Properties

Label 370.2.a.e.1.2
Level $370$
Weight $2$
Character 370.1
Self dual yes
Analytic conductor $2.954$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.95446487479\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 370.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.732051 q^{6} -4.73205 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.732051 q^{3} +1.00000 q^{4} +1.00000 q^{5} -0.732051 q^{6} -4.73205 q^{7} -1.00000 q^{8} -2.46410 q^{9} -1.00000 q^{10} -5.46410 q^{11} +0.732051 q^{12} -5.46410 q^{13} +4.73205 q^{14} +0.732051 q^{15} +1.00000 q^{16} +5.46410 q^{17} +2.46410 q^{18} +6.19615 q^{19} +1.00000 q^{20} -3.46410 q^{21} +5.46410 q^{22} -8.00000 q^{23} -0.732051 q^{24} +1.00000 q^{25} +5.46410 q^{26} -4.00000 q^{27} -4.73205 q^{28} +4.92820 q^{29} -0.732051 q^{30} +0.732051 q^{31} -1.00000 q^{32} -4.00000 q^{33} -5.46410 q^{34} -4.73205 q^{35} -2.46410 q^{36} +1.00000 q^{37} -6.19615 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +3.46410 q^{42} +6.92820 q^{43} -5.46410 q^{44} -2.46410 q^{45} +8.00000 q^{46} -4.73205 q^{47} +0.732051 q^{48} +15.3923 q^{49} -1.00000 q^{50} +4.00000 q^{51} -5.46410 q^{52} -6.00000 q^{53} +4.00000 q^{54} -5.46410 q^{55} +4.73205 q^{56} +4.53590 q^{57} -4.92820 q^{58} -10.1962 q^{59} +0.732051 q^{60} -4.92820 q^{61} -0.732051 q^{62} +11.6603 q^{63} +1.00000 q^{64} -5.46410 q^{65} +4.00000 q^{66} -3.66025 q^{67} +5.46410 q^{68} -5.85641 q^{69} +4.73205 q^{70} +2.92820 q^{71} +2.46410 q^{72} -0.928203 q^{73} -1.00000 q^{74} +0.732051 q^{75} +6.19615 q^{76} +25.8564 q^{77} +4.00000 q^{78} +8.73205 q^{79} +1.00000 q^{80} +4.46410 q^{81} +2.00000 q^{82} -8.73205 q^{83} -3.46410 q^{84} +5.46410 q^{85} -6.92820 q^{86} +3.60770 q^{87} +5.46410 q^{88} -2.00000 q^{89} +2.46410 q^{90} +25.8564 q^{91} -8.00000 q^{92} +0.535898 q^{93} +4.73205 q^{94} +6.19615 q^{95} -0.732051 q^{96} -2.00000 q^{97} -15.3923 q^{98} +13.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 2 q^{12} - 4 q^{13} + 6 q^{14} - 2 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 2 q^{19} + 2 q^{20} + 4 q^{22} - 16 q^{23} + 2 q^{24} + 2 q^{25} + 4 q^{26} - 8 q^{27} - 6 q^{28} - 4 q^{29} + 2 q^{30} - 2 q^{31} - 2 q^{32} - 8 q^{33} - 4 q^{34} - 6 q^{35} + 2 q^{36} + 2 q^{37} - 2 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} - 4 q^{44} + 2 q^{45} + 16 q^{46} - 6 q^{47} - 2 q^{48} + 10 q^{49} - 2 q^{50} + 8 q^{51} - 4 q^{52} - 12 q^{53} + 8 q^{54} - 4 q^{55} + 6 q^{56} + 16 q^{57} + 4 q^{58} - 10 q^{59} - 2 q^{60} + 4 q^{61} + 2 q^{62} + 6 q^{63} + 2 q^{64} - 4 q^{65} + 8 q^{66} + 10 q^{67} + 4 q^{68} + 16 q^{69} + 6 q^{70} - 8 q^{71} - 2 q^{72} + 12 q^{73} - 2 q^{74} - 2 q^{75} + 2 q^{76} + 24 q^{77} + 8 q^{78} + 14 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 14 q^{83} + 4 q^{85} + 28 q^{87} + 4 q^{88} - 4 q^{89} - 2 q^{90} + 24 q^{91} - 16 q^{92} + 8 q^{93} + 6 q^{94} + 2 q^{95} + 2 q^{96} - 4 q^{97} - 10 q^{98} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −0.732051 −0.298858
\(7\) −4.73205 −1.78855 −0.894274 0.447521i \(-0.852307\pi\)
−0.894274 + 0.447521i \(0.852307\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.46410 −0.821367
\(10\) −1.00000 −0.316228
\(11\) −5.46410 −1.64749 −0.823744 0.566961i \(-0.808119\pi\)
−0.823744 + 0.566961i \(0.808119\pi\)
\(12\) 0.732051 0.211325
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) 4.73205 1.26469
\(15\) 0.732051 0.189015
\(16\) 1.00000 0.250000
\(17\) 5.46410 1.32524 0.662620 0.748956i \(-0.269445\pi\)
0.662620 + 0.748956i \(0.269445\pi\)
\(18\) 2.46410 0.580794
\(19\) 6.19615 1.42149 0.710747 0.703447i \(-0.248357\pi\)
0.710747 + 0.703447i \(0.248357\pi\)
\(20\) 1.00000 0.223607
\(21\) −3.46410 −0.755929
\(22\) 5.46410 1.16495
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −0.732051 −0.149429
\(25\) 1.00000 0.200000
\(26\) 5.46410 1.07160
\(27\) −4.00000 −0.769800
\(28\) −4.73205 −0.894274
\(29\) 4.92820 0.915144 0.457572 0.889172i \(-0.348719\pi\)
0.457572 + 0.889172i \(0.348719\pi\)
\(30\) −0.732051 −0.133654
\(31\) 0.732051 0.131480 0.0657401 0.997837i \(-0.479059\pi\)
0.0657401 + 0.997837i \(0.479059\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) −5.46410 −0.937086
\(35\) −4.73205 −0.799863
\(36\) −2.46410 −0.410684
\(37\) 1.00000 0.164399
\(38\) −6.19615 −1.00515
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 3.46410 0.534522
\(43\) 6.92820 1.05654 0.528271 0.849076i \(-0.322841\pi\)
0.528271 + 0.849076i \(0.322841\pi\)
\(44\) −5.46410 −0.823744
\(45\) −2.46410 −0.367327
\(46\) 8.00000 1.17954
\(47\) −4.73205 −0.690241 −0.345120 0.938558i \(-0.612162\pi\)
−0.345120 + 0.938558i \(0.612162\pi\)
\(48\) 0.732051 0.105662
\(49\) 15.3923 2.19890
\(50\) −1.00000 −0.141421
\(51\) 4.00000 0.560112
\(52\) −5.46410 −0.757735
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 4.00000 0.544331
\(55\) −5.46410 −0.736779
\(56\) 4.73205 0.632347
\(57\) 4.53590 0.600794
\(58\) −4.92820 −0.647105
\(59\) −10.1962 −1.32743 −0.663713 0.747987i \(-0.731020\pi\)
