Properties

Label 1728.2.c.g.1727.8
Level $1728$
Weight $2$
Character 1728.1727
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.8
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.g.1727.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.14626i q^{5} +3.44949i q^{7} +O(q^{10})\) \(q+3.14626i q^{5} +3.44949i q^{7} +4.56048 q^{11} +6.89898 q^{13} +3.46410i q^{17} -4.89898i q^{19} +2.82843 q^{23} -4.89898 q^{25} -2.19275i q^{29} -2.55051i q^{31} -10.8530 q^{35} +4.89898 q^{37} +4.09978i q^{41} -2.89898i q^{43} -2.19275 q^{47} -4.89898 q^{49} +12.9029i q^{53} +14.3485i q^{55} -2.19275 q^{59} -4.00000 q^{61} +21.7060i q^{65} -14.8990i q^{67} -13.2207 q^{71} -7.89898 q^{73} +15.7313i q^{77} +2.00000i q^{79} +12.4101 q^{83} -10.8990 q^{85} -5.02118i q^{89} +23.7980i q^{91} +15.4135 q^{95} -5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 16 q^{13} - 32 q^{61} - 24 q^{73} - 48 q^{85} - 40 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) 3.14626i 1.40705i 0.710669 + 0.703526i \(0.248392\pi\)
−0.710669 + 0.703526i \(0.751608\pi\)
\(6\) 0 0
\(7\) 3.44949i 1.30378i 0.758312 + 0.651892i \(0.226025\pi\)
−0.758312 + 0.651892i \(0.773975\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.56048 1.37504 0.687518 0.726167i \(-0.258700\pi\)
0.687518 + 0.726167i \(0.258700\pi\)
\(12\) 0 0
\(13\) 6.89898 1.91343 0.956716 0.291022i \(-0.0939953\pi\)
0.956716 + 0.291022i \(0.0939953\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) − 4.89898i − 1.12390i −0.827170 0.561951i \(-0.810051\pi\)
0.827170 0.561951i \(-0.189949\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) −4.89898 −0.979796
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.19275i − 0.407184i −0.979056 0.203592i \(-0.934738\pi\)
0.979056 0.203592i \(-0.0652616\pi\)
\(30\) 0 0
\(31\) − 2.55051i − 0.458085i −0.973416 0.229043i \(-0.926440\pi\)
0.973416 0.229043i \(-0.0735595\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −10.8530 −1.83449
\(36\) 0 0
\(37\) 4.89898 0.805387 0.402694 0.915335i \(-0.368074\pi\)
0.402694 + 0.915335i \(0.368074\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.09978i 0.640277i 0.947371 + 0.320139i \(0.103730\pi\)
−0.947371 + 0.320139i \(0.896270\pi\)
\(42\) 0 0
\(43\) − 2.89898i − 0.442090i −0.975264 0.221045i \(-0.929053\pi\)
0.975264 0.221045i \(-0.0709468\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.19275 −0.319846 −0.159923 0.987130i \(-0.551125\pi\)
−0.159923 + 0.987130i \(0.551125\pi\)
\(48\) 0 0
\(49\) −4.89898 −0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.9029i 1.77235i 0.463352 + 0.886174i \(0.346647\pi\)
−0.463352 + 0.886174i \(0.653353\pi\)
\(54\) 0 0
\(55\) 14.3485i 1.93475i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.19275 −0.285472 −0.142736 0.989761i \(-0.545590\pi\)
−0.142736 + 0.989761i \(0.545590\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 21.7060i 2.69230i
\(66\) 0 0
\(67\) − 14.8990i − 1.82020i −0.414389 0.910100i \(-0.636005\pi\)
0.414389 0.910100i \(-0.363995\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.2207 −1.56901 −0.784506 0.620121i \(-0.787083\pi\)
−0.784506 + 0.620121i \(0.787083\pi\)
\(72\) 0 0
\(73\) −7.89898 −0.924506 −0.462253 0.886748i \(-0.652959\pi\)
−0.462253 + 0.886748i \(0.652959\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.7313i 1.79275i
\(78\) 0 0
\(79\) 2.00000i 0.225018i 0.993651 + 0.112509i \(0.0358886\pi\)
−0.993651 + 0.112509i \(0.964111\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.4101 1.36218 0.681092 0.732198i \(-0.261505\pi\)
0.681092 + 0.732198i \(0.261505\pi\)
\(84\) 0 0
\(85\) −10.8990 −1.18216
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 5.02118i − 0.532244i −0.963939 0.266122i \(-0.914258\pi\)
0.963939 0.266122i \(-0.0857424\pi\)
\(90\) 0 0
\(91\) 23.7980i 2.49470i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.4135 1.58139
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.58919i 0.158130i 0.996869 + 0.0790650i \(0.0251935\pi\)
