Properties

Label 3072.2.d.i.1537.3
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 1537.3
Root \(0.500000 + 0.0297061i\) of defining polynomial
Character \(\chi\) \(=\) 3072.1537
Dual form 3072.2.d.i.1537.6

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +0.473626i q^{5} +4.55765 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +0.473626i q^{5} +4.55765 q^{7} -1.00000 q^{9} +3.49824i q^{11} -0.0840215i q^{13} +0.473626 q^{15} -3.61706 q^{17} -3.61706i q^{19} -4.55765i q^{21} +2.82843 q^{23} +4.77568 q^{25} +1.00000i q^{27} +7.30205i q^{29} +0.557647 q^{31} +3.49824 q^{33} +2.15862i q^{35} -6.20285i q^{37} -0.0840215 q^{39} +9.27391 q^{41} +2.27744i q^{43} -0.473626i q^{45} +2.82843 q^{47} +13.7721 q^{49} +3.61706i q^{51} -0.697947i q^{53} -1.65685 q^{55} -3.61706 q^{57} +5.65685i q^{59} -3.85970i q^{61} -4.55765 q^{63} +0.0397948 q^{65} +5.33962i q^{67} -2.82843i q^{69} -9.11529 q^{71} +0.541560 q^{73} -4.77568i q^{75} +15.9437i q^{77} -10.9937 q^{79} +1.00000 q^{81} +15.0496i q^{83} -1.71313i q^{85} +7.30205 q^{87} +14.6533 q^{89} -0.382941i q^{91} -0.557647i q^{93} +1.71313 q^{95} +4.31724 q^{97} -3.49824i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} - 8q^{9} + O(q^{10}) \) \( 8q + 8q^{7} - 8q^{9} - 8q^{15} - 8q^{25} - 24q^{31} + 16q^{39} + 8q^{49} + 32q^{55} - 8q^{63} - 16q^{65} - 16q^{71} + 16q^{73} - 24q^{79} + 8q^{81} + 24q^{87} + 16q^{89} + 48q^{95} + O(q^{100}) \)

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
\(5\) 0.473626i 0.211812i 0.994376 + 0.105906i \(0.0337742\pi\)
−0.994376 + 0.105906i \(0.966226\pi\)
\(6\) 0 0
\(7\) 4.55765 1.72263 0.861314 0.508072i \(-0.169642\pi\)
0.861314 + 0.508072i \(0.169642\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.49824i 1.05476i 0.849630 + 0.527379i \(0.176825\pi\)
−0.849630 + 0.527379i \(0.823175\pi\)
\(12\) 0 0
\(13\) − 0.0840215i − 0.0233034i −0.999932 0.0116517i \(-0.996291\pi\)
0.999932 0.0116517i \(-0.00370893\pi\)
\(14\) 0 0
\(15\) 0.473626 0.122290
\(16\) 0 0
\(17\) −3.61706 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(18\) 0 0
\(19\) − 3.61706i − 0.829810i −0.909865 0.414905i \(-0.863815\pi\)
0.909865 0.414905i \(-0.136185\pi\)
\(20\) 0 0
\(21\) − 4.55765i − 0.994560i
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) 4.77568 0.955136
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 7.30205i 1.35596i 0.735082 + 0.677979i \(0.237144\pi\)
−0.735082 + 0.677979i \(0.762856\pi\)
\(30\) 0 0
\(31\) 0.557647 0.100156 0.0500782 0.998745i \(-0.484053\pi\)
0.0500782 + 0.998745i \(0.484053\pi\)
\(32\) 0 0
\(33\) 3.49824 0.608965
\(34\) 0 0
\(35\) 2.15862i 0.364873i
\(36\) 0 0
\(37\) − 6.20285i − 1.01974i −0.860251 0.509871i \(-0.829693\pi\)
0.860251 0.509871i \(-0.170307\pi\)
\(38\) 0 0
\(39\) −0.0840215 −0.0134542
\(40\) 0 0
\(41\) 9.27391 1.44834 0.724171 0.689620i \(-0.242223\pi\)
0.724171 + 0.689620i \(0.242223\pi\)
\(42\) 0 0
\(43\) 2.27744i 0.347307i 0.984807 + 0.173653i \(0.0555573\pi\)
−0.984807 + 0.173653i \(0.944443\pi\)
\(44\) 0 0
\(45\) − 0.473626i − 0.0706040i
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 13.7721 1.96745
\(50\) 0 0
\(51\) 3.61706i 0.506490i
\(52\) 0 0
\(53\) − 0.697947i − 0.0958704i −0.998850 0.0479352i \(-0.984736\pi\)
0.998850 0.0479352i \(-0.0152641\pi\)
\(54\) 0 0
\(55\) −1.65685 −0.223410
\(56\) 0 0
\(57\) −3.61706 −0.479091
\(58\) 0 0
\(59\) 5.65685i 0.736460i 0.929735 + 0.368230i \(0.120036\pi\)
−0.929735 + 0.368230i \(0.879964\pi\)
\(60\) 0 0
\(61\) − 3.85970i − 0.494184i −0.968992 0.247092i \(-0.920525\pi\)
0.968992 0.247092i \(-0.0794750\pi\)
\(62\) 0 0
\(63\) −4.55765 −0.574210
\(64\) 0 0
\(65\) 0.0397948 0.00493593
\(66\) 0 0
\(67\) 5.33962i 0.652338i 0.945311 + 0.326169i \(0.105758\pi\)
−0.945311 + 0.326169i \(0.894242\pi\)
\(68\) 0 0
\(69\) − 2.82843i − 0.340503i
\(70\) 0 0
\(71\) −9.11529 −1.08179 −0.540893 0.841091i \(-0.681914\pi\)
−0.540893 + 0.841091i \(0.681914\pi\)
\(72\) 0 0
\(73\) 0.541560 0.0633848 0.0316924 0.999498i \(-0.489910\pi\)
0.0316924 + 0.999498i \(0.489910\pi\)
\(74\) 0 0
\(75\) − 4.77568i − 0.551448i
\(76\) 0 0
\(77\) 15.9437i 1.81696i
\(78\) 0 0
\(79\) −10.9937 −1.23689 −0.618445 0.785828i \(-0.712237\pi\)
−0.618445 + 0.785828i \(0.712237\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.0496i 1.65191i 0.563738 + 0.825954i \(0.309363\pi\)
−0.563738 + 0.825954i \(0.690637\pi\)
\(84\) 0 0
\(85\) − 1.71313i − 0.185815i
