Properties

Label 4730.2.a.bf.1.10
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4730.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.7692401561\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-2.15443\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.15443 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.15443 q^{6} +5.20049 q^{7} +1.00000 q^{8} +1.64159 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.15443 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.15443 q^{6} +5.20049 q^{7} +1.00000 q^{8} +1.64159 q^{9} -1.00000 q^{10} +1.00000 q^{11} +2.15443 q^{12} -3.69356 q^{13} +5.20049 q^{14} -2.15443 q^{15} +1.00000 q^{16} +2.28911 q^{17} +1.64159 q^{18} +3.91291 q^{19} -1.00000 q^{20} +11.2041 q^{21} +1.00000 q^{22} -7.46477 q^{23} +2.15443 q^{24} +1.00000 q^{25} -3.69356 q^{26} -2.92661 q^{27} +5.20049 q^{28} +8.04319 q^{29} -2.15443 q^{30} -2.87929 q^{31} +1.00000 q^{32} +2.15443 q^{33} +2.28911 q^{34} -5.20049 q^{35} +1.64159 q^{36} +10.2217 q^{37} +3.91291 q^{38} -7.95753 q^{39} -1.00000 q^{40} +3.12737 q^{41} +11.2041 q^{42} +1.00000 q^{43} +1.00000 q^{44} -1.64159 q^{45} -7.46477 q^{46} -9.31889 q^{47} +2.15443 q^{48} +20.0451 q^{49} +1.00000 q^{50} +4.93173 q^{51} -3.69356 q^{52} -0.670647 q^{53} -2.92661 q^{54} -1.00000 q^{55} +5.20049 q^{56} +8.43011 q^{57} +8.04319 q^{58} +6.55249 q^{59} -2.15443 q^{60} +7.70264 q^{61} -2.87929 q^{62} +8.53706 q^{63} +1.00000 q^{64} +3.69356 q^{65} +2.15443 q^{66} -4.29126 q^{67} +2.28911 q^{68} -16.0824 q^{69} -5.20049 q^{70} -8.54294 q^{71} +1.64159 q^{72} -9.80369 q^{73} +10.2217 q^{74} +2.15443 q^{75} +3.91291 q^{76} +5.20049 q^{77} -7.95753 q^{78} +11.4945 q^{79} -1.00000 q^{80} -11.2300 q^{81} +3.12737 q^{82} -6.18517 q^{83} +11.2041 q^{84} -2.28911 q^{85} +1.00000 q^{86} +17.3285 q^{87} +1.00000 q^{88} -5.34397 q^{89} -1.64159 q^{90} -19.2083 q^{91} -7.46477 q^{92} -6.20325 q^{93} -9.31889 q^{94} -3.91291 q^{95} +2.15443 q^{96} +5.40206 q^{97} +20.0451 q^{98} +1.64159 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} + O(q^{10}) \) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} - 13q^{10} + 13q^{11} + 11q^{13} + 7q^{14} + 13q^{16} - 2q^{17} + 25q^{18} + 7q^{19} - 13q^{20} + 12q^{21} + 13q^{22} + 12q^{23} + 13q^{25} + 11q^{26} - 15q^{27} + 7q^{28} + 14q^{29} + 20q^{31} + 13q^{32} - 2q^{34} - 7q^{35} + 25q^{36} + 17q^{37} + 7q^{38} - 4q^{39} - 13q^{40} + 9q^{41} + 12q^{42} + 13q^{43} + 13q^{44} - 25q^{45} + 12q^{46} - 9q^{47} + 30q^{49} + 13q^{50} - 3q^{51} + 11q^{52} + 22q^{53} - 15q^{54} - 13q^{55} + 7q^{56} + 17q^{57} + 14q^{58} + 19q^{59} + 2q^{61} + 20q^{62} + 12q^{63} + 13q^{64} - 11q^{65} + 9q^{67} - 2q^{68} - 6q^{69} - 7q^{70} + 6q^{71} + 25q^{72} + 7q^{73} + 17q^{74} + 7q^{76} + 7q^{77} - 4q^{78} + 50q^{79} - 13q^{80} + 85q^{81} + 9q^{82} + q^{83} + 12q^{84} + 2q^{85} + 13q^{86} - 21q^{87} + 13q^{88} + 5q^{89} - 25q^{90} + 5q^{91} + 12q^{92} + 3q^{93} - 9q^{94} - 7q^{95} + 20q^{97} + 30q^{98} + 25q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.15443 1.24386 0.621932 0.783071i \(-0.286348\pi\)
0.621932 + 0.783071i \(0.286348\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.15443 0.879544
\(7\) 5.20049 1.96560 0.982800 0.184674i \(-0.0591230\pi\)
0.982800 + 0.184674i \(0.0591230\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.64159 0.547196
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 2.15443 0.621932
\(13\) −3.69356 −1.02441 −0.512204 0.858864i \(-0.671171\pi\)
−0.512204 + 0.858864i \(0.671171\pi\)
\(14\) 5.20049 1.38989
\(15\) −2.15443 −0.556273
\(16\) 1.00000 0.250000
\(17\) 2.28911 0.555190 0.277595 0.960698i \(-0.410463\pi\)
0.277595 + 0.960698i \(0.410463\pi\)
\(18\) 1.64159 0.386926
\(19\) 3.91291 0.897684 0.448842 0.893611i \(-0.351837\pi\)
0.448842 + 0.893611i \(0.351837\pi\)
\(20\) −1.00000 −0.223607
\(21\) 11.2041 2.44494
\(22\) 1.00000 0.213201
\(23\) −7.46477 −1.55651 −0.778256 0.627948i \(-0.783895\pi\)
−0.778256 + 0.627948i \(0.783895\pi\)
\(24\) 2.15443 0.439772
\(25\) 1.00000 0.200000
\(26\) −3.69356 −0.724366
\(27\) −2.92661 −0.563226
\(28\) 5.20049 0.982800
\(29\) 8.04319 1.49358 0.746791 0.665058i \(-0.231593\pi\)
0.746791 + 0.665058i \(0.231593\pi\)
\(30\) −2.15443 −0.393344
\(31\) −2.87929 −0.517136 −0.258568 0.965993i \(-0.583251\pi\)
−0.258568 + 0.965993i \(0.583251\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.15443 0.375039
\(34\) 2.28911 0.392579
\(35\) −5.20049 −0.879043
\(36\) 1.64159 0.273598
\(37\) 10.2217 1.68043 0.840216 0.542253i \(-0.182428\pi\)
0.840216 + 0.542253i \(0.182428\pi\)
\(38\) 3.91291 0.634758
\(39\) −7.95753 −1.27422
\(40\) −1.00000 −0.158114
\(41\) 3.12737 0.488413 0.244206 0.969723i \(-0.421473\pi\)
0.244206 + 0.969723i \(0.421473\pi\)
\(42\) 11.2041 1.72883
\(43\) 1.00000 0.152499
\(44\) 1.00000 0.150756
\(45\) −1.64159 −0.244714
\(46\) −7.46477 −1.10062
\(47\) −9.31889 −1.35930 −0.679650 0.733536i \(-0.737868\pi\)
−0.679650 + 0.733536i \(0.737868\pi\)
\(48\) 2.15443 0.310966
\(49\) 20.0451 2.86358
\(50\) 1.00000 0.141421
\(51\) 4.93173 0.690581
\(52\) −3.69356 −0.512204
\(53\) −0.670647 −0.0921205 −0.0460602 0.998939i \(-0.514667\pi\)
−0.0460602 + 0.998939i \(0.514667\pi\)