−0.996869 + 0.0790650i \(0.974807\pi\)
\(102\) 0 0
\(103\) − 10.0000i − 0.985329i −0.870219 0.492665i \(-0.836023\pi\)
0.870219 0.492665i \(-0.163977\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.66025 0.837218 0.418609 0.908166i \(-0.362518\pi\)
0.418609 + 0.908166i \(0.362518\pi\)
\(108\) 0 0
\(109\) −4.89898 −0.469237 −0.234619 0.972088i \(-0.575384\pi\)
−0.234619 + 0.972088i \(0.575384\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 14.1421i − 1.33038i −0.746674 0.665190i \(-0.768350\pi\)
0.746674 0.665190i \(-0.231650\pi\)
\(114\) 0 0
\(115\) 8.89898i 0.829834i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.9494 −1.09540
\(120\) 0 0
\(121\) 9.79796 0.890724
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.317837i 0.0284282i
\(126\) 0 0
\(127\) − 9.24745i − 0.820578i −0.911955 0.410289i \(-0.865428\pi\)
0.911955 0.410289i \(-0.134572\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −14.3171 −1.25089 −0.625446 0.780268i \(-0.715083\pi\)
−0.625446 + 0.780268i \(0.715083\pi\)
\(132\) 0 0
\(133\) 16.8990 1.46533
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.6062i 1.50420i 0.659048 + 0.752101i \(0.270960\pi\)
−0.659048 + 0.752101i \(0.729040\pi\)
\(138\) 0 0
\(139\) − 20.0000i − 1.69638i −0.529694 0.848189i \(-0.677693\pi\)
0.529694 0.848189i \(-0.322307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 31.4626 2.63104
\(144\) 0 0
\(145\) 6.89898 0.572929
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 6.61037i − 0.541542i −0.962644 0.270771i \(-0.912721\pi\)
0.962644 0.270771i \(-0.0872787\pi\)
\(150\) 0 0
\(151\) − 7.24745i − 0.589789i −0.955530 0.294895i \(-0.904715\pi\)
0.955530 0.294895i \(-0.0952845\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.02458 0.644550
\(156\) 0 0
\(157\) −5.79796 −0.462728 −0.231364 0.972867i \(-0.574319\pi\)
−0.231364 + 0.972867i \(0.574319\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 9.75663i 0.768930i
\(162\) 0 0
\(163\) 6.00000i 0.469956i 0.972001 + 0.234978i \(0.0755019\pi\)
−0.972001 + 0.234978i \(0.924498\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.6637 −0.902561 −0.451280 0.892382i \(-0.649033\pi\)
−0.451280 + 0.892382i \(0.649033\pi\)
\(168\) 0 0
\(169\) 34.5959 2.66122
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 8.80312i − 0.669289i −0.942344 0.334644i \(-0.891384\pi\)
0.942344 0.334644i \(-0.108616\pi\)
\(174\) 0 0
\(175\) − 16.8990i − 1.27744i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.66025 −0.647298 −0.323649 0.946177i \(-0.604910\pi\)
−0.323649 + 0.946177i \(0.604910\pi\)
\(180\) 0 0
\(181\) −9.79796 −0.728277 −0.364138 0.931345i \(-0.618636\pi\)
−0.364138 + 0.931345i \(0.618636\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 15.4135i 1.13322i
\(186\) 0 0
\(187\) 15.7980i 1.15526i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.6780 −0.772635 −0.386318 0.922366i \(-0.626253\pi\)
−0.386318 + 0.922366i \(0.626253\pi\)
\(192\) 0 0
\(193\) 11.8990 0.856507 0.428254 0.903659i \(-0.359129\pi\)
0.428254 + 0.903659i \(0.359129\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.87492i 0.133582i 0.997767 + 0.0667911i \(0.0212761\pi\)
−0.997767 + 0.0667911i \(0.978724\pi\)
\(198\) 0 0
\(199\) 5.65153i 0.400626i 0.979732 + 0.200313i \(0.0641960\pi\)
−0.979732 + 0.200313i \(0.935804\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.56388 0.530880
\(204\) 0 0
\(205\) −12.8990 −0.900904
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 22.3417i − 1.54541i
\(210\) 0 0
\(211\) 20.4949i 1.41093i 0.708746 + 0.705463i \(0.249261\pi\)
−0.708746 + 0.705463i \(0.750739\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.12096 0.622044
\(216\) 0 0
\(217\) 8.79796 0.597244
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 23.8988i 1.60760i
\(222\) 0 0
\(223\) 19.7980i 1.32577i 0.748721 + 0.662885i \(0.230668\pi\)
