Properties

Label 882.2.a.m.1.2
Level $882$
Weight $2$
Character 882.1
Self dual yes
Analytic conductor $7.043$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.04280545828\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} -1.00000 q^{8} -1.41421 q^{10} -4.00000 q^{11} -4.24264 q^{13} +1.00000 q^{16} -7.07107 q^{17} +5.65685 q^{19} +1.41421 q^{20} +4.00000 q^{22} -8.00000 q^{23} -3.00000 q^{25} +4.24264 q^{26} -2.00000 q^{29} -1.00000 q^{32} +7.07107 q^{34} +4.00000 q^{37} -5.65685 q^{38} -1.41421 q^{40} +9.89949 q^{41} -4.00000 q^{43} -4.00000 q^{44} +8.00000 q^{46} +5.65685 q^{47} +3.00000 q^{50} -4.24264 q^{52} -4.00000 q^{53} -5.65685 q^{55} +2.00000 q^{58} -11.3137 q^{59} -1.41421 q^{61} +1.00000 q^{64} -6.00000 q^{65} -12.0000 q^{67} -7.07107 q^{68} +15.5563 q^{73} -4.00000 q^{74} +5.65685 q^{76} -16.0000 q^{79} +1.41421 q^{80} -9.89949 q^{82} -5.65685 q^{83} -10.0000 q^{85} +4.00000 q^{86} +4.00000 q^{88} +7.07107 q^{89} -8.00000 q^{92} -5.65685 q^{94} +8.00000 q^{95} -7.07107 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 8q^{11} + 2q^{16} + 8q^{22} - 16q^{23} - 6q^{25} - 4q^{29} - 2q^{32} + 8q^{37} - 8q^{43} - 8q^{44} + 16q^{46} + 6q^{50} - 8q^{53} + 4q^{58} + 2q^{64} - 12q^{65} - 24q^{67} - 8q^{74} - 32q^{79} - 20q^{85} + 8q^{86} + 8q^{88} - 16q^{92} + 16q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.41421 −0.447214
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.07107 −1.71499 −0.857493 0.514496i \(-0.827979\pi\)
−0.857493 + 0.514496i \(0.827979\pi\)
\(18\) 0 0
\(19\) 5.65685 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) 1.41421 0.316228
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 4.24264 0.832050
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.07107 1.21268
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −5.65685 −0.917663
\(39\) 0 0
\(40\) −1.41421 −0.223607
\(41\) 9.89949 1.54604 0.773021 0.634381i \(-0.218745\pi\)
0.773021 + 0.634381i \(0.218745\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) 5.65685 0.825137 0.412568 0.910927i \(-0.364632\pi\)
0.412568 + 0.910927i \(0.364632\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 3.00000 0.424264
\(51\) 0 0
\(52\) −4.24264 −0.588348
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −11.3137 −1.47292 −0.736460 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) −1.41421 −0.181071 −0.0905357 0.995893i \(-0.528858\pi\)
−0.0905357 + 0.995893i \(0.528858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −7.07107 −0.857493
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 15.5563 1.82073 0.910366 0.413803i \(-0.135800\pi\)
0.910366 + 0.413803i \(0.135800\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) 5.65685 0.648886
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 1.41421 0.158114
\(81\) 0 0
\(82\) −9.89949 −1.09322
\(83\) −5.65685 −0.620920 −0.310460 0.950586i \(-0.600483\pi\)
−0.310460 + 0.950586i \(0.600483\pi\)
\(84\) 0 0
\(85\) −10.0000 −1.08465
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) −5.65685 −0.583460
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −7.07107 −0.717958 −0.358979 0.933346i \(-0.616875\pi\)
−0.358979 + 0.933346i \(0.616875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.00000 −0.300000
\(101\) 12.7279 1.26648 0.633238 0.773957i \(-0.281726\pi\)
0.633238 + 0.773957i \(0.281726\pi\)
\(102\) 0 0
\(103\) −5.65685 −0.557386 −0.278693 0.960380i \(-0.589901\pi\)
−0.278693 + 0.960380i \(0.589901\pi\)
\(104\) 4.24264 0.416025
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 5.65685 0.539360
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −11.3137 −1.05501
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 11.3137 1.04151
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 1.41421 0.128037
\(123\) 0 0
