Properties

Label 2352.2.a.be.1.2
Level $2352$
Weight $2$
Character 2352.1
Self dual yes
Analytic conductor $18.781$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.7808145554\)
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: no (minimal twist has level 147)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} +2.00000 q^{11} -5.41421 q^{13} -0.585786 q^{15} -6.24264 q^{17} +2.82843 q^{19} -3.65685 q^{23} -4.65685 q^{25} +1.00000 q^{27} -1.17157 q^{29} -6.82843 q^{31} +2.00000 q^{33} -4.00000 q^{37} -5.41421 q^{39} +2.24264 q^{41} +5.65685 q^{43} -0.585786 q^{45} +2.82843 q^{47} -6.24264 q^{51} -2.00000 q^{53} -1.17157 q^{55} +2.82843 q^{57} -6.82843 q^{59} -3.75736 q^{61} +3.17157 q^{65} -5.65685 q^{67} -3.65685 q^{69} +13.3137 q^{71} +5.89949 q^{73} -4.65685 q^{75} -2.34315 q^{79} +1.00000 q^{81} -15.3137 q^{83} +3.65685 q^{85} -1.17157 q^{87} +5.75736 q^{89} -6.82843 q^{93} -1.65685 q^{95} -5.41421 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{5} + 2q^{9} + 4q^{11} - 8q^{13} - 4q^{15} - 4q^{17} + 4q^{23} + 2q^{25} + 2q^{27} - 8q^{29} - 8q^{31} + 4q^{33} - 8q^{37} - 8q^{39} - 4q^{41} - 4q^{45} - 4q^{51} - 4q^{53} - 8q^{55} - 8q^{59} - 16q^{61} + 12q^{65} + 4q^{69} + 4q^{71} - 8q^{73} + 2q^{75} - 16q^{79} + 2q^{81} - 8q^{83} - 4q^{85} - 8q^{87} + 20q^{89} - 8q^{93} + 8q^{95} - 8q^{97} + 4q^{99} + O(q^{100}) \)

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
\(5\) −0.585786 −0.261972 −0.130986 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −5.41421 −1.50163 −0.750816 0.660511i \(-0.770340\pi\)
−0.750816 + 0.660511i \(0.770340\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) −6.24264 −1.51406 −0.757031 0.653379i \(-0.773351\pi\)
−0.757031 + 0.653379i \(0.773351\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.65685 −0.762507 −0.381253 0.924471i \(-0.624507\pi\)
−0.381253 + 0.924471i \(0.624507\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.17157 −0.217556 −0.108778 0.994066i \(-0.534694\pi\)
−0.108778 + 0.994066i \(0.534694\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) −5.41421 −0.866968
\(40\) 0 0
\(41\) 2.24264 0.350242 0.175121 0.984547i \(-0.443968\pi\)
0.175121 + 0.984547i \(0.443968\pi\)
\(42\) 0 0
\(43\) 5.65685 0.862662 0.431331 0.902194i \(-0.358044\pi\)
0.431331 + 0.902194i \(0.358044\pi\)
\(44\) 0 0
\(45\) −0.585786 −0.0873239
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6.24264 −0.874145
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −1.17157 −0.157975
\(56\) 0 0
\(57\) 2.82843 0.374634
\(58\) 0 0
\(59\) −6.82843 −0.888985 −0.444493 0.895782i \(-0.646616\pi\)
−0.444493 + 0.895782i \(0.646616\pi\)
\(60\) 0 0
\(61\) −3.75736 −0.481081 −0.240540 0.970639i \(-0.577325\pi\)
−0.240540 + 0.970639i \(0.577325\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.17157 0.393385
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) −3.65685 −0.440234
\(70\) 0 0
\(71\) 13.3137 1.58005 0.790023 0.613077i \(-0.210068\pi\)
0.790023 + 0.613077i \(0.210068\pi\)
\(72\) 0 0
\(73\) 5.89949 0.690484 0.345242 0.938514i \(-0.387797\pi\)
0.345242 + 0.938514i \(0.387797\pi\)
\(74\) 0 0
\(75\) −4.65685 −0.537727
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.34315 −0.263624 −0.131812 0.991275i \(-0.542080\pi\)
−0.131812 + 0.991275i \(0.542080\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.3137 −1.68090 −0.840449 0.541891i \(-0.817709\pi\)
−0.840449 + 0.541891i \(0.817709\pi\)
\(84\) 0 0
\(85\) 3.65685 0.396642
\(86\) 0 0
\(87\) −1.17157 −0.125606
\(88\) 0 0
\(89\) 5.75736 0.610279 0.305139 0.952308i \(-0.401297\pi\)
0.305139 + 0.952308i \(0.401297\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −6.82843 −0.708075
\(94\) 0 0
\(95\) −1.65685 −0.169990
\(96\) 0 0
\(97\) −5.41421 −0.549730 −0.274865 0.961483i \(-0.588633\pi\)
−0.274865 + 0.961483i \(0.588633\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −17.0711 −1.69863 −0.849317 0.527883i \(-0.822986\pi\)