\(86\) 0 0
\(87\) 7.30205 0.782862
\(88\) 0 0
\(89\) 14.6533 1.55325 0.776625 0.629964i \(-0.216930\pi\)
0.776625 + 0.629964i \(0.216930\pi\)
\(90\) 0 0
\(91\) − 0.382941i − 0.0401431i
\(92\) 0 0
\(93\) − 0.557647i − 0.0578253i
\(94\) 0 0
\(95\) 1.71313 0.175764
\(96\) 0 0
\(97\) 4.31724 0.438349 0.219175 0.975686i \(-0.429664\pi\)
0.219175 + 0.975686i \(0.429664\pi\)
\(98\) 0 0
\(99\) − 3.49824i − 0.351586i
\(100\) 0 0
\(101\) 0.641669i 0.0638484i 0.999490 + 0.0319242i \(0.0101635\pi\)
−0.999490 + 0.0319242i \(0.989836\pi\)
\(102\) 0 0
\(103\) −1.33686 −0.131724 −0.0658622 0.997829i \(-0.520980\pi\)
−0.0658622 + 0.997829i \(0.520980\pi\)
\(104\) 0 0
\(105\) 2.15862 0.210660
\(106\) 0 0
\(107\) − 8.57373i − 0.828854i −0.910083 0.414427i \(-0.863982\pi\)
0.910083 0.414427i \(-0.136018\pi\)
\(108\) 0 0
\(109\) 8.08402i 0.774309i 0.922015 + 0.387154i \(0.126542\pi\)
−0.922015 + 0.387154i \(0.873458\pi\)
\(110\) 0 0
\(111\) −6.20285 −0.588748
\(112\) 0 0
\(113\) 9.55136 0.898516 0.449258 0.893402i \(-0.351688\pi\)
0.449258 + 0.893402i \(0.351688\pi\)
\(114\) 0 0
\(115\) 1.33962i 0.124920i
\(116\) 0 0
\(117\) 0.0840215i 0.00776779i
\(118\) 0 0
\(119\) −16.4853 −1.51120
\(120\) 0 0
\(121\) −1.23765 −0.112514
\(122\) 0 0
\(123\) − 9.27391i − 0.836201i
\(124\) 0 0
\(125\) 4.63001i 0.414121i
\(126\) 0 0
\(127\) −5.09921 −0.452481 −0.226241 0.974071i \(-0.572644\pi\)
−0.226241 + 0.974071i \(0.572644\pi\)
\(128\) 0 0
\(129\) 2.27744 0.200518
\(130\) 0 0
\(131\) 2.99647i 0.261803i 0.991395 + 0.130901i \(0.0417871\pi\)
−0.991395 + 0.130901i \(0.958213\pi\)
\(132\) 0 0
\(133\) − 16.4853i − 1.42946i
\(134\) 0 0
\(135\) −0.473626 −0.0407632
\(136\) 0 0
\(137\) 3.37941 0.288723 0.144361 0.989525i \(-0.453887\pi\)
0.144361 + 0.989525i \(0.453887\pi\)
\(138\) 0 0
\(139\) − 8.31724i − 0.705459i −0.935725 0.352729i \(-0.885254\pi\)
0.935725 0.352729i \(-0.114746\pi\)
\(140\) 0 0
\(141\) − 2.82843i − 0.238197i
\(142\) 0 0
\(143\) 0.293927 0.0245794
\(144\) 0 0
\(145\) −3.45844 −0.287208
\(146\) 0 0
\(147\) − 13.7721i − 1.13591i
\(148\) 0 0
\(149\) − 14.1305i − 1.15761i −0.815465 0.578807i \(-0.803518\pi\)
0.815465 0.578807i \(-0.196482\pi\)
\(150\) 0 0
\(151\) 9.97685 0.811905 0.405952 0.913894i \(-0.366940\pi\)
0.405952 + 0.913894i \(0.366940\pi\)
\(152\) 0 0
\(153\) 3.61706 0.292422
\(154\) 0 0
\(155\) 0.264116i 0.0212143i
\(156\) 0 0
\(157\) 22.8562i 1.82412i 0.410056 + 0.912060i \(0.365509\pi\)
−0.410056 + 0.912060i \(0.634491\pi\)
\(158\) 0 0
\(159\) −0.697947 −0.0553508
\(160\) 0 0
\(161\) 12.8910 1.01595
\(162\) 0 0
\(163\) − 10.6135i − 0.831316i −0.909521 0.415658i \(-0.863551\pi\)
0.909521 0.415658i \(-0.136449\pi\)
\(164\) 0 0
\(165\) 1.65685i 0.128986i
\(166\) 0 0
\(167\) 5.83822 0.451775 0.225888 0.974153i \(-0.427472\pi\)
0.225888 + 0.974153i \(0.427472\pi\)
\(168\) 0 0
\(169\) 12.9929 0.999457
\(170\) 0 0
\(171\) 3.61706i 0.276603i
\(172\) 0 0
\(173\) − 5.12695i − 0.389795i −0.980824 0.194897i \(-0.937563\pi\)
0.980824 0.194897i \(-0.0624374\pi\)
\(174\) 0 0
\(175\) 21.7659 1.64534
\(176\) 0 0
\(177\) 5.65685 0.425195
\(178\) 0 0
\(179\) − 13.1286i − 0.981279i −0.871363 0.490640i \(-0.836763\pi\)
0.871363 0.490640i \(-0.163237\pi\)
\(180\) 0 0
\(181\) − 15.3181i − 1.13859i −0.822134 0.569294i \(-0.807217\pi\)
0.822134 0.569294i \(-0.192783\pi\)
\(182\) 0 0
\(183\) −3.85970 −0.285317
\(184\) 0 0
\(185\) 2.93783 0.215993
\(186\) 0 0
\(187\) − 12.6533i − 0.925303i
\(188\) 0 0
\(189\) 4.55765i 0.331520i
\(190\) 0 0
\(191\) 8.63001 0.624446 0.312223 0.950009i \(-0.398926\pi\)
0.312223 + 0.950009i \(0.398926\pi\)
\(192\) 0 0
\(193\) 11.4514 0.824288 0.412144 0.911119i \(-0.364780\pi\)
0.412144 + 0.911119i \(0.364780\pi\)
\(194\) 0 0
\(195\) − 0.0397948i − 0.00284976i
\(196\) 0 0
\(197\) − 10.5925i − 0.754681i −0.926075 0.377340i \(-0.876839\pi\)
0.926075 0.377340i \(-0.123161\pi\)
\(198\) 0 0
\(199\) 3.68000 0.260868 0.130434 0.991457i \(-0.458363\pi\)
0.130434 + 0.991457i \(0.458363\pi\)
\(200\) 0 0
\(201\) 5.33962 0.376627
\(202\) 0 0
\(203\) 33.2802i 2.33581i
\(204\) 0 0
\(205\) 4.39236i 0.306776i
\(206\) 0 0
\(207\) −2.82843 −0.196589
\(208\) 0 0
\(209\) 12.6533 0.875249
\(210\) 0 0
\(211\) − 14.3102i − 0.985153i −0.870269 0.492577i \(-0.836055\pi\)
0.870269 0.492577i \(-0.163945\pi\)
\(212\) 0 0
\(213\) 9.11529i 0.624570i
\(214\) 0 0
\(215\) −1.07866 −0.0735637
\(216\) 0 0
\(217\) 2.54156 0.172532
\(218\) 0 0
