Properties

Label 644.2.a.d.1.2
Level $644$
Weight $2$
Character 644.1
Self dual yes
Analytic conductor $5.142$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.14236589017\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.6963152.1
Defining polynomial: \( x^{5} - 2x^{4} - 10x^{3} + 10x^{2} + 29x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.76321\) of defining polynomial
Character \(\chi\) \(=\) 644.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.76321 q^{3} -3.11657 q^{5} -1.00000 q^{7} +0.108911 q^{9} +O(q^{10})\) \(q-1.76321 q^{3} -3.11657 q^{5} -1.00000 q^{7} +0.108911 q^{9} -5.23940 q^{11} +6.34832 q^{13} +5.49516 q^{15} -0.585105 q^{17} +8.48889 q^{19} +1.76321 q^{21} -1.00000 q^{23} +4.71298 q^{25} +5.09760 q^{27} -1.59780 q^{29} -5.97640 q^{31} +9.23817 q^{33} +3.11657 q^{35} +10.0153 q^{37} -11.1934 q^{39} +1.17811 q^{41} -2.96247 q^{43} -0.339428 q^{45} -2.55043 q^{47} +1.00000 q^{49} +1.03166 q^{51} +8.27107 q^{53} +16.3289 q^{55} -14.9677 q^{57} +14.6004 q^{59} +6.15410 q^{61} -0.108911 q^{63} -19.7849 q^{65} -8.52015 q^{67} +1.76321 q^{69} +8.38625 q^{71} -0.358396 q^{73} -8.30998 q^{75} +5.23940 q^{77} -11.0216 q^{79} -9.31487 q^{81} -10.7446 q^{83} +1.82352 q^{85} +2.81726 q^{87} +18.1369 q^{89} -6.34832 q^{91} +10.5376 q^{93} -26.4562 q^{95} -8.38723 q^{97} -0.570629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 3 q^{3} + 2 q^{5} - 5 q^{7} + 10 q^{9} + 2 q^{11} + 13 q^{13} + 4 q^{15} + 4 q^{17} + 12 q^{19} - 3 q^{21} - 5 q^{23} + 19 q^{25} + 15 q^{27} + 13 q^{29} - 3 q^{31} + 24 q^{33} - 2 q^{35} - 4 q^{37} + 3 q^{39} + q^{41} - 8 q^{43} - 16 q^{45} + 5 q^{47} + 5 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 12 q^{59} + 20 q^{61} - 10 q^{63} - 12 q^{65} - 12 q^{67} - 3 q^{69} + 9 q^{71} - 9 q^{73} + 35 q^{75} - 2 q^{77} - 8 q^{79} - 11 q^{81} - 28 q^{83} + 16 q^{85} - 15 q^{87} + 32 q^{89} - 13 q^{91} - 15 q^{93} - 36 q^{95} + 4 q^{97} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.76321 −1.01799 −0.508995 0.860769i \(-0.669983\pi\)
−0.508995 + 0.860769i \(0.669983\pi\)
\(4\) 0 0
\(5\) −3.11657 −1.39377 −0.696885 0.717183i \(-0.745431\pi\)
−0.696885 + 0.717183i \(0.745431\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0.108911 0.0363037
\(10\) 0 0
\(11\) −5.23940 −1.57974 −0.789870 0.613274i \(-0.789852\pi\)
−0.789870 + 0.613274i \(0.789852\pi\)
\(12\) 0 0
\(13\) 6.34832 1.76071 0.880353 0.474319i \(-0.157306\pi\)
0.880353 + 0.474319i \(0.157306\pi\)
\(14\) 0 0
\(15\) 5.49516 1.41884
\(16\) 0 0
\(17\) −0.585105 −0.141909 −0.0709544 0.997480i \(-0.522604\pi\)
−0.0709544 + 0.997480i \(0.522604\pi\)
\(18\) 0 0
\(19\) 8.48889 1.94748 0.973742 0.227653i \(-0.0731051\pi\)
0.973742 + 0.227653i \(0.0731051\pi\)
\(20\) 0 0
\(21\) 1.76321 0.384764
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 4.71298 0.942597
\(26\) 0 0
\(27\) 5.09760 0.981033
\(28\) 0 0
\(29\) −1.59780 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(30\) 0 0
\(31\) −5.97640 −1.07339 −0.536696 0.843776i \(-0.680328\pi\)
−0.536696 + 0.843776i \(0.680328\pi\)
\(32\) 0 0
\(33\) 9.23817 1.60816
\(34\) 0 0
\(35\) 3.11657 0.526796
\(36\) 0 0
\(37\) 10.0153 1.64651 0.823253 0.567674i \(-0.192157\pi\)
0.823253 + 0.567674i \(0.192157\pi\)
\(38\) 0 0
\(39\) −11.1934 −1.79238
\(40\) 0 0
\(41\) 1.17811 0.183989 0.0919946 0.995760i \(-0.470676\pi\)
0.0919946 + 0.995760i \(0.470676\pi\)
\(42\) 0 0
\(43\) −2.96247 −0.451772 −0.225886 0.974154i \(-0.572528\pi\)
−0.225886 + 0.974154i \(0.572528\pi\)
\(44\) 0 0
\(45\) −0.339428 −0.0505990
\(46\) 0 0
\(47\) −2.55043 −0.372018 −0.186009 0.982548i \(-0.559555\pi\)
−0.186009 + 0.982548i \(0.559555\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.03166 0.144462
\(52\) 0 0
\(53\) 8.27107 1.13612 0.568059 0.822988i \(-0.307694\pi\)
0.568059 + 0.822988i \(0.307694\pi\)
\(54\) 0 0
\(55\) 16.3289 2.20180
\(56\) 0 0
\(57\) −14.9677 −1.98252
\(58\) 0 0
\(59\) 14.6004 1.90081 0.950406 0.311012i \(-0.100668\pi\)
0.950406 + 0.311012i \(0.100668\pi\)
\(60\) 0 0
\(61\) 6.15410 0.787951 0.393976 0.919121i \(-0.371099\pi\)
0.393976 + 0.919121i \(0.371099\pi\)
\(62\) 0 0
\(63\) −0.108911 −0.0137215
\(64\) 0 0
\(65\) −19.7849 −2.45402