−0.663713 + 0.747987i \(0.731020\pi\)
\(60\) 0.732051 0.0945074
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) −0.732051 −0.0929705
\(63\) 11.6603 1.46905
\(64\) 1.00000 0.125000
\(65\) −5.46410 −0.677738
\(66\) 4.00000 0.492366
\(67\) −3.66025 −0.447171 −0.223586 0.974684i \(-0.571776\pi\)
−0.223586 + 0.974684i \(0.571776\pi\)
\(68\) 5.46410 0.662620
\(69\) −5.85641 −0.705028
\(70\) 4.73205 0.565588
\(71\) 2.92820 0.347514 0.173757 0.984789i \(-0.444409\pi\)
0.173757 + 0.984789i \(0.444409\pi\)
\(72\) 2.46410 0.290397
\(73\) −0.928203 −0.108638 −0.0543190 0.998524i \(-0.517299\pi\)
−0.0543190 + 0.998524i \(0.517299\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0.732051 0.0845299
\(76\) 6.19615 0.710747
\(77\) 25.8564 2.94661
\(78\) 4.00000 0.452911
\(79\) 8.73205 0.982432 0.491216 0.871038i \(-0.336552\pi\)
0.491216 + 0.871038i \(0.336552\pi\)
\(80\) 1.00000 0.111803
\(81\) 4.46410 0.496011
\(82\) 2.00000 0.220863
\(83\) −8.73205 −0.958467 −0.479234 0.877687i \(-0.659085\pi\)
−0.479234 + 0.877687i \(0.659085\pi\)
\(84\) −3.46410 −0.377964
\(85\) 5.46410 0.592665
\(86\) −6.92820 −0.747087
\(87\) 3.60770 0.386786
\(88\) 5.46410 0.582475
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 2.46410 0.259739
\(91\) 25.8564 2.71049
\(92\) −8.00000 −0.834058
\(93\) 0.535898 0.0555701
\(94\) 4.73205 0.488074
\(95\) 6.19615 0.635712
\(96\) −0.732051 −0.0747146
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −15.3923 −1.55486
\(99\) 13.4641 1.35319
\(100\) 1.00000 0.100000
\(101\) −9.46410 −0.941713 −0.470857 0.882210i \(-0.656055\pi\)
−0.470857 + 0.882210i \(0.656055\pi\)
\(102\) −4.00000 −0.396059
\(103\) −6.53590 −0.644001 −0.322001 0.946739i \(-0.604355\pi\)
−0.322001 + 0.946739i \(0.604355\pi\)
\(104\) 5.46410 0.535799
\(105\) −3.46410 −0.338062
\(106\) 6.00000 0.582772
\(107\) 3.26795 0.315925 0.157962 0.987445i \(-0.449508\pi\)
0.157962 + 0.987445i \(0.449508\pi\)
\(108\) −4.00000 −0.384900
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 5.46410 0.520982
\(111\) 0.732051 0.0694832
\(112\) −4.73205 −0.447137
\(113\) −10.5359 −0.991134 −0.495567 0.868570i \(-0.665040\pi\)
−0.495567 + 0.868570i \(0.665040\pi\)
\(114\) −4.53590 −0.424826
\(115\) −8.00000 −0.746004
\(116\) 4.92820 0.457572
\(117\) 13.4641 1.24476
\(118\) 10.1962 0.938632
\(119\) −25.8564 −2.37025
\(120\) −0.732051 −0.0668268
\(121\) 18.8564 1.71422
\(122\) 4.92820 0.446179
\(123\) −1.46410 −0.132014
\(124\) 0.732051 0.0657401
\(125\) 1.00000 0.0894427
\(126\) −11.6603 −1.03878
\(127\) 3.66025 0.324795 0.162398 0.986725i \(-0.448077\pi\)
0.162398 + 0.986725i \(0.448077\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.07180 0.446547
\(130\) 5.46410 0.479233
\(131\) −18.5885 −1.62408 −0.812041 0.583601i \(-0.801643\pi\)
−0.812041 + 0.583601i \(0.801643\pi\)
\(132\) −4.00000 −0.348155
\(133\) −29.3205 −2.54241
\(134\) 3.66025 0.316198
\(135\) −4.00000 −0.344265
\(136\) −5.46410 −0.468543
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 5.85641 0.498530
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) −4.73205 −0.399931
\(141\) −3.46410 −0.291730
\(142\) −2.92820 −0.245729
\(143\) 29.8564 2.49672
\(144\) −2.46410 −0.205342
\(145\) 4.92820 0.409265
\(146\) 0.928203 0.0768186
\(147\) 11.2679 0.929365
\(148\) 1.00000 0.0821995
\(149\) 4.39230 0.359832 0.179916 0.983682i \(-0.442417\pi\)
0.179916 + 0.983682i \(0.442417\pi\)
\(150\) −0.732051 −0.0597717
\(151\) −12.3923 −1.00847 −0.504236 0.863566i \(-0.668226\pi\)
−0.504236 + 0.863566i \(0.668226\pi\)
\(152\) −6.19615 −0.502574
\(153\) −13.4641 −1.08851
\(154\) −25.8564 −2.08357
\(155\) 0.732051 0.0587997
\(156\) −4.00000 −0.320256
\(157\) 3.07180 0.245156 0.122578 0.992459i \(-0.460884\pi\)
0.122578 + 0.992459i \(0.460884\pi\)
\(158\) −8.73205 −0.694685
\(159\) −4.39230 −0.348332
\(160\) −1.00000 −0.0790569
\(161\) 37.8564 2.98350
\(162\) −4.46410 −0.350733
\(163\) −11.3205 −0.886691 −0.443345 0.896351i \(-0.646208\pi\)
−0.443345 + 0.896351i \(0.646208\pi\)
\(164\) −2.00000 −0.156174
\(165\) −4.00000 −0.311400
\(166\) 8.73205 0.677739
\(167\) −1.46410 −0.113296 −0.0566478 0.998394i \(-0.518041\pi\)
−0.0566478 + 0.998394i \(0.518041\pi\)
\(168\) 3.46410 0.267261
\(169\) 16.8564 1.29665
\(170\) −5.46410 −0.419077
\(171\) −15.2679 −1.16757
\(172\) 6.92820 0.528271
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) −3.60770 −0.273499
\(175\) −4.73205 −0.357709
\(176\) −5.46410 −0.411872
\(177\) −7.46410 −0.561036
\(178\) 2.00000 0.149906
\(179\) −0.339746 −0.0253938 −0.0126969 0.999919i \(-0.504042\pi\)
−0.0126969 + 0.999919i \(0.504042\pi\)
\(180\) −2.46410 −0.183663
\(181\) 5.46410 0.406143 0.203072 0.979164i \(-0.434908\pi\)
0.203072 + 0.979164i \(0.434908\pi\)
\(182\) −25.8564 −1.91660
\(183\) −3.60770 −0.266688
\(184\) 8.00000 0.589768
\(185\) 1.00000 0.0735215
\(186\) −0.535898 −0.0392940
\(187\) −29.8564 −2.18332
\(188\) −4.73205 −0.345120
\(189\) 18.9282 1.37682
\(190\) −6.19615 −0.449516
\(191\) −8.73205 −0.631829 −0.315915 0.948788i \(-0.602311\pi\)