\(219\) − 0.541560i − 0.0365952i
\(220\) 0 0
\(221\) 0.303911i 0.0204433i
\(222\) 0 0
\(223\) 4.86156 0.325554 0.162777 0.986663i \(-0.447955\pi\)
0.162777 + 0.986663i \(0.447955\pi\)
\(224\) 0 0
\(225\) −4.77568 −0.318379
\(226\) 0 0
\(227\) 15.0496i 0.998877i 0.866349 + 0.499438i \(0.166460\pi\)
−0.866349 + 0.499438i \(0.833540\pi\)
\(228\) 0 0
\(229\) 28.5264i 1.88507i 0.334101 + 0.942537i \(0.391567\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(230\) 0 0
\(231\) 15.9437 1.04902
\(232\) 0 0
\(233\) −13.5702 −0.889014 −0.444507 0.895775i \(-0.646621\pi\)
−0.444507 + 0.895775i \(0.646621\pi\)
\(234\) 0 0
\(235\) 1.33962i 0.0873869i
\(236\) 0 0
\(237\) 10.9937i 0.714118i
\(238\) 0 0
\(239\) 29.3629 1.89933 0.949665 0.313267i \(-0.101424\pi\)
0.949665 + 0.313267i \(0.101424\pi\)
\(240\) 0 0
\(241\) −24.0063 −1.54638 −0.773190 0.634175i \(-0.781340\pi\)
−0.773190 + 0.634175i \(0.781340\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 6.52284i 0.416729i
\(246\) 0 0
\(247\) −0.303911 −0.0193374
\(248\) 0 0
\(249\) 15.0496 0.953729
\(250\) 0 0
\(251\) 22.2837i 1.40654i 0.710925 + 0.703268i \(0.248276\pi\)
−0.710925 + 0.703268i \(0.751724\pi\)
\(252\) 0 0
\(253\) 9.89450i 0.622062i
\(254\) 0 0
\(255\) −1.71313 −0.107281
\(256\) 0 0
\(257\) 8.66038 0.540220 0.270110 0.962829i \(-0.412940\pi\)
0.270110 + 0.962829i \(0.412940\pi\)
\(258\) 0 0
\(259\) − 28.2704i − 1.75664i
\(260\) 0 0
\(261\) − 7.30205i − 0.451986i
\(262\) 0 0
\(263\) −13.3208 −0.821394 −0.410697 0.911772i \(-0.634715\pi\)
−0.410697 + 0.911772i \(0.634715\pi\)
\(264\) 0 0
\(265\) 0.330566 0.0203065
\(266\) 0 0
\(267\) − 14.6533i − 0.896769i
\(268\) 0 0
\(269\) − 16.5058i − 1.00638i −0.864177 0.503188i \(-0.832160\pi\)
0.864177 0.503188i \(-0.167840\pi\)
\(270\) 0 0
\(271\) −21.9769 −1.33500 −0.667499 0.744610i \(-0.732635\pi\)
−0.667499 + 0.744610i \(0.732635\pi\)
\(272\) 0 0
\(273\) −0.382941 −0.0231766
\(274\) 0 0
\(275\) 16.7064i 1.00744i
\(276\) 0 0
\(277\) − 15.4862i − 0.930475i −0.885186 0.465237i \(-0.845969\pi\)
0.885186 0.465237i \(-0.154031\pi\)
\(278\) 0 0
\(279\) −0.557647 −0.0333855
\(280\) 0 0
\(281\) −22.8910 −1.36556 −0.682780 0.730624i \(-0.739229\pi\)
−0.682780 + 0.730624i \(0.739229\pi\)
\(282\) 0 0
\(283\) 6.34315i 0.377061i 0.982067 + 0.188530i \(0.0603724\pi\)
−0.982067 + 0.188530i \(0.939628\pi\)
\(284\) 0 0
\(285\) − 1.71313i − 0.101477i
\(286\) 0 0
\(287\) 42.2672 2.49496
\(288\) 0 0
\(289\) −3.91688 −0.230405
\(290\) 0 0
\(291\) − 4.31724i − 0.253081i
\(292\) 0 0
\(293\) − 30.5783i − 1.78641i −0.449654 0.893203i \(-0.648453\pi\)
0.449654 0.893203i \(-0.351547\pi\)
\(294\) 0 0
\(295\) −2.67923 −0.155991
\(296\) 0 0
\(297\) −3.49824 −0.202988
\(298\) 0 0
\(299\) − 0.237649i − 0.0137436i
\(300\) 0 0
\(301\) 10.3798i 0.598281i
\(302\) 0 0
\(303\) 0.641669 0.0368629
\(304\) 0 0
\(305\) 1.82805 0.104674
\(306\) 0 0
\(307\) 17.1286i 0.977582i 0.872401 + 0.488791i \(0.162562\pi\)
−0.872401 + 0.488791i \(0.837438\pi\)
\(308\) 0 0
\(309\) 1.33686i 0.0760511i
\(310\) 0 0
\(311\) −26.8651 −1.52338 −0.761689 0.647943i \(-0.775630\pi\)
−0.761689 + 0.647943i \(0.775630\pi\)
\(312\) 0 0
\(313\) 19.6890 1.11289 0.556445 0.830885i \(-0.312165\pi\)
0.556445 + 0.830885i \(0.312165\pi\)
\(314\) 0 0
\(315\) − 2.15862i − 0.121624i
\(316\) 0 0
\(317\) − 30.1860i − 1.69541i −0.530466 0.847706i \(-0.677983\pi\)
0.530466 0.847706i \(-0.322017\pi\)
\(318\) 0 0
\(319\) −25.5443 −1.43021
\(320\) 0 0
\(321\) −8.57373 −0.478539
\(322\) 0 0
\(323\) 13.0831i 0.727964i
\(324\) 0 0
\(325\) − 0.401260i − 0.0222579i
\(326\) 0 0
\(327\) 8.08402 0.447047
\(328\) 0 0
\(329\) 12.8910 0.710702
\(330\) 0 0
\(331\) − 20.7784i − 1.14209i −0.820920 0.571043i \(-0.806539\pi\)
0.820920 0.571043i \(-0.193461\pi\)
\(332\) 0 0
\(333\) 6.20285i 0.339914i
\(334\) 0 0
\(335\) −2.52898 −0.138173
\(336\) 0 0
\(337\) 23.0098 1.25342 0.626712 0.779251i \(-0.284400\pi\)
0.626712 + 0.779251i \(0.284400\pi\)
\(338\) 0 0
\(339\) − 9.55136i − 0.518759i
\(340\) 0 0
\(341\) 1.95078i 0.105641i
\(342\) 0 0
\(343\) 30.8651 1.66656
\(344\) 0 0
\(345\) 1.33962 0.0721225
\(346\) 0 0
\(347\) − 15.4186i − 0.827716i −0.910341 0.413858i \(-0.864181\pi\)
0.910341 0.413858i \(-0.135819\pi\)
\(348\) 0 0
\(349\) − 28.3638i − 1.51828i −0.650927 0.759140i \(-0.725620\pi\)
0.650927 0.759140i \(-0.274380\pi\)
\(350\) 0 0
\(351\) 0.0840215 0.00448474
\(352\) 0 0
