Properties

Label 4730.2.a.bf.1.3
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 1.3
Root \(2.61361\)
Character \(\chi\) = 4730.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.61361 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.61361 q^{6} +0.881450 q^{7} +1.00000 q^{8} +3.83093 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.61361 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.61361 q^{6} +0.881450 q^{7} +1.00000 q^{8} +3.83093 q^{9} -1.00000 q^{10} +1.00000 q^{11} -2.61361 q^{12} -6.07569 q^{13} +0.881450 q^{14} +2.61361 q^{15} +1.00000 q^{16} -6.61083 q^{17} +3.83093 q^{18} -1.59458 q^{19} -1.00000 q^{20} -2.30376 q^{21} +1.00000 q^{22} +1.90303 q^{23} -2.61361 q^{24} +1.00000 q^{25} -6.07569 q^{26} -2.17173 q^{27} +0.881450 q^{28} -3.16799 q^{29} +2.61361 q^{30} +7.78274 q^{31} +1.00000 q^{32} -2.61361 q^{33} -6.61083 q^{34} -0.881450 q^{35} +3.83093 q^{36} +3.74873 q^{37} -1.59458 q^{38} +15.8795 q^{39} -1.00000 q^{40} +7.16918 q^{41} -2.30376 q^{42} +1.00000 q^{43} +1.00000 q^{44} -3.83093 q^{45} +1.90303 q^{46} -12.3298 q^{47} -2.61361 q^{48} -6.22305 q^{49} +1.00000 q^{50} +17.2781 q^{51} -6.07569 q^{52} +8.84537 q^{53} -2.17173 q^{54} -1.00000 q^{55} +0.881450 q^{56} +4.16762 q^{57} -3.16799 q^{58} +4.82604 q^{59} +2.61361 q^{60} -6.29159 q^{61} +7.78274 q^{62} +3.37677 q^{63} +1.00000 q^{64} +6.07569 q^{65} -2.61361 q^{66} -7.78393 q^{67} -6.61083 q^{68} -4.97378 q^{69} -0.881450 q^{70} +4.25244 q^{71} +3.83093 q^{72} -6.44173 q^{73} +3.74873 q^{74} -2.61361 q^{75} -1.59458 q^{76} +0.881450 q^{77} +15.8795 q^{78} +5.91937 q^{79} -1.00000 q^{80} -5.81675 q^{81} +7.16918 q^{82} -8.39798 q^{83} -2.30376 q^{84} +6.61083 q^{85} +1.00000 q^{86} +8.27988 q^{87} +1.00000 q^{88} +14.5173 q^{89} -3.83093 q^{90} -5.35542 q^{91} +1.90303 q^{92} -20.3410 q^{93} -12.3298 q^{94} +1.59458 q^{95} -2.61361 q^{96} +3.95643 q^{97} -6.22305 q^{98} +3.83093 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} + O(q^{10}) \) \( 13q + 13q^{2} + 13q^{4} - 13q^{5} + 7q^{7} + 13q^{8} + 25q^{9} - 13q^{10} + 13q^{11} + 11q^{13} + 7q^{14} + 13q^{16} - 2q^{17} + 25q^{18} + 7q^{19} - 13q^{20} + 12q^{21} + 13q^{22} + 12q^{23} + 13q^{25} + 11q^{26} - 15q^{27} + 7q^{28} + 14q^{29} + 20q^{31} + 13q^{32} - 2q^{34} - 7q^{35} + 25q^{36} + 17q^{37} + 7q^{38} - 4q^{39} - 13q^{40} + 9q^{41} + 12q^{42} + 13q^{43} + 13q^{44} - 25q^{45} + 12q^{46} - 9q^{47} + 30q^{49} + 13q^{50} - 3q^{51} + 11q^{52} + 22q^{53} - 15q^{54} - 13q^{55} + 7q^{56} + 17q^{57} + 14q^{58} + 19q^{59} + 2q^{61} + 20q^{62} + 12q^{63} + 13q^{64} - 11q^{65} + 9q^{67} - 2q^{68} - 6q^{69} - 7q^{70} + 6q^{71} + 25q^{72} + 7q^{73} + 17q^{74} + 7q^{76} + 7q^{77} - 4q^{78} + 50q^{79} - 13q^{80} + 85q^{81} + 9q^{82} + q^{83} + 12q^{84} + 2q^{85} + 13q^{86} - 21q^{87} + 13q^{88} + 5q^{89} - 25q^{90} + 5q^{91} + 12q^{92} + 3q^{93} - 9q^{94} - 7q^{95} + 20q^{97} + 30q^{98} + 25q^{99} + O(q^{100}) \)

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
\(3\) −2.61361 −1.50897 −0.754483 0.656320i \(-0.772112\pi\)
−0.754483 + 0.656320i \(0.772112\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.61361 −1.06700
\(7\) 0.881450 0.333157 0.166578 0.986028i \(-0.446728\pi\)
0.166578 + 0.986028i \(0.446728\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.83093 1.27698
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −2.61361 −0.754483
\(13\) −6.07569 −1.68509 −0.842547 0.538623i \(-0.818945\pi\)
−0.842547 + 0.538623i \(0.818945\pi\)
\(14\) 0.881450 0.235577
\(15\) 2.61361 0.674830
\(16\) 1.00000 0.250000
\(17\) −6.61083 −1.60336 −0.801681 0.597752i \(-0.796061\pi\)
−0.801681 + 0.597752i \(0.796061\pi\)
\(18\) 3.83093 0.902959
\(19\) −1.59458 −0.365823 −0.182911 0.983129i \(-0.558552\pi\)
−0.182911 + 0.983129i \(0.558552\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.30376 −0.502722
\(22\) 1.00000 0.213201
\(23\) 1.90303 0.396810 0.198405 0.980120i \(-0.436424\pi\)
0.198405 + 0.980120i \(0.436424\pi\)
\(24\) −2.61361 −0.533500
\(25\) 1.00000 0.200000
\(26\) −6.07569 −1.19154
\(27\) −2.17173 −0.417949
\(28\) 0.881450 0.166578
\(29\) −3.16799 −0.588281 −0.294141 0.955762i \(-0.595033\pi\)
−0.294141 + 0.955762i \(0.595033\pi\)
\(30\) 2.61361 0.477177
\(31\) 7.78274 1.39782 0.698911 0.715209i \(-0.253669\pi\)
0.698911 + 0.715209i \(0.253669\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.61361 −0.454970
\(34\) −6.61083 −1.13375
\(35\) −0.881450 −0.148992
\(36\) 3.83093 0.638489
\(37\) 3.74873 0.616288 0.308144 0.951340i \(-0.400292\pi\)
0.308144 + 0.951340i \(0.400292\pi\)
\(38\) −1.59458 −0.258676
\(39\) 15.8795 2.54275
\(40\) −1.00000 −0.158114
\(41\) 7.16918 1.11964 0.559819 0.828615i \(-0.310871\pi\)
0.559819 + 0.828615i \(0.310871\pi\)
\(42\) −2.30376 −0.355478
\(43\) 1.00000 0.152499
\(44\) 1.00000 0.150756
\(45\) −3.83093 −0.571082
\(46\) 1.90303 0.280587
\(47\) −12.3298 −1.79849 −0.899244 0.437447i \(-0.855883\pi\)
−0.899244 + 0.437447i \(0.855883\pi\)
\(48\) −2.61361 −0.377241
\(49\) −6.22305 −0.889007
\(50\) 1.00000 0.141421
\(51\) 17.2781 2.41942
\(52\) −6.07569 −0.842547
\(53\) 8.84537 1.21501 0.607503 0.794318i \(-0.292171\pi\)
0.607503 + 0.794318i \(0.292171\pi\)
