Properties

Label 6048.2.c.d.3025.4
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.4
Root \(0.891007 - 0.453990i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.d.3025.13

$q$-expansion

\(f(q)\) \(=\) \(q-2.63506i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-2.63506i q^{5} +1.00000 q^{7} +4.29757i q^{11} +0.0480111i q^{13} +3.16228 q^{17} -0.726543i q^{19} -5.28146 q^{23} -1.94356 q^{25} -8.00208i q^{29} -4.90868 q^{31} -2.63506i q^{35} -9.10736i q^{37} -1.45412 q^{41} +8.85816i q^{43} -9.23016 q^{47} +1.00000 q^{49} -3.14726i q^{53} +11.3244 q^{55} -5.90515i q^{59} +2.19577i q^{61} +0.126512 q^{65} -8.55452i q^{67} -3.86725 q^{71} -7.56044 q^{73} +4.29757i q^{77} +14.6139 q^{79} -9.05240i q^{83} -8.33280i q^{85} -5.15394 q^{89} +0.0480111i q^{91} -1.91449 q^{95} +12.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{7} + O(q^{10}) \) \( 16q + 16q^{7} - 32q^{25} - 40q^{31} + 16q^{49} + 72q^{55} + 24q^{73} - 24q^{79} + 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(5\) − 2.63506i − 1.17844i −0.807974 0.589218i \(-0.799436\pi\)
0.807974 0.589218i \(-0.200564\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.29757i 1.29577i 0.761740 + 0.647883i \(0.224346\pi\)
−0.761740 + 0.647883i \(0.775654\pi\)
\(12\) 0 0
\(13\) 0.0480111i 0.0133159i 0.999978 + 0.00665794i \(0.00211930\pi\)
−0.999978 + 0.00665794i \(0.997881\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.16228 0.766965 0.383482 0.923548i \(-0.374725\pi\)
0.383482 + 0.923548i \(0.374725\pi\)
\(18\) 0 0
\(19\) − 0.726543i − 0.166680i −0.996521 0.0833401i \(-0.973441\pi\)
0.996521 0.0833401i \(-0.0265588\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.28146 −1.10126 −0.550631 0.834749i \(-0.685613\pi\)
−0.550631 + 0.834749i \(0.685613\pi\)
\(24\) 0 0
\(25\) −1.94356 −0.388712
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 8.00208i − 1.48595i −0.669319 0.742975i \(-0.733414\pi\)
0.669319 0.742975i \(-0.266586\pi\)
\(30\) 0 0
\(31\) −4.90868 −0.881625 −0.440812 0.897599i \(-0.645310\pi\)
−0.440812 + 0.897599i \(0.645310\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.63506i − 0.445407i
\(36\) 0 0
\(37\) − 9.10736i − 1.49724i −0.662999 0.748620i \(-0.730717\pi\)
0.662999 0.748620i \(-0.269283\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.45412 −0.227096 −0.113548 0.993533i \(-0.536222\pi\)
−0.113548 + 0.993533i \(0.536222\pi\)
\(42\) 0 0
\(43\) 8.85816i 1.35086i 0.737426 + 0.675428i \(0.236041\pi\)
−0.737426 + 0.675428i \(0.763959\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.23016 −1.34636 −0.673179 0.739480i \(-0.735072\pi\)
−0.673179 + 0.739480i \(0.735072\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 3.14726i − 0.432309i −0.976359 0.216154i \(-0.930649\pi\)
0.976359 0.216154i \(-0.0693515\pi\)
\(54\) 0 0
\(55\) 11.3244 1.52698
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 5.90515i − 0.768785i −0.923170 0.384392i \(-0.874411\pi\)
0.923170 0.384392i \(-0.125589\pi\)
\(60\) 0 0
\(61\) 2.19577i 0.281140i 0.990071 + 0.140570i \(0.0448935\pi\)
−0.990071 + 0.140570i \(0.955106\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.126512 0.0156919
\(66\) 0 0
\(67\) − 8.55452i − 1.04510i −0.852608 0.522550i \(-0.824981\pi\)
0.852608 0.522550i \(-0.175019\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.86725 −0.458958 −0.229479 0.973314i \(-0.573702\pi\)
−0.229479 + 0.973314i \(0.573702\pi\)
\(72\) 0 0
\(73\) −7.56044 −0.884883 −0.442441 0.896797i \(-0.645888\pi\)
−0.442441 + 0.896797i \(0.645888\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.29757i 0.489754i
\(78\) 0 0
\(79\) 14.6139 1.64419 0.822094 0.569352i \(-0.192806\pi\)
0.822094 + 0.569352i \(0.192806\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 9.05240i − 0.993631i −0.867856 0.496815i \(-0.834503\pi\)
0.867856 0.496815i \(-0.165497\pi\)
\(84\) 0 0
\(85\) − 8.33280i − 0.903819i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.15394 −0.546317 −0.273159 0.961969i \(-0.588068\pi\)
