Properties

Label 8470.2.a.cj.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.733.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.69639\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.42586 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.42586 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.966927 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.42586 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.42586 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.966927 q^{9} -1.00000 q^{10} +1.42586 q^{12} -3.96693 q^{13} -1.00000 q^{14} +1.42586 q^{15} +1.00000 q^{16} -0.425859 q^{17} +0.966927 q^{18} +3.54107 q^{19} +1.00000 q^{20} +1.42586 q^{21} +2.57414 q^{23} -1.42586 q^{24} +1.00000 q^{25} +3.96693 q^{26} -5.65628 q^{27} +1.00000 q^{28} -2.00000 q^{29} -1.42586 q^{30} +2.54107 q^{31} -1.00000 q^{32} +0.425859 q^{34} +1.00000 q^{35} -0.966927 q^{36} +4.85172 q^{37} -3.54107 q^{38} -5.65628 q^{39} -1.00000 q^{40} -1.39279 q^{41} -1.42586 q^{42} +1.57414 q^{43} -0.966927 q^{45} -2.57414 q^{46} +6.54107 q^{47} +1.42586 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.607214 q^{51} -3.96693 q^{52} +0.722424 q^{53} +5.65628 q^{54} -1.00000 q^{56} +5.04906 q^{57} +2.00000 q^{58} +11.2445 q^{59} +1.42586 q^{60} -5.81864 q^{61} -2.54107 q^{62} -0.966927 q^{63} +1.00000 q^{64} -3.96693 q^{65} +5.57414 q^{67} -0.425859 q^{68} +3.67036 q^{69} -1.00000 q^{70} +1.88479 q^{71} +0.966927 q^{72} -1.57414 q^{73} -4.85172 q^{74} +1.42586 q^{75} +3.54107 q^{76} +5.65628 q^{78} -1.54107 q^{79} +1.00000 q^{80} -5.16427 q^{81} +1.39279 q^{82} -0.574141 q^{83} +1.42586 q^{84} -0.425859 q^{85} -1.57414 q^{86} -2.85172 q^{87} -2.54107 q^{89} +0.966927 q^{90} -3.96693 q^{91} +2.57414 q^{92} +3.62320 q^{93} -6.54107 q^{94} +3.54107 q^{95} -1.42586 q^{96} -6.67036 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 13 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 13 q^{9} - 3 q^{10} + 2 q^{12} + 4 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} + q^{17} - 13 q^{18} - 3 q^{19} + 3 q^{20} + 2 q^{21} + 10 q^{23} - 2 q^{24} + 3 q^{25} - 4 q^{26} + 8 q^{27} + 3 q^{28} - 6 q^{29} - 2 q^{30} - 6 q^{31} - 3 q^{32} - q^{34} + 3 q^{35} + 13 q^{36} + 10 q^{37} + 3 q^{38} + 8 q^{39} - 3 q^{40} + 14 q^{41} - 2 q^{42} + 7 q^{43} + 13 q^{45} - 10 q^{46} + 6 q^{47} + 2 q^{48} + 3 q^{49} - 3 q^{50} - 20 q^{51} + 4 q^{52} + 9 q^{53} - 8 q^{54} - 3 q^{56} - 28 q^{57} + 6 q^{58} + 11 q^{59} + 2 q^{60} + 3 q^{61} + 6 q^{62} + 13 q^{63} + 3 q^{64} + 4 q^{65} + 19 q^{67} + q^{68} - 14 q^{69} - 3 q^{70} + 17 q^{71} - 13 q^{72} - 7 q^{73} - 10 q^{74} + 2 q^{75} - 3 q^{76} - 8 q^{78} + 9 q^{79} + 3 q^{80} + 39 q^{81} - 14 q^{82} - 4 q^{83} + 2 q^{84} + q^{85} - 7 q^{86} - 4 q^{87} + 6 q^{89} - 13 q^{90} + 4 q^{91} + 10 q^{92} - 30 q^{93} - 6 q^{94} - 3 q^{95} - 2 q^{96} + 5 q^{97} - 3 q^{98} + O(q^{100}) \)

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
\(3\) 1.42586 0.823220 0.411610 0.911360i \(-0.364967\pi\)
0.411610 + 0.911360i \(0.364967\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.42586 −0.582104
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −0.966927 −0.322309
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.42586 0.411610
\(13\) −3.96693 −1.10023 −0.550114 0.835090i \(-0.685416\pi\)
−0.550114 + 0.835090i \(0.685416\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.42586 0.368155
\(16\) 1.00000 0.250000
\(17\) −0.425859 −0.103286 −0.0516430 0.998666i \(-0.516446\pi\)
−0.0516430 + 0.998666i \(0.516446\pi\)
\(18\) 0.966927 0.227907
\(19\) 3.54107 0.812377 0.406188 0.913789i \(-0.366858\pi\)
0.406188 + 0.913789i \(0.366858\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.42586 0.311148
\(22\) 0 0
\(23\) 2.57414 0.536746 0.268373 0.963315i \(-0.413514\pi\)
0.268373 + 0.963315i \(0.413514\pi\)
\(24\) −1.42586 −0.291052
\(25\) 1.00000 0.200000
\(26\) 3.96693 0.777978
\(27\) −5.65628 −1.08855
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.42586 −0.260325
\(31\) 2.54107 0.456389 0.228195 0.973616i \(-0.426718\pi\)
0.228195 + 0.973616i \(0.426718\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0.425859 0.0730342
\(35\) 1.00000 0.169031
\(36\) −0.966927 −0.161154
\(37\) 4.85172 0.797617 0.398809 0.917034i \(-0.369424\pi\)
0.398809 + 0.917034i \(0.369424\pi\)
\(38\) −3.54107 −0.574437
\(39\) −5.65628 −0.905729
\(40\) −1.00000 −0.158114
\(41\) −1.39279 −0.217517 −0.108758 0.994068i \(-0.534687\pi\)
−0.108758 + 0.994068i \(0.534687\pi\)
\(42\) −1.42586 −0.220015
\(43\) 1.57414 0.240054 0.120027 0.992771i \(-0.461702\pi\)
0.120027 + 0.992771i \(0.461702\pi\)
\(44\) 0 0
\(45\) −0.966927 −0.144141
\(46\) −2.57414 −0.379536
\(47\) 6.54107 0.954113 0.477056 0.878873i \(-0.341704\pi\)
0.477056 + 0.878873i \(0.341704\pi\)
\(48\) 1.42586 0.205805
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −0.607214 −0.0850270
\(52\) −3.96693 −0.550114
\(53\) 0.722424 0.0992325 0.0496163 0.998768i \(-0.484200\pi\)
0.0496163 + 0.998768i \(0.484200\pi\)
\(54\) 5.65628 0.769722
