Properties

Label 9408.2.a.dn.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

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.41421 q^{5} +1.00000 q^{9} -2.82843 q^{11} -4.24264 q^{13} -1.41421 q^{15} -4.24264 q^{17} +8.00000 q^{19} -2.82843 q^{23} -3.00000 q^{25} -1.00000 q^{27} -4.00000 q^{31} +2.82843 q^{33} +4.00000 q^{37} +4.24264 q^{39} -7.07107 q^{41} -5.65685 q^{43} +1.41421 q^{45} +4.00000 q^{47} +4.24264 q^{51} +6.00000 q^{53} -4.00000 q^{55} -8.00000 q^{57} +8.00000 q^{59} -4.24264 q^{61} -6.00000 q^{65} +2.82843 q^{69} +14.1421 q^{71} +1.41421 q^{73} +3.00000 q^{75} -11.3137 q^{79} +1.00000 q^{81} +12.0000 q^{83} -6.00000 q^{85} +1.41421 q^{89} +4.00000 q^{93} +11.3137 q^{95} +12.7279 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 16q^{19} - 6q^{25} - 2q^{27} - 8q^{31} + 8q^{37} + 8q^{47} + 12q^{53} - 8q^{55} - 16q^{57} + 16q^{59} - 12q^{65} + 6q^{75} + 2q^{81} + 24q^{83} - 12q^{85} + 8q^{93} + O(q^{100}) \)

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
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) −4.24264 −1.02899 −0.514496 0.857493i \(-0.672021\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 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 \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 4.24264 0.679366
\(40\) 0 0
\(41\) −7.07107 −1.10432 −0.552158 0.833740i \(-0.686195\pi\)
−0.552158 + 0.833740i \(0.686195\pi\)
\(42\) 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
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4.24264 0.594089
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −4.24264 −0.543214 −0.271607 0.962408i \(-0.587555\pi\)
−0.271607 + 0.962408i \(0.587555\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) 14.1421 1.67836 0.839181 0.543852i \(-0.183035\pi\)
0.839181 + 0.543852i \(0.183035\pi\)
\(72\) 0 0
\(73\) 1.41421 0.165521 0.0827606 0.996569i \(-0.473626\pi\)
0.0827606 + 0.996569i \(0.473626\pi\)
\(74\) 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
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.41421 0.149906 0.0749532 0.997187i \(-0.476119\pi\)
0.0749532 + 0.997187i \(0.476119\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 11.3137 1.16076
\(96\) 0 0
\(97\) 12.7279 1.29232 0.646162 0.763200i \(-0.276373\pi\)
0.646162 + 0.763200i \(0.276373\pi\)
\(98\) 0 0
\(99\) −2.82843 −0.284268
\(100\) 0 0
\(101\) 4.24264 0.422159 0.211079 0.977469i \(-0.432302\pi\)
0.211079 + 0.977469i \(0.432302\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.48528 0.820303 0.410152 0.912017i \(-0.365476\pi\)
0.410152 + 0.912017i \(0.365476\pi\)
\(108\) 0 0
\(109\) −20.0000 −1.91565 −0.957826 0.287348i \(-0.907226\pi\)
−0.957826 + 0.287348i \(0.907226\pi\)
\(110\) 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
\(117\) −4.24264 −0.392232
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 7.07107 0.637577
\(124\) 0 0
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 5.65685 0.498058
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 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 \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 22.6274 1.84139 0.920697 0.390279i \(-0.127622\pi\)
0.920697 + 0.390279i \(0.127622\pi\)
\(152\) 0 0
\(153\) −4.24264 −0.342997
\(154\) 0 0
\(155\) −5.65685 −0.454369
\(156\) 0 0
\(157\) 21.2132 1.69300 0.846499 0.532390i \(-0.178706\pi\)
0.846499 + 0.532390i \(0.178706\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 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
