Properties

Label 3864.2.a.o.1.1
Level $3864$
Weight $2$
Character 3864.1
Self dual yes
Analytic conductor $30.854$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.8541953410\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.733.1
Defining polynomial: \(x^{3} - x^{2} - 7 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.69639\) of defining polynomial
Character \(\chi\) \(=\) 3864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.27053 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.27053 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.42586 q^{11} +4.69639 q^{13} -3.27053 q^{15} +3.96693 q^{17} -5.42586 q^{19} -1.00000 q^{21} +1.00000 q^{23} +5.69639 q^{25} +1.00000 q^{27} +3.96693 q^{29} -8.81864 q^{31} -1.42586 q^{33} +3.27053 q^{35} +4.81864 q^{37} +4.69639 q^{39} +2.57414 q^{41} -11.5481 q^{43} -3.27053 q^{45} +5.39279 q^{47} +1.00000 q^{49} +3.96693 q^{51} -12.6633 q^{53} +4.66332 q^{55} -5.42586 q^{57} +11.5150 q^{59} -13.2375 q^{61} -1.00000 q^{63} -15.3597 q^{65} +5.51504 q^{67} +1.00000 q^{69} -2.69639 q^{71} -9.35971 q^{73} +5.69639 q^{75} +1.42586 q^{77} -4.57414 q^{79} +1.00000 q^{81} -0.277576 q^{83} -12.9740 q^{85} +3.96693 q^{87} +11.4820 q^{89} -4.69639 q^{91} -8.81864 q^{93} +17.7455 q^{95} +7.96693 q^{97} -1.42586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} - 3 q^{5} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 5 q^{13} - 3 q^{15} - 4 q^{17} - 14 q^{19} - 3 q^{21} + 3 q^{23} + 8 q^{25} + 3 q^{27} - 4 q^{29} - 6 q^{31} - 2 q^{33} + 3 q^{35} - 6 q^{37} + 5 q^{39} + 10 q^{41} - 21 q^{43} - 3 q^{45} - 2 q^{47} + 3 q^{49} - 4 q^{51} - 13 q^{53} - 11 q^{55} - 14 q^{57} + 5 q^{59} - 17 q^{61} - 3 q^{63} - 12 q^{65} - 13 q^{67} + 3 q^{69} + q^{71} + 6 q^{73} + 8 q^{75} + 2 q^{77} - 16 q^{79} + 3 q^{81} + 6 q^{83} - 23 q^{85} - 4 q^{87} - 11 q^{89} - 5 q^{91} - 6 q^{93} + q^{95} + 8 q^{97} - 2 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −3.27053 −1.46263 −0.731314 0.682041i \(-0.761092\pi\)
−0.731314 + 0.682041i \(0.761092\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.42586 −0.429913 −0.214956 0.976624i \(-0.568961\pi\)
−0.214956 + 0.976624i \(0.568961\pi\)
\(12\) 0 0
\(13\) 4.69639 1.30254 0.651272 0.758844i \(-0.274235\pi\)
0.651272 + 0.758844i \(0.274235\pi\)
\(14\) 0 0
\(15\) −3.27053 −0.844448
\(16\) 0 0
\(17\) 3.96693 0.962121 0.481061 0.876687i \(-0.340252\pi\)
0.481061 + 0.876687i \(0.340252\pi\)
\(18\) 0 0
\(19\) −5.42586 −1.24478 −0.622389 0.782708i \(-0.713838\pi\)
−0.622389 + 0.782708i \(0.713838\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 5.69639 1.13928
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.96693 0.736640 0.368320 0.929699i \(-0.379933\pi\)
0.368320 + 0.929699i \(0.379933\pi\)
\(30\) 0 0
\(31\) −8.81864 −1.58388 −0.791938 0.610602i \(-0.790928\pi\)
−0.791938 + 0.610602i \(0.790928\pi\)
\(32\) 0 0
\(33\) −1.42586 −0.248210
\(34\) 0 0
\(35\) 3.27053 0.552821
\(36\) 0 0
\(37\) 4.81864 0.792180 0.396090 0.918212i \(-0.370367\pi\)
0.396090 + 0.918212i \(0.370367\pi\)
\(38\) 0 0
\(39\) 4.69639 0.752025
\(40\) 0 0
\(41\) 2.57414 0.402013 0.201007 0.979590i \(-0.435579\pi\)
0.201007 + 0.979590i \(0.435579\pi\)
\(42\) 0 0
\(43\) −11.5481 −1.76107 −0.880535 0.473981i \(-0.842817\pi\)
−0.880535 + 0.473981i \(0.842817\pi\)
\(44\) 0 0
\(45\) −3.27053 −0.487542
\(46\) 0 0
\(47\) 5.39279 0.786619 0.393309 0.919406i \(-0.371330\pi\)
0.393309 + 0.919406i \(0.371330\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.96693 0.555481
\(52\) 0 0
\(53\) −12.6633 −1.73944 −0.869720 0.493545i \(-0.835701\pi\)
−0.869720 + 0.493545i \(0.835701\pi\)
\(54\) 0 0
\(55\) 4.66332 0.628802
\(56\) 0 0
\(57\) −5.42586 −0.718673
\(58\) 0 0
\(59\) 11.5150 1.49913 0.749565 0.661931i \(-0.230263\pi\)
0.749565 + 0.661931i \(0.230263\pi\)
\(60\) 0 0
\(61\) −13.2375 −1.69488 −0.847442 0.530889i \(-0.821858\pi\)
−0.847442 + 0.530889i \(0.821858\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −15.3597 −1.90514
\(66\) 0 0
\(67\) 5.51504 0.673769 0.336885 0.941546i \(-0.390627\pi\)
0.336885 + 0.941546i \(0.390627\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −2.69639 −0.320003 −0.160001 0.987117i \(-0.551150\pi\)