−0.748721 + 0.662885i \(0.769332\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −22.9774 −1.52506 −0.762531 0.646952i \(-0.776043\pi\)
−0.762531 + 0.646952i \(0.776043\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 7.56388i − 0.495526i −0.968821 0.247763i \(-0.920305\pi\)
0.968821 0.247763i \(-0.0796954\pi\)
\(234\) 0 0
\(235\) − 6.89898i − 0.450040i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7.21393 −0.466630 −0.233315 0.972401i \(-0.574957\pi\)
−0.233315 + 0.972401i \(0.574957\pi\)
\(240\) 0 0
\(241\) −0.202041 −0.0130146 −0.00650730 0.999979i \(-0.502071\pi\)
−0.00650730 + 0.999979i \(0.502071\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 15.4135i − 0.984731i
\(246\) 0 0
\(247\) − 33.7980i − 2.15051i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.3205 1.09326 0.546630 0.837374i \(-0.315910\pi\)
0.546630 + 0.837374i \(0.315910\pi\)
\(252\) 0 0
\(253\) 12.8990 0.810952
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 22.9774i − 1.43329i −0.697439 0.716644i \(-0.745677\pi\)
0.697439 0.716644i \(-0.254323\pi\)
\(258\) 0 0
\(259\) 16.8990i 1.05005i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −30.5412 −1.88325 −0.941627 0.336659i \(-0.890703\pi\)
−0.941627 + 0.336659i \(0.890703\pi\)
\(264\) 0 0
\(265\) −40.5959 −2.49379
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 10.3923i − 0.633630i −0.948487 0.316815i \(-0.897387\pi\)
0.948487 0.316815i \(-0.102613\pi\)
\(270\) 0 0
\(271\) 17.4495i 1.05998i 0.848004 + 0.529991i \(0.177805\pi\)
−0.848004 + 0.529991i \(0.822195\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −22.3417 −1.34725
\(276\) 0 0
\(277\) −5.79796 −0.348366 −0.174183 0.984713i \(-0.555728\pi\)
−0.174183 + 0.984713i \(0.555728\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 21.7060i − 1.29487i −0.762120 0.647436i \(-0.775841\pi\)
0.762120 0.647436i \(-0.224159\pi\)
\(282\) 0 0
\(283\) − 12.8990i − 0.766765i −0.923590 0.383382i \(-0.874759\pi\)
0.923590 0.383382i \(-0.125241\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.1421 −0.834784
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 28.6342i − 1.67283i −0.548098 0.836414i \(-0.684648\pi\)
0.548098 0.836414i \(-0.315352\pi\)
\(294\) 0 0
\(295\) − 6.89898i − 0.401674i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 19.5133 1.12848
\(300\) 0 0
\(301\) 10.0000 0.576390
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 12.5851i − 0.720618i
\(306\) 0 0
\(307\) 18.0000i 1.02731i 0.857996 + 0.513657i \(0.171710\pi\)
−0.857996 + 0.513657i \(0.828290\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.75663 0.553248 0.276624 0.960978i \(-0.410784\pi\)
0.276624 + 0.960978i \(0.410784\pi\)
\(312\) 0 0
\(313\) −19.4949 −1.10192 −0.550958 0.834533i \(-0.685738\pi\)
−0.550958 + 0.834533i \(0.685738\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.4456i − 0.867511i −0.901031 0.433755i \(-0.857188\pi\)
0.901031 0.433755i \(-0.142812\pi\)
\(318\) 0 0
\(319\) − 10.0000i − 0.559893i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 16.9706 0.944267
\(324\) 0 0
\(325\) −33.7980 −1.87477
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 7.56388i − 0.417010i
\(330\) 0 0
\(331\) − 28.6969i − 1.57733i −0.614825 0.788663i \(-0.710774\pi\)
0.614825 0.788663i \(-0.289226\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 46.8761 2.56112
\(336\) 0 0
\(337\) 23.7980 1.29636 0.648179 0.761488i \(-0.275531\pi\)
0.648179 + 0.761488i \(0.275531\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 11.6315i − 0.629884i
\(342\) 0 0
\(343\) 7.24745i 0.391325i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.8024 1.22410 0.612048 0.790820i \(-0.290346\pi\)
0.612048 + 0.790820i \(0.290346\pi\)
\(348\) 0 0
\(349\) −15.1010 −0.808339 −0.404170 0.914684i \(-0.632439\pi\)
−0.404170 + 0.914684i \(0.632439\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.1421i 0.752710i 0.926476 + 0.376355i \(0.122823\pi\)
−0.926476 + 0.376355i \(0.877177\pi\)
