Properties

Label 4730.2.a.z.1.3
Level 4730
Weight 2
Character 4730.1
Self dual yes
Analytic conductor 37.769
Analytic rank 0
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.27147\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.27147 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.27147 q^{6} -0.316678 q^{7} -1.00000 q^{8} -1.38335 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.27147 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.27147 q^{6} -0.316678 q^{7} -1.00000 q^{8} -1.38335 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.27147 q^{12} -0.503303 q^{13} +0.316678 q^{14} -1.27147 q^{15} +1.00000 q^{16} -3.04381 q^{17} +1.38335 q^{18} -5.80916 q^{19} +1.00000 q^{20} +0.402648 q^{21} -1.00000 q^{22} -9.28218 q^{23} +1.27147 q^{24} +1.00000 q^{25} +0.503303 q^{26} +5.57332 q^{27} -0.316678 q^{28} -7.93811 q^{29} +1.27147 q^{30} +1.74862 q^{31} -1.00000 q^{32} -1.27147 q^{33} +3.04381 q^{34} -0.316678 q^{35} -1.38335 q^{36} +8.96829 q^{37} +5.80916 q^{38} +0.639936 q^{39} -1.00000 q^{40} -0.430252 q^{41} -0.402648 q^{42} +1.00000 q^{43} +1.00000 q^{44} -1.38335 q^{45} +9.28218 q^{46} +10.5211 q^{47} -1.27147 q^{48} -6.89972 q^{49} -1.00000 q^{50} +3.87013 q^{51} -0.503303 q^{52} +4.68904 q^{53} -5.57332 q^{54} +1.00000 q^{55} +0.316678 q^{56} +7.38620 q^{57} +7.93811 q^{58} -2.98409 q^{59} -1.27147 q^{60} -7.67431 q^{61} -1.74862 q^{62} +0.438077 q^{63} +1.00000 q^{64} -0.503303 q^{65} +1.27147 q^{66} +9.75924 q^{67} -3.04381 q^{68} +11.8021 q^{69} +0.316678 q^{70} +3.96679 q^{71} +1.38335 q^{72} +3.46098 q^{73} -8.96829 q^{74} -1.27147 q^{75} -5.80916 q^{76} -0.316678 q^{77} -0.639936 q^{78} +5.32454 q^{79} +1.00000 q^{80} -2.93627 q^{81} +0.430252 q^{82} -10.1475 q^{83} +0.402648 q^{84} -3.04381 q^{85} -1.00000 q^{86} +10.0931 q^{87} -1.00000 q^{88} +5.40924 q^{89} +1.38335 q^{90} +0.159385 q^{91} -9.28218 q^{92} -2.22333 q^{93} -10.5211 q^{94} -5.80916 q^{95} +1.27147 q^{96} -1.27905 q^{97} +6.89972 q^{98} -1.38335 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 10q^{2} + 8q^{3} + 10q^{4} + 10q^{5} - 8q^{6} + 3q^{7} - 10q^{8} + 14q^{9} + O(q^{10}) \) \( 10q - 10q^{2} + 8q^{3} + 10q^{4} + 10q^{5} - 8q^{6} + 3q^{7} - 10q^{8} + 14q^{9} - 10q^{10} + 10q^{11} + 8q^{12} + 7q^{13} - 3q^{14} + 8q^{15} + 10q^{16} + 2q^{17} - 14q^{18} - 7q^{19} + 10q^{20} - 2q^{21} - 10q^{22} + 12q^{23} - 8q^{24} + 10q^{25} - 7q^{26} + 23q^{27} + 3q^{28} - 12q^{29} - 8q^{30} + 16q^{31} - 10q^{32} + 8q^{33} - 2q^{34} + 3q^{35} + 14q^{36} + 19q^{37} + 7q^{38} + 6q^{39} - 10q^{40} + 9q^{41} + 2q^{42} + 10q^{43} + 10q^{44} + 14q^{45} - 12q^{46} + 29q^{47} + 8q^{48} + 23q^{49} - 10q^{50} - 7q^{51} + 7q^{52} + 6q^{53} - 23q^{54} + 10q^{55} - 3q^{56} + 23q^{57} + 12q^{58} + 29q^{59} + 8q^{60} - 4q^{61} - 16q^{62} + 10q^{64} + 7q^{65} - 8q^{66} + 45q^{67} + 2q^{68} + 24q^{69} - 3q^{70} - 18q^{71} - 14q^{72} + 3q^{73} - 19q^{74} + 8q^{75} - 7q^{76} + 3q^{77} - 6q^{78} - 14q^{79} + 10q^{80} + 6q^{81} - 9q^{82} + 23q^{83} - 2q^{84} + 2q^{85} - 10q^{86} + 25q^{87} - 10q^{88} + q^{89} - 14q^{90} + q^{91} + 12q^{92} + 35q^{93} - 29q^{94} - 7q^{95} - 8q^{96} + 30q^{97} - 23q^{98} + 14q^{99} + O(q^{100}) \)

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
\(3\) −1.27147 −0.734086 −0.367043 0.930204i \(-0.619630\pi\)
−0.367043 + 0.930204i \(0.619630\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.27147 0.519077
\(7\) −0.316678 −0.119693 −0.0598465 0.998208i \(-0.519061\pi\)
−0.0598465 + 0.998208i \(0.519061\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.38335 −0.461118
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −1.27147 −0.367043
\(13\) −0.503303 −0.139591 −0.0697955 0.997561i \(-0.522235\pi\)
−0.0697955 + 0.997561i \(0.522235\pi\)
\(14\) 0.316678 0.0846357
\(15\) −1.27147 −0.328293
\(16\) 1.00000 0.250000
\(17\) −3.04381 −0.738233 −0.369117 0.929383i \(-0.620340\pi\)
−0.369117 + 0.929383i \(0.620340\pi\)
\(18\) 1.38335 0.326060
\(19\) −5.80916 −1.33271 −0.666356 0.745633i \(-0.732147\pi\)
−0.666356 + 0.745633i \(0.732147\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.402648 0.0878649
\(22\) −1.00000 −0.213201
\(23\) −9.28218 −1.93547 −0.967734 0.251973i \(-0.918921\pi\)
−0.967734 + 0.251973i \(0.918921\pi\)
\(24\) 1.27147 0.259539
\(25\) 1.00000 0.200000
\(26\) 0.503303 0.0987058
\(27\) 5.57332 1.07259
\(28\) −0.316678 −0.0598465
\(29\) −7.93811 −1.47407 −0.737035 0.675854i \(-0.763775\pi\)
−0.737035 + 0.675854i \(0.763775\pi\)
\(30\) 1.27147 0.232138
\(31\) 1.74862 0.314062 0.157031 0.987594i \(-0.449808\pi\)
0.157031 + 0.987594i \(0.449808\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.27147 −0.221335
\(34\) 3.04381 0.522010
\(35\) −0.316678 −0.0535283
\(36\) −1.38335 −0.230559
\(37\) 8.96829 1.47438 0.737189 0.675686i \(-0.236153\pi\)
0.737189 + 0.675686i \(0.236153\pi\)
\(38\) 5.80916 0.942370
\(39\) 0.639936 0.102472
\(40\) −1.00000 −0.158114
\(41\) −0.430252 −0.0671940 −0.0335970 0.999435i \(-0.510696\pi\)
−0.0335970 + 0.999435i \(0.510696\pi\)
\(42\) −0.402648 −0.0621299
\(43\) 1.00000 0.152499
\(44\) 1.00000 0.150756
\(45\) −1.38335 −0.206218
\(46\) 9.28218 1.36858
\(47\) 10.5211 1.53466 0.767330 0.641252i \(-0.221585\pi\)
0.767330 + 0.641252i \(0.221585\pi\)
\(48\) −1.27147 −0.183521