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 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
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) 19.7990 1.47985 0.739923 0.672692i \(-0.234862\pi\)
0.739923 + 0.672692i \(0.234862\pi\)
\(180\) 0 0
\(181\) −1.41421 −0.105118 −0.0525588 0.998618i \(-0.516738\pi\)
−0.0525588 + 0.998618i \(0.516738\pi\)
\(182\) 0 0
\(183\) 4.24264 0.313625
\(184\) 0 0
\(185\) 5.65685 0.415900
\(186\) 0 0
\(187\) 12.0000 0.877527
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.7990 −1.43260 −0.716302 0.697790i \(-0.754167\pi\)
−0.716302 + 0.697790i \(0.754167\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 0 0
\(207\) −2.82843 −0.196589
\(208\) 0 0
\(209\) −22.6274 −1.56517
\(210\) 0 0
\(211\) −5.65685 −0.389434 −0.194717 0.980859i \(-0.562379\pi\)
−0.194717 + 0.980859i \(0.562379\pi\)
\(212\) 0 0
\(213\) −14.1421 −0.969003
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.41421 −0.0955637
\(220\) 0 0
\(221\) 18.0000 1.21081
\(222\) 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
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 0 0
\(229\) 21.2132 1.40181 0.700904 0.713256i \(-0.252780\pi\)
0.700904 + 0.713256i \(0.252780\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.0000 1.31024 0.655122 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) 0 0
\(235\) 5.65685 0.369012
\(236\) 0 0
\(237\) 11.3137 0.734904
\(238\) 0 0
\(239\) −25.4558 −1.64660 −0.823301 0.567605i \(-0.807870\pi\)
−0.823301 + 0.567605i \(0.807870\pi\)
\(240\) 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
\(247\) −33.9411 −2.15962
\(248\) 0 0
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 6.00000 0.375735
\(256\) 0 0
\(257\) 15.5563 0.970378 0.485189 0.874409i \(-0.338751\pi\)
0.485189 + 0.874409i \(0.338751\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.7990 1.22086 0.610429 0.792071i \(-0.290997\pi\)
0.610429 + 0.792071i \(0.290997\pi\)
\(264\) 0 0
\(265\) 8.48528 0.521247
\(266\) 0 0
\(267\) −1.41421 −0.0865485
\(268\) 0 0
\(269\) −24.0416 −1.46584 −0.732922 0.680313i \(-0.761844\pi\)
−0.732922 + 0.680313i \(0.761844\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.48528 0.511682
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 0 0
\(285\) −11.3137 −0.670166
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −12.7279 −0.746124
\(292\) 0 0
\(293\) 4.24264 0.247858 0.123929 0.992291i \(-0.460451\pi\)
0.123929 + 0.992291i \(0.460451\pi\)
\(294\) 0 0
\(295\) 11.3137 0.658710
\(296\) 0 0
\(297\) 2.82843 0.164122
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4.24264 −0.243733
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 32.0000 1.82634 0.913168 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) −21.2132 −1.19904 −0.599521 0.800359i \(-0.704642\pi\)
−0.599521 + 0.800359i \(0.704642\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 34.0000 1.90963 0.954815 0.297200i \(-0.0960529\pi\)
0.954815 + 0.297200i \(0.0960529\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −8.48528 −0.473602
\(322\) 0 0
\(323\) −33.9411 −1.88853
\(324\) 0 0
\(325\) 12.7279 0.706018
\(326\) 0 0
\(327\) 20.0000 1.10600
\(328\) 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
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 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
\(347\) −2.82843 −0.151838 −0.0759190 0.997114i \(-0.524189\pi\)
−0.0759190 + 0.997114i \(0.524189\pi\)
\(348\) 0 0
\(349\) −24.0416 −1.28692 −0.643459 0.765480i \(-0.722502\pi\)
−0.643459 + 0.765480i \(0.722502\pi\)
