Properties

Label 3072.2.d.i.1537.7
Level $3072$
Weight $2$
Character 3072.1537
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1537.7
Root \(0.500000 + 1.44392i\) of defining polynomial
Character \(\chi\) \(=\) 3072.1537
Dual form 3072.2.d.i.1537.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +2.47363i q^{5} -2.55765 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +2.47363i q^{5} -2.55765 q^{7} -1.00000 q^{9} +0.669808i q^{11} -4.08402i q^{13} -2.47363 q^{15} +6.44549 q^{17} -6.44549i q^{19} -2.55765i q^{21} +2.82843 q^{23} -1.11882 q^{25} -1.00000i q^{27} -4.35480i q^{29} -6.55765 q^{31} -0.669808 q^{33} -6.32666i q^{35} -3.85970i q^{37} +4.08402 q^{39} -0.788632 q^{41} -0.550984i q^{43} -2.47363i q^{45} +2.82843 q^{47} -0.458440 q^{49} +6.44549i q^{51} +3.64520i q^{53} -1.65685 q^{55} +6.44549 q^{57} -5.65685i q^{59} -6.20285i q^{61} +2.55765 q^{63} +10.1023 q^{65} +2.99647i q^{67} +2.82843i q^{69} +5.11529 q^{71} +14.7721 q^{73} -1.11882i q^{75} -1.71313i q^{77} -6.32000 q^{79} +1.00000 q^{81} +0.907457i q^{83} +15.9437i q^{85} +4.35480 q^{87} +6.31724 q^{89} +10.4455i q^{91} -6.55765i q^{93} +15.9437 q^{95} +12.6533 q^{97} -0.669808i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} - 8q^{9} + O(q^{10}) \) \( 8q + 8q^{7} - 8q^{9} - 8q^{15} - 8q^{25} - 24q^{31} + 16q^{39} + 8q^{49} + 32q^{55} - 8q^{63} - 16q^{65} - 16q^{71} + 16q^{73} - 24q^{79} + 8q^{81} + 24q^{87} + 16q^{89} + 48q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 2.47363i 1.10624i 0.833102 + 0.553120i \(0.186563\pi\)
−0.833102 + 0.553120i \(0.813437\pi\)
\(6\) 0 0
\(7\) −2.55765 −0.966700 −0.483350 0.875427i \(-0.660580\pi\)
−0.483350 + 0.875427i \(0.660580\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.669808i 0.201955i 0.994889 + 0.100977i \(0.0321970\pi\)
−0.994889 + 0.100977i \(0.967803\pi\)
\(12\) 0 0
\(13\) − 4.08402i − 1.13270i −0.824164 0.566352i \(-0.808354\pi\)
0.824164 0.566352i \(-0.191646\pi\)
\(14\) 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
\(19\) − 6.44549i − 1.47870i −0.673323 0.739348i \(-0.735134\pi\)
0.673323 0.739348i \(-0.264866\pi\)
\(20\) 0 0
\(21\) − 2.55765i − 0.558124i
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) −1.11882 −0.223765
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 4.35480i − 0.808666i −0.914612 0.404333i \(-0.867504\pi\)
0.914612 0.404333i \(-0.132496\pi\)
\(30\) 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
\(35\) − 6.32666i − 1.06940i
\(36\) 0 0
\(37\) − 3.85970i − 0.634531i −0.948337 0.317265i \(-0.897235\pi\)
0.948337 0.317265i \(-0.102765\pi\)
\(38\) 0 0
\(39\) 4.08402 0.653967
\(40\) 0 0
\(41\) −0.788632 −0.123164 −0.0615818 0.998102i \(-0.519615\pi\)
−0.0615818 + 0.998102i \(0.519615\pi\)
\(42\) 0 0
\(43\) − 0.550984i − 0.0840242i −0.999117 0.0420121i \(-0.986623\pi\)
0.999117 0.0420121i \(-0.0133768\pi\)
\(44\) 0 0
\(45\) − 2.47363i − 0.368746i
\(46\) 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
\(51\) 6.44549i 0.902549i
\(52\) 0 0
\(53\) 3.64520i 0.500707i 0.968155 + 0.250353i \(0.0805468\pi\)
−0.968155 + 0.250353i \(0.919453\pi\)
\(54\) 0 0
\(55\) −1.65685 −0.223410
\(56\) 0 0
\(57\) 6.44549 0.853726
\(58\) 0 0
\(59\) − 5.65685i − 0.736460i −0.929735 0.368230i \(-0.879964\pi\)
0.929735 0.368230i \(-0.120036\pi\)
\(60\) 0 0
\(61\) − 6.20285i − 0.794193i −0.917777 0.397097i \(-0.870018\pi\)
0.917777 0.397097i \(-0.129982\pi\)
\(62\) 0 0
\(63\) 2.55765 0.322233
\(64\) 0 0
\(65\) 10.1023 1.25304
\(66\) 0 0
\(67\) 2.99647i 0.366077i 0.983106 + 0.183039i \(0.0585933\pi\)
−0.983106 + 0.183039i \(0.941407\pi\)
\(68\) 0 0
\(69\) 2.82843i 0.340503i
\(70\) 0 0
\(71\) 5.11529 0.607074 0.303537 0.952820i \(-0.401832\pi\)
0.303537 + 0.952820i \(0.401832\pi\)
\(72\) 0 0
\(73\) 14.7721 1.72895 0.864475 0.502676i \(-0.167651\pi\)
0.864475 + 0.502676i \(0.167651\pi\)
\(74\) 0 0
\(75\) − 1.11882i − 0.129191i
\(76\) 0 0
\(77\) − 1.71313i − 0.195230i