\(49\) −6.89972 −0.985674
\(50\) −1.00000 −0.141421
\(51\) 3.87013 0.541927
\(52\) −0.503303 −0.0697955
\(53\) 4.68904 0.644089 0.322044 0.946725i \(-0.395630\pi\)
0.322044 + 0.946725i \(0.395630\pi\)
\(54\) −5.57332 −0.758433
\(55\) 1.00000 0.134840
\(56\) 0.316678 0.0423178
\(57\) 7.38620 0.978326
\(58\) 7.93811 1.04233
\(59\) −2.98409 −0.388496 −0.194248 0.980952i \(-0.562227\pi\)
−0.194248 + 0.980952i \(0.562227\pi\)
\(60\) −1.27147 −0.164147
\(61\) −7.67431 −0.982594 −0.491297 0.870992i \(-0.663477\pi\)
−0.491297 + 0.870992i \(0.663477\pi\)
\(62\) −1.74862 −0.222075
\(63\) 0.438077 0.0551926
\(64\) 1.00000 0.125000
\(65\) −0.503303 −0.0624270
\(66\) 1.27147 0.156508
\(67\) 9.75924 1.19228 0.596140 0.802880i \(-0.296700\pi\)
0.596140 + 0.802880i \(0.296700\pi\)
\(68\) −3.04381 −0.369117
\(69\) 11.8021 1.42080
\(70\) 0.316678 0.0378502
\(71\) 3.96679 0.470771 0.235385 0.971902i \(-0.424365\pi\)
0.235385 + 0.971902i \(0.424365\pi\)
\(72\) 1.38335 0.163030
\(73\) 3.46098 0.405077 0.202539 0.979274i \(-0.435081\pi\)
0.202539 + 0.979274i \(0.435081\pi\)
\(74\) −8.96829 −1.04254
\(75\) −1.27147 −0.146817
\(76\) −5.80916 −0.666356
\(77\) −0.316678 −0.0360888
\(78\) −0.639936 −0.0724585
\(79\) 5.32454 0.599058 0.299529 0.954087i \(-0.403171\pi\)
0.299529 + 0.954087i \(0.403171\pi\)
\(80\) 1.00000 0.111803
\(81\) −2.93627 −0.326252
\(82\) 0.430252 0.0475134
\(83\) −10.1475 −1.11383 −0.556914 0.830570i \(-0.688015\pi\)
−0.556914 + 0.830570i \(0.688015\pi\)
\(84\) 0.402648 0.0439324
\(85\) −3.04381 −0.330148
\(86\) −1.00000 −0.107833
\(87\) 10.0931 1.08209
\(88\) −1.00000 −0.106600
\(89\) 5.40924 0.573378 0.286689 0.958024i \(-0.407445\pi\)
0.286689 + 0.958024i \(0.407445\pi\)
\(90\) 1.38335 0.145818
\(91\) 0.159385 0.0167081
\(92\) −9.28218 −0.967734
\(93\) −2.22333 −0.230548
\(94\) −10.5211 −1.08517
\(95\) −5.80916 −0.596007
\(96\) 1.27147 0.129769
\(97\) −1.27905 −0.129868 −0.0649341 0.997890i \(-0.520684\pi\)
−0.0649341 + 0.997890i \(0.520684\pi\)
\(98\) 6.89972 0.696976
\(99\) −1.38335 −0.139032
\(100\) 1.00000 0.100000
\(101\) −7.34467 −0.730822 −0.365411 0.930846i \(-0.619072\pi\)
−0.365411 + 0.930846i \(0.619072\pi\)
\(102\) −3.87013 −0.383200
\(103\) −1.97820 −0.194918 −0.0974591 0.995240i \(-0.531072\pi\)
−0.0974591 + 0.995240i \(0.531072\pi\)
\(104\) 0.503303 0.0493529
\(105\) 0.402648 0.0392944
\(106\) −4.68904 −0.455440
\(107\) 10.5975 1.02450 0.512248 0.858837i \(-0.328813\pi\)
0.512248 + 0.858837i \(0.328813\pi\)
\(108\) 5.57332 0.536293
\(109\) 5.64929 0.541104 0.270552 0.962705i \(-0.412794\pi\)
0.270552 + 0.962705i \(0.412794\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −11.4030 −1.08232
\(112\) −0.316678 −0.0299232
\(113\) 15.4883 1.45701 0.728506 0.685039i \(-0.240215\pi\)
0.728506 + 0.685039i \(0.240215\pi\)
\(114\) −7.38620 −0.691781
\(115\) −9.28218 −0.865568
\(116\) −7.93811 −0.737035
\(117\) 0.696246 0.0643679
\(118\) 2.98409 0.274708
\(119\) 0.963908 0.0883613
\(120\) 1.27147 0.116069
\(121\) 1.00000 0.0909091
\(122\) 7.67431 0.694799
\(123\) 0.547054 0.0493262
\(124\) 1.74862 0.157031
\(125\) 1.00000 0.0894427
\(126\) −0.438077 −0.0390270
\(127\) 20.8483 1.84999 0.924994 0.379982i \(-0.124070\pi\)
0.924994 + 0.379982i \(0.124070\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.27147 −0.111947
\(130\) 0.503303 0.0441426
\(131\) −0.202318 −0.0176766 −0.00883829 0.999961i \(-0.502813\pi\)
−0.00883829 + 0.999961i \(0.502813\pi\)
\(132\) −1.27147 −0.110668
\(133\) 1.83963 0.159516
\(134\) −9.75924 −0.843070
\(135\) 5.57332 0.479675
\(136\) 3.04381 0.261005
\(137\) −15.8441 −1.35366 −0.676828 0.736141i \(-0.736646\pi\)
−0.676828 + 0.736141i \(0.736646\pi\)
\(138\) −11.8021 −1.00466
\(139\) −2.71120 −0.229961 −0.114980 0.993368i \(-0.536681\pi\)
−0.114980 + 0.993368i \(0.536681\pi\)
\(140\) −0.316678 −0.0267642
\(141\) −13.3773 −1.12657
\(142\) −3.96679 −0.332885
\(143\) −0.503303 −0.0420883
\(144\) −1.38335 −0.115279
\(145\) −7.93811 −0.659224
\(146\) −3.46098 −0.286433
\(147\) 8.77281 0.723569
\(148\) 8.96829 0.737189
\(149\) 10.9149 0.894187 0.447093 0.894487i \(-0.352459\pi\)
0.447093 + 0.894487i \(0.352459\pi\)
\(150\) 1.27147 0.103815
\(151\) −0.0709238 −0.00577169 −0.00288585 0.999996i \(-0.500919\pi\)
−0.00288585 + 0.999996i \(0.500919\pi\)
\(152\) 5.80916 0.471185
\(153\) 4.21067 0.340413
\(154\) 0.316678 0.0255186
\(155\) 1.74862 0.140453
\(156\) 0.639936 0.0512359
\(157\) −18.4300 −1.47087 −0.735437 0.677593i \(-0.763023\pi\)
−0.735437 + 0.677593i \(0.763023\pi\)
\(158\) −5.32454 −0.423598
\(159\) −5.96199 −0.472817
\(160\) −1.00000 −0.0790569
\(161\) 2.93946 0.231662
\(162\) 2.93627 0.230695
\(163\) 8.53284 0.668344 0.334172 0.942512i \(-0.391543\pi\)
0.334172 + 0.942512i \(0.391543\pi\)
\(164\) −0.430252 −0.0335970
\(165\) −1.27147 −0.0989841
\(166\) 10.1475 0.787595
\(167\) −5.40437 −0.418202 −0.209101 0.977894i \(-0.567054\pi\)
−0.209101 + 0.977894i \(0.567054\pi\)
\(168\) −0.402648 −0.0310649
\(169\) −12.7467 −0.980514
\(170\) 3.04381 0.233450
\(171\) 8.03612 0.614538
\(172\) 1.00000 0.0762493
\(173\) 6.57690 0.500033 0.250016 0.968242i \(-0.419564\pi\)
0.250016 + 0.968242i \(0.419564\pi\)
\(174\) −10.0931 −0.765156
\(175\) −0.316678 −0.0239386
\(176\) 1.00000 0.0753778