\(66\) 0 0
\(67\) −8.52015 −1.04090 −0.520451 0.853892i \(-0.674236\pi\)
−0.520451 + 0.853892i \(0.674236\pi\)
\(68\) 0 0
\(69\) 1.76321 0.212266
\(70\) 0 0
\(71\) 8.38625 0.995265 0.497632 0.867388i \(-0.334203\pi\)
0.497632 + 0.867388i \(0.334203\pi\)
\(72\) 0 0
\(73\) −0.358396 −0.0419471 −0.0209735 0.999780i \(-0.506677\pi\)
−0.0209735 + 0.999780i \(0.506677\pi\)
\(74\) 0 0
\(75\) −8.30998 −0.959554
\(76\) 0 0
\(77\) 5.23940 0.597086
\(78\) 0 0
\(79\) −11.0216 −1.24002 −0.620012 0.784592i \(-0.712872\pi\)
−0.620012 + 0.784592i \(0.712872\pi\)
\(80\) 0 0
\(81\) −9.31487 −1.03499
\(82\) 0 0
\(83\) −10.7446 −1.17938 −0.589689 0.807630i \(-0.700750\pi\)
−0.589689 + 0.807630i \(0.700750\pi\)
\(84\) 0 0
\(85\) 1.82352 0.197788
\(86\) 0 0
\(87\) 2.81726 0.302042
\(88\) 0 0
\(89\) 18.1369 1.92251 0.961255 0.275662i \(-0.0888970\pi\)
0.961255 + 0.275662i \(0.0888970\pi\)
\(90\) 0 0
\(91\) −6.34832 −0.665484
\(92\) 0 0
\(93\) 10.5376 1.09270
\(94\) 0 0
\(95\) −26.4562 −2.71435
\(96\) 0 0
\(97\) −8.38723 −0.851594 −0.425797 0.904819i \(-0.640006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(98\) 0 0
\(99\) −0.570629 −0.0573503
\(100\) 0 0
\(101\) 6.79708 0.676335 0.338168 0.941086i \(-0.390193\pi\)
0.338168 + 0.941086i \(0.390193\pi\)
\(102\) 0 0
\(103\) −9.75955 −0.961637 −0.480819 0.876820i \(-0.659660\pi\)
−0.480819 + 0.876820i \(0.659660\pi\)
\(104\) 0 0
\(105\) −5.49516 −0.536273
\(106\) 0 0
\(107\) −6.96247 −0.673087 −0.336544 0.941668i \(-0.609258\pi\)
−0.336544 + 0.941668i \(0.609258\pi\)
\(108\) 0 0
\(109\) −2.03794 −0.195199 −0.0975994 0.995226i \(-0.531116\pi\)
−0.0975994 + 0.995226i \(0.531116\pi\)
\(110\) 0 0
\(111\) −17.6591 −1.67613
\(112\) 0 0
\(113\) 3.20815 0.301797 0.150898 0.988549i \(-0.451783\pi\)
0.150898 + 0.988549i \(0.451783\pi\)
\(114\) 0 0
\(115\) 3.11657 0.290621
\(116\) 0 0
\(117\) 0.691401 0.0639201
\(118\) 0 0
\(119\) 0.585105 0.0536365
\(120\) 0 0
\(121\) 16.4514 1.49558
\(122\) 0 0
\(123\) −2.07725 −0.187299
\(124\) 0 0
\(125\) 0.894506 0.0800070
\(126\) 0 0
\(127\) 4.83720 0.429232 0.214616 0.976698i \(-0.431150\pi\)
0.214616 + 0.976698i \(0.431150\pi\)
\(128\) 0 0
\(129\) 5.22346 0.459900
\(130\) 0 0
\(131\) 13.9917 1.22246 0.611231 0.791453i \(-0.290675\pi\)
0.611231 + 0.791453i \(0.290675\pi\)
\(132\) 0 0
\(133\) −8.48889 −0.736080
\(134\) 0 0
\(135\) −15.8870 −1.36734
\(136\) 0 0
\(137\) 3.65910 0.312618 0.156309 0.987708i \(-0.450040\pi\)
0.156309 + 0.987708i \(0.450040\pi\)
\(138\) 0 0
\(139\) 16.4852 1.39826 0.699130 0.714995i \(-0.253571\pi\)
0.699130 + 0.714995i \(0.253571\pi\)
\(140\) 0 0
\(141\) 4.49694 0.378711
\(142\) 0 0
\(143\) −33.2614 −2.78146
\(144\) 0 0
\(145\) 4.97965 0.413537
\(146\) 0 0
\(147\) −1.76321 −0.145427
\(148\) 0 0
\(149\) −15.7596 −1.29107 −0.645536 0.763729i \(-0.723366\pi\)
−0.645536 + 0.763729i \(0.723366\pi\)
\(150\) 0 0
\(151\) 6.13049 0.498892 0.249446 0.968389i \(-0.419751\pi\)
0.249446 + 0.968389i \(0.419751\pi\)
\(152\) 0 0
\(153\) −0.0637243 −0.00515181
\(154\) 0 0
\(155\) 18.6258 1.49606
\(156\) 0 0
\(157\) 3.65811 0.291949 0.145974 0.989288i \(-0.453368\pi\)
0.145974 + 0.989288i \(0.453368\pi\)
\(158\) 0 0
\(159\) −14.5836 −1.15656
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −17.9381 −1.40502 −0.702509 0.711675i \(-0.747937\pi\)
−0.702509 + 0.711675i \(0.747937\pi\)
\(164\) 0 0
\(165\) −28.7914 −2.24141
\(166\) 0 0
\(167\) 0.419935 0.0324956 0.0162478 0.999868i \(-0.494828\pi\)
0.0162478 + 0.999868i \(0.494828\pi\)
\(168\) 0 0
\(169\) 27.3011 2.10009
\(170\) 0 0
\(171\) 0.924533 0.0707008
\(172\) 0 0
\(173\) 2.48889 0.189227 0.0946134 0.995514i \(-0.469839\pi\)
0.0946134 + 0.995514i \(0.469839\pi\)
\(174\) 0 0
\(175\) −4.71298 −0.356268
\(176\) 0 0
\(177\) −25.7436 −1.93501
\(178\) 0 0
\(179\) −2.64201 −0.197473 −0.0987365 0.995114i \(-0.531480\pi\)
−0.0987365 + 0.995114i \(0.531480\pi\)
\(180\) 0 0
\(181\) 1.38950 0.103281 0.0516405 0.998666i \(-0.483555\pi\)
0.0516405 + 0.998666i \(0.483555\pi\)
\(182\) 0 0
\(183\) −10.8510 −0.802127
\(184\) 0 0
\(185\) −31.2134 −2.29485
\(186\) 0 0
\(187\) 3.06560 0.224179
\(188\) 0 0
\(189\) −5.09760 −0.370796
\(190\) 0 0
\(191\) 13.0682 0.945578 0.472789 0.881176i \(-0.343247\pi\)
