Properties

Label 9408.2.a.dl.1.2
Level 9408
Weight 2
Character 9408.1
Self dual yes
Analytic conductor 75.123
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
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 4704)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} +2.82843 q^{11} +1.41421 q^{13} -1.41421 q^{15} +1.41421 q^{17} +2.82843 q^{23} -3.00000 q^{25} -1.00000 q^{27} +4.00000 q^{31} -2.82843 q^{33} -4.00000 q^{37} -1.41421 q^{39} -1.41421 q^{41} +5.65685 q^{43} +1.41421 q^{45} +12.0000 q^{47} -1.41421 q^{51} -10.0000 q^{53} +4.00000 q^{55} +1.41421 q^{61} +2.00000 q^{65} +11.3137 q^{67} -2.82843 q^{69} -2.82843 q^{71} +12.7279 q^{73} +3.00000 q^{75} +11.3137 q^{79} +1.00000 q^{81} -4.00000 q^{83} +2.00000 q^{85} +7.07107 q^{89} -4.00000 q^{93} -9.89949 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} - 6q^{25} - 2q^{27} + 8q^{31} - 8q^{37} + 24q^{47} - 20q^{53} + 8q^{55} + 4q^{65} + 6q^{75} + 2q^{81} - 8q^{83} + 4q^{85} - 8q^{93} + O(q^{100}) \)

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
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) 1.41421 0.392232 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 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
\(39\) −1.41421 −0.226455
\(40\) 0 0
\(41\) −1.41421 −0.220863 −0.110432 0.993884i \(-0.535223\pi\)
−0.110432 + 0.993884i \(0.535223\pi\)
\(42\) 0 0
\(43\) 5.65685 0.862662 0.431331 0.902194i \(-0.358044\pi\)
0.431331 + 0.902194i \(0.358044\pi\)
\(44\) 0 0
\(45\) 1.41421 0.210819
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −1.41421 −0.198030
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 1.41421 0.181071 0.0905357 0.995893i \(-0.471142\pi\)
0.0905357 + 0.995893i \(0.471142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 11.3137 1.38219 0.691095 0.722764i \(-0.257129\pi\)
0.691095 + 0.722764i \(0.257129\pi\)
\(68\) 0 0
\(69\) −2.82843 −0.340503
\(70\) 0 0
\(71\) −2.82843 −0.335673 −0.167836 0.985815i \(-0.553678\pi\)
−0.167836 + 0.985815i \(0.553678\pi\)
\(72\) 0 0
\(73\) 12.7279 1.48969 0.744845 0.667237i \(-0.232523\pi\)
0.744845 + 0.667237i \(0.232523\pi\)
\(74\) 0 0
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) 0 0
\(101\) −7.07107 −0.703598 −0.351799 0.936076i \(-0.614430\pi\)
−0.351799 + 0.936076i \(0.614430\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −19.7990 −1.91404 −0.957020 0.290021i \(-0.906338\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 1.41421 0.130744
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 1.41421 0.127515
\(124\) 0 0
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) −11.3137 −1.00393 −0.501965 0.864888i \(-0.667389\pi\)
−0.501965 + 0.864888i \(0.667389\pi\)
\(128\) 0 0
\(129\) −5.65685 −0.498058
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.41421 −0.121716
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 11.3137 0.920697 0.460348 0.887738i \(-0.347725\pi\)
0.460348 + 0.887738i \(0.347725\pi\)
\(152\) 0 0
\(153\) 1.41421 0.114332
\(154\) 0 0
\(155\) 5.65685 0.454369
\(156\) 0 0
\(157\) 4.24264 0.338600 0.169300 0.985565i \(-0.445849\pi\)
0.169300 + 0.985565i \(0.445849\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −16.9706 −1.32924 −0.664619 0.747183i \(-0.731406\pi\)
−0.664619 + 0.747183i \(0.731406\pi\)
\(164\) 0 0
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.07107 0.537603 0.268802 0.963196i \(-0.413372\pi\)
