Properties

Label 8470.2.a.co.1.3
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4400.1
Defining polynomial: \(x^{4} - 7 x^{2} + 11\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.54336\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.543362 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.543362 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.70476 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.543362 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.543362 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.70476 q^{9} -1.00000 q^{10} +0.543362 q^{12} -1.65861 q^{13} +1.00000 q^{14} +0.543362 q^{15} +1.00000 q^{16} +2.04615 q^{17} +2.70476 q^{18} +3.70476 q^{19} +1.00000 q^{20} -0.543362 q^{21} -4.99442 q^{23} -0.543362 q^{24} +1.00000 q^{25} +1.65861 q^{26} -3.09975 q^{27} -1.00000 q^{28} +7.13632 q^{29} -0.543362 q^{30} -10.1809 q^{31} -1.00000 q^{32} -2.04615 q^{34} -1.00000 q^{35} -2.70476 q^{36} +9.13632 q^{37} -3.70476 q^{38} -0.901224 q^{39} -1.00000 q^{40} +5.04615 q^{41} +0.543362 q^{42} +5.78688 q^{43} -2.70476 q^{45} +4.99442 q^{46} -4.56444 q^{47} +0.543362 q^{48} +1.00000 q^{49} -1.00000 q^{50} +1.11180 q^{51} -1.65861 q^{52} +8.88262 q^{53} +3.09975 q^{54} +1.00000 q^{56} +2.01302 q^{57} -7.13632 q^{58} -14.0846 q^{59} +0.543362 q^{60} -3.30516 q^{61} +10.1809 q^{62} +2.70476 q^{63} +1.00000 q^{64} -1.65861 q^{65} -3.18992 q^{67} +2.04615 q^{68} -2.71378 q^{69} +1.00000 q^{70} -7.80795 q^{71} +2.70476 q^{72} -10.9046 q^{73} -9.13632 q^{74} +0.543362 q^{75} +3.70476 q^{76} +0.901224 q^{78} +11.2054 q^{79} +1.00000 q^{80} +6.42999 q^{81} -5.04615 q^{82} +4.07812 q^{83} -0.543362 q^{84} +2.04615 q^{85} -5.78688 q^{86} +3.87760 q^{87} -3.32679 q^{89} +2.70476 q^{90} +1.65861 q^{91} -4.99442 q^{92} -5.53191 q^{93} +4.56444 q^{94} +3.70476 q^{95} -0.543362 q^{96} +9.19639 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{8} + 6 q^{9} - 4 q^{10} - 4 q^{12} + 14 q^{13} + 4 q^{14} - 4 q^{15} + 4 q^{16} + 12 q^{17} - 6 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} + 4 q^{24} + 4 q^{25} - 14 q^{26} - 22 q^{27} - 4 q^{28} + 10 q^{29} + 4 q^{30} - 18 q^{31} - 4 q^{32} - 12 q^{34} - 4 q^{35} + 6 q^{36} + 18 q^{37} + 2 q^{38} - 30 q^{39} - 4 q^{40} + 24 q^{41} - 4 q^{42} + 10 q^{43} + 6 q^{45} - 8 q^{47} - 4 q^{48} + 4 q^{49} - 4 q^{50} + 14 q^{52} + 20 q^{53} + 22 q^{54} + 4 q^{56} + 30 q^{57} - 10 q^{58} - 14 q^{59} - 4 q^{60} + 14 q^{61} + 18 q^{62} - 6 q^{63} + 4 q^{64} + 14 q^{65} + 12 q^{68} - 4 q^{69} + 4 q^{70} - 14 q^{71} - 6 q^{72} + 30 q^{73} - 18 q^{74} - 4 q^{75} - 2 q^{76} + 30 q^{78} + 8 q^{79} + 4 q^{80} + 16 q^{81} - 24 q^{82} + 8 q^{83} + 4 q^{84} + 12 q^{85} - 10 q^{86} + 2 q^{87} - 4 q^{89} - 6 q^{90} - 14 q^{91} - 2 q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 10 q^{97} - 4 q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.543362 0.313710 0.156855 0.987622i \(-0.449864\pi\)
0.156855 + 0.987622i \(0.449864\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −0.543362 −0.221827
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.70476 −0.901586
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 0.543362 0.156855
\(13\) −1.65861 −0.460015 −0.230008 0.973189i \(-0.573875\pi\)
−0.230008 + 0.973189i \(0.573875\pi\)
\(14\) 1.00000 0.267261
\(15\) 0.543362 0.140295
\(16\) 1.00000 0.250000
\(17\) 2.04615 0.496264 0.248132 0.968726i \(-0.420183\pi\)
0.248132 + 0.968726i \(0.420183\pi\)
\(18\) 2.70476 0.637518
\(19\) 3.70476 0.849930 0.424965 0.905210i \(-0.360286\pi\)
0.424965 + 0.905210i \(0.360286\pi\)
\(20\) 1.00000 0.223607
\(21\) −0.543362 −0.118571
\(22\) 0 0
\(23\) −4.99442 −1.04141 −0.520705 0.853737i \(-0.674331\pi\)
−0.520705 + 0.853737i \(0.674331\pi\)
\(24\) −0.543362 −0.110913
\(25\) 1.00000 0.200000
\(26\) 1.65861 0.325280
\(27\) −3.09975 −0.596547
\(28\) −1.00000 −0.188982
\(29\) 7.13632 1.32518 0.662591 0.748982i \(-0.269457\pi\)
0.662591 + 0.748982i \(0.269457\pi\)
\(30\) −0.543362 −0.0992039
\(31\) −10.1809 −1.82854 −0.914271 0.405102i \(-0.867236\pi\)
−0.914271 + 0.405102i \(0.867236\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.04615 −0.350912
\(35\) −1.00000 −0.169031
\(36\) −2.70476 −0.450793
\(37\) 9.13632 1.50200 0.751001 0.660301i \(-0.229571\pi\)
0.751001 + 0.660301i \(0.229571\pi\)
\(38\) −3.70476 −0.600991
\(39\) −0.901224 −0.144311
\(40\) −1.00000 −0.158114
\(41\) 5.04615 0.788076 0.394038 0.919094i \(-0.371078\pi\)
0.394038 + 0.919094i \(0.371078\pi\)
\(42\) 0.543362 0.0838426
\(43\) 5.78688 0.882491 0.441245 0.897387i \(-0.354537\pi\)
0.441245 + 0.897387i \(0.354537\pi\)
\(44\) 0 0
\(45\) −2.70476 −0.403201
\(46\) 4.99442 0.736388
\(47\) −4.56444 −0.665791 −0.332896 0.942964i \(-0.608026\pi\)
−0.332896 + 0.942964i \(0.608026\pi\)
\(48\) 0.543362 0.0784275
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 1.11180 0.155683
\(52\) −1.65861 −0.230008
\(53\) 8.88262 1.22012 0.610061 0.792354i \(-0.291145\pi\)
0.610061 + 0.792354i \(0.291145\pi\)
\(54\) 3.09975 0.421822