0.472789 + 0.881176i \(0.343247\pi\)
\(192\) 0 0
\(193\) 21.5964 1.55454 0.777270 0.629167i \(-0.216604\pi\)
0.777270 + 0.629167i \(0.216604\pi\)
\(194\) 0 0
\(195\) 34.8850 2.49817
\(196\) 0 0
\(197\) 10.2410 0.729643 0.364822 0.931077i \(-0.381130\pi\)
0.364822 + 0.931077i \(0.381130\pi\)
\(198\) 0 0
\(199\) −21.6744 −1.53646 −0.768229 0.640175i \(-0.778862\pi\)
−0.768229 + 0.640175i \(0.778862\pi\)
\(200\) 0 0
\(201\) 15.0228 1.05963
\(202\) 0 0
\(203\) 1.59780 0.112144
\(204\) 0 0
\(205\) −3.67164 −0.256439
\(206\) 0 0
\(207\) −0.108911 −0.00756983
\(208\) 0 0
\(209\) −44.4767 −3.07652
\(210\) 0 0
\(211\) 15.8499 1.09115 0.545577 0.838061i \(-0.316311\pi\)
0.545577 + 0.838061i \(0.316311\pi\)
\(212\) 0 0
\(213\) −14.7867 −1.01317
\(214\) 0 0
\(215\) 9.23273 0.629667
\(216\) 0 0
\(217\) 5.97640 0.405704
\(218\) 0 0
\(219\) 0.631927 0.0427017
\(220\) 0 0
\(221\) −3.71443 −0.249860
\(222\) 0 0
\(223\) 1.04860 0.0702197 0.0351098 0.999383i \(-0.488822\pi\)
0.0351098 + 0.999383i \(0.488822\pi\)
\(224\) 0 0
\(225\) 0.513296 0.0342197
\(226\) 0 0
\(227\) 27.9455 1.85481 0.927403 0.374063i \(-0.122036\pi\)
0.927403 + 0.374063i \(0.122036\pi\)
\(228\) 0 0
\(229\) −17.8239 −1.17783 −0.588917 0.808193i \(-0.700446\pi\)
−0.588917 + 0.808193i \(0.700446\pi\)
\(230\) 0 0
\(231\) −9.23817 −0.607827
\(232\) 0 0
\(233\) 9.53528 0.624677 0.312339 0.949971i \(-0.398888\pi\)
0.312339 + 0.949971i \(0.398888\pi\)
\(234\) 0 0
\(235\) 7.94858 0.518508
\(236\) 0 0
\(237\) 19.4334 1.26233
\(238\) 0 0
\(239\) 13.6190 0.880939 0.440469 0.897768i \(-0.354812\pi\)
0.440469 + 0.897768i \(0.354812\pi\)
\(240\) 0 0
\(241\) 16.1115 1.03783 0.518917 0.854824i \(-0.326335\pi\)
0.518917 + 0.854824i \(0.326335\pi\)
\(242\) 0 0
\(243\) 1.13128 0.0725718
\(244\) 0 0
\(245\) −3.11657 −0.199110
\(246\) 0 0
\(247\) 53.8901 3.42895
\(248\) 0 0
\(249\) 18.9451 1.20060
\(250\) 0 0
\(251\) 17.7822 1.12240 0.561201 0.827680i \(-0.310340\pi\)
0.561201 + 0.827680i \(0.310340\pi\)
\(252\) 0 0
\(253\) 5.23940 0.329399
\(254\) 0 0
\(255\) −3.21525 −0.201347
\(256\) 0 0
\(257\) −11.3733 −0.709447 −0.354724 0.934971i \(-0.615425\pi\)
−0.354724 + 0.934971i \(0.615425\pi\)
\(258\) 0 0
\(259\) −10.0153 −0.622321
\(260\) 0 0
\(261\) −0.174018 −0.0107714
\(262\) 0 0
\(263\) 8.22726 0.507315 0.253657 0.967294i \(-0.418366\pi\)
0.253657 + 0.967294i \(0.418366\pi\)
\(264\) 0 0
\(265\) −25.7773 −1.58349
\(266\) 0 0
\(267\) −31.9792 −1.95710
\(268\) 0 0
\(269\) 3.31078 0.201862 0.100931 0.994893i \(-0.467818\pi\)
0.100931 + 0.994893i \(0.467818\pi\)
\(270\) 0 0
\(271\) 24.3447 1.47883 0.739416 0.673248i \(-0.235102\pi\)
0.739416 + 0.673248i \(0.235102\pi\)
\(272\) 0 0
\(273\) 11.1934 0.677456
\(274\) 0 0
\(275\) −24.6932 −1.48906
\(276\) 0 0
\(277\) −14.1492 −0.850144 −0.425072 0.905160i \(-0.639751\pi\)
−0.425072 + 0.905160i \(0.639751\pi\)
\(278\) 0 0
\(279\) −0.650895 −0.0389681
\(280\) 0 0
\(281\) 10.2231 0.609856 0.304928 0.952375i \(-0.401368\pi\)
0.304928 + 0.952375i \(0.401368\pi\)
\(282\) 0 0
\(283\) 5.80480 0.345060 0.172530 0.985004i \(-0.444806\pi\)
0.172530 + 0.985004i \(0.444806\pi\)
\(284\) 0 0
\(285\) 46.6478 2.76318
\(286\) 0 0
\(287\) −1.17811 −0.0695414
\(288\) 0 0
\(289\) −16.6577 −0.979862
\(290\) 0 0
\(291\) 14.7885 0.866914
\(292\) 0 0
\(293\) −31.1117 −1.81757 −0.908783 0.417269i \(-0.862987\pi\)
−0.908783 + 0.417269i \(0.862987\pi\)
\(294\) 0 0
\(295\) −45.5032 −2.64930
\(296\) 0 0
\(297\) −26.7084 −1.54978
\(298\) 0 0
\(299\) −6.34832 −0.367133
\(300\) 0 0
\(301\) 2.96247 0.170754
\(302\) 0 0
\(303\) −11.9847 −0.688502
\(304\) 0 0
\(305\) −19.1797 −1.09822
\(306\) 0 0
\(307\) −4.55784 −0.260130 −0.130065 0.991505i \(-0.541519\pi\)
−0.130065 + 0.991505i \(0.541519\pi\)
\(308\) 0 0
\(309\) 17.2081 0.978937
\(310\) 0 0
\(311\) 1.08872 0.0617358 0.0308679 0.999523i \(-0.490173\pi\)
0.0308679 + 0.999523i \(0.490173\pi\)
\(312\) 0 0
\(313\) 16.9086 0.955731 0.477866 0.878433i \(-0.341411\pi\)
0.477866 + 0.878433i \(0.341411\pi\)
\(314\) 0 0
\(315\) 0.339428 0.0191246
\(316\) 0 0
\(317\) −4.26439 −0.239512 −0.119756 0.992803i \(-0.538211\pi\)
−0.119756 + 0.992803i \(0.538211\pi\)
\(318\) 0 0
\(319\) 8.37152 0.468715
