Properties

Label 9408.2.a.em.1.3
Level 9408
Weight 2
Character 9408.1
Self dual yes
Analytic conductor 75.123
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4704)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.814115\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.04244 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.04244 q^{5} +1.00000 q^{9} +5.93089 q^{11} +0.888450 q^{13} +1.04244 q^{15} +4.51668 q^{17} +3.47424 q^{19} +1.93089 q^{23} -3.91331 q^{25} +1.00000 q^{27} +8.38755 q^{29} -5.13109 q^{31} +5.93089 q^{33} -5.65685 q^{37} +0.888450 q^{39} +8.39663 q^{41} -0.743541 q^{43} +1.04244 q^{45} +4.21778 q^{47} +4.51668 q^{51} +6.91331 q^{53} +6.18262 q^{55} +3.47424 q^{57} +5.43908 q^{59} -10.9733 q^{61} +0.926159 q^{65} -6.20493 q^{67} +1.93089 q^{69} -3.72596 q^{71} -3.49910 q^{73} -3.91331 q^{75} -12.6053 q^{79} +1.00000 q^{81} -2.94847 q^{83} +4.70838 q^{85} +8.38755 q^{87} -5.00196 q^{89} -5.13109 q^{93} +3.62169 q^{95} +10.1578 q^{97} +5.93089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{5} + 4q^{9} + 4q^{11} - 8q^{13} - 4q^{15} + 4q^{17} + 8q^{19} - 12q^{23} + 12q^{25} + 4q^{27} + 8q^{31} + 4q^{33} - 8q^{39} + 20q^{41} - 8q^{43} - 4q^{45} + 16q^{47} + 4q^{51} + 8q^{55} + 8q^{57} - 16q^{61} - 8q^{65} - 8q^{67} - 12q^{69} - 12q^{71} + 8q^{73} + 12q^{75} - 16q^{79} + 4q^{81} + 8q^{85} + 28q^{89} + 8q^{93} - 24q^{95} + 40q^{97} + 4q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.04244 0.466195 0.233097 0.972453i \(-0.425114\pi\)
0.233097 + 0.972453i \(0.425114\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.93089 1.78823 0.894116 0.447836i \(-0.147805\pi\)
0.894116 + 0.447836i \(0.147805\pi\)
\(12\) 0 0
\(13\) 0.888450 0.246412 0.123206 0.992381i \(-0.460682\pi\)
0.123206 + 0.992381i \(0.460682\pi\)
\(14\) 0 0
\(15\) 1.04244 0.269158
\(16\) 0 0
\(17\) 4.51668 1.09546 0.547728 0.836657i \(-0.315493\pi\)
0.547728 + 0.836657i \(0.315493\pi\)
\(18\) 0 0
\(19\) 3.47424 0.797045 0.398522 0.917159i \(-0.369523\pi\)
0.398522 + 0.917159i \(0.369523\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.93089 0.402619 0.201310 0.979528i \(-0.435480\pi\)
0.201310 + 0.979528i \(0.435480\pi\)
\(24\) 0 0
\(25\) −3.91331 −0.782663
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.38755 1.55753 0.778764 0.627316i \(-0.215847\pi\)
0.778764 + 0.627316i \(0.215847\pi\)
\(30\) 0 0
\(31\) −5.13109 −0.921571 −0.460786 0.887511i \(-0.652432\pi\)
−0.460786 + 0.887511i \(0.652432\pi\)
\(32\) 0 0
\(33\) 5.93089 1.03244
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.65685 −0.929981 −0.464991 0.885316i \(-0.653942\pi\)
−0.464991 + 0.885316i \(0.653942\pi\)
\(38\) 0 0
\(39\) 0.888450 0.142266
\(40\) 0 0
\(41\) 8.39663 1.31133 0.655667 0.755050i \(-0.272388\pi\)
0.655667 + 0.755050i \(0.272388\pi\)
\(42\) 0 0
\(43\) −0.743541 −0.113389 −0.0566945 0.998392i \(-0.518056\pi\)
−0.0566945 + 0.998392i \(0.518056\pi\)
\(44\) 0 0
\(45\) 1.04244 0.155398
\(46\) 0 0
\(47\) 4.21778 0.615226 0.307613 0.951512i \(-0.400470\pi\)
0.307613 + 0.951512i \(0.400470\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4.51668 0.632462
\(52\) 0 0
\(53\) 6.91331 0.949617 0.474808 0.880089i \(-0.342517\pi\)
0.474808 + 0.880089i \(0.342517\pi\)
\(54\) 0 0
\(55\) 6.18262 0.833664
\(56\) 0 0
\(57\) 3.47424 0.460174
\(58\) 0 0
\(59\) 5.43908 0.708107 0.354054 0.935225i \(-0.384803\pi\)
0.354054 + 0.935225i \(0.384803\pi\)
\(60\) 0 0
\(61\) −10.9733 −1.40499 −0.702496 0.711688i \(-0.747931\pi\)
−0.702496 + 0.711688i \(0.747931\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.926159 0.114876
\(66\) 0 0
\(67\) −6.20493 −0.758053 −0.379027 0.925386i \(-0.623741\pi\)
−0.379027 + 0.925386i \(0.623741\pi\)
\(68\) 0 0
\(69\) 1.93089 0.232452
\(70\) 0 0
\(71\) −3.72596 −0.442190 −0.221095 0.975252i \(-0.570963\pi\)
−0.221095 + 0.975252i \(0.570963\pi\)
\(72\) 0 0
\(73\) −3.49910 −0.409539 −0.204769 0.978810i \(-0.565644\pi\)
−0.204769 + 0.978810i \(0.565644\pi\)
\(74\) 0 0
\(75\) −3.91331 −0.451870
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −12.6053 −1.41821 −0.709105 0.705103i \(-0.750901\pi\)