\(78\) 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
\(83\) 0.907457i 0.0996063i 0.998759 + 0.0498032i \(0.0158594\pi\)
−0.998759 + 0.0498032i \(0.984141\pi\)
\(84\) 0 0
\(85\) 15.9437i 1.72934i
\(86\) 0 0
\(87\) 4.35480 0.466884
\(88\) 0 0
\(89\) 6.31724 0.669626 0.334813 0.942285i \(-0.391327\pi\)
0.334813 + 0.942285i \(0.391327\pi\)
\(90\) 0 0
\(91\) 10.4455i 1.09498i
\(92\) 0 0
\(93\) − 6.55765i − 0.679996i
\(94\) 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
\(99\) − 0.669808i − 0.0673182i
\(100\) 0 0
\(101\) 10.6417i 1.05889i 0.848346 + 0.529443i \(0.177599\pi\)
−0.848346 + 0.529443i \(0.822401\pi\)
\(102\) 0 0
\(103\) 3.33686 0.328790 0.164395 0.986395i \(-0.447433\pi\)
0.164395 + 0.986395i \(0.447433\pi\)
\(104\) 0 0
\(105\) 6.32666 0.617419
\(106\) 0 0
\(107\) − 19.8874i − 1.92259i −0.275518 0.961296i \(-0.588849\pi\)
0.275518 0.961296i \(-0.411151\pi\)
\(108\) 0 0
\(109\) − 3.91598i − 0.375083i −0.982257 0.187541i \(-0.939948\pi\)
0.982257 0.187541i \(-0.0600519\pi\)
\(110\) 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
\(115\) 6.99647i 0.652424i
\(116\) 0 0
\(117\) 4.08402i 0.377568i
\(118\) 0 0
\(119\) −16.4853 −1.51120
\(120\) 0 0
\(121\) 10.5514 0.959214
\(122\) 0 0
\(123\) − 0.788632i − 0.0711086i
\(124\) 0 0
\(125\) 9.60058i 0.858702i
\(126\) 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
\(131\) 5.33962i 0.466524i 0.972414 + 0.233262i \(0.0749400\pi\)
−0.972414 + 0.233262i \(0.925060\pi\)
\(132\) 0 0
\(133\) 16.4853i 1.42946i
\(134\) 0 0
\(135\) 2.47363 0.212896
\(136\) 0 0
\(137\) 5.10587 0.436224 0.218112 0.975924i \(-0.430010\pi\)
0.218112 + 0.975924i \(0.430010\pi\)
\(138\) 0 0
\(139\) 16.6533i 1.41252i 0.707954 + 0.706258i \(0.249618\pi\)
−0.707954 + 0.706258i \(0.750382\pi\)
\(140\) 0 0
\(141\) 2.82843i 0.238197i
\(142\) 0 0
\(143\) 2.73551 0.228755
\(144\) 0 0
\(145\) 10.7721 0.894578
\(146\) 0 0
\(147\) − 0.458440i − 0.0378115i
\(148\) 0 0
\(149\) 11.1832i 0.916166i 0.888909 + 0.458083i \(0.151464\pi\)
−0.888909 + 0.458083i \(0.848536\pi\)
\(150\) 0 0
\(151\) 14.6506 1.19225 0.596123 0.802893i \(-0.296707\pi\)
0.596123 + 0.802893i \(0.296707\pi\)
\(152\) 0 0
\(153\) −6.44549 −0.521087
\(154\) 0 0
\(155\) − 16.2212i − 1.30292i
\(156\) 0 0
\(157\) − 4.45754i − 0.355750i −0.984053 0.177875i \(-0.943078\pi\)
0.984053 0.177875i \(-0.0569223\pi\)
\(158\) 0 0
\(159\) −3.64520 −0.289083
\(160\) 0 0
\(161\) −7.23412 −0.570128
\(162\) 0 0
\(163\) − 7.78510i − 0.609776i −0.952388 0.304888i \(-0.901381\pi\)
0.952388 0.304888i \(-0.0986191\pi\)
\(164\) 0 0
\(165\) − 1.65685i − 0.128986i
\(166\) 0 0
\(167\) −20.1814 −1.56168 −0.780841 0.624730i \(-0.785209\pi\)
−0.780841 + 0.624730i \(0.785209\pi\)
\(168\) 0 0
\(169\) −3.67923 −0.283018
\(170\) 0 0
\(171\) 6.44549i 0.492899i
\(172\) 0 0
\(173\) − 6.15639i − 0.468061i −0.972229 0.234031i \(-0.924808\pi\)
0.972229 0.234031i \(-0.0751916\pi\)
\(174\) 0 0
\(175\) 2.86156 0.216313
\(176\) 0 0
\(177\) 5.65685 0.425195
\(178\) 0 0
\(179\) − 18.7855i − 1.40409i −0.712131 0.702046i \(-0.752270\pi\)
0.712131 0.702046i \(-0.247730\pi\)
\(180\) 0 0
\(181\) − 8.97499i − 0.667106i −0.942731 0.333553i \(-0.891752\pi\)
0.942731 0.333553i \(-0.108248\pi\)
\(182\) 0 0
\(183\) 6.20285 0.458528
\(184\) 0 0
\(185\) 9.54745 0.701943
\(186\) 0 0
\(187\) 4.31724i 0.315708i
\(188\) 0 0
\(189\) 2.55765i 0.186041i
\(190\) 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
\(195\) 10.1023i 0.723444i
\(196\) 0 0
\(197\) 1.75070i 0.124732i 0.998053 + 0.0623659i \(0.0198646\pi\)
−0.998053 + 0.0623659i \(0.980135\pi\)
\(198\) 0 0
\(199\) −0.993710 −0.0704422 −0.0352211 0.999380i \(-0.511214\pi\)
−0.0352211 + 0.999380i \(0.511214\pi\)
\(200\) 0 0
\(201\) −2.99647 −0.211355
\(202\) 0 0
\(203\) 11.1380i 0.781738i
\(204\) 0 0
\(205\) − 1.95078i − 0.136248i
\(206\) 0 0
\(207\) −2.82843 −0.196589
\(208\) 0 0
\(209\) 4.31724 0.298630
\(210\) 0 0
\(211\) 5.97409i 0.411273i 0.978628 + 0.205637i \(0.0659265\pi\)