−0.849317 + 0.527883i \(0.822986\pi\)
\(102\) 0 0
\(103\) −12.4853 −1.23021 −0.615106 0.788445i \(-0.710887\pi\)
−0.615106 + 0.788445i \(0.710887\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.6569 1.12691 0.563455 0.826147i \(-0.309472\pi\)
0.563455 + 0.826147i \(0.309472\pi\)
\(108\) 0 0
\(109\) 5.65685 0.541828 0.270914 0.962604i \(-0.412674\pi\)
0.270914 + 0.962604i \(0.412674\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 17.3137 1.62874 0.814368 0.580348i \(-0.197084\pi\)
0.814368 + 0.580348i \(0.197084\pi\)
\(114\) 0 0
\(115\) 2.14214 0.199755
\(116\) 0 0
\(117\) −5.41421 −0.500544
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 2.24264 0.202212
\(124\) 0 0
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) −9.65685 −0.856907 −0.428454 0.903564i \(-0.640941\pi\)
−0.428454 + 0.903564i \(0.640941\pi\)
\(128\) 0 0
\(129\) 5.65685 0.498058
\(130\) 0 0
\(131\) 7.31371 0.639002 0.319501 0.947586i \(-0.396485\pi\)
0.319501 + 0.947586i \(0.396485\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.585786 −0.0504165
\(136\) 0 0
\(137\) −14.1421 −1.20824 −0.604122 0.796892i \(-0.706476\pi\)
−0.604122 + 0.796892i \(0.706476\pi\)
\(138\) 0 0
\(139\) −6.34315 −0.538019 −0.269009 0.963138i \(-0.586696\pi\)
−0.269009 + 0.963138i \(0.586696\pi\)
\(140\) 0 0
\(141\) 2.82843 0.238197
\(142\) 0 0
\(143\) −10.8284 −0.905519
\(144\) 0 0
\(145\) 0.686292 0.0569934
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.31371 −0.435316 −0.217658 0.976025i \(-0.569842\pi\)
−0.217658 + 0.976025i \(0.569842\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) −6.24264 −0.504688
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −20.2426 −1.61554 −0.807769 0.589499i \(-0.799325\pi\)
−0.807769 + 0.589499i \(0.799325\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −11.3137 −0.886158 −0.443079 0.896483i \(-0.646114\pi\)
−0.443079 + 0.896483i \(0.646114\pi\)
\(164\) 0 0
\(165\) −1.17157 −0.0912068
\(166\) 0 0
\(167\) 19.7990 1.53209 0.766046 0.642786i \(-0.222221\pi\)
0.766046 + 0.642786i \(0.222221\pi\)
\(168\) 0 0
\(169\) 16.3137 1.25490
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) −6.92893 −0.526797 −0.263398 0.964687i \(-0.584843\pi\)
−0.263398 + 0.964687i \(0.584843\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.82843 −0.513256
\(178\) 0 0
\(179\) 8.34315 0.623596 0.311798 0.950148i \(-0.399069\pi\)
0.311798 + 0.950148i \(0.399069\pi\)
\(180\) 0 0
\(181\) 5.41421 0.402435 0.201218 0.979547i \(-0.435510\pi\)
0.201218 + 0.979547i \(0.435510\pi\)
\(182\) 0 0
\(183\) −3.75736 −0.277752
\(184\) 0 0
\(185\) 2.34315 0.172272
\(186\) 0 0
\(187\) −12.4853 −0.913014
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) −17.3137 −1.24627 −0.623134 0.782115i \(-0.714141\pi\)
−0.623134 + 0.782115i \(0.714141\pi\)
\(194\) 0 0
\(195\) 3.17157 0.227121
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 0 0
\(201\) −5.65685 −0.399004
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.31371 −0.0917534
\(206\) 0 0
\(207\) −3.65685 −0.254169
\(208\) 0 0
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) 20.9706 1.44367 0.721837 0.692064i \(-0.243298\pi\)
0.721837 + 0.692064i \(0.243298\pi\)
\(212\) 0 0
\(213\) 13.3137 0.912240
\(214\) 0 0
\(215\) −3.31371 −0.225993
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 5.89949 0.398651
\(220\) 0 0
\(221\) 33.7990 2.27357
\(222\) 0 0
\(223\) −8.97056 −0.600713 −0.300357 0.953827i \(-0.597106\pi\)
−0.300357 + 0.953827i \(0.597106\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) 0 0
\(227\) −15.7990 −1.04862 −0.524308 0.851529i \(-0.675676\pi\)
−0.524308 + 0.851529i \(0.675676\pi\)
\(228\) 0 0
\(229\) −8.24264 −0.544689 −0.272345 0.962200i \(-0.587799\pi\)
−0.272345 + 0.962200i \(0.587799\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.1421 1.45058 0.725290 0.688444i \(-0.241706\pi\)
0.725290 + 0.688444i \(0.241706\pi\)
\(234\) 0 0
