Properties

Label 4608.2.a.ba.1.2
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1536)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.334904\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.08402 q^{5} -5.03127 q^{7} +O(q^{10})\) \(q-2.08402 q^{5} -5.03127 q^{7} -0.828427 q^{11} -2.94725 q^{13} -4.82843 q^{17} -2.82843 q^{19} -4.16804 q^{23} -0.656854 q^{25} -7.97852 q^{29} +5.03127 q^{31} +10.4853 q^{35} -7.11529 q^{37} -8.82843 q^{41} +12.4853 q^{43} -4.16804 q^{47} +18.3137 q^{49} +12.1466 q^{53} +1.72646 q^{55} -1.65685 q^{59} -7.11529 q^{61} +6.14214 q^{65} +2.34315 q^{67} -10.0625 q^{71} +4.00000 q^{73} +4.16804 q^{77} -5.03127 q^{79} +3.17157 q^{83} +10.0625 q^{85} -10.0000 q^{89} +14.8284 q^{91} +5.89450 q^{95} +0.343146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{11} - 8 q^{17} + 20 q^{25} + 8 q^{35} - 24 q^{41} + 16 q^{43} + 28 q^{49} + 16 q^{59} - 32 q^{65} + 32 q^{67} + 16 q^{73} + 24 q^{83} - 40 q^{89} + 48 q^{91} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −2.08402 −0.932003 −0.466001 0.884784i \(-0.654306\pi\)
−0.466001 + 0.884784i \(0.654306\pi\)
\(6\) 0 0
\(7\) −5.03127 −1.90164 −0.950821 0.309740i \(-0.899758\pi\)
−0.950821 + 0.309740i \(0.899758\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 0 0
\(13\) −2.94725 −0.817420 −0.408710 0.912664i \(-0.634021\pi\)
−0.408710 + 0.912664i \(0.634021\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.82843 −1.17107 −0.585533 0.810649i \(-0.699115\pi\)
−0.585533 + 0.810649i \(0.699115\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.16804 −0.869097 −0.434549 0.900648i \(-0.643092\pi\)
−0.434549 + 0.900648i \(0.643092\pi\)
\(24\) 0 0
\(25\) −0.656854 −0.131371
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.97852 −1.48157 −0.740787 0.671740i \(-0.765547\pi\)
−0.740787 + 0.671740i \(0.765547\pi\)
\(30\) 0 0
\(31\) 5.03127 0.903643 0.451822 0.892108i \(-0.350774\pi\)
0.451822 + 0.892108i \(0.350774\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.4853 1.77234
\(36\) 0 0
\(37\) −7.11529 −1.16975 −0.584874 0.811124i \(-0.698856\pi\)
−0.584874 + 0.811124i \(0.698856\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.82843 −1.37877 −0.689384 0.724396i \(-0.742119\pi\)
−0.689384 + 0.724396i \(0.742119\pi\)
\(42\) 0 0
\(43\) 12.4853 1.90399 0.951994 0.306117i \(-0.0990300\pi\)
0.951994 + 0.306117i \(0.0990300\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.16804 −0.607972 −0.303986 0.952677i \(-0.598318\pi\)
−0.303986 + 0.952677i \(0.598318\pi\)
\(48\) 0 0
\(49\) 18.3137 2.61624
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.1466 1.66846 0.834230 0.551417i \(-0.185913\pi\)
0.834230 + 0.551417i \(0.185913\pi\)
\(54\) 0 0
\(55\) 1.72646 0.232796
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.65685 −0.215704 −0.107852 0.994167i \(-0.534397\pi\)
−0.107852 + 0.994167i \(0.534397\pi\)
\(60\) 0 0
\(61\) −7.11529 −0.911020 −0.455510 0.890231i \(-0.650543\pi\)
−0.455510 + 0.890231i \(0.650543\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.14214 0.761838
\(66\) 0 0
\(67\) 2.34315 0.286261 0.143130 0.989704i \(-0.454283\pi\)
0.143130 + 0.989704i \(0.454283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.0625 −1.19420 −0.597102 0.802165i \(-0.703681\pi\)
−0.597102 + 0.802165i \(0.703681\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.16804 0.474993
\(78\) 0 0
\(79\) −5.03127 −0.566062 −0.283031 0.959111i \(-0.591340\pi\)
−0.283031 + 0.959111i \(0.591340\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.17157 0.348125 0.174063 0.984735i \(-0.444310\pi\)
0.174063 + 0.984735i \(0.444310\pi\)
\(84\) 0 0
\(85\) 10.0625 1.09144
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 14.8284 1.55444
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.89450 0.604763
\(96\) 0 0
