −27.3047 −1.84931
\(219\) 47.8135 3.23094
\(220\) −48.5133 −3.27077
\(221\) 8.76331 0.589484
\(222\) −3.06148 −0.205473
\(223\) −4.08663 −0.273661 −0.136831 0.990594i \(-0.543692\pi\)
−0.136831 + 0.990594i \(0.543692\pi\)
\(224\) 0 0
\(225\) 59.6977 3.97985
\(226\) 12.2133 0.812419
\(227\) 19.5310 1.29632 0.648158 0.761506i \(-0.275540\pi\)
0.648158 + 0.761506i \(0.275540\pi\)
\(228\) −0.478959 −0.0317198
\(229\) −23.0456 −1.52290 −0.761449 0.648225i \(-0.775512\pi\)
−0.761449 + 0.648225i \(0.775512\pi\)
\(230\) −34.4674 −2.27271
\(231\) 0 0
\(232\) −0.492390 −0.0323270
\(233\) 7.69494 0.504112 0.252056 0.967713i \(-0.418893\pi\)
0.252056 + 0.967713i \(0.418893\pi\)
\(234\) −18.1724 −1.18797
\(235\) 11.1957 0.730330
\(236\) −24.3212 −1.58317
\(237\) 44.1756 2.86951
\(238\) 0 0
\(239\) −12.2279 −0.790960 −0.395480 0.918475i \(-0.629422\pi\)
−0.395480 + 0.918475i \(0.629422\pi\)
\(240\) 43.3005 2.79503
\(241\) 3.68209 0.237184 0.118592 0.992943i \(-0.462162\pi\)
0.118592 + 0.992943i \(0.462162\pi\)
\(242\) 49.8798 3.20640
\(243\) −3.53112 −0.226521
\(244\) −6.26800 −0.401268
\(245\) 0 0
\(246\) −6.15703 −0.392558
\(247\) −0.107459 −0.00683748
\(248\) −0.451011 −0.0286393
\(249\) −43.9346 −2.78424
\(250\) 36.1241 2.28469
\(251\) 27.1519 1.71381 0.856907 0.515472i \(-0.172383\pi\)
0.856907 + 0.515472i \(0.172383\pi\)
\(252\) 0 0
\(253\) 26.4387 1.66219
\(254\) −12.7415 −0.799471
\(255\) 70.3183 4.40351
\(256\) 13.8011 0.862568
\(257\) 4.46344 0.278422 0.139211 0.990263i \(-0.455543\pi\)
0.139211 + 0.990263i \(0.455543\pi\)
\(258\) −25.2681 −1.57312
\(259\) 0 0
\(260\) −11.7623 −0.729464
\(261\) −11.9224 −0.737977
\(262\) −24.4522 −1.51066
\(263\) 16.6595 1.02727 0.513633 0.858010i \(-0.328299\pi\)
0.513633 + 0.858010i \(0.328299\pi\)
\(264\) 4.61946 0.284308
\(265\) 11.8041 0.725119
\(266\) 0 0
\(267\) 2.03121 0.124308
\(268\) −2.39129 −0.146071
\(269\) −3.43160 −0.209229 −0.104614 0.994513i \(-0.533361\pi\)
−0.104614 + 0.994513i \(0.533361\pi\)
\(270\) −75.1272 −4.57210
\(271\) −27.8083 −1.68924 −0.844618 0.535370i \(-0.820172\pi\)
−0.844618 + 0.535370i \(0.820172\pi\)
\(272\) 22.6246 1.37182
\(273\) 0 0
\(274\) 13.1358 0.793561
\(275\) −57.5242 −3.46884
\(276\) 28.5708 1.71976
\(277\) −13.5938 −0.816774 −0.408387 0.912809i \(-0.633909\pi\)
−0.408387 + 0.912809i \(0.633909\pi\)
\(278\) 17.5970 1.05540
\(279\) −10.9205 −0.653791
\(280\) 0 0
\(281\) 2.37494 0.141677 0.0708384 0.997488i \(-0.477433\pi\)
0.0708384 + 0.997488i \(0.477433\pi\)
\(282\) −18.0115 −1.07257
\(283\) 22.9301 1.36305 0.681526 0.731794i \(-0.261317\pi\)
0.681526 + 0.731794i \(0.261317\pi\)
\(284\) 12.4928 0.741314
\(285\) −0.862273 −0.0510767
\(286\) 17.5108 1.03543
\(287\) 0 0
\(288\) −50.0796 −2.95097
\(289\) 19.7415 1.16127
\(290\) −14.9770 −0.879478
\(291\) −32.0056 −1.87620
\(292\) −33.5321 −1.96232
\(293\) 7.30314 0.426654 0.213327 0.976981i \(-0.431570\pi\)
0.213327 + 0.976981i \(0.431570\pi\)
\(294\) 0 0
\(295\) −43.7856 −2.54930
\(296\) 0.127079 0.00738632
\(297\) 57.6274 3.34388
\(298\) 16.2903 0.943670
\(299\) 6.41016 0.370709
\(300\) −62.1632 −3.58900
\(301\) 0 0
\(302\) −9.81595 −0.564845
\(303\) −9.72860 −0.558893
\(304\) −0.277433 −0.0159118
\(305\) −11.2843 −0.646139
\(306\) −76.1906 −4.35553
\(307\) −16.2292 −0.926247 −0.463124 0.886294i \(-0.653271\pi\)
−0.463124 + 0.886294i \(0.653271\pi\)
\(308\) 0 0
\(309\) 17.5509 0.998439
\(310\) −13.7184 −0.779150
\(311\) −24.8162 −1.40720 −0.703599 0.710598i \(-0.748425\pi\)
−0.703599 + 0.710598i \(0.748425\pi\)