\(350\) 0 0
\(351\) 4.24264 0.226455
\(352\) 0 0
\(353\) −1.41421 −0.0752710 −0.0376355 0.999292i \(-0.511983\pi\)
−0.0376355 + 0.999292i \(0.511983\pi\)
\(354\) 0 0
\(355\) 20.0000 1.06149
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −31.1127 −1.64207 −0.821033 0.570881i \(-0.806602\pi\)
−0.821033 + 0.570881i \(0.806602\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) −7.07107 −0.368105
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 11.3137 0.584237
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −22.6274 −1.16229 −0.581146 0.813799i \(-0.697396\pi\)
−0.581146 + 0.813799i \(0.697396\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 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
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) −9.89949 −0.496841 −0.248421 0.968652i \(-0.579912\pi\)
−0.248421 + 0.968652i \(0.579912\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 16.9706 0.845364
\(404\) 0 0
\(405\) 1.41421 0.0702728
\(406\) 0 0
\(407\) −11.3137 −0.560800
\(408\) 0 0
\(409\) 7.07107 0.349642 0.174821 0.984600i \(-0.444065\pi\)
0.174821 + 0.984600i \(0.444065\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 16.9706 0.833052
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −38.0000 −1.85201 −0.926003 0.377515i \(-0.876779\pi\)
−0.926003 + 0.377515i \(0.876779\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) 12.7279 0.617395
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 19.7990 0.953684 0.476842 0.878989i \(-0.341781\pi\)
0.476842 + 0.878989i \(0.341781\pi\)
\(432\) 0 0
\(433\) 38.1838 1.83499 0.917497 0.397742i \(-0.130206\pi\)
0.917497 + 0.397742i \(0.130206\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −22.6274 −1.08242
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −14.1421 −0.671913 −0.335957 0.941877i \(-0.609060\pi\)
−0.335957 + 0.941877i \(0.609060\pi\)
\(444\) 0 0
\(445\) 2.00000 0.0948091
\(446\) 0 0
\(447\) 10.0000 0.472984
\(448\) 0 0
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) 20.0000 0.941763
\(452\) 0 0
\(453\) −22.6274 −1.06313
\(454\) 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
\(459\) 4.24264 0.198030
\(460\) 0 0
\(461\) 41.0122 1.91013 0.955064 0.296399i \(-0.0957859\pi\)
0.955064 + 0.296399i \(0.0957859\pi\)
\(462\) 0 0
\(463\) −5.65685 −0.262896 −0.131448 0.991323i \(-0.541963\pi\)
−0.131448 + 0.991323i \(0.541963\pi\)
\(464\) 0 0
\(465\) 5.65685 0.262330
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −21.2132 −0.977453
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −24.0000 −1.10120
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −16.9706 −0.773791
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18.0000 0.817338
\(486\) 0 0
\(487\) 22.6274 1.02535 0.512673 0.858584i \(-0.328655\pi\)
0.512673 + 0.858584i \(0.328655\pi\)
\(488\) 0 0
\(489\) −16.9706 −0.767435
\(490\) 0 0
\(491\) 2.82843 0.127645 0.0638226 0.997961i \(-0.479671\pi\)
0.0638226 + 0.997961i \(0.479671\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 22.6274 1.01294 0.506471 0.862257i \(-0.330950\pi\)
0.506471 + 0.862257i \(0.330950\pi\)
\(500\) 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
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −5.00000 −0.222058
\(508\) 0 0
\(509\) −1.41421 −0.0626839 −0.0313420 0.999509i \(-0.509978\pi\)
−0.0313420 + 0.999509i \(0.509978\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −8.00000 −0.353209
\(514\) 0 0
\(515\) 5.65685 0.249271
\(516\) 0 0
\(517\) −11.3137 −0.497576
\(518\) 0 0