\(54\) −2.17173 −0.295535
\(55\) −1.00000 −0.134840
\(56\) 0.881450 0.117789
\(57\) 4.16762 0.552014
\(58\) −3.16799 −0.415978
\(59\) 4.82604 0.628296 0.314148 0.949374i \(-0.398281\pi\)
0.314148 + 0.949374i \(0.398281\pi\)
\(60\) 2.61361 0.337415
\(61\) −6.29159 −0.805556 −0.402778 0.915298i \(-0.631955\pi\)
−0.402778 + 0.915298i \(0.631955\pi\)
\(62\) 7.78274 0.988409
\(63\) 3.37677 0.425434
\(64\) 1.00000 0.125000
\(65\) 6.07569 0.753597
\(66\) −2.61361 −0.321713
\(67\) −7.78393 −0.950958 −0.475479 0.879727i \(-0.657725\pi\)
−0.475479 + 0.879727i \(0.657725\pi\)
\(68\) −6.61083 −0.801681
\(69\) −4.97378 −0.598773
\(70\) −0.881450 −0.105353
\(71\) 4.25244 0.504672 0.252336 0.967640i \(-0.418801\pi\)
0.252336 + 0.967640i \(0.418801\pi\)
\(72\) 3.83093 0.451480
\(73\) −6.44173 −0.753948 −0.376974 0.926224i \(-0.623035\pi\)
−0.376974 + 0.926224i \(0.623035\pi\)
\(74\) 3.74873 0.435781
\(75\) −2.61361 −0.301793
\(76\) −1.59458 −0.182911
\(77\) 0.881450 0.100451
\(78\) 15.8795 1.79799
\(79\) 5.91937 0.665982 0.332991 0.942930i \(-0.391942\pi\)
0.332991 + 0.942930i \(0.391942\pi\)
\(80\) −1.00000 −0.111803
\(81\) −5.81675 −0.646306
\(82\) 7.16918 0.791704
\(83\) −8.39798 −0.921798 −0.460899 0.887453i \(-0.652473\pi\)
−0.460899 + 0.887453i \(0.652473\pi\)
\(84\) −2.30376 −0.251361
\(85\) 6.61083 0.717046
\(86\) 1.00000 0.107833
\(87\) 8.27988 0.887696
\(88\) 1.00000 0.106600
\(89\) 14.5173 1.53883 0.769414 0.638751i \(-0.220549\pi\)
0.769414 + 0.638751i \(0.220549\pi\)
\(90\) −3.83093 −0.403816
\(91\) −5.35542 −0.561400
\(92\) 1.90303 0.198405
\(93\) −20.3410 −2.10926
\(94\) −12.3298 −1.27172
\(95\) 1.59458 0.163601
\(96\) −2.61361 −0.266750
\(97\) 3.95643 0.401715 0.200857 0.979620i \(-0.435627\pi\)
0.200857 + 0.979620i \(0.435627\pi\)
\(98\) −6.22305 −0.628623
\(99\) 3.83093 0.385023
\(100\) 1.00000 0.100000
\(101\) −9.97999 −0.993047 −0.496523 0.868023i \(-0.665390\pi\)
−0.496523 + 0.868023i \(0.665390\pi\)
\(102\) 17.2781 1.71079
\(103\) 0.780011 0.0768568 0.0384284 0.999261i \(-0.487765\pi\)
0.0384284 + 0.999261i \(0.487765\pi\)
\(104\) −6.07569 −0.595771
\(105\) 2.30376 0.224824
\(106\) 8.84537 0.859139
\(107\) 13.7654 1.33075 0.665375 0.746509i \(-0.268272\pi\)
0.665375 + 0.746509i \(0.268272\pi\)
\(108\) −2.17173 −0.208975
\(109\) 15.6196 1.49608 0.748042 0.663651i \(-0.230994\pi\)
0.748042 + 0.663651i \(0.230994\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −9.79771 −0.929957
\(112\) 0.881450 0.0832892
\(113\) −6.48394 −0.609958 −0.304979 0.952359i \(-0.598649\pi\)
−0.304979 + 0.952359i \(0.598649\pi\)
\(114\) 4.16762 0.390333
\(115\) −1.90303 −0.177459
\(116\) −3.16799 −0.294141
\(117\) −23.2756 −2.15183
\(118\) 4.82604 0.444273
\(119\) −5.82712 −0.534171
\(120\) 2.61361 0.238588
\(121\) 1.00000 0.0909091
\(122\) −6.29159 −0.569614
\(123\) −18.7374 −1.68950
\(124\) 7.78274 0.698911
\(125\) −1.00000 −0.0894427
\(126\) 3.37677 0.300827
\(127\) 2.02587 0.179767 0.0898837 0.995952i \(-0.471350\pi\)
0.0898837 + 0.995952i \(0.471350\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.61361 −0.230115
\(130\) 6.07569 0.532873
\(131\) −12.5615 −1.09750 −0.548751 0.835986i \(-0.684896\pi\)
−0.548751 + 0.835986i \(0.684896\pi\)
\(132\) −2.61361 −0.227485
\(133\) −1.40555 −0.121876
\(134\) −7.78393 −0.672429
\(135\) 2.17173 0.186913
\(136\) −6.61083 −0.566874
\(137\) 19.0211 1.62508 0.812542 0.582902i \(-0.198083\pi\)
0.812542 + 0.582902i \(0.198083\pi\)
\(138\) −4.97378 −0.423396
\(139\) 11.1258 0.943677 0.471839 0.881685i \(-0.343590\pi\)
0.471839 + 0.881685i \(0.343590\pi\)
\(140\) −0.881450 −0.0744961
\(141\) 32.2253 2.71386
\(142\) 4.25244 0.356857
\(143\) −6.07569 −0.508075
\(144\) 3.83093 0.319244
\(145\) 3.16799 0.263087
\(146\) −6.44173 −0.533122
\(147\) 16.2646 1.34148
\(148\) 3.74873 0.308144
\(149\) −9.99029 −0.818436 −0.409218 0.912437i \(-0.634198\pi\)
−0.409218 + 0.912437i \(0.634198\pi\)
\(150\) −2.61361 −0.213400
\(151\) −1.15872 −0.0942957 −0.0471478 0.998888i \(-0.515013\pi\)
−0.0471478 + 0.998888i \(0.515013\pi\)
\(152\) −1.59458 −0.129338
\(153\) −25.3257 −2.04746
\(154\) 0.881450 0.0710293
\(155\) −7.78274 −0.625125
\(156\) 15.8795 1.27137
\(157\) 24.8105 1.98009 0.990046 0.140744i \(-0.0449494\pi\)
0.990046 + 0.140744i \(0.0449494\pi\)
\(158\) 5.91937 0.470920
\(159\) −23.1183 −1.83340
\(160\) −1.00000 −0.0790569
\(161\) 1.67743 0.132200
\(162\) −5.81675 −0.457007
\(163\) 12.5250 0.981031 0.490515 0.871433i \(-0.336809\pi\)
0.490515 + 0.871433i \(0.336809\pi\)
\(164\) 7.16918 0.559819
\(165\) 2.61361 0.203469
\(166\) −8.39798 −0.651809
\(167\) 24.0151 1.85835 0.929173 0.369645i \(-0.120521\pi\)
0.929173 + 0.369645i \(0.120521\pi\)
\(168\) −2.30376 −0.177739
\(169\) 23.9140 1.83954
\(170\) 6.61083 0.507028
\(171\) −6.10875 −0.467148
\(172\) 1.00000 0.0762493
\(173\) −17.8221 −1.35499 −0.677496 0.735526i \(-0.736935\pi\)
−0.677496 + 0.735526i \(0.736935\pi\)
\(174\) 8.27988 0.627696
\(175\) 0.881450 0.0666313
\(176\) 1.00000 0.0753778
\(177\) −12.6134 −0.948078
\(178\) 14.5173 1.08812
\(179\) 4.37125 0.326723 0.163361 0.986566i \(-0.447766\pi\)
0.163361 + 0.986566i \(0.447766\pi\)
\(180\) −3.83093 −0.285541
