Properties

Label 9408.2.a.dj.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 $1$

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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: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.27492 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.27492 q^{5} +1.00000 q^{9} -3.27492 q^{11} +6.27492 q^{13} -3.27492 q^{15} +4.00000 q^{17} -6.27492 q^{19} +4.00000 q^{23} +5.72508 q^{25} -1.00000 q^{27} -5.27492 q^{29} +1.00000 q^{31} +3.27492 q^{33} +2.27492 q^{37} -6.27492 q^{39} +4.54983 q^{41} -0.274917 q^{43} +3.27492 q^{45} +6.00000 q^{47} -4.00000 q^{51} -9.27492 q^{53} -10.7251 q^{55} +6.27492 q^{57} -1.27492 q^{59} +10.0000 q^{61} +20.5498 q^{65} +0.274917 q^{67} -4.00000 q^{69} +2.00000 q^{71} +4.27492 q^{73} -5.72508 q^{75} +11.5498 q^{79} +1.00000 q^{81} +7.27492 q^{83} +13.0997 q^{85} +5.27492 q^{87} +10.5498 q^{89} -1.00000 q^{93} -20.5498 q^{95} -8.72508 q^{97} -3.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - q^{5} + 2q^{9} + q^{11} + 5q^{13} + q^{15} + 8q^{17} - 5q^{19} + 8q^{23} + 19q^{25} - 2q^{27} - 3q^{29} + 2q^{31} - q^{33} - 3q^{37} - 5q^{39} - 6q^{41} + 7q^{43} - q^{45} + 12q^{47} - 8q^{51} - 11q^{53} - 29q^{55} + 5q^{57} + 5q^{59} + 20q^{61} + 26q^{65} - 7q^{67} - 8q^{69} + 4q^{71} + q^{73} - 19q^{75} + 8q^{79} + 2q^{81} + 7q^{83} - 4q^{85} + 3q^{87} + 6q^{89} - 2q^{93} - 26q^{95} - 25q^{97} + q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.27492 1.46459 0.732294 0.680989i \(-0.238450\pi\)
0.732294 + 0.680989i \(0.238450\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.27492 −0.987425 −0.493712 0.869625i \(-0.664360\pi\)
−0.493712 + 0.869625i \(0.664360\pi\)
\(12\) 0 0
\(13\) 6.27492 1.74035 0.870174 0.492744i \(-0.164006\pi\)
0.870174 + 0.492744i \(0.164006\pi\)
\(14\) 0 0
\(15\) −3.27492 −0.845580
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −6.27492 −1.43956 −0.719782 0.694200i \(-0.755758\pi\)
−0.719782 + 0.694200i \(0.755758\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 5.72508 1.14502
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.27492 −0.979528 −0.489764 0.871855i \(-0.662917\pi\)
−0.489764 + 0.871855i \(0.662917\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 3.27492 0.570090
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.27492 0.373994 0.186997 0.982360i \(-0.440125\pi\)
0.186997 + 0.982360i \(0.440125\pi\)
\(38\) 0 0
\(39\) −6.27492 −1.00479
\(40\) 0 0
\(41\) 4.54983 0.710565 0.355282 0.934759i \(-0.384385\pi\)
0.355282 + 0.934759i \(0.384385\pi\)
\(42\) 0 0
\(43\) −0.274917 −0.0419245 −0.0209622 0.999780i \(-0.506673\pi\)
−0.0209622 + 0.999780i \(0.506673\pi\)
\(44\) 0 0
\(45\) 3.27492 0.488196
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −9.27492 −1.27401 −0.637004 0.770861i \(-0.719827\pi\)
−0.637004 + 0.770861i \(0.719827\pi\)
\(54\) 0 0
\(55\) −10.7251 −1.44617
\(56\) 0 0
\(57\) 6.27492 0.831133
\(58\) 0 0
\(59\) −1.27492 −0.165980 −0.0829900 0.996550i \(-0.526447\pi\)
−0.0829900 + 0.996550i \(0.526447\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 20.5498 2.54889
\(66\) 0 0
\(67\) 0.274917 0.0335865 0.0167932 0.999859i \(-0.494654\pi\)
0.0167932 + 0.999859i \(0.494654\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 4.27492 0.500341 0.250171 0.968202i \(-0.419513\pi\)
0.250171 + 0.968202i \(0.419513\pi\)
\(74\) 0 0
\(75\) −5.72508 −0.661076
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.5498 1.29946 0.649729 0.760166i \(-0.274882\pi\)
0.649729 + 0.760166i \(0.274882\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.27492 0.798526 0.399263 0.916836i \(-0.369266\pi\)
0.399263 + 0.916836i \(0.369266\pi\)
\(84\) 0 0
\(85\) 13.0997 1.42086
\(86\) 0 0
\(87\) 5.27492 0.565530
\(88\) 0 0
\(89\) 10.5498 1.11828 0.559140 0.829073i \(-0.311131\pi\)
0.559140 + 0.829073i \(0.311131\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −20.5498 −2.10837
\(96\) 0 0
\(97\) −8.72508 −0.885898 −0.442949 0.896547i \(-0.646068\pi\)
−0.442949 + 0.896547i \(0.646068\pi\)