0.268802 + 0.963196i \(0.413372\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.1421 1.05703 0.528516 0.848923i \(-0.322748\pi\)
0.528516 + 0.848923i \(0.322748\pi\)
\(180\) 0 0
\(181\) 15.5563 1.15629 0.578147 0.815933i \(-0.303776\pi\)
0.578147 + 0.815933i \(0.303776\pi\)
\(182\) 0 0
\(183\) −1.41421 −0.104542
\(184\) 0 0
\(185\) −5.65685 −0.415900
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.48528 0.613973 0.306987 0.951714i \(-0.400679\pi\)
0.306987 + 0.951714i \(0.400679\pi\)
\(192\) 0 0
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) 0 0
\(195\) −2.00000 −0.143223
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −11.3137 −0.798007
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −5.65685 −0.389434 −0.194717 0.980859i \(-0.562379\pi\)
−0.194717 + 0.980859i \(0.562379\pi\)
\(212\) 0 0
\(213\) 2.82843 0.193801
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.7279 −0.860073
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −7.07107 −0.467269 −0.233635 0.972324i \(-0.575062\pi\)
−0.233635 + 0.972324i \(0.575062\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 16.9706 1.10704
\(236\) 0 0
\(237\) −11.3137 −0.734904
\(238\) 0 0
\(239\) −19.7990 −1.28069 −0.640345 0.768087i \(-0.721209\pi\)
−0.640345 + 0.768087i \(0.721209\pi\)
\(240\) 0 0
\(241\) −1.41421 −0.0910975 −0.0455488 0.998962i \(-0.514504\pi\)
−0.0455488 + 0.998962i \(0.514504\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) −2.00000 −0.125245
\(256\) 0 0
\(257\) 9.89949 0.617514 0.308757 0.951141i \(-0.400087\pi\)
0.308757 + 0.951141i \(0.400087\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 14.1421 0.872041 0.436021 0.899937i \(-0.356387\pi\)
0.436021 + 0.899937i \(0.356387\pi\)
\(264\) 0 0
\(265\) −14.1421 −0.868744
\(266\) 0 0
\(267\) −7.07107 −0.432742
\(268\) 0 0
\(269\) 9.89949 0.603583 0.301791 0.953374i \(-0.402415\pi\)
0.301791 + 0.953374i \(0.402415\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.48528 −0.511682
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) 9.89949 0.580319
\(292\) 0 0
\(293\) −29.6985 −1.73500 −0.867502 0.497434i \(-0.834276\pi\)
−0.867502 + 0.497434i \(0.834276\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.82843 −0.164122
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 7.07107 0.406222
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 12.7279 0.719425 0.359712 0.933063i \(-0.382875\pi\)
0.359712 + 0.933063i \(0.382875\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 19.7990 1.10507
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −4.24264 −0.235339
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −33.9411 −1.86557 −0.932786 0.360429i \(-0.882630\pi\)
−0.932786 + 0.360429i \(0.882630\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 11.3137 0.612672
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) 25.4558 1.36654 0.683271 0.730165i \(-0.260557\pi\)
0.683271 + 0.730165i \(0.260557\pi\)
\(348\) 0 0
\(349\) 26.8701 1.43832 0.719161 0.694844i \(-0.244527\pi\)
0.719161 + 0.694844i \(0.244527\pi\)
\(350\) 0 0
\(351\) −1.41421 −0.0754851
\(352\) 0 0
\(353\) −18.3848 −0.978523 −0.489261 0.872137i \(-0.662734\pi\)
−0.489261 + 0.872137i \(0.662734\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −14.1421 −0.746393 −0.373197 0.927752i \(-0.621738\pi\)
−0.373197 + 0.927752i \(0.621738\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) 18.0000 0.942163
\(366\) 0 0