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 2.01302 0.266632
\(58\) −7.13632 −0.937045
\(59\) −14.0846 −1.83366 −0.916829 0.399280i \(-0.869260\pi\)
−0.916829 + 0.399280i \(0.869260\pi\)
\(60\) 0.543362 0.0701477
\(61\) −3.30516 −0.423183 −0.211591 0.977358i \(-0.567865\pi\)
−0.211591 + 0.977358i \(0.567865\pi\)
\(62\) 10.1809 1.29298
\(63\) 2.70476 0.340767
\(64\) 1.00000 0.125000
\(65\) −1.65861 −0.205725
\(66\) 0 0
\(67\) −3.18992 −0.389711 −0.194855 0.980832i \(-0.562424\pi\)
−0.194855 + 0.980832i \(0.562424\pi\)
\(68\) 2.04615 0.248132
\(69\) −2.71378 −0.326701
\(70\) 1.00000 0.119523
\(71\) −7.80795 −0.926633 −0.463317 0.886193i \(-0.653341\pi\)
−0.463317 + 0.886193i \(0.653341\pi\)
\(72\) 2.70476 0.318759
\(73\) −10.9046 −1.27629 −0.638143 0.769918i \(-0.720297\pi\)
−0.638143 + 0.769918i \(0.720297\pi\)
\(74\) −9.13632 −1.06208
\(75\) 0.543362 0.0627420
\(76\) 3.70476 0.424965
\(77\) 0 0
\(78\) 0.901224 0.102044
\(79\) 11.2054 1.26071 0.630354 0.776308i \(-0.282910\pi\)
0.630354 + 0.776308i \(0.282910\pi\)
\(80\) 1.00000 0.111803
\(81\) 6.42999 0.714443
\(82\) −5.04615 −0.557254
\(83\) 4.07812 0.447632 0.223816 0.974631i \(-0.428149\pi\)
0.223816 + 0.974631i \(0.428149\pi\)
\(84\) −0.543362 −0.0592856
\(85\) 2.04615 0.221936
\(86\) −5.78688 −0.624015
\(87\) 3.87760 0.415723
\(88\) 0 0
\(89\) −3.32679 −0.352639 −0.176320 0.984333i \(-0.556419\pi\)
−0.176320 + 0.984333i \(0.556419\pi\)
\(90\) 2.70476 0.285107
\(91\) 1.65861 0.173869
\(92\) −4.99442 −0.520705
\(93\) −5.53191 −0.573632
\(94\) 4.56444 0.470786
\(95\) 3.70476 0.380100
\(96\) −0.543362 −0.0554566
\(97\) 9.19639 0.933752 0.466876 0.884323i \(-0.345379\pi\)
0.466876 + 0.884323i \(0.345379\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.73328 0.271972 0.135986 0.990711i \(-0.456580\pi\)
0.135986 + 0.990711i \(0.456580\pi\)
\(102\) −1.11180 −0.110085
\(103\) −1.54736 −0.152466 −0.0762332 0.997090i \(-0.524289\pi\)
−0.0762332 + 0.997090i \(0.524289\pi\)
\(104\) 1.65861 0.162640
\(105\) −0.543362 −0.0530267
\(106\) −8.88262 −0.862757
\(107\) 12.2089 1.18028 0.590138 0.807303i \(-0.299073\pi\)
0.590138 + 0.807303i \(0.299073\pi\)
\(108\) −3.09975 −0.298273
\(109\) −1.32376 −0.126794 −0.0633968 0.997988i \(-0.520193\pi\)
−0.0633968 + 0.997988i \(0.520193\pi\)
\(110\) 0 0
\(111\) 4.96433 0.471193
\(112\) −1.00000 −0.0944911
\(113\) 3.45851 0.325349 0.162675 0.986680i \(-0.447988\pi\)
0.162675 + 0.986680i \(0.447988\pi\)
\(114\) −2.01302 −0.188537
\(115\) −4.99442 −0.465732
\(116\) 7.13632 0.662591
\(117\) 4.48613 0.414743
\(118\) 14.0846 1.29659
\(119\) −2.04615 −0.187570
\(120\) −0.543362 −0.0496019
\(121\) 0 0
\(122\) 3.30516 0.299236
\(123\) 2.74189 0.247228
\(124\) −10.1809 −0.914271
\(125\) 1.00000 0.0894427
\(126\) −2.70476 −0.240959
\(127\) −18.2239 −1.61711 −0.808557 0.588418i \(-0.799751\pi\)
−0.808557 + 0.588418i \(0.799751\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.14437 0.276846
\(130\) 1.65861 0.145470
\(131\) −9.47001 −0.827398 −0.413699 0.910414i \(-0.635763\pi\)
−0.413699 + 0.910414i \(0.635763\pi\)
\(132\) 0 0
\(133\) −3.70476 −0.321243
\(134\) 3.18992 0.275567
\(135\) −3.09975 −0.266784
\(136\) −2.04615 −0.175456
\(137\) 11.4807 0.980866 0.490433 0.871479i \(-0.336839\pi\)
0.490433 + 0.871479i \(0.336839\pi\)
\(138\) 2.71378 0.231012
\(139\) −4.86312 −0.412485 −0.206242 0.978501i \(-0.566124\pi\)
−0.206242 + 0.978501i \(0.566124\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −2.48014 −0.208866
\(142\) 7.80795 0.655229
\(143\) 0 0
\(144\) −2.70476 −0.225396
\(145\) 7.13632 0.592639
\(146\) 10.9046 0.902471
\(147\) 0.543362 0.0448157
\(148\) 9.13632 0.751001
\(149\) 7.60901 0.623355 0.311677 0.950188i \(-0.399109\pi\)
0.311677 + 0.950188i \(0.399109\pi\)
\(150\) −0.543362 −0.0443653
\(151\) 10.8640 0.884102 0.442051 0.896990i \(-0.354251\pi\)
0.442051 + 0.896990i \(0.354251\pi\)
\(152\) −3.70476 −0.300496
\(153\) −5.53434 −0.447425
\(154\) 0 0
\(155\) −10.1809 −0.817749
\(156\) −0.901224 −0.0721557
\(157\) 9.67811 0.772397 0.386199 0.922416i \(-0.373788\pi\)
0.386199 + 0.922416i \(0.373788\pi\)
\(158\) −11.2054 −0.891455
\(159\) 4.82648 0.382765
\(160\) −1.00000 −0.0790569
\(161\) 4.99442 0.393616
\(162\) −6.42999 −0.505188
\(163\) 16.3076 1.27731 0.638656 0.769492i \(-0.279491\pi\)
0.638656 + 0.769492i \(0.279491\pi\)
\(164\) 5.04615 0.394038
\(165\) 0 0
\(166\) −4.07812 −0.316523
\(167\) −9.49351 −0.734630 −0.367315 0.930097i \(-0.619723\pi\)
−0.367315 + 0.930097i \(0.619723\pi\)
\(168\) 0.543362 0.0419213
\(169\) −10.2490 −0.788386
\(170\) −2.04615 −0.156933
\(171\) −10.0205 −0.766285
\(172\) 5.78688 0.441245
\(173\) 1.44017 0.109494 0.0547470 0.998500i \(-0.482565\pi\)
0.0547470 + 0.998500i \(0.482565\pi\)
\(174\) −3.87760 −0.293960
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −7.65303 −0.575237
\(178\) 3.32679 0.249354
\(179\) −9.52847 −0.712191 −0.356095 0.934450i \(-0.615892\pi\)
−0.356095 + 0.934450i \(0.615892\pi\)
\(180\) −2.70476 −0.201601
\(181\) −11.5431 −0.857992 −0.428996 0.903306i \(-0.641133\pi\)