\(177\) 3.79419 0.285189
\(178\) −5.40924 −0.405440
\(179\) 3.79891 0.283944 0.141972 0.989871i \(-0.454656\pi\)
0.141972 + 0.989871i \(0.454656\pi\)
\(180\) −1.38335 −0.103109
\(181\) 7.79219 0.579189 0.289594 0.957149i \(-0.406480\pi\)
0.289594 + 0.957149i \(0.406480\pi\)
\(182\) −0.159385 −0.0118144
\(183\) 9.75768 0.721308
\(184\) 9.28218 0.684291
\(185\) 8.96829 0.659362
\(186\) 2.22333 0.163022
\(187\) −3.04381 −0.222586
\(188\) 10.5211 0.767330
\(189\) −1.76495 −0.128381
\(190\) 5.80916 0.421441
\(191\) −18.6062 −1.34630 −0.673149 0.739507i \(-0.735059\pi\)
−0.673149 + 0.739507i \(0.735059\pi\)
\(192\) −1.27147 −0.0917607
\(193\) −8.43218 −0.606961 −0.303481 0.952838i \(-0.598149\pi\)
−0.303481 + 0.952838i \(0.598149\pi\)
\(194\) 1.27905 0.0918306
\(195\) 0.639936 0.0458268
\(196\) −6.89972 −0.492837
\(197\) 12.9260 0.920939 0.460470 0.887675i \(-0.347681\pi\)
0.460470 + 0.887675i \(0.347681\pi\)
\(198\) 1.38335 0.0983107
\(199\) 27.8773 1.97617 0.988084 0.153915i \(-0.0491883\pi\)
0.988084 + 0.153915i \(0.0491883\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −12.4086 −0.875236
\(202\) 7.34467 0.516769
\(203\) 2.51382 0.176436
\(204\) 3.87013 0.270963
\(205\) −0.430252 −0.0300501
\(206\) 1.97820 0.137828
\(207\) 12.8405 0.892479
\(208\) −0.503303 −0.0348978
\(209\) −5.80916 −0.401828
\(210\) −0.402648 −0.0277853
\(211\) −4.31767 −0.297241 −0.148620 0.988894i \(-0.547483\pi\)
−0.148620 + 0.988894i \(0.547483\pi\)
\(212\) 4.68904 0.322044
\(213\) −5.04367 −0.345586
\(214\) −10.5975 −0.724429
\(215\) 1.00000 0.0681994
\(216\) −5.57332 −0.379216
\(217\) −0.553750 −0.0375910
\(218\) −5.64929 −0.382618
\(219\) −4.40055 −0.297361
\(220\) 1.00000 0.0674200
\(221\) 1.53196 0.103051
\(222\) 11.4030 0.765316
\(223\) 11.4206 0.764777 0.382389 0.924002i \(-0.375102\pi\)
0.382389 + 0.924002i \(0.375102\pi\)
\(224\) 0.316678 0.0211589
\(225\) −1.38335 −0.0922236
\(226\) −15.4883 −1.03026
\(227\) 19.4623 1.29176 0.645878 0.763441i \(-0.276492\pi\)
0.645878 + 0.763441i \(0.276492\pi\)
\(228\) 7.38620 0.489163
\(229\) −10.7545 −0.710678 −0.355339 0.934738i \(-0.615635\pi\)
−0.355339 + 0.934738i \(0.615635\pi\)
\(230\) 9.28218 0.612049
\(231\) 0.402648 0.0264923
\(232\) 7.93811 0.521163
\(233\) −15.2663 −1.00013 −0.500065 0.865988i \(-0.666690\pi\)
−0.500065 + 0.865988i \(0.666690\pi\)
\(234\) −0.696246 −0.0455150
\(235\) 10.5211 0.686321
\(236\) −2.98409 −0.194248
\(237\) −6.77002 −0.439760
\(238\) −0.963908 −0.0624809
\(239\) −4.58174 −0.296368 −0.148184 0.988960i \(-0.547343\pi\)
−0.148184 + 0.988960i \(0.547343\pi\)
\(240\) −1.27147 −0.0820733
\(241\) 14.3755 0.926010 0.463005 0.886356i \(-0.346771\pi\)
0.463005 + 0.886356i \(0.346771\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −12.9866 −0.833089
\(244\) −7.67431 −0.491297
\(245\) −6.89972 −0.440807
\(246\) −0.547054 −0.0348789
\(247\) 2.92377 0.186035
\(248\) −1.74862 −0.111038
\(249\) 12.9022 0.817645
\(250\) −1.00000 −0.0632456
\(251\) 4.99686 0.315399 0.157700 0.987487i \(-0.449592\pi\)
0.157700 + 0.987487i \(0.449592\pi\)
\(252\) 0.438077 0.0275963
\(253\) −9.28218 −0.583566
\(254\) −20.8483 −1.30814
\(255\) 3.87013 0.242357
\(256\) 1.00000 0.0625000
\(257\) −18.2445 −1.13806 −0.569032 0.822316i \(-0.692682\pi\)
−0.569032 + 0.822316i \(0.692682\pi\)
\(258\) 1.27147 0.0791585
\(259\) −2.84006 −0.176473
\(260\) −0.503303 −0.0312135
\(261\) 10.9812 0.679720
\(262\) 0.202318 0.0124992
\(263\) −5.51612 −0.340139 −0.170069 0.985432i \(-0.554399\pi\)
−0.170069 + 0.985432i \(0.554399\pi\)
\(264\) 1.27147 0.0782538
\(265\) 4.68904 0.288045
\(266\) −1.83963 −0.112795
\(267\) −6.87771 −0.420909
\(268\) 9.75924 0.596140
\(269\) 8.01104 0.488441 0.244221 0.969720i \(-0.421468\pi\)
0.244221 + 0.969720i \(0.421468\pi\)
\(270\) −5.57332 −0.339181
\(271\) −29.9694 −1.82051 −0.910255 0.414048i \(-0.864115\pi\)
−0.910255 + 0.414048i \(0.864115\pi\)
\(272\) −3.04381 −0.184558
\(273\) −0.202654 −0.0122652
\(274\) 15.8441 0.957179
\(275\) 1.00000 0.0603023
\(276\) 11.8021 0.710400
\(277\) −2.09014 −0.125584 −0.0627921 0.998027i \(-0.520001\pi\)
−0.0627921 + 0.998027i \(0.520001\pi\)
\(278\) 2.71120 0.162607
\(279\) −2.41896 −0.144820
\(280\) 0.316678 0.0189251
\(281\) 31.3635 1.87099 0.935495 0.353340i \(-0.114954\pi\)
0.935495 + 0.353340i \(0.114954\pi\)
\(282\) 13.3773 0.796607
\(283\) 0.751951 0.0446988 0.0223494 0.999750i \(-0.492885\pi\)
0.0223494 + 0.999750i \(0.492885\pi\)
\(284\) 3.96679 0.235385
\(285\) 7.38620 0.437520
\(286\) 0.503303 0.0297609
\(287\) 0.136251 0.00804265
\(288\) 1.38335 0.0815149
\(289\) −7.73519 −0.455011
\(290\) 7.93811 0.466142
\(291\) 1.62628 0.0953343
\(292\) 3.46098 0.202539
\(293\) −16.8372 −0.983640 −0.491820 0.870697i \(-0.663668\pi\)
−0.491820 + 0.870697i \(0.663668\pi\)
\(294\) −8.77281 −0.511641
\(295\) −2.98409 −0.173740
\(296\) −8.96829 −0.521271
\(297\) 5.57332 0.323397
\(298\) −10.9149 −0.632285
\(299\) 4.67175 0.270174
\(300\) −1.27147 −0.0734086
\(301\) −0.316678 −0.0182530
\(302\) 0.0709238 0.00408120
\(303\) 9.33856 0.536486
\(304\) −5.80916 −0.333178
\(305\) −7.67431 −0.439429
\(306\) −4.21067 −0.240708
\(307\) −2.41881 −0.138049 −0.0690243 0.997615i \(-0.521989\pi\)
−0.0690243 + 0.997615i \(0.521989\pi\)
\(308\) −0.316678 −0.0180444
\(309\) 2.51523 0.143087
