Properties

Label 1216.2.b.d.607.5
Level $1216$
Weight $2$
Character 1216.607
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 607.5
Root \(-0.228425 + 1.39564i\) of defining polynomial
Character \(\chi\) \(=\) 1216.607
Dual form 1216.2.b.d.607.4

$q$-expansion

\(f(q)\) \(=\) \(q+2.64575i q^{3} -3.46410i q^{5} -1.00000i q^{7} -4.00000 q^{9} +O(q^{10})\) \(q+2.64575i q^{3} -3.46410i q^{5} -1.00000i q^{7} -4.00000 q^{9} -2.64575 q^{13} +9.16515 q^{15} -3.00000 q^{17} +(-3.46410 + 2.64575i) q^{19} +2.64575 q^{21} -3.00000i q^{23} -7.00000 q^{25} -2.64575i q^{27} -7.93725 q^{29} -3.46410 q^{35} -10.5830 q^{37} -7.00000i q^{39} -9.16515i q^{41} -10.3923 q^{43} +13.8564i q^{45} +6.00000 q^{49} -7.93725i q^{51} +7.93725 q^{53} +(-7.00000 - 9.16515i) q^{57} +7.93725i q^{59} +6.92820i q^{61} +4.00000i q^{63} +9.16515i q^{65} -13.2288i q^{67} +7.93725 q^{69} -9.16515 q^{71} +7.00000 q^{73} -18.5203i q^{75} +9.16515 q^{79} -5.00000 q^{81} +17.3205 q^{83} +10.3923i q^{85} -21.0000i q^{87} +9.16515i q^{89} +2.64575i q^{91} +(9.16515 + 12.0000i) q^{95} -9.16515i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 32q^{9} + O(q^{10}) \) \( 8q - 32q^{9} - 24q^{17} - 56q^{25} + 48q^{49} - 56q^{57} + 56q^{73} - 40q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 2.64575i 1.52753i 0.645497 + 0.763763i \(0.276650\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) 0 0
\(5\) 3.46410i 1.54919i −0.632456 0.774597i \(-0.717953\pi\)
0.632456 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 0 0
\(9\) −4.00000 −1.33333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.64575 −0.733799 −0.366900 0.930261i \(-0.619581\pi\)
−0.366900 + 0.930261i \(0.619581\pi\)
\(14\) 0 0
\(15\) 9.16515 2.36643
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −3.46410 + 2.64575i −0.794719 + 0.606977i
\(20\) 0 0
\(21\) 2.64575 0.577350
\(22\) 0 0
\(23\) 3.00000i 0.625543i −0.949828 0.312772i \(-0.898743\pi\)
0.949828 0.312772i \(-0.101257\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 2.64575i 0.509175i
\(28\) 0 0
\(29\) −7.93725 −1.47391 −0.736956 0.675941i \(-0.763737\pi\)
−0.736956 + 0.675941i \(0.763737\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.46410 −0.585540
\(36\) 0 0
\(37\) −10.5830 −1.73984 −0.869918 0.493197i \(-0.835828\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 7.00000i 1.12090i
\(40\) 0 0
\(41\) 9.16515i 1.43136i −0.698430 0.715678i \(-0.746118\pi\)
0.698430 0.715678i \(-0.253882\pi\)
\(42\) 0 0
\(43\) −10.3923 −1.58481 −0.792406 0.609994i \(-0.791172\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 13.8564i 2.06559i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 7.93725i 1.11144i
\(52\) 0 0
\(53\) 7.93725 1.09027 0.545133 0.838350i \(-0.316479\pi\)
0.545133 + 0.838350i \(0.316479\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −7.00000 9.16515i −0.927173 1.21395i
\(58\) 0 0
\(59\) 7.93725i 1.03334i 0.856184 + 0.516671i \(0.172829\pi\)
−0.856184 + 0.516671i \(0.827171\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i 0.896258 + 0.443533i \(0.146275\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) 9.16515i 1.13680i