\(519\) −7.07107 −0.310385
\(520\) 0 0
\(521\) −1.41421 −0.0619578 −0.0309789 0.999520i \(-0.509862\pi\)
−0.0309789 + 0.999520i \(0.509862\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.9706 0.739249
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 30.0000 1.29944
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) −19.7990 −0.854389
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 1.41421 0.0606897
\(544\) 0 0
\(545\) −28.2843 −1.21157
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) −4.24264 −0.181071
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −5.65685 −0.240120
\(556\) 0 0
\(557\) 34.0000 1.44063 0.720313 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −2.82843 −0.118993
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.00000 0.167689 0.0838444 0.996479i \(-0.473280\pi\)
0.0838444 + 0.996479i \(0.473280\pi\)
\(570\) 0 0
\(571\) 22.6274 0.946928 0.473464 0.880813i \(-0.343003\pi\)
0.473464 + 0.880813i \(0.343003\pi\)
\(572\) 0 0
\(573\) 19.7990 0.827115
\(574\) 0 0
\(575\) 8.48528 0.353861
\(576\) 0 0
\(577\) −24.0416 −1.00087 −0.500433 0.865775i \(-0.666826\pi\)
−0.500433 + 0.865775i \(0.666826\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −16.9706 −0.702849
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 0 0
\(593\) 35.3553 1.45187 0.725935 0.687763i \(-0.241407\pi\)
0.725935 + 0.687763i \(0.241407\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 2.82843 0.115566 0.0577832 0.998329i \(-0.481597\pi\)
0.0577832 + 0.998329i \(0.481597\pi\)
\(600\) 0 0
\(601\) 29.6985 1.21143 0.605713 0.795683i \(-0.292888\pi\)
0.605713 + 0.795683i \(0.292888\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.24264 −0.172488
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.9706 −0.686555
\(612\) 0 0
\(613\) 4.00000 0.161558 0.0807792 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 2.82843 0.113501
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 22.6274 0.903652
\(628\) 0 0
\(629\) −16.9706 −0.676661
\(630\) 0 0
\(631\) −39.5980 −1.57637 −0.788185 0.615438i \(-0.788979\pi\)
−0.788185 + 0.615438i \(0.788979\pi\)
\(632\) 0 0
\(633\) 5.65685 0.224840
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 14.1421 0.559454
\(640\) 0 0
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 4.00000 0.157256 0.0786281 0.996904i \(-0.474946\pi\)
0.0786281 + 0.996904i \(0.474946\pi\)
\(648\) 0 0
\(649\) −22.6274 −0.888204
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.00000 0.313064 0.156532 0.987673i \(-0.449969\pi\)
0.156532 + 0.987673i \(0.449969\pi\)
\(654\) 0 0
\(655\) 28.2843 1.10516
\(656\) 0 0
\(657\) 1.41421 0.0551737
\(658\) 0 0
\(659\) −2.82843 −0.110180 −0.0550899 0.998481i \(-0.517545\pi\)
−0.0550899 + 0.998481i \(0.517545\pi\)
\(660\) 0 0
\(661\) 21.2132 0.825098 0.412549 0.910935i \(-0.364639\pi\)
0.412549 + 0.910935i \(0.364639\pi\)
\(662\) 0 0
\(663\) −18.0000 −0.699062
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) 48.0000 1.85026 0.925132 0.379646i \(-0.123954\pi\)
0.925132 + 0.379646i \(0.123954\pi\)
\(674\) 0 0
\(675\) 3.00000 0.115470
\(676\) 0 0
\(677\) 24.0416 0.923995 0.461997 0.886881i \(-0.347133\pi\)
0.461997 + 0.886881i \(0.347133\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) 31.1127 1.19049 0.595247 0.803543i \(-0.297054\pi\)
0.595247 + 0.803543i \(0.297054\pi\)
\(684\) 0 0
\(685\) −16.9706 −0.648412
\(686\) 0 0
\(687\) −21.2132 −0.809334
\(688\) 0 0
\(689\) −25.4558 −0.969790
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.9706 0.643730
\(696\) 0 0
