Properties

Label 8016.2.a.o.1.4
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2777.1
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} + x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1002)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.50848\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.69113 q^{5} +2.24216 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.69113 q^{5} +2.24216 q^{7} +1.00000 q^{9} +5.56799 q^{13} +3.69113 q^{15} -1.01696 q^{17} +2.65167 q^{19} +2.24216 q^{21} -2.65167 q^{23} +8.62441 q^{25} +1.00000 q^{27} -1.38225 q^{29} -4.89383 q^{31} +8.27608 q^{35} -1.97751 q^{37} +5.56799 q^{39} +4.28639 q^{41} -0.551031 q^{43} +3.69113 q^{45} +5.62441 q^{47} -1.97273 q^{49} -1.01696 q^{51} +1.03945 q^{53} +2.65167 q^{57} +2.42790 q^{59} +5.38225 q^{61} +2.24216 q^{63} +20.5522 q^{65} -14.8610 q^{67} -2.65167 q^{69} +8.68560 q^{71} -1.75374 q^{73} +8.62441 q^{75} -3.74754 q^{79} +1.00000 q^{81} +5.43867 q^{83} -3.75374 q^{85} -1.38225 q^{87} -6.27608 q^{89} +12.4843 q^{91} -4.89383 q^{93} +9.78766 q^{95} -11.0067 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 5q^{5} - q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 5q^{5} - q^{7} + 4q^{9} + 8q^{13} + 5q^{15} + 10q^{17} + 2q^{19} - q^{21} - 2q^{23} + 5q^{25} + 4q^{27} + 14q^{29} - q^{31} - 5q^{35} + 5q^{37} + 8q^{39} + 14q^{41} - 2q^{43} + 5q^{45} - 7q^{47} + 13q^{49} + 10q^{51} + 3q^{53} + 2q^{57} + 5q^{59} + 2q^{61} - q^{63} + 6q^{65} + 7q^{67} - 2q^{69} - 2q^{71} + 2q^{73} + 5q^{75} + 10q^{79} + 4q^{81} - 13q^{83} - 6q^{85} + 14q^{87} + 13q^{89} + 30q^{91} - q^{93} + 2q^{95} + 5q^{97} + O(q^{100}) \)

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
\(5\) 3.69113 1.65072 0.825361 0.564606i \(-0.190972\pi\)
0.825361 + 0.564606i \(0.190972\pi\)
\(6\) 0 0
\(7\) 2.24216 0.847456 0.423728 0.905790i \(-0.360721\pi\)
0.423728 + 0.905790i \(0.360721\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 5.56799 1.54428 0.772142 0.635450i \(-0.219186\pi\)
0.772142 + 0.635450i \(0.219186\pi\)
\(14\) 0 0
\(15\) 3.69113 0.953045
\(16\) 0 0
\(17\) −1.01696 −0.246650 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(18\) 0 0
\(19\) 2.65167 0.608336 0.304168 0.952618i \(-0.401622\pi\)
0.304168 + 0.952618i \(0.401622\pi\)
\(20\) 0 0
\(21\) 2.24216 0.489279
\(22\) 0 0
\(23\) −2.65167 −0.552912 −0.276456 0.961027i \(-0.589160\pi\)
−0.276456 + 0.961027i \(0.589160\pi\)
\(24\) 0 0
\(25\) 8.62441 1.72488
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.38225 −0.256678 −0.128339 0.991730i \(-0.540964\pi\)
−0.128339 + 0.991730i \(0.540964\pi\)
\(30\) 0 0
\(31\) −4.89383 −0.878958 −0.439479 0.898253i \(-0.644837\pi\)
−0.439479 + 0.898253i \(0.644837\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.27608 1.39891
\(36\) 0 0
\(37\) −1.97751 −0.325101 −0.162550 0.986700i \(-0.551972\pi\)
−0.162550 + 0.986700i \(0.551972\pi\)
\(38\) 0 0
\(39\) 5.56799 0.891593
\(40\) 0 0
\(41\) 4.28639 0.669421 0.334710 0.942321i \(-0.391361\pi\)
0.334710 + 0.942321i \(0.391361\pi\)
\(42\) 0 0
\(43\) −0.551031 −0.0840314 −0.0420157 0.999117i \(-0.513378\pi\)
−0.0420157 + 0.999117i \(0.513378\pi\)
\(44\) 0 0
\(45\) 3.69113 0.550241
\(46\) 0 0
\(47\) 5.62441 0.820404 0.410202 0.911995i \(-0.365458\pi\)
0.410202 + 0.911995i \(0.365458\pi\)
\(48\) 0 0
\(49\) −1.97273 −0.281819
\(50\) 0 0
\(51\) −1.01696 −0.142403
\(52\) 0 0
\(53\) 1.03945 0.142780 0.0713898 0.997448i \(-0.477257\pi\)
0.0713898 + 0.997448i \(0.477257\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.65167 0.351223
\(58\) 0 0
\(59\) 2.42790 0.316086 0.158043 0.987432i \(-0.449482\pi\)
0.158043 + 0.987432i \(0.449482\pi\)
\(60\) 0 0
\(61\) 5.38225 0.689127 0.344563 0.938763i \(-0.388027\pi\)
0.344563 + 0.938763i \(0.388027\pi\)
\(62\) 0 0
\(63\) 2.24216 0.282485
\(64\) 0 0
\(65\) 20.5522 2.54918
\(66\) 0 0
\(67\) −14.8610 −1.81556 −0.907782 0.419442i \(-0.862226\pi\)
−0.907782 + 0.419442i \(0.862226\pi\)
\(68\) 0 0
\(69\) −2.65167 −0.319224
\(70\) 0 0
\(71\) 8.68560 1.03079 0.515395 0.856952i \(-0.327645\pi\)
0.515395 + 0.856952i \(0.327645\pi\)