\(124\) 0 0
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) −16.9706 −1.48272 −0.741362 0.671105i \(-0.765820\pi\)
−0.741362 + 0.671105i \(0.765820\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 7.07107 0.606339
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 5.65685 0.479808 0.239904 0.970797i \(-0.422884\pi\)
0.239904 + 0.970797i \(0.422884\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16.9706 1.41915
\(144\) 0 0
\(145\) −2.82843 −0.234888
\(146\) −15.5563 −1.28745
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −5.65685 −0.458831
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.3848 1.46726 0.733632 0.679546i \(-0.237823\pi\)
0.733632 + 0.679546i \(0.237823\pi\)
\(158\) 16.0000 1.27289
\(159\) 0 0
\(160\) −1.41421 −0.111803
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 9.89949 0.773021
\(165\) 0 0
\(166\) 5.65685 0.439057
\(167\) 11.3137 0.875481 0.437741 0.899101i \(-0.355779\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 10.0000 0.766965
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −12.7279 −0.967686 −0.483843 0.875155i \(-0.660759\pi\)
−0.483843 + 0.875155i \(0.660759\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −7.07107 −0.529999
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −12.7279 −0.946059 −0.473029 0.881047i \(-0.656840\pi\)
−0.473029 + 0.881047i \(0.656840\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 8.00000 0.589768
\(185\) 5.65685 0.415900
\(186\) 0 0
\(187\) 28.2843 2.06835
\(188\) 5.65685 0.412568
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 7.07107 0.507673
\(195\) 0 0
\(196\) 0 0
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 0 0
\(199\) −16.9706 −1.20301 −0.601506 0.798869i \(-0.705432\pi\)
−0.601506 + 0.798869i \(0.705432\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) −12.7279 −0.895533
\(203\) 0 0
\(204\) 0 0
\(205\) 14.0000 0.977802
\(206\) 5.65685 0.394132
\(207\) 0 0
\(208\) −4.24264 −0.294174
\(209\) −22.6274 −1.56517
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −4.00000 −0.274721
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −5.65685 −0.385794
\(216\) 0 0
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) −5.65685 −0.381385
\(221\) 30.0000 2.01802
\(222\) 0 0
\(223\) 16.9706 1.13643 0.568216 0.822879i \(-0.307634\pi\)
0.568216 + 0.822879i \(0.307634\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.9706 1.12638 0.563188 0.826329i \(-0.309575\pi\)
0.563188 + 0.826329i \(0.309575\pi\)
\(228\) 0 0
\(229\) −12.7279 −0.841085 −0.420542 0.907273i \(-0.638160\pi\)
−0.420542 + 0.907273i \(0.638160\pi\)
\(230\) 11.3137 0.746004
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) −11.3137 −0.736460
\(237\) 0 0
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −4.24264 −0.273293 −0.136646 0.990620i \(-0.543632\pi\)
−0.136646 + 0.990620i \(0.543632\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) −1.41421 −0.0905357
\(245\) 0 0
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 0 0
\(249\) 0 0
\(250\) 11.3137 0.715542
\(251\) −5.65685 −0.357057 −0.178529 0.983935i \(-0.557134\pi\)
−0.178529 + 0.983935i \(0.557134\pi\)
\(252\) 0 0
\(253\) 32.0000 2.01182
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −12.7279 −0.793946 −0.396973 0.917830i \(-0.629939\pi\)
−0.396973 + 0.917830i \(0.629939\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −6.00000 −0.372104
\(261\) 0 0
\(262\) 16.9706 1.04844
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −5.65685 −0.347498
\(266\) 0 0
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) −1.41421 −0.0862261 −0.0431131 0.999070i \(-0.513728\pi\)
−0.0431131 + 0.999070i \(0.513728\pi\)
\(270\) 0 0
\(271\) 5.65685 0.343629 0.171815 0.985129i \(-0.445037\pi\)
0.171815 + 0.985129i \(0.445037\pi\)