−0.273159 + 0.961969i \(0.588068\pi\)
\(90\) 0 0
\(91\) 0.0480111i 0.00503293i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.91449 −0.196422
\(96\) 0 0
\(97\) 12.1803 1.23673 0.618363 0.785893i \(-0.287796\pi\)
0.618363 + 0.785893i \(0.287796\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.35250i 0.433090i 0.976273 + 0.216545i \(0.0694788\pi\)
−0.976273 + 0.216545i \(0.930521\pi\)
\(102\) 0 0
\(103\) 12.3214 1.21406 0.607030 0.794679i \(-0.292361\pi\)
0.607030 + 0.794679i \(0.292361\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.69053i − 0.936819i −0.883512 0.468409i \(-0.844827\pi\)
0.883512 0.468409i \(-0.155173\pi\)
\(108\) 0 0
\(109\) − 9.08902i − 0.870570i −0.900293 0.435285i \(-0.856648\pi\)
0.900293 0.435285i \(-0.143352\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.5205 1.27190 0.635951 0.771729i \(-0.280608\pi\)
0.635951 + 0.771729i \(0.280608\pi\)
\(114\) 0 0
\(115\) 13.9170i 1.29777i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.16228 0.289886
\(120\) 0 0
\(121\) −7.46912 −0.679011
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 8.05391i − 0.720364i
\(126\) 0 0
\(127\) 10.9792 0.974242 0.487121 0.873334i \(-0.338047\pi\)
0.487121 + 0.873334i \(0.338047\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.770817i 0.0673466i 0.999433 + 0.0336733i \(0.0107206\pi\)
−0.999433 + 0.0336733i \(0.989279\pi\)
\(132\) 0 0
\(133\) − 0.726543i − 0.0629992i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.9782 −0.937933 −0.468967 0.883216i \(-0.655374\pi\)
−0.468967 + 0.883216i \(0.655374\pi\)
\(138\) 0 0
\(139\) − 22.0189i − 1.86762i −0.357767 0.933811i \(-0.616462\pi\)
0.357767 0.933811i \(-0.383538\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.206331 −0.0172543
\(144\) 0 0
\(145\) −21.0860 −1.75110
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 5.06643i − 0.415058i −0.978229 0.207529i \(-0.933458\pi\)
0.978229 0.207529i \(-0.0665422\pi\)
\(150\) 0 0
\(151\) 6.94054 0.564813 0.282407 0.959295i \(-0.408867\pi\)
0.282407 + 0.959295i \(0.408867\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12.9347i 1.03894i
\(156\) 0 0
\(157\) 18.4181i 1.46992i 0.678109 + 0.734962i \(0.262800\pi\)
−0.678109 + 0.734962i \(0.737200\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.28146 −0.416238
\(162\) 0 0
\(163\) − 10.5949i − 0.829859i −0.909854 0.414929i \(-0.863806\pi\)
0.909854 0.414929i \(-0.136194\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −17.3800 −1.34490 −0.672451 0.740142i \(-0.734758\pi\)
−0.672451 + 0.740142i \(0.734758\pi\)
\(168\) 0 0
\(169\) 12.9977 0.999823
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 22.9078i − 1.74165i −0.491595 0.870824i \(-0.663586\pi\)
0.491595 0.870824i \(-0.336414\pi\)
\(174\) 0 0
\(175\) −1.94356 −0.146919
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 21.6267i − 1.61646i −0.588869 0.808228i \(-0.700427\pi\)
0.588869 0.808228i \(-0.299573\pi\)
\(180\) 0 0
\(181\) 12.6144i 0.937619i 0.883299 + 0.468810i \(0.155317\pi\)
−0.883299 + 0.468810i \(0.844683\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −23.9985 −1.76440
\(186\) 0 0
\(187\) 13.5901i 0.993808i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −13.5578 −0.981006 −0.490503 0.871439i \(-0.663187\pi\)
−0.490503 + 0.871439i \(0.663187\pi\)
\(192\) 0 0
\(193\) −26.0614 −1.87594 −0.937971 0.346713i \(-0.887298\pi\)
−0.937971 + 0.346713i \(0.887298\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.18922i 0.227222i 0.993525 + 0.113611i \(0.0362418\pi\)
−0.993525 + 0.113611i \(0.963758\pi\)
\(198\) 0 0
\(199\) −25.2129 −1.78730 −0.893648 0.448768i \(-0.851863\pi\)
−0.893648 + 0.448768i \(0.851863\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 8.00208i − 0.561636i
\(204\) 0 0
\(205\) 3.83171i 0.267618i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.12237 0.215979
\(210\) 0 0
\(211\) − 5.25731i − 0.361928i −0.983490 0.180964i \(-0.942078\pi\)
0.983490 0.180964i \(-0.0579218\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 23.3418 1.59190