\(353\) −12.2117 −0.649965 −0.324983 0.945720i \(-0.605358\pi\)
−0.324983 + 0.945720i \(0.605358\pi\)
\(354\) 0 0
\(355\) − 4.31724i − 0.229135i
\(356\) 0 0
\(357\) 16.4853i 0.872494i
\(358\) 0 0
\(359\) −33.4780 −1.76690 −0.883452 0.468522i \(-0.844786\pi\)
−0.883452 + 0.468522i \(0.844786\pi\)
\(360\) 0 0
\(361\) 5.91688 0.311415
\(362\) 0 0
\(363\) 1.23765i 0.0649597i
\(364\) 0 0
\(365\) 0.256497i 0.0134256i
\(366\) 0 0
\(367\) −0.702379 −0.0366639 −0.0183319 0.999832i \(-0.505836\pi\)
−0.0183319 + 0.999832i \(0.505836\pi\)
\(368\) 0 0
\(369\) −9.27391 −0.482781
\(370\) 0 0
\(371\) − 3.18100i − 0.165149i
\(372\) 0 0
\(373\) 26.8132i 1.38834i 0.719813 + 0.694168i \(0.244227\pi\)
−0.719813 + 0.694168i \(0.755773\pi\)
\(374\) 0 0
\(375\) 4.63001 0.239093
\(376\) 0 0
\(377\) 0.613530 0.0315984
\(378\) 0 0
\(379\) − 2.51509i − 0.129192i −0.997912 0.0645958i \(-0.979424\pi\)
0.997912 0.0645958i \(-0.0205758\pi\)
\(380\) 0 0
\(381\) 5.09921i 0.261240i
\(382\) 0 0
\(383\) −25.4880 −1.30238 −0.651188 0.758916i \(-0.725729\pi\)
−0.651188 + 0.758916i \(0.725729\pi\)
\(384\) 0 0
\(385\) −7.55136 −0.384853
\(386\) 0 0
\(387\) − 2.27744i − 0.115769i
\(388\) 0 0
\(389\) 16.5532i 0.839281i 0.907690 + 0.419641i \(0.137844\pi\)
−0.907690 + 0.419641i \(0.862156\pi\)
\(390\) 0 0
\(391\) −10.2306 −0.517383
\(392\) 0 0
\(393\) 2.99647 0.151152
\(394\) 0 0
\(395\) − 5.20690i − 0.261988i
\(396\) 0 0
\(397\) − 12.7936i − 0.642094i −0.947063 0.321047i \(-0.895965\pi\)
0.947063 0.321047i \(-0.104035\pi\)
\(398\) 0 0
\(399\) −16.4853 −0.825296
\(400\) 0 0
\(401\) 18.0853 0.903137 0.451568 0.892237i \(-0.350865\pi\)
0.451568 + 0.892237i \(0.350865\pi\)
\(402\) 0 0
\(403\) − 0.0468544i − 0.00233398i
\(404\) 0 0
\(405\) 0.473626i 0.0235347i
\(406\) 0 0
\(407\) 21.6990 1.07558
\(408\) 0 0
\(409\) −25.2271 −1.24740 −0.623699 0.781665i \(-0.714371\pi\)
−0.623699 + 0.781665i \(0.714371\pi\)
\(410\) 0 0
\(411\) − 3.37941i − 0.166694i
\(412\) 0 0
\(413\) 25.7819i 1.26865i
\(414\) 0 0
\(415\) −7.12787 −0.349894
\(416\) 0 0
\(417\) −8.31724 −0.407297
\(418\) 0 0
\(419\) 10.2571i 0.501090i 0.968105 + 0.250545i \(0.0806098\pi\)
−0.968105 + 0.250545i \(0.919390\pi\)
\(420\) 0 0
\(421\) − 3.38775i − 0.165109i −0.996587 0.0825543i \(-0.973692\pi\)
0.996587 0.0825543i \(-0.0263078\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) −17.2739 −0.837908
\(426\) 0 0
\(427\) − 17.5912i − 0.851296i
\(428\) 0 0
\(429\) − 0.293927i − 0.0141909i
\(430\) 0 0
\(431\) −4.42454 −0.213123 −0.106561 0.994306i \(-0.533984\pi\)
−0.106561 + 0.994306i \(0.533984\pi\)
\(432\) 0 0
\(433\) −7.31371 −0.351474 −0.175737 0.984437i \(-0.556231\pi\)
−0.175737 + 0.984437i \(0.556231\pi\)
\(434\) 0 0
\(435\) 3.45844i 0.165820i
\(436\) 0 0
\(437\) − 10.2306i − 0.489395i
\(438\) 0 0
\(439\) −29.6533 −1.41527 −0.707637 0.706576i \(-0.750239\pi\)
−0.707637 + 0.706576i \(0.750239\pi\)
\(440\) 0 0
\(441\) −13.7721 −0.655817
\(442\) 0 0
\(443\) 14.5743i 0.692446i 0.938152 + 0.346223i \(0.112536\pi\)
−0.938152 + 0.346223i \(0.887464\pi\)
\(444\) 0 0
\(445\) 6.94019i 0.328997i
\(446\) 0 0
\(447\) −14.1305 −0.668349
\(448\) 0 0
\(449\) −6.48844 −0.306208 −0.153104 0.988210i \(-0.548927\pi\)
−0.153104 + 0.988210i \(0.548927\pi\)
\(450\) 0 0
\(451\) 32.4423i 1.52765i
\(452\) 0 0
\(453\) − 9.97685i − 0.468753i
\(454\) 0 0
\(455\) 0.181370 0.00850278
\(456\) 0 0
\(457\) 9.00353 0.421167 0.210584 0.977576i \(-0.432464\pi\)
0.210584 + 0.977576i \(0.432464\pi\)
\(458\) 0 0
\(459\) − 3.61706i − 0.168830i
\(460\) 0 0
\(461\) 20.6783i 0.963085i 0.876423 + 0.481542i \(0.159923\pi\)
−0.876423 + 0.481542i \(0.840077\pi\)
\(462\) 0 0
\(463\) −18.6435 −0.866437 −0.433219 0.901289i \(-0.642622\pi\)
−0.433219 + 0.901289i \(0.642622\pi\)
\(464\) 0 0
\(465\) 0.264116 0.0122481
\(466\) 0 0
\(467\) 33.2535i 1.53879i 0.638773 + 0.769395i \(0.279442\pi\)
−0.638773 + 0.769395i \(0.720558\pi\)
\(468\) 0 0
\(469\) 24.3361i 1.12374i
\(470\) 0 0
\(471\) 22.8562 1.05316
\(472\) 0 0
\(473\) −7.96703 −0.366325
\(474\) 0 0
\(475\) − 17.2739i − 0.792582i
\(476\) 0 0
\(477\) 0.697947i 0.0319568i
\(478\) 0 0
\(479\) 1.08864 0.0497412 0.0248706 0.999691i \(-0.492083\pi\)
0.0248706 + 0.999691i \(0.492083\pi\)
\(480\) 0 0
\(481\) −0.521173 −0.0237634
\(482\) 0 0
\(483\) − 12.8910i − 0.586560i
\(484\) 0 0
\(485\) 2.04476i 0.0928476i
\(486\) 0 0
\(487\) 35.3298 1.60095 0.800473 0.599369i \(-0.204582\pi\)
0.800473 + 0.599369i \(0.204582\pi\)