\(272\) −7.07107 −0.428746
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −5.65685 −0.339276
\(279\) 0 0
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −22.6274 −1.34506 −0.672530 0.740070i \(-0.734792\pi\)
−0.672530 + 0.740070i \(0.734792\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −16.9706 −1.00349
\(287\) 0 0
\(288\) 0 0
\(289\) 33.0000 1.94118
\(290\) 2.82843 0.166091
\(291\) 0 0
\(292\) 15.5563 0.910366
\(293\) 24.0416 1.40453 0.702264 0.711917i \(-0.252173\pi\)
0.702264 + 0.711917i \(0.252173\pi\)
\(294\) 0 0
\(295\) −16.0000 −0.931556
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) −20.0000 −1.15857
\(299\) 33.9411 1.96287
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) 5.65685 0.324443
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −5.65685 −0.322854 −0.161427 0.986885i \(-0.551610\pi\)
−0.161427 + 0.986885i \(0.551610\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5.65685 −0.320771 −0.160385 0.987054i \(-0.551274\pi\)
−0.160385 + 0.987054i \(0.551274\pi\)
\(312\) 0 0
\(313\) −21.2132 −1.19904 −0.599521 0.800359i \(-0.704642\pi\)
−0.599521 + 0.800359i \(0.704642\pi\)
\(314\) −18.3848 −1.03751
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −28.0000 −1.57264 −0.786318 0.617822i \(-0.788015\pi\)
−0.786318 + 0.617822i \(0.788015\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 1.41421 0.0790569
\(321\) 0 0
\(322\) 0 0
\(323\) −40.0000 −2.22566
\(324\) 0 0
\(325\) 12.7279 0.706018
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −9.89949 −0.546608
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −5.65685 −0.310460
\(333\) 0 0
\(334\) −11.3137 −0.619059
\(335\) −16.9706 −0.927201
\(336\) 0 0
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) −5.00000 −0.271964
\(339\) 0 0
\(340\) −10.0000 −0.542326
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 12.7279 0.684257
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 29.6985 1.58972 0.794862 0.606791i \(-0.207543\pi\)
0.794862 + 0.606791i \(0.207543\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 1.41421 0.0752710 0.0376355 0.999292i \(-0.488017\pi\)
0.0376355 + 0.999292i \(0.488017\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.07107 0.374766
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 12.7279 0.668965
\(363\) 0 0
\(364\) 0 0
\(365\) 22.0000 1.15153
\(366\) 0 0
\(367\) −5.65685 −0.295285 −0.147643 0.989041i \(-0.547169\pi\)
−0.147643 + 0.989041i \(0.547169\pi\)
\(368\) −8.00000 −0.417029
\(369\) 0 0
\(370\) −5.65685 −0.294086
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −28.2843 −1.46254
\(375\) 0 0
\(376\) −5.65685 −0.291730
\(377\) 8.48528 0.437014
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −5.65685 −0.289052 −0.144526 0.989501i \(-0.546166\pi\)
−0.144526 + 0.989501i \(0.546166\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) −7.07107 −0.358979
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 56.5685 2.86079
\(392\) 0 0
\(393\) 0 0
\(394\) −4.00000 −0.201517
\(395\) −22.6274 −1.13851
\(396\) 0 0
\(397\) −7.07107 −0.354887 −0.177443 0.984131i \(-0.556783\pi\)
−0.177443 + 0.984131i \(0.556783\pi\)
\(398\) 16.9706 0.850657
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 12.7279 0.633238
\(405\) 0 0
\(406\) 0 0
\(407\) −16.0000 −0.793091
\(408\) 0 0
\(409\) 21.2132 1.04893 0.524463 0.851433i \(-0.324266\pi\)
0.524463 + 0.851433i \(0.324266\pi\)
\(410\) −14.0000 −0.691411
\(411\) 0 0
\(412\) −5.65685 −0.278693
\(413\) 0 0
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) 4.24264 0.208013
\(417\) 0 0
\(418\) 22.6274 1.10674
\(419\) 22.6274 1.10542 0.552711 0.833373i \(-0.313593\pi\)
0.552711 + 0.833373i \(0.313593\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) 4.00000 0.194257
\(425\) 21.2132 1.02899