\(216\) 0 0
\(217\) −4.90868 −0.333223
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.151824i 0.0102128i
\(222\) 0 0
\(223\) −2.27095 −0.152074 −0.0760370 0.997105i \(-0.524227\pi\)
−0.0760370 + 0.997105i \(0.524227\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.44780i 0.0960940i 0.998845 + 0.0480470i \(0.0152997\pi\)
−0.998845 + 0.0480470i \(0.984700\pi\)
\(228\) 0 0
\(229\) 13.1128i 0.866516i 0.901270 + 0.433258i \(0.142636\pi\)
−0.901270 + 0.433258i \(0.857364\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.6984 0.831903 0.415951 0.909387i \(-0.363449\pi\)
0.415951 + 0.909387i \(0.363449\pi\)
\(234\) 0 0
\(235\) 24.3221i 1.58660i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.1437 −0.979564 −0.489782 0.871845i \(-0.662924\pi\)
−0.489782 + 0.871845i \(0.662924\pi\)
\(240\) 0 0
\(241\) −17.0920 −1.10099 −0.550497 0.834837i \(-0.685562\pi\)
−0.550497 + 0.834837i \(0.685562\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 2.63506i − 0.168348i
\(246\) 0 0
\(247\) 0.0348821 0.00221949
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 11.7844i − 0.743822i −0.928268 0.371911i \(-0.878703\pi\)
0.928268 0.371911i \(-0.121297\pi\)
\(252\) 0 0
\(253\) − 22.6975i − 1.42698i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.8905 −0.928847 −0.464423 0.885613i \(-0.653738\pi\)
−0.464423 + 0.885613i \(0.653738\pi\)
\(258\) 0 0
\(259\) − 9.10736i − 0.565904i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 17.1032 1.05463 0.527315 0.849670i \(-0.323199\pi\)
0.527315 + 0.849670i \(0.323199\pi\)
\(264\) 0 0
\(265\) −8.29322 −0.509448
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.32461i 0.141734i 0.997486 + 0.0708670i \(0.0225766\pi\)
−0.997486 + 0.0708670i \(0.977423\pi\)
\(270\) 0 0
\(271\) −12.4721 −0.757628 −0.378814 0.925473i \(-0.623668\pi\)
−0.378814 + 0.925473i \(0.623668\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 8.35259i − 0.503680i
\(276\) 0 0
\(277\) − 15.6257i − 0.938858i −0.882970 0.469429i \(-0.844460\pi\)
0.882970 0.469429i \(-0.155540\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −26.0872 −1.55623 −0.778115 0.628122i \(-0.783824\pi\)
−0.778115 + 0.628122i \(0.783824\pi\)
\(282\) 0 0
\(283\) 17.4083i 1.03482i 0.855739 + 0.517408i \(0.173103\pi\)
−0.855739 + 0.517408i \(0.826897\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.45412 −0.0858342
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 13.6647i − 0.798299i −0.916886 0.399149i \(-0.869305\pi\)
0.916886 0.399149i \(-0.130695\pi\)
\(294\) 0 0
\(295\) −15.5604 −0.905964
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 0.253569i − 0.0146643i
\(300\) 0 0
\(301\) 8.85816i 0.510576i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.78600 0.331306
\(306\) 0 0
\(307\) 8.43102i 0.481184i 0.970626 + 0.240592i \(0.0773415\pi\)
−0.970626 + 0.240592i \(0.922659\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 25.4195 1.44141 0.720703 0.693244i \(-0.243819\pi\)
0.720703 + 0.693244i \(0.243819\pi\)
\(312\) 0 0
\(313\) −24.3183 −1.37455 −0.687277 0.726396i \(-0.741194\pi\)
−0.687277 + 0.726396i \(0.741194\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.2593i − 0.857047i −0.903531 0.428523i \(-0.859034\pi\)
0.903531 0.428523i \(-0.140966\pi\)
\(318\) 0 0
\(319\) 34.3895 1.92544
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 2.29753i − 0.127838i
\(324\) 0 0
\(325\) − 0.0933124i − 0.00517604i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.23016 −0.508875
\(330\) 0 0
\(331\) 22.9170i 1.25963i 0.776744 + 0.629816i \(0.216870\pi\)
−0.776744 + 0.629816i \(0.783130\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −22.5417 −1.23158
\(336\) 0 0
\(337\) 34.0474 1.85468 0.927340 0.374221i \(-0.122090\pi\)
0.927340 + 0.374221i \(0.122090\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 21.0954i − 1.14238i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.72245i 0.0924662i 0.998931 + 0.0462331i \(0.0147217\pi\)
