Properties

Label 1850.2.a.x.1.1
Level $1850$
Weight $2$
Character 1850.1
Self dual yes
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
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: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1850.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} +4.73205 q^{7} +1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -0.732051 q^{6} +4.73205 q^{7} +1.00000 q^{8} -2.46410 q^{9} -5.46410 q^{11} -0.732051 q^{12} +5.46410 q^{13} +4.73205 q^{14} +1.00000 q^{16} -5.46410 q^{17} -2.46410 q^{18} +6.19615 q^{19} -3.46410 q^{21} -5.46410 q^{22} +8.00000 q^{23} -0.732051 q^{24} +5.46410 q^{26} +4.00000 q^{27} +4.73205 q^{28} +4.92820 q^{29} +0.732051 q^{31} +1.00000 q^{32} +4.00000 q^{33} -5.46410 q^{34} -2.46410 q^{36} -1.00000 q^{37} +6.19615 q^{38} -4.00000 q^{39} -2.00000 q^{41} -3.46410 q^{42} -6.92820 q^{43} -5.46410 q^{44} +8.00000 q^{46} +4.73205 q^{47} -0.732051 q^{48} +15.3923 q^{49} +4.00000 q^{51} +5.46410 q^{52} +6.00000 q^{53} +4.00000 q^{54} +4.73205 q^{56} -4.53590 q^{57} +4.92820 q^{58} -10.1962 q^{59} -4.92820 q^{61} +0.732051 q^{62} -11.6603 q^{63} +1.00000 q^{64} +4.00000 q^{66} +3.66025 q^{67} -5.46410 q^{68} -5.85641 q^{69} +2.92820 q^{71} -2.46410 q^{72} +0.928203 q^{73} -1.00000 q^{74} +6.19615 q^{76} -25.8564 q^{77} -4.00000 q^{78} +8.73205 q^{79} +4.46410 q^{81} -2.00000 q^{82} +8.73205 q^{83} -3.46410 q^{84} -6.92820 q^{86} -3.60770 q^{87} -5.46410 q^{88} -2.00000 q^{89} +25.8564 q^{91} +8.00000 q^{92} -0.535898 q^{93} +4.73205 q^{94} -0.732051 q^{96} +2.00000 q^{97} +15.3923 q^{98} +13.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} - 4 q^{11} + 2 q^{12} + 4 q^{13} + 6 q^{14} + 2 q^{16} - 4 q^{17} + 2 q^{18} + 2 q^{19} - 4 q^{22} + 16 q^{23} + 2 q^{24} + 4 q^{26} + 8 q^{27} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 2 q^{32} + 8 q^{33} - 4 q^{34} + 2 q^{36} - 2 q^{37} + 2 q^{38} - 8 q^{39} - 4 q^{41} - 4 q^{44} + 16 q^{46} + 6 q^{47} + 2 q^{48} + 10 q^{49} + 8 q^{51} + 4 q^{52} + 12 q^{53} + 8 q^{54} + 6 q^{56} - 16 q^{57} - 4 q^{58} - 10 q^{59} + 4 q^{61} - 2 q^{62} - 6 q^{63} + 2 q^{64} + 8 q^{66} - 10 q^{67} - 4 q^{68} + 16 q^{69} - 8 q^{71} + 2 q^{72} - 12 q^{73} - 2 q^{74} + 2 q^{76} - 24 q^{77} - 8 q^{78} + 14 q^{79} + 2 q^{81} - 4 q^{82} + 14 q^{83} - 28 q^{87} - 4 q^{88} - 4 q^{89} + 24 q^{91} + 16 q^{92} - 8 q^{93} + 6 q^{94} + 2 q^{96} + 4 q^{97} + 10 q^{98} + 20 q^{99}+O(q^{100}) \) 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 \(-0.567778\pi\)
−0.211325 + 0.977416i \(0.567778\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −0.732051 −0.298858
\(7\) 4.73205 1.78855 0.894274 0.447521i \(-0.147693\pi\)
0.894274 + 0.447521i \(0.147693\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) −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 \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) 4.73205 1.26469
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.46410 −1.32524 −0.662620 0.748956i \(-0.730555\pi\)
−0.662620 + 0.748956i \(0.730555\pi\)
\(18\) −2.46410 −0.580794
\(19\) 6.19615 1.42149 0.710747 0.703447i \(-0.248357\pi\)
0.710747 + 0.703447i \(0.248357\pi\)
\(20\) 0 0
\(21\) −3.46410 −0.755929
\(22\) −5.46410 −1.16495
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −0.732051 −0.149429
\(25\) 0 0
\(26\) 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 0
\(31\) 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
\(35\) 0 0
\(36\) −2.46410 −0.410684
\(37\) −1.00000 −0.164399
\(38\) 6.19615 1.00515
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −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 \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) −5.46410 −0.823744
