Properties

Label 9408.2.a.ei.1.1
Level $9408$
Weight $2$
Character 9408.1
Self dual yes
Analytic conductor $75.123$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.621.1
Defining polynomial: \(x^{3} - 6 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.523976\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.20147 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.20147 q^{5} +1.00000 q^{9} -4.24943 q^{11} -3.15352 q^{13} -3.20147 q^{15} +4.40294 q^{17} +3.15352 q^{19} +4.40294 q^{23} +5.24943 q^{25} +1.00000 q^{27} -7.20147 q^{29} -2.04795 q^{31} -4.24943 q^{33} +9.65237 q^{37} -3.15352 q^{39} +10.4989 q^{41} -0.750575 q^{43} -3.20147 q^{45} +2.40294 q^{47} +4.40294 q^{51} +3.29738 q^{53} +13.6044 q^{55} +3.15352 q^{57} +8.24943 q^{59} -8.09591 q^{61} +10.0959 q^{65} -5.15352 q^{67} +4.40294 q^{69} +6.40294 q^{71} -15.2494 q^{73} +5.24943 q^{75} -16.4509 q^{79} +1.00000 q^{81} +14.5565 q^{83} -14.0959 q^{85} -7.20147 q^{87} -2.40294 q^{89} -2.04795 q^{93} -10.0959 q^{95} -4.24943 q^{97} -4.24943 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} + 3q^{9} - 3q^{13} - 6q^{17} + 3q^{19} - 6q^{23} + 3q^{25} + 3q^{27} - 12q^{29} - 3q^{31} - 3q^{37} - 3q^{39} + 6q^{41} - 15q^{43} - 12q^{47} - 6q^{51} - 6q^{53} + 12q^{55} + 3q^{57} + 12q^{59} - 18q^{61} + 24q^{65} - 9q^{67} - 6q^{69} - 33q^{73} + 3q^{75} - 27q^{79} + 3q^{81} + 18q^{83} - 36q^{85} - 12q^{87} + 12q^{89} - 3q^{93} - 24q^{95} + O(q^{100}) \)

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
\(5\) −3.20147 −1.43174 −0.715871 0.698233i \(-0.753970\pi\)
−0.715871 + 0.698233i \(0.753970\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.24943 −1.28125 −0.640625 0.767854i \(-0.721325\pi\)
−0.640625 + 0.767854i \(0.721325\pi\)
\(12\) 0 0
\(13\) −3.15352 −0.874629 −0.437314 0.899309i \(-0.644070\pi\)
−0.437314 + 0.899309i \(0.644070\pi\)
\(14\) 0 0
\(15\) −3.20147 −0.826617
\(16\) 0 0
\(17\) 4.40294 1.06787 0.533935 0.845525i \(-0.320713\pi\)
0.533935 + 0.845525i \(0.320713\pi\)
\(18\) 0 0
\(19\) 3.15352 0.723467 0.361734 0.932282i \(-0.382185\pi\)
0.361734 + 0.932282i \(0.382185\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.40294 0.918077 0.459039 0.888416i \(-0.348194\pi\)
0.459039 + 0.888416i \(0.348194\pi\)
\(24\) 0 0
\(25\) 5.24943 1.04989
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.20147 −1.33728 −0.668640 0.743586i \(-0.733123\pi\)
−0.668640 + 0.743586i \(0.733123\pi\)
\(30\) 0 0
\(31\) −2.04795 −0.367823 −0.183912 0.982943i \(-0.558876\pi\)
−0.183912 + 0.982943i \(0.558876\pi\)
\(32\) 0 0
\(33\) −4.24943 −0.739730
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.65237 1.58684 0.793420 0.608675i \(-0.208299\pi\)
0.793420 + 0.608675i \(0.208299\pi\)
\(38\) 0 0
\(39\) −3.15352 −0.504967
\(40\) 0 0
\(41\) 10.4989 1.63964 0.819822 0.572618i \(-0.194072\pi\)
0.819822 + 0.572618i \(0.194072\pi\)
\(42\) 0 0
\(43\) −0.750575 −0.114462 −0.0572308 0.998361i \(-0.518227\pi\)
−0.0572308 + 0.998361i \(0.518227\pi\)
\(44\) 0 0
\(45\) −3.20147 −0.477247
\(46\) 0 0
\(47\) 2.40294 0.350506 0.175253 0.984523i \(-0.443926\pi\)
0.175253 + 0.984523i \(0.443926\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4.40294 0.616536
\(52\) 0 0
\(53\) 3.29738 0.452930 0.226465 0.974019i \(-0.427283\pi\)
0.226465 + 0.974019i \(0.427283\pi\)
\(54\) 0 0
\(55\) 13.6044 1.83442
\(56\) 0 0
\(57\) 3.15352 0.417694
\(58\) 0 0
\(59\) 8.24943 1.07398 0.536992 0.843587i \(-0.319561\pi\)
0.536992 + 0.843587i \(0.319561\pi\)
\(60\) 0 0
\(61\) −8.09591 −1.03657 −0.518287 0.855207i \(-0.673430\pi\)
−0.518287 + 0.855207i \(0.673430\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.0959 1.25224
\(66\) 0 0
\(67\) −5.15352 −0.629603 −0.314801 0.949158i \(-0.601938\pi\)
−0.314801 + 0.949158i \(0.601938\pi\)
\(68\) 0 0
\(69\) 4.40294 0.530052
\(70\) 0 0
\(71\) 6.40294 0.759890 0.379945 0.925009i \(-0.375943\pi\)
0.379945 + 0.925009i \(0.375943\pi\)
\(72\) 0 0
\(73\) −15.2494 −1.78481 −0.892405 0.451235i \(-0.850984\pi\)
−0.892405 + 0.451235i \(0.850984\pi\)
\(74\) 0 0
\(75\) 5.24943 0.606151
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −16.4509 −1.85087 −0.925435 0.378906i \(-0.876300\pi\)
