Properties

Label 3072.2.a.o.1.2
Level $3072$
Weight $2$
Character 3072.1
Self dual yes
Analytic conductor $24.530$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5300435009\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.74912\) of defining polynomial
Character \(\chi\) \(=\) 3072.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.47363 q^{5} +2.55765 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.47363 q^{5} +2.55765 q^{7} +1.00000 q^{9} -0.669808 q^{11} -4.08402 q^{13} -2.47363 q^{15} +6.44549 q^{17} -6.44549 q^{19} +2.55765 q^{21} -2.82843 q^{23} +1.11882 q^{25} +1.00000 q^{27} -4.35480 q^{29} -6.55765 q^{31} -0.669808 q^{33} -6.32666 q^{35} +3.85970 q^{37} -4.08402 q^{39} +0.788632 q^{41} +0.550984 q^{43} -2.47363 q^{45} +2.82843 q^{47} -0.458440 q^{49} +6.44549 q^{51} -3.64520 q^{53} +1.65685 q^{55} -6.44549 q^{57} +5.65685 q^{59} -6.20285 q^{61} +2.55765 q^{63} +10.1023 q^{65} +2.99647 q^{67} -2.82843 q^{69} -5.11529 q^{71} -14.7721 q^{73} +1.11882 q^{75} -1.71313 q^{77} -6.32000 q^{79} +1.00000 q^{81} +0.907457 q^{83} -15.9437 q^{85} -4.35480 q^{87} -6.31724 q^{89} -10.4455 q^{91} -6.55765 q^{93} +15.9437 q^{95} +12.6533 q^{97} -0.669808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} - 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} - 4q^{5} - 4q^{7} + 4q^{9} - 8q^{13} - 4q^{15} - 4q^{21} + 4q^{25} + 4q^{27} - 12q^{29} - 12q^{31} - 16q^{37} - 8q^{39} - 4q^{45} + 4q^{49} - 20q^{53} - 16q^{55} - 16q^{61} - 4q^{63} - 8q^{65} - 16q^{67} + 8q^{71} - 8q^{73} + 4q^{75} - 24q^{77} - 12q^{79} + 4q^{81} - 24q^{85} - 12q^{87} - 8q^{89} - 16q^{91} - 12q^{93} + 24q^{95} + O(q^{100}) \)

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
\(5\) −2.47363 −1.10624 −0.553120 0.833102i \(-0.686563\pi\)
−0.553120 + 0.833102i \(0.686563\pi\)
\(6\) 0 0
\(7\) 2.55765 0.966700 0.483350 0.875427i \(-0.339420\pi\)
0.483350 + 0.875427i \(0.339420\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.669808 −0.201955 −0.100977 0.994889i \(-0.532197\pi\)
−0.100977 + 0.994889i \(0.532197\pi\)
\(12\) 0 0
\(13\) −4.08402 −1.13270 −0.566352 0.824164i \(-0.691646\pi\)
−0.566352 + 0.824164i \(0.691646\pi\)
\(14\) 0 0
\(15\) −2.47363 −0.638687
\(16\) 0 0
\(17\) 6.44549 1.56326 0.781630 0.623742i \(-0.214389\pi\)
0.781630 + 0.623742i \(0.214389\pi\)
\(18\) 0 0
\(19\) −6.44549 −1.47870 −0.739348 0.673323i \(-0.764866\pi\)
−0.739348 + 0.673323i \(0.764866\pi\)
\(20\) 0 0
\(21\) 2.55765 0.558124
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 1.11882 0.223765
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.35480 −0.808666 −0.404333 0.914612i \(-0.632496\pi\)
−0.404333 + 0.914612i \(0.632496\pi\)
\(30\) 0 0
\(31\) −6.55765 −1.17779 −0.588894 0.808210i \(-0.700437\pi\)
−0.588894 + 0.808210i \(0.700437\pi\)
\(32\) 0 0
\(33\) −0.669808 −0.116599
\(34\) 0 0
\(35\) −6.32666 −1.06940
\(36\) 0 0
\(37\) 3.85970 0.634531 0.317265 0.948337i \(-0.397235\pi\)
0.317265 + 0.948337i \(0.397235\pi\)
\(38\) 0 0
\(39\) −4.08402 −0.653967
\(40\) 0 0
\(41\) 0.788632 0.123164 0.0615818 0.998102i \(-0.480385\pi\)
0.0615818 + 0.998102i \(0.480385\pi\)
\(42\) 0 0
\(43\) 0.550984 0.0840242 0.0420121 0.999117i \(-0.486623\pi\)
0.0420121 + 0.999117i \(0.486623\pi\)
\(44\) 0 0
\(45\) −2.47363 −0.368746
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) −0.458440 −0.0654915
\(50\) 0 0
\(51\) 6.44549 0.902549
\(52\) 0 0
\(53\) −3.64520 −0.500707 −0.250353 0.968155i \(-0.580547\pi\)
−0.250353 + 0.968155i \(0.580547\pi\)
\(54\) 0 0
\(55\) 1.65685 0.223410
\(56\) 0 0
\(57\) −6.44549 −0.853726
\(58\) 0 0
\(59\) 5.65685 0.736460 0.368230 0.929735i \(-0.379964\pi\)
0.368230 + 0.929735i \(0.379964\pi\)
\(60\) 0 0
\(61\) −6.20285 −0.794193 −0.397097 0.917777i \(-0.629982\pi\)
−0.397097 + 0.917777i \(0.629982\pi\)
\(62\) 0 0
\(63\) 2.55765 0.322233
\(64\) 0 0
\(65\) 10.1023 1.25304
\(66\) 0 0
\(67\) 2.99647 0.366077 0.183039 0.983106i \(-0.441407\pi\)
0.183039 + 0.983106i \(0.441407\pi\)
\(68\) 0 0
\(69\) −2.82843 −0.340503
\(70\) 0 0
\(71\) −5.11529 −0.607074 −0.303537 0.952820i \(-0.598168\pi\)
−0.303537 + 0.952820i \(0.598168\pi\)
\(72\) 0 0
\(73\) −14.7721 −1.72895 −0.864475 0.502676i \(-0.832349\pi\)
−0.864475 + 0.502676i \(0.832349\pi\)
\(74\) 0 0
\(75\) 1.11882 0.129191
\(76\) 0 0
\(77\) −1.71313 −0.195230
\(78\) 0 0
