Properties

Label 7728.2.a.bp.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.11491 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.11491 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.94567 q^{11} -5.47283 q^{13} -1.11491 q^{15} -0.926221 q^{17} +2.28415 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.75698 q^{25} -1.00000 q^{27} +2.58774 q^{29} -8.81756 q^{31} -3.94567 q^{33} -1.11491 q^{35} +1.87189 q^{37} +5.47283 q^{39} -4.28415 q^{41} +9.70265 q^{43} +1.11491 q^{45} +7.81756 q^{47} +1.00000 q^{49} +0.926221 q^{51} +4.70265 q^{53} +4.39905 q^{55} -2.28415 q^{57} +6.93246 q^{59} -13.0606 q^{61} -1.00000 q^{63} -6.10170 q^{65} +8.77643 q^{67} -1.00000 q^{69} +12.6483 q^{71} -7.87189 q^{73} +3.75698 q^{75} -3.94567 q^{77} -8.81756 q^{79} +1.00000 q^{81} -7.04737 q^{83} -1.03265 q^{85} -2.58774 q^{87} +0.0411284 q^{89} +5.47283 q^{91} +8.81756 q^{93} +2.54661 q^{95} +16.5613 q^{97} +3.94567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + q^{11} - 11 q^{13} + 3 q^{15} + 5 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{27} - 4 q^{29} - 2 q^{31} - q^{33} + 3 q^{35} - 8 q^{37} + 11 q^{39} - 11 q^{41} + 11 q^{43} - 3 q^{45} - q^{47} + 3 q^{49} - 4 q^{53} + 5 q^{55} - 5 q^{57} - 10 q^{59} - 22 q^{61} - 3 q^{63} + 8 q^{65} + 11 q^{67} - 3 q^{69} + 9 q^{71} - 10 q^{73} + 4 q^{75} - q^{77} - 2 q^{79} + 3 q^{81} + 16 q^{83} - 15 q^{85} + 4 q^{87} - 9 q^{89} + 11 q^{91} + 2 q^{93} + 5 q^{95} - 2 q^{97} + q^{99} + O(q^{100}) \)

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
\(5\) 1.11491 0.498602 0.249301 0.968426i \(-0.419799\pi\)
0.249301 + 0.968426i \(0.419799\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.94567 1.18966 0.594832 0.803850i \(-0.297219\pi\)
0.594832 + 0.803850i \(0.297219\pi\)
\(12\) 0 0
\(13\) −5.47283 −1.51789 −0.758946 0.651154i \(-0.774285\pi\)
−0.758946 + 0.651154i \(0.774285\pi\)
\(14\) 0 0
\(15\) −1.11491 −0.287868
\(16\) 0 0
\(17\) −0.926221 −0.224642 −0.112321 0.993672i \(-0.535828\pi\)
−0.112321 + 0.993672i \(0.535828\pi\)
\(18\) 0 0
\(19\) 2.28415 0.524019 0.262010 0.965065i \(-0.415615\pi\)
0.262010 + 0.965065i \(0.415615\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.75698 −0.751396
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.58774 0.480532 0.240266 0.970707i \(-0.422765\pi\)
0.240266 + 0.970707i \(0.422765\pi\)
\(30\) 0 0
\(31\) −8.81756 −1.58368 −0.791840 0.610729i \(-0.790877\pi\)
−0.791840 + 0.610729i \(0.790877\pi\)
\(32\) 0 0
\(33\) −3.94567 −0.686853
\(34\) 0 0
\(35\) −1.11491 −0.188454
\(36\) 0 0
\(37\) 1.87189 0.307737 0.153868 0.988091i \(-0.450827\pi\)
0.153868 + 0.988091i \(0.450827\pi\)
\(38\) 0 0
\(39\) 5.47283 0.876355
\(40\) 0 0
\(41\) −4.28415 −0.669071 −0.334536 0.942383i \(-0.608579\pi\)
−0.334536 + 0.942383i \(0.608579\pi\)
\(42\) 0 0
\(43\) 9.70265 1.47964 0.739820 0.672805i \(-0.234911\pi\)
0.739820 + 0.672805i \(0.234911\pi\)
\(44\) 0 0
\(45\) 1.11491 0.166201
\(46\) 0 0
\(47\) 7.81756 1.14031 0.570154 0.821538i \(-0.306884\pi\)
0.570154 + 0.821538i \(0.306884\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0.926221 0.129697
\(52\) 0 0
\(53\) 4.70265 0.645959 0.322979 0.946406i \(-0.395316\pi\)
0.322979 + 0.946406i \(0.395316\pi\)
\(54\) 0 0
\(55\) 4.39905 0.593168
\(56\) 0 0
\(57\) −2.28415 −0.302543
\(58\) 0 0
\(59\) 6.93246 0.902530 0.451265 0.892390i \(-0.350973\pi\)
0.451265 + 0.892390i \(0.350973\pi\)
\(60\) 0 0
\(61\) −13.0606 −1.67224 −0.836118 0.548550i \(-0.815180\pi\)
−0.836118 + 0.548550i \(0.815180\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −6.10170 −0.756823
\(66\) 0 0
\(67\) 8.77643 1.07221 0.536106 0.844151i \(-0.319895\pi\)
0.536106 + 0.844151i \(0.319895\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 12.6483 1.50108 0.750540 0.660826i \(-0.229794\pi\)
0.750540 + 0.660826i \(0.229794\pi\)
\(72\) 0 0
\(73\) −7.87189 −0.921335 −0.460667 0.887573i \(-0.652390\pi\)
−0.460667 + 0.887573i \(0.652390\pi\)
\(74\) 0 0
\(75\) 3.75698 0.433819
\(76\) 0 0
\(77\) −3.94567 −0.449651
\(78\) 0 0
\(79\) −8.81756 −0.992053 −0.496026 0.868307i \(-0.665208\pi\)