−0.925435 + 0.378906i \(0.876300\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.5565 1.59778 0.798890 0.601477i \(-0.205421\pi\)
0.798890 + 0.601477i \(0.205421\pi\)
\(84\) 0 0
\(85\) −14.0959 −1.52892
\(86\) 0 0
\(87\) −7.20147 −0.772079
\(88\) 0 0
\(89\) −2.40294 −0.254712 −0.127356 0.991857i \(-0.540649\pi\)
−0.127356 + 0.991857i \(0.540649\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.04795 −0.212363
\(94\) 0 0
\(95\) −10.0959 −1.03582
\(96\) 0 0
\(97\) −4.24943 −0.431464 −0.215732 0.976453i \(-0.569214\pi\)
−0.215732 + 0.976453i \(0.569214\pi\)
\(98\) 0 0
\(99\) −4.24943 −0.427083
\(100\) 0 0
\(101\) 3.90409 0.388472 0.194236 0.980955i \(-0.437777\pi\)
0.194236 + 0.980955i \(0.437777\pi\)
\(102\) 0 0
\(103\) −9.24943 −0.911373 −0.455686 0.890140i \(-0.650606\pi\)
−0.455686 + 0.890140i \(0.650606\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.5565 −1.60057 −0.800287 0.599617i \(-0.795319\pi\)
−0.800287 + 0.599617i \(0.795319\pi\)
\(108\) 0 0
\(109\) −20.0553 −1.92095 −0.960475 0.278365i \(-0.910208\pi\)
−0.960475 + 0.278365i \(0.910208\pi\)
\(110\) 0 0
\(111\) 9.65237 0.916162
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) −14.0959 −1.31445
\(116\) 0 0
\(117\) −3.15352 −0.291543
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.05761 0.641601
\(122\) 0 0
\(123\) 10.4989 0.946649
\(124\) 0 0
\(125\) −0.798528 −0.0714225
\(126\) 0 0
\(127\) 12.4509 1.10484 0.552419 0.833566i \(-0.313705\pi\)
0.552419 + 0.833566i \(0.313705\pi\)
\(128\) 0 0
\(129\) −0.750575 −0.0660844
\(130\) 0 0
\(131\) 4.24943 0.371274 0.185637 0.982618i \(-0.440565\pi\)
0.185637 + 0.982618i \(0.440565\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −3.20147 −0.275539
\(136\) 0 0
\(137\) −10.4029 −0.888784 −0.444392 0.895833i \(-0.646580\pi\)
−0.444392 + 0.895833i \(0.646580\pi\)
\(138\) 0 0
\(139\) 11.6524 0.988341 0.494171 0.869365i \(-0.335472\pi\)
0.494171 + 0.869365i \(0.335472\pi\)
\(140\) 0 0
\(141\) 2.40294 0.202364
\(142\) 0 0
\(143\) 13.4006 1.12062
\(144\) 0 0
\(145\) 23.0553 1.91464
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.4029 −0.852242 −0.426121 0.904666i \(-0.640120\pi\)
−0.426121 + 0.904666i \(0.640120\pi\)
\(150\) 0 0
\(151\) −20.0074 −1.62818 −0.814088 0.580742i \(-0.802763\pi\)
−0.814088 + 0.580742i \(0.802763\pi\)
\(152\) 0 0
\(153\) 4.40294 0.355957
\(154\) 0 0
\(155\) 6.55646 0.526628
\(156\) 0 0
\(157\) 14.4989 1.15713 0.578567 0.815635i \(-0.303612\pi\)
0.578567 + 0.815635i \(0.303612\pi\)
\(158\) 0 0
\(159\) 3.29738 0.261499
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.7100 0.838871 0.419435 0.907785i \(-0.362228\pi\)
0.419435 + 0.907785i \(0.362228\pi\)
\(164\) 0 0
\(165\) 13.6044 1.05910
\(166\) 0 0
\(167\) −17.3047 −1.33908 −0.669540 0.742776i \(-0.733509\pi\)
−0.669540 + 0.742776i \(0.733509\pi\)
\(168\) 0 0
\(169\) −3.05531 −0.235024
\(170\) 0 0
\(171\) 3.15352 0.241156
\(172\) 0 0
\(173\) −3.90409 −0.296823 −0.148411 0.988926i \(-0.547416\pi\)
−0.148411 + 0.988926i \(0.547416\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.24943 0.620065
\(178\) 0 0
\(179\) −20.8059 −1.55511 −0.777553 0.628818i \(-0.783539\pi\)
−0.777553 + 0.628818i \(0.783539\pi\)
\(180\) 0 0
\(181\) −13.2494 −0.984822 −0.492411 0.870363i \(-0.663884\pi\)
−0.492411 + 0.870363i \(0.663884\pi\)
\(182\) 0 0
\(183\) −8.09591 −0.598467
\(184\) 0 0
\(185\) −30.9018 −2.27195
\(186\) 0 0
\(187\) −18.7100 −1.36821
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.19181 0.303309 0.151654 0.988434i \(-0.451540\pi\)
0.151654 + 0.988434i \(0.451540\pi\)
\(192\) 0 0
\(193\) −18.3047 −1.31760 −0.658802 0.752316i \(-0.728936\pi\)
−0.658802 + 0.752316i \(0.728936\pi\)
\(194\) 0 0
\(195\) 10.0959 0.722983
\(196\) 0 0
\(197\) −11.9041 −0.848132 −0.424066 0.905631i \(-0.639397\pi\)
−0.424066 + 0.905631i \(0.639397\pi\)
\(198\) 0 0
\(199\) −0.805889 −0.0571280 −0.0285640 0.999592i \(-0.509093\pi\)
−0.0285640 + 0.999592i \(0.509093\pi\)
\(200\) 0 0
\(201\) −5.15352 −0.363501
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −33.6118 −2.34755
\(206\) 0 0
\(207\) 4.40294 0.306026
\(208\) 0 0
\(209\) −13.4006 −0.926942
\(210\) 0 0