−0.315915 + 0.948788i \(0.602311\pi\)
\(192\) 0.732051 0.0528312
\(193\) 15.8564 1.14137 0.570685 0.821169i \(-0.306678\pi\)
0.570685 + 0.821169i \(0.306678\pi\)
\(194\) 2.00000 0.143592
\(195\) −4.00000 −0.286446
\(196\) 15.3923 1.09945
\(197\) −22.7846 −1.62334 −0.811668 0.584119i \(-0.801440\pi\)
−0.811668 + 0.584119i \(0.801440\pi\)
\(198\) −13.4641 −0.956852
\(199\) −12.0526 −0.854383 −0.427192 0.904161i \(-0.640497\pi\)
−0.427192 + 0.904161i \(0.640497\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −2.67949 −0.188997
\(202\) 9.46410 0.665892
\(203\) −23.3205 −1.63678
\(204\) 4.00000 0.280056
\(205\) −2.00000 −0.139686
\(206\) 6.53590 0.455378
\(207\) 19.7128 1.37014
\(208\) −5.46410 −0.378867
\(209\) −33.8564 −2.34190
\(210\) 3.46410 0.239046
\(211\) −17.8564 −1.22929 −0.614643 0.788806i \(-0.710700\pi\)
−0.614643 + 0.788806i \(0.710700\pi\)
\(212\) −6.00000 −0.412082
\(213\) 2.14359 0.146877
\(214\) −3.26795 −0.223392
\(215\) 6.92820 0.472500
\(216\) 4.00000 0.272166
\(217\) −3.46410 −0.235159
\(218\) −2.00000 −0.135457
\(219\) −0.679492 −0.0459158
\(220\) −5.46410 −0.368390
\(221\) −29.8564 −2.00836
\(222\) −0.732051 −0.0491320
\(223\) 16.0526 1.07496 0.537479 0.843277i \(-0.319377\pi\)
0.537479 + 0.843277i \(0.319377\pi\)
\(224\) 4.73205 0.316173
\(225\) −2.46410 −0.164273
\(226\) 10.5359 0.700838
\(227\) 24.3923 1.61897 0.809487 0.587138i \(-0.199745\pi\)
0.809487 + 0.587138i \(0.199745\pi\)
\(228\) 4.53590 0.300397
\(229\) −11.8564 −0.783493 −0.391747 0.920073i \(-0.628129\pi\)
−0.391747 + 0.920073i \(0.628129\pi\)
\(230\) 8.00000 0.527504
\(231\) 18.9282 1.24538
\(232\) −4.92820 −0.323552
\(233\) 28.9282 1.89515 0.947575 0.319534i \(-0.103526\pi\)
0.947575 + 0.319534i \(0.103526\pi\)
\(234\) −13.4641 −0.880176
\(235\) −4.73205 −0.308685
\(236\) −10.1962 −0.663713
\(237\) 6.39230 0.415225
\(238\) 25.8564 1.67602
\(239\) −20.7321 −1.34104 −0.670522 0.741889i \(-0.733930\pi\)
−0.670522 + 0.741889i \(0.733930\pi\)
\(240\) 0.732051 0.0472537
\(241\) 4.92820 0.317453 0.158727 0.987323i \(-0.449261\pi\)
0.158727 + 0.987323i \(0.449261\pi\)
\(242\) −18.8564 −1.21214
\(243\) 15.2679 0.979439
\(244\) −4.92820 −0.315496
\(245\) 15.3923 0.983378
\(246\) 1.46410 0.0933477
\(247\) −33.8564 −2.15423
\(248\) −0.732051 −0.0464853
\(249\) −6.39230 −0.405096
\(250\) −1.00000 −0.0632456
\(251\) 19.2679 1.21618 0.608091 0.793867i \(-0.291936\pi\)
0.608091 + 0.793867i \(0.291936\pi\)
\(252\) 11.6603 0.734527
\(253\) 43.7128 2.74820
\(254\) −3.66025 −0.229665
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −18.5359 −1.15624 −0.578119 0.815953i \(-0.696213\pi\)
−0.578119 + 0.815953i \(0.696213\pi\)
\(258\) −5.07180 −0.315756
\(259\) −4.73205 −0.294035
\(260\) −5.46410 −0.338869
\(261\) −12.1436 −0.751670
\(262\) 18.5885 1.14840
\(263\) −8.05256 −0.496542 −0.248271 0.968691i \(-0.579862\pi\)
−0.248271 + 0.968691i \(0.579862\pi\)
\(264\) 4.00000 0.246183
\(265\) −6.00000 −0.368577
\(266\) 29.3205 1.79776
\(267\) −1.46410 −0.0896016
\(268\) −3.66025 −0.223586
\(269\) 20.3923 1.24334 0.621670 0.783279i \(-0.286454\pi\)
0.621670 + 0.783279i \(0.286454\pi\)
\(270\) 4.00000 0.243432
\(271\) −24.7846 −1.50556 −0.752779 0.658273i \(-0.771287\pi\)
−0.752779 + 0.658273i \(0.771287\pi\)
\(272\) 5.46410 0.331310
\(273\) 18.9282 1.14559
\(274\) −2.00000 −0.120824
\(275\) −5.46410 −0.329498
\(276\) −5.85641 −0.352514
\(277\) 22.2487 1.33680 0.668398 0.743804i \(-0.266980\pi\)
0.668398 + 0.743804i \(0.266980\pi\)
\(278\) −6.92820 −0.415526
\(279\) −1.80385 −0.107994
\(280\) 4.73205 0.282794
\(281\) −8.92820 −0.532612 −0.266306 0.963889i \(-0.585803\pi\)
−0.266306 + 0.963889i \(0.585803\pi\)
\(282\) 3.46410 0.206284
\(283\) 16.3923 0.974421 0.487211 0.873284i \(-0.338014\pi\)
0.487211 + 0.873284i \(0.338014\pi\)
\(284\) 2.92820 0.173757
\(285\) 4.53590 0.268683
\(286\) −29.8564 −1.76545
\(287\) 9.46410 0.558648
\(288\) 2.46410 0.145199
\(289\) 12.8564 0.756259
\(290\) −4.92820 −0.289394
\(291\) −1.46410 −0.0858272
\(292\) −0.928203 −0.0543190
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −11.2679 −0.657160
\(295\) −10.1962 −0.593643
\(296\) −1.00000 −0.0581238
\(297\) 21.8564 1.26824
\(298\) −4.39230 −0.254439
\(299\) 43.7128 2.52798
\(300\) 0.732051 0.0422650
\(301\) −32.7846 −1.88967
\(302\) 12.3923 0.713097
\(303\) −6.92820 −0.398015
\(304\) 6.19615 0.355374
\(305\) −4.92820 −0.282188
\(306\) 13.4641 0.769691
\(307\) 18.5885 1.06090 0.530450 0.847716i \(-0.322023\pi\)
0.530450 + 0.847716i \(0.322023\pi\)
\(308\) 25.8564 1.47331
\(309\) −4.78461 −0.272187
\(310\) −0.732051 −0.0415777
\(311\) 2.87564 0.163063 0.0815314 0.996671i \(-0.474019\pi\)
0.0815314 + 0.996671i \(0.474019\pi\)
\(312\) 4.00000 0.226455
\(313\) −23.8564 −1.34844 −0.674222 0.738529i \(-0.735521\pi\)
−0.674222 + 0.738529i \(0.735521\pi\)
\(314\) −3.07180 −0.173352
\(315\) 11.6603 0.656981
\(316\) 8.73205 0.491216
\(317\) −4.14359 −0.232727 −0.116364 0.993207i \(-0.537124\pi\)
−0.116364 + 0.993207i \(0.537124\pi\)
\(318\) 4.39230 0.246308
