Properties

Label 1216.2.b.a.607.4
Level $1216$
Weight $2$
Character 1216.607
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 607.4
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1216.607
Dual form 1216.2.b.a.607.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} +1.73205i q^{5} +1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} +1.73205i q^{5} +1.00000i q^{7} -1.00000 q^{9} -5.19615 q^{11} -4.00000 q^{13} -3.46410 q^{15} +3.00000 q^{17} +(-1.73205 - 4.00000i) q^{19} -2.00000 q^{21} +2.00000 q^{25} +4.00000i q^{27} -6.00000 q^{29} -10.3923 q^{31} -10.3923i q^{33} -1.73205 q^{35} +2.00000 q^{37} -8.00000i q^{39} -3.46410i q^{41} +5.19615 q^{43} -1.73205i q^{45} +9.00000i q^{47} +6.00000 q^{49} +6.00000i q^{51} +6.00000 q^{53} -9.00000i q^{55} +(8.00000 - 3.46410i) q^{57} +1.73205i q^{61} -1.00000i q^{63} -6.92820i q^{65} +2.00000i q^{67} -6.92820 q^{71} -11.0000 q^{73} +4.00000i q^{75} -5.19615i q^{77} -3.46410 q^{79} -11.0000 q^{81} +3.46410 q^{83} +5.19615i q^{85} -12.0000i q^{87} +3.46410i q^{89} -4.00000i q^{91} -20.7846i q^{93} +(6.92820 - 3.00000i) q^{95} -13.8564i q^{97} +5.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 16q^{13} + 12q^{17} - 8q^{21} + 8q^{25} - 24q^{29} + 8q^{37} + 24q^{49} + 24q^{53} + 32q^{57} - 44q^{73} - 44q^{81} + O(q^{100}) \)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 1.73205i 0.774597i 0.921954 + 0.387298i \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −5.19615 −1.56670 −0.783349 0.621582i \(-0.786490\pi\)
−0.783349 + 0.621582i \(0.786490\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −1.73205 4.00000i −0.397360 0.917663i
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −10.3923 −1.86651 −0.933257 0.359211i \(-0.883046\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) 10.3923i 1.80907i
\(34\) 0 0
\(35\) −1.73205 −0.292770
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 8.00000i 1.28103i
\(40\) 0 0
\(41\) 3.46410i 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) 5.19615 0.792406 0.396203 0.918163i \(-0.370328\pi\)
0.396203 + 0.918163i \(0.370328\pi\)
\(44\) 0 0
\(45\) 1.73205i 0.258199i
\(46\) 0 0
\(47\) 9.00000i 1.31278i 0.754420 + 0.656392i \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 6.00000i 0.840168i
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 9.00000i 1.21356i
\(56\) 0 0
\(57\) 8.00000 3.46410i 1.05963 0.458831i
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.73205i 0.221766i 0.993833 + 0.110883i \(0.0353679\pi\)
−0.993833 + 0.110883i \(0.964632\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 6.92820i 0.859338i
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.92820 −0.822226 −0.411113 0.911584i \(-0.634860\pi\)
−0.411113 + 0.911584i \(0.634860\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 4.00000i 0.461880i
\(76\) 0 0
\(77\) 5.19615i 0.592157i
\(78\) 0 0
\(79\) −3.46410 −0.389742 −0.194871 0.980829i \(-0.562429\pi\)
−0.194871 + 0.980829i \(0.562429\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 3.46410 0.380235 0.190117 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) 0 0
\(85\) 5.19615i 0.563602i
\(86\) 0 0
\(87\) 12.0000i 1.28654i
\(88\) 0 0