−0.998931 + 0.0462331i \(0.985278\pi\)
\(348\) 0 0
\(349\) − 7.95586i − 0.425868i −0.977067 0.212934i \(-0.931698\pi\)
0.977067 0.212934i \(-0.0683019\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −23.1711 −1.23327 −0.616637 0.787247i \(-0.711505\pi\)
−0.616637 + 0.787247i \(0.711505\pi\)
\(354\) 0 0
\(355\) 10.1905i 0.540853i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.7889 −0.991640 −0.495820 0.868425i \(-0.665133\pi\)
−0.495820 + 0.868425i \(0.665133\pi\)
\(360\) 0 0
\(361\) 18.4721 0.972218
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 19.9222i 1.04278i
\(366\) 0 0
\(367\) −25.1394 −1.31227 −0.656134 0.754645i \(-0.727809\pi\)
−0.656134 + 0.754645i \(0.727809\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 3.14726i − 0.163397i
\(372\) 0 0
\(373\) 25.0530i 1.29719i 0.761132 + 0.648597i \(0.224644\pi\)
−0.761132 + 0.648597i \(0.775356\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.384189 0.0197867
\(378\) 0 0
\(379\) − 15.2282i − 0.782222i −0.920344 0.391111i \(-0.872091\pi\)
0.920344 0.391111i \(-0.127909\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −29.5703 −1.51097 −0.755487 0.655164i \(-0.772600\pi\)
−0.755487 + 0.655164i \(0.772600\pi\)
\(384\) 0 0
\(385\) 11.3244 0.577143
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.90315i 0.0964934i 0.998835 + 0.0482467i \(0.0153634\pi\)
−0.998835 + 0.0482467i \(0.984637\pi\)
\(390\) 0 0
\(391\) −16.7015 −0.844629
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 38.5085i − 1.93757i
\(396\) 0 0
\(397\) 16.7660i 0.841462i 0.907185 + 0.420731i \(0.138226\pi\)
−0.907185 + 0.420731i \(0.861774\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −17.2567 −0.861759 −0.430880 0.902409i \(-0.641797\pi\)
−0.430880 + 0.902409i \(0.641797\pi\)
\(402\) 0 0
\(403\) − 0.235671i − 0.0117396i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 39.1395 1.94007
\(408\) 0 0
\(409\) 1.55671 0.0769744 0.0384872 0.999259i \(-0.487746\pi\)
0.0384872 + 0.999259i \(0.487746\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 5.90515i − 0.290573i
\(414\) 0 0
\(415\) −23.8537 −1.17093
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.1570i 1.27785i 0.769268 + 0.638926i \(0.220621\pi\)
−0.769268 + 0.638926i \(0.779379\pi\)
\(420\) 0 0
\(421\) 0.696870i 0.0339634i 0.999856 + 0.0169817i \(0.00540570\pi\)
−0.999856 + 0.0169817i \(0.994594\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.14608 −0.298128
\(426\) 0 0
\(427\) 2.19577i 0.106261i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.65116 −0.368543 −0.184272 0.982875i \(-0.558993\pi\)
−0.184272 + 0.982875i \(0.558993\pi\)
\(432\) 0 0
\(433\) 17.0860 0.821101 0.410550 0.911838i \(-0.365337\pi\)
0.410550 + 0.911838i \(0.365337\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.83721i 0.183559i
\(438\) 0 0
\(439\) 3.23837 0.154559 0.0772796 0.997009i \(-0.475377\pi\)
0.0772796 + 0.997009i \(0.475377\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.42501i 0.162727i 0.996684 + 0.0813636i \(0.0259275\pi\)
−0.996684 + 0.0813636i \(0.974073\pi\)
\(444\) 0 0
\(445\) 13.5810i 0.643800i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 39.7647 1.87661 0.938305 0.345808i \(-0.112395\pi\)
0.938305 + 0.345808i \(0.112395\pi\)
\(450\) 0 0
\(451\) − 6.24920i − 0.294263i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.126512 0.00593098
\(456\) 0 0
\(457\) 7.46983 0.349424 0.174712 0.984620i \(-0.444101\pi\)
0.174712 + 0.984620i \(0.444101\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 15.0075i − 0.698967i −0.936942 0.349484i \(-0.886357\pi\)
0.936942 0.349484i \(-0.113643\pi\)
\(462\) 0 0
\(463\) 4.29553 0.199630 0.0998150 0.995006i \(-0.468175\pi\)
0.0998150 + 0.995006i \(0.468175\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 26.3945i − 1.22139i −0.791865 0.610696i \(-0.790890\pi\)
0.791865 0.610696i \(-0.209110\pi\)
\(468\) 0 0
\(469\) − 8.55452i − 0.395011i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −38.0686 −1.75039
\(474\) 0 0