−0.978628 + 0.205637i \(0.934073\pi\)
\(212\) 0 0
\(213\) 5.11529i 0.350494i
\(214\) 0 0
\(215\) 1.36293 0.0929509
\(216\) 0 0
\(217\) 16.7721 1.13857
\(218\) 0 0
\(219\) 14.7721i 0.998209i
\(220\) 0 0
\(221\) − 26.3235i − 1.77071i
\(222\) 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
\(227\) 0.907457i 0.0602300i 0.999546 + 0.0301150i \(0.00958735\pi\)
−0.999546 + 0.0301150i \(0.990413\pi\)
\(228\) 0 0
\(229\) 7.55579i 0.499301i 0.968336 + 0.249650i \(0.0803157\pi\)
−0.968336 + 0.249650i \(0.919684\pi\)
\(230\) 0 0
\(231\) 1.71313 0.112716
\(232\) 0 0
\(233\) 23.2271 1.52166 0.760828 0.648954i \(-0.224793\pi\)
0.760828 + 0.648954i \(0.224793\pi\)
\(234\) 0 0
\(235\) 6.99647i 0.456399i
\(236\) 0 0
\(237\) − 6.32000i − 0.410528i
\(238\) 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
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) − 1.13401i − 0.0724492i
\(246\) 0 0
\(247\) −26.3235 −1.67492
\(248\) 0 0
\(249\) −0.907457 −0.0575077
\(250\) 0 0
\(251\) 13.7984i 0.870949i 0.900201 + 0.435475i \(0.143419\pi\)
−0.900201 + 0.435475i \(0.856581\pi\)
\(252\) 0 0
\(253\) 1.89450i 0.119106i
\(254\) 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
\(259\) 9.87175i 0.613401i
\(260\) 0 0
\(261\) 4.35480i 0.269555i
\(262\) 0 0
\(263\) −29.9929 −1.84944 −0.924722 0.380643i \(-0.875703\pi\)
−0.924722 + 0.380643i \(0.875703\pi\)
\(264\) 0 0
\(265\) −9.01686 −0.553901
\(266\) 0 0
\(267\) 6.31724i 0.386609i
\(268\) 0 0
\(269\) − 29.1332i − 1.77628i −0.459569 0.888142i \(-0.651996\pi\)
0.459569 0.888142i \(-0.348004\pi\)
\(270\) 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
\(275\) − 0.749397i − 0.0451904i
\(276\) 0 0
\(277\) − 17.1430i − 1.03003i −0.857183 0.515013i \(-0.827787\pi\)
0.857183 0.515013i \(-0.172213\pi\)
\(278\) 0 0
\(279\) 6.55765 0.392596
\(280\) 0 0
\(281\) −2.76588 −0.164999 −0.0824993 0.996591i \(-0.526290\pi\)
−0.0824993 + 0.996591i \(0.526290\pi\)
\(282\) 0 0
\(283\) − 6.34315i − 0.377061i −0.982067 0.188530i \(-0.939628\pi\)
0.982067 0.188530i \(-0.0603724\pi\)
\(284\) 0 0
\(285\) 15.9437i 0.944425i
\(286\) 0 0
\(287\) 2.01704 0.119062
\(288\) 0 0
\(289\) 24.5443 1.44378
\(290\) 0 0
\(291\) 12.6533i 0.741751i
\(292\) 0 0
\(293\) − 11.6078i − 0.678133i −0.940762 0.339067i \(-0.889889\pi\)
0.940762 0.339067i \(-0.110111\pi\)
\(294\) 0 0
\(295\) 13.9929 0.814700
\(296\) 0 0
\(297\) 0.669808 0.0388662
\(298\) 0 0
\(299\) − 11.5514i − 0.668032i
\(300\) 0 0
\(301\) 1.40922i 0.0812262i
\(302\) 0 0
\(303\) −10.6417 −0.611348
\(304\) 0 0
\(305\) 15.3435 0.878567
\(306\) 0 0
\(307\) 14.7855i 0.843852i 0.906630 + 0.421926i \(0.138646\pi\)
−0.906630 + 0.421926i \(0.861354\pi\)
\(308\) 0 0
\(309\) 3.33686i 0.189827i
\(310\) 0 0
\(311\) −15.0761 −0.854885 −0.427442 0.904043i \(-0.640585\pi\)
−0.427442 + 0.904043i \(0.640585\pi\)
\(312\) 0 0
\(313\) −23.0027 −1.30019 −0.650096 0.759852i \(-0.725271\pi\)
−0.650096 + 0.759852i \(0.725271\pi\)
\(314\) 0 0
\(315\) 6.32666i 0.356467i
\(316\) 0 0
\(317\) − 9.55855i − 0.536862i −0.963299 0.268431i \(-0.913495\pi\)
0.963299 0.268431i \(-0.0865051\pi\)
\(318\) 0 0
\(319\) 2.91688 0.163314
\(320\) 0 0
\(321\) 19.8874 1.11001
\(322\) 0 0
\(323\) − 41.5443i − 2.31159i
\(324\) 0 0
\(325\) 4.56930i 0.253459i
\(326\) 0 0
\(327\) 3.91598 0.216554
\(328\) 0 0
\(329\) −7.23412 −0.398830
\(330\) 0 0
\(331\) − 27.8079i − 1.52846i −0.644945 0.764229i \(-0.723120\pi\)
0.644945 0.764229i \(-0.276880\pi\)
\(332\) 0 0
\(333\) 3.85970i 0.211510i
\(334\) 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
\(339\) − 2.23765i − 0.121532i
\(340\) 0 0
\(341\) − 4.39236i − 0.237860i
\(342\) 0 0
\(343\) 19.0761 1.03001
\(344\) 0 0
\(345\) −6.99647 −0.376677
\(346\) 0 0
\(347\) − 8.87449i − 0.476408i −0.971215 0.238204i \(-0.923441\pi\)
0.971215 0.238204i \(-0.0765586\pi\)
\(348\) 0 0
\(349\) − 6.70698i − 0.359016i −0.983757 0.179508i \(-0.942549\pi\)
0.983757 0.179508i \(-0.0574506\pi\)
