Properties

Label 171.2.a.e.1.3
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 1.3
Root \(-1.82405\) 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 \(-0.351042\pi\)
0.451071 + 0.892488i \(0.351042\pi\)
\(3\) 0 0
\(4\) −0.372281 −0.186141
\(5\) 2.15121 0.962052 0.481026 0.876706i \(-0.340264\pi\)
0.481026 + 0.876706i \(0.340264\pi\)
\(6\) 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 \(-0.750845\pi\)
−0.708982 + 0.705227i \(0.750845\pi\)
\(12\) 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 \(-0.415990\pi\)
0.260873 + 0.965373i \(0.415990\pi\)
\(18\) 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 \(-0.753385\pi\)
−0.714586 + 0.699548i \(0.753385\pi\)
\(24\) 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 \(-0.719571\pi\)
−0.636384 + 0.771372i \(0.719571\pi\)
\(30\) 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 \(-0.563850\pi\)
−0.199250 + 0.979949i \(0.563850\pi\)
\(42\) 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 \(-0.271919\pi\)
0.656778 + 0.754084i \(0.271919\pi\)
\(48\) 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 \(-0.193286\pi\)
0.821234 + 0.570591i \(0.193286\pi\)
\(54\) 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 \(-0.392211\pi\)
0.332195 + 0.943211i \(0.392211\pi\)
\(60\) 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 \(-0.802401\pi\)
−0.813428 + 0.581665i \(0.802401\pi\)
\(72\) 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 \(-0.469367\pi\)
0.0960868 + 0.995373i \(0.469367\pi\)
\(84\) 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 \(-0.281520\pi\)
0.633738 + 0.773547i \(0.281520\pi\)
\(90\) 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 \(-0.640820\pi\)
−0.428107 + 0.903728i \(0.640820\pi\)
\(102\) 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 \(-0.349767\pi\)
0.454642 + 0.890674i \(0.349767\pi\)
\(108\) 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 \(-0.473757\pi\)
0.0823500 + 0.996603i \(0.473757\pi\)
\(114\) 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 \(-0.445475\pi\)
0.170460 + 0.985365i \(0.445475\pi\)
\(132\) 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 \(-0.247931\pi\)
0.711689 + 0.702495i \(0.247931\pi\)
\(138\) 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 \(-0.517616\pi\)
−0.0553128 + 0.998469i \(0.517616\pi\)
\(150\) 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 \(-0.446764\pi\)
0.166466 + 0.986047i \(0.446764\pi\)
\(168\) 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 \(-0.416095\pi\)
0.260553 + 0.965460i \(0.416095\pi\)
\(174\) 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 \(-0.614331\pi\)
−0.351508 + 0.936185i \(0.614331\pi\)
\(180\) 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 \(-0.404072\pi\)
0.296826 + 0.954932i \(0.404072\pi\)
\(192\) 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 \(-0.460191\pi\)
0.124738 + 0.992190i \(0.460191\pi\)
\(198\) 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 \(-0.640903\pi\)
−0.428343 + 0.903616i \(0.640903\pi\)
\(228\) 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 \(-0.576370\pi\)
−0.237629 + 0.971356i \(0.576370\pi\)
\(234\) 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 \(-0.161678\pi\)
0.873755 + 0.486367i \(0.161678\pi\)
\(240\) 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 \(-0.344060\pi\)
0.470537 + 0.882380i \(0.344060\pi\)
\(252\) 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 \(-0.474641\pi\)
0.0795835 + 0.996828i \(0.474641\pi\)
\(258\) 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 \(-0.402233\pi\)
0.302336 + 0.953201i \(0.402233\pi\)
\(264\) 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 \(-0.213982\pi\)
0.782426 + 0.622743i \(0.213982\pi\)
\(270\) 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 \(-0.599955\pi\)
−0.308883 + 0.951100i \(0.599955\pi\)
\(282\) 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 \(-0.781290\pi\)
−0.773090 + 0.634296i \(0.781290\pi\)
\(294\) 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 \(-0.777738\pi\)
−0.765964 + 0.642883i \(0.777738\pi\)
\(312\) 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 \(-0.944609\pi\)
−0.984897 + 0.173139i \(0.944609\pi\)
\(318\) 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 \(-0.526526\pi\)
−0.0832390 + 0.996530i \(0.526526\pi\)
\(348\) 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 \(-0.702373\pi\)
−0.593800 + 0.804612i \(0.702373\pi\)
\(354\) 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 \(-0.762374\pi\)
−0.734055 + 0.679090i \(0.762374\pi\)
\(360\) 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 \(-0.506513\pi\)
−0.0204609 + 0.999791i \(0.506513\pi\)
\(384\) 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 \(-0.361204\pi\)
0.422353 + 0.906431i \(0.361204\pi\)
\(390\) 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 \(-0.366928\pi\)
0.405987 + 0.913879i \(0.366928\pi\)
\(402\) 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 \(-0.513617\pi\)
−0.0427657 + 0.999085i \(0.513617\pi\)
\(420\) 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 \(-0.439815\pi\)
0.187953 + 0.982178i \(0.439815\pi\)
\(432\) 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 \(-0.403977\pi\)
0.297109 + 0.954843i \(0.403977\pi\)
\(444\) 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 \(-0.789796\pi\)
−0.789761 + 0.613414i \(0.789796\pi\)
\(450\) 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 \(-0.457976\pi\)
0.131638 + 0.991298i \(0.457976\pi\)