\(312\) 1.12001 0.0634079
\(313\) −1.00459 −0.0567828 −0.0283914 0.999597i \(-0.509038\pi\)
−0.0283914 + 0.999597i \(0.509038\pi\)
\(314\) 20.7421 1.17055
\(315\) 0 0
\(316\) −30.9808 −1.74281
\(317\) −29.9777 −1.68372 −0.841859 0.539697i \(-0.818539\pi\)
−0.841859 + 0.539697i \(0.818539\pi\)
\(318\) −18.9902 −1.06492
\(319\) 11.4883 0.643221
\(320\) −34.3406 −1.91970
\(321\) 17.8886 0.998443
\(322\) 0 0
\(323\) −0.450540 −0.0250687
\(324\) 22.8094 1.26719
\(325\) −13.9470 −0.773639
\(326\) −39.1930 −2.17070
\(327\) 40.7473 2.25333
\(328\) 0.255573 0.0141116
\(329\) 0 0
\(330\) 140.510 7.73480
\(331\) −25.1471 −1.38221 −0.691106 0.722754i \(-0.742876\pi\)
−0.691106 + 0.722754i \(0.742876\pi\)
\(332\) 30.8118 1.69102
\(333\) 3.07700 0.168619
\(334\) −4.92359 −0.269407
\(335\) −4.30506 −0.235210
\(336\) 0 0
\(337\) 1.81608 0.0989285 0.0494642 0.998776i \(-0.484249\pi\)
0.0494642 + 0.998776i \(0.484249\pi\)
\(338\) −22.1602 −1.20536
\(339\) −18.2262 −0.989909
\(340\) −49.3150 −2.67448
\(341\) 10.5229 0.569845
\(342\) 0.934282 0.0505202
\(343\) 0 0
\(344\) 1.04885 0.0565504
\(345\) 51.4363 2.76924
\(346\) 17.8622 0.960280
\(347\) 23.5712 1.26537 0.632685 0.774409i \(-0.281953\pi\)
0.632685 + 0.774409i \(0.281953\pi\)
\(348\) 12.4148 0.665501
\(349\) −22.3613 −1.19697 −0.598486 0.801133i \(-0.704231\pi\)
−0.598486 + 0.801133i \(0.704231\pi\)
\(350\) 0 0
\(351\) 13.9720 0.745769
\(352\) 48.2562 2.57207
\(353\) −12.0297 −0.640277 −0.320139 0.947371i \(-0.603729\pi\)
−0.320139 + 0.947371i \(0.603729\pi\)
\(354\) 70.4416 3.74393
\(355\) 22.4910 1.19370
\(356\) −1.42451 −0.0754990
\(357\) 0 0
\(358\) 29.4182 1.55480
\(359\) −29.0810 −1.53483 −0.767417 0.641148i \(-0.778459\pi\)
−0.767417 + 0.641148i \(0.778459\pi\)
\(360\) 6.05279 0.319010
\(361\) −18.9945 −0.999709
\(362\) 6.24677 0.328323
\(363\) −74.4365 −3.90690
\(364\) 0 0
\(365\) −60.3681 −3.15981
\(366\) 18.1541 0.948928
\(367\) 34.9839 1.82615 0.913073 0.407796i \(-0.133702\pi\)
0.913073 + 0.407796i \(0.133702\pi\)
\(368\) 16.5494 0.862697
\(369\) 6.18825 0.322147
\(370\) 3.86535 0.200950
\(371\) 0 0
\(372\) 11.3715 0.589583
\(373\) 25.0688 1.29801 0.649006 0.760783i \(-0.275185\pi\)
0.649006 + 0.760783i \(0.275185\pi\)
\(374\) 73.4166 3.79628
\(375\) −53.9086 −2.78383
\(376\) 0.747642 0.0385567
\(377\) 2.78538 0.143455
\(378\) 0 0
\(379\) 16.2692 0.835695 0.417848 0.908517i \(-0.362785\pi\)
0.417848 + 0.908517i \(0.362785\pi\)
\(380\) 0.604722 0.0310216
\(381\) 19.0143 0.974133
\(382\) 9.97235 0.510230
\(383\) 7.88034 0.402667 0.201333 0.979523i \(-0.435473\pi\)
0.201333 + 0.979523i \(0.435473\pi\)
\(384\) 6.18529 0.315642
\(385\) 0 0
\(386\) 22.9629 1.16878
\(387\) 25.3962 1.29096
\(388\) 22.4459 1.13952
\(389\) 28.0477 1.42208 0.711038 0.703154i \(-0.248225\pi\)
0.711038 + 0.703154i \(0.248225\pi\)
\(390\) 34.0671 1.72505
\(391\) 26.8756 1.35916
\(392\) 0 0
\(393\) 36.4904 1.84070
\(394\) −36.2762 −1.82757
\(395\) −55.7750 −2.80634
\(396\) −78.4432 −3.94192
\(397\) −12.8210 −0.643466 −0.321733 0.946830i \(-0.604265\pi\)
−0.321733 + 0.946830i \(0.604265\pi\)
\(398\) −27.6915 −1.38805
\(399\) 0 0
\(400\) −36.0075 −1.80037
\(401\) −1.81684 −0.0907285 −0.0453643 0.998971i \(-0.514445\pi\)
−0.0453643 + 0.998971i \(0.514445\pi\)
\(402\) 6.92591 0.345433
\(403\) 2.55131 0.127090
\(404\) 6.82277 0.339446
\(405\) 41.0638 2.04048
\(406\) 0 0
\(407\) −2.96497 −0.146968
\(408\) 4.69580 0.232477
\(409\) 35.2528 1.74314 0.871569 0.490273i \(-0.163103\pi\)
0.871569 + 0.490273i \(0.163103\pi\)