\(79\) −6.32000 −0.711055 −0.355528 0.934666i \(-0.615699\pi\)
−0.355528 + 0.934666i \(0.615699\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.907457 0.0996063 0.0498032 0.998759i \(-0.484141\pi\)
0.0498032 + 0.998759i \(0.484141\pi\)
\(84\) 0 0
\(85\) −15.9437 −1.72934
\(86\) 0 0
\(87\) −4.35480 −0.466884
\(88\) 0 0
\(89\) −6.31724 −0.669626 −0.334813 0.942285i \(-0.608673\pi\)
−0.334813 + 0.942285i \(0.608673\pi\)
\(90\) 0 0
\(91\) −10.4455 −1.09498
\(92\) 0 0
\(93\) −6.55765 −0.679996
\(94\) 0 0
\(95\) 15.9437 1.63579
\(96\) 0 0
\(97\) 12.6533 1.28475 0.642375 0.766390i \(-0.277949\pi\)
0.642375 + 0.766390i \(0.277949\pi\)
\(98\) 0 0
\(99\) −0.669808 −0.0673182
\(100\) 0 0
\(101\) −10.6417 −1.05889 −0.529443 0.848346i \(-0.677599\pi\)
−0.529443 + 0.848346i \(0.677599\pi\)
\(102\) 0 0
\(103\) −3.33686 −0.328790 −0.164395 0.986395i \(-0.552567\pi\)
−0.164395 + 0.986395i \(0.552567\pi\)
\(104\) 0 0
\(105\) −6.32666 −0.617419
\(106\) 0 0
\(107\) 19.8874 1.92259 0.961296 0.275518i \(-0.0888494\pi\)
0.961296 + 0.275518i \(0.0888494\pi\)
\(108\) 0 0
\(109\) −3.91598 −0.375083 −0.187541 0.982257i \(-0.560052\pi\)
−0.187541 + 0.982257i \(0.560052\pi\)
\(110\) 0 0
\(111\) 3.85970 0.366347
\(112\) 0 0
\(113\) −2.23765 −0.210500 −0.105250 0.994446i \(-0.533564\pi\)
−0.105250 + 0.994446i \(0.533564\pi\)
\(114\) 0 0
\(115\) 6.99647 0.652424
\(116\) 0 0
\(117\) −4.08402 −0.377568
\(118\) 0 0
\(119\) 16.4853 1.51120
\(120\) 0 0
\(121\) −10.5514 −0.959214
\(122\) 0 0
\(123\) 0.788632 0.0711086
\(124\) 0 0
\(125\) 9.60058 0.858702
\(126\) 0 0
\(127\) −12.2145 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(128\) 0 0
\(129\) 0.550984 0.0485114
\(130\) 0 0
\(131\) 5.33962 0.466524 0.233262 0.972414i \(-0.425060\pi\)
0.233262 + 0.972414i \(0.425060\pi\)
\(132\) 0 0
\(133\) −16.4853 −1.42946
\(134\) 0 0
\(135\) −2.47363 −0.212896
\(136\) 0 0
\(137\) −5.10587 −0.436224 −0.218112 0.975924i \(-0.569990\pi\)
−0.218112 + 0.975924i \(0.569990\pi\)
\(138\) 0 0
\(139\) −16.6533 −1.41252 −0.706258 0.707954i \(-0.749618\pi\)
−0.706258 + 0.707954i \(0.749618\pi\)
\(140\) 0 0
\(141\) 2.82843 0.238197
\(142\) 0 0
\(143\) 2.73551 0.228755
\(144\) 0 0
\(145\) 10.7721 0.894578
\(146\) 0 0
\(147\) −0.458440 −0.0378115
\(148\) 0 0
\(149\) −11.1832 −0.916166 −0.458083 0.888909i \(-0.651464\pi\)
−0.458083 + 0.888909i \(0.651464\pi\)
\(150\) 0 0
\(151\) −14.6506 −1.19225 −0.596123 0.802893i \(-0.703293\pi\)
−0.596123 + 0.802893i \(0.703293\pi\)
\(152\) 0 0
\(153\) 6.44549 0.521087
\(154\) 0 0
\(155\) 16.2212 1.30292
\(156\) 0 0
\(157\) −4.45754 −0.355750 −0.177875 0.984053i \(-0.556922\pi\)
−0.177875 + 0.984053i \(0.556922\pi\)
\(158\) 0 0
\(159\) −3.64520 −0.289083
\(160\) 0 0
\(161\) −7.23412 −0.570128
\(162\) 0 0
\(163\) −7.78510 −0.609776 −0.304888 0.952388i \(-0.598619\pi\)
−0.304888 + 0.952388i \(0.598619\pi\)
\(164\) 0 0
\(165\) 1.65685 0.128986
\(166\) 0 0
\(167\) 20.1814 1.56168 0.780841 0.624730i \(-0.214791\pi\)
0.780841 + 0.624730i \(0.214791\pi\)
\(168\) 0 0
\(169\) 3.67923 0.283018
\(170\) 0 0
\(171\) −6.44549 −0.492899
\(172\) 0 0
\(173\) −6.15639 −0.468061 −0.234031 0.972229i \(-0.575192\pi\)
−0.234031 + 0.972229i \(0.575192\pi\)
\(174\) 0 0
\(175\) 2.86156 0.216313
\(176\) 0 0
\(177\) 5.65685 0.425195
\(178\) 0 0
\(179\) −18.7855 −1.40409 −0.702046 0.712131i \(-0.747730\pi\)
−0.702046 + 0.712131i \(0.747730\pi\)
\(180\) 0 0
\(181\) 8.97499 0.667106 0.333553 0.942731i \(-0.391752\pi\)
0.333553 + 0.942731i \(0.391752\pi\)
\(182\) 0 0
\(183\) −6.20285 −0.458528
\(184\) 0 0
\(185\) −9.54745 −0.701943
\(186\) 0 0
\(187\) −4.31724 −0.315708
\(188\) 0 0
\(189\) 2.55765 0.186041
\(190\) 0 0
\(191\) −5.60058 −0.405243 −0.202622 0.979257i \(-0.564946\pi\)
−0.202622 + 0.979257i \(0.564946\pi\)
\(192\) 0 0
\(193\) −19.4514 −1.40014 −0.700071 0.714074i \(-0.746848\pi\)
−0.700071 + 0.714074i \(0.746848\pi\)
\(194\) 0 0
\(195\) 10.1023 0.723444
\(196\) 0 0
\(197\) −1.75070 −0.124732 −0.0623659 0.998053i \(-0.519865\pi\)
−0.0623659 + 0.998053i \(0.519865\pi\)
\(198\) 0 0
\(199\) 0.993710 0.0704422 0.0352211 0.999380i \(-0.488786\pi\)
0.0352211 + 0.999380i \(0.488786\pi\)
\(200\) 0 0
\(201\) 2.99647 0.211355
\(202\) 0 0
\(203\) −11.1380 −0.781738
\(204\) 0 0
\(205\) −1.95078 −0.136248
\(206\) 0 0
\(207\) −2.82843 −0.196589
\(208\) 0 0
\(209\) 4.31724 0.298630