\(697\) 30.0000 1.13633
\(698\) 0 0
\(699\) −20.0000 −0.756469
\(700\) 0 0
\(701\) 48.0000 1.81293 0.906467 0.422276i \(-0.138769\pi\)
0.906467 + 0.422276i \(0.138769\pi\)
\(702\) 0 0
\(703\) 32.0000 1.20690
\(704\) 0 0
\(705\) −5.65685 −0.213049
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 20.0000 0.751116 0.375558 0.926799i \(-0.377451\pi\)
0.375558 + 0.926799i \(0.377451\pi\)
\(710\) 0 0
\(711\) −11.3137 −0.424297
\(712\) 0 0
\(713\) 11.3137 0.423702
\(714\) 0 0
\(715\) 16.9706 0.634663
\(716\) 0 0
\(717\) 25.4558 0.950666
\(718\) 0 0
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 1.41421 0.0525952
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −36.0000 −1.33517 −0.667583 0.744535i \(-0.732671\pi\)
−0.667583 + 0.744535i \(0.732671\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 9.89949 0.365646 0.182823 0.983146i \(-0.441476\pi\)
0.182823 + 0.983146i \(0.441476\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −5.65685 −0.208091 −0.104045 0.994573i \(-0.533179\pi\)
−0.104045 + 0.994573i \(0.533179\pi\)
\(740\) 0 0
\(741\) 33.9411 1.24686
\(742\) 0 0
\(743\) 31.1127 1.14141 0.570707 0.821154i \(-0.306669\pi\)
0.570707 + 0.821154i \(0.306669\pi\)
\(744\) 0 0
\(745\) −14.1421 −0.518128
\(746\) 0 0
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 39.5980 1.44495 0.722475 0.691397i \(-0.243004\pi\)
0.722475 + 0.691397i \(0.243004\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) 12.0000 0.436147 0.218074 0.975932i \(-0.430023\pi\)
0.218074 + 0.975932i \(0.430023\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 52.3259 1.89681 0.948406 0.317058i \(-0.102695\pi\)
0.948406 + 0.317058i \(0.102695\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) −33.9411 −1.22554
\(768\) 0 0
\(769\) 38.1838 1.37694 0.688471 0.725264i \(-0.258282\pi\)
0.688471 + 0.725264i \(0.258282\pi\)
\(770\) 0 0
\(771\) −15.5563 −0.560248
\(772\) 0 0
\(773\) −26.8701 −0.966449 −0.483224 0.875497i \(-0.660534\pi\)
−0.483224 + 0.875497i \(0.660534\pi\)
\(774\) 0 0
\(775\) 12.0000 0.431053
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −56.5685 −2.02678
\(780\) 0 0
\(781\) −40.0000 −1.43131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) −19.7990 −0.704863
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 18.0000 0.639199
\(794\) 0 0
\(795\) −8.48528 −0.300942
\(796\) 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
\(801\) 1.41421 0.0499688
\(802\) 0 0
\(803\) −4.00000 −0.141157
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.0416 0.846305
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) −45.2548 −1.58327
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −11.3137 −0.394371 −0.197186 0.980366i \(-0.563180\pi\)
−0.197186 + 0.980366i \(0.563180\pi\)
\(824\) 0 0
\(825\) −8.48528 −0.295420
\(826\) 0 0
\(827\) 53.7401 1.86873 0.934363 0.356321i \(-0.115969\pi\)
0.934363 + 0.356321i \(0.115969\pi\)
\(828\) 0 0
\(829\) 35.3553 1.22794 0.613971 0.789329i \(-0.289571\pi\)
0.613971 + 0.789329i \(0.289571\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −28.2843 −0.978818
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −20.0000 −0.688837
\(844\) 0 0
\(845\) 7.07107 0.243252
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) −11.3137 −0.387829
\(852\) 0 0
\(853\) 4.24264 0.145265 0.0726326 0.997359i \(-0.476860\pi\)
0.0726326 + 0.997359i \(0.476860\pi\)
\(854\) 0 0
\(855\) 11.3137 0.386921
\(856\) 0 0
\(857\) 18.3848 0.628012 0.314006 0.949421i \(-0.398329\pi\)