\(54\) −2.92661 −0.398261
\(55\) −1.00000 −0.134840
\(56\) 5.20049 0.694944
\(57\) 8.43011 1.11660
\(58\) 8.04319 1.05612
\(59\) 6.55249 0.853062 0.426531 0.904473i \(-0.359736\pi\)
0.426531 + 0.904473i \(0.359736\pi\)
\(60\) −2.15443 −0.278136
\(61\) 7.70264 0.986221 0.493111 0.869967i \(-0.335860\pi\)
0.493111 + 0.869967i \(0.335860\pi\)
\(62\) −2.87929 −0.365671
\(63\) 8.53706 1.07557
\(64\) 1.00000 0.125000
\(65\) 3.69356 0.458129
\(66\) 2.15443 0.265193
\(67\) −4.29126 −0.524261 −0.262131 0.965032i \(-0.584425\pi\)
−0.262131 + 0.965032i \(0.584425\pi\)
\(68\) 2.28911 0.277595
\(69\) −16.0824 −1.93609
\(70\) −5.20049 −0.621577
\(71\) −8.54294 −1.01386 −0.506930 0.861987i \(-0.669220\pi\)
−0.506930 + 0.861987i \(0.669220\pi\)
\(72\) 1.64159 0.193463
\(73\) −9.80369 −1.14743 −0.573717 0.819053i \(-0.694499\pi\)
−0.573717 + 0.819053i \(0.694499\pi\)
\(74\) 10.2217 1.18824
\(75\) 2.15443 0.248773
\(76\) 3.91291 0.448842
\(77\) 5.20049 0.592651
\(78\) −7.95753 −0.901012
\(79\) 11.4945 1.29323 0.646614 0.762818i \(-0.276185\pi\)
0.646614 + 0.762818i \(0.276185\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.2300 −1.24777
\(82\) 3.12737 0.345360
\(83\) −6.18517 −0.678910 −0.339455 0.940622i \(-0.610243\pi\)
−0.339455 + 0.940622i \(0.610243\pi\)
\(84\) 11.2041 1.22247
\(85\) −2.28911 −0.248289
\(86\) 1.00000 0.107833
\(87\) 17.3285 1.85781
\(88\) 1.00000 0.106600
\(89\) −5.34397 −0.566460 −0.283230 0.959052i \(-0.591406\pi\)
−0.283230 + 0.959052i \(0.591406\pi\)
\(90\) −1.64159 −0.173039
\(91\) −19.2083 −2.01358
\(92\) −7.46477 −0.778256
\(93\) −6.20325 −0.643247
\(94\) −9.31889 −0.961170
\(95\) −3.91291 −0.401456
\(96\) 2.15443 0.219886
\(97\) 5.40206 0.548496 0.274248 0.961659i \(-0.411571\pi\)
0.274248 + 0.961659i \(0.411571\pi\)
\(98\) 20.0451 2.02486
\(99\) 1.64159 0.164986
\(100\) 1.00000 0.100000
\(101\) 16.4334 1.63518 0.817591 0.575799i \(-0.195309\pi\)
0.817591 + 0.575799i \(0.195309\pi\)
\(102\) 4.93173 0.488314
\(103\) −12.9541 −1.27640 −0.638201 0.769870i \(-0.720321\pi\)
−0.638201 + 0.769870i \(0.720321\pi\)
\(104\) −3.69356 −0.362183
\(105\) −11.2041 −1.09341
\(106\) −0.670647 −0.0651390
\(107\) −16.3514 −1.58075 −0.790373 0.612626i \(-0.790113\pi\)
−0.790373 + 0.612626i \(0.790113\pi\)
\(108\) −2.92661 −0.281613
\(109\) 16.8177 1.61084 0.805419 0.592705i \(-0.201940\pi\)
0.805419 + 0.592705i \(0.201940\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 22.0219 2.09023
\(112\) 5.20049 0.491400
\(113\) −5.65787 −0.532248 −0.266124 0.963939i \(-0.585743\pi\)
−0.266124 + 0.963939i \(0.585743\pi\)
\(114\) 8.43011 0.789553
\(115\) 7.46477 0.696093
\(116\) 8.04319 0.746791
\(117\) −6.06330 −0.560552
\(118\) 6.55249 0.603206
\(119\) 11.9045 1.09128
\(120\) −2.15443 −0.196672
\(121\) 1.00000 0.0909091
\(122\) 7.70264 0.697364
\(123\) 6.73771 0.607519
\(124\) −2.87929 −0.258568
\(125\) −1.00000 −0.0894427
\(126\) 8.53706 0.760542
\(127\) −17.6856 −1.56934 −0.784670 0.619914i \(-0.787167\pi\)
−0.784670 + 0.619914i \(0.787167\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.15443 0.189687
\(130\) 3.69356 0.323946
\(131\) −16.4398 −1.43635 −0.718176 0.695861i \(-0.755023\pi\)
−0.718176 + 0.695861i \(0.755023\pi\)
\(132\) 2.15443 0.187519
\(133\) 20.3491 1.76449
\(134\) −4.29126 −0.370709
\(135\) 2.92661 0.251882
\(136\) 2.28911 0.196289
\(137\) −4.54096 −0.387960 −0.193980 0.981005i \(-0.562140\pi\)
−0.193980 + 0.981005i \(0.562140\pi\)
\(138\) −16.0824 −1.36902
\(139\) 17.7324 1.50404 0.752021 0.659139i \(-0.229079\pi\)
0.752021 + 0.659139i \(0.229079\pi\)
\(140\) −5.20049 −0.439521
\(141\) −20.0769 −1.69078
\(142\) −8.54294 −0.716908
\(143\) −3.69356 −0.308871
\(144\) 1.64159 0.136799
\(145\) −8.04319 −0.667950
\(146\) −9.80369 −0.811359
\(147\) 43.1858 3.56190
\(148\) 10.2217 0.840216
\(149\) −2.59038 −0.212212 −0.106106 0.994355i \(-0.533838\pi\)
−0.106106 + 0.994355i \(0.533838\pi\)
\(150\) 2.15443 0.175909
\(151\) 20.9220 1.70261 0.851304 0.524673i \(-0.175813\pi\)
0.851304 + 0.524673i \(0.175813\pi\)
\(152\) 3.91291 0.317379
\(153\) 3.75777 0.303798
\(154\) 5.20049 0.419067
\(155\) 2.87929 0.231270
\(156\) −7.95753 −0.637112
\(157\) −13.6347 −1.08817 −0.544083 0.839031i \(-0.683122\pi\)
−0.544083 + 0.839031i \(0.683122\pi\)
\(158\) 11.4945 0.914450
\(159\) −1.44487 −0.114585
\(160\) −1.00000 −0.0790569
\(161\) −38.8204 −3.05948
\(162\) −11.2300 −0.882308
\(163\) −18.6569 −1.46133 −0.730663 0.682739i \(-0.760789\pi\)
−0.730663 + 0.682739i \(0.760789\pi\)
\(164\) 3.12737 0.244206
\(165\) −2.15443 −0.167723
\(166\) −6.18517 −0.480062
\(167\) 2.47973 0.191887 0.0959435 0.995387i \(-0.469413\pi\)
0.0959435 + 0.995387i \(0.469413\pi\)
\(168\) 11.2041 0.864416
\(169\) 0.642357 0.0494121
\(170\) −2.28911 −0.175567
\(171\) 6.42339 0.491209
\(172\) 1.00000 0.0762493
\(173\) −2.91913 −0.221937 −0.110968 0.993824i \(-0.535395\pi\)
−0.110968 + 0.993824i \(0.535395\pi\)
\(174\) 17.3285 1.31367
\(175\) 5.20049 0.393120
\(176\) 1.00000 0.0753778
\(177\) 14.1169 1.06109
\(178\) −5.34397 −0.400548
\(179\) −13.7699 −1.02921 −0.514606 0.857427i \(-0.672062\pi\)
−0.514606 + 0.857427i \(0.672062\pi\)
\(180\) −1.64159 −0.122357