\(181\) 10.6388 0.790773 0.395386 0.918515i \(-0.370611\pi\)
0.395386 + 0.918515i \(0.370611\pi\)
\(182\) −5.35542 −0.396970
\(183\) 16.4437 1.21556
\(184\) 1.90303 0.140294
\(185\) −3.74873 −0.275612
\(186\) −20.3410 −1.49148
\(187\) −6.61083 −0.483432
\(188\) −12.3298 −0.899244
\(189\) −1.91427 −0.139243
\(190\) 1.59458 0.115683
\(191\) 19.9082 1.44050 0.720252 0.693712i \(-0.244026\pi\)
0.720252 + 0.693712i \(0.244026\pi\)
\(192\) −2.61361 −0.188621
\(193\) 10.7455 0.773480 0.386740 0.922189i \(-0.373601\pi\)
0.386740 + 0.922189i \(0.373601\pi\)
\(194\) 3.95643 0.284055
\(195\) −15.8795 −1.13715
\(196\) −6.22305 −0.444503
\(197\) −6.53034 −0.465267 −0.232634 0.972564i \(-0.574734\pi\)
−0.232634 + 0.972564i \(0.574734\pi\)
\(198\) 3.83093 0.272253
\(199\) 18.5722 1.31655 0.658273 0.752779i \(-0.271287\pi\)
0.658273 + 0.752779i \(0.271287\pi\)
\(200\) 1.00000 0.0707107
\(201\) 20.3441 1.43496
\(202\) −9.97999 −0.702190
\(203\) −2.79243 −0.195990
\(204\) 17.2781 1.20971
\(205\) −7.16918 −0.500717
\(206\) 0.780011 0.0543460
\(207\) 7.29040 0.506718
\(208\) −6.07569 −0.421273
\(209\) −1.59458 −0.110300
\(210\) 2.30376 0.158975
\(211\) −23.6147 −1.62571 −0.812853 0.582469i \(-0.802087\pi\)
−0.812853 + 0.582469i \(0.802087\pi\)
\(212\) 8.84537 0.607503
\(213\) −11.1142 −0.761532
\(214\) 13.7654 0.940982
\(215\) −1.00000 −0.0681994
\(216\) −2.17173 −0.147767
\(217\) 6.86009 0.465694
\(218\) 15.6196 1.05789
\(219\) 16.8362 1.13768
\(220\) −1.00000 −0.0674200
\(221\) 40.1654 2.70182
\(222\) −9.79771 −0.657579
\(223\) −11.8260 −0.791929 −0.395965 0.918266i \(-0.629590\pi\)
−0.395965 + 0.918266i \(0.629590\pi\)
\(224\) 0.881450 0.0588943
\(225\) 3.83093 0.255395
\(226\) −6.48394 −0.431306
\(227\) 20.5501 1.36396 0.681981 0.731370i \(-0.261119\pi\)
0.681981 + 0.731370i \(0.261119\pi\)
\(228\) 4.16762 0.276007
\(229\) 21.7488 1.43720 0.718600 0.695424i \(-0.244783\pi\)
0.718600 + 0.695424i \(0.244783\pi\)
\(230\) −1.90303 −0.125482
\(231\) −2.30376 −0.151576
\(232\) −3.16799 −0.207989
\(233\) 7.30511 0.478574 0.239287 0.970949i \(-0.423086\pi\)
0.239287 + 0.970949i \(0.423086\pi\)
\(234\) −23.2756 −1.52157
\(235\) 12.3298 0.804308
\(236\) 4.82604 0.314148
\(237\) −15.4709 −1.00494
\(238\) −5.82712 −0.377716
\(239\) −15.1275 −0.978518 −0.489259 0.872139i \(-0.662733\pi\)
−0.489259 + 0.872139i \(0.662733\pi\)
\(240\) 2.61361 0.168707
\(241\) 23.2924 1.50040 0.750198 0.661213i \(-0.229958\pi\)
0.750198 + 0.661213i \(0.229958\pi\)
\(242\) 1.00000 0.0642824
\(243\) 21.7179 1.39320
\(244\) −6.29159 −0.402778
\(245\) 6.22305 0.397576
\(246\) −18.7374 −1.19465
\(247\) 9.68820 0.616446
\(248\) 7.78274 0.494204
\(249\) 21.9490 1.39096
\(250\) −1.00000 −0.0632456
\(251\) −6.47298 −0.408571 −0.204286 0.978911i \(-0.565487\pi\)
−0.204286 + 0.978911i \(0.565487\pi\)
\(252\) 3.37677 0.212717
\(253\) 1.90303 0.119643
\(254\) 2.02587 0.127115
\(255\) −17.2781 −1.08200
\(256\) 1.00000 0.0625000
\(257\) 19.8928 1.24088 0.620440 0.784254i \(-0.286954\pi\)
0.620440 + 0.784254i \(0.286954\pi\)
\(258\) −2.61361 −0.162716
\(259\) 3.30432 0.205320
\(260\) 6.07569 0.376798
\(261\) −12.1364 −0.751222
\(262\) −12.5615 −0.776050
\(263\) −23.5082 −1.44958 −0.724790 0.688970i \(-0.758063\pi\)
−0.724790 + 0.688970i \(0.758063\pi\)
\(264\) −2.61361 −0.160856
\(265\) −8.84537 −0.543367
\(266\) −1.40555 −0.0861796
\(267\) −37.9424 −2.32204
\(268\) −7.78393 −0.475479
\(269\) −0.123729 −0.00754387 −0.00377194 0.999993i \(-0.501201\pi\)
−0.00377194 + 0.999993i \(0.501201\pi\)
\(270\) 2.17173 0.132167
\(271\) −0.206038 −0.0125159 −0.00625797 0.999980i \(-0.501992\pi\)
−0.00625797 + 0.999980i \(0.501992\pi\)
\(272\) −6.61083 −0.400841
\(273\) 13.9969 0.847134
\(274\) 19.0211 1.14911
\(275\) 1.00000 0.0603023
\(276\) −4.97378 −0.299386
\(277\) −15.6731 −0.941704 −0.470852 0.882212i \(-0.656053\pi\)
−0.470852 + 0.882212i \(0.656053\pi\)
\(278\) 11.1258 0.667281
\(279\) 29.8151 1.78499
\(280\) −0.881450 −0.0526767
\(281\) 29.8787 1.78241 0.891207 0.453596i \(-0.149859\pi\)
0.891207 + 0.453596i \(0.149859\pi\)
\(282\) 32.2253 1.91899
\(283\) −9.07865 −0.539670 −0.269835 0.962907i \(-0.586969\pi\)
−0.269835 + 0.962907i \(0.586969\pi\)
\(284\) 4.25244 0.252336
\(285\) −4.16762 −0.246868
\(286\) −6.07569 −0.359263
\(287\) 6.31927 0.373015
\(288\) 3.83093 0.225740
\(289\) 26.7031 1.57077
\(290\) 3.16799 0.186031
\(291\) −10.3406 −0.606174
\(292\) −6.44173 −0.376974
\(293\) −12.6194 −0.737232 −0.368616 0.929582i \(-0.620168\pi\)
−0.368616 + 0.929582i \(0.620168\pi\)
\(294\) 16.2646 0.948570
\(295\) −4.82604 −0.280983
\(296\) 3.74873 0.217891
\(297\) −2.17173 −0.126017
\(298\) −9.99029 −0.578722
\(299\) −11.5623 −0.668662
\(300\) −2.61361 −0.150897
\(301\) 0.881450 0.0508059
\(302\) −1.15872 −0.0666771
\(303\) 26.0838 1.49847
\(304\) −1.59458 −0.0914557
\(305\) 6.29159 0.360256
\(306\) −25.3257 −1.44777
\(307\) 5.09549 0.290815 0.145408 0.989372i \(-0.453551\pi\)
0.145408 + 0.989372i \(0.453551\pi\)
\(308\) 0.881450 0.0502253
\(309\) −2.03864 −0.115974
\(310\) −7.78274 −0.442030
\(311\) 26.9660 1.52910 0.764550 0.644565i \(-0.222961\pi\)
0.764550 + 0.644565i \(0.222961\pi\)
\(312\) 15.8795 0.898997
