Properties

Label 5184.2.f.e.2591.8
Level $5184$
Weight $2$
Character 5184.2591
Analytic conductor $41.394$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 5184 = 2^{6} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5184.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.3944484078\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.8
Root \(-2.04058 - 1.17813i\) of defining polynomial
Character \(\chi\) \(=\) 5184.2591
Dual form 5184.2.f.e.2591.7

$q$-expansion

\(f(q)\) \(=\) \(q-0.206954 q^{5} +3.03957i q^{7} +O(q^{10})\) \(q-0.206954 q^{5} +3.03957i q^{7} +5.02319i q^{11} -4.03957i q^{13} -4.81624i q^{17} -1.03957 q^{19} -5.43710 q^{23} -4.95717 q^{25} -2.90487 q^{29} -1.23898i q^{31} -0.629051i q^{35} -8.77162i q^{37} +0.979317i q^{41} +3.18555 q^{43} +3.26332 q^{47} -2.23898 q^{49} +7.91987 q^{53} -1.03957i q^{55} -14.8544i q^{59} -9.53264i q^{61} +0.836005i q^{65} +0.735311 q^{67} -2.33127 q^{71} -10.0363 q^{73} -15.2683 q^{77} -14.1929i q^{79} +4.23665i q^{83} +0.996740i q^{85} +11.6053i q^{89} +12.2786 q^{91} +0.215143 q^{95} +4.84016 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{19} + 16 q^{25} + 24 q^{43} - 48 q^{49} + 120 q^{67} + 16 q^{73} + 168 q^{91} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −0.206954 −0.0925526 −0.0462763 0.998929i \(-0.514735\pi\)
−0.0462763 + 0.998929i \(0.514735\pi\)
\(6\) 0 0
\(7\) 3.03957i 1.14885i 0.818557 + 0.574425i \(0.194774\pi\)
−0.818557 + 0.574425i \(0.805226\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.02319i 1.51455i 0.653096 + 0.757275i \(0.273470\pi\)
−0.653096 + 0.757275i \(0.726530\pi\)
\(12\) 0 0
\(13\) − 4.03957i − 1.12037i −0.828366 0.560187i \(-0.810729\pi\)
0.828366 0.560187i \(-0.189271\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.81624i − 1.16811i −0.811714 0.584055i \(-0.801465\pi\)
0.811714 0.584055i \(-0.198535\pi\)
\(18\) 0 0
\(19\) −1.03957 −0.238494 −0.119247 0.992865i \(-0.538048\pi\)
−0.119247 + 0.992865i \(0.538048\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.43710 −1.13371 −0.566857 0.823816i \(-0.691841\pi\)
−0.566857 + 0.823816i \(0.691841\pi\)
\(24\) 0 0
\(25\) −4.95717 −0.991434
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.90487 −0.539421 −0.269710 0.962941i \(-0.586928\pi\)
−0.269710 + 0.962941i \(0.586928\pi\)
\(30\) 0 0
\(31\) − 1.23898i − 0.222528i −0.993791 0.111264i \(-0.964510\pi\)
0.993791 0.111264i \(-0.0354899\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 0.629051i − 0.106329i
\(36\) 0 0
\(37\) − 8.77162i − 1.44205i −0.692911 0.721023i \(-0.743672\pi\)
0.692911 0.721023i \(-0.256328\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.979317i 0.152944i 0.997072 + 0.0764718i \(0.0243655\pi\)
−0.997072 + 0.0764718i \(0.975634\pi\)
\(42\) 0 0
\(43\) 3.18555 0.485792 0.242896 0.970052i \(-0.421903\pi\)
0.242896 + 0.970052i \(0.421903\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.26332 0.476005 0.238002 0.971265i \(-0.423507\pi\)
0.238002 + 0.971265i \(0.423507\pi\)
\(48\) 0 0
\(49\) −2.23898 −0.319855
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.91987 1.08788 0.543939 0.839125i \(-0.316932\pi\)
0.543939 + 0.839125i \(0.316932\pi\)
\(54\) 0 0
\(55\) − 1.03957i − 0.140176i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 14.8544i − 1.93388i −0.254999 0.966941i \(-0.582075\pi\)
0.254999 0.966941i \(-0.417925\pi\)
\(60\) 0 0
\(61\) − 9.53264i − 1.22053i −0.792198 0.610265i \(-0.791063\pi\)
0.792198 0.610265i \(-0.208937\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.836005i 0.103694i
\(66\) 0 0
\(67\) 0.735311 0.0898326 0.0449163 0.998991i \(-0.485698\pi\)
0.0449163 + 0.998991i \(0.485698\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.33127 −0.276671 −0.138336 0.990385i \(-0.544175\pi\)
−0.138336 + 0.990385i \(0.544175\pi\)
\(72\) 0 0
\(73\) −10.0363 −1.17466 −0.587331 0.809347i \(-0.699821\pi\)
−0.587331 + 0.809347i \(0.699821\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −15.2683 −1.73999
\(78\) 0 0
\(79\) − 14.1929i − 1.59683i −0.602111 0.798413i \(-0.705673\pi\)