\(310\) −1.74862 −0.0993151
\(311\) 29.5823 1.67746 0.838728 0.544551i \(-0.183300\pi\)
0.838728 + 0.544551i \(0.183300\pi\)
\(312\) −0.639936 −0.0362293
\(313\) 26.2434 1.48336 0.741681 0.670753i \(-0.234029\pi\)
0.741681 + 0.670753i \(0.234029\pi\)
\(314\) 18.4300 1.04007
\(315\) 0.438077 0.0246829
\(316\) 5.32454 0.299529
\(317\) −10.6549 −0.598439 −0.299219 0.954184i \(-0.596726\pi\)
−0.299219 + 0.954184i \(0.596726\pi\)
\(318\) 5.96199 0.334332
\(319\) −7.93811 −0.444449
\(320\) 1.00000 0.0559017
\(321\) −13.4744 −0.752069
\(322\) −2.93946 −0.163810
\(323\) 17.6820 0.983853
\(324\) −2.93627 −0.163126
\(325\) −0.503303 −0.0279182
\(326\) −8.53284 −0.472590
\(327\) −7.18293 −0.397217
\(328\) 0.430252 0.0237567
\(329\) −3.33180 −0.183688
\(330\) 1.27147 0.0699923
\(331\) 14.8383 0.815587 0.407794 0.913074i \(-0.366298\pi\)
0.407794 + 0.913074i \(0.366298\pi\)
\(332\) −10.1475 −0.556914
\(333\) −12.4063 −0.679862
\(334\) 5.40437 0.295714
\(335\) 9.75924 0.533204
\(336\) 0.402648 0.0219662
\(337\) 31.7045 1.72706 0.863528 0.504300i \(-0.168250\pi\)
0.863528 + 0.504300i \(0.168250\pi\)
\(338\) 12.7467 0.693328
\(339\) −19.6929 −1.06957
\(340\) −3.04381 −0.165074
\(341\) 1.74862 0.0946932
\(342\) −8.03612 −0.434544
\(343\) 4.40173 0.237671
\(344\) −1.00000 −0.0539164
\(345\) 11.8021 0.635401
\(346\) −6.57690 −0.353577
\(347\) −14.8944 −0.799573 −0.399786 0.916608i \(-0.630916\pi\)
−0.399786 + 0.916608i \(0.630916\pi\)
\(348\) 10.0931 0.541047
\(349\) 1.05540 0.0564945 0.0282472 0.999601i \(-0.491007\pi\)
0.0282472 + 0.999601i \(0.491007\pi\)
\(350\) 0.316678 0.0169271
\(351\) −2.80507 −0.149723
\(352\) −1.00000 −0.0533002
\(353\) −3.97273 −0.211447 −0.105724 0.994396i \(-0.533716\pi\)
−0.105724 + 0.994396i \(0.533716\pi\)
\(354\) −3.79419 −0.201659
\(355\) 3.96679 0.210535
\(356\) 5.40924 0.286689
\(357\) −1.22558 −0.0648648
\(358\) −3.79891 −0.200779
\(359\) 13.7528 0.725847 0.362924 0.931819i \(-0.381779\pi\)
0.362924 + 0.931819i \(0.381779\pi\)
\(360\) 1.38335 0.0729091
\(361\) 14.7463 0.776123
\(362\) −7.79219 −0.409548
\(363\) −1.27147 −0.0667351
\(364\) 0.159385 0.00835403
\(365\) 3.46098 0.181156
\(366\) −9.75768 −0.510042
\(367\) 17.0155 0.888204 0.444102 0.895976i \(-0.353523\pi\)
0.444102 + 0.895976i \(0.353523\pi\)
\(368\) −9.28218 −0.483867
\(369\) 0.595191 0.0309844
\(370\) −8.96829 −0.466239
\(371\) −1.48491 −0.0770929
\(372\) −2.22333 −0.115274
\(373\) 23.1769 1.20006 0.600028 0.799979i \(-0.295156\pi\)
0.600028 + 0.799979i \(0.295156\pi\)
\(374\) 3.04381 0.157392
\(375\) −1.27147 −0.0656586
\(376\) −10.5211 −0.542584
\(377\) 3.99527 0.205767
\(378\) 1.76495 0.0907791
\(379\) −2.91256 −0.149608 −0.0748042 0.997198i \(-0.523833\pi\)
−0.0748042 + 0.997198i \(0.523833\pi\)
\(380\) −5.80916 −0.298004
\(381\) −26.5081 −1.35805
\(382\) 18.6062 0.951976
\(383\) 27.9708 1.42924 0.714621 0.699512i \(-0.246599\pi\)
0.714621 + 0.699512i \(0.246599\pi\)
\(384\) 1.27147 0.0648846
\(385\) −0.316678 −0.0161394
\(386\) 8.43218 0.429186
\(387\) −1.38335 −0.0703198
\(388\) −1.27905 −0.0649341
\(389\) 26.3452 1.33575 0.667877 0.744272i \(-0.267203\pi\)
0.667877 + 0.744272i \(0.267203\pi\)
\(390\) −0.639936 −0.0324044
\(391\) 28.2532 1.42883
\(392\) 6.89972 0.348488
\(393\) 0.257242 0.0129761
\(394\) −12.9260 −0.651202
\(395\) 5.32454 0.267907
\(396\) −1.38335 −0.0695161
\(397\) 9.86779 0.495250 0.247625 0.968856i \(-0.420350\pi\)
0.247625 + 0.968856i \(0.420350\pi\)
\(398\) −27.8773 −1.39736
\(399\) −2.33904 −0.117099
\(400\) 1.00000 0.0500000
\(401\) −8.21715 −0.410345 −0.205172 0.978726i \(-0.565775\pi\)
−0.205172 + 0.978726i \(0.565775\pi\)
\(402\) 12.4086 0.618886
\(403\) −0.880086 −0.0438402
\(404\) −7.34467 −0.365411
\(405\) −2.93627 −0.145904
\(406\) −2.51382 −0.124759
\(407\) 8.96829 0.444542
\(408\) −3.87013 −0.191600
\(409\) 24.7206 1.22236 0.611179 0.791493i \(-0.290696\pi\)
0.611179 + 0.791493i \(0.290696\pi\)
\(410\) 0.430252 0.0212486
\(411\) 20.1454 0.993700
\(412\) −1.97820 −0.0974591
\(413\) 0.944995 0.0465002
\(414\) −12.8405 −0.631078
\(415\) −10.1475 −0.498119
\(416\) 0.503303 0.0246764
\(417\) 3.44722 0.168811
\(418\) 5.80916 0.284135
\(419\) −10.3159 −0.503963 −0.251981 0.967732i \(-0.581082\pi\)
−0.251981 + 0.967732i \(0.581082\pi\)
\(420\) 0.402648 0.0196472
\(421\) −29.5355 −1.43947 −0.719735 0.694249i \(-0.755737\pi\)
−0.719735 + 0.694249i \(0.755737\pi\)
\(422\) 4.31767 0.210181
\(423\) −14.5544 −0.707660
\(424\) −4.68904 −0.227720
\(425\) −3.04381 −0.147647
\(426\) 5.04367 0.244366
\(427\) 2.43028 0.117610
\(428\) 10.5975 0.512248
\(429\) 0.639936 0.0308964
\(430\) −1.00000 −0.0482243
\(431\) −8.54566 −0.411630 −0.205815 0.978591i \(-0.565985\pi\)
−0.205815 + 0.978591i \(0.565985\pi\)
\(432\) 5.57332 0.268147
\(433\) 37.1283 1.78427 0.892137 0.451765i \(-0.149205\pi\)
0.892137 + 0.451765i \(0.149205\pi\)
\(434\) 0.553750 0.0265808
\(435\) 10.0931 0.483927
\(436\) 5.64929 0.270552
\(437\) 53.9217 2.57942
\(438\) 4.40055 0.210266
\(439\) −10.0716 −0.480691 −0.240346 0.970687i \(-0.577261\pi\)
−0.240346 + 0.970687i \(0.577261\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 9.54475 0.454512
\(442\) −1.53196 −0.0728679
\(443\) 35.7867 1.70028 0.850140 0.526557i \(-0.176517\pi\)
0.850140 + 0.526557i \(0.176517\pi\)