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 4.73205 0.690241 0.345120 0.938558i \(-0.387838\pi\)
0.345120 + 0.938558i \(0.387838\pi\)
\(48\) −0.732051 −0.105662
\(49\) 15.3923 2.19890
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 5.46410 0.757735
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 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 0
\(61\) −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
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 3.66025 0.447171 0.223586 0.974684i \(-0.428224\pi\)
0.223586 + 0.974684i \(0.428224\pi\)
\(68\) −5.46410 −0.662620
\(69\) −5.85641 −0.705028
\(70\) 0 0
\(71\) 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 \(-0.482701\pi\)
0.0543190 + 0.998524i \(0.482701\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 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 \(0.336552\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) −2.00000 −0.220863
\(83\) 8.73205 0.958467 0.479234 0.877687i \(-0.340915\pi\)
0.479234 + 0.877687i \(0.340915\pi\)
\(84\) −3.46410 −0.377964
\(85\) 0 0
\(86\) −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 \(0.533804\pi\)
\(90\) 0 0
\(91\) 25.8564 2.71049
\(92\) 8.00000 0.834058
\(93\) −0.535898 −0.0555701
\(94\) 4.73205 0.488074
\(95\) 0 0
\(96\) −0.732051 −0.0747146
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 15.3923 1.55486
\(99\) 13.4641 1.35319
\(100\) 0 0
\(101\) −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 \(-0.395645\pi\)
0.322001 + 0.946739i \(0.395645\pi\)
\(104\) 5.46410 0.535799
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −3.26795 −0.315925 −0.157962 0.987445i \(-0.550492\pi\)
−0.157962 + 0.987445i \(0.550492\pi\)
\(108\) 4.00000 0.384900
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0.732051 0.0694832
\(112\) 4.73205 0.447137
\(113\) 10.5359 0.991134 0.495567 0.868570i \(-0.334960\pi\)
0.495567 + 0.868570i \(0.334960\pi\)
\(114\) −4.53590 −0.424826
\(115\) 0 0
\(116\) 4.92820 0.457572
\(117\) −13.4641 −1.24476
\(118\) −10.1962 −0.938632
\(119\) −25.8564 −2.37025
\(120\) 0 0
\(121\) 18.8564 1.71422
\(122\) −4.92820 −0.446179
\(123\) 1.46410 0.132014
\(124\) 0.732051 0.0657401
\(125\) 0 0
\(126\) −11.6603 −1.03878
\(127\) −3.66025 −0.324795 −0.162398 0.986725i \(-0.551923\pi\)
−0.162398 + 0.986725i \(0.551923\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.07180 0.446547
\(130\) 0 0
\(131\) −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
\(135\) 0 0
\(136\) −5.46410 −0.468543
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −5.85641 −0.498530
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) −3.46410 −0.291730
\(142\) 2.92820 0.245729
\(143\) −29.8564 −2.49672
\(144\) −2.46410 −0.205342
\(145\) 0 0
\(146\) 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 0
\(151\) −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 0
\(156\) −4.00000 −0.320256
\(157\) −3.07180 −0.245156 −0.122578 0.992459i \(-0.539116\pi\)
−0.122578 + 0.992459i \(0.539116\pi\)
\(158\) 8.73205 0.694685
\(159\) −4.39230 −0.348332
\(160\) 0 0
\(161\) 37.8564 2.98350
\(162\) 4.46410 0.350733
\(163\) 11.3205 0.886691 0.443345 0.896351i \(-0.353792\pi\)
0.443345 + 0.896351i \(0.353792\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 8.73205 0.677739
\(167\) 1.46410 0.113296 0.0566478 0.998394i \(-0.481959\pi\)
0.0566478 + 0.998394i \(0.481959\pi\)
\(168\) −3.46410 −0.267261
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) −15.2679 −1.16757
\(172\) −6.92820 −0.528271
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) −3.60770 −0.273499
\(175\) 0 0
\(176\) −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 \(0.504042\pi\)