\(320\) 0 0
\(321\) 12.2763 0.685196
\(322\) 0 0
\(323\) −4.96689 −0.276365
\(324\) 0 0
\(325\) 29.9195 1.65964
\(326\) 0 0
\(327\) 3.59331 0.198710
\(328\) 0 0
\(329\) 2.55043 0.140610
\(330\) 0 0
\(331\) 16.9450 0.931380 0.465690 0.884948i \(-0.345806\pi\)
0.465690 + 0.884948i \(0.345806\pi\)
\(332\) 0 0
\(333\) 1.09078 0.0597742
\(334\) 0 0
\(335\) 26.5536 1.45078
\(336\) 0 0
\(337\) 10.7547 0.585847 0.292924 0.956136i \(-0.405372\pi\)
0.292924 + 0.956136i \(0.405372\pi\)
\(338\) 0 0
\(339\) −5.65663 −0.307226
\(340\) 0 0
\(341\) 31.3128 1.69568
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −5.49516 −0.295850
\(346\) 0 0
\(347\) 30.2105 1.62178 0.810892 0.585195i \(-0.198982\pi\)
0.810892 + 0.585195i \(0.198982\pi\)
\(348\) 0 0
\(349\) −20.2309 −1.08294 −0.541469 0.840721i \(-0.682132\pi\)
−0.541469 + 0.840721i \(0.682132\pi\)
\(350\) 0 0
\(351\) 32.3612 1.72731
\(352\) 0 0
\(353\) 14.0953 0.750218 0.375109 0.926981i \(-0.377605\pi\)
0.375109 + 0.926981i \(0.377605\pi\)
\(354\) 0 0
\(355\) −26.1363 −1.38717
\(356\) 0 0
\(357\) −1.03166 −0.0546014
\(358\) 0 0
\(359\) −16.0086 −0.844903 −0.422452 0.906385i \(-0.638830\pi\)
−0.422452 + 0.906385i \(0.638830\pi\)
\(360\) 0 0
\(361\) 53.0612 2.79270
\(362\) 0 0
\(363\) −29.0072 −1.52248
\(364\) 0 0
\(365\) 1.11696 0.0584646
\(366\) 0 0
\(367\) −28.8277 −1.50479 −0.752397 0.658710i \(-0.771102\pi\)
−0.752397 + 0.658710i \(0.771102\pi\)
\(368\) 0 0
\(369\) 0.128309 0.00667948
\(370\) 0 0
\(371\) −8.27107 −0.429412
\(372\) 0 0
\(373\) −18.6273 −0.964484 −0.482242 0.876038i \(-0.660177\pi\)
−0.482242 + 0.876038i \(0.660177\pi\)
\(374\) 0 0
\(375\) −1.57720 −0.0814464
\(376\) 0 0
\(377\) −10.1433 −0.522409
\(378\) 0 0
\(379\) 14.8138 0.760936 0.380468 0.924794i \(-0.375763\pi\)
0.380468 + 0.924794i \(0.375763\pi\)
\(380\) 0 0
\(381\) −8.52901 −0.436954
\(382\) 0 0
\(383\) 0.611968 0.0312701 0.0156351 0.999878i \(-0.495023\pi\)
0.0156351 + 0.999878i \(0.495023\pi\)
\(384\) 0 0
\(385\) −16.3289 −0.832200
\(386\) 0 0
\(387\) −0.322645 −0.0164010
\(388\) 0 0
\(389\) 11.4514 0.580607 0.290303 0.956935i \(-0.406244\pi\)
0.290303 + 0.956935i \(0.406244\pi\)
\(390\) 0 0
\(391\) 0.585105 0.0295900
\(392\) 0 0
\(393\) −24.6703 −1.24445
\(394\) 0 0
\(395\) 34.3495 1.72831
\(396\) 0 0
\(397\) 15.9228 0.799140 0.399570 0.916703i \(-0.369159\pi\)
0.399570 + 0.916703i \(0.369159\pi\)
\(398\) 0 0
\(399\) 14.9677 0.749322
\(400\) 0 0
\(401\) −35.1157 −1.75359 −0.876797 0.480861i \(-0.840324\pi\)
−0.876797 + 0.480861i \(0.840324\pi\)
\(402\) 0 0
\(403\) −37.9400 −1.88993
\(404\) 0 0
\(405\) 29.0304 1.44253
\(406\) 0 0
\(407\) −52.4743 −2.60105
\(408\) 0 0
\(409\) 24.6726 1.21998 0.609991 0.792408i \(-0.291173\pi\)
0.609991 + 0.792408i \(0.291173\pi\)
\(410\) 0 0
\(411\) −6.45176 −0.318242
\(412\) 0 0
\(413\) −14.6004 −0.718439
\(414\) 0 0
\(415\) 33.4864 1.64378
\(416\) 0 0
\(417\) −29.0669 −1.42341
\(418\) 0 0
\(419\) 23.0411 1.12563 0.562816 0.826582i \(-0.309718\pi\)
0.562816 + 0.826582i \(0.309718\pi\)
\(420\) 0 0
\(421\) 11.8048 0.575331 0.287665 0.957731i \(-0.407121\pi\)
0.287665 + 0.957731i \(0.407121\pi\)
\(422\) 0 0
\(423\) −0.277770 −0.0135056
\(424\) 0 0
\(425\) −2.75759 −0.133763
\(426\) 0 0
\(427\) −6.15410 −0.297818
\(428\) 0 0
\(429\) 58.6468 2.83150
\(430\) 0 0
\(431\) −3.56873 −0.171899 −0.0859497 0.996299i \(-0.527392\pi\)
−0.0859497 + 0.996299i \(0.527392\pi\)
\(432\) 0 0
\(433\) −41.0992 −1.97510 −0.987550 0.157305i \(-0.949719\pi\)
−0.987550 + 0.157305i \(0.949719\pi\)
\(434\) 0 0
\(435\) −8.78017 −0.420977
\(436\) 0 0
\(437\) −8.48889 −0.406079
\(438\) 0 0
\(439\) 31.6361 1.50991 0.754954 0.655778i \(-0.227659\pi\)
0.754954 + 0.655778i \(0.227659\pi\)
\(440\) 0 0
\(441\) 0.108911 0.00518624
\(442\) 0 0
\(443\) 3.30070 0.156821 0.0784106 0.996921i \(-0.475015\pi\)
0.0784106 + 0.996921i \(0.475015\pi\)
\(444\) 0 0
\(445\) −56.5249 −2.67954
\(446\) 0 0
\(447\) 27.7874 1.31430
\(448\) 0 0
\(449\) −19.8820 −0.938289 −0.469145 0.883121i \(-0.655438\pi\)
−0.469145 + 0.883121i \(0.655438\pi\)
\(450\) 0 0
\(451\) −6.17257 −0.290655
\(452\) 0 0
\(453\) −10.8093 −0.507868
\(454\) 0 0
\(455\) 19.7849 0.927532
\(456\) 0 0
