Properties

Label 171.2.a.e.1.2
Level $171$
Weight $2$
Character 171.1
Self dual yes
Analytic conductor $1.365$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.548230\) of defining polynomial
Character \(\chi\) \(=\) 171.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.27582 q^{2} -0.372281 q^{4} -2.15121 q^{5} +3.37228 q^{7} +3.02661 q^{8} +O(q^{10})\) \(q-1.27582 q^{2} -0.372281 q^{4} -2.15121 q^{5} +3.37228 q^{7} +3.02661 q^{8} +2.74456 q^{10} +4.70285 q^{11} +2.00000 q^{13} -4.30243 q^{14} -3.11684 q^{16} -2.15121 q^{17} +1.00000 q^{19} +0.800857 q^{20} -6.00000 q^{22} +6.85407 q^{23} -0.372281 q^{25} -2.55164 q^{26} -1.25544 q^{28} +6.85407 q^{29} -6.74456 q^{31} -2.07668 q^{32} +2.74456 q^{34} -7.25450 q^{35} -0.744563 q^{37} -1.27582 q^{38} -6.51087 q^{40} +2.55164 q^{41} +6.11684 q^{43} -1.75079 q^{44} -8.74456 q^{46} -9.00528 q^{47} +4.37228 q^{49} +0.474964 q^{50} -0.744563 q^{52} -11.9574 q^{53} -10.1168 q^{55} +10.2066 q^{56} -8.74456 q^{58} -5.10328 q^{59} +12.1168 q^{61} +8.60485 q^{62} +8.88316 q^{64} -4.30243 q^{65} -4.00000 q^{67} +0.800857 q^{68} +9.25544 q^{70} +13.7081 q^{71} +12.1168 q^{73} +0.949929 q^{74} -0.372281 q^{76} +15.8593 q^{77} -4.00000 q^{79} +6.70500 q^{80} -3.25544 q^{82} -1.75079 q^{83} +4.62772 q^{85} -7.80400 q^{86} +14.2337 q^{88} -11.9574 q^{89} +6.74456 q^{91} -2.55164 q^{92} +11.4891 q^{94} -2.15121 q^{95} -15.4891 q^{97} -5.57825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} + 2 q^{7} - 12 q^{10} + 8 q^{13} + 22 q^{16} + 4 q^{19} - 24 q^{22} + 10 q^{25} - 28 q^{28} - 4 q^{31} - 12 q^{34} + 20 q^{37} - 72 q^{40} - 10 q^{43} - 12 q^{46} + 6 q^{49} + 20 q^{52} - 6 q^{55} - 12 q^{58} + 14 q^{61} + 70 q^{64} - 16 q^{67} + 60 q^{70} + 14 q^{73} + 10 q^{76} - 16 q^{79} - 36 q^{82} + 30 q^{85} - 12 q^{88} + 4 q^{91} - 16 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.27582 −0.902142 −0.451071 0.892488i \(-0.648958\pi\)
−0.451071 + 0.892488i \(0.648958\pi\)
\(3\) 0 0
\(4\) −0.372281 −0.186141
\(5\) −2.15121 −0.962052 −0.481026 0.876706i \(-0.659736\pi\)
−0.481026 + 0.876706i \(0.659736\pi\)
\(6\) 0 0
\(7\) 3.37228 1.27460 0.637301 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(8\) 3.02661 1.07007
\(9\) 0 0
\(10\) 2.74456 0.867907
\(11\) 4.70285 1.41796 0.708982 0.705227i \(-0.249155\pi\)
0.708982 + 0.705227i \(0.249155\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −4.30243 −1.14987
\(15\) 0 0
\(16\) −3.11684 −0.779211
\(17\) −2.15121 −0.521746 −0.260873 0.965373i \(-0.584010\pi\)
−0.260873 + 0.965373i \(0.584010\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0.800857 0.179077
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 6.85407 1.42917 0.714586 0.699548i \(-0.246615\pi\)
0.714586 + 0.699548i \(0.246615\pi\)
\(24\) 0 0
\(25\) −0.372281 −0.0744563
\(26\) −2.55164 −0.500418
\(27\) 0 0
\(28\) −1.25544 −0.237255
\(29\) 6.85407 1.27277 0.636384 0.771372i \(-0.280429\pi\)
0.636384 + 0.771372i \(0.280429\pi\)
\(30\) 0 0
\(31\) −6.74456 −1.21136 −0.605680 0.795709i \(-0.707099\pi\)
−0.605680 + 0.795709i \(0.707099\pi\)
\(32\) −2.07668 −0.367108
\(33\) 0 0
\(34\) 2.74456 0.470689
\(35\) −7.25450 −1.22623
\(36\) 0 0
\(37\) −0.744563 −0.122405 −0.0612027 0.998125i \(-0.519494\pi\)
−0.0612027 + 0.998125i \(0.519494\pi\)
\(38\) −1.27582 −0.206965
\(39\) 0 0
\(40\) −6.51087 −1.02946
\(41\) 2.55164 0.398499 0.199250 0.979949i \(-0.436150\pi\)
0.199250 + 0.979949i \(0.436150\pi\)
\(42\) 0 0
\(43\) 6.11684 0.932810 0.466405 0.884571i \(-0.345549\pi\)
0.466405 + 0.884571i \(0.345549\pi\)
\(44\) −1.75079 −0.263941
\(45\) 0 0
\(46\) −8.74456 −1.28932
\(47\) −9.00528 −1.31356 −0.656778 0.754084i \(-0.728081\pi\)
−0.656778 + 0.754084i \(0.728081\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0.474964 0.0671701
\(51\) 0 0
\(52\) −0.744563 −0.103252
\(53\) −11.9574 −1.64247 −0.821234 0.570591i \(-0.806714\pi\)
−0.821234 + 0.570591i \(0.806714\pi\)
\(54\) 0 0
\(55\) −10.1168 −1.36415
\(56\) 10.2066 1.36391
\(57\) 0 0
\(58\) −8.74456 −1.14822
\(59\) −5.10328 −0.664391 −0.332195 0.943211i \(-0.607789\pi\)
−0.332195 + 0.943211i \(0.607789\pi\)
\(60\) 0 0
\(61\) 12.1168 1.55140 0.775701 0.631100i \(-0.217396\pi\)
0.775701 + 0.631100i \(0.217396\pi\)
\(62\) 8.60485 1.09282
\(63\) 0 0
\(64\) 8.88316 1.11039
\(65\) −4.30243 −0.533650
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0.800857 0.0971181
\(69\) 0 0
\(70\) 9.25544 1.10624