\(235\) −1.65685 −0.108081
\(236\) 0 0
\(237\) −2.34315 −0.152204
\(238\) 0 0
\(239\) 4.34315 0.280935 0.140467 0.990085i \(-0.455139\pi\)
0.140467 + 0.990085i \(0.455139\pi\)
\(240\) 0 0
\(241\) 7.75736 0.499695 0.249848 0.968285i \(-0.419619\pi\)
0.249848 + 0.968285i \(0.419619\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −15.3137 −0.974388
\(248\) 0 0
\(249\) −15.3137 −0.970467
\(250\) 0 0
\(251\) 4.48528 0.283108 0.141554 0.989931i \(-0.454790\pi\)
0.141554 + 0.989931i \(0.454790\pi\)
\(252\) 0 0
\(253\) −7.31371 −0.459809
\(254\) 0 0
\(255\) 3.65685 0.229001
\(256\) 0 0
\(257\) 19.2132 1.19849 0.599243 0.800567i \(-0.295468\pi\)
0.599243 + 0.800567i \(0.295468\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.17157 −0.0725185
\(262\) 0 0
\(263\) 17.3137 1.06761 0.533805 0.845608i \(-0.320762\pi\)
0.533805 + 0.845608i \(0.320762\pi\)
\(264\) 0 0
\(265\) 1.17157 0.0719691
\(266\) 0 0
\(267\) 5.75736 0.352345
\(268\) 0 0
\(269\) −10.7279 −0.654093 −0.327046 0.945008i \(-0.606053\pi\)
−0.327046 + 0.945008i \(0.606053\pi\)
\(270\) 0 0
\(271\) 18.1421 1.10206 0.551028 0.834487i \(-0.314236\pi\)
0.551028 + 0.834487i \(0.314236\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9.31371 −0.561638
\(276\) 0 0
\(277\) 13.3137 0.799943 0.399972 0.916528i \(-0.369020\pi\)
0.399972 + 0.916528i \(0.369020\pi\)
\(278\) 0 0
\(279\) −6.82843 −0.408807
\(280\) 0 0
\(281\) −16.4853 −0.983429 −0.491715 0.870756i \(-0.663630\pi\)
−0.491715 + 0.870756i \(0.663630\pi\)
\(282\) 0 0
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) 0 0
\(285\) −1.65685 −0.0981436
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 21.9706 1.29239
\(290\) 0 0
\(291\) −5.41421 −0.317387
\(292\) 0 0
\(293\) −19.4142 −1.13419 −0.567095 0.823652i \(-0.691933\pi\)
−0.567095 + 0.823652i \(0.691933\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 19.7990 1.14501
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −17.0711 −0.980707
\(304\) 0 0
\(305\) 2.20101 0.126029
\(306\) 0 0
\(307\) 1.85786 0.106034 0.0530170 0.998594i \(-0.483116\pi\)
0.0530170 + 0.998594i \(0.483116\pi\)
\(308\) 0 0
\(309\) −12.4853 −0.710263
\(310\) 0 0
\(311\) −22.1421 −1.25557 −0.627783 0.778389i \(-0.716037\pi\)
−0.627783 + 0.778389i \(0.716037\pi\)
\(312\) 0 0
\(313\) 17.8995 1.01174 0.505870 0.862610i \(-0.331172\pi\)
0.505870 + 0.862610i \(0.331172\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) −2.34315 −0.131191
\(320\) 0 0
\(321\) 11.6569 0.650622
\(322\) 0 0
\(323\) −17.6569 −0.982454
\(324\) 0 0
\(325\) 25.2132 1.39858
\(326\) 0 0
\(327\) 5.65685 0.312825
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 3.31371 0.181047
\(336\) 0 0
\(337\) −18.3431 −0.999215 −0.499607 0.866252i \(-0.666522\pi\)
−0.499607 + 0.866252i \(0.666522\pi\)
\(338\) 0 0
\(339\) 17.3137 0.940352
\(340\) 0 0
\(341\) −13.6569 −0.739560
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.14214 0.115329
\(346\) 0 0
\(347\) −10.6863 −0.573670 −0.286835 0.957980i \(-0.592603\pi\)
−0.286835 + 0.957980i \(0.592603\pi\)
\(348\) 0 0
\(349\) −9.89949 −0.529908 −0.264954 0.964261i \(-0.585357\pi\)
−0.264954 + 0.964261i \(0.585357\pi\)
\(350\) 0 0
\(351\) −5.41421 −0.288989
\(352\) 0 0
\(353\) −10.7279 −0.570990 −0.285495 0.958380i \(-0.592158\pi\)
−0.285495 + 0.958380i \(0.592158\pi\)
\(354\) 0 0
\(355\) −7.79899 −0.413927
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.6569 0.615225 0.307613 0.951512i \(-0.400470\pi\)
0.307613 + 0.951512i \(0.400470\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −3.45584 −0.180887
\(366\) 0 0
\(367\) 19.3137 1.00817 0.504084 0.863655i \(-0.331830\pi\)
0.504084 + 0.863655i \(0.331830\pi\)
\(368\) 0 0
\(369\) 2.24264 0.116747
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −33.3137 −1.72492 −0.862459 0.506127i \(-0.831077\pi\)
−0.862459 + 0.506127i \(0.831077\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) 6.34315 0.326689