−0.428996 + 0.903306i \(0.641133\pi\)
\(182\) −1.65861 −0.122944
\(183\) −1.79590 −0.132757
\(184\) 4.99442 0.368194
\(185\) 9.13632 0.671716
\(186\) 5.53191 0.405619
\(187\) 0 0
\(188\) −4.56444 −0.332896
\(189\) 3.09975 0.225474
\(190\) −3.70476 −0.268771
\(191\) 25.8516 1.87055 0.935276 0.353918i \(-0.115151\pi\)
0.935276 + 0.353918i \(0.115151\pi\)
\(192\) 0.543362 0.0392138
\(193\) 25.6289 1.84481 0.922403 0.386230i \(-0.126223\pi\)
0.922403 + 0.386230i \(0.126223\pi\)
\(194\) −9.19639 −0.660263
\(195\) −0.901224 −0.0645380
\(196\) 1.00000 0.0714286
\(197\) 10.5223 0.749682 0.374841 0.927089i \(-0.377697\pi\)
0.374841 + 0.927089i \(0.377697\pi\)
\(198\) 0 0
\(199\) 18.3423 1.30025 0.650125 0.759827i \(-0.274716\pi\)
0.650125 + 0.759827i \(0.274716\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −1.73328 −0.122256
\(202\) −2.73328 −0.192313
\(203\) −7.13632 −0.500871
\(204\) 1.11180 0.0778416
\(205\) 5.04615 0.352438
\(206\) 1.54736 0.107810
\(207\) 13.5087 0.938920
\(208\) −1.65861 −0.115004
\(209\) 0 0
\(210\) 0.543362 0.0374955
\(211\) −23.6304 −1.62679 −0.813393 0.581715i \(-0.802382\pi\)
−0.813393 + 0.581715i \(0.802382\pi\)
\(212\) 8.88262 0.610061
\(213\) −4.24254 −0.290694
\(214\) −12.2089 −0.834581
\(215\) 5.78688 0.394662
\(216\) 3.09975 0.210911
\(217\) 10.1809 0.691124
\(218\) 1.32376 0.0896566
\(219\) −5.92514 −0.400384
\(220\) 0 0
\(221\) −3.39376 −0.228289
\(222\) −4.96433 −0.333184
\(223\) 2.76854 0.185395 0.0926974 0.995694i \(-0.470451\pi\)
0.0926974 + 0.995694i \(0.470451\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.70476 −0.180317
\(226\) −3.45851 −0.230057
\(227\) 1.59296 0.105728 0.0528642 0.998602i \(-0.483165\pi\)
0.0528642 + 0.998602i \(0.483165\pi\)
\(228\) 2.01302 0.133316
\(229\) 23.5747 1.55786 0.778929 0.627113i \(-0.215763\pi\)
0.778929 + 0.627113i \(0.215763\pi\)
\(230\) 4.99442 0.329323
\(231\) 0 0
\(232\) −7.13632 −0.468522
\(233\) 3.72239 0.243862 0.121931 0.992539i \(-0.461091\pi\)
0.121931 + 0.992539i \(0.461091\pi\)
\(234\) −4.48613 −0.293268
\(235\) −4.56444 −0.297751
\(236\) −14.0846 −0.916829
\(237\) 6.08860 0.395497
\(238\) 2.04615 0.132632
\(239\) 15.0088 0.970836 0.485418 0.874282i \(-0.338667\pi\)
0.485418 + 0.874282i \(0.338667\pi\)
\(240\) 0.543362 0.0350739
\(241\) 23.4709 1.51189 0.755947 0.654633i \(-0.227177\pi\)
0.755947 + 0.654633i \(0.227177\pi\)
\(242\) 0 0
\(243\) 12.7931 0.820675
\(244\) −3.30516 −0.211591
\(245\) 1.00000 0.0638877
\(246\) −2.74189 −0.174816
\(247\) −6.14474 −0.390980
\(248\) 10.1809 0.646488
\(249\) 2.21589 0.140427
\(250\) −1.00000 −0.0632456
\(251\) 23.8705 1.50669 0.753346 0.657624i \(-0.228438\pi\)
0.753346 + 0.657624i \(0.228438\pi\)
\(252\) 2.70476 0.170384
\(253\) 0 0
\(254\) 18.2239 1.14347
\(255\) 1.11180 0.0696236
\(256\) 1.00000 0.0625000
\(257\) 3.13935 0.195827 0.0979136 0.995195i \(-0.468783\pi\)
0.0979136 + 0.995195i \(0.468783\pi\)
\(258\) −3.14437 −0.195760
\(259\) −9.13632 −0.567703
\(260\) −1.65861 −0.102862
\(261\) −19.3020 −1.19476
\(262\) 9.47001 0.585059
\(263\) 18.0997 1.11608 0.558039 0.829815i \(-0.311554\pi\)
0.558039 + 0.829815i \(0.311554\pi\)
\(264\) 0 0
\(265\) 8.88262 0.545655
\(266\) 3.70476 0.227153
\(267\) −1.80765 −0.110627
\(268\) −3.18992 −0.194855
\(269\) −5.65771 −0.344957 −0.172478 0.985013i \(-0.555177\pi\)
−0.172478 + 0.985013i \(0.555177\pi\)
\(270\) 3.09975 0.188645
\(271\) 31.7788 1.93042 0.965211 0.261472i \(-0.0842080\pi\)
0.965211 + 0.261472i \(0.0842080\pi\)
\(272\) 2.04615 0.124066
\(273\) 0.901224 0.0545446
\(274\) −11.4807 −0.693577
\(275\) 0 0
\(276\) −2.71378 −0.163350
\(277\) −2.40678 −0.144610 −0.0723048 0.997383i \(-0.523035\pi\)
−0.0723048 + 0.997383i \(0.523035\pi\)
\(278\) 4.86312 0.291671
\(279\) 27.5369 1.64859
\(280\) 1.00000 0.0597614
\(281\) 11.2228 0.669495 0.334748 0.942308i \(-0.391349\pi\)
0.334748 + 0.942308i \(0.391349\pi\)
\(282\) 2.48014 0.147690
\(283\) 31.2559 1.85797 0.928986 0.370116i \(-0.120682\pi\)
0.928986 + 0.370116i \(0.120682\pi\)
\(284\) −7.80795 −0.463317
\(285\) 2.01302 0.119241
\(286\) 0 0
\(287\) −5.04615 −0.297865
\(288\) 2.70476 0.159379
\(289\) −12.8133 −0.753722
\(290\) −7.13632 −0.419059
\(291\) 4.99697 0.292928
\(292\) −10.9046 −0.638143
\(293\) 2.27230 0.132749 0.0663745 0.997795i \(-0.478857\pi\)
0.0663745 + 0.997795i \(0.478857\pi\)
\(294\) −0.543362 −0.0316895
\(295\) −14.0846 −0.820037
\(296\) −9.13632 −0.531038
\(297\) 0 0
\(298\) −7.60901 −0.440778
\(299\) 8.28379 0.479064
\(300\) 0.543362 0.0313710
\(301\) −5.78688 −0.333550
\(302\) −10.8640 −0.625154
\(303\) 1.48516 0.0853202
\(304\) 3.70476 0.212482
\(305\) −3.30516 −0.189253
\(306\) 5.53434 0.316377
\(307\) 5.55583 0.317088 0.158544 0.987352i \(-0.449320\pi\)
0.158544 + 0.987352i \(0.449320\pi\)
\(308\) 0 0
\(309\) −0.840779 −0.0478302
\(310\) 10.1809 0.578236
\(311\) −20.3446 −1.15364 −0.576818 0.816873i \(-0.695706\pi\)
−0.576818 + 0.816873i \(0.695706\pi\)
\(312\) 0.901224 0.0510218
\(313\) 28.0901 1.58775 0.793873 0.608083i \(-0.208061\pi\)