\(319\) −26.9282 −1.50769
\(320\) 1.00000 0.0559017
\(321\) 2.39230 0.133525
\(322\) −37.8564 −2.10966
\(323\) 33.8564 1.88382
\(324\) 4.46410 0.248006
\(325\) −5.46410 −0.303094
\(326\) 11.3205 0.626985
\(327\) 1.46410 0.0809650
\(328\) 2.00000 0.110432
\(329\) 22.3923 1.23453
\(330\) 4.00000 0.220193
\(331\) −33.1244 −1.82068 −0.910340 0.413862i \(-0.864180\pi\)
−0.910340 + 0.413862i \(0.864180\pi\)
\(332\) −8.73205 −0.479234
\(333\) −2.46410 −0.135032
\(334\) 1.46410 0.0801121
\(335\) −3.66025 −0.199981
\(336\) −3.46410 −0.188982
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −16.8564 −0.916868
\(339\) −7.71281 −0.418902
\(340\) 5.46410 0.296333
\(341\) −4.00000 −0.216612
\(342\) 15.2679 0.825596
\(343\) −39.7128 −2.14429
\(344\) −6.92820 −0.373544
\(345\) −5.85641 −0.315298
\(346\) −10.0000 −0.537603
\(347\) 30.9282 1.66031 0.830156 0.557530i \(-0.188251\pi\)
0.830156 + 0.557530i \(0.188251\pi\)
\(348\) 3.60770 0.193393
\(349\) −15.3205 −0.820088 −0.410044 0.912066i \(-0.634487\pi\)
−0.410044 + 0.912066i \(0.634487\pi\)
\(350\) 4.73205 0.252939
\(351\) 21.8564 1.16661
\(352\) 5.46410 0.291238
\(353\) −11.8564 −0.631053 −0.315526 0.948917i \(-0.602181\pi\)
−0.315526 + 0.948917i \(0.602181\pi\)
\(354\) 7.46410 0.396713
\(355\) 2.92820 0.155413
\(356\) −2.00000 −0.106000
\(357\) −18.9282 −1.00179
\(358\) 0.339746 0.0179561
\(359\) −12.3923 −0.654041 −0.327020 0.945017i \(-0.606045\pi\)
−0.327020 + 0.945017i \(0.606045\pi\)
\(360\) 2.46410 0.129870
\(361\) 19.3923 1.02065
\(362\) −5.46410 −0.287187
\(363\) 13.8038 0.724514
\(364\) 25.8564 1.35524
\(365\) −0.928203 −0.0485844
\(366\) 3.60770 0.188577
\(367\) −4.33975 −0.226533 −0.113266 0.993565i \(-0.536131\pi\)
−0.113266 + 0.993565i \(0.536131\pi\)
\(368\) −8.00000 −0.417029
\(369\) 4.92820 0.256552
\(370\) −1.00000 −0.0519875
\(371\) 28.3923 1.47406
\(372\) 0.535898 0.0277850
\(373\) −10.7846 −0.558406 −0.279203 0.960232i \(-0.590070\pi\)
−0.279203 + 0.960232i \(0.590070\pi\)
\(374\) 29.8564 1.54384
\(375\) 0.732051 0.0378029
\(376\) 4.73205 0.244037
\(377\) −26.9282 −1.38687
\(378\) −18.9282 −0.973562
\(379\) 32.3923 1.66388 0.831940 0.554865i \(-0.187230\pi\)
0.831940 + 0.554865i \(0.187230\pi\)
\(380\) 6.19615 0.317856
\(381\) 2.67949 0.137275
\(382\) 8.73205 0.446771
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) −0.732051 −0.0373573
\(385\) 25.8564 1.31776
\(386\) −15.8564 −0.807070
\(387\) −17.0718 −0.867808
\(388\) −2.00000 −0.101535
\(389\) −11.8564 −0.601144 −0.300572 0.953759i \(-0.597178\pi\)
−0.300572 + 0.953759i \(0.597178\pi\)
\(390\) 4.00000 0.202548
\(391\) −43.7128 −2.21065
\(392\) −15.3923 −0.777429
\(393\) −13.6077 −0.686417
\(394\) 22.7846 1.14787
\(395\) 8.73205 0.439357
\(396\) 13.4641 0.676597
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 12.0526 0.604140
\(399\) −21.4641 −1.07455
\(400\) 1.00000 0.0500000
\(401\) −32.9282 −1.64436 −0.822178 0.569230i \(-0.807241\pi\)
−0.822178 + 0.569230i \(0.807241\pi\)
\(402\) 2.67949 0.133641
\(403\) −4.00000 −0.199254
\(404\) −9.46410 −0.470857
\(405\) 4.46410 0.221823
\(406\) 23.3205 1.15738
\(407\) −5.46410 −0.270845
\(408\) −4.00000 −0.198030
\(409\) 3.07180 0.151891 0.0759453 0.997112i \(-0.475803\pi\)
0.0759453 + 0.997112i \(0.475803\pi\)
\(410\) 2.00000 0.0987730
\(411\) 1.46410 0.0722188
\(412\) −6.53590 −0.322001
\(413\) 48.2487 2.37416
\(414\) −19.7128 −0.968832
\(415\) −8.73205 −0.428640
\(416\) 5.46410 0.267900
\(417\) 5.07180 0.248367
\(418\) 33.8564 1.65597
\(419\) 14.2487 0.696095 0.348048 0.937477i \(-0.386845\pi\)
0.348048 + 0.937477i \(0.386845\pi\)
\(420\) −3.46410 −0.169031
\(421\) −0.143594 −0.00699832 −0.00349916 0.999994i \(-0.501114\pi\)
−0.00349916 + 0.999994i \(0.501114\pi\)
\(422\) 17.8564 0.869236
\(423\) 11.6603 0.566941
\(424\) 6.00000 0.291386
\(425\) 5.46410 0.265048
\(426\) −2.14359 −0.103857
\(427\) 23.3205 1.12856
\(428\) 3.26795 0.157962
\(429\) 21.8564 1.05524
\(430\) −6.92820 −0.334108
\(431\) −2.19615 −0.105785 −0.0528925 0.998600i \(-0.516844\pi\)
−0.0528925 + 0.998600i \(0.516844\pi\)
\(432\) −4.00000 −0.192450
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 3.46410 0.166282
\(435\) 3.60770 0.172976
\(436\) 2.00000 0.0957826
\(437\) −49.5692 −2.37122
\(438\) 0.679492 0.0324674
\(439\) 13.5167 0.645115 0.322558 0.946550i \(-0.395457\pi\)
0.322558 + 0.946550i \(0.395457\pi\)
\(440\) 5.46410 0.260491
\(441\) −37.9282 −1.80610
\(442\) 29.8564 1.42012
\(443\) −14.8756 −0.706763 −0.353382 0.935479i \(-0.614968\pi\)
−0.353382 + 0.935479i \(0.614968\pi\)
\(444\) 0.732051 0.0347416
\(445\) −2.00000 −0.0948091
\(446\) −16.0526 −0.760111
\(447\) 3.21539 0.152083
\(448\) −4.73205 −0.223568
\(449\) −21.7128 −1.02469 −0.512345 0.858779i \(-0.671223\pi\)
−0.512345 + 0.858779i \(0.671223\pi\)
\(450\) 2.46410 0.116159
\(451\) 10.9282 0.514589
\(452\) −10.5359 −0.495567
\(453\) −9.07180 −0.426230
\(454\) −24.3923 −1.14479
\(455\) 25.8564 1.21217
\(456\) −4.53590 −0.212413
\(457\) −31.8564 −1.49018 −0.745090 0.666964i \(-0.767593\pi\)
