Properties

Label 6864.2.a.ca.1.3
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.70164.1
Defining polynomial: \(x^{4} - x^{3} - 10 x^{2} + 10 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.699291\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.699291 q^{5} -3.79091 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.699291 q^{5} -3.79091 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.699291 q^{15} -5.09162 q^{17} +0.538544 q^{19} -3.79091 q^{21} +7.18950 q^{23} -4.51099 q^{25} +1.00000 q^{27} -9.14116 q^{29} -1.95166 q^{31} -1.00000 q^{33} -2.65095 q^{35} -1.39858 q^{37} -1.00000 q^{39} -1.79091 q^{41} +10.8612 q^{43} +0.699291 q^{45} +13.5818 q^{47} +7.37103 q^{49} -5.09162 q^{51} +9.90332 q^{53} -0.699291 q^{55} +0.538544 q^{57} -5.72804 q^{59} +11.3514 q^{61} -3.79091 q^{63} -0.699291 q^{65} +13.3998 q^{67} +7.18950 q^{69} +9.90332 q^{71} +3.83249 q^{73} -4.51099 q^{75} +3.79091 q^{77} -1.37155 q^{79} +1.00000 q^{81} +4.32150 q^{83} -3.56053 q^{85} -9.14116 q^{87} +5.95166 q^{89} +3.79091 q^{91} -1.95166 q^{93} +0.376599 q^{95} +1.95843 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} - 4q^{11} - 4q^{13} + q^{15} - 6q^{17} + q^{21} + 9q^{23} + q^{25} + 4q^{27} - q^{29} + 8q^{31} - 4q^{33} + 7q^{35} - 2q^{37} - 4q^{39} + 9q^{41} + 5q^{43} + q^{45} + 22q^{47} + 9q^{49} - 6q^{51} + 8q^{53} - q^{55} - q^{59} - 11q^{61} + q^{63} - q^{65} + 13q^{67} + 9q^{69} + 8q^{71} - 3q^{73} + q^{75} - q^{77} + 6q^{79} + 4q^{81} + 18q^{83} + 26q^{85} - q^{87} + 8q^{89} - q^{91} + 8q^{93} + 36q^{95} + 10q^{97} - 4q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.699291 0.312733 0.156366 0.987699i \(-0.450022\pi\)
0.156366 + 0.987699i \(0.450022\pi\)
\(6\) 0 0
\(7\) −3.79091 −1.43283 −0.716415 0.697674i \(-0.754218\pi\)
−0.716415 + 0.697674i \(0.754218\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0.699291 0.180556
\(16\) 0 0
\(17\) −5.09162 −1.23490 −0.617450 0.786610i \(-0.711834\pi\)
−0.617450 + 0.786610i \(0.711834\pi\)
\(18\) 0 0
\(19\) 0.538544 0.123550 0.0617752 0.998090i \(-0.480324\pi\)
0.0617752 + 0.998090i \(0.480324\pi\)
\(20\) 0 0
\(21\) −3.79091 −0.827245
\(22\) 0 0
\(23\) 7.18950 1.49911 0.749557 0.661940i \(-0.230267\pi\)
0.749557 + 0.661940i \(0.230267\pi\)
\(24\) 0 0
\(25\) −4.51099 −0.902198
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −9.14116 −1.69747 −0.848735 0.528818i \(-0.822635\pi\)
−0.848735 + 0.528818i \(0.822635\pi\)
\(30\) 0 0
\(31\) −1.95166 −0.350529 −0.175264 0.984521i \(-0.556078\pi\)
−0.175264 + 0.984521i \(0.556078\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −2.65095 −0.448093
\(36\) 0 0
\(37\) −1.39858 −0.229926 −0.114963 0.993370i \(-0.536675\pi\)
−0.114963 + 0.993370i \(0.536675\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −1.79091 −0.279694 −0.139847 0.990173i \(-0.544661\pi\)
−0.139847 + 0.990173i \(0.544661\pi\)
\(42\) 0 0
\(43\) 10.8612 1.65632 0.828161 0.560490i \(-0.189387\pi\)
0.828161 + 0.560490i \(0.189387\pi\)
\(44\) 0 0
\(45\) 0.699291 0.104244
\(46\) 0 0
\(47\) 13.5818 1.98111 0.990557 0.137104i \(-0.0437795\pi\)
0.990557 + 0.137104i \(0.0437795\pi\)
\(48\) 0 0
\(49\) 7.37103 1.05300
\(50\) 0 0
\(51\) −5.09162 −0.712970
\(52\) 0 0
\(53\) 9.90332 1.36033 0.680163 0.733061i \(-0.261909\pi\)
0.680163 + 0.733061i \(0.261909\pi\)
\(54\) 0 0
\(55\) −0.699291 −0.0942924
\(56\) 0 0
\(57\) 0.538544 0.0713318
\(58\) 0 0
\(59\) −5.72804 −0.745727 −0.372864 0.927886i \(-0.621624\pi\)
−0.372864 + 0.927886i \(0.621624\pi\)
\(60\) 0 0
\(61\) 11.3514 1.45340 0.726702 0.686953i \(-0.241052\pi\)
0.726702 + 0.686953i \(0.241052\pi\)
\(62\) 0 0
\(63\) −3.79091 −0.477610
\(64\) 0 0
\(65\) −0.699291 −0.0867364
\(66\) 0 0
\(67\) 13.3998 1.63704 0.818522 0.574475i \(-0.194794\pi\)
0.818522 + 0.574475i \(0.194794\pi\)
\(68\) 0 0
\(69\) 7.18950 0.865514
\(70\) 0 0
\(71\) 9.90332 1.17531 0.587654 0.809112i \(-0.300052\pi\)
0.587654 + 0.809112i \(0.300052\pi\)
\(72\) 0 0
\(73\) 3.83249 0.448559 0.224279 0.974525i \(-0.427997\pi\)
0.224279 + 0.974525i \(0.427997\pi\)
\(74\) 0 0
\(75\) −4.51099 −0.520884
\(76\) 0 0