\(426\) 0 0
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 5.65685 0.272798
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 4.24264 0.203888 0.101944 0.994790i \(-0.467494\pi\)
0.101944 + 0.994790i \(0.467494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) −45.2548 −2.16483
\(438\) 0 0
\(439\) −33.9411 −1.61992 −0.809961 0.586484i \(-0.800512\pi\)
−0.809961 + 0.586484i \(0.800512\pi\)
\(440\) 5.65685 0.269680
\(441\) 0 0
\(442\) −30.0000 −1.42695
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) −16.9706 −0.803579
\(447\) 0 0
\(448\) 0 0
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) −39.5980 −1.86460
\(452\) 0 0
\(453\) 0 0
\(454\) −16.9706 −0.796468
\(455\) 0 0
\(456\) 0 0
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) 12.7279 0.594737
\(459\) 0 0
\(460\) −11.3137 −0.527504
\(461\) −1.41421 −0.0658665 −0.0329332 0.999458i \(-0.510485\pi\)
−0.0329332 + 0.999458i \(0.510485\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 5.65685 0.261768 0.130884 0.991398i \(-0.458218\pi\)
0.130884 + 0.991398i \(0.458218\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 11.3137 0.520756
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −16.9706 −0.778663
\(476\) 0 0
\(477\) 0 0
\(478\) −24.0000 −1.09773
\(479\) −28.2843 −1.29234 −0.646171 0.763193i \(-0.723631\pi\)
−0.646171 + 0.763193i \(0.723631\pi\)
\(480\) 0 0
\(481\) −16.9706 −0.773791
\(482\) 4.24264 0.193247
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 1.41421 0.0640184
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 14.1421 0.636930
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −11.3137 −0.505964
\(501\) 0 0
\(502\) 5.65685 0.252478
\(503\) 28.2843 1.26113 0.630567 0.776135i \(-0.282823\pi\)
0.630567 + 0.776135i \(0.282823\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) −32.0000 −1.42257
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 32.5269 1.44173 0.720865 0.693075i \(-0.243745\pi\)
0.720865 + 0.693075i \(0.243745\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 12.7279 0.561405
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) −22.6274 −0.995153
\(518\) 0 0
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) 1.41421 0.0619578 0.0309789 0.999520i \(-0.490138\pi\)
0.0309789 + 0.999520i \(0.490138\pi\)
\(522\) 0 0
\(523\) 33.9411 1.48414 0.742071 0.670321i \(-0.233844\pi\)
0.742071 + 0.670321i \(0.233844\pi\)
\(524\) −16.9706 −0.741362
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 5.65685 0.245718
\(531\) 0 0
\(532\) 0 0
\(533\) −42.0000 −1.81922
\(534\) 0 0
\(535\) 5.65685 0.244567
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 1.41421 0.0609711
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −5.65685 −0.242983
\(543\) 0 0
\(544\) 7.07107 0.303170
\(545\) 5.65685 0.242313
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) −12.0000 −0.511682
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 5.65685 0.239904
\(557\) −36.0000 −1.52537 −0.762684 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 0 0
\(559\) 16.9706 0.717778
\(560\) 0 0
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −11.3137 −0.476816 −0.238408 0.971165i \(-0.576626\pi\)
−0.238408 + 0.971165i \(0.576626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 22.6274 0.951101
\(567\) 0 0
\(568\) 0 0
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 16.9706 0.709575
\(573\) 0 0
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) −12.7279 −0.529870 −0.264935 0.964266i \(-0.585351\pi\)
−0.264935 + 0.964266i \(0.585351\pi\)
\(578\) −33.0000 −1.37262
\(579\) 0 0
\(580\) −2.82843 −0.117444
\(581\) 0 0
\(582\) 0 0
\(583\) 16.0000 0.662652
\(584\) −15.5563 −0.643726
\(585\) 0 0
\(586\) −24.0416 −0.993151
\(587\) −16.9706 −0.700450 −0.350225 0.936666i \(-0.613895\pi\)