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 5.04906 0.668765
\(58\) 2.00000 0.262613
\(59\) 11.2445 1.46391 0.731955 0.681353i \(-0.238608\pi\)
0.731955 + 0.681353i \(0.238608\pi\)
\(60\) 1.42586 0.184078
\(61\) −5.81864 −0.745001 −0.372501 0.928032i \(-0.621500\pi\)
−0.372501 + 0.928032i \(0.621500\pi\)
\(62\) −2.54107 −0.322716
\(63\) −0.966927 −0.121821
\(64\) 1.00000 0.125000
\(65\) −3.96693 −0.492037
\(66\) 0 0
\(67\) 5.57414 0.680990 0.340495 0.940246i \(-0.389405\pi\)
0.340495 + 0.940246i \(0.389405\pi\)
\(68\) −0.425859 −0.0516430
\(69\) 3.67036 0.441860
\(70\) −1.00000 −0.119523
\(71\) 1.88479 0.223684 0.111842 0.993726i \(-0.464325\pi\)
0.111842 + 0.993726i \(0.464325\pi\)
\(72\) 0.966927 0.113953
\(73\) −1.57414 −0.184239 −0.0921196 0.995748i \(-0.529364\pi\)
−0.0921196 + 0.995748i \(0.529364\pi\)
\(74\) −4.85172 −0.564001
\(75\) 1.42586 0.164644
\(76\) 3.54107 0.406188
\(77\) 0 0
\(78\) 5.65628 0.640447
\(79\) −1.54107 −0.173384 −0.0866919 0.996235i \(-0.527630\pi\)
−0.0866919 + 0.996235i \(0.527630\pi\)
\(80\) 1.00000 0.111803
\(81\) −5.16427 −0.573808
\(82\) 1.39279 0.153807
\(83\) −0.574141 −0.0630202 −0.0315101 0.999503i \(-0.510032\pi\)
−0.0315101 + 0.999503i \(0.510032\pi\)
\(84\) 1.42586 0.155574
\(85\) −0.425859 −0.0461909
\(86\) −1.57414 −0.169744
\(87\) −2.85172 −0.305736
\(88\) 0 0
\(89\) −2.54107 −0.269353 −0.134676 0.990890i \(-0.542999\pi\)
−0.134676 + 0.990890i \(0.542999\pi\)
\(90\) 0.966927 0.101923
\(91\) −3.96693 −0.415847
\(92\) 2.57414 0.268373
\(93\) 3.62320 0.375709
\(94\) −6.54107 −0.674660
\(95\) 3.54107 0.363306
\(96\) −1.42586 −0.145526
\(97\) −6.67036 −0.677273 −0.338636 0.940917i \(-0.609966\pi\)
−0.338636 + 0.940917i \(0.609966\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −1.11521 −0.110967 −0.0554837 0.998460i \(-0.517670\pi\)
−0.0554837 + 0.998460i \(0.517670\pi\)
\(102\) 0.607214 0.0601232
\(103\) 17.5080 1.72511 0.862557 0.505960i \(-0.168862\pi\)
0.862557 + 0.505960i \(0.168862\pi\)
\(104\) 3.96693 0.388989
\(105\) 1.42586 0.139150
\(106\) −0.722424 −0.0701680
\(107\) 1.03307 0.0998710 0.0499355 0.998752i \(-0.484098\pi\)
0.0499355 + 0.998752i \(0.484098\pi\)
\(108\) −5.65628 −0.544276
\(109\) −9.32664 −0.893330 −0.446665 0.894701i \(-0.647388\pi\)
−0.446665 + 0.894701i \(0.647388\pi\)
\(110\) 0 0
\(111\) 6.91786 0.656615
\(112\) 1.00000 0.0944911
\(113\) 16.7525 1.57594 0.787971 0.615712i \(-0.211132\pi\)
0.787971 + 0.615712i \(0.211132\pi\)
\(114\) −5.04906 −0.472888
\(115\) 2.57414 0.240040
\(116\) −2.00000 −0.185695
\(117\) 3.83573 0.354613
\(118\) −11.2445 −1.03514
\(119\) −0.425859 −0.0390384
\(120\) −1.42586 −0.130163
\(121\) 0 0
\(122\) 5.81864 0.526795
\(123\) −1.98592 −0.179064
\(124\) 2.54107 0.228195
\(125\) 1.00000 0.0894427
\(126\) 0.966927 0.0861407
\(127\) 4.21143 0.373704 0.186852 0.982388i \(-0.440172\pi\)
0.186852 + 0.982388i \(0.440172\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.24450 0.197617
\(130\) 3.96693 0.347922
\(131\) 6.27758 0.548474 0.274237 0.961662i \(-0.411575\pi\)
0.274237 + 0.961662i \(0.411575\pi\)
\(132\) 0 0
\(133\) 3.54107 0.307050
\(134\) −5.57414 −0.481532
\(135\) −5.65628 −0.486815
\(136\) 0.425859 0.0365171
\(137\) −13.9008 −1.18762 −0.593812 0.804604i \(-0.702378\pi\)
−0.593812 + 0.804604i \(0.702378\pi\)
\(138\) −3.67036 −0.312442
\(139\) 6.68935 0.567383 0.283692 0.958916i \(-0.408441\pi\)
0.283692 + 0.958916i \(0.408441\pi\)
\(140\) 1.00000 0.0845154
\(141\) 9.32664 0.785445
\(142\) −1.88479 −0.158168
\(143\) 0 0
\(144\) −0.966927 −0.0805772
\(145\) −2.00000 −0.166091
\(146\) 1.57414 0.130277
\(147\) 1.42586 0.117603
\(148\) 4.85172 0.398809
\(149\) 17.0821 1.39942 0.699712 0.714425i \(-0.253312\pi\)
0.699712 + 0.714425i \(0.253312\pi\)
\(150\) −1.42586 −0.116421
\(151\) −12.5741 −1.02327 −0.511635 0.859203i \(-0.670960\pi\)
−0.511635 + 0.859203i \(0.670960\pi\)
\(152\) −3.54107 −0.287219
\(153\) 0.411774 0.0332900
\(154\) 0 0
\(155\) 2.54107 0.204103
\(156\) −5.65628 −0.452865
\(157\) −12.7525 −1.01776 −0.508880 0.860837i \(-0.669940\pi\)
−0.508880 + 0.860837i \(0.669940\pi\)
\(158\) 1.54107 0.122601
\(159\) 1.03007 0.0816902
\(160\) −1.00000 −0.0790569
\(161\) 2.57414 0.202871
\(162\) 5.16427 0.405744
\(163\) 3.26349 0.255616 0.127808 0.991799i \(-0.459206\pi\)
0.127808 + 0.991799i \(0.459206\pi\)
\(164\) −1.39279 −0.108758
\(165\) 0 0
\(166\) 0.574141 0.0445620
\(167\) 3.27758 0.253626 0.126813 0.991927i \(-0.459525\pi\)
0.126813 + 0.991927i \(0.459525\pi\)
\(168\) −1.42586 −0.110007
\(169\) 2.73651 0.210501
\(170\) 0.425859 0.0326619
\(171\) −3.42395 −0.261836
\(172\) 1.57414 0.120027
\(173\) 11.9339 0.907314 0.453657 0.891176i \(-0.350119\pi\)
0.453657 + 0.891176i \(0.350119\pi\)
\(174\) 2.85172 0.216188
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 16.0331 1.20512
\(178\) 2.54107 0.190461
\(179\) −4.31065 −0.322193 −0.161097 0.986939i \(-0.551503\pi\)
−0.161097 + 0.986939i \(0.551503\pi\)
\(180\) −0.966927 −0.0720705
\(181\) −0.214429 −0.0159384 −0.00796919 0.999968i \(-0.502537\pi\)