\(313\) −3.49719 −0.197673 −0.0988366 0.995104i \(-0.531512\pi\)
−0.0988366 + 0.995104i \(0.531512\pi\)
\(314\) 24.8105 1.40014
\(315\) −3.37677 −0.190260
\(316\) 5.91937 0.332991
\(317\) 2.46634 0.138523 0.0692617 0.997599i \(-0.477936\pi\)
0.0692617 + 0.997599i \(0.477936\pi\)
\(318\) −23.1183 −1.29641
\(319\) −3.16799 −0.177373
\(320\) −1.00000 −0.0559017
\(321\) −35.9773 −2.00806
\(322\) 1.67743 0.0934795
\(323\) 10.5415 0.586547
\(324\) −5.81675 −0.323153
\(325\) −6.07569 −0.337019
\(326\) 12.5250 0.693693
\(327\) −40.8234 −2.25754
\(328\) 7.16918 0.395852
\(329\) −10.8681 −0.599178
\(330\) 2.61361 0.143874
\(331\) −9.52802 −0.523708 −0.261854 0.965108i \(-0.584334\pi\)
−0.261854 + 0.965108i \(0.584334\pi\)
\(332\) −8.39798 −0.460899
\(333\) 14.3611 0.786986
\(334\) 24.0151 1.31405
\(335\) 7.78393 0.425281
\(336\) −2.30376 −0.125681
\(337\) 12.4590 0.678685 0.339342 0.940663i \(-0.389795\pi\)
0.339342 + 0.940663i \(0.389795\pi\)
\(338\) 23.9140 1.30075
\(339\) 16.9465 0.920406
\(340\) 6.61083 0.358523
\(341\) 7.78274 0.421459
\(342\) −6.10875 −0.330323
\(343\) −11.6555 −0.629335
\(344\) 1.00000 0.0539164
\(345\) 4.97378 0.267779
\(346\) −17.8221 −0.958124
\(347\) 14.7880 0.793863 0.396931 0.917848i \(-0.370075\pi\)
0.396931 + 0.917848i \(0.370075\pi\)
\(348\) 8.27988 0.443848
\(349\) 11.1748 0.598171 0.299086 0.954226i \(-0.403318\pi\)
0.299086 + 0.954226i \(0.403318\pi\)
\(350\) 0.881450 0.0471155
\(351\) 13.1948 0.704284
\(352\) 1.00000 0.0533002
\(353\) −32.8622 −1.74908 −0.874539 0.484955i \(-0.838836\pi\)
−0.874539 + 0.484955i \(0.838836\pi\)
\(354\) −12.6134 −0.670392
\(355\) −4.25244 −0.225696
\(356\) 14.5173 0.769414
\(357\) 15.2298 0.806046
\(358\) 4.37125 0.231028
\(359\) 18.5947 0.981391 0.490696 0.871331i \(-0.336743\pi\)
0.490696 + 0.871331i \(0.336743\pi\)
\(360\) −3.83093 −0.201908
\(361\) −16.4573 −0.866174
\(362\) 10.6388 0.559161
\(363\) −2.61361 −0.137179
\(364\) −5.35542 −0.280700
\(365\) 6.44173 0.337176
\(366\) 16.4437 0.859528
\(367\) 27.1813 1.41885 0.709425 0.704781i \(-0.248955\pi\)
0.709425 + 0.704781i \(0.248955\pi\)
\(368\) 1.90303 0.0992025
\(369\) 27.4646 1.42975
\(370\) −3.74873 −0.194887
\(371\) 7.79675 0.404787
\(372\) −20.3410 −1.05463
\(373\) −10.9314 −0.566005 −0.283003 0.959119i \(-0.591331\pi\)
−0.283003 + 0.959119i \(0.591331\pi\)
\(374\) −6.61083 −0.341838
\(375\) 2.61361 0.134966
\(376\) −12.3298 −0.635862
\(377\) 19.2477 0.991309
\(378\) −1.91427 −0.0984594
\(379\) −35.8223 −1.84007 −0.920034 0.391838i \(-0.871839\pi\)
−0.920034 + 0.391838i \(0.871839\pi\)
\(380\) 1.59458 0.0818005
\(381\) −5.29484 −0.271263
\(382\) 19.9082 1.01859
\(383\) −11.8569 −0.605859 −0.302929 0.953013i \(-0.597965\pi\)
−0.302929 + 0.953013i \(0.597965\pi\)
\(384\) −2.61361 −0.133375
\(385\) −0.881450 −0.0449228
\(386\) 10.7455 0.546933
\(387\) 3.83093 0.194737
\(388\) 3.95643 0.200857
\(389\) 2.65249 0.134486 0.0672432 0.997737i \(-0.478580\pi\)
0.0672432 + 0.997737i \(0.478580\pi\)
\(390\) −15.8795 −0.804088
\(391\) −12.5806 −0.636231
\(392\) −6.22305 −0.314311
\(393\) 32.8307 1.65609
\(394\) −6.53034 −0.328994
\(395\) −5.91937 −0.297836
\(396\) 3.83093 0.192512
\(397\) 23.4796 1.17841 0.589205 0.807984i \(-0.299441\pi\)
0.589205 + 0.807984i \(0.299441\pi\)
\(398\) 18.5722 0.930939
\(399\) 3.67354 0.183907
\(400\) 1.00000 0.0500000
\(401\) 14.8649 0.742319 0.371159 0.928569i \(-0.378960\pi\)
0.371159 + 0.928569i \(0.378960\pi\)
\(402\) 20.3441 1.01467
\(403\) −47.2855 −2.35546
\(404\) −9.97999 −0.496523
\(405\) 5.81675 0.289037
\(406\) −2.79243 −0.138586
\(407\) 3.74873 0.185818
\(408\) 17.2781 0.855394
\(409\) 13.3349 0.659370 0.329685 0.944091i \(-0.393057\pi\)
0.329685 + 0.944091i \(0.393057\pi\)
\(410\) −7.16918 −0.354061
\(411\) −49.7137 −2.45220
\(412\) 0.780011 0.0384284
\(413\) 4.25391 0.209321
\(414\) 7.29040 0.358303
\(415\) 8.39798 0.412240
\(416\) −6.07569 −0.297885
\(417\) −29.0784 −1.42398
\(418\) −1.59458 −0.0779937
\(419\) 13.4409 0.656633 0.328316 0.944568i \(-0.393519\pi\)
0.328316 + 0.944568i \(0.393519\pi\)
\(420\) 2.30376 0.112412
\(421\) 9.78705 0.476992 0.238496 0.971144i \(-0.423346\pi\)
0.238496 + 0.971144i \(0.423346\pi\)
\(422\) −23.6147 −1.14955
\(423\) −47.2347 −2.29663
\(424\) 8.84537 0.429569
\(425\) −6.61083 −0.320673
\(426\) −11.1142 −0.538484
\(427\) −5.54572 −0.268376
\(428\) 13.7654 0.665375
\(429\) 15.8795 0.766667
\(430\) −1.00000 −0.0482243
\(431\) −39.2910 −1.89258 −0.946291 0.323316i \(-0.895202\pi\)
−0.946291 + 0.323316i \(0.895202\pi\)
\(432\) −2.17173 −0.104487
\(433\) 34.9697 1.68054 0.840268 0.542172i \(-0.182398\pi\)
0.840268 + 0.542172i \(0.182398\pi\)
\(434\) 6.86009 0.329295
\(435\) −8.27988 −0.396990
\(436\) 15.6196 0.748042
\(437\) −3.03455 −0.145162
\(438\) 16.8362 0.804462
\(439\) −1.14756 −0.0547702 −0.0273851 0.999625i \(-0.508718\pi\)
−0.0273851 + 0.999625i \(0.508718\pi\)
\(440\) −1.00000 −0.0476731
\(441\) −23.8401 −1.13524
\(442\) 40.1654 1.91047
\(443\) 2.40564 0.114296 0.0571478 0.998366i \(-0.481799\pi\)
0.0571478 + 0.998366i \(0.481799\pi\)
\(444\) −9.79771 −0.464979
\(445\) −14.5173 −0.688184
\(446\) −11.8260 −0.559979
\(447\) 26.1107 1.23499