\(378\) 0 0
\(379\) −31.3137 −1.60848 −0.804239 0.594307i \(-0.797427\pi\)
−0.804239 + 0.594307i \(0.797427\pi\)
\(380\) 0 0
\(381\) −9.65685 −0.494736
\(382\) 0 0
\(383\) −29.6569 −1.51539 −0.757697 0.652606i \(-0.773676\pi\)
−0.757697 + 0.652606i \(0.773676\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.65685 0.287554
\(388\) 0 0
\(389\) 10.1421 0.514227 0.257113 0.966381i \(-0.417229\pi\)
0.257113 + 0.966381i \(0.417229\pi\)
\(390\) 0 0
\(391\) 22.8284 1.15448
\(392\) 0 0
\(393\) 7.31371 0.368928
\(394\) 0 0
\(395\) 1.37258 0.0690621
\(396\) 0 0
\(397\) −34.3848 −1.72572 −0.862861 0.505441i \(-0.831330\pi\)
−0.862861 + 0.505441i \(0.831330\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.1421 1.10573 0.552863 0.833272i \(-0.313535\pi\)
0.552863 + 0.833272i \(0.313535\pi\)
\(402\) 0 0
\(403\) 36.9706 1.84163
\(404\) 0 0
\(405\) −0.585786 −0.0291080
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 18.5858 0.919008 0.459504 0.888176i \(-0.348027\pi\)
0.459504 + 0.888176i \(0.348027\pi\)
\(410\) 0 0
\(411\) −14.1421 −0.697580
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 8.97056 0.440348
\(416\) 0 0
\(417\) −6.34315 −0.310625
\(418\) 0 0
\(419\) 38.8284 1.89689 0.948446 0.316938i \(-0.102655\pi\)
0.948446 + 0.316938i \(0.102655\pi\)
\(420\) 0 0
\(421\) −28.6274 −1.39521 −0.697607 0.716480i \(-0.745752\pi\)
−0.697607 + 0.716480i \(0.745752\pi\)
\(422\) 0 0
\(423\) 2.82843 0.137523
\(424\) 0 0
\(425\) 29.0711 1.41015
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −10.8284 −0.522801
\(430\) 0 0
\(431\) −6.97056 −0.335760 −0.167880 0.985807i \(-0.553692\pi\)
−0.167880 + 0.985807i \(0.553692\pi\)
\(432\) 0 0
\(433\) 11.7574 0.565023 0.282511 0.959264i \(-0.408833\pi\)
0.282511 + 0.959264i \(0.408833\pi\)
\(434\) 0 0
\(435\) 0.686292 0.0329052
\(436\) 0 0
\(437\) −10.3431 −0.494780
\(438\) 0 0
\(439\) 35.3137 1.68543 0.842716 0.538359i \(-0.180956\pi\)
0.842716 + 0.538359i \(0.180956\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.02944 −0.0489100 −0.0244550 0.999701i \(-0.507785\pi\)
−0.0244550 + 0.999701i \(0.507785\pi\)
\(444\) 0 0
\(445\) −3.37258 −0.159876
\(446\) 0 0
\(447\) −5.31371 −0.251330
\(448\) 0 0
\(449\) 17.3137 0.817084 0.408542 0.912739i \(-0.366037\pi\)
0.408542 + 0.912739i \(0.366037\pi\)
\(450\) 0 0
\(451\) 4.48528 0.211204
\(452\) 0 0
\(453\) −12.0000 −0.563809
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) −6.24264 −0.291382
\(460\) 0 0
\(461\) 19.4142 0.904210 0.452105 0.891965i \(-0.350673\pi\)
0.452105 + 0.891965i \(0.350673\pi\)
\(462\) 0 0
\(463\) −18.6274 −0.865689 −0.432845 0.901468i \(-0.642490\pi\)
−0.432845 + 0.901468i \(0.642490\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) 39.7990 1.84168 0.920839 0.389943i \(-0.127505\pi\)
0.920839 + 0.389943i \(0.127505\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −20.2426 −0.932732
\(472\) 0 0
\(473\) 11.3137 0.520205
\(474\) 0 0
\(475\) −13.1716 −0.604353
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) 30.1421 1.37723 0.688615 0.725127i \(-0.258219\pi\)
0.688615 + 0.725127i \(0.258219\pi\)
\(480\) 0 0
\(481\) 21.6569 0.987468
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.17157 0.144014
\(486\) 0 0
\(487\) 18.6274 0.844089 0.422044 0.906575i \(-0.361313\pi\)
0.422044 + 0.906575i \(0.361313\pi\)
\(488\) 0 0
\(489\) −11.3137 −0.511624
\(490\) 0 0
\(491\) −38.9706 −1.75872 −0.879358 0.476160i \(-0.842028\pi\)
−0.879358 + 0.476160i \(0.842028\pi\)
\(492\) 0 0
\(493\) 7.31371 0.329393
\(494\) 0 0
\(495\) −1.17157 −0.0526583
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 19.3137 0.864600 0.432300 0.901730i \(-0.357702\pi\)
0.432300 + 0.901730i \(0.357702\pi\)
\(500\) 0 0
\(501\) 19.7990 0.884554
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 16.3137 0.724517
\(508\) 0 0
\(509\) 25.5563 1.13277 0.566383 0.824142i \(-0.308342\pi\)
0.566383 + 0.824142i \(0.308342\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2.82843 0.124878