−0.160001 + 0.987117i \(0.551150\pi\)
\(72\) 0 0
\(73\) −9.35971 −1.09547 −0.547736 0.836651i \(-0.684510\pi\)
−0.547736 + 0.836651i \(0.684510\pi\)
\(74\) 0 0
\(75\) 5.69639 0.657763
\(76\) 0 0
\(77\) 1.42586 0.162492
\(78\) 0 0
\(79\) −4.57414 −0.514631 −0.257316 0.966327i \(-0.582838\pi\)
−0.257316 + 0.966327i \(0.582838\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.277576 −0.0304680 −0.0152340 0.999884i \(-0.504849\pi\)
−0.0152340 + 0.999884i \(0.504849\pi\)
\(84\) 0 0
\(85\) −12.9740 −1.40722
\(86\) 0 0
\(87\) 3.96693 0.425299
\(88\) 0 0
\(89\) 11.4820 1.21709 0.608543 0.793521i \(-0.291754\pi\)
0.608543 + 0.793521i \(0.291754\pi\)
\(90\) 0 0
\(91\) −4.69639 −0.492316
\(92\) 0 0
\(93\) −8.81864 −0.914451
\(94\) 0 0
\(95\) 17.7455 1.82065
\(96\) 0 0
\(97\) 7.96693 0.808919 0.404459 0.914556i \(-0.367460\pi\)
0.404459 + 0.914556i \(0.367460\pi\)
\(98\) 0 0
\(99\) −1.42586 −0.143304
\(100\) 0 0
\(101\) 6.97397 0.693936 0.346968 0.937877i \(-0.387211\pi\)
0.346968 + 0.937877i \(0.387211\pi\)
\(102\) 0 0
\(103\) −19.8677 −1.95762 −0.978812 0.204763i \(-0.934358\pi\)
−0.978812 + 0.204763i \(0.934358\pi\)
\(104\) 0 0
\(105\) 3.27053 0.319171
\(106\) 0 0
\(107\) −10.3998 −1.00539 −0.502695 0.864464i \(-0.667658\pi\)
−0.502695 + 0.864464i \(0.667658\pi\)
\(108\) 0 0
\(109\) −12.3527 −1.18317 −0.591586 0.806242i \(-0.701498\pi\)
−0.591586 + 0.806242i \(0.701498\pi\)
\(110\) 0 0
\(111\) 4.81864 0.457365
\(112\) 0 0
\(113\) −2.41882 −0.227543 −0.113772 0.993507i \(-0.536293\pi\)
−0.113772 + 0.993507i \(0.536293\pi\)
\(114\) 0 0
\(115\) −3.27053 −0.304979
\(116\) 0 0
\(117\) 4.69639 0.434182
\(118\) 0 0
\(119\) −3.96693 −0.363648
\(120\) 0 0
\(121\) −8.96693 −0.815175
\(122\) 0 0
\(123\) 2.57414 0.232102
\(124\) 0 0
\(125\) −2.27758 −0.203713
\(126\) 0 0
\(127\) −16.9268 −1.50201 −0.751006 0.660296i \(-0.770431\pi\)
−0.751006 + 0.660296i \(0.770431\pi\)
\(128\) 0 0
\(129\) −11.5481 −1.01675
\(130\) 0 0
\(131\) −19.2936 −1.68569 −0.842843 0.538159i \(-0.819120\pi\)
−0.842843 + 0.538159i \(0.819120\pi\)
\(132\) 0 0
\(133\) 5.42586 0.470482
\(134\) 0 0
\(135\) −3.27053 −0.281483
\(136\) 0 0
\(137\) 4.27758 0.365458 0.182729 0.983163i \(-0.441507\pi\)
0.182729 + 0.983163i \(0.441507\pi\)
\(138\) 0 0
\(139\) −2.18840 −0.185617 −0.0928087 0.995684i \(-0.529585\pi\)
−0.0928087 + 0.995684i \(0.529585\pi\)
\(140\) 0 0
\(141\) 5.39279 0.454154
\(142\) 0 0
\(143\) −6.69639 −0.559980
\(144\) 0 0
\(145\) −12.9740 −1.07743
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −15.6894 −1.28532 −0.642661 0.766151i \(-0.722170\pi\)
−0.642661 + 0.766151i \(0.722170\pi\)
\(150\) 0 0
\(151\) 11.0962 0.902998 0.451499 0.892272i \(-0.350890\pi\)
0.451499 + 0.892272i \(0.350890\pi\)
\(152\) 0 0
\(153\) 3.96693 0.320707
\(154\) 0 0
\(155\) 28.8417 2.31662
\(156\) 0 0
\(157\) −6.54107 −0.522034 −0.261017 0.965334i \(-0.584058\pi\)
−0.261017 + 0.965334i \(0.584058\pi\)
\(158\) 0 0
\(159\) −12.6633 −1.00427
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 3.51504 0.275319 0.137659 0.990480i \(-0.456042\pi\)
0.137659 + 0.990480i \(0.456042\pi\)
\(164\) 0 0
\(165\) 4.66332 0.363039
\(166\) 0 0
\(167\) −2.27758 −0.176244 −0.0881221 0.996110i \(-0.528087\pi\)
−0.0881221 + 0.996110i \(0.528087\pi\)
\(168\) 0 0
\(169\) 9.05611 0.696623
\(170\) 0 0
\(171\) −5.42586 −0.414926
\(172\) 0 0
\(173\) −9.35971 −0.711606 −0.355803 0.934561i \(-0.615793\pi\)
−0.355803 + 0.934561i \(0.615793\pi\)
\(174\) 0 0
\(175\) −5.69639 −0.430607
\(176\) 0 0
\(177\) 11.5150 0.865523
\(178\) 0 0
\(179\) −4.66332 −0.348553 −0.174276 0.984697i \(-0.555759\pi\)
−0.174276 + 0.984697i \(0.555759\pi\)
\(180\) 0 0
\(181\) −17.3738 −1.29138 −0.645692 0.763598i \(-0.723431\pi\)
−0.645692 + 0.763598i \(0.723431\pi\)
\(182\) 0 0
\(183\) −13.2375 −0.978541
\(184\) 0 0
\(185\) −15.7595 −1.15866
\(186\) 0 0
\(187\) −5.65628 −0.413628
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −11.6894 −0.845812 −0.422906 0.906174i \(-0.638990\pi\)
−0.422906 + 0.906174i \(0.638990\pi\)
\(192\) 0 0
\(193\) 4.60721 0.331635 0.165817 0.986156i \(-0.446974\pi\)