0.793873 + 0.608083i \(0.208061\pi\)
\(314\) −9.67811 −0.546167
\(315\) 2.70476 0.152396
\(316\) 11.2054 0.630354
\(317\) 7.61646 0.427783 0.213892 0.976857i \(-0.431386\pi\)
0.213892 + 0.976857i \(0.431386\pi\)
\(318\) −4.82648 −0.270656
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 6.63383 0.370264
\(322\) −4.99442 −0.278328
\(323\) 7.58049 0.421790
\(324\) 6.42999 0.357222
\(325\) −1.65861 −0.0920030
\(326\) −16.3076 −0.903197
\(327\) −0.719283 −0.0397764
\(328\) −5.04615 −0.278627
\(329\) 4.56444 0.251645
\(330\) 0 0
\(331\) 9.86518 0.542239 0.271120 0.962546i \(-0.412606\pi\)
0.271120 + 0.962546i \(0.412606\pi\)
\(332\) 4.07812 0.223816
\(333\) −24.7115 −1.35418
\(334\) 9.49351 0.519462
\(335\) −3.18992 −0.174284
\(336\) −0.543362 −0.0296428
\(337\) −22.8921 −1.24701 −0.623507 0.781818i \(-0.714293\pi\)
−0.623507 + 0.781818i \(0.714293\pi\)
\(338\) 10.2490 0.557473
\(339\) 1.87922 0.102065
\(340\) 2.04615 0.110968
\(341\) 0 0
\(342\) 10.0205 0.541845
\(343\) −1.00000 −0.0539949
\(344\) −5.78688 −0.312008
\(345\) −2.71378 −0.146105
\(346\) −1.44017 −0.0774239
\(347\) −8.07715 −0.433604 −0.216802 0.976216i \(-0.569563\pi\)
−0.216802 + 0.976216i \(0.569563\pi\)
\(348\) 3.87760 0.207861
\(349\) 16.3618 0.875827 0.437913 0.899017i \(-0.355718\pi\)
0.437913 + 0.899017i \(0.355718\pi\)
\(350\) 1.00000 0.0534522
\(351\) 5.14127 0.274421
\(352\) 0 0
\(353\) −18.5642 −0.988072 −0.494036 0.869442i \(-0.664479\pi\)
−0.494036 + 0.869442i \(0.664479\pi\)
\(354\) 7.65303 0.406754
\(355\) −7.80795 −0.414403
\(356\) −3.32679 −0.176320
\(357\) −1.11180 −0.0588427
\(358\) 9.52847 0.503595
\(359\) 4.59048 0.242276 0.121138 0.992636i \(-0.461346\pi\)
0.121138 + 0.992636i \(0.461346\pi\)
\(360\) 2.70476 0.142553
\(361\) −5.27477 −0.277619
\(362\) 11.5431 0.606692
\(363\) 0 0
\(364\) 1.65861 0.0869347
\(365\) −10.9046 −0.570773
\(366\) 1.79590 0.0938732
\(367\) 12.7694 0.666559 0.333280 0.942828i \(-0.391845\pi\)
0.333280 + 0.942828i \(0.391845\pi\)
\(368\) −4.99442 −0.260352
\(369\) −13.6486 −0.710518
\(370\) −9.13632 −0.474975
\(371\) −8.88262 −0.461163
\(372\) −5.53191 −0.286816
\(373\) −22.4340 −1.16159 −0.580795 0.814050i \(-0.697258\pi\)
−0.580795 + 0.814050i \(0.697258\pi\)
\(374\) 0 0
\(375\) 0.543362 0.0280591
\(376\) 4.56444 0.235393
\(377\) −11.8364 −0.609603
\(378\) −3.09975 −0.159434
\(379\) −30.1347 −1.54791 −0.773957 0.633238i \(-0.781725\pi\)
−0.773957 + 0.633238i \(0.781725\pi\)
\(380\) 3.70476 0.190050
\(381\) −9.90220 −0.507305
\(382\) −25.8516 −1.32268
\(383\) 27.9554 1.42845 0.714226 0.699915i \(-0.246779\pi\)
0.714226 + 0.699915i \(0.246779\pi\)
\(384\) −0.543362 −0.0277283
\(385\) 0 0
\(386\) −25.6289 −1.30447
\(387\) −15.6521 −0.795641
\(388\) 9.19639 0.466876
\(389\) 4.57649 0.232037 0.116019 0.993247i \(-0.462987\pi\)
0.116019 + 0.993247i \(0.462987\pi\)
\(390\) 0.901224 0.0456353
\(391\) −10.2193 −0.516814
\(392\) −1.00000 −0.0505076
\(393\) −5.14564 −0.259563
\(394\) −10.5223 −0.530105
\(395\) 11.2054 0.563806
\(396\) 0 0
\(397\) −37.9390 −1.90411 −0.952053 0.305933i \(-0.901032\pi\)
−0.952053 + 0.305933i \(0.901032\pi\)
\(398\) −18.3423 −0.919416
\(399\) −2.01302 −0.100777
\(400\) 1.00000 0.0500000
\(401\) 15.9557 0.796791 0.398395 0.917214i \(-0.369567\pi\)
0.398395 + 0.917214i \(0.369567\pi\)
\(402\) 1.73328 0.0864482
\(403\) 16.8861 0.841157
\(404\) 2.73328 0.135986
\(405\) 6.42999 0.319509
\(406\) 7.13632 0.354170
\(407\) 0 0
\(408\) −1.11180 −0.0550423
\(409\) −24.2684 −1.19999 −0.599997 0.800002i \(-0.704832\pi\)
−0.599997 + 0.800002i \(0.704832\pi\)
\(410\) −5.04615 −0.249212
\(411\) 6.23820 0.307708
\(412\) −1.54736 −0.0762332
\(413\) 14.0846 0.693058
\(414\) −13.5087 −0.663917
\(415\) 4.07812 0.200187
\(416\) 1.65861 0.0813199
\(417\) −2.64244 −0.129401
\(418\) 0 0
\(419\) −25.6838 −1.25473 −0.627367 0.778724i \(-0.715867\pi\)
−0.627367 + 0.778724i \(0.715867\pi\)
\(420\) −0.543362 −0.0265133
\(421\) −9.36362 −0.456355 −0.228178 0.973620i \(-0.573277\pi\)
−0.228178 + 0.973620i \(0.573277\pi\)
\(422\) 23.6304 1.15031
\(423\) 12.3457 0.600268
\(424\) −8.88262 −0.431378
\(425\) 2.04615 0.0992528
\(426\) 4.24254 0.205552
\(427\) 3.30516 0.159948
\(428\) 12.2089 0.590138
\(429\) 0 0
\(430\) −5.78688 −0.279068
\(431\) 16.9902 0.818387 0.409194 0.912448i \(-0.365810\pi\)
0.409194 + 0.912448i \(0.365810\pi\)
\(432\) −3.09975 −0.149137
\(433\) 6.70663 0.322300 0.161150 0.986930i \(-0.448480\pi\)
0.161150 + 0.986930i \(0.448480\pi\)
\(434\) −10.1809 −0.488699
\(435\) 3.87760 0.185917
\(436\) −1.32376 −0.0633968
\(437\) −18.5031 −0.885125
\(438\) 5.92514 0.283114
\(439\) 12.8474 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(440\) 0 0
\(441\) −2.70476 −0.128798
\(442\) 3.39376 0.161425
\(443\) 0.678365 0.0322301 0.0161151 0.999870i \(-0.494870\pi\)
0.0161151 + 0.999870i \(0.494870\pi\)
\(444\) 4.96433 0.235597
\(445\) −3.32679 −0.157705
\(446\) −2.76854 −0.131094
\(447\) 4.13445 0.195553
\(448\) −1.00000 −0.0472456
\(449\) 7.59525 0.358442 0.179221 0.983809i \(-0.442642\pi\)
0.179221 + 0.983809i \(0.442642\pi\)