\(89\) 3.46410i 0.367194i 0.983002 + 0.183597i \(0.0587741\pi\)
−0.983002 + 0.183597i \(0.941226\pi\)
\(90\) 0 0
\(91\) 4.00000i 0.419314i
\(92\) 0 0
\(93\) 20.7846i 2.15526i
\(94\) 0 0
\(95\) 6.92820 3.00000i 0.710819 0.307794i
\(96\) 0 0
\(97\) 13.8564i 1.40690i −0.710742 0.703452i \(-0.751641\pi\)
0.710742 0.703452i \(-0.248359\pi\)
\(98\) 0 0
\(99\) 5.19615 0.522233
\(100\) 0 0
\(101\) 13.8564i 1.37876i −0.724398 0.689382i \(-0.757882\pi\)
0.724398 0.689382i \(-0.242118\pi\)
\(102\) 0 0
\(103\) −6.92820 −0.682656 −0.341328 0.939944i \(-0.610877\pi\)
−0.341328 + 0.939944i \(0.610877\pi\)
\(104\) 0 0
\(105\) 3.46410i 0.338062i
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 4.00000i 0.379663i
\(112\) 0 0
\(113\) 17.3205i 1.62938i 0.579899 + 0.814688i \(0.303092\pi\)
−0.579899 + 0.814688i \(0.696908\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 3.00000i 0.275010i
\(120\) 0 0
\(121\) 16.0000 1.45455
\(122\) 0 0
\(123\) 6.92820 0.624695
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) −10.3923 −0.922168 −0.461084 0.887357i \(-0.652539\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 0 0
\(129\) 10.3923i 0.914991i
\(130\) 0 0
\(131\) −1.73205 −0.151330 −0.0756650 0.997133i \(-0.524108\pi\)
−0.0756650 + 0.997133i \(0.524108\pi\)
\(132\) 0 0
\(133\) 4.00000 1.73205i 0.346844 0.150188i
\(134\) 0 0
\(135\) −6.92820 −0.596285
\(136\) 0 0
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 0 0
\(139\) −1.73205 −0.146911 −0.0734553 0.997299i \(-0.523403\pi\)
−0.0734553 + 0.997299i \(0.523403\pi\)
\(140\) 0 0
\(141\) −18.0000 −1.51587
\(142\) 0 0
\(143\) 20.7846 1.73810
\(144\) 0 0
\(145\) 10.3923i 0.863034i
\(146\) 0 0
\(147\) 12.0000i 0.989743i
\(148\) 0 0
\(149\) 19.0526i 1.56085i 0.625252 + 0.780423i \(0.284996\pi\)
−0.625252 + 0.780423i \(0.715004\pi\)
\(150\) 0 0
\(151\) 13.8564 1.12762 0.563809 0.825905i \(-0.309335\pi\)
0.563809 + 0.825905i \(0.309335\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 18.0000i 1.44579i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 12.0000i 0.951662i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −17.3205 −1.35665 −0.678323 0.734763i \(-0.737293\pi\)
−0.678323 + 0.734763i \(0.737293\pi\)
\(164\) 0 0
\(165\) 18.0000 1.40130
\(166\) 0 0
\(167\) 17.3205 1.34030 0.670151 0.742225i \(-0.266230\pi\)
0.670151 + 0.742225i \(0.266230\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 1.73205 + 4.00000i 0.132453 + 0.305888i
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 2.00000i 0.151186i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000i 0.448461i −0.974536 0.224231i \(-0.928013\pi\)
0.974536 0.224231i \(-0.0719869\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) −3.46410 −0.256074
\(184\) 0 0
\(185\) 3.46410i 0.254686i
\(186\) 0 0
\(187\) −15.5885 −1.13994
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 9.00000i 0.651217i −0.945505 0.325609i \(-0.894431\pi\)
0.945505 0.325609i \(-0.105569\pi\)
\(192\) 0 0
\(193\) 17.3205i 1.24676i 0.781920 + 0.623379i \(0.214240\pi\)
−0.781920 + 0.623379i \(0.785760\pi\)
\(194\) 0 0
\(195\) 13.8564 0.992278
\(196\) 0 0
\(197\) 20.7846i 1.48084i 0.672143 + 0.740421i \(0.265374\pi\)