\(66\) 0 0
\(67\) 13.2288i 1.61615i −0.589080 0.808075i \(-0.700510\pi\)
0.589080 0.808075i \(-0.299490\pi\)
\(68\) 0 0
\(69\) 7.93725 0.955533
\(70\) 0 0
\(71\) −9.16515 −1.08770 −0.543852 0.839181i \(-0.683035\pi\)
−0.543852 + 0.839181i \(0.683035\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 18.5203i 2.13854i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.16515 1.03116 0.515580 0.856841i \(-0.327576\pi\)
0.515580 + 0.856841i \(0.327576\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 17.3205 1.90117 0.950586 0.310460i \(-0.100483\pi\)
0.950586 + 0.310460i \(0.100483\pi\)
\(84\) 0 0
\(85\) 10.3923i 1.12720i
\(86\) 0 0
\(87\) 21.0000i 2.25144i
\(88\) 0 0
\(89\) 9.16515i 0.971504i 0.874097 + 0.485752i \(0.161454\pi\)
−0.874097 + 0.485752i \(0.838546\pi\)
\(90\) 0 0
\(91\) 2.64575i 0.277350i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.16515 + 12.0000i 0.940325 + 1.23117i
\(96\) 0 0
\(97\) 9.16515i 0.930580i −0.885158 0.465290i \(-0.845950\pi\)
0.885158 0.465290i \(-0.154050\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.8564i 1.37876i −0.724398 0.689382i \(-0.757882\pi\)
0.724398 0.689382i \(-0.242118\pi\)
\(102\) 0 0
\(103\) −9.16515 −0.903069 −0.451535 0.892254i \(-0.649123\pi\)
−0.451535 + 0.892254i \(0.649123\pi\)
\(104\) 0 0
\(105\) 9.16515i 0.894427i
\(106\) 0 0
\(107\) 7.93725i 0.767323i 0.923474 + 0.383662i \(0.125337\pi\)
−0.923474 + 0.383662i \(0.874663\pi\)
\(108\) 0 0
\(109\) 2.64575 0.253417 0.126709 0.991940i \(-0.459559\pi\)
0.126709 + 0.991940i \(0.459559\pi\)
\(110\) 0 0
\(111\) 28.0000i 2.65764i
\(112\) 0 0
\(113\) 9.16515i 0.862185i −0.902308 0.431092i \(-0.858128\pi\)
0.902308 0.431092i \(-0.141872\pi\)
\(114\) 0 0
\(115\) −10.3923 −0.969087
\(116\) 0 0
\(117\) 10.5830 0.978399
\(118\) 0 0
\(119\) 3.00000i 0.275010i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 24.2487 2.18643
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 27.4955i 2.42084i
\(130\) 0 0
\(131\) −3.46410 −0.302660 −0.151330 0.988483i \(-0.548356\pi\)
−0.151330 + 0.988483i \(0.548356\pi\)
\(132\) 0 0
\(133\) 2.64575 + 3.46410i 0.229416 + 0.300376i
\(134\) 0 0
\(135\) −9.16515 −0.788811
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) −13.8564 −1.17529 −0.587643 0.809121i \(-0.699944\pi\)
−0.587643 + 0.809121i \(0.699944\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 27.4955i 2.28337i
\(146\) 0 0
\(147\) 15.8745i 1.30931i
\(148\) 0 0
\(149\) 17.3205i 1.41895i −0.704730 0.709476i \(-0.748932\pi\)
0.704730 0.709476i \(-0.251068\pi\)
\(150\) 0 0
\(151\) 18.3303 1.49170 0.745849 0.666115i \(-0.232044\pi\)
0.745849 + 0.666115i \(0.232044\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.3923i 0.829396i −0.909959 0.414698i \(-0.863887\pi\)
0.909959 0.414698i \(-0.136113\pi\)
\(158\) 0 0
\(159\) 21.0000i 1.66541i
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) 6.92820 0.542659 0.271329 0.962487i \(-0.412537\pi\)