−0.496026 + 0.868307i \(0.665208\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −7.04737 −0.773550 −0.386775 0.922174i \(-0.626411\pi\)
−0.386775 + 0.922174i \(0.626411\pi\)
\(84\) 0 0
\(85\) −1.03265 −0.112007
\(86\) 0 0
\(87\) −2.58774 −0.277435
\(88\) 0 0
\(89\) 0.0411284 0.00435961 0.00217980 0.999998i \(-0.499306\pi\)
0.00217980 + 0.999998i \(0.499306\pi\)
\(90\) 0 0
\(91\) 5.47283 0.573709
\(92\) 0 0
\(93\) 8.81756 0.914338
\(94\) 0 0
\(95\) 2.54661 0.261277
\(96\) 0 0
\(97\) 16.5613 1.68155 0.840774 0.541386i \(-0.182100\pi\)
0.840774 + 0.541386i \(0.182100\pi\)
\(98\) 0 0
\(99\) 3.94567 0.396555
\(100\) 0 0
\(101\) 12.1079 1.20479 0.602393 0.798200i \(-0.294214\pi\)
0.602393 + 0.798200i \(0.294214\pi\)
\(102\) 0 0
\(103\) 8.87189 0.874173 0.437087 0.899419i \(-0.356010\pi\)
0.437087 + 0.899419i \(0.356010\pi\)
\(104\) 0 0
\(105\) 1.11491 0.108804
\(106\) 0 0
\(107\) 3.93246 0.380166 0.190083 0.981768i \(-0.439124\pi\)
0.190083 + 0.981768i \(0.439124\pi\)
\(108\) 0 0
\(109\) −16.5202 −1.58235 −0.791174 0.611591i \(-0.790530\pi\)
−0.791174 + 0.611591i \(0.790530\pi\)
\(110\) 0 0
\(111\) −1.87189 −0.177672
\(112\) 0 0
\(113\) 15.9130 1.49697 0.748485 0.663151i \(-0.230781\pi\)
0.748485 + 0.663151i \(0.230781\pi\)
\(114\) 0 0
\(115\) 1.11491 0.103966
\(116\) 0 0
\(117\) −5.47283 −0.505964
\(118\) 0 0
\(119\) 0.926221 0.0849065
\(120\) 0 0
\(121\) 4.56829 0.415300
\(122\) 0 0
\(123\) 4.28415 0.386289
\(124\) 0 0
\(125\) −9.76322 −0.873249
\(126\) 0 0
\(127\) −3.06058 −0.271582 −0.135791 0.990737i \(-0.543358\pi\)
−0.135791 + 0.990737i \(0.543358\pi\)
\(128\) 0 0
\(129\) −9.70265 −0.854271
\(130\) 0 0
\(131\) 10.8913 0.951580 0.475790 0.879559i \(-0.342162\pi\)
0.475790 + 0.879559i \(0.342162\pi\)
\(132\) 0 0
\(133\) −2.28415 −0.198061
\(134\) 0 0
\(135\) −1.11491 −0.0959560
\(136\) 0 0
\(137\) 11.3774 0.972035 0.486017 0.873949i \(-0.338449\pi\)
0.486017 + 0.873949i \(0.338449\pi\)
\(138\) 0 0
\(139\) −11.8044 −1.00123 −0.500616 0.865669i \(-0.666893\pi\)
−0.500616 + 0.865669i \(0.666893\pi\)
\(140\) 0 0
\(141\) −7.81756 −0.658357
\(142\) 0 0
\(143\) −21.5940 −1.80578
\(144\) 0 0
\(145\) 2.88509 0.239594
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 3.35793 0.275092 0.137546 0.990495i \(-0.456079\pi\)
0.137546 + 0.990495i \(0.456079\pi\)
\(150\) 0 0
\(151\) −6.99304 −0.569085 −0.284543 0.958663i \(-0.591842\pi\)
−0.284543 + 0.958663i \(0.591842\pi\)
\(152\) 0 0
\(153\) −0.926221 −0.0748805
\(154\) 0 0
\(155\) −9.83076 −0.789626
\(156\) 0 0
\(157\) −13.2104 −1.05430 −0.527151 0.849772i \(-0.676740\pi\)
−0.527151 + 0.849772i \(0.676740\pi\)
\(158\) 0 0
\(159\) −4.70265 −0.372944
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 4.47283 0.350339 0.175170 0.984538i \(-0.443953\pi\)
0.175170 + 0.984538i \(0.443953\pi\)
\(164\) 0 0
\(165\) −4.39905 −0.342466
\(166\) 0 0
\(167\) −6.06682 −0.469465 −0.234732 0.972060i \(-0.575421\pi\)
−0.234732 + 0.972060i \(0.575421\pi\)
\(168\) 0 0
\(169\) 16.9519 1.30399
\(170\) 0 0
\(171\) 2.28415 0.174673
\(172\) 0 0
\(173\) −0.735300 −0.0559038 −0.0279519 0.999609i \(-0.508899\pi\)
−0.0279519 + 0.999609i \(0.508899\pi\)
\(174\) 0 0
\(175\) 3.75698 0.284001
\(176\) 0 0
\(177\) −6.93246 −0.521076
\(178\) 0 0
\(179\) −12.1428 −0.907598 −0.453799 0.891104i \(-0.649932\pi\)
−0.453799 + 0.891104i \(0.649932\pi\)
\(180\) 0 0
\(181\) 5.99304 0.445459 0.222730 0.974880i \(-0.428503\pi\)
0.222730 + 0.974880i \(0.428503\pi\)
\(182\) 0 0
\(183\) 13.0606 0.965466
\(184\) 0 0
\(185\) 2.08698 0.153438
\(186\) 0 0
\(187\) −3.65456 −0.267248
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 13.5613 0.981264 0.490632 0.871367i \(-0.336766\pi\)
0.490632 + 0.871367i \(0.336766\pi\)
\(192\) 0 0
\(193\) −7.24926 −0.521813 −0.260907 0.965364i \(-0.584021\pi\)
−0.260907 + 0.965364i \(0.584021\pi\)
\(194\) 0 0
\(195\) 6.10170 0.436952
\(196\) 0 0
\(197\) 3.62039 0.257942 0.128971 0.991648i \(-0.458833\pi\)
0.128971 + 0.991648i \(0.458833\pi\)
\(198\) 0 0
\(199\) 5.41850 0.384107 0.192054 0.981384i \(-0.438485\pi\)
0.192054 + 0.981384i \(0.438485\pi\)