−0.350225 + 0.936666i \(0.613895\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 16.0000 0.658710
\(591\) 0 0
\(592\) 4.00000 0.164399
\(593\) −35.3553 −1.45187 −0.725935 0.687763i \(-0.758593\pi\)
−0.725935 + 0.687763i \(0.758593\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 0 0
\(598\) −33.9411 −1.38796
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −4.24264 −0.173061 −0.0865305 0.996249i \(-0.527578\pi\)
−0.0865305 + 0.996249i \(0.527578\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 7.07107 0.287480
\(606\) 0 0
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) −5.65685 −0.229416
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 12.0000 0.484675 0.242338 0.970192i \(-0.422086\pi\)
0.242338 + 0.970192i \(0.422086\pi\)
\(614\) 5.65685 0.228292
\(615\) 0 0
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) −11.3137 −0.454736 −0.227368 0.973809i \(-0.573012\pi\)
−0.227368 + 0.973809i \(0.573012\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 5.65685 0.226819
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 21.2132 0.847850
\(627\) 0 0
\(628\) 18.3848 0.733632
\(629\) −28.2843 −1.12777
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 16.0000 0.636446
\(633\) 0 0
\(634\) 28.0000 1.11202
\(635\) −11.3137 −0.448971
\(636\) 0 0
\(637\) 0 0
\(638\) −8.00000 −0.316723
\(639\) 0 0
\(640\) −1.41421 −0.0559017
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −11.3137 −0.446169 −0.223085 0.974799i \(-0.571613\pi\)
−0.223085 + 0.974799i \(0.571613\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 40.0000 1.57378
\(647\) 33.9411 1.33436 0.667182 0.744895i \(-0.267500\pi\)
0.667182 + 0.744895i \(0.267500\pi\)
\(648\) 0 0
\(649\) 45.2548 1.77641
\(650\) −12.7279 −0.499230
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) 9.89949 0.386510
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −4.24264 −0.165020 −0.0825098 0.996590i \(-0.526294\pi\)
−0.0825098 + 0.996590i \(0.526294\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 5.65685 0.219529
\(665\) 0 0
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 11.3137 0.437741
\(669\) 0 0
\(670\) 16.9706 0.655630
\(671\) 5.65685 0.218380
\(672\) 0 0
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) 16.0000 0.616297
\(675\) 0 0
\(676\) 5.00000 0.192308
\(677\) 4.24264 0.163058 0.0815290 0.996671i \(-0.474020\pi\)
0.0815290 + 0.996671i \(0.474020\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 10.0000 0.383482
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −8.48528 −0.324206
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 16.9706 0.646527
\(690\) 0 0
\(691\) 50.9117 1.93677 0.968386 0.249457i \(-0.0802520\pi\)
0.968386 + 0.249457i \(0.0802520\pi\)
\(692\) −12.7279 −0.483843
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) −70.0000 −2.65144
\(698\) −29.6985 −1.12410
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 22.6274 0.853409
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −1.41421 −0.0532246
\(707\) 0 0
\(708\) 0 0
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −7.07107 −0.264999
\(713\) 0 0
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) 39.5980 1.47676 0.738378 0.674387i \(-0.235592\pi\)
0.738378 + 0.674387i \(0.235592\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −13.0000 −0.483810
\(723\) 0 0
\(724\) −12.7279 −0.473029
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −28.2843 −1.04901 −0.524503 0.851409i \(-0.675749\pi\)
−0.524503 + 0.851409i \(0.675749\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −22.0000 −0.814257
\(731\) 28.2843 1.04613
\(732\) 0 0
\(733\) −12.7279 −0.470117 −0.235058 0.971981i \(-0.575528\pi\)
−0.235058 + 0.971981i \(0.575528\pi\)
\(734\) 5.65685 0.208798
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 5.65685 0.207950