\(457\) 15.2707 0.714332 0.357166 0.934041i \(-0.383743\pi\)
0.357166 + 0.934041i \(0.383743\pi\)
\(458\) 0 0
\(459\) −2.98263 −0.139217
\(460\) 0 0
\(461\) −8.68922 −0.404697 −0.202349 0.979314i \(-0.564857\pi\)
−0.202349 + 0.979314i \(0.564857\pi\)
\(462\) 0 0
\(463\) 8.57355 0.398447 0.199223 0.979954i \(-0.436158\pi\)
0.199223 + 0.979954i \(0.436158\pi\)
\(464\) 0 0
\(465\) −32.8413 −1.52298
\(466\) 0 0
\(467\) 30.1327 1.39437 0.697187 0.716889i \(-0.254435\pi\)
0.697187 + 0.716889i \(0.254435\pi\)
\(468\) 0 0
\(469\) 8.52015 0.393424
\(470\) 0 0
\(471\) −6.45001 −0.297201
\(472\) 0 0
\(473\) 15.5216 0.713683
\(474\) 0 0
\(475\) 40.0080 1.83569
\(476\) 0 0
\(477\) 0.900810 0.0412452
\(478\) 0 0
\(479\) 4.74988 0.217027 0.108514 0.994095i \(-0.465391\pi\)
0.108514 + 0.994095i \(0.465391\pi\)
\(480\) 0 0
\(481\) 63.5803 2.89901
\(482\) 0 0
\(483\) −1.76321 −0.0802289
\(484\) 0 0
\(485\) 26.1394 1.18693
\(486\) 0 0
\(487\) −18.9728 −0.859741 −0.429870 0.902891i \(-0.641441\pi\)
−0.429870 + 0.902891i \(0.641441\pi\)
\(488\) 0 0
\(489\) 31.6286 1.43029
\(490\) 0 0
\(491\) 6.18017 0.278907 0.139453 0.990229i \(-0.455465\pi\)
0.139453 + 0.990229i \(0.455465\pi\)
\(492\) 0 0
\(493\) 0.934881 0.0421049
\(494\) 0 0
\(495\) 1.77840 0.0799332
\(496\) 0 0
\(497\) −8.38625 −0.376175
\(498\) 0 0
\(499\) −20.9756 −0.938997 −0.469498 0.882933i \(-0.655565\pi\)
−0.469498 + 0.882933i \(0.655565\pi\)
\(500\) 0 0
\(501\) −0.740435 −0.0330802
\(502\) 0 0
\(503\) −3.42597 −0.152756 −0.0763782 0.997079i \(-0.524336\pi\)
−0.0763782 + 0.997079i \(0.524336\pi\)
\(504\) 0 0
\(505\) −21.1836 −0.942656
\(506\) 0 0
\(507\) −48.1376 −2.13787
\(508\) 0 0
\(509\) −35.6250 −1.57905 −0.789526 0.613718i \(-0.789673\pi\)
−0.789526 + 0.613718i \(0.789673\pi\)
\(510\) 0 0
\(511\) 0.358396 0.0158545
\(512\) 0 0
\(513\) 43.2729 1.91055
\(514\) 0 0
\(515\) 30.4163 1.34030
\(516\) 0 0
\(517\) 13.3627 0.587692
\(518\) 0 0
\(519\) −4.38844 −0.192631
\(520\) 0 0
\(521\) −28.6482 −1.25510 −0.627551 0.778576i \(-0.715942\pi\)
−0.627551 + 0.778576i \(0.715942\pi\)
\(522\) 0 0
\(523\) −14.7717 −0.645921 −0.322961 0.946412i \(-0.604678\pi\)
−0.322961 + 0.946412i \(0.604678\pi\)
\(524\) 0 0
\(525\) 8.30998 0.362677
\(526\) 0 0
\(527\) 3.49682 0.152324
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 1.59015 0.0690064
\(532\) 0 0
\(533\) 7.47899 0.323951
\(534\) 0 0
\(535\) 21.6990 0.938129
\(536\) 0 0
\(537\) 4.65842 0.201025
\(538\) 0 0
\(539\) −5.23940 −0.225677
\(540\) 0 0
\(541\) −12.3001 −0.528824 −0.264412 0.964410i \(-0.585178\pi\)
−0.264412 + 0.964410i \(0.585178\pi\)
\(542\) 0 0
\(543\) −2.44999 −0.105139
\(544\) 0 0
\(545\) 6.35136 0.272062
\(546\) 0 0
\(547\) −20.4059 −0.872495 −0.436247 0.899827i \(-0.643693\pi\)
−0.436247 + 0.899827i \(0.643693\pi\)
\(548\) 0 0
\(549\) 0.670249 0.0286055
\(550\) 0 0
\(551\) −13.5635 −0.577827
\(552\) 0 0
\(553\) 11.0216 0.468685
\(554\) 0 0
\(555\) 55.0357 2.33614
\(556\) 0 0
\(557\) 13.9455 0.590889 0.295444 0.955360i \(-0.404532\pi\)
0.295444 + 0.955360i \(0.404532\pi\)
\(558\) 0 0
\(559\) −18.8067 −0.795438
\(560\) 0 0
\(561\) −5.40530 −0.228212
\(562\) 0 0
\(563\) −22.3868 −0.943492 −0.471746 0.881734i \(-0.656376\pi\)
−0.471746 + 0.881734i \(0.656376\pi\)
\(564\) 0 0
\(565\) −9.99840 −0.420636
\(566\) 0 0
\(567\) 9.31487 0.391188
\(568\) 0 0
\(569\) −23.0581 −0.966645 −0.483322 0.875442i \(-0.660570\pi\)
−0.483322 + 0.875442i \(0.660570\pi\)
\(570\) 0 0
\(571\) 0.606257 0.0253711 0.0126855 0.999920i \(-0.495962\pi\)
0.0126855 + 0.999920i \(0.495962\pi\)
\(572\) 0 0
\(573\) −23.0419 −0.962589
\(574\) 0 0
\(575\) −4.71298 −0.196545
\(576\) 0 0
\(577\) 5.57828 0.232227 0.116113 0.993236i \(-0.462956\pi\)
0.116113 + 0.993236i \(0.462956\pi\)
\(578\) 0 0
\(579\) −38.0789 −1.58251
\(580\) 0 0
\(581\) 10.7446 0.445763
\(582\) 0 0
\(583\) −43.3355 −1.79477
\(584\) 0 0
\(585\) −2.15480 −0.0890899
\(586\) 0 0
\(587\) −15.5804 −0.643071 −0.321536 0.946898i \(-0.604199\pi\)
−0.321536 + 0.946898i \(0.604199\pi\)
\(588\) 0 0
\(589\) −50.7330 −2.09042
\(590\) 0 0
\(591\) −18.0571 −0.742769
\(592\) 0 0
\(593\) −12.4788 −0.512443 −0.256222 0.966618i \(-0.582478\pi\)
−0.256222 + 0.966618i \(0.582478\pi\)