\(488\) 0 0
\(489\) −10.6135 −0.479960
\(490\) 0 0
\(491\) 18.2306i 0.822735i 0.911470 + 0.411367i \(0.134949\pi\)
−0.911470 + 0.411367i \(0.865051\pi\)
\(492\) 0 0
\(493\) − 26.4120i − 1.18953i
\(494\) 0 0
\(495\) 1.65685 0.0744701
\(496\) 0 0
\(497\) −41.5443 −1.86352
\(498\) 0 0
\(499\) − 20.3361i − 0.910368i −0.890397 0.455184i \(-0.849573\pi\)
0.890397 0.455184i \(-0.150427\pi\)
\(500\) 0 0
\(501\) − 5.83822i − 0.260833i
\(502\) 0 0
\(503\) −30.2969 −1.35087 −0.675435 0.737420i \(-0.736044\pi\)
−0.675435 + 0.737420i \(0.736044\pi\)
\(504\) 0 0
\(505\) −0.303911 −0.0135239
\(506\) 0 0
\(507\) − 12.9929i − 0.577037i
\(508\) 0 0
\(509\) − 14.9660i − 0.663355i −0.943393 0.331677i \(-0.892385\pi\)
0.943393 0.331677i \(-0.107615\pi\)
\(510\) 0 0
\(511\) 2.46824 0.109188
\(512\) 0 0
\(513\) 3.61706 0.159697
\(514\) 0 0
\(515\) − 0.633169i − 0.0279008i
\(516\) 0 0
\(517\) 9.89450i 0.435160i
\(518\) 0 0
\(519\) −5.12695 −0.225048
\(520\) 0 0
\(521\) −24.9049 −1.09110 −0.545551 0.838078i \(-0.683680\pi\)
−0.545551 + 0.838078i \(0.683680\pi\)
\(522\) 0 0
\(523\) 18.2445i 0.797775i 0.917000 + 0.398888i \(0.130604\pi\)
−0.917000 + 0.398888i \(0.869396\pi\)
\(524\) 0 0
\(525\) − 21.7659i − 0.949940i
\(526\) 0 0
\(527\) −2.01704 −0.0878638
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) − 5.65685i − 0.245487i
\(532\) 0 0
\(533\) − 0.779208i − 0.0337513i
\(534\) 0 0
\(535\) 4.06074 0.175561
\(536\) 0 0
\(537\) −13.1286 −0.566542
\(538\) 0 0
\(539\) 48.1782i 2.07518i
\(540\) 0 0
\(541\) − 25.8471i − 1.11125i −0.831432 0.555627i \(-0.812478\pi\)
0.831432 0.555627i \(-0.187522\pi\)
\(542\) 0 0
\(543\) −15.3181 −0.657364
\(544\) 0 0
\(545\) −3.82880 −0.164008
\(546\) 0 0
\(547\) 19.4249i 0.830549i 0.909696 + 0.415275i \(0.136315\pi\)
−0.909696 + 0.415275i \(0.863685\pi\)
\(548\) 0 0
\(549\) 3.85970i 0.164728i
\(550\) 0 0
\(551\) 26.4120 1.12519
\(552\) 0 0
\(553\) −50.1055 −2.13070
\(554\) 0 0
\(555\) − 2.93783i − 0.124704i
\(556\) 0 0
\(557\) − 38.9652i − 1.65101i −0.564397 0.825504i \(-0.690891\pi\)
0.564397 0.825504i \(-0.309109\pi\)
\(558\) 0 0
\(559\) 0.191354 0.00809342
\(560\) 0 0
\(561\) −12.6533 −0.534224
\(562\) 0 0
\(563\) 28.1327i 1.18565i 0.805330 + 0.592826i \(0.201988\pi\)
−0.805330 + 0.592826i \(0.798012\pi\)
\(564\) 0 0
\(565\) 4.52377i 0.190316i
\(566\) 0 0
\(567\) 4.55765 0.191403
\(568\) 0 0
\(569\) 13.4849 0.565317 0.282658 0.959221i \(-0.408784\pi\)
0.282658 + 0.959221i \(0.408784\pi\)
\(570\) 0 0
\(571\) − 20.9706i − 0.877591i −0.898587 0.438795i \(-0.855405\pi\)
0.898587 0.438795i \(-0.144595\pi\)
\(572\) 0 0
\(573\) − 8.63001i − 0.360524i
\(574\) 0 0
\(575\) 13.5077 0.563308
\(576\) 0 0
\(577\) −11.6176 −0.483648 −0.241824 0.970320i \(-0.577746\pi\)
−0.241824 + 0.970320i \(0.577746\pi\)
\(578\) 0 0
\(579\) − 11.4514i − 0.475903i
\(580\) 0 0
\(581\) 68.5907i 2.84562i
\(582\) 0 0
\(583\) 2.44158 0.101120
\(584\) 0 0
\(585\) −0.0397948 −0.00164531
\(586\) 0 0
\(587\) 24.0796i 0.993871i 0.867787 + 0.496936i \(0.165541\pi\)
−0.867787 + 0.496936i \(0.834459\pi\)
\(588\) 0 0
\(589\) − 2.01704i − 0.0831108i
\(590\) 0 0
\(591\) −10.5925 −0.435715
\(592\) 0 0
\(593\) −41.5372 −1.70573 −0.852865 0.522132i \(-0.825137\pi\)
−0.852865 + 0.522132i \(0.825137\pi\)
\(594\) 0 0
\(595\) − 7.80785i − 0.320091i
\(596\) 0 0
\(597\) − 3.68000i − 0.150612i
\(598\) 0 0
\(599\) −6.43160 −0.262788 −0.131394 0.991330i \(-0.541945\pi\)
−0.131394 + 0.991330i \(0.541945\pi\)
\(600\) 0 0
\(601\) 3.45844 0.141073 0.0705364 0.997509i \(-0.477529\pi\)
0.0705364 + 0.997509i \(0.477529\pi\)
\(602\) 0 0
\(603\) − 5.33962i − 0.217446i
\(604\) 0 0
\(605\) − 0.586182i − 0.0238317i
\(606\) 0 0
\(607\) 30.1019 1.22180 0.610900 0.791708i \(-0.290808\pi\)
0.610900 + 0.791708i \(0.290808\pi\)
\(608\) 0 0
\(609\) 33.2802 1.34858
\(610\) 0 0
\(611\) − 0.237649i − 0.00961424i
\(612\) 0 0
\(613\) − 3.54246i − 0.143079i −0.997438 0.0715393i \(-0.977209\pi\)
0.997438 0.0715393i \(-0.0227911\pi\)
\(614\) 0 0
\(615\) 4.39236 0.177117
\(616\) 0 0
\(617\) 22.9098 0.922315 0.461157 0.887318i \(-0.347434\pi\)
0.461157 + 0.887318i \(0.347434\pi\)
\(618\) 0 0
\(619\) 40.4612i 1.62627i 0.582074 + 0.813136i \(0.302242\pi\)
−0.582074 + 0.813136i \(0.697758\pi\)
\(620\) 0 0
\(621\) 2.82843i 0.113501i
\(622\) 0 0
\(623\) 66.7847 2.67567
\(624\) 0 0
\(625\) 21.6855 0.867420
\(626\) 0 0
\(627\) − 12.6533i − 0.505325i
\(628\) 0 0