\(210\) 0 0
\(211\) 5.97409 0.411273 0.205637 0.978628i \(-0.434073\pi\)
0.205637 + 0.978628i \(0.434073\pi\)
\(212\) 0 0
\(213\) −5.11529 −0.350494
\(214\) 0 0
\(215\) −1.36293 −0.0929509
\(216\) 0 0
\(217\) −16.7721 −1.13857
\(218\) 0 0
\(219\) −14.7721 −0.998209
\(220\) 0 0
\(221\) −26.3235 −1.77071
\(222\) 0 0
\(223\) 23.7659 1.59148 0.795740 0.605639i \(-0.207082\pi\)
0.795740 + 0.605639i \(0.207082\pi\)
\(224\) 0 0
\(225\) 1.11882 0.0745883
\(226\) 0 0
\(227\) 0.907457 0.0602300 0.0301150 0.999546i \(-0.490413\pi\)
0.0301150 + 0.999546i \(0.490413\pi\)
\(228\) 0 0
\(229\) −7.55579 −0.499301 −0.249650 0.968336i \(-0.580316\pi\)
−0.249650 + 0.968336i \(0.580316\pi\)
\(230\) 0 0
\(231\) −1.71313 −0.112716
\(232\) 0 0
\(233\) −23.2271 −1.52166 −0.760828 0.648954i \(-0.775207\pi\)
−0.760828 + 0.648954i \(0.775207\pi\)
\(234\) 0 0
\(235\) −6.99647 −0.456399
\(236\) 0 0
\(237\) −6.32000 −0.410528
\(238\) 0 0
\(239\) 26.9213 1.74140 0.870698 0.491817i \(-0.163667\pi\)
0.870698 + 0.491817i \(0.163667\pi\)
\(240\) 0 0
\(241\) 10.3494 0.666664 0.333332 0.942809i \(-0.391827\pi\)
0.333332 + 0.942809i \(0.391827\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.13401 0.0724492
\(246\) 0 0
\(247\) 26.3235 1.67492
\(248\) 0 0
\(249\) 0.907457 0.0575077
\(250\) 0 0
\(251\) −13.7984 −0.870949 −0.435475 0.900201i \(-0.643419\pi\)
−0.435475 + 0.900201i \(0.643419\pi\)
\(252\) 0 0
\(253\) 1.89450 0.119106
\(254\) 0 0
\(255\) −15.9437 −0.998435
\(256\) 0 0
\(257\) 16.9965 1.06021 0.530105 0.847932i \(-0.322152\pi\)
0.530105 + 0.847932i \(0.322152\pi\)
\(258\) 0 0
\(259\) 9.87175 0.613401
\(260\) 0 0
\(261\) −4.35480 −0.269555
\(262\) 0 0
\(263\) 29.9929 1.84944 0.924722 0.380643i \(-0.124297\pi\)
0.924722 + 0.380643i \(0.124297\pi\)
\(264\) 0 0
\(265\) 9.01686 0.553901
\(266\) 0 0
\(267\) −6.31724 −0.386609
\(268\) 0 0
\(269\) −29.1332 −1.77628 −0.888142 0.459569i \(-0.848004\pi\)
−0.888142 + 0.459569i \(0.848004\pi\)
\(270\) 0 0
\(271\) −26.6506 −1.61891 −0.809453 0.587184i \(-0.800236\pi\)
−0.809453 + 0.587184i \(0.800236\pi\)
\(272\) 0 0
\(273\) −10.4455 −0.632190
\(274\) 0 0
\(275\) −0.749397 −0.0451904
\(276\) 0 0
\(277\) 17.1430 1.03003 0.515013 0.857183i \(-0.327787\pi\)
0.515013 + 0.857183i \(0.327787\pi\)
\(278\) 0 0
\(279\) −6.55765 −0.392596
\(280\) 0 0
\(281\) 2.76588 0.164999 0.0824993 0.996591i \(-0.473710\pi\)
0.0824993 + 0.996591i \(0.473710\pi\)
\(282\) 0 0
\(283\) 6.34315 0.377061 0.188530 0.982067i \(-0.439628\pi\)
0.188530 + 0.982067i \(0.439628\pi\)
\(284\) 0 0
\(285\) 15.9437 0.944425
\(286\) 0 0
\(287\) 2.01704 0.119062
\(288\) 0 0
\(289\) 24.5443 1.44378
\(290\) 0 0
\(291\) 12.6533 0.741751
\(292\) 0 0
\(293\) 11.6078 0.678133 0.339067 0.940762i \(-0.389889\pi\)
0.339067 + 0.940762i \(0.389889\pi\)
\(294\) 0 0
\(295\) −13.9929 −0.814700
\(296\) 0 0
\(297\) −0.669808 −0.0388662
\(298\) 0 0
\(299\) 11.5514 0.668032
\(300\) 0 0
\(301\) 1.40922 0.0812262
\(302\) 0 0
\(303\) −10.6417 −0.611348
\(304\) 0 0
\(305\) 15.3435 0.878567
\(306\) 0 0
\(307\) 14.7855 0.843852 0.421926 0.906630i \(-0.361354\pi\)
0.421926 + 0.906630i \(0.361354\pi\)
\(308\) 0 0
\(309\) −3.33686 −0.189827
\(310\) 0 0
\(311\) 15.0761 0.854885 0.427442 0.904043i \(-0.359415\pi\)
0.427442 + 0.904043i \(0.359415\pi\)
\(312\) 0 0
\(313\) 23.0027 1.30019 0.650096 0.759852i \(-0.274729\pi\)
0.650096 + 0.759852i \(0.274729\pi\)
\(314\) 0 0
\(315\) −6.32666 −0.356467
\(316\) 0 0
\(317\) −9.55855 −0.536862 −0.268431 0.963299i \(-0.586505\pi\)
−0.268431 + 0.963299i \(0.586505\pi\)
\(318\) 0 0
\(319\) 2.91688 0.163314
\(320\) 0 0
\(321\) 19.8874 1.11001
\(322\) 0 0
\(323\) −41.5443 −2.31159
\(324\) 0 0
\(325\) −4.56930 −0.253459
\(326\) 0 0
\(327\) −3.91598 −0.216554
\(328\) 0 0
\(329\) 7.23412 0.398830
\(330\) 0 0
\(331\) 27.8079 1.52846 0.764229 0.644945i \(-0.223120\pi\)
0.764229 + 0.644945i \(0.223120\pi\)
\(332\) 0 0
\(333\) 3.85970 0.211510
\(334\) 0 0
\(335\) −7.41215 −0.404969
\(336\) 0 0
\(337\) −3.00980 −0.163954 −0.0819771 0.996634i \(-0.526123\pi\)
−0.0819771 + 0.996634i \(0.526123\pi\)
\(338\) 0 0
\(339\) −2.23765 −0.121532
\(340\) 0 0
\(341\) 4.39236 0.237860
\(342\) 0 0
\(343\) −19.0761 −1.03001
\(344\) 0 0
\(345\) 6.99647 0.376677
\(346\) 0 0
\(347\) 8.87449 0.476408 0.238204 0.971215i \(-0.423441\pi\)
0.238204 + 0.971215i \(0.423441\pi\)