\(354\) 0 0
\(355\) − 41.5959i − 2.20768i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.349945 −0.0184694 −0.00923470 0.999957i \(-0.502940\pi\)
−0.00923470 + 0.999957i \(0.502940\pi\)
\(360\) 0 0
\(361\) −5.00000 −0.263158
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 24.8523i − 1.30083i
\(366\) 0 0
\(367\) 13.4495i 0.702058i 0.936365 + 0.351029i \(0.114168\pi\)
−0.936365 + 0.351029i \(0.885832\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −44.5084 −2.31076
\(372\) 0 0
\(373\) 14.4949 0.750517 0.375259 0.926920i \(-0.377554\pi\)
0.375259 + 0.926920i \(0.377554\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 15.1278i − 0.779119i
\(378\) 0 0
\(379\) 8.00000i 0.410932i 0.978664 + 0.205466i \(0.0658711\pi\)
−0.978664 + 0.205466i \(0.934129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.5344 1.25365 0.626826 0.779160i \(-0.284354\pi\)
0.626826 + 0.779160i \(0.284354\pi\)
\(384\) 0 0
\(385\) −49.4949 −2.52249
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 21.6739i 1.09891i 0.835523 + 0.549455i \(0.185165\pi\)
−0.835523 + 0.549455i \(0.814835\pi\)
\(390\) 0 0
\(391\) 9.79796i 0.495504i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.29253 −0.316611
\(396\) 0 0
\(397\) 20.6969 1.03875 0.519375 0.854547i \(-0.326165\pi\)
0.519375 + 0.854547i \(0.326165\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.6274i 1.12996i 0.825105 + 0.564980i \(0.191116\pi\)
−0.825105 + 0.564980i \(0.808884\pi\)
\(402\) 0 0
\(403\) − 17.5959i − 0.876515i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 22.3417 1.10744
\(408\) 0 0
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 7.56388i − 0.372194i
\(414\) 0 0
\(415\) 39.0454i 1.91666i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 19.1633 0.936189 0.468095 0.883678i \(-0.344941\pi\)
0.468095 + 0.883678i \(0.344941\pi\)
\(420\) 0 0
\(421\) 10.2020 0.497217 0.248609 0.968604i \(-0.420027\pi\)
0.248609 + 0.968604i \(0.420027\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 16.9706i − 0.823193i
\(426\) 0 0
\(427\) − 13.7980i − 0.667730i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.57826 −0.316864 −0.158432 0.987370i \(-0.550644\pi\)
−0.158432 + 0.987370i \(0.550644\pi\)
\(432\) 0 0
\(433\) −26.3939 −1.26841 −0.634204 0.773165i \(-0.718672\pi\)
−0.634204 + 0.773165i \(0.718672\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 13.8564i − 0.662842i
\(438\) 0 0
\(439\) 5.24745i 0.250447i 0.992129 + 0.125224i \(0.0399648\pi\)
−0.992129 + 0.125224i \(0.960035\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.00680 0.285392 0.142696 0.989767i \(-0.454423\pi\)
0.142696 + 0.989767i \(0.454423\pi\)
\(444\) 0 0
\(445\) 15.7980 0.748895
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 5.37113i − 0.253479i −0.991936 0.126740i \(-0.959549\pi\)
0.991936 0.126740i \(-0.0404512\pi\)
\(450\) 0 0
\(451\) 18.6969i 0.880404i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −74.8747 −3.51018
\(456\) 0 0
\(457\) 25.0000 1.16945 0.584725 0.811231i \(-0.301202\pi\)
0.584725 + 0.811231i \(0.301202\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.2672i 0.571341i 0.958328 + 0.285671i \(0.0922164\pi\)
−0.958328 + 0.285671i \(0.907784\pi\)
\(462\) 0 0
\(463\) 17.0454i 0.792167i 0.918214 + 0.396084i \(0.129631\pi\)
−0.918214 + 0.396084i \(0.870369\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17.1455 −0.793401 −0.396700 0.917948i \(-0.629845\pi\)
−0.396700 + 0.917948i \(0.629845\pi\)
\(468\) 0 0
\(469\) 51.3939 2.37315
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 13.2207i − 0.607890i
\(474\) 0 0
\(475\) 24.0000i 1.10120i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19.1633 0.875594 0.437797 0.899074i \(-0.355759\pi\)
0.437797 + 0.899074i \(0.355759\pi\)
\(480\) 0 0
\(481\) 33.7980 1.54105
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 15.7313i − 0.714323i
\(486\) 0 0
\(487\) − 15.7980i − 0.715874i −0.933746 0.357937i \(-0.883480\pi\)