\(180\) 0 0
\(181\) 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
\(185\) 0 0
\(186\) −0.535898 −0.0392940
\(187\) 29.8564 2.18332
\(188\) 4.73205 0.345120
\(189\) 18.9282 1.37682
\(190\) 0 0
\(191\) −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 \(-0.693322\pi\)
−0.570685 + 0.821169i \(0.693322\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 15.3923 1.09945
\(197\) 22.7846 1.62334 0.811668 0.584119i \(-0.198560\pi\)
0.811668 + 0.584119i \(0.198560\pi\)
\(198\) 13.4641 0.956852
\(199\) −12.0526 −0.854383 −0.427192 0.904161i \(-0.640497\pi\)
−0.427192 + 0.904161i \(0.640497\pi\)
\(200\) 0 0
\(201\) −2.67949 −0.188997
\(202\) −9.46410 −0.665892
\(203\) 23.3205 1.63678
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 6.53590 0.455378
\(207\) −19.7128 −1.37014
\(208\) 5.46410 0.378867
\(209\) −33.8564 −2.34190
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 3.46410 0.235159
\(218\) 2.00000 0.135457
\(219\) −0.679492 −0.0459158
\(220\) 0 0
\(221\) −29.8564 −2.00836
\(222\) 0.732051 0.0491320
\(223\) −16.0526 −1.07496 −0.537479 0.843277i \(-0.680623\pi\)
−0.537479 + 0.843277i \(0.680623\pi\)
\(224\) 4.73205 0.316173
\(225\) 0 0
\(226\) 10.5359 0.700838
\(227\) −24.3923 −1.61897 −0.809487 0.587138i \(-0.800255\pi\)
−0.809487 + 0.587138i \(0.800255\pi\)
\(228\) −4.53590 −0.300397
\(229\) −11.8564 −0.783493 −0.391747 0.920073i \(-0.628129\pi\)
−0.391747 + 0.920073i \(0.628129\pi\)
\(230\) 0 0
\(231\) 18.9282 1.24538
\(232\) 4.92820 0.323552
\(233\) −28.9282 −1.89515 −0.947575 0.319534i \(-0.896474\pi\)
−0.947575 + 0.319534i \(0.896474\pi\)
\(234\) −13.4641 −0.880176
\(235\) 0 0
\(236\) −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 0
\(241\) 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
\(245\) 0 0
\(246\) 1.46410 0.0933477
\(247\) 33.8564 2.15423
\(248\) 0.732051 0.0464853
\(249\) −6.39230 −0.405096
\(250\) 0 0
\(251\) 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
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.5359 1.15624 0.578119 0.815953i \(-0.303787\pi\)
0.578119 + 0.815953i \(0.303787\pi\)
\(258\) 5.07180 0.315756
\(259\) −4.73205 −0.294035
\(260\) 0 0
\(261\) −12.1436 −0.751670
\(262\) −18.5885 −1.14840
\(263\) 8.05256 0.496542 0.248271 0.968691i \(-0.420138\pi\)
0.248271 + 0.968691i \(0.420138\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 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 \(0.286454\pi\)
\(270\) 0 0
\(271\) −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
\(275\) 0 0
\(276\) −5.85641 −0.352514
\(277\) −22.2487 −1.33680 −0.668398 0.743804i \(-0.733020\pi\)
−0.668398 + 0.743804i \(0.733020\pi\)
\(278\) 6.92820 0.415526
\(279\) −1.80385 −0.107994
\(280\) 0 0
\(281\) −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 \(-0.661986\pi\)
−0.487211 + 0.873284i \(0.661986\pi\)
\(284\) 2.92820 0.173757
\(285\) 0 0
\(286\) −29.8564 −1.76545
\(287\) −9.46410 −0.558648
\(288\) −2.46410 −0.145199
\(289\) 12.8564 0.756259
\(290\) 0 0
\(291\) −1.46410 −0.0858272
\(292\) 0.928203 0.0543190
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −11.2679 −0.657160
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −21.8564 −1.26824
\(298\) 4.39230 0.254439
\(299\) 43.7128 2.52798
\(300\) 0 0
\(301\) −32.7846 −1.88967
\(302\) −12.3923 −0.713097
\(303\) 6.92820 0.398015
\(304\) 6.19615 0.355374
\(305\) 0 0
\(306\) 13.4641 0.769691
\(307\) −18.5885 −1.06090 −0.530450 0.847716i \(-0.677977\pi\)
−0.530450 + 0.847716i \(0.677977\pi\)
\(308\) −25.8564 −1.47331
\(309\) −4.78461 −0.272187
\(310\) 0 0
\(311\) 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 \(-0.264479\pi\)
0.674222 + 0.738529i \(0.264479\pi\)