\(72\) 0 0
\(73\) −1.75374 −0.205259 −0.102630 0.994720i \(-0.532726\pi\)
−0.102630 + 0.994720i \(0.532726\pi\)
\(74\) 0 0
\(75\) 8.62441 0.995861
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.74754 −0.421631 −0.210816 0.977526i \(-0.567612\pi\)
−0.210816 + 0.977526i \(0.567612\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.43867 0.596971 0.298486 0.954414i \(-0.403519\pi\)
0.298486 + 0.954414i \(0.403519\pi\)
\(84\) 0 0
\(85\) −3.75374 −0.407150
\(86\) 0 0
\(87\) −1.38225 −0.148193
\(88\) 0 0
\(89\) −6.27608 −0.665263 −0.332632 0.943057i \(-0.607937\pi\)
−0.332632 + 0.943057i \(0.607937\pi\)
\(90\) 0 0
\(91\) 12.4843 1.30871
\(92\) 0 0
\(93\) −4.89383 −0.507467
\(94\) 0 0
\(95\) 9.78766 1.00419
\(96\) 0 0
\(97\) −11.0067 −1.11756 −0.558778 0.829317i \(-0.688730\pi\)
−0.558778 + 0.829317i \(0.688730\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.3114 1.52354 0.761772 0.647845i \(-0.224330\pi\)
0.761772 + 0.647845i \(0.224330\pi\)
\(102\) 0 0
\(103\) 13.2972 1.31021 0.655104 0.755539i \(-0.272625\pi\)
0.655104 + 0.755539i \(0.272625\pi\)
\(104\) 0 0
\(105\) 8.27608 0.807663
\(106\) 0 0
\(107\) −18.3848 −1.77733 −0.888663 0.458561i \(-0.848365\pi\)
−0.888663 + 0.458561i \(0.848365\pi\)
\(108\) 0 0
\(109\) −18.1651 −1.73990 −0.869952 0.493136i \(-0.835850\pi\)
−0.869952 + 0.493136i \(0.835850\pi\)
\(110\) 0 0
\(111\) −1.97751 −0.187697
\(112\) 0 0
\(113\) 14.4331 1.35776 0.678878 0.734251i \(-0.262467\pi\)
0.678878 + 0.734251i \(0.262467\pi\)
\(114\) 0 0
\(115\) −9.78766 −0.912704
\(116\) 0 0
\(117\) 5.56799 0.514761
\(118\) 0 0
\(119\) −2.28019 −0.209025
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 4.28639 0.386490
\(124\) 0 0
\(125\) 13.3781 1.19658
\(126\) 0 0
\(127\) 14.4434 1.28165 0.640824 0.767688i \(-0.278593\pi\)
0.640824 + 0.767688i \(0.278593\pi\)
\(128\) 0 0
\(129\) −0.551031 −0.0485156
\(130\) 0 0
\(131\) −11.4346 −0.999042 −0.499521 0.866302i \(-0.666491\pi\)
−0.499521 + 0.866302i \(0.666491\pi\)
\(132\) 0 0
\(133\) 5.94547 0.515537
\(134\) 0 0
\(135\) 3.69113 0.317682
\(136\) 0 0
\(137\) −17.6134 −1.50481 −0.752405 0.658701i \(-0.771106\pi\)
−0.752405 + 0.658701i \(0.771106\pi\)
\(138\) 0 0
\(139\) 14.9060 1.26431 0.632156 0.774841i \(-0.282170\pi\)
0.632156 + 0.774841i \(0.282170\pi\)
\(140\) 0 0
\(141\) 5.62441 0.473661
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −5.10206 −0.423703
\(146\) 0 0
\(147\) −1.97273 −0.162708
\(148\) 0 0
\(149\) 0.308874 0.0253040 0.0126520 0.999920i \(-0.495973\pi\)
0.0126520 + 0.999920i \(0.495973\pi\)
\(150\) 0 0
\(151\) 4.61155 0.375283 0.187641 0.982238i \(-0.439916\pi\)
0.187641 + 0.982238i \(0.439916\pi\)
\(152\) 0 0
\(153\) −1.01696 −0.0822165
\(154\) 0 0
\(155\) −18.0637 −1.45091
\(156\) 0 0
\(157\) −7.30335 −0.582871 −0.291435 0.956591i \(-0.594133\pi\)
−0.291435 + 0.956591i \(0.594133\pi\)
\(158\) 0 0
\(159\) 1.03945 0.0824339
\(160\) 0 0
\(161\) −5.94547 −0.468569
\(162\) 0 0
\(163\) 1.73610 0.135982 0.0679911 0.997686i \(-0.478341\pi\)
0.0679911 + 0.997686i \(0.478341\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 18.0026 1.38481
\(170\) 0 0
\(171\) 2.65167 0.202779
\(172\) 0 0
\(173\) −5.38225 −0.409205 −0.204602 0.978845i \(-0.565590\pi\)
−0.204602 + 0.978845i \(0.565590\pi\)
\(174\) 0 0
\(175\) 19.3373 1.46176
\(176\) 0 0
\(177\) 2.42790 0.182492
\(178\) 0 0
\(179\) −1.30335 −0.0974168 −0.0487084 0.998813i \(-0.515510\pi\)
−0.0487084 + 0.998813i \(0.515510\pi\)
\(180\) 0 0
\(181\) −16.5632 −1.23113 −0.615567 0.788084i \(-0.711073\pi\)
−0.615567 + 0.788084i \(0.711073\pi\)
\(182\) 0 0
\(183\) 5.38225 0.397867
\(184\) 0 0
\(185\) −7.29924 −0.536651
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.24216 0.163093
\(190\) 0 0
\(191\) 21.3332 1.54361 0.771807 0.635857i \(-0.219353\pi\)
0.771807 + 0.635857i \(0.219353\pi\)
\(192\) 0 0
\(193\) −13.1360 −0.945549 −0.472775 0.881183i \(-0.656747\pi\)
−0.472775 + 0.881183i \(0.656747\pi\)
\(194\) 0 0
\(195\) 20.5522 1.47177
\(196\) 0 0
\(197\) −6.55103 −0.466742 −0.233371 0.972388i \(-0.574976\pi\)
−0.233371 + 0.972388i \(0.574976\pi\)