\(348\) 0 0
\(349\) −6.70698 −0.359016 −0.179508 0.983757i \(-0.557451\pi\)
−0.179508 + 0.983757i \(0.557451\pi\)
\(350\) 0 0
\(351\) −4.08402 −0.217989
\(352\) 0 0
\(353\) −8.75882 −0.466185 −0.233093 0.972455i \(-0.574884\pi\)
−0.233093 + 0.972455i \(0.574884\pi\)
\(354\) 0 0
\(355\) 12.6533 0.671569
\(356\) 0 0
\(357\) 16.4853 0.872494
\(358\) 0 0
\(359\) −32.7917 −1.73068 −0.865341 0.501184i \(-0.832898\pi\)
−0.865341 + 0.501184i \(0.832898\pi\)
\(360\) 0 0
\(361\) 22.5443 1.18654
\(362\) 0 0
\(363\) −10.5514 −0.553803
\(364\) 0 0
\(365\) 36.5408 1.91263
\(366\) 0 0
\(367\) 20.6435 1.07758 0.538791 0.842439i \(-0.318881\pi\)
0.538791 + 0.842439i \(0.318881\pi\)
\(368\) 0 0
\(369\) 0.788632 0.0410546
\(370\) 0 0
\(371\) −9.32313 −0.484033
\(372\) 0 0
\(373\) −23.4995 −1.21676 −0.608379 0.793646i \(-0.708180\pi\)
−0.608379 + 0.793646i \(0.708180\pi\)
\(374\) 0 0
\(375\) 9.60058 0.495772
\(376\) 0 0
\(377\) 17.7851 0.915979
\(378\) 0 0
\(379\) 11.0004 0.565051 0.282526 0.959260i \(-0.408828\pi\)
0.282526 + 0.959260i \(0.408828\pi\)
\(380\) 0 0
\(381\) −12.2145 −0.625768
\(382\) 0 0
\(383\) 17.2037 0.879070 0.439535 0.898225i \(-0.355143\pi\)
0.439535 + 0.898225i \(0.355143\pi\)
\(384\) 0 0
\(385\) 4.23765 0.215971
\(386\) 0 0
\(387\) 0.550984 0.0280081
\(388\) 0 0
\(389\) 33.7311 1.71023 0.855116 0.518436i \(-0.173486\pi\)
0.855116 + 0.518436i \(0.173486\pi\)
\(390\) 0 0
\(391\) −18.2306 −0.921961
\(392\) 0 0
\(393\) 5.33962 0.269348
\(394\) 0 0
\(395\) 15.6333 0.786597
\(396\) 0 0
\(397\) 14.5201 0.728742 0.364371 0.931254i \(-0.381284\pi\)
0.364371 + 0.931254i \(0.381284\pi\)
\(398\) 0 0
\(399\) −16.4853 −0.825296
\(400\) 0 0
\(401\) −32.2274 −1.60936 −0.804681 0.593708i \(-0.797663\pi\)
−0.804681 + 0.593708i \(0.797663\pi\)
\(402\) 0 0
\(403\) 26.7816 1.33409
\(404\) 0 0
\(405\) −2.47363 −0.122915
\(406\) 0 0
\(407\) −2.58526 −0.128146
\(408\) 0 0
\(409\) −11.5702 −0.572110 −0.286055 0.958213i \(-0.592344\pi\)
−0.286055 + 0.958213i \(0.592344\pi\)
\(410\) 0 0
\(411\) −5.10587 −0.251854
\(412\) 0 0
\(413\) 14.4682 0.711935
\(414\) 0 0
\(415\) −2.24471 −0.110188
\(416\) 0 0
\(417\) −16.6533 −0.815517
\(418\) 0 0
\(419\) −9.54193 −0.466154 −0.233077 0.972458i \(-0.574879\pi\)
−0.233077 + 0.972458i \(0.574879\pi\)
\(420\) 0 0
\(421\) 24.3583 1.18715 0.593576 0.804778i \(-0.297716\pi\)
0.593576 + 0.804778i \(0.297716\pi\)
\(422\) 0 0
\(423\) 2.82843 0.137523
\(424\) 0 0
\(425\) 7.21137 0.349803
\(426\) 0 0
\(427\) −15.8647 −0.767746
\(428\) 0 0
\(429\) 2.73551 0.132072
\(430\) 0 0
\(431\) 40.7088 1.96087 0.980437 0.196832i \(-0.0630654\pi\)
0.980437 + 0.196832i \(0.0630654\pi\)
\(432\) 0 0
\(433\) −7.31371 −0.351474 −0.175737 0.984437i \(-0.556231\pi\)
−0.175737 + 0.984437i \(0.556231\pi\)
\(434\) 0 0
\(435\) 10.7721 0.516485
\(436\) 0 0
\(437\) 18.2306 0.872087
\(438\) 0 0
\(439\) −17.7122 −0.845356 −0.422678 0.906280i \(-0.638910\pi\)
−0.422678 + 0.906280i \(0.638910\pi\)
\(440\) 0 0
\(441\) −0.458440 −0.0218305
\(442\) 0 0
\(443\) 22.1953 1.05453 0.527264 0.849701i \(-0.323218\pi\)
0.527264 + 0.849701i \(0.323218\pi\)
\(444\) 0 0
\(445\) 15.6265 0.740766
\(446\) 0 0
\(447\) −11.1832 −0.528949
\(448\) 0 0
\(449\) −28.3400 −1.33745 −0.668723 0.743511i \(-0.733159\pi\)
−0.668723 + 0.743511i \(0.733159\pi\)
\(450\) 0 0
\(451\) −0.528232 −0.0248735
\(452\) 0 0
\(453\) −14.6506 −0.688344
\(454\) 0 0
\(455\) 25.8382 1.21131
\(456\) 0 0
\(457\) −17.3396 −0.811113 −0.405557 0.914070i \(-0.632922\pi\)
−0.405557 + 0.914070i \(0.632922\pi\)
\(458\) 0 0
\(459\) 6.44549 0.300850
\(460\) 0 0
\(461\) 2.39404 0.111501 0.0557507 0.998445i \(-0.482245\pi\)
0.0557507 + 0.998445i \(0.482245\pi\)
\(462\) 0 0
\(463\) 2.70238 0.125590 0.0627951 0.998026i \(-0.479999\pi\)
0.0627951 + 0.998026i \(0.479999\pi\)
\(464\) 0 0
\(465\) 16.2212 0.752239
\(466\) 0 0
\(467\) −24.2023 −1.11995 −0.559975 0.828510i \(-0.689189\pi\)
−0.559975 + 0.828510i \(0.689189\pi\)
\(468\) 0 0
\(469\) 7.66391 0.353887
\(470\) 0 0
\(471\) −4.45754 −0.205393
\(472\) 0 0
\(473\) −0.369053 −0.0169691
\(474\) 0 0
\(475\) −7.21137 −0.330880
\(476\) 0 0
\(477\) −3.64520 −0.166902
\(478\) 0 0
\(479\) 22.2251 1.01549 0.507745 0.861508i \(-0.330479\pi\)
0.507745 + 0.861508i \(0.330479\pi\)
\(480\) 0 0
\(481\) −15.7631 −0.718735
\(482\) 0 0
\(483\) −7.23412 −0.329164