\(71\) 13.7081 1.62686 0.813428 0.581665i \(-0.197599\pi\)
0.813428 + 0.581665i \(0.197599\pi\)
\(72\) 0 0
\(73\) 12.1168 1.41817 0.709085 0.705123i \(-0.249108\pi\)
0.709085 + 0.705123i \(0.249108\pi\)
\(74\) 0.949929 0.110427
\(75\) 0 0
\(76\) −0.372281 −0.0427036
\(77\) 15.8593 1.80734
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 6.70500 0.749641
\(81\) 0 0
\(82\) −3.25544 −0.359503
\(83\) −1.75079 −0.192174 −0.0960868 0.995373i \(-0.530633\pi\)
−0.0960868 + 0.995373i \(0.530633\pi\)
\(84\) 0 0
\(85\) 4.62772 0.501947
\(86\) −7.80400 −0.841527
\(87\) 0 0
\(88\) 14.2337 1.51732
\(89\) −11.9574 −1.26748 −0.633738 0.773547i \(-0.718480\pi\)
−0.633738 + 0.773547i \(0.718480\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) −2.55164 −0.266027
\(93\) 0 0
\(94\) 11.4891 1.18501
\(95\) −2.15121 −0.220710
\(96\) 0 0
\(97\) −15.4891 −1.57268 −0.786341 0.617792i \(-0.788027\pi\)
−0.786341 + 0.617792i \(0.788027\pi\)
\(98\) −5.57825 −0.563488
\(99\) 0 0
\(100\) 0.138593 0.0138593
\(101\) 8.60485 0.856215 0.428107 0.903728i \(-0.359180\pi\)
0.428107 + 0.903728i \(0.359180\pi\)
\(102\) 0 0
\(103\) −18.7446 −1.84696 −0.923478 0.383651i \(-0.874667\pi\)
−0.923478 + 0.383651i \(0.874667\pi\)
\(104\) 6.05321 0.593566
\(105\) 0 0
\(106\) 15.2554 1.48174
\(107\) −9.40571 −0.909284 −0.454642 0.890674i \(-0.650233\pi\)
−0.454642 + 0.890674i \(0.650233\pi\)
\(108\) 0 0
\(109\) 16.7446 1.60384 0.801919 0.597433i \(-0.203812\pi\)
0.801919 + 0.597433i \(0.203812\pi\)
\(110\) 12.9073 1.23066
\(111\) 0 0
\(112\) −10.5109 −0.993184
\(113\) −1.75079 −0.164700 −0.0823500 0.996603i \(-0.526243\pi\)
−0.0823500 + 0.996603i \(0.526243\pi\)
\(114\) 0 0
\(115\) −14.7446 −1.37494
\(116\) −2.55164 −0.236914
\(117\) 0 0
\(118\) 6.51087 0.599375
\(119\) −7.25450 −0.665019
\(120\) 0 0
\(121\) 11.1168 1.01062
\(122\) −15.4589 −1.39958
\(123\) 0 0
\(124\) 2.51087 0.225483
\(125\) 11.5569 1.03368
\(126\) 0 0
\(127\) −1.25544 −0.111402 −0.0557010 0.998447i \(-0.517739\pi\)
−0.0557010 + 0.998447i \(0.517739\pi\)
\(128\) −7.17996 −0.634625
\(129\) 0 0
\(130\) 5.48913 0.481428
\(131\) −3.90200 −0.340919 −0.170460 0.985365i \(-0.554525\pi\)
−0.170460 + 0.985365i \(0.554525\pi\)
\(132\) 0 0
\(133\) 3.37228 0.292414
\(134\) 5.10328 0.440857
\(135\) 0 0
\(136\) −6.51087 −0.558303
\(137\) −16.6602 −1.42338 −0.711689 0.702495i \(-0.752069\pi\)
−0.711689 + 0.702495i \(0.752069\pi\)
\(138\) 0 0
\(139\) −14.1168 −1.19738 −0.598688 0.800983i \(-0.704311\pi\)
−0.598688 + 0.800983i \(0.704311\pi\)
\(140\) 2.70071 0.228252
\(141\) 0 0
\(142\) −17.4891 −1.46765
\(143\) 9.40571 0.786545
\(144\) 0 0
\(145\) −14.7446 −1.22447
\(146\) −15.4589 −1.27939
\(147\) 0 0
\(148\) 0.277187 0.0227846
\(149\) 1.35036 0.110626 0.0553128 0.998469i \(-0.482384\pi\)
0.0553128 + 0.998469i \(0.482384\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 3.02661 0.245490
\(153\) 0 0
\(154\) −20.2337 −1.63048
\(155\) 14.5090 1.16539
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 5.10328 0.405995
\(159\) 0 0
\(160\) 4.46738 0.353177
\(161\) 23.1138 1.82163
\(162\) 0 0
\(163\) 1.48913 0.116637 0.0583186 0.998298i \(-0.481426\pi\)
0.0583186 + 0.998298i \(0.481426\pi\)
\(164\) −0.949929 −0.0741770
\(165\) 0 0
\(166\) 2.23369 0.173368
\(167\) −4.30243 −0.332932 −0.166466 0.986047i \(-0.553236\pi\)
−0.166466 + 0.986047i \(0.553236\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −5.90414 −0.452827
\(171\) 0 0
\(172\) −2.27719 −0.173634
\(173\) −6.85407 −0.521105 −0.260553 0.965460i \(-0.583905\pi\)
−0.260553 + 0.965460i \(0.583905\pi\)
\(174\) 0 0
\(175\) −1.25544 −0.0949021
\(176\) −14.6581 −1.10489
\(177\) 0 0
\(178\) 15.2554 1.14344
\(179\) 9.40571 0.703016 0.351508 0.936185i \(-0.385669\pi\)
0.351508 + 0.936185i \(0.385669\pi\)
\(180\) 0 0
\(181\) −3.48913 −0.259345 −0.129672 0.991557i \(-0.541393\pi\)
−0.129672 + 0.991557i \(0.541393\pi\)
\(182\) −8.60485 −0.637834
\(183\) 0 0
\(184\) 20.7446 1.52931
\(185\) 1.60171 0.117760
\(186\) 0 0
\(187\) −10.1168 −0.739817
\(188\) 3.35250 0.244506
\(189\) 0 0
\(190\) 2.74456 0.199112
\(191\) −8.20442 −0.593651 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(192\) 0 0
\(193\) −18.2337 −1.31249 −0.656245 0.754548i \(-0.727856\pi\)
−0.656245 + 0.754548i \(0.727856\pi\)
\(194\) 19.7613 1.41878
\(195\) 0 0
\(196\) −1.62772 −0.116266
\(197\) −3.50157 −0.249477 −0.124738 0.992190i \(-0.539809\pi\)