\(77\) 3.79091 0.432015
\(78\) 0 0
\(79\) −1.37155 −0.154311 −0.0771555 0.997019i \(-0.524584\pi\)
−0.0771555 + 0.997019i \(0.524584\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.32150 0.474346 0.237173 0.971467i \(-0.423779\pi\)
0.237173 + 0.971467i \(0.423779\pi\)
\(84\) 0 0
\(85\) −3.56053 −0.386193
\(86\) 0 0
\(87\) −9.14116 −0.980035
\(88\) 0 0
\(89\) 5.95166 0.630875 0.315437 0.948946i \(-0.397849\pi\)
0.315437 + 0.948946i \(0.397849\pi\)
\(90\) 0 0
\(91\) 3.79091 0.397396
\(92\) 0 0
\(93\) −1.95166 −0.202378
\(94\) 0 0
\(95\) 0.376599 0.0386382
\(96\) 0 0
\(97\) 1.95843 0.198848 0.0994241 0.995045i \(-0.468300\pi\)
0.0994241 + 0.995045i \(0.468300\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −1.93036 −0.192078 −0.0960390 0.995378i \(-0.530617\pi\)
−0.0960390 + 0.995378i \(0.530617\pi\)
\(102\) 0 0
\(103\) −7.35144 −0.724359 −0.362180 0.932108i \(-0.617967\pi\)
−0.362180 + 0.932108i \(0.617967\pi\)
\(104\) 0 0
\(105\) −2.65095 −0.258707
\(106\) 0 0
\(107\) −11.0299 −1.06631 −0.533153 0.846019i \(-0.678993\pi\)
−0.533153 + 0.846019i \(0.678993\pi\)
\(108\) 0 0
\(109\) 2.44187 0.233888 0.116944 0.993138i \(-0.462690\pi\)
0.116944 + 0.993138i \(0.462690\pi\)
\(110\) 0 0
\(111\) −1.39858 −0.132748
\(112\) 0 0
\(113\) −6.65095 −0.625669 −0.312835 0.949808i \(-0.601279\pi\)
−0.312835 + 0.949808i \(0.601279\pi\)
\(114\) 0 0
\(115\) 5.02755 0.468822
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 19.3019 1.76940
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −1.79091 −0.161481
\(124\) 0 0
\(125\) −6.65095 −0.594879
\(126\) 0 0
\(127\) 5.41312 0.480337 0.240168 0.970731i \(-0.422797\pi\)
0.240168 + 0.970731i \(0.422797\pi\)
\(128\) 0 0
\(129\) 10.8612 0.956279
\(130\) 0 0
\(131\) −8.55428 −0.747391 −0.373695 0.927552i \(-0.621909\pi\)
−0.373695 + 0.927552i \(0.621909\pi\)
\(132\) 0 0
\(133\) −2.04157 −0.177027
\(134\) 0 0
\(135\) 0.699291 0.0601854
\(136\) 0 0
\(137\) −1.11292 −0.0950835 −0.0475418 0.998869i \(-0.515139\pi\)
−0.0475418 + 0.998869i \(0.515139\pi\)
\(138\) 0 0
\(139\) 3.79211 0.321643 0.160821 0.986984i \(-0.448586\pi\)
0.160821 + 0.986984i \(0.448586\pi\)
\(140\) 0 0
\(141\) 13.5818 1.14380
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −6.39233 −0.530854
\(146\) 0 0
\(147\) 7.37103 0.607952
\(148\) 0 0
\(149\) 4.25862 0.348880 0.174440 0.984668i \(-0.444189\pi\)
0.174440 + 0.984668i \(0.444189\pi\)
\(150\) 0 0
\(151\) 7.04328 0.573174 0.286587 0.958054i \(-0.407479\pi\)
0.286587 + 0.958054i \(0.407479\pi\)
\(152\) 0 0
\(153\) −5.09162 −0.411633
\(154\) 0 0
\(155\) −1.36478 −0.109622
\(156\) 0 0
\(157\) 16.3452 1.30449 0.652244 0.758009i \(-0.273828\pi\)
0.652244 + 0.758009i \(0.273828\pi\)
\(158\) 0 0
\(159\) 9.90332 0.785385
\(160\) 0 0
\(161\) −27.2548 −2.14798
\(162\) 0 0
\(163\) 2.97921 0.233350 0.116675 0.993170i \(-0.462776\pi\)
0.116675 + 0.993170i \(0.462776\pi\)
\(164\) 0 0
\(165\) −0.699291 −0.0544397
\(166\) 0 0
\(167\) 19.2524 1.48979 0.744897 0.667180i \(-0.232499\pi\)
0.744897 + 0.667180i \(0.232499\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0.538544 0.0411835
\(172\) 0 0
\(173\) −0.440670 −0.0335035 −0.0167518 0.999860i \(-0.505333\pi\)
−0.0167518 + 0.999860i \(0.505333\pi\)
\(174\) 0 0
\(175\) 17.1008 1.29270
\(176\) 0 0
\(177\) −5.72804 −0.430546
\(178\) 0 0
\(179\) 21.7022 1.62210 0.811049 0.584978i \(-0.198897\pi\)
0.811049 + 0.584978i \(0.198897\pi\)
\(180\) 0 0
\(181\) −1.74138 −0.129436 −0.0647178 0.997904i \(-0.520615\pi\)
−0.0647178 + 0.997904i \(0.520615\pi\)
\(182\) 0 0
\(183\) 11.3514 0.839123
\(184\) 0 0
\(185\) −0.978016 −0.0719052
\(186\) 0 0
\(187\) 5.09162 0.372336
\(188\) 0 0
\(189\) −3.79091 −0.275748
\(190\) 0 0
\(191\) −18.6118 −1.34670 −0.673350 0.739324i \(-0.735145\pi\)
−0.673350 + 0.739324i \(0.735145\pi\)
\(192\) 0 0
\(193\) −20.9040 −1.50470 −0.752352 0.658762i \(-0.771081\pi\)
−0.752352 + 0.658762i \(0.771081\pi\)
\(194\) 0 0
\(195\) −0.699291 −0.0500773
\(196\) 0 0
\(197\) 18.1228 1.29119 0.645597 0.763678i \(-0.276609\pi\)
0.645597 + 0.763678i \(0.276609\pi\)
\(198\) 0 0
\(199\) −12.6533 −0.896972 −0.448486 0.893790i \(-0.648037\pi\)