\(200\) 0 0
\(201\) −8.77643 −0.619042
\(202\) 0 0
\(203\) −2.58774 −0.181624
\(204\) 0 0
\(205\) −4.77643 −0.333600
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 9.01249 0.623407
\(210\) 0 0
\(211\) 22.5140 1.54993 0.774963 0.632007i \(-0.217769\pi\)
0.774963 + 0.632007i \(0.217769\pi\)
\(212\) 0 0
\(213\) −12.6483 −0.866648
\(214\) 0 0
\(215\) 10.8176 0.737751
\(216\) 0 0
\(217\) 8.81756 0.598575
\(218\) 0 0
\(219\) 7.87189 0.531933
\(220\) 0 0
\(221\) 5.06905 0.340981
\(222\) 0 0
\(223\) 17.7221 1.18676 0.593380 0.804923i \(-0.297793\pi\)
0.593380 + 0.804923i \(0.297793\pi\)
\(224\) 0 0
\(225\) −3.75698 −0.250465
\(226\) 0 0
\(227\) −9.00624 −0.597765 −0.298883 0.954290i \(-0.596614\pi\)
−0.298883 + 0.954290i \(0.596614\pi\)
\(228\) 0 0
\(229\) 6.51173 0.430307 0.215154 0.976580i \(-0.430975\pi\)
0.215154 + 0.976580i \(0.430975\pi\)
\(230\) 0 0
\(231\) 3.94567 0.259606
\(232\) 0 0
\(233\) 7.93942 0.520129 0.260065 0.965591i \(-0.416256\pi\)
0.260065 + 0.965591i \(0.416256\pi\)
\(234\) 0 0
\(235\) 8.71585 0.568560
\(236\) 0 0
\(237\) 8.81756 0.572762
\(238\) 0 0
\(239\) 7.42698 0.480411 0.240206 0.970722i \(-0.422785\pi\)
0.240206 + 0.970722i \(0.422785\pi\)
\(240\) 0 0
\(241\) 2.05433 0.132331 0.0661656 0.997809i \(-0.478923\pi\)
0.0661656 + 0.997809i \(0.478923\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.11491 0.0712288
\(246\) 0 0
\(247\) −12.5008 −0.795404
\(248\) 0 0
\(249\) 7.04737 0.446609
\(250\) 0 0
\(251\) 16.7283 1.05588 0.527942 0.849281i \(-0.322964\pi\)
0.527942 + 0.849281i \(0.322964\pi\)
\(252\) 0 0
\(253\) 3.94567 0.248062
\(254\) 0 0
\(255\) 1.03265 0.0646671
\(256\) 0 0
\(257\) 7.67000 0.478441 0.239221 0.970965i \(-0.423108\pi\)
0.239221 + 0.970965i \(0.423108\pi\)
\(258\) 0 0
\(259\) −1.87189 −0.116314
\(260\) 0 0
\(261\) 2.58774 0.160177
\(262\) 0 0
\(263\) −5.35097 −0.329955 −0.164977 0.986297i \(-0.552755\pi\)
−0.164977 + 0.986297i \(0.552755\pi\)
\(264\) 0 0
\(265\) 5.24302 0.322076
\(266\) 0 0
\(267\) −0.0411284 −0.00251702
\(268\) 0 0
\(269\) −13.5117 −0.823825 −0.411912 0.911223i \(-0.635139\pi\)
−0.411912 + 0.911223i \(0.635139\pi\)
\(270\) 0 0
\(271\) 10.4402 0.634196 0.317098 0.948393i \(-0.397292\pi\)
0.317098 + 0.948393i \(0.397292\pi\)
\(272\) 0 0
\(273\) −5.47283 −0.331231
\(274\) 0 0
\(275\) −14.8238 −0.893909
\(276\) 0 0
\(277\) 15.4379 0.927576 0.463788 0.885946i \(-0.346490\pi\)
0.463788 + 0.885946i \(0.346490\pi\)
\(278\) 0 0
\(279\) −8.81756 −0.527893
\(280\) 0 0
\(281\) 7.80908 0.465851 0.232925 0.972495i \(-0.425170\pi\)
0.232925 + 0.972495i \(0.425170\pi\)
\(282\) 0 0
\(283\) 10.5466 0.626931 0.313466 0.949600i \(-0.398510\pi\)
0.313466 + 0.949600i \(0.398510\pi\)
\(284\) 0 0
\(285\) −2.54661 −0.150848
\(286\) 0 0
\(287\) 4.28415 0.252885
\(288\) 0 0
\(289\) −16.1421 −0.949536
\(290\) 0 0
\(291\) −16.5613 −0.970843
\(292\) 0 0
\(293\) −13.2144 −0.771992 −0.385996 0.922500i \(-0.626142\pi\)
−0.385996 + 0.922500i \(0.626142\pi\)
\(294\) 0 0
\(295\) 7.72906 0.450003
\(296\) 0 0
\(297\) −3.94567 −0.228951
\(298\) 0 0
\(299\) −5.47283 −0.316502
\(300\) 0 0
\(301\) −9.70265 −0.559251
\(302\) 0 0
\(303\) −12.1079 −0.695583
\(304\) 0 0
\(305\) −14.5613 −0.833780
\(306\) 0 0
\(307\) −8.92622 −0.509446 −0.254723 0.967014i \(-0.581984\pi\)
−0.254723 + 0.967014i \(0.581984\pi\)
\(308\) 0 0
\(309\) −8.87189 −0.504704
\(310\) 0 0
\(311\) 15.1817 0.860877 0.430438 0.902620i \(-0.358359\pi\)
0.430438 + 0.902620i \(0.358359\pi\)
\(312\) 0 0
\(313\) 22.1560 1.25233 0.626167 0.779689i \(-0.284623\pi\)
0.626167 + 0.779689i \(0.284623\pi\)
\(314\) 0 0
\(315\) −1.11491 −0.0628179
\(316\) 0 0
\(317\) 12.2081 0.685677 0.342839 0.939394i \(-0.388612\pi\)
0.342839 + 0.939394i \(0.388612\pi\)
\(318\) 0 0
\(319\) 10.2104 0.571671
\(320\) 0 0
\(321\) −3.93246 −0.219489
\(322\) 0 0
\(323\) −2.11562 −0.117717
\(324\) 0 0
\(325\) 20.5613 1.14054
\(326\) 0 0
\(327\) 16.5202 0.913569
\(328\) 0 0
\(329\) −7.81756 −0.430996
\(330\) 0 0
\(331\) 15.4791 0.850807 0.425404 0.905004i \(-0.360132\pi\)
0.425404 + 0.905004i \(0.360132\pi\)
\(332\) 0 0
\(333\) 1.87189 0.102579