0.165817 + 0.986156i \(0.446974\pi\)
\(194\) 0 0
\(195\) −15.3597 −1.09993
\(196\) 0 0
\(197\) −24.8747 −1.77225 −0.886126 0.463444i \(-0.846614\pi\)
−0.886126 + 0.463444i \(0.846614\pi\)
\(198\) 0 0
\(199\) 7.44889 0.528038 0.264019 0.964517i \(-0.414952\pi\)
0.264019 + 0.964517i \(0.414952\pi\)
\(200\) 0 0
\(201\) 5.51504 0.389001
\(202\) 0 0
\(203\) −3.96693 −0.278424
\(204\) 0 0
\(205\) −8.41882 −0.587996
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 7.73651 0.535145
\(210\) 0 0
\(211\) 11.9810 0.824807 0.412403 0.911001i \(-0.364689\pi\)
0.412403 + 0.911001i \(0.364689\pi\)
\(212\) 0 0
\(213\) −2.69639 −0.184754
\(214\) 0 0
\(215\) 37.7685 2.57579
\(216\) 0 0
\(217\) 8.81864 0.598649
\(218\) 0 0
\(219\) −9.35971 −0.632471
\(220\) 0 0
\(221\) 18.6302 1.25321
\(222\) 0 0
\(223\) −10.1884 −0.682266 −0.341133 0.940015i \(-0.610811\pi\)
−0.341133 + 0.940015i \(0.610811\pi\)
\(224\) 0 0
\(225\) 5.69639 0.379760
\(226\) 0 0
\(227\) −20.0561 −1.33117 −0.665585 0.746322i \(-0.731818\pi\)
−0.665585 + 0.746322i \(0.731818\pi\)
\(228\) 0 0
\(229\) 9.28462 0.613545 0.306772 0.951783i \(-0.400751\pi\)
0.306772 + 0.951783i \(0.400751\pi\)
\(230\) 0 0
\(231\) 1.42586 0.0938146
\(232\) 0 0
\(233\) 5.20439 0.340951 0.170475 0.985362i \(-0.445470\pi\)
0.170475 + 0.985362i \(0.445470\pi\)
\(234\) 0 0
\(235\) −17.6373 −1.15053
\(236\) 0 0
\(237\) −4.57414 −0.297122
\(238\) 0 0
\(239\) −23.1382 −1.49669 −0.748344 0.663311i \(-0.769151\pi\)
−0.748344 + 0.663311i \(0.769151\pi\)
\(240\) 0 0
\(241\) 1.11521 0.0718369 0.0359185 0.999355i \(-0.488564\pi\)
0.0359185 + 0.999355i \(0.488564\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.27053 −0.208947
\(246\) 0 0
\(247\) −25.4820 −1.62138
\(248\) 0 0
\(249\) −0.277576 −0.0175907
\(250\) 0 0
\(251\) 23.8818 1.50741 0.753703 0.657216i \(-0.228266\pi\)
0.753703 + 0.657216i \(0.228266\pi\)
\(252\) 0 0
\(253\) −1.42586 −0.0896430
\(254\) 0 0
\(255\) −12.9740 −0.812461
\(256\) 0 0
\(257\) −22.1122 −1.37932 −0.689661 0.724132i \(-0.742240\pi\)
−0.689661 + 0.724132i \(0.742240\pi\)
\(258\) 0 0
\(259\) −4.81864 −0.299416
\(260\) 0 0
\(261\) 3.96693 0.245547
\(262\) 0 0
\(263\) 16.1973 0.998771 0.499386 0.866380i \(-0.333559\pi\)
0.499386 + 0.866380i \(0.333559\pi\)
\(264\) 0 0
\(265\) 41.4158 2.54415
\(266\) 0 0
\(267\) 11.4820 0.702685
\(268\) 0 0
\(269\) 20.3196 1.23891 0.619454 0.785033i \(-0.287354\pi\)
0.619454 + 0.785033i \(0.287354\pi\)
\(270\) 0 0
\(271\) −16.5080 −1.00279 −0.501395 0.865219i \(-0.667180\pi\)
−0.501395 + 0.865219i \(0.667180\pi\)
\(272\) 0 0
\(273\) −4.69639 −0.284239
\(274\) 0 0
\(275\) −8.12225 −0.489790
\(276\) 0 0
\(277\) −3.30361 −0.198495 −0.0992473 0.995063i \(-0.531643\pi\)
−0.0992473 + 0.995063i \(0.531643\pi\)
\(278\) 0 0
\(279\) −8.81864 −0.527958
\(280\) 0 0
\(281\) 32.4749 1.93729 0.968646 0.248446i \(-0.0799199\pi\)
0.968646 + 0.248446i \(0.0799199\pi\)
\(282\) 0 0
\(283\) 6.66332 0.396093 0.198047 0.980193i \(-0.436540\pi\)
0.198047 + 0.980193i \(0.436540\pi\)
\(284\) 0 0
\(285\) 17.7455 1.05115
\(286\) 0 0
\(287\) −2.57414 −0.151947
\(288\) 0 0
\(289\) −1.26349 −0.0743230
\(290\) 0 0
\(291\) 7.96693 0.467030
\(292\) 0 0
\(293\) −1.08214 −0.0632191 −0.0316095 0.999500i \(-0.510063\pi\)
−0.0316095 + 0.999500i \(0.510063\pi\)
\(294\) 0 0
\(295\) −37.6603 −2.19267
\(296\) 0 0
\(297\) −1.42586 −0.0827367
\(298\) 0 0
\(299\) 4.69639 0.271599
\(300\) 0 0
\(301\) 11.5481 0.665622
\(302\) 0 0
\(303\) 6.97397 0.400644
\(304\) 0 0
\(305\) 43.2936 2.47898
\(306\) 0 0
\(307\) −8.88479 −0.507082 −0.253541 0.967325i \(-0.581595\pi\)
−0.253541 + 0.967325i \(0.581595\pi\)
\(308\) 0 0
\(309\) −19.8677 −1.13023
\(310\) 0 0
\(311\) 15.2515 0.864836 0.432418 0.901673i \(-0.357661\pi\)
0.432418 + 0.901673i \(0.357661\pi\)
\(312\) 0 0
\(313\) 33.3126 1.88294 0.941468 0.337101i \(-0.109446\pi\)
0.941468 + 0.337101i \(0.109446\pi\)
\(314\) 0 0
\(315\) 3.27053 0.184274
\(316\) 0 0
\(317\) −17.2044 −0.966295 −0.483147 0.875539i \(-0.660506\pi\)
−0.483147 + 0.875539i \(0.660506\pi\)
\(318\) 0 0
\(319\) −5.65628 −0.316691
\(320\) 0 0
\(321\) −10.3998 −0.580462