\(97\) 0.343146 0.0348412 0.0174206 0.999848i \(-0.494455\pi\)
0.0174206 + 0.999848i \(0.494455\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −12.1466 −1.20863 −0.604314 0.796746i \(-0.706553\pi\)
−0.604314 + 0.796746i \(0.706553\pi\)
\(102\) 0 0
\(103\) 3.30481 0.325633 0.162816 0.986656i \(-0.447942\pi\)
0.162816 + 0.986656i \(0.447942\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 8.84175 0.846886 0.423443 0.905923i \(-0.360821\pi\)
0.423443 + 0.905923i \(0.360821\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.6569 −1.09658 −0.548292 0.836287i \(-0.684722\pi\)
−0.548292 + 0.836287i \(0.684722\pi\)
\(114\) 0 0
\(115\) 8.68629 0.810001
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 24.2931 2.22695
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.7890 1.05444
\(126\) 0 0
\(127\) −13.3674 −1.18616 −0.593081 0.805143i \(-0.702088\pi\)
−0.593081 + 0.805143i \(0.702088\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.3137 1.33796 0.668982 0.743278i \(-0.266730\pi\)
0.668982 + 0.743278i \(0.266730\pi\)
\(132\) 0 0
\(133\) 14.2306 1.23395
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.1421 −1.03737 −0.518686 0.854965i \(-0.673579\pi\)
−0.518686 + 0.854965i \(0.673579\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.44158 0.204175
\(144\) 0 0
\(145\) 16.6274 1.38083
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.08402 −0.170730 −0.0853648 0.996350i \(-0.527206\pi\)
−0.0853648 + 0.996350i \(0.527206\pi\)
\(150\) 0 0
\(151\) 13.3674 1.08782 0.543910 0.839143i \(-0.316943\pi\)
0.543910 + 0.839143i \(0.316943\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −10.4853 −0.842198
\(156\) 0 0
\(157\) 15.4514 1.23315 0.616577 0.787294i \(-0.288519\pi\)
0.616577 + 0.787294i \(0.288519\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 20.9706 1.65271
\(162\) 0 0
\(163\) 18.1421 1.42100 0.710501 0.703696i \(-0.248468\pi\)
0.710501 + 0.703696i \(0.248468\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.2306 1.10120 0.550598 0.834771i \(-0.314400\pi\)
0.550598 + 0.834771i \(0.314400\pi\)
\(168\) 0 0
\(169\) −4.31371 −0.331824
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.25206 −0.475336 −0.237668 0.971346i \(-0.576383\pi\)
−0.237668 + 0.971346i \(0.576383\pi\)
\(174\) 0 0
\(175\) 3.30481 0.249820
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −20.9706 −1.56741 −0.783707 0.621131i \(-0.786673\pi\)
−0.783707 + 0.621131i \(0.786673\pi\)
\(180\) 0 0
\(181\) −8.84175 −0.657202 −0.328601 0.944469i \(-0.606577\pi\)
−0.328601 + 0.944469i \(0.606577\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 14.8284 1.09021
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −26.0196 −1.88271 −0.941356 0.337415i \(-0.890447\pi\)
−0.941356 + 0.337415i \(0.890447\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.4827 1.45933 0.729664 0.683806i \(-0.239676\pi\)
0.729664 + 0.683806i \(0.239676\pi\)
\(198\) 0 0
\(199\) −5.03127 −0.356657 −0.178329 0.983971i \(-0.557069\pi\)
−0.178329 + 0.983971i \(0.557069\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 40.1421 2.81743
\(204\) 0 0
\(205\) 18.3986 1.28502
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.34315 0.162079
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −26.0196 −1.77452
\(216\) 0 0
\(217\) −25.3137 −1.71841
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 14.2306 0.957253
\(222\) 0 0
\(223\) 13.3674 0.895145 0.447572 0.894248i \(-0.352289\pi\)
0.447572 + 0.894248i \(0.352289\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.1421 1.07139 0.535696 0.844411i \(-0.320049\pi\)
0.535696 + 0.844411i \(0.320049\pi\)
\(228\) 0 0
\(229\) −5.38883 −0.356104 −0.178052 0.984021i \(-0.556980\pi\)
−0.178052 + 0.984021i \(0.556980\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.31371 −0.0860639 −0.0430320 0.999074i \(-0.513702\pi\)