\(475\) 1.41208i 0.0647906i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.66267 −0.304425 −0.152213 0.988348i \(-0.548640\pi\)
−0.152213 + 0.988348i \(0.548640\pi\)
\(480\) 0 0
\(481\) 0.437254 0.0199371
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 32.0960i − 1.45740i
\(486\) 0 0
\(487\) 36.6813 1.66219 0.831095 0.556131i \(-0.187715\pi\)
0.831095 + 0.556131i \(0.187715\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 4.38460i − 0.197874i −0.995094 0.0989371i \(-0.968456\pi\)
0.995094 0.0989371i \(-0.0315443\pi\)
\(492\) 0 0
\(493\) − 25.3048i − 1.13967i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.86725 −0.173470
\(498\) 0 0
\(499\) − 35.6754i − 1.59705i −0.601961 0.798525i \(-0.705614\pi\)
0.601961 0.798525i \(-0.294386\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 11.3657 0.506770 0.253385 0.967365i \(-0.418456\pi\)
0.253385 + 0.967365i \(0.418456\pi\)
\(504\) 0 0
\(505\) 11.4691 0.510369
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.78884i 0.123613i 0.998088 + 0.0618066i \(0.0196862\pi\)
−0.998088 + 0.0618066i \(0.980314\pi\)
\(510\) 0 0
\(511\) −7.56044 −0.334454
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 32.4676i − 1.43069i
\(516\) 0 0
\(517\) − 39.6673i − 1.74457i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 40.3056 1.76582 0.882910 0.469542i \(-0.155581\pi\)
0.882910 + 0.469542i \(0.155581\pi\)
\(522\) 0 0
\(523\) − 35.9726i − 1.57297i −0.617608 0.786486i \(-0.711898\pi\)
0.617608 0.786486i \(-0.288102\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.5226 −0.676175
\(528\) 0 0
\(529\) 4.89387 0.212777
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 0.0698140i − 0.00302398i
\(534\) 0 0
\(535\) −25.5352 −1.10398
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.29757i 0.185109i
\(540\) 0 0
\(541\) 10.1931i 0.438234i 0.975699 + 0.219117i \(0.0703177\pi\)
−0.975699 + 0.219117i \(0.929682\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −23.9501 −1.02591
\(546\) 0 0
\(547\) − 12.3063i − 0.526182i −0.964771 0.263091i \(-0.915258\pi\)
0.964771 0.263091i \(-0.0847419\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.81385 −0.247679
\(552\) 0 0
\(553\) 14.6139 0.621445
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.2392i 1.15416i 0.816687 + 0.577082i \(0.195809\pi\)
−0.816687 + 0.577082i \(0.804191\pi\)
\(558\) 0 0
\(559\) −0.425290 −0.0179878
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.9187i 1.38736i 0.720285 + 0.693678i \(0.244011\pi\)
−0.720285 + 0.693678i \(0.755989\pi\)
\(564\) 0 0
\(565\) − 35.6274i − 1.49886i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.9554 1.08811 0.544053 0.839051i \(-0.316889\pi\)
0.544053 + 0.839051i \(0.316889\pi\)
\(570\) 0 0
\(571\) 0.527977i 0.0220951i 0.999939 + 0.0110476i \(0.00351662\pi\)
−0.999939 + 0.0110476i \(0.996483\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 10.2648 0.428073
\(576\) 0 0
\(577\) 43.1860 1.79786 0.898929 0.438094i \(-0.144346\pi\)
0.898929 + 0.438094i \(0.144346\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 9.05240i − 0.375557i
\(582\) 0 0
\(583\) 13.5256 0.560171
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 20.5375i − 0.847675i −0.905738 0.423837i \(-0.860683\pi\)
0.905738 0.423837i \(-0.139317\pi\)
\(588\) 0 0
\(589\) 3.56636i 0.146949i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −47.4930 −1.95030 −0.975152 0.221537i \(-0.928893\pi\)
−0.975152 + 0.221537i \(0.928893\pi\)
\(594\) 0 0
\(595\) − 8.33280i − 0.341612i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.31521 0.217174 0.108587 0.994087i \(-0.465367\pi\)
0.108587 + 0.994087i \(0.465367\pi\)
\(600\) 0 0
\(601\) 29.9137 1.22020 0.610102 0.792323i \(-0.291129\pi\)
0.610102 + 0.792323i \(0.291129\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 19.6816i 0.800171i
\(606\) 0 0
\(607\) 17.3429 0.703927 0.351964 0.936014i \(-0.385514\pi\)
0.351964 + 0.936014i \(0.385514\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 0.443150i − 0.0179279i