0.271329 + 0.962487i \(0.412537\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.16515 0.709221 0.354610 0.935014i \(-0.384614\pi\)
0.354610 + 0.935014i \(0.384614\pi\)
\(168\) 0 0
\(169\) −6.00000 −0.461538
\(170\) 0 0
\(171\) 13.8564 10.5830i 1.05963 0.809303i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 7.00000i 0.529150i
\(176\) 0 0
\(177\) −21.0000 −1.57846
\(178\) 0 0
\(179\) 15.8745i 1.18652i 0.805012 + 0.593258i \(0.202159\pi\)
−0.805012 + 0.593258i \(0.797841\pi\)
\(180\) 0 0
\(181\) −10.5830 −0.786629 −0.393314 0.919404i \(-0.628672\pi\)
−0.393314 + 0.919404i \(0.628672\pi\)
\(182\) 0 0
\(183\) −18.3303 −1.35501
\(184\) 0 0
\(185\) 36.6606i 2.69534i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −2.64575 −0.192450
\(190\) 0 0
\(191\) 3.00000i 0.217072i 0.994092 + 0.108536i \(0.0346163\pi\)
−0.994092 + 0.108536i \(0.965384\pi\)
\(192\) 0 0
\(193\) 18.3303i 1.31944i 0.751510 + 0.659722i \(0.229326\pi\)
−0.751510 + 0.659722i \(0.770674\pi\)
\(194\) 0 0
\(195\) −24.2487 −1.73649
\(196\) 0 0
\(197\) 10.3923i 0.740421i 0.928948 + 0.370211i \(0.120714\pi\)
−0.928948 + 0.370211i \(0.879286\pi\)
\(198\) 0 0
\(199\) 17.0000i 1.20510i 0.798082 + 0.602549i \(0.205848\pi\)
−0.798082 + 0.602549i \(0.794152\pi\)
\(200\) 0 0
\(201\) 35.0000 2.46871
\(202\) 0 0
\(203\) 7.93725i 0.557086i
\(204\) 0 0
\(205\) −31.7490 −2.21745
\(206\) 0 0
\(207\) 12.0000i 0.834058i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2.64575i 0.182141i 0.995844 + 0.0910705i \(0.0290289\pi\)
−0.995844 + 0.0910705i \(0.970971\pi\)
\(212\) 0 0
\(213\) 24.2487i 1.66149i
\(214\) 0 0
\(215\) 36.0000i 2.45518i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 18.5203i 1.25148i
\(220\) 0 0
\(221\) 7.93725 0.533917
\(222\) 0 0
\(223\) −27.4955 −1.84123 −0.920616 0.390469i \(-0.872313\pi\)
−0.920616 + 0.390469i \(0.872313\pi\)
\(224\) 0 0
\(225\) 28.0000 1.86667
\(226\) 0 0
\(227\) 7.93725i 0.526814i 0.964685 + 0.263407i \(0.0848462\pi\)
−0.964685 + 0.263407i \(0.915154\pi\)
\(228\) 0 0
\(229\) 20.7846i 1.37349i −0.726900 0.686743i \(-0.759040\pi\)
0.726900 0.686743i \(-0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 24.2487i 1.57512i
\(238\) 0 0
\(239\) 15.0000i 0.970269i −0.874439 0.485135i \(-0.838771\pi\)
0.874439 0.485135i \(-0.161229\pi\)
\(240\) 0 0
\(241\) 9.16515i 0.590379i −0.955439 0.295190i \(-0.904617\pi\)
0.955439 0.295190i \(-0.0953828\pi\)
\(242\) 0 0
\(243\) 21.1660i 1.35780i
\(244\) 0 0
\(245\) 20.7846i 1.32788i
\(246\) 0 0
\(247\) 9.16515 7.00000i 0.583165 0.445399i
\(248\) 0 0
\(249\) 45.8258i 2.90409i
\(250\) 0 0
\(251\) −17.3205 −1.09326 −0.546630 0.837374i \(-0.684090\pi\)
−0.546630 + 0.837374i \(0.684090\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −27.4955 −1.72183
\(256\) 0 0
\(257\) 9.16515i 0.571706i −0.958273 0.285853i \(-0.907723\pi\)
0.958273 0.285853i \(-0.0922770\pi\)
\(258\) 0 0
\(259\) 10.5830i 0.657596i
\(260\) 0 0
\(261\) 31.7490 1.96521
\(262\) 0 0
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 0 0
\(265\) 27.4955i 1.68903i
\(266\) 0 0
\(267\) −24.2487 −1.48400