\(322\) 0 0
\(323\) −21.5240 −1.19763
\(324\) 0 0
\(325\) 26.7525 1.48396
\(326\) 0 0
\(327\) −12.3527 −0.683104
\(328\) 0 0
\(329\) −5.39279 −0.297314
\(330\) 0 0
\(331\) 5.14828 0.282975 0.141488 0.989940i \(-0.454811\pi\)
0.141488 + 0.989940i \(0.454811\pi\)
\(332\) 0 0
\(333\) 4.81864 0.264060
\(334\) 0 0
\(335\) −18.0371 −0.985473
\(336\) 0 0
\(337\) −6.71048 −0.365543 −0.182771 0.983155i \(-0.558507\pi\)
−0.182771 + 0.983155i \(0.558507\pi\)
\(338\) 0 0
\(339\) −2.41882 −0.131372
\(340\) 0 0
\(341\) 12.5741 0.680928
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −3.27053 −0.176080
\(346\) 0 0
\(347\) −6.83273 −0.366800 −0.183400 0.983038i \(-0.558710\pi\)
−0.183400 + 0.983038i \(0.558710\pi\)
\(348\) 0 0
\(349\) 15.7455 0.842835 0.421417 0.906867i \(-0.361533\pi\)
0.421417 + 0.906867i \(0.361533\pi\)
\(350\) 0 0
\(351\) 4.69639 0.250675
\(352\) 0 0
\(353\) 3.78857 0.201645 0.100823 0.994904i \(-0.467853\pi\)
0.100823 + 0.994904i \(0.467853\pi\)
\(354\) 0 0
\(355\) 8.81864 0.468045
\(356\) 0 0
\(357\) −3.96693 −0.209952
\(358\) 0 0
\(359\) 34.8276 1.83813 0.919065 0.394106i \(-0.128946\pi\)
0.919065 + 0.394106i \(0.128946\pi\)
\(360\) 0 0
\(361\) 10.4399 0.549471
\(362\) 0 0
\(363\) −8.96693 −0.470642
\(364\) 0 0
\(365\) 30.6113 1.60227
\(366\) 0 0
\(367\) 0.926811 0.0483792 0.0241896 0.999707i \(-0.492299\pi\)
0.0241896 + 0.999707i \(0.492299\pi\)
\(368\) 0 0
\(369\) 2.57414 0.134004
\(370\) 0 0
\(371\) 12.6633 0.657447
\(372\) 0 0
\(373\) 7.60422 0.393731 0.196866 0.980430i \(-0.436924\pi\)
0.196866 + 0.980430i \(0.436924\pi\)
\(374\) 0 0
\(375\) −2.27758 −0.117614
\(376\) 0 0
\(377\) 18.6302 0.959507
\(378\) 0 0
\(379\) 32.7194 1.68068 0.840342 0.542057i \(-0.182354\pi\)
0.840342 + 0.542057i \(0.182354\pi\)
\(380\) 0 0
\(381\) −16.9268 −0.867187
\(382\) 0 0
\(383\) −1.78857 −0.0913917 −0.0456958 0.998955i \(-0.514551\pi\)
−0.0456958 + 0.998955i \(0.514551\pi\)
\(384\) 0 0
\(385\) −4.66332 −0.237665
\(386\) 0 0
\(387\) −11.5481 −0.587023
\(388\) 0 0
\(389\) −30.4559 −1.54418 −0.772089 0.635515i \(-0.780788\pi\)
−0.772089 + 0.635515i \(0.780788\pi\)
\(390\) 0 0
\(391\) 3.96693 0.200616
\(392\) 0 0
\(393\) −19.2936 −0.973232
\(394\) 0 0
\(395\) 14.9599 0.752713
\(396\) 0 0
\(397\) 26.7053 1.34030 0.670151 0.742225i \(-0.266229\pi\)
0.670151 + 0.742225i \(0.266229\pi\)
\(398\) 0 0
\(399\) 5.42586 0.271633
\(400\) 0 0
\(401\) −15.9669 −0.797350 −0.398675 0.917092i \(-0.630530\pi\)
−0.398675 + 0.917092i \(0.630530\pi\)
\(402\) 0 0
\(403\) −41.4158 −2.06307
\(404\) 0 0
\(405\) −3.27053 −0.162514
\(406\) 0 0
\(407\) −6.87071 −0.340568
\(408\) 0 0
\(409\) 0.560057 0.0276930 0.0138465 0.999904i \(-0.495592\pi\)
0.0138465 + 0.999904i \(0.495592\pi\)
\(410\) 0 0
\(411\) 4.27758 0.210997
\(412\) 0 0
\(413\) −11.5150 −0.566618
\(414\) 0 0
\(415\) 0.907823 0.0445633
\(416\) 0 0
\(417\) −2.18840 −0.107166
\(418\) 0 0
\(419\) −16.8937 −0.825313 −0.412657 0.910887i \(-0.635399\pi\)
−0.412657 + 0.910887i \(0.635399\pi\)
\(420\) 0 0
\(421\) 12.0892 0.589191 0.294595 0.955622i \(-0.404815\pi\)
0.294595 + 0.955622i \(0.404815\pi\)
\(422\) 0 0
\(423\) 5.39279 0.262206
\(424\) 0 0
\(425\) 22.5972 1.09612
\(426\) 0 0
\(427\) 13.2375 0.640606
\(428\) 0 0
\(429\) −6.69639 −0.323305
\(430\) 0 0
\(431\) 10.1412 0.488486 0.244243 0.969714i \(-0.421460\pi\)
0.244243 + 0.969714i \(0.421460\pi\)
\(432\) 0 0
\(433\) 40.4749 1.94510 0.972550 0.232693i \(-0.0747536\pi\)
0.972550 + 0.232693i \(0.0747536\pi\)
\(434\) 0 0
\(435\) −12.9740 −0.622054
\(436\) 0 0
\(437\) −5.42586 −0.259554
\(438\) 0 0
\(439\) −6.03307 −0.287943 −0.143971 0.989582i \(-0.545987\pi\)
−0.143971 + 0.989582i \(0.545987\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 0.621299 0.0295188 0.0147594 0.999891i \(-0.495302\pi\)
0.0147594 + 0.999891i \(0.495302\pi\)
\(444\) 0 0
\(445\) −37.5522 −1.78014
\(446\) 0 0
\(447\) −15.6894 −0.742081
\(448\) 0 0
\(449\) 31.7455 1.49816 0.749080 0.662479i \(-0.230496\pi\)
0.749080 + 0.662479i \(0.230496\pi\)
\(450\) 0 0
\(451\) −3.67036 −0.172831
\(452\) 0 0
\(453\) 11.0962 0.521346
\(454\) 0 0
\(455\) 15.3597 0.720074
\(456\) 0 0