−0.124738 + 0.992190i \(0.539809\pi\)
\(198\) 0 0
\(199\) 18.1168 1.28427 0.642135 0.766592i \(-0.278049\pi\)
0.642135 + 0.766592i \(0.278049\pi\)
\(200\) −1.12675 −0.0796732
\(201\) 0 0
\(202\) −10.9783 −0.772427
\(203\) 23.1138 1.62227
\(204\) 0 0
\(205\) −5.48913 −0.383377
\(206\) 23.9147 1.66622
\(207\) 0 0
\(208\) −6.23369 −0.432228
\(209\) 4.70285 0.325303
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 4.45150 0.305730
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −13.1586 −0.897412
\(216\) 0 0
\(217\) −22.7446 −1.54400
\(218\) −21.3631 −1.44689
\(219\) 0 0
\(220\) 3.76631 0.253925
\(221\) −4.30243 −0.289413
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −7.00314 −0.467917
\(225\) 0 0
\(226\) 2.23369 0.148583
\(227\) 12.9073 0.856686 0.428343 0.903616i \(-0.359097\pi\)
0.428343 + 0.903616i \(0.359097\pi\)
\(228\) 0 0
\(229\) 21.3723 1.41232 0.706160 0.708052i \(-0.250426\pi\)
0.706160 + 0.708052i \(0.250426\pi\)
\(230\) 18.8114 1.24039
\(231\) 0 0
\(232\) 20.7446 1.36195
\(233\) 7.25450 0.475258 0.237629 0.971356i \(-0.423630\pi\)
0.237629 + 0.971356i \(0.423630\pi\)
\(234\) 0 0
\(235\) 19.3723 1.26371
\(236\) 1.89986 0.123670
\(237\) 0 0
\(238\) 9.25544 0.599941
\(239\) −27.0158 −1.74751 −0.873755 0.486367i \(-0.838322\pi\)
−0.873755 + 0.486367i \(0.838322\pi\)
\(240\) 0 0
\(241\) 10.2337 0.659210 0.329605 0.944119i \(-0.393084\pi\)
0.329605 + 0.944119i \(0.393084\pi\)
\(242\) −14.1831 −0.911724
\(243\) 0 0
\(244\) −4.51087 −0.288779
\(245\) −9.40571 −0.600909
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) −20.4131 −1.29624
\(249\) 0 0
\(250\) −14.7446 −0.932528
\(251\) −14.9094 −0.941074 −0.470537 0.882380i \(-0.655940\pi\)
−0.470537 + 0.882380i \(0.655940\pi\)
\(252\) 0 0
\(253\) 32.2337 2.02651
\(254\) 1.60171 0.100500
\(255\) 0 0
\(256\) −8.60597 −0.537873
\(257\) −2.55164 −0.159167 −0.0795835 0.996828i \(-0.525359\pi\)
−0.0795835 + 0.996828i \(0.525359\pi\)
\(258\) 0 0
\(259\) −2.51087 −0.156018
\(260\) 1.60171 0.0993340
\(261\) 0 0
\(262\) 4.97825 0.307557
\(263\) −9.80614 −0.604672 −0.302336 0.953201i \(-0.597767\pi\)
−0.302336 + 0.953201i \(0.597767\pi\)
\(264\) 0 0
\(265\) 25.7228 1.58014
\(266\) −4.30243 −0.263799
\(267\) 0 0
\(268\) 1.48913 0.0909628
\(269\) −25.6655 −1.56485 −0.782426 0.622743i \(-0.786018\pi\)
−0.782426 + 0.622743i \(0.786018\pi\)
\(270\) 0 0
\(271\) 2.51087 0.152525 0.0762624 0.997088i \(-0.475701\pi\)
0.0762624 + 0.997088i \(0.475701\pi\)
\(272\) 6.70500 0.406550
\(273\) 0 0
\(274\) 21.2554 1.28409
\(275\) −1.75079 −0.105576
\(276\) 0 0
\(277\) −8.11684 −0.487694 −0.243847 0.969814i \(-0.578409\pi\)
−0.243847 + 0.969814i \(0.578409\pi\)
\(278\) 18.0106 1.08020
\(279\) 0 0
\(280\) −21.9565 −1.31215
\(281\) 10.3556 0.617766 0.308883 0.951100i \(-0.400045\pi\)
0.308883 + 0.951100i \(0.400045\pi\)
\(282\) 0 0
\(283\) 0.627719 0.0373140 0.0186570 0.999826i \(-0.494061\pi\)
0.0186570 + 0.999826i \(0.494061\pi\)
\(284\) −5.10328 −0.302824
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 8.60485 0.507928
\(288\) 0 0
\(289\) −12.3723 −0.727781
\(290\) 18.8114 1.10464
\(291\) 0 0
\(292\) −4.51087 −0.263979
\(293\) 26.4663 1.54618 0.773090 0.634296i \(-0.218710\pi\)
0.773090 + 0.634296i \(0.218710\pi\)
\(294\) 0 0
\(295\) 10.9783 0.639178
\(296\) −2.25350 −0.130982
\(297\) 0 0
\(298\) −1.72281 −0.0997999
\(299\) 13.7081 0.792762
\(300\) 0 0
\(301\) 20.6277 1.18896
\(302\) 5.10328 0.293661
\(303\) 0 0
\(304\) −3.11684 −0.178763
\(305\) −26.0659 −1.49253
\(306\) 0 0
\(307\) 2.51087 0.143303 0.0716516 0.997430i \(-0.477173\pi\)
0.0716516 + 0.997430i \(0.477173\pi\)
\(308\) −5.90414 −0.336420
\(309\) 0 0
\(310\) −18.5109 −1.05135
\(311\) 27.0158 1.53193 0.765964 0.642883i \(-0.222262\pi\)
0.765964 + 0.642883i \(0.222262\pi\)
\(312\) 0 0
\(313\) −15.4891 −0.875497 −0.437749 0.899097i \(-0.644224\pi\)
−0.437749 + 0.899097i \(0.644224\pi\)
\(314\) −2.55164 −0.143997
\(315\) 0 0
\(316\) 1.48913 0.0837698
\(317\) 35.0712 1.96979 0.984897 0.173139i \(-0.0553911\pi\)
0.984897 + 0.173139i \(0.0553911\pi\)
\(318\) 0 0
\(319\) 32.2337 1.80474
\(320\) −19.1096 −1.06826
\(321\) 0 0
\(322\) −29.4891 −1.64336
\(323\) −2.15121 −0.119697
\(324\) 0 0
\(325\) −0.744563 −0.0413009
\(326\) −1.89986 −0.105223
\(327\) 0 0
\(328\) 7.72281 0.426421
\(329\) −30.3683 −1.67426
\(330\) 0 0
\(331\) −26.9783 −1.48286 −0.741429 0.671031i \(-0.765852\pi\)
−0.741429 + 0.671031i \(0.765852\pi\)