\(450\) 2.70476 0.127504
\(451\) 0 0
\(452\) 3.45851 0.162675
\(453\) 5.90310 0.277352
\(454\) −1.59296 −0.0747612
\(455\) 1.65861 0.0777567
\(456\) −2.01302 −0.0942685
\(457\) −33.0502 −1.54602 −0.773012 0.634391i \(-0.781251\pi\)
−0.773012 + 0.634391i \(0.781251\pi\)
\(458\) −23.5747 −1.10157
\(459\) −6.34255 −0.296045
\(460\) −4.99442 −0.232866
\(461\) 20.2500 0.943136 0.471568 0.881830i \(-0.343688\pi\)
0.471568 + 0.881830i \(0.343688\pi\)
\(462\) 0 0
\(463\) 39.3213 1.82742 0.913708 0.406371i \(-0.133206\pi\)
0.913708 + 0.406371i \(0.133206\pi\)
\(464\) 7.13632 0.331295
\(465\) −5.53191 −0.256536
\(466\) −3.72239 −0.172436
\(467\) −18.8167 −0.870734 −0.435367 0.900253i \(-0.643381\pi\)
−0.435367 + 0.900253i \(0.643381\pi\)
\(468\) 4.48613 0.207372
\(469\) 3.18992 0.147297
\(470\) 4.56444 0.210542
\(471\) 5.25872 0.242309
\(472\) 14.0846 0.648296
\(473\) 0 0
\(474\) −6.08860 −0.279658
\(475\) 3.70476 0.169986
\(476\) −2.04615 −0.0937851
\(477\) −24.0253 −1.10005
\(478\) −15.0088 −0.686485
\(479\) −7.56631 −0.345713 −0.172857 0.984947i \(-0.555300\pi\)
−0.172857 + 0.984947i \(0.555300\pi\)
\(480\) −0.543362 −0.0248010
\(481\) −15.1536 −0.690943
\(482\) −23.4709 −1.06907
\(483\) 2.71378 0.123481
\(484\) 0 0
\(485\) 9.19639 0.417587
\(486\) −12.7931 −0.580305
\(487\) 7.92350 0.359048 0.179524 0.983754i \(-0.442544\pi\)
0.179524 + 0.983754i \(0.442544\pi\)
\(488\) 3.30516 0.149618
\(489\) 8.86095 0.400706
\(490\) −1.00000 −0.0451754
\(491\) −15.1101 −0.681909 −0.340954 0.940080i \(-0.610750\pi\)
−0.340954 + 0.940080i \(0.610750\pi\)
\(492\) 2.74189 0.123614
\(493\) 14.6020 0.657640
\(494\) 6.14474 0.276465
\(495\) 0 0
\(496\) −10.1809 −0.457136
\(497\) 7.80795 0.350235
\(498\) −2.21589 −0.0992966
\(499\) −5.96549 −0.267052 −0.133526 0.991045i \(-0.542630\pi\)
−0.133526 + 0.991045i \(0.542630\pi\)
\(500\) 1.00000 0.0447214
\(501\) −5.15841 −0.230461
\(502\) −23.8705 −1.06539
\(503\) −25.5217 −1.13795 −0.568977 0.822353i \(-0.692661\pi\)
−0.568977 + 0.822353i \(0.692661\pi\)
\(504\) −2.70476 −0.120479
\(505\) 2.73328 0.121629
\(506\) 0 0
\(507\) −5.56893 −0.247325
\(508\) −18.2239 −0.808557
\(509\) −27.7639 −1.23061 −0.615306 0.788289i \(-0.710967\pi\)
−0.615306 + 0.788289i \(0.710967\pi\)
\(510\) −1.11180 −0.0492313
\(511\) 10.9046 0.482391
\(512\) −1.00000 −0.0441942
\(513\) −11.4838 −0.507023
\(514\) −3.13935 −0.138471
\(515\) −1.54736 −0.0681850
\(516\) 3.14437 0.138423
\(517\) 0 0
\(518\) 9.13632 0.401427
\(519\) 0.782532 0.0343494
\(520\) 1.65861 0.0727348
\(521\) 23.8699 1.04576 0.522880 0.852407i \(-0.324858\pi\)
0.522880 + 0.852407i \(0.324858\pi\)
\(522\) 19.3020 0.844826
\(523\) 2.38789 0.104415 0.0522075 0.998636i \(-0.483374\pi\)
0.0522075 + 0.998636i \(0.483374\pi\)
\(524\) −9.47001 −0.413699
\(525\) −0.543362 −0.0237143
\(526\) −18.0997 −0.789187
\(527\) −20.8316 −0.907440
\(528\) 0 0
\(529\) 1.94427 0.0845336
\(530\) −8.88262 −0.385837
\(531\) 38.0954 1.65320
\(532\) −3.70476 −0.160622
\(533\) −8.36958 −0.362527
\(534\) 1.80765 0.0782248
\(535\) 12.2089 0.527835
\(536\) 3.18992 0.137783
\(537\) −5.17741 −0.223422
\(538\) 5.65771 0.243921
\(539\) 0 0
\(540\) −3.09975 −0.133392
\(541\) 24.8483 1.06831 0.534155 0.845386i \(-0.320630\pi\)
0.534155 + 0.845386i \(0.320630\pi\)
\(542\) −31.7788 −1.36501
\(543\) −6.27208 −0.269161
\(544\) −2.04615 −0.0877280
\(545\) −1.32376 −0.0567038
\(546\) −0.901224 −0.0385688
\(547\) 33.1504 1.41741 0.708704 0.705506i \(-0.249280\pi\)
0.708704 + 0.705506i \(0.249280\pi\)
\(548\) 11.4807 0.490433
\(549\) 8.93967 0.381536
\(550\) 0 0
\(551\) 26.4383 1.12631
\(552\) 2.71378 0.115506
\(553\) −11.2054 −0.476503
\(554\) 2.40678 0.102254
\(555\) 4.96433 0.210724
\(556\) −4.86312 −0.206242
\(557\) 28.0256 1.18748 0.593741 0.804657i \(-0.297651\pi\)
0.593741 + 0.804657i \(0.297651\pi\)
\(558\) −27.5369 −1.16573
\(559\) −9.59816 −0.405959
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −11.2228 −0.473405
\(563\) 3.17711 0.133899 0.0669495 0.997756i \(-0.478673\pi\)
0.0669495 + 0.997756i \(0.478673\pi\)
\(564\) −2.48014 −0.104433
\(565\) 3.45851 0.145501
\(566\) −31.2559 −1.31378
\(567\) −6.42999 −0.270034
\(568\) 7.80795 0.327614
\(569\) 18.7562 0.786302 0.393151 0.919474i \(-0.371385\pi\)
0.393151 + 0.919474i \(0.371385\pi\)
\(570\) −2.01302 −0.0843163
\(571\) −20.5329 −0.859276 −0.429638 0.903001i \(-0.641359\pi\)
−0.429638 + 0.903001i \(0.641359\pi\)
\(572\) 0 0
\(573\) 14.0468 0.586811
\(574\) 5.04615 0.210622
\(575\) −4.99442 −0.208282
\(576\) −2.70476 −0.112698
\(577\) 16.7729 0.698264 0.349132 0.937074i \(-0.386476\pi\)
0.349132 + 0.937074i \(0.386476\pi\)
\(578\) 12.8133 0.532962
\(579\) 13.9257 0.578734
\(580\) 7.13632 0.296320
\(581\) −4.07812 −0.169189
\(582\) −4.99697 −0.207131
\(583\) 0 0
\(584\) 10.9046 0.451235
\(585\) 4.48613 0.185479
\(586\) −2.27230 −0.0938677
\(587\) 25.0595 1.03432 0.517158 0.855890i \(-0.326990\pi\)
0.517158 + 0.855890i \(0.326990\pi\)
\(588\) 0.543362 0.0224079
\(589\) −37.7178 −1.55413
\(590\) 14.0846 0.579854