\(181\) −9.50722 −0.706666 −0.353333 0.935498i \(-0.614952\pi\)
−0.353333 + 0.935498i \(0.614952\pi\)
\(182\) −19.2083 −1.42381
\(183\) 16.5948 1.22672
\(184\) −7.46477 −0.550310
\(185\) −10.2217 −0.751512
\(186\) −6.20325 −0.454844
\(187\) 2.28911 0.167396
\(188\) −9.31889 −0.679650
\(189\) −15.2198 −1.10708
\(190\) −3.91291 −0.283873
\(191\) 22.3373 1.61627 0.808134 0.588998i \(-0.200477\pi\)
0.808134 + 0.588998i \(0.200477\pi\)
\(192\) 2.15443 0.155483
\(193\) 5.77134 0.415430 0.207715 0.978189i \(-0.433397\pi\)
0.207715 + 0.978189i \(0.433397\pi\)
\(194\) 5.40206 0.387845
\(195\) 7.95753 0.569850
\(196\) 20.0451 1.43179
\(197\) 7.78147 0.554407 0.277203 0.960811i \(-0.410592\pi\)
0.277203 + 0.960811i \(0.410592\pi\)
\(198\) 1.64159 0.116663
\(199\) −17.2211 −1.22077 −0.610387 0.792104i \(-0.708986\pi\)
−0.610387 + 0.792104i \(0.708986\pi\)
\(200\) 1.00000 0.0707107
\(201\) −9.24525 −0.652109
\(202\) 16.4334 1.15625
\(203\) 41.8285 2.93579
\(204\) 4.93173 0.345290
\(205\) −3.12737 −0.218425
\(206\) −12.9541 −0.902552
\(207\) −12.2541 −0.851717
\(208\) −3.69356 −0.256102
\(209\) 3.91291 0.270662
\(210\) −11.2041 −0.773157
\(211\) −2.25780 −0.155434 −0.0777168 0.996975i \(-0.524763\pi\)
−0.0777168 + 0.996975i \(0.524763\pi\)
\(212\) −0.670647 −0.0460602
\(213\) −18.4052 −1.26110
\(214\) −16.3514 −1.11776
\(215\) −1.00000 −0.0681994
\(216\) −2.92661 −0.199131
\(217\) −14.9737 −1.01648
\(218\) 16.8177 1.13904
\(219\) −21.1214 −1.42725
\(220\) −1.00000 −0.0674200
\(221\) −8.45495 −0.568741
\(222\) 22.0219 1.47801
\(223\) −5.70292 −0.381896 −0.190948 0.981600i \(-0.561156\pi\)
−0.190948 + 0.981600i \(0.561156\pi\)
\(224\) 5.20049 0.347472
\(225\) 1.64159 0.109439
\(226\) −5.65787 −0.376356
\(227\) −1.85378 −0.123040 −0.0615199 0.998106i \(-0.519595\pi\)
−0.0615199 + 0.998106i \(0.519595\pi\)
\(228\) 8.43011 0.558298
\(229\) −12.0610 −0.797014 −0.398507 0.917165i \(-0.630472\pi\)
−0.398507 + 0.917165i \(0.630472\pi\)
\(230\) 7.46477 0.492212
\(231\) 11.2041 0.737176
\(232\) 8.04319 0.528061
\(233\) 6.65456 0.435955 0.217977 0.975954i \(-0.430054\pi\)
0.217977 + 0.975954i \(0.430054\pi\)
\(234\) −6.06330 −0.396370
\(235\) 9.31889 0.607898
\(236\) 6.55249 0.426531
\(237\) 24.7641 1.60860
\(238\) 11.9045 0.771653
\(239\) −15.6396 −1.01164 −0.505819 0.862639i \(-0.668810\pi\)
−0.505819 + 0.862639i \(0.668810\pi\)
\(240\) −2.15443 −0.139068
\(241\) −0.896388 −0.0577414 −0.0288707 0.999583i \(-0.509191\pi\)
−0.0288707 + 0.999583i \(0.509191\pi\)
\(242\) 1.00000 0.0642824
\(243\) −15.4144 −0.988832
\(244\) 7.70264 0.493111
\(245\) −20.0451 −1.28063
\(246\) 6.73771 0.429581
\(247\) −14.4526 −0.919595
\(248\) −2.87929 −0.182835
\(249\) −13.3255 −0.844471
\(250\) −1.00000 −0.0632456
\(251\) 3.71338 0.234387 0.117193 0.993109i \(-0.462610\pi\)
0.117193 + 0.993109i \(0.462610\pi\)
\(252\) 8.53706 0.537784
\(253\) −7.46477 −0.469306
\(254\) −17.6856 −1.10969
\(255\) −4.93173 −0.308837
\(256\) 1.00000 0.0625000
\(257\) 5.07711 0.316701 0.158351 0.987383i \(-0.449382\pi\)
0.158351 + 0.987383i \(0.449382\pi\)
\(258\) 2.15443 0.134129
\(259\) 53.1576 3.30305
\(260\) 3.69356 0.229065
\(261\) 13.2036 0.817283
\(262\) −16.4398 −1.01565
\(263\) 27.1172 1.67212 0.836058 0.548641i \(-0.184855\pi\)
0.836058 + 0.548641i \(0.184855\pi\)
\(264\) 2.15443 0.132596
\(265\) 0.670647 0.0411975
\(266\) 20.3491 1.24768
\(267\) −11.5132 −0.704599
\(268\) −4.29126 −0.262131
\(269\) −2.78067 −0.169540 −0.0847702 0.996401i \(-0.527016\pi\)
−0.0847702 + 0.996401i \(0.527016\pi\)
\(270\) 2.92661 0.178108
\(271\) −12.6460 −0.768189 −0.384094 0.923294i \(-0.625486\pi\)
−0.384094 + 0.923294i \(0.625486\pi\)
\(272\) 2.28911 0.138798
\(273\) −41.3830 −2.50461
\(274\) −4.54096 −0.274329
\(275\) 1.00000 0.0603023
\(276\) −16.0824 −0.968044
\(277\) −6.74725 −0.405403 −0.202701 0.979241i \(-0.564972\pi\)
−0.202701 + 0.979241i \(0.564972\pi\)
\(278\) 17.7324 1.06352
\(279\) −4.72661 −0.282975
\(280\) −5.20049 −0.310789
\(281\) −5.29405 −0.315816 −0.157908 0.987454i \(-0.550475\pi\)
−0.157908 + 0.987454i \(0.550475\pi\)
\(282\) −20.0769 −1.19556
\(283\) −0.938443 −0.0557846 −0.0278923 0.999611i \(-0.508880\pi\)
−0.0278923 + 0.999611i \(0.508880\pi\)
\(284\) −8.54294 −0.506930
\(285\) −8.43011 −0.499357
\(286\) −3.69356 −0.218405
\(287\) 16.2638 0.960024
\(288\) 1.64159 0.0967315
\(289\) −11.7600 −0.691764
\(290\) −8.04319 −0.472312
\(291\) 11.6384 0.682254
\(292\) −9.80369 −0.573717
\(293\) 25.1151 1.46724 0.733620 0.679560i \(-0.237829\pi\)
0.733620 + 0.679560i \(0.237829\pi\)
\(294\) 43.1858 2.51865
\(295\) −6.55249 −0.381501
\(296\) 10.2217 0.594122
\(297\) −2.92661 −0.169819
\(298\) −2.59038 −0.150057
\(299\) 27.5715 1.59450
\(300\) 2.15443 0.124386
\(301\) 5.20049 0.299751
\(302\) 20.9220 1.20393
\(303\) 35.4046 2.03394
\(304\) 3.91291 0.224421
\(305\) −7.70264 −0.441052
\(306\) 3.75777 0.214818
\(307\) −11.3935 −0.650262 −0.325131 0.945669i \(-0.605408\pi\)
−0.325131 + 0.945669i \(0.605408\pi\)
\(308\) 5.20049 0.296325
\(309\) −27.9087 −1.58767
\(310\) 2.87929 0.163533
\(311\) −4.64254 −0.263254 −0.131627 0.991299i \(-0.542020\pi\)
−0.131627 + 0.991299i \(0.542020\pi\)
\(312\) −7.95753 −0.450506