\(98\) 0 0
\(99\) −3.27492 −0.329142
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −10.8248 −1.06659 −0.533297 0.845928i \(-0.679047\pi\)
−0.533297 + 0.845928i \(0.679047\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.8248 1.52984 0.764918 0.644127i \(-0.222779\pi\)
0.764918 + 0.644127i \(0.222779\pi\)
\(108\) 0 0
\(109\) −16.8248 −1.61152 −0.805759 0.592243i \(-0.798243\pi\)
−0.805759 + 0.592243i \(0.798243\pi\)
\(110\) 0 0
\(111\) −2.27492 −0.215926
\(112\) 0 0
\(113\) −4.54983 −0.428012 −0.214006 0.976832i \(-0.568651\pi\)
−0.214006 + 0.976832i \(0.568651\pi\)
\(114\) 0 0
\(115\) 13.0997 1.22155
\(116\) 0 0
\(117\) 6.27492 0.580116
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.274917 −0.0249925
\(122\) 0 0
\(123\) −4.54983 −0.410245
\(124\) 0 0
\(125\) 2.37459 0.212389
\(126\) 0 0
\(127\) −6.45017 −0.572360 −0.286180 0.958176i \(-0.592385\pi\)
−0.286180 + 0.958176i \(0.592385\pi\)
\(128\) 0 0
\(129\) 0.274917 0.0242051
\(130\) 0 0
\(131\) 7.27492 0.635612 0.317806 0.948156i \(-0.397054\pi\)
0.317806 + 0.948156i \(0.397054\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.27492 −0.281860
\(136\) 0 0
\(137\) 1.45017 0.123896 0.0619480 0.998079i \(-0.480269\pi\)
0.0619480 + 0.998079i \(0.480269\pi\)
\(138\) 0 0
\(139\) 8.27492 0.701869 0.350935 0.936400i \(-0.385864\pi\)
0.350935 + 0.936400i \(0.385864\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) −20.5498 −1.71846
\(144\) 0 0
\(145\) −17.2749 −1.43460
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.5498 1.19197 0.595984 0.802996i \(-0.296762\pi\)
0.595984 + 0.802996i \(0.296762\pi\)
\(150\) 0 0
\(151\) 22.3746 1.82082 0.910409 0.413709i \(-0.135767\pi\)
0.910409 + 0.413709i \(0.135767\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 3.27492 0.263048
\(156\) 0 0
\(157\) −0.549834 −0.0438816 −0.0219408 0.999759i \(-0.506985\pi\)
−0.0219408 + 0.999759i \(0.506985\pi\)
\(158\) 0 0
\(159\) 9.27492 0.735549
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) 10.7251 0.834947
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 26.3746 2.02881
\(170\) 0 0
\(171\) −6.27492 −0.479855
\(172\) 0 0
\(173\) −22.5498 −1.71443 −0.857216 0.514957i \(-0.827808\pi\)
−0.857216 + 0.514957i \(0.827808\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.27492 0.0958286
\(178\) 0 0
\(179\) 16.5498 1.23699 0.618496 0.785788i \(-0.287742\pi\)
0.618496 + 0.785788i \(0.287742\pi\)
\(180\) 0 0
\(181\) −18.8248 −1.39923 −0.699616 0.714519i \(-0.746646\pi\)
−0.699616 + 0.714519i \(0.746646\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 7.45017 0.547747
\(186\) 0 0
\(187\) −13.0997 −0.957943
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.54983 0.329214 0.164607 0.986359i \(-0.447364\pi\)
0.164607 + 0.986359i \(0.447364\pi\)
\(192\) 0 0
\(193\) 0.450166 0.0324036 0.0162018 0.999869i \(-0.494843\pi\)
0.0162018 + 0.999869i \(0.494843\pi\)
\(194\) 0 0
\(195\) −20.5498 −1.47160
\(196\) 0 0
\(197\) 1.45017 0.103320 0.0516600 0.998665i \(-0.483549\pi\)
0.0516600 + 0.998665i \(0.483549\pi\)
\(198\) 0 0
\(199\) 5.09967 0.361506 0.180753 0.983529i \(-0.442147\pi\)
0.180753 + 0.983529i \(0.442147\pi\)
\(200\) 0 0
\(201\) −0.274917 −0.0193912
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 14.9003 1.04068
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 20.5498 1.42146
\(210\) 0 0
\(211\) 27.6495 1.90347 0.951735 0.306921i \(-0.0992986\pi\)
0.951735 + 0.306921i \(0.0992986\pi\)
\(212\) 0 0
\(213\) −2.00000 −0.137038
\(214\) 0 0
\(215\) −0.900331 −0.0614021
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.27492 −0.288872
\(220\) 0 0
\(221\) 25.0997 1.68839
\(222\) 0 0
\(223\) 1.27492 0.0853748 0.0426874 0.999088i \(-0.486408\pi\)
0.0426874 + 0.999088i \(0.486408\pi\)
\(224\) 0 0
\(225\) 5.72508 0.381672
\(226\) 0 0
\(227\) −11.2749 −0.748343 −0.374171 0.927360i \(-0.622073\pi\)
−0.374171 + 0.927360i \(0.622073\pi\)
\(228\) 0 0
\(229\) 18.2749 1.20764 0.603820 0.797121i \(-0.293644\pi\)