\(612\) 0 0
\(613\) − 17.8469i − 0.720830i −0.932792 0.360415i \(-0.882635\pi\)
0.932792 0.360415i \(-0.117365\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.51870 0.141658 0.0708289 0.997488i \(-0.477436\pi\)
0.0708289 + 0.997488i \(0.477436\pi\)
\(618\) 0 0
\(619\) 5.33557i 0.214455i 0.994235 + 0.107227i \(0.0341973\pi\)
−0.994235 + 0.107227i \(0.965803\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5.15394 −0.206488
\(624\) 0 0
\(625\) −30.9404 −1.23761
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 28.8000i − 1.14833i
\(630\) 0 0
\(631\) −13.1520 −0.523574 −0.261787 0.965126i \(-0.584312\pi\)
−0.261787 + 0.965126i \(0.584312\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 28.9308i − 1.14808i
\(636\) 0 0
\(637\) 0.0480111i 0.00190227i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.20177 0.323950 0.161975 0.986795i \(-0.448214\pi\)
0.161975 + 0.986795i \(0.448214\pi\)
\(642\) 0 0
\(643\) − 32.8167i − 1.29416i −0.762421 0.647082i \(-0.775989\pi\)
0.762421 0.647082i \(-0.224011\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 38.7207 1.52227 0.761134 0.648595i \(-0.224643\pi\)
0.761134 + 0.648595i \(0.224643\pi\)
\(648\) 0 0
\(649\) 25.3778 0.996166
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28.7630i 1.12558i 0.826598 + 0.562792i \(0.190273\pi\)
−0.826598 + 0.562792i \(0.809727\pi\)
\(654\) 0 0
\(655\) 2.03115 0.0793637
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 30.6124i 1.19249i 0.802803 + 0.596244i \(0.203341\pi\)
−0.802803 + 0.596244i \(0.796659\pi\)
\(660\) 0 0
\(661\) 44.1975i 1.71908i 0.511066 + 0.859541i \(0.329251\pi\)
−0.511066 + 0.859541i \(0.670749\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.91449 −0.0742406
\(666\) 0 0
\(667\) 42.2627i 1.63642i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.43649 −0.364292
\(672\) 0 0
\(673\) 17.0117 0.655754 0.327877 0.944720i \(-0.393667\pi\)
0.327877 + 0.944720i \(0.393667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 45.4750i 1.74775i 0.486155 + 0.873873i \(0.338399\pi\)
−0.486155 + 0.873873i \(0.661601\pi\)
\(678\) 0 0
\(679\) 12.1803 0.467439
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.4287i 1.31738i 0.752416 + 0.658689i \(0.228889\pi\)
−0.752416 + 0.658689i \(0.771111\pi\)
\(684\) 0 0
\(685\) 28.9283i 1.10529i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.151103 0.00575657
\(690\) 0 0
\(691\) − 28.9692i − 1.10204i −0.834491 0.551021i \(-0.814238\pi\)
0.834491 0.551021i \(-0.185762\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −58.0213 −2.20087
\(696\) 0 0
\(697\) −4.59834 −0.174175
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.9846i 0.490421i 0.969470 + 0.245211i \(0.0788571\pi\)
−0.969470 + 0.245211i \(0.921143\pi\)
\(702\) 0 0
\(703\) −6.61688 −0.249560
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.35250i 0.163693i
\(708\) 0 0
\(709\) 5.62511i 0.211255i 0.994406 + 0.105628i \(0.0336852\pi\)
−0.994406 + 0.105628i \(0.966315\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 25.9250 0.970899
\(714\) 0 0
\(715\) 0.543695i 0.0203330i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.3194 0.496732 0.248366 0.968666i \(-0.420106\pi\)
0.248366 + 0.968666i \(0.420106\pi\)
\(720\) 0 0
\(721\) 12.3214 0.458871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.5525i 0.577606i
\(726\) 0 0
\(727\) −20.0614 −0.744037 −0.372019 0.928225i \(-0.621334\pi\)
−0.372019 + 0.928225i \(0.621334\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 28.0120i 1.03606i
\(732\) 0 0
\(733\) − 4.06695i − 0.150216i −0.997175 0.0751081i \(-0.976070\pi\)
0.997175 0.0751081i \(-0.0239302\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.7637 1.35421
\(738\) 0 0
\(739\) 2.22283i 0.0817680i 0.999164 + 0.0408840i \(0.0130174\pi\)
−0.999164 + 0.0408840i \(0.986983\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −28.6763 −1.05203 −0.526015 0.850475i \(-0.676314\pi\)
−0.526015 + 0.850475i \(0.676314\pi\)
\(744\) 0 0