\(211\) −8.49885 −0.585085 −0.292542 0.956253i \(-0.594501\pi\)
−0.292542 + 0.956253i \(0.594501\pi\)
\(212\) 0 0
\(213\) 6.40294 0.438723
\(214\) 0 0
\(215\) 2.40294 0.163879
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −15.2494 −1.03046
\(220\) 0 0
\(221\) −13.8848 −0.933991
\(222\) 0 0
\(223\) 16.7985 1.12491 0.562456 0.826827i \(-0.309856\pi\)
0.562456 + 0.826827i \(0.309856\pi\)
\(224\) 0 0
\(225\) 5.24943 0.349962
\(226\) 0 0
\(227\) −16.8635 −1.11927 −0.559635 0.828739i \(-0.689059\pi\)
−0.559635 + 0.828739i \(0.689059\pi\)
\(228\) 0 0
\(229\) 20.5542 1.35826 0.679129 0.734019i \(-0.262358\pi\)
0.679129 + 0.734019i \(0.262358\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.90409 0.648839 0.324419 0.945913i \(-0.394831\pi\)
0.324419 + 0.945913i \(0.394831\pi\)
\(234\) 0 0
\(235\) −7.69296 −0.501833
\(236\) 0 0
\(237\) −16.4509 −1.06860
\(238\) 0 0
\(239\) 4.40294 0.284803 0.142401 0.989809i \(-0.454518\pi\)
0.142401 + 0.989809i \(0.454518\pi\)
\(240\) 0 0
\(241\) −20.8635 −1.34394 −0.671968 0.740580i \(-0.734551\pi\)
−0.671968 + 0.740580i \(0.734551\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −9.94469 −0.632765
\(248\) 0 0
\(249\) 14.5565 0.922478
\(250\) 0 0
\(251\) 3.44354 0.217354 0.108677 0.994077i \(-0.465339\pi\)
0.108677 + 0.994077i \(0.465339\pi\)
\(252\) 0 0
\(253\) −18.7100 −1.17629
\(254\) 0 0
\(255\) −14.0959 −0.882720
\(256\) 0 0
\(257\) 5.59706 0.349135 0.174567 0.984645i \(-0.444147\pi\)
0.174567 + 0.984645i \(0.444147\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −7.20147 −0.445760
\(262\) 0 0
\(263\) −1.59706 −0.0984787 −0.0492393 0.998787i \(-0.515680\pi\)
−0.0492393 + 0.998787i \(0.515680\pi\)
\(264\) 0 0
\(265\) −10.5565 −0.648478
\(266\) 0 0
\(267\) −2.40294 −0.147058
\(268\) 0 0
\(269\) 17.7003 1.07921 0.539604 0.841919i \(-0.318574\pi\)
0.539604 + 0.841919i \(0.318574\pi\)
\(270\) 0 0
\(271\) 2.39558 0.145521 0.0727607 0.997349i \(-0.476819\pi\)
0.0727607 + 0.997349i \(0.476819\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −22.3070 −1.34517
\(276\) 0 0
\(277\) 5.24943 0.315407 0.157704 0.987486i \(-0.449591\pi\)
0.157704 + 0.987486i \(0.449591\pi\)
\(278\) 0 0
\(279\) −2.04795 −0.122608
\(280\) 0 0
\(281\) 20.0959 1.19882 0.599411 0.800442i \(-0.295402\pi\)
0.599411 + 0.800442i \(0.295402\pi\)
\(282\) 0 0
\(283\) −23.2494 −1.38203 −0.691017 0.722838i \(-0.742837\pi\)
−0.691017 + 0.722838i \(0.742837\pi\)
\(284\) 0 0
\(285\) −10.0959 −0.598030
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.38592 0.140348
\(290\) 0 0
\(291\) −4.24943 −0.249106
\(292\) 0 0
\(293\) 11.2015 0.654397 0.327199 0.944956i \(-0.393895\pi\)
0.327199 + 0.944956i \(0.393895\pi\)
\(294\) 0 0
\(295\) −26.4103 −1.53767
\(296\) 0 0
\(297\) −4.24943 −0.246577
\(298\) 0 0
\(299\) −13.8848 −0.802977
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3.90409 0.224284
\(304\) 0 0
\(305\) 25.9188 1.48411
\(306\) 0 0
\(307\) −28.9571 −1.65267 −0.826335 0.563179i \(-0.809578\pi\)
−0.826335 + 0.563179i \(0.809578\pi\)
\(308\) 0 0
\(309\) −9.24943 −0.526181
\(310\) 0 0
\(311\) −27.3047 −1.54831 −0.774155 0.632996i \(-0.781825\pi\)
−0.774155 + 0.632996i \(0.781825\pi\)
\(312\) 0 0
\(313\) 5.49885 0.310813 0.155407 0.987851i \(-0.450331\pi\)
0.155407 + 0.987851i \(0.450331\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.60442 0.539438 0.269719 0.962939i \(-0.413069\pi\)
0.269719 + 0.962939i \(0.413069\pi\)
\(318\) 0 0
\(319\) 30.6021 1.71339
\(320\) 0 0
\(321\) −16.5565 −0.924092
\(322\) 0 0
\(323\) 13.8848 0.772569
\(324\) 0 0
\(325\) −16.5542 −0.918260
\(326\) 0 0
\(327\) −20.0553 −1.10906
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −15.5565 −0.855061 −0.427530 0.904001i \(-0.640616\pi\)
−0.427530 + 0.904001i \(0.640616\pi\)
\(332\) 0 0
\(333\) 9.65237 0.528947
\(334\) 0 0
\(335\) 16.4989 0.901428
\(336\) 0 0
\(337\) 17.9977 0.980397 0.490199 0.871611i \(-0.336924\pi\)
0.490199 + 0.871611i \(0.336924\pi\)
\(338\) 0 0
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) 8.70262 0.471273
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −14.0959 −0.758898
\(346\) 0 0
\(347\) −5.69296 −0.305614 −0.152807 0.988256i \(-0.548831\pi\)