\(332\) 0.651785 0.0357713
\(333\) 0 0
\(334\) 5.48913 0.300352
\(335\) 8.60485 0.470133
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 11.4824 0.624560
\(339\) 0 0
\(340\) −1.72281 −0.0934327
\(341\) −31.7187 −1.71766
\(342\) 0 0
\(343\) −8.86141 −0.478471
\(344\) 18.5133 0.998169
\(345\) 0 0
\(346\) 8.74456 0.470111
\(347\) 3.10114 0.166478 0.0832390 0.996530i \(-0.473474\pi\)
0.0832390 + 0.996530i \(0.473474\pi\)
\(348\) 0 0
\(349\) −10.8614 −0.581398 −0.290699 0.956815i \(-0.593888\pi\)
−0.290699 + 0.956815i \(0.593888\pi\)
\(350\) 1.60171 0.0856152
\(351\) 0 0
\(352\) −9.76631 −0.520546
\(353\) 22.3130 1.18760 0.593800 0.804612i \(-0.297627\pi\)
0.593800 + 0.804612i \(0.297627\pi\)
\(354\) 0 0
\(355\) −29.4891 −1.56512
\(356\) 4.45150 0.235929
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 27.8167 1.46811 0.734055 0.679090i \(-0.237626\pi\)
0.734055 + 0.679090i \(0.237626\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 4.45150 0.233966
\(363\) 0 0
\(364\) −2.51087 −0.131606
\(365\) −26.0659 −1.36435
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −21.3631 −1.11363
\(369\) 0 0
\(370\) −2.04350 −0.106236
\(371\) −40.3236 −2.09349
\(372\) 0 0
\(373\) 10.2337 0.529880 0.264940 0.964265i \(-0.414648\pi\)
0.264940 + 0.964265i \(0.414648\pi\)
\(374\) 12.9073 0.667420
\(375\) 0 0
\(376\) −27.2554 −1.40559
\(377\) 13.7081 0.706005
\(378\) 0 0
\(379\) 17.2554 0.886352 0.443176 0.896435i \(-0.353852\pi\)
0.443176 + 0.896435i \(0.353852\pi\)
\(380\) 0.800857 0.0410831
\(381\) 0 0
\(382\) 10.4674 0.535558
\(383\) 0.800857 0.0409219 0.0204609 0.999791i \(-0.493487\pi\)
0.0204609 + 0.999791i \(0.493487\pi\)
\(384\) 0 0
\(385\) −34.1168 −1.73876
\(386\) 23.2629 1.18405
\(387\) 0 0
\(388\) 5.76631 0.292740
\(389\) −16.6602 −0.844706 −0.422353 0.906431i \(-0.638796\pi\)
−0.422353 + 0.906431i \(0.638796\pi\)
\(390\) 0 0
\(391\) −14.7446 −0.745665
\(392\) 13.2332 0.668376
\(393\) 0 0
\(394\) 4.46738 0.225063
\(395\) 8.60485 0.432957
\(396\) 0 0
\(397\) 0.116844 0.00586423 0.00293212 0.999996i \(-0.499067\pi\)
0.00293212 + 0.999996i \(0.499067\pi\)
\(398\) −23.1138 −1.15859
\(399\) 0 0
\(400\) 1.16034 0.0580171
\(401\) −16.2598 −0.811975 −0.405987 0.913879i \(-0.633072\pi\)
−0.405987 + 0.913879i \(0.633072\pi\)
\(402\) 0 0
\(403\) −13.4891 −0.671941
\(404\) −3.20343 −0.159376
\(405\) 0 0
\(406\) −29.4891 −1.46352
\(407\) −3.50157 −0.173566
\(408\) 0 0
\(409\) −15.4891 −0.765888 −0.382944 0.923772i \(-0.625090\pi\)
−0.382944 + 0.923772i \(0.625090\pi\)
\(410\) 7.00314 0.345860
\(411\) 0 0
\(412\) 6.97825 0.343794
\(413\) −17.2097 −0.846834
\(414\) 0 0
\(415\) 3.76631 0.184881
\(416\) −4.15335 −0.203635
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) 1.75079 0.0855314 0.0427657 0.999085i \(-0.486383\pi\)
0.0427657 + 0.999085i \(0.486383\pi\)
\(420\) 0 0
\(421\) 36.9783 1.80221 0.901105 0.433601i \(-0.142757\pi\)
0.901105 + 0.433601i \(0.142757\pi\)
\(422\) 5.10328 0.248424
\(423\) 0 0
\(424\) −36.1902 −1.75755
\(425\) 0.800857 0.0388472
\(426\) 0 0
\(427\) 40.8614 1.97742
\(428\) 3.50157 0.169255
\(429\) 0 0
\(430\) 16.7881 0.809592
\(431\) −7.80400 −0.375905 −0.187953 0.982178i \(-0.560185\pi\)
−0.187953 + 0.982178i \(0.560185\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 29.0180 1.39291
\(435\) 0 0
\(436\) −6.23369 −0.298540
\(437\) 6.85407 0.327875
\(438\) 0 0
\(439\) 16.2337 0.774792 0.387396 0.921913i \(-0.373375\pi\)
0.387396 + 0.921913i \(0.373375\pi\)
\(440\) −30.6197 −1.45974
\(441\) 0 0
\(442\) 5.48913 0.261091
\(443\) −12.5069 −0.594218 −0.297109 0.954843i \(-0.596023\pi\)
−0.297109 + 0.954843i \(0.596023\pi\)
\(444\) 0 0
\(445\) 25.7228 1.21938
\(446\) 5.10328 0.241647
\(447\) 0 0
\(448\) 29.9565 1.41531
\(449\) 33.4695 1.57952 0.789761 0.613414i \(-0.210204\pi\)
0.789761 + 0.613414i \(0.210204\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0.651785 0.0306574
\(453\) 0 0
\(454\) −16.4674 −0.772852
\(455\) −14.5090 −0.680192
\(456\) 0 0
\(457\) −2.62772 −0.122919 −0.0614597 0.998110i \(-0.519576\pi\)
−0.0614597 + 0.998110i \(0.519576\pi\)
\(458\) −27.2672 −1.27411
\(459\) 0 0
\(460\) 5.48913 0.255932
\(461\) −5.65278 −0.263276 −0.131638 0.991298i \(-0.542024\pi\)
−0.131638 + 0.991298i \(0.542024\pi\)
\(462\) 0 0
\(463\) 3.37228 0.156723 0.0783616 0.996925i \(-0.475031\pi\)
0.0783616 + 0.996925i \(0.475031\pi\)
\(464\) −21.3631 −0.991755
\(465\) 0 0