\(350\) 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
\(355\) 12.6533i 0.671569i
\(356\) 0 0
\(357\) − 16.4853i − 0.872494i
\(358\) 0 0
\(359\) 32.7917 1.73068 0.865341 0.501184i \(-0.167102\pi\)
0.865341 + 0.501184i \(0.167102\pi\)
\(360\) 0 0
\(361\) −22.5443 −1.18654
\(362\) 0 0
\(363\) 10.5514i 0.553803i
\(364\) 0 0
\(365\) 36.5408i 1.91263i
\(366\) 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
\(371\) − 9.32313i − 0.484033i
\(372\) 0 0
\(373\) 23.4995i 1.21676i 0.793646 + 0.608379i \(0.208180\pi\)
−0.793646 + 0.608379i \(0.791820\pi\)
\(374\) 0 0
\(375\) −9.60058 −0.495772
\(376\) 0 0
\(377\) −17.7851 −0.915979
\(378\) 0 0
\(379\) − 11.0004i − 0.565051i −0.959260 0.282526i \(-0.908828\pi\)
0.959260 0.282526i \(-0.0911722\pi\)
\(380\) 0 0
\(381\) − 12.2145i − 0.625768i
\(382\) 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
\(387\) 0.550984i 0.0280081i
\(388\) 0 0
\(389\) − 33.7311i − 1.71023i −0.518436 0.855116i \(-0.673486\pi\)
0.518436 0.855116i \(-0.326514\pi\)
\(390\) 0 0
\(391\) 18.2306 0.921961
\(392\) 0 0
\(393\) −5.33962 −0.269348
\(394\) 0 0
\(395\) − 15.6333i − 0.786597i
\(396\) 0 0
\(397\) 14.5201i 0.728742i 0.931254 + 0.364371i \(0.118716\pi\)
−0.931254 + 0.364371i \(0.881284\pi\)
\(398\) 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
\(403\) 26.7816i 1.33409i
\(404\) 0 0
\(405\) 2.47363i 0.122915i
\(406\) 0 0
\(407\) 2.58526 0.128146
\(408\) 0 0
\(409\) 11.5702 0.572110 0.286055 0.958213i \(-0.407656\pi\)
0.286055 + 0.958213i \(0.407656\pi\)
\(410\) 0 0
\(411\) 5.10587i 0.251854i
\(412\) 0 0
\(413\) 14.4682i 0.711935i
\(414\) 0 0
\(415\) −2.24471 −0.110188
\(416\) 0 0
\(417\) −16.6533 −0.815517
\(418\) 0 0
\(419\) − 9.54193i − 0.466154i −0.972458 0.233077i \(-0.925121\pi\)
0.972458 0.233077i \(-0.0748794\pi\)
\(420\) 0 0
\(421\) − 24.3583i − 1.18715i −0.804778 0.593576i \(-0.797716\pi\)
0.804778 0.593576i \(-0.202284\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) −7.21137 −0.349803
\(426\) 0 0
\(427\) 15.8647i 0.767746i
\(428\) 0 0
\(429\) 2.73551i 0.132072i
\(430\) 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
\(435\) 10.7721i 0.516485i
\(436\) 0 0
\(437\) − 18.2306i − 0.872087i
\(438\) 0 0
\(439\) 17.7122 0.845356 0.422678 0.906280i \(-0.361090\pi\)
0.422678 + 0.906280i \(0.361090\pi\)
\(440\) 0 0
\(441\) 0.458440 0.0218305
\(442\) 0 0
\(443\) − 22.1953i − 1.05453i −0.849701 0.527264i \(-0.823218\pi\)
0.849701 0.527264i \(-0.176782\pi\)
\(444\) 0 0
\(445\) 15.6265i 0.740766i
\(446\) 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
\(451\) − 0.528232i − 0.0248735i
\(452\) 0 0
\(453\) 14.6506i 0.688344i
\(454\) 0 0
\(455\) −25.8382 −1.21131
\(456\) 0 0
\(457\) 17.3396 0.811113 0.405557 0.914070i \(-0.367078\pi\)
0.405557 + 0.914070i \(0.367078\pi\)
\(458\) 0 0
\(459\) − 6.44549i − 0.300850i
\(460\) 0 0
\(461\) 2.39404i 0.111501i 0.998445 + 0.0557507i \(0.0177552\pi\)
−0.998445 + 0.0557507i \(0.982245\pi\)
\(462\) 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
\(467\) − 24.2023i − 1.11995i −0.828510 0.559975i \(-0.810811\pi\)
0.828510 0.559975i \(-0.189189\pi\)
\(468\) 0 0
\(469\) − 7.66391i − 0.353887i
\(470\) 0 0
\(471\) 4.45754 0.205393
\(472\) 0 0
\(473\) 0.369053 0.0169691
\(474\) 0 0
\(475\) 7.21137i 0.330880i
\(476\) 0 0
\(477\) − 3.64520i − 0.166902i
\(478\) 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
\(483\) − 7.23412i − 0.329164i
\(484\) 0 0
\(485\) 31.2996i 1.42124i
\(486\) 0 0
\(487\) 13.9839 0.633672 0.316836 0.948480i \(-0.397380\pi\)
0.316836 + 0.948480i \(0.397380\pi\)
\(488\) 0 0
\(489\) 7.78510 0.352055
\(490\) 0 0
\(491\) 10.2306i 0.461700i 0.972989 + 0.230850i \(0.0741507\pi\)
−0.972989 + 0.230850i \(0.925849\pi\)
\(492\) 0 0
\(493\) − 28.0688i − 1.26416i
\(494\) 0 0
\(495\) 1.65685 0.0744701
\(496\) 0 0
\(497\) −13.0831 −0.586858
\(498\) 0 0
\(499\) 3.66391i 0.164019i 0.996632 + 0.0820097i \(0.0261338\pi\)
−0.996632 + 0.0820097i \(0.973866\pi\)