\(514\) 0 0
\(515\) 7.31371 0.322281
\(516\) 0 0
\(517\) 5.65685 0.248788
\(518\) 0 0
\(519\) −6.92893 −0.304146
\(520\) 0 0
\(521\) 32.5858 1.42761 0.713805 0.700345i \(-0.246970\pi\)
0.713805 + 0.700345i \(0.246970\pi\)
\(522\) 0 0
\(523\) −14.3431 −0.627182 −0.313591 0.949558i \(-0.601532\pi\)
−0.313591 + 0.949558i \(0.601532\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42.6274 1.85688
\(528\) 0 0
\(529\) −9.62742 −0.418583
\(530\) 0 0
\(531\) −6.82843 −0.296328
\(532\) 0 0
\(533\) −12.1421 −0.525934
\(534\) 0 0
\(535\) −6.82843 −0.295219
\(536\) 0 0
\(537\) 8.34315 0.360033
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.31371 −0.228454 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(542\) 0 0
\(543\) 5.41421 0.232346
\(544\) 0 0
\(545\) −3.31371 −0.141944
\(546\) 0 0
\(547\) 3.02944 0.129529 0.0647647 0.997901i \(-0.479370\pi\)
0.0647647 + 0.997901i \(0.479370\pi\)
\(548\) 0 0
\(549\) −3.75736 −0.160360
\(550\) 0 0
\(551\) −3.31371 −0.141169
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 2.34315 0.0994610
\(556\) 0 0
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 0 0
\(559\) −30.6274 −1.29540
\(560\) 0 0
\(561\) −12.4853 −0.527129
\(562\) 0 0
\(563\) −6.82843 −0.287784 −0.143892 0.989593i \(-0.545962\pi\)
−0.143892 + 0.989593i \(0.545962\pi\)
\(564\) 0 0
\(565\) −10.1421 −0.426683
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.485281 0.0203441 0.0101720 0.999948i \(-0.496762\pi\)
0.0101720 + 0.999948i \(0.496762\pi\)
\(570\) 0 0
\(571\) −33.6569 −1.40850 −0.704248 0.709954i \(-0.748716\pi\)
−0.704248 + 0.709954i \(0.748716\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) 17.0294 0.710177
\(576\) 0 0
\(577\) 14.1005 0.587012 0.293506 0.955957i \(-0.405178\pi\)
0.293506 + 0.955957i \(0.405178\pi\)
\(578\) 0 0
\(579\) −17.3137 −0.719533
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 3.17157 0.131128
\(586\) 0 0
\(587\) −17.1716 −0.708747 −0.354373 0.935104i \(-0.615306\pi\)
−0.354373 + 0.935104i \(0.615306\pi\)
\(588\) 0 0
\(589\) −19.3137 −0.795807
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 0 0
\(593\) 21.0711 0.865285 0.432643 0.901566i \(-0.357581\pi\)
0.432643 + 0.901566i \(0.357581\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.3431 0.423317
\(598\) 0 0
\(599\) 2.00000 0.0817178 0.0408589 0.999165i \(-0.486991\pi\)
0.0408589 + 0.999165i \(0.486991\pi\)
\(600\) 0 0
\(601\) 0.928932 0.0378919 0.0189460 0.999821i \(-0.493969\pi\)
0.0189460 + 0.999821i \(0.493969\pi\)
\(602\) 0 0
\(603\) −5.65685 −0.230365
\(604\) 0 0
\(605\) 4.10051 0.166709
\(606\) 0 0
\(607\) −29.6569 −1.20373 −0.601867 0.798596i \(-0.705576\pi\)
−0.601867 + 0.798596i \(0.705576\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −15.3137 −0.619526
\(612\) 0 0
\(613\) 27.3137 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(614\) 0 0
\(615\) −1.31371 −0.0529738
\(616\) 0 0
\(617\) −7.51472 −0.302531 −0.151266 0.988493i \(-0.548335\pi\)
−0.151266 + 0.988493i \(0.548335\pi\)
\(618\) 0 0
\(619\) −4.97056 −0.199784 −0.0998919 0.994998i \(-0.531850\pi\)
−0.0998919 + 0.994998i \(0.531850\pi\)
\(620\) 0 0
\(621\) −3.65685 −0.146745
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) 5.65685 0.225913
\(628\) 0 0
\(629\) 24.9706 0.995642
\(630\) 0 0
\(631\) −0.686292 −0.0273208 −0.0136604 0.999907i \(-0.504348\pi\)
−0.0136604 + 0.999907i \(0.504348\pi\)
\(632\) 0 0
\(633\) 20.9706 0.833505
\(634\) 0 0
\(635\) 5.65685 0.224485
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 13.3137 0.526682
\(640\) 0 0
\(641\) −5.17157 −0.204265 −0.102132 0.994771i \(-0.532567\pi\)
−0.102132 + 0.994771i \(0.532567\pi\)
\(642\) 0 0
\(643\) 50.4264 1.98862 0.994312 0.106510i \(-0.0339675\pi\)
0.994312 + 0.106510i \(0.0339675\pi\)
\(644\) 0 0
\(645\) −3.31371 −0.130477
\(646\) 0 0
\(647\) 21.1716 0.832340 0.416170 0.909287i \(-0.363372\pi\)