\(410\) 7.77372 0.383917
\(411\) −19.6027 −0.966932
\(412\) −12.3087 −0.606405
\(413\) 0 0
\(414\) −55.7317 −2.73906
\(415\) 55.4707 2.72295
\(416\) 11.6999 0.573636
\(417\) −26.2603 −1.28597
\(418\) −0.900266 −0.0440334
\(419\) −16.1804 −0.790464 −0.395232 0.918581i \(-0.629336\pi\)
−0.395232 + 0.918581i \(0.629336\pi\)
\(420\) 0 0
\(421\) −8.95139 −0.436264 −0.218132 0.975919i \(-0.569996\pi\)
−0.218132 + 0.975919i \(0.569996\pi\)
\(422\) 45.5927 2.21942
\(423\) 18.1028 0.880191
\(424\) 0.788266 0.0382816
\(425\) −58.4748 −2.83644
\(426\) −36.1831 −1.75308
\(427\) 0 0
\(428\) −12.5455 −0.606408
\(429\) −26.1316 −1.26165
\(430\) 31.9028 1.53849
\(431\) 12.5790 0.605907 0.302954 0.953005i \(-0.402027\pi\)
0.302954 + 0.953005i \(0.402027\pi\)
\(432\) 36.0720 1.73552
\(433\) −16.4245 −0.789309 −0.394654 0.918830i \(-0.629136\pi\)
−0.394654 + 0.918830i \(0.629136\pi\)
\(434\) 0 0
\(435\) 22.3504 1.07162
\(436\) −28.5765 −1.36857
\(437\) −0.329560 −0.0157650
\(438\) 97.1193 4.64054
\(439\) 25.2770 1.20640 0.603202 0.797589i \(-0.293891\pi\)
0.603202 + 0.797589i \(0.293891\pi\)
\(440\) −5.83242 −0.278050
\(441\) 0 0
\(442\) 17.8002 0.846667
\(443\) −0.450540 −0.0214058 −0.0107029 0.999943i \(-0.503407\pi\)
−0.0107029 + 0.999943i \(0.503407\pi\)
\(444\) −3.20408 −0.152059
\(445\) −2.56456 −0.121572
\(446\) −8.30082 −0.393055
\(447\) −24.3102 −1.14983
\(448\) 0 0
\(449\) −18.2531 −0.861418 −0.430709 0.902491i \(-0.641736\pi\)
−0.430709 + 0.902491i \(0.641736\pi\)
\(450\) 121.259 5.71619
\(451\) −5.96294 −0.280784
\(452\) 12.7822 0.601224
\(453\) 14.6485 0.688247
\(454\) 39.6716 1.86188
\(455\) 0 0
\(456\) −0.0575818 −0.00269652
\(457\) 28.5308 1.33461 0.667307 0.744783i \(-0.267447\pi\)
0.667307 + 0.744783i \(0.267447\pi\)
\(458\) −46.8105 −2.18731
\(459\) 58.5797 2.73426
\(460\) −36.0728 −1.68190
\(461\) 7.30314 0.340141 0.170071 0.985432i \(-0.445600\pi\)
0.170071 + 0.985432i \(0.445600\pi\)
\(462\) 0 0
\(463\) −16.5369 −0.768535 −0.384268 0.923222i \(-0.625546\pi\)
−0.384268 + 0.923222i \(0.625546\pi\)
\(464\) 7.19114 0.333840
\(465\) 20.4721 0.949373
\(466\) 15.6301 0.724048
\(467\) 16.6909 0.772363 0.386181 0.922423i \(-0.373794\pi\)
0.386181 + 0.922423i \(0.373794\pi\)
\(468\) −19.0189 −0.879147
\(469\) 0 0
\(470\) 22.7409 1.04896
\(471\) −30.9538 −1.42628
\(472\) −2.92396 −0.134586
\(473\) −24.4715 −1.12520
\(474\) 89.7300 4.12143
\(475\) 0.717043 0.0329002
\(476\) 0 0
\(477\) 19.0865 0.873911
\(478\) −24.8376 −1.13604
\(479\) −7.64133 −0.349141 −0.174571 0.984645i \(-0.555854\pi\)
−0.174571 + 0.984645i \(0.555854\pi\)
\(480\) 93.8821 4.28511
\(481\) −0.718869 −0.0327776
\(482\) 7.47911 0.340664
\(483\) 0 0
\(484\) 52.2031 2.37287
\(485\) 40.4095 1.83490
\(486\) −7.17245 −0.325349
\(487\) −11.6260 −0.526825 −0.263412 0.964683i \(-0.584848\pi\)
−0.263412 + 0.964683i \(0.584848\pi\)
\(488\) −0.753557 −0.0341119
\(489\) 58.4884 2.64494
\(490\) 0 0
\(491\) 25.9788 1.17241 0.586204 0.810164i \(-0.300622\pi\)
0.586204 + 0.810164i \(0.300622\pi\)
\(492\) −6.44382 −0.290510
\(493\) 11.6781 0.525957
\(494\) −0.218273 −0.00982056
\(495\) −141.222 −6.34745
\(496\) 6.58682 0.295757
\(497\) 0 0
\(498\) −89.2404 −3.99896
\(499\) −23.1920 −1.03821 −0.519107 0.854709i \(-0.673736\pi\)
−0.519107 + 0.854709i \(0.673736\pi\)
\(500\) 37.8067 1.69077
\(501\) 7.34756 0.328265
\(502\) 55.1513 2.46152
\(503\) −10.6434 −0.474564 −0.237282 0.971441i \(-0.576257\pi\)
−0.237282 + 0.971441i \(0.576257\pi\)
\(504\) 0 0
\(505\) 12.2831 0.546590
\(506\) 53.7026 2.38737