\(500\) 0 0
\(501\) − 20.1814i − 0.901637i
\(502\) 0 0
\(503\) −39.6443 −1.76765 −0.883825 0.467817i \(-0.845041\pi\)
−0.883825 + 0.467817i \(0.845041\pi\)
\(504\) 0 0
\(505\) −26.3235 −1.17138
\(506\) 0 0
\(507\) − 3.67923i − 0.163400i
\(508\) 0 0
\(509\) 28.6909i 1.27170i 0.771812 + 0.635851i \(0.219351\pi\)
−0.771812 + 0.635851i \(0.780649\pi\)
\(510\) 0 0
\(511\) −37.7819 −1.67137
\(512\) 0 0
\(513\) −6.44549 −0.284575
\(514\) 0 0
\(515\) 8.25413i 0.363721i
\(516\) 0 0
\(517\) 1.89450i 0.0833201i
\(518\) 0 0
\(519\) 6.15639 0.270235
\(520\) 0 0
\(521\) −23.1784 −1.01546 −0.507732 0.861515i \(-0.669516\pi\)
−0.507732 + 0.861515i \(0.669516\pi\)
\(522\) 0 0
\(523\) − 8.18193i − 0.357771i −0.983870 0.178885i \(-0.942751\pi\)
0.983870 0.178885i \(-0.0572491\pi\)
\(524\) 0 0
\(525\) 2.86156i 0.124889i
\(526\) 0 0
\(527\) −42.2672 −1.84119
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 5.65685i 0.245487i
\(532\) 0 0
\(533\) 3.22079i 0.139508i
\(534\) 0 0
\(535\) 49.1941 2.12685
\(536\) 0 0
\(537\) 18.7855 0.810653
\(538\) 0 0
\(539\) − 0.307067i − 0.0132263i
\(540\) 0 0
\(541\) 6.43715i 0.276755i 0.990380 + 0.138377i \(0.0441887\pi\)
−0.990380 + 0.138377i \(0.955811\pi\)
\(542\) 0 0
\(543\) 8.97499 0.385154
\(544\) 0 0
\(545\) 9.68667 0.414931
\(546\) 0 0
\(547\) 39.2239i 1.67709i 0.544830 + 0.838546i \(0.316594\pi\)
−0.544830 + 0.838546i \(0.683406\pi\)
\(548\) 0 0
\(549\) 6.20285i 0.264731i
\(550\) 0 0
\(551\) −28.0688 −1.19577
\(552\) 0 0
\(553\) 16.1643 0.687377
\(554\) 0 0
\(555\) 9.54745i 0.405267i
\(556\) 0 0
\(557\) 1.66224i 0.0704315i 0.999380 + 0.0352157i \(0.0112118\pi\)
−0.999380 + 0.0352157i \(0.988788\pi\)
\(558\) 0 0
\(559\) −2.25023 −0.0951745
\(560\) 0 0
\(561\) −4.31724 −0.182274
\(562\) 0 0
\(563\) − 40.6368i − 1.71264i −0.516447 0.856319i \(-0.672746\pi\)
0.516447 0.856319i \(-0.327254\pi\)
\(564\) 0 0
\(565\) − 5.53511i − 0.232864i
\(566\) 0 0
\(567\) −2.55765 −0.107411
\(568\) 0 0
\(569\) 27.0004 1.13191 0.565957 0.824435i \(-0.308507\pi\)
0.565957 + 0.824435i \(0.308507\pi\)
\(570\) 0 0
\(571\) 20.9706i 0.877591i 0.898587 + 0.438795i \(0.144595\pi\)
−0.898587 + 0.438795i \(0.855405\pi\)
\(572\) 0 0
\(573\) − 5.60058i − 0.233967i
\(574\) 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
\(579\) − 19.4514i − 0.808372i
\(580\) 0 0
\(581\) − 2.32095i − 0.0962894i
\(582\) 0 0
\(583\) −2.44158 −0.101120
\(584\) 0 0
\(585\) −10.1023 −0.417680
\(586\) 0 0
\(587\) − 44.2047i − 1.82452i −0.409609 0.912261i \(-0.634335\pi\)
0.409609 0.912261i \(-0.365665\pi\)
\(588\) 0 0
\(589\) 42.2672i 1.74159i
\(590\) 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
\(595\) − 40.7784i − 1.67175i
\(596\) 0 0
\(597\) − 0.993710i − 0.0406698i
\(598\) 0 0
\(599\) 22.0296 0.900104 0.450052 0.893002i \(-0.351405\pi\)
0.450052 + 0.893002i \(0.351405\pi\)
\(600\) 0 0
\(601\) −10.7721 −0.439405 −0.219703 0.975567i \(-0.570509\pi\)
−0.219703 + 0.975567i \(0.570509\pi\)
\(602\) 0 0
\(603\) − 2.99647i − 0.122026i
\(604\) 0 0
\(605\) 26.1001i 1.06112i
\(606\) 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
\(611\) − 11.5514i − 0.467318i
\(612\) 0 0
\(613\) − 14.8562i − 0.600035i −0.953934 0.300018i \(-0.903007\pi\)
0.953934 0.300018i \(-0.0969925\pi\)
\(614\) 0 0
\(615\) 1.95078 0.0786631
\(616\) 0 0
\(617\) −22.2235 −0.894686 −0.447343 0.894363i \(-0.647630\pi\)
−0.447343 + 0.894363i \(0.647630\pi\)
\(618\) 0 0
\(619\) 16.4612i 0.661631i 0.943696 + 0.330815i \(0.107324\pi\)
−0.943696 + 0.330815i \(0.892676\pi\)
\(620\) 0 0
\(621\) − 2.82843i − 0.113501i
\(622\) 0 0
\(623\) −16.1573 −0.647327
\(624\) 0 0
\(625\) −29.3424 −1.17369
\(626\) 0 0
\(627\) 4.31724i 0.172414i
\(628\) 0 0
\(629\) − 24.8776i − 0.991937i
\(630\) 0 0
\(631\) −4.06977 −0.162015 −0.0810075 0.996713i \(-0.525814\pi\)
−0.0810075 + 0.996713i \(0.525814\pi\)
\(632\) 0 0
\(633\) −5.97409 −0.237449
\(634\) 0 0
\(635\) − 30.2141i − 1.19901i
\(636\) 0 0