−0.0430320 + 0.999074i \(0.513702\pi\)
\(234\) 0 0
\(235\) 8.68629 0.566631
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.2306 0.920500 0.460250 0.887789i \(-0.347760\pi\)
0.460250 + 0.887789i \(0.347760\pi\)
\(240\) 0 0
\(241\) 16.9706 1.09317 0.546585 0.837404i \(-0.315928\pi\)
0.546585 + 0.837404i \(0.315928\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −38.1662 −2.43835
\(246\) 0 0
\(247\) 8.33609 0.530412
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.1716 0.957621 0.478811 0.877918i \(-0.341068\pi\)
0.478811 + 0.877918i \(0.341068\pi\)
\(252\) 0 0
\(253\) 3.45292 0.217083
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.34315 0.520431 0.260216 0.965551i \(-0.416206\pi\)
0.260216 + 0.965551i \(0.416206\pi\)
\(258\) 0 0
\(259\) 35.7990 2.22444
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.33609 0.514025 0.257013 0.966408i \(-0.417262\pi\)
0.257013 + 0.966408i \(0.417262\pi\)
\(264\) 0 0
\(265\) −25.3137 −1.55501
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.97852 0.486459 0.243230 0.969969i \(-0.421793\pi\)
0.243230 + 0.969969i \(0.421793\pi\)
\(270\) 0 0
\(271\) −3.30481 −0.200753 −0.100377 0.994950i \(-0.532005\pi\)
−0.100377 + 0.994950i \(0.532005\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.544156 0.0328138
\(276\) 0 0
\(277\) −25.5139 −1.53298 −0.766492 0.642254i \(-0.777999\pi\)
−0.766492 + 0.642254i \(0.777999\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −11.6569 −0.695390 −0.347695 0.937608i \(-0.613035\pi\)
−0.347695 + 0.937608i \(0.613035\pi\)
\(282\) 0 0
\(283\) −4.97056 −0.295469 −0.147735 0.989027i \(-0.547198\pi\)
−0.147735 + 0.989027i \(0.547198\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 44.4182 2.62193
\(288\) 0 0
\(289\) 6.31371 0.371395
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.1466 0.709610 0.354805 0.934940i \(-0.384547\pi\)
0.354805 + 0.934940i \(0.384547\pi\)
\(294\) 0 0
\(295\) 3.45292 0.201037
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.2843 0.710418
\(300\) 0 0
\(301\) −62.8169 −3.62070
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 14.8284 0.849073
\(306\) 0 0
\(307\) 10.3431 0.590315 0.295157 0.955449i \(-0.404628\pi\)
0.295157 + 0.955449i \(0.404628\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −15.3137 −0.865582 −0.432791 0.901494i \(-0.642471\pi\)
−0.432791 + 0.901494i \(0.642471\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −23.9356 −1.34436 −0.672178 0.740390i \(-0.734641\pi\)
−0.672178 + 0.740390i \(0.734641\pi\)
\(318\) 0 0
\(319\) 6.60963 0.370068
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 13.6569 0.759888
\(324\) 0 0
\(325\) 1.93591 0.107385
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 20.9706 1.15614
\(330\) 0 0
\(331\) −21.6569 −1.19037 −0.595184 0.803589i \(-0.702921\pi\)
−0.595184 + 0.803589i \(0.702921\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.88317 −0.266796
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.16804 −0.225712
\(342\) 0 0
\(343\) −56.9224 −3.07352
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.82843 −0.259204 −0.129602 0.991566i \(-0.541370\pi\)
−0.129602 + 0.991566i \(0.541370\pi\)
\(348\) 0 0
\(349\) −27.2404 −1.45814 −0.729072 0.684437i \(-0.760048\pi\)
−0.729072 + 0.684437i \(0.760048\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.3137 1.13441 0.567207 0.823575i \(-0.308024\pi\)
0.567207 + 0.823575i \(0.308024\pi\)
\(354\) 0 0
\(355\) 20.9706 1.11300
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.60963 −0.348843 −0.174421 0.984671i \(-0.555805\pi\)
−0.174421 + 0.984671i \(0.555805\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.33609 −0.436331
\(366\) 0 0
\(367\) 16.8203 0.878011 0.439006 0.898484i \(-0.355331\pi\)