\(594\) 0 0
\(595\) −1.82352 −0.0747569
\(596\) 0 0
\(597\) 38.2165 1.56410
\(598\) 0 0
\(599\) −25.6897 −1.04965 −0.524827 0.851209i \(-0.675870\pi\)
−0.524827 + 0.851209i \(0.675870\pi\)
\(600\) 0 0
\(601\) 14.1712 0.578055 0.289028 0.957321i \(-0.406668\pi\)
0.289028 + 0.957321i \(0.406668\pi\)
\(602\) 0 0
\(603\) −0.927938 −0.0377885
\(604\) 0 0
\(605\) −51.2717 −2.08449
\(606\) 0 0
\(607\) −38.6010 −1.56677 −0.783383 0.621539i \(-0.786508\pi\)
−0.783383 + 0.621539i \(0.786508\pi\)
\(608\) 0 0
\(609\) −2.81726 −0.114161
\(610\) 0 0
\(611\) −16.1909 −0.655015
\(612\) 0 0
\(613\) −2.40423 −0.0971058 −0.0485529 0.998821i \(-0.515461\pi\)
−0.0485529 + 0.998821i \(0.515461\pi\)
\(614\) 0 0
\(615\) 6.47388 0.261052
\(616\) 0 0
\(617\) 24.5675 0.989051 0.494526 0.869163i \(-0.335342\pi\)
0.494526 + 0.869163i \(0.335342\pi\)
\(618\) 0 0
\(619\) 31.0657 1.24864 0.624318 0.781171i \(-0.285377\pi\)
0.624318 + 0.781171i \(0.285377\pi\)
\(620\) 0 0
\(621\) −5.09760 −0.204560
\(622\) 0 0
\(623\) −18.1369 −0.726640
\(624\) 0 0
\(625\) −26.3527 −1.05411
\(626\) 0 0
\(627\) 78.4218 3.13187
\(628\) 0 0
\(629\) −5.86001 −0.233654
\(630\) 0 0
\(631\) −15.9461 −0.634805 −0.317402 0.948291i \(-0.602811\pi\)
−0.317402 + 0.948291i \(0.602811\pi\)
\(632\) 0 0
\(633\) −27.9468 −1.11078
\(634\) 0 0
\(635\) −15.0755 −0.598252
\(636\) 0 0
\(637\) 6.34832 0.251529
\(638\) 0 0
\(639\) 0.913355 0.0361317
\(640\) 0 0
\(641\) −34.1109 −1.34730 −0.673650 0.739051i \(-0.735274\pi\)
−0.673650 + 0.739051i \(0.735274\pi\)
\(642\) 0 0
\(643\) 22.7215 0.896050 0.448025 0.894021i \(-0.352128\pi\)
0.448025 + 0.894021i \(0.352128\pi\)
\(644\) 0 0
\(645\) −16.2792 −0.640995
\(646\) 0 0
\(647\) −4.53740 −0.178384 −0.0891918 0.996014i \(-0.528428\pi\)
−0.0891918 + 0.996014i \(0.528428\pi\)
\(648\) 0 0
\(649\) −76.4975 −3.00279
\(650\) 0 0
\(651\) −10.5376 −0.413003
\(652\) 0 0
\(653\) −38.9525 −1.52433 −0.762164 0.647384i \(-0.775863\pi\)
−0.762164 + 0.647384i \(0.775863\pi\)
\(654\) 0 0
\(655\) −43.6061 −1.70383
\(656\) 0 0
\(657\) −0.0390332 −0.00152283
\(658\) 0 0
\(659\) −14.8983 −0.580357 −0.290179 0.956973i \(-0.593715\pi\)
−0.290179 + 0.956973i \(0.593715\pi\)
\(660\) 0 0
\(661\) 17.2118 0.669461 0.334731 0.942314i \(-0.391355\pi\)
0.334731 + 0.942314i \(0.391355\pi\)
\(662\) 0 0
\(663\) 6.54932 0.254355
\(664\) 0 0
\(665\) 26.4562 1.02593
\(666\) 0 0
\(667\) 1.59780 0.0618671
\(668\) 0 0
\(669\) −1.84891 −0.0714829
\(670\) 0 0
\(671\) −32.2438 −1.24476
\(672\) 0 0
\(673\) 19.7685 0.762018 0.381009 0.924571i \(-0.375577\pi\)
0.381009 + 0.924571i \(0.375577\pi\)
\(674\) 0 0
\(675\) 24.0249 0.924719
\(676\) 0 0
\(677\) −36.0117 −1.38404 −0.692020 0.721878i \(-0.743279\pi\)
−0.692020 + 0.721878i \(0.743279\pi\)
\(678\) 0 0
\(679\) 8.38723 0.321872
\(680\) 0 0
\(681\) −49.2738 −1.88817
\(682\) 0 0
\(683\) 11.0796 0.423949 0.211975 0.977275i \(-0.432011\pi\)
0.211975 + 0.977275i \(0.432011\pi\)
\(684\) 0 0
\(685\) −11.4038 −0.435718
\(686\) 0 0
\(687\) 31.4272 1.19902
\(688\) 0 0
\(689\) 52.5073 2.00037
\(690\) 0 0
\(691\) −16.5328 −0.628936 −0.314468 0.949268i \(-0.601826\pi\)
−0.314468 + 0.949268i \(0.601826\pi\)
\(692\) 0 0
\(693\) 0.570629 0.0216764
\(694\) 0 0
\(695\) −51.3773 −1.94885
\(696\) 0 0
\(697\) −0.689315 −0.0261097
\(698\) 0 0
\(699\) −16.8127 −0.635915
\(700\) 0 0
\(701\) 17.2642 0.652058 0.326029 0.945360i \(-0.394289\pi\)
0.326029 + 0.945360i \(0.394289\pi\)
\(702\) 0 0
\(703\) 85.0189 3.20655
\(704\) 0 0
\(705\) −14.0150 −0.527836
\(706\) 0 0
\(707\) −6.79708 −0.255631
\(708\) 0 0
\(709\) −3.98260 −0.149570 −0.0747849 0.997200i \(-0.523827\pi\)
−0.0747849 + 0.997200i \(0.523827\pi\)
\(710\) 0 0
\(711\) −1.20037 −0.0450174
\(712\) 0 0
\(713\) 5.97640 0.223818
\(714\) 0 0
\(715\) 103.661 3.87671
\(716\) 0 0
\(717\) −24.0131 −0.896787
\(718\) 0 0
\(719\) −7.33439 −0.273527 −0.136763 0.990604i \(-0.543670\pi\)
−0.136763 + 0.990604i \(0.543670\pi\)
\(720\) 0 0
\(721\) 9.75955 0.363465
\(722\) 0 0
\(723\) −28.4080 −1.05651
\(724\) 0 0
\(725\) −7.53041 −0.279672
\(726\) 0 0
\(727\) 4.06498 0.150762 0.0753809 0.997155i \(-0.475983\pi\)
0.0753809 + 0.997155i \(0.475983\pi\)
\(728\) 0 0
\(729\) 25.9499 0.961108
\(730\) 0 0