\(745\) −13.3504 −0.489120
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 9.69053i − 0.354084i
\(750\) 0 0
\(751\) −8.61386 −0.314324 −0.157162 0.987573i \(-0.550235\pi\)
−0.157162 + 0.987573i \(0.550235\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 18.2888i − 0.665596i
\(756\) 0 0
\(757\) 27.8835i 1.01344i 0.862109 + 0.506722i \(0.169143\pi\)
−0.862109 + 0.506722i \(0.830857\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.20889 0.152572 0.0762861 0.997086i \(-0.475694\pi\)
0.0762861 + 0.997086i \(0.475694\pi\)
\(762\) 0 0
\(763\) − 9.08902i − 0.329045i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.283512 0.0102370
\(768\) 0 0
\(769\) 21.5187 0.775986 0.387993 0.921662i \(-0.373168\pi\)
0.387993 + 0.921662i \(0.373168\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 30.1139i − 1.08312i −0.840661 0.541562i \(-0.817833\pi\)
0.840661 0.541562i \(-0.182167\pi\)
\(774\) 0 0
\(775\) 9.54031 0.342698
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.05648i 0.0378524i
\(780\) 0 0
\(781\) − 16.6198i − 0.594703i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 48.5328 1.73221
\(786\) 0 0
\(787\) − 48.9845i − 1.74611i −0.487624 0.873054i \(-0.662136\pi\)
0.487624 0.873054i \(-0.337864\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 13.5205 0.480734
\(792\) 0 0
\(793\) −0.105421 −0.00374363
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 47.5262i − 1.68347i −0.539894 0.841733i \(-0.681536\pi\)
0.539894 0.841733i \(-0.318464\pi\)
\(798\) 0 0
\(799\) −29.1883 −1.03261
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 32.4915i − 1.14660i
\(804\) 0 0
\(805\) 13.9170i 0.490510i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.1063 1.05848 0.529240 0.848472i \(-0.322477\pi\)
0.529240 + 0.848472i \(0.322477\pi\)
\(810\) 0 0
\(811\) 30.2492i 1.06219i 0.847311 + 0.531096i \(0.178220\pi\)
−0.847311 + 0.531096i \(0.821780\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −27.9183 −0.977935
\(816\) 0 0
\(817\) 6.43583 0.225161
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 15.4111i − 0.537851i −0.963161 0.268926i \(-0.913331\pi\)
0.963161 0.268926i \(-0.0866686\pi\)
\(822\) 0 0
\(823\) 20.0591 0.699217 0.349608 0.936896i \(-0.386315\pi\)
0.349608 + 0.936896i \(0.386315\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35.3363i 1.22876i 0.789009 + 0.614382i \(0.210594\pi\)
−0.789009 + 0.614382i \(0.789406\pi\)
\(828\) 0 0
\(829\) 52.3976i 1.81985i 0.414778 + 0.909923i \(0.363859\pi\)
−0.414778 + 0.909923i \(0.636141\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.16228 0.109566
\(834\) 0 0
\(835\) 45.7973i 1.58488i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −43.5132 −1.50224 −0.751121 0.660164i \(-0.770487\pi\)
−0.751121 + 0.660164i \(0.770487\pi\)
\(840\) 0 0
\(841\) −35.0334 −1.20805
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 34.2498i − 1.17823i
\(846\) 0 0
\(847\) −7.46912 −0.256642
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 48.1002i 1.64885i
\(852\) 0 0
\(853\) 28.8647i 0.988310i 0.869374 + 0.494155i \(0.164522\pi\)
−0.869374 + 0.494155i \(0.835478\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10.7856 −0.368429 −0.184214 0.982886i \(-0.558974\pi\)
−0.184214 + 0.982886i \(0.558974\pi\)
\(858\) 0 0
\(859\) 47.7595i 1.62953i 0.579789 + 0.814767i \(0.303135\pi\)
−0.579789 + 0.814767i \(0.696865\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 41.2620 1.40457 0.702287 0.711894i \(-0.252162\pi\)
0.702287 + 0.711894i \(0.252162\pi\)
\(864\) 0 0
\(865\) −60.3635 −2.05242
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 62.8041i 2.13048i
\(870\) 0 0
\(871\) 0.410712 0.0139164
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 8.05391i − 0.272272i
\(876\) 0 0
\(877\) − 34.8776i − 1.17773i −0.808230 0.588867i \(-0.799574\pi\)
0.808230 0.588867i \(-0.200426\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.15123 −0.207240 −0.103620 0.994617i \(-0.533043\pi\)
−0.103620 + 0.994617i \(0.533043\pi\)
\(882\) 0 0