\(637\) 1.87228i 0.0741824i
\(638\) 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
\(643\) 10.4266i 0.411186i 0.978638 + 0.205593i \(0.0659124\pi\)
−0.978638 + 0.205593i \(0.934088\pi\)
\(644\) 0 0
\(645\) 1.36293i 0.0536652i
\(646\) 0 0
\(647\) 11.6132 0.456560 0.228280 0.973595i \(-0.426690\pi\)
0.228280 + 0.973595i \(0.426690\pi\)
\(648\) 0 0
\(649\) 3.78901 0.148731
\(650\) 0 0
\(651\) 16.7721i 0.657352i
\(652\) 0 0
\(653\) 2.73012i 0.106838i 0.998572 + 0.0534190i \(0.0170119\pi\)
−0.998572 + 0.0534190i \(0.982988\pi\)
\(654\) 0 0
\(655\) −13.2082 −0.516088
\(656\) 0 0
\(657\) −14.7721 −0.576316
\(658\) 0 0
\(659\) 31.5514i 1.22907i 0.788891 + 0.614533i \(0.210656\pi\)
−0.788891 + 0.614533i \(0.789344\pi\)
\(660\) 0 0
\(661\) − 15.1368i − 0.588752i −0.955690 0.294376i \(-0.904888\pi\)
0.955690 0.294376i \(-0.0951118\pi\)
\(662\) 0 0
\(663\) 26.3235 1.02232
\(664\) 0 0
\(665\) −40.7784 −1.58132
\(666\) 0 0
\(667\) − 12.3172i − 0.476925i
\(668\) 0 0
\(669\) 23.7659i 0.918841i
\(670\) 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
\(675\) 1.11882i 0.0430636i
\(676\) 0 0
\(677\) 37.9357i 1.45799i 0.684520 + 0.728994i \(0.260012\pi\)
−0.684520 + 0.728994i \(0.739988\pi\)
\(678\) 0 0
\(679\) −32.3627 −1.24197
\(680\) 0 0
\(681\) −0.907457 −0.0347738
\(682\) 0 0
\(683\) 18.2471i 0.698205i 0.937085 + 0.349102i \(0.113513\pi\)
−0.937085 + 0.349102i \(0.886487\pi\)
\(684\) 0 0
\(685\) 12.6300i 0.482568i
\(686\) 0 0
\(687\) −7.55579 −0.288271
\(688\) 0 0
\(689\) 14.8871 0.567152
\(690\) 0 0
\(691\) − 30.2533i − 1.15089i −0.817840 0.575446i \(-0.804829\pi\)
0.817840 0.575446i \(-0.195171\pi\)
\(692\) 0 0
\(693\) 1.71313i 0.0650765i
\(694\) 0 0
\(695\) −41.1941 −1.56258
\(696\) 0 0
\(697\) −5.08312 −0.192537
\(698\) 0 0
\(699\) 23.2271i 0.878528i
\(700\) 0 0
\(701\) 20.0875i 0.758696i 0.925254 + 0.379348i \(0.123852\pi\)
−0.925254 + 0.379348i \(0.876148\pi\)
\(702\) 0 0
\(703\) −24.8776 −0.938278
\(704\) 0 0
\(705\) −6.99647 −0.263502
\(706\) 0 0
\(707\) − 27.2176i − 1.02362i
\(708\) 0 0
\(709\) − 41.7864i − 1.56932i −0.619926 0.784660i \(-0.712837\pi\)
0.619926 0.784660i \(-0.287163\pi\)
\(710\) 0 0
\(711\) 6.32000 0.237018
\(712\) 0 0
\(713\) −18.5478 −0.694622
\(714\) 0 0
\(715\) 6.76663i 0.253058i
\(716\) 0 0
\(717\) 26.9213i 1.00540i
\(718\) 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
\(723\) 10.3494i 0.384899i
\(724\) 0 0
\(725\) 4.87226i 0.180951i
\(726\) 0 0
\(727\) −20.4843 −0.759722 −0.379861 0.925044i \(-0.624028\pi\)
−0.379861 + 0.925044i \(0.624028\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 3.55136i − 0.131352i
\(732\) 0 0
\(733\) 48.0777i 1.77579i 0.460045 + 0.887895i \(0.347833\pi\)
−0.460045 + 0.887895i \(0.652167\pi\)
\(734\) 0 0
\(735\) 1.13401 0.0418286
\(736\) 0 0
\(737\) −2.00706 −0.0739310
\(738\) 0 0
\(739\) − 21.4459i − 0.788899i −0.918918 0.394449i \(-0.870935\pi\)
0.918918 0.394449i \(-0.129065\pi\)
\(740\) 0 0
\(741\) − 26.3235i − 0.967018i
\(742\) 0 0
\(743\) 2.17431 0.0797677 0.0398839 0.999204i \(-0.487301\pi\)
0.0398839 + 0.999204i \(0.487301\pi\)
\(744\) 0 0
\(745\) −27.6631 −1.01350
\(746\) 0 0
\(747\) − 0.907457i − 0.0332021i
\(748\) 0 0
\(749\) 50.8651i 1.85857i
\(750\) 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
\(755\) 36.2400i 1.31891i
\(756\) 0 0
\(757\) 21.6791i 0.787939i 0.919123 + 0.393970i \(0.128899\pi\)
−0.919123 + 0.393970i \(0.871101\pi\)
\(758\) 0 0
\(759\) −1.89450 −0.0687661
\(760\) 0 0
\(761\) −4.29449 −0.155675 −0.0778375 0.996966i \(-0.524802\pi\)
−0.0778375 + 0.996966i \(0.524802\pi\)
\(762\) 0 0
\(763\) 10.0157i 0.362592i
\(764\) 0 0
\(765\) − 15.9437i − 0.576446i
\(766\) 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
\(771\) 16.9965i 0.612113i
\(772\) 0 0
\(773\) − 49.5300i − 1.78147i −0.454521 0.890736i \(-0.650190\pi\)
0.454521 0.890736i \(-0.349810\pi\)
\(774\) 0 0
\(775\) 7.33686 0.263548
\(776\) 0 0
\(777\) −9.87175 −0.354147
\(778\) 0 0
\(779\) 5.08312i 0.182122i