0.602111 0.798413i \(-0.294327\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.23665i 0.465032i 0.972593 + 0.232516i \(0.0746959\pi\)
−0.972593 + 0.232516i \(0.925304\pi\)
\(84\) 0 0
\(85\) 0.996740i 0.108112i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.6053i 1.23016i 0.788465 + 0.615079i \(0.210876\pi\)
−0.788465 + 0.615079i \(0.789124\pi\)
\(90\) 0 0
\(91\) 12.2786 1.28714
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.215143 0.0220732
\(96\) 0 0
\(97\) 4.84016 0.491443 0.245722 0.969340i \(-0.420975\pi\)
0.245722 + 0.969340i \(0.420975\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.3949 1.73085 0.865426 0.501036i \(-0.167048\pi\)
0.865426 + 0.501036i \(0.167048\pi\)
\(102\) 0 0
\(103\) − 2.00000i − 0.197066i −0.995134 0.0985329i \(-0.968585\pi\)
0.995134 0.0985329i \(-0.0314150\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.4456i 1.10649i 0.833019 + 0.553244i \(0.186610\pi\)
−0.833019 + 0.553244i \(0.813390\pi\)
\(108\) 0 0
\(109\) 7.80793i 0.747864i 0.927456 + 0.373932i \(0.121991\pi\)
−0.927456 + 0.373932i \(0.878009\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 19.6707i 1.85046i 0.379405 + 0.925231i \(0.376129\pi\)
−0.379405 + 0.925231i \(0.623871\pi\)
\(114\) 0 0
\(115\) 1.12523 0.104928
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 14.6393 1.34198
\(120\) 0 0
\(121\) −14.2325 −1.29386
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.06068 0.184312
\(126\) 0 0
\(127\) − 7.35274i − 0.652450i −0.945292 0.326225i \(-0.894223\pi\)
0.945292 0.326225i \(-0.105777\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 11.0730i − 0.967449i −0.875220 0.483725i \(-0.839284\pi\)
0.875220 0.483725i \(-0.160716\pi\)
\(132\) 0 0
\(133\) − 3.15984i − 0.273993i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 16.1103i − 1.37640i −0.725521 0.688200i \(-0.758401\pi\)
0.725521 0.688200i \(-0.241599\pi\)
\(138\) 0 0
\(139\) 21.1444 1.79345 0.896723 0.442592i \(-0.145941\pi\)
0.896723 + 0.442592i \(0.145941\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.2915 1.69686
\(144\) 0 0
\(145\) 0.601174 0.0499248
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.3437 −1.33893 −0.669464 0.742844i \(-0.733476\pi\)
−0.669464 + 0.742844i \(0.733476\pi\)
\(150\) 0 0
\(151\) − 6.07914i − 0.494713i −0.968924 0.247357i \(-0.920438\pi\)
0.968924 0.247357i \(-0.0795619\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.256412i 0.0205955i
\(156\) 0 0
\(157\) − 14.3817i − 1.14778i −0.818931 0.573892i \(-0.805433\pi\)
0.818931 0.573892i \(-0.194567\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 16.5264i − 1.30247i
\(162\) 0 0
\(163\) 13.8263 1.08296 0.541479 0.840714i \(-0.317864\pi\)
0.541479 + 0.840714i \(0.317864\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.97675 −0.772024 −0.386012 0.922494i \(-0.626148\pi\)
−0.386012 + 0.922494i \(0.626148\pi\)
\(168\) 0 0
\(169\) −3.31812 −0.255240
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.8401 1.73650 0.868252 0.496123i \(-0.165244\pi\)
0.868252 + 0.496123i \(0.165244\pi\)
\(174\) 0 0
\(175\) − 15.0677i − 1.13901i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 15.4258i 1.15298i 0.817104 + 0.576491i \(0.195578\pi\)
−0.817104 + 0.576491i \(0.804422\pi\)
\(180\) 0 0
\(181\) − 14.4502i − 1.07408i −0.843557 0.537039i \(-0.819543\pi\)
0.843557 0.537039i \(-0.180457\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.81532i 0.133465i
\(186\) 0 0
\(187\) 24.1929 1.76916
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −25.4190 −1.83925 −0.919626 0.392796i \(-0.871508\pi\)
−0.919626 + 0.392796i \(0.871508\pi\)
\(192\) 0 0
\(193\) 15.1583 1.09112 0.545558 0.838073i \(-0.316318\pi\)
0.545558 + 0.838073i \(0.316318\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.06068 0.146817 0.0734085 0.997302i \(-0.476612\pi\)
0.0734085 + 0.997302i \(0.476612\pi\)
\(198\) 0 0
\(199\) 0.0791389i 0.00561001i 0.999996 + 0.00280500i \(0.000892861\pi\)
−0.999996 + 0.00280500i \(0.999107\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 8.82955i − 0.619713i