\(334\) 0 0
\(335\) 9.78491 0.534607
\(336\) 0 0
\(337\) −10.6483 −0.580051 −0.290025 0.957019i \(-0.593664\pi\)
−0.290025 + 0.957019i \(0.593664\pi\)
\(338\) 0 0
\(339\) −15.9130 −0.864276
\(340\) 0 0
\(341\) −34.7911 −1.88405
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.11491 −0.0600246
\(346\) 0 0
\(347\) 29.5334 1.58544 0.792718 0.609588i \(-0.208665\pi\)
0.792718 + 0.609588i \(0.208665\pi\)
\(348\) 0 0
\(349\) −13.4270 −0.718730 −0.359365 0.933197i \(-0.617007\pi\)
−0.359365 + 0.933197i \(0.617007\pi\)
\(350\) 0 0
\(351\) 5.47283 0.292118
\(352\) 0 0
\(353\) 25.3594 1.34975 0.674873 0.737933i \(-0.264198\pi\)
0.674873 + 0.737933i \(0.264198\pi\)
\(354\) 0 0
\(355\) 14.1017 0.748441
\(356\) 0 0
\(357\) −0.926221 −0.0490208
\(358\) 0 0
\(359\) −13.9130 −0.734301 −0.367150 0.930162i \(-0.619667\pi\)
−0.367150 + 0.930162i \(0.619667\pi\)
\(360\) 0 0
\(361\) −13.7827 −0.725404
\(362\) 0 0
\(363\) −4.56829 −0.239773
\(364\) 0 0
\(365\) −8.77643 −0.459379
\(366\) 0 0
\(367\) 13.2081 0.689459 0.344729 0.938702i \(-0.387971\pi\)
0.344729 + 0.938702i \(0.387971\pi\)
\(368\) 0 0
\(369\) −4.28415 −0.223024
\(370\) 0 0
\(371\) −4.70265 −0.244149
\(372\) 0 0
\(373\) −8.93871 −0.462829 −0.231414 0.972855i \(-0.574335\pi\)
−0.231414 + 0.972855i \(0.574335\pi\)
\(374\) 0 0
\(375\) 9.76322 0.504171
\(376\) 0 0
\(377\) −14.1623 −0.729394
\(378\) 0 0
\(379\) −8.60719 −0.442122 −0.221061 0.975260i \(-0.570952\pi\)
−0.221061 + 0.975260i \(0.570952\pi\)
\(380\) 0 0
\(381\) 3.06058 0.156798
\(382\) 0 0
\(383\) −20.8998 −1.06793 −0.533965 0.845506i \(-0.679299\pi\)
−0.533965 + 0.845506i \(0.679299\pi\)
\(384\) 0 0
\(385\) −4.39905 −0.224197
\(386\) 0 0
\(387\) 9.70265 0.493213
\(388\) 0 0
\(389\) −18.7911 −0.952749 −0.476375 0.879242i \(-0.658049\pi\)
−0.476375 + 0.879242i \(0.658049\pi\)
\(390\) 0 0
\(391\) −0.926221 −0.0468410
\(392\) 0 0
\(393\) −10.8913 −0.549395
\(394\) 0 0
\(395\) −9.83076 −0.494639
\(396\) 0 0
\(397\) 37.9038 1.90234 0.951169 0.308670i \(-0.0998839\pi\)
0.951169 + 0.308670i \(0.0998839\pi\)
\(398\) 0 0
\(399\) 2.28415 0.114350
\(400\) 0 0
\(401\) 31.0558 1.55086 0.775428 0.631437i \(-0.217534\pi\)
0.775428 + 0.631437i \(0.217534\pi\)
\(402\) 0 0
\(403\) 48.2570 2.40385
\(404\) 0 0
\(405\) 1.11491 0.0554002
\(406\) 0 0
\(407\) 7.38585 0.366103
\(408\) 0 0
\(409\) 1.53341 0.0758222 0.0379111 0.999281i \(-0.487930\pi\)
0.0379111 + 0.999281i \(0.487930\pi\)
\(410\) 0 0
\(411\) −11.3774 −0.561204
\(412\) 0 0
\(413\) −6.93246 −0.341124
\(414\) 0 0
\(415\) −7.85717 −0.385693
\(416\) 0 0
\(417\) 11.8044 0.578062
\(418\) 0 0
\(419\) −23.2361 −1.13516 −0.567578 0.823320i \(-0.692119\pi\)
−0.567578 + 0.823320i \(0.692119\pi\)
\(420\) 0 0
\(421\) 25.4031 1.23807 0.619035 0.785364i \(-0.287524\pi\)
0.619035 + 0.785364i \(0.287524\pi\)
\(422\) 0 0
\(423\) 7.81756 0.380103
\(424\) 0 0
\(425\) 3.47979 0.168795
\(426\) 0 0
\(427\) 13.0606 0.632046
\(428\) 0 0
\(429\) 21.5940 1.04257
\(430\) 0 0
\(431\) −25.8712 −1.24617 −0.623085 0.782154i \(-0.714121\pi\)
−0.623085 + 0.782154i \(0.714121\pi\)
\(432\) 0 0
\(433\) 23.5613 1.13229 0.566143 0.824307i \(-0.308435\pi\)
0.566143 + 0.824307i \(0.308435\pi\)
\(434\) 0 0
\(435\) −2.88509 −0.138330
\(436\) 0 0
\(437\) 2.28415 0.109266
\(438\) 0 0
\(439\) −6.10170 −0.291218 −0.145609 0.989342i \(-0.546514\pi\)
−0.145609 + 0.989342i \(0.546514\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −10.3510 −0.491789 −0.245895 0.969297i \(-0.579082\pi\)
−0.245895 + 0.969297i \(0.579082\pi\)
\(444\) 0 0
\(445\) 0.0458544 0.00217371
\(446\) 0 0
\(447\) −3.35793 −0.158824
\(448\) 0 0
\(449\) 6.02168 0.284181 0.142090 0.989854i \(-0.454618\pi\)
0.142090 + 0.989854i \(0.454618\pi\)
\(450\) 0 0
\(451\) −16.9038 −0.795970
\(452\) 0 0
\(453\) 6.99304 0.328562
\(454\) 0 0
\(455\) 6.10170 0.286052
\(456\) 0 0
\(457\) −30.8448 −1.44286 −0.721429 0.692489i \(-0.756514\pi\)
−0.721429 + 0.692489i \(0.756514\pi\)
\(458\) 0 0
\(459\) 0.926221 0.0432323
\(460\) 0 0
\(461\) 24.3726 1.13515 0.567574 0.823323i \(-0.307882\pi\)
0.567574 + 0.823323i \(0.307882\pi\)