−0.745090 + 0.666964i \(0.767593\pi\)
\(458\) 11.8564 0.554013
\(459\) −21.8564 −1.02017
\(460\) −8.00000 −0.373002
\(461\) 14.7846 0.688588 0.344294 0.938862i \(-0.388118\pi\)
0.344294 + 0.938862i \(0.388118\pi\)
\(462\) −18.9282 −0.880620
\(463\) 18.9282 0.879668 0.439834 0.898079i \(-0.355037\pi\)
0.439834 + 0.898079i \(0.355037\pi\)
\(464\) 4.92820 0.228786
\(465\) 0.535898 0.0248517
\(466\) −28.9282 −1.34007
\(467\) −25.1769 −1.16505 −0.582524 0.812813i \(-0.697935\pi\)
−0.582524 + 0.812813i \(0.697935\pi\)
\(468\) 13.4641 0.622378
\(469\) 17.3205 0.799787
\(470\) 4.73205 0.218273
\(471\) 2.24871 0.103615
\(472\) 10.1962 0.469316
\(473\) −37.8564 −1.74064
\(474\) −6.39230 −0.293608
\(475\) 6.19615 0.284299
\(476\) −25.8564 −1.18513
\(477\) 14.7846 0.676941
\(478\) 20.7321 0.948262
\(479\) 4.05256 0.185166 0.0925831 0.995705i \(-0.470488\pi\)
0.0925831 + 0.995705i \(0.470488\pi\)
\(480\) −0.732051 −0.0334134
\(481\) −5.46410 −0.249142
\(482\) −4.92820 −0.224474
\(483\) 27.7128 1.26098
\(484\) 18.8564 0.857109
\(485\) −2.00000 −0.0908153
\(486\) −15.2679 −0.692568
\(487\) 4.39230 0.199034 0.0995172 0.995036i \(-0.468270\pi\)
0.0995172 + 0.995036i \(0.468270\pi\)
\(488\) 4.92820 0.223089
\(489\) −8.28719 −0.374760
\(490\) −15.3923 −0.695353
\(491\) −14.9282 −0.673700 −0.336850 0.941558i \(-0.609362\pi\)
−0.336850 + 0.941558i \(0.609362\pi\)
\(492\) −1.46410 −0.0660068
\(493\) 26.9282 1.21279
\(494\) 33.8564 1.52327
\(495\) 13.4641 0.605166
\(496\) 0.732051 0.0328701
\(497\) −13.8564 −0.621545
\(498\) 6.39230 0.286446
\(499\) 9.41154 0.421319 0.210659 0.977560i \(-0.432439\pi\)
0.210659 + 0.977560i \(0.432439\pi\)
\(500\) 1.00000 0.0447214
\(501\) −1.07180 −0.0478843
\(502\) −19.2679 −0.859971
\(503\) −18.9282 −0.843967 −0.421983 0.906604i \(-0.638666\pi\)
−0.421983 + 0.906604i \(0.638666\pi\)
\(504\) −11.6603 −0.519389
\(505\) −9.46410 −0.421147
\(506\) −43.7128 −1.94327
\(507\) 12.3397 0.548027
\(508\) 3.66025 0.162398
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) −4.00000 −0.177123
\(511\) 4.39230 0.194304
\(512\) −1.00000 −0.0441942
\(513\) −24.7846 −1.09427
\(514\) 18.5359 0.817583
\(515\) −6.53590 −0.288006
\(516\) 5.07180 0.223273
\(517\) 25.8564 1.13716
\(518\) 4.73205 0.207914
\(519\) 7.32051 0.321335
\(520\) 5.46410 0.239617
\(521\) 26.5359 1.16256 0.581279 0.813704i \(-0.302552\pi\)
0.581279 + 0.813704i \(0.302552\pi\)
\(522\) 12.1436 0.531511
\(523\) 36.7846 1.60848 0.804239 0.594306i \(-0.202573\pi\)
0.804239 + 0.594306i \(0.202573\pi\)
\(524\) −18.5885 −0.812041
\(525\) −3.46410 −0.151186
\(526\) 8.05256 0.351108
\(527\) 4.00000 0.174243
\(528\) −4.00000 −0.174078
\(529\) 41.0000 1.78261
\(530\) 6.00000 0.260623
\(531\) 25.1244 1.09030
\(532\) −29.3205 −1.27121
\(533\) 10.9282 0.473353
\(534\) 1.46410 0.0633579
\(535\) 3.26795 0.141286
\(536\) 3.66025 0.158099
\(537\) −0.248711 −0.0107327
\(538\) −20.3923 −0.879175
\(539\) −84.1051 −3.62266
\(540\) −4.00000 −0.172133
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 24.7846 1.06459
\(543\) 4.00000 0.171656
\(544\) −5.46410 −0.234271
\(545\) 2.00000 0.0856706
\(546\) −18.9282 −0.810052
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 2.00000 0.0854358
\(549\) 12.1436 0.518276
\(550\) 5.46410 0.232990
\(551\) 30.5359 1.30087
\(552\) 5.85641 0.249265
\(553\) −41.3205 −1.75713
\(554\) −22.2487 −0.945257
\(555\) 0.732051 0.0310738
\(556\) 6.92820 0.293821
\(557\) 44.1051 1.86879 0.934397 0.356234i \(-0.115939\pi\)
0.934397 + 0.356234i \(0.115939\pi\)
\(558\) 1.80385 0.0763630
\(559\) −37.8564 −1.60116
\(560\) −4.73205 −0.199966
\(561\) −21.8564 −0.922778
\(562\) 8.92820 0.376614
\(563\) 30.9282 1.30347 0.651734 0.758447i \(-0.274042\pi\)
0.651734 + 0.758447i \(0.274042\pi\)
\(564\) −3.46410 −0.145865
\(565\) −10.5359 −0.443249
\(566\) −16.3923 −0.689020
\(567\) −21.1244 −0.887140
\(568\) −2.92820 −0.122865
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −4.53590 −0.189988
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 29.8564 1.24836
\(573\) −6.39230 −0.267042
\(574\) −9.46410 −0.395024
\(575\) −8.00000 −0.333623
\(576\) −2.46410 −0.102671
\(577\) −24.3923 −1.01546 −0.507732 0.861515i \(-0.669516\pi\)
−0.507732 + 0.861515i \(0.669516\pi\)
\(578\) −12.8564 −0.534756
\(579\) 11.6077 0.482399
\(580\) 4.92820 0.204633
\(581\) 41.3205 1.71426
\(582\) 1.46410 0.0606890
\(583\) 32.7846 1.35780
\(584\) 0.928203 0.0384093
\(585\) 13.4641 0.556672
\(586\) −6.00000 −0.247858
\(587\) 44.7846 1.84846 0.924229 0.381838i \(-0.124709\pi\)
0.924229 + 0.381838i \(0.124709\pi\)
\(588\) 11.2679 0.464682
\(589\) 4.53590 0.186898
\(590\) 10.1962 0.419769
\(591\) −16.6795 −0.686103
\(592\) 1.00000 0.0410997
\(593\) −34.7846 −1.42843 −0.714216 0.699925i \(-0.753217\pi\)
−0.714216 + 0.699925i \(0.753217\pi\)
\(594\) −21.8564 −0.896779
\(595\) −25.8564 −1.06001
\(596\) 4.39230 0.179916
\(597\) −8.82309 −0.361105
\(598\) −43.7128 −1.78755
\(599\) 9.46410 0.386693 0.193346 0.981131i \(-0.438066\pi\)
0.193346 + 0.981131i \(0.438066\pi\)