\(484\) 0 0
\(485\) −31.2996 −1.42124
\(486\) 0 0
\(487\) −13.9839 −0.633672 −0.316836 0.948480i \(-0.602620\pi\)
−0.316836 + 0.948480i \(0.602620\pi\)
\(488\) 0 0
\(489\) −7.78510 −0.352055
\(490\) 0 0
\(491\) −10.2306 −0.461700 −0.230850 0.972989i \(-0.574151\pi\)
−0.230850 + 0.972989i \(0.574151\pi\)
\(492\) 0 0
\(493\) −28.0688 −1.26416
\(494\) 0 0
\(495\) 1.65685 0.0744701
\(496\) 0 0
\(497\) −13.0831 −0.586858
\(498\) 0 0
\(499\) 3.66391 0.164019 0.0820097 0.996632i \(-0.473866\pi\)
0.0820097 + 0.996632i \(0.473866\pi\)
\(500\) 0 0
\(501\) 20.1814 0.901637
\(502\) 0 0
\(503\) 39.6443 1.76765 0.883825 0.467817i \(-0.154959\pi\)
0.883825 + 0.467817i \(0.154959\pi\)
\(504\) 0 0
\(505\) 26.3235 1.17138
\(506\) 0 0
\(507\) 3.67923 0.163400
\(508\) 0 0
\(509\) 28.6909 1.27170 0.635851 0.771812i \(-0.280649\pi\)
0.635851 + 0.771812i \(0.280649\pi\)
\(510\) 0 0
\(511\) −37.7819 −1.67137
\(512\) 0 0
\(513\) −6.44549 −0.284575
\(514\) 0 0
\(515\) 8.25413 0.363721
\(516\) 0 0
\(517\) −1.89450 −0.0833201
\(518\) 0 0
\(519\) −6.15639 −0.270235
\(520\) 0 0
\(521\) 23.1784 1.01546 0.507732 0.861515i \(-0.330484\pi\)
0.507732 + 0.861515i \(0.330484\pi\)
\(522\) 0 0
\(523\) 8.18193 0.357771 0.178885 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(524\) 0 0
\(525\) 2.86156 0.124889
\(526\) 0 0
\(527\) −42.2672 −1.84119
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 5.65685 0.245487
\(532\) 0 0
\(533\) −3.22079 −0.139508
\(534\) 0 0
\(535\) −49.1941 −2.12685
\(536\) 0 0
\(537\) −18.7855 −0.810653
\(538\) 0 0
\(539\) 0.307067 0.0132263
\(540\) 0 0
\(541\) 6.43715 0.276755 0.138377 0.990380i \(-0.455811\pi\)
0.138377 + 0.990380i \(0.455811\pi\)
\(542\) 0 0
\(543\) 8.97499 0.385154
\(544\) 0 0
\(545\) 9.68667 0.414931
\(546\) 0 0
\(547\) 39.2239 1.67709 0.838546 0.544830i \(-0.183406\pi\)
0.838546 + 0.544830i \(0.183406\pi\)
\(548\) 0 0
\(549\) −6.20285 −0.264731
\(550\) 0 0
\(551\) 28.0688 1.19577
\(552\) 0 0
\(553\) −16.1643 −0.687377
\(554\) 0 0
\(555\) −9.54745 −0.405267
\(556\) 0 0
\(557\) 1.66224 0.0704315 0.0352157 0.999380i \(-0.488788\pi\)
0.0352157 + 0.999380i \(0.488788\pi\)
\(558\) 0 0
\(559\) −2.25023 −0.0951745
\(560\) 0 0
\(561\) −4.31724 −0.182274
\(562\) 0 0
\(563\) −40.6368 −1.71264 −0.856319 0.516447i \(-0.827254\pi\)
−0.856319 + 0.516447i \(0.827254\pi\)
\(564\) 0 0
\(565\) 5.53511 0.232864
\(566\) 0 0
\(567\) 2.55765 0.107411
\(568\) 0 0
\(569\) −27.0004 −1.13191 −0.565957 0.824435i \(-0.691493\pi\)
−0.565957 + 0.824435i \(0.691493\pi\)
\(570\) 0 0
\(571\) −20.9706 −0.877591 −0.438795 0.898587i \(-0.644595\pi\)
−0.438795 + 0.898587i \(0.644595\pi\)
\(572\) 0 0
\(573\) −5.60058 −0.233967
\(574\) 0 0
\(575\) −3.16451 −0.131969
\(576\) 0 0
\(577\) −37.6372 −1.56686 −0.783429 0.621481i \(-0.786531\pi\)
−0.783429 + 0.621481i \(0.786531\pi\)
\(578\) 0 0
\(579\) −19.4514 −0.808372
\(580\) 0 0
\(581\) 2.32095 0.0962894
\(582\) 0 0
\(583\) 2.44158 0.101120
\(584\) 0 0
\(585\) 10.1023 0.417680
\(586\) 0 0
\(587\) 44.2047 1.82452 0.912261 0.409609i \(-0.134335\pi\)
0.912261 + 0.409609i \(0.134335\pi\)
\(588\) 0 0
\(589\) 42.2672 1.74159
\(590\) 0 0
\(591\) −1.75070 −0.0720140
\(592\) 0 0
\(593\) 3.59611 0.147675 0.0738373 0.997270i \(-0.476475\pi\)
0.0738373 + 0.997270i \(0.476475\pi\)
\(594\) 0 0
\(595\) −40.7784 −1.67175
\(596\) 0 0
\(597\) 0.993710 0.0406698
\(598\) 0 0
\(599\) −22.0296 −0.900104 −0.450052 0.893002i \(-0.648595\pi\)
−0.450052 + 0.893002i \(0.648595\pi\)
\(600\) 0 0
\(601\) 10.7721 0.439405 0.219703 0.975567i \(-0.429491\pi\)
0.219703 + 0.975567i \(0.429491\pi\)
\(602\) 0 0
\(603\) 2.99647 0.122026
\(604\) 0 0
\(605\) 26.1001 1.06112
\(606\) 0 0
\(607\) −5.47453 −0.222204 −0.111102 0.993809i \(-0.535438\pi\)
−0.111102 + 0.993809i \(0.535438\pi\)
\(608\) 0 0
\(609\) −11.1380 −0.451336
\(610\) 0 0
\(611\) −11.5514 −0.467318
\(612\) 0 0
\(613\) 14.8562 0.600035 0.300018 0.953934i \(-0.403007\pi\)
0.300018 + 0.953934i \(0.403007\pi\)
\(614\) 0 0
\(615\) −1.95078 −0.0786631
\(616\) 0 0
\(617\) 22.2235 0.894686 0.447343 0.894363i \(-0.352370\pi\)
0.447343 + 0.894363i \(0.352370\pi\)
\(618\) 0 0
\(619\) −16.4612 −0.661631 −0.330815 0.943696i \(-0.607324\pi\)
−0.330815 + 0.943696i \(0.607324\pi\)
\(620\) 0 0
\(621\) −2.82843 −0.113501
\(622\) 0 0
\(623\) −16.1573 −0.647327
\(624\) 0 0