\(731\) 1.73335 0.0641104
\(732\) 0 0
\(733\) 17.2247 0.636207 0.318104 0.948056i \(-0.396954\pi\)
0.318104 + 0.948056i \(0.396954\pi\)
\(734\) 0 0
\(735\) 5.49516 0.202692
\(736\) 0 0
\(737\) 44.6405 1.64435
\(738\) 0 0
\(739\) 13.0829 0.481262 0.240631 0.970617i \(-0.422646\pi\)
0.240631 + 0.970617i \(0.422646\pi\)
\(740\) 0 0
\(741\) −95.0197 −3.49063
\(742\) 0 0
\(743\) −9.58553 −0.351659 −0.175830 0.984421i \(-0.556261\pi\)
−0.175830 + 0.984421i \(0.556261\pi\)
\(744\) 0 0
\(745\) 49.1157 1.79946
\(746\) 0 0
\(747\) −1.17021 −0.0428157
\(748\) 0 0
\(749\) 6.96247 0.254403
\(750\) 0 0
\(751\) 39.3151 1.43463 0.717314 0.696750i \(-0.245371\pi\)
0.717314 + 0.696750i \(0.245371\pi\)
\(752\) 0 0
\(753\) −31.3537 −1.14259
\(754\) 0 0
\(755\) −19.1061 −0.695342
\(756\) 0 0
\(757\) −30.7971 −1.11934 −0.559670 0.828716i \(-0.689072\pi\)
−0.559670 + 0.828716i \(0.689072\pi\)
\(758\) 0 0
\(759\) −9.23817 −0.335324
\(760\) 0 0
\(761\) 19.0651 0.691110 0.345555 0.938399i \(-0.387691\pi\)
0.345555 + 0.938399i \(0.387691\pi\)
\(762\) 0 0
\(763\) 2.03794 0.0737782
\(764\) 0 0
\(765\) 0.198601 0.00718044
\(766\) 0 0
\(767\) 92.6880 3.34677
\(768\) 0 0
\(769\) 14.0861 0.507959 0.253980 0.967210i \(-0.418260\pi\)
0.253980 + 0.967210i \(0.418260\pi\)
\(770\) 0 0
\(771\) 20.0535 0.722210
\(772\) 0 0
\(773\) −5.46538 −0.196576 −0.0982879 0.995158i \(-0.531337\pi\)
−0.0982879 + 0.995158i \(0.531337\pi\)
\(774\) 0 0
\(775\) −28.1667 −1.01178
\(776\) 0 0
\(777\) 17.6591 0.633517
\(778\) 0 0
\(779\) 10.0008 0.358316
\(780\) 0 0
\(781\) −43.9390 −1.57226
\(782\) 0 0
\(783\) −8.14494 −0.291077
\(784\) 0 0
\(785\) −11.4007 −0.406910
\(786\) 0 0
\(787\) 36.3921 1.29724 0.648618 0.761114i \(-0.275347\pi\)
0.648618 + 0.761114i \(0.275347\pi\)
\(788\) 0 0
\(789\) −14.5064 −0.516441
\(790\) 0 0
\(791\) −3.20815 −0.114069
\(792\) 0 0
\(793\) 39.0682 1.38735
\(794\) 0 0
\(795\) 45.4509 1.61198
\(796\) 0 0
\(797\) 43.5810 1.54372 0.771858 0.635794i \(-0.219327\pi\)
0.771858 + 0.635794i \(0.219327\pi\)
\(798\) 0 0
\(799\) 1.49227 0.0527927
\(800\) 0 0
\(801\) 1.97531 0.0697941
\(802\) 0 0
\(803\) 1.87778 0.0662654
\(804\) 0 0
\(805\) −3.11657 −0.109845
\(806\) 0 0
\(807\) −5.83761 −0.205494
\(808\) 0 0
\(809\) 5.11013 0.179663 0.0898313 0.995957i \(-0.471367\pi\)
0.0898313 + 0.995957i \(0.471367\pi\)
\(810\) 0 0
\(811\) −2.06953 −0.0726712 −0.0363356 0.999340i \(-0.511569\pi\)
−0.0363356 + 0.999340i \(0.511569\pi\)
\(812\) 0 0
\(813\) −42.9248 −1.50544
\(814\) 0 0
\(815\) 55.9052 1.95827
\(816\) 0 0
\(817\) −25.1481 −0.879819
\(818\) 0 0
\(819\) −0.691401 −0.0241595
\(820\) 0 0
\(821\) 2.87359 0.100289 0.0501446 0.998742i \(-0.484032\pi\)
0.0501446 + 0.998742i \(0.484032\pi\)
\(822\) 0 0
\(823\) 3.81222 0.132886 0.0664428 0.997790i \(-0.478835\pi\)
0.0664428 + 0.997790i \(0.478835\pi\)
\(824\) 0 0
\(825\) 43.5394 1.51585
\(826\) 0 0
\(827\) −6.87210 −0.238966 −0.119483 0.992836i \(-0.538124\pi\)
−0.119483 + 0.992836i \(0.538124\pi\)
\(828\) 0 0
\(829\) 5.53374 0.192195 0.0960973 0.995372i \(-0.469364\pi\)
0.0960973 + 0.995372i \(0.469364\pi\)
\(830\) 0 0
\(831\) 24.9480 0.865438
\(832\) 0 0
\(833\) −0.585105 −0.0202727
\(834\) 0 0
\(835\) −1.30876 −0.0452914
\(836\) 0 0
\(837\) −30.4653 −1.05303
\(838\) 0 0
\(839\) 1.16338 0.0401642 0.0200821 0.999798i \(-0.493607\pi\)
0.0200821 + 0.999798i \(0.493607\pi\)
\(840\) 0 0
\(841\) −26.4470 −0.911967
\(842\) 0 0
\(843\) −18.0254 −0.620827
\(844\) 0 0
\(845\) −85.0857 −2.92704
\(846\) 0 0
\(847\) −16.4514 −0.565275
\(848\) 0 0
\(849\) −10.2351 −0.351267
\(850\) 0 0
\(851\) −10.0153 −0.343320
\(852\) 0 0
\(853\) 5.30694 0.181706 0.0908531 0.995864i \(-0.471041\pi\)
0.0908531 + 0.995864i \(0.471041\pi\)
\(854\) 0 0
\(855\) −2.88137 −0.0985407
\(856\) 0 0
\(857\) −1.84412 −0.0629938 −0.0314969 0.999504i \(-0.510027\pi\)
−0.0314969 + 0.999504i \(0.510027\pi\)
\(858\) 0 0
\(859\) 45.1077 1.53905 0.769527 0.638615i \(-0.220492\pi\)
0.769527 + 0.638615i \(0.220492\pi\)
\(860\) 0 0
\(861\) 2.07725 0.0707924
\(862\) 0 0
\(863\) 2.75173 0.0936701 0.0468351 0.998903i \(-0.485086\pi\)
0.0468351 + 0.998903i \(0.485086\pi\)
\(864\) 0 0
\(865\) −7.75679 −0.263739
\(866\) 0 0