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 0 0
\(369\) −1.41421 −0.0736210
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 11.3137 0.584237
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 11.3137 0.579619
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.65685 0.287554
\(388\) 0 0
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) −38.1838 −1.91639 −0.958194 0.286119i \(-0.907635\pi\)
−0.958194 + 0.286119i \(0.907635\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 5.65685 0.281788
\(404\) 0 0
\(405\) 1.41421 0.0702728
\(406\) 0 0
\(407\) −11.3137 −0.560800
\(408\) 0 0
\(409\) 29.6985 1.46850 0.734248 0.678882i \(-0.237535\pi\)
0.734248 + 0.678882i \(0.237535\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −5.65685 −0.277684
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 0 0
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) −4.24264 −0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 2.82843 0.136241 0.0681203 0.997677i \(-0.478300\pi\)
0.0681203 + 0.997677i \(0.478300\pi\)
\(432\) 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
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −19.7990 −0.940678 −0.470339 0.882486i \(-0.655868\pi\)
−0.470339 + 0.882486i \(0.655868\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) −11.3137 −0.531564
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) −1.41421 −0.0660098
\(460\) 0 0
\(461\) −4.24264 −0.197599 −0.0987997 0.995107i \(-0.531500\pi\)
−0.0987997 + 0.995107i \(0.531500\pi\)
\(462\) 0 0
\(463\) −39.5980 −1.84027 −0.920137 0.391596i \(-0.871923\pi\)
−0.920137 + 0.391596i \(0.871923\pi\)
\(464\) 0 0
\(465\) −5.65685 −0.262330
\(466\) 0 0
\(467\) 16.0000 0.740392 0.370196 0.928954i \(-0.379291\pi\)
0.370196 + 0.928954i \(0.379291\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −4.24264 −0.195491
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) −5.65685 −0.257930
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) 11.3137 0.512673 0.256337 0.966588i \(-0.417484\pi\)
0.256337 + 0.966588i \(0.417484\pi\)
\(488\) 0 0
\(489\) 16.9706 0.767435
\(490\) 0 0
\(491\) −14.1421 −0.638226 −0.319113 0.947717i \(-0.603385\pi\)
−0.319113 + 0.947717i \(0.603385\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 11.3137 0.506471 0.253236 0.967405i \(-0.418505\pi\)
0.253236 + 0.967405i \(0.418505\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 0 0
\(507\) 11.0000 0.488527
\(508\) 0 0
\(509\) 32.5269 1.44173 0.720865 0.693075i \(-0.243745\pi\)
0.720865 + 0.693075i \(0.243745\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.9706 0.747812
\(516\) 0 0
\(517\) 33.9411 1.49273
\(518\) 0 0
\(519\) −7.07107 −0.310385
\(520\) 0 0
\(521\) −29.6985 −1.30111 −0.650557 0.759457i \(-0.725465\pi\)
−0.650557 + 0.759457i \(0.725465\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.65685 0.246416
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 0 0
\(535\) −28.0000 −1.21055
\(536\) 0 0
\(537\) −14.1421 −0.610278
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) −15.5563 −0.667587
\(544\) 0 0
\(545\) 5.65685 0.242313
\(546\) 0 0
\(547\) −22.6274 −0.967478 −0.483739 0.875212i \(-0.660722\pi\)
−0.483739 + 0.875212i \(0.660722\pi\)
\(548\) 0 0
\(549\) 1.41421 0.0603572
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 5.65685 0.240120
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 16.0000 0.674320 0.337160 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(564\) 0 0