\(600\) −0.732051 −0.0298858
\(601\) 27.6077 1.12614 0.563071 0.826409i \(-0.309620\pi\)
0.563071 + 0.826409i \(0.309620\pi\)
\(602\) 32.7846 1.33620
\(603\) 9.01924 0.367292
\(604\) −12.3923 −0.504236
\(605\) 18.8564 0.766622
\(606\) 6.92820 0.281439
\(607\) 0.784610 0.0318463 0.0159232 0.999873i \(-0.494931\pi\)
0.0159232 + 0.999873i \(0.494931\pi\)
\(608\) −6.19615 −0.251287
\(609\) −17.0718 −0.691784
\(610\) 4.92820 0.199537
\(611\) 25.8564 1.04604
\(612\) −13.4641 −0.544254
\(613\) 16.9282 0.683724 0.341862 0.939750i \(-0.388943\pi\)
0.341862 + 0.939750i \(0.388943\pi\)
\(614\) −18.5885 −0.750169
\(615\) −1.46410 −0.0590383
\(616\) −25.8564 −1.04178
\(617\) −0.928203 −0.0373681 −0.0186840 0.999825i \(-0.505948\pi\)
−0.0186840 + 0.999825i \(0.505948\pi\)
\(618\) 4.78461 0.192465
\(619\) −17.8564 −0.717710 −0.358855 0.933393i \(-0.616833\pi\)
−0.358855 + 0.933393i \(0.616833\pi\)
\(620\) 0.732051 0.0293999
\(621\) 32.0000 1.28412
\(622\) −2.87564 −0.115303
\(623\) 9.46410 0.379171
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 23.8564 0.953494
\(627\) −24.7846 −0.989802
\(628\) 3.07180 0.122578
\(629\) 5.46410 0.217868
\(630\) −11.6603 −0.464556
\(631\) 30.9808 1.23332 0.616662 0.787228i \(-0.288484\pi\)
0.616662 + 0.787228i \(0.288484\pi\)
\(632\) −8.73205 −0.347342
\(633\) −13.0718 −0.519557
\(634\) 4.14359 0.164563
\(635\) 3.66025 0.145253
\(636\) −4.39230 −0.174166
\(637\) −84.1051 −3.33237
\(638\) 26.9282 1.06610
\(639\) −7.21539 −0.285436
\(640\) −1.00000 −0.0395285
\(641\) −6.53590 −0.258152 −0.129076 0.991635i \(-0.541201\pi\)
−0.129076 + 0.991635i \(0.541201\pi\)
\(642\) −2.39230 −0.0944167
\(643\) 34.5359 1.36196 0.680981 0.732301i \(-0.261553\pi\)
0.680981 + 0.732301i \(0.261553\pi\)
\(644\) 37.8564 1.49175
\(645\) 5.07180 0.199702
\(646\) −33.8564 −1.33206
\(647\) −35.7128 −1.40402 −0.702008 0.712169i \(-0.747713\pi\)
−0.702008 + 0.712169i \(0.747713\pi\)
\(648\) −4.46410 −0.175366
\(649\) 55.7128 2.18692
\(650\) 5.46410 0.214320
\(651\) −2.53590 −0.0993897
\(652\) −11.3205 −0.443345
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −1.46410 −0.0572509
\(655\) −18.5885 −0.726311
\(656\) −2.00000 −0.0780869
\(657\) 2.28719 0.0892317
\(658\) −22.3923 −0.872943
\(659\) −6.92820 −0.269884 −0.134942 0.990853i \(-0.543085\pi\)
−0.134942 + 0.990853i \(0.543085\pi\)
\(660\) −4.00000 −0.155700
\(661\) −31.0718 −1.20855 −0.604276 0.796775i \(-0.706538\pi\)
−0.604276 + 0.796775i \(0.706538\pi\)
\(662\) 33.1244 1.28741
\(663\) −21.8564 −0.848832
\(664\) 8.73205 0.338869
\(665\) −29.3205 −1.13700
\(666\) 2.46410 0.0954820
\(667\) −39.4256 −1.52657
\(668\) −1.46410 −0.0566478
\(669\) 11.7513 0.454331
\(670\) 3.66025 0.141408
\(671\) 26.9282 1.03955
\(672\) 3.46410 0.133631
\(673\) −32.9282 −1.26929 −0.634644 0.772804i \(-0.718853\pi\)
−0.634644 + 0.772804i \(0.718853\pi\)
\(674\) −26.0000 −1.00148
\(675\) −4.00000 −0.153960
\(676\) 16.8564 0.648323
\(677\) −4.14359 −0.159251 −0.0796256 0.996825i \(-0.525372\pi\)
−0.0796256 + 0.996825i \(0.525372\pi\)
\(678\) 7.71281 0.296209
\(679\) 9.46410 0.363199
\(680\) −5.46410 −0.209539
\(681\) 17.8564 0.684259
\(682\) 4.00000 0.153168
\(683\) −4.78461 −0.183078 −0.0915390 0.995801i \(-0.529179\pi\)
−0.0915390 + 0.995801i \(0.529179\pi\)
\(684\) −15.2679 −0.583785
\(685\) 2.00000 0.0764161
\(686\) 39.7128 1.51624
\(687\) −8.67949 −0.331143
\(688\) 6.92820 0.264135
\(689\) 32.7846 1.24899
\(690\) 5.85641 0.222950
\(691\) −0.392305 −0.0149240 −0.00746199 0.999972i \(-0.502375\pi\)
−0.00746199 + 0.999972i \(0.502375\pi\)
\(692\) 10.0000 0.380143
\(693\) −63.7128 −2.42025
\(694\) −30.9282 −1.17402
\(695\) 6.92820 0.262802
\(696\) −3.60770 −0.136749
\(697\) −10.9282 −0.413935
\(698\) 15.3205 0.579890
\(699\) 21.1769 0.800984
\(700\) −4.73205 −0.178855
\(701\) −11.8564 −0.447810 −0.223905 0.974611i \(-0.571881\pi\)
−0.223905 + 0.974611i \(0.571881\pi\)
\(702\) −21.8564 −0.824917
\(703\) 6.19615 0.233692
\(704\) −5.46410 −0.205936
\(705\) −3.46410 −0.130466
\(706\) 11.8564 0.446222
\(707\) 44.7846 1.68430
\(708\) −7.46410 −0.280518
\(709\) 43.8564 1.64706 0.823531 0.567271i \(-0.192001\pi\)
0.823531 + 0.567271i \(0.192001\pi\)
\(710\) −2.92820 −0.109894
\(711\) −21.5167 −0.806938
\(712\) 2.00000 0.0749532
\(713\) −5.85641 −0.219324
\(714\) 18.9282 0.708370
\(715\) 29.8564 1.11657
\(716\) −0.339746 −0.0126969
\(717\) −15.1769 −0.566792
\(718\) 12.3923 0.462477
\(719\) −12.3923 −0.462155 −0.231077 0.972935i \(-0.574225\pi\)
−0.231077 + 0.972935i \(0.574225\pi\)
\(720\) −2.46410 −0.0918316
\(721\) 30.9282 1.15183
\(722\) −19.3923 −0.721707
\(723\) 3.60770 0.134172
\(724\) 5.46410 0.203072
\(725\) 4.92820 0.183029
\(726\) −13.8038 −0.512309
\(727\) 8.78461 0.325803 0.162902 0.986642i \(-0.447915\pi\)
0.162902 + 0.986642i \(0.447915\pi\)
\(728\) −25.8564 −0.958302
\(729\) −2.21539 −0.0820515
\(730\) 0.928203 0.0343543
\(731\) 37.8564 1.40017
\(732\) −3.60770 −0.133344
\(733\) 27.8564 1.02890 0.514450 0.857520i \(-0.327996\pi\)
0.514450 + 0.857520i \(0.327996\pi\)