−0.709105 + 0.705103i \(0.750901\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.94847 −0.323637 −0.161819 0.986821i \(-0.551736\pi\)
−0.161819 + 0.986821i \(0.551736\pi\)
\(84\) 0 0
\(85\) 4.70838 0.510696
\(86\) 0 0
\(87\) 8.38755 0.899240
\(88\) 0 0
\(89\) −5.00196 −0.530207 −0.265103 0.964220i \(-0.585406\pi\)
−0.265103 + 0.964220i \(0.585406\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.13109 −0.532069
\(94\) 0 0
\(95\) 3.62169 0.371578
\(96\) 0 0
\(97\) 10.1578 1.03136 0.515682 0.856780i \(-0.327539\pi\)
0.515682 + 0.856780i \(0.327539\pi\)
\(98\) 0 0
\(99\) 5.93089 0.596077
\(100\) 0 0
\(101\) −1.78598 −0.177712 −0.0888560 0.996044i \(-0.528321\pi\)
−0.0888560 + 0.996044i \(0.528321\pi\)
\(102\) 0 0
\(103\) 18.5925 1.83197 0.915986 0.401211i \(-0.131410\pi\)
0.915986 + 0.401211i \(0.131410\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.23888 0.603135 0.301568 0.953445i \(-0.402490\pi\)
0.301568 + 0.953445i \(0.402490\pi\)
\(108\) 0 0
\(109\) 4.74354 0.454349 0.227174 0.973854i \(-0.427051\pi\)
0.227174 + 0.973854i \(0.427051\pi\)
\(110\) 0 0
\(111\) −5.65685 −0.536925
\(112\) 0 0
\(113\) −13.3137 −1.25245 −0.626224 0.779643i \(-0.715401\pi\)
−0.626224 + 0.779643i \(0.715401\pi\)
\(114\) 0 0
\(115\) 2.01285 0.187699
\(116\) 0 0
\(117\) 0.888450 0.0821373
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 24.1755 2.19777
\(122\) 0 0
\(123\) 8.39663 0.757099
\(124\) 0 0
\(125\) −9.29162 −0.831068
\(126\) 0 0
\(127\) 0.307985 0.0273292 0.0136646 0.999907i \(-0.495650\pi\)
0.0136646 + 0.999907i \(0.495650\pi\)
\(128\) 0 0
\(129\) −0.743541 −0.0654652
\(130\) 0 0
\(131\) 9.46139 0.826646 0.413323 0.910585i \(-0.364368\pi\)
0.413323 + 0.910585i \(0.364368\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.04244 0.0897192
\(136\) 0 0
\(137\) −1.87463 −0.160161 −0.0800803 0.996788i \(-0.525518\pi\)
−0.0800803 + 0.996788i \(0.525518\pi\)
\(138\) 0 0
\(139\) −7.31371 −0.620341 −0.310170 0.950681i \(-0.600386\pi\)
−0.310170 + 0.950681i \(0.600386\pi\)
\(140\) 0 0
\(141\) 4.21778 0.355201
\(142\) 0 0
\(143\) 5.26930 0.440641
\(144\) 0 0
\(145\) 8.74354 0.726112
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −22.2270 −1.82091 −0.910454 0.413610i \(-0.864268\pi\)
−0.910454 + 0.413610i \(0.864268\pi\)
\(150\) 0 0
\(151\) 14.4004 1.17189 0.585944 0.810352i \(-0.300724\pi\)
0.585944 + 0.810352i \(0.300724\pi\)
\(152\) 0 0
\(153\) 4.51668 0.365152
\(154\) 0 0
\(155\) −5.34887 −0.429632
\(156\) 0 0
\(157\) −17.9715 −1.43428 −0.717142 0.696927i \(-0.754550\pi\)
−0.717142 + 0.696927i \(0.754550\pi\)
\(158\) 0 0
\(159\) 6.91331 0.548261
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −23.1755 −1.81524 −0.907622 0.419787i \(-0.862105\pi\)
−0.907622 + 0.419787i \(0.862105\pi\)
\(164\) 0 0
\(165\) 6.18262 0.481316
\(166\) 0 0
\(167\) −12.9929 −1.00542 −0.502710 0.864455i \(-0.667663\pi\)
−0.502710 + 0.864455i \(0.667663\pi\)
\(168\) 0 0
\(169\) −12.2107 −0.939281
\(170\) 0 0
\(171\) 3.47424 0.265682
\(172\) 0 0
\(173\) −23.1846 −1.76269 −0.881345 0.472472i \(-0.843362\pi\)
−0.881345 + 0.472472i \(0.843362\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.43908 0.408826
\(178\) 0 0
\(179\) 24.9366 1.86385 0.931925 0.362651i \(-0.118128\pi\)
0.931925 + 0.362651i \(0.118128\pi\)
\(180\) 0 0
\(181\) −22.3498 −1.66125 −0.830625 0.556832i \(-0.812017\pi\)
−0.830625 + 0.556832i \(0.812017\pi\)
\(182\) 0 0
\(183\) −10.9733 −0.811172
\(184\) 0 0
\(185\) −5.89695 −0.433552
\(186\) 0 0
\(187\) 26.7879 1.95893
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.03967 −0.509373 −0.254686 0.967024i \(-0.581972\pi\)
−0.254686 + 0.967024i \(0.581972\pi\)
\(192\) 0 0
\(193\) 12.3747 0.890751 0.445375 0.895344i \(-0.353070\pi\)
0.445375 + 0.895344i \(0.353070\pi\)
\(194\) 0 0
\(195\) 0.926159 0.0663236
\(196\) 0 0
\(197\) −13.4262 −0.956579 −0.478290 0.878202i \(-0.658743\pi\)
−0.478290 + 0.878202i \(0.658743\pi\)
\(198\) 0 0
\(199\) −9.82663 −0.696591 −0.348296 0.937385i \(-0.613239\pi\)
−0.348296 + 0.937385i \(0.613239\pi\)