−0.152807 + 0.988256i \(0.548831\pi\)
\(348\) 0 0
\(349\) −18.1918 −0.973785 −0.486893 0.873462i \(-0.661870\pi\)
−0.486893 + 0.873462i \(0.661870\pi\)
\(350\) 0 0
\(351\) −3.15352 −0.168322
\(352\) 0 0
\(353\) 20.9977 1.11759 0.558797 0.829304i \(-0.311263\pi\)
0.558797 + 0.829304i \(0.311263\pi\)
\(354\) 0 0
\(355\) −20.4989 −1.08797
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.6907 −1.30312 −0.651562 0.758596i \(-0.725886\pi\)
−0.651562 + 0.758596i \(0.725886\pi\)
\(360\) 0 0
\(361\) −9.05531 −0.476595
\(362\) 0 0
\(363\) 7.05761 0.370429
\(364\) 0 0
\(365\) 48.8206 2.55539
\(366\) 0 0
\(367\) 5.85614 0.305688 0.152844 0.988250i \(-0.451157\pi\)
0.152844 + 0.988250i \(0.451157\pi\)
\(368\) 0 0
\(369\) 10.4989 0.546548
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.635347 −0.0328970 −0.0164485 0.999865i \(-0.505236\pi\)
−0.0164485 + 0.999865i \(0.505236\pi\)
\(374\) 0 0
\(375\) −0.798528 −0.0412358
\(376\) 0 0
\(377\) 22.7100 1.16962
\(378\) 0 0
\(379\) −10.7506 −0.552220 −0.276110 0.961126i \(-0.589045\pi\)
−0.276110 + 0.961126i \(0.589045\pi\)
\(380\) 0 0
\(381\) 12.4509 0.637879
\(382\) 0 0
\(383\) −14.9977 −0.766347 −0.383173 0.923676i \(-0.625169\pi\)
−0.383173 + 0.923676i \(0.625169\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.750575 −0.0381539
\(388\) 0 0
\(389\) −6.21113 −0.314917 −0.157458 0.987526i \(-0.550330\pi\)
−0.157458 + 0.987526i \(0.550330\pi\)
\(390\) 0 0
\(391\) 19.3859 0.980388
\(392\) 0 0
\(393\) 4.24943 0.214355
\(394\) 0 0
\(395\) 52.6671 2.64997
\(396\) 0 0
\(397\) 1.24943 0.0627068 0.0313534 0.999508i \(-0.490018\pi\)
0.0313534 + 0.999508i \(0.490018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.11523 0.105629 0.0528147 0.998604i \(-0.483181\pi\)
0.0528147 + 0.998604i \(0.483181\pi\)
\(402\) 0 0
\(403\) 6.45826 0.321709
\(404\) 0 0
\(405\) −3.20147 −0.159082
\(406\) 0 0
\(407\) −41.0170 −2.03314
\(408\) 0 0
\(409\) 16.1129 0.796733 0.398367 0.917226i \(-0.369577\pi\)
0.398367 + 0.917226i \(0.369577\pi\)
\(410\) 0 0
\(411\) −10.4029 −0.513139
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −46.6021 −2.28761
\(416\) 0 0
\(417\) 11.6524 0.570619
\(418\) 0 0
\(419\) −20.1106 −0.982469 −0.491234 0.871027i \(-0.663454\pi\)
−0.491234 + 0.871027i \(0.663454\pi\)
\(420\) 0 0
\(421\) −4.96171 −0.241819 −0.120909 0.992664i \(-0.538581\pi\)
−0.120909 + 0.992664i \(0.538581\pi\)
\(422\) 0 0
\(423\) 2.40294 0.116835
\(424\) 0 0
\(425\) 23.1129 1.12114
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 13.4006 0.646989
\(430\) 0 0
\(431\) −20.8059 −1.00218 −0.501092 0.865394i \(-0.667068\pi\)
−0.501092 + 0.865394i \(0.667068\pi\)
\(432\) 0 0
\(433\) 8.05531 0.387114 0.193557 0.981089i \(-0.437998\pi\)
0.193557 + 0.981089i \(0.437998\pi\)
\(434\) 0 0
\(435\) 23.0553 1.10542
\(436\) 0 0
\(437\) 13.8848 0.664199
\(438\) 0 0
\(439\) 16.1992 0.773144 0.386572 0.922259i \(-0.373659\pi\)
0.386572 + 0.922259i \(0.373659\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −15.4435 −0.733745 −0.366872 0.930271i \(-0.619571\pi\)
−0.366872 + 0.930271i \(0.619571\pi\)
\(444\) 0 0
\(445\) 7.69296 0.364681
\(446\) 0 0
\(447\) −10.4029 −0.492042
\(448\) 0 0
\(449\) 31.5159 1.48733 0.743663 0.668555i \(-0.233087\pi\)
0.743663 + 0.668555i \(0.233087\pi\)
\(450\) 0 0
\(451\) −44.6141 −2.10079
\(452\) 0 0
\(453\) −20.0074 −0.940028
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11.1152 −0.519948 −0.259974 0.965616i \(-0.583714\pi\)
−0.259974 + 0.965616i \(0.583714\pi\)
\(458\) 0 0
\(459\) 4.40294 0.205512
\(460\) 0 0
\(461\) 13.5971 0.633278 0.316639 0.948546i \(-0.397446\pi\)
0.316639 + 0.948546i \(0.397446\pi\)
\(462\) 0 0
\(463\) 2.55876 0.118916 0.0594579 0.998231i \(-0.481063\pi\)
0.0594579 + 0.998231i \(0.481063\pi\)
\(464\) 0 0
\(465\) 6.55646 0.304049
\(466\) 0 0
\(467\) −0.614078 −0.0284161 −0.0142081 0.999899i \(-0.504523\pi\)
−0.0142081 + 0.999899i \(0.504523\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 14.4989 0.668072
\(472\) 0 0
\(473\) 3.18951 0.146654
\(474\) 0 0
\(475\) 16.5542 0.759557
\(476\) 0 0
\(477\) 3.29738 0.150977
\(478\) 0 0
\(479\) −20.0959 −0.918205 −0.459103 0.888383i \(-0.651829\pi\)