−0.672143 + 0.740421i \(0.734626\pi\)
\(198\) 0 0
\(199\) 19.0000i 1.34687i 0.739244 + 0.673437i \(0.235183\pi\)
−0.739244 + 0.673437i \(0.764817\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.00000 + 20.7846i 0.622543 + 1.43770i
\(210\) 0 0
\(211\) 4.00000i 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 0 0
\(213\) 13.8564i 0.949425i
\(214\) 0 0
\(215\) 9.00000i 0.613795i
\(216\) 0 0
\(217\) 10.3923i 0.705476i
\(218\) 0 0
\(219\) 22.0000i 1.48662i
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −10.3923 −0.695920 −0.347960 0.937509i \(-0.613126\pi\)
−0.347960 + 0.937509i \(0.613126\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) 24.0000i 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 0 0
\(229\) 25.9808i 1.71686i 0.512933 + 0.858429i \(0.328559\pi\)
−0.512933 + 0.858429i \(0.671441\pi\)
\(230\) 0 0
\(231\) 10.3923 0.683763
\(232\) 0 0
\(233\) −9.00000 −0.589610 −0.294805 0.955557i \(-0.595255\pi\)
−0.294805 + 0.955557i \(0.595255\pi\)
\(234\) 0 0
\(235\) −15.5885 −1.01688
\(236\) 0 0
\(237\) 6.92820i 0.450035i
\(238\) 0 0
\(239\) 9.00000i 0.582162i 0.956698 + 0.291081i \(0.0940149\pi\)
−0.956698 + 0.291081i \(0.905985\pi\)
\(240\) 0 0
\(241\) 3.46410i 0.223142i −0.993756 0.111571i \(-0.964412\pi\)
0.993756 0.111571i \(-0.0355883\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) 10.3923i 0.663940i
\(246\) 0 0
\(247\) 6.92820 + 16.0000i 0.440831 + 1.01806i
\(248\) 0 0
\(249\) 6.92820i 0.439057i
\(250\) 0 0
\(251\) 12.1244 0.765283 0.382641 0.923897i \(-0.375015\pi\)
0.382641 + 0.923897i \(0.375015\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −10.3923 −0.650791
\(256\) 0 0
\(257\) 27.7128i 1.72868i 0.502910 + 0.864339i \(0.332263\pi\)
−0.502910 + 0.864339i \(0.667737\pi\)
\(258\) 0 0
\(259\) 2.00000i 0.124274i
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 3.00000i 0.184988i −0.995713 0.0924940i \(-0.970516\pi\)
0.995713 0.0924940i \(-0.0294839\pi\)
\(264\) 0 0
\(265\) 10.3923i 0.638394i
\(266\) 0 0
\(267\) −6.92820 −0.423999
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 28.0000i 1.70088i −0.526073 0.850439i \(-0.676336\pi\)
0.526073 0.850439i \(-0.323664\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 0 0
\(275\) −10.3923 −0.626680
\(276\) 0 0
\(277\) 5.19615i 0.312207i −0.987741 0.156103i \(-0.950107\pi\)
0.987741 0.156103i \(-0.0498933\pi\)
\(278\) 0 0
\(279\) 10.3923 0.622171
\(280\) 0 0
\(281\) 27.7128i 1.65321i 0.562784 + 0.826604i \(0.309730\pi\)
−0.562784 + 0.826604i \(0.690270\pi\)
\(282\) 0 0
\(283\) 25.9808 1.54440 0.772198 0.635382i \(-0.219157\pi\)
0.772198 + 0.635382i \(0.219157\pi\)
\(284\) 0 0
\(285\) 6.00000 + 13.8564i 0.355409 + 0.820783i
\(286\) 0 0
\(287\) 3.46410 0.204479
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 27.7128 1.62455
\(292\) 0 0
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 20.7846i 1.20605i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 5.19615i 0.299501i
\(302\) 0 0
\(303\) 27.7128 1.59206
\(304\) 0 0
\(305\) −3.00000 −0.171780
\(306\) 0 0
\(307\) 22.0000i 1.25561i 0.778372 + 0.627803i \(0.216046\pi\)
−0.778372 + 0.627803i \(0.783954\pi\)
\(308\) 0 0
\(309\) 13.8564i 0.788263i
\(310\) 0 0