\(466\) −9.25544 −0.428750
\(467\) −30.5174 −1.41218 −0.706089 0.708123i \(-0.749542\pi\)
−0.706089 + 0.708123i \(0.749542\pi\)
\(468\) 0 0
\(469\) −13.4891 −0.622870
\(470\) −24.7156 −1.14004
\(471\) 0 0
\(472\) −15.4456 −0.710943
\(473\) 28.7666 1.32269
\(474\) 0 0
\(475\) −0.372281 −0.0170814
\(476\) 2.70071 0.123787
\(477\) 0 0
\(478\) 34.4674 1.57650
\(479\) −8.45578 −0.386355 −0.193177 0.981164i \(-0.561879\pi\)
−0.193177 + 0.981164i \(0.561879\pi\)
\(480\) 0 0
\(481\) −1.48913 −0.0678983
\(482\) −13.0564 −0.594701
\(483\) 0 0
\(484\) −4.13859 −0.188118
\(485\) 33.3204 1.51300
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 36.6729 1.66010
\(489\) 0 0
\(490\) 12.0000 0.542105
\(491\) −6.85407 −0.309320 −0.154660 0.987968i \(-0.549428\pi\)
−0.154660 + 0.987968i \(0.549428\pi\)
\(492\) 0 0
\(493\) −14.7446 −0.664062
\(494\) −2.55164 −0.114804
\(495\) 0 0
\(496\) 21.0217 0.943904
\(497\) 46.2277 2.07360
\(498\) 0 0
\(499\) 6.11684 0.273828 0.136914 0.990583i \(-0.456282\pi\)
0.136914 + 0.990583i \(0.456282\pi\)
\(500\) −4.30243 −0.192410
\(501\) 0 0
\(502\) 19.0217 0.848982
\(503\) 22.1639 0.988240 0.494120 0.869394i \(-0.335490\pi\)
0.494120 + 0.869394i \(0.335490\pi\)
\(504\) 0 0
\(505\) −18.5109 −0.823723
\(506\) −41.1244 −1.82820
\(507\) 0 0
\(508\) 0.467376 0.0207365
\(509\) 10.3556 0.459006 0.229503 0.973308i \(-0.426290\pi\)
0.229503 + 0.973308i \(0.426290\pi\)
\(510\) 0 0
\(511\) 40.8614 1.80760
\(512\) 25.3396 1.11986
\(513\) 0 0
\(514\) 3.25544 0.143591
\(515\) 40.3236 1.77687
\(516\) 0 0
\(517\) −42.3505 −1.86257
\(518\) 3.20343 0.140750
\(519\) 0 0
\(520\) −13.0217 −0.571041
\(521\) −7.65492 −0.335368 −0.167684 0.985841i \(-0.553629\pi\)
−0.167684 + 0.985841i \(0.553629\pi\)
\(522\) 0 0
\(523\) 16.2337 0.709850 0.354925 0.934895i \(-0.384506\pi\)
0.354925 + 0.934895i \(0.384506\pi\)
\(524\) 1.45264 0.0634589
\(525\) 0 0
\(526\) 12.5109 0.545500
\(527\) 14.5090 0.632022
\(528\) 0 0
\(529\) 23.9783 1.04253
\(530\) −32.8177 −1.42551
\(531\) 0 0
\(532\) −1.25544 −0.0544301
\(533\) 5.10328 0.221048
\(534\) 0 0
\(535\) 20.2337 0.874779
\(536\) −12.1064 −0.522918
\(537\) 0 0
\(538\) 32.7446 1.41172
\(539\) 20.5622 0.885677
\(540\) 0 0
\(541\) 3.88316 0.166950 0.0834750 0.996510i \(-0.473398\pi\)
0.0834750 + 0.996510i \(0.473398\pi\)
\(542\) −3.20343 −0.137599
\(543\) 0 0
\(544\) 4.46738 0.191537
\(545\) −36.0211 −1.54298
\(546\) 0 0
\(547\) −12.2337 −0.523075 −0.261537 0.965193i \(-0.584229\pi\)
−0.261537 + 0.965193i \(0.584229\pi\)
\(548\) 6.20228 0.264948
\(549\) 0 0
\(550\) 2.23369 0.0952448
\(551\) 6.85407 0.291993
\(552\) 0 0
\(553\) −13.4891 −0.573616
\(554\) 10.3556 0.439969
\(555\) 0 0
\(556\) 5.25544 0.222880
\(557\) −1.35036 −0.0572165 −0.0286082 0.999591i \(-0.509108\pi\)
−0.0286082 + 0.999591i \(0.509108\pi\)
\(558\) 0 0
\(559\) 12.2337 0.517430
\(560\) 22.6111 0.955495
\(561\) 0 0
\(562\) −13.2119 −0.557312
\(563\) −25.8146 −1.08795 −0.543977 0.839100i \(-0.683082\pi\)
−0.543977 + 0.839100i \(0.683082\pi\)
\(564\) 0 0
\(565\) 3.76631 0.158450
\(566\) −0.800857 −0.0336625
\(567\) 0 0
\(568\) 41.4891 1.74085
\(569\) 20.5622 0.862012 0.431006 0.902349i \(-0.358159\pi\)
0.431006 + 0.902349i \(0.358159\pi\)
\(570\) 0 0
\(571\) 25.4891 1.06669 0.533343 0.845899i \(-0.320935\pi\)
0.533343 + 0.845899i \(0.320935\pi\)
\(572\) −3.50157 −0.146408
\(573\) 0 0
\(574\) −10.9783 −0.458223
\(575\) −2.55164 −0.106411
\(576\) 0 0
\(577\) −23.8832 −0.994269 −0.497134 0.867674i \(-0.665614\pi\)
−0.497134 + 0.867674i \(0.665614\pi\)
\(578\) 15.7848 0.656562
\(579\) 0 0
\(580\) 5.48913 0.227924
\(581\) −5.90414 −0.244945
\(582\) 0 0
\(583\) −56.2337 −2.32896
\(584\) 36.6729 1.51754
\(585\) 0 0
\(586\) −33.7663 −1.39487
\(587\) −32.1191 −1.32570 −0.662849 0.748753i \(-0.730653\pi\)
−0.662849 + 0.748753i \(0.730653\pi\)
\(588\) 0 0
\(589\) −6.74456 −0.277905
\(590\) −14.0063 −0.576629
\(591\) 0 0
\(592\) 2.32069 0.0953796
\(593\) 27.4163 1.12585 0.562926 0.826508i \(-0.309676\pi\)
0.562926 + 0.826508i \(0.309676\pi\)
\(594\) 0 0
\(595\) 15.6060 0.639782
\(596\) −0.502713 −0.0205919
\(597\) 0 0
\(598\) −17.4891 −0.715184
\(599\) 8.60485 0.351585 0.175792 0.984427i \(-0.443751\pi\)
0.175792 + 0.984427i \(0.443751\pi\)
\(600\) 0 0
\(601\) −36.7446 −1.49884 −0.749421 0.662094i \(-0.769668\pi\)
−0.749421 + 0.662094i \(0.769668\pi\)
\(602\) −26.3173 −1.07261
\(603\) 0 0
\(604\) 1.48913 0.0605916