0.933746 0.357937i \(-0.116520\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.00340 0.135542 0.0677708 0.997701i \(-0.478411\pi\)
0.0677708 + 0.997701i \(0.478411\pi\)
\(492\) 0 0
\(493\) 7.59592 0.342103
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 45.6048i − 2.04565i
\(498\) 0 0
\(499\) 5.10102i 0.228353i 0.993460 + 0.114177i \(0.0364229\pi\)
−0.993460 + 0.114177i \(0.963577\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −31.1127 −1.38725 −0.693623 0.720338i \(-0.743987\pi\)
−0.693623 + 0.720338i \(0.743987\pi\)
\(504\) 0 0
\(505\) −5.00000 −0.222497
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 44.0156i − 1.95096i −0.220095 0.975478i \(-0.570637\pi\)
0.220095 0.975478i \(-0.429363\pi\)
\(510\) 0 0
\(511\) − 27.2474i − 1.20536i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 31.4626 1.38641
\(516\) 0 0
\(517\) −10.0000 −0.439799
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 0.921404i − 0.0403674i −0.999796 0.0201837i \(-0.993575\pi\)
0.999796 0.0201837i \(-0.00642511\pi\)
\(522\) 0 0
\(523\) − 31.3939i − 1.37276i −0.727244 0.686379i \(-0.759199\pi\)
0.727244 0.686379i \(-0.240801\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.83523 0.384869
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 28.2843i 1.22513i
\(534\) 0 0
\(535\) 27.2474i 1.17801i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −22.3417 −0.962325
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 15.4135i − 0.660241i
\(546\) 0 0
\(547\) − 4.20204i − 0.179666i −0.995957 0.0898332i \(-0.971367\pi\)
0.995957 0.0898332i \(-0.0286334\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10.7423 −0.457635
\(552\) 0 0
\(553\) −6.89898 −0.293374
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 29.8735i − 1.26578i −0.774242 0.632890i \(-0.781869\pi\)
0.774242 0.632890i \(-0.218131\pi\)
\(558\) 0 0
\(559\) − 20.0000i − 0.845910i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.48188 0.231034 0.115517 0.993306i \(-0.463148\pi\)
0.115517 + 0.993306i \(0.463148\pi\)
\(564\) 0 0
\(565\) 44.4949 1.87191
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 19.5133i − 0.818038i −0.912526 0.409019i \(-0.865871\pi\)
0.912526 0.409019i \(-0.134129\pi\)
\(570\) 0 0
\(571\) − 0.202041i − 0.00845515i −0.999991 0.00422758i \(-0.998654\pi\)
0.999991 0.00422758i \(-0.00134568\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −13.8564 −0.577852
\(576\) 0 0
\(577\) 15.7980 0.657678 0.328839 0.944386i \(-0.393343\pi\)
0.328839 + 0.944386i \(0.393343\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 42.8085i 1.77599i
\(582\) 0 0
\(583\) 58.8434i 2.43704i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −33.4804 −1.38188 −0.690942 0.722910i \(-0.742804\pi\)
−0.690942 + 0.722910i \(0.742804\pi\)
\(588\) 0 0
\(589\) −12.4949 −0.514843
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 42.7764i 1.75661i 0.478097 + 0.878307i \(0.341327\pi\)
−0.478097 + 0.878307i \(0.658673\pi\)
\(594\) 0 0
\(595\) − 37.5959i − 1.54128i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 36.8338 1.50499 0.752493 0.658600i \(-0.228851\pi\)
0.752493 + 0.658600i \(0.228851\pi\)
\(600\) 0 0
\(601\) 39.6969 1.61927 0.809636 0.586932i \(-0.199665\pi\)
0.809636 + 0.586932i \(0.199665\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 30.8270i 1.25329i
\(606\) 0 0
\(607\) 4.20204i 0.170556i 0.996357 + 0.0852778i \(0.0271778\pi\)
−0.996357 + 0.0852778i \(0.972822\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −15.1278 −0.612003
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.285729i 0.0115030i 0.999983 + 0.00575151i \(0.00183077\pi\)
−0.999983 + 0.00575151i \(0.998169\pi\)
\(618\) 0 0
\(619\) 17.5959i 0.707240i 0.935389 + 0.353620i \(0.115049\pi\)
−0.935389 + 0.353620i \(0.884951\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 17.3205 0.693932
\(624\) 0 0
\(625\) −25.4949 −1.01980