−0.448486 + 0.893790i \(0.648037\pi\)
\(200\) 0 0
\(201\) 13.3998 0.945148
\(202\) 0 0
\(203\) 34.6533 2.43219
\(204\) 0 0
\(205\) −1.25237 −0.0874694
\(206\) 0 0
\(207\) 7.18950 0.499705
\(208\) 0 0
\(209\) −0.538544 −0.0372518
\(210\) 0 0
\(211\) −15.8337 −1.09004 −0.545018 0.838424i \(-0.683477\pi\)
−0.545018 + 0.838424i \(0.683477\pi\)
\(212\) 0 0
\(213\) 9.90332 0.678565
\(214\) 0 0
\(215\) 7.59517 0.517986
\(216\) 0 0
\(217\) 7.39858 0.502249
\(218\) 0 0
\(219\) 3.83249 0.258975
\(220\) 0 0
\(221\) 5.09162 0.342500
\(222\) 0 0
\(223\) −4.65215 −0.311531 −0.155766 0.987794i \(-0.549784\pi\)
−0.155766 + 0.987794i \(0.549784\pi\)
\(224\) 0 0
\(225\) −4.51099 −0.300733
\(226\) 0 0
\(227\) 5.84045 0.387644 0.193822 0.981037i \(-0.437912\pi\)
0.193822 + 0.981037i \(0.437912\pi\)
\(228\) 0 0
\(229\) 15.9113 1.05145 0.525724 0.850655i \(-0.323795\pi\)
0.525724 + 0.850655i \(0.323795\pi\)
\(230\) 0 0
\(231\) 3.79091 0.249424
\(232\) 0 0
\(233\) −8.90838 −0.583607 −0.291804 0.956478i \(-0.594255\pi\)
−0.291804 + 0.956478i \(0.594255\pi\)
\(234\) 0 0
\(235\) 9.49765 0.619559
\(236\) 0 0
\(237\) −1.37155 −0.0890914
\(238\) 0 0
\(239\) −14.1699 −0.916575 −0.458288 0.888804i \(-0.651537\pi\)
−0.458288 + 0.888804i \(0.651537\pi\)
\(240\) 0 0
\(241\) −6.46317 −0.416329 −0.208165 0.978094i \(-0.566749\pi\)
−0.208165 + 0.978094i \(0.566749\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 5.15450 0.329309
\(246\) 0 0
\(247\) −0.538544 −0.0342667
\(248\) 0 0
\(249\) 4.32150 0.273864
\(250\) 0 0
\(251\) 11.1187 0.701804 0.350902 0.936412i \(-0.385875\pi\)
0.350902 + 0.936412i \(0.385875\pi\)
\(252\) 0 0
\(253\) −7.18950 −0.452000
\(254\) 0 0
\(255\) −3.56053 −0.222969
\(256\) 0 0
\(257\) 3.63136 0.226518 0.113259 0.993565i \(-0.463871\pi\)
0.113259 + 0.993565i \(0.463871\pi\)
\(258\) 0 0
\(259\) 5.30191 0.329444
\(260\) 0 0
\(261\) −9.14116 −0.565824
\(262\) 0 0
\(263\) 18.7005 1.15312 0.576561 0.817054i \(-0.304394\pi\)
0.576561 + 0.817054i \(0.304394\pi\)
\(264\) 0 0
\(265\) 6.92531 0.425418
\(266\) 0 0
\(267\) 5.95166 0.364236
\(268\) 0 0
\(269\) −16.9253 −1.03195 −0.515977 0.856602i \(-0.672571\pi\)
−0.515977 + 0.856602i \(0.672571\pi\)
\(270\) 0 0
\(271\) 17.7438 1.07786 0.538929 0.842351i \(-0.318829\pi\)
0.538929 + 0.842351i \(0.318829\pi\)
\(272\) 0 0
\(273\) 3.79091 0.229437
\(274\) 0 0
\(275\) 4.51099 0.272023
\(276\) 0 0
\(277\) 18.6533 1.12077 0.560386 0.828232i \(-0.310653\pi\)
0.560386 + 0.828232i \(0.310653\pi\)
\(278\) 0 0
\(279\) −1.95166 −0.116843
\(280\) 0 0
\(281\) −8.26658 −0.493143 −0.246572 0.969125i \(-0.579304\pi\)
−0.246572 + 0.969125i \(0.579304\pi\)
\(282\) 0 0
\(283\) 3.92582 0.233366 0.116683 0.993169i \(-0.462774\pi\)
0.116683 + 0.993169i \(0.462774\pi\)
\(284\) 0 0
\(285\) 0.376599 0.0223078
\(286\) 0 0
\(287\) 6.78920 0.400754
\(288\) 0 0
\(289\) 8.92462 0.524978
\(290\) 0 0
\(291\) 1.95843 0.114805
\(292\) 0 0
\(293\) −16.3036 −0.952468 −0.476234 0.879319i \(-0.657998\pi\)
−0.476234 + 0.879319i \(0.657998\pi\)
\(294\) 0 0
\(295\) −4.00557 −0.233213
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −7.18950 −0.415779
\(300\) 0 0
\(301\) −41.1740 −2.37323
\(302\) 0 0
\(303\) −1.93036 −0.110896
\(304\) 0 0
\(305\) 7.93796 0.454526
\(306\) 0 0
\(307\) −6.77586 −0.386719 −0.193359 0.981128i \(-0.561938\pi\)
−0.193359 + 0.981128i \(0.561938\pi\)
\(308\) 0 0
\(309\) −7.35144 −0.418209
\(310\) 0 0
\(311\) 13.8820 0.787177 0.393589 0.919287i \(-0.371233\pi\)
0.393589 + 0.919287i \(0.371233\pi\)
\(312\) 0 0
\(313\) −19.1344 −1.08154 −0.540770 0.841171i \(-0.681867\pi\)
−0.540770 + 0.841171i \(0.681867\pi\)
\(314\) 0 0
\(315\) −2.65095 −0.149364
\(316\) 0 0
\(317\) 16.1429 0.906674 0.453337 0.891339i \(-0.350233\pi\)
0.453337 + 0.891339i \(0.350233\pi\)
\(318\) 0 0
\(319\) 9.14116 0.511807
\(320\) 0 0
\(321\) −11.0299 −0.615632
\(322\) 0 0
\(323\) −2.74206 −0.152572
\(324\) 0 0
\(325\) 4.51099 0.250225
\(326\) 0 0
\(327\) 2.44187 0.135036
\(328\) 0 0
\(329\) −51.4875 −2.83860
\(330\) 0 0
\(331\) −7.00829 −0.385210 −0.192605 0.981276i \(-0.561694\pi\)