\(457\) −41.3166 −1.93271 −0.966354 0.257214i \(-0.917195\pi\)
−0.966354 + 0.257214i \(0.917195\pi\)
\(458\) 0 0
\(459\) 3.96693 0.185160
\(460\) 0 0
\(461\) 13.7595 0.640846 0.320423 0.947275i \(-0.396175\pi\)
0.320423 + 0.947275i \(0.396175\pi\)
\(462\) 0 0
\(463\) 20.1312 0.935576 0.467788 0.883841i \(-0.345051\pi\)
0.467788 + 0.883841i \(0.345051\pi\)
\(464\) 0 0
\(465\) 28.8417 1.33750
\(466\) 0 0
\(467\) −25.2936 −1.17045 −0.585223 0.810872i \(-0.698993\pi\)
−0.585223 + 0.810872i \(0.698993\pi\)
\(468\) 0 0
\(469\) −5.51504 −0.254661
\(470\) 0 0
\(471\) −6.54107 −0.301397
\(472\) 0 0
\(473\) 16.4660 0.757106
\(474\) 0 0
\(475\) −30.9078 −1.41815
\(476\) 0 0
\(477\) −12.6633 −0.579814
\(478\) 0 0
\(479\) 24.6864 1.12795 0.563974 0.825792i \(-0.309272\pi\)
0.563974 + 0.825792i \(0.309272\pi\)
\(480\) 0 0
\(481\) 22.6302 1.03185
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −26.0561 −1.18315
\(486\) 0 0
\(487\) −37.2415 −1.68757 −0.843787 0.536678i \(-0.819679\pi\)
−0.843787 + 0.536678i \(0.819679\pi\)
\(488\) 0 0
\(489\) 3.51504 0.158955
\(490\) 0 0
\(491\) −39.1193 −1.76543 −0.882714 0.469911i \(-0.844286\pi\)
−0.882714 + 0.469911i \(0.844286\pi\)
\(492\) 0 0
\(493\) 15.7365 0.708737
\(494\) 0 0
\(495\) 4.66332 0.209601
\(496\) 0 0
\(497\) 2.69639 0.120950
\(498\) 0 0
\(499\) −19.3367 −0.865629 −0.432814 0.901483i \(-0.642479\pi\)
−0.432814 + 0.901483i \(0.642479\pi\)
\(500\) 0 0
\(501\) −2.27758 −0.101755
\(502\) 0 0
\(503\) 2.95988 0.131975 0.0659874 0.997820i \(-0.478980\pi\)
0.0659874 + 0.997820i \(0.478980\pi\)
\(504\) 0 0
\(505\) −22.8086 −1.01497
\(506\) 0 0
\(507\) 9.05611 0.402196
\(508\) 0 0
\(509\) −22.2445 −0.985970 −0.492985 0.870038i \(-0.664094\pi\)
−0.492985 + 0.870038i \(0.664094\pi\)
\(510\) 0 0
\(511\) 9.35971 0.414049
\(512\) 0 0
\(513\) −5.42586 −0.239558
\(514\) 0 0
\(515\) 64.9780 2.86327
\(516\) 0 0
\(517\) −7.68935 −0.338177
\(518\) 0 0
\(519\) −9.35971 −0.410846
\(520\) 0 0
\(521\) 42.4890 1.86148 0.930739 0.365685i \(-0.119165\pi\)
0.930739 + 0.365685i \(0.119165\pi\)
\(522\) 0 0
\(523\) 37.7354 1.65005 0.825027 0.565093i \(-0.191160\pi\)
0.825027 + 0.565093i \(0.191160\pi\)
\(524\) 0 0
\(525\) −5.69639 −0.248611
\(526\) 0 0
\(527\) −34.9829 −1.52388
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 11.5150 0.499710
\(532\) 0 0
\(533\) 12.0892 0.523640
\(534\) 0 0
\(535\) 34.0130 1.47051
\(536\) 0 0
\(537\) −4.66332 −0.201237
\(538\) 0 0
\(539\) −1.42586 −0.0614161
\(540\) 0 0
\(541\) −23.9479 −1.02960 −0.514801 0.857310i \(-0.672134\pi\)
−0.514801 + 0.857310i \(0.672134\pi\)
\(542\) 0 0
\(543\) −17.3738 −0.745581
\(544\) 0 0
\(545\) 40.3998 1.73054
\(546\) 0 0
\(547\) 3.44889 0.147464 0.0737320 0.997278i \(-0.476509\pi\)
0.0737320 + 0.997278i \(0.476509\pi\)
\(548\) 0 0
\(549\) −13.2375 −0.564961
\(550\) 0 0
\(551\) −21.5240 −0.916953
\(552\) 0 0
\(553\) 4.57414 0.194512
\(554\) 0 0
\(555\) −15.7595 −0.668955
\(556\) 0 0
\(557\) −19.0301 −0.806330 −0.403165 0.915127i \(-0.632090\pi\)
−0.403165 + 0.915127i \(0.632090\pi\)
\(558\) 0 0
\(559\) −54.2345 −2.29387
\(560\) 0 0
\(561\) −5.65628 −0.238808
\(562\) 0 0
\(563\) −3.76444 −0.158652 −0.0793262 0.996849i \(-0.525277\pi\)
−0.0793262 + 0.996849i \(0.525277\pi\)
\(564\) 0 0
\(565\) 7.91082 0.332811
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −33.9339 −1.42258 −0.711291 0.702898i \(-0.751889\pi\)
−0.711291 + 0.702898i \(0.751889\pi\)
\(570\) 0 0
\(571\) 24.0791 1.00768 0.503840 0.863797i \(-0.331920\pi\)
0.503840 + 0.863797i \(0.331920\pi\)
\(572\) 0 0
\(573\) −11.6894 −0.488330
\(574\) 0 0
\(575\) 5.69639 0.237556
\(576\) 0 0
\(577\) 28.1924 1.17367 0.586833 0.809708i \(-0.300374\pi\)
0.586833 + 0.809708i \(0.300374\pi\)
\(578\) 0 0
\(579\) 4.60721 0.191469
\(580\) 0 0
\(581\) 0.277576 0.0115158
\(582\) 0 0
\(583\) 18.0561 0.747807
\(584\) 0 0
\(585\) −15.3597 −0.635046
\(586\) 0 0
\(587\) 24.8888 1.02727 0.513636 0.858008i \(-0.328298\pi\)
0.513636 + 0.858008i \(0.328298\pi\)
\(588\) 0 0
\(589\) 47.8487 1.97157
\(590\) 0 0
\(591\) −24.8747 −1.02321
\(592\) 0 0
\(593\) 19.3928 0.796366 0.398183 0.917306i \(-0.369641\pi\)