\(629\) 22.4361i 0.894584i
\(630\) 0 0
\(631\) −11.1851 −0.445270 −0.222635 0.974902i \(-0.571466\pi\)
−0.222635 + 0.974902i \(0.571466\pi\)
\(632\) 0 0
\(633\) −14.3102 −0.568779
\(634\) 0 0
\(635\) − 2.41512i − 0.0958409i
\(636\) 0 0
\(637\) − 1.15716i − 0.0458482i
\(638\) 0 0
\(639\) 9.11529 0.360595
\(640\) 0 0
\(641\) −6.69312 −0.264362 −0.132181 0.991226i \(-0.542198\pi\)
−0.132181 + 0.991226i \(0.542198\pi\)
\(642\) 0 0
\(643\) − 25.3724i − 1.00059i −0.865856 0.500294i \(-0.833225\pi\)
0.865856 0.500294i \(-0.166775\pi\)
\(644\) 0 0
\(645\) 1.07866i 0.0424720i
\(646\) 0 0
\(647\) 6.72999 0.264583 0.132292 0.991211i \(-0.457766\pi\)
0.132292 + 0.991211i \(0.457766\pi\)
\(648\) 0 0
\(649\) −19.7890 −0.776786
\(650\) 0 0
\(651\) − 2.54156i − 0.0996116i
\(652\) 0 0
\(653\) 37.0144i 1.44849i 0.689545 + 0.724243i \(0.257810\pi\)
−0.689545 + 0.724243i \(0.742190\pi\)
\(654\) 0 0
\(655\) −1.41921 −0.0554529
\(656\) 0 0
\(657\) −0.541560 −0.0211283
\(658\) 0 0
\(659\) − 19.7624i − 0.769832i −0.922952 0.384916i \(-0.874230\pi\)
0.922952 0.384916i \(-0.125770\pi\)
\(660\) 0 0
\(661\) 16.8632i 0.655904i 0.944694 + 0.327952i \(0.106358\pi\)
−0.944694 + 0.327952i \(0.893642\pi\)
\(662\) 0 0
\(663\) 0.303911 0.0118029
\(664\) 0 0
\(665\) 7.80785 0.302776
\(666\) 0 0
\(667\) 20.6533i 0.799700i
\(668\) 0 0
\(669\) − 4.86156i − 0.187959i
\(670\) 0 0
\(671\) 13.5021 0.521244
\(672\) 0 0
\(673\) −37.3066 −1.43807 −0.719033 0.694976i \(-0.755415\pi\)
−0.719033 + 0.694976i \(0.755415\pi\)
\(674\) 0 0
\(675\) 4.77568i 0.183816i
\(676\) 0 0
\(677\) − 0.632805i − 0.0243207i −0.999926 0.0121603i \(-0.996129\pi\)
0.999926 0.0121603i \(-0.00387085\pi\)
\(678\) 0 0
\(679\) 19.6764 0.755113
\(680\) 0 0
\(681\) 15.0496 0.576702
\(682\) 0 0
\(683\) 6.04606i 0.231346i 0.993287 + 0.115673i \(0.0369025\pi\)
−0.993287 + 0.115673i \(0.963098\pi\)
\(684\) 0 0
\(685\) 1.60058i 0.0611549i
\(686\) 0 0
\(687\) 28.5264 1.08835
\(688\) 0 0
\(689\) −0.0586426 −0.00223410
\(690\) 0 0
\(691\) − 28.3955i − 1.08021i −0.841596 0.540107i \(-0.818384\pi\)
0.841596 0.540107i \(-0.181616\pi\)
\(692\) 0 0
\(693\) − 15.9437i − 0.605652i
\(694\) 0 0
\(695\) 3.93926 0.149425
\(696\) 0 0
\(697\) −33.5443 −1.27058
\(698\) 0 0
\(699\) 13.5702i 0.513272i
\(700\) 0 0
\(701\) 14.7738i 0.558000i 0.960291 + 0.279000i \(0.0900029\pi\)
−0.960291 + 0.279000i \(0.909997\pi\)
\(702\) 0 0
\(703\) −22.4361 −0.846192
\(704\) 0 0
\(705\) 1.33962 0.0504529
\(706\) 0 0
\(707\) 2.92450i 0.109987i
\(708\) 0 0
\(709\) − 22.7569i − 0.854655i −0.904097 0.427327i \(-0.859455\pi\)
0.904097 0.427327i \(-0.140545\pi\)
\(710\) 0 0
\(711\) 10.9937 0.412296
\(712\) 0 0
\(713\) 1.57726 0.0590690
\(714\) 0 0
\(715\) 0.139211i 0.00520621i
\(716\) 0 0
\(717\) − 29.3629i − 1.09658i
\(718\) 0 0
\(719\) −30.9957 −1.15594 −0.577972 0.816057i \(-0.696156\pi\)
−0.577972 + 0.816057i \(0.696156\pi\)
\(720\) 0 0
\(721\) −6.09292 −0.226912
\(722\) 0 0
\(723\) 24.0063i 0.892803i
\(724\) 0 0
\(725\) 34.8723i 1.29512i
\(726\) 0 0
\(727\) 41.1117 1.52475 0.762375 0.647135i \(-0.224033\pi\)
0.762375 + 0.647135i \(0.224033\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 8.23765i − 0.304680i
\(732\) 0 0
\(733\) − 0.206562i − 0.00762954i −0.999993 0.00381477i \(-0.998786\pi\)
0.999993 0.00381477i \(-0.00121428\pi\)
\(734\) 0 0
\(735\) 6.52284 0.240599
\(736\) 0 0
\(737\) −18.6792 −0.688058
\(738\) 0 0
\(739\) − 2.13215i − 0.0784325i −0.999231 0.0392162i \(-0.987514\pi\)
0.999231 0.0392162i \(-0.0124861\pi\)
\(740\) 0 0
\(741\) 0.303911i 0.0111644i
\(742\) 0 0
\(743\) −40.5175 −1.48644 −0.743221 0.669046i \(-0.766703\pi\)
−0.743221 + 0.669046i \(0.766703\pi\)
\(744\) 0 0
\(745\) 6.69256 0.245196
\(746\) 0 0
\(747\) − 15.0496i − 0.550636i
\(748\) 0 0
\(749\) − 39.0761i − 1.42781i
\(750\) 0 0
\(751\) −12.5843 −0.459208 −0.229604 0.973284i \(-0.573743\pi\)
−0.229604 + 0.973284i \(0.573743\pi\)
\(752\) 0 0
\(753\) 22.2837 0.812064
\(754\) 0 0
\(755\) 4.72529i 0.171971i
\(756\) 0 0
\(757\) − 10.6052i − 0.385452i −0.981253 0.192726i \(-0.938267\pi\)
0.981253 0.192726i \(-0.0617329\pi\)
\(758\) 0 0
\(759\) 9.89450 0.359148
\(760\) 0 0
\(761\) −42.8182 −1.55216 −0.776079 0.630635i \(-0.782794\pi\)
−0.776079 + 0.630635i \(0.782794\pi\)
\(762\) 0 0
\(763\) 36.8441i 1.33385i
\(764\) 0 0
\(765\) 1.71313i 0.0619384i
\(766\) 0 0
\(767\) 0.475298 0.0171620
\(768\) 0 0
\(769\) 12.7455 0.459614 0.229807 0.973236i \(-0.426190\pi\)