0.439006 + 0.898484i \(0.355331\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −61.1127 −3.17281
\(372\) 0 0
\(373\) 18.9043 0.978828 0.489414 0.872052i \(-0.337211\pi\)
0.489414 + 0.872052i \(0.337211\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 23.5147 1.21107
\(378\) 0 0
\(379\) −16.4853 −0.846792 −0.423396 0.905945i \(-0.639162\pi\)
−0.423396 + 0.905945i \(0.639162\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.89450 −0.301195 −0.150598 0.988595i \(-0.548120\pi\)
−0.150598 + 0.988595i \(0.548120\pi\)
\(384\) 0 0
\(385\) −8.68629 −0.442694
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.08402 0.105664 0.0528320 0.998603i \(-0.483175\pi\)
0.0528320 + 0.998603i \(0.483175\pi\)
\(390\) 0 0
\(391\) 20.1251 1.01777
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 10.4853 0.527572
\(396\) 0 0
\(397\) 15.4514 0.775483 0.387741 0.921768i \(-0.373255\pi\)
0.387741 + 0.921768i \(0.373255\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.14214 0.206848 0.103424 0.994637i \(-0.467020\pi\)
0.103424 + 0.994637i \(0.467020\pi\)
\(402\) 0 0
\(403\) −14.8284 −0.738657
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.89450 0.292180
\(408\) 0 0
\(409\) 7.65685 0.378607 0.189304 0.981919i \(-0.439377\pi\)
0.189304 + 0.981919i \(0.439377\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.33609 0.410192
\(414\) 0 0
\(415\) −6.60963 −0.324454
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.8284 1.01754 0.508768 0.860904i \(-0.330101\pi\)
0.508768 + 0.860904i \(0.330101\pi\)
\(420\) 0 0
\(421\) −5.38883 −0.262636 −0.131318 0.991340i \(-0.541921\pi\)
−0.131318 + 0.991340i \(0.541921\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.17157 0.153844
\(426\) 0 0
\(427\) 35.7990 1.73243
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −15.9570 −0.768624 −0.384312 0.923203i \(-0.625561\pi\)
−0.384312 + 0.923203i \(0.625561\pi\)
\(432\) 0 0
\(433\) −20.6274 −0.991290 −0.495645 0.868525i \(-0.665068\pi\)
−0.495645 + 0.868525i \(0.665068\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 11.7890 0.563945
\(438\) 0 0
\(439\) −8.48419 −0.404928 −0.202464 0.979290i \(-0.564895\pi\)
−0.202464 + 0.979290i \(0.564895\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.51472 −0.0719665 −0.0359832 0.999352i \(-0.511456\pi\)
−0.0359832 + 0.999352i \(0.511456\pi\)
\(444\) 0 0
\(445\) 20.8402 0.987921
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −34.4853 −1.62746 −0.813731 0.581242i \(-0.802567\pi\)
−0.813731 + 0.581242i \(0.802567\pi\)
\(450\) 0 0
\(451\) 7.31371 0.344389
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −30.9028 −1.44874
\(456\) 0 0
\(457\) −12.3431 −0.577388 −0.288694 0.957421i \(-0.593221\pi\)
−0.288694 + 0.957421i \(0.593221\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −29.8301 −1.38933 −0.694663 0.719336i \(-0.744446\pi\)
−0.694663 + 0.719336i \(0.744446\pi\)
\(462\) 0 0
\(463\) 23.4299 1.08888 0.544440 0.838800i \(-0.316742\pi\)
0.544440 + 0.838800i \(0.316742\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.79899 −0.0832473 −0.0416237 0.999133i \(-0.513253\pi\)
−0.0416237 + 0.999133i \(0.513253\pi\)
\(468\) 0 0
\(469\) −11.7890 −0.544366
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.3431 −0.475578
\(474\) 0 0
\(475\) 1.85786 0.0852447
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −41.9766 −1.91796 −0.958981 0.283471i \(-0.908514\pi\)
−0.958981 + 0.283471i \(0.908514\pi\)
\(480\) 0 0
\(481\) 20.9706 0.956175
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.715123 −0.0324721
\(486\) 0 0
\(487\) 6.75773 0.306222 0.153111 0.988209i \(-0.451071\pi\)
0.153111 + 0.988209i \(0.451071\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.65685 −0.435808 −0.217904 0.975970i \(-0.569922\pi\)
−0.217904 + 0.975970i \(0.569922\pi\)
\(492\) 0 0
\(493\) 38.5237 1.73502