\(204\) 0 0
\(205\) − 0.202673i − 0.0141553i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 5.22196i − 0.361210i
\(210\) 0 0
\(211\) 21.3527 1.46998 0.734991 0.678076i \(-0.237186\pi\)
0.734991 + 0.678076i \(0.237186\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.659262 −0.0449613
\(216\) 0 0
\(217\) 3.76597 0.255651
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −19.4555 −1.30872
\(222\) 0 0
\(223\) 13.4018i 0.897448i 0.893670 + 0.448724i \(0.148121\pi\)
−0.893670 + 0.448724i \(0.851879\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.25984i 0.614597i 0.951613 + 0.307299i \(0.0994250\pi\)
−0.951613 + 0.307299i \(0.900575\pi\)
\(228\) 0 0
\(229\) − 11.1717i − 0.738246i −0.929381 0.369123i \(-0.879658\pi\)
0.929381 0.369123i \(-0.120342\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 14.1233i − 0.925251i −0.886554 0.462625i \(-0.846907\pi\)
0.886554 0.462625i \(-0.153093\pi\)
\(234\) 0 0
\(235\) −0.675358 −0.0440555
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −23.2106 −1.50137 −0.750684 0.660661i \(-0.770276\pi\)
−0.750684 + 0.660661i \(0.770276\pi\)
\(240\) 0 0
\(241\) −14.5143 −0.934947 −0.467473 0.884007i \(-0.654836\pi\)
−0.467473 + 0.884007i \(0.654836\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.463366 0.0296034
\(246\) 0 0
\(247\) 4.19941i 0.267202i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 4.03788i − 0.254869i −0.991847 0.127434i \(-0.959326\pi\)
0.991847 0.127434i \(-0.0406742\pi\)
\(252\) 0 0
\(253\) − 27.3116i − 1.71707i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.77579i 0.609797i 0.952385 + 0.304898i \(0.0986225\pi\)
−0.952385 + 0.304898i \(0.901377\pi\)
\(258\) 0 0
\(259\) 26.6619 1.65669
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.5814 1.51576 0.757878 0.652396i \(-0.226236\pi\)
0.757878 + 0.652396i \(0.226236\pi\)
\(264\) 0 0
\(265\) −1.63905 −0.100686
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.02008 0.123167 0.0615833 0.998102i \(-0.480385\pi\)
0.0615833 + 0.998102i \(0.480385\pi\)
\(270\) 0 0
\(271\) − 28.6643i − 1.74123i −0.491961 0.870617i \(-0.663720\pi\)
0.491961 0.870617i \(-0.336280\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 24.9008i − 1.50158i
\(276\) 0 0
\(277\) − 14.2162i − 0.854169i −0.904212 0.427085i \(-0.859541\pi\)
0.904212 0.427085i \(-0.140459\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 7.49178i − 0.446922i −0.974713 0.223461i \(-0.928264\pi\)
0.974713 0.223461i \(-0.0717356\pi\)
\(282\) 0 0
\(283\) −27.6159 −1.64159 −0.820796 0.571221i \(-0.806470\pi\)
−0.820796 + 0.571221i \(0.806470\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.97670 −0.175709
\(288\) 0 0
\(289\) −6.19615 −0.364480
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −26.3495 −1.53935 −0.769677 0.638433i \(-0.779583\pi\)
−0.769677 + 0.638433i \(0.779583\pi\)
\(294\) 0 0
\(295\) 3.07418i 0.178986i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 21.9635i 1.27018i
\(300\) 0 0
\(301\) 9.68270i 0.558102i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.97282i 0.112963i
\(306\) 0 0
\(307\) 9.07418 0.517891 0.258946 0.965892i \(-0.416625\pi\)
0.258946 + 0.965892i \(0.416625\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −21.0661 −1.19455 −0.597274 0.802037i \(-0.703750\pi\)
−0.597274 + 0.802037i \(0.703750\pi\)
\(312\) 0 0
\(313\) −18.2325 −1.03056 −0.515280 0.857022i \(-0.672312\pi\)
−0.515280 + 0.857022i \(0.672312\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.60278 −0.314684 −0.157342 0.987544i \(-0.550292\pi\)
−0.157342 + 0.987544i \(0.550292\pi\)
\(318\) 0 0
\(319\) − 14.5917i − 0.816979i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.00681i 0.278587i
\(324\) 0 0
\(325\) 20.0248i 1.11078i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.91910i 0.546858i
\(330\) 0 0
\(331\) −29.5836 −1.62606 −0.813032 0.582219i \(-0.802184\pi\)
−0.813032 + 0.582219i \(0.802184\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.152176 −0.00831424
\(336\) 0 0
\(337\) −19.3116 −1.05197 −0.525985 0.850494i \(-0.676303\pi\)