\(314\) −3.07180 −0.173352
\(315\) 0 0
\(316\) 8.73205 0.491216
\(317\) 4.14359 0.232727 0.116364 0.993207i \(-0.462876\pi\)
0.116364 + 0.993207i \(0.462876\pi\)
\(318\) −4.39230 −0.246308
\(319\) −26.9282 −1.50769
\(320\) 0 0
\(321\) 2.39230 0.133525
\(322\) 37.8564 2.10966
\(323\) −33.8564 −1.88382
\(324\) 4.46410 0.248006
\(325\) 0 0
\(326\) 11.3205 0.626985
\(327\) −1.46410 −0.0809650
\(328\) −2.00000 −0.110432
\(329\) 22.3923 1.23453
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) −3.46410 −0.188982
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 16.8564 0.916868
\(339\) −7.71281 −0.418902
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) −15.2679 −0.825596
\(343\) 39.7128 2.14429
\(344\) −6.92820 −0.373544
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −30.9282 −1.66031 −0.830156 0.557530i \(-0.811749\pi\)
−0.830156 + 0.557530i \(0.811749\pi\)
\(348\) −3.60770 −0.193393
\(349\) −15.3205 −0.820088 −0.410044 0.912066i \(-0.634487\pi\)
−0.410044 + 0.912066i \(0.634487\pi\)
\(350\) 0 0
\(351\) 21.8564 1.16661
\(352\) −5.46410 −0.291238
\(353\) 11.8564 0.631053 0.315526 0.948917i \(-0.397819\pi\)
0.315526 + 0.948917i \(0.397819\pi\)
\(354\) 7.46410 0.396713
\(355\) 0 0
\(356\) −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 \(0.606045\pi\)
\(360\) 0 0
\(361\) 19.3923 1.02065
\(362\) 5.46410 0.287187
\(363\) −13.8038 −0.724514
\(364\) 25.8564 1.35524
\(365\) 0 0
\(366\) 3.60770 0.188577
\(367\) 4.33975 0.226533 0.113266 0.993565i \(-0.463869\pi\)
0.113266 + 0.993565i \(0.463869\pi\)
\(368\) 8.00000 0.417029
\(369\) 4.92820 0.256552
\(370\) 0 0
\(371\) 28.3923 1.47406
\(372\) −0.535898 −0.0277850
\(373\) 10.7846 0.558406 0.279203 0.960232i \(-0.409930\pi\)
0.279203 + 0.960232i \(0.409930\pi\)
\(374\) 29.8564 1.54384
\(375\) 0 0
\(376\) 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 \(0.187230\pi\)
\(380\) 0 0
\(381\) 2.67949 0.137275
\(382\) −8.73205 −0.446771
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −0.732051 −0.0373573
\(385\) 0 0
\(386\) −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 \(0.597178\pi\)
\(390\) 0 0
\(391\) −43.7128 −2.21065
\(392\) 15.3923 0.777429
\(393\) 13.6077 0.686417
\(394\) 22.7846 1.14787
\(395\) 0 0
\(396\) 13.4641 0.676597
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −12.0526 −0.604140
\(399\) −21.4641 −1.07455
\(400\) 0 0
\(401\) −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
\(405\) 0 0
\(406\) 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 \(0.475803\pi\)
\(410\) 0 0
\(411\) 1.46410 0.0722188
\(412\) 6.53590 0.322001
\(413\) −48.2487 −2.37416
\(414\) −19.7128 −0.968832
\(415\) 0 0
\(416\) 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 \(0.386845\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) −2.14359 −0.103857
\(427\) −23.3205 −1.12856
\(428\) −3.26795 −0.157962
\(429\) 21.8564 1.05524
\(430\) 0 0
\(431\) −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 \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 3.46410 0.166282
\(435\) 0 0
\(436\) 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 \(0.395457\pi\)
\(440\) 0 0
\(441\) −37.9282 −1.80610
\(442\) −29.8564 −1.42012
\(443\) 14.8756 0.706763 0.353382 0.935479i \(-0.385032\pi\)
0.353382 + 0.935479i \(0.385032\pi\)
\(444\) 0.732051 0.0347416
\(445\) 0 0
\(446\) −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 \(0.671223\pi\)
\(450\) 0 0
\(451\) 10.9282 0.514589
\(452\) 10.5359 0.495567
\(453\) 9.07180 0.426230
\(454\) −24.3923 −1.14479
\(455\) 0 0
\(456\) −4.53590 −0.212413
\(457\) 31.8564 1.49018 0.745090 0.666964i \(-0.232407\pi\)
0.745090 + 0.666964i \(0.232407\pi\)
\(458\) −11.8564 −0.554013
\(459\) −21.8564 −1.02017