\(741\) 0 0
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 28.2843 1.03626
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 28.2843 1.03418
\(749\) 0 0
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 5.65685 0.206284
\(753\) 0 0
\(754\) −8.48528 −0.309016
\(755\) −22.6274 −0.823496
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) −9.89949 −0.358856 −0.179428 0.983771i \(-0.557425\pi\)
−0.179428 + 0.983771i \(0.557425\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 5.65685 0.204390
\(767\) 48.0000 1.73318
\(768\) 0 0
\(769\) 4.24264 0.152994 0.0764968 0.997070i \(-0.475627\pi\)
0.0764968 + 0.997070i \(0.475627\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 32.5269 1.16991 0.584956 0.811065i \(-0.301112\pi\)
0.584956 + 0.811065i \(0.301112\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 7.07107 0.253837
\(777\) 0 0
\(778\) 26.0000 0.932145
\(779\) 56.0000 2.00641
\(780\) 0 0
\(781\) 0 0
\(782\) −56.5685 −2.02289
\(783\) 0 0
\(784\) 0 0
\(785\) 26.0000 0.927980
\(786\) 0 0
\(787\) −5.65685 −0.201645 −0.100823 0.994904i \(-0.532147\pi\)
−0.100823 + 0.994904i \(0.532147\pi\)
\(788\) 4.00000 0.142494
\(789\) 0 0
\(790\) 22.6274 0.805047
\(791\) 0 0
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 7.07107 0.250943
\(795\) 0 0
\(796\) −16.9706 −0.601506
\(797\) 12.7279 0.450846 0.225423 0.974261i \(-0.427624\pi\)
0.225423 + 0.974261i \(0.427624\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 3.00000 0.106066
\(801\) 0 0
\(802\) 18.0000 0.635602
\(803\) −62.2254 −2.19589
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −12.7279 −0.447767
\(809\) 40.0000 1.40633 0.703163 0.711029i \(-0.251771\pi\)
0.703163 + 0.711029i \(0.251771\pi\)
\(810\) 0 0
\(811\) −33.9411 −1.19183 −0.595917 0.803046i \(-0.703211\pi\)
−0.595917 + 0.803046i \(0.703211\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 16.0000 0.560800
\(815\) 5.65685 0.198151
\(816\) 0 0
\(817\) −22.6274 −0.791633
\(818\) −21.2132 −0.741702
\(819\) 0 0
\(820\) 14.0000 0.488901
\(821\) 36.0000 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 5.65685 0.197066
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −43.8406 −1.52265 −0.761324 0.648372i \(-0.775450\pi\)
−0.761324 + 0.648372i \(0.775450\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) −4.24264 −0.147087
\(833\) 0 0
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) −22.6274 −0.782586
\(837\) 0 0
\(838\) −22.6274 −0.781651
\(839\) −45.2548 −1.56237 −0.781185 0.624299i \(-0.785385\pi\)
−0.781185 + 0.624299i \(0.785385\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 6.00000 0.206774
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) 7.07107 0.243252
\(846\) 0 0
\(847\) 0 0
\(848\) −4.00000 −0.137361
\(849\) 0 0
\(850\) −21.2132 −0.727607
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) −21.2132 −0.726326 −0.363163 0.931726i \(-0.618303\pi\)
−0.363163 + 0.931726i \(0.618303\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) 24.0416 0.821246 0.410623 0.911805i \(-0.365311\pi\)
0.410623 + 0.911805i \(0.365311\pi\)
\(858\) 0 0
\(859\) −5.65685 −0.193009 −0.0965047 0.995333i \(-0.530766\pi\)
−0.0965047 + 0.995333i \(0.530766\pi\)
\(860\) −5.65685 −0.192897
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −8.00000 −0.272323 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) −4.24264 −0.144171
\(867\) 0 0
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) 50.9117 1.72508
\(872\) −4.00000 −0.135457
\(873\) 0 0
\(874\) 45.2548 1.53077
\(875\) 0 0
\(876\) 0 0
\(877\) −28.0000 −0.945493 −0.472746 0.881199i \(-0.656737\pi\)
−0.472746 + 0.881199i \(0.656737\pi\)
\(878\) 33.9411 1.14546
\(879\) 0 0
\(880\) −5.65685 −0.190693
\(881\) 21.2132 0.714691 0.357345 0.933972i \(-0.383682\pi\)