\(268\) 0 0
\(269\) −31.7490 −1.93577 −0.967886 0.251390i \(-0.919112\pi\)
−0.967886 + 0.251390i \(0.919112\pi\)
\(270\) 0 0
\(271\) 1.00000i 0.0607457i 0.999539 + 0.0303728i \(0.00966946\pi\)
−0.999539 + 0.0303728i \(0.990331\pi\)
\(272\) 0 0
\(273\) −7.00000 −0.423659
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.3303i 1.09349i 0.837298 + 0.546747i \(0.184134\pi\)
−0.837298 + 0.546747i \(0.815866\pi\)
\(282\) 0 0
\(283\) 10.3923 0.617758 0.308879 0.951101i \(-0.400046\pi\)
0.308879 + 0.951101i \(0.400046\pi\)
\(284\) 0 0
\(285\) −31.7490 + 24.2487i −1.88065 + 1.43637i
\(286\) 0 0
\(287\) −9.16515 −0.541002
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 24.2487 1.42148
\(292\) 0 0
\(293\) 23.8118 1.39110 0.695549 0.718479i \(-0.255161\pi\)
0.695549 + 0.718479i \(0.255161\pi\)
\(294\) 0 0
\(295\) 27.4955 1.60085
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.93725i 0.459023i
\(300\) 0 0
\(301\) 10.3923i 0.599002i
\(302\) 0 0
\(303\) 36.6606 2.10610
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 26.4575i 1.51001i −0.655719 0.755005i \(-0.727634\pi\)
0.655719 0.755005i \(-0.272366\pi\)
\(308\) 0 0
\(309\) 24.2487i 1.37946i
\(310\) 0 0
\(311\) 3.00000i 0.170114i −0.996376 0.0850572i \(-0.972893\pi\)
0.996376 0.0850572i \(-0.0271073\pi\)
\(312\) 0 0
\(313\) −13.0000 −0.734803 −0.367402 0.930062i \(-0.619753\pi\)
−0.367402 + 0.930062i \(0.619753\pi\)
\(314\) 0 0
\(315\) 13.8564 0.780720
\(316\) 0 0
\(317\) 23.8118 1.33740 0.668701 0.743532i \(-0.266851\pi\)
0.668701 + 0.743532i \(0.266851\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −21.0000 −1.17211
\(322\) 0 0
\(323\) 10.3923 7.93725i 0.578243 0.441641i
\(324\) 0 0
\(325\) 18.5203 1.02732
\(326\) 0 0
\(327\) 7.00000i 0.387101i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 18.5203i 1.01797i 0.860777 + 0.508983i \(0.169978\pi\)
−0.860777 + 0.508983i \(0.830022\pi\)
\(332\) 0 0
\(333\) 42.3320 2.31978
\(334\) 0 0
\(335\) −45.8258 −2.50373
\(336\) 0 0
\(337\) 9.16515i 0.499258i 0.968342 + 0.249629i \(0.0803086\pi\)
−0.968342 + 0.249629i \(0.919691\pi\)
\(338\) 0 0
\(339\) 24.2487 1.31701
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 27.4955i 1.48031i
\(346\) 0 0
\(347\) 20.7846 1.11578 0.557888 0.829916i \(-0.311612\pi\)
0.557888 + 0.829916i \(0.311612\pi\)
\(348\) 0 0
\(349\) 17.3205i 0.927146i 0.886059 + 0.463573i \(0.153433\pi\)
−0.886059 + 0.463573i \(0.846567\pi\)
\(350\) 0 0
\(351\) 7.00000i 0.373632i
\(352\) 0 0
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) 31.7490i 1.68506i
\(356\) 0 0
\(357\) −7.93725 −0.420084
\(358\) 0 0
\(359\) 27.0000i 1.42501i 0.701669 + 0.712503i \(0.252438\pi\)
−0.701669 + 0.712503i \(0.747562\pi\)
\(360\) 0 0
\(361\) 5.00000 18.3303i 0.263158 0.964753i
\(362\) 0 0
\(363\) 29.1033i 1.52753i
\(364\) 0 0
\(365\) 24.2487i 1.26924i
\(366\) 0 0
\(367\) 28.0000i 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 0 0
\(369\) 36.6606i 1.90847i
\(370\) 0 0
\(371\) 7.93725i 0.412082i
\(372\) 0 0
\(373\) 13.2288 0.684959 0.342480 0.939525i \(-0.388733\pi\)