\(313\) −7.23924 −0.409186 −0.204593 0.978847i \(-0.565587\pi\)
−0.204593 + 0.978847i \(0.565587\pi\)
\(314\) −13.6347 −0.769450
\(315\) −8.53706 −0.481009
\(316\) 11.4945 0.646614
\(317\) 13.1816 0.740353 0.370176 0.928961i \(-0.379297\pi\)
0.370176 + 0.928961i \(0.379297\pi\)
\(318\) −1.44487 −0.0810240
\(319\) 8.04319 0.450332
\(320\) −1.00000 −0.0559017
\(321\) −35.2280 −1.96623
\(322\) −38.8204 −2.16338
\(323\) 8.95708 0.498385
\(324\) −11.2300 −0.623886
\(325\) −3.69356 −0.204882
\(326\) −18.6569 −1.03331
\(327\) 36.2325 2.00366
\(328\) 3.12737 0.172680
\(329\) −48.4628 −2.67184
\(330\) −2.15443 −0.118598
\(331\) 5.45659 0.299921 0.149961 0.988692i \(-0.452085\pi\)
0.149961 + 0.988692i \(0.452085\pi\)
\(332\) −6.18517 −0.339455
\(333\) 16.7798 0.919525
\(334\) 2.47973 0.135685
\(335\) 4.29126 0.234457
\(336\) 11.2041 0.611234
\(337\) 23.7542 1.29397 0.646987 0.762501i \(-0.276029\pi\)
0.646987 + 0.762501i \(0.276029\pi\)
\(338\) 0.642357 0.0349396
\(339\) −12.1895 −0.662044
\(340\) −2.28911 −0.124144
\(341\) −2.87929 −0.155922
\(342\) 6.42339 0.347337
\(343\) 67.8407 3.66306
\(344\) 1.00000 0.0539164
\(345\) 16.0824 0.865845
\(346\) −2.91913 −0.156933
\(347\) −27.5506 −1.47899 −0.739497 0.673160i \(-0.764937\pi\)
−0.739497 + 0.673160i \(0.764937\pi\)
\(348\) 17.3285 0.928906
\(349\) 19.1046 1.02264 0.511322 0.859389i \(-0.329156\pi\)
0.511322 + 0.859389i \(0.329156\pi\)
\(350\) 5.20049 0.277978
\(351\) 10.8096 0.576973
\(352\) 1.00000 0.0533002
\(353\) −19.9199 −1.06023 −0.530114 0.847926i \(-0.677851\pi\)
−0.530114 + 0.847926i \(0.677851\pi\)
\(354\) 14.1169 0.750306
\(355\) 8.54294 0.453412
\(356\) −5.34397 −0.283230
\(357\) 25.6474 1.35741
\(358\) −13.7699 −0.727763
\(359\) −6.69249 −0.353216 −0.176608 0.984281i \(-0.556512\pi\)
−0.176608 + 0.984281i \(0.556512\pi\)
\(360\) −1.64159 −0.0865193
\(361\) −3.68911 −0.194164
\(362\) −9.50722 −0.499688
\(363\) 2.15443 0.113078
\(364\) −19.2083 −1.00679
\(365\) 9.80369 0.513149
\(366\) 16.5948 0.867425
\(367\) −10.2267 −0.533827 −0.266914 0.963720i \(-0.586004\pi\)
−0.266914 + 0.963720i \(0.586004\pi\)
\(368\) −7.46477 −0.389128
\(369\) 5.13385 0.267258
\(370\) −10.2217 −0.531399
\(371\) −3.48769 −0.181072
\(372\) −6.20325 −0.321623
\(373\) 18.4363 0.954594 0.477297 0.878742i \(-0.341617\pi\)
0.477297 + 0.878742i \(0.341617\pi\)
\(374\) 2.28911 0.118367
\(375\) −2.15443 −0.111255
\(376\) −9.31889 −0.480585
\(377\) −29.7080 −1.53004
\(378\) −15.2198 −0.782822
\(379\) 2.83712 0.145733 0.0728666 0.997342i \(-0.476785\pi\)
0.0728666 + 0.997342i \(0.476785\pi\)
\(380\) −3.91291 −0.200728
\(381\) −38.1024 −1.95204
\(382\) 22.3373 1.14287
\(383\) 13.9972 0.715223 0.357612 0.933870i \(-0.383591\pi\)
0.357612 + 0.933870i \(0.383591\pi\)
\(384\) 2.15443 0.109943
\(385\) −5.20049 −0.265041
\(386\) 5.77134 0.293753
\(387\) 1.64159 0.0834466
\(388\) 5.40206 0.274248
\(389\) −3.40004 −0.172389 −0.0861944 0.996278i \(-0.527471\pi\)
−0.0861944 + 0.996278i \(0.527471\pi\)
\(390\) 7.95753 0.402945
\(391\) −17.0877 −0.864160
\(392\) 20.0451 1.01243
\(393\) −35.4185 −1.78663
\(394\) 7.78147 0.392025
\(395\) −11.4945 −0.578349
\(396\) 1.64159 0.0824929
\(397\) 19.0532 0.956251 0.478125 0.878292i \(-0.341316\pi\)
0.478125 + 0.878292i \(0.341316\pi\)
\(398\) −17.2211 −0.863217
\(399\) 43.8407 2.19478
\(400\) 1.00000 0.0500000
\(401\) 6.62751 0.330962 0.165481 0.986213i \(-0.447082\pi\)
0.165481 + 0.986213i \(0.447082\pi\)
\(402\) −9.24525 −0.461111
\(403\) 10.6348 0.529759
\(404\) 16.4334 0.817591
\(405\) 11.2300 0.558021
\(406\) 41.8285 2.07591
\(407\) 10.2217 0.506669
\(408\) 4.93173 0.244157
\(409\) −26.0155 −1.28638 −0.643192 0.765705i \(-0.722390\pi\)
−0.643192 + 0.765705i \(0.722390\pi\)
\(410\) −3.12737 −0.154450
\(411\) −9.78319 −0.482569
\(412\) −12.9541 −0.638201
\(413\) 34.0762 1.67678
\(414\) −12.2541 −0.602255
\(415\) 6.18517 0.303618
\(416\) −3.69356 −0.181091
\(417\) 38.2033 1.87082
\(418\) 3.91291 0.191387
\(419\) 17.7419 0.866747 0.433373 0.901214i \(-0.357323\pi\)
0.433373 + 0.901214i \(0.357323\pi\)
\(420\) −11.2041 −0.546705
\(421\) 7.77410 0.378886 0.189443 0.981892i \(-0.439332\pi\)
0.189443 + 0.981892i \(0.439332\pi\)
\(422\) −2.25780 −0.109908
\(423\) −15.2978 −0.743804
\(424\) −0.670647 −0.0325695
\(425\) 2.28911 0.111038
\(426\) −18.4052 −0.891735
\(427\) 40.0575 1.93852
\(428\) −16.3514 −0.790373
\(429\) −7.95753 −0.384193
\(430\) −1.00000 −0.0482243
\(431\) 24.8823 1.19854 0.599270 0.800547i \(-0.295458\pi\)
0.599270 + 0.800547i \(0.295458\pi\)
\(432\) −2.92661 −0.140807
\(433\) 32.3003 1.55226 0.776128 0.630576i \(-0.217181\pi\)
0.776128 + 0.630576i \(0.217181\pi\)
\(434\) −14.9737 −0.718762
\(435\) −17.3285 −0.830839
\(436\) 16.8177 0.805419
\(437\) −29.2090 −1.39725
\(438\) −21.1214 −1.00922
\(439\) −33.5844 −1.60290 −0.801448 0.598064i \(-0.795937\pi\)
−0.801448 + 0.598064i \(0.795937\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 32.9058 1.56694
\(442\) −8.45495 −0.402161
\(443\) −26.5635 −1.26207 −0.631036 0.775754i \(-0.717370\pi\)
−0.631036 + 0.775754i \(0.717370\pi\)
\(444\) 22.0219 1.04511
\(445\) 5.34397 0.253329
\(446\) −5.70292 −0.270041
\(447\) −5.58080 −0.263963