\(507\) 33.0700 1.46869
\(508\) −13.3350 −0.591643
\(509\) −1.33530 −0.0591860 −0.0295930 0.999562i \(-0.509421\pi\)
−0.0295930 + 0.999562i \(0.509421\pi\)
\(510\) 142.831 6.32468
\(511\) 0 0
\(512\) 32.1140 1.41925
\(513\) −0.718329 −0.0317150
\(514\) 9.06619 0.399893
\(515\) −22.1594 −0.976460
\(516\) −26.4450 −1.16418
\(517\) −17.4438 −0.767175
\(518\) 0 0
\(519\) −26.6561 −1.17007
\(520\) −1.41409 −0.0620120
\(521\) −26.5594 −1.16359 −0.581795 0.813335i \(-0.697649\pi\)
−0.581795 + 0.813335i \(0.697649\pi\)
\(522\) −24.2169 −1.05994
\(523\) 9.41507 0.411692 0.205846 0.978584i \(-0.434005\pi\)
0.205846 + 0.978584i \(0.434005\pi\)
\(524\) −25.5911 −1.11795
\(525\) 0 0
\(526\) 33.8389 1.47545
\(527\) 10.6968 0.465958
\(528\) −67.4651 −2.93604
\(529\) −3.34111 −0.145266
\(530\) 23.9766 1.04148
\(531\) −70.7987 −3.07240
\(532\) 0 0
\(533\) −1.44574 −0.0626219
\(534\) 4.12582 0.178542
\(535\) −22.5857 −0.976464
\(536\) −0.287488 −0.0124176
\(537\) −43.9013 −1.89448
\(538\) −6.97032 −0.300512
\(539\) 0 0
\(540\) −78.6265 −3.38355
\(541\) −8.71416 −0.374651 −0.187325 0.982298i \(-0.559982\pi\)
−0.187325 + 0.982298i \(0.559982\pi\)
\(542\) −56.4846 −2.42622
\(543\) −9.32217 −0.400052
\(544\) 49.0537 2.10316
\(545\) −51.4465 −2.20373
\(546\) 0 0
\(547\) −16.6541 −0.712079 −0.356040 0.934471i \(-0.615873\pi\)
−0.356040 + 0.934471i \(0.615873\pi\)
\(548\) 13.7476 0.587269
\(549\) −18.2461 −0.778724
\(550\) −116.844 −4.98224
\(551\) −0.143202 −0.00610062
\(552\) 3.43487 0.146198
\(553\) 0 0
\(554\) −27.6119 −1.17312
\(555\) −5.76833 −0.244852
\(556\) 18.4167 0.781040
\(557\) 8.98351 0.380643 0.190322 0.981722i \(-0.439047\pi\)
0.190322 + 0.981722i \(0.439047\pi\)
\(558\) −22.1818 −0.939029
\(559\) −5.93321 −0.250948
\(560\) 0 0
\(561\) −109.561 −4.62566
\(562\) 4.82400 0.203488
\(563\) 6.35931 0.268013 0.134007 0.990980i \(-0.457216\pi\)
0.134007 + 0.990980i \(0.457216\pi\)
\(564\) −18.8505 −0.793749
\(565\) 23.0119 0.968118
\(566\) 46.5759 1.95773
\(567\) 0 0
\(568\) 1.50193 0.0630194
\(569\) −5.92160 −0.248246 −0.124123 0.992267i \(-0.539612\pi\)
−0.124123 + 0.992267i \(0.539612\pi\)
\(570\) −1.75146 −0.0733606
\(571\) 45.8163 1.91735 0.958676 0.284502i \(-0.0918281\pi\)
0.958676 + 0.284502i \(0.0918281\pi\)
\(572\) 18.3264 0.766266
\(573\) −14.8819 −0.621701
\(574\) 0 0
\(575\) −42.7730 −1.78376
\(576\) −55.5267 −2.31361
\(577\) −37.9400 −1.57946 −0.789731 0.613453i \(-0.789780\pi\)
−0.789731 + 0.613453i \(0.789780\pi\)
\(578\) 40.0992 1.66791
\(579\) −34.2679 −1.42413
\(580\) −15.6746 −0.650852
\(581\) 0 0
\(582\) −65.0102 −2.69476
\(583\) −18.3916 −0.761702
\(584\) −4.03133 −0.166818
\(585\) −34.2398 −1.41564
\(586\) 14.8342 0.612796
\(587\) −5.81661 −0.240077 −0.120039 0.992769i \(-0.538302\pi\)
−0.120039 + 0.992769i \(0.538302\pi\)
\(588\) 0 0
\(589\) −0.131168 −0.00540469
\(590\) −88.9379 −3.66151
\(591\) 54.1357 2.22684
\(592\) −1.85593 −0.0762784
\(593\) 16.7778 0.688980 0.344490 0.938790i \(-0.388052\pi\)
0.344490 + 0.938790i \(0.388052\pi\)
\(594\) 117.053 4.80276
\(595\) 0 0
\(596\) 17.0490 0.698356
\(597\) 41.3245 1.69130
\(598\) 13.0204 0.532444
\(599\) −13.8344 −0.565260 −0.282630 0.959229i \(-0.591207\pi\)
−0.282630 + 0.959229i \(0.591207\pi\)
\(600\) −7.47344 −0.305102
\(601\) −22.2486 −0.907539 −0.453769 0.891119i \(-0.649921\pi\)
−0.453769 + 0.891119i \(0.649921\pi\)
\(602\) 0 0
\(603\) −6.96102 −0.283475
\(604\) −10.2732 −0.418009
\(605\) 93.9817 3.82090
\(606\) −19.7608 −0.802729
\(607\) 2.39107 0.0970505 0.0485252 0.998822i \(-0.484548\pi\)