0.314006 + 0.949421i \(0.398329\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.4558 0.866527 0.433264 0.901267i \(-0.357362\pi\)
0.433264 + 0.901267i \(0.357362\pi\)
\(864\) 0 0
\(865\) 10.0000 0.340010
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 12.7279 0.430775
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 12.0000 0.405211 0.202606 0.979260i \(-0.435059\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) 0 0
\(879\) −4.24264 −0.143101
\(880\) 0 0
\(881\) −9.89949 −0.333522 −0.166761 0.985997i \(-0.553331\pi\)
−0.166761 + 0.985997i \(0.553331\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) −11.3137 −0.380306
\(886\) 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
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 28.0000 0.935937
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −25.4558 −0.848057
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) 22.6274 0.751331 0.375666 0.926755i \(-0.377414\pi\)
0.375666 + 0.926755i \(0.377414\pi\)
\(908\) 0 0
\(909\) 4.24264 0.140720
\(910\) 0 0
\(911\) −25.4558 −0.843390 −0.421695 0.906738i \(-0.638565\pi\)
−0.421695 + 0.906738i \(0.638565\pi\)
\(912\) 0 0
\(913\) −33.9411 −1.12329
\(914\) 0 0
\(915\) 6.00000 0.198354
\(916\) 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
\(921\) −32.0000 −1.05444
\(922\) 0 0
\(923\) −60.0000 −1.97492
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) 46.6690 1.53116 0.765581 0.643340i \(-0.222452\pi\)
0.765581 + 0.643340i \(0.222452\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) 16.9706 0.554997
\(936\) 0 0
\(937\) −4.24264 −0.138601 −0.0693005 0.997596i \(-0.522077\pi\)
−0.0693005 + 0.997596i \(0.522077\pi\)
\(938\) 0 0
\(939\) 21.2132 0.692267
\(940\) 0 0
\(941\) 15.5563 0.507122 0.253561 0.967319i \(-0.418398\pi\)
0.253561 + 0.967319i \(0.418398\pi\)
\(942\) 0 0
\(943\) 20.0000 0.651290
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.1421 0.459558 0.229779 0.973243i \(-0.426200\pi\)
0.229779 + 0.973243i \(0.426200\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) −34.0000 −1.10253
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −28.0000 −0.906059
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 8.48528 0.273434
\(964\) 0 0
\(965\) 8.48528 0.273151
\(966\) 0 0
\(967\) 11.3137 0.363824 0.181912 0.983315i \(-0.441771\pi\)
0.181912 + 0.983315i \(0.441771\pi\)
\(968\) 0 0
\(969\) 33.9411 1.09035
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −12.7279 −0.407620
\(976\) 0 0
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) −20.0000 −0.638551
\(982\) 0 0
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) 14.1421 0.450606
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −50.9117 −1.61726 −0.808632 0.588315i \(-0.799791\pi\)
−0.808632 + 0.588315i \(0.799791\pi\)
\(992\) 0 0
\(993\) 33.9411 1.07709
\(994\) 0 0
\(995\) 11.3137 0.358669
\(996\) 0 0
\(997\) −35.3553 −1.11971 −0.559857 0.828589i \(-0.689144\pi\)
−0.559857 + 0.828589i \(0.689144\pi\)
\(998\) 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.dn.1.2 2
4.3 odd 2 9408.2.a.dy.1.2 2
7.6 odd 2 9408.2.a.dy.1.1 2
8.3 odd 2 4704.2.a.bl.1.1 2
8.5 even 2 4704.2.a.bo.1.1 yes 2
28.27 even 2 inner 9408.2.a.dn.1.1 2
56.13 odd 2 4704.2.a.bl.1.2 yes 2
56.27 even 2 4704.2.a.bo.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bl.1.1 2 8.3 odd 2
4704.2.a.bl.1.2 yes 2 56.13 odd 2
4704.2.a.bo.1.1 yes 2 8.5 even 2
4704.2.a.bo.1.2 yes 2 56.27 even 2
9408.2.a.dn.1.1 2 28.27 even 2 inner
9408.2.a.dn.1.2 2 1.1 even 1 trivial
9408.2.a.dy.1.1 2 7.6 odd 2
9408.2.a.dy.1.2 2 4.3 odd 2