0.603820 + 0.797121i \(0.293644\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.549834 −0.0360209 −0.0180104 0.999838i \(-0.505733\pi\)
−0.0180104 + 0.999838i \(0.505733\pi\)
\(234\) 0 0
\(235\) 19.6495 1.28179
\(236\) 0 0
\(237\) −11.5498 −0.750242
\(238\) 0 0
\(239\) 15.4502 0.999388 0.499694 0.866202i \(-0.333446\pi\)
0.499694 + 0.866202i \(0.333446\pi\)
\(240\) 0 0
\(241\) −9.82475 −0.632868 −0.316434 0.948615i \(-0.602486\pi\)
−0.316434 + 0.948615i \(0.602486\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −39.3746 −2.50534
\(248\) 0 0
\(249\) −7.27492 −0.461029
\(250\) 0 0
\(251\) −18.3746 −1.15979 −0.579897 0.814690i \(-0.696907\pi\)
−0.579897 + 0.814690i \(0.696907\pi\)
\(252\) 0 0
\(253\) −13.0997 −0.823569
\(254\) 0 0
\(255\) −13.0997 −0.820333
\(256\) 0 0
\(257\) −11.0997 −0.692378 −0.346189 0.938165i \(-0.612524\pi\)
−0.346189 + 0.938165i \(0.612524\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −5.27492 −0.326509
\(262\) 0 0
\(263\) −9.45017 −0.582722 −0.291361 0.956613i \(-0.594108\pi\)
−0.291361 + 0.956613i \(0.594108\pi\)
\(264\) 0 0
\(265\) −30.3746 −1.86590
\(266\) 0 0
\(267\) −10.5498 −0.645639
\(268\) 0 0
\(269\) 20.7251 1.26363 0.631815 0.775119i \(-0.282310\pi\)
0.631815 + 0.775119i \(0.282310\pi\)
\(270\) 0 0
\(271\) −1.27492 −0.0774457 −0.0387229 0.999250i \(-0.512329\pi\)
−0.0387229 + 0.999250i \(0.512329\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.7492 −1.13062
\(276\) 0 0
\(277\) 26.8248 1.61174 0.805872 0.592090i \(-0.201697\pi\)
0.805872 + 0.592090i \(0.201697\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) −26.5498 −1.58383 −0.791915 0.610631i \(-0.790916\pi\)
−0.791915 + 0.610631i \(0.790916\pi\)
\(282\) 0 0
\(283\) −25.9244 −1.54105 −0.770523 0.637412i \(-0.780005\pi\)
−0.770523 + 0.637412i \(0.780005\pi\)
\(284\) 0 0
\(285\) 20.5498 1.21727
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 8.72508 0.511473
\(292\) 0 0
\(293\) −27.8248 −1.62554 −0.812770 0.582585i \(-0.802041\pi\)
−0.812770 + 0.582585i \(0.802041\pi\)
\(294\) 0 0
\(295\) −4.17525 −0.243092
\(296\) 0 0
\(297\) 3.27492 0.190030
\(298\) 0 0
\(299\) 25.0997 1.45155
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) 32.7492 1.87521
\(306\) 0 0
\(307\) 11.3746 0.649182 0.324591 0.945854i \(-0.394773\pi\)
0.324591 + 0.945854i \(0.394773\pi\)
\(308\) 0 0
\(309\) 10.8248 0.615799
\(310\) 0 0
\(311\) 4.54983 0.257997 0.128999 0.991645i \(-0.458824\pi\)
0.128999 + 0.991645i \(0.458824\pi\)
\(312\) 0 0
\(313\) 19.5498 1.10502 0.552511 0.833506i \(-0.313670\pi\)
0.552511 + 0.833506i \(0.313670\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.8248 1.56279 0.781397 0.624034i \(-0.214507\pi\)
0.781397 + 0.624034i \(0.214507\pi\)
\(318\) 0 0
\(319\) 17.2749 0.967210
\(320\) 0 0
\(321\) −15.8248 −0.883252
\(322\) 0 0
\(323\) −25.0997 −1.39658
\(324\) 0 0
\(325\) 35.9244 1.99273
\(326\) 0 0
\(327\) 16.8248 0.930411
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.17525 −0.0645975 −0.0322987 0.999478i \(-0.510283\pi\)
−0.0322987 + 0.999478i \(0.510283\pi\)
\(332\) 0 0
\(333\) 2.27492 0.124665
\(334\) 0 0
\(335\) 0.900331 0.0491903
\(336\) 0 0
\(337\) −24.0997 −1.31279 −0.656396 0.754416i \(-0.727920\pi\)
−0.656396 + 0.754416i \(0.727920\pi\)
\(338\) 0 0
\(339\) 4.54983 0.247113
\(340\) 0 0
\(341\) −3.27492 −0.177347
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −13.0997 −0.705262
\(346\) 0 0
\(347\) −30.1993 −1.62119 −0.810593 0.585610i \(-0.800855\pi\)
−0.810593 + 0.585610i \(0.800855\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) −6.27492 −0.334930
\(352\) 0 0
\(353\) 20.5498 1.09376 0.546879 0.837212i \(-0.315816\pi\)
0.546879 + 0.837212i \(0.315816\pi\)
\(354\) 0 0
\(355\) 6.54983 0.347629
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.6495 −1.03706 −0.518531 0.855059i \(-0.673521\pi\)
−0.518531 + 0.855059i \(0.673521\pi\)
\(360\) 0 0
\(361\) 20.3746 1.07235
\(362\) 0 0
\(363\) 0.274917 0.0144294
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −22.0997 −1.15359 −0.576797 0.816888i \(-0.695698\pi\)