−0.525985 + 0.850494i \(0.676303\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.22365 0.337029
\(342\) 0 0
\(343\) 14.4714i 0.781385i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 14.2254i − 0.763659i −0.924233 0.381829i \(-0.875294\pi\)
0.924233 0.381829i \(-0.124706\pi\)
\(348\) 0 0
\(349\) − 12.1672i − 0.651295i −0.945491 0.325647i \(-0.894418\pi\)
0.945491 0.325647i \(-0.105582\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 0.419903i − 0.0223492i −0.999938 0.0111746i \(-0.996443\pi\)
0.999938 0.0111746i \(-0.00355705\pi\)
\(354\) 0 0
\(355\) 0.482466 0.0256066
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −14.0799 −0.743107 −0.371554 0.928411i \(-0.621175\pi\)
−0.371554 + 0.928411i \(0.621175\pi\)
\(360\) 0 0
\(361\) −17.9193 −0.943121
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.07705 0.108718
\(366\) 0 0
\(367\) − 31.6248i − 1.65080i −0.564549 0.825400i \(-0.690950\pi\)
0.564549 0.825400i \(-0.309050\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 24.0730i 1.24981i
\(372\) 0 0
\(373\) − 33.1077i − 1.71425i −0.515108 0.857126i \(-0.672248\pi\)
0.515108 0.857126i \(-0.327752\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.7344i 0.604353i
\(378\) 0 0
\(379\) −0.167186 −0.00858775 −0.00429387 0.999991i \(-0.501367\pi\)
−0.00429387 + 0.999991i \(0.501367\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.7432 1.67310 0.836550 0.547890i \(-0.184569\pi\)
0.836550 + 0.547890i \(0.184569\pi\)
\(384\) 0 0
\(385\) 3.15984 0.161041
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.4016 −0.780893 −0.390447 0.920626i \(-0.627679\pi\)
−0.390447 + 0.920626i \(0.627679\pi\)
\(390\) 0 0
\(391\) 26.1864i 1.32430i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.93727i 0.147790i
\(396\) 0 0
\(397\) − 25.3010i − 1.26982i −0.772586 0.634910i \(-0.781037\pi\)
0.772586 0.634910i \(-0.218963\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 13.0391i − 0.651142i −0.945518 0.325571i \(-0.894443\pi\)
0.945518 0.325571i \(-0.105557\pi\)
\(402\) 0 0
\(403\) −5.00496 −0.249315
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 44.0615 2.18405
\(408\) 0 0
\(409\) −7.75326 −0.383374 −0.191687 0.981456i \(-0.561396\pi\)
−0.191687 + 0.981456i \(0.561396\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 45.1511 2.22174
\(414\) 0 0
\(415\) − 0.876791i − 0.0430400i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 30.2966i − 1.48009i −0.672558 0.740044i \(-0.734805\pi\)
0.672558 0.740044i \(-0.265195\pi\)
\(420\) 0 0
\(421\) − 18.7427i − 0.913461i −0.889605 0.456731i \(-0.849020\pi\)
0.889605 0.456731i \(-0.150980\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 23.8749i 1.15810i
\(426\) 0 0
\(427\) 28.9751 1.40220
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.8198 0.713846 0.356923 0.934134i \(-0.383826\pi\)
0.356923 + 0.934134i \(0.383826\pi\)
\(432\) 0 0
\(433\) −9.83364 −0.472574 −0.236287 0.971683i \(-0.575931\pi\)
−0.236287 + 0.971683i \(0.575931\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.65224 0.270383
\(438\) 0 0
\(439\) 9.79609i 0.467542i 0.972292 + 0.233771i \(0.0751065\pi\)
−0.972292 + 0.233771i \(0.924893\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 12.2734i − 0.583128i −0.956551 0.291564i \(-0.905824\pi\)
0.956551 0.291564i \(-0.0941756\pi\)
\(444\) 0 0
\(445\) − 2.40176i − 0.113854i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.46106i 0.399302i 0.979867 + 0.199651i \(0.0639809\pi\)
−0.979867 + 0.199651i \(0.936019\pi\)
\(450\) 0 0
\(451\) −4.91930 −0.231641
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.54109 −0.119128
\(456\) 0 0
\(457\) 23.5127 1.09988 0.549939 0.835205i \(-0.314651\pi\)
0.549939 + 0.835205i \(0.314651\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.7941 0.875327 0.437664 0.899139i \(-0.355806\pi\)
0.437664 + 0.899139i \(0.355806\pi\)
\(462\) 0 0
\(463\) − 23.7896i − 1.10559i −0.833316 0.552797i \(-0.813560\pi\)
0.833316 0.552797i \(-0.186440\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 0.827816i − 0.0383067i −0.999817 0.0191534i \(-0.993903\pi\)