−0.459103 + 0.888383i \(0.651829\pi\)
\(480\) 0 0
\(481\) −30.4389 −1.38790
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.6044 0.617745
\(486\) 0 0
\(487\) 23.3720 1.05909 0.529544 0.848283i \(-0.322363\pi\)
0.529544 + 0.848283i \(0.322363\pi\)
\(488\) 0 0
\(489\) 10.7100 0.484322
\(490\) 0 0
\(491\) −22.7483 −1.02662 −0.513308 0.858205i \(-0.671580\pi\)
−0.513308 + 0.858205i \(0.671580\pi\)
\(492\) 0 0
\(493\) −31.7077 −1.42804
\(494\) 0 0
\(495\) 13.6044 0.611473
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 8.53944 0.382278 0.191139 0.981563i \(-0.438782\pi\)
0.191139 + 0.981563i \(0.438782\pi\)
\(500\) 0 0
\(501\) −17.3047 −0.773119
\(502\) 0 0
\(503\) −7.59706 −0.338736 −0.169368 0.985553i \(-0.554173\pi\)
−0.169368 + 0.985553i \(0.554173\pi\)
\(504\) 0 0
\(505\) −12.4989 −0.556192
\(506\) 0 0
\(507\) −3.05531 −0.135691
\(508\) 0 0
\(509\) −4.99034 −0.221193 −0.110596 0.993865i \(-0.535276\pi\)
−0.110596 + 0.993865i \(0.535276\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.15352 0.139231
\(514\) 0 0
\(515\) 29.6118 1.30485
\(516\) 0 0
\(517\) −10.2111 −0.449085
\(518\) 0 0
\(519\) −3.90409 −0.171371
\(520\) 0 0
\(521\) 12.4989 0.547585 0.273792 0.961789i \(-0.411722\pi\)
0.273792 + 0.961789i \(0.411722\pi\)
\(522\) 0 0
\(523\) 44.2618 1.93544 0.967718 0.252036i \(-0.0811003\pi\)
0.967718 + 0.252036i \(0.0811003\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.01702 −0.392788
\(528\) 0 0
\(529\) −3.61408 −0.157134
\(530\) 0 0
\(531\) 8.24943 0.357995
\(532\) 0 0
\(533\) −33.1083 −1.43408
\(534\) 0 0
\(535\) 53.0051 2.29161
\(536\) 0 0
\(537\) −20.8059 −0.897840
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −25.9594 −1.11608 −0.558041 0.829813i \(-0.688447\pi\)
−0.558041 + 0.829813i \(0.688447\pi\)
\(542\) 0 0
\(543\) −13.2494 −0.568587
\(544\) 0 0
\(545\) 64.2065 2.75031
\(546\) 0 0
\(547\) −22.9018 −0.979210 −0.489605 0.871944i \(-0.662859\pi\)
−0.489605 + 0.871944i \(0.662859\pi\)
\(548\) 0 0
\(549\) −8.09591 −0.345525
\(550\) 0 0
\(551\) −22.7100 −0.967478
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −30.9018 −1.31171
\(556\) 0 0
\(557\) 6.50621 0.275677 0.137839 0.990455i \(-0.455984\pi\)
0.137839 + 0.990455i \(0.455984\pi\)
\(558\) 0 0
\(559\) 2.36695 0.100111
\(560\) 0 0
\(561\) −18.7100 −0.789936
\(562\) 0 0
\(563\) −44.2448 −1.86470 −0.932349 0.361561i \(-0.882244\pi\)
−0.932349 + 0.361561i \(0.882244\pi\)
\(564\) 0 0
\(565\) −25.6118 −1.07750
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.0170 −0.545702 −0.272851 0.962056i \(-0.587967\pi\)
−0.272851 + 0.962056i \(0.587967\pi\)
\(570\) 0 0
\(571\) −30.1705 −1.26260 −0.631299 0.775540i \(-0.717478\pi\)
−0.631299 + 0.775540i \(0.717478\pi\)
\(572\) 0 0
\(573\) 4.19181 0.175115
\(574\) 0 0
\(575\) 23.1129 0.963876
\(576\) 0 0
\(577\) 29.8059 1.24084 0.620418 0.784272i \(-0.286963\pi\)
0.620418 + 0.784272i \(0.286963\pi\)
\(578\) 0 0
\(579\) −18.3047 −0.760719
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −14.0120 −0.580316
\(584\) 0 0
\(585\) 10.0959 0.417414
\(586\) 0 0
\(587\) 18.2494 0.753234 0.376617 0.926369i \(-0.377087\pi\)
0.376617 + 0.926369i \(0.377087\pi\)
\(588\) 0 0
\(589\) −6.45826 −0.266108
\(590\) 0 0
\(591\) −11.9041 −0.489669
\(592\) 0 0
\(593\) 23.7889 0.976892 0.488446 0.872594i \(-0.337564\pi\)
0.488446 + 0.872594i \(0.337564\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.805889 −0.0329829
\(598\) 0 0
\(599\) −15.5011 −0.633360 −0.316680 0.948532i \(-0.602568\pi\)
−0.316680 + 0.948532i \(0.602568\pi\)
\(600\) 0 0
\(601\) −15.8059 −0.644736 −0.322368 0.946614i \(-0.604479\pi\)
−0.322368 + 0.946614i \(0.604479\pi\)
\(602\) 0 0
\(603\) −5.15352 −0.209868
\(604\) 0 0
\(605\) −22.5948 −0.918607
\(606\) 0 0
\(607\) −5.95205 −0.241586 −0.120793 0.992678i \(-0.538544\pi\)
−0.120793 + 0.992678i \(0.538544\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.57773 −0.306562
\(612\) 0 0
\(613\) 28.4029 1.14718 0.573592 0.819141i \(-0.305549\pi\)
0.573592 + 0.819141i \(0.305549\pi\)
\(614\) 0 0
\(615\) −33.6118 −1.35536
\(616\) 0 0
\(617\) −10.1918 −0.410307 −0.205153 0.978730i \(-0.565769\pi\)
−0.205153 + 0.978730i \(0.565769\pi\)