\(591\) 5.71741 0.235183
\(592\) 9.13632 0.375500
\(593\) 44.2586 1.81748 0.908741 0.417360i \(-0.137044\pi\)
0.908741 + 0.417360i \(0.137044\pi\)
\(594\) 0 0
\(595\) −2.04615 −0.0838840
\(596\) 7.60901 0.311677
\(597\) 9.96650 0.407902
\(598\) −8.28379 −0.338749
\(599\) 2.39133 0.0977072 0.0488536 0.998806i \(-0.484443\pi\)
0.0488536 + 0.998806i \(0.484443\pi\)
\(600\) −0.543362 −0.0221827
\(601\) −34.6639 −1.41397 −0.706984 0.707229i \(-0.749945\pi\)
−0.706984 + 0.707229i \(0.749945\pi\)
\(602\) 5.78688 0.235856
\(603\) 8.62796 0.351358
\(604\) 10.8640 0.442051
\(605\) 0 0
\(606\) −1.48516 −0.0603305
\(607\) 27.2676 1.10676 0.553379 0.832929i \(-0.313338\pi\)
0.553379 + 0.832929i \(0.313338\pi\)
\(608\) −3.70476 −0.150248
\(609\) −3.87760 −0.157128
\(610\) 3.30516 0.133822
\(611\) 7.57061 0.306274
\(612\) −5.53434 −0.223712
\(613\) 0.453748 0.0183267 0.00916336 0.999958i \(-0.497083\pi\)
0.00916336 + 0.999958i \(0.497083\pi\)
\(614\) −5.55583 −0.224215
\(615\) 2.74189 0.110564
\(616\) 0 0
\(617\) 35.9399 1.44688 0.723442 0.690385i \(-0.242559\pi\)
0.723442 + 0.690385i \(0.242559\pi\)
\(618\) 0.840779 0.0338211
\(619\) −44.4805 −1.78782 −0.893912 0.448243i \(-0.852050\pi\)
−0.893912 + 0.448243i \(0.852050\pi\)
\(620\) −10.1809 −0.408875
\(621\) 15.4815 0.621249
\(622\) 20.3446 0.815743
\(623\) 3.32679 0.133285
\(624\) −0.901224 −0.0360778
\(625\) 1.00000 0.0400000
\(626\) −28.0901 −1.12271
\(627\) 0 0
\(628\) 9.67811 0.386199
\(629\) 18.6943 0.745390
\(630\) −2.70476 −0.107760
\(631\) 19.3612 0.770756 0.385378 0.922759i \(-0.374071\pi\)
0.385378 + 0.922759i \(0.374071\pi\)
\(632\) −11.2054 −0.445728
\(633\) −12.8399 −0.510339
\(634\) −7.61646 −0.302488
\(635\) −18.2239 −0.723195
\(636\) 4.82648 0.191382
\(637\) −1.65861 −0.0657164
\(638\) 0 0
\(639\) 21.1186 0.835440
\(640\) −1.00000 −0.0395285
\(641\) 24.0572 0.950203 0.475102 0.879931i \(-0.342411\pi\)
0.475102 + 0.879931i \(0.342411\pi\)
\(642\) −6.63383 −0.261816
\(643\) −14.4357 −0.569286 −0.284643 0.958634i \(-0.591875\pi\)
−0.284643 + 0.958634i \(0.591875\pi\)
\(644\) 4.99442 0.196808
\(645\) 3.14437 0.123809
\(646\) −7.58049 −0.298250
\(647\) −14.6590 −0.576306 −0.288153 0.957584i \(-0.593041\pi\)
−0.288153 + 0.957584i \(0.593041\pi\)
\(648\) −6.42999 −0.252594
\(649\) 0 0
\(650\) 1.65861 0.0650560
\(651\) 5.53191 0.216813
\(652\) 16.3076 0.638656
\(653\) 24.1399 0.944668 0.472334 0.881420i \(-0.343412\pi\)
0.472334 + 0.881420i \(0.343412\pi\)
\(654\) 0.719283 0.0281262
\(655\) −9.47001 −0.370024
\(656\) 5.04615 0.197019
\(657\) 29.4943 1.15068
\(658\) −4.56444 −0.177940
\(659\) −25.1109 −0.978182 −0.489091 0.872233i \(-0.662671\pi\)
−0.489091 + 0.872233i \(0.662671\pi\)
\(660\) 0 0
\(661\) −29.7048 −1.15538 −0.577692 0.816255i \(-0.696047\pi\)
−0.577692 + 0.816255i \(0.696047\pi\)
\(662\) −9.86518 −0.383421
\(663\) −1.84404 −0.0716166
\(664\) −4.07812 −0.158262
\(665\) −3.70476 −0.143664
\(666\) 24.7115 0.957552
\(667\) −35.6418 −1.38006
\(668\) −9.49351 −0.367315
\(669\) 1.50432 0.0581602
\(670\) 3.18992 0.123237
\(671\) 0 0
\(672\) 0.543362 0.0209606
\(673\) −49.0571 −1.89101 −0.945505 0.325607i \(-0.894431\pi\)
−0.945505 + 0.325607i \(0.894431\pi\)
\(674\) 22.8921 0.881772
\(675\) −3.09975 −0.119309
\(676\) −10.2490 −0.394193
\(677\) 27.6246 1.06170 0.530849 0.847466i \(-0.321873\pi\)
0.530849 + 0.847466i \(0.321873\pi\)
\(678\) −1.87922 −0.0721711
\(679\) −9.19639 −0.352925
\(680\) −2.04615 −0.0784663
\(681\) 0.865553 0.0331681
\(682\) 0 0
\(683\) 6.50934 0.249073 0.124536 0.992215i \(-0.460256\pi\)
0.124536 + 0.992215i \(0.460256\pi\)
\(684\) −10.0205 −0.383142
\(685\) 11.4807 0.438657
\(686\) 1.00000 0.0381802
\(687\) 12.8096 0.488716
\(688\) 5.78688 0.220623
\(689\) −14.7328 −0.561275
\(690\) 2.71378 0.103312
\(691\) 19.8453 0.754949 0.377475 0.926020i \(-0.376793\pi\)
0.377475 + 0.926020i \(0.376793\pi\)
\(692\) 1.44017 0.0547470
\(693\) 0 0
\(694\) 8.07715 0.306604
\(695\) −4.86312 −0.184469
\(696\) −3.87760 −0.146980
\(697\) 10.3252 0.391094
\(698\) −16.3618 −0.619303
\(699\) 2.02260 0.0765019
\(700\) −1.00000 −0.0377964
\(701\) −9.40172 −0.355098 −0.177549 0.984112i \(-0.556817\pi\)
−0.177549 + 0.984112i \(0.556817\pi\)
\(702\) −5.14127 −0.194045
\(703\) 33.8479 1.27660
\(704\) 0 0
\(705\) −2.48014 −0.0934075
\(706\) 18.5642 0.698672
\(707\) −2.73328 −0.102796
\(708\) −7.65303 −0.287619
\(709\) −26.9787 −1.01321 −0.506603 0.862180i \(-0.669099\pi\)
−0.506603 + 0.862180i \(0.669099\pi\)
\(710\) 7.80795 0.293027
\(711\) −30.3079 −1.13664
\(712\) 3.32679 0.124677
\(713\) 50.8477 1.90426
\(714\) 1.11180 0.0416081
\(715\) 0 0
\(716\) −9.52847 −0.356095
\(717\) 8.15519 0.304561
\(718\) −4.59048 −0.171315
\(719\) −23.7054 −0.884061 −0.442030 0.897000i \(-0.645742\pi\)
−0.442030 + 0.897000i \(0.645742\pi\)
\(720\) −2.70476 −0.100800
\(721\) 1.54736 0.0576269
\(722\) 5.27477 0.196307
\(723\) 12.7532 0.474296
\(724\) −11.5431 −0.428996
\(725\) 7.13632 0.265036
\(726\) 0 0
\(727\) −42.5192 −1.57695 −0.788475 0.615066i \(-0.789129\pi\)
−0.788475 + 0.615066i \(0.789129\pi\)