0.0485252 + 0.998822i \(0.484548\pi\)
\(608\) −0.601517 −0.0243947
\(609\) 0 0
\(610\) −22.9209 −0.928038
\(611\) −4.22931 −0.171099
\(612\) −79.7395 −3.22328
\(613\) 38.0777 1.53795 0.768973 0.639282i \(-0.220768\pi\)
0.768973 + 0.639282i \(0.220768\pi\)
\(614\) −32.9649 −1.33035
\(615\) −11.6009 −0.467792
\(616\) 0 0
\(617\) −22.6988 −0.913819 −0.456910 0.889513i \(-0.651044\pi\)
−0.456910 + 0.889513i \(0.651044\pi\)
\(618\) 35.6497 1.43404
\(619\) 33.3077 1.33875 0.669376 0.742924i \(-0.266562\pi\)
0.669376 + 0.742924i \(0.266562\pi\)
\(620\) −14.3573 −0.576605
\(621\) 42.8497 1.71950
\(622\) −50.4070 −2.02113
\(623\) 0 0
\(624\) −16.3572 −0.654811
\(625\) 19.8289 0.793158
\(626\) −2.04053 −0.0815562
\(627\) 1.34348 0.0536535
\(628\) 21.7083 0.866254
\(629\) −3.01397 −0.120175
\(630\) 0 0
\(631\) −5.52118 −0.219795 −0.109897 0.993943i \(-0.535052\pi\)
−0.109897 + 0.993943i \(0.535052\pi\)
\(632\) −3.72461 −0.148157
\(633\) −68.0387 −2.70430
\(634\) −60.8912 −2.41830
\(635\) −24.0070 −0.952689
\(636\) −19.8748 −0.788086
\(637\) 0 0
\(638\) 23.3352 0.923848
\(639\) 36.3665 1.43864
\(640\) −7.80940 −0.308694
\(641\) 12.7799 0.504775 0.252387 0.967626i \(-0.418784\pi\)
0.252387 + 0.967626i \(0.418784\pi\)
\(642\) 36.3355 1.43405
\(643\) −19.0687 −0.751998 −0.375999 0.926620i \(-0.622700\pi\)
−0.375999 + 0.926620i \(0.622700\pi\)
\(644\) 0 0
\(645\) −47.6091 −1.87461
\(646\) −0.915143 −0.0360058
\(647\) −20.9933 −0.825331 −0.412665 0.910883i \(-0.635402\pi\)
−0.412665 + 0.910883i \(0.635402\pi\)
\(648\) 2.74221 0.107724
\(649\) 68.2210 2.67791
\(650\) −28.3293 −1.11116
\(651\) 0 0
\(652\) −41.0185 −1.60641
\(653\) −36.4523 −1.42649 −0.713244 0.700916i \(-0.752775\pi\)
−0.713244 + 0.700916i \(0.752775\pi\)
\(654\) 82.7663 3.23642
\(655\) −46.0719 −1.80018
\(656\) −3.73252 −0.145731
\(657\) −97.6117 −3.80819
\(658\) 0 0
\(659\) −20.9058 −0.814373 −0.407187 0.913345i \(-0.633490\pi\)
−0.407187 + 0.913345i \(0.633490\pi\)
\(660\) 147.054 5.72408
\(661\) −51.0624 −1.98610 −0.993048 0.117709i \(-0.962445\pi\)
−0.993048 + 0.117709i \(0.962445\pi\)
\(662\) −51.0792 −1.98525
\(663\) −26.5635 −1.03164
\(664\) 3.70428 0.143754
\(665\) 0 0
\(666\) 6.25004 0.242184
\(667\) 8.54230 0.330759
\(668\) −5.15293 −0.199373
\(669\) 12.3875 0.478927
\(670\) −8.74449 −0.337829
\(671\) 17.5818 0.678737
\(672\) 0 0
\(673\) 42.4278 1.63547 0.817736 0.575594i \(-0.195229\pi\)
0.817736 + 0.575594i \(0.195229\pi\)
\(674\) 3.68885 0.142089
\(675\) −93.2306 −3.58845
\(676\) −23.1924 −0.892015
\(677\) −15.1486 −0.582209 −0.291105 0.956691i \(-0.594023\pi\)
−0.291105 + 0.956691i \(0.594023\pi\)
\(678\) −37.0212 −1.42179
\(679\) 0 0
\(680\) −5.92880 −0.227359
\(681\) −59.2025 −2.26865
\(682\) 21.3742 0.818459
\(683\) 8.69113 0.332557 0.166278 0.986079i \(-0.446825\pi\)
0.166278 + 0.986079i \(0.446825\pi\)
\(684\) 0.977799 0.0373871
\(685\) 24.7499 0.945647
\(686\) 0 0
\(687\) 69.8562 2.66518
\(688\) −15.3180 −0.583994
\(689\) −4.45911 −0.169879
\(690\) 104.478 3.97741
\(691\) −24.8701 −0.946102 −0.473051 0.881035i \(-0.656847\pi\)
−0.473051 + 0.881035i \(0.656847\pi\)
\(692\) 18.6942 0.710648
\(693\) 0 0
\(694\) 47.8782 1.81743
\(695\) 33.1556 1.25767
\(696\) 1.49254 0.0565746
\(697\) −6.06148 −0.229595
\(698\) −45.4205 −1.71919
\(699\) −23.3250 −0.882233
\(700\) 0 0
\(701\) 38.5821 1.45723 0.728613 0.684926i \(-0.240165\pi\)
0.728613 + 0.684926i \(0.240165\pi\)
\(702\) 28.3801 1.07114
\(703\) 0.0369585 0.00139392
\(704\) 53.5050 2.01655