\(462\) 0 0
\(463\) 26.8913 1.24975 0.624873 0.780726i \(-0.285151\pi\)
0.624873 + 0.780726i \(0.285151\pi\)
\(464\) 0 0
\(465\) 9.83076 0.455891
\(466\) 0 0
\(467\) −27.5459 −1.27467 −0.637336 0.770586i \(-0.719964\pi\)
−0.637336 + 0.770586i \(0.719964\pi\)
\(468\) 0 0
\(469\) −8.77643 −0.405258
\(470\) 0 0
\(471\) 13.2104 0.608702
\(472\) 0 0
\(473\) 38.2834 1.76027
\(474\) 0 0
\(475\) −8.58150 −0.393746
\(476\) 0 0
\(477\) 4.70265 0.215320
\(478\) 0 0
\(479\) 31.7368 1.45009 0.725046 0.688700i \(-0.241818\pi\)
0.725046 + 0.688700i \(0.241818\pi\)
\(480\) 0 0
\(481\) −10.2445 −0.467111
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 18.4644 0.838423
\(486\) 0 0
\(487\) 24.7911 1.12339 0.561697 0.827343i \(-0.310149\pi\)
0.561697 + 0.827343i \(0.310149\pi\)
\(488\) 0 0
\(489\) −4.47283 −0.202269
\(490\) 0 0
\(491\) −35.9450 −1.62217 −0.811086 0.584926i \(-0.801123\pi\)
−0.811086 + 0.584926i \(0.801123\pi\)
\(492\) 0 0
\(493\) −2.39682 −0.107947
\(494\) 0 0
\(495\) 4.39905 0.197723
\(496\) 0 0
\(497\) −12.6483 −0.567355
\(498\) 0 0
\(499\) −30.4938 −1.36509 −0.682545 0.730844i \(-0.739127\pi\)
−0.682545 + 0.730844i \(0.739127\pi\)
\(500\) 0 0
\(501\) 6.06682 0.271045
\(502\) 0 0
\(503\) 9.65679 0.430575 0.215288 0.976551i \(-0.430931\pi\)
0.215288 + 0.976551i \(0.430931\pi\)
\(504\) 0 0
\(505\) 13.4992 0.600708
\(506\) 0 0
\(507\) −16.9519 −0.752861
\(508\) 0 0
\(509\) 31.9123 1.41449 0.707244 0.706970i \(-0.249938\pi\)
0.707244 + 0.706970i \(0.249938\pi\)
\(510\) 0 0
\(511\) 7.87189 0.348232
\(512\) 0 0
\(513\) −2.28415 −0.100848
\(514\) 0 0
\(515\) 9.89134 0.435864
\(516\) 0 0
\(517\) 30.8455 1.35658
\(518\) 0 0
\(519\) 0.735300 0.0322761
\(520\) 0 0
\(521\) −0.108664 −0.00476067 −0.00238034 0.999997i \(-0.500758\pi\)
−0.00238034 + 0.999997i \(0.500758\pi\)
\(522\) 0 0
\(523\) 24.2508 1.06041 0.530206 0.847869i \(-0.322114\pi\)
0.530206 + 0.847869i \(0.322114\pi\)
\(524\) 0 0
\(525\) −3.75698 −0.163968
\(526\) 0 0
\(527\) 8.16701 0.355760
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.93246 0.300843
\(532\) 0 0
\(533\) 23.4464 1.01558
\(534\) 0 0
\(535\) 4.38433 0.189551
\(536\) 0 0
\(537\) 12.1428 0.524002
\(538\) 0 0
\(539\) 3.94567 0.169952
\(540\) 0 0
\(541\) 14.5334 0.624840 0.312420 0.949944i \(-0.398860\pi\)
0.312420 + 0.949944i \(0.398860\pi\)
\(542\) 0 0
\(543\) −5.99304 −0.257186
\(544\) 0 0
\(545\) −18.4185 −0.788962
\(546\) 0 0
\(547\) 41.0187 1.75383 0.876917 0.480642i \(-0.159596\pi\)
0.876917 + 0.480642i \(0.159596\pi\)
\(548\) 0 0
\(549\) −13.0606 −0.557412
\(550\) 0 0
\(551\) 5.91078 0.251808
\(552\) 0 0
\(553\) 8.81756 0.374961
\(554\) 0 0
\(555\) −2.08698 −0.0885875
\(556\) 0 0
\(557\) −8.65055 −0.366536 −0.183268 0.983063i \(-0.558668\pi\)
−0.183268 + 0.983063i \(0.558668\pi\)
\(558\) 0 0
\(559\) −53.1010 −2.24593
\(560\) 0 0
\(561\) 3.65456 0.154296
\(562\) 0 0
\(563\) 17.0257 0.717547 0.358774 0.933425i \(-0.383195\pi\)
0.358774 + 0.933425i \(0.383195\pi\)
\(564\) 0 0
\(565\) 17.7415 0.746392
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 29.8440 1.25112 0.625562 0.780174i \(-0.284870\pi\)
0.625562 + 0.780174i \(0.284870\pi\)
\(570\) 0 0
\(571\) 12.7228 0.532433 0.266217 0.963913i \(-0.414226\pi\)
0.266217 + 0.963913i \(0.414226\pi\)
\(572\) 0 0
\(573\) −13.5613 −0.566533
\(574\) 0 0
\(575\) −3.75698 −0.156677
\(576\) 0 0
\(577\) 16.5683 0.689747 0.344874 0.938649i \(-0.387922\pi\)
0.344874 + 0.938649i \(0.387922\pi\)
\(578\) 0 0
\(579\) 7.24926 0.301269
\(580\) 0 0
\(581\) 7.04737 0.292374
\(582\) 0 0
\(583\) 18.5551 0.768473
\(584\) 0 0
\(585\) −6.10170 −0.252274
\(586\) 0 0
\(587\) 30.7779 1.27034 0.635171 0.772372i \(-0.280930\pi\)
0.635171 + 0.772372i \(0.280930\pi\)
\(588\) 0 0
\(589\) −20.1406 −0.829879
\(590\) 0 0
\(591\) −3.62039 −0.148923
\(592\) 0 0
\(593\) −25.5698 −1.05003 −0.525013 0.851094i \(-0.675940\pi\)
−0.525013 + 0.851094i \(0.675940\pi\)
\(594\) 0 0
\(595\) 1.03265 0.0423345
\(596\) 0 0
\(597\) −5.41850 −0.221765
\(598\) 0 0
\(599\) 0.974310 0.0398092 0.0199046 0.999802i \(-0.493664\pi\)
0.0199046 + 0.999802i \(0.493664\pi\)
