Properties

Label 9408.2.a.ea.1.2
Level 9408
Weight 2
Character 9408.1
Self dual yes
Analytic conductor 75.123
Analytic rank 1
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: \(1\)
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 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 \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 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 \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 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 \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 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 \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 0 0
\(45\) 1.41421 0.210819
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 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 \(-0.742871\pi\)
−0.691095 + 0.722764i \(0.742871\pi\)
\(68\) 0 0
\(69\) −2.82843 −0.340503
\(70\) 0 0
\(71\) 2.82843 0.335673 0.167836 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(72\) 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 \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 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 \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.7990 1.91404 0.957020 0.290021i \(-0.0936623\pi\)
0.957020 + 0.290021i \(0.0936623\pi\)
\(108\) 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 \(-0.332611\pi\)
0.501965 + 0.864888i \(0.332611\pi\)
\(128\) 0 0
\(129\) −5.65685 −0.498058
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 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 \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 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 \(-0.652275\pi\)
−0.460348 + 0.887738i \(0.652275\pi\)
\(152\) 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 \(-0.268594\pi\)
0.664619 + 0.747183i \(0.268594\pi\)
\(164\) 0 0
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) −20.0000 −1.54765 −0.773823 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 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 \(-0.677252\pi\)
−0.528516 + 0.848923i \(0.677252\pi\)
\(180\) 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 \(-0.599321\pi\)
−0.306987 + 0.951714i \(0.599321\pi\)
\(192\) 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 \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 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 \(-0.437621\pi\)
0.194717 + 0.980859i \(0.437621\pi\)
\(212\) 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 \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 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 \(-0.278791\pi\)
0.640345 + 0.768087i \(0.278791\pi\)
\(240\) 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 \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 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 \(-0.643613\pi\)
−0.436021 + 0.899937i \(0.643613\pi\)
\(264\) 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 \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 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 \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 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 \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 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 \(-0.117370\pi\)
0.932786 + 0.360429i \(0.117370\pi\)
\(332\) 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 \(-0.739443\pi\)
−0.683271 + 0.730165i \(0.739443\pi\)
\(348\) 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 \(-0.378262\pi\)
0.373197 + 0.927752i \(0.378262\pi\)
\(360\) 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 \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 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 \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 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 \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 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 \(-0.521700\pi\)
−0.0681203 + 0.997677i \(0.521700\pi\)
\(432\) 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 \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.7990 0.940678 0.470339 0.882486i \(-0.344132\pi\)
0.470339 + 0.882486i \(0.344132\pi\)
\(444\) 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 \(-0.128077\pi\)
0.920137 + 0.391596i \(0.128077\pi\)
\(464\) 0 0
\(465\) −5.65685 −0.262330
\(466\) 0 0
\(467\) −16.0000 −0.740392 −0.370196 0.928954i \(-0.620709\pi\)
−0.370196 + 0.928954i \(0.620709\pi\)
\(468\) 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 \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 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 \(-0.582516\pi\)
−0.256337 + 0.966588i \(0.582516\pi\)
\(488\) 0 0
\(489\) 16.9706 0.767435
\(490\) 0 0
\(491\) 14.1421 0.638226 0.319113 0.947717i \(-0.396615\pi\)
0.319113 + 0.947717i \(0.396615\pi\)
\(492\) 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 \(-0.581495\pi\)
−0.253236 + 0.967405i \(0.581495\pi\)
\(500\) 0 0
\(501\) −20.0000 −0.893534
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 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 \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 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 \(-0.339278\pi\)
0.483739 + 0.875212i \(0.339278\pi\)
\(548\) 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 \(-0.609466\pi\)
−0.337160 + 0.941447i \(0.609466\pi\)
\(564\) 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 \(-0.656997\pi\)
−0.473464 + 0.880813i \(0.656997\pi\)
\(572\) 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 \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 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 \(-0.719253\pi\)
−0.635615 + 0.772006i \(0.719253\pi\)
\(600\) 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 \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 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 \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 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 \(-0.609681\pi\)
−0.337794 + 0.941220i \(0.609681\pi\)
\(632\) 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 \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 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 \(-0.373984\pi\)
0.385630 + 0.922654i \(0.373984\pi\)
\(660\) 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 \(-0.376339\pi\)
0.378794 + 0.925481i \(0.376339\pi\)
\(684\) 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 \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) 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 \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 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 \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 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 \(-0.398955\pi\)
0.312136 + 0.950037i \(0.398955\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.7696 −1.34894 −0.674472 0.738300i \(-0.735629\pi\)
−0.674472 + 0.738300i \(0.735629\pi\)
\(744\) 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 \(-0.327400\pi\)
0.516054 + 0.856556i \(0.327400\pi\)
\(752\) 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 \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 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 \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 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 \(-0.701486\pi\)
−0.591557 + 0.806263i \(0.701486\pi\)
\(824\) 0 0
\(825\) 8.48528 0.295420
\(826\) 0 0
\(827\) 19.7990 0.688478 0.344239 0.938882i \(-0.388137\pi\)
0.344239 + 0.938882i \(0.388137\pi\)
\(828\) 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 \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 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 \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.48528 −0.288842 −0.144421 0.989516i \(-0.546132\pi\)
−0.144421 + 0.989516i \(0.546132\pi\)
\(864\) 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 \(-0.439032\pi\)
0.190368 + 0.981713i \(0.439032\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 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 \(-0.439854\pi\)
0.187833 + 0.982201i \(0.439854\pi\)
\(908\) 0 0
\(909\) −7.07107 −0.234533
\(910\) 0 0
\(911\) −25.4558 −0.843390 −0.421695 0.906738i \(-0.638565\pi\)
−0.421695 + 0.906738i \(0.638565\pi\)
\(912\) 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 \(-0.409697\pi\)
0.279904 + 0.960028i \(0.409697\pi\)
\(920\) 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 \(-0.837933\pi\)
−0.873160 + 0.487435i \(0.837933\pi\)
\(948\) 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 \(-0.759387\pi\)
−0.727649 + 0.685950i \(0.759387\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 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 \(-0.222501\pi\)
0.765481 + 0.643458i \(0.222501\pi\)
\(984\) 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 \(-0.528638\pi\)
−0.0898479 + 0.995955i \(0.528638\pi\)
\(992\) 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.ea.1.2 2
4.3 odd 2 9408.2.a.dl.1.2 2
7.6 odd 2 9408.2.a.dl.1.1 2
8.3 odd 2 4704.2.a.bq.1.1 yes 2
8.5 even 2 4704.2.a.bj.1.1 2
28.27 even 2 inner 9408.2.a.ea.1.1 2
56.13 odd 2 4704.2.a.bq.1.2 yes 2
56.27 even 2 4704.2.a.bj.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bj.1.1 2 8.5 even 2
4704.2.a.bj.1.2 yes 2 56.27 even 2
4704.2.a.bq.1.1 yes 2 8.3 odd 2
4704.2.a.bq.1.2 yes 2 56.13 odd 2
9408.2.a.dl.1.1 2 7.6 odd 2
9408.2.a.dl.1.2 2 4.3 odd 2
9408.2.a.ea.1.1 2 28.27 even 2 inner
9408.2.a.ea.1.2 2 1.1 even 1 trivial