\(565\) −2.82843 −0.118993
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.0000 1.17382 0.586911 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) 22.6274 0.946928 0.473464 0.880813i \(-0.343003\pi\)
0.473464 + 0.880813i \(0.343003\pi\)
\(572\) 0 0
\(573\) −8.48528 −0.354478
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) 21.2132 0.883117 0.441559 0.897232i \(-0.354426\pi\)
0.441559 + 0.897232i \(0.354426\pi\)
\(578\) 0 0
\(579\) 26.0000 1.08052
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −28.2843 −1.17141
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 8.00000 0.330195 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 29.6985 1.21957 0.609785 0.792567i \(-0.291256\pi\)
0.609785 + 0.792567i \(0.291256\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 31.1127 1.27123 0.635615 0.772006i \(-0.280747\pi\)
0.635615 + 0.772006i \(0.280747\pi\)
\(600\) 0 0
\(601\) −15.5563 −0.634557 −0.317278 0.948332i \(-0.602769\pi\)
−0.317278 + 0.948332i \(0.602769\pi\)
\(602\) 0 0
\(603\) 11.3137 0.460730
\(604\) 0 0
\(605\) −4.24264 −0.172488
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.9706 0.686555
\(612\) 0 0
\(613\) 44.0000 1.77714 0.888572 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 36.0000 1.44931 0.724653 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −2.82843 −0.113501
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.65685 −0.225554
\(630\) 0 0
\(631\) 16.9706 0.675587 0.337794 0.941220i \(-0.390319\pi\)
0.337794 + 0.941220i \(0.390319\pi\)
\(632\) 0 0
\(633\) 5.65685 0.224840
\(634\) 0 0
\(635\) −16.0000 −0.634941
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.82843 −0.111891
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 0 0
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −8.00000 −0.313064 −0.156532 0.987673i \(-0.550031\pi\)
−0.156532 + 0.987673i \(0.550031\pi\)
\(654\) 0 0
\(655\) 5.65685 0.221032
\(656\) 0 0
\(657\) 12.7279 0.496564
\(658\) 0 0
\(659\) −19.7990 −0.771259 −0.385630 0.922654i \(-0.626016\pi\)
−0.385630 + 0.922654i \(0.626016\pi\)
\(660\) 0 0
\(661\) −29.6985 −1.15514 −0.577569 0.816342i \(-0.695998\pi\)
−0.577569 + 0.816342i \(0.695998\pi\)
\(662\) 0 0
\(663\) −2.00000 −0.0776736
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −32.0000 −1.23351 −0.616755 0.787155i \(-0.711553\pi\)
−0.616755 + 0.787155i \(0.711553\pi\)
\(674\) 0 0
\(675\) 3.00000 0.115470
\(676\) 0 0
\(677\) 46.6690 1.79364 0.896819 0.442398i \(-0.145872\pi\)
0.896819 + 0.442398i \(0.145872\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −19.7990 −0.757587 −0.378794 0.925481i \(-0.623661\pi\)
−0.378794 + 0.925481i \(0.623661\pi\)
\(684\) 0 0
\(685\) 16.9706 0.648412
\(686\) 0 0
\(687\) 7.07107 0.269778
\(688\) 0 0
\(689\) −14.1421 −0.538772
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −28.2843 −1.07288
\(696\) 0 0
\(697\) −2.00000 −0.0757554
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 16.0000 0.604312 0.302156 0.953259i \(-0.402294\pi\)
0.302156 + 0.953259i \(0.402294\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −16.9706 −0.639148
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −36.0000 −1.35201 −0.676004 0.736898i \(-0.736290\pi\)
−0.676004 + 0.736898i \(0.736290\pi\)
\(710\) 0 0
\(711\) 11.3137 0.424297
\(712\) 0 0
\(713\) 11.3137 0.423702
\(714\) 0 0
\(715\) 5.65685 0.211554