\(600\) 0 0
\(601\) −23.0885 −0.941800 −0.470900 0.882187i \(-0.656071\pi\)
−0.470900 + 0.882187i \(0.656071\pi\)
\(602\) 0 0
\(603\) 8.77643 0.357404
\(604\) 0 0
\(605\) 5.09323 0.207069
\(606\) 0 0
\(607\) 6.93095 0.281318 0.140659 0.990058i \(-0.455078\pi\)
0.140659 + 0.990058i \(0.455078\pi\)
\(608\) 0 0
\(609\) 2.58774 0.104861
\(610\) 0 0
\(611\) −42.7842 −1.73086
\(612\) 0 0
\(613\) 31.5893 1.27588 0.637939 0.770087i \(-0.279787\pi\)
0.637939 + 0.770087i \(0.279787\pi\)
\(614\) 0 0
\(615\) 4.77643 0.192604
\(616\) 0 0
\(617\) 35.5676 1.43190 0.715948 0.698153i \(-0.245995\pi\)
0.715948 + 0.698153i \(0.245995\pi\)
\(618\) 0 0
\(619\) 32.8128 1.31886 0.659430 0.751766i \(-0.270798\pi\)
0.659430 + 0.751766i \(0.270798\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −0.0411284 −0.00164778
\(624\) 0 0
\(625\) 7.89981 0.315993
\(626\) 0 0
\(627\) −9.01249 −0.359924
\(628\) 0 0
\(629\) −1.73378 −0.0691304
\(630\) 0 0
\(631\) 32.6825 1.30107 0.650535 0.759477i \(-0.274545\pi\)
0.650535 + 0.759477i \(0.274545\pi\)
\(632\) 0 0
\(633\) −22.5140 −0.894850
\(634\) 0 0
\(635\) −3.41226 −0.135411
\(636\) 0 0
\(637\) −5.47283 −0.216842
\(638\) 0 0
\(639\) 12.6483 0.500360
\(640\) 0 0
\(641\) −15.6942 −0.619882 −0.309941 0.950756i \(-0.600309\pi\)
−0.309941 + 0.950756i \(0.600309\pi\)
\(642\) 0 0
\(643\) 7.62887 0.300853 0.150427 0.988621i \(-0.451935\pi\)
0.150427 + 0.988621i \(0.451935\pi\)
\(644\) 0 0
\(645\) −10.8176 −0.425941
\(646\) 0 0
\(647\) −3.64431 −0.143273 −0.0716363 0.997431i \(-0.522822\pi\)
−0.0716363 + 0.997431i \(0.522822\pi\)
\(648\) 0 0
\(649\) 27.3532 1.07371
\(650\) 0 0
\(651\) −8.81756 −0.345587
\(652\) 0 0
\(653\) 34.9673 1.36838 0.684189 0.729305i \(-0.260156\pi\)
0.684189 + 0.729305i \(0.260156\pi\)
\(654\) 0 0
\(655\) 12.1428 0.474460
\(656\) 0 0
\(657\) −7.87189 −0.307112
\(658\) 0 0
\(659\) −43.7368 −1.70374 −0.851872 0.523750i \(-0.824533\pi\)
−0.851872 + 0.523750i \(0.824533\pi\)
\(660\) 0 0
\(661\) −46.9986 −1.82803 −0.914016 0.405678i \(-0.867036\pi\)
−0.914016 + 0.405678i \(0.867036\pi\)
\(662\) 0 0
\(663\) −5.06905 −0.196866
\(664\) 0 0
\(665\) −2.54661 −0.0987534
\(666\) 0 0
\(667\) 2.58774 0.100198
\(668\) 0 0
\(669\) −17.7221 −0.685176
\(670\) 0 0
\(671\) −51.5327 −1.98940
\(672\) 0 0
\(673\) 0.245253 0.00945382 0.00472691 0.999989i \(-0.498495\pi\)
0.00472691 + 0.999989i \(0.498495\pi\)
\(674\) 0 0
\(675\) 3.75698 0.144606
\(676\) 0 0
\(677\) 17.2819 0.664198 0.332099 0.943245i \(-0.392243\pi\)
0.332099 + 0.943245i \(0.392243\pi\)
\(678\) 0 0
\(679\) −16.5613 −0.635566
\(680\) 0 0
\(681\) 9.00624 0.345120
\(682\) 0 0
\(683\) −15.1127 −0.578270 −0.289135 0.957288i \(-0.593368\pi\)
−0.289135 + 0.957288i \(0.593368\pi\)
\(684\) 0 0
\(685\) 12.6847 0.484658
\(686\) 0 0
\(687\) −6.51173 −0.248438
\(688\) 0 0
\(689\) −25.7368 −0.980495
\(690\) 0 0
\(691\) −14.1234 −0.537279 −0.268639 0.963241i \(-0.586574\pi\)
−0.268639 + 0.963241i \(0.586574\pi\)
\(692\) 0 0
\(693\) −3.94567 −0.149884
\(694\) 0 0
\(695\) −13.1608 −0.499216
\(696\) 0 0
\(697\) 3.96807 0.150301
\(698\) 0 0
\(699\) −7.93942 −0.300297
\(700\) 0 0
\(701\) −48.1204 −1.81748 −0.908742 0.417359i \(-0.862956\pi\)
−0.908742 + 0.417359i \(0.862956\pi\)
\(702\) 0 0
\(703\) 4.27567 0.161260
\(704\) 0 0
\(705\) −8.71585 −0.328258
\(706\) 0 0
\(707\) −12.1079 −0.455366
\(708\) 0 0
\(709\) −3.83772 −0.144129 −0.0720643 0.997400i \(-0.522959\pi\)
−0.0720643 + 0.997400i \(0.522959\pi\)
\(710\) 0 0
\(711\) −8.81756 −0.330684
\(712\) 0 0
\(713\) −8.81756 −0.330220
\(714\) 0 0
\(715\) −24.0753 −0.900365
\(716\) 0 0
\(717\) −7.42698 −0.277366
\(718\) 0 0
\(719\) 53.4108 1.99189 0.995944 0.0899774i \(-0.0286795\pi\)
0.995944 + 0.0899774i \(0.0286795\pi\)
\(720\) 0 0
\(721\) −8.87189 −0.330406
\(722\) 0 0
\(723\) −2.05433 −0.0764014
\(724\) 0 0
\(725\) −9.72210 −0.361070
\(726\) 0 0
\(727\) −3.20341 −0.118808 −0.0594039 0.998234i \(-0.518920\pi\)
−0.0594039 + 0.998234i \(0.518920\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.98680 −0.332389
\(732\) 0 0
\(733\) −46.6197 −1.72194 −0.860968 0.508658i \(-0.830142\pi\)