−0.00796919 + 0.999968i \(0.502537\pi\)
\(182\) 3.96693 0.294048
\(183\) −8.29656 −0.613300
\(184\) −2.57414 −0.189768
\(185\) 4.85172 0.356705
\(186\) −3.62320 −0.265666
\(187\) 0 0
\(188\) 6.54107 0.477056
\(189\) −5.65628 −0.411434
\(190\) −3.54107 −0.256896
\(191\) 24.9970 1.80872 0.904360 0.426771i \(-0.140349\pi\)
0.904360 + 0.426771i \(0.140349\pi\)
\(192\) 1.42586 0.102902
\(193\) 20.7525 1.49380 0.746899 0.664938i \(-0.231542\pi\)
0.746899 + 0.664938i \(0.231542\pi\)
\(194\) 6.67036 0.478904
\(195\) −5.65628 −0.405054
\(196\) 1.00000 0.0714286
\(197\) 7.26349 0.517502 0.258751 0.965944i \(-0.416689\pi\)
0.258751 + 0.965944i \(0.416689\pi\)
\(198\) 0 0
\(199\) 19.5711 1.38736 0.693681 0.720283i \(-0.255988\pi\)
0.693681 + 0.720283i \(0.255988\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 7.94794 0.560604
\(202\) 1.11521 0.0784659
\(203\) −2.00000 −0.140372
\(204\) −0.607214 −0.0425135
\(205\) −1.39279 −0.0972764
\(206\) −17.5080 −1.21984
\(207\) −2.48901 −0.172998
\(208\) −3.96693 −0.275057
\(209\) 0 0
\(210\) −1.42586 −0.0983936
\(211\) 9.63729 0.663458 0.331729 0.943375i \(-0.392368\pi\)
0.331729 + 0.943375i \(0.392368\pi\)
\(212\) 0.722424 0.0496163
\(213\) 2.68745 0.184141
\(214\) −1.03307 −0.0706194
\(215\) 1.57414 0.107356
\(216\) 5.65628 0.384861
\(217\) 2.54107 0.172499
\(218\) 9.32664 0.631680
\(219\) −2.24450 −0.151669
\(220\) 0 0
\(221\) 1.68935 0.113638
\(222\) −6.91786 −0.464297
\(223\) 4.35971 0.291948 0.145974 0.989288i \(-0.453368\pi\)
0.145974 + 0.989288i \(0.453368\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −0.966927 −0.0644618
\(226\) −16.7525 −1.11436
\(227\) 22.7856 1.51233 0.756166 0.654380i \(-0.227070\pi\)
0.756166 + 0.654380i \(0.227070\pi\)
\(228\) 5.04906 0.334382
\(229\) 15.2635 1.00864 0.504320 0.863517i \(-0.331743\pi\)
0.504320 + 0.863517i \(0.331743\pi\)
\(230\) −2.57414 −0.169734
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −3.73651 −0.244787 −0.122393 0.992482i \(-0.539057\pi\)
−0.122393 + 0.992482i \(0.539057\pi\)
\(234\) −3.83573 −0.250749
\(235\) 6.54107 0.426692
\(236\) 11.2445 0.731955
\(237\) −2.19735 −0.142733
\(238\) 0.425859 0.0276043
\(239\) −1.78857 −0.115693 −0.0578465 0.998325i \(-0.518423\pi\)
−0.0578465 + 0.998325i \(0.518423\pi\)
\(240\) 1.42586 0.0920388
\(241\) 17.6373 1.13612 0.568059 0.822988i \(-0.307695\pi\)
0.568059 + 0.822988i \(0.307695\pi\)
\(242\) 0 0
\(243\) 9.60531 0.616181
\(244\) −5.81864 −0.372501
\(245\) 1.00000 0.0638877
\(246\) 1.98592 0.126617
\(247\) −14.0472 −0.893799
\(248\) −2.54107 −0.161358
\(249\) −0.818644 −0.0518795
\(250\) −1.00000 −0.0632456
\(251\) 6.96693 0.439749 0.219874 0.975528i \(-0.429435\pi\)
0.219874 + 0.975528i \(0.429435\pi\)
\(252\) −0.966927 −0.0609107
\(253\) 0 0
\(254\) −4.21143 −0.264249
\(255\) −0.607214 −0.0380252
\(256\) 1.00000 0.0625000
\(257\) 17.2114 1.07362 0.536810 0.843703i \(-0.319629\pi\)
0.536810 + 0.843703i \(0.319629\pi\)
\(258\) −2.24450 −0.139737
\(259\) 4.85172 0.301471
\(260\) −3.96693 −0.246018
\(261\) 1.93385 0.119703
\(262\) −6.27758 −0.387830
\(263\) 3.65628 0.225456 0.112728 0.993626i \(-0.464041\pi\)
0.112728 + 0.993626i \(0.464041\pi\)
\(264\) 0 0
\(265\) 0.722424 0.0443781
\(266\) −3.54107 −0.217117
\(267\) −3.62320 −0.221736
\(268\) 5.57414 0.340495
\(269\) −11.4890 −0.700497 −0.350249 0.936657i \(-0.613903\pi\)
−0.350249 + 0.936657i \(0.613903\pi\)
\(270\) 5.65628 0.344230
\(271\) 6.55515 0.398197 0.199099 0.979979i \(-0.436199\pi\)
0.199099 + 0.979979i \(0.436199\pi\)
\(272\) −0.425859 −0.0258215
\(273\) −5.65628 −0.342333
\(274\) 13.9008 0.839777
\(275\) 0 0
\(276\) 3.67036 0.220930
\(277\) 5.62320 0.337866 0.168933 0.985628i \(-0.445968\pi\)
0.168933 + 0.985628i \(0.445968\pi\)
\(278\) −6.68935 −0.401201
\(279\) −2.45703 −0.147098
\(280\) −1.00000 −0.0597614
\(281\) −9.23042 −0.550641 −0.275320 0.961353i \(-0.588784\pi\)
−0.275320 + 0.961353i \(0.588784\pi\)
\(282\) −9.32664 −0.555393
\(283\) 4.50799 0.267972 0.133986 0.990983i \(-0.457222\pi\)
0.133986 + 0.990983i \(0.457222\pi\)
\(284\) 1.88479 0.111842
\(285\) 5.04906 0.299081
\(286\) 0 0
\(287\) −1.39279 −0.0822135
\(288\) 0.966927 0.0569767
\(289\) −16.8186 −0.989332
\(290\) 2.00000 0.117444
\(291\) −9.51099 −0.557544
\(292\) −1.57414 −0.0921196
\(293\) −25.8346 −1.50928 −0.754638 0.656142i \(-0.772187\pi\)
−0.754638 + 0.656142i \(0.772187\pi\)
\(294\) −1.42586 −0.0831578
\(295\) 11.2445 0.654680
\(296\) −4.85172 −0.282000
\(297\) 0 0
\(298\) −17.0821 −0.989542
\(299\) −10.2114 −0.590542
\(300\) 1.42586 0.0823220
\(301\) 1.57414 0.0907320
\(302\) 12.5741 0.723561
\(303\) −1.59013 −0.0913506
\(304\) 3.54107 0.203094
\(305\) −5.81864 −0.333175
\(306\) −0.411774 −0.0235396
\(307\) −11.1483 −0.636266 −0.318133 0.948046i \(-0.603056\pi\)
−0.318133 + 0.948046i \(0.603056\pi\)
\(308\) 0 0
\(309\) 24.9639 1.42015
\(310\) −2.54107 −0.144323
\(311\) −13.9479 −0.790915 −0.395458 0.918484i \(-0.629414\pi\)
−0.395458 + 0.918484i \(0.629414\pi\)
\(312\) 5.65628 0.320224
\(313\) −34.4890 −1.94943 −0.974717 0.223443i \(-0.928270\pi\)