\(444\) −11.4030 −0.541160
\(445\) 5.40924 0.256423
\(446\) −11.4206 −0.540779
\(447\) −13.8781 −0.656410
\(448\) −0.316678 −0.0149616
\(449\) −32.6197 −1.53942 −0.769709 0.638395i \(-0.779599\pi\)
−0.769709 + 0.638395i \(0.779599\pi\)
\(450\) 1.38335 0.0652119
\(451\) −0.430252 −0.0202598
\(452\) 15.4883 0.728506
\(453\) 0.0901777 0.00423692
\(454\) −19.4623 −0.913409
\(455\) 0.159385 0.00747207
\(456\) −7.38620 −0.345890
\(457\) 3.22695 0.150950 0.0754751 0.997148i \(-0.475953\pi\)
0.0754751 + 0.997148i \(0.475953\pi\)
\(458\) 10.7545 0.502525
\(459\) −16.9642 −0.791819
\(460\) −9.28218 −0.432784
\(461\) −11.1272 −0.518245 −0.259123 0.965844i \(-0.583433\pi\)
−0.259123 + 0.965844i \(0.583433\pi\)
\(462\) −0.402648 −0.0187329
\(463\) 24.5307 1.14004 0.570020 0.821631i \(-0.306935\pi\)
0.570020 + 0.821631i \(0.306935\pi\)
\(464\) −7.93811 −0.368518
\(465\) −2.22333 −0.103104
\(466\) 15.2663 0.707198
\(467\) 13.5931 0.629012 0.314506 0.949256i \(-0.398161\pi\)
0.314506 + 0.949256i \(0.398161\pi\)
\(468\) 0.696246 0.0321840
\(469\) −3.09053 −0.142708
\(470\) −10.5211 −0.485302
\(471\) 23.4333 1.07975
\(472\) 2.98409 0.137354
\(473\) 1.00000 0.0459800
\(474\) 6.77002 0.310957
\(475\) −5.80916 −0.266543
\(476\) 0.963908 0.0441807
\(477\) −6.48660 −0.297001
\(478\) 4.58174 0.209564
\(479\) 37.2117 1.70025 0.850124 0.526583i \(-0.176527\pi\)
0.850124 + 0.526583i \(0.176527\pi\)
\(480\) 1.27147 0.0580346
\(481\) −4.51377 −0.205810
\(482\) −14.3755 −0.654788
\(483\) −3.73745 −0.170060
\(484\) 1.00000 0.0454545
\(485\) −1.27905 −0.0580788
\(486\) 12.9866 0.589083
\(487\) 12.2209 0.553782 0.276891 0.960901i \(-0.410696\pi\)
0.276891 + 0.960901i \(0.410696\pi\)
\(488\) 7.67431 0.347399
\(489\) −10.8493 −0.490622
\(490\) 6.89972 0.311697
\(491\) −12.1137 −0.546682 −0.273341 0.961917i \(-0.588129\pi\)
−0.273341 + 0.961917i \(0.588129\pi\)
\(492\) 0.547054 0.0246631
\(493\) 24.1621 1.08821
\(494\) −2.92377 −0.131546
\(495\) −1.38335 −0.0621771
\(496\) 1.74862 0.0785154
\(497\) −1.25619 −0.0563480
\(498\) −12.9022 −0.578162
\(499\) 2.61092 0.116881 0.0584404 0.998291i \(-0.481387\pi\)
0.0584404 + 0.998291i \(0.481387\pi\)
\(500\) 1.00000 0.0447214
\(501\) 6.87151 0.306996
\(502\) −4.99686 −0.223021
\(503\) 17.7741 0.792506 0.396253 0.918141i \(-0.370310\pi\)
0.396253 + 0.918141i \(0.370310\pi\)
\(504\) −0.438077 −0.0195135
\(505\) −7.34467 −0.326834
\(506\) 9.28218 0.412643
\(507\) 16.2071 0.719782
\(508\) 20.8483 0.924994
\(509\) −26.4370 −1.17180 −0.585900 0.810384i \(-0.699259\pi\)
−0.585900 + 0.810384i \(0.699259\pi\)
\(510\) −3.87013 −0.171372
\(511\) −1.09602 −0.0484849
\(512\) −1.00000 −0.0441942
\(513\) −32.3763 −1.42945
\(514\) 18.2445 0.804732
\(515\) −1.97820 −0.0871701
\(516\) −1.27147 −0.0559735
\(517\) 10.5211 0.462718
\(518\) 2.84006 0.124785
\(519\) −8.36236 −0.367067
\(520\) 0.503303 0.0220713
\(521\) 2.97501 0.130337 0.0651687 0.997874i \(-0.479241\pi\)
0.0651687 + 0.997874i \(0.479241\pi\)
\(522\) −10.9812 −0.480635
\(523\) −24.3954 −1.06674 −0.533368 0.845883i \(-0.679074\pi\)
−0.533368 + 0.845883i \(0.679074\pi\)
\(524\) −0.202318 −0.00883829
\(525\) 0.402648 0.0175730
\(526\) 5.51612 0.240514
\(527\) −5.32248 −0.231851
\(528\) −1.27147 −0.0553338
\(529\) 63.1589 2.74604
\(530\) −4.68904 −0.203679
\(531\) 4.12805 0.179142
\(532\) 1.83963 0.0797582
\(533\) 0.216547 0.00937969
\(534\) 6.87771 0.297627
\(535\) 10.5975 0.458169
\(536\) −9.75924 −0.421535
\(537\) −4.83021 −0.208439
\(538\) −8.01104 −0.345380
\(539\) −6.89972 −0.297192
\(540\) 5.57332 0.239838
\(541\) −1.92969 −0.0829639 −0.0414819 0.999139i \(-0.513208\pi\)
−0.0414819 + 0.999139i \(0.513208\pi\)
\(542\) 29.9694 1.28730
\(543\) −9.90757 −0.425174
\(544\) 3.04381 0.130502
\(545\) 5.64929 0.241989
\(546\) 0.202654 0.00867277
\(547\) 42.4996 1.81715 0.908577 0.417718i \(-0.137170\pi\)
0.908577 + 0.417718i \(0.137170\pi\)
\(548\) −15.8441 −0.676828
\(549\) 10.6163 0.453092
\(550\) −1.00000 −0.0426401
\(551\) 46.1138 1.96451
\(552\) −11.8021 −0.502329
\(553\) −1.68616 −0.0717030
\(554\) 2.09014 0.0888015
\(555\) −11.4030 −0.484028
\(556\) −2.71120 −0.114980
\(557\) 4.90565 0.207859 0.103929 0.994585i \(-0.466858\pi\)
0.103929 + 0.994585i \(0.466858\pi\)
\(558\) 2.41896 0.102403
\(559\) −0.503303 −0.0212874
\(560\) −0.316678 −0.0133821
\(561\) 3.87013 0.163397
\(562\) −31.3635 −1.32299
\(563\) 26.8339 1.13091 0.565457 0.824778i \(-0.308700\pi\)
0.565457 + 0.824778i \(0.308700\pi\)
\(564\) −13.3773 −0.563286
\(565\) 15.4883 0.651596
\(566\) −0.751951 −0.0316068
\(567\) 0.929852 0.0390501
\(568\) −3.96679 −0.166443
\(569\) −2.23408 −0.0936574 −0.0468287 0.998903i \(-0.514911\pi\)
−0.0468287 + 0.998903i \(0.514911\pi\)
\(570\) −7.38620 −0.309374
\(571\) 8.07422 0.337896 0.168948 0.985625i \(-0.445963\pi\)
0.168948 + 0.985625i \(0.445963\pi\)
\(572\) −0.503303 −0.0210441
\(573\) 23.6573 0.988298
\(574\) −0.136251 −0.00568701
\(575\) −9.28218 −0.387094
\(576\) −1.38335 −0.0576397
\(577\) 12.0013 0.499622 0.249811 0.968295i \(-0.419632\pi\)
0.249811 + 0.968295i \(0.419632\pi\)
\(578\) 7.73519 0.321742
\(579\) 10.7213 0.445562
\(580\) −7.93811 −0.329612
\(581\) 3.21347 0.133317
\(582\) −1.62628 −0.0674116
\(583\) 4.68904 0.194200
\(584\) −3.46098 −0.143216