\(780\) 0 0
\(781\) 3.42627i 0.122601i
\(782\) 0 0
\(783\) −4.35480 −0.155628
\(784\) 0 0
\(785\) 11.0263 0.393545
\(786\) 0 0
\(787\) 34.0953i 1.21537i 0.794180 + 0.607683i \(0.207901\pi\)
−0.794180 + 0.607683i \(0.792099\pi\)
\(788\) 0 0
\(789\) − 29.9929i − 1.06778i
\(790\) 0 0
\(791\) 5.72312 0.203491
\(792\) 0 0
\(793\) −25.3326 −0.899585
\(794\) 0 0
\(795\) − 9.01686i − 0.319795i
\(796\) 0 0
\(797\) − 40.6901i − 1.44132i −0.693290 0.720659i \(-0.743840\pi\)
0.693290 0.720659i \(-0.256160\pi\)
\(798\) 0 0
\(799\) 18.2306 0.644952
\(800\) 0 0
\(801\) −6.31724 −0.223209
\(802\) 0 0
\(803\) 9.89450i 0.349169i
\(804\) 0 0
\(805\) − 17.8945i − 0.630698i
\(806\) 0 0
\(807\) 29.1332 1.02554
\(808\) 0 0
\(809\) 10.9926 0.386478 0.193239 0.981152i \(-0.438101\pi\)
0.193239 + 0.981152i \(0.438101\pi\)
\(810\) 0 0
\(811\) − 21.2498i − 0.746182i −0.927795 0.373091i \(-0.878298\pi\)
0.927795 0.373091i \(-0.121702\pi\)
\(812\) 0 0
\(813\) − 26.6506i − 0.934676i
\(814\) 0 0
\(815\) 19.2574 0.674558
\(816\) 0 0
\(817\) −3.55136 −0.124246
\(818\) 0 0
\(819\) − 10.4455i − 0.364995i
\(820\) 0 0
\(821\) 30.0572i 1.04900i 0.851410 + 0.524501i \(0.175748\pi\)
−0.851410 + 0.524501i \(0.824252\pi\)
\(822\) 0 0
\(823\) −55.0851 −1.92015 −0.960073 0.279751i \(-0.909748\pi\)
−0.960073 + 0.279751i \(0.909748\pi\)
\(824\) 0 0
\(825\) 0.749397 0.0260907
\(826\) 0 0
\(827\) 34.5478i 1.20135i 0.799495 + 0.600673i \(0.205101\pi\)
−0.799495 + 0.600673i \(0.794899\pi\)
\(828\) 0 0
\(829\) − 31.0046i − 1.07683i −0.842679 0.538417i \(-0.819023\pi\)
0.842679 0.538417i \(-0.180977\pi\)
\(830\) 0 0
\(831\) 17.1430 0.594685
\(832\) 0 0
\(833\) −2.95487 −0.102380
\(834\) 0 0
\(835\) − 49.9212i − 1.72759i
\(836\) 0 0
\(837\) 6.55765i 0.226665i
\(838\) 0 0
\(839\) 5.14195 0.177520 0.0887599 0.996053i \(-0.471710\pi\)
0.0887599 + 0.996053i \(0.471710\pi\)
\(840\) 0 0
\(841\) 10.0357 0.346059
\(842\) 0 0
\(843\) − 2.76588i − 0.0952620i
\(844\) 0 0
\(845\) − 9.10104i − 0.313085i
\(846\) 0 0
\(847\) −26.9867 −0.927272
\(848\) 0 0
\(849\) 6.34315 0.217696
\(850\) 0 0
\(851\) − 10.9169i − 0.374226i
\(852\) 0 0
\(853\) − 18.5060i − 0.633632i −0.948487 0.316816i \(-0.897386\pi\)
0.948487 0.316816i \(-0.102614\pi\)
\(854\) 0 0
\(855\) −15.9437 −0.545264
\(856\) 0 0
\(857\) −22.8878 −0.781833 −0.390916 0.920426i \(-0.627842\pi\)
−0.390916 + 0.920426i \(0.627842\pi\)
\(858\) 0 0
\(859\) 35.5286i 1.21222i 0.795381 + 0.606110i \(0.207271\pi\)
−0.795381 + 0.606110i \(0.792729\pi\)
\(860\) 0 0
\(861\) 2.01704i 0.0687407i
\(862\) 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
\(867\) 24.5443i 0.833568i
\(868\) 0 0
\(869\) − 4.23319i − 0.143601i
\(870\) 0 0
\(871\) 12.2376 0.414657
\(872\) 0 0
\(873\) −12.6533 −0.428250
\(874\) 0 0
\(875\) − 24.5549i − 0.830107i
\(876\) 0 0
\(877\) − 21.5773i − 0.728614i −0.931279 0.364307i \(-0.881306\pi\)
0.931279 0.364307i \(-0.118694\pi\)
\(878\) 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
\(883\) − 0.0834930i − 0.00280976i −0.999999 0.00140488i \(-0.999553\pi\)
0.999999 0.00140488i \(-0.000447188\pi\)
\(884\) 0 0
\(885\) 13.9929i 0.470368i
\(886\) 0 0
\(887\) 30.8043 1.03431 0.517154 0.855892i \(-0.326991\pi\)
0.517154 + 0.855892i \(0.326991\pi\)
\(888\) 0 0
\(889\) 31.2404 1.04777
\(890\) 0 0
\(891\) 0.669808i 0.0224394i
\(892\) 0 0
\(893\) − 18.2306i − 0.610063i
\(894\) 0 0
\(895\) 46.4682 1.55326
\(896\) 0 0
\(897\) 11.5514 0.385689
\(898\) 0 0
\(899\) 28.5573i 0.952438i
\(900\) 0 0
\(901\) 23.4951i 0.782735i
\(902\) 0 0
\(903\) −1.40922 −0.0468960
\(904\) 0 0
\(905\) 22.2008 0.737979
\(906\) 0 0
\(907\) 49.5215i 1.64434i 0.569245 + 0.822168i \(0.307236\pi\)
−0.569245 + 0.822168i \(0.692764\pi\)
\(908\) 0 0
\(909\) − 10.6417i − 0.352962i
\(910\) 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
\(915\) 15.3435i 0.507241i
\(916\) 0 0
\(917\) − 13.6569i − 0.450989i
\(918\) 0 0
\(919\) 20.1161 0.663568 0.331784 0.943355i \(-0.392350\pi\)