−0.860968 + 0.508658i \(0.830142\pi\)
\(734\) 0 0
\(735\) −1.11491 −0.0411240
\(736\) 0 0
\(737\) 34.6289 1.27557
\(738\) 0 0
\(739\) 34.7787 1.27935 0.639677 0.768644i \(-0.279068\pi\)
0.639677 + 0.768644i \(0.279068\pi\)
\(740\) 0 0
\(741\) 12.5008 0.459227
\(742\) 0 0
\(743\) −1.09394 −0.0401329 −0.0200664 0.999799i \(-0.506388\pi\)
−0.0200664 + 0.999799i \(0.506388\pi\)
\(744\) 0 0
\(745\) 3.74378 0.137161
\(746\) 0 0
\(747\) −7.04737 −0.257850
\(748\) 0 0
\(749\) −3.93246 −0.143689
\(750\) 0 0
\(751\) −11.6009 −0.423325 −0.211662 0.977343i \(-0.567888\pi\)
−0.211662 + 0.977343i \(0.567888\pi\)
\(752\) 0 0
\(753\) −16.7283 −0.609615
\(754\) 0 0
\(755\) −7.79659 −0.283747
\(756\) 0 0
\(757\) 26.6825 0.969791 0.484896 0.874572i \(-0.338858\pi\)
0.484896 + 0.874572i \(0.338858\pi\)
\(758\) 0 0
\(759\) −3.94567 −0.143219
\(760\) 0 0
\(761\) 20.9497 0.759425 0.379713 0.925105i \(-0.376023\pi\)
0.379713 + 0.925105i \(0.376023\pi\)
\(762\) 0 0
\(763\) 16.5202 0.598072
\(764\) 0 0
\(765\) −1.03265 −0.0373356
\(766\) 0 0
\(767\) −37.9402 −1.36994
\(768\) 0 0
\(769\) 11.5918 0.418009 0.209005 0.977915i \(-0.432978\pi\)
0.209005 + 0.977915i \(0.432978\pi\)
\(770\) 0 0
\(771\) −7.67000 −0.276228
\(772\) 0 0
\(773\) −45.9277 −1.65191 −0.825953 0.563739i \(-0.809362\pi\)
−0.825953 + 0.563739i \(0.809362\pi\)
\(774\) 0 0
\(775\) 33.1274 1.18997
\(776\) 0 0
\(777\) 1.87189 0.0671536
\(778\) 0 0
\(779\) −9.78562 −0.350606
\(780\) 0 0
\(781\) 49.9061 1.78578
\(782\) 0 0
\(783\) −2.58774 −0.0924783
\(784\) 0 0
\(785\) −14.7283 −0.525677
\(786\) 0 0
\(787\) 32.7585 1.16771 0.583857 0.811856i \(-0.301543\pi\)
0.583857 + 0.811856i \(0.301543\pi\)
\(788\) 0 0
\(789\) 5.35097 0.190499
\(790\) 0 0
\(791\) −15.9130 −0.565802
\(792\) 0 0
\(793\) 71.4784 2.53827
\(794\) 0 0
\(795\) −5.24302 −0.185951
\(796\) 0 0
\(797\) −24.1770 −0.856393 −0.428197 0.903686i \(-0.640851\pi\)
−0.428197 + 0.903686i \(0.640851\pi\)
\(798\) 0 0
\(799\) −7.24078 −0.256161
\(800\) 0 0
\(801\) 0.0411284 0.00145320
\(802\) 0 0
\(803\) −31.0599 −1.09608
\(804\) 0 0
\(805\) −1.11491 −0.0392953
\(806\) 0 0
\(807\) 13.5117 0.475635
\(808\) 0 0
\(809\) −29.1777 −1.02583 −0.512917 0.858438i \(-0.671435\pi\)
−0.512917 + 0.858438i \(0.671435\pi\)
\(810\) 0 0
\(811\) −16.9915 −0.596653 −0.298327 0.954464i \(-0.596428\pi\)
−0.298327 + 0.954464i \(0.596428\pi\)
\(812\) 0 0
\(813\) −10.4402 −0.366153
\(814\) 0 0
\(815\) 4.98680 0.174680
\(816\) 0 0
\(817\) 22.1623 0.775360
\(818\) 0 0
\(819\) 5.47283 0.191236
\(820\) 0 0
\(821\) 37.0793 1.29408 0.647038 0.762457i \(-0.276007\pi\)
0.647038 + 0.762457i \(0.276007\pi\)
\(822\) 0 0
\(823\) 32.9450 1.14839 0.574194 0.818719i \(-0.305315\pi\)
0.574194 + 0.818719i \(0.305315\pi\)
\(824\) 0 0
\(825\) 14.8238 0.516098
\(826\) 0 0
\(827\) −30.8936 −1.07427 −0.537137 0.843495i \(-0.680494\pi\)
−0.537137 + 0.843495i \(0.680494\pi\)
\(828\) 0 0
\(829\) 20.1406 0.699512 0.349756 0.936841i \(-0.386265\pi\)
0.349756 + 0.936841i \(0.386265\pi\)
\(830\) 0 0
\(831\) −15.4379 −0.535537
\(832\) 0 0
\(833\) −0.926221 −0.0320917
\(834\) 0 0
\(835\) −6.76394 −0.234076
\(836\) 0 0
\(837\) 8.81756 0.304779
\(838\) 0 0
\(839\) 32.2401 1.11305 0.556525 0.830831i \(-0.312134\pi\)
0.556525 + 0.830831i \(0.312134\pi\)
\(840\) 0 0
\(841\) −22.3036 −0.769089
\(842\) 0 0
\(843\) −7.80908 −0.268959
\(844\) 0 0
\(845\) 18.8998 0.650173
\(846\) 0 0
\(847\) −4.56829 −0.156968
\(848\) 0 0
\(849\) −10.5466 −0.361959
\(850\) 0 0
\(851\) 1.87189 0.0641675
\(852\) 0 0
\(853\) −51.8510 −1.77534 −0.887672 0.460476i \(-0.847679\pi\)
−0.887672 + 0.460476i \(0.847679\pi\)
\(854\) 0 0
\(855\) 2.54661 0.0870923
\(856\) 0 0
\(857\) 49.8425 1.70259 0.851294 0.524689i \(-0.175818\pi\)
0.851294 + 0.524689i \(0.175818\pi\)
\(858\) 0 0
\(859\) 42.3121 1.44367 0.721835 0.692066i \(-0.243299\pi\)
0.721835 + 0.692066i \(0.243299\pi\)
\(860\) 0 0
\(861\) −4.28415 −0.146003
\(862\) 0 0
\(863\) −8.48604 −0.288868 −0.144434 0.989514i \(-0.546136\pi\)
−0.144434 + 0.989514i \(0.546136\pi\)
\(864\) 0 0
\(865\) −0.819791 −0.0278737
\(866\) 0 0