0.398183 + 0.917306i \(0.369641\pi\)
\(594\) 0 0
\(595\) 12.9740 0.531881
\(596\) 0 0
\(597\) 7.44889 0.304863
\(598\) 0 0
\(599\) 4.15532 0.169782 0.0848910 0.996390i \(-0.472946\pi\)
0.0848910 + 0.996390i \(0.472946\pi\)
\(600\) 0 0
\(601\) −0.300608 −0.0122621 −0.00613103 0.999981i \(-0.501952\pi\)
−0.00613103 + 0.999981i \(0.501952\pi\)
\(602\) 0 0
\(603\) 5.51504 0.224590
\(604\) 0 0
\(605\) 29.3266 1.19230
\(606\) 0 0
\(607\) −29.3497 −1.19127 −0.595633 0.803257i \(-0.703099\pi\)
−0.595633 + 0.803257i \(0.703099\pi\)
\(608\) 0 0
\(609\) −3.96693 −0.160748
\(610\) 0 0
\(611\) 25.3266 1.02461
\(612\) 0 0
\(613\) −5.72242 −0.231127 −0.115563 0.993300i \(-0.536867\pi\)
−0.115563 + 0.993300i \(0.536867\pi\)
\(614\) 0 0
\(615\) −8.41882 −0.339479
\(616\) 0 0
\(617\) 24.6964 0.994239 0.497120 0.867682i \(-0.334391\pi\)
0.497120 + 0.867682i \(0.334391\pi\)
\(618\) 0 0
\(619\) −40.7425 −1.63758 −0.818789 0.574095i \(-0.805354\pi\)
−0.818789 + 0.574095i \(0.805354\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −11.4820 −0.460015
\(624\) 0 0
\(625\) −21.0331 −0.841323
\(626\) 0 0
\(627\) 7.73651 0.308966
\(628\) 0 0
\(629\) 19.1152 0.762173
\(630\) 0 0
\(631\) −10.5221 −0.418877 −0.209439 0.977822i \(-0.567164\pi\)
−0.209439 + 0.977822i \(0.567164\pi\)
\(632\) 0 0
\(633\) 11.9810 0.476202
\(634\) 0 0
\(635\) 55.3597 2.19688
\(636\) 0 0
\(637\) 4.69639 0.186078
\(638\) 0 0
\(639\) −2.69639 −0.106668
\(640\) 0 0
\(641\) −2.64433 −0.104445 −0.0522224 0.998635i \(-0.516630\pi\)
−0.0522224 + 0.998635i \(0.516630\pi\)
\(642\) 0 0
\(643\) 5.18540 0.204492 0.102246 0.994759i \(-0.467397\pi\)
0.102246 + 0.994759i \(0.467397\pi\)
\(644\) 0 0
\(645\) 37.7685 1.48713
\(646\) 0 0
\(647\) 17.2564 0.678421 0.339211 0.940710i \(-0.389840\pi\)
0.339211 + 0.940710i \(0.389840\pi\)
\(648\) 0 0
\(649\) −16.4188 −0.644495
\(650\) 0 0
\(651\) 8.81864 0.345630
\(652\) 0 0
\(653\) −31.1523 −1.21908 −0.609542 0.792754i \(-0.708647\pi\)
−0.609542 + 0.792754i \(0.708647\pi\)
\(654\) 0 0
\(655\) 63.1003 2.46553
\(656\) 0 0
\(657\) −9.35971 −0.365157
\(658\) 0 0
\(659\) 5.84872 0.227834 0.113917 0.993490i \(-0.463660\pi\)
0.113917 + 0.993490i \(0.463660\pi\)
\(660\) 0 0
\(661\) −7.64219 −0.297247 −0.148623 0.988894i \(-0.547484\pi\)
−0.148623 + 0.988894i \(0.547484\pi\)
\(662\) 0 0
\(663\) 18.6302 0.723539
\(664\) 0 0
\(665\) −17.7455 −0.688139
\(666\) 0 0
\(667\) 3.96693 0.153600
\(668\) 0 0
\(669\) −10.1884 −0.393906
\(670\) 0 0
\(671\) 18.8747 0.728652
\(672\) 0 0
\(673\) 2.26349 0.0872512 0.0436256 0.999048i \(-0.486109\pi\)
0.0436256 + 0.999048i \(0.486109\pi\)
\(674\) 0 0
\(675\) 5.69639 0.219254
\(676\) 0 0
\(677\) 22.8747 0.879148 0.439574 0.898206i \(-0.355129\pi\)
0.439574 + 0.898206i \(0.355129\pi\)
\(678\) 0 0
\(679\) −7.96693 −0.305743
\(680\) 0 0
\(681\) −20.0561 −0.768552
\(682\) 0 0
\(683\) 10.3437 0.395792 0.197896 0.980223i \(-0.436589\pi\)
0.197896 + 0.980223i \(0.436589\pi\)
\(684\) 0 0
\(685\) −13.9900 −0.534529
\(686\) 0 0
\(687\) 9.28462 0.354230
\(688\) 0 0
\(689\) −59.4719 −2.26570
\(690\) 0 0
\(691\) −37.6603 −1.43267 −0.716333 0.697759i \(-0.754181\pi\)
−0.716333 + 0.697759i \(0.754181\pi\)
\(692\) 0 0
\(693\) 1.42586 0.0541639
\(694\) 0 0
\(695\) 7.15723 0.271489
\(696\) 0 0
\(697\) 10.2114 0.386785
\(698\) 0 0
\(699\) 5.20439 0.196848
\(700\) 0 0
\(701\) −26.1223 −0.986624 −0.493312 0.869852i \(-0.664214\pi\)
−0.493312 + 0.869852i \(0.664214\pi\)
\(702\) 0 0
\(703\) −26.1453 −0.986088
\(704\) 0 0
\(705\) −17.6373 −0.664259
\(706\) 0 0
\(707\) −6.97397 −0.262283
\(708\) 0 0
\(709\) 18.1412 0.681309 0.340654 0.940189i \(-0.389351\pi\)
0.340654 + 0.940189i \(0.389351\pi\)
\(710\) 0 0
\(711\) −4.57414 −0.171544
\(712\) 0 0
\(713\) −8.81864 −0.330261
\(714\) 0 0
\(715\) 21.9008 0.819043
\(716\) 0 0
\(717\) −23.1382 −0.864113
\(718\) 0 0
\(719\) −18.4559 −0.688290 −0.344145 0.938916i \(-0.611831\pi\)
−0.344145 + 0.938916i \(0.611831\pi\)
\(720\) 0 0
\(721\) 19.8677 0.739912
\(722\) 0 0
\(723\) 1.11521 0.0414751
\(724\) 0 0
\(725\) 22.5972 0.839238
\(726\) 0 0
\(727\) 17.6563 0.654835 0.327418 0.944880i \(-0.393822\pi\)