0.342480 + 0.939525i \(0.388733\pi\)
\(374\) 0 0
\(375\) −18.3303 −0.946573
\(376\) 0 0
\(377\) 21.0000 1.08156
\(378\) 0 0
\(379\) 13.2288i 0.679516i 0.940513 + 0.339758i \(0.110345\pi\)
−0.940513 + 0.339758i \(0.889655\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −27.4955 −1.40495 −0.702476 0.711707i \(-0.747922\pi\)
−0.702476 + 0.711707i \(0.747922\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 41.5692 2.11308
\(388\) 0 0
\(389\) 20.7846i 1.05382i −0.849921 0.526911i \(-0.823350\pi\)
0.849921 0.526911i \(-0.176650\pi\)
\(390\) 0 0
\(391\) 9.00000i 0.455150i
\(392\) 0 0
\(393\) 9.16515i 0.462321i
\(394\) 0 0
\(395\) 31.7490i 1.59747i
\(396\) 0 0
\(397\) 20.7846i 1.04315i −0.853206 0.521575i \(-0.825345\pi\)
0.853206 0.521575i \(-0.174655\pi\)
\(398\) 0 0
\(399\) −9.16515 + 7.00000i −0.458831 + 0.350438i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 17.3205i 0.860663i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 18.3303i 0.906375i −0.891415 0.453188i \(-0.850287\pi\)
0.891415 0.453188i \(-0.149713\pi\)
\(410\) 0 0
\(411\) 7.93725i 0.391516i
\(412\) 0 0
\(413\) 7.93725 0.390567
\(414\) 0 0
\(415\) 60.0000i 2.94528i
\(416\) 0 0
\(417\) 36.6606i 1.79528i
\(418\) 0 0
\(419\) 31.1769 1.52309 0.761546 0.648111i \(-0.224441\pi\)
0.761546 + 0.648111i \(0.224441\pi\)
\(420\) 0 0
\(421\) −2.64575 −0.128946 −0.0644730 0.997919i \(-0.520537\pi\)
−0.0644730 + 0.997919i \(0.520537\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 21.0000 1.01865
\(426\) 0 0
\(427\) 6.92820 0.335279
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −18.3303 −0.882940 −0.441470 0.897276i \(-0.645543\pi\)
−0.441470 + 0.897276i \(0.645543\pi\)
\(432\) 0 0
\(433\) 27.4955i 1.32135i −0.750673 0.660674i \(-0.770271\pi\)
0.750673 0.660674i \(-0.229729\pi\)
\(434\) 0 0
\(435\) −72.7461 −3.48791
\(436\) 0 0
\(437\) 7.93725 + 10.3923i 0.379690 + 0.497131i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −24.0000 −1.14286
\(442\) 0 0
\(443\) −6.92820 −0.329169 −0.164584 0.986363i \(-0.552628\pi\)
−0.164584 + 0.986363i \(0.552628\pi\)
\(444\) 0 0
\(445\) 31.7490 1.50505
\(446\) 0 0
\(447\) 45.8258 2.16748
\(448\) 0 0
\(449\) 36.6606i 1.73012i 0.501668 + 0.865060i \(0.332720\pi\)
−0.501668 + 0.865060i \(0.667280\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 48.4974i 2.27861i
\(454\) 0 0
\(455\) 9.16515 0.429669
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 0 0
\(459\) 7.93725i 0.370479i
\(460\) 0 0
\(461\) 13.8564i 0.645357i 0.946509 + 0.322679i \(0.104583\pi\)
−0.946509 + 0.322679i \(0.895417\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −31.1769 −1.44270 −0.721348 0.692573i \(-0.756477\pi\)
−0.721348 + 0.692573i \(0.756477\pi\)
\(468\) 0 0
\(469\) −13.2288 −0.610847
\(470\) 0 0
\(471\) 27.4955 1.26692
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 24.2487 18.5203i 1.11261 0.849768i
\(476\) 0 0
\(477\) −31.7490 −1.45369
\(478\) 0 0
\(479\) 24.0000i 1.09659i 0.836286 + 0.548294i \(0.184723\pi\)
−0.836286 + 0.548294i \(0.815277\pi\)