−0.974717 + 0.223443i \(0.928270\pi\)
\(314\) 12.7525 0.719665
\(315\) −0.966927 −0.0544802
\(316\) −1.54107 −0.0866919
\(317\) −13.5080 −0.758685 −0.379342 0.925256i \(-0.623850\pi\)
−0.379342 + 0.925256i \(0.623850\pi\)
\(318\) −1.03007 −0.0577637
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 1.47302 0.0822158
\(322\) −2.57414 −0.143451
\(323\) −1.50799 −0.0839071
\(324\) −5.16427 −0.286904
\(325\) −3.96693 −0.220046
\(326\) −3.26349 −0.180748
\(327\) −13.2985 −0.735407
\(328\) 1.39279 0.0769037
\(329\) 6.54107 0.360621
\(330\) 0 0
\(331\) 30.1122 1.65512 0.827558 0.561380i \(-0.189729\pi\)
0.827558 + 0.561380i \(0.189729\pi\)
\(332\) −0.574141 −0.0315101
\(333\) −4.69126 −0.257079
\(334\) −3.27758 −0.179341
\(335\) 5.57414 0.304548
\(336\) 1.42586 0.0777870
\(337\) 12.5221 0.682121 0.341061 0.940041i \(-0.389214\pi\)
0.341061 + 0.940041i \(0.389214\pi\)
\(338\) −2.73651 −0.148846
\(339\) 23.8867 1.29735
\(340\) −0.425859 −0.0230954
\(341\) 0 0
\(342\) 3.42395 0.185146
\(343\) 1.00000 0.0539949
\(344\) −1.57414 −0.0848720
\(345\) 3.67036 0.197606
\(346\) −11.9339 −0.641568
\(347\) 20.3738 1.09372 0.546861 0.837223i \(-0.315822\pi\)
0.546861 + 0.837223i \(0.315822\pi\)
\(348\) −2.85172 −0.152868
\(349\) −18.2635 −0.977622 −0.488811 0.872390i \(-0.662569\pi\)
−0.488811 + 0.872390i \(0.662569\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 22.4380 1.19765
\(352\) 0 0
\(353\) 11.5080 0.612509 0.306254 0.951950i \(-0.400924\pi\)
0.306254 + 0.951950i \(0.400924\pi\)
\(354\) −16.0331 −0.852148
\(355\) 1.88479 0.100034
\(356\) −2.54107 −0.134676
\(357\) −0.607214 −0.0321372
\(358\) 4.31065 0.227825
\(359\) −25.2635 −1.33336 −0.666678 0.745346i \(-0.732284\pi\)
−0.666678 + 0.745346i \(0.732284\pi\)
\(360\) 0.966927 0.0509615
\(361\) −6.46084 −0.340044
\(362\) 0.214429 0.0112701
\(363\) 0 0
\(364\) −3.96693 −0.207923
\(365\) −1.57414 −0.0823943
\(366\) 8.29656 0.433668
\(367\) 20.1293 1.05074 0.525370 0.850874i \(-0.323927\pi\)
0.525370 + 0.850874i \(0.323927\pi\)
\(368\) 2.57414 0.134186
\(369\) 1.34672 0.0701075
\(370\) −4.85172 −0.252229
\(371\) 0.722424 0.0375064
\(372\) 3.62320 0.187854
\(373\) −22.9669 −1.18918 −0.594591 0.804028i \(-0.702686\pi\)
−0.594591 + 0.804028i \(0.702686\pi\)
\(374\) 0 0
\(375\) 1.42586 0.0736310
\(376\) −6.54107 −0.337330
\(377\) 7.93385 0.408614
\(378\) 5.65628 0.290928
\(379\) −19.8677 −1.02054 −0.510268 0.860016i \(-0.670454\pi\)
−0.510268 + 0.860016i \(0.670454\pi\)
\(380\) 3.54107 0.181653
\(381\) 6.00490 0.307641
\(382\) −24.9970 −1.27896
\(383\) 16.0631 0.820788 0.410394 0.911908i \(-0.365391\pi\)
0.410394 + 0.911908i \(0.365391\pi\)
\(384\) −1.42586 −0.0727631
\(385\) 0 0
\(386\) −20.7525 −1.05627
\(387\) −1.52208 −0.0773716
\(388\) −6.67036 −0.338636
\(389\) −16.7997 −0.851776 −0.425888 0.904776i \(-0.640038\pi\)
−0.425888 + 0.904776i \(0.640038\pi\)
\(390\) 5.65628 0.286417
\(391\) −1.09622 −0.0554383
\(392\) −1.00000 −0.0505076
\(393\) 8.95094 0.451515
\(394\) −7.26349 −0.365929
\(395\) −1.54107 −0.0775395
\(396\) 0 0
\(397\) 7.93385 0.398189 0.199094 0.979980i \(-0.436200\pi\)
0.199094 + 0.979980i \(0.436200\pi\)
\(398\) −19.5711 −0.981013
\(399\) 5.04906 0.252769
\(400\) 1.00000 0.0500000
\(401\) −23.7034 −1.18369 −0.591847 0.806051i \(-0.701601\pi\)
−0.591847 + 0.806051i \(0.701601\pi\)
\(402\) −7.94794 −0.396407
\(403\) −10.0802 −0.502132
\(404\) −1.11521 −0.0554837
\(405\) −5.16427 −0.256615
\(406\) 2.00000 0.0992583
\(407\) 0 0
\(408\) 0.607214 0.0300616
\(409\) −13.6373 −0.674321 −0.337160 0.941447i \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(410\) 1.39279 0.0687848
\(411\) −19.8205 −0.977676
\(412\) 17.5080 0.862557
\(413\) 11.2445 0.553306
\(414\) 2.48901 0.122328
\(415\) −0.574141 −0.0281835
\(416\) 3.96693 0.194495
\(417\) 9.53807 0.467081
\(418\) 0 0
\(419\) 25.7715 1.25902 0.629510 0.776993i \(-0.283256\pi\)
0.629510 + 0.776993i \(0.283256\pi\)
\(420\) 1.42586 0.0695748
\(421\) 29.5711 1.44121 0.720605 0.693346i \(-0.243864\pi\)
0.720605 + 0.693346i \(0.243864\pi\)
\(422\) −9.63729 −0.469136
\(423\) −6.32473 −0.307519
\(424\) −0.722424 −0.0350840
\(425\) −0.425859 −0.0206572
\(426\) −2.68745 −0.130207
\(427\) −5.81864 −0.281584
\(428\) 1.03307 0.0499355
\(429\) 0 0
\(430\) −1.57414 −0.0759118
\(431\) 21.7997 1.05005 0.525026 0.851086i \(-0.324056\pi\)
0.525026 + 0.851086i \(0.324056\pi\)
\(432\) −5.65628 −0.272138
\(433\) −25.5571 −1.22819 −0.614097 0.789231i \(-0.710480\pi\)
−0.614097 + 0.789231i \(0.710480\pi\)
\(434\) −2.54107 −0.121975
\(435\) −2.85172 −0.136729
\(436\) −9.32664 −0.446665
\(437\) 9.11521 0.436040
\(438\) 2.24450 0.107247
\(439\) −9.82164 −0.468761 −0.234381 0.972145i \(-0.575306\pi\)
−0.234381 + 0.972145i \(0.575306\pi\)
\(440\) 0 0
\(441\) −0.966927 −0.0460441
\(442\) −1.68935 −0.0803542
\(443\) 3.37870 0.160527 0.0802635 0.996774i \(-0.474424\pi\)
0.0802635 + 0.996774i \(0.474424\pi\)
\(444\) 6.91786 0.328307
\(445\) −2.54107 −0.120458
\(446\) −4.35971 −0.206438
\(447\) 24.3567 1.15203
\(448\) 1.00000 0.0472456