−0.576797 + 0.816888i \(0.695698\pi\)
\(368\) 0 0
\(369\) 4.54983 0.236855
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.27492 0.324903 0.162451 0.986717i \(-0.448060\pi\)
0.162451 + 0.986717i \(0.448060\pi\)
\(374\) 0 0
\(375\) −2.37459 −0.122623
\(376\) 0 0
\(377\) −33.0997 −1.70472
\(378\) 0 0
\(379\) −13.1752 −0.676767 −0.338384 0.941008i \(-0.609880\pi\)
−0.338384 + 0.941008i \(0.609880\pi\)
\(380\) 0 0
\(381\) 6.45017 0.330452
\(382\) 0 0
\(383\) −10.5498 −0.539071 −0.269536 0.962990i \(-0.586870\pi\)
−0.269536 + 0.962990i \(0.586870\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.274917 −0.0139748
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) −7.27492 −0.366971
\(394\) 0 0
\(395\) 37.8248 1.90317
\(396\) 0 0
\(397\) −3.37459 −0.169366 −0.0846828 0.996408i \(-0.526988\pi\)
−0.0846828 + 0.996408i \(0.526988\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 6.27492 0.312576
\(404\) 0 0
\(405\) 3.27492 0.162732
\(406\) 0 0
\(407\) −7.45017 −0.369291
\(408\) 0 0
\(409\) −10.4502 −0.516727 −0.258364 0.966048i \(-0.583183\pi\)
−0.258364 + 0.966048i \(0.583183\pi\)
\(410\) 0 0
\(411\) −1.45017 −0.0715314
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 23.8248 1.16951
\(416\) 0 0
\(417\) −8.27492 −0.405224
\(418\) 0 0
\(419\) 28.5498 1.39475 0.697375 0.716706i \(-0.254351\pi\)
0.697375 + 0.716706i \(0.254351\pi\)
\(420\) 0 0
\(421\) −8.82475 −0.430092 −0.215046 0.976604i \(-0.568990\pi\)
−0.215046 + 0.976604i \(0.568990\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 22.9003 1.11083
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 20.5498 0.992155
\(430\) 0 0
\(431\) −17.6495 −0.850147 −0.425073 0.905159i \(-0.639752\pi\)
−0.425073 + 0.905159i \(0.639752\pi\)
\(432\) 0 0
\(433\) −3.17525 −0.152593 −0.0762963 0.997085i \(-0.524310\pi\)
−0.0762963 + 0.997085i \(0.524310\pi\)
\(434\) 0 0
\(435\) 17.2749 0.828269
\(436\) 0 0
\(437\) −25.0997 −1.20068
\(438\) 0 0
\(439\) −17.2749 −0.824487 −0.412243 0.911074i \(-0.635255\pi\)
−0.412243 + 0.911074i \(0.635255\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.37459 0.302866 0.151433 0.988468i \(-0.451611\pi\)
0.151433 + 0.988468i \(0.451611\pi\)
\(444\) 0 0
\(445\) 34.5498 1.63782
\(446\) 0 0
\(447\) −14.5498 −0.688184
\(448\) 0 0
\(449\) 20.5498 0.969807 0.484903 0.874568i \(-0.338855\pi\)
0.484903 + 0.874568i \(0.338855\pi\)
\(450\) 0 0
\(451\) −14.9003 −0.701629
\(452\) 0 0
\(453\) −22.3746 −1.05125
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 36.6495 1.71439 0.857196 0.514991i \(-0.172205\pi\)
0.857196 + 0.514991i \(0.172205\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) −3.64950 −0.169974 −0.0849872 0.996382i \(-0.527085\pi\)
−0.0849872 + 0.996382i \(0.527085\pi\)
\(462\) 0 0
\(463\) 13.1752 0.612306 0.306153 0.951982i \(-0.400958\pi\)
0.306153 + 0.951982i \(0.400958\pi\)
\(464\) 0 0
\(465\) −3.27492 −0.151871
\(466\) 0 0
\(467\) 40.5498 1.87642 0.938211 0.346063i \(-0.112482\pi\)
0.938211 + 0.346063i \(0.112482\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.549834 0.0253350
\(472\) 0 0
\(473\) 0.900331 0.0413973
\(474\) 0 0
\(475\) −35.9244 −1.64833
\(476\) 0 0
\(477\) −9.27492 −0.424669
\(478\) 0 0
\(479\) 5.45017 0.249024 0.124512 0.992218i \(-0.460263\pi\)
0.124512 + 0.992218i \(0.460263\pi\)
\(480\) 0 0
\(481\) 14.2749 0.650880
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −28.5739 −1.29748
\(486\) 0 0
\(487\) 1.00000 0.0453143 0.0226572 0.999743i \(-0.492787\pi\)
0.0226572 + 0.999743i \(0.492787\pi\)
\(488\) 0 0
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −36.9244 −1.66638 −0.833188 0.552990i \(-0.813487\pi\)
−0.833188 + 0.552990i \(0.813487\pi\)
\(492\) 0 0
\(493\) −21.0997 −0.950281
\(494\) 0 0
\(495\) −10.7251 −0.482057
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −32.2749 −1.44482 −0.722412 0.691463i \(-0.756966\pi\)
−0.722412 + 0.691463i \(0.756966\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) 37.6495 1.67871 0.839354 0.543585i \(-0.182933\pi\)