\(448\) 0.881450 0.0416446
\(449\) 15.0760 0.711479 0.355740 0.934585i \(-0.384229\pi\)
0.355740 + 0.934585i \(0.384229\pi\)
\(450\) 3.83093 0.180592
\(451\) 7.16918 0.337584
\(452\) −6.48394 −0.304979
\(453\) 3.02845 0.142289
\(454\) 20.5501 0.964466
\(455\) 5.35542 0.251066
\(456\) 4.16762 0.195166
\(457\) 8.86295 0.414591 0.207296 0.978278i \(-0.433534\pi\)
0.207296 + 0.978278i \(0.433534\pi\)
\(458\) 21.7488 1.01625
\(459\) 14.3569 0.670125
\(460\) −1.90303 −0.0887295
\(461\) −30.0315 −1.39870 −0.699352 0.714777i \(-0.746528\pi\)
−0.699352 + 0.714777i \(0.746528\pi\)
\(462\) −2.30376 −0.107181
\(463\) 5.52756 0.256887 0.128444 0.991717i \(-0.459002\pi\)
0.128444 + 0.991717i \(0.459002\pi\)
\(464\) −3.16799 −0.147070
\(465\) 20.3410 0.943292
\(466\) 7.30511 0.338403
\(467\) −36.2153 −1.67585 −0.837923 0.545789i \(-0.816230\pi\)
−0.837923 + 0.545789i \(0.816230\pi\)
\(468\) −23.2756 −1.07591
\(469\) −6.86114 −0.316818
\(470\) 12.3298 0.568732
\(471\) −64.8448 −2.98789
\(472\) 4.82604 0.222136
\(473\) 1.00000 0.0459800
\(474\) −15.4709 −0.710602
\(475\) −1.59458 −0.0731646
\(476\) −5.82712 −0.267086
\(477\) 33.8860 1.55153
\(478\) −15.1275 −0.691917
\(479\) −37.7653 −1.72554 −0.862770 0.505597i \(-0.831272\pi\)
−0.862770 + 0.505597i \(0.831272\pi\)
\(480\) 2.61361 0.119294
\(481\) −22.7761 −1.03850
\(482\) 23.2924 1.06094
\(483\) −4.38414 −0.199485
\(484\) 1.00000 0.0454545
\(485\) −3.95643 −0.179652
\(486\) 21.7179 0.985143
\(487\) −25.7226 −1.16560 −0.582800 0.812615i \(-0.698043\pi\)
−0.582800 + 0.812615i \(0.698043\pi\)
\(488\) −6.29159 −0.284807
\(489\) −32.7353 −1.48034
\(490\) 6.22305 0.281129
\(491\) −1.78560 −0.0805828 −0.0402914 0.999188i \(-0.512829\pi\)
−0.0402914 + 0.999188i \(0.512829\pi\)
\(492\) −18.7374 −0.844748
\(493\) 20.9431 0.943228
\(494\) 9.68820 0.435893
\(495\) −3.83093 −0.172188
\(496\) 7.78274 0.349455
\(497\) 3.74831 0.168135
\(498\) 21.9490 0.983558
\(499\) −35.4149 −1.58539 −0.792695 0.609618i \(-0.791323\pi\)
−0.792695 + 0.609618i \(0.791323\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −62.7660 −2.80418
\(502\) −6.47298 −0.288903
\(503\) 22.0363 0.982550 0.491275 0.871005i \(-0.336531\pi\)
0.491275 + 0.871005i \(0.336531\pi\)
\(504\) 3.37677 0.150413
\(505\) 9.97999 0.444104
\(506\) 1.90303 0.0846002
\(507\) −62.5018 −2.77580
\(508\) 2.02587 0.0898837
\(509\) −4.04984 −0.179506 −0.0897529 0.995964i \(-0.528608\pi\)
−0.0897529 + 0.995964i \(0.528608\pi\)
\(510\) −17.2781 −0.765088
\(511\) −5.67807 −0.251183
\(512\) 1.00000 0.0441942
\(513\) 3.46301 0.152895
\(514\) 19.8928 0.877435
\(515\) −0.780011 −0.0343714
\(516\) −2.61361 −0.115058
\(517\) −12.3298 −0.542265
\(518\) 3.30432 0.145183
\(519\) 46.5801 2.04464
\(520\) 6.07569 0.266437
\(521\) −10.0001 −0.438111 −0.219056 0.975712i \(-0.570298\pi\)
−0.219056 + 0.975712i \(0.570298\pi\)
\(522\) −12.1364 −0.531194
\(523\) 6.76617 0.295864 0.147932 0.988998i \(-0.452738\pi\)
0.147932 + 0.988998i \(0.452738\pi\)
\(524\) −12.5615 −0.548751
\(525\) −2.30376 −0.100544
\(526\) −23.5082 −1.02501
\(527\) −51.4504 −2.24121
\(528\) −2.61361 −0.113743
\(529\) −19.3785 −0.842542
\(530\) −8.84537 −0.384218
\(531\) 18.4882 0.802320
\(532\) −1.40555 −0.0609382
\(533\) −43.5577 −1.88669
\(534\) −37.9424 −1.64193
\(535\) −13.7654 −0.595130
\(536\) −7.78393 −0.336214
\(537\) −11.4247 −0.493014
\(538\) −0.123729 −0.00533432
\(539\) −6.22305 −0.268046
\(540\) 2.17173 0.0934563
\(541\) −17.8195 −0.766122 −0.383061 0.923723i \(-0.625130\pi\)
−0.383061 + 0.923723i \(0.625130\pi\)
\(542\) −0.206038 −0.00885011
\(543\) −27.8055 −1.19325
\(544\) −6.61083 −0.283437
\(545\) −15.6196 −0.669069
\(546\) 13.9969 0.599014
\(547\) 19.5478 0.835802 0.417901 0.908493i \(-0.362766\pi\)
0.417901 + 0.908493i \(0.362766\pi\)
\(548\) 19.0211 0.812542
\(549\) −24.1027 −1.02868
\(550\) 1.00000 0.0426401
\(551\) 5.05163 0.215207
\(552\) −4.97378 −0.211698
\(553\) 5.21763 0.221876
\(554\) −15.6731 −0.665885
\(555\) 9.79771 0.415889
\(556\) 11.1258 0.471839
\(557\) 42.8182 1.81426 0.907132 0.420846i \(-0.138267\pi\)
0.907132 + 0.420846i \(0.138267\pi\)
\(558\) 29.8151 1.26218
\(559\) −6.07569 −0.256974
\(560\) −0.881450 −0.0372481
\(561\) 17.2781 0.729482
\(562\) 29.8787 1.26036
\(563\) −42.3419 −1.78450 −0.892250 0.451543i \(-0.850874\pi\)
−0.892250 + 0.451543i \(0.850874\pi\)
\(564\) 32.2253 1.35693
\(565\) 6.48394 0.272782
\(566\) −9.07865 −0.381604
\(567\) −5.12718 −0.215321
\(568\) 4.25244 0.178428
\(569\) 15.7452 0.660072 0.330036 0.943968i \(-0.392939\pi\)
0.330036 + 0.943968i \(0.392939\pi\)
\(570\) −4.16762 −0.174562
\(571\) 7.66332 0.320700 0.160350 0.987060i \(-0.448738\pi\)
0.160350 + 0.987060i \(0.448738\pi\)
\(572\) −6.07569 −0.254037
\(573\) −52.0321 −2.17367
\(574\) 6.31927 0.263761
\(575\) 1.90303 0.0793620
\(576\) 3.83093 0.159622
\(577\) −42.0284 −1.74967 −0.874833 0.484424i \(-0.839029\pi\)
−0.874833 + 0.484424i \(0.839029\pi\)
\(578\) 26.7031 1.11070
\(579\) −28.0846 −1.16715
\(580\) 3.16799 0.131544
\(581\) −7.40239 −0.307103
\(582\) −10.3406 −0.428630
\(583\) 8.84537 0.366338
\(584\) −6.44173 −0.266561
\(585\) 23.2756 0.962326
\(586\) −12.6194 −0.521302