0.999817 0.0191534i \(-0.00609708\pi\)
\(468\) 0 0
\(469\) 2.23503i 0.103204i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.0016i 0.735756i
\(474\) 0 0
\(475\) 5.15332 0.236451
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −22.9671 −1.04939 −0.524696 0.851290i \(-0.675821\pi\)
−0.524696 + 0.851290i \(0.675821\pi\)
\(480\) 0 0
\(481\) −35.4336 −1.61563
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.00169 −0.0454844
\(486\) 0 0
\(487\) 33.4665i 1.51651i 0.651957 + 0.758256i \(0.273948\pi\)
−0.651957 + 0.758256i \(0.726052\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19.8776i 0.897065i 0.893767 + 0.448532i \(0.148053\pi\)
−0.893767 + 0.448532i \(0.851947\pi\)
\(492\) 0 0
\(493\) 13.9905i 0.630102i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 7.08606i − 0.317853i
\(498\) 0 0
\(499\) 6.04452 0.270590 0.135295 0.990805i \(-0.456802\pi\)
0.135295 + 0.990805i \(0.456802\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −0.252024 −0.0112372 −0.00561859 0.999984i \(-0.501788\pi\)
−0.00561859 + 0.999984i \(0.501788\pi\)
\(504\) 0 0
\(505\) −3.59993 −0.160195
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 27.1848 1.20495 0.602473 0.798139i \(-0.294182\pi\)
0.602473 + 0.798139i \(0.294182\pi\)
\(510\) 0 0
\(511\) − 30.5061i − 1.34951i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.413908i 0.0182390i
\(516\) 0 0
\(517\) 16.3923i 0.720933i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 37.6451i − 1.64926i −0.565671 0.824631i \(-0.691383\pi\)
0.565671 0.824631i \(-0.308617\pi\)
\(522\) 0 0
\(523\) 2.20180 0.0962780 0.0481390 0.998841i \(-0.484671\pi\)
0.0481390 + 0.998841i \(0.484671\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.96723 −0.259937
\(528\) 0 0
\(529\) 6.56206 0.285307
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.95602 0.171354
\(534\) 0 0
\(535\) − 2.36871i − 0.102408i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 11.2468i − 0.484436i
\(540\) 0 0
\(541\) 8.04196i 0.345751i 0.984944 + 0.172875i \(0.0553058\pi\)
−0.984944 + 0.172875i \(0.944694\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 1.61588i − 0.0692168i
\(546\) 0 0
\(547\) −21.1745 −0.905357 −0.452679 0.891674i \(-0.649532\pi\)
−0.452679 + 0.891674i \(0.649532\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.01981 0.128648
\(552\) 0 0
\(553\) 43.1403 1.83451
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.5871 −0.787562 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(558\) 0 0
\(559\) − 12.8682i − 0.544269i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.0085i 0.801115i 0.916272 + 0.400557i \(0.131183\pi\)
−0.916272 + 0.400557i \(0.868817\pi\)
\(564\) 0 0
\(565\) − 4.07092i − 0.171265i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.82654i 0.160417i 0.996778 + 0.0802084i \(0.0255586\pi\)
−0.996778 + 0.0802084i \(0.974441\pi\)
\(570\) 0 0
\(571\) −14.6174 −0.611720 −0.305860 0.952076i \(-0.598944\pi\)
−0.305860 + 0.952076i \(0.598944\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 26.9526 1.12400
\(576\) 0 0
\(577\) −1.28305 −0.0534142 −0.0267071 0.999643i \(-0.508502\pi\)
−0.0267071 + 0.999643i \(0.508502\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.8776 −0.534252
\(582\) 0 0
\(583\) 39.7830i 1.64765i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 5.45348i − 0.225089i −0.993647 0.112545i \(-0.964100\pi\)
0.993647 0.112545i \(-0.0359001\pi\)
\(588\) 0 0
\(589\) 1.28801i 0.0530715i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 25.7774i 1.05855i 0.848450 + 0.529276i \(0.177536\pi\)
−0.848450 + 0.529276i \(0.822464\pi\)
\(594\) 0 0
\(595\) −3.02966 −0.124204
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −39.6141 −1.61859 −0.809295 0.587403i \(-0.800151\pi\)
−0.809295 + 0.587403i \(0.800151\pi\)
\(600\) 0 0
\(601\) 3.44785 0.140641 0.0703204 0.997524i \(-0.477598\pi\)
0.0703204 + 0.997524i \(0.477598\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.94546 0.119750
\(606\) 0 0