\(480\) 0 0
\(481\) 28.0000 1.27669
\(482\) 0 0
\(483\) 7.93725i 0.361158i
\(484\) 0 0
\(485\) −31.7490 −1.44165
\(486\) 0 0
\(487\) −9.16515 −0.415313 −0.207656 0.978202i \(-0.566584\pi\)
−0.207656 + 0.978202i \(0.566584\pi\)
\(488\) 0 0
\(489\) 18.3303i 0.828925i
\(490\) 0 0
\(491\) 13.8564 0.625331 0.312665 0.949863i \(-0.398778\pi\)
0.312665 + 0.949863i \(0.398778\pi\)
\(492\) 0 0
\(493\) 23.8118 1.07243
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.16515i 0.411113i
\(498\) 0 0
\(499\) −31.1769 −1.39567 −0.697835 0.716258i \(-0.745853\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(500\) 0 0
\(501\) 24.2487i 1.08335i
\(502\) 0 0
\(503\) 9.00000i 0.401290i −0.979664 0.200645i \(-0.935696\pi\)
0.979664 0.200645i \(-0.0643038\pi\)
\(504\) 0 0
\(505\) −48.0000 −2.13597
\(506\) 0 0
\(507\) 15.8745i 0.705012i
\(508\) 0 0
\(509\) −31.7490 −1.40725 −0.703625 0.710571i \(-0.748437\pi\)
−0.703625 + 0.710571i \(0.748437\pi\)
\(510\) 0 0
\(511\) 7.00000i 0.309662i
\(512\) 0 0
\(513\) 7.00000 + 9.16515i 0.309058 + 0.404651i
\(514\) 0 0
\(515\) 31.7490i 1.39903i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.16515i 0.401533i 0.979639 + 0.200766i \(0.0643432\pi\)
−0.979639 + 0.200766i \(0.935657\pi\)
\(522\) 0 0
\(523\) 29.1033i 1.27260i −0.771443 0.636298i \(-0.780465\pi\)
0.771443 0.636298i \(-0.219535\pi\)
\(524\) 0 0
\(525\) −18.5203 −0.808290
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 31.7490i 1.37779i
\(532\) 0 0
\(533\) 24.2487i 1.05033i
\(534\) 0 0
\(535\) 27.4955 1.18873
\(536\) 0 0
\(537\) −42.0000 −1.81243
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17.3205i 0.744667i −0.928099 0.372333i \(-0.878558\pi\)
0.928099 0.372333i \(-0.121442\pi\)
\(542\) 0 0
\(543\) 28.0000i 1.20160i
\(544\) 0 0
\(545\) 9.16515i 0.392592i
\(546\) 0 0
\(547\) 5.29150i 0.226248i −0.993581 0.113124i \(-0.963914\pi\)
0.993581 0.113124i \(-0.0360858\pi\)
\(548\) 0 0
\(549\) 27.7128i 1.18275i
\(550\) 0 0
\(551\) 27.4955 21.0000i 1.17135 0.894630i
\(552\) 0 0
\(553\) 9.16515i 0.389742i
\(554\) 0 0
\(555\) −96.9948 −4.11720
\(556\) 0 0
\(557\) 3.46410i 0.146779i 0.997303 + 0.0733893i \(0.0233816\pi\)
−0.997303 + 0.0733893i \(0.976618\pi\)
\(558\) 0 0
\(559\) 27.4955 1.16293
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.8745i 0.669031i 0.942390 + 0.334515i \(0.108573\pi\)
−0.942390 + 0.334515i \(0.891427\pi\)
\(564\) 0 0
\(565\) −31.7490 −1.33569
\(566\) 0 0
\(567\) 5.00000i 0.209980i
\(568\) 0 0
\(569\) 9.16515i 0.384223i −0.981373 0.192112i \(-0.938466\pi\)
0.981373 0.192112i \(-0.0615335\pi\)
\(570\) 0 0
\(571\) −3.46410 −0.144968 −0.0724841 0.997370i \(-0.523093\pi\)
−0.0724841 + 0.997370i \(0.523093\pi\)
\(572\) 0 0
\(573\) −7.93725 −0.331584
\(574\) 0 0
\(575\) 21.0000i 0.875761i
\(576\) 0 0
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 0 0
\(579\) −48.4974 −2.01548
\(580\) 0 0
\(581\) 17.3205i 0.718576i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 36.6606i 1.51573i
\(586\) 0 0
\(587\) −6.92820 −0.285958 −0.142979 0.989726i \(-0.545668\pi\)