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.9706i 0.676661i
\(630\) 0 0
\(631\) − 27.7423i − 1.10441i −0.833710 0.552203i \(-0.813787\pi\)
0.833710 0.552203i \(-0.186213\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 29.0949 1.15460
\(636\) 0 0
\(637\) −33.7980 −1.33912
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.4989i 0.809657i 0.914393 + 0.404829i \(0.132669\pi\)
−0.914393 + 0.404829i \(0.867331\pi\)
\(642\) 0 0
\(643\) − 21.3939i − 0.843692i −0.906667 0.421846i \(-0.861382\pi\)
0.906667 0.421846i \(-0.138618\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.7846 −0.817127 −0.408564 0.912730i \(-0.633970\pi\)
−0.408564 + 0.912730i \(0.633970\pi\)
\(648\) 0 0
\(649\) −10.0000 −0.392534
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.23924i 0.0484952i 0.999706 + 0.0242476i \(0.00771901\pi\)
−0.999706 + 0.0242476i \(0.992281\pi\)
\(654\) 0 0
\(655\) − 45.0454i − 1.76007i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −17.4313 −0.679026 −0.339513 0.940601i \(-0.610262\pi\)
−0.339513 + 0.940601i \(0.610262\pi\)
\(660\) 0 0
\(661\) −44.4949 −1.73065 −0.865325 0.501210i \(-0.832888\pi\)
−0.865325 + 0.501210i \(0.832888\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 53.1687i 2.06179i
\(666\) 0 0
\(667\) − 6.20204i − 0.240144i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −18.2419 −0.704221
\(672\) 0 0
\(673\) 2.59592 0.100065 0.0500326 0.998748i \(-0.484067\pi\)
0.0500326 + 0.998748i \(0.484067\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 45.6048i − 1.75273i −0.481644 0.876367i \(-0.659960\pi\)
0.481644 0.876367i \(-0.340040\pi\)
\(678\) 0 0
\(679\) − 17.2474i − 0.661896i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.3923 0.397650 0.198825 0.980035i \(-0.436287\pi\)
0.198825 + 0.980035i \(0.436287\pi\)
\(684\) 0 0
\(685\) −55.3939 −2.11649
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 89.0168i 3.39127i
\(690\) 0 0
\(691\) − 4.00000i − 0.152167i −0.997101 0.0760836i \(-0.975758\pi\)
0.997101 0.0760836i \(-0.0242416\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 62.9253 2.38689
\(696\) 0 0
\(697\) −14.2020 −0.537941
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.2528i 0.500553i 0.968174 + 0.250276i \(0.0805215\pi\)
−0.968174 + 0.250276i \(0.919478\pi\)
\(702\) 0 0
\(703\) − 24.0000i − 0.905177i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.48188 −0.206167
\(708\) 0 0
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 7.21393i − 0.270164i
\(714\) 0 0
\(715\) 98.9898i 3.70201i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.92820 0.258378 0.129189 0.991620i \(-0.458763\pi\)
0.129189 + 0.991620i \(0.458763\pi\)
\(720\) 0 0
\(721\) 34.4949 1.28466
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 10.7423i 0.398957i
\(726\) 0 0
\(727\) 17.2474i 0.639672i 0.947473 + 0.319836i \(0.103628\pi\)
−0.947473 + 0.319836i \(0.896372\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 10.0424 0.371430
\(732\) 0 0
\(733\) 44.4949 1.64346 0.821728 0.569880i \(-0.193010\pi\)
0.821728 + 0.569880i \(0.193010\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 67.9465i − 2.50284i
\(738\) 0 0
\(739\) 39.3939i 1.44913i 0.689208 + 0.724564i \(0.257959\pi\)
−0.689208 + 0.724564i \(0.742041\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.17837 −0.116603 −0.0583016 0.998299i \(-0.518569\pi\)
−0.0583016 + 0.998299i \(0.518569\pi\)
\(744\) 0 0
\(745\) 20.7980 0.761978
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 29.8735i 1.09155i
\(750\) 0 0
\(751\) − 25.7423i − 0.939352i −0.882839 0.469676i \(-0.844371\pi\)
0.882839 0.469676i \(-0.155629\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 22.8024 0.829864
\(756\) 0 0
\(757\) −19.3939 −0.704882 −0.352441 0.935834i \(-0.614648\pi\)
−0.352441 + 0.935834i \(0.614648\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.2555i 1.09676i 0.836229 + 0.548381i \(0.184756\pi\)
−0.836229 + 0.548381i \(0.815244\pi\)