\(625\) −29.3424 −1.17369
\(626\) 0 0
\(627\) 4.31724 0.172414
\(628\) 0 0
\(629\) 24.8776 0.991937
\(630\) 0 0
\(631\) 4.06977 0.162015 0.0810075 0.996713i \(-0.474186\pi\)
0.0810075 + 0.996713i \(0.474186\pi\)
\(632\) 0 0
\(633\) 5.97409 0.237449
\(634\) 0 0
\(635\) 30.2141 1.19901
\(636\) 0 0
\(637\) 1.87228 0.0741824
\(638\) 0 0
\(639\) −5.11529 −0.202358
\(640\) 0 0
\(641\) −8.41958 −0.332553 −0.166277 0.986079i \(-0.553174\pi\)
−0.166277 + 0.986079i \(0.553174\pi\)
\(642\) 0 0
\(643\) 10.4266 0.411186 0.205593 0.978638i \(-0.434088\pi\)
0.205593 + 0.978638i \(0.434088\pi\)
\(644\) 0 0
\(645\) −1.36293 −0.0536652
\(646\) 0 0
\(647\) −11.6132 −0.456560 −0.228280 0.973595i \(-0.573310\pi\)
−0.228280 + 0.973595i \(0.573310\pi\)
\(648\) 0 0
\(649\) −3.78901 −0.148731
\(650\) 0 0
\(651\) −16.7721 −0.657352
\(652\) 0 0
\(653\) 2.73012 0.106838 0.0534190 0.998572i \(-0.482988\pi\)
0.0534190 + 0.998572i \(0.482988\pi\)
\(654\) 0 0
\(655\) −13.2082 −0.516088
\(656\) 0 0
\(657\) −14.7721 −0.576316
\(658\) 0 0
\(659\) 31.5514 1.22907 0.614533 0.788891i \(-0.289344\pi\)
0.614533 + 0.788891i \(0.289344\pi\)
\(660\) 0 0
\(661\) 15.1368 0.588752 0.294376 0.955690i \(-0.404888\pi\)
0.294376 + 0.955690i \(0.404888\pi\)
\(662\) 0 0
\(663\) −26.3235 −1.02232
\(664\) 0 0
\(665\) 40.7784 1.58132
\(666\) 0 0
\(667\) 12.3172 0.476925
\(668\) 0 0
\(669\) 23.7659 0.918841
\(670\) 0 0
\(671\) 4.15472 0.160391
\(672\) 0 0
\(673\) −20.6345 −0.795401 −0.397700 0.917515i \(-0.630192\pi\)
−0.397700 + 0.917515i \(0.630192\pi\)
\(674\) 0 0
\(675\) 1.11882 0.0430636
\(676\) 0 0
\(677\) −37.9357 −1.45799 −0.728994 0.684520i \(-0.760012\pi\)
−0.728994 + 0.684520i \(0.760012\pi\)
\(678\) 0 0
\(679\) 32.3627 1.24197
\(680\) 0 0
\(681\) 0.907457 0.0347738
\(682\) 0 0
\(683\) −18.2471 −0.698205 −0.349102 0.937085i \(-0.613513\pi\)
−0.349102 + 0.937085i \(0.613513\pi\)
\(684\) 0 0
\(685\) 12.6300 0.482568
\(686\) 0 0
\(687\) −7.55579 −0.288271
\(688\) 0 0
\(689\) 14.8871 0.567152
\(690\) 0 0
\(691\) −30.2533 −1.15089 −0.575446 0.817840i \(-0.695171\pi\)
−0.575446 + 0.817840i \(0.695171\pi\)
\(692\) 0 0
\(693\) −1.71313 −0.0650765
\(694\) 0 0
\(695\) 41.1941 1.56258
\(696\) 0 0
\(697\) 5.08312 0.192537
\(698\) 0 0
\(699\) −23.2271 −0.878528
\(700\) 0 0
\(701\) 20.0875 0.758696 0.379348 0.925254i \(-0.376148\pi\)
0.379348 + 0.925254i \(0.376148\pi\)
\(702\) 0 0
\(703\) −24.8776 −0.938278
\(704\) 0 0
\(705\) −6.99647 −0.263502
\(706\) 0 0
\(707\) −27.2176 −1.02362
\(708\) 0 0
\(709\) 41.7864 1.56932 0.784660 0.619926i \(-0.212837\pi\)
0.784660 + 0.619926i \(0.212837\pi\)
\(710\) 0 0
\(711\) −6.32000 −0.237018
\(712\) 0 0
\(713\) 18.5478 0.694622
\(714\) 0 0
\(715\) −6.76663 −0.253058
\(716\) 0 0
\(717\) 26.9213 1.00540
\(718\) 0 0
\(719\) 28.3683 1.05796 0.528979 0.848635i \(-0.322575\pi\)
0.528979 + 0.848635i \(0.322575\pi\)
\(720\) 0 0
\(721\) −8.53450 −0.317841
\(722\) 0 0
\(723\) 10.3494 0.384899
\(724\) 0 0
\(725\) −4.87226 −0.180951
\(726\) 0 0
\(727\) 20.4843 0.759722 0.379861 0.925044i \(-0.375972\pi\)
0.379861 + 0.925044i \(0.375972\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 3.55136 0.131352
\(732\) 0 0
\(733\) 48.0777 1.77579 0.887895 0.460045i \(-0.152167\pi\)
0.887895 + 0.460045i \(0.152167\pi\)
\(734\) 0 0
\(735\) 1.13401 0.0418286
\(736\) 0 0
\(737\) −2.00706 −0.0739310
\(738\) 0 0
\(739\) −21.4459 −0.788899 −0.394449 0.918918i \(-0.629065\pi\)
−0.394449 + 0.918918i \(0.629065\pi\)
\(740\) 0 0
\(741\) 26.3235 0.967018
\(742\) 0 0
\(743\) −2.17431 −0.0797677 −0.0398839 0.999204i \(-0.512699\pi\)
−0.0398839 + 0.999204i \(0.512699\pi\)
\(744\) 0 0
\(745\) 27.6631 1.01350
\(746\) 0 0
\(747\) 0.907457 0.0332021
\(748\) 0 0
\(749\) 50.8651 1.85857
\(750\) 0 0
\(751\) 29.8980 1.09099 0.545497 0.838113i \(-0.316341\pi\)
0.545497 + 0.838113i \(0.316341\pi\)
\(752\) 0 0
\(753\) −13.7984 −0.502843
\(754\) 0 0
\(755\) 36.2400 1.31891
\(756\) 0 0
\(757\) −21.6791 −0.787939 −0.393970 0.919123i \(-0.628899\pi\)
−0.393970 + 0.919123i \(0.628899\pi\)
\(758\) 0 0
\(759\) 1.89450 0.0687661
\(760\) 0 0
\(761\) 4.29449 0.155675 0.0778375 0.996966i \(-0.475198\pi\)
0.0778375 + 0.996966i \(0.475198\pi\)
\(762\) 0 0
\(763\) −10.0157 −0.362592
\(764\) 0 0
\(765\) −15.9437 −0.576446
\(766\) 0 0
\(767\) −23.1027 −0.834191
\(768\) 0 0