\(449\) 7.29656 0.344346 0.172173 0.985067i \(-0.444921\pi\)
0.172173 + 0.985067i \(0.444921\pi\)
\(450\) 0.966927 0.0455814
\(451\) 0 0
\(452\) 16.7525 0.787971
\(453\) −17.9289 −0.842376
\(454\) −22.7856 −1.06938
\(455\) −3.96693 −0.185972
\(456\) −5.04906 −0.236444
\(457\) 35.5381 1.66240 0.831200 0.555973i \(-0.187654\pi\)
0.831200 + 0.555973i \(0.187654\pi\)
\(458\) −15.2635 −0.713216
\(459\) 2.40878 0.112432
\(460\) 2.57414 0.120020
\(461\) 26.6704 1.24216 0.621081 0.783746i \(-0.286694\pi\)
0.621081 + 0.783746i \(0.286694\pi\)
\(462\) 0 0
\(463\) 17.9528 0.834339 0.417170 0.908829i \(-0.363022\pi\)
0.417170 + 0.908829i \(0.363022\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 3.62320 0.168022
\(466\) 3.73651 0.173090
\(467\) −18.8327 −0.871475 −0.435737 0.900074i \(-0.643512\pi\)
−0.435737 + 0.900074i \(0.643512\pi\)
\(468\) 3.83573 0.177307
\(469\) 5.57414 0.257390
\(470\) −6.54107 −0.301717
\(471\) −18.1833 −0.837840
\(472\) −11.2445 −0.517570
\(473\) 0 0
\(474\) 2.19735 0.100927
\(475\) 3.54107 0.162475
\(476\) −0.425859 −0.0195192
\(477\) −0.698531 −0.0319835
\(478\) 1.78857 0.0818073
\(479\) −0.310649 −0.0141939 −0.00709697 0.999975i \(-0.502259\pi\)
−0.00709697 + 0.999975i \(0.502259\pi\)
\(480\) −1.42586 −0.0650813
\(481\) −19.2464 −0.877561
\(482\) −17.6373 −0.803356
\(483\) 3.67036 0.167007
\(484\) 0 0
\(485\) −6.67036 −0.302886
\(486\) −9.60531 −0.435706
\(487\) −12.5362 −0.568068 −0.284034 0.958814i \(-0.591673\pi\)
−0.284034 + 0.958814i \(0.591673\pi\)
\(488\) 5.81864 0.263398
\(489\) 4.65328 0.210429
\(490\) −1.00000 −0.0451754
\(491\) −7.71752 −0.348287 −0.174143 0.984720i \(-0.555716\pi\)
−0.174143 + 0.984720i \(0.555716\pi\)
\(492\) −1.98592 −0.0895320
\(493\) 0.851718 0.0383594
\(494\) 14.0472 0.632012
\(495\) 0 0
\(496\) 2.54107 0.114097
\(497\) 1.88479 0.0845444
\(498\) 0.818644 0.0366843
\(499\) −18.9179 −0.846880 −0.423440 0.905924i \(-0.639178\pi\)
−0.423440 + 0.905924i \(0.639178\pi\)
\(500\) 1.00000 0.0447214
\(501\) 4.67336 0.208790
\(502\) −6.96693 −0.310949
\(503\) 16.4259 0.732393 0.366196 0.930538i \(-0.380660\pi\)
0.366196 + 0.930538i \(0.380660\pi\)
\(504\) 0.966927 0.0430703
\(505\) −1.11521 −0.0496262
\(506\) 0 0
\(507\) 3.90187 0.173288
\(508\) 4.21143 0.186852
\(509\) −7.71942 −0.342157 −0.171079 0.985257i \(-0.554725\pi\)
−0.171079 + 0.985257i \(0.554725\pi\)
\(510\) 0.607214 0.0268879
\(511\) −1.57414 −0.0696359
\(512\) −1.00000 −0.0441942
\(513\) −20.0293 −0.884314
\(514\) −17.2114 −0.759164
\(515\) 17.5080 0.771494
\(516\) 2.24450 0.0988087
\(517\) 0 0
\(518\) −4.85172 −0.213172
\(519\) 17.0160 0.746919
\(520\) 3.96693 0.173961
\(521\) 31.0821 1.36173 0.680867 0.732407i \(-0.261603\pi\)
0.680867 + 0.732407i \(0.261603\pi\)
\(522\) −1.93385 −0.0846425
\(523\) 34.6723 1.51611 0.758056 0.652189i \(-0.226149\pi\)
0.758056 + 0.652189i \(0.226149\pi\)
\(524\) 6.27758 0.274237
\(525\) 1.42586 0.0622296
\(526\) −3.65628 −0.159421
\(527\) −1.08214 −0.0471386
\(528\) 0 0
\(529\) −16.3738 −0.711904
\(530\) −0.722424 −0.0313801
\(531\) −10.8726 −0.471831
\(532\) 3.54107 0.153525
\(533\) 5.52508 0.239318
\(534\) 3.62320 0.156791
\(535\) 1.03307 0.0446636
\(536\) −5.57414 −0.240766
\(537\) −6.14638 −0.265236
\(538\) 11.4890 0.495326
\(539\) 0 0
\(540\) −5.65628 −0.243407
\(541\) 18.4890 0.794904 0.397452 0.917623i \(-0.369894\pi\)
0.397452 + 0.917623i \(0.369894\pi\)
\(542\) −6.55515 −0.281568
\(543\) −0.305745 −0.0131208
\(544\) 0.425859 0.0182585
\(545\) −9.32664 −0.399509
\(546\) 5.65628 0.242066
\(547\) −29.5741 −1.26450 −0.632249 0.774765i \(-0.717868\pi\)
−0.632249 + 0.774765i \(0.717868\pi\)
\(548\) −13.9008 −0.593812
\(549\) 5.62620 0.240120
\(550\) 0 0
\(551\) −7.08214 −0.301709
\(552\) −3.67036 −0.156221
\(553\) −1.54107 −0.0655329
\(554\) −5.62320 −0.238907
\(555\) 6.91786 0.293647
\(556\) 6.68935 0.283692
\(557\) −22.7715 −0.964859 −0.482429 0.875935i \(-0.660246\pi\)
−0.482429 + 0.875935i \(0.660246\pi\)
\(558\) 2.45703 0.104014
\(559\) −6.24450 −0.264114
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 9.23042 0.389362
\(563\) 18.4418 0.777231 0.388616 0.921400i \(-0.372953\pi\)
0.388616 + 0.921400i \(0.372953\pi\)
\(564\) 9.32664 0.392722
\(565\) 16.7525 0.704783
\(566\) −4.50799 −0.189485
\(567\) −5.16427 −0.216879
\(568\) −1.88479 −0.0790841
\(569\) −31.3567 −1.31454 −0.657271 0.753654i \(-0.728289\pi\)
−0.657271 + 0.753654i \(0.728289\pi\)
\(570\) −5.04906 −0.211482
\(571\) 24.7194 1.03448 0.517238 0.855842i \(-0.326960\pi\)
0.517238 + 0.855842i \(0.326960\pi\)
\(572\) 0 0
\(573\) 35.6422 1.48897
\(574\) 1.39279 0.0581337
\(575\) 2.57414 0.107349
\(576\) −0.966927 −0.0402886
\(577\) −35.9970 −1.49857 −0.749287 0.662245i \(-0.769604\pi\)
−0.749287 + 0.662245i \(0.769604\pi\)
\(578\) 16.8186 0.699563
\(579\) 29.5901 1.22972
\(580\) −2.00000 −0.0830455
\(581\) −0.574141 −0.0238194
\(582\) 9.51099 0.394243
\(583\) 0 0
\(584\) 1.57414 0.0651384
\(585\) 3.83573 0.158588
\(586\) 25.8346 1.06722
\(587\) −34.3757 −1.41884 −0.709419 0.704787i \(-0.751042\pi\)
−0.709419 + 0.704787i \(0.751042\pi\)