\(448\) 5.20049 0.245700
\(449\) 36.7589 1.73476 0.867379 0.497648i \(-0.165803\pi\)
0.867379 + 0.497648i \(0.165803\pi\)
\(450\) 1.64159 0.0773852
\(451\) 3.12737 0.147262
\(452\) −5.65787 −0.266124
\(453\) 45.0751 2.11781
\(454\) −1.85378 −0.0870022
\(455\) 19.2083 0.900499
\(456\) 8.43011 0.394776
\(457\) 24.9793 1.16848 0.584241 0.811581i \(-0.301392\pi\)
0.584241 + 0.811581i \(0.301392\pi\)
\(458\) −12.0610 −0.563574
\(459\) −6.69932 −0.312698
\(460\) 7.46477 0.348047
\(461\) −28.6874 −1.33610 −0.668052 0.744115i \(-0.732872\pi\)
−0.668052 + 0.744115i \(0.732872\pi\)
\(462\) 11.2041 0.521262
\(463\) −6.94892 −0.322944 −0.161472 0.986877i \(-0.551624\pi\)
−0.161472 + 0.986877i \(0.551624\pi\)
\(464\) 8.04319 0.373396
\(465\) 6.20325 0.287669
\(466\) 6.65456 0.308267
\(467\) −25.0640 −1.15982 −0.579911 0.814680i \(-0.696913\pi\)
−0.579911 + 0.814680i \(0.696913\pi\)
\(468\) −6.06330 −0.280276
\(469\) −22.3167 −1.03049
\(470\) 9.31889 0.429848
\(471\) −29.3750 −1.35353
\(472\) 6.55249 0.301603
\(473\) 1.00000 0.0459800
\(474\) 24.7641 1.13745
\(475\) 3.91291 0.179537
\(476\) 11.9045 0.545641
\(477\) −1.10093 −0.0504080
\(478\) −15.6396 −0.715337
\(479\) −24.7063 −1.12886 −0.564429 0.825482i \(-0.690904\pi\)
−0.564429 + 0.825482i \(0.690904\pi\)
\(480\) −2.15443 −0.0983360
\(481\) −37.7543 −1.72145
\(482\) −0.896388 −0.0408294
\(483\) −83.6361 −3.80557
\(484\) 1.00000 0.0454545
\(485\) −5.40206 −0.245295
\(486\) −15.4144 −0.699210
\(487\) −27.4194 −1.24249 −0.621245 0.783617i \(-0.713373\pi\)
−0.621245 + 0.783617i \(0.713373\pi\)
\(488\) 7.70264 0.348682
\(489\) −40.1952 −1.81769
\(490\) −20.0451 −0.905544
\(491\) −26.8303 −1.21084 −0.605418 0.795907i \(-0.706994\pi\)
−0.605418 + 0.795907i \(0.706994\pi\)
\(492\) 6.73771 0.303759
\(493\) 18.4117 0.829223
\(494\) −14.4526 −0.650252
\(495\) −1.64159 −0.0737839
\(496\) −2.87929 −0.129284
\(497\) −44.4275 −1.99284
\(498\) −13.3255 −0.597131
\(499\) −19.6464 −0.879495 −0.439748 0.898121i \(-0.644932\pi\)
−0.439748 + 0.898121i \(0.644932\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 5.34241 0.238681
\(502\) 3.71338 0.165736
\(503\) −21.2482 −0.947409 −0.473705 0.880684i \(-0.657084\pi\)
−0.473705 + 0.880684i \(0.657084\pi\)
\(504\) 8.53706 0.380271
\(505\) −16.4334 −0.731276
\(506\) −7.46477 −0.331849
\(507\) 1.38392 0.0614619
\(508\) −17.6856 −0.784670
\(509\) −13.9210 −0.617036 −0.308518 0.951219i \(-0.599833\pi\)
−0.308518 + 0.951219i \(0.599833\pi\)
\(510\) −4.93173 −0.218381
\(511\) −50.9840 −2.25540
\(512\) 1.00000 0.0441942
\(513\) −11.4516 −0.505599
\(514\) 5.07711 0.223942
\(515\) 12.9541 0.570824
\(516\) 2.15443 0.0948437
\(517\) −9.31889 −0.409844
\(518\) 53.1576 2.33561
\(519\) −6.28906 −0.276059
\(520\) 3.69356 0.161973
\(521\) 40.9486 1.79399 0.896995 0.442041i \(-0.145745\pi\)
0.896995 + 0.442041i \(0.145745\pi\)
\(522\) 13.2036 0.577906
\(523\) 21.5173 0.940887 0.470444 0.882430i \(-0.344094\pi\)
0.470444 + 0.882430i \(0.344094\pi\)
\(524\) −16.4398 −0.718176
\(525\) 11.2041 0.488987
\(526\) 27.1172 1.18236
\(527\) −6.59101 −0.287109
\(528\) 2.15443 0.0937597
\(529\) 32.7227 1.42273
\(530\) 0.670647 0.0291310
\(531\) 10.7565 0.466792
\(532\) 20.3491 0.882243
\(533\) −11.5511 −0.500334
\(534\) −11.5132 −0.498227
\(535\) 16.3514 0.706931
\(536\) −4.29126 −0.185354
\(537\) −29.6664 −1.28020
\(538\) −2.78067 −0.119883
\(539\) 20.0451 0.863402
\(540\) 2.92661 0.125941
\(541\) 25.3884 1.09153 0.545766 0.837938i \(-0.316239\pi\)
0.545766 + 0.837938i \(0.316239\pi\)
\(542\) −12.6460 −0.543191
\(543\) −20.4827 −0.878996
\(544\) 2.28911 0.0981447
\(545\) −16.8177 −0.720389
\(546\) −41.3830 −1.77103
\(547\) −27.2451 −1.16492 −0.582458 0.812861i \(-0.697909\pi\)
−0.582458 + 0.812861i \(0.697909\pi\)
\(548\) −4.54096 −0.193980
\(549\) 12.6446 0.539657
\(550\) 1.00000 0.0426401
\(551\) 31.4723 1.34076
\(552\) −16.0824 −0.684510
\(553\) 59.7768 2.54197
\(554\) −6.74725 −0.286663
\(555\) −22.0219 −0.934778
\(556\) 17.7324 0.752021
\(557\) −38.6556 −1.63789 −0.818945 0.573872i \(-0.805441\pi\)
−0.818945 + 0.573872i \(0.805441\pi\)
\(558\) −4.72661 −0.200094
\(559\) −3.69356 −0.156221
\(560\) −5.20049 −0.219761
\(561\) 4.93173 0.208218
\(562\) −5.29405 −0.223316
\(563\) −10.9047 −0.459579 −0.229790 0.973240i \(-0.573804\pi\)
−0.229790 + 0.973240i \(0.573804\pi\)
\(564\) −20.0769 −0.845392
\(565\) 5.65787 0.238028
\(566\) −0.938443 −0.0394457
\(567\) −58.4012 −2.45262
\(568\) −8.54294 −0.358454
\(569\) −42.4101 −1.77792 −0.888961 0.457982i \(-0.848572\pi\)
−0.888961 + 0.457982i \(0.848572\pi\)
\(570\) −8.43011 −0.353099
\(571\) 0.618946 0.0259021 0.0129510 0.999916i \(-0.495877\pi\)
0.0129510 + 0.999916i \(0.495877\pi\)
\(572\) −3.69356 −0.154435
\(573\) 48.1242 2.01042
\(574\) 16.2638 0.678839
\(575\) −7.46477 −0.311302
\(576\) 1.64159 0.0683995
\(577\) −27.3660 −1.13926 −0.569630 0.821901i \(-0.692914\pi\)
−0.569630 + 0.821901i \(0.692914\pi\)
\(578\) −11.7600 −0.489151
\(579\) 12.4340 0.516738
\(580\) −8.04319 −0.333975
\(581\) −32.1659 −1.33447
\(582\) 11.6384 0.482426
\(583\) −0.670647 −0.0277754
\(584\) −9.80369 −0.405680
\(585\) 6.06330 0.250687
\(586\) 25.1151 1.03750
\(587\) 3.63605 0.150076 0.0750380 0.997181i \(-0.476092\pi\)