−0.142979 + 0.989726i \(0.545668\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −27.4955 −1.13101
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 10.3923 0.426043
\(596\) 0 0
\(597\) −44.9778 −1.84082
\(598\) 0 0
\(599\) −18.3303 −0.748956 −0.374478 0.927236i \(-0.622178\pi\)
−0.374478 + 0.927236i \(0.622178\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 52.9150i 2.15487i
\(604\) 0 0
\(605\) 38.1051i 1.54919i
\(606\) 0 0
\(607\) −9.16515 −0.372002 −0.186001 0.982550i \(-0.559553\pi\)
−0.186001 + 0.982550i \(0.559553\pi\)
\(608\) 0 0
\(609\) −21.0000 −0.850963
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 3.46410i 0.139914i −0.997550 0.0699569i \(-0.977714\pi\)
0.997550 0.0699569i \(-0.0222862\pi\)
\(614\) 0 0
\(615\) 84.0000i 3.38721i
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 13.8564 0.556936 0.278468 0.960446i \(-0.410173\pi\)
0.278468 + 0.960446i \(0.410173\pi\)
\(620\) 0 0
\(621\) −7.93725 −0.318511
\(622\) 0 0
\(623\) 9.16515 0.367194
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 31.7490 1.26592
\(630\) 0 0
\(631\) 20.0000i 0.796187i 0.917345 + 0.398094i \(0.130328\pi\)
−0.917345 + 0.398094i \(0.869672\pi\)
\(632\) 0 0
\(633\) −7.00000 −0.278225
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −15.8745 −0.628971
\(638\) 0 0
\(639\) 36.6606 1.45027
\(640\) 0 0
\(641\) 9.16515i 0.362002i 0.983483 + 0.181001i \(0.0579337\pi\)
−0.983483 + 0.181001i \(0.942066\pi\)
\(642\) 0 0
\(643\) 31.1769 1.22950 0.614749 0.788723i \(-0.289257\pi\)
0.614749 + 0.788723i \(0.289257\pi\)
\(644\) 0 0
\(645\) −95.2470 −3.75035
\(646\) 0 0
\(647\) 3.00000i 0.117942i −0.998260 0.0589711i \(-0.981218\pi\)
0.998260 0.0589711i \(-0.0187820\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.92820i 0.271122i −0.990769 0.135561i \(-0.956716\pi\)
0.990769 0.135561i \(-0.0432836\pi\)
\(654\) 0 0
\(655\) 12.0000i 0.468879i
\(656\) 0 0
\(657\) −28.0000 −1.09238
\(658\) 0 0
\(659\) 7.93725i 0.309192i −0.987978 0.154596i \(-0.950592\pi\)
0.987978 0.154596i \(-0.0494075\pi\)
\(660\) 0 0
\(661\) 44.9778 1.74943 0.874716 0.484635i \(-0.161048\pi\)
0.874716 + 0.484635i \(0.161048\pi\)
\(662\) 0 0
\(663\) 21.0000i 0.815572i
\(664\) 0 0
\(665\) 12.0000 9.16515i 0.465340 0.355409i
\(666\) 0 0
\(667\) 23.8118i 0.921995i
\(668\) 0 0
\(669\) 72.7461i 2.81253i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 9.16515i 0.353291i −0.984275 0.176645i \(-0.943475\pi\)
0.984275 0.176645i \(-0.0565246\pi\)
\(674\) 0 0
\(675\) 18.5203i 0.712845i
\(676\) 0 0
\(677\) 23.8118 0.915160 0.457580 0.889168i \(-0.348716\pi\)
0.457580 + 0.889168i \(0.348716\pi\)
\(678\) 0 0
\(679\) −9.16515 −0.351726
\(680\) 0 0
\(681\) −21.0000 −0.804722
\(682\) 0 0
\(683\) 15.8745i 0.607421i −0.952764 0.303711i \(-0.901774\pi\)
0.952764 0.303711i \(-0.0982256\pi\)
\(684\) 0 0
\(685\) 10.3923i 0.397070i
\(686\) 0 0
\(687\) 54.9909 2.09803
\(688\) 0 0
\(689\) −21.0000 −0.800036
\(690\) 0 0
\(691\) 20.7846 0.790684 0.395342 0.918534i \(-0.370626\pi\)