0.416170 + 0.909287i \(0.363372\pi\)
\(648\) 0 0
\(649\) −13.6569 −0.536078
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.5147 0.763670 0.381835 0.924231i \(-0.375292\pi\)
0.381835 + 0.924231i \(0.375292\pi\)
\(654\) 0 0
\(655\) −4.28427 −0.167400
\(656\) 0 0
\(657\) 5.89949 0.230161
\(658\) 0 0
\(659\) 13.3137 0.518628 0.259314 0.965793i \(-0.416503\pi\)
0.259314 + 0.965793i \(0.416503\pi\)
\(660\) 0 0
\(661\) −7.55635 −0.293908 −0.146954 0.989143i \(-0.546947\pi\)
−0.146954 + 0.989143i \(0.546947\pi\)
\(662\) 0 0
\(663\) 33.7990 1.31264
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.28427 0.165888
\(668\) 0 0
\(669\) −8.97056 −0.346822
\(670\) 0 0
\(671\) −7.51472 −0.290102
\(672\) 0 0
\(673\) 0.686292 0.0264546 0.0132273 0.999913i \(-0.495789\pi\)
0.0132273 + 0.999913i \(0.495789\pi\)
\(674\) 0 0
\(675\) −4.65685 −0.179242
\(676\) 0 0
\(677\) −28.5858 −1.09864 −0.549321 0.835612i \(-0.685113\pi\)
−0.549321 + 0.835612i \(0.685113\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −15.7990 −0.605419
\(682\) 0 0
\(683\) 8.34315 0.319242 0.159621 0.987178i \(-0.448973\pi\)
0.159621 + 0.987178i \(0.448973\pi\)
\(684\) 0 0
\(685\) 8.28427 0.316526
\(686\) 0 0
\(687\) −8.24264 −0.314476
\(688\) 0 0
\(689\) 10.8284 0.412530
\(690\) 0 0
\(691\) −23.3137 −0.886895 −0.443448 0.896300i \(-0.646245\pi\)
−0.443448 + 0.896300i \(0.646245\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.71573 0.140946
\(696\) 0 0
\(697\) −14.0000 −0.530288
\(698\) 0 0
\(699\) 22.1421 0.837492
\(700\) 0 0
\(701\) −22.8284 −0.862218 −0.431109 0.902300i \(-0.641878\pi\)
−0.431109 + 0.902300i \(0.641878\pi\)
\(702\) 0 0
\(703\) −11.3137 −0.426705
\(704\) 0 0
\(705\) −1.65685 −0.0624007
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 20.2843 0.761792 0.380896 0.924618i \(-0.375616\pi\)
0.380896 + 0.924618i \(0.375616\pi\)
\(710\) 0 0
\(711\) −2.34315 −0.0878748
\(712\) 0 0
\(713\) 24.9706 0.935155
\(714\) 0 0
\(715\) 6.34315 0.237220
\(716\) 0 0
\(717\) 4.34315 0.162198
\(718\) 0 0
\(719\) −25.9411 −0.967441 −0.483720 0.875223i \(-0.660715\pi\)
−0.483720 + 0.875223i \(0.660715\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 7.75736 0.288499
\(724\) 0 0
\(725\) 5.45584 0.202625
\(726\) 0 0
\(727\) −4.48528 −0.166350 −0.0831749 0.996535i \(-0.526506\pi\)
−0.0831749 + 0.996535i \(0.526506\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −35.3137 −1.30612
\(732\) 0 0
\(733\) 9.69848 0.358222 0.179111 0.983829i \(-0.442678\pi\)
0.179111 + 0.983829i \(0.442678\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.3137 −0.416746
\(738\) 0 0
\(739\) −27.3137 −1.00475 −0.502376 0.864650i \(-0.667540\pi\)
−0.502376 + 0.864650i \(0.667540\pi\)
\(740\) 0 0
\(741\) −15.3137 −0.562563
\(742\) 0 0
\(743\) 17.0294 0.624749 0.312375 0.949959i \(-0.398876\pi\)
0.312375 + 0.949959i \(0.398876\pi\)
\(744\) 0 0
\(745\) 3.11270 0.114040
\(746\) 0 0
\(747\) −15.3137 −0.560299
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.34315 −0.0855026 −0.0427513 0.999086i \(-0.513612\pi\)
−0.0427513 + 0.999086i \(0.513612\pi\)
\(752\) 0 0
\(753\) 4.48528 0.163453
\(754\) 0 0
\(755\) 7.02944 0.255827
\(756\) 0 0
\(757\) 37.6569 1.36866 0.684331 0.729172i \(-0.260094\pi\)
0.684331 + 0.729172i \(0.260094\pi\)
\(758\) 0 0
\(759\) −7.31371 −0.265471
\(760\) 0 0
\(761\) −46.5269 −1.68660 −0.843300 0.537444i \(-0.819390\pi\)
−0.843300 + 0.537444i \(0.819390\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3.65685 0.132214
\(766\) 0 0
\(767\) 36.9706 1.33493
\(768\) 0 0
\(769\) 29.6985 1.07095 0.535477 0.844550i \(-0.320132\pi\)
0.535477 + 0.844550i \(0.320132\pi\)
\(770\) 0 0
\(771\) 19.2132 0.691947
\(772\) 0 0
\(773\) −21.5563 −0.775328 −0.387664 0.921801i \(-0.626718\pi\)
−0.387664 + 0.921801i \(0.626718\pi\)
\(774\) 0 0
\(775\) 31.7990 1.14225
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.34315 0.227267
\(780\) 0 0
\(781\) 26.6274 0.952804
\(782\) 0 0