0.0750380 + 0.997181i \(0.476092\pi\)
\(588\) 43.1858 1.78095
\(589\) −11.2664 −0.464225
\(590\) −6.55249 −0.269762
\(591\) 16.7647 0.689606
\(592\) 10.2217 0.420108
\(593\) 2.40157 0.0986206 0.0493103 0.998784i \(-0.484298\pi\)
0.0493103 + 0.998784i \(0.484298\pi\)
\(594\) −2.92661 −0.120080
\(595\) −11.9045 −0.488036
\(596\) −2.59038 −0.106106
\(597\) −37.1018 −1.51848
\(598\) 27.5715 1.12748
\(599\) −21.5690 −0.881284 −0.440642 0.897683i \(-0.645249\pi\)
−0.440642 + 0.897683i \(0.645249\pi\)
\(600\) 2.15443 0.0879544
\(601\) 25.5815 1.04349 0.521745 0.853101i \(-0.325281\pi\)
0.521745 + 0.853101i \(0.325281\pi\)
\(602\) 5.20049 0.211956
\(603\) −7.04449 −0.286874
\(604\) 20.9220 0.851304
\(605\) −1.00000 −0.0406558
\(606\) 35.4046 1.43822
\(607\) −16.2877 −0.661097 −0.330548 0.943789i \(-0.607234\pi\)
−0.330548 + 0.943789i \(0.607234\pi\)
\(608\) 3.91291 0.158690
\(609\) 90.1168 3.65172
\(610\) −7.70264 −0.311871
\(611\) 34.4199 1.39248
\(612\) 3.75777 0.151899
\(613\) −4.91861 −0.198661 −0.0993304 0.995055i \(-0.531670\pi\)
−0.0993304 + 0.995055i \(0.531670\pi\)
\(614\) −11.3935 −0.459805
\(615\) −6.73771 −0.271691
\(616\) 5.20049 0.209534
\(617\) −18.4337 −0.742114 −0.371057 0.928610i \(-0.621004\pi\)
−0.371057 + 0.928610i \(0.621004\pi\)
\(618\) −27.9087 −1.12265
\(619\) 26.7959 1.07702 0.538509 0.842620i \(-0.318988\pi\)
0.538509 + 0.842620i \(0.318988\pi\)
\(620\) 2.87929 0.115635
\(621\) 21.8464 0.876668
\(622\) −4.64254 −0.186149
\(623\) −27.7913 −1.11343
\(624\) −7.95753 −0.318556
\(625\) 1.00000 0.0400000
\(626\) −7.23924 −0.289338
\(627\) 8.43011 0.336666
\(628\) −13.6347 −0.544083
\(629\) 23.3985 0.932959
\(630\) −8.53706 −0.340125
\(631\) −3.76933 −0.150055 −0.0750273 0.997181i \(-0.523904\pi\)
−0.0750273 + 0.997181i \(0.523904\pi\)
\(632\) 11.4945 0.457225
\(633\) −4.86429 −0.193338
\(634\) 13.1816 0.523508
\(635\) 17.6856 0.701830
\(636\) −1.44487 −0.0572926
\(637\) −74.0376 −2.93348
\(638\) 8.04319 0.318433
\(639\) −14.0240 −0.554781
\(640\) −1.00000 −0.0395285
\(641\) 33.5915 1.32678 0.663392 0.748272i \(-0.269116\pi\)
0.663392 + 0.748272i \(0.269116\pi\)
\(642\) −35.2280 −1.39034
\(643\) −25.2666 −0.996417 −0.498208 0.867057i \(-0.666009\pi\)
−0.498208 + 0.867057i \(0.666009\pi\)
\(644\) −38.8204 −1.52974
\(645\) −2.15443 −0.0848308
\(646\) 8.95708 0.352412
\(647\) 38.3656 1.50831 0.754154 0.656698i \(-0.228047\pi\)
0.754154 + 0.656698i \(0.228047\pi\)
\(648\) −11.2300 −0.441154
\(649\) 6.55249 0.257208
\(650\) −3.69356 −0.144873
\(651\) −32.2599 −1.26437
\(652\) −18.6569 −0.730663
\(653\) −13.4101 −0.524777 −0.262388 0.964962i \(-0.584510\pi\)
−0.262388 + 0.964962i \(0.584510\pi\)
\(654\) 36.2325 1.41680
\(655\) 16.4398 0.642357
\(656\) 3.12737 0.122103
\(657\) −16.0936 −0.627872
\(658\) −48.4628 −1.88928
\(659\) −13.5960 −0.529624 −0.264812 0.964300i \(-0.585310\pi\)
−0.264812 + 0.964300i \(0.585310\pi\)
\(660\) −2.15443 −0.0838613
\(661\) −14.5741 −0.566868 −0.283434 0.958992i \(-0.591474\pi\)
−0.283434 + 0.958992i \(0.591474\pi\)
\(662\) 5.45659 0.212076
\(663\) −18.2156 −0.707437
\(664\) −6.18517 −0.240031
\(665\) −20.3491 −0.789102
\(666\) 16.7798 0.650203
\(667\) −60.0405 −2.32478
\(668\) 2.47973 0.0959435
\(669\) −12.2866 −0.475026
\(670\) 4.29126 0.165786
\(671\) 7.70264 0.297357
\(672\) 11.2041 0.432208
\(673\) 20.4531 0.788408 0.394204 0.919023i \(-0.371020\pi\)
0.394204 + 0.919023i \(0.371020\pi\)
\(674\) 23.7542 0.914978
\(675\) −2.92661 −0.112645
\(676\) 0.642357 0.0247061
\(677\) −23.6276 −0.908082 −0.454041 0.890981i \(-0.650018\pi\)
−0.454041 + 0.890981i \(0.650018\pi\)
\(678\) −12.1895 −0.468136
\(679\) 28.0933 1.07812
\(680\) −2.28911 −0.0877833
\(681\) −3.99385 −0.153045
\(682\) −2.87929 −0.110254
\(683\) −20.3160 −0.777371 −0.388685 0.921371i \(-0.627071\pi\)
−0.388685 + 0.921371i \(0.627071\pi\)
\(684\) 6.42339 0.245605
\(685\) 4.54096 0.173501
\(686\) 67.8407 2.59017
\(687\) −25.9847 −0.991376
\(688\) 1.00000 0.0381246
\(689\) 2.47707 0.0943690
\(690\) 16.0824 0.612245
\(691\) −33.7100 −1.28239 −0.641194 0.767379i \(-0.721561\pi\)
−0.641194 + 0.767379i \(0.721561\pi\)
\(692\) −2.91913 −0.110968
\(693\) 8.53706 0.324296
\(694\) −27.5506 −1.04581
\(695\) −17.7324 −0.672628
\(696\) 17.3285 0.656836
\(697\) 7.15888 0.271162
\(698\) 19.1046 0.723119
\(699\) 14.3368 0.542268
\(700\) 5.20049 0.196560
\(701\) −25.8681 −0.977025 −0.488513 0.872557i \(-0.662460\pi\)
−0.488513 + 0.872557i \(0.662460\pi\)
\(702\) 10.8096 0.407982
\(703\) 39.9965 1.50850
\(704\) 1.00000 0.0376889
\(705\) 20.0769 0.756141
\(706\) −19.9199 −0.749694
\(707\) 85.4616 3.21411
\(708\) 14.1169 0.530546
\(709\) 31.4230 1.18012 0.590058 0.807361i \(-0.299105\pi\)
0.590058 + 0.807361i \(0.299105\pi\)
\(710\) 8.54294 0.320611
\(711\) 18.8692 0.707649
\(712\) −5.34397 −0.200274
\(713\) 21.4933 0.804929
\(714\) 25.6474 0.959831
\(715\) 3.69356 0.138131
\(716\) −13.7699 −0.514606
\(717\) −33.6944 −1.25834
\(718\) −6.69249 −0.249761
\(719\) −9.03068 −0.336787 −0.168394 0.985720i \(-0.553858\pi\)
−0.168394 + 0.985720i \(0.553858\pi\)
\(720\) −1.64159 −0.0611784
\(721\) −67.3674 −2.50889
\(722\) −3.68911 −0.137295