\(605\) −23.9147 −0.972271
\(606\) 0 0
\(607\) 17.2554 0.700377 0.350188 0.936679i \(-0.386118\pi\)
0.350188 + 0.936679i \(0.386118\pi\)
\(608\) −2.07668 −0.0842204
\(609\) 0 0
\(610\) 33.2554 1.34647
\(611\) −18.0106 −0.728629
\(612\) 0 0
\(613\) −37.6060 −1.51889 −0.759445 0.650571i \(-0.774530\pi\)
−0.759445 + 0.650571i \(0.774530\pi\)
\(614\) −3.20343 −0.129280
\(615\) 0 0
\(616\) 48.0000 1.93398
\(617\) 2.15121 0.0866046 0.0433023 0.999062i \(-0.486212\pi\)
0.0433023 + 0.999062i \(0.486212\pi\)
\(618\) 0 0
\(619\) −38.9783 −1.56667 −0.783334 0.621601i \(-0.786483\pi\)
−0.783334 + 0.621601i \(0.786483\pi\)
\(620\) −5.40143 −0.216927
\(621\) 0 0
\(622\) −34.4674 −1.38202
\(623\) −40.3236 −1.61553
\(624\) 0 0
\(625\) −23.0000 −0.920000
\(626\) 19.7613 0.789822
\(627\) 0 0
\(628\) −0.744563 −0.0297113
\(629\) 1.60171 0.0638645
\(630\) 0 0
\(631\) −2.11684 −0.0842702 −0.0421351 0.999112i \(-0.513416\pi\)
−0.0421351 + 0.999112i \(0.513416\pi\)
\(632\) −12.1064 −0.481568
\(633\) 0 0
\(634\) −44.7446 −1.77703
\(635\) 2.70071 0.107175
\(636\) 0 0
\(637\) 8.74456 0.346472
\(638\) −41.1244 −1.62813
\(639\) 0 0
\(640\) 15.4456 0.610542
\(641\) −10.3556 −0.409023 −0.204512 0.978864i \(-0.565561\pi\)
−0.204512 + 0.978864i \(0.565561\pi\)
\(642\) 0 0
\(643\) −17.8832 −0.705243 −0.352621 0.935766i \(-0.614710\pi\)
−0.352621 + 0.935766i \(0.614710\pi\)
\(644\) −8.60485 −0.339079
\(645\) 0 0
\(646\) 2.74456 0.107983
\(647\) −17.6101 −0.692326 −0.346163 0.938174i \(-0.612516\pi\)
−0.346163 + 0.938174i \(0.612516\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0.949929 0.0372593
\(651\) 0 0
\(652\) −0.554374 −0.0217109
\(653\) −2.95207 −0.115523 −0.0577617 0.998330i \(-0.518396\pi\)
−0.0577617 + 0.998330i \(0.518396\pi\)
\(654\) 0 0
\(655\) 8.39403 0.327982
\(656\) −7.95307 −0.310515
\(657\) 0 0
\(658\) 38.7446 1.51042
\(659\) −41.9253 −1.63318 −0.816588 0.577221i \(-0.804137\pi\)
−0.816588 + 0.577221i \(0.804137\pi\)
\(660\) 0 0
\(661\) 4.74456 0.184542 0.0922710 0.995734i \(-0.470587\pi\)
0.0922710 + 0.995734i \(0.470587\pi\)
\(662\) 34.4194 1.33775
\(663\) 0 0
\(664\) −5.29894 −0.205639
\(665\) −7.25450 −0.281317
\(666\) 0 0
\(667\) 46.9783 1.81901
\(668\) 1.60171 0.0619721
\(669\) 0 0
\(670\) −10.9783 −0.424127
\(671\) 56.9838 2.19983
\(672\) 0 0
\(673\) −36.7446 −1.41640 −0.708199 0.706012i \(-0.750492\pi\)
−0.708199 + 0.706012i \(0.750492\pi\)
\(674\) −17.8615 −0.687999
\(675\) 0 0
\(676\) 3.35053 0.128867
\(677\) −13.5591 −0.521117 −0.260559 0.965458i \(-0.583907\pi\)
−0.260559 + 0.965458i \(0.583907\pi\)
\(678\) 0 0
\(679\) −52.2337 −2.00454
\(680\) 14.0063 0.537116
\(681\) 0 0
\(682\) 40.4674 1.54958
\(683\) 35.2203 1.34767 0.673833 0.738884i \(-0.264647\pi\)
0.673833 + 0.738884i \(0.264647\pi\)
\(684\) 0 0
\(685\) 35.8397 1.36936
\(686\) 11.3056 0.431649
\(687\) 0 0
\(688\) −19.0652 −0.726856
\(689\) −23.9147 −0.911078
\(690\) 0 0
\(691\) 9.88316 0.375973 0.187986 0.982172i \(-0.439804\pi\)
0.187986 + 0.982172i \(0.439804\pi\)
\(692\) 2.55164 0.0969989
\(693\) 0 0
\(694\) −3.95650 −0.150187
\(695\) 30.3683 1.15194
\(696\) 0 0
\(697\) −5.48913 −0.207915
\(698\) 13.8572 0.524503
\(699\) 0 0
\(700\) 0.467376 0.0176651
\(701\) 29.0180 1.09599 0.547997 0.836480i \(-0.315390\pi\)
0.547997 + 0.836480i \(0.315390\pi\)
\(702\) 0 0
\(703\) −0.744563 −0.0280817
\(704\) 41.7762 1.57450
\(705\) 0 0
\(706\) −28.4674 −1.07138
\(707\) 29.0180 1.09133
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 37.6228 1.41196
\(711\) 0 0
\(712\) −36.1902 −1.35628
\(713\) −46.2277 −1.73124
\(714\) 0 0
\(715\) −20.2337 −0.756697
\(716\) −3.50157 −0.130860
\(717\) 0 0
\(718\) −35.4891 −1.32444
\(719\) −7.40357 −0.276107 −0.138053 0.990425i \(-0.544085\pi\)
−0.138053 + 0.990425i \(0.544085\pi\)
\(720\) 0 0
\(721\) −63.2119 −2.35414
\(722\) −1.27582 −0.0474811
\(723\) 0 0
\(724\) 1.29894 0.0482746
\(725\) −2.55164 −0.0947656
\(726\) 0 0
\(727\) 14.3505 0.532232 0.266116 0.963941i \(-0.414260\pi\)
0.266116 + 0.963941i \(0.414260\pi\)
\(728\) 20.4131 0.756561
\(729\) 0 0
\(730\) 33.2554 1.23084
\(731\) −13.1586 −0.486690
\(732\) 0 0
\(733\) −50.4674 −1.86406 −0.932028 0.362387i \(-0.881962\pi\)
−0.932028 + 0.362387i \(0.881962\pi\)
\(734\) −10.2066 −0.376731
\(735\) 0 0
\(736\) −14.2337 −0.524661
\(737\) −18.8114 −0.692928
\(738\) 0 0
\(739\) 9.88316 0.363558 0.181779 0.983339i \(-0.441814\pi\)
0.181779 + 0.983339i \(0.441814\pi\)