\(883\) − 24.2713i − 0.816796i −0.912804 0.408398i \(-0.866088\pi\)
0.912804 0.408398i \(-0.133912\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.8061 0.429987 0.214994 0.976615i \(-0.431027\pi\)
0.214994 + 0.976615i \(0.431027\pi\)
\(888\) 0 0
\(889\) 10.9792 0.368229
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.70611i 0.224411i
\(894\) 0 0
\(895\) −56.9878 −1.90489
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 39.2797i 1.31005i
\(900\) 0 0
\(901\) − 9.95250i − 0.331566i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 33.2397 1.10492
\(906\) 0 0
\(907\) − 26.7650i − 0.888717i −0.895849 0.444358i \(-0.853432\pi\)
0.895849 0.444358i \(-0.146568\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 34.9594 1.15826 0.579128 0.815237i \(-0.303393\pi\)
0.579128 + 0.815237i \(0.303393\pi\)
\(912\) 0 0
\(913\) 38.9034 1.28751
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0.770817i 0.0254546i
\(918\) 0 0
\(919\) −17.3058 −0.570867 −0.285433 0.958399i \(-0.592138\pi\)
−0.285433 + 0.958399i \(0.592138\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 0.185671i − 0.00611143i
\(924\) 0 0
\(925\) 17.7007i 0.581995i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −8.69526 −0.285282 −0.142641 0.989774i \(-0.545559\pi\)
−0.142641 + 0.989774i \(0.545559\pi\)
\(930\) 0 0
\(931\) − 0.726543i − 0.0238115i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 35.8108 1.17114
\(936\) 0 0
\(937\) 24.8633 0.812249 0.406125 0.913818i \(-0.366880\pi\)
0.406125 + 0.913818i \(0.366880\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 15.1562i 0.494078i 0.969006 + 0.247039i \(0.0794575\pi\)
−0.969006 + 0.247039i \(0.920542\pi\)
\(942\) 0 0
\(943\) 7.67990 0.250092
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 25.6626i 0.833922i 0.908924 + 0.416961i \(0.136905\pi\)
−0.908924 + 0.416961i \(0.863095\pi\)
\(948\) 0 0
\(949\) − 0.362985i − 0.0117830i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.66478 0.313073 0.156536 0.987672i \(-0.449967\pi\)
0.156536 + 0.987672i \(0.449967\pi\)
\(954\) 0 0
\(955\) 35.7256i 1.15605i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.9782 −0.354505
\(960\) 0 0
\(961\) −6.90488 −0.222738
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 68.6735i 2.21068i
\(966\) 0 0
\(967\) −6.28949 −0.202256 −0.101128 0.994873i \(-0.532245\pi\)
−0.101128 + 0.994873i \(0.532245\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 28.3396i − 0.909462i −0.890629 0.454731i \(-0.849735\pi\)
0.890629 0.454731i \(-0.150265\pi\)
\(972\) 0 0
\(973\) − 22.0189i − 0.705895i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28.0973 0.898911 0.449455 0.893303i \(-0.351618\pi\)
0.449455 + 0.893303i \(0.351618\pi\)
\(978\) 0 0
\(979\) − 22.1494i − 0.707899i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 38.4571 1.22659 0.613296 0.789853i \(-0.289843\pi\)
0.613296 + 0.789853i \(0.289843\pi\)
\(984\) 0 0
\(985\) 8.40380 0.267767
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 46.7841i − 1.48765i
\(990\) 0 0
\(991\) 30.2448 0.960759 0.480380 0.877061i \(-0.340499\pi\)
0.480380 + 0.877061i \(0.340499\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 66.4376i 2.10621i
\(996\) 0 0
\(997\) − 17.2007i − 0.544751i −0.962191 0.272375i \(-0.912191\pi\)
0.962191 0.272375i \(-0.0878093\pi\)
\(998\) 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 6048.2.c.d.3025.4 16
3.2 odd 2 inner 6048.2.c.d.3025.14 16
4.3 odd 2 1512.2.c.d.757.15 yes 16
8.3 odd 2 1512.2.c.d.757.14 yes 16
8.5 even 2 inner 6048.2.c.d.3025.13 16
12.11 even 2 1512.2.c.d.757.2 16
24.5 odd 2 inner 6048.2.c.d.3025.3 16
24.11 even 2 1512.2.c.d.757.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.d.757.2 16 12.11 even 2
1512.2.c.d.757.3 yes 16 24.11 even 2
1512.2.c.d.757.14 yes 16 8.3 odd 2
1512.2.c.d.757.15 yes 16 4.3 odd 2
6048.2.c.d.3025.3 16 24.5 odd 2 inner
6048.2.c.d.3025.4 16 1.1 even 1 trivial
6048.2.c.d.3025.13 16 8.5 even 2 inner
6048.2.c.d.3025.14 16 3.2 odd 2 inner