\(200\) 0 0
\(201\) −6.20493 −0.437662
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 8.75301 0.611337
\(206\) 0 0
\(207\) 1.93089 0.134206
\(208\) 0 0
\(209\) 20.6053 1.42530
\(210\) 0 0
\(211\) −20.1698 −1.38854 −0.694272 0.719713i \(-0.744274\pi\)
−0.694272 + 0.719713i \(0.744274\pi\)
\(212\) 0 0
\(213\) −3.72596 −0.255299
\(214\) 0 0
\(215\) −0.775099 −0.0528613
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.49910 −0.236447
\(220\) 0 0
\(221\) 4.01285 0.269933
\(222\) 0 0
\(223\) 27.0373 1.81055 0.905275 0.424826i \(-0.139665\pi\)
0.905275 + 0.424826i \(0.139665\pi\)
\(224\) 0 0
\(225\) −3.91331 −0.260888
\(226\) 0 0
\(227\) 8.02231 0.532460 0.266230 0.963910i \(-0.414222\pi\)
0.266230 + 0.963910i \(0.414222\pi\)
\(228\) 0 0
\(229\) 12.4920 0.825493 0.412747 0.910846i \(-0.364570\pi\)
0.412747 + 0.910846i \(0.364570\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.7013 1.29067 0.645336 0.763899i \(-0.276717\pi\)
0.645336 + 0.763899i \(0.276717\pi\)
\(234\) 0 0
\(235\) 4.39679 0.286815
\(236\) 0 0
\(237\) −12.6053 −0.818804
\(238\) 0 0
\(239\) 1.93089 0.124899 0.0624496 0.998048i \(-0.480109\pi\)
0.0624496 + 0.998048i \(0.480109\pi\)
\(240\) 0 0
\(241\) −15.1026 −0.972846 −0.486423 0.873724i \(-0.661699\pi\)
−0.486423 + 0.873724i \(0.661699\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.08669 0.196401
\(248\) 0 0
\(249\) −2.94847 −0.186852
\(250\) 0 0
\(251\) 29.5982 1.86822 0.934111 0.356982i \(-0.116194\pi\)
0.934111 + 0.356982i \(0.116194\pi\)
\(252\) 0 0
\(253\) 11.4519 0.719976
\(254\) 0 0
\(255\) 4.70838 0.294850
\(256\) 0 0
\(257\) 13.8580 0.864440 0.432220 0.901768i \(-0.357730\pi\)
0.432220 + 0.901768i \(0.357730\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 8.38755 0.519176
\(262\) 0 0
\(263\) −2.78696 −0.171851 −0.0859255 0.996302i \(-0.527385\pi\)
−0.0859255 + 0.996302i \(0.527385\pi\)
\(264\) 0 0
\(265\) 7.20673 0.442706
\(266\) 0 0
\(267\) −5.00196 −0.306115
\(268\) 0 0
\(269\) 6.44104 0.392717 0.196358 0.980532i \(-0.437088\pi\)
0.196358 + 0.980532i \(0.437088\pi\)
\(270\) 0 0
\(271\) −20.8547 −1.26683 −0.633415 0.773812i \(-0.718348\pi\)
−0.633415 + 0.773812i \(0.718348\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −23.2094 −1.39958
\(276\) 0 0
\(277\) 27.0724 1.62663 0.813313 0.581827i \(-0.197662\pi\)
0.813313 + 0.581827i \(0.197662\pi\)
\(278\) 0 0
\(279\) −5.13109 −0.307190
\(280\) 0 0
\(281\) 6.28450 0.374902 0.187451 0.982274i \(-0.439977\pi\)
0.187451 + 0.982274i \(0.439977\pi\)
\(282\) 0 0
\(283\) 5.57729 0.331535 0.165768 0.986165i \(-0.446990\pi\)
0.165768 + 0.986165i \(0.446990\pi\)
\(284\) 0 0
\(285\) 3.62169 0.214531
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.40040 0.200023
\(290\) 0 0
\(291\) 10.1578 0.595458
\(292\) 0 0
\(293\) 12.6460 0.738785 0.369393 0.929273i \(-0.379566\pi\)
0.369393 + 0.929273i \(0.379566\pi\)
\(294\) 0 0
\(295\) 5.66993 0.330116
\(296\) 0 0
\(297\) 5.93089 0.344145
\(298\) 0 0
\(299\) 1.71550 0.0992101
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1.78598 −0.102602
\(304\) 0 0
\(305\) −11.4391 −0.655000
\(306\) 0 0
\(307\) −16.2750 −0.928865 −0.464432 0.885609i \(-0.653742\pi\)
−0.464432 + 0.885609i \(0.653742\pi\)
\(308\) 0 0
\(309\) 18.5925 1.05769
\(310\) 0 0
\(311\) 28.9929 1.64404 0.822018 0.569462i \(-0.192848\pi\)
0.822018 + 0.569462i \(0.192848\pi\)
\(312\) 0 0
\(313\) 13.5840 0.767812 0.383906 0.923372i \(-0.374579\pi\)
0.383906 + 0.923372i \(0.374579\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −24.1240 −1.35494 −0.677469 0.735552i \(-0.736923\pi\)
−0.677469 + 0.735552i \(0.736923\pi\)
\(318\) 0 0
\(319\) 49.7457 2.78522
\(320\) 0 0
\(321\) 6.23888 0.348220
\(322\) 0 0
\(323\) 15.6920 0.873127
\(324\) 0 0
\(325\) −3.47678 −0.192857
\(326\) 0 0
\(327\) 4.74354 0.262318
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.72357 0.424526 0.212263 0.977213i \(-0.431917\pi\)
0.212263 + 0.977213i \(0.431917\pi\)
\(332\) 0 0
\(333\) −5.65685 −0.309994
\(334\) 0 0
\(335\) −6.46829 −0.353400
\(336\) 0 0