−0.192605 + 0.981276i \(0.561694\pi\)
\(332\) 0 0
\(333\) −1.39858 −0.0766418
\(334\) 0 0
\(335\) 9.37035 0.511957
\(336\) 0 0
\(337\) 18.4632 1.00575 0.502876 0.864358i \(-0.332275\pi\)
0.502876 + 0.864358i \(0.332275\pi\)
\(338\) 0 0
\(339\) −6.65095 −0.361230
\(340\) 0 0
\(341\) 1.95166 0.105688
\(342\) 0 0
\(343\) −1.40655 −0.0759463
\(344\) 0 0
\(345\) 5.02755 0.270674
\(346\) 0 0
\(347\) −25.3594 −1.36136 −0.680682 0.732579i \(-0.738316\pi\)
−0.680682 + 0.732579i \(0.738316\pi\)
\(348\) 0 0
\(349\) 23.2052 1.24215 0.621074 0.783752i \(-0.286697\pi\)
0.621074 + 0.783752i \(0.286697\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 21.2986 1.13361 0.566804 0.823853i \(-0.308180\pi\)
0.566804 + 0.823853i \(0.308180\pi\)
\(354\) 0 0
\(355\) 6.92531 0.367557
\(356\) 0 0
\(357\) 19.3019 1.02157
\(358\) 0 0
\(359\) 18.7298 0.988519 0.494259 0.869315i \(-0.335439\pi\)
0.494259 + 0.869315i \(0.335439\pi\)
\(360\) 0 0
\(361\) −18.7100 −0.984735
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 2.68002 0.140279
\(366\) 0 0
\(367\) 6.33400 0.330632 0.165316 0.986241i \(-0.447136\pi\)
0.165316 + 0.986241i \(0.447136\pi\)
\(368\) 0 0
\(369\) −1.79091 −0.0932313
\(370\) 0 0
\(371\) −37.5427 −1.94912
\(372\) 0 0
\(373\) −6.13371 −0.317591 −0.158796 0.987311i \(-0.550761\pi\)
−0.158796 + 0.987311i \(0.550761\pi\)
\(374\) 0 0
\(375\) −6.65095 −0.343454
\(376\) 0 0
\(377\) 9.14116 0.470794
\(378\) 0 0
\(379\) −17.8966 −0.919284 −0.459642 0.888104i \(-0.652022\pi\)
−0.459642 + 0.888104i \(0.652022\pi\)
\(380\) 0 0
\(381\) 5.41312 0.277322
\(382\) 0 0
\(383\) −17.7225 −0.905576 −0.452788 0.891618i \(-0.649571\pi\)
−0.452788 + 0.891618i \(0.649571\pi\)
\(384\) 0 0
\(385\) 2.65095 0.135105
\(386\) 0 0
\(387\) 10.8612 0.552108
\(388\) 0 0
\(389\) 20.0575 1.01696 0.508478 0.861075i \(-0.330208\pi\)
0.508478 + 0.861075i \(0.330208\pi\)
\(390\) 0 0
\(391\) −36.6062 −1.85126
\(392\) 0 0
\(393\) −8.55428 −0.431506
\(394\) 0 0
\(395\) −0.959110 −0.0482580
\(396\) 0 0
\(397\) −7.02995 −0.352823 −0.176411 0.984317i \(-0.556449\pi\)
−0.176411 + 0.984317i \(0.556449\pi\)
\(398\) 0 0
\(399\) −2.04157 −0.102206
\(400\) 0 0
\(401\) 7.19607 0.359355 0.179677 0.983726i \(-0.442495\pi\)
0.179677 + 0.983726i \(0.442495\pi\)
\(402\) 0 0
\(403\) 1.95166 0.0972192
\(404\) 0 0
\(405\) 0.699291 0.0347481
\(406\) 0 0
\(407\) 1.39858 0.0693252
\(408\) 0 0
\(409\) 4.93625 0.244082 0.122041 0.992525i \(-0.461056\pi\)
0.122041 + 0.992525i \(0.461056\pi\)
\(410\) 0 0
\(411\) −1.11292 −0.0548965
\(412\) 0 0
\(413\) 21.7145 1.06850
\(414\) 0 0
\(415\) 3.02198 0.148343
\(416\) 0 0
\(417\) 3.79211 0.185700
\(418\) 0 0
\(419\) 24.5825 1.20093 0.600467 0.799649i \(-0.294981\pi\)
0.600467 + 0.799649i \(0.294981\pi\)
\(420\) 0 0
\(421\) 21.6189 1.05364 0.526819 0.849977i \(-0.323384\pi\)
0.526819 + 0.849977i \(0.323384\pi\)
\(422\) 0 0
\(423\) 13.5818 0.660371
\(424\) 0 0
\(425\) 22.9683 1.11412
\(426\) 0 0
\(427\) −43.0323 −2.08248
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 25.7201 1.23889 0.619446 0.785039i \(-0.287357\pi\)
0.619446 + 0.785039i \(0.287357\pi\)
\(432\) 0 0
\(433\) −13.5874 −0.652969 −0.326484 0.945203i \(-0.605864\pi\)
−0.326484 + 0.945203i \(0.605864\pi\)
\(434\) 0 0
\(435\) −6.39233 −0.306489
\(436\) 0 0
\(437\) 3.87186 0.185216
\(438\) 0 0
\(439\) 32.1170 1.53286 0.766431 0.642327i \(-0.222031\pi\)
0.766431 + 0.642327i \(0.222031\pi\)
\(440\) 0 0
\(441\) 7.37103 0.351001
\(442\) 0 0
\(443\) 6.79716 0.322943 0.161472 0.986877i \(-0.448376\pi\)
0.161472 + 0.986877i \(0.448376\pi\)
\(444\) 0 0
\(445\) 4.16194 0.197295
\(446\) 0 0
\(447\) 4.25862 0.201426
\(448\) 0 0
\(449\) −3.95406 −0.186603 −0.0933017 0.995638i \(-0.529742\pi\)
−0.0933017 + 0.995638i \(0.529742\pi\)
\(450\) 0 0
\(451\) 1.79091 0.0843309
\(452\) 0 0
\(453\) 7.04328 0.330922
\(454\) 0 0
\(455\) 2.65095 0.124279
\(456\) 0 0
\(457\) 20.1486 0.942512 0.471256 0.881996i \(-0.343801\pi\)
0.471256 + 0.881996i \(0.343801\pi\)
\(458\) 0 0
\(459\) −5.09162 −0.237657
\(460\) 0 0
\(461\) −2.49594 −0.116248 −0.0581238 0.998309i \(-0.518512\pi\)
−0.0581238 + 0.998309i \(0.518512\pi\)