\(460\) 0 0
\(461\) 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 \(-0.644963\pi\)
−0.439834 + 0.898079i \(0.644963\pi\)
\(464\) 4.92820 0.228786
\(465\) 0 0
\(466\) −28.9282 −1.34007
\(467\) 25.1769 1.16505 0.582524 0.812813i \(-0.302065\pi\)
0.582524 + 0.812813i \(0.302065\pi\)
\(468\) −13.4641 −0.622378
\(469\) 17.3205 0.799787
\(470\) 0 0
\(471\) 2.24871 0.103615
\(472\) −10.1962 −0.469316
\(473\) 37.8564 1.74064
\(474\) −6.39230 −0.293608
\(475\) 0 0
\(476\) −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 0
\(481\) −5.46410 −0.249142
\(482\) 4.92820 0.224474
\(483\) −27.7128 −1.26098
\(484\) 18.8564 0.857109
\(485\) 0 0
\(486\) −15.2679 −0.692568
\(487\) −4.39230 −0.199034 −0.0995172 0.995036i \(-0.531730\pi\)
−0.0995172 + 0.995036i \(0.531730\pi\)
\(488\) −4.92820 −0.223089
\(489\) −8.28719 −0.374760
\(490\) 0 0
\(491\) −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
\(495\) 0 0
\(496\) 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 \(0.432439\pi\)
\(500\) 0 0
\(501\) −1.07180 −0.0478843
\(502\) 19.2679 0.859971
\(503\) 18.9282 0.843967 0.421983 0.906604i \(-0.361334\pi\)
0.421983 + 0.906604i \(0.361334\pi\)
\(504\) −11.6603 −0.519389
\(505\) 0 0
\(506\) −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 \(0.514113\pi\)
\(510\) 0 0
\(511\) 4.39230 0.194304
\(512\) 1.00000 0.0441942
\(513\) 24.7846 1.09427
\(514\) 18.5359 0.817583
\(515\) 0 0
\(516\) 5.07180 0.223273
\(517\) −25.8564 −1.13716
\(518\) −4.73205 −0.207914
\(519\) 7.32051 0.321335
\(520\) 0 0
\(521\) 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 \(-0.797427\pi\)
−0.804239 + 0.594306i \(0.797427\pi\)
\(524\) −18.5885 −0.812041
\(525\) 0 0
\(526\) 8.05256 0.351108
\(527\) −4.00000 −0.174243
\(528\) 4.00000 0.174078
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 25.1244 1.09030
\(532\) 29.3205 1.27121
\(533\) −10.9282 −0.473353
\(534\) 1.46410 0.0633579
\(535\) 0 0
\(536\) 3.66025 0.158099
\(537\) 0.248711 0.0107327
\(538\) 20.3923 0.879175
\(539\) −84.1051 −3.62266
\(540\) 0 0
\(541\) −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
\(545\) 0 0
\(546\) −18.9282 −0.810052
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 12.1436 0.518276
\(550\) 0 0
\(551\) 30.5359 1.30087
\(552\) −5.85641 −0.249265
\(553\) 41.3205 1.75713
\(554\) −22.2487 −0.945257
\(555\) 0 0
\(556\) 6.92820 0.293821
\(557\) −44.1051 −1.86879 −0.934397 0.356234i \(-0.884061\pi\)
−0.934397 + 0.356234i \(0.884061\pi\)
\(558\) −1.80385 −0.0763630
\(559\) −37.8564 −1.60116
\(560\) 0 0
\(561\) −21.8564 −0.922778
\(562\) −8.92820 −0.376614
\(563\) −30.9282 −1.30347 −0.651734 0.758447i \(-0.725958\pi\)
−0.651734 + 0.758447i \(0.725958\pi\)
\(564\) −3.46410 −0.145865
\(565\) 0 0
\(566\) −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 \(0.652561\pi\)
\(570\) 0 0
\(571\) −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
\(575\) 0 0
\(576\) −2.46410 −0.102671
\(577\) 24.3923 1.01546 0.507732 0.861515i \(-0.330484\pi\)
0.507732 + 0.861515i \(0.330484\pi\)
\(578\) 12.8564 0.534756
\(579\) 11.6077 0.482399
\(580\) 0 0
\(581\) 41.3205 1.71426
\(582\) −1.46410 −0.0606890
\(583\) −32.7846 −1.35780
\(584\) 0.928203 0.0384093
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −44.7846 −1.84846 −0.924229 0.381838i \(-0.875291\pi\)
−0.924229 + 0.381838i \(0.875291\pi\)
\(588\) −11.2679 −0.464682
\(589\) 4.53590 0.186898
\(590\) 0 0
\(591\) −16.6795 −0.686103
\(592\) −1.00000 −0.0410997
\(593\) 34.7846 1.42843 0.714216 0.699925i \(-0.246783\pi\)
0.714216 + 0.699925i \(0.246783\pi\)
\(594\) −21.8564 −0.896779
\(595\) 0 0
\(596\) 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 0
\(601\) 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