\(723\) −1.93121 −0.0718225
\(724\) −9.50722 −0.353333
\(725\) 8.04319 0.298717
\(726\) 2.15443 0.0799586
\(727\) −3.98733 −0.147882 −0.0739410 0.997263i \(-0.523558\pi\)
−0.0739410 + 0.997263i \(0.523558\pi\)
\(728\) −19.2083 −0.711907
\(729\) 0.480600 0.0178000
\(730\) 9.80369 0.362851
\(731\) 2.28911 0.0846657
\(732\) 16.5948 0.613362
\(733\) −22.6962 −0.838303 −0.419151 0.907916i \(-0.637672\pi\)
−0.419151 + 0.907916i \(0.637672\pi\)
\(734\) −10.2267 −0.377473
\(735\) −43.1858 −1.59293
\(736\) −7.46477 −0.275155
\(737\) −4.29126 −0.158071
\(738\) 5.13385 0.188980
\(739\) 33.6588 1.23816 0.619079 0.785328i \(-0.287506\pi\)
0.619079 + 0.785328i \(0.287506\pi\)
\(740\) −10.2217 −0.375756
\(741\) −31.1371 −1.14385
\(742\) −3.48769 −0.128037
\(743\) −9.34944 −0.342998 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(744\) −6.20325 −0.227422
\(745\) 2.59038 0.0949042
\(746\) 18.4363 0.675000
\(747\) −10.1535 −0.371497
\(748\) 2.28911 0.0836981
\(749\) −85.0351 −3.10711
\(750\) −2.15443 −0.0786688
\(751\) 52.3542 1.91043 0.955216 0.295908i \(-0.0956223\pi\)
0.955216 + 0.295908i \(0.0956223\pi\)
\(752\) −9.31889 −0.339825
\(753\) 8.00024 0.291545
\(754\) −29.7080 −1.08190
\(755\) −20.9220 −0.761429
\(756\) −15.2198 −0.553539
\(757\) 38.6890 1.40618 0.703088 0.711103i \(-0.251804\pi\)
0.703088 + 0.711103i \(0.251804\pi\)
\(758\) 2.83712 0.103049
\(759\) −16.0824 −0.583752
\(760\) −3.91291 −0.141936
\(761\) −20.6832 −0.749765 −0.374882 0.927072i \(-0.622317\pi\)
−0.374882 + 0.927072i \(0.622317\pi\)
\(762\) −38.1024 −1.38030
\(763\) 87.4600 3.16626
\(764\) 22.3373 0.808134
\(765\) −3.75777 −0.135863
\(766\) 13.9972 0.505739
\(767\) −24.2020 −0.873884
\(768\) 2.15443 0.0777415
\(769\) 6.62323 0.238840 0.119420 0.992844i \(-0.461897\pi\)
0.119420 + 0.992844i \(0.461897\pi\)
\(770\) −5.20049 −0.187413
\(771\) 10.9383 0.393933
\(772\) 5.77134 0.207715
\(773\) −44.9087 −1.61525 −0.807627 0.589694i \(-0.799248\pi\)
−0.807627 + 0.589694i \(0.799248\pi\)
\(774\) 1.64159 0.0590057
\(775\) −2.87929 −0.103427
\(776\) 5.40206 0.193923
\(777\) 114.525 4.10855
\(778\) −3.40004 −0.121897
\(779\) 12.2371 0.438440
\(780\) 7.95753 0.284925
\(781\) −8.54294 −0.305690
\(782\) −17.0877 −0.611053
\(783\) −23.5393 −0.841225
\(784\) 20.0451 0.715895
\(785\) 13.6347 0.486643
\(786\) −35.4185 −1.26334
\(787\) 10.4312 0.371832 0.185916 0.982566i \(-0.440475\pi\)
0.185916 + 0.982566i \(0.440475\pi\)
\(788\) 7.78147 0.277203
\(789\) 58.4222 2.07988
\(790\) −11.4945 −0.408954
\(791\) −29.4237 −1.04619
\(792\) 1.64159 0.0583313
\(793\) −28.4501 −1.01029
\(794\) 19.0532 0.676172
\(795\) 1.44487 0.0512441
\(796\) −17.2211 −0.610387
\(797\) 32.0697 1.13597 0.567983 0.823040i \(-0.307724\pi\)
0.567983 + 0.823040i \(0.307724\pi\)
\(798\) 43.8407 1.55194
\(799\) −21.3320 −0.754670
\(800\) 1.00000 0.0353553
\(801\) −8.77260 −0.309965
\(802\) 6.62751 0.234026
\(803\) −9.80369 −0.345965
\(804\) −9.24525 −0.326055
\(805\) 38.8204 1.36824
\(806\) 10.6348 0.374596
\(807\) −5.99077 −0.210885
\(808\) 16.4334 0.578124
\(809\) −10.9308 −0.384308 −0.192154 0.981365i \(-0.561547\pi\)
−0.192154 + 0.981365i \(0.561547\pi\)
\(810\) 11.2300 0.394580
\(811\) 9.65584 0.339063 0.169531 0.985525i \(-0.445775\pi\)
0.169531 + 0.985525i \(0.445775\pi\)
\(812\) 41.8285 1.46789
\(813\) −27.2449 −0.955522
\(814\) 10.2217 0.358269
\(815\) 18.6569 0.653524
\(816\) 4.93173 0.172645
\(817\) 3.91291 0.136895
\(818\) −26.0155 −0.909611
\(819\) −31.5321 −1.10182
\(820\) −3.12737 −0.109212
\(821\) −27.9243 −0.974565 −0.487283 0.873244i \(-0.662012\pi\)
−0.487283 + 0.873244i \(0.662012\pi\)
\(822\) −9.78319 −0.341228
\(823\) −18.5996 −0.648341 −0.324170 0.945999i \(-0.605085\pi\)
−0.324170 + 0.945999i \(0.605085\pi\)
\(824\) −12.9541 −0.451276
\(825\) 2.15443 0.0750078
\(826\) 34.0762 1.18566
\(827\) 45.5295 1.58322 0.791608 0.611030i \(-0.209244\pi\)
0.791608 + 0.611030i \(0.209244\pi\)
\(828\) −12.2541 −0.425859
\(829\) 44.6185 1.54967 0.774833 0.632166i \(-0.217834\pi\)
0.774833 + 0.632166i \(0.217834\pi\)
\(830\) 6.18517 0.214690
\(831\) −14.5365 −0.504266
\(832\) −3.69356 −0.128051
\(833\) 45.8853 1.58983
\(834\) 38.2033 1.32287
\(835\) −2.47973 −0.0858145
\(836\) 3.91291 0.135331
\(837\) 8.42656 0.291265
\(838\) 17.7419 0.612883
\(839\) −6.02623 −0.208048 −0.104024 0.994575i \(-0.533172\pi\)
−0.104024 + 0.994575i \(0.533172\pi\)
\(840\) −11.2041 −0.386579
\(841\) 35.6929 1.23079
\(842\) 7.77410 0.267913
\(843\) −11.4057 −0.392832
\(844\) −2.25780 −0.0777168
\(845\) −0.642357 −0.0220978
\(846\) −15.2978 −0.525949
\(847\) 5.20049 0.178691
\(848\) −0.670647 −0.0230301
\(849\) −2.02181 −0.0693884
\(850\) 2.28911 0.0785158
\(851\) −76.3023 −2.61561
\(852\) −18.4052 −0.630552
\(853\) 37.9163 1.29823 0.649116 0.760690i \(-0.275139\pi\)
0.649116 + 0.760690i \(0.275139\pi\)
\(854\) 40.0575 1.37074
\(855\) −6.42339 −0.219675
\(856\) −16.3514 −0.558878
\(857\) 9.15268 0.312650 0.156325 0.987706i \(-0.450035\pi\)
0.156325 + 0.987706i \(0.450035\pi\)
\(858\) −7.95753 −0.271665
\(859\) −6.50679 −0.222009 −0.111004 0.993820i \(-0.535407\pi\)
−0.111004 + 0.993820i \(0.535407\pi\)
\(860\) −1.00000 −0.0340997
\(861\) 35.0394 1.19414