\(462\) 0 0
\(463\) 14.3147 0.665262 0.332631 0.943057i \(-0.392064\pi\)
0.332631 + 0.943057i \(0.392064\pi\)
\(464\) 0 0
\(465\) −1.36478 −0.0632902
\(466\) 0 0
\(467\) 18.9591 0.877323 0.438661 0.898652i \(-0.355453\pi\)
0.438661 + 0.898652i \(0.355453\pi\)
\(468\) 0 0
\(469\) −50.7974 −2.34561
\(470\) 0 0
\(471\) 16.3452 0.753147
\(472\) 0 0
\(473\) −10.8612 −0.499400
\(474\) 0 0
\(475\) −2.42937 −0.111467
\(476\) 0 0
\(477\) 9.90332 0.453442
\(478\) 0 0
\(479\) 37.1402 1.69698 0.848490 0.529212i \(-0.177512\pi\)
0.848490 + 0.529212i \(0.177512\pi\)
\(480\) 0 0
\(481\) 1.39858 0.0637699
\(482\) 0 0
\(483\) −27.2548 −1.24013
\(484\) 0 0
\(485\) 1.36951 0.0621863
\(486\) 0 0
\(487\) −34.7488 −1.57462 −0.787310 0.616558i \(-0.788527\pi\)
−0.787310 + 0.616558i \(0.788527\pi\)
\(488\) 0 0
\(489\) 2.97921 0.134725
\(490\) 0 0
\(491\) −42.4735 −1.91680 −0.958402 0.285423i \(-0.907866\pi\)
−0.958402 + 0.285423i \(0.907866\pi\)
\(492\) 0 0
\(493\) 46.5433 2.09621
\(494\) 0 0
\(495\) −0.699291 −0.0314308
\(496\) 0 0
\(497\) −37.5427 −1.68402
\(498\) 0 0
\(499\) 22.1844 0.993112 0.496556 0.868005i \(-0.334598\pi\)
0.496556 + 0.868005i \(0.334598\pi\)
\(500\) 0 0
\(501\) 19.2524 0.860132
\(502\) 0 0
\(503\) −30.5071 −1.36025 −0.680123 0.733098i \(-0.738074\pi\)
−0.680123 + 0.733098i \(0.738074\pi\)
\(504\) 0 0
\(505\) −1.34988 −0.0600691
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 5.67277 0.251441 0.125721 0.992066i \(-0.459876\pi\)
0.125721 + 0.992066i \(0.459876\pi\)
\(510\) 0 0
\(511\) −14.5286 −0.642709
\(512\) 0 0
\(513\) 0.538544 0.0237773
\(514\) 0 0
\(515\) −5.14080 −0.226531
\(516\) 0 0
\(517\) −13.5818 −0.597328
\(518\) 0 0
\(519\) −0.440670 −0.0193433
\(520\) 0 0
\(521\) −11.6154 −0.508882 −0.254441 0.967088i \(-0.581891\pi\)
−0.254441 + 0.967088i \(0.581891\pi\)
\(522\) 0 0
\(523\) −1.14673 −0.0501429 −0.0250714 0.999686i \(-0.507981\pi\)
−0.0250714 + 0.999686i \(0.507981\pi\)
\(524\) 0 0
\(525\) 17.1008 0.746339
\(526\) 0 0
\(527\) 9.93713 0.432868
\(528\) 0 0
\(529\) 28.6889 1.24734
\(530\) 0 0
\(531\) −5.72804 −0.248576
\(532\) 0 0
\(533\) 1.79091 0.0775731
\(534\) 0 0
\(535\) −7.71314 −0.333468
\(536\) 0 0
\(537\) 21.7022 0.936519
\(538\) 0 0
\(539\) −7.37103 −0.317493
\(540\) 0 0
\(541\) −35.8506 −1.54134 −0.770669 0.637235i \(-0.780078\pi\)
−0.770669 + 0.637235i \(0.780078\pi\)
\(542\) 0 0
\(543\) −1.74138 −0.0747297
\(544\) 0 0
\(545\) 1.70758 0.0731445
\(546\) 0 0
\(547\) −29.7159 −1.27056 −0.635280 0.772282i \(-0.719115\pi\)
−0.635280 + 0.772282i \(0.719115\pi\)
\(548\) 0 0
\(549\) 11.3514 0.484468
\(550\) 0 0
\(551\) −4.92291 −0.209723
\(552\) 0 0
\(553\) 5.19941 0.221101
\(554\) 0 0
\(555\) −0.978016 −0.0415145
\(556\) 0 0
\(557\) 27.7978 1.17783 0.588916 0.808194i \(-0.299555\pi\)
0.588916 + 0.808194i \(0.299555\pi\)
\(558\) 0 0
\(559\) −10.8612 −0.459381
\(560\) 0 0
\(561\) 5.09162 0.214968
\(562\) 0 0
\(563\) 41.4436 1.74664 0.873319 0.487148i \(-0.161963\pi\)
0.873319 + 0.487148i \(0.161963\pi\)
\(564\) 0 0
\(565\) −4.65095 −0.195667
\(566\) 0 0
\(567\) −3.79091 −0.159203
\(568\) 0 0
\(569\) 35.7945 1.50058 0.750292 0.661107i \(-0.229913\pi\)
0.750292 + 0.661107i \(0.229913\pi\)
\(570\) 0 0
\(571\) −34.0766 −1.42606 −0.713030 0.701133i \(-0.752678\pi\)
−0.713030 + 0.701133i \(0.752678\pi\)
\(572\) 0 0
\(573\) −18.6118 −0.777518
\(574\) 0 0
\(575\) −32.4318 −1.35250
\(576\) 0 0
\(577\) −41.2201 −1.71602 −0.858008 0.513636i \(-0.828298\pi\)
−0.858008 + 0.513636i \(0.828298\pi\)
\(578\) 0 0
\(579\) −20.9040 −0.868741
\(580\) 0 0
\(581\) −16.3824 −0.679657
\(582\) 0 0
\(583\) −9.90332 −0.410154
\(584\) 0 0
\(585\) −0.699291 −0.0289121
\(586\) 0 0
\(587\) 12.5252 0.516971 0.258485 0.966015i \(-0.416777\pi\)
0.258485 + 0.966015i \(0.416777\pi\)
\(588\) 0 0
\(589\) −1.05106 −0.0433080
\(590\) 0 0
\(591\) 18.1228 0.745471
\(592\) 0 0
\(593\) −9.13996 −0.375333 −0.187667 0.982233i \(-0.560092\pi\)
−0.187667 + 0.982233i \(0.560092\pi\)
\(594\) 0 0
\(595\) 13.4977 0.553350
\(596\) 0 0
\(597\) −12.6533 −0.517867
\(598\) 0 0
\(599\) −5.39301 −0.220353 −0.110176 0.993912i \(-0.535142\pi\)