\(337\) 1.05153 0.0572803 0.0286401 0.999590i \(-0.490882\pi\)
0.0286401 + 0.999590i \(0.490882\pi\)
\(338\) 0 0
\(339\) −13.3137 −0.723101
\(340\) 0 0
\(341\) −30.4320 −1.64798
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.01285 0.108368
\(346\) 0 0
\(347\) 1.52103 0.0816531 0.0408265 0.999166i \(-0.487001\pi\)
0.0408265 + 0.999166i \(0.487001\pi\)
\(348\) 0 0
\(349\) −18.1670 −0.972457 −0.486229 0.873832i \(-0.661628\pi\)
−0.486229 + 0.873832i \(0.661628\pi\)
\(350\) 0 0
\(351\) 0.888450 0.0474220
\(352\) 0 0
\(353\) 11.2069 0.596483 0.298241 0.954490i \(-0.403600\pi\)
0.298241 + 0.954490i \(0.403600\pi\)
\(354\) 0 0
\(355\) −3.88410 −0.206147
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.44381 −0.445647 −0.222824 0.974859i \(-0.571527\pi\)
−0.222824 + 0.974859i \(0.571527\pi\)
\(360\) 0 0
\(361\) −6.92968 −0.364720
\(362\) 0 0
\(363\) 24.1755 1.26888
\(364\) 0 0
\(365\) −3.64761 −0.190925
\(366\) 0 0
\(367\) 23.2880 1.21562 0.607812 0.794081i \(-0.292047\pi\)
0.607812 + 0.794081i \(0.292047\pi\)
\(368\) 0 0
\(369\) 8.39663 0.437111
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −35.5502 −1.84072 −0.920360 0.391073i \(-0.872104\pi\)
−0.920360 + 0.391073i \(0.872104\pi\)
\(374\) 0 0
\(375\) −9.29162 −0.479817
\(376\) 0 0
\(377\) 7.45192 0.383794
\(378\) 0 0
\(379\) −13.4168 −0.689173 −0.344586 0.938755i \(-0.611981\pi\)
−0.344586 + 0.938755i \(0.611981\pi\)
\(380\) 0 0
\(381\) 0.307985 0.0157785
\(382\) 0 0
\(383\) 35.6533 1.82180 0.910898 0.412632i \(-0.135390\pi\)
0.910898 + 0.412632i \(0.135390\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.743541 −0.0377963
\(388\) 0 0
\(389\) −23.3360 −1.18318 −0.591592 0.806238i \(-0.701500\pi\)
−0.591592 + 0.806238i \(0.701500\pi\)
\(390\) 0 0
\(391\) 8.72123 0.441051
\(392\) 0 0
\(393\) 9.46139 0.477264
\(394\) 0 0
\(395\) −13.1403 −0.661162
\(396\) 0 0
\(397\) 9.63199 0.483416 0.241708 0.970349i \(-0.422292\pi\)
0.241708 + 0.970349i \(0.422292\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.61245 −0.380148 −0.190074 0.981770i \(-0.560873\pi\)
−0.190074 + 0.981770i \(0.560873\pi\)
\(402\) 0 0
\(403\) −4.55872 −0.227086
\(404\) 0 0
\(405\) 1.04244 0.0517994
\(406\) 0 0
\(407\) −33.5502 −1.66302
\(408\) 0 0
\(409\) 34.6564 1.71365 0.856825 0.515607i \(-0.172434\pi\)
0.856825 + 0.515607i \(0.172434\pi\)
\(410\) 0 0
\(411\) −1.87463 −0.0924688
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −3.07362 −0.150878
\(416\) 0 0
\(417\) −7.31371 −0.358154
\(418\) 0 0
\(419\) −6.53523 −0.319267 −0.159633 0.987176i \(-0.551031\pi\)
−0.159633 + 0.987176i \(0.551031\pi\)
\(420\) 0 0
\(421\) 10.4778 0.510655 0.255327 0.966855i \(-0.417817\pi\)
0.255327 + 0.966855i \(0.417817\pi\)
\(422\) 0 0
\(423\) 4.21778 0.205075
\(424\) 0 0
\(425\) −17.6752 −0.857372
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5.26930 0.254404
\(430\) 0 0
\(431\) 28.9682 1.39535 0.697674 0.716415i \(-0.254218\pi\)
0.697674 + 0.716415i \(0.254218\pi\)
\(432\) 0 0
\(433\) −7.31116 −0.351352 −0.175676 0.984448i \(-0.556211\pi\)
−0.175676 + 0.984448i \(0.556211\pi\)
\(434\) 0 0
\(435\) 8.74354 0.419221
\(436\) 0 0
\(437\) 6.70838 0.320905
\(438\) 0 0
\(439\) 12.6863 0.605484 0.302742 0.953073i \(-0.402098\pi\)
0.302742 + 0.953073i \(0.402098\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.6136 −0.789335 −0.394668 0.918824i \(-0.629140\pi\)
−0.394668 + 0.918824i \(0.629140\pi\)
\(444\) 0 0
\(445\) −5.21426 −0.247180
\(446\) 0 0
\(447\) −22.2270 −1.05130
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 49.7995 2.34497
\(452\) 0 0
\(453\) 14.4004 0.676590
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.72357 −0.454850 −0.227425 0.973796i \(-0.573031\pi\)
−0.227425 + 0.973796i \(0.573031\pi\)
\(458\) 0 0
\(459\) 4.51668 0.210821
\(460\) 0 0
\(461\) 17.9451 0.835787 0.417894 0.908496i \(-0.362768\pi\)
0.417894 + 0.908496i \(0.362768\pi\)
\(462\) 0 0
\(463\) −2.37470 −0.110362 −0.0551809 0.998476i \(-0.517574\pi\)
−0.0551809 + 0.998476i \(0.517574\pi\)
\(464\) 0 0
\(465\) −5.34887 −0.248048
\(466\) 0 0
\(467\) 15.3617 0.710855 0.355428 0.934704i \(-0.384335\pi\)