0.229807 + 0.973236i \(0.426190\pi\)
\(770\) 0 0
\(771\) − 8.66038i − 0.311896i
\(772\) 0 0
\(773\) 32.3522i 1.16363i 0.813322 + 0.581814i \(0.197657\pi\)
−0.813322 + 0.581814i \(0.802343\pi\)
\(774\) 0 0
\(775\) 2.66314 0.0956630
\(776\) 0 0
\(777\) −28.2704 −1.01419
\(778\) 0 0
\(779\) − 33.5443i − 1.20185i
\(780\) 0 0
\(781\) − 31.8874i − 1.14102i
\(782\) 0 0
\(783\) −7.30205 −0.260954
\(784\) 0 0
\(785\) −10.8253 −0.386370
\(786\) 0 0
\(787\) − 7.36056i − 0.262376i −0.991358 0.131188i \(-0.958121\pi\)
0.991358 0.131188i \(-0.0418791\pi\)
\(788\) 0 0
\(789\) 13.3208i 0.474232i
\(790\) 0 0
\(791\) 43.5317 1.54781
\(792\) 0 0
\(793\) −0.324298 −0.0115162
\(794\) 0 0
\(795\) − 0.330566i − 0.0117240i
\(796\) 0 0
\(797\) − 24.0627i − 0.852344i −0.904642 0.426172i \(-0.859862\pi\)
0.904642 0.426172i \(-0.140138\pi\)
\(798\) 0 0
\(799\) −10.2306 −0.361932
\(800\) 0 0
\(801\) −14.6533 −0.517750
\(802\) 0 0
\(803\) 1.89450i 0.0668556i
\(804\) 0 0
\(805\) 6.10550i 0.215190i
\(806\) 0 0
\(807\) −16.5058 −0.581032
\(808\) 0 0
\(809\) 7.83586 0.275494 0.137747 0.990467i \(-0.456014\pi\)
0.137747 + 0.990467i \(0.456014\pi\)
\(810\) 0 0
\(811\) − 45.7351i − 1.60598i −0.595995 0.802988i \(-0.703242\pi\)
0.595995 0.802988i \(-0.296758\pi\)
\(812\) 0 0
\(813\) 21.9769i 0.770762i
\(814\) 0 0
\(815\) 5.02684 0.176083
\(816\) 0 0
\(817\) 8.23765 0.288199
\(818\) 0 0
\(819\) 0.382941i 0.0133810i
\(820\) 0 0
\(821\) 27.3709i 0.955250i 0.878564 + 0.477625i \(0.158502\pi\)
−0.878564 + 0.477625i \(0.841498\pi\)
\(822\) 0 0
\(823\) −28.8560 −1.00586 −0.502929 0.864328i \(-0.667744\pi\)
−0.502929 + 0.864328i \(0.667744\pi\)
\(824\) 0 0
\(825\) 16.7064 0.581644
\(826\) 0 0
\(827\) − 14.4227i − 0.501528i −0.968048 0.250764i \(-0.919318\pi\)
0.968048 0.250764i \(-0.0806818\pi\)
\(828\) 0 0
\(829\) − 21.7497i − 0.755400i −0.925928 0.377700i \(-0.876715\pi\)
0.925928 0.377700i \(-0.123285\pi\)
\(830\) 0 0
\(831\) −15.4862 −0.537210
\(832\) 0 0
\(833\) −49.8147 −1.72598
\(834\) 0 0
\(835\) 2.76513i 0.0956914i
\(836\) 0 0
\(837\) 0.557647i 0.0192751i
\(838\) 0 0
\(839\) −44.4557 −1.53478 −0.767390 0.641181i \(-0.778445\pi\)
−0.767390 + 0.641181i \(0.778445\pi\)
\(840\) 0 0
\(841\) −24.3200 −0.838620
\(842\) 0 0
\(843\) 22.8910i 0.788407i
\(844\) 0 0
\(845\) 6.15379i 0.211697i
\(846\) 0 0
\(847\) −5.64077 −0.193819
\(848\) 0 0
\(849\) 6.34315 0.217696
\(850\) 0 0
\(851\) − 17.5443i − 0.601411i
\(852\) 0 0
\(853\) − 16.5648i − 0.567169i −0.958947 0.283585i \(-0.908476\pi\)
0.958947 0.283585i \(-0.0915237\pi\)
\(854\) 0 0
\(855\) −1.71313 −0.0585879
\(856\) 0 0
\(857\) 19.0888 0.652062 0.326031 0.945359i \(-0.394289\pi\)
0.326031 + 0.945359i \(0.394289\pi\)
\(858\) 0 0
\(859\) − 53.9272i − 1.83997i −0.391949 0.919987i \(-0.628199\pi\)
0.391949 0.919987i \(-0.371801\pi\)
\(860\) 0 0
\(861\) − 42.2672i − 1.44046i
\(862\) 0 0
\(863\) −3.64533 −0.124089 −0.0620443 0.998073i \(-0.519762\pi\)
−0.0620443 + 0.998073i \(0.519762\pi\)
\(864\) 0 0
\(865\) 2.42826 0.0825632
\(866\) 0 0
\(867\) 3.91688i 0.133024i
\(868\) 0 0
\(869\) − 38.4586i − 1.30462i
\(870\) 0 0
\(871\) 0.448643 0.0152017
\(872\) 0 0
\(873\) −4.31724 −0.146116
\(874\) 0 0
\(875\) 21.1020i 0.713377i
\(876\) 0 0
\(877\) 56.6481i 1.91287i 0.291945 + 0.956435i \(0.405698\pi\)
−0.291945 + 0.956435i \(0.594302\pi\)
\(878\) 0 0
\(879\) −30.5783 −1.03138
\(880\) 0 0
\(881\) 20.0118 0.674214 0.337107 0.941466i \(-0.390552\pi\)
0.337107 + 0.941466i \(0.390552\pi\)
\(882\) 0 0
\(883\) 15.0292i 0.505773i 0.967496 + 0.252887i \(0.0813799\pi\)
−0.967496 + 0.252887i \(0.918620\pi\)
\(884\) 0 0
\(885\) 2.67923i 0.0900614i
\(886\) 0 0
\(887\) −26.1180 −0.876958 −0.438479 0.898742i \(-0.644483\pi\)
−0.438479 + 0.898742i \(0.644483\pi\)
\(888\) 0 0
\(889\) −23.2404 −0.779458
\(890\) 0 0
\(891\) 3.49824i 0.117195i
\(892\) 0 0
\(893\) − 10.2306i − 0.342354i
\(894\) 0 0
\(895\) 6.21805 0.207847
\(896\) 0 0
\(897\) −0.237649 −0.00793486
\(898\) 0 0
\(899\) 4.07197i 0.135808i
\(900\) 0 0
\(901\) 2.52452i 0.0841038i
\(902\) 0 0
\(903\) 10.3798 0.345418
\(904\) 0 0
\(905\) 7.25507 0.241167
\(906\) 0 0
\(907\) − 51.2480i − 1.70166i −0.525439 0.850831i \(-0.676099\pi\)
0.525439 0.850831i \(-0.323901\pi\)
\(908\) 0 0
\(909\) − 0.641669i − 0.0212828i
\(910\) 0 0
\(911\) −21.0535 −0.697533 −0.348767 0.937210i \(-0.613399\pi\)
−0.348767 + 0.937210i \(0.613399\pi\)
\(912\) 0 0
\(913\) −52.6470 −1.74236