\(198\) 0 0
\(199\) −7.13599 −0.505857 −0.252928 0.967485i \(-0.581394\pi\)
−0.252928 + 0.967485i \(0.581394\pi\)
\(200\) 0 0
\(201\) −14.8610 −1.04822
\(202\) 0 0
\(203\) −3.09922 −0.217523
\(204\) 0 0
\(205\) 15.8216 1.10503
\(206\) 0 0
\(207\) −2.65167 −0.184304
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 18.6517 1.28403 0.642017 0.766690i \(-0.278098\pi\)
0.642017 + 0.766690i \(0.278098\pi\)
\(212\) 0 0
\(213\) 8.68560 0.595127
\(214\) 0 0
\(215\) −2.03392 −0.138713
\(216\) 0 0
\(217\) −10.9727 −0.744878
\(218\) 0 0
\(219\) −1.75374 −0.118506
\(220\) 0 0
\(221\) −5.66244 −0.380897
\(222\) 0 0
\(223\) −2.24216 −0.150146 −0.0750730 0.997178i \(-0.523919\pi\)
−0.0750730 + 0.997178i \(0.523919\pi\)
\(224\) 0 0
\(225\) 8.62441 0.574961
\(226\) 0 0
\(227\) 18.3829 1.22012 0.610059 0.792356i \(-0.291146\pi\)
0.610059 + 0.792356i \(0.291146\pi\)
\(228\) 0 0
\(229\) −11.2149 −0.741101 −0.370550 0.928812i \(-0.620831\pi\)
−0.370550 + 0.928812i \(0.620831\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −17.5249 −1.14809 −0.574047 0.818822i \(-0.694627\pi\)
−0.574047 + 0.818822i \(0.694627\pi\)
\(234\) 0 0
\(235\) 20.7604 1.35426
\(236\) 0 0
\(237\) −3.74754 −0.243429
\(238\) 0 0
\(239\) 7.13599 0.461589 0.230794 0.973003i \(-0.425868\pi\)
0.230794 + 0.973003i \(0.425868\pi\)
\(240\) 0 0
\(241\) −5.86656 −0.377899 −0.188949 0.981987i \(-0.560508\pi\)
−0.188949 + 0.981987i \(0.560508\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −7.28161 −0.465205
\(246\) 0 0
\(247\) 14.7645 0.939443
\(248\) 0 0
\(249\) 5.43867 0.344661
\(250\) 0 0
\(251\) −17.0815 −1.07817 −0.539086 0.842251i \(-0.681230\pi\)
−0.539086 + 0.842251i \(0.681230\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −3.75374 −0.235068
\(256\) 0 0
\(257\) 11.0959 0.692141 0.346071 0.938208i \(-0.387516\pi\)
0.346071 + 0.938208i \(0.387516\pi\)
\(258\) 0 0
\(259\) −4.43389 −0.275508
\(260\) 0 0
\(261\) −1.38225 −0.0855592
\(262\) 0 0
\(263\) −23.7836 −1.46656 −0.733278 0.679929i \(-0.762011\pi\)
−0.733278 + 0.679929i \(0.762011\pi\)
\(264\) 0 0
\(265\) 3.83675 0.235689
\(266\) 0 0
\(267\) −6.27608 −0.384090
\(268\) 0 0
\(269\) −24.8404 −1.51455 −0.757274 0.653097i \(-0.773469\pi\)
−0.757274 + 0.653097i \(0.773469\pi\)
\(270\) 0 0
\(271\) 16.6918 1.01395 0.506977 0.861959i \(-0.330763\pi\)
0.506977 + 0.861959i \(0.330763\pi\)
\(272\) 0 0
\(273\) 12.4843 0.755585
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 30.8888 1.85593 0.927963 0.372672i \(-0.121558\pi\)
0.927963 + 0.372672i \(0.121558\pi\)
\(278\) 0 0
\(279\) −4.89383 −0.292986
\(280\) 0 0
\(281\) 25.0406 1.49380 0.746898 0.664939i \(-0.231542\pi\)
0.746898 + 0.664939i \(0.231542\pi\)
\(282\) 0 0
\(283\) −27.6964 −1.64638 −0.823189 0.567767i \(-0.807807\pi\)
−0.823189 + 0.567767i \(0.807807\pi\)
\(284\) 0 0
\(285\) 9.78766 0.579771
\(286\) 0 0
\(287\) 9.61075 0.567304
\(288\) 0 0
\(289\) −15.9658 −0.939164
\(290\) 0 0
\(291\) −11.0067 −0.645222
\(292\) 0 0
\(293\) 21.3823 1.24916 0.624582 0.780959i \(-0.285269\pi\)
0.624582 + 0.780959i \(0.285269\pi\)
\(294\) 0 0
\(295\) 8.96168 0.521769
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −14.7645 −0.853853
\(300\) 0 0
\(301\) −1.23550 −0.0712129
\(302\) 0 0
\(303\) 15.3114 0.879618
\(304\) 0 0
\(305\) 19.8666 1.13756
\(306\) 0 0
\(307\) −5.35640 −0.305706 −0.152853 0.988249i \(-0.548846\pi\)
−0.152853 + 0.988249i \(0.548846\pi\)
\(308\) 0 0
\(309\) 13.2972 0.756449
\(310\) 0 0
\(311\) −11.0980 −0.629307 −0.314654 0.949207i \(-0.601888\pi\)
−0.314654 + 0.949207i \(0.601888\pi\)
\(312\) 0 0
\(313\) −20.4527 −1.15605 −0.578026 0.816018i \(-0.696177\pi\)
−0.578026 + 0.816018i \(0.696177\pi\)
\(314\) 0 0
\(315\) 8.27608 0.466304
\(316\) 0 0
\(317\) 0.889723 0.0499718 0.0249859 0.999688i \(-0.492046\pi\)
0.0249859 + 0.999688i \(0.492046\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −18.3848 −1.02614
\(322\) 0 0
\(323\) −2.69665 −0.150046
\(324\) 0 0
\(325\) 48.0206 2.66371
\(326\) 0 0
\(327\) −18.1651 −1.00453
\(328\) 0 0
\(329\) 12.6108 0.695256
\(330\) 0 0