0.355428 + 0.934704i \(0.384335\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −17.9715 −0.828085
\(472\) 0 0
\(473\) −4.40986 −0.202766
\(474\) 0 0
\(475\) −13.5958 −0.623817
\(476\) 0 0
\(477\) 6.91331 0.316539
\(478\) 0 0
\(479\) 14.0444 0.641705 0.320853 0.947129i \(-0.396031\pi\)
0.320853 + 0.947129i \(0.396031\pi\)
\(480\) 0 0
\(481\) −5.02583 −0.229158
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.5889 0.480816
\(486\) 0 0
\(487\) 30.4004 1.37757 0.688787 0.724964i \(-0.258144\pi\)
0.688787 + 0.724964i \(0.258144\pi\)
\(488\) 0 0
\(489\) −23.1755 −1.04803
\(490\) 0 0
\(491\) −25.5526 −1.15317 −0.576586 0.817036i \(-0.695615\pi\)
−0.576586 + 0.817036i \(0.695615\pi\)
\(492\) 0 0
\(493\) 37.8839 1.70620
\(494\) 0 0
\(495\) 6.18262 0.277888
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −16.2270 −0.726421 −0.363211 0.931707i \(-0.618319\pi\)
−0.363211 + 0.931707i \(0.618319\pi\)
\(500\) 0 0
\(501\) −12.9929 −0.580479
\(502\) 0 0
\(503\) 39.2880 1.75177 0.875883 0.482523i \(-0.160280\pi\)
0.875883 + 0.482523i \(0.160280\pi\)
\(504\) 0 0
\(505\) −1.86179 −0.0828484
\(506\) 0 0
\(507\) −12.2107 −0.542294
\(508\) 0 0
\(509\) 2.93187 0.129953 0.0649763 0.997887i \(-0.479303\pi\)
0.0649763 + 0.997887i \(0.479303\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.47424 0.153391
\(514\) 0 0
\(515\) 19.3816 0.854055
\(516\) 0 0
\(517\) 25.0152 1.10017
\(518\) 0 0
\(519\) −23.1846 −1.01769
\(520\) 0 0
\(521\) 26.3508 1.15445 0.577225 0.816585i \(-0.304135\pi\)
0.577225 + 0.816585i \(0.304135\pi\)
\(522\) 0 0
\(523\) 40.7048 1.77990 0.889948 0.456062i \(-0.150741\pi\)
0.889948 + 0.456062i \(0.150741\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −23.1755 −1.00954
\(528\) 0 0
\(529\) −19.2717 −0.837898
\(530\) 0 0
\(531\) 5.43908 0.236036
\(532\) 0 0
\(533\) 7.45999 0.323128
\(534\) 0 0
\(535\) 6.50367 0.281178
\(536\) 0 0
\(537\) 24.9366 1.07609
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 15.3489 0.659899 0.329950 0.943999i \(-0.392968\pi\)
0.329950 + 0.943999i \(0.392968\pi\)
\(542\) 0 0
\(543\) −22.3498 −0.959123
\(544\) 0 0
\(545\) 4.94487 0.211815
\(546\) 0 0
\(547\) −34.2258 −1.46339 −0.731696 0.681631i \(-0.761271\pi\)
−0.731696 + 0.681631i \(0.761271\pi\)
\(548\) 0 0
\(549\) −10.9733 −0.468331
\(550\) 0 0
\(551\) 29.1403 1.24142
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −5.89695 −0.250311
\(556\) 0 0
\(557\) 18.6369 0.789670 0.394835 0.918752i \(-0.370802\pi\)
0.394835 + 0.918752i \(0.370802\pi\)
\(558\) 0 0
\(559\) −0.660600 −0.0279404
\(560\) 0 0
\(561\) 26.7879 1.13099
\(562\) 0 0
\(563\) −2.90047 −0.122240 −0.0611201 0.998130i \(-0.519467\pi\)
−0.0611201 + 0.998130i \(0.519467\pi\)
\(564\) 0 0
\(565\) −13.8788 −0.583885
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −27.0853 −1.13547 −0.567737 0.823210i \(-0.692181\pi\)
−0.567737 + 0.823210i \(0.692181\pi\)
\(570\) 0 0
\(571\) −3.39467 −0.142063 −0.0710313 0.997474i \(-0.522629\pi\)
−0.0710313 + 0.997474i \(0.522629\pi\)
\(572\) 0 0
\(573\) −7.03967 −0.294086
\(574\) 0 0
\(575\) −7.55619 −0.315115
\(576\) 0 0
\(577\) −22.5545 −0.938958 −0.469479 0.882944i \(-0.655558\pi\)
−0.469479 + 0.882944i \(0.655558\pi\)
\(578\) 0 0
\(579\) 12.3747 0.514275
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 41.0021 1.69813
\(584\) 0 0
\(585\) 0.926159 0.0382920
\(586\) 0 0
\(587\) 29.4391 1.21508 0.607540 0.794289i \(-0.292156\pi\)
0.607540 + 0.794289i \(0.292156\pi\)
\(588\) 0 0
\(589\) −17.8266 −0.734533
\(590\) 0 0
\(591\) −13.4262 −0.552281
\(592\) 0 0
\(593\) −33.9153 −1.39273 −0.696367 0.717686i \(-0.745201\pi\)
−0.696367 + 0.717686i \(0.745201\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9.82663 −0.402177
\(598\) 0 0
\(599\) −28.0655 −1.14673 −0.573363 0.819302i \(-0.694361\pi\)
−0.573363 + 0.819302i \(0.694361\pi\)
\(600\) 0 0
\(601\) −17.0434 −0.695216 −0.347608 0.937640i \(-0.613006\pi\)
−0.347608 + 0.937640i \(0.613006\pi\)
\(602\) 0 0
\(603\) −6.20493 −0.252684
\(604\) 0 0
\(605\) 25.2016 1.02459
\(606\) 0 0