\(740\) −0.596288 −0.0219200
\(741\) 0 0
\(742\) 51.4456 1.88863
\(743\) −42.7261 −1.56747 −0.783735 0.621096i \(-0.786688\pi\)
−0.783735 + 0.621096i \(0.786688\pi\)
\(744\) 0 0
\(745\) −2.90491 −0.106428
\(746\) −13.0564 −0.478027
\(747\) 0 0
\(748\) 3.76631 0.137710
\(749\) −31.7187 −1.15898
\(750\) 0 0
\(751\) −44.4674 −1.62264 −0.811319 0.584604i \(-0.801250\pi\)
−0.811319 + 0.584604i \(0.801250\pi\)
\(752\) 28.0681 1.02354
\(753\) 0 0
\(754\) −17.4891 −0.636916
\(755\) 8.60485 0.313163
\(756\) 0 0
\(757\) 12.1168 0.440394 0.220197 0.975455i \(-0.429330\pi\)
0.220197 + 0.975455i \(0.429330\pi\)
\(758\) −22.0148 −0.799615
\(759\) 0 0
\(760\) −6.51087 −0.236174
\(761\) 25.2651 0.915858 0.457929 0.888989i \(-0.348591\pi\)
0.457929 + 0.888989i \(0.348591\pi\)
\(762\) 0 0
\(763\) 56.4674 2.04426
\(764\) 3.05435 0.110503
\(765\) 0 0
\(766\) −1.02175 −0.0369173
\(767\) −10.2066 −0.368538
\(768\) 0 0
\(769\) 24.1168 0.869676 0.434838 0.900509i \(-0.356806\pi\)
0.434838 + 0.900509i \(0.356806\pi\)
\(770\) 43.5270 1.56860
\(771\) 0 0
\(772\) 6.78806 0.244308
\(773\) −2.55164 −0.0917762 −0.0458881 0.998947i \(-0.514612\pi\)
−0.0458881 + 0.998947i \(0.514612\pi\)
\(774\) 0 0
\(775\) 2.51087 0.0901933
\(776\) −46.8795 −1.68288
\(777\) 0 0
\(778\) 21.2554 0.762044
\(779\) 2.55164 0.0914220
\(780\) 0 0
\(781\) 64.4674 2.30682
\(782\) 18.8114 0.672695
\(783\) 0 0
\(784\) −13.6277 −0.486704
\(785\) −4.30243 −0.153560
\(786\) 0 0
\(787\) 41.9565 1.49559 0.747794 0.663931i \(-0.231113\pi\)
0.747794 + 0.663931i \(0.231113\pi\)
\(788\) 1.30357 0.0464377
\(789\) 0 0
\(790\) −10.9783 −0.390589
\(791\) −5.90414 −0.209927
\(792\) 0 0
\(793\) 24.2337 0.860563
\(794\) −0.149072 −0.00529037
\(795\) 0 0
\(796\) −6.74456 −0.239055
\(797\) −11.1565 −0.395183 −0.197592 0.980284i \(-0.563312\pi\)
−0.197592 + 0.980284i \(0.563312\pi\)
\(798\) 0 0
\(799\) 19.3723 0.685342
\(800\) 0.773108 0.0273335
\(801\) 0 0
\(802\) 20.7446 0.732516
\(803\) 56.9838 2.01091
\(804\) 0 0
\(805\) −49.7228 −1.75250
\(806\) 17.2097 0.606186
\(807\) 0 0
\(808\) 26.0435 0.916207
\(809\) 34.6708 1.21896 0.609480 0.792802i \(-0.291378\pi\)
0.609480 + 0.792802i \(0.291378\pi\)
\(810\) 0 0
\(811\) −25.2554 −0.886838 −0.443419 0.896314i \(-0.646235\pi\)
−0.443419 + 0.896314i \(0.646235\pi\)
\(812\) −8.60485 −0.301971
\(813\) 0 0
\(814\) 4.46738 0.156581
\(815\) −3.20343 −0.112211
\(816\) 0 0
\(817\) 6.11684 0.214001
\(818\) 19.7613 0.690939
\(819\) 0 0
\(820\) 2.04350 0.0713621
\(821\) 17.4611 0.609395 0.304698 0.952449i \(-0.401445\pi\)
0.304698 + 0.952449i \(0.401445\pi\)
\(822\) 0 0
\(823\) −31.6060 −1.10171 −0.550857 0.834599i \(-0.685699\pi\)
−0.550857 + 0.834599i \(0.685699\pi\)
\(824\) −56.7324 −1.97637
\(825\) 0 0
\(826\) 21.9565 0.763964
\(827\) 42.7261 1.48573 0.742866 0.669440i \(-0.233466\pi\)
0.742866 + 0.669440i \(0.233466\pi\)
\(828\) 0 0
\(829\) −12.7446 −0.442637 −0.221318 0.975202i \(-0.571036\pi\)
−0.221318 + 0.975202i \(0.571036\pi\)
\(830\) −4.80514 −0.166789
\(831\) 0 0
\(832\) 17.7663 0.615936
\(833\) −9.40571 −0.325889
\(834\) 0 0
\(835\) 9.25544 0.320298
\(836\) −1.75079 −0.0605522
\(837\) 0 0
\(838\) −2.23369 −0.0771615
\(839\) 7.80400 0.269424 0.134712 0.990885i \(-0.456989\pi\)
0.134712 + 0.990885i \(0.456989\pi\)
\(840\) 0 0
\(841\) 17.9783 0.619940
\(842\) −47.1776 −1.62585
\(843\) 0 0
\(844\) 1.48913 0.0512578
\(845\) 19.3609 0.666036
\(846\) 0 0
\(847\) 37.4891 1.28814
\(848\) 37.2692 1.27983
\(849\) 0 0
\(850\) −1.02175 −0.0350457
\(851\) −5.10328 −0.174938
\(852\) 0 0
\(853\) 30.4674 1.04318 0.521592 0.853195i \(-0.325339\pi\)
0.521592 + 0.853195i \(0.325339\pi\)
\(854\) −52.1318 −1.78391
\(855\) 0 0
\(856\) −28.4674 −0.972995
\(857\) −42.0743 −1.43723 −0.718616 0.695407i \(-0.755224\pi\)
−0.718616 + 0.695407i \(0.755224\pi\)
\(858\) 0 0
\(859\) −14.1168 −0.481661 −0.240830 0.970567i \(-0.577420\pi\)
−0.240830 + 0.970567i \(0.577420\pi\)
\(860\) 4.89871 0.167045
\(861\) 0 0
\(862\) 9.95650 0.339120
\(863\) −22.3130 −0.759543 −0.379771 0.925080i \(-0.623997\pi\)
−0.379771 + 0.925080i \(0.623997\pi\)
\(864\) 0 0
\(865\) 14.7446 0.501330
\(866\) 28.0681 0.953791
\(867\) 0 0
\(868\) 8.46738 0.287401
\(869\) −18.8114 −0.638134
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 50.6792 1.71621
\(873\) 0 0
\(874\) −8.74456 −0.295789
\(875\) 38.9732 1.31753
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −20.7113 −0.698972