\(734\) 4.33975 0.160183
\(735\) 11.2679 0.415625
\(736\) 8.00000 0.294884
\(737\) 20.0000 0.736709
\(738\) −4.92820 −0.181410
\(739\) 6.92820 0.254858 0.127429 0.991848i \(-0.459327\pi\)
0.127429 + 0.991848i \(0.459327\pi\)
\(740\) 1.00000 0.0367607
\(741\) −24.7846 −0.910485
\(742\) −28.3923 −1.04231
\(743\) 17.1244 0.628232 0.314116 0.949385i \(-0.398292\pi\)
0.314116 + 0.949385i \(0.398292\pi\)
\(744\) −0.535898 −0.0196470
\(745\) 4.39230 0.160922
\(746\) 10.7846 0.394853
\(747\) 21.5167 0.787253
\(748\) −29.8564 −1.09166
\(749\) −15.4641 −0.565046
\(750\) −0.732051 −0.0267307
\(751\) −20.3923 −0.744126 −0.372063 0.928208i \(-0.621349\pi\)
−0.372063 + 0.928208i \(0.621349\pi\)
\(752\) −4.73205 −0.172560
\(753\) 14.1051 0.514019
\(754\) 26.9282 0.980667
\(755\) −12.3923 −0.451002
\(756\) 18.9282 0.688412
\(757\) 1.71281 0.0622532 0.0311266 0.999515i \(-0.490090\pi\)
0.0311266 + 0.999515i \(0.490090\pi\)
\(758\) −32.3923 −1.17654
\(759\) 32.0000 1.16153
\(760\) −6.19615 −0.224758
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) −2.67949 −0.0970678
\(763\) −9.46410 −0.342623
\(764\) −8.73205 −0.315915
\(765\) −13.4641 −0.486796
\(766\) 8.00000 0.289052
\(767\) 55.7128 2.01167
\(768\) 0.732051 0.0264156
\(769\) 7.07180 0.255016 0.127508 0.991838i \(-0.459302\pi\)
0.127508 + 0.991838i \(0.459302\pi\)
\(770\) −25.8564 −0.931800
\(771\) −13.5692 −0.488683
\(772\) 15.8564 0.570685
\(773\) 2.78461 0.100155 0.0500777 0.998745i \(-0.484053\pi\)
0.0500777 + 0.998745i \(0.484053\pi\)
\(774\) 17.0718 0.613633
\(775\) 0.732051 0.0262960
\(776\) 2.00000 0.0717958
\(777\) −3.46410 −0.124274
\(778\) 11.8564 0.425073
\(779\) −12.3923 −0.444000
\(780\) −4.00000 −0.143223
\(781\) −16.0000 −0.572525
\(782\) 43.7128 1.56317
\(783\) −19.7128 −0.704478
\(784\) 15.3923 0.549725
\(785\) 3.07180 0.109637
\(786\) 13.6077 0.485370
\(787\) 22.1962 0.791207 0.395604 0.918421i \(-0.370535\pi\)
0.395604 + 0.918421i \(0.370535\pi\)
\(788\) −22.7846 −0.811668
\(789\) −5.89488 −0.209863
\(790\) −8.73205 −0.310672
\(791\) 49.8564 1.77269
\(792\) −13.4641 −0.478426
\(793\) 26.9282 0.956249
\(794\) 22.0000 0.780751
\(795\) −4.39230 −0.155779
\(796\) −12.0526 −0.427192
\(797\) 10.5359 0.373201 0.186600 0.982436i \(-0.440253\pi\)
0.186600 + 0.982436i \(0.440253\pi\)
\(798\) 21.4641 0.759821
\(799\) −25.8564 −0.914734
\(800\) −1.00000 −0.0353553
\(801\) 4.92820 0.174129
\(802\) 32.9282 1.16274
\(803\) 5.07180 0.178980
\(804\) −2.67949 −0.0944984
\(805\) 37.8564 1.33426
\(806\) 4.00000 0.140894
\(807\) 14.9282 0.525498
\(808\) 9.46410 0.332946
\(809\) 43.5692 1.53181 0.765906 0.642952i \(-0.222291\pi\)
0.765906 + 0.642952i \(0.222291\pi\)
\(810\) −4.46410 −0.156853
\(811\) −41.8564 −1.46978 −0.734889 0.678188i \(-0.762766\pi\)
−0.734889 + 0.678188i \(0.762766\pi\)
\(812\) −23.3205 −0.818389
\(813\) −18.1436 −0.636324
\(814\) 5.46410 0.191517
\(815\) −11.3205 −0.396540
\(816\) 4.00000 0.140028
\(817\) 42.9282 1.50187
\(818\) −3.07180 −0.107403
\(819\) −63.7128 −2.22631
\(820\) −2.00000 −0.0698430
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) −1.46410 −0.0510664
\(823\) 48.4449 1.68868 0.844341 0.535806i \(-0.179992\pi\)
0.844341 + 0.535806i \(0.179992\pi\)
\(824\) 6.53590 0.227689
\(825\) −4.00000 −0.139262
\(826\) −48.2487 −1.67879
\(827\) −16.3923 −0.570016 −0.285008 0.958525i \(-0.591996\pi\)
−0.285008 + 0.958525i \(0.591996\pi\)
\(828\) 19.7128 0.685068
\(829\) −51.5692 −1.79107 −0.895537 0.444988i \(-0.853208\pi\)
−0.895537 + 0.444988i \(0.853208\pi\)
\(830\) 8.73205 0.303094
\(831\) 16.2872 0.564996
\(832\) −5.46410 −0.189434
\(833\) 84.1051 2.91407
\(834\) −5.07180 −0.175622
\(835\) −1.46410 −0.0506673
\(836\) −33.8564 −1.17095
\(837\) −2.92820 −0.101214
\(838\) −14.2487 −0.492214
\(839\) −8.78461 −0.303278 −0.151639 0.988436i \(-0.548455\pi\)
−0.151639 + 0.988436i \(0.548455\pi\)
\(840\) 3.46410 0.119523
\(841\) −4.71281 −0.162511
\(842\) 0.143594 0.00494856
\(843\) −6.53590 −0.225108
\(844\) −17.8564 −0.614643
\(845\) 16.8564 0.579878
\(846\) −11.6603 −0.400888
\(847\) −89.2295 −3.06596
\(848\) −6.00000 −0.206041
\(849\) 12.0000 0.411839
\(850\) −5.46410 −0.187417
\(851\) −8.00000 −0.274236
\(852\) 2.14359 0.0734383
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) −23.3205 −0.798011
\(855\) −15.2679 −0.522153
\(856\) −3.26795 −0.111696
\(857\) −31.8564 −1.08819 −0.544097 0.839022i \(-0.683128\pi\)
−0.544097 + 0.839022i \(0.683128\pi\)
\(858\) −21.8564 −0.746165
\(859\) 44.4449 1.51644 0.758220 0.651999i \(-0.226069\pi\)
0.758220 + 0.651999i \(0.226069\pi\)
\(860\) 6.92820 0.236250
\(861\) 6.92820 0.236113
\(862\) 2.19615 0.0748012
\(863\) −20.4449 −0.695951 −0.347976 0.937504i \(-0.613131\pi\)
−0.347976 + 0.937504i \(0.613131\pi\)
\(864\) 4.00000 0.136083
\(865\) 10.0000 0.340010
\(866\) 34.0000 1.15537
\(867\) 9.41154 0.319633
\(868\) −3.46410 −0.117579
\(869\) −47.7128 −1.61855
\(870\) −3.60770 −0.122312
\(871\) 20.0000 0.677674
\(872\) −2.00000 −0.0677285
\(873\) 4.92820 0.166794
\(874\) 49.5692 1.67670