−0.110176 + 0.993912i \(0.535142\pi\)
\(600\) 0 0
\(601\) −2.65892 −0.108459 −0.0542297 0.998528i \(-0.517270\pi\)
−0.0542297 + 0.998528i \(0.517270\pi\)
\(602\) 0 0
\(603\) 13.3998 0.545681
\(604\) 0 0
\(605\) 0.699291 0.0284302
\(606\) 0 0
\(607\) −39.7945 −1.61521 −0.807605 0.589724i \(-0.799236\pi\)
−0.807605 + 0.589724i \(0.799236\pi\)
\(608\) 0 0
\(609\) 34.6533 1.40422
\(610\) 0 0
\(611\) −13.5818 −0.549462
\(612\) 0 0
\(613\) −26.1654 −1.05681 −0.528405 0.848993i \(-0.677210\pi\)
−0.528405 + 0.848993i \(0.677210\pi\)
\(614\) 0 0
\(615\) −1.25237 −0.0505005
\(616\) 0 0
\(617\) −23.4101 −0.942457 −0.471228 0.882011i \(-0.656189\pi\)
−0.471228 + 0.882011i \(0.656189\pi\)
\(618\) 0 0
\(619\) −36.8974 −1.48303 −0.741517 0.670935i \(-0.765893\pi\)
−0.741517 + 0.670935i \(0.765893\pi\)
\(620\) 0 0
\(621\) 7.18950 0.288505
\(622\) 0 0
\(623\) −22.5622 −0.903937
\(624\) 0 0
\(625\) 17.9040 0.716160
\(626\) 0 0
\(627\) −0.538544 −0.0215074
\(628\) 0 0
\(629\) 7.12105 0.283935
\(630\) 0 0
\(631\) −21.8834 −0.871165 −0.435582 0.900149i \(-0.643458\pi\)
−0.435582 + 0.900149i \(0.643458\pi\)
\(632\) 0 0
\(633\) −15.8337 −0.629332
\(634\) 0 0
\(635\) 3.78535 0.150217
\(636\) 0 0
\(637\) −7.37103 −0.292051
\(638\) 0 0
\(639\) 9.90332 0.391769
\(640\) 0 0
\(641\) 20.5982 0.813582 0.406791 0.913521i \(-0.366648\pi\)
0.406791 + 0.913521i \(0.366648\pi\)
\(642\) 0 0
\(643\) −36.0518 −1.42174 −0.710871 0.703322i \(-0.751699\pi\)
−0.710871 + 0.703322i \(0.751699\pi\)
\(644\) 0 0
\(645\) 7.59517 0.299059
\(646\) 0 0
\(647\) 29.4300 1.15701 0.578507 0.815677i \(-0.303635\pi\)
0.578507 + 0.815677i \(0.303635\pi\)
\(648\) 0 0
\(649\) 5.72804 0.224845
\(650\) 0 0
\(651\) 7.39858 0.289973
\(652\) 0 0
\(653\) −49.5301 −1.93826 −0.969132 0.246541i \(-0.920706\pi\)
−0.969132 + 0.246541i \(0.920706\pi\)
\(654\) 0 0
\(655\) −5.98193 −0.233733
\(656\) 0 0
\(657\) 3.83249 0.149520
\(658\) 0 0
\(659\) 26.7895 1.04357 0.521784 0.853077i \(-0.325267\pi\)
0.521784 + 0.853077i \(0.325267\pi\)
\(660\) 0 0
\(661\) 21.2177 0.825274 0.412637 0.910896i \(-0.364608\pi\)
0.412637 + 0.910896i \(0.364608\pi\)
\(662\) 0 0
\(663\) 5.09162 0.197742
\(664\) 0 0
\(665\) −1.42765 −0.0553620
\(666\) 0 0
\(667\) −65.7203 −2.54470
\(668\) 0 0
\(669\) −4.65215 −0.179863
\(670\) 0 0
\(671\) −11.3514 −0.438218
\(672\) 0 0
\(673\) 24.1015 0.929043 0.464522 0.885562i \(-0.346226\pi\)
0.464522 + 0.885562i \(0.346226\pi\)
\(674\) 0 0
\(675\) −4.51099 −0.173628
\(676\) 0 0
\(677\) −8.67242 −0.333308 −0.166654 0.986015i \(-0.553296\pi\)
−0.166654 + 0.986015i \(0.553296\pi\)
\(678\) 0 0
\(679\) −7.42423 −0.284916
\(680\) 0 0
\(681\) 5.84045 0.223807
\(682\) 0 0
\(683\) 22.8917 0.875926 0.437963 0.898993i \(-0.355700\pi\)
0.437963 + 0.898993i \(0.355700\pi\)
\(684\) 0 0
\(685\) −0.778258 −0.0297357
\(686\) 0 0
\(687\) 15.9113 0.607053
\(688\) 0 0
\(689\) −9.90332 −0.377287
\(690\) 0 0
\(691\) 37.4943 1.42635 0.713175 0.700986i \(-0.247256\pi\)
0.713175 + 0.700986i \(0.247256\pi\)
\(692\) 0 0
\(693\) 3.79091 0.144005
\(694\) 0 0
\(695\) 2.65179 0.100588
\(696\) 0 0
\(697\) 9.11866 0.345394
\(698\) 0 0
\(699\) −8.90838 −0.336946
\(700\) 0 0
\(701\) −5.59183 −0.211200 −0.105600 0.994409i \(-0.533676\pi\)
−0.105600 + 0.994409i \(0.533676\pi\)
\(702\) 0 0
\(703\) −0.753198 −0.0284074
\(704\) 0 0
\(705\) 9.49765 0.357702
\(706\) 0 0
\(707\) 7.31783 0.275215
\(708\) 0 0
\(709\) 7.22569 0.271367 0.135683 0.990752i \(-0.456677\pi\)
0.135683 + 0.990752i \(0.456677\pi\)
\(710\) 0 0
\(711\) −1.37155 −0.0514370
\(712\) 0 0
\(713\) −14.0315 −0.525483
\(714\) 0 0
\(715\) 0.699291 0.0261520
\(716\) 0 0
\(717\) −14.1699 −0.529185
\(718\) 0 0
\(719\) 43.2998 1.61481 0.807404 0.589999i \(-0.200872\pi\)
0.807404 + 0.589999i \(0.200872\pi\)
\(720\) 0 0
\(721\) 27.8687 1.03788
\(722\) 0 0
\(723\) −6.46317 −0.240368
\(724\) 0 0
\(725\) 41.2357 1.53146
\(726\) 0 0
\(727\) −1.79175 −0.0664524 −0.0332262 0.999448i \(-0.510578\pi\)
−0.0332262 + 0.999448i \(0.510578\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −55.3013 −2.04539
\(732\) 0 0