\(705\) −33.9367 −1.27813
\(706\) −24.4349 −0.919620
\(707\) 0 0
\(708\) 73.7227 2.77067
\(709\) 13.3306 0.500642 0.250321 0.968163i \(-0.419464\pi\)
0.250321 + 0.968163i \(0.419464\pi\)
\(710\) 45.6839 1.71449
\(711\) −90.1848 −3.38220
\(712\) −0.171259 −0.00641820
\(713\) 7.82443 0.293027
\(714\) 0 0
\(715\) 32.9932 1.23387
\(716\) 30.7885 1.15062
\(717\) 37.0655 1.38424
\(718\) −59.0696 −2.20446
\(719\) −19.0012 −0.708625 −0.354312 0.935127i \(-0.615285\pi\)
−0.354312 + 0.935127i \(0.615285\pi\)
\(720\) −88.3983 −3.29441
\(721\) 0 0
\(722\) −38.5818 −1.43587
\(723\) −11.1612 −0.415090
\(724\) 6.53774 0.242973
\(725\) −18.5860 −0.690266
\(726\) −151.196 −5.61142
\(727\) −19.0336 −0.705916 −0.352958 0.935639i \(-0.614824\pi\)
−0.352958 + 0.935639i \(0.614824\pi\)
\(728\) 0 0
\(729\) −21.4854 −0.795756
\(730\) −122.620 −4.53839
\(731\) −24.8759 −0.920069
\(732\) 18.9996 0.702247
\(733\) −50.7508 −1.87452 −0.937261 0.348628i \(-0.886648\pi\)
−0.937261 + 0.348628i \(0.886648\pi\)
\(734\) 71.0598 2.62286
\(735\) 0 0
\(736\) 35.8817 1.32262
\(737\) 6.70758 0.247077
\(738\) 12.5696 0.462695
\(739\) −6.70443 −0.246626 −0.123313 0.992368i \(-0.539352\pi\)
−0.123313 + 0.992368i \(0.539352\pi\)
\(740\) 4.04539 0.148712
\(741\) 0.325732 0.0119661
\(742\) 0 0
\(743\) 30.7597 1.12846 0.564232 0.825617i \(-0.309173\pi\)
0.564232 + 0.825617i \(0.309173\pi\)
\(744\) 1.36711 0.0501207
\(745\) 30.6935 1.12452
\(746\) 50.9200 1.86432
\(747\) 89.6928 3.28169
\(748\) 76.8363 2.80941
\(749\) 0 0
\(750\) −109.500 −3.99837
\(751\) −38.8738 −1.41852 −0.709262 0.704945i \(-0.750972\pi\)
−0.709262 + 0.704945i \(0.750972\pi\)
\(752\) −10.9190 −0.398174
\(753\) −82.3032 −2.99930
\(754\) 5.65770 0.206041
\(755\) −18.4948 −0.673096
\(756\) 0 0
\(757\) 3.89914 0.141717 0.0708583 0.997486i \(-0.477426\pi\)
0.0708583 + 0.997486i \(0.477426\pi\)
\(758\) 33.0463 1.20030
\(759\) −80.1413 −2.90894
\(760\) 0.0727014 0.00263716
\(761\) −13.9575 −0.505959 −0.252979 0.967472i \(-0.581410\pi\)
−0.252979 + 0.967472i \(0.581410\pi\)
\(762\) 38.6221 1.39913
\(763\) 0 0
\(764\) 10.4369 0.377592
\(765\) −143.556 −5.19026
\(766\) 16.0067 0.578344
\(767\) 16.5405 0.597241
\(768\) −41.8340 −1.50956
\(769\) −9.40607 −0.339192 −0.169596 0.985514i \(-0.554246\pi\)
−0.169596 + 0.985514i \(0.554246\pi\)
\(770\) 0 0
\(771\) −13.5296 −0.487258
\(772\) 24.0325 0.864948
\(773\) −6.38151 −0.229527 −0.114764 0.993393i \(-0.536611\pi\)
−0.114764 + 0.993393i \(0.536611\pi\)
\(774\) 51.5850 1.85418
\(775\) −17.0241 −0.611523
\(776\) 2.69851 0.0968709
\(777\) 0 0
\(778\) 56.9709 2.04250
\(779\) 0.0743284 0.00266309
\(780\) 35.6539 1.27661
\(781\) −35.0425 −1.25392
\(782\) 54.5900 1.95214
\(783\) 18.6193 0.665400
\(784\) 0 0
\(785\) 39.0815 1.39488
\(786\) 74.1197 2.64376
\(787\) 31.4514 1.12112 0.560561 0.828113i \(-0.310586\pi\)
0.560561 + 0.828113i \(0.310586\pi\)
\(788\) −37.9659 −1.35248
\(789\) −50.4984 −1.79779
\(790\) −113.291 −4.03071
\(791\) 0 0
\(792\) −9.43067 −0.335104
\(793\) 4.26277 0.151375
\(794\) −26.0421 −0.924200
\(795\) −35.7807 −1.26901
\(796\) −28.9814 −1.02722
\(797\) 12.7701 0.452339 0.226170 0.974088i \(-0.427380\pi\)
0.226170 + 0.974088i \(0.427380\pi\)
\(798\) 0 0
\(799\) −17.7320 −0.627313
\(800\) −78.0698 −2.76018
\(801\) −4.14674 −0.146518
\(802\) −3.69038 −0.130312
\(803\) 94.0578 3.31923
\(804\) 7.24851 0.255635
\(805\) 0 0
\(806\) 5.18225 0.182537
\(807\) 10.4019 0.366165
\(808\) 0.820254 0.0288564
\(809\) 37.8116 1.32939 0.664693 0.747117i \(-0.268563\pi\)