\(331\) −7.31553 −0.402098 −0.201049 0.979581i \(-0.564435\pi\)
−0.201049 + 0.979581i \(0.564435\pi\)
\(332\) 0 0
\(333\) −1.97751 −0.108367
\(334\) 0 0
\(335\) −54.8540 −2.99699
\(336\) 0 0
\(337\) 6.11827 0.333284 0.166642 0.986017i \(-0.446708\pi\)
0.166642 + 0.986017i \(0.446708\pi\)
\(338\) 0 0
\(339\) 14.4331 0.783900
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.1183 −1.08628
\(344\) 0 0
\(345\) −9.78766 −0.526950
\(346\) 0 0
\(347\) −17.6400 −0.946962 −0.473481 0.880804i \(-0.657003\pi\)
−0.473481 + 0.880804i \(0.657003\pi\)
\(348\) 0 0
\(349\) 12.7420 0.682064 0.341032 0.940052i \(-0.389223\pi\)
0.341032 + 0.940052i \(0.389223\pi\)
\(350\) 0 0
\(351\) 5.56799 0.297198
\(352\) 0 0
\(353\) 33.0406 1.75857 0.879286 0.476293i \(-0.158020\pi\)
0.879286 + 0.476293i \(0.158020\pi\)
\(354\) 0 0
\(355\) 32.0596 1.70155
\(356\) 0 0
\(357\) −2.28019 −0.120680
\(358\) 0 0
\(359\) 10.4054 0.549177 0.274588 0.961562i \(-0.411458\pi\)
0.274588 + 0.961562i \(0.411458\pi\)
\(360\) 0 0
\(361\) −11.9686 −0.629928
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −6.47326 −0.338826
\(366\) 0 0
\(367\) −6.89794 −0.360069 −0.180035 0.983660i \(-0.557621\pi\)
−0.180035 + 0.983660i \(0.557621\pi\)
\(368\) 0 0
\(369\) 4.28639 0.223140
\(370\) 0 0
\(371\) 2.33061 0.120999
\(372\) 0 0
\(373\) 9.15026 0.473783 0.236891 0.971536i \(-0.423871\pi\)
0.236891 + 0.971536i \(0.423871\pi\)
\(374\) 0 0
\(375\) 13.3781 0.690844
\(376\) 0 0
\(377\) −7.69636 −0.396383
\(378\) 0 0
\(379\) −0.229971 −0.0118128 −0.00590641 0.999983i \(-0.501880\pi\)
−0.00590641 + 0.999983i \(0.501880\pi\)
\(380\) 0 0
\(381\) 14.4434 0.739960
\(382\) 0 0
\(383\) −21.6992 −1.10878 −0.554389 0.832258i \(-0.687048\pi\)
−0.554389 + 0.832258i \(0.687048\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.551031 −0.0280105
\(388\) 0 0
\(389\) −15.1917 −0.770247 −0.385124 0.922865i \(-0.625841\pi\)
−0.385124 + 0.922865i \(0.625841\pi\)
\(390\) 0 0
\(391\) 2.69665 0.136376
\(392\) 0 0
\(393\) −11.4346 −0.576797
\(394\) 0 0
\(395\) −13.8326 −0.695996
\(396\) 0 0
\(397\) 8.45039 0.424113 0.212056 0.977257i \(-0.431984\pi\)
0.212056 + 0.977257i \(0.431984\pi\)
\(398\) 0 0
\(399\) 5.94547 0.297646
\(400\) 0 0
\(401\) −2.56657 −0.128169 −0.0640843 0.997944i \(-0.520413\pi\)
−0.0640843 + 0.997944i \(0.520413\pi\)
\(402\) 0 0
\(403\) −27.2488 −1.35736
\(404\) 0 0
\(405\) 3.69113 0.183414
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 18.6650 0.922924 0.461462 0.887160i \(-0.347325\pi\)
0.461462 + 0.887160i \(0.347325\pi\)
\(410\) 0 0
\(411\) −17.6134 −0.868803
\(412\) 0 0
\(413\) 5.44373 0.267868
\(414\) 0 0
\(415\) 20.0748 0.985433
\(416\) 0 0
\(417\) 14.9060 0.729951
\(418\) 0 0
\(419\) 1.83264 0.0895303 0.0447651 0.998998i \(-0.485746\pi\)
0.0447651 + 0.998998i \(0.485746\pi\)
\(420\) 0 0
\(421\) −38.2174 −1.86260 −0.931302 0.364248i \(-0.881326\pi\)
−0.931302 + 0.364248i \(0.881326\pi\)
\(422\) 0 0
\(423\) 5.62441 0.273468
\(424\) 0 0
\(425\) −8.77070 −0.425441
\(426\) 0 0
\(427\) 12.0678 0.584004
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.31029 0.0631146 0.0315573 0.999502i \(-0.489953\pi\)
0.0315573 + 0.999502i \(0.489953\pi\)
\(432\) 0 0
\(433\) 23.0434 1.10740 0.553698 0.832717i \(-0.313216\pi\)
0.553698 + 0.832717i \(0.313216\pi\)
\(434\) 0 0
\(435\) −5.10206 −0.244625
\(436\) 0 0
\(437\) −7.03137 −0.336356
\(438\) 0 0
\(439\) −38.4154 −1.83347 −0.916733 0.399501i \(-0.869183\pi\)
−0.916733 + 0.399501i \(0.869183\pi\)
\(440\) 0 0
\(441\) −1.97273 −0.0939397
\(442\) 0 0
\(443\) 36.8127 1.74902 0.874512 0.485004i \(-0.161182\pi\)
0.874512 + 0.485004i \(0.161182\pi\)
\(444\) 0 0
\(445\) −23.1658 −1.09816
\(446\) 0 0
\(447\) 0.308874 0.0146093
\(448\) 0 0
\(449\) 19.8257 0.935632 0.467816 0.883826i \(-0.345041\pi\)
0.467816 + 0.883826i \(0.345041\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.61155 0.216670
\(454\) 0 0
\(455\) 46.0812 2.16032
\(456\) 0 0
\(457\) −3.25987 −0.152490 −0.0762451 0.997089i \(-0.524293\pi\)
−0.0762451 + 0.997089i \(0.524293\pi\)
\(458\) 0 0
\(459\) −1.01696 −0.0474677
\(460\) 0 0