\(607\) −36.8639 −1.49626 −0.748130 0.663552i \(-0.769048\pi\)
−0.748130 + 0.663552i \(0.769048\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.74729 0.151599
\(612\) 0 0
\(613\) −25.4483 −1.02785 −0.513924 0.857836i \(-0.671809\pi\)
−0.513924 + 0.857836i \(0.671809\pi\)
\(614\) 0 0
\(615\) 8.75301 0.352955
\(616\) 0 0
\(617\) −35.3103 −1.42154 −0.710770 0.703424i \(-0.751653\pi\)
−0.710770 + 0.703424i \(0.751653\pi\)
\(618\) 0 0
\(619\) 40.9599 1.64632 0.823159 0.567811i \(-0.192209\pi\)
0.823159 + 0.567811i \(0.192209\pi\)
\(620\) 0 0
\(621\) 1.93089 0.0774841
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.88058 0.395223
\(626\) 0 0
\(627\) 20.6053 0.822898
\(628\) 0 0
\(629\) −25.5502 −1.01875
\(630\) 0 0
\(631\) 21.9648 0.874406 0.437203 0.899363i \(-0.355969\pi\)
0.437203 + 0.899363i \(0.355969\pi\)
\(632\) 0 0
\(633\) −20.1698 −0.801676
\(634\) 0 0
\(635\) 0.321057 0.0127407
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −3.72596 −0.147397
\(640\) 0 0
\(641\) −31.9520 −1.26203 −0.631014 0.775772i \(-0.717361\pi\)
−0.631014 + 0.775772i \(0.717361\pi\)
\(642\) 0 0
\(643\) −41.0501 −1.61886 −0.809430 0.587217i \(-0.800223\pi\)
−0.809430 + 0.587217i \(0.800223\pi\)
\(644\) 0 0
\(645\) −0.775099 −0.0305195
\(646\) 0 0
\(647\) −9.67917 −0.380527 −0.190264 0.981733i \(-0.560934\pi\)
−0.190264 + 0.981733i \(0.560934\pi\)
\(648\) 0 0
\(649\) 32.2586 1.26626
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.5982 0.688671 0.344336 0.938847i \(-0.388104\pi\)
0.344336 + 0.938847i \(0.388104\pi\)
\(654\) 0 0
\(655\) 9.86296 0.385378
\(656\) 0 0
\(657\) −3.49910 −0.136513
\(658\) 0 0
\(659\) 4.10427 0.159880 0.0799398 0.996800i \(-0.474527\pi\)
0.0799398 + 0.996800i \(0.474527\pi\)
\(660\) 0 0
\(661\) −47.0124 −1.82857 −0.914286 0.405070i \(-0.867247\pi\)
−0.914286 + 0.405070i \(0.867247\pi\)
\(662\) 0 0
\(663\) 4.01285 0.155846
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.1955 0.627091
\(668\) 0 0
\(669\) 27.0373 1.04532
\(670\) 0 0
\(671\) −65.0817 −2.51245
\(672\) 0 0
\(673\) 21.7587 0.838738 0.419369 0.907816i \(-0.362251\pi\)
0.419369 + 0.907816i \(0.362251\pi\)
\(674\) 0 0
\(675\) −3.91331 −0.150623
\(676\) 0 0
\(677\) 23.6829 0.910209 0.455105 0.890438i \(-0.349602\pi\)
0.455105 + 0.890438i \(0.349602\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 8.02231 0.307416
\(682\) 0 0
\(683\) 32.0655 1.22695 0.613476 0.789713i \(-0.289771\pi\)
0.613476 + 0.789713i \(0.289771\pi\)
\(684\) 0 0
\(685\) −1.95420 −0.0746660
\(686\) 0 0
\(687\) 12.4920 0.476599
\(688\) 0 0
\(689\) 6.14214 0.233997
\(690\) 0 0
\(691\) −14.8782 −0.565992 −0.282996 0.959121i \(-0.591328\pi\)
−0.282996 + 0.959121i \(0.591328\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.62412 −0.289199
\(696\) 0 0
\(697\) 37.9249 1.43651
\(698\) 0 0
\(699\) 19.7013 0.745170
\(700\) 0 0
\(701\) −31.0150 −1.17142 −0.585710 0.810521i \(-0.699184\pi\)
−0.585710 + 0.810521i \(0.699184\pi\)
\(702\) 0 0
\(703\) −19.6533 −0.741236
\(704\) 0 0
\(705\) 4.39679 0.165593
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 51.7105 1.94203 0.971014 0.239021i \(-0.0768266\pi\)
0.971014 + 0.239021i \(0.0768266\pi\)
\(710\) 0 0
\(711\) −12.6053 −0.472737
\(712\) 0 0
\(713\) −9.90759 −0.371042
\(714\) 0 0
\(715\) 5.49295 0.205425
\(716\) 0 0
\(717\) 1.93089 0.0721105
\(718\) 0 0
\(719\) −19.7493 −0.736523 −0.368262 0.929722i \(-0.620047\pi\)
−0.368262 + 0.929722i \(0.620047\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −15.1026 −0.561673
\(724\) 0 0
\(725\) −32.8231 −1.21902
\(726\) 0 0
\(727\) 51.3230 1.90346 0.951731 0.306932i \(-0.0993024\pi\)
0.951731 + 0.306932i \(0.0993024\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.35834 −0.124213
\(732\) 0 0
\(733\) 29.1230 1.07568 0.537841 0.843046i \(-0.319240\pi\)
0.537841 + 0.843046i \(0.319240\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −36.8008 −1.35557
\(738\) 0 0
\(739\) 9.27855 0.341317 0.170658 0.985330i \(-0.445411\pi\)
0.170658 + 0.985330i \(0.445411\pi\)
\(740\) 0 0
\(741\) 3.08669 0.113392
\(742\) 0 0