\(585\) 0.696246 0.0287862
\(586\) 16.8372 0.695539
\(587\) −40.9786 −1.69137 −0.845685 0.533683i \(-0.820808\pi\)
−0.845685 + 0.533683i \(0.820808\pi\)
\(588\) 8.77281 0.361785
\(589\) −10.1580 −0.418554
\(590\) 2.98409 0.122853
\(591\) −16.4351 −0.676049
\(592\) 8.96829 0.368595
\(593\) −30.3713 −1.24720 −0.623599 0.781744i \(-0.714330\pi\)
−0.623599 + 0.781744i \(0.714330\pi\)
\(594\) −5.57332 −0.228676
\(595\) 0.963908 0.0395164
\(596\) 10.9149 0.447093
\(597\) −35.4452 −1.45068
\(598\) −4.67175 −0.191042
\(599\) −8.00755 −0.327180 −0.163590 0.986528i \(-0.552307\pi\)
−0.163590 + 0.986528i \(0.552307\pi\)
\(600\) 1.27147 0.0519077
\(601\) 6.53459 0.266552 0.133276 0.991079i \(-0.457450\pi\)
0.133276 + 0.991079i \(0.457450\pi\)
\(602\) 0.316678 0.0129068
\(603\) −13.5005 −0.549782
\(604\) −0.0709238 −0.00288585
\(605\) 1.00000 0.0406558
\(606\) −9.33856 −0.379353
\(607\) 10.5037 0.426332 0.213166 0.977016i \(-0.431622\pi\)
0.213166 + 0.977016i \(0.431622\pi\)
\(608\) 5.80916 0.235593
\(609\) −3.19626 −0.129519
\(610\) 7.67431 0.310724
\(611\) −5.29530 −0.214225
\(612\) 4.21067 0.170206
\(613\) 0.0240921 0.000973071 0 0.000486536 1.00000i \(-0.499845\pi\)
0.000486536 1.00000i \(0.499845\pi\)
\(614\) 2.41881 0.0976150
\(615\) 0.547054 0.0220593
\(616\) 0.316678 0.0127593
\(617\) 11.7746 0.474028 0.237014 0.971506i \(-0.423831\pi\)
0.237014 + 0.971506i \(0.423831\pi\)
\(618\) −2.51523 −0.101178
\(619\) 0.621622 0.0249851 0.0124925 0.999922i \(-0.496023\pi\)
0.0124925 + 0.999922i \(0.496023\pi\)
\(620\) 1.74862 0.0702263
\(621\) −51.7326 −2.07596
\(622\) −29.5823 −1.18614
\(623\) −1.71299 −0.0686293
\(624\) 0.639936 0.0256180
\(625\) 1.00000 0.0400000
\(626\) −26.2434 −1.04890
\(627\) 7.38620 0.294976
\(628\) −18.4300 −0.735437
\(629\) −27.2978 −1.08844
\(630\) −0.438077 −0.0174534
\(631\) −1.38820 −0.0552633 −0.0276316 0.999618i \(-0.508797\pi\)
−0.0276316 + 0.999618i \(0.508797\pi\)
\(632\) −5.32454 −0.211799
\(633\) 5.48981 0.218200
\(634\) 10.6549 0.423160
\(635\) 20.8483 0.827340
\(636\) −5.96199 −0.236408
\(637\) 3.47265 0.137591
\(638\) 7.93811 0.314273
\(639\) −5.48747 −0.217081
\(640\) −1.00000 −0.0395285
\(641\) −3.82545 −0.151096 −0.0755481 0.997142i \(-0.524071\pi\)
−0.0755481 + 0.997142i \(0.524071\pi\)
\(642\) 13.4744 0.531793
\(643\) 14.7052 0.579918 0.289959 0.957039i \(-0.406358\pi\)
0.289959 + 0.957039i \(0.406358\pi\)
\(644\) 2.93946 0.115831
\(645\) −1.27147 −0.0500642
\(646\) −17.6820 −0.695689
\(647\) 24.8603 0.977359 0.488679 0.872464i \(-0.337479\pi\)
0.488679 + 0.872464i \(0.337479\pi\)
\(648\) 2.93627 0.115348
\(649\) −2.98409 −0.117136
\(650\) 0.503303 0.0197412
\(651\) 0.704078 0.0275950
\(652\) 8.53284 0.334172
\(653\) −41.6244 −1.62889 −0.814445 0.580241i \(-0.802958\pi\)
−0.814445 + 0.580241i \(0.802958\pi\)
\(654\) 7.18293 0.280875
\(655\) −0.202318 −0.00790520
\(656\) −0.430252 −0.0167985
\(657\) −4.78776 −0.186788
\(658\) 3.33180 0.129887
\(659\) 13.6060 0.530016 0.265008 0.964246i \(-0.414625\pi\)
0.265008 + 0.964246i \(0.414625\pi\)
\(660\) −1.27147 −0.0494921
\(661\) 38.6568 1.50358 0.751788 0.659404i \(-0.229191\pi\)
0.751788 + 0.659404i \(0.229191\pi\)
\(662\) −14.8383 −0.576707
\(663\) −1.94785 −0.0756481
\(664\) 10.1475 0.393798
\(665\) 1.83963 0.0713379
\(666\) 12.4063 0.480735
\(667\) 73.6830 2.85302
\(668\) −5.40437 −0.209101
\(669\) −14.5209 −0.561412
\(670\) −9.75924 −0.377032
\(671\) −7.67431 −0.296263
\(672\) −0.402648 −0.0155325
\(673\) −38.2515 −1.47449 −0.737243 0.675627i \(-0.763873\pi\)
−0.737243 + 0.675627i \(0.763873\pi\)
\(674\) −31.7045 −1.22121
\(675\) 5.57332 0.214517
\(676\) −12.7467 −0.490257
\(677\) 24.3649 0.936419 0.468209 0.883618i \(-0.344899\pi\)
0.468209 + 0.883618i \(0.344899\pi\)
\(678\) 19.6929 0.756302
\(679\) 0.405048 0.0155443
\(680\) 3.04381 0.116725
\(681\) −24.7458 −0.948259
\(682\) −1.74862 −0.0669582
\(683\) −7.77868 −0.297643 −0.148821 0.988864i \(-0.547548\pi\)
−0.148821 + 0.988864i \(0.547548\pi\)
\(684\) 8.03612 0.307269
\(685\) −15.8441 −0.605373
\(686\) −4.40173 −0.168059
\(687\) 13.6741 0.521699
\(688\) 1.00000 0.0381246
\(689\) −2.36001 −0.0899091
\(690\) −11.8021 −0.449296
\(691\) −6.48628 −0.246750 −0.123375 0.992360i \(-0.539372\pi\)
−0.123375 + 0.992360i \(0.539372\pi\)
\(692\) 6.57690 0.250016
\(693\) 0.438077 0.0166412
\(694\) 14.8944 0.565383
\(695\) −2.71120 −0.102842
\(696\) −10.0931 −0.382578
\(697\) 1.30961 0.0496049
\(698\) −1.05540 −0.0399476
\(699\) 19.4107 0.734181
\(700\) −0.316678 −0.0119693
\(701\) −15.3333 −0.579130 −0.289565 0.957158i \(-0.593511\pi\)
−0.289565 + 0.957158i \(0.593511\pi\)
\(702\) 2.80507 0.105870
\(703\) −52.0982 −1.96492
\(704\) 1.00000 0.0376889
\(705\) −13.3773 −0.503819
\(706\) 3.97273 0.149516
\(707\) 2.32589 0.0874743
\(708\) 3.79419 0.142595
\(709\) 3.93645 0.147836 0.0739182 0.997264i \(-0.476450\pi\)
0.0739182 + 0.997264i \(0.476450\pi\)
\(710\) −3.96679 −0.148871
\(711\) −7.36573 −0.276236
\(712\) −5.40924 −0.202720
\(713\) −16.2310 −0.607857
\(714\) 1.22558 0.0458663
\(715\) −0.503303 −0.0188225
\(716\) 3.79891 0.141972
\(717\) 5.82556 0.217560
\(718\) −13.7528 −0.513251
\(719\) −11.8565 −0.442171 −0.221086 0.975254i \(-0.570960\pi\)
−0.221086 + 0.975254i \(0.570960\pi\)
\(720\) −1.38335 −0.0515546