\(461\) −6.97684 −0.324944 −0.162472 0.986713i \(-0.551947\pi\)
−0.162472 + 0.986713i \(0.551947\pi\)
\(462\) 0 0
\(463\) −37.8943 −1.76110 −0.880549 0.473956i \(-0.842826\pi\)
−0.880549 + 0.473956i \(0.842826\pi\)
\(464\) 0 0
\(465\) −18.0637 −0.837686
\(466\) 0 0
\(467\) −4.32651 −0.200207 −0.100103 0.994977i \(-0.531917\pi\)
−0.100103 + 0.994977i \(0.531917\pi\)
\(468\) 0 0
\(469\) −33.3208 −1.53861
\(470\) 0 0
\(471\) −7.30335 −0.336520
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 22.8691 1.04931
\(476\) 0 0
\(477\) 1.03945 0.0475932
\(478\) 0 0
\(479\) 41.4323 1.89309 0.946546 0.322569i \(-0.104547\pi\)
0.946546 + 0.322569i \(0.104547\pi\)
\(480\) 0 0
\(481\) −11.0108 −0.502048
\(482\) 0 0
\(483\) −5.94547 −0.270528
\(484\) 0 0
\(485\) −40.6270 −1.84478
\(486\) 0 0
\(487\) 16.2319 0.735535 0.367768 0.929918i \(-0.380122\pi\)
0.367768 + 0.929918i \(0.380122\pi\)
\(488\) 0 0
\(489\) 1.73610 0.0785093
\(490\) 0 0
\(491\) −17.6868 −0.798195 −0.399097 0.916909i \(-0.630676\pi\)
−0.399097 + 0.916909i \(0.630676\pi\)
\(492\) 0 0
\(493\) 1.40570 0.0633094
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 19.4745 0.873549
\(498\) 0 0
\(499\) 4.88575 0.218716 0.109358 0.994002i \(-0.465120\pi\)
0.109358 + 0.994002i \(0.465120\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −36.4637 −1.62584 −0.812918 0.582378i \(-0.802122\pi\)
−0.812918 + 0.582378i \(0.802122\pi\)
\(504\) 0 0
\(505\) 56.5164 2.51495
\(506\) 0 0
\(507\) 18.0026 0.799521
\(508\) 0 0
\(509\) −7.71981 −0.342175 −0.171087 0.985256i \(-0.554728\pi\)
−0.171087 + 0.985256i \(0.554728\pi\)
\(510\) 0 0
\(511\) −3.93215 −0.173948
\(512\) 0 0
\(513\) 2.65167 0.117074
\(514\) 0 0
\(515\) 49.0815 2.16279
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −5.38225 −0.236255
\(520\) 0 0
\(521\) 32.2658 1.41359 0.706795 0.707419i \(-0.250140\pi\)
0.706795 + 0.707419i \(0.250140\pi\)
\(522\) 0 0
\(523\) −18.9782 −0.829858 −0.414929 0.909854i \(-0.636194\pi\)
−0.414929 + 0.909854i \(0.636194\pi\)
\(524\) 0 0
\(525\) 19.3373 0.843948
\(526\) 0 0
\(527\) 4.97684 0.216795
\(528\) 0 0
\(529\) −15.9686 −0.694288
\(530\) 0 0
\(531\) 2.42790 0.105362
\(532\) 0 0
\(533\) 23.8666 1.03378
\(534\) 0 0
\(535\) −67.8606 −2.93387
\(536\) 0 0
\(537\) −1.30335 −0.0562436
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −41.9037 −1.80158 −0.900791 0.434254i \(-0.857012\pi\)
−0.900791 + 0.434254i \(0.857012\pi\)
\(542\) 0 0
\(543\) −16.5632 −0.710796
\(544\) 0 0
\(545\) −67.0498 −2.87210
\(546\) 0 0
\(547\) −2.27211 −0.0971484 −0.0485742 0.998820i \(-0.515468\pi\)
−0.0485742 + 0.998820i \(0.515468\pi\)
\(548\) 0 0
\(549\) 5.38225 0.229709
\(550\) 0 0
\(551\) −3.66528 −0.156146
\(552\) 0 0
\(553\) −8.40257 −0.357314
\(554\) 0 0
\(555\) −7.29924 −0.309835
\(556\) 0 0
\(557\) 1.58354 0.0670966 0.0335483 0.999437i \(-0.489319\pi\)
0.0335483 + 0.999437i \(0.489319\pi\)
\(558\) 0 0
\(559\) −3.06814 −0.129768
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.85580 −0.373227 −0.186614 0.982433i \(-0.559751\pi\)
−0.186614 + 0.982433i \(0.559751\pi\)
\(564\) 0 0
\(565\) 53.2745 2.24128
\(566\) 0 0
\(567\) 2.24216 0.0941617
\(568\) 0 0
\(569\) 44.1552 1.85108 0.925541 0.378646i \(-0.123610\pi\)
0.925541 + 0.378646i \(0.123610\pi\)
\(570\) 0 0
\(571\) 29.3114 1.22664 0.613322 0.789833i \(-0.289833\pi\)
0.613322 + 0.789833i \(0.289833\pi\)
\(572\) 0 0
\(573\) 21.3332 0.891206
\(574\) 0 0
\(575\) −22.8691 −0.953708
\(576\) 0 0
\(577\) 7.78221 0.323978 0.161989 0.986793i \(-0.448209\pi\)
0.161989 + 0.986793i \(0.448209\pi\)
\(578\) 0 0
\(579\) −13.1360 −0.545913
\(580\) 0 0
\(581\) 12.1943 0.505906
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 20.5522 0.849727
\(586\) 0 0
\(587\) 25.7895 1.06445 0.532224 0.846603i \(-0.321356\pi\)
0.532224 + 0.846603i \(0.321356\pi\)
\(588\) 0 0
\(589\) −12.9768 −0.534701
\(590\) 0 0
\(591\) −6.55103 −0.269473
\(592\) 0 0
\(593\) 12.9720 0.532696 0.266348 0.963877i \(-0.414183\pi\)
0.266348 + 0.963877i \(0.414183\pi\)
\(594\) 0 0
\(595\) −8.41646 −0.345041
\(596\) 0 0
\(597\) −7.13599 −0.292057
\(598\) 0 0