\(607\) 23.0029i 0.933660i 0.884347 + 0.466830i \(0.154604\pi\)
−0.884347 + 0.466830i \(0.845396\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 13.1824i − 0.533304i
\(612\) 0 0
\(613\) 6.82786i 0.275775i 0.990448 + 0.137887i \(0.0440312\pi\)
−0.990448 + 0.137887i \(0.955969\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.9201i 0.882469i 0.897392 + 0.441235i \(0.145459\pi\)
−0.897392 + 0.441235i \(0.854541\pi\)
\(618\) 0 0
\(619\) 37.0131 1.48768 0.743841 0.668356i \(-0.233002\pi\)
0.743841 + 0.668356i \(0.233002\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −35.2751 −1.41327
\(624\) 0 0
\(625\) 24.3594 0.974375
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −42.2462 −1.68447
\(630\) 0 0
\(631\) − 25.2815i − 1.00644i −0.864158 0.503220i \(-0.832149\pi\)
0.864158 0.503220i \(-0.167851\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.52168i 0.0603859i
\(636\) 0 0
\(637\) 9.04452i 0.358357i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 19.6465i − 0.775988i −0.921662 0.387994i \(-0.873168\pi\)
0.921662 0.387994i \(-0.126832\pi\)
\(642\) 0 0
\(643\) −17.2089 −0.678652 −0.339326 0.940669i \(-0.610199\pi\)
−0.339326 + 0.940669i \(0.610199\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −33.9988 −1.33663 −0.668315 0.743879i \(-0.732984\pi\)
−0.668315 + 0.743879i \(0.732984\pi\)
\(648\) 0 0
\(649\) 74.6167 2.92896
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −20.5636 −0.804718 −0.402359 0.915482i \(-0.631810\pi\)
−0.402359 + 0.915482i \(0.631810\pi\)
\(654\) 0 0
\(655\) 2.29159i 0.0895400i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 22.4586i − 0.874864i −0.899251 0.437432i \(-0.855888\pi\)
0.899251 0.437432i \(-0.144112\pi\)
\(660\) 0 0
\(661\) 17.1362i 0.666521i 0.942835 + 0.333260i \(0.108149\pi\)
−0.942835 + 0.333260i \(0.891851\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.653942i 0.0253588i
\(666\) 0 0
\(667\) 15.7941 0.611549
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 47.8843 1.84855
\(672\) 0 0
\(673\) −30.7595 −1.18569 −0.592845 0.805317i \(-0.701995\pi\)
−0.592845 + 0.805317i \(0.701995\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.3157 0.511764 0.255882 0.966708i \(-0.417634\pi\)
0.255882 + 0.966708i \(0.417634\pi\)
\(678\) 0 0
\(679\) 14.7120i 0.564594i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 27.4006i 1.04846i 0.851578 + 0.524228i \(0.175646\pi\)
−0.851578 + 0.524228i \(0.824354\pi\)
\(684\) 0 0
\(685\) 3.33410i 0.127389i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 31.9929i − 1.21883i
\(690\) 0 0
\(691\) 17.1790 0.653521 0.326761 0.945107i \(-0.394043\pi\)
0.326761 + 0.945107i \(0.394043\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.37592 −0.165988
\(696\) 0 0
\(697\) 4.71662 0.178655
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −42.7770 −1.61567 −0.807833 0.589412i \(-0.799360\pi\)
−0.807833 + 0.589412i \(0.799360\pi\)
\(702\) 0 0
\(703\) 9.11871i 0.343919i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 52.8729i 1.98849i
\(708\) 0 0
\(709\) 51.9018i 1.94921i 0.223928 + 0.974606i \(0.428112\pi\)
−0.223928 + 0.974606i \(0.571888\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.73647i 0.252283i
\(714\) 0 0
\(715\) −4.19941 −0.157049
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 29.6166 1.10451 0.552257 0.833674i \(-0.313767\pi\)
0.552257 + 0.833674i \(0.313767\pi\)
\(720\) 0 0
\(721\) 6.07914 0.226399
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.3999 0.534800
\(726\) 0 0
\(727\) 31.8177i 1.18005i 0.807384 + 0.590026i \(0.200882\pi\)
−0.807384 + 0.590026i \(0.799118\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 15.3424i − 0.567458i
\(732\) 0 0
\(733\) 23.4062i 0.864527i 0.901747 + 0.432263i \(0.142285\pi\)
−0.901747 + 0.432263i \(0.857715\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.69361i 0.136056i
\(738\) 0 0
\(739\) 14.0424 0.516558 0.258279 0.966070i \(-0.416845\pi\)
0.258279 + 0.966070i \(0.416845\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.8612 0.471832 0.235916 0.971773i \(-0.424191\pi\)