\(588\) 1.42586 0.0588014
\(589\) 8.99809 0.370760
\(590\) −11.2445 −0.462929
\(591\) 10.3567 0.426018
\(592\) 4.85172 0.199404
\(593\) −8.53807 −0.350616 −0.175308 0.984514i \(-0.556092\pi\)
−0.175308 + 0.984514i \(0.556092\pi\)
\(594\) 0 0
\(595\) −0.425859 −0.0174585
\(596\) 17.0821 0.699712
\(597\) 27.9057 1.14210
\(598\) 10.2114 0.417576
\(599\) 6.75550 0.276022 0.138011 0.990431i \(-0.455929\pi\)
0.138011 + 0.990431i \(0.455929\pi\)
\(600\) −1.42586 −0.0582104
\(601\) −4.17836 −0.170439 −0.0852194 0.996362i \(-0.527159\pi\)
−0.0852194 + 0.996362i \(0.527159\pi\)
\(602\) −1.57414 −0.0641572
\(603\) −5.38979 −0.219489
\(604\) −12.5741 −0.511635
\(605\) 0 0
\(606\) 1.59013 0.0645947
\(607\) −42.4058 −1.72120 −0.860599 0.509284i \(-0.829910\pi\)
−0.860599 + 0.509284i \(0.829910\pi\)
\(608\) −3.54107 −0.143609
\(609\) −2.85172 −0.115557
\(610\) 5.81864 0.235590
\(611\) −25.9479 −1.04974
\(612\) 0.411774 0.0166450
\(613\) 25.7384 1.03956 0.519782 0.854299i \(-0.326013\pi\)
0.519782 + 0.854299i \(0.326013\pi\)
\(614\) 11.1483 0.449908
\(615\) −1.98592 −0.0800798
\(616\) 0 0
\(617\) −28.1643 −1.13385 −0.566925 0.823769i \(-0.691867\pi\)
−0.566925 + 0.823769i \(0.691867\pi\)
\(618\) −24.9639 −1.00420
\(619\) −29.6183 −1.19046 −0.595230 0.803555i \(-0.702939\pi\)
−0.595230 + 0.803555i \(0.702939\pi\)
\(620\) 2.54107 0.102052
\(621\) −14.5601 −0.584275
\(622\) 13.9479 0.559261
\(623\) −2.54107 −0.101806
\(624\) −5.65628 −0.226432
\(625\) 1.00000 0.0400000
\(626\) 34.4890 1.37846
\(627\) 0 0
\(628\) −12.7525 −0.508880
\(629\) −2.06615 −0.0823827
\(630\) 0.966927 0.0385233
\(631\) −9.52208 −0.379068 −0.189534 0.981874i \(-0.560698\pi\)
−0.189534 + 0.981874i \(0.560698\pi\)
\(632\) 1.54107 0.0613004
\(633\) 13.7414 0.546172
\(634\) 13.5080 0.536471
\(635\) 4.21143 0.167125
\(636\) 1.03007 0.0408451
\(637\) −3.96693 −0.157175
\(638\) 0 0
\(639\) −1.82245 −0.0720952
\(640\) −1.00000 −0.0395285
\(641\) 32.8016 1.29558 0.647792 0.761817i \(-0.275693\pi\)
0.647792 + 0.761817i \(0.275693\pi\)
\(642\) −1.47302 −0.0581353
\(643\) 40.9780 1.61602 0.808008 0.589172i \(-0.200546\pi\)
0.808008 + 0.589172i \(0.200546\pi\)
\(644\) 2.57414 0.101435
\(645\) 2.24450 0.0883772
\(646\) 1.50799 0.0593313
\(647\) 28.6563 1.12659 0.563297 0.826254i \(-0.309533\pi\)
0.563297 + 0.826254i \(0.309533\pi\)
\(648\) 5.16427 0.202872
\(649\) 0 0
\(650\) 3.96693 0.155596
\(651\) 3.62320 0.142005
\(652\) 3.26349 0.127808
\(653\) 5.21143 0.203939 0.101970 0.994788i \(-0.467486\pi\)
0.101970 + 0.994788i \(0.467486\pi\)
\(654\) 13.2985 0.520011
\(655\) 6.27758 0.245285
\(656\) −1.39279 −0.0543791
\(657\) 1.52208 0.0593820
\(658\) −6.54107 −0.254997
\(659\) 44.6674 1.73999 0.869997 0.493058i \(-0.164121\pi\)
0.869997 + 0.493058i \(0.164121\pi\)
\(660\) 0 0
\(661\) −39.2415 −1.52632 −0.763159 0.646211i \(-0.776353\pi\)
−0.763159 + 0.646211i \(0.776353\pi\)
\(662\) −30.1122 −1.17034
\(663\) 2.40878 0.0935491
\(664\) 0.574141 0.0222810
\(665\) 3.54107 0.137317
\(666\) 4.69126 0.181782
\(667\) −5.14828 −0.199342
\(668\) 3.27758 0.126813
\(669\) 6.21633 0.240337
\(670\) −5.57414 −0.215348
\(671\) 0 0
\(672\) −1.42586 −0.0550037
\(673\) 3.11521 0.120082 0.0600412 0.998196i \(-0.480877\pi\)
0.0600412 + 0.998196i \(0.480877\pi\)
\(674\) −12.5221 −0.482332
\(675\) −5.65628 −0.217710
\(676\) 2.73651 0.105250
\(677\) 23.0772 0.886930 0.443465 0.896292i \(-0.353749\pi\)
0.443465 + 0.896292i \(0.353749\pi\)
\(678\) −23.8867 −0.917363
\(679\) −6.67036 −0.255985
\(680\) 0.425859 0.0163309
\(681\) 32.4890 1.24498
\(682\) 0 0
\(683\) −28.9669 −1.10839 −0.554194 0.832387i \(-0.686974\pi\)
−0.554194 + 0.832387i \(0.686974\pi\)
\(684\) −3.42395 −0.130918
\(685\) −13.9008 −0.531122
\(686\) −1.00000 −0.0381802
\(687\) 21.7636 0.830332
\(688\) 1.57414 0.0600136
\(689\) −2.86580 −0.109178
\(690\) −3.67036 −0.139728
\(691\) −23.1973 −0.882469 −0.441234 0.897392i \(-0.645459\pi\)
−0.441234 + 0.897392i \(0.645459\pi\)
\(692\) 11.9339 0.453657
\(693\) 0 0
\(694\) −20.3738 −0.773379
\(695\) 6.68935 0.253742
\(696\) 2.85172 0.108094
\(697\) 0.593130 0.0224664
\(698\) 18.2635 0.691283
\(699\) −5.32773 −0.201513
\(700\) 1.00000 0.0377964
\(701\) 14.5411 0.549209 0.274604 0.961557i \(-0.411453\pi\)
0.274604 + 0.961557i \(0.411453\pi\)
\(702\) −22.4380 −0.846869
\(703\) 17.1803 0.647966
\(704\) 0 0
\(705\) 9.32664 0.351262
\(706\) −11.5080 −0.433109
\(707\) −1.11521 −0.0419418
\(708\) 16.0331 0.602560
\(709\) −20.1263 −0.755859 −0.377929 0.925834i \(-0.623364\pi\)
−0.377929 + 0.925834i \(0.623364\pi\)
\(710\) −1.88479 −0.0707349
\(711\) 1.49010 0.0558831
\(712\) 2.54107 0.0952306
\(713\) 6.54107 0.244965
\(714\) 0.607214 0.0227244
\(715\) 0 0
\(716\) −4.31065 −0.161097
\(717\) −2.55025 −0.0952408
\(718\) 25.2635 0.942825
\(719\) −49.8016 −1.85728 −0.928642 0.370976i \(-0.879023\pi\)
−0.928642 + 0.370976i \(0.879023\pi\)
\(720\) −0.966927 −0.0360352
\(721\) 17.5080 0.652032
\(722\) 6.46084 0.240447
\(723\) 25.1483 0.935275
\(724\) −0.214429 −0.00796919