\(728\) −1.65861 −0.0614721
\(729\) −12.3387 −0.456989
\(730\) 10.9046 0.403597
\(731\) 11.8408 0.437949
\(732\) −1.79590 −0.0663784
\(733\) −4.52655 −0.167192 −0.0835959 0.996500i \(-0.526641\pi\)
−0.0835959 + 0.996500i \(0.526641\pi\)
\(734\) −12.7694 −0.471328
\(735\) 0.543362 0.0200422
\(736\) 4.99442 0.184097
\(737\) 0 0
\(738\) 13.6486 0.502412
\(739\) 31.4047 1.15524 0.577619 0.816306i \(-0.303982\pi\)
0.577619 + 0.816306i \(0.303982\pi\)
\(740\) 9.13632 0.335858
\(741\) −3.33882 −0.122655
\(742\) 8.88262 0.326091
\(743\) 9.01673 0.330792 0.165396 0.986227i \(-0.447110\pi\)
0.165396 + 0.986227i \(0.447110\pi\)
\(744\) 5.53191 0.202810
\(745\) 7.60901 0.278773
\(746\) 22.4340 0.821368
\(747\) −11.0303 −0.403578
\(748\) 0 0
\(749\) −12.2089 −0.446102
\(750\) −0.543362 −0.0198408
\(751\) 8.52599 0.311118 0.155559 0.987827i \(-0.450282\pi\)
0.155559 + 0.987827i \(0.450282\pi\)
\(752\) −4.56444 −0.166448
\(753\) 12.9703 0.472665
\(754\) 11.8364 0.431055
\(755\) 10.8640 0.395382
\(756\) 3.09975 0.112737
\(757\) 31.4517 1.14313 0.571566 0.820556i \(-0.306336\pi\)
0.571566 + 0.820556i \(0.306336\pi\)
\(758\) 30.1347 1.09454
\(759\) 0 0
\(760\) −3.70476 −0.134386
\(761\) 49.4950 1.79419 0.897096 0.441835i \(-0.145672\pi\)
0.897096 + 0.441835i \(0.145672\pi\)
\(762\) 9.90220 0.358719
\(763\) 1.32376 0.0479235
\(764\) 25.8516 0.935276
\(765\) −5.53434 −0.200094
\(766\) −27.9554 −1.01007
\(767\) 23.3608 0.843510
\(768\) 0.543362 0.0196069
\(769\) −32.2787 −1.16400 −0.582000 0.813189i \(-0.697730\pi\)
−0.582000 + 0.813189i \(0.697730\pi\)
\(770\) 0 0
\(771\) 1.70580 0.0614330
\(772\) 25.6289 0.922403
\(773\) 5.16410 0.185740 0.0928699 0.995678i \(-0.470396\pi\)
0.0928699 + 0.995678i \(0.470396\pi\)
\(774\) 15.6521 0.562603
\(775\) −10.1809 −0.365709
\(776\) −9.19639 −0.330131
\(777\) −4.96433 −0.178094
\(778\) −4.57649 −0.164075
\(779\) 18.6948 0.669809
\(780\) −0.901224 −0.0322690
\(781\) 0 0
\(782\) 10.2193 0.365443
\(783\) −22.1208 −0.790533
\(784\) 1.00000 0.0357143
\(785\) 9.67811 0.345426
\(786\) 5.14564 0.183539
\(787\) −14.6338 −0.521638 −0.260819 0.965388i \(-0.583993\pi\)
−0.260819 + 0.965388i \(0.583993\pi\)
\(788\) 10.5223 0.374841
\(789\) 9.83471 0.350125
\(790\) −11.2054 −0.398671
\(791\) −3.45851 −0.122970
\(792\) 0 0
\(793\) 5.48197 0.194671
\(794\) 37.9390 1.34641
\(795\) 4.82648 0.171178
\(796\) 18.3423 0.650125
\(797\) 29.1014 1.03082 0.515412 0.856942i \(-0.327639\pi\)
0.515412 + 0.856942i \(0.327639\pi\)
\(798\) 2.01302 0.0712603
\(799\) −9.33952 −0.330408
\(800\) −1.00000 −0.0353553
\(801\) 8.99817 0.317935
\(802\) −15.9557 −0.563416
\(803\) 0 0
\(804\) −1.73328 −0.0611281
\(805\) 4.99442 0.176030
\(806\) −16.8861 −0.594788
\(807\) −3.07418 −0.108216
\(808\) −2.73328 −0.0961565
\(809\) −29.3769 −1.03284 −0.516418 0.856337i \(-0.672735\pi\)
−0.516418 + 0.856337i \(0.672735\pi\)
\(810\) −6.42999 −0.225927
\(811\) 27.8818 0.979064 0.489532 0.871985i \(-0.337168\pi\)
0.489532 + 0.871985i \(0.337168\pi\)
\(812\) −7.13632 −0.250436
\(813\) 17.2674 0.605593
\(814\) 0 0
\(815\) 16.3076 0.571232
\(816\) 1.11180 0.0389208
\(817\) 21.4390 0.750055
\(818\) 24.2684 0.848524
\(819\) −4.48613 −0.156758
\(820\) 5.04615 0.176219
\(821\) 5.70412 0.199075 0.0995375 0.995034i \(-0.468264\pi\)
0.0995375 + 0.995034i \(0.468264\pi\)
\(822\) −6.23820 −0.217582
\(823\) −11.2756 −0.393044 −0.196522 0.980499i \(-0.562965\pi\)
−0.196522 + 0.980499i \(0.562965\pi\)
\(824\) 1.54736 0.0539050
\(825\) 0 0
\(826\) −14.0846 −0.490066
\(827\) 10.9311 0.380110 0.190055 0.981773i \(-0.439133\pi\)
0.190055 + 0.981773i \(0.439133\pi\)
\(828\) 13.5087 0.469460
\(829\) 0.763376 0.0265131 0.0132566 0.999912i \(-0.495780\pi\)
0.0132566 + 0.999912i \(0.495780\pi\)
\(830\) −4.07812 −0.141554
\(831\) −1.30776 −0.0453655
\(832\) −1.65861 −0.0575019
\(833\) 2.04615 0.0708949
\(834\) 2.64244 0.0915001
\(835\) −9.49351 −0.328536
\(836\) 0 0
\(837\) 31.5582 1.09081
\(838\) 25.6838 0.887231
\(839\) −27.3649 −0.944740 −0.472370 0.881400i \(-0.656601\pi\)
−0.472370 + 0.881400i \(0.656601\pi\)
\(840\) 0.543362 0.0187478
\(841\) 21.9271 0.756106
\(842\) 9.36362 0.322692
\(843\) 6.09803 0.210027
\(844\) −23.6304 −0.813393
\(845\) −10.2490 −0.352577
\(846\) −12.3457 −0.424454
\(847\) 0 0
\(848\) 8.88262 0.305031
\(849\) 16.9833 0.582864
\(850\) −2.04615 −0.0701824
\(851\) −45.6307 −1.56420
\(852\) −4.24254 −0.145347
\(853\) 36.3022 1.24296 0.621482 0.783428i \(-0.286531\pi\)
0.621482 + 0.783428i \(0.286531\pi\)
\(854\) −3.30516 −0.113100
\(855\) −10.0205 −0.342693
\(856\) −12.2089 −0.417290
\(857\) 30.3248 1.03587 0.517937 0.855419i \(-0.326700\pi\)
0.517937 + 0.855419i \(0.326700\pi\)
\(858\) 0 0
\(859\) −45.7487 −1.56093 −0.780463 0.625202i \(-0.785017\pi\)
−0.780463 + 0.625202i \(0.785017\pi\)
\(860\) 5.78688 0.197331
\(861\) −2.74189 −0.0934432
\(862\) −16.9902 −0.578687
\(863\) −4.79812 −0.163330 −0.0816649 0.996660i \(-0.526024\pi\)
−0.0816649 + 0.996660i \(0.526024\pi\)
\(864\) 3.09975 0.105456
\(865\) 1.44017 0.0489672