\(914\) 0 0
\(915\) − 1.82805i − 0.0604336i
\(916\) 0 0
\(917\) 13.6569i 0.450989i
\(918\) 0 0
\(919\) 17.8839 0.589937 0.294968 0.955507i \(-0.404691\pi\)
0.294968 + 0.955507i \(0.404691\pi\)
\(920\) 0 0
\(921\) 17.1286 0.564407
\(922\) 0 0
\(923\) 0.765881i 0.0252093i
\(924\) 0 0
\(925\) − 29.6228i − 0.973992i
\(926\) 0 0
\(927\) 1.33686 0.0439081
\(928\) 0 0
\(929\) 10.2774 0.337192 0.168596 0.985685i \(-0.446077\pi\)
0.168596 + 0.985685i \(0.446077\pi\)
\(930\) 0 0
\(931\) − 49.8147i − 1.63261i
\(932\) 0 0
\(933\) 26.8651i 0.879523i
\(934\) 0 0
\(935\) 5.99294 0.195990
\(936\) 0 0
\(937\) 13.5780 0.443574 0.221787 0.975095i \(-0.428811\pi\)
0.221787 + 0.975095i \(0.428811\pi\)
\(938\) 0 0
\(939\) − 19.6890i − 0.642527i
\(940\) 0 0
\(941\) − 5.59890i − 0.182519i −0.995827 0.0912595i \(-0.970911\pi\)
0.995827 0.0912595i \(-0.0290893\pi\)
\(942\) 0 0
\(943\) 26.2306 0.854186
\(944\) 0 0
\(945\) −2.15862 −0.0702199
\(946\) 0 0
\(947\) 46.9106i 1.52439i 0.647348 + 0.762194i \(0.275878\pi\)
−0.647348 + 0.762194i \(0.724122\pi\)
\(948\) 0 0
\(949\) − 0.0455027i − 0.00147708i
\(950\) 0 0
\(951\) −30.1860 −0.978847
\(952\) 0 0
\(953\) −5.59115 −0.181115 −0.0905576 0.995891i \(-0.528865\pi\)
−0.0905576 + 0.995891i \(0.528865\pi\)
\(954\) 0 0
\(955\) 4.08740i 0.132265i
\(956\) 0 0
\(957\) 25.5443i 0.825730i
\(958\) 0 0
\(959\) 15.4022 0.497362
\(960\) 0 0
\(961\) −30.6890 −0.989969
\(962\) 0 0
\(963\) 8.57373i 0.276285i
\(964\) 0 0
\(965\) 5.42367i 0.174594i
\(966\) 0 0
\(967\) −30.7561 −0.989048 −0.494524 0.869164i \(-0.664658\pi\)
−0.494524 + 0.869164i \(0.664658\pi\)
\(968\) 0 0
\(969\) 13.0831 0.420290
\(970\) 0 0
\(971\) − 11.3668i − 0.364779i −0.983226 0.182389i \(-0.941617\pi\)
0.983226 0.182389i \(-0.0583832\pi\)
\(972\) 0 0
\(973\) − 37.9070i − 1.21524i
\(974\) 0 0
\(975\) −0.401260 −0.0128506
\(976\) 0 0
\(977\) −22.8323 −0.730471 −0.365235 0.930915i \(-0.619012\pi\)
−0.365235 + 0.930915i \(0.619012\pi\)
\(978\) 0 0
\(979\) 51.2608i 1.63830i
\(980\) 0 0
\(981\) − 8.08402i − 0.258103i
\(982\) 0 0
\(983\) −46.3557 −1.47852 −0.739258 0.673422i \(-0.764824\pi\)
−0.739258 + 0.673422i \(0.764824\pi\)
\(984\) 0 0
\(985\) 5.01686 0.159850
\(986\) 0 0
\(987\) − 12.8910i − 0.410324i
\(988\) 0 0
\(989\) 6.44158i 0.204830i
\(990\) 0 0
\(991\) −3.43683 −0.109175 −0.0545873 0.998509i \(-0.517384\pi\)
−0.0545873 + 0.998509i \(0.517384\pi\)
\(992\) 0 0
\(993\) −20.7784 −0.659383
\(994\) 0 0
\(995\) 1.74294i 0.0552550i
\(996\) 0 0
\(997\) 31.0320i 0.982794i 0.870936 + 0.491397i \(0.163514\pi\)
−0.870936 + 0.491397i \(0.836486\pi\)
\(998\) 0 0
\(999\) 6.20285 0.196249
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.1537.3 8
4.3 odd 2 3072.2.d.f.1537.7 8
8.3 odd 2 3072.2.d.f.1537.2 8
8.5 even 2 inner 3072.2.d.i.1537.6 8
16.3 odd 4 3072.2.a.i.1.3 4
16.5 even 4 3072.2.a.n.1.2 4
16.11 odd 4 3072.2.a.t.1.2 4
16.13 even 4 3072.2.a.o.1.3 4
32.3 odd 8 48.2.j.a.37.4 yes 8
32.5 even 8 192.2.j.a.145.3 8
32.11 odd 8 384.2.j.b.289.4 8
32.13 even 8 384.2.j.a.97.2 8
32.19 odd 8 384.2.j.b.97.4 8
32.21 even 8 384.2.j.a.289.2 8
32.27 odd 8 48.2.j.a.13.4 8
32.29 even 8 192.2.j.a.49.3 8
48.5 odd 4 9216.2.a.x.1.3 4
48.11 even 4 9216.2.a.y.1.3 4
48.29 odd 4 9216.2.a.bn.1.2 4
48.35 even 4 9216.2.a.bo.1.2 4
96.5 odd 8 576.2.k.b.145.3 8
96.11 even 8 1152.2.k.c.289.2 8
96.29 odd 8 576.2.k.b.433.3 8
96.35 even 8 144.2.k.b.37.1 8
96.53 odd 8 1152.2.k.f.289.2 8
96.59 even 8 144.2.k.b.109.1 8
96.77 odd 8 1152.2.k.f.865.2 8
96.83 even 8 1152.2.k.c.865.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.4 8 32.27 odd 8
48.2.j.a.37.4 yes 8 32.3 odd 8
144.2.k.b.37.1 8 96.35 even 8
144.2.k.b.109.1 8 96.59 even 8
192.2.j.a.49.3 8 32.29 even 8
192.2.j.a.145.3 8 32.5 even 8
384.2.j.a.97.2 8 32.13 even 8
384.2.j.a.289.2 8 32.21 even 8
384.2.j.b.97.4 8 32.19 odd 8
384.2.j.b.289.4 8 32.11 odd 8
576.2.k.b.145.3 8 96.5 odd 8
576.2.k.b.433.3 8 96.29 odd 8
1152.2.k.c.289.2 8 96.11 even 8
1152.2.k.c.865.2 8 96.83 even 8
1152.2.k.f.289.2 8 96.53 odd 8
1152.2.k.f.865.2 8 96.77 odd 8
3072.2.a.i.1.3 4 16.3 odd 4
3072.2.a.n.1.2 4 16.5 even 4
3072.2.a.o.1.3 4 16.13 even 4
3072.2.a.t.1.2 4 16.11 odd 4
3072.2.d.f.1537.2 8 8.3 odd 2
3072.2.d.f.1537.7 8 4.3 odd 2
3072.2.d.i.1537.3 8 1.1 even 1 trivial
3072.2.d.i.1537.6 8 8.5 even 2 inner
9216.2.a.x.1.3 4 48.5 odd 4
9216.2.a.y.1.3 4 48.11 even 4
9216.2.a.bn.1.2 4 48.29 odd 4
9216.2.a.bo.1.2 4 48.35 even 4