\(618\) 0 0
\(619\) 20.8612 0.838483 0.419241 0.907875i \(-0.362296\pi\)
0.419241 + 0.907875i \(0.362296\pi\)
\(620\) 0 0
\(621\) 4.40294 0.176684
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −23.6907 −0.947626
\(626\) 0 0
\(627\) −13.4006 −0.535170
\(628\) 0 0
\(629\) 42.4989 1.69454
\(630\) 0 0
\(631\) 41.6191 1.65683 0.828416 0.560113i \(-0.189242\pi\)
0.828416 + 0.560113i \(0.189242\pi\)
\(632\) 0 0
\(633\) −8.49885 −0.337799
\(634\) 0 0
\(635\) −39.8612 −1.58184
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.40294 0.253297
\(640\) 0 0
\(641\) −48.8013 −1.92754 −0.963768 0.266744i \(-0.914052\pi\)
−0.963768 + 0.266744i \(0.914052\pi\)
\(642\) 0 0
\(643\) 11.8442 0.467089 0.233544 0.972346i \(-0.424968\pi\)
0.233544 + 0.972346i \(0.424968\pi\)
\(644\) 0 0
\(645\) 2.40294 0.0946159
\(646\) 0 0
\(647\) −33.5159 −1.31764 −0.658822 0.752298i \(-0.728945\pi\)
−0.658822 + 0.752298i \(0.728945\pi\)
\(648\) 0 0
\(649\) −35.0553 −1.37604
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 46.5062 1.81993 0.909964 0.414686i \(-0.136109\pi\)
0.909964 + 0.414686i \(0.136109\pi\)
\(654\) 0 0
\(655\) −13.6044 −0.531569
\(656\) 0 0
\(657\) −15.2494 −0.594937
\(658\) 0 0
\(659\) 10.1152 0.394033 0.197017 0.980400i \(-0.436875\pi\)
0.197017 + 0.980400i \(0.436875\pi\)
\(660\) 0 0
\(661\) 33.2641 1.29383 0.646913 0.762564i \(-0.276060\pi\)
0.646913 + 0.762564i \(0.276060\pi\)
\(662\) 0 0
\(663\) −13.8848 −0.539240
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −31.7077 −1.22773
\(668\) 0 0
\(669\) 16.7985 0.649469
\(670\) 0 0
\(671\) 34.4029 1.32811
\(672\) 0 0
\(673\) −50.4966 −1.94650 −0.973249 0.229751i \(-0.926209\pi\)
−0.973249 + 0.229751i \(0.926209\pi\)
\(674\) 0 0
\(675\) 5.24943 0.202050
\(676\) 0 0
\(677\) 28.7985 1.10682 0.553409 0.832910i \(-0.313327\pi\)
0.553409 + 0.832910i \(0.313327\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −16.8635 −0.646211
\(682\) 0 0
\(683\) 13.2471 0.506887 0.253444 0.967350i \(-0.418437\pi\)
0.253444 + 0.967350i \(0.418437\pi\)
\(684\) 0 0
\(685\) 33.3047 1.27251
\(686\) 0 0
\(687\) 20.5542 0.784190
\(688\) 0 0
\(689\) −10.3983 −0.396145
\(690\) 0 0
\(691\) −39.6671 −1.50901 −0.754504 0.656296i \(-0.772122\pi\)
−0.754504 + 0.656296i \(0.772122\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −37.3047 −1.41505
\(696\) 0 0
\(697\) 46.2259 1.75093
\(698\) 0 0
\(699\) 9.90409 0.374607
\(700\) 0 0
\(701\) 2.10787 0.0796130 0.0398065 0.999207i \(-0.487326\pi\)
0.0398065 + 0.999207i \(0.487326\pi\)
\(702\) 0 0
\(703\) 30.4389 1.14803
\(704\) 0 0
\(705\) −7.69296 −0.289734
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 14.9977 0.563250 0.281625 0.959524i \(-0.409127\pi\)
0.281625 + 0.959524i \(0.409127\pi\)
\(710\) 0 0
\(711\) −16.4509 −0.616957
\(712\) 0 0
\(713\) −9.01702 −0.337690
\(714\) 0 0
\(715\) −42.9018 −1.60444
\(716\) 0 0
\(717\) 4.40294 0.164431
\(718\) 0 0
\(719\) −22.8059 −0.850516 −0.425258 0.905072i \(-0.639817\pi\)
−0.425258 + 0.905072i \(0.639817\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −20.8635 −0.775922
\(724\) 0 0
\(725\) −37.8036 −1.40399
\(726\) 0 0
\(727\) −16.9691 −0.629348 −0.314674 0.949200i \(-0.601895\pi\)
−0.314674 + 0.949200i \(0.601895\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.30474 −0.122230
\(732\) 0 0
\(733\) 1.34533 0.0496909 0.0248455 0.999691i \(-0.492091\pi\)
0.0248455 + 0.999691i \(0.492091\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21.8995 0.806678
\(738\) 0 0
\(739\) −45.2641 −1.66507 −0.832534 0.553973i \(-0.813111\pi\)
−0.832534 + 0.553973i \(0.813111\pi\)
\(740\) 0 0
\(741\) −9.94469 −0.365327
\(742\) 0 0
\(743\) −27.8036 −1.02001 −0.510007 0.860170i \(-0.670357\pi\)
−0.510007 + 0.860170i \(0.670357\pi\)
\(744\) 0 0
\(745\) 33.3047 1.22019
\(746\) 0 0
\(747\) 14.5565 0.532593
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 30.0627 1.09700 0.548501 0.836150i \(-0.315199\pi\)
0.548501 + 0.836150i \(0.315199\pi\)
\(752\) 0 0
\(753\) 3.44354 0.125489
\(754\) 0 0
\(755\) 64.0530 2.33113
\(756\) 0 0
\(757\) 17.3241 0.629654 0.314827 0.949149i \(-0.398054\pi\)
0.314827 + 0.949149i \(0.398054\pi\)
\(758\) 0 0
\(759\) −18.7100 −0.679129