\(783\) −1.17157 −0.0418686
\(784\) 0 0
\(785\) 11.8579 0.423225
\(786\) 0 0
\(787\) 47.3137 1.68655 0.843276 0.537481i \(-0.180624\pi\)
0.843276 + 0.537481i \(0.180624\pi\)
\(788\) 0 0
\(789\) 17.3137 0.616384
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 20.3431 0.722406
\(794\) 0 0
\(795\) 1.17157 0.0415514
\(796\) 0 0
\(797\) −28.3848 −1.00544 −0.502720 0.864449i \(-0.667667\pi\)
−0.502720 + 0.864449i \(0.667667\pi\)
\(798\) 0 0
\(799\) −17.6569 −0.624655
\(800\) 0 0
\(801\) 5.75736 0.203426
\(802\) 0 0
\(803\) 11.7990 0.416377
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.7279 −0.377641
\(808\) 0 0
\(809\) −47.9411 −1.68552 −0.842760 0.538289i \(-0.819071\pi\)
−0.842760 + 0.538289i \(0.819071\pi\)
\(810\) 0 0
\(811\) −6.34315 −0.222738 −0.111369 0.993779i \(-0.535524\pi\)
−0.111369 + 0.993779i \(0.535524\pi\)
\(812\) 0 0
\(813\) 18.1421 0.636272
\(814\) 0 0
\(815\) 6.62742 0.232148
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −33.3137 −1.16266 −0.581328 0.813669i \(-0.697467\pi\)
−0.581328 + 0.813669i \(0.697467\pi\)
\(822\) 0 0
\(823\) 24.9706 0.870419 0.435210 0.900329i \(-0.356674\pi\)
0.435210 + 0.900329i \(0.356674\pi\)
\(824\) 0 0
\(825\) −9.31371 −0.324262
\(826\) 0 0
\(827\) −36.3431 −1.26378 −0.631888 0.775060i \(-0.717720\pi\)
−0.631888 + 0.775060i \(0.717720\pi\)
\(828\) 0 0
\(829\) −24.7279 −0.858836 −0.429418 0.903106i \(-0.641281\pi\)
−0.429418 + 0.903106i \(0.641281\pi\)
\(830\) 0 0
\(831\) 13.3137 0.461847
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −11.5980 −0.401365
\(836\) 0 0
\(837\) −6.82843 −0.236025
\(838\) 0 0
\(839\) 45.1716 1.55950 0.779748 0.626094i \(-0.215347\pi\)
0.779748 + 0.626094i \(0.215347\pi\)
\(840\) 0 0
\(841\) −27.6274 −0.952670
\(842\) 0 0
\(843\) −16.4853 −0.567783
\(844\) 0 0
\(845\) −9.55635 −0.328748
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 8.48528 0.291214
\(850\) 0 0
\(851\) 14.6274 0.501421
\(852\) 0 0
\(853\) −49.4975 −1.69476 −0.847381 0.530986i \(-0.821822\pi\)
−0.847381 + 0.530986i \(0.821822\pi\)
\(854\) 0 0
\(855\) −1.65685 −0.0566632
\(856\) 0 0
\(857\) −12.5858 −0.429922 −0.214961 0.976623i \(-0.568962\pi\)
−0.214961 + 0.976623i \(0.568962\pi\)
\(858\) 0 0
\(859\) 6.54416 0.223284 0.111642 0.993749i \(-0.464389\pi\)
0.111642 + 0.993749i \(0.464389\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 5.31371 0.180881 0.0904404 0.995902i \(-0.471173\pi\)
0.0904404 + 0.995902i \(0.471173\pi\)
\(864\) 0 0
\(865\) 4.05887 0.138006
\(866\) 0 0
\(867\) 21.9706 0.746159
\(868\) 0 0
\(869\) −4.68629 −0.158972
\(870\) 0 0
\(871\) 30.6274 1.03777
\(872\) 0 0
\(873\) −5.41421 −0.183243
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11.3137 0.382037 0.191018 0.981586i \(-0.438821\pi\)
0.191018 + 0.981586i \(0.438821\pi\)
\(878\) 0 0
\(879\) −19.4142 −0.654825
\(880\) 0 0
\(881\) −30.2426 −1.01890 −0.509450 0.860500i \(-0.670151\pi\)
−0.509450 + 0.860500i \(0.670151\pi\)
\(882\) 0 0
\(883\) 27.3137 0.919179 0.459590 0.888131i \(-0.347996\pi\)
0.459590 + 0.888131i \(0.347996\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) 2.82843 0.0949693 0.0474846 0.998872i \(-0.484879\pi\)
0.0474846 + 0.998872i \(0.484879\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) −4.88730 −0.163364
\(896\) 0 0
\(897\) 19.7990 0.661069
\(898\) 0 0
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 12.4853 0.415945
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.17157 −0.105427
\(906\) 0 0
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) 0 0
\(909\) −17.0711 −0.566212
\(910\) 0 0
\(911\) 34.9706 1.15863 0.579313 0.815105i \(-0.303321\pi\)
0.579313 + 0.815105i \(0.303321\pi\)
\(912\) 0 0
\(913\) −30.6274 −1.01362
\(914\) 0 0
\(915\) 2.20101 0.0727632
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −48.2843 −1.59275 −0.796376 0.604802i \(-0.793252\pi\)
−0.796376 + 0.604802i \(0.793252\pi\)
\(920\) 0 0
\(921\) 1.85786 0.0612187