0.327418 + 0.944880i \(0.393822\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −45.8105 −1.69436
\(732\) 0 0
\(733\) −36.1122 −1.33383 −0.666917 0.745132i \(-0.732387\pi\)
−0.666917 + 0.745132i \(0.732387\pi\)
\(734\) 0 0
\(735\) −3.27053 −0.120635
\(736\) 0 0
\(737\) −7.86366 −0.289662
\(738\) 0 0
\(739\) −19.9149 −0.732580 −0.366290 0.930501i \(-0.619372\pi\)
−0.366290 + 0.930501i \(0.619372\pi\)
\(740\) 0 0
\(741\) −25.4820 −0.936103
\(742\) 0 0
\(743\) −40.2865 −1.47797 −0.738985 0.673722i \(-0.764694\pi\)
−0.738985 + 0.673722i \(0.764694\pi\)
\(744\) 0 0
\(745\) 51.3126 1.87995
\(746\) 0 0
\(747\) −0.277576 −0.0101560
\(748\) 0 0
\(749\) 10.3998 0.380001
\(750\) 0 0
\(751\) −49.6793 −1.81282 −0.906412 0.422395i \(-0.861190\pi\)
−0.906412 + 0.422395i \(0.861190\pi\)
\(752\) 0 0
\(753\) 23.8818 0.870301
\(754\) 0 0
\(755\) −36.2906 −1.32075
\(756\) 0 0
\(757\) −37.7685 −1.37272 −0.686360 0.727262i \(-0.740792\pi\)
−0.686360 + 0.727262i \(0.740792\pi\)
\(758\) 0 0
\(759\) −1.42586 −0.0517554
\(760\) 0 0
\(761\) 3.14828 0.114125 0.0570626 0.998371i \(-0.481827\pi\)
0.0570626 + 0.998371i \(0.481827\pi\)
\(762\) 0 0
\(763\) 12.3527 0.447197
\(764\) 0 0
\(765\) −12.9740 −0.469075
\(766\) 0 0
\(767\) 54.0791 1.95268
\(768\) 0 0
\(769\) 19.2605 0.694551 0.347276 0.937763i \(-0.387107\pi\)
0.347276 + 0.937763i \(0.387107\pi\)
\(770\) 0 0
\(771\) −22.1122 −0.796352
\(772\) 0 0
\(773\) −13.0631 −0.469849 −0.234924 0.972014i \(-0.575484\pi\)
−0.234924 + 0.972014i \(0.575484\pi\)
\(774\) 0 0
\(775\) −50.2345 −1.80448
\(776\) 0 0
\(777\) −4.81864 −0.172868
\(778\) 0 0
\(779\) −13.9669 −0.500417
\(780\) 0 0
\(781\) 3.84468 0.137573
\(782\) 0 0
\(783\) 3.96693 0.141766
\(784\) 0 0
\(785\) 21.3928 0.763541
\(786\) 0 0
\(787\) −45.7595 −1.63115 −0.815576 0.578650i \(-0.803580\pi\)
−0.815576 + 0.578650i \(0.803580\pi\)
\(788\) 0 0
\(789\) 16.1973 0.576641
\(790\) 0 0
\(791\) 2.41882 0.0860032
\(792\) 0 0
\(793\) −62.1683 −2.20766
\(794\) 0 0
\(795\) 41.4158 1.46887
\(796\) 0 0
\(797\) 10.4749 0.371041 0.185520 0.982640i \(-0.440603\pi\)
0.185520 + 0.982640i \(0.440603\pi\)
\(798\) 0 0
\(799\) 21.3928 0.756822
\(800\) 0 0
\(801\) 11.4820 0.405695
\(802\) 0 0
\(803\) 13.3456 0.470957
\(804\) 0 0
\(805\) 3.27053 0.115271
\(806\) 0 0
\(807\) 20.3196 0.715284
\(808\) 0 0
\(809\) 14.0230 0.493024 0.246512 0.969140i \(-0.420716\pi\)
0.246512 + 0.969140i \(0.420716\pi\)
\(810\) 0 0
\(811\) 30.0791 1.05622 0.528111 0.849176i \(-0.322901\pi\)
0.528111 + 0.849176i \(0.322901\pi\)
\(812\) 0 0
\(813\) −16.5080 −0.578961
\(814\) 0 0
\(815\) −11.4960 −0.402689
\(816\) 0 0
\(817\) 62.6584 2.19214
\(818\) 0 0
\(819\) −4.69639 −0.164105
\(820\) 0 0
\(821\) 44.4088 1.54988 0.774938 0.632037i \(-0.217781\pi\)
0.774938 + 0.632037i \(0.217781\pi\)
\(822\) 0 0
\(823\) −14.8276 −0.516857 −0.258429 0.966030i \(-0.583205\pi\)
−0.258429 + 0.966030i \(0.583205\pi\)
\(824\) 0 0
\(825\) −8.12225 −0.282781
\(826\) 0 0
\(827\) −27.0401 −0.940277 −0.470138 0.882593i \(-0.655796\pi\)
−0.470138 + 0.882593i \(0.655796\pi\)
\(828\) 0 0
\(829\) −11.9669 −0.415629 −0.207814 0.978168i \(-0.566635\pi\)
−0.207814 + 0.978168i \(0.566635\pi\)
\(830\) 0 0
\(831\) −3.30361 −0.114601
\(832\) 0 0
\(833\) 3.96693 0.137446
\(834\) 0 0
\(835\) 7.44889 0.257779
\(836\) 0 0
\(837\) −8.81864 −0.304817
\(838\) 0 0
\(839\) −42.2535 −1.45875 −0.729376 0.684114i \(-0.760189\pi\)
−0.729376 + 0.684114i \(0.760189\pi\)
\(840\) 0 0
\(841\) −13.2635 −0.457362
\(842\) 0 0
\(843\) 32.4749 1.11850
\(844\) 0 0
\(845\) −29.6183 −1.01890
\(846\) 0 0
\(847\) 8.96693 0.308107
\(848\) 0 0
\(849\) 6.66332 0.228685
\(850\) 0 0
\(851\) 4.81864 0.165181
\(852\) 0 0
\(853\) 20.4749 0.701048 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(854\) 0 0
\(855\) 17.7455 0.606882
\(856\) 0 0
\(857\) 17.6894 0.604257 0.302128 0.953267i \(-0.402303\pi\)
0.302128 + 0.953267i \(0.402303\pi\)
\(858\) 0 0
\(859\) −5.01599 −0.171143 −0.0855717 0.996332i \(-0.527272\pi\)
−0.0855717 + 0.996332i \(0.527272\pi\)
\(860\) 0 0
\(861\) −2.57414 −0.0877265
\(862\) 0 0
\(863\) 16.9699 0.577663 0.288831 0.957380i \(-0.406733\pi\)
0.288831 + 0.957380i \(0.406733\pi\)