0.839354 + 0.543585i \(0.182933\pi\)
\(504\) 0 0
\(505\) −19.6495 −0.874391
\(506\) 0 0
\(507\) −26.3746 −1.17134
\(508\) 0 0
\(509\) −11.2749 −0.499752 −0.249876 0.968278i \(-0.580390\pi\)
−0.249876 + 0.968278i \(0.580390\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.27492 0.277044
\(514\) 0 0
\(515\) −35.4502 −1.56212
\(516\) 0 0
\(517\) −19.6495 −0.864184
\(518\) 0 0
\(519\) 22.5498 0.989828
\(520\) 0 0
\(521\) 14.5498 0.637440 0.318720 0.947849i \(-0.396747\pi\)
0.318720 + 0.947849i \(0.396747\pi\)
\(522\) 0 0
\(523\) −17.7251 −0.775064 −0.387532 0.921856i \(-0.626672\pi\)
−0.387532 + 0.921856i \(0.626672\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −1.27492 −0.0553267
\(532\) 0 0
\(533\) 28.5498 1.23663
\(534\) 0 0
\(535\) 51.8248 2.24058
\(536\) 0 0
\(537\) −16.5498 −0.714178
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.27492 0.355766 0.177883 0.984052i \(-0.443075\pi\)
0.177883 + 0.984052i \(0.443075\pi\)
\(542\) 0 0
\(543\) 18.8248 0.807847
\(544\) 0 0
\(545\) −55.0997 −2.36021
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 33.0997 1.41009
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −7.45017 −0.316242
\(556\) 0 0
\(557\) 29.8248 1.26372 0.631858 0.775084i \(-0.282293\pi\)
0.631858 + 0.775084i \(0.282293\pi\)
\(558\) 0 0
\(559\) −1.72508 −0.0729632
\(560\) 0 0
\(561\) 13.0997 0.553068
\(562\) 0 0
\(563\) 15.2749 0.643761 0.321881 0.946780i \(-0.395685\pi\)
0.321881 + 0.946780i \(0.395685\pi\)
\(564\) 0 0
\(565\) −14.9003 −0.626862
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.4502 0.480016 0.240008 0.970771i \(-0.422850\pi\)
0.240008 + 0.970771i \(0.422850\pi\)
\(570\) 0 0
\(571\) −8.27492 −0.346295 −0.173147 0.984896i \(-0.555394\pi\)
−0.173147 + 0.984896i \(0.555394\pi\)
\(572\) 0 0
\(573\) −4.54983 −0.190072
\(574\) 0 0
\(575\) 22.9003 0.955010
\(576\) 0 0
\(577\) 25.0000 1.04076 0.520382 0.853934i \(-0.325790\pi\)
0.520382 + 0.853934i \(0.325790\pi\)
\(578\) 0 0
\(579\) −0.450166 −0.0187082
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 30.3746 1.25799
\(584\) 0 0
\(585\) 20.5498 0.849631
\(586\) 0 0
\(587\) 9.27492 0.382817 0.191408 0.981510i \(-0.438695\pi\)
0.191408 + 0.981510i \(0.438695\pi\)
\(588\) 0 0
\(589\) −6.27492 −0.258553
\(590\) 0 0
\(591\) −1.45017 −0.0596518
\(592\) 0 0
\(593\) 0.549834 0.0225790 0.0112895 0.999936i \(-0.496406\pi\)
0.0112895 + 0.999936i \(0.496406\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −5.09967 −0.208716
\(598\) 0 0
\(599\) 22.5498 0.921361 0.460681 0.887566i \(-0.347605\pi\)
0.460681 + 0.887566i \(0.347605\pi\)
\(600\) 0 0
\(601\) −4.09967 −0.167229 −0.0836145 0.996498i \(-0.526646\pi\)
−0.0836145 + 0.996498i \(0.526646\pi\)
\(602\) 0 0
\(603\) 0.274917 0.0111955
\(604\) 0 0
\(605\) −0.900331 −0.0366037
\(606\) 0 0
\(607\) 7.00000 0.284121 0.142061 0.989858i \(-0.454627\pi\)
0.142061 + 0.989858i \(0.454627\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 37.6495 1.52314
\(612\) 0 0
\(613\) 8.54983 0.345325 0.172662 0.984981i \(-0.444763\pi\)
0.172662 + 0.984981i \(0.444763\pi\)
\(614\) 0 0
\(615\) −14.9003 −0.600839
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 28.8248 1.15856 0.579282 0.815127i \(-0.303333\pi\)
0.579282 + 0.815127i \(0.303333\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −20.8488 −0.833954
\(626\) 0 0
\(627\) −20.5498 −0.820681
\(628\) 0 0
\(629\) 9.09967 0.362828
\(630\) 0 0
\(631\) 19.8248 0.789211 0.394605 0.918851i \(-0.370881\pi\)
0.394605 + 0.918851i \(0.370881\pi\)
\(632\) 0 0
\(633\) −27.6495 −1.09897
\(634\) 0 0
\(635\) −21.1238 −0.838271
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) −3.64950 −0.144147 −0.0720734 0.997399i \(-0.522962\pi\)
−0.0720734 + 0.997399i \(0.522962\pi\)
\(642\) 0 0
\(643\) 5.37459 0.211953 0.105976 0.994369i \(-0.466203\pi\)
0.105976 + 0.994369i \(0.466203\pi\)
\(644\) 0 0
\(645\) 0.900331 0.0354505
\(646\) 0 0
\(647\) 34.0000 1.33668 0.668339 0.743857i \(-0.267006\pi\)
0.668339 + 0.743857i \(0.267006\pi\)