\(733\) −27.9371 −1.03188 −0.515941 0.856624i \(-0.672557\pi\)
−0.515941 + 0.856624i \(0.672557\pi\)
\(734\) 0 0
\(735\) 5.15450 0.190126
\(736\) 0 0
\(737\) −13.3998 −0.493587
\(738\) 0 0
\(739\) 47.9318 1.76320 0.881600 0.471997i \(-0.156467\pi\)
0.881600 + 0.471997i \(0.156467\pi\)
\(740\) 0 0
\(741\) −0.538544 −0.0197839
\(742\) 0 0
\(743\) −0.216856 −0.00795568 −0.00397784 0.999992i \(-0.501266\pi\)
−0.00397784 + 0.999992i \(0.501266\pi\)
\(744\) 0 0
\(745\) 2.97802 0.109106
\(746\) 0 0
\(747\) 4.32150 0.158115
\(748\) 0 0
\(749\) 41.8136 1.52783
\(750\) 0 0
\(751\) 24.5002 0.894025 0.447013 0.894528i \(-0.352488\pi\)
0.447013 + 0.894528i \(0.352488\pi\)
\(752\) 0 0
\(753\) 11.1187 0.405186
\(754\) 0 0
\(755\) 4.92531 0.179250
\(756\) 0 0
\(757\) −31.3831 −1.14064 −0.570319 0.821423i \(-0.693180\pi\)
−0.570319 + 0.821423i \(0.693180\pi\)
\(758\) 0 0
\(759\) −7.18950 −0.260962
\(760\) 0 0
\(761\) 28.5340 1.03436 0.517178 0.855878i \(-0.326982\pi\)
0.517178 + 0.855878i \(0.326982\pi\)
\(762\) 0 0
\(763\) −9.25691 −0.335123
\(764\) 0 0
\(765\) −3.56053 −0.128731
\(766\) 0 0
\(767\) 5.72804 0.206828
\(768\) 0 0
\(769\) 19.4491 0.701354 0.350677 0.936496i \(-0.385951\pi\)
0.350677 + 0.936496i \(0.385951\pi\)
\(770\) 0 0
\(771\) 3.63136 0.130780
\(772\) 0 0
\(773\) 44.2881 1.59293 0.796465 0.604684i \(-0.206701\pi\)
0.796465 + 0.604684i \(0.206701\pi\)
\(774\) 0 0
\(775\) 8.80393 0.316247
\(776\) 0 0
\(777\) 5.30191 0.190205
\(778\) 0 0
\(779\) −0.964485 −0.0345563
\(780\) 0 0
\(781\) −9.90332 −0.354369
\(782\) 0 0
\(783\) −9.14116 −0.326678
\(784\) 0 0
\(785\) 11.4300 0.407956
\(786\) 0 0
\(787\) −8.55447 −0.304934 −0.152467 0.988309i \(-0.548722\pi\)
−0.152467 + 0.988309i \(0.548722\pi\)
\(788\) 0 0
\(789\) 18.7005 0.665755
\(790\) 0 0
\(791\) 25.2132 0.896478
\(792\) 0 0
\(793\) −11.3514 −0.403102
\(794\) 0 0
\(795\) 6.92531 0.245615
\(796\) 0 0
\(797\) −37.9483 −1.34420 −0.672099 0.740461i \(-0.734607\pi\)
−0.672099 + 0.740461i \(0.734607\pi\)
\(798\) 0 0
\(799\) −69.1536 −2.44648
\(800\) 0 0
\(801\) 5.95166 0.210292
\(802\) 0 0
\(803\) −3.83249 −0.135246
\(804\) 0 0
\(805\) −19.0590 −0.671742
\(806\) 0 0
\(807\) −16.9253 −0.595799
\(808\) 0 0
\(809\) 27.9047 0.981077 0.490539 0.871419i \(-0.336800\pi\)
0.490539 + 0.871419i \(0.336800\pi\)
\(810\) 0 0
\(811\) 32.1252 1.12807 0.564033 0.825752i \(-0.309249\pi\)
0.564033 + 0.825752i \(0.309249\pi\)
\(812\) 0 0
\(813\) 17.7438 0.622301
\(814\) 0 0
\(815\) 2.08334 0.0729761
\(816\) 0 0
\(817\) 5.84925 0.204639
\(818\) 0 0
\(819\) 3.79091 0.132465
\(820\) 0 0
\(821\) −18.3428 −0.640168 −0.320084 0.947389i \(-0.603711\pi\)
−0.320084 + 0.947389i \(0.603711\pi\)
\(822\) 0 0
\(823\) −32.3585 −1.12795 −0.563974 0.825793i \(-0.690728\pi\)
−0.563974 + 0.825793i \(0.690728\pi\)
\(824\) 0 0
\(825\) 4.51099 0.157053
\(826\) 0 0
\(827\) 6.65179 0.231305 0.115653 0.993290i \(-0.463104\pi\)
0.115653 + 0.993290i \(0.463104\pi\)
\(828\) 0 0
\(829\) −10.3915 −0.360912 −0.180456 0.983583i \(-0.557757\pi\)
−0.180456 + 0.983583i \(0.557757\pi\)
\(830\) 0 0
\(831\) 18.6533 0.647077
\(832\) 0 0
\(833\) −37.5305 −1.30036
\(834\) 0 0
\(835\) 13.4630 0.465907
\(836\) 0 0
\(837\) −1.95166 −0.0674593
\(838\) 0 0
\(839\) 49.4324 1.70660 0.853299 0.521422i \(-0.174598\pi\)
0.853299 + 0.521422i \(0.174598\pi\)
\(840\) 0 0
\(841\) 54.5608 1.88141
\(842\) 0 0
\(843\) −8.26658 −0.284716
\(844\) 0 0
\(845\) 0.699291 0.0240563
\(846\) 0 0
\(847\) −3.79091 −0.130257
\(848\) 0 0
\(849\) 3.92582 0.134734
\(850\) 0 0
\(851\) −10.0551 −0.344685
\(852\) 0 0
\(853\) −20.6555 −0.707231 −0.353615 0.935391i \(-0.615048\pi\)
−0.353615 + 0.935391i \(0.615048\pi\)
\(854\) 0 0
\(855\) 0.376599 0.0128794
\(856\) 0 0
\(857\) 27.4605 0.938033 0.469017 0.883189i \(-0.344608\pi\)
0.469017 + 0.883189i \(0.344608\pi\)
\(858\) 0 0
\(859\) −18.0149 −0.614660 −0.307330 0.951603i \(-0.599436\pi\)
−0.307330 + 0.951603i \(0.599436\pi\)
\(860\) 0 0
\(861\) 6.78920 0.231375
\(862\) 0 0
\(863\) 34.2664 1.16644 0.583221 0.812314i \(-0.301792\pi\)
0.583221 + 0.812314i \(0.301792\pi\)
\(864\) 0 0
\(865\) −0.308157 −0.0104776
\(866\) 0 0