\(866\) −6.70663 −0.227901
\(867\) −6.96224 −0.236450
\(868\) 10.1809 0.345562
\(869\) 0 0
\(870\) −3.87760 −0.131463
\(871\) 5.29082 0.179273
\(872\) 1.32376 0.0448283
\(873\) −24.8740 −0.841858
\(874\) 18.5031 0.625878
\(875\) −1.00000 −0.0338062
\(876\) −5.92514 −0.200192
\(877\) 32.5055 1.09763 0.548816 0.835943i \(-0.315079\pi\)
0.548816 + 0.835943i \(0.315079\pi\)
\(878\) −12.8474 −0.433578
\(879\) 1.23468 0.0416447
\(880\) 0 0
\(881\) −45.5066 −1.53316 −0.766578 0.642151i \(-0.778042\pi\)
−0.766578 + 0.642151i \(0.778042\pi\)
\(882\) 2.70476 0.0910739
\(883\) −16.9399 −0.570071 −0.285036 0.958517i \(-0.592005\pi\)
−0.285036 + 0.958517i \(0.592005\pi\)
\(884\) −3.39376 −0.114145
\(885\) −7.65303 −0.257254
\(886\) −0.678365 −0.0227901
\(887\) −1.79960 −0.0604248 −0.0302124 0.999544i \(-0.509618\pi\)
−0.0302124 + 0.999544i \(0.509618\pi\)
\(888\) −4.96433 −0.166592
\(889\) 18.2239 0.611211
\(890\) 3.32679 0.111514
\(891\) 0 0
\(892\) 2.76854 0.0926974
\(893\) −16.9101 −0.565876
\(894\) −4.13445 −0.138277
\(895\) −9.52847 −0.318501
\(896\) 1.00000 0.0334077
\(897\) 4.50110 0.150287
\(898\) −7.59525 −0.253457
\(899\) −72.6541 −2.42315
\(900\) −2.70476 −0.0901586
\(901\) 18.1752 0.605503
\(902\) 0 0
\(903\) −3.14437 −0.104638
\(904\) −3.45851 −0.115028
\(905\) −11.5431 −0.383706
\(906\) −5.90310 −0.196117
\(907\) 18.0878 0.600595 0.300297 0.953846i \(-0.402914\pi\)
0.300297 + 0.953846i \(0.402914\pi\)
\(908\) 1.59296 0.0528642
\(909\) −7.39286 −0.245206
\(910\) −1.65861 −0.0549823
\(911\) 20.4160 0.676412 0.338206 0.941072i \(-0.390180\pi\)
0.338206 + 0.941072i \(0.390180\pi\)
\(912\) 2.01302 0.0666579
\(913\) 0 0
\(914\) 33.0502 1.09320
\(915\) −1.79590 −0.0593706
\(916\) 23.5747 0.778929
\(917\) 9.47001 0.312727
\(918\) 6.34255 0.209335
\(919\) 41.4420 1.36704 0.683522 0.729930i \(-0.260447\pi\)
0.683522 + 0.729930i \(0.260447\pi\)
\(920\) 4.99442 0.164661
\(921\) 3.01883 0.0994737
\(922\) −20.2500 −0.666898
\(923\) 12.9503 0.426265
\(924\) 0 0
\(925\) 9.13632 0.300400
\(926\) −39.3213 −1.29218
\(927\) 4.18525 0.137461
\(928\) −7.13632 −0.234261
\(929\) −45.4529 −1.49126 −0.745630 0.666360i \(-0.767851\pi\)
−0.745630 + 0.666360i \(0.767851\pi\)
\(930\) 5.53191 0.181399
\(931\) 3.70476 0.121419
\(932\) 3.72239 0.121931
\(933\) −11.0545 −0.361907
\(934\) 18.8167 0.615702
\(935\) 0 0
\(936\) −4.48613 −0.146634
\(937\) −24.2812 −0.793233 −0.396616 0.917984i \(-0.629816\pi\)
−0.396616 + 0.917984i \(0.629816\pi\)
\(938\) −3.18992 −0.104155
\(939\) 15.2631 0.498092
\(940\) −4.56444 −0.148875
\(941\) −28.6929 −0.935361 −0.467680 0.883898i \(-0.654910\pi\)
−0.467680 + 0.883898i \(0.654910\pi\)
\(942\) −5.25872 −0.171338
\(943\) −25.2026 −0.820710
\(944\) −14.0846 −0.458414
\(945\) 3.09975 0.100835
\(946\) 0 0
\(947\) 50.0233 1.62554 0.812770 0.582585i \(-0.197959\pi\)
0.812770 + 0.582585i \(0.197959\pi\)
\(948\) 6.08860 0.197748
\(949\) 18.0865 0.587111
\(950\) −3.70476 −0.120198
\(951\) 4.13849 0.134200
\(952\) 2.04615 0.0663161
\(953\) 59.7306 1.93486 0.967431 0.253133i \(-0.0814612\pi\)
0.967431 + 0.253133i \(0.0814612\pi\)
\(954\) 24.0253 0.777849
\(955\) 25.8516 0.836537
\(956\) 15.0088 0.485418
\(957\) 0 0
\(958\) 7.56631 0.244456
\(959\) −11.4807 −0.370732
\(960\) 0.543362 0.0175369
\(961\) 72.6506 2.34357
\(962\) 15.1536 0.488571
\(963\) −33.0220 −1.06412
\(964\) 23.4709 0.755947
\(965\) 25.6289 0.825022
\(966\) −2.71378 −0.0873144
\(967\) −14.2933 −0.459640 −0.229820 0.973233i \(-0.573814\pi\)
−0.229820 + 0.973233i \(0.573814\pi\)
\(968\) 0 0
\(969\) 4.11895 0.132320
\(970\) −9.19639 −0.295278
\(971\) 17.7082 0.568283 0.284142 0.958782i \(-0.408291\pi\)
0.284142 + 0.958782i \(0.408291\pi\)
\(972\) 12.7931 0.410337
\(973\) 4.86312 0.155905
\(974\) −7.92350 −0.253885
\(975\) −0.901224 −0.0288623
\(976\) −3.30516 −0.105796
\(977\) −24.9354 −0.797755 −0.398878 0.917004i \(-0.630600\pi\)
−0.398878 + 0.917004i \(0.630600\pi\)
\(978\) −8.86095 −0.283342
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 3.58046 0.114315
\(982\) 15.1101 0.482182
\(983\) −50.1552 −1.59970 −0.799851 0.600199i \(-0.795088\pi\)
−0.799851 + 0.600199i \(0.795088\pi\)
\(984\) −2.74189 −0.0874081
\(985\) 10.5223 0.335268
\(986\) −14.6020 −0.465022
\(987\) 2.48014 0.0789437
\(988\) −6.14474 −0.195490
\(989\) −28.9021 −0.919034
\(990\) 0 0
\(991\) −5.56911 −0.176909 −0.0884543 0.996080i \(-0.528193\pi\)
−0.0884543 + 0.996080i \(0.528193\pi\)
\(992\) 10.1809 0.323244
\(993\) 5.36036 0.170106
\(994\) −7.80795 −0.247653
\(995\) 18.3423 0.581490
\(996\) 2.21589 0.0702133
\(997\) −12.2104 −0.386708 −0.193354 0.981129i \(-0.561937\pi\)
−0.193354 + 0.981129i \(0.561937\pi\)
\(998\) 5.96549 0.188834
\(999\) −28.3203 −0.896014
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 8470.2.a.co.1.3 4
11.5 even 5 770.2.n.f.421.2 8
11.9 even 5 770.2.n.f.631.2 yes 8
11.10 odd 2 8470.2.a.cs.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.f.421.2 8 11.5 even 5
770.2.n.f.631.2 yes 8 11.9 even 5
8470.2.a.co.1.3 4 1.1 even 1 trivial
8470.2.a.cs.1.3 4 11.10 odd 2