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 50.6274 2.27095
\(498\) 0 0
\(499\) 41.6569 1.86482 0.932408 0.361406i \(-0.117703\pi\)
0.932408 + 0.361406i \(0.117703\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −18.3986 −0.820354 −0.410177 0.912006i \(-0.634533\pi\)
−0.410177 + 0.912006i \(0.634533\pi\)
\(504\) 0 0
\(505\) 25.3137 1.12645
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −16.3146 −0.723132 −0.361566 0.932346i \(-0.617758\pi\)
−0.361566 + 0.932346i \(0.617758\pi\)
\(510\) 0 0
\(511\) −20.1251 −0.890282
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.88730 −0.303491
\(516\) 0 0
\(517\) 3.45292 0.151859
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −20.8284 −0.912510 −0.456255 0.889849i \(-0.650810\pi\)
−0.456255 + 0.889849i \(0.650810\pi\)
\(522\) 0 0
\(523\) −18.8284 −0.823310 −0.411655 0.911340i \(-0.635049\pi\)
−0.411655 + 0.911340i \(0.635049\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −24.2931 −1.05823
\(528\) 0 0
\(529\) −5.62742 −0.244670
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 26.0196 1.12703
\(534\) 0 0
\(535\) 8.33609 0.360400
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −15.1716 −0.653486
\(540\) 0 0
\(541\) 11.2833 0.485109 0.242554 0.970138i \(-0.422015\pi\)
0.242554 + 0.970138i \(0.422015\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −18.4264 −0.789301
\(546\) 0 0
\(547\) −31.7990 −1.35963 −0.679813 0.733385i \(-0.737939\pi\)
−0.679813 + 0.733385i \(0.737939\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 22.5667 0.961373
\(552\) 0 0
\(553\) 25.3137 1.07645
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.5882 0.618120 0.309060 0.951043i \(-0.399986\pi\)
0.309060 + 0.951043i \(0.399986\pi\)
\(558\) 0 0
\(559\) −36.7973 −1.55636
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −0.828427 −0.0349140 −0.0174570 0.999848i \(-0.505557\pi\)
−0.0174570 + 0.999848i \(0.505557\pi\)
\(564\) 0 0
\(565\) 24.2931 1.02202
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −23.4558 −0.983320 −0.491660 0.870787i \(-0.663610\pi\)
−0.491660 + 0.870787i \(0.663610\pi\)
\(570\) 0 0
\(571\) −23.3137 −0.975648 −0.487824 0.872942i \(-0.662209\pi\)
−0.487824 + 0.872942i \(0.662209\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.73780 0.114174
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −15.9570 −0.662010
\(582\) 0 0
\(583\) −10.0625 −0.416748
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.97056 −0.205157 −0.102579 0.994725i \(-0.532709\pi\)
−0.102579 + 0.994725i \(0.532709\pi\)
\(588\) 0 0
\(589\) −14.2306 −0.586361
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.97056 0.121986 0.0609932 0.998138i \(-0.480573\pi\)
0.0609932 + 0.998138i \(0.480573\pi\)
\(594\) 0 0
\(595\) −50.6274 −2.07552
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0.715123 0.0292191 0.0146096 0.999893i \(-0.495349\pi\)
0.0146096 + 0.999893i \(0.495349\pi\)
\(600\) 0 0
\(601\) −4.68629 −0.191158 −0.0955789 0.995422i \(-0.530470\pi\)
−0.0955789 + 0.995422i \(0.530470\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 21.4940 0.873855
\(606\) 0 0
\(607\) −25.1564 −1.02107 −0.510533 0.859858i \(-0.670552\pi\)
−0.510533 + 0.859858i \(0.670552\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.2843 0.496968
\(612\) 0 0
\(613\) 10.5682 0.426846 0.213423 0.976960i \(-0.431539\pi\)
0.213423 + 0.976960i \(0.431539\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 43.9411 1.76900 0.884502 0.466537i \(-0.154499\pi\)
0.884502 + 0.466537i \(0.154499\pi\)
\(618\) 0 0
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 50.3127 2.01574
\(624\) 0 0
\(625\) −21.2843 −0.851371
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 34.3557 1.36985
\(630\) 0 0
\(631\) 35.2189 1.40204 0.701021 0.713140i \(-0.252728\pi\)