\(762\) 0 0
\(763\) − 16.8990i − 0.611784i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −15.1278 −0.546232
\(768\) 0 0
\(769\) −37.4949 −1.35210 −0.676050 0.736855i \(-0.736310\pi\)
−0.676050 + 0.736855i \(0.736310\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.349945i 0.0125867i 0.999980 + 0.00629333i \(0.00200324\pi\)
−0.999980 + 0.00629333i \(0.997997\pi\)
\(774\) 0 0
\(775\) 12.4949i 0.448830i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.0847 0.719610
\(780\) 0 0
\(781\) −60.2929 −2.15745
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 18.2419i − 0.651082i
\(786\) 0 0
\(787\) 43.1918i 1.53962i 0.638271 + 0.769811i \(0.279650\pi\)
−0.638271 + 0.769811i \(0.720350\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 48.7832 1.73453
\(792\) 0 0
\(793\) −27.5959 −0.979960
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 10.0745i − 0.356856i −0.983953 0.178428i \(-0.942899\pi\)
0.983953 0.178428i \(-0.0571012\pi\)
\(798\) 0 0
\(799\) − 7.59592i − 0.268724i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −36.0231 −1.27123
\(804\) 0 0
\(805\) −30.6969 −1.08192
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 5.02118i − 0.176535i −0.996097 0.0882676i \(-0.971867\pi\)
0.996097 0.0882676i \(-0.0281331\pi\)
\(810\) 0 0
\(811\) 24.4949i 0.860132i 0.902797 + 0.430066i \(0.141510\pi\)
−0.902797 + 0.430066i \(0.858490\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −18.8776 −0.661253
\(816\) 0 0
\(817\) −14.2020 −0.496867
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 11.6637i 0.407064i 0.979068 + 0.203532i \(0.0652421\pi\)
−0.979068 + 0.203532i \(0.934758\pi\)
\(822\) 0 0
\(823\) 44.3485i 1.54589i 0.634473 + 0.772945i \(0.281217\pi\)
−0.634473 + 0.772945i \(0.718783\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 51.2616 1.78254 0.891271 0.453471i \(-0.149815\pi\)
0.891271 + 0.453471i \(0.149815\pi\)
\(828\) 0 0
\(829\) −1.59592 −0.0554285 −0.0277143 0.999616i \(-0.508823\pi\)
−0.0277143 + 0.999616i \(0.508823\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 16.9706i − 0.587995i
\(834\) 0 0
\(835\) − 36.6969i − 1.26995i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −31.4626 −1.08621 −0.543106 0.839664i \(-0.682752\pi\)
−0.543106 + 0.839664i \(0.682752\pi\)
\(840\) 0 0
\(841\) 24.1918 0.834201
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 108.848i 3.74448i
\(846\) 0 0
\(847\) 33.7980i 1.16131i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.8564 0.474991
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22.9774i 0.784892i 0.919775 + 0.392446i \(0.128371\pi\)
−0.919775 + 0.392446i \(0.871629\pi\)
\(858\) 0 0
\(859\) 0.404082i 0.0137871i 0.999976 + 0.00689355i \(0.00219430\pi\)
−0.999976 + 0.00689355i \(0.997806\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.92820 −0.235839 −0.117919 0.993023i \(-0.537622\pi\)
−0.117919 + 0.993023i \(0.537622\pi\)
\(864\) 0 0
\(865\) 27.6969 0.941724
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9.12096i 0.309407i
\(870\) 0 0
\(871\) − 102.788i − 3.48283i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.09638 −0.0370643
\(876\) 0 0
\(877\) 35.7980 1.20881 0.604406 0.796677i \(-0.293411\pi\)
0.604406 + 0.796677i \(0.293411\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 41.2193i 1.38871i 0.719631 + 0.694356i \(0.244311\pi\)
−0.719631 + 0.694356i \(0.755689\pi\)
\(882\) 0 0
\(883\) − 43.1010i − 1.45046i −0.688504 0.725232i \(-0.741732\pi\)
0.688504 0.725232i \(-0.258268\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 17.0348 0.571972 0.285986 0.958234i \(-0.407679\pi\)
0.285986 + 0.958234i \(0.407679\pi\)
\(888\) 0 0
\(889\) 31.8990 1.06986
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 10.7423i 0.359476i
\(894\) 0 0
\(895\) − 27.2474i − 0.910782i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −5.59264 −0.186525
\(900\) 0 0
\(901\) −44.6969 −1.48907
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 30.8270i − 1.02472i