0.664693 + 0.747117i \(0.268563\pi\)
\(810\) 83.4094 2.93071
\(811\) 29.0566 1.02031 0.510157 0.860081i \(-0.329587\pi\)
0.510157 + 0.860081i \(0.329587\pi\)
\(812\) 0 0
\(813\) 84.2929 2.95628
\(814\) −6.02248 −0.211088
\(815\) −73.8460 −2.58671
\(816\) −68.5800 −2.40078
\(817\) 0.305039 0.0106720
\(818\) 71.6059 2.50364
\(819\) 0 0
\(820\) 8.13581 0.284115
\(821\) −22.1598 −0.773381 −0.386691 0.922209i \(-0.626382\pi\)
−0.386691 + 0.922209i \(0.626382\pi\)
\(822\) −39.8173 −1.38879
\(823\) −44.5895 −1.55429 −0.777147 0.629319i \(-0.783334\pi\)
−0.777147 + 0.629319i \(0.783334\pi\)
\(824\) −1.47978 −0.0515507
\(825\) 174.368 6.07072
\(826\) 0 0
\(827\) 17.7093 0.615814 0.307907 0.951416i \(-0.400371\pi\)
0.307907 + 0.951416i \(0.400371\pi\)
\(828\) −58.3276 −2.02702
\(829\) −29.8270 −1.03594 −0.517968 0.855400i \(-0.673311\pi\)
−0.517968 + 0.855400i \(0.673311\pi\)
\(830\) 112.673 3.91093
\(831\) 41.2058 1.42941
\(832\) 12.9725 0.449741
\(833\) 0 0
\(834\) −53.3403 −1.84702
\(835\) −9.27685 −0.321039
\(836\) −0.942199 −0.0325866
\(837\) 17.0546 0.589493
\(838\) −32.8658 −1.13533
\(839\) −31.3123 −1.08102 −0.540511 0.841337i \(-0.681769\pi\)
−0.540511 + 0.841337i \(0.681769\pi\)
\(840\) 0 0
\(841\) −25.2881 −0.872005
\(842\) −18.1822 −0.626599
\(843\) −7.19894 −0.247945
\(844\) 47.7163 1.64246
\(845\) −41.7534 −1.43636
\(846\) 36.7707 1.26420
\(847\) 0 0
\(848\) −11.5123 −0.395333
\(849\) −69.5060 −2.38544
\(850\) −118.775 −4.07394
\(851\) −2.20465 −0.0755744
\(852\) −37.8685 −1.29735
\(853\) −33.0949 −1.13315 −0.566574 0.824011i \(-0.691731\pi\)
−0.566574 + 0.824011i \(0.691731\pi\)
\(854\) 0 0
\(855\) 1.76034 0.0602023
\(856\) −1.50825 −0.0515510
\(857\) 20.0025 0.683272 0.341636 0.939832i \(-0.389019\pi\)
0.341636 + 0.939832i \(0.389019\pi\)
\(858\) −53.0789 −1.81208
\(859\) 37.3721 1.27512 0.637559 0.770401i \(-0.279944\pi\)
0.637559 + 0.770401i \(0.279944\pi\)
\(860\) 33.3888 1.13855
\(861\) 0 0
\(862\) 25.5505 0.870255
\(863\) 46.4668 1.58175 0.790875 0.611978i \(-0.209626\pi\)
0.790875 + 0.611978i \(0.209626\pi\)
\(864\) 78.2098 2.66075
\(865\) 33.6554 1.14432
\(866\) −33.3615 −1.13367
\(867\) −59.8408 −2.03230
\(868\) 0 0
\(869\) 86.9013 2.94793
\(870\) 45.3984 1.53915
\(871\) 1.62628 0.0551044
\(872\) −3.43555 −0.116342
\(873\) 65.3397 2.21142
\(874\) −0.669406 −0.0226430
\(875\) 0 0
\(876\) 101.643 3.43420
\(877\) −10.5546 −0.356404 −0.178202 0.983994i \(-0.557028\pi\)
−0.178202 + 0.983994i \(0.557028\pi\)
\(878\) 51.3429 1.73274
\(879\) −22.1374 −0.746675
\(880\) 85.1798 2.87141
\(881\) 25.0214 0.842993 0.421497 0.906830i \(-0.361505\pi\)
0.421497 + 0.906830i \(0.361505\pi\)
\(882\) 0 0
\(883\) 14.4626 0.486704 0.243352 0.969938i \(-0.421753\pi\)
0.243352 + 0.969938i \(0.421753\pi\)
\(884\) 18.6293 0.626570
\(885\) 132.723 4.46145
\(886\) −0.915143 −0.0307448
\(887\) 5.65795 0.189975 0.0949877 0.995478i \(-0.469719\pi\)
0.0949877 + 0.995478i \(0.469719\pi\)
\(888\) −0.385204 −0.0129266
\(889\) 0 0
\(890\) −5.20916 −0.174612
\(891\) −63.9803 −2.14342
\(892\) −8.68746 −0.290878
\(893\) 0.217437 0.00727626
\(894\) −49.3792 −1.65149
\(895\) 55.4288 1.85278
\(896\) 0 0
\(897\) −19.4306 −0.648768
\(898\) −37.0759 −1.23724
\(899\) 3.39992 0.113394
\(900\) 126.907 4.23023
\(901\) −18.6955 −0.622838
\(902\) −12.1120 −0.403285
\(903\) 0 0
\(904\) 1.53671 0.0511104
\(905\) 11.7699 0.391246
\(906\) 29.7542 0.988518
\(907\) −45.9050 −1.52425 −0.762126 0.647429i \(-0.775844\pi\)
−0.762126 + 0.647429i \(0.775844\pi\)
\(908\) 41.5194 1.37787