\(867\) 8.92462 0.303096
\(868\) 0 0
\(869\) 1.37155 0.0465265
\(870\) 0 0
\(871\) −13.3998 −0.454034
\(872\) 0 0
\(873\) 1.95843 0.0662827
\(874\) 0 0
\(875\) 25.2132 0.852361
\(876\) 0 0
\(877\) 20.9679 0.708036 0.354018 0.935239i \(-0.384815\pi\)
0.354018 + 0.935239i \(0.384815\pi\)
\(878\) 0 0
\(879\) −16.3036 −0.549907
\(880\) 0 0
\(881\) 5.04348 0.169919 0.0849595 0.996384i \(-0.472924\pi\)
0.0849595 + 0.996384i \(0.472924\pi\)
\(882\) 0 0
\(883\) −46.0173 −1.54861 −0.774303 0.632816i \(-0.781899\pi\)
−0.774303 + 0.632816i \(0.781899\pi\)
\(884\) 0 0
\(885\) −4.00557 −0.134646
\(886\) 0 0
\(887\) −9.19033 −0.308581 −0.154291 0.988026i \(-0.549309\pi\)
−0.154291 + 0.988026i \(0.549309\pi\)
\(888\) 0 0
\(889\) −20.5207 −0.688241
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 7.31441 0.244767
\(894\) 0 0
\(895\) 15.1762 0.507283
\(896\) 0 0
\(897\) −7.18950 −0.240050
\(898\) 0 0
\(899\) 17.8404 0.595012
\(900\) 0 0
\(901\) −50.4240 −1.67987
\(902\) 0 0
\(903\) −41.1740 −1.37019
\(904\) 0 0
\(905\) −1.21773 −0.0404787
\(906\) 0 0
\(907\) −27.9081 −0.926674 −0.463337 0.886182i \(-0.653348\pi\)
−0.463337 + 0.886182i \(0.653348\pi\)
\(908\) 0 0
\(909\) −1.93036 −0.0640260
\(910\) 0 0
\(911\) 6.82624 0.226163 0.113082 0.993586i \(-0.463928\pi\)
0.113082 + 0.993586i \(0.463928\pi\)
\(912\) 0 0
\(913\) −4.32150 −0.143021
\(914\) 0 0
\(915\) 7.93796 0.262421
\(916\) 0 0
\(917\) 32.4285 1.07088
\(918\) 0 0
\(919\) 27.3899 0.903508 0.451754 0.892142i \(-0.350798\pi\)
0.451754 + 0.892142i \(0.350798\pi\)
\(920\) 0 0
\(921\) −6.77586 −0.223272
\(922\) 0 0
\(923\) −9.90332 −0.325972
\(924\) 0 0
\(925\) 6.30899 0.207438
\(926\) 0 0
\(927\) −7.35144 −0.241453
\(928\) 0 0
\(929\) 14.9185 0.489461 0.244731 0.969591i \(-0.421300\pi\)
0.244731 + 0.969591i \(0.421300\pi\)
\(930\) 0 0
\(931\) 3.96962 0.130099
\(932\) 0 0
\(933\) 13.8820 0.454477
\(934\) 0 0
\(935\) 3.56053 0.116442
\(936\) 0 0
\(937\) −41.0954 −1.34253 −0.671264 0.741218i \(-0.734248\pi\)
−0.671264 + 0.741218i \(0.734248\pi\)
\(938\) 0 0
\(939\) −19.1344 −0.624427
\(940\) 0 0
\(941\) −15.0558 −0.490805 −0.245402 0.969421i \(-0.578920\pi\)
−0.245402 + 0.969421i \(0.578920\pi\)
\(942\) 0 0
\(943\) −12.8758 −0.419293
\(944\) 0 0
\(945\) −2.65095 −0.0862355
\(946\) 0 0
\(947\) 49.7394 1.61631 0.808157 0.588967i \(-0.200465\pi\)
0.808157 + 0.588967i \(0.200465\pi\)
\(948\) 0 0
\(949\) −3.83249 −0.124408
\(950\) 0 0
\(951\) 16.1429 0.523468
\(952\) 0 0
\(953\) 5.48415 0.177649 0.0888245 0.996047i \(-0.471689\pi\)
0.0888245 + 0.996047i \(0.471689\pi\)
\(954\) 0 0
\(955\) −13.0151 −0.421157
\(956\) 0 0
\(957\) 9.14116 0.295492
\(958\) 0 0
\(959\) 4.21900 0.136239
\(960\) 0 0
\(961\) −27.1910 −0.877130
\(962\) 0 0
\(963\) −11.0299 −0.355435
\(964\) 0 0
\(965\) −14.6180 −0.470570
\(966\) 0 0
\(967\) −6.30919 −0.202890 −0.101445 0.994841i \(-0.532347\pi\)
−0.101445 + 0.994841i \(0.532347\pi\)
\(968\) 0 0
\(969\) −2.74206 −0.0880877
\(970\) 0 0
\(971\) −37.6572 −1.20848 −0.604239 0.796803i \(-0.706523\pi\)
−0.604239 + 0.796803i \(0.706523\pi\)
\(972\) 0 0
\(973\) −14.3756 −0.460860
\(974\) 0 0
\(975\) 4.51099 0.144467
\(976\) 0 0
\(977\) 10.8063 0.345725 0.172862 0.984946i \(-0.444698\pi\)
0.172862 + 0.984946i \(0.444698\pi\)
\(978\) 0 0
\(979\) −5.95166 −0.190216
\(980\) 0 0
\(981\) 2.44187 0.0779628
\(982\) 0 0
\(983\) 61.7408 1.96923 0.984613 0.174751i \(-0.0559119\pi\)
0.984613 + 0.174751i \(0.0559119\pi\)
\(984\) 0 0
\(985\) 12.6731 0.403798
\(986\) 0 0
\(987\) −51.4875 −1.63887
\(988\) 0 0
\(989\) 78.0868 2.48302
\(990\) 0 0
\(991\) −16.8917 −0.536582 −0.268291 0.963338i \(-0.586459\pi\)
−0.268291 + 0.963338i \(0.586459\pi\)
\(992\) 0 0
\(993\) −7.00829 −0.222401
\(994\) 0 0
\(995\) −8.84837 −0.280512
\(996\) 0 0
\(997\) 31.9012 1.01032 0.505160 0.863026i \(-0.331433\pi\)
0.505160 + 0.863026i \(0.331433\pi\)
\(998\) 0 0
\(999\) −1.39858 −0.0442492
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 6864.2.a.ca.1.3 4
4.3 odd 2 3432.2.a.r.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.r.1.3 4 4.3 odd 2
6864.2.a.ca.1.3 4 1.1 even 1 trivial