\(867\) 16.1421 0.548215
\(868\) 0 0
\(869\) −34.7911 −1.18021
\(870\) 0 0
\(871\) −48.0319 −1.62750
\(872\) 0 0
\(873\) 16.5613 0.560516
\(874\) 0 0
\(875\) 9.76322 0.330057
\(876\) 0 0
\(877\) −17.5488 −0.592582 −0.296291 0.955098i \(-0.595750\pi\)
−0.296291 + 0.955098i \(0.595750\pi\)
\(878\) 0 0
\(879\) 13.2144 0.445710
\(880\) 0 0
\(881\) 19.7313 0.664764 0.332382 0.943145i \(-0.392148\pi\)
0.332382 + 0.943145i \(0.392148\pi\)
\(882\) 0 0
\(883\) −46.1873 −1.55432 −0.777162 0.629300i \(-0.783342\pi\)
−0.777162 + 0.629300i \(0.783342\pi\)
\(884\) 0 0
\(885\) −7.72906 −0.259809
\(886\) 0 0
\(887\) 39.8113 1.33673 0.668367 0.743832i \(-0.266994\pi\)
0.668367 + 0.743832i \(0.266994\pi\)
\(888\) 0 0
\(889\) 3.06058 0.102648
\(890\) 0 0
\(891\) 3.94567 0.132185
\(892\) 0 0
\(893\) 17.8565 0.597543
\(894\) 0 0
\(895\) −13.5381 −0.452530
\(896\) 0 0
\(897\) 5.47283 0.182733
\(898\) 0 0
\(899\) −22.8176 −0.761008
\(900\) 0 0
\(901\) −4.35569 −0.145109
\(902\) 0 0
\(903\) 9.70265 0.322884
\(904\) 0 0
\(905\) 6.68168 0.222107
\(906\) 0 0
\(907\) −51.5481 −1.71163 −0.855814 0.517284i \(-0.826943\pi\)
−0.855814 + 0.517284i \(0.826943\pi\)
\(908\) 0 0
\(909\) 12.1079 0.401595
\(910\) 0 0
\(911\) −16.7842 −0.556085 −0.278042 0.960569i \(-0.589686\pi\)
−0.278042 + 0.960569i \(0.589686\pi\)
\(912\) 0 0
\(913\) −27.8066 −0.920264
\(914\) 0 0
\(915\) 14.5613 0.481383
\(916\) 0 0
\(917\) −10.8913 −0.359664
\(918\) 0 0
\(919\) 27.2508 0.898920 0.449460 0.893300i \(-0.351616\pi\)
0.449460 + 0.893300i \(0.351616\pi\)
\(920\) 0 0
\(921\) 8.92622 0.294129
\(922\) 0 0
\(923\) −69.2221 −2.27847
\(924\) 0 0
\(925\) −7.03265 −0.231232
\(926\) 0 0
\(927\) 8.87189 0.291391
\(928\) 0 0
\(929\) 5.04514 0.165526 0.0827628 0.996569i \(-0.473626\pi\)
0.0827628 + 0.996569i \(0.473626\pi\)
\(930\) 0 0
\(931\) 2.28415 0.0748599
\(932\) 0 0
\(933\) −15.1817 −0.497027
\(934\) 0 0
\(935\) −4.07450 −0.133250
\(936\) 0 0
\(937\) 2.64903 0.0865402 0.0432701 0.999063i \(-0.486222\pi\)
0.0432701 + 0.999063i \(0.486222\pi\)
\(938\) 0 0
\(939\) −22.1560 −0.723035
\(940\) 0 0
\(941\) −32.7019 −1.06605 −0.533026 0.846099i \(-0.678945\pi\)
−0.533026 + 0.846099i \(0.678945\pi\)
\(942\) 0 0
\(943\) −4.28415 −0.139511
\(944\) 0 0
\(945\) 1.11491 0.0362679
\(946\) 0 0
\(947\) 15.8216 0.514132 0.257066 0.966394i \(-0.417244\pi\)
0.257066 + 0.966394i \(0.417244\pi\)
\(948\) 0 0
\(949\) 43.0815 1.39849
\(950\) 0 0
\(951\) −12.2081 −0.395876
\(952\) 0 0
\(953\) 26.7655 0.867018 0.433509 0.901149i \(-0.357275\pi\)
0.433509 + 0.901149i \(0.357275\pi\)
\(954\) 0 0
\(955\) 15.1196 0.489260
\(956\) 0 0
\(957\) −10.2104 −0.330054
\(958\) 0 0
\(959\) −11.3774 −0.367395
\(960\) 0 0
\(961\) 46.7493 1.50804
\(962\) 0 0
\(963\) 3.93246 0.126722
\(964\) 0 0
\(965\) −8.08226 −0.260177
\(966\) 0 0
\(967\) −29.3400 −0.943511 −0.471755 0.881729i \(-0.656379\pi\)
−0.471755 + 0.881729i \(0.656379\pi\)
\(968\) 0 0
\(969\) 2.11562 0.0679637
\(970\) 0 0
\(971\) −20.9309 −0.671706 −0.335853 0.941914i \(-0.609025\pi\)
−0.335853 + 0.941914i \(0.609025\pi\)
\(972\) 0 0
\(973\) 11.8044 0.378430
\(974\) 0 0
\(975\) −20.5613 −0.658490
\(976\) 0 0
\(977\) 26.0496 0.833401 0.416700 0.909044i \(-0.363186\pi\)
0.416700 + 0.909044i \(0.363186\pi\)
\(978\) 0 0
\(979\) 0.162279 0.00518646
\(980\) 0 0
\(981\) −16.5202 −0.527450
\(982\) 0 0
\(983\) −59.7882 −1.90695 −0.953474 0.301476i \(-0.902521\pi\)
−0.953474 + 0.301476i \(0.902521\pi\)
\(984\) 0 0
\(985\) 4.03640 0.128610
\(986\) 0 0
\(987\) 7.81756 0.248836
\(988\) 0 0
\(989\) 9.70265 0.308526
\(990\) 0 0
\(991\) −30.0062 −0.953180 −0.476590 0.879126i \(-0.658127\pi\)
−0.476590 + 0.879126i \(0.658127\pi\)
\(992\) 0 0
\(993\) −15.4791 −0.491214
\(994\) 0 0
\(995\) 6.04113 0.191517
\(996\) 0 0
\(997\) 7.82853 0.247932 0.123966 0.992286i \(-0.460439\pi\)
0.123966 + 0.992286i \(0.460439\pi\)
\(998\) 0 0
\(999\) −1.87189 −0.0592239
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 7728.2.a.bp.1.3 3
4.3 odd 2 3864.2.a.p.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.p.1.3 3 4.3 odd 2
7728.2.a.bp.1.3 3 1.1 even 1 trivial