0.395342 + 0.918534i \(0.370626\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 48.0000i 1.82074i
\(696\) 0 0
\(697\) 27.4955i 1.04146i
\(698\) 0 0
\(699\) 47.6235i 1.80129i
\(700\) 0 0
\(701\) 13.8564i 0.523349i 0.965156 + 0.261675i \(0.0842747\pi\)
−0.965156 + 0.261675i \(0.915725\pi\)
\(702\) 0 0
\(703\) 36.6606 28.0000i 1.38268 1.05604i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −13.8564 −0.521124
\(708\) 0 0
\(709\) 6.92820i 0.260194i −0.991501 0.130097i \(-0.958471\pi\)
0.991501 0.130097i \(-0.0415289\pi\)
\(710\) 0 0
\(711\) −36.6606 −1.37488
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 39.6863 1.48211
\(718\) 0 0
\(719\) 15.0000i 0.559406i 0.960087 + 0.279703i \(0.0902359\pi\)
−0.960087 + 0.279703i \(0.909764\pi\)
\(720\) 0 0
\(721\) 9.16515i 0.341328i
\(722\) 0 0
\(723\) 24.2487 0.901819
\(724\) 0 0
\(725\) 55.5608 2.06348
\(726\) 0 0
\(727\) 47.0000i 1.74313i −0.490277 0.871567i \(-0.663104\pi\)
0.490277 0.871567i \(-0.336896\pi\)
\(728\) 0 0
\(729\) 41.0000 1.51852
\(730\) 0 0
\(731\) 31.1769 1.15312
\(732\) 0 0
\(733\) 31.1769i 1.15155i 0.817610 + 0.575773i \(0.195299\pi\)
−0.817610 + 0.575773i \(0.804701\pi\)
\(734\) 0 0
\(735\) 54.9909 2.02837
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.7846 −0.764574 −0.382287 0.924044i \(-0.624863\pi\)
−0.382287 + 0.924044i \(0.624863\pi\)
\(740\) 0 0
\(741\) 18.5203 + 24.2487i 0.680359 + 0.890799i
\(742\) 0 0
\(743\) 18.3303 0.672474 0.336237 0.941777i \(-0.390846\pi\)
0.336237 + 0.941777i \(0.390846\pi\)
\(744\) 0 0
\(745\) −60.0000 −2.19823
\(746\) 0 0
\(747\) −69.2820 −2.53490
\(748\) 0 0
\(749\) 7.93725 0.290021
\(750\) 0 0
\(751\) −45.8258 −1.67221 −0.836103 0.548573i \(-0.815171\pi\)
−0.836103 + 0.548573i \(0.815171\pi\)
\(752\) 0 0
\(753\) 45.8258i 1.66998i
\(754\) 0 0
\(755\) 63.4980i 2.31093i
\(756\) 0 0
\(757\) 48.4974i 1.76267i 0.472493 + 0.881334i \(0.343354\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 0.543750 0.271875 0.962333i \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) 0 0
\(763\) 2.64575i 0.0957826i
\(764\) 0 0
\(765\) 41.5692i 1.50294i
\(766\) 0 0
\(767\) 21.0000i 0.758266i
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 24.2487 0.873296
\(772\) 0 0
\(773\) −39.6863 −1.42742 −0.713708 0.700443i \(-0.752986\pi\)
−0.713708 + 0.700443i \(0.752986\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −28.0000 −1.00449
\(778\) 0 0
\(779\) 24.2487 + 31.7490i 0.868800 + 1.13753i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 21.0000i 0.750479i
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) 2.64575i 0.0943108i −0.998888 0.0471554i \(-0.984984\pi\)
0.998888 0.0471554i \(-0.0150156\pi\)
\(788\) 0 0
\(789\) −31.7490 −1.13029
\(790\) 0 0
\(791\) −9.16515 −0.325875
\(792\) 0 0
\(793\) 18.3303i 0.650928i
\(794\) 0 0
\(795\) 72.7461 2.58004
\(796\) 0 0
\(797\) 23.8118 0.843456 0.421728 0.906722i \(-0.361424\pi\)
0.421728 + 0.906722i \(0.361424\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 36.6606i 1.29534i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 10.3923i 0.366281i
\(806\) 0