\(605\) 0 0
\(606\) 6.92820 0.281439
\(607\) −0.784610 −0.0318463 −0.0159232 0.999873i \(-0.505069\pi\)
−0.0159232 + 0.999873i \(0.505069\pi\)
\(608\) 6.19615 0.251287
\(609\) −17.0718 −0.691784
\(610\) 0 0
\(611\) 25.8564 1.04604
\(612\) 13.4641 0.544254
\(613\) −16.9282 −0.683724 −0.341862 0.939750i \(-0.611057\pi\)
−0.341862 + 0.939750i \(0.611057\pi\)
\(614\) −18.5885 −0.750169
\(615\) 0 0
\(616\) −25.8564 −1.04178
\(617\) 0.928203 0.0373681 0.0186840 0.999825i \(-0.494052\pi\)
0.0186840 + 0.999825i \(0.494052\pi\)
\(618\) −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 0
\(621\) 32.0000 1.28412
\(622\) 2.87564 0.115303
\(623\) −9.46410 −0.379171
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 23.8564 0.953494
\(627\) 24.7846 0.989802
\(628\) −3.07180 −0.122578
\(629\) 5.46410 0.217868
\(630\) 0 0
\(631\) 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
\(635\) 0 0
\(636\) −4.39230 −0.174166
\(637\) 84.1051 3.33237
\(638\) −26.9282 −1.06610
\(639\) −7.21539 −0.285436
\(640\) 0 0
\(641\) −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 \(-0.738447\pi\)
−0.680981 + 0.732301i \(0.738447\pi\)
\(644\) 37.8564 1.49175
\(645\) 0 0
\(646\) −33.8564 −1.33206
\(647\) 35.7128 1.40402 0.702008 0.712169i \(-0.252287\pi\)
0.702008 + 0.712169i \(0.252287\pi\)
\(648\) 4.46410 0.175366
\(649\) 55.7128 2.18692
\(650\) 0 0
\(651\) −2.53590 −0.0993897
\(652\) 11.3205 0.443345
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −1.46410 −0.0572509
\(655\) 0 0
\(656\) −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 \(0.543085\pi\)
\(660\) 0 0
\(661\) −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
\(665\) 0 0
\(666\) 2.46410 0.0954820
\(667\) 39.4256 1.52657
\(668\) 1.46410 0.0566478
\(669\) 11.7513 0.454331
\(670\) 0 0
\(671\) 26.9282 1.03955
\(672\) −3.46410 −0.133631
\(673\) 32.9282 1.26929 0.634644 0.772804i \(-0.281147\pi\)
0.634644 + 0.772804i \(0.281147\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 4.14359 0.159251 0.0796256 0.996825i \(-0.474628\pi\)
0.0796256 + 0.996825i \(0.474628\pi\)
\(678\) −7.71281 −0.296209
\(679\) 9.46410 0.363199
\(680\) 0 0
\(681\) 17.8564 0.684259
\(682\) −4.00000 −0.153168
\(683\) 4.78461 0.183078 0.0915390 0.995801i \(-0.470821\pi\)
0.0915390 + 0.995801i \(0.470821\pi\)
\(684\) −15.2679 −0.583785
\(685\) 0 0
\(686\) 39.7128 1.51624
\(687\) 8.67949 0.331143
\(688\) −6.92820 −0.264135
\(689\) 32.7846 1.24899
\(690\) 0 0
\(691\) −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
\(695\) 0 0
\(696\) −3.60770 −0.136749
\(697\) 10.9282 0.413935
\(698\) −15.3205 −0.579890
\(699\) 21.1769 0.800984
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) 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 \(0.192001\pi\)
\(710\) 0 0
\(711\) −21.5167 −0.806938
\(712\) −2.00000 −0.0749532
\(713\) 5.85641 0.219324
\(714\) 18.9282 0.708370
\(715\) 0 0
\(716\) −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 \(0.574225\pi\)
\(720\) 0 0
\(721\) 30.9282 1.15183
\(722\) 19.3923 0.721707
\(723\) −3.60770 −0.134172
\(724\) 5.46410 0.203072
\(725\) 0 0
\(726\) −13.8038 −0.512309
\(727\) −8.78461 −0.325803 −0.162902 0.986642i \(-0.552085\pi\)
−0.162902 + 0.986642i \(0.552085\pi\)
\(728\) 25.8564 0.958302
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) 37.8564 1.40017
\(732\) 3.60770 0.133344
\(733\) −27.8564 −1.02890 −0.514450 0.857520i \(-0.672004\pi\)
−0.514450 + 0.857520i \(0.672004\pi\)
\(734\) 4.33975 0.160183
\(735\) 0 0
\(736\) 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 \(0.459327\pi\)
\(740\) 0 0
\(741\) −24.7846 −0.910485
\(742\) 28.3923 1.04231
\(743\) −17.1244 −0.628232 −0.314116 0.949385i \(-0.601708\pi\)