\(587\) 1.15858 0.0478196 0.0239098 0.999714i \(-0.492389\pi\)
0.0239098 + 0.999714i \(0.492389\pi\)
\(588\) 16.2646 0.670740
\(589\) −12.4102 −0.511355
\(590\) −4.82604 −0.198685
\(591\) 17.0677 0.702072
\(592\) 3.74873 0.154072
\(593\) −4.13863 −0.169953 −0.0849766 0.996383i \(-0.527082\pi\)
−0.0849766 + 0.996383i \(0.527082\pi\)
\(594\) −2.17173 −0.0891071
\(595\) 5.82712 0.238889
\(596\) −9.99029 −0.409218
\(597\) −48.5403 −1.98662
\(598\) −11.5623 −0.472816
\(599\) 20.5290 0.838794 0.419397 0.907803i \(-0.362242\pi\)
0.419397 + 0.907803i \(0.362242\pi\)
\(600\) −2.61361 −0.106700
\(601\) 19.7931 0.807375 0.403688 0.914897i \(-0.367728\pi\)
0.403688 + 0.914897i \(0.367728\pi\)
\(602\) 0.881450 0.0359252
\(603\) −29.8197 −1.21435
\(604\) −1.15872 −0.0471478
\(605\) −1.00000 −0.0406558
\(606\) 26.0838 1.05958
\(607\) −20.7049 −0.840384 −0.420192 0.907435i \(-0.638037\pi\)
−0.420192 + 0.907435i \(0.638037\pi\)
\(608\) −1.59458 −0.0646690
\(609\) 7.29830 0.295742
\(610\) 6.29159 0.254739
\(611\) 74.9121 3.03062
\(612\) −25.3257 −1.02373
\(613\) 35.5051 1.43404 0.717019 0.697054i \(-0.245506\pi\)
0.717019 + 0.697054i \(0.245506\pi\)
\(614\) 5.09549 0.205637
\(615\) 18.7374 0.755565
\(616\) 0.881450 0.0355146
\(617\) 34.8611 1.40345 0.701727 0.712446i \(-0.252413\pi\)
0.701727 + 0.712446i \(0.252413\pi\)
\(618\) −2.03864 −0.0820062
\(619\) 26.5928 1.06886 0.534428 0.845214i \(-0.320527\pi\)
0.534428 + 0.845214i \(0.320527\pi\)
\(620\) −7.78274 −0.312562
\(621\) −4.13288 −0.165847
\(622\) 26.9660 1.08124
\(623\) 12.7962 0.512671
\(624\) 15.8795 0.635687
\(625\) 1.00000 0.0400000
\(626\) −3.49719 −0.139776
\(627\) 4.16762 0.166439
\(628\) 24.8105 0.990046
\(629\) −24.7822 −0.988133
\(630\) −3.37677 −0.134534
\(631\) 8.75865 0.348676 0.174338 0.984686i \(-0.444221\pi\)
0.174338 + 0.984686i \(0.444221\pi\)
\(632\) 5.91937 0.235460
\(633\) 61.7196 2.45313
\(634\) 2.46634 0.0979509
\(635\) −2.02587 −0.0803944
\(636\) −23.1183 −0.916701
\(637\) 37.8093 1.49806
\(638\) −3.16799 −0.125422
\(639\) 16.2908 0.644454
\(640\) −1.00000 −0.0395285
\(641\) 18.4370 0.728217 0.364108 0.931357i \(-0.381374\pi\)
0.364108 + 0.931357i \(0.381374\pi\)
\(642\) −35.9773 −1.41991
\(643\) 20.2150 0.797204 0.398602 0.917124i \(-0.369496\pi\)
0.398602 + 0.917124i \(0.369496\pi\)
\(644\) 1.67743 0.0661000
\(645\) 2.61361 0.102911
\(646\) 10.5415 0.414751
\(647\) −39.5194 −1.55367 −0.776834 0.629706i \(-0.783176\pi\)
−0.776834 + 0.629706i \(0.783176\pi\)
\(648\) −5.81675 −0.228504
\(649\) 4.82604 0.189439
\(650\) −6.07569 −0.238308
\(651\) −17.9296 −0.702716
\(652\) 12.5250 0.490515
\(653\) −5.61823 −0.219858 −0.109929 0.993939i \(-0.535062\pi\)
−0.109929 + 0.993939i \(0.535062\pi\)
\(654\) −40.8234 −1.59632
\(655\) 12.5615 0.490817
\(656\) 7.16918 0.279909
\(657\) −24.6778 −0.962775
\(658\) −10.8681 −0.423683
\(659\) −20.6197 −0.803228 −0.401614 0.915809i \(-0.631551\pi\)
−0.401614 + 0.915809i \(0.631551\pi\)
\(660\) 2.61361 0.101734
\(661\) −9.71150 −0.377734 −0.188867 0.982003i \(-0.560481\pi\)
−0.188867 + 0.982003i \(0.560481\pi\)
\(662\) −9.52802 −0.370317
\(663\) −104.976 −4.07695
\(664\) −8.39798 −0.325905
\(665\) 1.40555 0.0545048
\(666\) 14.3611 0.556483
\(667\) −6.02880 −0.233436
\(668\) 24.0151 0.929173
\(669\) 30.9086 1.19499
\(670\) 7.78393 0.300719
\(671\) −6.29159 −0.242884
\(672\) −2.30376 −0.0888695
\(673\) 34.2384 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(674\) 12.4590 0.479903
\(675\) −2.17173 −0.0835899
\(676\) 23.9140 0.919770
\(677\) −25.2026 −0.968615 −0.484308 0.874898i \(-0.660928\pi\)
−0.484308 + 0.874898i \(0.660928\pi\)
\(678\) 16.9465 0.650825
\(679\) 3.48740 0.133834
\(680\) 6.61083 0.253514
\(681\) −53.7100 −2.05817
\(682\) 7.78274 0.298016
\(683\) −30.4342 −1.16453 −0.582267 0.812998i \(-0.697834\pi\)
−0.582267 + 0.812998i \(0.697834\pi\)
\(684\) −6.10875 −0.233574
\(685\) −19.0211 −0.726760
\(686\) −11.6555 −0.445007
\(687\) −56.8427 −2.16868
\(688\) 1.00000 0.0381246
\(689\) −53.7418 −2.04740
\(690\) 4.97378 0.189349
\(691\) −38.7300 −1.47336 −0.736679 0.676242i \(-0.763607\pi\)
−0.736679 + 0.676242i \(0.763607\pi\)
\(692\) −17.8221 −0.677496
\(693\) 3.37677 0.128273
\(694\) 14.7880 0.561346
\(695\) −11.1258 −0.422025
\(696\) 8.27988 0.313848
\(697\) −47.3943 −1.79519
\(698\) 11.1748 0.422971
\(699\) −19.0927 −0.722152
\(700\) 0.881450 0.0333157
\(701\) −7.01983 −0.265135 −0.132568 0.991174i \(-0.542322\pi\)
−0.132568 + 0.991174i \(0.542322\pi\)
\(702\) 13.1948 0.498004
\(703\) −5.97767 −0.225452
\(704\) 1.00000 0.0376889
\(705\) −32.2253 −1.21367
\(706\) −32.8622 −1.23679
\(707\) −8.79686 −0.330840
\(708\) −12.6134 −0.474039
\(709\) 24.1643 0.907508 0.453754 0.891127i \(-0.350084\pi\)
0.453754 + 0.891127i \(0.350084\pi\)
\(710\) −4.25244 −0.159591
\(711\) 22.6767 0.850443
\(712\) 14.5173 0.544058
\(713\) 14.8108 0.554670
\(714\) 15.2298 0.569960
\(715\) 6.07569 0.227218
\(716\) 4.37125 0.163361
\(717\) 39.5374 1.47655
\(718\) 18.5947 0.693948
\(719\) 47.7856 1.78210 0.891051 0.453903i \(-0.149969\pi\)
0.891051 + 0.453903i \(0.149969\pi\)
\(720\) −3.83093 −0.142770
\(721\) 0.687541 0.0256054
\(722\) −16.4573 −0.612477
\(723\) −60.8772 −2.26405