\(743\) −26.5935 −0.975620 −0.487810 0.872950i \(-0.662204\pi\)
−0.487810 + 0.872950i \(0.662204\pi\)
\(744\) 0 0
\(745\) −23.1704 −0.848898
\(746\) 0 0
\(747\) −2.94847 −0.107879
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −28.2843 −1.03211 −0.516054 0.856556i \(-0.672600\pi\)
−0.516054 + 0.856556i \(0.672600\pi\)
\(752\) 0 0
\(753\) 29.5982 1.07862
\(754\) 0 0
\(755\) 15.0116 0.546328
\(756\) 0 0
\(757\) −17.8839 −0.650001 −0.325000 0.945714i \(-0.605364\pi\)
−0.325000 + 0.945714i \(0.605364\pi\)
\(758\) 0 0
\(759\) 11.4519 0.415678
\(760\) 0 0
\(761\) −49.5263 −1.79533 −0.897664 0.440681i \(-0.854737\pi\)
−0.897664 + 0.440681i \(0.854737\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 4.70838 0.170232
\(766\) 0 0
\(767\) 4.83235 0.174486
\(768\) 0 0
\(769\) −45.0675 −1.62517 −0.812587 0.582840i \(-0.801942\pi\)
−0.812587 + 0.582840i \(0.801942\pi\)
\(770\) 0 0
\(771\) 13.8580 0.499085
\(772\) 0 0
\(773\) −4.45051 −0.160074 −0.0800368 0.996792i \(-0.525504\pi\)
−0.0800368 + 0.996792i \(0.525504\pi\)
\(774\) 0 0
\(775\) 20.0796 0.721279
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 29.1719 1.04519
\(780\) 0 0
\(781\) −22.0983 −0.790739
\(782\) 0 0
\(783\) 8.38755 0.299747
\(784\) 0 0
\(785\) −18.7343 −0.668656
\(786\) 0 0
\(787\) 18.1919 0.648470 0.324235 0.945977i \(-0.394893\pi\)
0.324235 + 0.945977i \(0.394893\pi\)
\(788\) 0 0
\(789\) −2.78696 −0.0992183
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.74926 −0.346207
\(794\) 0 0
\(795\) 7.20673 0.255597
\(796\) 0 0
\(797\) 23.8005 0.843059 0.421529 0.906815i \(-0.361493\pi\)
0.421529 + 0.906815i \(0.361493\pi\)
\(798\) 0 0
\(799\) 19.0504 0.673953
\(800\) 0 0
\(801\) −5.00196 −0.176736
\(802\) 0 0
\(803\) −20.7528 −0.732350
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.44104 0.226735
\(808\) 0 0
\(809\) 51.8897 1.82435 0.912173 0.409805i \(-0.134403\pi\)
0.912173 + 0.409805i \(0.134403\pi\)
\(810\) 0 0
\(811\) −13.6462 −0.479183 −0.239592 0.970874i \(-0.577014\pi\)
−0.239592 + 0.970874i \(0.577014\pi\)
\(812\) 0 0
\(813\) −20.8547 −0.731405
\(814\) 0 0
\(815\) −24.1591 −0.846257
\(816\) 0 0
\(817\) −2.58324 −0.0903761
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.69576 −0.163883 −0.0819416 0.996637i \(-0.526112\pi\)
−0.0819416 + 0.996637i \(0.526112\pi\)
\(822\) 0 0
\(823\) 15.8045 0.550912 0.275456 0.961314i \(-0.411171\pi\)
0.275456 + 0.961314i \(0.411171\pi\)
\(824\) 0 0
\(825\) −23.2094 −0.808049
\(826\) 0 0
\(827\) 32.8221 1.14134 0.570668 0.821181i \(-0.306684\pi\)
0.570668 + 0.821181i \(0.306684\pi\)
\(828\) 0 0
\(829\) −3.71688 −0.129092 −0.0645462 0.997915i \(-0.520560\pi\)
−0.0645462 + 0.997915i \(0.520560\pi\)
\(830\) 0 0
\(831\) 27.0724 0.939133
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −13.5443 −0.468721
\(836\) 0 0
\(837\) −5.13109 −0.177356
\(838\) 0 0
\(839\) −3.16625 −0.109311 −0.0546556 0.998505i \(-0.517406\pi\)
−0.0546556 + 0.998505i \(0.517406\pi\)
\(840\) 0 0
\(841\) 41.3510 1.42590
\(842\) 0 0
\(843\) 6.28450 0.216450
\(844\) 0 0
\(845\) −12.7289 −0.437888
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 5.57729 0.191412
\(850\) 0 0
\(851\) −10.9228 −0.374428
\(852\) 0 0
\(853\) −8.30404 −0.284325 −0.142162 0.989843i \(-0.545406\pi\)
−0.142162 + 0.989843i \(0.545406\pi\)
\(854\) 0 0
\(855\) 3.62169 0.123859
\(856\) 0 0
\(857\) 14.3117 0.488880 0.244440 0.969664i \(-0.421396\pi\)
0.244440 + 0.969664i \(0.421396\pi\)
\(858\) 0 0
\(859\) 34.1463 1.16506 0.582528 0.812811i \(-0.302064\pi\)
0.582528 + 0.812811i \(0.302064\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −39.0397 −1.32893 −0.664463 0.747321i \(-0.731340\pi\)
−0.664463 + 0.747321i \(0.731340\pi\)
\(864\) 0 0
\(865\) −24.1686 −0.821757
\(866\) 0 0
\(867\) 3.40040 0.115483
\(868\) 0 0
\(869\) −74.7609 −2.53609
\(870\) 0 0
\(871\) −5.51277 −0.186793
\(872\) 0 0
\(873\) 10.1578 0.343788
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 12.4225 0.419478 0.209739 0.977757i \(-0.432739\pi\)
0.209739 + 0.977757i \(0.432739\pi\)
\(878\) 0 0
\(879\) 12.6460 0.426538
\(880\) 0 0