0.331784 + 0.943355i \(0.392350\pi\)
\(920\) 0 0
\(921\) −14.7855 −0.487198
\(922\) 0 0
\(923\) − 20.8910i − 0.687635i
\(924\) 0 0
\(925\) 4.31833i 0.141986i
\(926\) 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
\(931\) 2.95487i 0.0968420i
\(932\) 0 0
\(933\) − 15.0761i − 0.493568i
\(934\) 0 0
\(935\) −10.6792 −0.349248
\(936\) 0 0
\(937\) −33.5780 −1.09695 −0.548473 0.836168i \(-0.684791\pi\)
−0.548473 + 0.836168i \(0.684791\pi\)
\(938\) 0 0
\(939\) − 23.0027i − 0.750666i
\(940\) 0 0
\(941\) 11.9991i 0.391159i 0.980688 + 0.195579i \(0.0626587\pi\)
−0.980688 + 0.195579i \(0.937341\pi\)
\(942\) 0 0
\(943\) −2.23059 −0.0726380
\(944\) 0 0
\(945\) −6.32666 −0.205806
\(946\) 0 0
\(947\) 25.2537i 0.820635i 0.911943 + 0.410318i \(0.134582\pi\)
−0.911943 + 0.410318i \(0.865418\pi\)
\(948\) 0 0
\(949\) − 60.3298i − 1.95839i
\(950\) 0 0
\(951\) 9.55855 0.309957
\(952\) 0 0
\(953\) −3.86469 −0.125190 −0.0625948 0.998039i \(-0.519938\pi\)
−0.0625948 + 0.998039i \(0.519938\pi\)
\(954\) 0 0
\(955\) − 13.8537i − 0.448296i
\(956\) 0 0
\(957\) 2.91688i 0.0942894i
\(958\) 0 0
\(959\) −13.0590 −0.421698
\(960\) 0 0
\(961\) 12.0027 0.387185
\(962\) 0 0
\(963\) 19.8874i 0.640864i
\(964\) 0 0
\(965\) − 48.1154i − 1.54889i
\(966\) 0 0
\(967\) −37.8714 −1.21786 −0.608930 0.793224i \(-0.708401\pi\)
−0.608930 + 0.793224i \(0.708401\pi\)
\(968\) 0 0
\(969\) 41.5443 1.33460
\(970\) 0 0
\(971\) 3.74587i 0.120211i 0.998192 + 0.0601053i \(0.0191437\pi\)
−0.998192 + 0.0601053i \(0.980856\pi\)
\(972\) 0 0
\(973\) − 42.5933i − 1.36548i
\(974\) 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
\(979\) 4.23134i 0.135234i
\(980\) 0 0
\(981\) 3.91598i 0.125028i
\(982\) 0 0
\(983\) 22.3557 0.713035 0.356518 0.934289i \(-0.383964\pi\)
0.356518 + 0.934289i \(0.383964\pi\)
\(984\) 0 0
\(985\) −4.33057 −0.137983
\(986\) 0 0
\(987\) − 7.23412i − 0.230265i
\(988\) 0 0
\(989\) − 1.55842i − 0.0495548i
\(990\) 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
\(995\) − 2.45807i − 0.0779260i
\(996\) 0 0
\(997\) 6.06146i 0.191968i 0.995383 + 0.0959841i \(0.0305998\pi\)
−0.995383 + 0.0959841i \(0.969400\pi\)
\(998\) 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.d.i.1537.7 8
4.3 odd 2 3072.2.d.f.1537.3 8
8.3 odd 2 3072.2.d.f.1537.6 8
8.5 even 2 inner 3072.2.d.i.1537.2 8
16.3 odd 4 3072.2.a.t.1.3 4
16.5 even 4 3072.2.a.o.1.2 4
16.11 odd 4 3072.2.a.i.1.2 4
16.13 even 4 3072.2.a.n.1.3 4
32.3 odd 8 384.2.j.b.97.3 8
32.5 even 8 384.2.j.a.289.1 8
32.11 odd 8 48.2.j.a.13.2 8
32.13 even 8 192.2.j.a.49.4 8
32.19 odd 8 48.2.j.a.37.2 yes 8
32.21 even 8 192.2.j.a.145.4 8
32.27 odd 8 384.2.j.b.289.3 8
32.29 even 8 384.2.j.a.97.1 8
48.5 odd 4 9216.2.a.bn.1.3 4
48.11 even 4 9216.2.a.bo.1.3 4
48.29 odd 4 9216.2.a.x.1.2 4
48.35 even 4 9216.2.a.y.1.2 4
96.5 odd 8 1152.2.k.f.289.4 8
96.11 even 8 144.2.k.b.109.3 8
96.29 odd 8 1152.2.k.f.865.4 8
96.35 even 8 1152.2.k.c.865.4 8
96.53 odd 8 576.2.k.b.145.1 8
96.59 even 8 1152.2.k.c.289.4 8
96.77 odd 8 576.2.k.b.433.1 8
96.83 even 8 144.2.k.b.37.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.2 8 32.11 odd 8
48.2.j.a.37.2 yes 8 32.19 odd 8
144.2.k.b.37.3 8 96.83 even 8
144.2.k.b.109.3 8 96.11 even 8
192.2.j.a.49.4 8 32.13 even 8
192.2.j.a.145.4 8 32.21 even 8
384.2.j.a.97.1 8 32.29 even 8
384.2.j.a.289.1 8 32.5 even 8
384.2.j.b.97.3 8 32.3 odd 8
384.2.j.b.289.3 8 32.27 odd 8
576.2.k.b.145.1 8 96.53 odd 8
576.2.k.b.433.1 8 96.77 odd 8
1152.2.k.c.289.4 8 96.59 even 8
1152.2.k.c.865.4 8 96.35 even 8
1152.2.k.f.289.4 8 96.5 odd 8
1152.2.k.f.865.4 8 96.29 odd 8
3072.2.a.i.1.2 4 16.11 odd 4
3072.2.a.n.1.3 4 16.13 even 4
3072.2.a.o.1.2 4 16.5 even 4
3072.2.a.t.1.3 4 16.3 odd 4
3072.2.d.f.1537.3 8 4.3 odd 2
3072.2.d.f.1537.6 8 8.3 odd 2
3072.2.d.i.1537.2 8 8.5 even 2 inner
3072.2.d.i.1537.7 8 1.1 even 1 trivial
9216.2.a.x.1.2 4 48.29 odd 4
9216.2.a.y.1.2 4 48.35 even 4
9216.2.a.bn.1.3 4 48.5 odd 4
9216.2.a.bo.1.3 4 48.11 even 4