0.235916 + 0.971773i \(0.424191\pi\)
\(744\) 0 0
\(745\) 3.38239 0.123921
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −34.7897 −1.27119
\(750\) 0 0
\(751\) − 39.4254i − 1.43865i −0.694673 0.719326i \(-0.744451\pi\)
0.694673 0.719326i \(-0.255549\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.25810i 0.0457870i
\(756\) 0 0
\(757\) 14.4992i 0.526981i 0.964662 + 0.263491i \(0.0848738\pi\)
−0.964662 + 0.263491i \(0.915126\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 46.6306i − 1.69036i −0.534482 0.845180i \(-0.679493\pi\)
0.534482 0.845180i \(-0.320507\pi\)
\(762\) 0 0
\(763\) −23.7327 −0.859183
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −60.0055 −2.16667
\(768\) 0 0
\(769\) 23.7055 0.854841 0.427421 0.904053i \(-0.359422\pi\)
0.427421 + 0.904053i \(0.359422\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 24.1631 0.869085 0.434542 0.900651i \(-0.356910\pi\)
0.434542 + 0.900651i \(0.356910\pi\)
\(774\) 0 0
\(775\) 6.14185i 0.220622i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 1.01807i − 0.0364761i
\(780\) 0 0
\(781\) − 11.7104i − 0.419032i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.97635i 0.106230i
\(786\) 0 0
\(787\) −27.5199 −0.980979 −0.490490 0.871447i \(-0.663182\pi\)
−0.490490 + 0.871447i \(0.663182\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −59.7904 −2.12590
\(792\) 0 0
\(793\) −38.5078 −1.36745
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.6199 0.447019 0.223510 0.974702i \(-0.428249\pi\)
0.223510 + 0.974702i \(0.428249\pi\)
\(798\) 0 0
\(799\) − 15.7169i − 0.556026i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 50.4143i − 1.77908i
\(804\) 0 0
\(805\) 3.42021i 0.120547i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 15.5226i − 0.545744i −0.962050 0.272872i \(-0.912026\pi\)
0.962050 0.272872i \(-0.0879736\pi\)
\(810\) 0 0
\(811\) −48.4347 −1.70077 −0.850386 0.526159i \(-0.823632\pi\)
−0.850386 + 0.526159i \(0.823632\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.86141 −0.100231
\(816\) 0 0
\(817\) −3.31160 −0.115858
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0891 −0.421912 −0.210956 0.977496i \(-0.567658\pi\)
−0.210956 + 0.977496i \(0.567658\pi\)
\(822\) 0 0
\(823\) 5.10430i 0.177925i 0.996035 + 0.0889623i \(0.0283551\pi\)
−0.996035 + 0.0889623i \(0.971645\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 6.40603i − 0.222760i −0.993778 0.111380i \(-0.964473\pi\)
0.993778 0.111380i \(-0.0355270\pi\)
\(828\) 0 0
\(829\) 42.1518i 1.46399i 0.681309 + 0.731996i \(0.261411\pi\)
−0.681309 + 0.731996i \(0.738589\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 10.7835i 0.373625i
\(834\) 0 0
\(835\) 2.06473 0.0714528
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 52.3893 1.80868 0.904340 0.426813i \(-0.140364\pi\)
0.904340 + 0.426813i \(0.140364\pi\)
\(840\) 0 0
\(841\) −20.5617 −0.709025
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.686698 0.0236231
\(846\) 0 0
\(847\) − 43.2606i − 1.48645i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 47.6922i 1.63487i
\(852\) 0 0
\(853\) 28.2920i 0.968698i 0.874875 + 0.484349i \(0.160944\pi\)
−0.874875 + 0.484349i \(0.839056\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 32.5139i − 1.11065i −0.831632 0.555327i \(-0.812593\pi\)
0.831632 0.555327i \(-0.187407\pi\)
\(858\) 0 0
\(859\) 40.9453 1.39703 0.698517 0.715593i \(-0.253843\pi\)
0.698517 + 0.715593i \(0.253843\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.23907 0.314502 0.157251 0.987559i \(-0.449737\pi\)
0.157251 + 0.987559i \(0.449737\pi\)
\(864\) 0 0
\(865\) −4.72686 −0.160718
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 71.2936 2.41847
\(870\) 0 0
\(871\) − 2.97034i − 0.100646i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 6.26357i 0.211747i
\(876\) 0 0
\(877\) − 1.01556i − 0.0342930i −0.999853 0.0171465i \(-0.994542\pi\)
0.999853 0.0171465i \(-0.00545816\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 49.9531i 1.68296i 0.540285 + 0.841482i \(0.318316\pi\)
−0.540285 + 0.841482i \(0.681684\pi\)