\(599\) −23.0406 −0.941413 −0.470706 0.882290i \(-0.656001\pi\)
−0.470706 + 0.882290i \(0.656001\pi\)
\(600\) 0 0
\(601\) 19.0884 0.778632 0.389316 0.921104i \(-0.372711\pi\)
0.389316 + 0.921104i \(0.372711\pi\)
\(602\) 0 0
\(603\) −14.8610 −0.605188
\(604\) 0 0
\(605\) −40.6024 −1.65072
\(606\) 0 0
\(607\) 30.3704 1.23270 0.616348 0.787474i \(-0.288611\pi\)
0.616348 + 0.787474i \(0.288611\pi\)
\(608\) 0 0
\(609\) −3.09922 −0.125587
\(610\) 0 0
\(611\) 31.3167 1.26694
\(612\) 0 0
\(613\) 36.6678 1.48100 0.740500 0.672057i \(-0.234589\pi\)
0.740500 + 0.672057i \(0.234589\pi\)
\(614\) 0 0
\(615\) 15.8216 0.637988
\(616\) 0 0
\(617\) 27.1331 1.09234 0.546170 0.837675i \(-0.316085\pi\)
0.546170 + 0.837675i \(0.316085\pi\)
\(618\) 0 0
\(619\) 31.2867 1.25752 0.628760 0.777600i \(-0.283563\pi\)
0.628760 + 0.777600i \(0.283563\pi\)
\(620\) 0 0
\(621\) −2.65167 −0.106408
\(622\) 0 0
\(623\) −14.0720 −0.563781
\(624\) 0 0
\(625\) 6.25837 0.250335
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.01105 0.0801860
\(630\) 0 0
\(631\) −2.37559 −0.0945708 −0.0472854 0.998881i \(-0.515057\pi\)
−0.0472854 + 0.998881i \(0.515057\pi\)
\(632\) 0 0
\(633\) 18.6517 0.741337
\(634\) 0 0
\(635\) 53.3126 2.11564
\(636\) 0 0
\(637\) −10.9842 −0.435209
\(638\) 0 0
\(639\) 8.68560 0.343597
\(640\) 0 0
\(641\) −1.27144 −0.0502189 −0.0251095 0.999685i \(-0.507993\pi\)
−0.0251095 + 0.999685i \(0.507993\pi\)
\(642\) 0 0
\(643\) 0.789080 0.0311183 0.0155591 0.999879i \(-0.495047\pi\)
0.0155591 + 0.999879i \(0.495047\pi\)
\(644\) 0 0
\(645\) −2.03392 −0.0800857
\(646\) 0 0
\(647\) 20.6774 0.812912 0.406456 0.913670i \(-0.366764\pi\)
0.406456 + 0.913670i \(0.366764\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −10.9727 −0.430055
\(652\) 0 0
\(653\) −22.8434 −0.893932 −0.446966 0.894551i \(-0.647495\pi\)
−0.446966 + 0.894551i \(0.647495\pi\)
\(654\) 0 0
\(655\) −42.2064 −1.64914
\(656\) 0 0
\(657\) −1.75374 −0.0684198
\(658\) 0 0
\(659\) 10.5040 0.409176 0.204588 0.978848i \(-0.434414\pi\)
0.204588 + 0.978848i \(0.434414\pi\)
\(660\) 0 0
\(661\) 15.0168 0.584084 0.292042 0.956405i \(-0.405665\pi\)
0.292042 + 0.956405i \(0.405665\pi\)
\(662\) 0 0
\(663\) −5.66244 −0.219911
\(664\) 0 0
\(665\) 21.9455 0.851009
\(666\) 0 0
\(667\) 3.66528 0.141920
\(668\) 0 0
\(669\) −2.24216 −0.0866868
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −16.9370 −0.652872 −0.326436 0.945219i \(-0.605848\pi\)
−0.326436 + 0.945219i \(0.605848\pi\)
\(674\) 0 0
\(675\) 8.62441 0.331954
\(676\) 0 0
\(677\) −34.3509 −1.32021 −0.660106 0.751173i \(-0.729489\pi\)
−0.660106 + 0.751173i \(0.729489\pi\)
\(678\) 0 0
\(679\) −24.6787 −0.947080
\(680\) 0 0
\(681\) 18.3829 0.704435
\(682\) 0 0
\(683\) −3.52452 −0.134862 −0.0674309 0.997724i \(-0.521480\pi\)
−0.0674309 + 0.997724i \(0.521480\pi\)
\(684\) 0 0
\(685\) −65.0131 −2.48402
\(686\) 0 0
\(687\) −11.2149 −0.427875
\(688\) 0 0
\(689\) 5.78766 0.220492
\(690\) 0 0
\(691\) −2.91968 −0.111070 −0.0555349 0.998457i \(-0.517686\pi\)
−0.0555349 + 0.998457i \(0.517686\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 55.0200 2.08703
\(696\) 0 0
\(697\) −4.35909 −0.165112
\(698\) 0 0
\(699\) −17.5249 −0.662852
\(700\) 0 0
\(701\) 42.3944 1.60121 0.800606 0.599191i \(-0.204511\pi\)
0.800606 + 0.599191i \(0.204511\pi\)
\(702\) 0 0
\(703\) −5.24371 −0.197770
\(704\) 0 0
\(705\) 20.7604 0.781882
\(706\) 0 0
\(707\) 34.3306 1.29114
\(708\) 0 0
\(709\) −48.2207 −1.81096 −0.905482 0.424384i \(-0.860491\pi\)
−0.905482 + 0.424384i \(0.860491\pi\)
\(710\) 0 0
\(711\) −3.74754 −0.140544
\(712\) 0 0
\(713\) 12.9768 0.485987
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.13599 0.266498
\(718\) 0 0
\(719\) −27.9344 −1.04178 −0.520889 0.853624i \(-0.674399\pi\)
−0.520889 + 0.853624i \(0.674399\pi\)
\(720\) 0 0
\(721\) 29.8143 1.11034
\(722\) 0 0
\(723\) −5.86656 −0.218180
\(724\) 0 0
\(725\) −11.9211 −0.442738
\(726\) 0 0
\(727\) −4.65653 −0.172701 −0.0863506 0.996265i \(-0.527521\pi\)
−0.0863506 + 0.996265i \(0.527521\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.560378 0.0207263
\(732\) 0 0