\(867\) 29.3709 0.997490
\(868\) 0 0
\(869\) 57.7465 1.95892
\(870\) 0 0
\(871\) −54.0886 −1.83272
\(872\) 0 0
\(873\) −0.913461 −0.0309160
\(874\) 0 0
\(875\) −0.894506 −0.0302398
\(876\) 0 0
\(877\) 32.9340 1.11210 0.556052 0.831148i \(-0.312316\pi\)
0.556052 + 0.831148i \(0.312316\pi\)
\(878\) 0 0
\(879\) 54.8565 1.85026
\(880\) 0 0
\(881\) 40.6201 1.36853 0.684264 0.729235i \(-0.260124\pi\)
0.684264 + 0.729235i \(0.260124\pi\)
\(882\) 0 0
\(883\) 0.817246 0.0275025 0.0137513 0.999905i \(-0.495623\pi\)
0.0137513 + 0.999905i \(0.495623\pi\)
\(884\) 0 0
\(885\) 80.2316 2.69696
\(886\) 0 0
\(887\) 4.29164 0.144099 0.0720496 0.997401i \(-0.477046\pi\)
0.0720496 + 0.997401i \(0.477046\pi\)
\(888\) 0 0
\(889\) −4.83720 −0.162235
\(890\) 0 0
\(891\) 48.8044 1.63501
\(892\) 0 0
\(893\) −21.6503 −0.724500
\(894\) 0 0
\(895\) 8.23399 0.275232
\(896\) 0 0
\(897\) 11.1934 0.373737
\(898\) 0 0
\(899\) 9.54909 0.318480
\(900\) 0 0
\(901\) −4.83944 −0.161225
\(902\) 0 0
\(903\) −5.22346 −0.173826
\(904\) 0 0
\(905\) −4.33048 −0.143950
\(906\) 0 0
\(907\) −43.7845 −1.45384 −0.726920 0.686723i \(-0.759049\pi\)
−0.726920 + 0.686723i \(0.759049\pi\)
\(908\) 0 0
\(909\) 0.740277 0.0245534
\(910\) 0 0
\(911\) −48.8009 −1.61684 −0.808422 0.588603i \(-0.799678\pi\)
−0.808422 + 0.588603i \(0.799678\pi\)
\(912\) 0 0
\(913\) 56.2955 1.86311
\(914\) 0 0
\(915\) 33.8178 1.11798
\(916\) 0 0
\(917\) −13.9917 −0.462047
\(918\) 0 0
\(919\) 10.5677 0.348596 0.174298 0.984693i \(-0.444234\pi\)
0.174298 + 0.984693i \(0.444234\pi\)
\(920\) 0 0
\(921\) 8.03644 0.264810
\(922\) 0 0
\(923\) 53.2386 1.75237
\(924\) 0 0
\(925\) 47.2020 1.55199
\(926\) 0 0
\(927\) −1.06292 −0.0349109
\(928\) 0 0
\(929\) 5.51933 0.181083 0.0905417 0.995893i \(-0.471140\pi\)
0.0905417 + 0.995893i \(0.471140\pi\)
\(930\) 0 0
\(931\) 8.48889 0.278212
\(932\) 0 0
\(933\) −1.91965 −0.0628465
\(934\) 0 0
\(935\) −9.55415 −0.312454
\(936\) 0 0
\(937\) −51.4761 −1.68165 −0.840826 0.541305i \(-0.817930\pi\)
−0.840826 + 0.541305i \(0.817930\pi\)
\(938\) 0 0
\(939\) −29.8134 −0.972925
\(940\) 0 0
\(941\) 23.0595 0.751718 0.375859 0.926677i \(-0.377348\pi\)
0.375859 + 0.926677i \(0.377348\pi\)
\(942\) 0 0
\(943\) −1.17811 −0.0383644
\(944\) 0 0
\(945\) 15.8870 0.516804
\(946\) 0 0
\(947\) 27.0207 0.878054 0.439027 0.898474i \(-0.355323\pi\)
0.439027 + 0.898474i \(0.355323\pi\)
\(948\) 0 0
\(949\) −2.27521 −0.0738564
\(950\) 0 0
\(951\) 7.51902 0.243821
\(952\) 0 0
\(953\) 29.6466 0.960348 0.480174 0.877173i \(-0.340574\pi\)
0.480174 + 0.877173i \(0.340574\pi\)
\(954\) 0 0
\(955\) −40.7278 −1.31792
\(956\) 0 0
\(957\) −14.7608 −0.477147
\(958\) 0 0
\(959\) −3.65910 −0.118158
\(960\) 0 0
\(961\) 4.71731 0.152171
\(962\) 0 0
\(963\) −0.758289 −0.0244355
\(964\) 0 0
\(965\) −67.3065 −2.16667
\(966\) 0 0
\(967\) −41.6275 −1.33865 −0.669324 0.742970i \(-0.733416\pi\)
−0.669324 + 0.742970i \(0.733416\pi\)
\(968\) 0 0
\(969\) 8.75767 0.281337
\(970\) 0 0
\(971\) 37.4436 1.20162 0.600811 0.799391i \(-0.294845\pi\)
0.600811 + 0.799391i \(0.294845\pi\)
\(972\) 0 0
\(973\) −16.4852 −0.528492
\(974\) 0 0
\(975\) −52.7544 −1.68949
\(976\) 0 0
\(977\) 25.8568 0.827231 0.413616 0.910452i \(-0.364266\pi\)
0.413616 + 0.910452i \(0.364266\pi\)
\(978\) 0 0
\(979\) −95.0266 −3.03706
\(980\) 0 0
\(981\) −0.221953 −0.00708643
\(982\) 0 0
\(983\) 33.8201 1.07869 0.539347 0.842084i \(-0.318671\pi\)
0.539347 + 0.842084i \(0.318671\pi\)
\(984\) 0 0
\(985\) −31.9168 −1.01696
\(986\) 0 0
\(987\) −4.49694 −0.143139
\(988\) 0 0
\(989\) 2.96247 0.0942010
\(990\) 0 0
\(991\) −26.9278 −0.855390 −0.427695 0.903923i \(-0.640674\pi\)
−0.427695 + 0.903923i \(0.640674\pi\)
\(992\) 0 0
\(993\) −29.8776 −0.948136
\(994\) 0 0
\(995\) 67.5497 2.14147
\(996\) 0 0
\(997\) 4.87456 0.154379 0.0771894 0.997016i \(-0.475405\pi\)
0.0771894 + 0.997016i \(0.475405\pi\)
\(998\) 0 0
\(999\) 51.0540 1.61528
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 644.2.a.d.1.2 5
3.2 odd 2 5796.2.a.t.1.5 5
4.3 odd 2 2576.2.a.bb.1.4 5
7.6 odd 2 4508.2.a.f.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
644.2.a.d.1.2 5 1.1 even 1 trivial
2576.2.a.bb.1.4 5 4.3 odd 2
4508.2.a.f.1.4 5 7.6 odd 2
5796.2.a.t.1.5 5 3.2 odd 2