\(648\) 0 0
\(649\) 4.17525 0.163893
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.92442 0.270974 0.135487 0.990779i \(-0.456740\pi\)
0.135487 + 0.990779i \(0.456740\pi\)
\(654\) 0 0
\(655\) 23.8248 0.930910
\(656\) 0 0
\(657\) 4.27492 0.166780
\(658\) 0 0
\(659\) 42.1993 1.64385 0.821926 0.569594i \(-0.192899\pi\)
0.821926 + 0.569594i \(0.192899\pi\)
\(660\) 0 0
\(661\) 7.17525 0.279085 0.139542 0.990216i \(-0.455437\pi\)
0.139542 + 0.990216i \(0.455437\pi\)
\(662\) 0 0
\(663\) −25.0997 −0.974790
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −21.0997 −0.816982
\(668\) 0 0
\(669\) −1.27492 −0.0492911
\(670\) 0 0
\(671\) −32.7492 −1.26427
\(672\) 0 0
\(673\) 41.5498 1.60163 0.800814 0.598913i \(-0.204400\pi\)
0.800814 + 0.598913i \(0.204400\pi\)
\(674\) 0 0
\(675\) −5.72508 −0.220359
\(676\) 0 0
\(677\) −13.2749 −0.510197 −0.255098 0.966915i \(-0.582108\pi\)
−0.255098 + 0.966915i \(0.582108\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 11.2749 0.432056
\(682\) 0 0
\(683\) −12.1752 −0.465873 −0.232936 0.972492i \(-0.574833\pi\)
−0.232936 + 0.972492i \(0.574833\pi\)
\(684\) 0 0
\(685\) 4.74917 0.181457
\(686\) 0 0
\(687\) −18.2749 −0.697232
\(688\) 0 0
\(689\) −58.1993 −2.21722
\(690\) 0 0
\(691\) −10.8248 −0.411793 −0.205896 0.978574i \(-0.566011\pi\)
−0.205896 + 0.978574i \(0.566011\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 27.0997 1.02795
\(696\) 0 0
\(697\) 18.1993 0.689349
\(698\) 0 0
\(699\) 0.549834 0.0207966
\(700\) 0 0
\(701\) 12.9244 0.488149 0.244074 0.969757i \(-0.421516\pi\)
0.244074 + 0.969757i \(0.421516\pi\)
\(702\) 0 0
\(703\) −14.2749 −0.538389
\(704\) 0 0
\(705\) −19.6495 −0.740043
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 28.1993 1.05905 0.529524 0.848295i \(-0.322370\pi\)
0.529524 + 0.848295i \(0.322370\pi\)
\(710\) 0 0
\(711\) 11.5498 0.433153
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) −67.2990 −2.51684
\(716\) 0 0
\(717\) −15.4502 −0.576997
\(718\) 0 0
\(719\) −28.1993 −1.05166 −0.525829 0.850590i \(-0.676245\pi\)
−0.525829 + 0.850590i \(0.676245\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 9.82475 0.365386
\(724\) 0 0
\(725\) −30.1993 −1.12158
\(726\) 0 0
\(727\) −31.5498 −1.17012 −0.585059 0.810991i \(-0.698929\pi\)
−0.585059 + 0.810991i \(0.698929\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −1.09967 −0.0406727
\(732\) 0 0
\(733\) 5.92442 0.218823 0.109412 0.993997i \(-0.465103\pi\)
0.109412 + 0.993997i \(0.465103\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.900331 −0.0331641
\(738\) 0 0
\(739\) −1.37459 −0.0505650 −0.0252825 0.999680i \(-0.508049\pi\)
−0.0252825 + 0.999680i \(0.508049\pi\)
\(740\) 0 0
\(741\) 39.3746 1.44646
\(742\) 0 0
\(743\) −44.1993 −1.62152 −0.810758 0.585381i \(-0.800945\pi\)
−0.810758 + 0.585381i \(0.800945\pi\)
\(744\) 0 0
\(745\) 47.6495 1.74574
\(746\) 0 0
\(747\) 7.27492 0.266175
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.4502 −0.381332 −0.190666 0.981655i \(-0.561065\pi\)
−0.190666 + 0.981655i \(0.561065\pi\)
\(752\) 0 0
\(753\) 18.3746 0.669607
\(754\) 0 0
\(755\) 73.2749 2.66675
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 13.0997 0.475488
\(760\) 0 0
\(761\) 25.0997 0.909862 0.454931 0.890527i \(-0.349664\pi\)
0.454931 + 0.890527i \(0.349664\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 13.0997 0.473620
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 32.6495 1.17737 0.588686 0.808362i \(-0.299646\pi\)
0.588686 + 0.808362i \(0.299646\pi\)
\(770\) 0 0
\(771\) 11.0997 0.399745
\(772\) 0 0
\(773\) 3.09967 0.111487 0.0557437 0.998445i \(-0.482247\pi\)
0.0557437 + 0.998445i \(0.482247\pi\)
\(774\) 0 0
\(775\) 5.72508 0.205651
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −28.5498 −1.02290
\(780\) 0 0
\(781\) −6.54983 −0.234372
\(782\) 0 0
\(783\) 5.27492 0.188510
\(784\) 0 0
\(785\) −1.80066 −0.0642684
\(786\) 0 0
\(787\) 2.54983 0.0908918 0.0454459 0.998967i \(-0.485529\pi\)
0.0454459 + 0.998967i \(0.485529\pi\)
\(788\) 0 0
\(789\) 9.45017 0.336435