\(769\) 33.8819 1.22181 0.610907 0.791703i \(-0.290805\pi\)
0.610907 + 0.791703i \(0.290805\pi\)
\(770\) 0 0
\(771\) 16.9965 0.612113
\(772\) 0 0
\(773\) 49.5300 1.78147 0.890736 0.454521i \(-0.150190\pi\)
0.890736 + 0.454521i \(0.150190\pi\)
\(774\) 0 0
\(775\) −7.33686 −0.263548
\(776\) 0 0
\(777\) 9.87175 0.354147
\(778\) 0 0
\(779\) −5.08312 −0.182122
\(780\) 0 0
\(781\) 3.42627 0.122601
\(782\) 0 0
\(783\) −4.35480 −0.155628
\(784\) 0 0
\(785\) 11.0263 0.393545
\(786\) 0 0
\(787\) 34.0953 1.21537 0.607683 0.794180i \(-0.292099\pi\)
0.607683 + 0.794180i \(0.292099\pi\)
\(788\) 0 0
\(789\) 29.9929 1.06778
\(790\) 0 0
\(791\) −5.72312 −0.203491
\(792\) 0 0
\(793\) 25.3326 0.899585
\(794\) 0 0
\(795\) 9.01686 0.319795
\(796\) 0 0
\(797\) −40.6901 −1.44132 −0.720659 0.693290i \(-0.756160\pi\)
−0.720659 + 0.693290i \(0.756160\pi\)
\(798\) 0 0
\(799\) 18.2306 0.644952
\(800\) 0 0
\(801\) −6.31724 −0.223209
\(802\) 0 0
\(803\) 9.89450 0.349169
\(804\) 0 0
\(805\) 17.8945 0.630698
\(806\) 0 0
\(807\) −29.1332 −1.02554
\(808\) 0 0
\(809\) −10.9926 −0.386478 −0.193239 0.981152i \(-0.561899\pi\)
−0.193239 + 0.981152i \(0.561899\pi\)
\(810\) 0 0
\(811\) 21.2498 0.746182 0.373091 0.927795i \(-0.378298\pi\)
0.373091 + 0.927795i \(0.378298\pi\)
\(812\) 0 0
\(813\) −26.6506 −0.934676
\(814\) 0 0
\(815\) 19.2574 0.674558
\(816\) 0 0
\(817\) −3.55136 −0.124246
\(818\) 0 0
\(819\) −10.4455 −0.364995
\(820\) 0 0
\(821\) −30.0572 −1.04900 −0.524501 0.851410i \(-0.675748\pi\)
−0.524501 + 0.851410i \(0.675748\pi\)
\(822\) 0 0
\(823\) 55.0851 1.92015 0.960073 0.279751i \(-0.0902518\pi\)
0.960073 + 0.279751i \(0.0902518\pi\)
\(824\) 0 0
\(825\) −0.749397 −0.0260907
\(826\) 0 0
\(827\) −34.5478 −1.20135 −0.600673 0.799495i \(-0.705101\pi\)
−0.600673 + 0.799495i \(0.705101\pi\)
\(828\) 0 0
\(829\) −31.0046 −1.07683 −0.538417 0.842679i \(-0.680977\pi\)
−0.538417 + 0.842679i \(0.680977\pi\)
\(830\) 0 0
\(831\) 17.1430 0.594685
\(832\) 0 0
\(833\) −2.95487 −0.102380
\(834\) 0 0
\(835\) −49.9212 −1.72759
\(836\) 0 0
\(837\) −6.55765 −0.226665
\(838\) 0 0
\(839\) −5.14195 −0.177520 −0.0887599 0.996053i \(-0.528290\pi\)
−0.0887599 + 0.996053i \(0.528290\pi\)
\(840\) 0 0
\(841\) −10.0357 −0.346059
\(842\) 0 0
\(843\) 2.76588 0.0952620
\(844\) 0 0
\(845\) −9.10104 −0.313085
\(846\) 0 0
\(847\) −26.9867 −0.927272
\(848\) 0 0
\(849\) 6.34315 0.217696
\(850\) 0 0
\(851\) −10.9169 −0.374226
\(852\) 0 0
\(853\) 18.5060 0.633632 0.316816 0.948487i \(-0.397386\pi\)
0.316816 + 0.948487i \(0.397386\pi\)
\(854\) 0 0
\(855\) 15.9437 0.545264
\(856\) 0 0
\(857\) 22.8878 0.781833 0.390916 0.920426i \(-0.372158\pi\)
0.390916 + 0.920426i \(0.372158\pi\)
\(858\) 0 0
\(859\) −35.5286 −1.21222 −0.606110 0.795381i \(-0.707271\pi\)
−0.606110 + 0.795381i \(0.707271\pi\)
\(860\) 0 0
\(861\) 2.01704 0.0687407
\(862\) 0 0
\(863\) 43.9296 1.49538 0.747691 0.664047i \(-0.231163\pi\)
0.747691 + 0.664047i \(0.231163\pi\)
\(864\) 0 0
\(865\) 15.2286 0.517788
\(866\) 0 0
\(867\) 24.5443 0.833568
\(868\) 0 0
\(869\) 4.23319 0.143601
\(870\) 0 0
\(871\) −12.2376 −0.414657
\(872\) 0 0
\(873\) 12.6533 0.428250
\(874\) 0 0
\(875\) 24.5549 0.830107
\(876\) 0 0
\(877\) −21.5773 −0.728614 −0.364307 0.931279i \(-0.618694\pi\)
−0.364307 + 0.931279i \(0.618694\pi\)
\(878\) 0 0
\(879\) 11.6078 0.391520
\(880\) 0 0
\(881\) −21.6686 −0.730035 −0.365018 0.931001i \(-0.618937\pi\)
−0.365018 + 0.931001i \(0.618937\pi\)
\(882\) 0 0
\(883\) −0.0834930 −0.00280976 −0.00140488 0.999999i \(-0.500447\pi\)
−0.00140488 + 0.999999i \(0.500447\pi\)
\(884\) 0 0
\(885\) −13.9929 −0.470368
\(886\) 0 0
\(887\) −30.8043 −1.03431 −0.517154 0.855892i \(-0.673009\pi\)
−0.517154 + 0.855892i \(0.673009\pi\)
\(888\) 0 0
\(889\) −31.2404 −1.04777
\(890\) 0 0
\(891\) −0.669808 −0.0224394
\(892\) 0 0
\(893\) −18.2306 −0.610063
\(894\) 0 0
\(895\) 46.4682 1.55326
\(896\) 0 0
\(897\) 11.5514 0.385689
\(898\) 0 0
\(899\) 28.5573 0.952438
\(900\) 0 0
\(901\) −23.4951 −0.782735
\(902\) 0 0
\(903\) 1.40922 0.0468960
\(904\) 0 0
\(905\) −22.2008 −0.737979
\(906\) 0 0
\(907\) −49.5215 −1.64434 −0.822168 0.569245i \(-0.807236\pi\)
−0.822168 + 0.569245i \(0.807236\pi\)
\(908\) 0 0
\(909\) −10.6417 −0.352962
\(910\) 0 0
\(911\) 0.0829331 0.00274770 0.00137385 0.999999i \(-0.499563\pi\)
0.00137385 + 0.999999i \(0.499563\pi\)
\(912\) 0 0
\(913\) −0.607822 −0.0201160