\(724\) 10.6388 0.395386
\(725\) −3.16799 −0.117656
\(726\) −2.61361 −0.0970000
\(727\) 19.1015 0.708437 0.354218 0.935163i \(-0.384747\pi\)
0.354218 + 0.935163i \(0.384747\pi\)
\(728\) −5.35542 −0.198485
\(729\) −39.3117 −1.45599
\(730\) 6.44173 0.238419
\(731\) −6.61083 −0.244511
\(732\) 16.4437 0.607778
\(733\) 34.1410 1.26103 0.630514 0.776178i \(-0.282844\pi\)
0.630514 + 0.776178i \(0.282844\pi\)
\(734\) 27.1813 1.00328
\(735\) −16.2646 −0.599928
\(736\) 1.90303 0.0701468
\(737\) −7.78393 −0.286725
\(738\) 27.4646 1.01099
\(739\) −31.2299 −1.14881 −0.574406 0.818571i \(-0.694767\pi\)
−0.574406 + 0.818571i \(0.694767\pi\)
\(740\) −3.74873 −0.137806
\(741\) −25.3211 −0.930195
\(742\) 7.79675 0.286228
\(743\) 17.6911 0.649022 0.324511 0.945882i \(-0.394800\pi\)
0.324511 + 0.945882i \(0.394800\pi\)
\(744\) −20.3410 −0.745738
\(745\) 9.99029 0.366016
\(746\) −10.9314 −0.400226
\(747\) −32.1721 −1.17711
\(748\) −6.61083 −0.241716
\(749\) 12.1335 0.443348
\(750\) 2.61361 0.0954354
\(751\) 20.7215 0.756137 0.378068 0.925778i \(-0.376588\pi\)
0.378068 + 0.925778i \(0.376588\pi\)
\(752\) −12.3298 −0.449622
\(753\) 16.9178 0.616520
\(754\) 19.2477 0.700961
\(755\) 1.15872 0.0421703
\(756\) −1.91427 −0.0696213
\(757\) −39.3806 −1.43131 −0.715657 0.698452i \(-0.753873\pi\)
−0.715657 + 0.698452i \(0.753873\pi\)
\(758\) −35.8223 −1.30113
\(759\) −4.97378 −0.180537
\(760\) 1.59458 0.0578417
\(761\) 25.5078 0.924659 0.462329 0.886708i \(-0.347014\pi\)
0.462329 + 0.886708i \(0.347014\pi\)
\(762\) −5.29484 −0.191812
\(763\) 13.7679 0.498431
\(764\) 19.9082 0.720252
\(765\) 25.3257 0.915651
\(766\) −11.8569 −0.428407
\(767\) −29.3215 −1.05874
\(768\) −2.61361 −0.0943104
\(769\) 45.6956 1.64782 0.823912 0.566718i \(-0.191787\pi\)
0.823912 + 0.566718i \(0.191787\pi\)
\(770\) −0.881450 −0.0317652
\(771\) −51.9920 −1.87245
\(772\) 10.7455 0.386740
\(773\) −9.19292 −0.330646 −0.165323 0.986239i \(-0.552867\pi\)
−0.165323 + 0.986239i \(0.552867\pi\)
\(774\) 3.83093 0.137700
\(775\) 7.78274 0.279564
\(776\) 3.95643 0.142028
\(777\) −8.63619 −0.309821
\(778\) 2.65249 0.0950962
\(779\) −11.4319 −0.409589
\(780\) −15.8795 −0.568576
\(781\) 4.25244 0.152164
\(782\) −12.5806 −0.449883
\(783\) 6.88002 0.245872
\(784\) −6.22305 −0.222252
\(785\) −24.8105 −0.885524
\(786\) 32.8307 1.17103
\(787\) 24.4487 0.871503 0.435752 0.900067i \(-0.356483\pi\)
0.435752 + 0.900067i \(0.356483\pi\)
\(788\) −6.53034 −0.232634
\(789\) 61.4413 2.18737
\(790\) −5.91937 −0.210602
\(791\) −5.71527 −0.203212
\(792\) 3.83093 0.136126
\(793\) 38.2258 1.35744
\(794\) 23.4796 0.833261
\(795\) 23.1183 0.819922
\(796\) 18.5722 0.658273
\(797\) −9.11415 −0.322840 −0.161420 0.986886i \(-0.551607\pi\)
−0.161420 + 0.986886i \(0.551607\pi\)
\(798\) 3.67354 0.130042
\(799\) 81.5104 2.88363
\(800\) 1.00000 0.0353553
\(801\) 55.6147 1.96505
\(802\) 14.8649 0.524899
\(803\) −6.44173 −0.227324
\(804\) 20.3441 0.717482
\(805\) −1.67743 −0.0591216
\(806\) −47.2855 −1.66556
\(807\) 0.323378 0.0113834
\(808\) −9.97999 −0.351095
\(809\) −37.2246 −1.30875 −0.654374 0.756171i \(-0.727068\pi\)
−0.654374 + 0.756171i \(0.727068\pi\)
\(810\) 5.81675 0.204380
\(811\) −15.8260 −0.555725 −0.277862 0.960621i \(-0.589626\pi\)
−0.277862 + 0.960621i \(0.589626\pi\)
\(812\) −2.79243 −0.0979949
\(813\) 0.538503 0.0188861
\(814\) 3.74873 0.131393
\(815\) −12.5250 −0.438730
\(816\) 17.2781 0.604855
\(817\) −1.59458 −0.0557875
\(818\) 13.3349 0.466245
\(819\) −20.5162 −0.716895
\(820\) −7.16918 −0.250359
\(821\) 11.5666 0.403678 0.201839 0.979419i \(-0.435308\pi\)
0.201839 + 0.979419i \(0.435308\pi\)
\(822\) −49.7137 −1.73397
\(823\) −39.0398 −1.36084 −0.680422 0.732821i \(-0.738203\pi\)
−0.680422 + 0.732821i \(0.738203\pi\)
\(824\) 0.780011 0.0271730
\(825\) −2.61361 −0.0909941
\(826\) 4.25391 0.148012
\(827\) 23.9556 0.833017 0.416508 0.909132i \(-0.363254\pi\)
0.416508 + 0.909132i \(0.363254\pi\)
\(828\) 7.29040 0.253359
\(829\) 19.1803 0.666158 0.333079 0.942899i \(-0.391912\pi\)
0.333079 + 0.942899i \(0.391912\pi\)
\(830\) 8.39798 0.291498
\(831\) 40.9632 1.42100
\(832\) −6.07569 −0.210637
\(833\) 41.1395 1.42540
\(834\) −29.0784 −1.00690
\(835\) −24.0151 −0.831078
\(836\) −1.59458 −0.0551499
\(837\) −16.9020 −0.584219
\(838\) 13.4409 0.464310
\(839\) 42.9410 1.48249 0.741244 0.671236i \(-0.234236\pi\)
0.741244 + 0.671236i \(0.234236\pi\)
\(840\) 2.30376 0.0794873
\(841\) −18.9638 −0.653925
\(842\) 9.78705 0.337284
\(843\) −78.0912 −2.68960
\(844\) −23.6147 −0.812853
\(845\) −23.9140 −0.822667
\(846\) −47.2347 −1.62396
\(847\) 0.881450 0.0302870
\(848\) 8.84537 0.303751
\(849\) 23.7280 0.814343
\(850\) −6.61083 −0.226750
\(851\) 7.13397 0.244549
\(852\) −11.1142 −0.380766
\(853\) −30.1347 −1.03179 −0.515896 0.856651i \(-0.672541\pi\)
−0.515896 + 0.856651i \(0.672541\pi\)
\(854\) −5.54572 −0.189771
\(855\) 6.10875 0.208915
\(856\) 13.7654 0.470491
\(857\) −9.14189 −0.312281 −0.156140 0.987735i \(-0.549905\pi\)
−0.156140 + 0.987735i \(0.549905\pi\)
\(858\) 15.8795 0.542116
\(859\) 35.2103 1.20136 0.600680 0.799490i \(-0.294897\pi\)
0.600680 + 0.799490i \(0.294897\pi\)
\(860\) −1.00000 −0.0340997
\(861\) −16.5161 −0.562867