0.701021 + 0.713140i \(0.252728\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 27.8579 1.10551
\(636\) 0 0
\(637\) −53.9751 −2.13857
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −32.8284 −1.29664 −0.648322 0.761366i \(-0.724529\pi\)
−0.648322 + 0.761366i \(0.724529\pi\)
\(642\) 0 0
\(643\) −3.51472 −0.138607 −0.0693035 0.997596i \(-0.522078\pi\)
−0.0693035 + 0.997596i \(0.522078\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 44.4182 1.74626 0.873130 0.487487i \(-0.162086\pi\)
0.873130 + 0.487487i \(0.162086\pi\)
\(648\) 0 0
\(649\) 1.37258 0.0538786
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 30.5452 1.19533 0.597663 0.801747i \(-0.296096\pi\)
0.597663 + 0.801747i \(0.296096\pi\)
\(654\) 0 0
\(655\) −31.9141 −1.24699
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.65685 −0.376178 −0.188089 0.982152i \(-0.560229\pi\)
−0.188089 + 0.982152i \(0.560229\pi\)
\(660\) 0 0
\(661\) 27.2404 1.05953 0.529764 0.848145i \(-0.322280\pi\)
0.529764 + 0.848145i \(0.322280\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −29.6569 −1.15004
\(666\) 0 0
\(667\) 33.2548 1.28763
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.89450 0.227555
\(672\) 0 0
\(673\) 37.3137 1.43834 0.719169 0.694835i \(-0.244523\pi\)
0.719169 + 0.694835i \(0.244523\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −22.2091 −0.853566 −0.426783 0.904354i \(-0.640353\pi\)
−0.426783 + 0.904354i \(0.640353\pi\)
\(678\) 0 0
\(679\) −1.72646 −0.0662555
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.4853 −0.860375 −0.430188 0.902739i \(-0.641553\pi\)
−0.430188 + 0.902739i \(0.641553\pi\)
\(684\) 0 0
\(685\) 25.3045 0.966834
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −35.7990 −1.36383
\(690\) 0 0
\(691\) −33.1716 −1.26191 −0.630953 0.775821i \(-0.717336\pi\)
−0.630953 + 0.775821i \(0.717336\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.6722 −0.632412
\(696\) 0 0
\(697\) 42.6274 1.61463
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.52560 0.170930 0.0854649 0.996341i \(-0.472762\pi\)
0.0854649 + 0.996341i \(0.472762\pi\)
\(702\) 0 0
\(703\) 20.1251 0.759032
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 61.1127 2.29838
\(708\) 0 0
\(709\) 19.6194 0.736823 0.368411 0.929663i \(-0.379902\pi\)
0.368411 + 0.929663i \(0.379902\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −20.9706 −0.785354
\(714\) 0 0
\(715\) −5.08831 −0.190292
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.60963 0.246497 0.123249 0.992376i \(-0.460669\pi\)
0.123249 + 0.992376i \(0.460669\pi\)
\(720\) 0 0
\(721\) −16.6274 −0.619237
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.24073 0.194636
\(726\) 0 0
\(727\) 36.9454 1.37023 0.685114 0.728436i \(-0.259752\pi\)
0.685114 + 0.728436i \(0.259752\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −60.2843 −2.22969
\(732\) 0 0
\(733\) 43.1974 1.59553 0.797767 0.602966i \(-0.206015\pi\)
0.797767 + 0.602966i \(0.206015\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.94113 −0.0715023
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −17.6835 −0.648745 −0.324373 0.945929i \(-0.605153\pi\)
−0.324373 + 0.945929i \(0.605153\pi\)
\(744\) 0 0
\(745\) 4.34315 0.159121
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 20.1251 0.735355
\(750\) 0 0
\(751\) −5.03127 −0.183594 −0.0917969 0.995778i \(-0.529261\pi\)
−0.0917969 + 0.995778i \(0.529261\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −27.8579 −1.01385
\(756\) 0 0
\(757\) −17.1778 −0.624339 −0.312170 0.950026i \(-0.601056\pi\)
−0.312170 + 0.950026i \(0.601056\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.7696 1.55040 0.775198 0.631719i \(-0.217650\pi\)
0.775198 + 0.631719i \(0.217650\pi\)
\(762\) 0 0
\(763\) −44.4853 −1.61048
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.88317 0.176321
\(768\) 0 0