\(875\) −4.73205 −0.159973
\(876\) −0.679492 −0.0229579
\(877\) −9.21539 −0.311182 −0.155591 0.987822i \(-0.549728\pi\)
−0.155591 + 0.987822i \(0.549728\pi\)
\(878\) −13.5167 −0.456165
\(879\) 4.39230 0.148149
\(880\) −5.46410 −0.184195
\(881\) 12.6795 0.427183 0.213591 0.976923i \(-0.431484\pi\)
0.213591 + 0.976923i \(0.431484\pi\)
\(882\) 37.9282 1.27711
\(883\) 14.2487 0.479507 0.239754 0.970834i \(-0.422933\pi\)
0.239754 + 0.970834i \(0.422933\pi\)
\(884\) −29.8564 −1.00418
\(885\) −7.46410 −0.250903
\(886\) 14.8756 0.499757
\(887\) 9.80385 0.329181 0.164590 0.986362i \(-0.447370\pi\)
0.164590 + 0.986362i \(0.447370\pi\)
\(888\) −0.732051 −0.0245660
\(889\) −17.3205 −0.580911
\(890\) 2.00000 0.0670402
\(891\) −24.3923 −0.817173
\(892\) 16.0526 0.537479
\(893\) −29.3205 −0.981173
\(894\) −3.21539 −0.107539
\(895\) −0.339746 −0.0113565
\(896\) 4.73205 0.158087
\(897\) 32.0000 1.06845
\(898\) 21.7128 0.724566
\(899\) 3.60770 0.120323
\(900\) −2.46410 −0.0821367
\(901\) −32.7846 −1.09221
\(902\) −10.9282 −0.363869
\(903\) −24.0000 −0.798670
\(904\) 10.5359 0.350419
\(905\) 5.46410 0.181633
\(906\) 9.07180 0.301390
\(907\) 54.2487 1.80130 0.900649 0.434546i \(-0.143091\pi\)
0.900649 + 0.434546i \(0.143091\pi\)
\(908\) 24.3923 0.809487
\(909\) 23.3205 0.773492
\(910\) −25.8564 −0.857132
\(911\) 37.9090 1.25598 0.627990 0.778221i \(-0.283878\pi\)
0.627990 + 0.778221i \(0.283878\pi\)
\(912\) 4.53590 0.150199
\(913\) 47.7128 1.57906
\(914\) 31.8564 1.05372
\(915\) −3.60770 −0.119267
\(916\) −11.8564 −0.391747
\(917\) 87.9615 2.90475
\(918\) 21.8564 0.721369
\(919\) −16.7321 −0.551939 −0.275970 0.961166i \(-0.588999\pi\)
−0.275970 + 0.961166i \(0.588999\pi\)
\(920\) 8.00000 0.263752
\(921\) 13.6077 0.448389
\(922\) −14.7846 −0.486905
\(923\) −16.0000 −0.526646
\(924\) 18.9282 0.622692
\(925\) 1.00000 0.0328798
\(926\) −18.9282 −0.622019
\(927\) 16.1051 0.528961
\(928\) −4.92820 −0.161776
\(929\) 11.8564 0.388996 0.194498 0.980903i \(-0.437692\pi\)
0.194498 + 0.980903i \(0.437692\pi\)
\(930\) −0.535898 −0.0175728
\(931\) 95.3731 3.12573
\(932\) 28.9282 0.947575
\(933\) 2.10512 0.0689185
\(934\) 25.1769 0.823814
\(935\) −29.8564 −0.976409
\(936\) −13.4641 −0.440088
\(937\) −23.8564 −0.779355 −0.389677 0.920951i \(-0.627413\pi\)
−0.389677 + 0.920951i \(0.627413\pi\)
\(938\) −17.3205 −0.565535
\(939\) −17.4641 −0.569919
\(940\) −4.73205 −0.154342
\(941\) 7.60770 0.248004 0.124002 0.992282i \(-0.460427\pi\)
0.124002 + 0.992282i \(0.460427\pi\)
\(942\) −2.24871 −0.0732670
\(943\) 16.0000 0.521032
\(944\) −10.1962 −0.331856
\(945\) 18.9282 0.615734
\(946\) 37.8564 1.23082
\(947\) 17.1769 0.558175 0.279087 0.960266i \(-0.409968\pi\)
0.279087 + 0.960266i \(0.409968\pi\)
\(948\) 6.39230 0.207612
\(949\) 5.07180 0.164637
\(950\) −6.19615 −0.201030
\(951\) −3.03332 −0.0983622
\(952\) 25.8564 0.838011
\(953\) 15.8564 0.513639 0.256820 0.966459i \(-0.417325\pi\)
0.256820 + 0.966459i \(0.417325\pi\)
\(954\) −14.7846 −0.478669
\(955\) −8.73205 −0.282563
\(956\) −20.7321 −0.670522
\(957\) −19.7128 −0.637225
\(958\) −4.05256 −0.130932
\(959\) −9.46410 −0.305612
\(960\) 0.732051 0.0236268
\(961\) −30.4641 −0.982713
\(962\) 5.46410 0.176170
\(963\) −8.05256 −0.259490
\(964\) 4.92820 0.158727
\(965\) 15.8564 0.510436
\(966\) −27.7128 −0.891645
\(967\) −44.3923 −1.42756 −0.713780 0.700370i \(-0.753018\pi\)
−0.713780 + 0.700370i \(0.753018\pi\)
\(968\) −18.8564 −0.606068
\(969\) 24.7846 0.796196
\(970\) 2.00000 0.0642161
\(971\) 1.07180 0.0343956 0.0171978 0.999852i \(-0.494526\pi\)
0.0171978 + 0.999852i \(0.494526\pi\)
\(972\) 15.2679 0.489720
\(973\) −32.7846 −1.05103
\(974\) −4.39230 −0.140739
\(975\) −4.00000 −0.128103
\(976\) −4.92820 −0.157748
\(977\) −34.0000 −1.08776 −0.543878 0.839164i \(-0.683045\pi\)
−0.543878 + 0.839164i \(0.683045\pi\)
\(978\) 8.28719 0.264995
\(979\) 10.9282 0.349267
\(980\) 15.3923 0.491689
\(981\) −4.92820 −0.157345
\(982\) 14.9282 0.476378
\(983\) 20.4449 0.652090 0.326045 0.945354i \(-0.394284\pi\)
0.326045 + 0.945354i \(0.394284\pi\)
\(984\) 1.46410 0.0466739
\(985\) −22.7846 −0.725978
\(986\) −26.9282 −0.857569
\(987\) 16.3923 0.521773
\(988\) −33.8564 −1.07712
\(989\) −55.4256 −1.76243
\(990\) −13.4641 −0.427917
\(991\) 44.0526 1.39938 0.699688 0.714449i \(-0.253322\pi\)
0.699688 + 0.714449i \(0.253322\pi\)
\(992\) −0.732051 −0.0232426
\(993\) −24.2487 −0.769510
\(994\) 13.8564 0.439499
\(995\) −12.0526 −0.382092
\(996\) −6.39230 −0.202548
\(997\) −31.8564 −1.00890 −0.504451 0.863440i \(-0.668305\pi\)
−0.504451 + 0.863440i \(0.668305\pi\)
\(998\) −9.41154 −0.297917
\(999\) −4.00000 −0.126554
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 370.2.a.e.1.2 2
3.2 odd 2 3330.2.a.bd.1.1 2
4.3 odd 2 2960.2.a.q.1.1 2
5.2 odd 4 1850.2.b.l.149.1 4
5.3 odd 4 1850.2.b.l.149.4 4
5.4 even 2 1850.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.2 2 1.1 even 1 trivial
1850.2.a.x.1.1 2 5.4 even 2
1850.2.b.l.149.1 4 5.2 odd 4
1850.2.b.l.149.4 4 5.3 odd 4
2960.2.a.q.1.1 2 4.3 odd 2
3330.2.a.bd.1.1 2 3.2 odd 2