\(881\) −32.4282 −1.09253 −0.546267 0.837611i \(-0.683952\pi\)
−0.546267 + 0.837611i \(0.683952\pi\)
\(882\) 0 0
\(883\) 21.8475 0.735228 0.367614 0.929978i \(-0.380175\pi\)
0.367614 + 0.929978i \(0.380175\pi\)
\(884\) 0 0
\(885\) 5.66993 0.190592
\(886\) 0 0
\(887\) 54.2035 1.81998 0.909988 0.414634i \(-0.136090\pi\)
0.909988 + 0.414634i \(0.136090\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5.93089 0.198692
\(892\) 0 0
\(893\) 14.6536 0.490363
\(894\) 0 0
\(895\) 25.9950 0.868917
\(896\) 0 0
\(897\) 1.71550 0.0572790
\(898\) 0 0
\(899\) −43.0373 −1.43537
\(900\) 0 0
\(901\) 31.2252 1.04026
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −23.2984 −0.774466
\(906\) 0 0
\(907\) −19.8172 −0.658018 −0.329009 0.944327i \(-0.606715\pi\)
−0.329009 + 0.944327i \(0.606715\pi\)
\(908\) 0 0
\(909\) −1.78598 −0.0592374
\(910\) 0 0
\(911\) 32.5583 1.07870 0.539352 0.842080i \(-0.318669\pi\)
0.539352 + 0.842080i \(0.318669\pi\)
\(912\) 0 0
\(913\) −17.4871 −0.578738
\(914\) 0 0
\(915\) −11.4391 −0.378164
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.95420 0.0644630 0.0322315 0.999480i \(-0.489739\pi\)
0.0322315 + 0.999480i \(0.489739\pi\)
\(920\) 0 0
\(921\) −16.2750 −0.536280
\(922\) 0 0
\(923\) −3.31033 −0.108961
\(924\) 0 0
\(925\) 22.1370 0.727861
\(926\) 0 0
\(927\) 18.5925 0.610657
\(928\) 0 0
\(929\) −23.2641 −0.763272 −0.381636 0.924313i \(-0.624639\pi\)
−0.381636 + 0.924313i \(0.624639\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 28.9929 0.949184
\(934\) 0 0
\(935\) 27.9249 0.913242
\(936\) 0 0
\(937\) −25.1244 −0.820778 −0.410389 0.911911i \(-0.634607\pi\)
−0.410389 + 0.911911i \(0.634607\pi\)
\(938\) 0 0
\(939\) 13.5840 0.443297
\(940\) 0 0
\(941\) −4.67721 −0.152473 −0.0762363 0.997090i \(-0.524290\pi\)
−0.0762363 + 0.997090i \(0.524290\pi\)
\(942\) 0 0
\(943\) 16.2130 0.527968
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.24460 −0.0404441 −0.0202221 0.999796i \(-0.506437\pi\)
−0.0202221 + 0.999796i \(0.506437\pi\)
\(948\) 0 0
\(949\) −3.10878 −0.100915
\(950\) 0 0
\(951\) −24.1240 −0.782273
\(952\) 0 0
\(953\) −33.0373 −1.07018 −0.535091 0.844794i \(-0.679723\pi\)
−0.535091 + 0.844794i \(0.679723\pi\)
\(954\) 0 0
\(955\) −7.33845 −0.237467
\(956\) 0 0
\(957\) 49.7457 1.60805
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −4.67190 −0.150707
\(962\) 0 0
\(963\) 6.23888 0.201045
\(964\) 0 0
\(965\) 12.8999 0.415263
\(966\) 0 0
\(967\) −13.9799 −0.449563 −0.224781 0.974409i \(-0.572167\pi\)
−0.224781 + 0.974409i \(0.572167\pi\)
\(968\) 0 0
\(969\) 15.6920 0.504100
\(970\) 0 0
\(971\) 5.48708 0.176089 0.0880444 0.996117i \(-0.471938\pi\)
0.0880444 + 0.996117i \(0.471938\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −3.47678 −0.111346
\(976\) 0 0
\(977\) −0.324434 −0.0103796 −0.00518978 0.999987i \(-0.501652\pi\)
−0.00518978 + 0.999987i \(0.501652\pi\)
\(978\) 0 0
\(979\) −29.6661 −0.948133
\(980\) 0 0
\(981\) 4.74354 0.151450
\(982\) 0 0
\(983\) 22.5129 0.718051 0.359025 0.933328i \(-0.383109\pi\)
0.359025 + 0.933328i \(0.383109\pi\)
\(984\) 0 0
\(985\) −13.9961 −0.445952
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.43570 −0.0456526
\(990\) 0 0
\(991\) 43.1649 1.37118 0.685588 0.727989i \(-0.259545\pi\)
0.685588 + 0.727989i \(0.259545\pi\)
\(992\) 0 0
\(993\) 7.72357 0.245100
\(994\) 0 0
\(995\) −10.2437 −0.324747
\(996\) 0 0
\(997\) 24.7743 0.784609 0.392305 0.919835i \(-0.371678\pi\)
0.392305 + 0.919835i \(0.371678\pi\)
\(998\) 0 0
\(999\) −5.65685 −0.178975
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9408.2.a.em.1.3 4
4.3 odd 2 9408.2.a.ek.1.3 4
7.6 odd 2 9408.2.a.el.1.2 4
8.3 odd 2 4704.2.a.bz.1.2 yes 4
8.5 even 2 4704.2.a.bx.1.2 yes 4
28.27 even 2 9408.2.a.en.1.2 4
56.13 odd 2 4704.2.a.by.1.3 yes 4
56.27 even 2 4704.2.a.bw.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bw.1.3 4 56.27 even 2
4704.2.a.bx.1.2 yes 4 8.5 even 2
4704.2.a.by.1.3 yes 4 56.13 odd 2
4704.2.a.bz.1.2 yes 4 8.3 odd 2
9408.2.a.ek.1.3 4 4.3 odd 2
9408.2.a.el.1.2 4 7.6 odd 2
9408.2.a.em.1.3 4 1.1 even 1 trivial
9408.2.a.en.1.2 4 28.27 even 2