\(733\) 3.63957 0.134431 0.0672153 0.997738i \(-0.478589\pi\)
0.0672153 + 0.997738i \(0.478589\pi\)
\(734\) 0 0
\(735\) −7.28161 −0.268586
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −40.1562 −1.47717 −0.738584 0.674161i \(-0.764505\pi\)
−0.738584 + 0.674161i \(0.764505\pi\)
\(740\) 0 0
\(741\) 14.7645 0.542388
\(742\) 0 0
\(743\) 23.5167 0.862743 0.431372 0.902174i \(-0.358030\pi\)
0.431372 + 0.902174i \(0.358030\pi\)
\(744\) 0 0
\(745\) 1.14009 0.0417698
\(746\) 0 0
\(747\) 5.43867 0.198990
\(748\) 0 0
\(749\) −41.2216 −1.50620
\(750\) 0 0
\(751\) 9.49844 0.346603 0.173301 0.984869i \(-0.444557\pi\)
0.173301 + 0.984869i \(0.444557\pi\)
\(752\) 0 0
\(753\) −17.0815 −0.622483
\(754\) 0 0
\(755\) 17.0218 0.619487
\(756\) 0 0
\(757\) 40.6663 1.47804 0.739021 0.673682i \(-0.235288\pi\)
0.739021 + 0.673682i \(0.235288\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −16.3018 −0.590939 −0.295470 0.955352i \(-0.595476\pi\)
−0.295470 + 0.955352i \(0.595476\pi\)
\(762\) 0 0
\(763\) −40.7291 −1.47449
\(764\) 0 0
\(765\) −3.75374 −0.135717
\(766\) 0 0
\(767\) 13.5185 0.488126
\(768\) 0 0
\(769\) −28.9452 −1.04379 −0.521895 0.853010i \(-0.674775\pi\)
−0.521895 + 0.853010i \(0.674775\pi\)
\(770\) 0 0
\(771\) 11.0959 0.399608
\(772\) 0 0
\(773\) 25.6351 0.922030 0.461015 0.887392i \(-0.347485\pi\)
0.461015 + 0.887392i \(0.347485\pi\)
\(774\) 0 0
\(775\) −42.2064 −1.51610
\(776\) 0 0
\(777\) −4.43389 −0.159065
\(778\) 0 0
\(779\) 11.3661 0.407233
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.38225 −0.0493976
\(784\) 0 0
\(785\) −26.9576 −0.962157
\(786\) 0 0
\(787\) −12.1428 −0.432843 −0.216422 0.976300i \(-0.569439\pi\)
−0.216422 + 0.976300i \(0.569439\pi\)
\(788\) 0 0
\(789\) −23.7836 −0.846717
\(790\) 0 0
\(791\) 32.3614 1.15064
\(792\) 0 0
\(793\) 29.9683 1.06421
\(794\) 0 0
\(795\) 3.83675 0.136075
\(796\) 0 0
\(797\) 15.6163 0.553159 0.276579 0.960991i \(-0.410799\pi\)
0.276579 + 0.960991i \(0.410799\pi\)
\(798\) 0 0
\(799\) −5.71981 −0.202352
\(800\) 0 0
\(801\) −6.27608 −0.221754
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −21.9455 −0.773476
\(806\) 0 0
\(807\) −24.8404 −0.874425
\(808\) 0 0
\(809\) −2.93881 −0.103323 −0.0516615 0.998665i \(-0.516452\pi\)
−0.0516615 + 0.998665i \(0.516452\pi\)
\(810\) 0 0
\(811\) 38.3561 1.34687 0.673433 0.739249i \(-0.264819\pi\)
0.673433 + 0.739249i \(0.264819\pi\)
\(812\) 0 0
\(813\) 16.6918 0.585407
\(814\) 0 0
\(815\) 6.40818 0.224469
\(816\) 0 0
\(817\) −1.46115 −0.0511193
\(818\) 0 0
\(819\) 12.4843 0.436237
\(820\) 0 0
\(821\) −16.7552 −0.584759 −0.292379 0.956302i \(-0.594447\pi\)
−0.292379 + 0.956302i \(0.594447\pi\)
\(822\) 0 0
\(823\) 13.9827 0.487408 0.243704 0.969850i \(-0.421637\pi\)
0.243704 + 0.969850i \(0.421637\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −0.468770 −0.0163007 −0.00815037 0.999967i \(-0.502594\pi\)
−0.00815037 + 0.999967i \(0.502594\pi\)
\(828\) 0 0
\(829\) 54.6669 1.89866 0.949329 0.314283i \(-0.101764\pi\)
0.949329 + 0.314283i \(0.101764\pi\)
\(830\) 0 0
\(831\) 30.8888 1.07152
\(832\) 0 0
\(833\) 2.00620 0.0695106
\(834\) 0 0
\(835\) 3.69113 0.127737
\(836\) 0 0
\(837\) −4.89383 −0.169156
\(838\) 0 0
\(839\) 31.9399 1.10269 0.551343 0.834279i \(-0.314115\pi\)
0.551343 + 0.834279i \(0.314115\pi\)
\(840\) 0 0
\(841\) −27.0894 −0.934117
\(842\) 0 0
\(843\) 25.0406 0.862444
\(844\) 0 0
\(845\) 66.4497 2.28594
\(846\) 0 0
\(847\) −24.6637 −0.847456
\(848\) 0 0
\(849\) −27.6964 −0.950537
\(850\) 0 0
\(851\) 5.24371 0.179752
\(852\) 0 0
\(853\) 13.0108 0.445480 0.222740 0.974878i \(-0.428500\pi\)
0.222740 + 0.974878i \(0.428500\pi\)
\(854\) 0 0
\(855\) 9.78766 0.334731
\(856\) 0 0
\(857\) 16.1331 0.551096 0.275548 0.961287i \(-0.411141\pi\)
0.275548 + 0.961287i \(0.411141\pi\)
\(858\) 0 0
\(859\) 6.77689 0.231225 0.115612 0.993294i \(-0.463117\pi\)
0.115612 + 0.993294i \(0.463117\pi\)
\(860\) 0 0
\(861\) 9.61075 0.327533
\(862\) 0 0
\(863\) 16.4517 0.560023 0.280012 0.959997i \(-0.409662\pi\)
0.280012 + 0.959997i \(0.409662\pi\)
\(864\) 0 0
\(865\) −19.8666 −0.675483
\(866\) 0 0
\(867\) −15.9658 −0.542227