\(721\) 0.626453 0.0233303
\(722\) −14.7463 −0.548802
\(723\) −18.2781 −0.679771
\(724\) 7.79219 0.289594
\(725\) −7.93811 −0.294814
\(726\) 1.27147 0.0471888
\(727\) 13.4119 0.497421 0.248710 0.968578i \(-0.419993\pi\)
0.248710 + 0.968578i \(0.419993\pi\)
\(728\) −0.159385 −0.00590719
\(729\) 25.3209 0.937811
\(730\) −3.46098 −0.128097
\(731\) −3.04381 −0.112580
\(732\) 9.75768 0.360654
\(733\) −11.6652 −0.430862 −0.215431 0.976519i \(-0.569116\pi\)
−0.215431 + 0.976519i \(0.569116\pi\)
\(734\) −17.0155 −0.628055
\(735\) 8.77281 0.323590
\(736\) 9.28218 0.342146
\(737\) 9.75924 0.359486
\(738\) −0.595191 −0.0219093
\(739\) −27.7796 −1.02189 −0.510944 0.859614i \(-0.670704\pi\)
−0.510944 + 0.859614i \(0.670704\pi\)
\(740\) 8.96829 0.329681
\(741\) −3.71749 −0.136566
\(742\) 1.48491 0.0545129
\(743\) −26.0764 −0.956651 −0.478325 0.878183i \(-0.658756\pi\)
−0.478325 + 0.878183i \(0.658756\pi\)
\(744\) 2.22333 0.0815111
\(745\) 10.9149 0.399892
\(746\) −23.1769 −0.848567
\(747\) 14.0375 0.513606
\(748\) −3.04381 −0.111293
\(749\) −3.35599 −0.122625
\(750\) 1.27147 0.0464277
\(751\) −15.2244 −0.555547 −0.277773 0.960647i \(-0.589596\pi\)
−0.277773 + 0.960647i \(0.589596\pi\)
\(752\) 10.5211 0.383665
\(753\) −6.35338 −0.231530
\(754\) −3.99527 −0.145499
\(755\) −0.0709238 −0.00258118
\(756\) −1.76495 −0.0641905
\(757\) −6.43416 −0.233854 −0.116927 0.993141i \(-0.537304\pi\)
−0.116927 + 0.993141i \(0.537304\pi\)
\(758\) 2.91256 0.105789
\(759\) 11.8021 0.428387
\(760\) 5.80916 0.210720
\(761\) −54.8295 −1.98757 −0.993785 0.111319i \(-0.964492\pi\)
−0.993785 + 0.111319i \(0.964492\pi\)
\(762\) 26.5081 0.960286
\(763\) −1.78901 −0.0647664
\(764\) −18.6062 −0.673149
\(765\) 4.21067 0.152237
\(766\) −27.9708 −1.01063
\(767\) 1.50190 0.0542305
\(768\) −1.27147 −0.0458804
\(769\) −33.4535 −1.20637 −0.603183 0.797603i \(-0.706101\pi\)
−0.603183 + 0.797603i \(0.706101\pi\)
\(770\) 0.316678 0.0114123
\(771\) 23.1975 0.835436
\(772\) −8.43218 −0.303481
\(773\) 0.553428 0.0199054 0.00995271 0.999950i \(-0.496832\pi\)
0.00995271 + 0.999950i \(0.496832\pi\)
\(774\) 1.38335 0.0497236
\(775\) 1.74862 0.0628124
\(776\) 1.27905 0.0459153
\(777\) 3.61106 0.129546
\(778\) −26.3452 −0.944520
\(779\) 2.49940 0.0895504
\(780\) 0.639936 0.0229134
\(781\) 3.96679 0.141943
\(782\) −28.2532 −1.01033
\(783\) −44.2416 −1.58107
\(784\) −6.89972 −0.246418
\(785\) −18.4300 −0.657795
\(786\) −0.257242 −0.00917550
\(787\) 48.5017 1.72890 0.864450 0.502719i \(-0.167667\pi\)
0.864450 + 0.502719i \(0.167667\pi\)
\(788\) 12.9260 0.460470
\(789\) 7.01360 0.249691
\(790\) −5.32454 −0.189439
\(791\) −4.90479 −0.174394
\(792\) 1.38335 0.0491553
\(793\) 3.86250 0.137161
\(794\) −9.86779 −0.350195
\(795\) −5.96199 −0.211450
\(796\) 27.8773 0.988084
\(797\) −39.0165 −1.38204 −0.691018 0.722837i \(-0.742838\pi\)
−0.691018 + 0.722837i \(0.742838\pi\)
\(798\) 2.33904 0.0828013
\(799\) −32.0243 −1.13294
\(800\) −1.00000 −0.0353553
\(801\) −7.48289 −0.264395
\(802\) 8.21715 0.290157
\(803\) 3.46098 0.122135
\(804\) −12.4086 −0.437618
\(805\) 2.93946 0.103602
\(806\) 0.880086 0.0309997
\(807\) −10.1858 −0.358558
\(808\) 7.34467 0.258385
\(809\) −32.7857 −1.15268 −0.576341 0.817209i \(-0.695520\pi\)
−0.576341 + 0.817209i \(0.695520\pi\)
\(810\) 2.93627 0.103170
\(811\) 22.0619 0.774698 0.387349 0.921933i \(-0.373391\pi\)
0.387349 + 0.921933i \(0.373391\pi\)
\(812\) 2.51382 0.0882179
\(813\) 38.1053 1.33641
\(814\) −8.96829 −0.314339
\(815\) 8.53284 0.298892
\(816\) 3.87013 0.135482
\(817\) −5.80916 −0.203237
\(818\) −24.7206 −0.864337
\(819\) −0.220486 −0.00770439
\(820\) −0.430252 −0.0150250
\(821\) −12.5094 −0.436580 −0.218290 0.975884i \(-0.570048\pi\)
−0.218290 + 0.975884i \(0.570048\pi\)
\(822\) −20.1454 −0.702652
\(823\) 6.34676 0.221234 0.110617 0.993863i \(-0.464717\pi\)
0.110617 + 0.993863i \(0.464717\pi\)
\(824\) 1.97820 0.0689140
\(825\) −1.27147 −0.0442670
\(826\) −0.944995 −0.0328806
\(827\) −48.8755 −1.69957 −0.849783 0.527133i \(-0.823267\pi\)
−0.849783 + 0.527133i \(0.823267\pi\)
\(828\) 12.8405 0.446240
\(829\) −11.1419 −0.386974 −0.193487 0.981103i \(-0.561980\pi\)
−0.193487 + 0.981103i \(0.561980\pi\)
\(830\) 10.1475 0.352223
\(831\) 2.65756 0.0921897
\(832\) −0.503303 −0.0174489
\(833\) 21.0015 0.727657
\(834\) −3.44722 −0.119367
\(835\) −5.40437 −0.187026
\(836\) −5.80916 −0.200914
\(837\) 9.74563 0.336858
\(838\) 10.3159 0.356356
\(839\) 4.40504 0.152079 0.0760394 0.997105i \(-0.475773\pi\)
0.0760394 + 0.997105i \(0.475773\pi\)
\(840\) −0.402648 −0.0138927
\(841\) 34.0136 1.17288
\(842\) 29.5355 1.01786
\(843\) −39.8779 −1.37347
\(844\) −4.31767 −0.148620
\(845\) −12.7467 −0.438499
\(846\) 14.5544 0.500391
\(847\) −0.316678 −0.0108812
\(848\) 4.68904 0.161022
\(849\) −0.956086 −0.0328128
\(850\) 3.04381 0.104402
\(851\) −83.2453 −2.85361
\(852\) −5.04367 −0.172793
\(853\) 32.1697 1.10147 0.550735 0.834680i \(-0.314348\pi\)
0.550735 + 0.834680i \(0.314348\pi\)
\(854\) −2.43028 −0.0831625
\(855\) 8.03612 0.274830
\(856\) −10.5975 −0.362214
\(857\) −55.9742 −1.91204 −0.956020 0.293300i \(-0.905246\pi\)
−0.956020 + 0.293300i \(0.905246\pi\)
\(858\) −0.639936 −0.0218471
\(859\) 38.0150 1.29706 0.648528 0.761191i \(-0.275385\pi\)
0.648528 + 0.761191i \(0.275385\pi\)