\(882\) 0 0
\(883\) −40.0726 −1.34855 −0.674275 0.738480i \(-0.735544\pi\)
−0.674275 + 0.738480i \(0.735544\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 29.4591 0.989141 0.494570 0.869138i \(-0.335325\pi\)
0.494570 + 0.869138i \(0.335325\pi\)
\(888\) 0 0
\(889\) 22.3491 0.749566
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.39245 −0.113524
\(894\) 0 0
\(895\) − 3.19244i − 0.106711i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.59908i 0.120036i
\(900\) 0 0
\(901\) − 38.1440i − 1.27076i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.99053i 0.0994087i
\(906\) 0 0
\(907\) −36.3557 −1.20717 −0.603585 0.797299i \(-0.706262\pi\)
−0.603585 + 0.797299i \(0.706262\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28.3465 −0.939162 −0.469581 0.882889i \(-0.655595\pi\)
−0.469581 + 0.882889i \(0.655595\pi\)
\(912\) 0 0
\(913\) −21.2815 −0.704314
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 33.6570 1.11145
\(918\) 0 0
\(919\) 58.7435i 1.93777i 0.247514 + 0.968884i \(0.420386\pi\)
−0.247514 + 0.968884i \(0.579614\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9.41733i 0.309975i
\(924\) 0 0
\(925\) 43.4824i 1.42969i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 24.7981i 0.813599i 0.913517 + 0.406800i \(0.133355\pi\)
−0.913517 + 0.406800i \(0.866645\pi\)
\(930\) 0 0
\(931\) 2.32758 0.0762833
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.00681 −0.163740
\(936\) 0 0
\(937\) 25.1819 0.822656 0.411328 0.911487i \(-0.365065\pi\)
0.411328 + 0.911487i \(0.365065\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 36.6929 1.19615 0.598077 0.801439i \(-0.295932\pi\)
0.598077 + 0.801439i \(0.295932\pi\)
\(942\) 0 0
\(943\) − 5.32464i − 0.173394i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 21.7134i − 0.705590i −0.935701 0.352795i \(-0.885231\pi\)
0.935701 0.352795i \(-0.114769\pi\)
\(948\) 0 0
\(949\) 40.5424i 1.31606i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 16.5469i 0.536005i 0.963418 + 0.268003i \(0.0863636\pi\)
−0.963418 + 0.268003i \(0.913636\pi\)
\(954\) 0 0
\(955\) 5.26055 0.170228
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 48.9685 1.58128
\(960\) 0 0
\(961\) 29.4649 0.950481
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3.13707 −0.100986
\(966\) 0 0
\(967\) 27.0566i 0.870083i 0.900410 + 0.435041i \(0.143266\pi\)
−0.900410 + 0.435041i \(0.856734\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 41.0968i − 1.31886i −0.751766 0.659430i \(-0.770798\pi\)
0.751766 0.659430i \(-0.229202\pi\)
\(972\) 0 0
\(973\) 64.2699i 2.06040i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 46.7497i − 1.49566i −0.663892 0.747828i \(-0.731097\pi\)
0.663892 0.747828i \(-0.268903\pi\)
\(978\) 0 0
\(979\) −58.2956 −1.86314
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 45.5184 1.45181 0.725906 0.687794i \(-0.241421\pi\)
0.725906 + 0.687794i \(0.241421\pi\)
\(984\) 0 0
\(985\) −0.426465 −0.0135883
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −17.3202 −0.550749
\(990\) 0 0
\(991\) 22.8381i 0.725477i 0.931891 + 0.362739i \(0.118158\pi\)
−0.931891 + 0.362739i \(0.881842\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 0.0163781i 0 0.000519221i
\(996\) 0 0
\(997\) − 43.7357i − 1.38512i −0.721358 0.692562i \(-0.756482\pi\)
0.721358 0.692562i \(-0.243518\pi\)
\(998\) 0 0
\(999\) 0 0
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 5184.2.f.e.2591.8 yes 16
3.2 odd 2 inner 5184.2.f.e.2591.10 yes 16
4.3 odd 2 5184.2.f.b.2591.7 16
8.3 odd 2 inner 5184.2.f.e.2591.9 yes 16
8.5 even 2 5184.2.f.b.2591.10 yes 16
12.11 even 2 5184.2.f.b.2591.9 yes 16
24.5 odd 2 5184.2.f.b.2591.8 yes 16
24.11 even 2 inner 5184.2.f.e.2591.7 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5184.2.f.b.2591.7 16 4.3 odd 2
5184.2.f.b.2591.8 yes 16 24.5 odd 2
5184.2.f.b.2591.9 yes 16 12.11 even 2
5184.2.f.b.2591.10 yes 16 8.5 even 2
5184.2.f.e.2591.7 yes 16 24.11 even 2 inner
5184.2.f.e.2591.8 yes 16 1.1 even 1 trivial
5184.2.f.e.2591.9 yes 16 8.3 odd 2 inner
5184.2.f.e.2591.10 yes 16 3.2 odd 2 inner