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −82.7462 −2.80375
\(872\) 0 0
\(873\) −11.0067 −0.372519
\(874\) 0 0
\(875\) 29.9959 1.01405
\(876\) 0 0
\(877\) 3.08145 0.104053 0.0520267 0.998646i \(-0.483432\pi\)
0.0520267 + 0.998646i \(0.483432\pi\)
\(878\) 0 0
\(879\) 21.3823 0.721205
\(880\) 0 0
\(881\) 26.5909 0.895872 0.447936 0.894066i \(-0.352159\pi\)
0.447936 + 0.894066i \(0.352159\pi\)
\(882\) 0 0
\(883\) 40.9997 1.37975 0.689875 0.723928i \(-0.257665\pi\)
0.689875 + 0.723928i \(0.257665\pi\)
\(884\) 0 0
\(885\) 8.96168 0.301244
\(886\) 0 0
\(887\) −24.5233 −0.823413 −0.411707 0.911316i \(-0.635067\pi\)
−0.411707 + 0.911316i \(0.635067\pi\)
\(888\) 0 0
\(889\) 32.3845 1.08614
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 14.9141 0.499081
\(894\) 0 0
\(895\) −4.81082 −0.160808
\(896\) 0 0
\(897\) −14.7645 −0.492972
\(898\) 0 0
\(899\) 6.76450 0.225609
\(900\) 0 0
\(901\) −1.05708 −0.0352166
\(902\) 0 0
\(903\) −1.23550 −0.0411148
\(904\) 0 0
\(905\) −61.1369 −2.03226
\(906\) 0 0
\(907\) −8.73058 −0.289894 −0.144947 0.989439i \(-0.546301\pi\)
−0.144947 + 0.989439i \(0.546301\pi\)
\(908\) 0 0
\(909\) 15.3114 0.507848
\(910\) 0 0
\(911\) −43.0841 −1.42744 −0.713719 0.700432i \(-0.752991\pi\)
−0.713719 + 0.700432i \(0.752991\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 19.8666 0.656768
\(916\) 0 0
\(917\) −25.6381 −0.846644
\(918\) 0 0
\(919\) 40.3233 1.33014 0.665072 0.746779i \(-0.268401\pi\)
0.665072 + 0.746779i \(0.268401\pi\)
\(920\) 0 0
\(921\) −5.35640 −0.176500
\(922\) 0 0
\(923\) 48.3614 1.59183
\(924\) 0 0
\(925\) −17.0549 −0.560760
\(926\) 0 0
\(927\) 13.2972 0.436736
\(928\) 0 0
\(929\) −6.18762 −0.203009 −0.101505 0.994835i \(-0.532366\pi\)
−0.101505 + 0.994835i \(0.532366\pi\)
\(930\) 0 0
\(931\) −5.23105 −0.171441
\(932\) 0 0
\(933\) −11.0980 −0.363331
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 6.88688 0.224985 0.112492 0.993653i \(-0.464117\pi\)
0.112492 + 0.993653i \(0.464117\pi\)
\(938\) 0 0
\(939\) −20.4527 −0.667447
\(940\) 0 0
\(941\) 48.3203 1.57520 0.787599 0.616188i \(-0.211324\pi\)
0.787599 + 0.616188i \(0.211324\pi\)
\(942\) 0 0
\(943\) −11.3661 −0.370131
\(944\) 0 0
\(945\) 8.27608 0.269221
\(946\) 0 0
\(947\) 13.9867 0.454506 0.227253 0.973836i \(-0.427026\pi\)
0.227253 + 0.973836i \(0.427026\pi\)
\(948\) 0 0
\(949\) −9.76479 −0.316978
\(950\) 0 0
\(951\) 0.889723 0.0288512
\(952\) 0 0
\(953\) 49.2220 1.59446 0.797229 0.603677i \(-0.206299\pi\)
0.797229 + 0.603677i \(0.206299\pi\)
\(954\) 0 0
\(955\) 78.7434 2.54808
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −39.4919 −1.27526
\(960\) 0 0
\(961\) −7.05042 −0.227433
\(962\) 0 0
\(963\) −18.3848 −0.592442
\(964\) 0 0
\(965\) −48.4866 −1.56084
\(966\) 0 0
\(967\) −5.91692 −0.190275 −0.0951376 0.995464i \(-0.530329\pi\)
−0.0951376 + 0.995464i \(0.530329\pi\)
\(968\) 0 0
\(969\) −2.69665 −0.0866290
\(970\) 0 0
\(971\) 27.8006 0.892164 0.446082 0.894992i \(-0.352819\pi\)
0.446082 + 0.894992i \(0.352819\pi\)
\(972\) 0 0
\(973\) 33.4216 1.07145
\(974\) 0 0
\(975\) 48.0206 1.53789
\(976\) 0 0
\(977\) −34.4493 −1.10213 −0.551065 0.834462i \(-0.685778\pi\)
−0.551065 + 0.834462i \(0.685778\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −18.1651 −0.579968
\(982\) 0 0
\(983\) 7.58504 0.241925 0.120963 0.992657i \(-0.461402\pi\)
0.120963 + 0.992657i \(0.461402\pi\)
\(984\) 0 0
\(985\) −24.1807 −0.770460
\(986\) 0 0
\(987\) 12.6108 0.401406
\(988\) 0 0
\(989\) 1.46115 0.0464620
\(990\) 0 0
\(991\) 30.1199 0.956792 0.478396 0.878144i \(-0.341218\pi\)
0.478396 + 0.878144i \(0.341218\pi\)
\(992\) 0 0
\(993\) −7.31553 −0.232151
\(994\) 0 0
\(995\) −26.3398 −0.835029
\(996\) 0 0
\(997\) −17.5318 −0.555239 −0.277620 0.960691i \(-0.589545\pi\)
−0.277620 + 0.960691i \(0.589545\pi\)
\(998\) 0 0
\(999\) −1.97751 −0.0625657
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 8016.2.a.o.1.4 4
4.3 odd 2 1002.2.a.i.1.4 4
12.11 even 2 3006.2.a.s.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1002.2.a.i.1.4 4 4.3 odd 2
3006.2.a.s.1.1 4 12.11 even 2
8016.2.a.o.1.4 4 1.1 even 1 trivial