\(860\) 1.00000 0.0340997
\(861\) −0.173240 −0.00590400
\(862\) 8.54566 0.291066
\(863\) −16.9960 −0.578551 −0.289276 0.957246i \(-0.593414\pi\)
−0.289276 + 0.957246i \(0.593414\pi\)
\(864\) −5.57332 −0.189608
\(865\) 6.57690 0.223621
\(866\) −37.1283 −1.26167
\(867\) 9.83510 0.334017
\(868\) −0.553750 −0.0187955
\(869\) 5.32454 0.180623
\(870\) −10.0931 −0.342188
\(871\) −4.91185 −0.166432
\(872\) −5.64929 −0.191309
\(873\) 1.76938 0.0598845
\(874\) −53.9217 −1.82393
\(875\) −0.316678 −0.0107057
\(876\) −4.40055 −0.148681
\(877\) 22.1512 0.747993 0.373997 0.927430i \(-0.377987\pi\)
0.373997 + 0.927430i \(0.377987\pi\)
\(878\) 10.0716 0.339900
\(879\) 21.4081 0.722076
\(880\) 1.00000 0.0337100
\(881\) 11.7140 0.394656 0.197328 0.980338i \(-0.436774\pi\)
0.197328 + 0.980338i \(0.436774\pi\)
\(882\) −9.54475 −0.321388
\(883\) 12.4442 0.418779 0.209390 0.977832i \(-0.432852\pi\)
0.209390 + 0.977832i \(0.432852\pi\)
\(884\) 1.53196 0.0515254
\(885\) 3.79419 0.127540
\(886\) −35.7867 −1.20228
\(887\) −23.9978 −0.805767 −0.402884 0.915251i \(-0.631992\pi\)
−0.402884 + 0.915251i \(0.631992\pi\)
\(888\) 11.4030 0.382658
\(889\) −6.60219 −0.221431
\(890\) −5.40924 −0.181318
\(891\) −2.93627 −0.0983688
\(892\) 11.4206 0.382389
\(893\) −61.1188 −2.04526
\(894\) 13.8781 0.464152
\(895\) 3.79891 0.126983
\(896\) 0.316678 0.0105795
\(897\) −5.94000 −0.198331
\(898\) 32.6197 1.08853
\(899\) −13.8808 −0.462949
\(900\) −1.38335 −0.0461118
\(901\) −14.2726 −0.475488
\(902\) 0.430252 0.0143258
\(903\) 0.402648 0.0133993
\(904\) −15.4883 −0.515132
\(905\) 7.79219 0.259021
\(906\) −0.0901777 −0.00299595
\(907\) 10.8665 0.360815 0.180407 0.983592i \(-0.442258\pi\)
0.180407 + 0.983592i \(0.442258\pi\)
\(908\) 19.4623 0.645878
\(909\) 10.1603 0.336995
\(910\) −0.159385 −0.00528355
\(911\) −6.94084 −0.229960 −0.114980 0.993368i \(-0.536680\pi\)
−0.114980 + 0.993368i \(0.536680\pi\)
\(912\) 7.38620 0.244581
\(913\) −10.1475 −0.335832
\(914\) −3.22695 −0.106738
\(915\) 9.75768 0.322579
\(916\) −10.7545 −0.355339
\(917\) 0.0640695 0.00211576
\(918\) 16.9642 0.559900
\(919\) 51.7330 1.70651 0.853256 0.521492i \(-0.174624\pi\)
0.853256 + 0.521492i \(0.174624\pi\)
\(920\) 9.28218 0.306024
\(921\) 3.07545 0.101339
\(922\) 11.1272 0.366455
\(923\) −1.99649 −0.0657154
\(924\) 0.402648 0.0132461
\(925\) 8.96829 0.294876
\(926\) −24.5307 −0.806130
\(927\) 2.73656 0.0898803
\(928\) 7.93811 0.260581
\(929\) −6.02374 −0.197632 −0.0988162 0.995106i \(-0.531506\pi\)
−0.0988162 + 0.995106i \(0.531506\pi\)
\(930\) 2.22333 0.0729058
\(931\) 40.0815 1.31362
\(932\) −15.2663 −0.500065
\(933\) −37.6131 −1.23140
\(934\) −13.5931 −0.444778
\(935\) −3.04381 −0.0995434
\(936\) −0.696246 −0.0227575
\(937\) −32.6799 −1.06761 −0.533804 0.845609i \(-0.679238\pi\)
−0.533804 + 0.845609i \(0.679238\pi\)
\(938\) 3.09053 0.100910
\(939\) −33.3677 −1.08892
\(940\) 10.5211 0.343161
\(941\) −32.4452 −1.05768 −0.528841 0.848721i \(-0.677373\pi\)
−0.528841 + 0.848721i \(0.677373\pi\)
\(942\) −23.4333 −0.763497
\(943\) 3.99368 0.130052
\(944\) −2.98409 −0.0971239
\(945\) −1.76495 −0.0574137
\(946\) −1.00000 −0.0325128
\(947\) 15.9886 0.519559 0.259780 0.965668i \(-0.416350\pi\)
0.259780 + 0.965668i \(0.416350\pi\)
\(948\) −6.77002 −0.219880
\(949\) −1.74192 −0.0565451
\(950\) 5.80916 0.188474
\(951\) 13.5474 0.439305
\(952\) −0.963908 −0.0312404
\(953\) −56.9964 −1.84629 −0.923147 0.384448i \(-0.874392\pi\)
−0.923147 + 0.384448i \(0.874392\pi\)
\(954\) 6.48660 0.210011
\(955\) −18.6062 −0.602082
\(956\) −4.58174 −0.148184
\(957\) 10.0931 0.326264
\(958\) −37.2117 −1.20226
\(959\) 5.01749 0.162023
\(960\) −1.27147 −0.0410366
\(961\) −27.9423 −0.901365
\(962\) 4.51377 0.145530
\(963\) −14.6601 −0.472414
\(964\) 14.3755 0.463005
\(965\) −8.43218 −0.271441
\(966\) 3.73745 0.120250
\(967\) 24.9638 0.802783 0.401391 0.915907i \(-0.368527\pi\)
0.401391 + 0.915907i \(0.368527\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −22.4822 −0.722233
\(970\) 1.27905 0.0410679
\(971\) 50.1432 1.60917 0.804586 0.593836i \(-0.202387\pi\)
0.804586 + 0.593836i \(0.202387\pi\)
\(972\) −12.9866 −0.416544
\(973\) 0.858576 0.0275247
\(974\) −12.2209 −0.391583
\(975\) 0.639936 0.0204944
\(976\) −7.67431 −0.245649
\(977\) 5.59808 0.179098 0.0895492 0.995982i \(-0.471457\pi\)
0.0895492 + 0.995982i \(0.471457\pi\)
\(978\) 10.8493 0.346922
\(979\) 5.40924 0.172880
\(980\) −6.89972 −0.220403
\(981\) −7.81497 −0.249513
\(982\) 12.1137 0.386563
\(983\) 2.72248 0.0868337 0.0434168 0.999057i \(-0.486176\pi\)
0.0434168 + 0.999057i \(0.486176\pi\)
\(984\) −0.547054 −0.0174394
\(985\) 12.9260 0.411857
\(986\) −24.1621 −0.769479
\(987\) 4.23630 0.134843
\(988\) 2.92377 0.0930174
\(989\) −9.28218 −0.295156
\(990\) 1.38335 0.0439659
\(991\) 28.1134 0.893052 0.446526 0.894771i \(-0.352661\pi\)
0.446526 + 0.894771i \(0.352661\pi\)
\(992\) −1.74862 −0.0555188
\(993\) −18.8665 −0.598711
\(994\) 1.25619 0.0398440
\(995\) 27.8773 0.883769
\(996\) 12.9022 0.408823
\(997\) 26.4197 0.836721 0.418361 0.908281i \(-0.362605\pi\)
0.418361 + 0.908281i \(0.362605\pi\)
\(998\) −2.61092 −0.0826473
\(999\) 49.9832 1.58140
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.z.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.z.1.3 10 1.1 even 1 trivial