\(914\) 0 0
\(915\) 15.3435 0.507241
\(916\) 0 0
\(917\) 13.6569 0.450989
\(918\) 0 0
\(919\) −20.1161 −0.663568 −0.331784 0.943355i \(-0.607650\pi\)
−0.331784 + 0.943355i \(0.607650\pi\)
\(920\) 0 0
\(921\) 14.7855 0.487198
\(922\) 0 0
\(923\) 20.8910 0.687635
\(924\) 0 0
\(925\) 4.31833 0.141986
\(926\) 0 0
\(927\) −3.33686 −0.109597
\(928\) 0 0
\(929\) 8.55098 0.280549 0.140274 0.990113i \(-0.455202\pi\)
0.140274 + 0.990113i \(0.455202\pi\)
\(930\) 0 0
\(931\) 2.95487 0.0968420
\(932\) 0 0
\(933\) 15.0761 0.493568
\(934\) 0 0
\(935\) 10.6792 0.349248
\(936\) 0 0
\(937\) 33.5780 1.09695 0.548473 0.836168i \(-0.315209\pi\)
0.548473 + 0.836168i \(0.315209\pi\)
\(938\) 0 0
\(939\) 23.0027 0.750666
\(940\) 0 0
\(941\) 11.9991 0.391159 0.195579 0.980688i \(-0.437341\pi\)
0.195579 + 0.980688i \(0.437341\pi\)
\(942\) 0 0
\(943\) −2.23059 −0.0726380
\(944\) 0 0
\(945\) −6.32666 −0.205806
\(946\) 0 0
\(947\) 25.2537 0.820635 0.410318 0.911943i \(-0.365418\pi\)
0.410318 + 0.911943i \(0.365418\pi\)
\(948\) 0 0
\(949\) 60.3298 1.95839
\(950\) 0 0
\(951\) −9.55855 −0.309957
\(952\) 0 0
\(953\) 3.86469 0.125190 0.0625948 0.998039i \(-0.480062\pi\)
0.0625948 + 0.998039i \(0.480062\pi\)
\(954\) 0 0
\(955\) 13.8537 0.448296
\(956\) 0 0
\(957\) 2.91688 0.0942894
\(958\) 0 0
\(959\) −13.0590 −0.421698
\(960\) 0 0
\(961\) 12.0027 0.387185
\(962\) 0 0
\(963\) 19.8874 0.640864
\(964\) 0 0
\(965\) 48.1154 1.54889
\(966\) 0 0
\(967\) 37.8714 1.21786 0.608930 0.793224i \(-0.291599\pi\)
0.608930 + 0.793224i \(0.291599\pi\)
\(968\) 0 0
\(969\) −41.5443 −1.33460
\(970\) 0 0
\(971\) −3.74587 −0.120211 −0.0601053 0.998192i \(-0.519144\pi\)
−0.0601053 + 0.998192i \(0.519144\pi\)
\(972\) 0 0
\(973\) −42.5933 −1.36548
\(974\) 0 0
\(975\) −4.56930 −0.146335
\(976\) 0 0
\(977\) −17.6530 −0.564768 −0.282384 0.959301i \(-0.591125\pi\)
−0.282384 + 0.959301i \(0.591125\pi\)
\(978\) 0 0
\(979\) 4.23134 0.135234
\(980\) 0 0
\(981\) −3.91598 −0.125028
\(982\) 0 0
\(983\) −22.3557 −0.713035 −0.356518 0.934289i \(-0.616036\pi\)
−0.356518 + 0.934289i \(0.616036\pi\)
\(984\) 0 0
\(985\) 4.33057 0.137983
\(986\) 0 0
\(987\) 7.23412 0.230265
\(988\) 0 0
\(989\) −1.55842 −0.0495548
\(990\) 0 0
\(991\) −17.8769 −0.567878 −0.283939 0.958842i \(-0.591641\pi\)
−0.283939 + 0.958842i \(0.591641\pi\)
\(992\) 0 0
\(993\) 27.8079 0.882456
\(994\) 0 0
\(995\) −2.45807 −0.0779260
\(996\) 0 0
\(997\) −6.06146 −0.191968 −0.0959841 0.995383i \(-0.530600\pi\)
−0.0959841 + 0.995383i \(0.530600\pi\)
\(998\) 0 0
\(999\) 3.85970 0.122116
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 3072.2.a.o.1.2 4
3.2 odd 2 9216.2.a.bn.1.3 4
4.3 odd 2 3072.2.a.i.1.2 4
8.3 odd 2 3072.2.a.t.1.3 4
8.5 even 2 3072.2.a.n.1.3 4
12.11 even 2 9216.2.a.bo.1.3 4
16.3 odd 4 3072.2.d.f.1537.3 8
16.5 even 4 3072.2.d.i.1537.2 8
16.11 odd 4 3072.2.d.f.1537.6 8
16.13 even 4 3072.2.d.i.1537.7 8
24.5 odd 2 9216.2.a.x.1.2 4
24.11 even 2 9216.2.a.y.1.2 4
32.3 odd 8 48.2.j.a.13.2 8
32.5 even 8 384.2.j.a.97.1 8
32.11 odd 8 48.2.j.a.37.2 yes 8
32.13 even 8 384.2.j.a.289.1 8
32.19 odd 8 384.2.j.b.289.3 8
32.21 even 8 192.2.j.a.49.4 8
32.27 odd 8 384.2.j.b.97.3 8
32.29 even 8 192.2.j.a.145.4 8
96.5 odd 8 1152.2.k.f.865.4 8
96.11 even 8 144.2.k.b.37.3 8
96.29 odd 8 576.2.k.b.145.1 8
96.35 even 8 144.2.k.b.109.3 8
96.53 odd 8 576.2.k.b.433.1 8
96.59 even 8 1152.2.k.c.865.4 8
96.77 odd 8 1152.2.k.f.289.4 8
96.83 even 8 1152.2.k.c.289.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.2 8 32.3 odd 8
48.2.j.a.37.2 yes 8 32.11 odd 8
144.2.k.b.37.3 8 96.11 even 8
144.2.k.b.109.3 8 96.35 even 8
192.2.j.a.49.4 8 32.21 even 8
192.2.j.a.145.4 8 32.29 even 8
384.2.j.a.97.1 8 32.5 even 8
384.2.j.a.289.1 8 32.13 even 8
384.2.j.b.97.3 8 32.27 odd 8
384.2.j.b.289.3 8 32.19 odd 8
576.2.k.b.145.1 8 96.29 odd 8
576.2.k.b.433.1 8 96.53 odd 8
1152.2.k.c.289.4 8 96.83 even 8
1152.2.k.c.865.4 8 96.59 even 8
1152.2.k.f.289.4 8 96.77 odd 8
1152.2.k.f.865.4 8 96.5 odd 8
3072.2.a.i.1.2 4 4.3 odd 2
3072.2.a.n.1.3 4 8.5 even 2
3072.2.a.o.1.2 4 1.1 even 1 trivial
3072.2.a.t.1.3 4 8.3 odd 2
3072.2.d.f.1537.3 8 16.3 odd 4
3072.2.d.f.1537.6 8 16.11 odd 4
3072.2.d.i.1537.2 8 16.5 even 4
3072.2.d.i.1537.7 8 16.13 even 4
9216.2.a.x.1.2 4 24.5 odd 2
9216.2.a.y.1.2 4 24.11 even 2
9216.2.a.bn.1.3 4 3.2 odd 2
9216.2.a.bo.1.3 4 12.11 even 2