\(769\) 12.0000 0.432731 0.216366 0.976312i \(-0.430580\pi\)
0.216366 + 0.976312i \(0.430580\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −32.2717 −1.16073 −0.580365 0.814356i \(-0.697090\pi\)
−0.580365 + 0.814356i \(0.697090\pi\)
\(774\) 0 0
\(775\) −3.30481 −0.118712
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.9706 0.894663
\(780\) 0 0
\(781\) 8.33609 0.298289
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −32.2010 −1.14930
\(786\) 0 0
\(787\) −52.7696 −1.88103 −0.940516 0.339750i \(-0.889657\pi\)
−0.940516 + 0.339750i \(0.889657\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 58.6488 2.08531
\(792\) 0 0
\(793\) 20.9706 0.744687
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.5452 −1.08197 −0.540983 0.841033i \(-0.681948\pi\)
−0.540983 + 0.841033i \(0.681948\pi\)
\(798\) 0 0
\(799\) 20.1251 0.711975
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.31371 −0.116938
\(804\) 0 0
\(805\) −43.7031 −1.54033
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.45584 0.121501 0.0607505 0.998153i \(-0.480651\pi\)
0.0607505 + 0.998153i \(0.480651\pi\)
\(810\) 0 0
\(811\) −49.4558 −1.73663 −0.868315 0.496014i \(-0.834797\pi\)
−0.868315 + 0.496014i \(0.834797\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −37.8086 −1.32438
\(816\) 0 0
\(817\) −35.3137 −1.23547
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 36.4397 1.27175 0.635877 0.771790i \(-0.280638\pi\)
0.635877 + 0.771790i \(0.280638\pi\)
\(822\) 0 0
\(823\) −13.3674 −0.465957 −0.232978 0.972482i \(-0.574847\pi\)
−0.232978 + 0.972482i \(0.574847\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −53.9411 −1.87572 −0.937858 0.347018i \(-0.887194\pi\)
−0.937858 + 0.347018i \(0.887194\pi\)
\(828\) 0 0
\(829\) −2.94725 −0.102362 −0.0511811 0.998689i \(-0.516299\pi\)
−0.0511811 + 0.998689i \(0.516299\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −88.4264 −3.06379
\(834\) 0 0
\(835\) −29.6569 −1.02632
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −41.9766 −1.44919 −0.724597 0.689172i \(-0.757974\pi\)
−0.724597 + 0.689172i \(0.757974\pi\)
\(840\) 0 0
\(841\) 34.6569 1.19506
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8.98986 0.309261
\(846\) 0 0
\(847\) 51.8911 1.78300
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 29.6569 1.01662
\(852\) 0 0
\(853\) 15.4514 0.529045 0.264523 0.964379i \(-0.414786\pi\)
0.264523 + 0.964379i \(0.414786\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.4853 −1.04136 −0.520679 0.853753i \(-0.674321\pi\)
−0.520679 + 0.853753i \(0.674321\pi\)
\(858\) 0 0
\(859\) −21.4558 −0.732064 −0.366032 0.930602i \(-0.619284\pi\)
−0.366032 + 0.930602i \(0.619284\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −14.2306 −0.484415 −0.242207 0.970224i \(-0.577871\pi\)
−0.242207 + 0.970224i \(0.577871\pi\)
\(864\) 0 0
\(865\) 13.0294 0.443014
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.16804 0.141391
\(870\) 0 0
\(871\) −6.90584 −0.233995
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −59.3137 −2.00517
\(876\) 0 0
\(877\) −22.3572 −0.754950 −0.377475 0.926020i \(-0.623208\pi\)
−0.377475 + 0.926020i \(0.623208\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 38.9706 1.31295 0.656476 0.754347i \(-0.272046\pi\)
0.656476 + 0.754347i \(0.272046\pi\)
\(882\) 0 0
\(883\) −24.7696 −0.833562 −0.416781 0.909007i \(-0.636842\pi\)
−0.416781 + 0.909007i \(0.636842\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.33609 0.279898 0.139949 0.990159i \(-0.455306\pi\)
0.139949 + 0.990159i \(0.455306\pi\)
\(888\) 0 0
\(889\) 67.2548 2.25565
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.7890 0.394504
\(894\) 0 0
\(895\) 43.7031 1.46083
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −40.1421 −1.33882
\(900\) 0 0
\(901\) −58.6488 −1.95388
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.4264 0.612514