\(922\) 0 0
\(923\) −72.0833 −2.37265
\(924\) 0 0
\(925\) 18.6274 0.612466
\(926\) 0 0
\(927\) −12.4853 −0.410070
\(928\) 0 0
\(929\) −3.21320 −0.105422 −0.0527109 0.998610i \(-0.516786\pi\)
−0.0527109 + 0.998610i \(0.516786\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −22.1421 −0.724901
\(934\) 0 0
\(935\) 7.31371 0.239184
\(936\) 0 0
\(937\) −33.4142 −1.09159 −0.545797 0.837917i \(-0.683773\pi\)
−0.545797 + 0.837917i \(0.683773\pi\)
\(938\) 0 0
\(939\) 17.8995 0.584128
\(940\) 0 0
\(941\) 7.21320 0.235144 0.117572 0.993064i \(-0.462489\pi\)
0.117572 + 0.993064i \(0.462489\pi\)
\(942\) 0 0
\(943\) −8.20101 −0.267062
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −53.3137 −1.73246 −0.866231 0.499643i \(-0.833465\pi\)
−0.866231 + 0.499643i \(0.833465\pi\)
\(948\) 0 0
\(949\) −31.9411 −1.03685
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) 0 0
\(953\) 2.00000 0.0647864 0.0323932 0.999475i \(-0.489687\pi\)
0.0323932 + 0.999475i \(0.489687\pi\)
\(954\) 0 0
\(955\) −10.5442 −0.341201
\(956\) 0 0
\(957\) −2.34315 −0.0757431
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) 0 0
\(963\) 11.6569 0.375637
\(964\) 0 0
\(965\) 10.1421 0.326487
\(966\) 0 0
\(967\) −22.3431 −0.718507 −0.359254 0.933240i \(-0.616969\pi\)
−0.359254 + 0.933240i \(0.616969\pi\)
\(968\) 0 0
\(969\) −17.6569 −0.567220
\(970\) 0 0
\(971\) −5.37258 −0.172414 −0.0862072 0.996277i \(-0.527475\pi\)
−0.0862072 + 0.996277i \(0.527475\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 25.2132 0.807469
\(976\) 0 0
\(977\) −26.8284 −0.858317 −0.429159 0.903229i \(-0.641190\pi\)
−0.429159 + 0.903229i \(0.641190\pi\)
\(978\) 0 0
\(979\) 11.5147 0.368012
\(980\) 0 0
\(981\) 5.65685 0.180609
\(982\) 0 0
\(983\) 37.2548 1.18824 0.594122 0.804375i \(-0.297499\pi\)
0.594122 + 0.804375i \(0.297499\pi\)
\(984\) 0 0
\(985\) −1.17157 −0.0373294
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −20.6863 −0.657786
\(990\) 0 0
\(991\) −20.9706 −0.666152 −0.333076 0.942900i \(-0.608087\pi\)
−0.333076 + 0.942900i \(0.608087\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) −6.05887 −0.192079
\(996\) 0 0
\(997\) 10.3848 0.328889 0.164445 0.986386i \(-0.447417\pi\)
0.164445 + 0.986386i \(0.447417\pi\)
\(998\) 0 0
\(999\) −4.00000 −0.126554
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 2352.2.a.be.1.2 2
3.2 odd 2 7056.2.a.cv.1.1 2
4.3 odd 2 147.2.a.d.1.1 2
7.2 even 3 2352.2.q.bb.1537.1 4
7.3 odd 6 2352.2.q.bd.961.2 4
7.4 even 3 2352.2.q.bb.961.1 4
7.5 odd 6 2352.2.q.bd.1537.2 4
7.6 odd 2 2352.2.a.bc.1.1 2
8.3 odd 2 9408.2.a.ef.1.1 2
8.5 even 2 9408.2.a.dq.1.1 2
12.11 even 2 441.2.a.j.1.2 2
20.19 odd 2 3675.2.a.bf.1.2 2
21.20 even 2 7056.2.a.cf.1.2 2
28.3 even 6 147.2.e.d.79.2 4
28.11 odd 6 147.2.e.e.79.2 4
28.19 even 6 147.2.e.d.67.2 4
28.23 odd 6 147.2.e.e.67.2 4
28.27 even 2 147.2.a.e.1.1 yes 2
56.13 odd 2 9408.2.a.dt.1.2 2
56.27 even 2 9408.2.a.di.1.2 2
84.11 even 6 441.2.e.f.226.1 4
84.23 even 6 441.2.e.f.361.1 4
84.47 odd 6 441.2.e.g.361.1 4
84.59 odd 6 441.2.e.g.226.1 4
84.83 odd 2 441.2.a.i.1.2 2
140.139 even 2 3675.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.2.a.d.1.1 2 4.3 odd 2
147.2.a.e.1.1 yes 2 28.27 even 2
147.2.e.d.67.2 4 28.19 even 6
147.2.e.d.79.2 4 28.3 even 6
147.2.e.e.67.2 4 28.23 odd 6
147.2.e.e.79.2 4 28.11 odd 6
441.2.a.i.1.2 2 84.83 odd 2
441.2.a.j.1.2 2 12.11 even 2
441.2.e.f.226.1 4 84.11 even 6
441.2.e.f.361.1 4 84.23 even 6
441.2.e.g.226.1 4 84.59 odd 6
441.2.e.g.361.1 4 84.47 odd 6
2352.2.a.bc.1.1 2 7.6 odd 2
2352.2.a.be.1.2 2 1.1 even 1 trivial
2352.2.q.bb.961.1 4 7.4 even 3
2352.2.q.bb.1537.1 4 7.2 even 3
2352.2.q.bd.961.2 4 7.3 odd 6
2352.2.q.bd.1537.2 4 7.5 odd 6
3675.2.a.bd.1.2 2 140.139 even 2
3675.2.a.bf.1.2 2 20.19 odd 2
7056.2.a.cf.1.2 2 21.20 even 2
7056.2.a.cv.1.1 2 3.2 odd 2
9408.2.a.di.1.2 2 56.27 even 2
9408.2.a.dq.1.1 2 8.5 even 2
9408.2.a.dt.1.2 2 56.13 odd 2
9408.2.a.ef.1.1 2 8.3 odd 2