\(864\) 0 0
\(865\) 30.6113 1.04081
\(866\) 0 0
\(867\) −1.26349 −0.0429104
\(868\) 0 0
\(869\) 6.52208 0.221246
\(870\) 0 0
\(871\) 25.9008 0.877614
\(872\) 0 0
\(873\) 7.96693 0.269640
\(874\) 0 0
\(875\) 2.27758 0.0769961
\(876\) 0 0
\(877\) 0.851718 0.0287605 0.0143802 0.999897i \(-0.495422\pi\)
0.0143802 + 0.999897i \(0.495422\pi\)
\(878\) 0 0
\(879\) −1.08214 −0.0364995
\(880\) 0 0
\(881\) −44.8316 −1.51042 −0.755208 0.655485i \(-0.772464\pi\)
−0.755208 + 0.655485i \(0.772464\pi\)
\(882\) 0 0
\(883\) 51.6462 1.73803 0.869017 0.494782i \(-0.164752\pi\)
0.869017 + 0.494782i \(0.164752\pi\)
\(884\) 0 0
\(885\) −37.6603 −1.26594
\(886\) 0 0
\(887\) 54.9870 1.84628 0.923141 0.384462i \(-0.125613\pi\)
0.923141 + 0.384462i \(0.125613\pi\)
\(888\) 0 0
\(889\) 16.9268 0.567707
\(890\) 0 0
\(891\) −1.42586 −0.0477681
\(892\) 0 0
\(893\) −29.2605 −0.979165
\(894\) 0 0
\(895\) 15.2515 0.509803
\(896\) 0 0
\(897\) 4.69639 0.156808
\(898\) 0 0
\(899\) −34.9829 −1.16675
\(900\) 0 0
\(901\) −50.2345 −1.67355
\(902\) 0 0
\(903\) 11.5481 0.384297
\(904\) 0 0
\(905\) 56.8216 1.88881
\(906\) 0 0
\(907\) −27.6132 −0.916880 −0.458440 0.888725i \(-0.651592\pi\)
−0.458440 + 0.888725i \(0.651592\pi\)
\(908\) 0 0
\(909\) 6.97397 0.231312
\(910\) 0 0
\(911\) 25.3928 0.841301 0.420650 0.907223i \(-0.361802\pi\)
0.420650 + 0.907223i \(0.361802\pi\)
\(912\) 0 0
\(913\) 0.395785 0.0130986
\(914\) 0 0
\(915\) 43.2936 1.43124
\(916\) 0 0
\(917\) 19.2936 0.637130
\(918\) 0 0
\(919\) 52.6202 1.73578 0.867890 0.496756i \(-0.165476\pi\)
0.867890 + 0.496756i \(0.165476\pi\)
\(920\) 0 0
\(921\) −8.88479 −0.292764
\(922\) 0 0
\(923\) −12.6633 −0.416818
\(924\) 0 0
\(925\) 27.4489 0.902514
\(926\) 0 0
\(927\) −19.8677 −0.652541
\(928\) 0 0
\(929\) −39.9900 −1.31203 −0.656014 0.754749i \(-0.727759\pi\)
−0.656014 + 0.754749i \(0.727759\pi\)
\(930\) 0 0
\(931\) −5.42586 −0.177825
\(932\) 0 0
\(933\) 15.2515 0.499313
\(934\) 0 0
\(935\) 18.4990 0.604984
\(936\) 0 0
\(937\) −36.6061 −1.19587 −0.597935 0.801545i \(-0.704012\pi\)
−0.597935 + 0.801545i \(0.704012\pi\)
\(938\) 0 0
\(939\) 33.3126 1.08711
\(940\) 0 0
\(941\) 31.4069 1.02383 0.511917 0.859035i \(-0.328935\pi\)
0.511917 + 0.859035i \(0.328935\pi\)
\(942\) 0 0
\(943\) 2.57414 0.0838256
\(944\) 0 0
\(945\) 3.27053 0.106390
\(946\) 0 0
\(947\) −28.0661 −0.912027 −0.456014 0.889973i \(-0.650723\pi\)
−0.456014 + 0.889973i \(0.650723\pi\)
\(948\) 0 0
\(949\) −43.9569 −1.42690
\(950\) 0 0
\(951\) −17.2044 −0.557890
\(952\) 0 0
\(953\) 51.8105 1.67831 0.839153 0.543895i \(-0.183051\pi\)
0.839153 + 0.543895i \(0.183051\pi\)
\(954\) 0 0
\(955\) 38.2304 1.23711
\(956\) 0 0
\(957\) −5.65628 −0.182841
\(958\) 0 0
\(959\) −4.27758 −0.138130
\(960\) 0 0
\(961\) 46.7685 1.50866
\(962\) 0 0
\(963\) −10.3998 −0.335130
\(964\) 0 0
\(965\) −15.0681 −0.485058
\(966\) 0 0
\(967\) 25.0160 0.804460 0.402230 0.915539i \(-0.368235\pi\)
0.402230 + 0.915539i \(0.368235\pi\)
\(968\) 0 0
\(969\) −21.5240 −0.691450
\(970\) 0 0
\(971\) 38.5782 1.23803 0.619016 0.785378i \(-0.287531\pi\)
0.619016 + 0.785378i \(0.287531\pi\)
\(972\) 0 0
\(973\) 2.18840 0.0701568
\(974\) 0 0
\(975\) 26.7525 0.856766
\(976\) 0 0
\(977\) −10.0230 −0.320665 −0.160333 0.987063i \(-0.551257\pi\)
−0.160333 + 0.987063i \(0.551257\pi\)
\(978\) 0 0
\(979\) −16.3717 −0.523240
\(980\) 0 0
\(981\) −12.3527 −0.394390
\(982\) 0 0
\(983\) 10.3437 0.329914 0.164957 0.986301i \(-0.447252\pi\)
0.164957 + 0.986301i \(0.447252\pi\)
\(984\) 0 0
\(985\) 81.3537 2.59214
\(986\) 0 0
\(987\) −5.39279 −0.171654
\(988\) 0 0
\(989\) −11.5481 −0.367209
\(990\) 0 0
\(991\) 38.6302 1.22713 0.613565 0.789644i \(-0.289735\pi\)
0.613565 + 0.789644i \(0.289735\pi\)
\(992\) 0 0
\(993\) 5.14828 0.163376
\(994\) 0 0
\(995\) −24.3619 −0.772323
\(996\) 0 0
\(997\) 28.0331 0.887816 0.443908 0.896072i \(-0.353592\pi\)
0.443908 + 0.896072i \(0.353592\pi\)
\(998\) 0 0
\(999\) 4.81864 0.152455
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 3864.2.a.o.1.1 3
4.3 odd 2 7728.2.a.bq.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.o.1.1 3 1.1 even 1 trivial
7728.2.a.bq.1.1 3 4.3 odd 2