\(862\) −39.2910 −1.33826
\(863\) −18.9005 −0.643380 −0.321690 0.946845i \(-0.604251\pi\)
−0.321690 + 0.946845i \(0.604251\pi\)
\(864\) −2.17173 −0.0738837
\(865\) 17.8221 0.605971
\(866\) 34.9697 1.18832
\(867\) −69.7914 −2.37024
\(868\) 6.86009 0.232847
\(869\) 5.91937 0.200801
\(870\) −8.27988 −0.280714
\(871\) 47.2927 1.60245
\(872\) 15.6196 0.528946
\(873\) 15.1568 0.512981
\(874\) −3.03455 −0.102645
\(875\) −0.881450 −0.0297984
\(876\) 16.8362 0.568841
\(877\) 23.3674 0.789062 0.394531 0.918883i \(-0.370907\pi\)
0.394531 + 0.918883i \(0.370907\pi\)
\(878\) −1.14756 −0.0387284
\(879\) 32.9821 1.11246
\(880\) −1.00000 −0.0337100
\(881\) 55.5887 1.87283 0.936415 0.350895i \(-0.114122\pi\)
0.936415 + 0.350895i \(0.114122\pi\)
\(882\) −23.8401 −0.802737
\(883\) −19.3782 −0.652130 −0.326065 0.945347i \(-0.605723\pi\)
−0.326065 + 0.945347i \(0.605723\pi\)
\(884\) 40.1654 1.35091
\(885\) 12.6134 0.423993
\(886\) 2.40564 0.0808192
\(887\) −20.8706 −0.700765 −0.350382 0.936607i \(-0.613948\pi\)
−0.350382 + 0.936607i \(0.613948\pi\)
\(888\) −9.79771 −0.328789
\(889\) 1.78571 0.0598907
\(890\) −14.5173 −0.486620
\(891\) −5.81675 −0.194869
\(892\) −11.8260 −0.395965
\(893\) 19.6609 0.657928
\(894\) 26.1107 0.873271
\(895\) −4.37125 −0.146115
\(896\) 0.881450 0.0294472
\(897\) 30.2192 1.00899
\(898\) 15.0760 0.503092
\(899\) −24.6556 −0.822312
\(900\) 3.83093 0.127698
\(901\) −58.4753 −1.94809
\(902\) 7.16918 0.238708
\(903\) −2.30376 −0.0766644
\(904\) −6.48394 −0.215653
\(905\) −10.6388 −0.353644
\(906\) 3.02845 0.100613
\(907\) 34.0028 1.12904 0.564522 0.825418i \(-0.309060\pi\)
0.564522 + 0.825418i \(0.309060\pi\)
\(908\) 20.5501 0.681981
\(909\) −38.2327 −1.26810
\(910\) 5.35542 0.177530
\(911\) 10.8221 0.358552 0.179276 0.983799i \(-0.442624\pi\)
0.179276 + 0.983799i \(0.442624\pi\)
\(912\) 4.16762 0.138004
\(913\) −8.39798 −0.277932
\(914\) 8.86295 0.293160
\(915\) −16.4437 −0.543613
\(916\) 21.7488 0.718600
\(917\) −11.0723 −0.365640
\(918\) 14.3569 0.473850
\(919\) 35.9048 1.18439 0.592194 0.805795i \(-0.298262\pi\)
0.592194 + 0.805795i \(0.298262\pi\)
\(920\) −1.90303 −0.0627412
\(921\) −13.3176 −0.438830
\(922\) −30.0315 −0.989033
\(923\) −25.8365 −0.850419
\(924\) −2.30376 −0.0757882
\(925\) 3.74873 0.123258
\(926\) 5.52756 0.181647
\(927\) 2.98817 0.0981444
\(928\) −3.16799 −0.103994
\(929\) −23.9691 −0.786401 −0.393200 0.919453i \(-0.628632\pi\)
−0.393200 + 0.919453i \(0.628632\pi\)
\(930\) 20.3410 0.667008
\(931\) 9.92317 0.325219
\(932\) 7.30511 0.239287
\(933\) −70.4784 −2.30736
\(934\) −36.2153 −1.18500
\(935\) 6.61083 0.216197
\(936\) −23.2756 −0.760786
\(937\) −38.1709 −1.24699 −0.623494 0.781828i \(-0.714287\pi\)
−0.623494 + 0.781828i \(0.714287\pi\)
\(938\) −6.86114 −0.224024
\(939\) 9.14029 0.298282
\(940\) 12.3298 0.402154
\(941\) 33.3013 1.08559 0.542795 0.839865i \(-0.317366\pi\)
0.542795 + 0.839865i \(0.317366\pi\)
\(942\) −64.8448 −2.11276
\(943\) 13.6432 0.444284
\(944\) 4.82604 0.157074
\(945\) 1.91427 0.0622712
\(946\) 1.00000 0.0325128
\(947\) −27.6796 −0.899467 −0.449733 0.893163i \(-0.648481\pi\)
−0.449733 + 0.893163i \(0.648481\pi\)
\(948\) −15.4709 −0.502472
\(949\) 39.1380 1.27047
\(950\) −1.59458 −0.0517352
\(951\) −6.44604 −0.209027
\(952\) −5.82712 −0.188858
\(953\) −26.6247 −0.862459 −0.431230 0.902242i \(-0.641920\pi\)
−0.431230 + 0.902242i \(0.641920\pi\)
\(954\) 33.8860 1.09710
\(955\) −19.9082 −0.644213
\(956\) −15.1275 −0.489259
\(957\) 8.27988 0.267650
\(958\) −37.7653 −1.22014
\(959\) 16.7662 0.541408
\(960\) 2.61361 0.0843537
\(961\) 29.5710 0.953904
\(962\) −22.7761 −0.734332
\(963\) 52.7343 1.69934
\(964\) 23.2924 0.750198
\(965\) −10.7455 −0.345911
\(966\) −4.38414 −0.141057
\(967\) 38.3834 1.23433 0.617164 0.786835i \(-0.288282\pi\)
0.617164 + 0.786835i \(0.288282\pi\)
\(968\) 1.00000 0.0321412
\(969\) −27.5514 −0.885079
\(970\) −3.95643 −0.127033
\(971\) −4.13789 −0.132791 −0.0663955 0.997793i \(-0.521150\pi\)
−0.0663955 + 0.997793i \(0.521150\pi\)
\(972\) 21.7179 0.696602
\(973\) 9.80683 0.314392
\(974\) −25.7226 −0.824204
\(975\) 15.8795 0.508550
\(976\) −6.29159 −0.201389
\(977\) 21.1621 0.677034 0.338517 0.940960i \(-0.390075\pi\)
0.338517 + 0.940960i \(0.390075\pi\)
\(978\) −32.7353 −1.04676
\(979\) 14.5173 0.463974
\(980\) 6.22305 0.198788
\(981\) 59.8376 1.91047
\(982\) −1.78560 −0.0569807
\(983\) −26.0831 −0.831923 −0.415961 0.909382i \(-0.636555\pi\)
−0.415961 + 0.909382i \(0.636555\pi\)
\(984\) −18.7374 −0.597327
\(985\) 6.53034 0.208074
\(986\) 20.9431 0.666963
\(987\) 28.4050 0.904140
\(988\) 9.68820 0.308223
\(989\) 1.90303 0.0605130
\(990\) −3.83093 −0.121755
\(991\) 25.7153 0.816874 0.408437 0.912786i \(-0.366074\pi\)
0.408437 + 0.912786i \(0.366074\pi\)
\(992\) 7.78274 0.247102
\(993\) 24.9025 0.790257
\(994\) 3.74831 0.118889
\(995\) −18.5722 −0.588777
\(996\) 21.9490 0.695480
\(997\) −8.66972 −0.274573 −0.137286 0.990531i \(-0.543838\pi\)
−0.137286 + 0.990531i \(0.543838\pi\)
\(998\) −35.4149 −1.12104
\(999\) −8.14123 −0.257577
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.1.3 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.bf.1.3 13 1.1 even 1 trivial