\(906\) 0 0
\(907\) 23.7990 0.790232 0.395116 0.918631i \(-0.370704\pi\)
0.395116 + 0.918631i \(0.370704\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 34.3557 1.13825 0.569127 0.822249i \(-0.307281\pi\)
0.569127 + 0.822249i \(0.307281\pi\)
\(912\) 0 0
\(913\) −2.62742 −0.0869548
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −77.0474 −2.54433
\(918\) 0 0
\(919\) 15.0938 0.497899 0.248950 0.968516i \(-0.419915\pi\)
0.248950 + 0.968516i \(0.419915\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 29.6569 0.976167
\(924\) 0 0
\(925\) 4.67371 0.153671
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 17.7990 0.583966 0.291983 0.956424i \(-0.405685\pi\)
0.291983 + 0.956424i \(0.405685\pi\)
\(930\) 0 0
\(931\) −51.7990 −1.69764
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.33609 −0.272619
\(936\) 0 0
\(937\) 4.62742 0.151171 0.0755856 0.997139i \(-0.475917\pi\)
0.0755856 + 0.997139i \(0.475917\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −51.3854 −1.67512 −0.837558 0.546348i \(-0.816018\pi\)
−0.837558 + 0.546348i \(0.816018\pi\)
\(942\) 0 0
\(943\) 36.7973 1.19828
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.97056 0.161522 0.0807608 0.996734i \(-0.474265\pi\)
0.0807608 + 0.996734i \(0.474265\pi\)
\(948\) 0 0
\(949\) −11.7890 −0.382687
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.79899 0.317420 0.158710 0.987325i \(-0.449266\pi\)
0.158710 + 0.987325i \(0.449266\pi\)
\(954\) 0 0
\(955\) 54.2254 1.75469
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 61.0904 1.97271
\(960\) 0 0
\(961\) −5.68629 −0.183429
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 8.33609 0.268348
\(966\) 0 0
\(967\) −18.5467 −0.596423 −0.298211 0.954500i \(-0.596390\pi\)
−0.298211 + 0.954500i \(0.596390\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 35.4558 1.13783 0.568916 0.822396i \(-0.307363\pi\)
0.568916 + 0.822396i \(0.307363\pi\)
\(972\) 0 0
\(973\) −40.2502 −1.29036
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −30.7696 −0.984405 −0.492203 0.870481i \(-0.663808\pi\)
−0.492203 + 0.870481i \(0.663808\pi\)
\(978\) 0 0
\(979\) 8.28427 0.264766
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.45292 0.110131 0.0550655 0.998483i \(-0.482463\pi\)
0.0550655 + 0.998483i \(0.482463\pi\)
\(984\) 0 0
\(985\) −42.6863 −1.36010
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −52.0392 −1.65475
\(990\) 0 0
\(991\) −28.6093 −0.908804 −0.454402 0.890797i \(-0.650147\pi\)
−0.454402 + 0.890797i \(0.650147\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.4853 0.332406
\(996\) 0 0
\(997\) 9.55688 0.302669 0.151335 0.988483i \(-0.451643\pi\)
0.151335 + 0.988483i \(0.451643\pi\)
\(998\) 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 4608.2.a.ba.1.2 4
3.2 odd 2 1536.2.a.m.1.3 yes 4
4.3 odd 2 4608.2.a.t.1.2 4
8.3 odd 2 inner 4608.2.a.ba.1.3 4
8.5 even 2 4608.2.a.t.1.3 4
12.11 even 2 1536.2.a.n.1.3 yes 4
16.3 odd 4 4608.2.d.p.2305.5 8
16.5 even 4 4608.2.d.p.2305.4 8
16.11 odd 4 4608.2.d.p.2305.3 8
16.13 even 4 4608.2.d.p.2305.6 8
24.5 odd 2 1536.2.a.n.1.2 yes 4
24.11 even 2 1536.2.a.m.1.2 4
48.5 odd 4 1536.2.d.g.769.7 8
48.11 even 4 1536.2.d.g.769.3 8
48.29 odd 4 1536.2.d.g.769.2 8
48.35 even 4 1536.2.d.g.769.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.m.1.2 4 24.11 even 2
1536.2.a.m.1.3 yes 4 3.2 odd 2
1536.2.a.n.1.2 yes 4 24.5 odd 2
1536.2.a.n.1.3 yes 4 12.11 even 2
1536.2.d.g.769.2 8 48.29 odd 4
1536.2.d.g.769.3 8 48.11 even 4
1536.2.d.g.769.6 8 48.35 even 4
1536.2.d.g.769.7 8 48.5 odd 4
4608.2.a.t.1.2 4 4.3 odd 2
4608.2.a.t.1.3 4 8.5 even 2
4608.2.a.ba.1.2 4 1.1 even 1 trivial
4608.2.a.ba.1.3 4 8.3 odd 2 inner
4608.2.d.p.2305.3 8 16.11 odd 4
4608.2.d.p.2305.4 8 16.5 even 4
4608.2.d.p.2305.5 8 16.3 odd 4
4608.2.d.p.2305.6 8 16.13 even 4