\(862\) 24.8823 0.847495
\(863\) −12.9822 −0.441918 −0.220959 0.975283i \(-0.570919\pi\)
−0.220959 + 0.975283i \(0.570919\pi\)
\(864\) −2.92661 −0.0995653
\(865\) 2.91913 0.0992532
\(866\) 32.3003 1.09761
\(867\) −25.3361 −0.860460
\(868\) −14.9737 −0.508242
\(869\) 11.4945 0.389923
\(870\) −17.3285 −0.587492
\(871\) 15.8500 0.537057
\(872\) 16.8177 0.569518
\(873\) 8.86796 0.300135
\(874\) −29.2090 −0.988008
\(875\) −5.20049 −0.175809
\(876\) −21.1214 −0.713626
\(877\) −15.6834 −0.529592 −0.264796 0.964305i \(-0.585305\pi\)
−0.264796 + 0.964305i \(0.585305\pi\)
\(878\) −33.5844 −1.13342
\(879\) 54.1089 1.82505
\(880\) −1.00000 −0.0337100
\(881\) −33.4494 −1.12694 −0.563469 0.826137i \(-0.690533\pi\)
−0.563469 + 0.826137i \(0.690533\pi\)
\(882\) 32.9058 1.10799
\(883\) −57.8360 −1.94634 −0.973168 0.230098i \(-0.926095\pi\)
−0.973168 + 0.230098i \(0.926095\pi\)
\(884\) −8.45495 −0.284371
\(885\) −14.1169 −0.474535
\(886\) −26.5635 −0.892419
\(887\) 9.92486 0.333244 0.166622 0.986021i \(-0.446714\pi\)
0.166622 + 0.986021i \(0.446714\pi\)
\(888\) 22.0219 0.739007
\(889\) −91.9735 −3.08469
\(890\) 5.34397 0.179130
\(891\) −11.2300 −0.376218
\(892\) −5.70292 −0.190948
\(893\) −36.4640 −1.22022
\(894\) −5.58080 −0.186650
\(895\) 13.7699 0.460278
\(896\) 5.20049 0.173736
\(897\) 59.4011 1.98334
\(898\) 36.7589 1.22666
\(899\) −23.1587 −0.772386
\(900\) 1.64159 0.0547196
\(901\) −1.53518 −0.0511444
\(902\) 3.12737 0.104130
\(903\) 11.2041 0.372849
\(904\) −5.65787 −0.188178
\(905\) 9.50722 0.316031
\(906\) 45.0751 1.49752
\(907\) 48.1403 1.59847 0.799236 0.601017i \(-0.205238\pi\)
0.799236 + 0.601017i \(0.205238\pi\)
\(908\) −1.85378 −0.0615199
\(909\) 26.9768 0.894765
\(910\) 19.2083 0.636749
\(911\) 40.6760 1.34766 0.673829 0.738888i \(-0.264649\pi\)
0.673829 + 0.738888i \(0.264649\pi\)
\(912\) 8.43011 0.279149
\(913\) −6.18517 −0.204699
\(914\) 24.9793 0.826241
\(915\) −16.5948 −0.548608
\(916\) −12.0610 −0.398507
\(917\) −85.4950 −2.82329
\(918\) −6.69932 −0.221111
\(919\) 33.9679 1.12050 0.560249 0.828324i \(-0.310705\pi\)
0.560249 + 0.828324i \(0.310705\pi\)
\(920\) 7.46477 0.246106
\(921\) −24.5466 −0.808837
\(922\) −28.6874 −0.944768
\(923\) 31.5538 1.03861
\(924\) 11.2041 0.368588
\(925\) 10.2217 0.336086
\(926\) −6.94892 −0.228356
\(927\) −21.2652 −0.698442
\(928\) 8.04319 0.264031
\(929\) 10.5337 0.345600 0.172800 0.984957i \(-0.444718\pi\)
0.172800 + 0.984957i \(0.444718\pi\)
\(930\) 6.20325 0.203413
\(931\) 78.4346 2.57059
\(932\) 6.65456 0.217977
\(933\) −10.0021 −0.327453
\(934\) −25.0640 −0.820118
\(935\) −2.28911 −0.0748618
\(936\) −6.06330 −0.198185
\(937\) 5.91694 0.193298 0.0966490 0.995319i \(-0.469188\pi\)
0.0966490 + 0.995319i \(0.469188\pi\)
\(938\) −22.3167 −0.728665
\(939\) −15.5965 −0.508972
\(940\) 9.31889 0.303949
\(941\) −26.3145 −0.857829 −0.428914 0.903345i \(-0.641104\pi\)
−0.428914 + 0.903345i \(0.641104\pi\)
\(942\) −29.3750 −0.957090
\(943\) −23.3451 −0.760220
\(944\) 6.55249 0.213265
\(945\) 15.2198 0.495100
\(946\) 1.00000 0.0325128
\(947\) −44.4225 −1.44354 −0.721769 0.692134i \(-0.756671\pi\)
−0.721769 + 0.692134i \(0.756671\pi\)
\(948\) 24.7641 0.804299
\(949\) 36.2105 1.17544
\(950\) 3.91291 0.126952
\(951\) 28.3989 0.920898
\(952\) 11.9045 0.385826
\(953\) 31.1009 1.00746 0.503729 0.863862i \(-0.331961\pi\)
0.503729 + 0.863862i \(0.331961\pi\)
\(954\) −1.10093 −0.0356438
\(955\) −22.3373 −0.722817
\(956\) −15.6396 −0.505819
\(957\) 17.3285 0.560152
\(958\) −24.7063 −0.798223
\(959\) −23.6152 −0.762574
\(960\) −2.15443 −0.0695341
\(961\) −22.7097 −0.732570
\(962\) −37.7543 −1.21725
\(963\) −26.8422 −0.864979
\(964\) −0.896388 −0.0288707
\(965\) −5.77134 −0.185786
\(966\) −83.6361 −2.69095
\(967\) 42.4769 1.36596 0.682982 0.730435i \(-0.260683\pi\)
0.682982 + 0.730435i \(0.260683\pi\)
\(968\) 1.00000 0.0321412
\(969\) 19.2974 0.619923
\(970\) −5.40206 −0.173450
\(971\) −17.2183 −0.552562 −0.276281 0.961077i \(-0.589102\pi\)
−0.276281 + 0.961077i \(0.589102\pi\)
\(972\) −15.4144 −0.494416
\(973\) 92.2171 2.95635
\(974\) −27.4194 −0.878573
\(975\) −7.95753 −0.254845
\(976\) 7.70264 0.246555
\(977\) 9.75852 0.312203 0.156101 0.987741i \(-0.450107\pi\)
0.156101 + 0.987741i \(0.450107\pi\)
\(978\) −40.1952 −1.28530
\(979\) −5.34397 −0.170794
\(980\) −20.0451 −0.640316
\(981\) 27.6077 0.881445
\(982\) −26.8303 −0.856191
\(983\) 45.8749 1.46318 0.731591 0.681744i \(-0.238778\pi\)
0.731591 + 0.681744i \(0.238778\pi\)
\(984\) 6.73771 0.214790
\(985\) −7.78147 −0.247938
\(986\) 18.4117 0.586349
\(987\) −104.410 −3.32340
\(988\) −14.4526 −0.459797
\(989\) −7.46477 −0.237366
\(990\) −1.64159 −0.0521731
\(991\) −14.7351 −0.468077 −0.234039 0.972227i \(-0.575194\pi\)
−0.234039 + 0.972227i \(0.575194\pi\)
\(992\) −2.87929 −0.0914177
\(993\) 11.7559 0.373061
\(994\) −44.4275 −1.40915
\(995\) 17.2211 0.545946
\(996\) −13.3255 −0.422236
\(997\) 11.3414 0.359185 0.179592 0.983741i \(-0.442522\pi\)
0.179592 + 0.983741i \(0.442522\pi\)
\(998\) −19.6464 −0.621897
\(999\) −29.9148 −0.946463
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 4730.2.a.bf.1.10 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.bf.1.10 13 1.1 even 1 trivial