−0.314116 + 0.949385i \(0.601708\pi\)
\(744\) −0.535898 −0.0196470
\(745\) 0 0
\(746\) 10.7846 0.394853
\(747\) −21.5167 −0.787253
\(748\) 29.8564 1.09166
\(749\) −15.4641 −0.565046
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 18.9282 0.688412
\(757\) −1.71281 −0.0622532 −0.0311266 0.999515i \(-0.509910\pi\)
−0.0311266 + 0.999515i \(0.509910\pi\)
\(758\) 32.3923 1.17654
\(759\) 32.0000 1.16153
\(760\) 0 0
\(761\) 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
\(765\) 0 0
\(766\) 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 \(0.459302\pi\)
\(770\) 0 0
\(771\) −13.5692 −0.488683
\(772\) −15.8564 −0.570685
\(773\) −2.78461 −0.100155 −0.0500777 0.998745i \(-0.515947\pi\)
−0.0500777 + 0.998745i \(0.515947\pi\)
\(774\) 17.0718 0.613633
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 3.46410 0.124274
\(778\) −11.8564 −0.425073
\(779\) −12.3923 −0.444000
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) −43.7128 −1.56317
\(783\) 19.7128 0.704478
\(784\) 15.3923 0.549725
\(785\) 0 0
\(786\) 13.6077 0.485370
\(787\) −22.1962 −0.791207 −0.395604 0.918421i \(-0.629465\pi\)
−0.395604 + 0.918421i \(0.629465\pi\)
\(788\) 22.7846 0.811668
\(789\) −5.89488 −0.209863
\(790\) 0 0
\(791\) 49.8564 1.77269
\(792\) 13.4641 0.478426
\(793\) −26.9282 −0.956249
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −12.0526 −0.427192
\(797\) −10.5359 −0.373201 −0.186600 0.982436i \(-0.559747\pi\)
−0.186600 + 0.982436i \(0.559747\pi\)
\(798\) −21.4641 −0.759821
\(799\) −25.8564 −0.914734
\(800\) 0 0
\(801\) 4.92820 0.174129
\(802\) −32.9282 −1.16274
\(803\) −5.07180 −0.178980
\(804\) −2.67949 −0.0944984
\(805\) 0 0
\(806\) 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 \(0.222291\pi\)
\(810\) 0 0
\(811\) −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
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) −42.9282 −1.50187
\(818\) 3.07180 0.107403
\(819\) −63.7128 −2.22631
\(820\) 0 0
\(821\) 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 \(-0.820008\pi\)
−0.844341 + 0.535806i \(0.820008\pi\)
\(824\) 6.53590 0.227689
\(825\) 0 0
\(826\) −48.2487 −1.67879
\(827\) 16.3923 0.570016 0.285008 0.958525i \(-0.408004\pi\)
0.285008 + 0.958525i \(0.408004\pi\)
\(828\) −19.7128 −0.685068
\(829\) −51.5692 −1.79107 −0.895537 0.444988i \(-0.853208\pi\)
−0.895537 + 0.444988i \(0.853208\pi\)
\(830\) 0 0
\(831\) 16.2872 0.564996
\(832\) 5.46410 0.189434
\(833\) −84.1051 −2.91407
\(834\) −5.07180 −0.175622
\(835\) 0 0
\(836\) −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 \(0.548455\pi\)
\(840\) 0 0
\(841\) −4.71281 −0.162511
\(842\) −0.143594 −0.00494856
\(843\) 6.53590 0.225108
\(844\) −17.8564 −0.614643
\(845\) 0 0
\(846\) −11.6603 −0.400888
\(847\) 89.2295 3.06596
\(848\) 6.00000 0.206041
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) −2.14359 −0.0734383
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) −23.3205 −0.798011
\(855\) 0 0
\(856\) −3.26795 −0.111696
\(857\) 31.8564 1.08819 0.544097 0.839022i \(-0.316872\pi\)
0.544097 + 0.839022i \(0.316872\pi\)
\(858\) 21.8564 0.746165
\(859\) 44.4449 1.51644 0.758220 0.651999i \(-0.226069\pi\)
0.758220 + 0.651999i \(0.226069\pi\)
\(860\) 0 0
\(861\) 6.92820 0.236113
\(862\) −2.19615 −0.0748012
\(863\) 20.4449 0.695951 0.347976 0.937504i \(-0.386869\pi\)
0.347976 + 0.937504i \(0.386869\pi\)
\(864\) 4.00000 0.136083
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) −9.41154 −0.319633
\(868\) 3.46410 0.117579
\(869\) −47.7128 −1.61855
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 2.00000 0.0677285
\(873\) −4.92820 −0.166794
\(874\) 49.5692 1.67670
\(875\) 0 0
\(876\) −0.679492 −0.0229579