\(716\) 0 0
\(717\) 19.7990 0.739407
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.41421 0.0525952
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −29.6985 −1.09694 −0.548469 0.836171i \(-0.684789\pi\)
−0.548469 + 0.836171i \(0.684789\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) −16.9706 −0.624272 −0.312136 0.950037i \(-0.601045\pi\)
−0.312136 + 0.950037i \(0.601045\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.7696 1.34894 0.674472 0.738300i \(-0.264371\pi\)
0.674472 + 0.738300i \(0.264371\pi\)
\(744\) 0 0
\(745\) 8.48528 0.310877
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −28.2843 −1.03211 −0.516054 0.856556i \(-0.672600\pi\)
−0.516054 + 0.856556i \(0.672600\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −28.0000 −1.01768 −0.508839 0.860862i \(-0.669925\pi\)
−0.508839 + 0.860862i \(0.669925\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −43.8406 −1.58922 −0.794611 0.607119i \(-0.792325\pi\)
−0.794611 + 0.607119i \(0.792325\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −18.3848 −0.662972 −0.331486 0.943460i \(-0.607550\pi\)
−0.331486 + 0.943460i \(0.607550\pi\)
\(770\) 0 0
\(771\) −9.89949 −0.356522
\(772\) 0 0
\(773\) −15.5563 −0.559523 −0.279761 0.960070i \(-0.590255\pi\)
−0.279761 + 0.960070i \(0.590255\pi\)
\(774\) 0 0
\(775\) −12.0000 −0.431053
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.00000 0.214149
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 0 0
\(789\) −14.1421 −0.503473
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) 14.1421 0.501570
\(796\) 0 0
\(797\) −18.3848 −0.651222 −0.325611 0.945504i \(-0.605570\pi\)
−0.325611 + 0.945504i \(0.605570\pi\)
\(798\) 0 0
\(799\) 16.9706 0.600375
\(800\) 0 0
\(801\) 7.07107 0.249844
\(802\) 0 0
\(803\) 36.0000 1.27041
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −9.89949 −0.348479
\(808\) 0 0
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) −20.0000 −0.701431
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.0000 0.907406 0.453703 0.891153i \(-0.350103\pi\)
0.453703 + 0.891153i \(0.350103\pi\)
\(822\) 0 0
\(823\) 33.9411 1.18311 0.591557 0.806263i \(-0.298514\pi\)
0.591557 + 0.806263i \(0.298514\pi\)
\(824\) 0 0
\(825\) 8.48528 0.295420
\(826\) 0 0
\(827\) −19.7990 −0.688478 −0.344239 0.938882i \(-0.611863\pi\)
−0.344239 + 0.938882i \(0.611863\pi\)
\(828\) 0 0
\(829\) −38.1838 −1.32618 −0.663089 0.748541i \(-0.730755\pi\)
−0.663089 + 0.748541i \(0.730755\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 28.2843 0.978818
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 0 0
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −12.0000 −0.413302
\(844\) 0 0
\(845\) −15.5563 −0.535155
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −11.3137 −0.387829
\(852\) 0 0
\(853\) 55.1543 1.88845 0.944224 0.329304i \(-0.106814\pi\)
0.944224 + 0.329304i \(0.106814\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.89949 −0.338160 −0.169080 0.985602i \(-0.554080\pi\)
−0.169080 + 0.985602i \(0.554080\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.48528 0.288842 0.144421 0.989516i \(-0.453868\pi\)
0.144421 + 0.989516i \(0.453868\pi\)
\(864\) 0 0
\(865\) 10.0000 0.340010
\(866\) 0 0
\(867\) 15.0000 0.509427
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) −9.89949 −0.335047
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.00000 0.135070 0.0675352 0.997717i \(-0.478487\pi\)