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −21.5050 −0.797576 −0.398788 0.917043i \(-0.630569\pi\)
−0.398788 + 0.917043i \(0.630569\pi\)
\(728\) 3.96693 0.147024
\(729\) 29.1886 1.08106
\(730\) 1.57414 0.0582616
\(731\) −0.670362 −0.0247942
\(732\) −8.29656 −0.306650
\(733\) −2.45593 −0.0907120 −0.0453560 0.998971i \(-0.514442\pi\)
−0.0453560 + 0.998971i \(0.514442\pi\)
\(734\) −20.1293 −0.742986
\(735\) 1.42586 0.0525936
\(736\) −2.57414 −0.0948841
\(737\) 0 0
\(738\) −1.34672 −0.0495735
\(739\) −0.310649 −0.0114274 −0.00571371 0.999984i \(-0.501819\pi\)
−0.00571371 + 0.999984i \(0.501819\pi\)
\(740\) 4.85172 0.178353
\(741\) −20.0293 −0.735793
\(742\) −0.722424 −0.0265210
\(743\) 13.3407 0.489424 0.244712 0.969596i \(-0.421307\pi\)
0.244712 + 0.969596i \(0.421307\pi\)
\(744\) −3.62320 −0.132833
\(745\) 17.0821 0.625841
\(746\) 22.9669 0.840879
\(747\) 0.555153 0.0203120
\(748\) 0 0
\(749\) 1.03307 0.0377477
\(750\) −1.42586 −0.0520650
\(751\) 5.10112 0.186143 0.0930713 0.995659i \(-0.470332\pi\)
0.0930713 + 0.995659i \(0.470332\pi\)
\(752\) 6.54107 0.238528
\(753\) 9.93385 0.362010
\(754\) −7.93385 −0.288934
\(755\) −12.5741 −0.457620
\(756\) −5.65628 −0.205717
\(757\) 10.6844 0.388333 0.194166 0.980969i \(-0.437800\pi\)
0.194166 + 0.980969i \(0.437800\pi\)
\(758\) 19.8677 0.721628
\(759\) 0 0
\(760\) −3.54107 −0.128448
\(761\) 33.9198 1.22959 0.614795 0.788687i \(-0.289239\pi\)
0.614795 + 0.788687i \(0.289239\pi\)
\(762\) −6.00490 −0.217535
\(763\) −9.32664 −0.337647
\(764\) 24.9970 0.904360
\(765\) 0.411774 0.0148877
\(766\) −16.0631 −0.580385
\(767\) −44.6061 −1.61063
\(768\) 1.42586 0.0514512
\(769\) −36.7194 −1.32414 −0.662068 0.749444i \(-0.730321\pi\)
−0.662068 + 0.749444i \(0.730321\pi\)
\(770\) 0 0
\(771\) 24.5411 0.883825
\(772\) 20.7525 0.746899
\(773\) −13.3077 −0.478643 −0.239321 0.970940i \(-0.576925\pi\)
−0.239321 + 0.970940i \(0.576925\pi\)
\(774\) 1.52208 0.0547100
\(775\) 2.54107 0.0912779
\(776\) 6.67036 0.239452
\(777\) 6.91786 0.248177
\(778\) 16.7997 0.602297
\(779\) −4.93195 −0.176705
\(780\) −5.65628 −0.202527
\(781\) 0 0
\(782\) 1.09622 0.0392008
\(783\) 11.3126 0.404278
\(784\) 1.00000 0.0357143
\(785\) −12.7525 −0.455156
\(786\) −8.95094 −0.319269
\(787\) −31.5050 −1.12303 −0.561516 0.827466i \(-0.689782\pi\)
−0.561516 + 0.827466i \(0.689782\pi\)
\(788\) 7.26349 0.258751
\(789\) 5.21334 0.185600
\(790\) 1.54107 0.0548287
\(791\) 16.7525 0.595650
\(792\) 0 0
\(793\) 23.0821 0.819671
\(794\) −7.93385 −0.281562
\(795\) 1.03007 0.0365330
\(796\) 19.5711 0.693681
\(797\) 14.2635 0.505239 0.252619 0.967566i \(-0.418708\pi\)
0.252619 + 0.967566i \(0.418708\pi\)
\(798\) −5.04906 −0.178735
\(799\) −2.78557 −0.0985464
\(800\) −1.00000 −0.0353553
\(801\) 2.45703 0.0868148
\(802\) 23.7034 0.836997
\(803\) 0 0
\(804\) 7.94794 0.280302
\(805\) 2.57414 0.0907266
\(806\) 10.0802 0.355061
\(807\) −16.3817 −0.576663
\(808\) 1.11521 0.0392329
\(809\) 11.3898 0.400444 0.200222 0.979751i \(-0.435834\pi\)
0.200222 + 0.979751i \(0.435834\pi\)
\(810\) 5.16427 0.181454
\(811\) 10.1924 0.357905 0.178953 0.983858i \(-0.442729\pi\)
0.178953 + 0.983858i \(0.442729\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 9.34672 0.327804
\(814\) 0 0
\(815\) 3.26349 0.114315
\(816\) −0.607214 −0.0212568
\(817\) 5.57414 0.195015
\(818\) 13.6373 0.476817
\(819\) 3.83573 0.134031
\(820\) −1.39279 −0.0486382
\(821\) −15.4730 −0.540012 −0.270006 0.962859i \(-0.587026\pi\)
−0.270006 + 0.962859i \(0.587026\pi\)
\(822\) 19.8205 0.691321
\(823\) 37.3126 1.30063 0.650317 0.759663i \(-0.274636\pi\)
0.650317 + 0.759663i \(0.274636\pi\)
\(824\) −17.5080 −0.609920
\(825\) 0 0
\(826\) −11.2445 −0.391246
\(827\) 13.0851 0.455015 0.227507 0.973776i \(-0.426942\pi\)
0.227507 + 0.973776i \(0.426942\pi\)
\(828\) −2.48901 −0.0864989
\(829\) −1.77449 −0.0616304 −0.0308152 0.999525i \(-0.509810\pi\)
−0.0308152 + 0.999525i \(0.509810\pi\)
\(830\) 0.574141 0.0199287
\(831\) 8.01789 0.278138
\(832\) −3.96693 −0.137528
\(833\) −0.425859 −0.0147551
\(834\) −9.53807 −0.330276
\(835\) 3.27758 0.113425
\(836\) 0 0
\(837\) −14.3730 −0.496803
\(838\) −25.7715 −0.890261
\(839\) −26.6674 −0.920660 −0.460330 0.887748i \(-0.652269\pi\)
−0.460330 + 0.887748i \(0.652269\pi\)
\(840\) −1.42586 −0.0491968
\(841\) −25.0000 −0.862069
\(842\) −29.5711 −1.01909
\(843\) −13.1613 −0.453298
\(844\) 9.63729 0.331729
\(845\) 2.73651 0.0941387
\(846\) 6.32473 0.217449
\(847\) 0 0
\(848\) 0.722424 0.0248081
\(849\) 6.42776 0.220600
\(850\) 0.425859 0.0146068
\(851\) 12.4890 0.428118
\(852\) 2.68745 0.0920704
\(853\) 48.8506 1.67261 0.836307 0.548262i \(-0.184710\pi\)
0.836307 + 0.548262i \(0.184710\pi\)
\(854\) 5.81864 0.199110
\(855\) −3.42395 −0.117097
\(856\) −1.03307 −0.0353097
\(857\) −11.0442 −0.377261 −0.188631 0.982048i \(-0.560405\pi\)
−0.188631 + 0.982048i \(0.560405\pi\)
\(858\) 0 0
\(859\) −1.39578 −0.0476236 −0.0238118 0.999716i \(-0.507580\pi\)
−0.0238118 + 0.999716i \(0.507580\pi\)
\(860\) 1.57414 0.0536778
\(861\) −1.98592 −0.0676798
\(862\) −21.7997 −0.742499