0.357345 + 0.933972i \(0.383682\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 30.0000 1.00901
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) −22.6274 −0.759754 −0.379877 0.925037i \(-0.624034\pi\)
−0.379877 + 0.925037i \(0.624034\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −10.0000 −0.335201
\(891\) 0 0
\(892\) 16.9706 0.568216
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 16.9706 0.567263
\(896\) 0 0
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) 0 0
\(900\) 0 0
\(901\) 28.2843 0.942286
\(902\) 39.5980 1.31847
\(903\) 0 0
\(904\) 0 0
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 16.9706 0.563188
\(909\) 0 0
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) 22.6274 0.748858
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −12.7279 −0.420542
\(917\) 0 0
\(918\) 0 0
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 11.3137 0.373002
\(921\) 0 0
\(922\) 1.41421 0.0465746
\(923\) 0 0
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 32.0000 1.05159
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 18.3848 0.603185 0.301592 0.953437i \(-0.402482\pi\)
0.301592 + 0.953437i \(0.402482\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −5.65685 −0.185098
\(935\) 40.0000 1.30814
\(936\) 0 0
\(937\) −15.5563 −0.508204 −0.254102 0.967177i \(-0.581780\pi\)
−0.254102 + 0.967177i \(0.581780\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 26.8701 0.875939 0.437969 0.898990i \(-0.355698\pi\)
0.437969 + 0.898990i \(0.355698\pi\)
\(942\) 0 0
\(943\) −79.1960 −2.57898
\(944\) −11.3137 −0.368230
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −66.0000 −2.14245
\(950\) 16.9706 0.550598
\(951\) 0 0
\(952\) 0 0
\(953\) 8.00000 0.259145 0.129573 0.991570i \(-0.458639\pi\)
0.129573 + 0.991570i \(0.458639\pi\)
\(954\) 0 0
\(955\) 22.6274 0.732206
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 28.2843 0.913823
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 16.9706 0.547153
\(963\) 0 0
\(964\) −4.24264 −0.136646
\(965\) 19.7990 0.637352
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) −11.3137 −0.363074 −0.181537 0.983384i \(-0.558107\pi\)
−0.181537 + 0.983384i \(0.558107\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) −1.41421 −0.0452679
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) −28.2843 −0.903969
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 11.3137 0.360851 0.180426 0.983589i \(-0.442252\pi\)
0.180426 + 0.983589i \(0.442252\pi\)
\(984\) 0 0
\(985\) 5.65685 0.180242
\(986\) −14.1421 −0.450377
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) 0 0
\(997\) 7.07107 0.223943 0.111971 0.993711i \(-0.464283\pi\)
0.111971 + 0.993711i \(0.464283\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 882.2.a.m.1.2 yes 2
3.2 odd 2 882.2.a.o.1.1 yes 2
4.3 odd 2 7056.2.a.cs.1.2 2
7.2 even 3 882.2.g.m.361.1 4
7.3 odd 6 882.2.g.m.667.2 4
7.4 even 3 882.2.g.m.667.1 4
7.5 odd 6 882.2.g.m.361.2 4
7.6 odd 2 inner 882.2.a.m.1.1 2
12.11 even 2 7056.2.a.ci.1.1 2
21.2 odd 6 882.2.g.k.361.2 4
21.5 even 6 882.2.g.k.361.1 4
21.11 odd 6 882.2.g.k.667.2 4
21.17 even 6 882.2.g.k.667.1 4
21.20 even 2 882.2.a.o.1.2 yes 2
28.27 even 2 7056.2.a.cs.1.1 2
84.83 odd 2 7056.2.a.ci.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
882.2.a.m.1.1 2 7.6 odd 2 inner
882.2.a.m.1.2 yes 2 1.1 even 1 trivial
882.2.a.o.1.1 yes 2 3.2 odd 2
882.2.a.o.1.2 yes 2 21.20 even 2
882.2.g.k.361.1 4 21.5 even 6
882.2.g.k.361.2 4 21.2 odd 6
882.2.g.k.667.1 4 21.17 even 6
882.2.g.k.667.2 4 21.11 odd 6
882.2.g.m.361.1 4 7.2 even 3
882.2.g.m.361.2 4 7.5 odd 6
882.2.g.m.667.1 4 7.4 even 3
882.2.g.m.667.2 4 7.3 odd 6
7056.2.a.ci.1.1 2 12.11 even 2
7056.2.a.ci.1.2 2 84.83 odd 2
7056.2.a.cs.1.1 2 28.27 even 2
7056.2.a.cs.1.2 2 4.3 odd 2