\(877\) 9.21539 0.311182 0.155591 0.987822i \(-0.450272\pi\)
0.155591 + 0.987822i \(0.450272\pi\)
\(878\) 13.5167 0.456165
\(879\) 4.39230 0.148149
\(880\) 0 0
\(881\) 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 \(-0.577067\pi\)
−0.239754 + 0.970834i \(0.577067\pi\)
\(884\) −29.8564 −1.00418
\(885\) 0 0
\(886\) 14.8756 0.499757
\(887\) −9.80385 −0.329181 −0.164590 0.986362i \(-0.552630\pi\)
−0.164590 + 0.986362i \(0.552630\pi\)
\(888\) 0.732051 0.0245660
\(889\) −17.3205 −0.580911
\(890\) 0 0
\(891\) −24.3923 −0.817173
\(892\) −16.0526 −0.537479
\(893\) 29.3205 0.981173
\(894\) −3.21539 −0.107539
\(895\) 0 0
\(896\) 4.73205 0.158087
\(897\) −32.0000 −1.06845
\(898\) −21.7128 −0.724566
\(899\) 3.60770 0.120323
\(900\) 0 0
\(901\) −32.7846 −1.09221
\(902\) 10.9282 0.363869
\(903\) 24.0000 0.798670
\(904\) 10.5359 0.350419
\(905\) 0 0
\(906\) 9.07180 0.301390
\(907\) −54.2487 −1.80130 −0.900649 0.434546i \(-0.856909\pi\)
−0.900649 + 0.434546i \(0.856909\pi\)
\(908\) −24.3923 −0.809487
\(909\) 23.3205 0.773492
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) −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 \(0.588999\pi\)
\(920\) 0 0
\(921\) 13.6077 0.448389
\(922\) 14.7846 0.486905
\(923\) 16.0000 0.526646
\(924\) 18.9282 0.622692
\(925\) 0 0
\(926\) −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 0
\(931\) 95.3731 3.12573
\(932\) −28.9282 −0.947575
\(933\) −2.10512 −0.0689185
\(934\) 25.1769 0.823814
\(935\) 0 0
\(936\) −13.4641 −0.440088
\(937\) 23.8564 0.779355 0.389677 0.920951i \(-0.372587\pi\)
0.389677 + 0.920951i \(0.372587\pi\)
\(938\) 17.3205 0.565535
\(939\) −17.4641 −0.569919
\(940\) 0 0
\(941\) 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
\(945\) 0 0
\(946\) 37.8564 1.23082
\(947\) −17.1769 −0.558175 −0.279087 0.960266i \(-0.590032\pi\)
−0.279087 + 0.960266i \(0.590032\pi\)
\(948\) −6.39230 −0.207612
\(949\) 5.07180 0.164637
\(950\) 0 0
\(951\) −3.03332 −0.0983622
\(952\) −25.8564 −0.838011
\(953\) −15.8564 −0.513639 −0.256820 0.966459i \(-0.582675\pi\)
−0.256820 + 0.966459i \(0.582675\pi\)
\(954\) −14.7846 −0.478669
\(955\) 0 0
\(956\) −20.7321 −0.670522
\(957\) 19.7128 0.637225
\(958\) 4.05256 0.130932
\(959\) −9.46410 −0.305612
\(960\) 0 0
\(961\) −30.4641 −0.982713
\(962\) −5.46410 −0.176170
\(963\) 8.05256 0.259490
\(964\) 4.92820 0.158727
\(965\) 0 0
\(966\) −27.7128 −0.891645
\(967\) 44.3923 1.42756 0.713780 0.700370i \(-0.246982\pi\)
0.713780 + 0.700370i \(0.246982\pi\)
\(968\) 18.8564 0.606068
\(969\) 24.7846 0.796196
\(970\) 0 0
\(971\) 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
\(975\) 0 0
\(976\) −4.92820 −0.157748
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) −8.28719 −0.264995
\(979\) 10.9282 0.349267
\(980\) 0 0
\(981\) −4.92820 −0.157345
\(982\) −14.9282 −0.476378
\(983\) −20.4449 −0.652090 −0.326045 0.945354i \(-0.605716\pi\)
−0.326045 + 0.945354i \(0.605716\pi\)
\(984\) 1.46410 0.0466739
\(985\) 0 0
\(986\) −26.9282 −0.857569
\(987\) −16.3923 −0.521773
\(988\) 33.8564 1.07712
\(989\) −55.4256 −1.76243
\(990\) 0 0
\(991\) 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
\(995\) 0 0
\(996\) −6.39230 −0.202548
\(997\) 31.8564 1.00890 0.504451 0.863440i \(-0.331695\pi\)
0.504451 + 0.863440i \(0.331695\pi\)
\(998\) 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 1850.2.a.x.1.1 2
5.2 odd 4 1850.2.b.l.149.4 4
5.3 odd 4 1850.2.b.l.149.1 4
5.4 even 2 370.2.a.e.1.2 2
15.14 odd 2 3330.2.a.bd.1.1 2
20.19 odd 2 2960.2.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.e.1.2 2 5.4 even 2
1850.2.a.x.1.1 2 1.1 even 1 trivial
1850.2.b.l.149.1 4 5.3 odd 4
1850.2.b.l.149.4 4 5.2 odd 4
2960.2.a.q.1.1 2 20.19 odd 2
3330.2.a.bd.1.1 2 15.14 odd 2