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 62.7492 2.22829
\(794\) 0 0
\(795\) 30.3746 1.07728
\(796\) 0 0
\(797\) −31.4743 −1.11488 −0.557438 0.830219i \(-0.688215\pi\)
−0.557438 + 0.830219i \(0.688215\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 10.5498 0.372760
\(802\) 0 0
\(803\) −14.0000 −0.494049
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −20.7251 −0.729557
\(808\) 0 0
\(809\) −29.6495 −1.04242 −0.521211 0.853428i \(-0.674519\pi\)
−0.521211 + 0.853428i \(0.674519\pi\)
\(810\) 0 0
\(811\) −42.5498 −1.49413 −0.747063 0.664753i \(-0.768537\pi\)
−0.747063 + 0.664753i \(0.768537\pi\)
\(812\) 0 0
\(813\) 1.27492 0.0447133
\(814\) 0 0
\(815\) 39.2990 1.37658
\(816\) 0 0
\(817\) 1.72508 0.0603530
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −12.7251 −0.444108 −0.222054 0.975034i \(-0.571276\pi\)
−0.222054 + 0.975034i \(0.571276\pi\)
\(822\) 0 0
\(823\) −34.1993 −1.19211 −0.596057 0.802942i \(-0.703267\pi\)
−0.596057 + 0.802942i \(0.703267\pi\)
\(824\) 0 0
\(825\) 18.7492 0.652762
\(826\) 0 0
\(827\) 44.0241 1.53087 0.765434 0.643515i \(-0.222524\pi\)
0.765434 + 0.643515i \(0.222524\pi\)
\(828\) 0 0
\(829\) 30.2749 1.05149 0.525746 0.850642i \(-0.323786\pi\)
0.525746 + 0.850642i \(0.323786\pi\)
\(830\) 0 0
\(831\) −26.8248 −0.930540
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 19.6495 0.679999
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) 30.1993 1.04260 0.521298 0.853374i \(-0.325448\pi\)
0.521298 + 0.853374i \(0.325448\pi\)
\(840\) 0 0
\(841\) −1.17525 −0.0405258
\(842\) 0 0
\(843\) 26.5498 0.914425
\(844\) 0 0
\(845\) 86.3746 2.97138
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 25.9244 0.889724
\(850\) 0 0
\(851\) 9.09967 0.311933
\(852\) 0 0
\(853\) 13.3746 0.457937 0.228969 0.973434i \(-0.426465\pi\)
0.228969 + 0.973434i \(0.426465\pi\)
\(854\) 0 0
\(855\) −20.5498 −0.702790
\(856\) 0 0
\(857\) 13.4502 0.459449 0.229724 0.973256i \(-0.426218\pi\)
0.229724 + 0.973256i \(0.426218\pi\)
\(858\) 0 0
\(859\) −55.6495 −1.89874 −0.949368 0.314165i \(-0.898275\pi\)
−0.949368 + 0.314165i \(0.898275\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 19.0997 0.650160 0.325080 0.945686i \(-0.394609\pi\)
0.325080 + 0.945686i \(0.394609\pi\)
\(864\) 0 0
\(865\) −73.8488 −2.51094
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −37.8248 −1.28312
\(870\) 0 0
\(871\) 1.72508 0.0584522
\(872\) 0 0
\(873\) −8.72508 −0.295299
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −46.7492 −1.57861 −0.789304 0.614003i \(-0.789558\pi\)
−0.789304 + 0.614003i \(0.789558\pi\)
\(878\) 0 0
\(879\) 27.8248 0.938506
\(880\) 0 0
\(881\) −9.45017 −0.318384 −0.159192 0.987248i \(-0.550889\pi\)
−0.159192 + 0.987248i \(0.550889\pi\)
\(882\) 0 0
\(883\) −7.37459 −0.248175 −0.124087 0.992271i \(-0.539600\pi\)
−0.124087 + 0.992271i \(0.539600\pi\)
\(884\) 0 0
\(885\) 4.17525 0.140349
\(886\) 0 0
\(887\) −33.6495 −1.12984 −0.564920 0.825146i \(-0.691093\pi\)
−0.564920 + 0.825146i \(0.691093\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3.27492 −0.109714
\(892\) 0 0
\(893\) −37.6495 −1.25989
\(894\) 0 0
\(895\) 54.1993 1.81168
\(896\) 0 0
\(897\) −25.0997 −0.838054
\(898\) 0 0
\(899\) −5.27492 −0.175928
\(900\) 0 0
\(901\) −37.0997 −1.23597
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −61.6495 −2.04930
\(906\) 0 0
\(907\) 29.7251 0.987005 0.493503 0.869744i \(-0.335716\pi\)
0.493503 + 0.869744i \(0.335716\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 53.8488 1.78409 0.892046 0.451945i \(-0.149270\pi\)
0.892046 + 0.451945i \(0.149270\pi\)
\(912\) 0 0
\(913\) −23.8248 −0.788484
\(914\) 0 0
\(915\) −32.7492 −1.08265
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −2.82475 −0.0931800 −0.0465900 0.998914i \(-0.514835\pi\)
−0.0465900 + 0.998914i \(0.514835\pi\)
\(920\) 0 0
\(921\) −11.3746 −0.374805
\(922\) 0 0
\(923\) 12.5498 0.413083
\(924\) 0 0
\(925\) 13.0241 0.428229
\(926\) 0 0
\(927\) −10.8248 −0.355531
\(928\) 0 0
\(929\) 40.1993 1.31890 0.659449 0.751750i \(-0.270790\pi\)