\(863\) 27.0442 0.920594 0.460297 0.887765i \(-0.347743\pi\)
0.460297 + 0.887765i \(0.347743\pi\)
\(864\) 5.65628 0.192430
\(865\) 11.9339 0.405763
\(866\) 25.5571 0.868464
\(867\) −23.9810 −0.814438
\(868\) 2.54107 0.0862495
\(869\) 0 0
\(870\) 2.85172 0.0966823
\(871\) −22.1122 −0.749244
\(872\) 9.32664 0.315840
\(873\) 6.44975 0.218291
\(874\) −9.11521 −0.308327
\(875\) 1.00000 0.0338062
\(876\) −2.24450 −0.0758347
\(877\) −17.3437 −0.585656 −0.292828 0.956165i \(-0.594596\pi\)
−0.292828 + 0.956165i \(0.594596\pi\)
\(878\) 9.82164 0.331464
\(879\) −36.8365 −1.24247
\(880\) 0 0
\(881\) 9.67527 0.325968 0.162984 0.986629i \(-0.447888\pi\)
0.162984 + 0.986629i \(0.447888\pi\)
\(882\) 0.966927 0.0325581
\(883\) 32.3077 1.08724 0.543620 0.839332i \(-0.317053\pi\)
0.543620 + 0.839332i \(0.317053\pi\)
\(884\) 1.68935 0.0568190
\(885\) 16.0331 0.538946
\(886\) −3.37870 −0.113510
\(887\) 38.4088 1.28964 0.644820 0.764334i \(-0.276932\pi\)
0.644820 + 0.764334i \(0.276932\pi\)
\(888\) −6.91786 −0.232148
\(889\) 4.21143 0.141247
\(890\) 2.54107 0.0851768
\(891\) 0 0
\(892\) 4.35971 0.145974
\(893\) 23.1624 0.775099
\(894\) −24.3567 −0.814610
\(895\) −4.31065 −0.144089
\(896\) −1.00000 −0.0334077
\(897\) −14.5601 −0.486146
\(898\) −7.29656 −0.243490
\(899\) −5.08214 −0.169499
\(900\) −0.966927 −0.0322309
\(901\) −0.307650 −0.0102493
\(902\) 0 0
\(903\) 2.24450 0.0746924
\(904\) −16.7525 −0.557180
\(905\) −0.214429 −0.00712786
\(906\) 17.9289 0.595649
\(907\) −11.9509 −0.396824 −0.198412 0.980119i \(-0.563578\pi\)
−0.198412 + 0.980119i \(0.563578\pi\)
\(908\) 22.7856 0.756166
\(909\) 1.07833 0.0357658
\(910\) 3.96693 0.131502
\(911\) 29.9639 0.992749 0.496375 0.868108i \(-0.334664\pi\)
0.496375 + 0.868108i \(0.334664\pi\)
\(912\) 5.04906 0.167191
\(913\) 0 0
\(914\) −35.5381 −1.17549
\(915\) −8.29656 −0.274276
\(916\) 15.2635 0.504320
\(917\) 6.27758 0.207304
\(918\) −2.40878 −0.0795014
\(919\) −49.4388 −1.63084 −0.815419 0.578872i \(-0.803493\pi\)
−0.815419 + 0.578872i \(0.803493\pi\)
\(920\) −2.57414 −0.0848669
\(921\) −15.8959 −0.523787
\(922\) −26.6704 −0.878342
\(923\) −7.47683 −0.246103
\(924\) 0 0
\(925\) 4.85172 0.159523
\(926\) −17.9528 −0.589967
\(927\) −16.9289 −0.556020
\(928\) 2.00000 0.0656532
\(929\) −9.83573 −0.322700 −0.161350 0.986897i \(-0.551585\pi\)
−0.161350 + 0.986897i \(0.551585\pi\)
\(930\) −3.62320 −0.118810
\(931\) 3.54107 0.116054
\(932\) −3.73651 −0.122393
\(933\) −19.8878 −0.651097
\(934\) 18.8327 0.616226
\(935\) 0 0
\(936\) −3.83573 −0.125375
\(937\) 13.2746 0.433662 0.216831 0.976209i \(-0.430428\pi\)
0.216831 + 0.976209i \(0.430428\pi\)
\(938\) −5.57414 −0.182002
\(939\) −49.1765 −1.60481
\(940\) 6.54107 0.213346
\(941\) −45.9449 −1.49776 −0.748881 0.662704i \(-0.769409\pi\)
−0.748881 + 0.662704i \(0.769409\pi\)
\(942\) 18.1833 0.592443
\(943\) −3.58523 −0.116751
\(944\) 11.2445 0.365977
\(945\) −5.65628 −0.183999
\(946\) 0 0
\(947\) 60.5351 1.96713 0.983563 0.180567i \(-0.0577932\pi\)
0.983563 + 0.180567i \(0.0577932\pi\)
\(948\) −2.19735 −0.0713665
\(949\) 6.24450 0.202705
\(950\) −3.54107 −0.114887
\(951\) −19.2605 −0.624564
\(952\) 0.425859 0.0138022
\(953\) −49.0491 −1.58885 −0.794427 0.607359i \(-0.792229\pi\)
−0.794427 + 0.607359i \(0.792229\pi\)
\(954\) 0.698531 0.0226158
\(955\) 24.9970 0.808884
\(956\) −1.78857 −0.0578465
\(957\) 0 0
\(958\) 0.310649 0.0100366
\(959\) −13.9008 −0.448880
\(960\) 1.42586 0.0460194
\(961\) −24.5430 −0.791709
\(962\) 19.2464 0.620529
\(963\) −0.998906 −0.0321893
\(964\) 17.6373 0.568059
\(965\) 20.7525 0.668047
\(966\) −3.67036 −0.118092
\(967\) −26.4229 −0.849702 −0.424851 0.905263i \(-0.639673\pi\)
−0.424851 + 0.905263i \(0.639673\pi\)
\(968\) 0 0
\(969\) −2.15019 −0.0690740
\(970\) 6.67036 0.214172
\(971\) −26.6232 −0.854379 −0.427190 0.904162i \(-0.640496\pi\)
−0.427190 + 0.904162i \(0.640496\pi\)
\(972\) 9.60531 0.308090
\(973\) 6.68935 0.214451
\(974\) 12.5362 0.401685
\(975\) −5.65628 −0.181146
\(976\) −5.81864 −0.186250
\(977\) 35.7967 1.14524 0.572618 0.819822i \(-0.305928\pi\)
0.572618 + 0.819822i \(0.305928\pi\)
\(978\) −4.65328 −0.148795
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 9.01818 0.287928
\(982\) 7.71752 0.246276
\(983\) 47.1423 1.50361 0.751803 0.659388i \(-0.229185\pi\)
0.751803 + 0.659388i \(0.229185\pi\)
\(984\) 1.98592 0.0633087
\(985\) 7.26349 0.231434
\(986\) −0.851718 −0.0271242
\(987\) 9.32664 0.296870
\(988\) −14.0472 −0.446900
\(989\) 4.05206 0.128848
\(990\) 0 0
\(991\) −17.0141 −0.540470 −0.270235 0.962794i \(-0.587101\pi\)
−0.270235 + 0.962794i \(0.587101\pi\)
\(992\) −2.54107 −0.0806790
\(993\) 42.9358 1.36253
\(994\) −1.88479 −0.0597819
\(995\) 19.5711 0.620447
\(996\) −0.818644 −0.0259397
\(997\) 29.9008 0.946967 0.473484 0.880803i \(-0.342996\pi\)
0.473484 + 0.880803i \(0.342996\pi\)
\(998\) 18.9179 0.598835
\(999\) −27.4427 −0.868247
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.cj.1.2 3
11.10 odd 2 8470.2.a.cm.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cj.1.2 3 1.1 even 1 trivial
8470.2.a.cm.1.2 yes 3 11.10 odd 2