\(906\) 0 0
\(907\) 44.4949i 1.47743i 0.674019 + 0.738714i \(0.264567\pi\)
−0.674019 + 0.738714i \(0.735433\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −21.7060 −0.719152 −0.359576 0.933116i \(-0.617079\pi\)
−0.359576 + 0.933116i \(0.617079\pi\)
\(912\) 0 0
\(913\) 56.5959 1.87305
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 49.3867i − 1.63089i
\(918\) 0 0
\(919\) − 27.2474i − 0.898810i −0.893328 0.449405i \(-0.851636\pi\)
0.893328 0.449405i \(-0.148364\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −91.2096 −3.00220
\(924\) 0 0
\(925\) −24.0000 −0.789115
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 26.4415i 0.867516i 0.901029 + 0.433758i \(0.142813\pi\)
−0.901029 + 0.433758i \(0.857187\pi\)
\(930\) 0 0
\(931\) 24.0000i 0.786568i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −49.7046 −1.62551
\(936\) 0 0
\(937\) −15.0000 −0.490029 −0.245014 0.969519i \(-0.578793\pi\)
−0.245014 + 0.969519i \(0.578793\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 10.4244i 0.339826i 0.985459 + 0.169913i \(0.0543487\pi\)
−0.985459 + 0.169913i \(0.945651\pi\)
\(942\) 0 0
\(943\) 11.5959i 0.377615i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.4804 1.08797 0.543984 0.839096i \(-0.316915\pi\)
0.543984 + 0.839096i \(0.316915\pi\)
\(948\) 0 0
\(949\) −54.4949 −1.76898
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 44.6834i 1.44744i 0.690096 + 0.723718i \(0.257568\pi\)
−0.690096 + 0.723718i \(0.742432\pi\)
\(954\) 0 0
\(955\) − 33.5959i − 1.08714i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −60.7325 −1.96116
\(960\) 0 0
\(961\) 24.4949 0.790158
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 37.4373i 1.20515i
\(966\) 0 0
\(967\) 24.7526i 0.795988i 0.917388 + 0.397994i \(0.130294\pi\)
−0.917388 + 0.397994i \(0.869706\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 8.94598 0.287090 0.143545 0.989644i \(-0.454150\pi\)
0.143545 + 0.989644i \(0.454150\pi\)
\(972\) 0 0
\(973\) 68.9898 2.21171
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.17837i 0.101685i 0.998707 + 0.0508426i \(0.0161907\pi\)
−0.998707 + 0.0508426i \(0.983809\pi\)
\(978\) 0 0
\(979\) − 22.8990i − 0.731855i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.03556 −0.128714 −0.0643572 0.997927i \(-0.520500\pi\)
−0.0643572 + 0.997927i \(0.520500\pi\)
\(984\) 0 0
\(985\) −5.89898 −0.187957
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 8.19955i − 0.260731i
\(990\) 0 0
\(991\) − 33.2474i − 1.05614i −0.849201 0.528070i \(-0.822916\pi\)
0.849201 0.528070i \(-0.177084\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −17.7812 −0.563702
\(996\) 0 0
\(997\) 31.3031 0.991378 0.495689 0.868500i \(-0.334916\pi\)
0.495689 + 0.868500i \(0.334916\pi\)
\(998\) 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 1728.2.c.g.1727.8 8
3.2 odd 2 inner 1728.2.c.g.1727.2 8
4.3 odd 2 inner 1728.2.c.g.1727.7 8
8.3 odd 2 864.2.c.a.863.1 8
8.5 even 2 864.2.c.a.863.2 yes 8
12.11 even 2 inner 1728.2.c.g.1727.1 8
24.5 odd 2 864.2.c.a.863.8 yes 8
24.11 even 2 864.2.c.a.863.7 yes 8
72.5 odd 6 2592.2.s.h.1727.3 8
72.11 even 6 2592.2.s.a.863.1 8
72.13 even 6 2592.2.s.a.1727.1 8
72.29 odd 6 2592.2.s.a.863.2 8
72.43 odd 6 2592.2.s.h.863.3 8
72.59 even 6 2592.2.s.h.1727.4 8
72.61 even 6 2592.2.s.h.863.4 8
72.67 odd 6 2592.2.s.a.1727.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.c.a.863.1 8 8.3 odd 2
864.2.c.a.863.2 yes 8 8.5 even 2
864.2.c.a.863.7 yes 8 24.11 even 2
864.2.c.a.863.8 yes 8 24.5 odd 2
1728.2.c.g.1727.1 8 12.11 even 2 inner
1728.2.c.g.1727.2 8 3.2 odd 2 inner
1728.2.c.g.1727.7 8 4.3 odd 2 inner
1728.2.c.g.1727.8 8 1.1 even 1 trivial
2592.2.s.a.863.1 8 72.11 even 6
2592.2.s.a.863.2 8 72.29 odd 6
2592.2.s.a.1727.1 8 72.13 even 6
2592.2.s.a.1727.2 8 72.67 odd 6
2592.2.s.h.863.3 8 72.43 odd 6
2592.2.s.h.863.4 8 72.61 even 6
2592.2.s.h.1727.3 8 72.5 odd 6
2592.2.s.h.1727.4 8 72.59 even 6