0.659449 + 0.751750i \(0.270790\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −4.54983 −0.148955
\(934\) 0 0
\(935\) −42.9003 −1.40299
\(936\) 0 0
\(937\) 6.09967 0.199267 0.0996337 0.995024i \(-0.468233\pi\)
0.0996337 + 0.995024i \(0.468233\pi\)
\(938\) 0 0
\(939\) −19.5498 −0.637985
\(940\) 0 0
\(941\) 51.8248 1.68944 0.844719 0.535210i \(-0.179767\pi\)
0.844719 + 0.535210i \(0.179767\pi\)
\(942\) 0 0
\(943\) 18.1993 0.592652
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 19.4502 0.632045 0.316023 0.948752i \(-0.397652\pi\)
0.316023 + 0.948752i \(0.397652\pi\)
\(948\) 0 0
\(949\) 26.8248 0.870768
\(950\) 0 0
\(951\) −27.8248 −0.902279
\(952\) 0 0
\(953\) 55.6495 1.80266 0.901332 0.433129i \(-0.142590\pi\)
0.901332 + 0.433129i \(0.142590\pi\)
\(954\) 0 0
\(955\) 14.9003 0.482163
\(956\) 0 0
\(957\) −17.2749 −0.558419
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 15.8248 0.509945
\(964\) 0 0
\(965\) 1.47425 0.0474579
\(966\) 0 0
\(967\) −53.5498 −1.72205 −0.861023 0.508566i \(-0.830176\pi\)
−0.861023 + 0.508566i \(0.830176\pi\)
\(968\) 0 0
\(969\) 25.0997 0.806318
\(970\) 0 0
\(971\) 56.5739 1.81554 0.907772 0.419464i \(-0.137782\pi\)
0.907772 + 0.419464i \(0.137782\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −35.9244 −1.15050
\(976\) 0 0
\(977\) 8.90033 0.284747 0.142373 0.989813i \(-0.454527\pi\)
0.142373 + 0.989813i \(0.454527\pi\)
\(978\) 0 0
\(979\) −34.5498 −1.10422
\(980\) 0 0
\(981\) −16.8248 −0.537173
\(982\) 0 0
\(983\) −39.2990 −1.25344 −0.626722 0.779243i \(-0.715604\pi\)
−0.626722 + 0.779243i \(0.715604\pi\)
\(984\) 0 0
\(985\) 4.74917 0.151321
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.09967 −0.0349674
\(990\) 0 0
\(991\) −14.0997 −0.447891 −0.223945 0.974602i \(-0.571894\pi\)
−0.223945 + 0.974602i \(0.571894\pi\)
\(992\) 0 0
\(993\) 1.17525 0.0372954
\(994\) 0 0
\(995\) 16.7010 0.529457
\(996\) 0 0
\(997\) −44.8248 −1.41961 −0.709807 0.704396i \(-0.751218\pi\)
−0.709807 + 0.704396i \(0.751218\pi\)
\(998\) 0 0
\(999\) −2.27492 −0.0719752
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.dj.1.2 2
4.3 odd 2 9408.2.a.dw.1.2 2
7.3 odd 6 1344.2.q.w.961.2 4
7.5 odd 6 1344.2.q.w.193.2 4
7.6 odd 2 9408.2.a.ec.1.1 2
8.3 odd 2 2352.2.a.ba.1.1 2
8.5 even 2 1176.2.a.n.1.1 2
24.5 odd 2 3528.2.a.bd.1.2 2
24.11 even 2 7056.2.a.ch.1.2 2
28.3 even 6 1344.2.q.x.961.2 4
28.19 even 6 1344.2.q.x.193.2 4
28.27 even 2 9408.2.a.dp.1.1 2
56.3 even 6 336.2.q.g.289.1 4
56.5 odd 6 168.2.q.c.25.1 4
56.11 odd 6 2352.2.q.bf.961.2 4
56.13 odd 2 1176.2.a.k.1.2 2
56.19 even 6 336.2.q.g.193.1 4
56.27 even 2 2352.2.a.bf.1.2 2
56.37 even 6 1176.2.q.l.361.2 4
56.45 odd 6 168.2.q.c.121.1 yes 4
56.51 odd 6 2352.2.q.bf.1537.2 4
56.53 even 6 1176.2.q.l.961.2 4
168.5 even 6 504.2.s.i.361.2 4
168.53 odd 6 3528.2.s.bk.3313.1 4
168.59 odd 6 1008.2.s.r.289.2 4
168.83 odd 2 7056.2.a.cu.1.1 2
168.101 even 6 504.2.s.i.289.2 4
168.125 even 2 3528.2.a.bk.1.1 2
168.131 odd 6 1008.2.s.r.865.2 4
168.149 odd 6 3528.2.s.bk.361.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.c.25.1 4 56.5 odd 6
168.2.q.c.121.1 yes 4 56.45 odd 6
336.2.q.g.193.1 4 56.19 even 6
336.2.q.g.289.1 4 56.3 even 6
504.2.s.i.289.2 4 168.101 even 6
504.2.s.i.361.2 4 168.5 even 6
1008.2.s.r.289.2 4 168.59 odd 6
1008.2.s.r.865.2 4 168.131 odd 6
1176.2.a.k.1.2 2 56.13 odd 2
1176.2.a.n.1.1 2 8.5 even 2
1176.2.q.l.361.2 4 56.37 even 6
1176.2.q.l.961.2 4 56.53 even 6
1344.2.q.w.193.2 4 7.5 odd 6
1344.2.q.w.961.2 4 7.3 odd 6
1344.2.q.x.193.2 4 28.19 even 6
1344.2.q.x.961.2 4 28.3 even 6
2352.2.a.ba.1.1 2 8.3 odd 2
2352.2.a.bf.1.2 2 56.27 even 2
2352.2.q.bf.961.2 4 56.11 odd 6
2352.2.q.bf.1537.2 4 56.51 odd 6
3528.2.a.bd.1.2 2 24.5 odd 2
3528.2.a.bk.1.1 2 168.125 even 2
3528.2.s.bk.361.1 4 168.149 odd 6
3528.2.s.bk.3313.1 4 168.53 odd 6
7056.2.a.ch.1.2 2 24.11 even 2
7056.2.a.cu.1.1 2 168.83 odd 2
9408.2.a.dj.1.2 2 1.1 even 1 trivial
9408.2.a.dp.1.1 2 28.27 even 2
9408.2.a.dw.1.2 2 4.3 odd 2
9408.2.a.ec.1.1 2 7.6 odd 2