\(879\) 0 0
\(880\) 31.5326 1.06296
\(881\) −25.2651 −0.851201 −0.425601 0.904911i \(-0.639937\pi\)
−0.425601 + 0.904911i \(0.639937\pi\)
\(882\) 0 0
\(883\) −14.1168 −0.475070 −0.237535 0.971379i \(-0.576339\pi\)
−0.237535 + 0.971379i \(0.576339\pi\)
\(884\) 1.60171 0.0538714
\(885\) 0 0
\(886\) 15.9565 0.536069
\(887\) −10.2066 −0.342703 −0.171351 0.985210i \(-0.554813\pi\)
−0.171351 + 0.985210i \(0.554813\pi\)
\(888\) 0 0
\(889\) −4.23369 −0.141993
\(890\) −32.8177 −1.10005
\(891\) 0 0
\(892\) 1.48913 0.0498596
\(893\) −9.00528 −0.301350
\(894\) 0 0
\(895\) −20.2337 −0.676338
\(896\) −24.2128 −0.808894
\(897\) 0 0
\(898\) −42.7011 −1.42495
\(899\) −46.2277 −1.54178
\(900\) 0 0
\(901\) 25.7228 0.856951
\(902\) −15.3098 −0.509762
\(903\) 0 0
\(904\) −5.29894 −0.176240
\(905\) 7.50585 0.249503
\(906\) 0 0
\(907\) 34.7446 1.15367 0.576837 0.816859i \(-0.304287\pi\)
0.576837 + 0.816859i \(0.304287\pi\)
\(908\) −4.80514 −0.159464
\(909\) 0 0
\(910\) 18.5109 0.613630
\(911\) 24.7156 0.818863 0.409432 0.912341i \(-0.365727\pi\)
0.409432 + 0.912341i \(0.365727\pi\)
\(912\) 0 0
\(913\) −8.23369 −0.272495
\(914\) 3.35250 0.110891
\(915\) 0 0
\(916\) −7.95650 −0.262890
\(917\) −13.1586 −0.434536
\(918\) 0 0
\(919\) −26.9783 −0.889930 −0.444965 0.895548i \(-0.646784\pi\)
−0.444965 + 0.895548i \(0.646784\pi\)
\(920\) −44.6260 −1.47127
\(921\) 0 0
\(922\) 7.21194 0.237513
\(923\) 27.4163 0.902418
\(924\) 0 0
\(925\) 0.277187 0.00911384
\(926\) −4.30243 −0.141387
\(927\) 0 0
\(928\) −14.2337 −0.467244
\(929\) −46.2277 −1.51668 −0.758341 0.651858i \(-0.773990\pi\)
−0.758341 + 0.651858i \(0.773990\pi\)
\(930\) 0 0
\(931\) 4.37228 0.143296
\(932\) −2.70071 −0.0884648
\(933\) 0 0
\(934\) 38.9348 1.27398
\(935\) 21.7635 0.711742
\(936\) 0 0
\(937\) 12.1168 0.395840 0.197920 0.980218i \(-0.436581\pi\)
0.197920 + 0.980218i \(0.436581\pi\)
\(938\) 17.2097 0.561917
\(939\) 0 0
\(940\) −7.21194 −0.235227
\(941\) 10.3556 0.337584 0.168792 0.985652i \(-0.446013\pi\)
0.168792 + 0.985652i \(0.446013\pi\)
\(942\) 0 0
\(943\) 17.4891 0.569524
\(944\) 15.9061 0.517701
\(945\) 0 0
\(946\) −36.7011 −1.19325
\(947\) 11.9574 0.388562 0.194281 0.980946i \(-0.437763\pi\)
0.194281 + 0.980946i \(0.437763\pi\)
\(948\) 0 0
\(949\) 24.2337 0.786659
\(950\) 0.474964 0.0154099
\(951\) 0 0
\(952\) −21.9565 −0.711614
\(953\) 24.8646 0.805444 0.402722 0.915322i \(-0.368064\pi\)
0.402722 + 0.915322i \(0.368064\pi\)
\(954\) 0 0
\(955\) 17.6495 0.571123
\(956\) 10.0575 0.325283
\(957\) 0 0
\(958\) 10.7881 0.348546
\(959\) −56.1829 −1.81424
\(960\) 0 0
\(961\) 14.4891 0.467391
\(962\) 1.89986 0.0612538
\(963\) 0 0
\(964\) −3.80981 −0.122706
\(965\) 39.2246 1.26268
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 33.6463 1.08143
\(969\) 0 0
\(970\) −42.5109 −1.36494
\(971\) −34.4194 −1.10457 −0.552286 0.833655i \(-0.686244\pi\)
−0.552286 + 0.833655i \(0.686244\pi\)
\(972\) 0 0
\(973\) −47.6060 −1.52618
\(974\) −10.2066 −0.327039
\(975\) 0 0
\(976\) −37.7663 −1.20887
\(977\) 17.0606 0.545818 0.272909 0.962040i \(-0.412014\pi\)
0.272909 + 0.962040i \(0.412014\pi\)
\(978\) 0 0
\(979\) −56.2337 −1.79724
\(980\) 3.50157 0.111854
\(981\) 0 0
\(982\) 8.74456 0.279050
\(983\) −39.2246 −1.25107 −0.625534 0.780197i \(-0.715119\pi\)
−0.625534 + 0.780197i \(0.715119\pi\)
\(984\) 0 0
\(985\) 7.53262 0.240009
\(986\) 18.8114 0.599078
\(987\) 0 0
\(988\) −0.744563 −0.0236877
\(989\) 41.9253 1.33315
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 14.0063 0.444700
\(993\) 0 0
\(994\) −58.9783 −1.87068
\(995\) −38.9732 −1.23553
\(996\) 0 0
\(997\) 32.3505 1.02455 0.512276 0.858821i \(-0.328803\pi\)
0.512276 + 0.858821i \(0.328803\pi\)
\(998\) −7.80400 −0.247031
\(999\) 0 0
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 171.2.a.e.1.2 4
3.2 odd 2 inner 171.2.a.e.1.3 yes 4
4.3 odd 2 2736.2.a.bf.1.2 4
5.4 even 2 4275.2.a.bp.1.3 4
7.6 odd 2 8379.2.a.bw.1.2 4
12.11 even 2 2736.2.a.bf.1.3 4
15.14 odd 2 4275.2.a.bp.1.2 4
19.18 odd 2 3249.2.a.bf.1.3 4
21.20 even 2 8379.2.a.bw.1.3 4
57.56 even 2 3249.2.a.bf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.2 4 1.1 even 1 trivial
171.2.a.e.1.3 yes 4 3.2 odd 2 inner
2736.2.a.bf.1.2 4 4.3 odd 2
2736.2.a.bf.1.3 4 12.11 even 2
3249.2.a.bf.1.2 4 57.56 even 2
3249.2.a.bf.1.3 4 19.18 odd 2
4275.2.a.bp.1.2 4 15.14 odd 2
4275.2.a.bp.1.3 4 5.4 even 2
8379.2.a.bw.1.2 4 7.6 odd 2
8379.2.a.bw.1.3 4 21.20 even 2