0.0675352 + 0.997717i \(0.478487\pi\)
\(878\) 0 0
\(879\) 29.6985 1.00171
\(880\) 0 0
\(881\) −15.5563 −0.524107 −0.262053 0.965053i \(-0.584400\pi\)
−0.262053 + 0.965053i \(0.584400\pi\)
\(882\) 0 0
\(883\) −11.3137 −0.380737 −0.190368 0.981713i \(-0.560968\pi\)
−0.190368 + 0.981713i \(0.560968\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.82843 0.0947559
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 0 0
\(897\) −4.00000 −0.133556
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −14.1421 −0.471143
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 22.0000 0.731305
\(906\) 0 0
\(907\) −11.3137 −0.375666 −0.187833 0.982201i \(-0.560146\pi\)
−0.187833 + 0.982201i \(0.560146\pi\)
\(908\) 0 0
\(909\) −7.07107 −0.234533
\(910\) 0 0
\(911\) 25.4558 0.843390 0.421695 0.906738i \(-0.361435\pi\)
0.421695 + 0.906738i \(0.361435\pi\)
\(912\) 0 0
\(913\) −11.3137 −0.374429
\(914\) 0 0
\(915\) −2.00000 −0.0661180
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16.9706 −0.559807 −0.279904 0.960028i \(-0.590303\pi\)
−0.279904 + 0.960028i \(0.590303\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 0 0
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 0 0
\(927\) 12.0000 0.394132
\(928\) 0 0
\(929\) 41.0122 1.34557 0.672783 0.739840i \(-0.265099\pi\)
0.672783 + 0.739840i \(0.265099\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) 5.65685 0.184999
\(936\) 0 0
\(937\) −26.8701 −0.877807 −0.438903 0.898534i \(-0.644633\pi\)
−0.438903 + 0.898534i \(0.644633\pi\)
\(938\) 0 0
\(939\) −12.7279 −0.415360
\(940\) 0 0
\(941\) −29.6985 −0.968143 −0.484071 0.875028i \(-0.660843\pi\)
−0.484071 + 0.875028i \(0.660843\pi\)
\(942\) 0 0
\(943\) −4.00000 −0.130258
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 53.7401 1.74632 0.873160 0.487435i \(-0.162067\pi\)
0.873160 + 0.487435i \(0.162067\pi\)
\(948\) 0 0
\(949\) 18.0000 0.584305
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 58.0000 1.87880 0.939402 0.342817i \(-0.111381\pi\)
0.939402 + 0.342817i \(0.111381\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −19.7990 −0.638014
\(964\) 0 0
\(965\) −36.7696 −1.18365
\(966\) 0 0
\(967\) 45.2548 1.45530 0.727649 0.685950i \(-0.240613\pi\)
0.727649 + 0.685950i \(0.240613\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 4.24264 0.135873
\(976\) 0 0
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) −48.0000 −1.53096 −0.765481 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) −8.48528 −0.270364
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 5.65685 0.179696 0.0898479 0.995955i \(-0.471362\pi\)
0.0898479 + 0.995955i \(0.471362\pi\)
\(992\) 0 0
\(993\) 33.9411 1.07709
\(994\) 0 0
\(995\) 11.3137 0.358669
\(996\) 0 0
\(997\) 4.24264 0.134366 0.0671829 0.997741i \(-0.478599\pi\)
0.0671829 + 0.997741i \(0.478599\pi\)
\(998\) 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 9408.2.a.dl.1.2 2
4.3 odd 2 9408.2.a.ea.1.2 2
7.6 odd 2 9408.2.a.ea.1.1 2
8.3 odd 2 4704.2.a.bj.1.1 2
8.5 even 2 4704.2.a.bq.1.1 yes 2
28.27 even 2 inner 9408.2.a.dl.1.1 2
56.13 odd 2 4704.2.a.bj.1.2 yes 2
56.27 even 2 4704.2.a.bq.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bj.1.1 2 8.3 odd 2
4704.2.a.bj.1.2 yes 2 56.13 odd 2
4704.2.a.bq.1.1 yes 2 8.5 even 2
4704.2.a.bq.1.2 yes 2 56.27 even 2
9408.2.a.dl.1.1 2 28.27 even 2 inner
9408.2.a.dl.1.2 2 1.1 even 1 trivial
9408.2.a.ea.1.1 2 7.6 odd 2
9408.2.a.ea.1.2 2 4.3 odd 2