\(462\) 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 \(-0.250458\pi\)
0.706089 + 0.708123i \(0.250458\pi\)
\(468\) 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 \(-0.438121\pi\)
0.193177 + 0.981164i \(0.438121\pi\)
\(480\) 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 \(-0.450572\pi\)
0.154660 + 0.987968i \(0.450572\pi\)
\(492\) 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 \(-0.664510\pi\)
−0.494120 + 0.869394i \(0.664510\pi\)
\(504\) 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 \(-0.573710\pi\)
−0.229503 + 0.973308i \(0.573710\pi\)
\(510\) 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 \(-0.446371\pi\)
0.167684 + 0.985841i \(0.446371\pi\)
\(522\) 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 \(-0.490892\pi\)
0.0286082 + 0.999591i \(0.490892\pi\)
\(558\) 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 \(-0.316918\pi\)
0.543977 + 0.839100i \(0.316918\pi\)
\(564\) 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 \(-0.641841\pi\)
−0.431006 + 0.902349i \(0.641841\pi\)
\(570\) 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 \(-0.269347\pi\)
0.662849 + 0.748753i \(0.269347\pi\)
\(588\) 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 \(-0.690324\pi\)
−0.562926 + 0.826508i \(0.690324\pi\)
\(594\) 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 \(-0.556249\pi\)
−0.175792 + 0.984427i \(0.556249\pi\)
\(600\) 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 \(-0.513788\pi\)
−0.0433023 + 0.999062i \(0.513788\pi\)
\(618\) 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 \(-0.434439\pi\)
0.204512 + 0.978864i \(0.434439\pi\)
\(642\) 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 \(-0.387484\pi\)
0.346163 + 0.938174i \(0.387484\pi\)
\(648\) 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 \(-0.481604\pi\)
0.0577617 + 0.998330i \(0.481604\pi\)
\(654\) 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 \(-0.195863\pi\)
0.816588 + 0.577221i \(0.195863\pi\)
\(660\) 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 \(-0.416093\pi\)
0.260559 + 0.965458i \(0.416093\pi\)
\(678\) 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 \(-0.735353\pi\)
−0.673833 + 0.738884i \(0.735353\pi\)
\(684\) 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 \(-0.684610\pi\)
−0.547997 + 0.836480i \(0.684610\pi\)
\(702\) 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 \(-0.455915\pi\)
0.138053 + 0.990425i \(0.455915\pi\)
\(720\) 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 \(-0.213312\pi\)
0.783735 + 0.621096i \(0.213312\pi\)
\(744\) 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 \(-0.651409\pi\)
−0.457929 + 0.888989i \(0.651409\pi\)
\(762\) 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 \(-0.485388\pi\)
0.0458881 + 0.998947i \(0.485388\pi\)
\(774\) 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 \(-0.436688\pi\)
0.197592 + 0.980284i \(0.436688\pi\)
\(798\) 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 \(-0.708622\pi\)
−0.609480 + 0.792802i \(0.708622\pi\)
\(810\) 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 \(-0.598555\pi\)
−0.304698 + 0.952449i \(0.598555\pi\)
\(822\) 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 \(-0.766534\pi\)
−0.742866 + 0.669440i \(0.766534\pi\)
\(828\) 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 \(-0.543011\pi\)
−0.134712 + 0.990885i \(0.543011\pi\)
\(840\) 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 \(-0.244776\pi\)
0.718616 + 0.695407i \(0.244776\pi\)
\(858\) 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 \(-0.376003\pi\)
0.379771 + 0.925080i \(0.376003\pi\)
\(864\) 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 \(-0.360063\pi\)
0.425601 + 0.904911i \(0.360063\pi\)
\(882\) 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 \(-0.445187\pi\)
0.171351 + 0.985210i \(0.445187\pi\)
\(888\) 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 \(-0.634273\pi\)
−0.409432 + 0.912341i \(0.634273\pi\)
\(912\) 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 \(-0.226010\pi\)
0.758341 + 0.651858i \(0.226010\pi\)
\(930\) 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 \(-0.553987\pi\)
−0.168792 + 0.985652i \(0.553987\pi\)
\(942\) 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 \(-0.562237\pi\)
−0.194281 + 0.980946i \(0.562237\pi\)
\(948\) 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 \(-0.631936\pi\)
−0.402722 + 0.915322i \(0.631936\pi\)
\(954\) 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 \(-0.313756\pi\)
0.552286 + 0.833655i \(0.313756\pi\)
\(972\) 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 \(-0.587986\pi\)
−0.272909 + 0.962040i \(0.587986\pi\)
\(978\) 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 \(-0.284881\pi\)
0.625534 + 0.780197i \(0.284881\pi\)
\(984\) 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.1.3 yes 4
3.2 odd 2 inner 171.2.a.e.1.2 4
4.3 odd 2 2736.2.a.bf.1.3 4
5.4 even 2 4275.2.a.bp.1.2 4
7.6 odd 2 8379.2.a.bw.1.3 4
12.11 even 2 2736.2.a.bf.1.2 4
15.14 odd 2 4275.2.a.bp.1.3 4
19.18 odd 2 3249.2.a.bf.1.2 4
21.20 even 2 8379.2.a.bw.1.2 4
57.56 even 2 3249.2.a.bf.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.2 4 3.2 odd 2 inner
171.2.a.e.1.3 yes 4 1.1 even 1 trivial
2736.2.a.bf.1.2 4 12.11 even 2
2736.2.a.bf.1.3 4 4.3 odd 2
3249.2.a.bf.1.2 4 19.18 odd 2
3249.2.a.bf.1.3 4 57.56 even 2
4275.2.a.bp.1.2 4 5.4 even 2
4275.2.a.bp.1.3 4 15.14 odd 2
8379.2.a.bw.1.2 4 21.20 even 2
8379.2.a.bw.1.3 4 7.6 odd 2