\(760\) 0 0
\(761\) −12.2111 −0.442653 −0.221327 0.975200i \(-0.571039\pi\)
−0.221327 + 0.975200i \(0.571039\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −14.0959 −0.509639
\(766\) 0 0
\(767\) −26.0147 −0.939337
\(768\) 0 0
\(769\) −11.1152 −0.400825 −0.200413 0.979712i \(-0.564228\pi\)
−0.200413 + 0.979712i \(0.564228\pi\)
\(770\) 0 0
\(771\) 5.59706 0.201573
\(772\) 0 0
\(773\) −0.287717 −0.0103485 −0.00517423 0.999987i \(-0.501647\pi\)
−0.00517423 + 0.999987i \(0.501647\pi\)
\(774\) 0 0
\(775\) −10.7506 −0.386172
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 33.1083 1.18623
\(780\) 0 0
\(781\) −27.2088 −0.973609
\(782\) 0 0
\(783\) −7.20147 −0.257360
\(784\) 0 0
\(785\) −46.4177 −1.65672
\(786\) 0 0
\(787\) −11.5011 −0.409972 −0.204986 0.978765i \(-0.565715\pi\)
−0.204986 + 0.978765i \(0.565715\pi\)
\(788\) 0 0
\(789\) −1.59706 −0.0568567
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 25.5306 0.906618
\(794\) 0 0
\(795\) −10.5565 −0.374399
\(796\) 0 0
\(797\) −18.6980 −0.662318 −0.331159 0.943575i \(-0.607440\pi\)
−0.331159 + 0.943575i \(0.607440\pi\)
\(798\) 0 0
\(799\) 10.5800 0.374295
\(800\) 0 0
\(801\) −2.40294 −0.0849039
\(802\) 0 0
\(803\) 64.8013 2.28679
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 17.7003 0.623081
\(808\) 0 0
\(809\) −53.8036 −1.89163 −0.945817 0.324701i \(-0.894736\pi\)
−0.945817 + 0.324701i \(0.894736\pi\)
\(810\) 0 0
\(811\) −34.7866 −1.22152 −0.610761 0.791815i \(-0.709136\pi\)
−0.610761 + 0.791815i \(0.709136\pi\)
\(812\) 0 0
\(813\) 2.39558 0.0840168
\(814\) 0 0
\(815\) −34.2877 −1.20105
\(816\) 0 0
\(817\) −2.36695 −0.0828092
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −37.8110 −1.31961 −0.659806 0.751436i \(-0.729361\pi\)
−0.659806 + 0.751436i \(0.729361\pi\)
\(822\) 0 0
\(823\) −54.4177 −1.89688 −0.948440 0.316956i \(-0.897339\pi\)
−0.948440 + 0.316956i \(0.897339\pi\)
\(824\) 0 0
\(825\) −22.3070 −0.776631
\(826\) 0 0
\(827\) 7.86120 0.273361 0.136680 0.990615i \(-0.456357\pi\)
0.136680 + 0.990615i \(0.456357\pi\)
\(828\) 0 0
\(829\) 6.45826 0.224305 0.112152 0.993691i \(-0.464226\pi\)
0.112152 + 0.993691i \(0.464226\pi\)
\(830\) 0 0
\(831\) 5.24943 0.182101
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 55.4006 1.91722
\(836\) 0 0
\(837\) −2.04795 −0.0707876
\(838\) 0 0
\(839\) 28.3218 0.977776 0.488888 0.872347i \(-0.337403\pi\)
0.488888 + 0.872347i \(0.337403\pi\)
\(840\) 0 0
\(841\) 22.8612 0.788317
\(842\) 0 0
\(843\) 20.0959 0.692140
\(844\) 0 0
\(845\) 9.78150 0.336494
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −23.2494 −0.797918
\(850\) 0 0
\(851\) 42.4989 1.45684
\(852\) 0 0
\(853\) 8.75057 0.299614 0.149807 0.988715i \(-0.452135\pi\)
0.149807 + 0.988715i \(0.452135\pi\)
\(854\) 0 0
\(855\) −10.0959 −0.345273
\(856\) 0 0
\(857\) 24.6907 0.843417 0.421708 0.906731i \(-0.361431\pi\)
0.421708 + 0.906731i \(0.361431\pi\)
\(858\) 0 0
\(859\) −8.49885 −0.289977 −0.144989 0.989433i \(-0.546315\pi\)
−0.144989 + 0.989433i \(0.546315\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.6141 0.633631 0.316815 0.948487i \(-0.397386\pi\)
0.316815 + 0.948487i \(0.397386\pi\)
\(864\) 0 0
\(865\) 12.4989 0.424974
\(866\) 0 0
\(867\) 2.38592 0.0810302
\(868\) 0 0
\(869\) 69.9069 2.37143
\(870\) 0 0
\(871\) 16.2517 0.550669
\(872\) 0 0
\(873\) −4.24943 −0.143821
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 44.1106 1.48951 0.744755 0.667338i \(-0.232566\pi\)
0.744755 + 0.667338i \(0.232566\pi\)
\(878\) 0 0
\(879\) 11.2015 0.377816
\(880\) 0 0
\(881\) 11.2900 0.380370 0.190185 0.981748i \(-0.439091\pi\)
0.190185 + 0.981748i \(0.439091\pi\)
\(882\) 0 0
\(883\) 42.2471 1.42173 0.710864 0.703329i \(-0.248304\pi\)
0.710864 + 0.703329i \(0.248304\pi\)
\(884\) 0 0
\(885\) −26.4103 −0.887773
\(886\) 0 0
\(887\) 3.98068 0.133658 0.0668290 0.997764i \(-0.478712\pi\)
0.0668290 + 0.997764i \(0.478712\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.24943 −0.142361
\(892\) 0 0
\(893\) 7.57773 0.253579
\(894\) 0 0
\(895\) 66.6095 2.22651
\(896\) 0 0
\(897\) −13.8848 −0.463599
\(898\) 0 0
\(899\) 14.7483 0.491883
\(900\) 0 0
\(901\) 14.5182 0.483670