\(909\) 19.8610 0.658748
\(910\) 0 0
\(911\) 42.5495 1.40973 0.704864 0.709342i \(-0.251008\pi\)
0.704864 + 0.709342i \(0.251008\pi\)
\(912\) 0.840957 0.0278469
\(913\) −86.4272 −2.86032
\(914\) 57.9521 1.91688
\(915\) 34.2052 1.13079
\(916\) −48.9909 −1.61870
\(917\) 0 0
\(918\) 118.988 3.92718
\(919\) −25.1999 −0.831269 −0.415634 0.909532i \(-0.636440\pi\)
−0.415634 + 0.909532i \(0.636440\pi\)
\(920\) −4.33678 −0.142979
\(921\) 49.1940 1.62100
\(922\) 14.8342 0.488539
\(923\) −8.49618 −0.279655
\(924\) 0 0
\(925\) 4.79678 0.157717
\(926\) −33.5900 −1.10384
\(927\) −35.8304 −1.17683
\(928\) 15.5915 0.511816
\(929\) −37.0410 −1.21528 −0.607639 0.794214i \(-0.707883\pi\)
−0.607639 + 0.794214i \(0.707883\pi\)
\(930\) 41.5833 1.36357
\(931\) 0 0
\(932\) 16.3581 0.535827
\(933\) 75.2231 2.46270
\(934\) 33.9027 1.10933
\(935\) 138.329 4.52384
\(936\) −2.28650 −0.0747367
\(937\) −24.4307 −0.798115 −0.399058 0.916926i \(-0.630663\pi\)
−0.399058 + 0.916926i \(0.630663\pi\)
\(938\) 0 0
\(939\) 3.04512 0.0993739
\(940\) 23.8002 0.776276
\(941\) 42.4153 1.38270 0.691350 0.722520i \(-0.257016\pi\)
0.691350 + 0.722520i \(0.257016\pi\)
\(942\) −62.8738 −2.04854
\(943\) −4.43383 −0.144386
\(944\) 42.7032 1.38987
\(945\) 0 0
\(946\) −49.7068 −1.61611
\(947\) −10.0783 −0.327501 −0.163750 0.986502i \(-0.552359\pi\)
−0.163750 + 0.986502i \(0.552359\pi\)
\(948\) 93.9095 3.05004
\(949\) 22.8047 0.740271
\(950\) 1.45647 0.0472540
\(951\) 90.8689 2.94663
\(952\) 0 0
\(953\) −50.1799 −1.62549 −0.812743 0.582623i \(-0.802026\pi\)
−0.812743 + 0.582623i \(0.802026\pi\)
\(954\) 38.7687 1.25518
\(955\) 18.7895 0.608015
\(956\) −25.9944 −0.840720
\(957\) −34.8235 −1.12568
\(958\) −15.5212 −0.501466
\(959\) 0 0
\(960\) 104.094 3.35961
\(961\) −27.8858 −0.899542
\(962\) −1.46017 −0.0470779
\(963\) −36.5197 −1.17683
\(964\) 7.82748 0.252106
\(965\) 43.2659 1.39278
\(966\) 0 0
\(967\) 33.4107 1.07441 0.537207 0.843450i \(-0.319479\pi\)
0.537207 + 0.843450i \(0.319479\pi\)
\(968\) 6.27601 0.201719
\(969\) 1.36568 0.0438721
\(970\) 82.0803 2.63544
\(971\) 57.1446 1.83386 0.916929 0.399051i \(-0.130660\pi\)
0.916929 + 0.399051i \(0.130660\pi\)
\(972\) −7.50653 −0.240772
\(973\) 0 0
\(974\) −23.6149 −0.756670
\(975\) 42.2762 1.35392
\(976\) 11.0054 0.352273
\(977\) −5.17315 −0.165504 −0.0827519 0.996570i \(-0.526371\pi\)
−0.0827519 + 0.996570i \(0.526371\pi\)
\(978\) 118.802 3.79888
\(979\) 3.99576 0.127705
\(980\) 0 0
\(981\) −83.1859 −2.65592
\(982\) 52.7685 1.68391
\(983\) −53.4989 −1.70635 −0.853175 0.521624i \(-0.825326\pi\)
−0.853175 + 0.521624i \(0.825326\pi\)
\(984\) −0.774695 −0.0246964
\(985\) −68.3503 −2.17782
\(986\) 23.7208 0.755424
\(987\) 0 0
\(988\) −0.228440 −0.00726764
\(989\) −18.1962 −0.578604
\(990\) −286.852 −9.11674
\(991\) 16.6924 0.530252 0.265126 0.964214i \(-0.414587\pi\)
0.265126 + 0.964214i \(0.414587\pi\)
\(992\) 14.2813 0.453430
\(993\) 76.2263 2.41897
\(994\) 0 0
\(995\) −52.1754 −1.65407
\(996\) −93.3971 −2.95940
\(997\) −0.818514 −0.0259226 −0.0129613 0.999916i \(-0.504126\pi\)
−0.0129613 + 0.999916i \(0.504126\pi\)
\(998\) −47.1078 −1.49117
\(999\) −4.80538 −0.152036
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.1.5 5
7.6 odd 2 287.2.a.e.1.5 5
21.20 even 2 2583.2.a.r.1.1 5
28.27 even 2 4592.2.a.bb.1.1 5
35.34 odd 2 7175.2.a.n.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
287.2.a.e.1.5 5 7.6 odd 2
2009.2.a.n.1.5 5 1.1 even 1 trivial
2583.2.a.r.1.1 5 21.20 even 2
4592.2.a.bb.1.1 5 28.27 even 2
7175.2.a.n.1.1 5 35.34 odd 2