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 42.4177 1.41001
\(906\) 0 0
\(907\) 12.5735 0.417496 0.208748 0.977969i \(-0.433061\pi\)
0.208748 + 0.977969i \(0.433061\pi\)
\(908\) 0 0
\(909\) 3.90409 0.129491
\(910\) 0 0
\(911\) 7.09821 0.235174 0.117587 0.993063i \(-0.462484\pi\)
0.117587 + 0.993063i \(0.462484\pi\)
\(912\) 0 0
\(913\) −61.8566 −2.04715
\(914\) 0 0
\(915\) 25.9188 0.856850
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −37.4412 −1.23507 −0.617536 0.786542i \(-0.711869\pi\)
−0.617536 + 0.786542i \(0.711869\pi\)
\(920\) 0 0
\(921\) −28.9571 −0.954169
\(922\) 0 0
\(923\) −20.1918 −0.664622
\(924\) 0 0
\(925\) 50.6694 1.66600
\(926\) 0 0
\(927\) −9.24943 −0.303791
\(928\) 0 0
\(929\) −7.88477 −0.258691 −0.129345 0.991600i \(-0.541288\pi\)
−0.129345 + 0.991600i \(0.541288\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −27.3047 −0.893917
\(934\) 0 0
\(935\) 59.8995 1.95892
\(936\) 0 0
\(937\) −36.9954 −1.20859 −0.604294 0.796762i \(-0.706545\pi\)
−0.604294 + 0.796762i \(0.706545\pi\)
\(938\) 0 0
\(939\) 5.49885 0.179448
\(940\) 0 0
\(941\) −12.7026 −0.414094 −0.207047 0.978331i \(-0.566385\pi\)
−0.207047 + 0.978331i \(0.566385\pi\)
\(942\) 0 0
\(943\) 46.2259 1.50532
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23.1129 0.751069 0.375535 0.926808i \(-0.377459\pi\)
0.375535 + 0.926808i \(0.377459\pi\)
\(948\) 0 0
\(949\) 48.0894 1.56105
\(950\) 0 0
\(951\) 9.60442 0.311445
\(952\) 0 0
\(953\) −43.4200 −1.40651 −0.703255 0.710937i \(-0.748271\pi\)
−0.703255 + 0.710937i \(0.748271\pi\)
\(954\) 0 0
\(955\) −13.4200 −0.434260
\(956\) 0 0
\(957\) 30.6021 0.989226
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −26.8059 −0.864706
\(962\) 0 0
\(963\) −16.5565 −0.533525
\(964\) 0 0
\(965\) 58.6021 1.88647
\(966\) 0 0
\(967\) −11.6450 −0.374478 −0.187239 0.982314i \(-0.559954\pi\)
−0.187239 + 0.982314i \(0.559954\pi\)
\(968\) 0 0
\(969\) 13.8848 0.446043
\(970\) 0 0
\(971\) 33.3624 1.07065 0.535324 0.844647i \(-0.320189\pi\)
0.535324 + 0.844647i \(0.320189\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −16.5542 −0.530158
\(976\) 0 0
\(977\) 19.1895 0.613927 0.306963 0.951721i \(-0.400687\pi\)
0.306963 + 0.951721i \(0.400687\pi\)
\(978\) 0 0
\(979\) 10.2111 0.326349
\(980\) 0 0
\(981\) −20.0553 −0.640317
\(982\) 0 0
\(983\) −6.07658 −0.193813 −0.0969065 0.995293i \(-0.530895\pi\)
−0.0969065 + 0.995293i \(0.530895\pi\)
\(984\) 0 0
\(985\) 38.1106 1.21431
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3.30474 −0.105085
\(990\) 0 0
\(991\) 31.6597 1.00570 0.502852 0.864372i \(-0.332284\pi\)
0.502852 + 0.864372i \(0.332284\pi\)
\(992\) 0 0
\(993\) −15.5565 −0.493669
\(994\) 0 0
\(995\) 2.58003 0.0817925
\(996\) 0 0
\(997\) 15.3601 0.486458 0.243229 0.969969i \(-0.421793\pi\)
0.243229 + 0.969969i \(0.421793\pi\)
\(998\) 0 0
\(999\) 9.65237 0.305387
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.ei.1.1 3
4.3 odd 2 9408.2.a.eg.1.1 3
7.3 odd 6 1344.2.q.z.961.1 6
7.5 odd 6 1344.2.q.z.193.1 6
7.6 odd 2 9408.2.a.eh.1.3 3
8.3 odd 2 4704.2.a.bv.1.3 3
8.5 even 2 4704.2.a.bt.1.3 3
28.3 even 6 1344.2.q.y.961.1 6
28.19 even 6 1344.2.q.y.193.1 6
28.27 even 2 9408.2.a.ej.1.3 3
56.3 even 6 672.2.q.l.289.3 yes 6
56.5 odd 6 672.2.q.k.193.3 6
56.13 odd 2 4704.2.a.bu.1.1 3
56.19 even 6 672.2.q.l.193.3 yes 6
56.27 even 2 4704.2.a.bs.1.1 3
56.45 odd 6 672.2.q.k.289.3 yes 6
168.5 even 6 2016.2.s.u.865.1 6
168.59 odd 6 2016.2.s.v.289.1 6
168.101 even 6 2016.2.s.u.289.1 6
168.131 odd 6 2016.2.s.v.865.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.k.193.3 6 56.5 odd 6
672.2.q.k.289.3 yes 6 56.45 odd 6
672.2.q.l.193.3 yes 6 56.19 even 6
672.2.q.l.289.3 yes 6 56.3 even 6
1344.2.q.y.193.1 6 28.19 even 6
1344.2.q.y.961.1 6 28.3 even 6
1344.2.q.z.193.1 6 7.5 odd 6
1344.2.q.z.961.1 6 7.3 odd 6
2016.2.s.u.289.1 6 168.101 even 6
2016.2.s.u.865.1 6 168.5 even 6
2016.2.s.v.289.1 6 168.59 odd 6
2016.2.s.v.865.1 6 168.131 odd 6
4704.2.a.bs.1.1 3 56.27 even 2
4704.2.a.bt.1.3 3 8.5 even 2
4704.2.a.bu.1.1 3 56.13 odd 2
4704.2.a.bv.1.3 3 8.3 odd 2
9408.2.a.eg.1.1 3 4.3 odd 2
9408.2.a.eh.1.3 3 7.6 odd 2
9408.2.a.ei.1.1 3 1.1 even 1 trivial
9408.2.a.ej.1.3 3 28.27 even 2