\(311\) 33.0000i 1.87126i 0.352985 + 0.935629i \(0.385167\pi\)
−0.352985 + 0.935629i \(0.614833\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 1.73205 0.0975900
\(316\) 0 0
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) 31.1769 1.74557
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) −5.19615 12.0000i −0.289122 0.667698i
\(324\) 0 0
\(325\) −8.00000 −0.443760
\(326\) 0 0
\(327\) 28.0000i 1.54840i
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) 26.0000i 1.42909i 0.699590 + 0.714545i \(0.253366\pi\)
−0.699590 + 0.714545i \(0.746634\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −3.46410 −0.189264
\(336\) 0 0
\(337\) 27.7128i 1.50961i −0.655947 0.754807i \(-0.727731\pi\)
0.655947 0.754807i \(-0.272269\pi\)
\(338\) 0 0
\(339\) −34.6410 −1.88144
\(340\) 0 0
\(341\) 54.0000 2.92426
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.5885 0.836832 0.418416 0.908255i \(-0.362585\pi\)
0.418416 + 0.908255i \(0.362585\pi\)
\(348\) 0 0
\(349\) 8.66025i 0.463573i −0.972767 0.231786i \(-0.925543\pi\)
0.972767 0.231786i \(-0.0744570\pi\)
\(350\) 0 0
\(351\) 16.0000i 0.854017i
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 12.0000i 0.636894i
\(356\) 0 0
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) 15.0000i 0.791670i 0.918322 + 0.395835i \(0.129545\pi\)
−0.918322 + 0.395835i \(0.870455\pi\)
\(360\) 0 0
\(361\) −13.0000 + 13.8564i −0.684211 + 0.729285i
\(362\) 0 0
\(363\) 32.0000i 1.67956i
\(364\) 0 0
\(365\) 19.0526i 0.997257i
\(366\) 0 0
\(367\) 8.00000i 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 0 0
\(369\) 3.46410i 0.180334i
\(370\) 0 0
\(371\) 6.00000i 0.311504i
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) −24.2487 −1.25220
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 20.0000i 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 0 0
\(381\) 20.7846i 1.06483i
\(382\) 0 0
\(383\) 31.1769 1.59307 0.796533 0.604595i \(-0.206665\pi\)
0.796533 + 0.604595i \(0.206665\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) 0 0
\(387\) −5.19615 −0.264135
\(388\) 0 0
\(389\) 5.19615i 0.263455i −0.991286 0.131728i \(-0.957948\pi\)
0.991286 0.131728i \(-0.0420524\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 3.46410i 0.174741i
\(394\) 0 0
\(395\) 6.00000i 0.301893i
\(396\) 0 0
\(397\) 25.9808i 1.30394i −0.758246 0.651969i \(-0.773943\pi\)
0.758246 0.651969i \(-0.226057\pi\)
\(398\) 0 0
\(399\) 3.46410 + 8.00000i 0.173422 + 0.400501i
\(400\) 0 0
\(401\) 10.3923i 0.518967i 0.965748 + 0.259483i \(0.0835523\pi\)
−0.965748 + 0.259483i \(0.916448\pi\)
\(402\) 0 0
\(403\) 41.5692 2.07071
\(404\) 0 0
\(405\) 19.0526i 0.946729i
\(406\) 0 0
\(407\) −10.3923 −0.515127
\(408\) 0 0
\(409\) 24.2487i 1.19902i 0.800367 + 0.599511i \(0.204638\pi\)
−0.800367 + 0.599511i \(0.795362\pi\)
\(410\) 0 0
\(411\) 18.0000i 0.887875i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 6.00000i 0.294528i
\(416\) 0 0
\(417\) 3.46410i 0.169638i
\(418\) 0 0
\(419\) 10.3923 0.507697 0.253849 0.967244i \(-0.418303\pi\)
0.253849 + 0.967244i \(0.418303\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 9.00000i 0.437595i
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −1.73205 −0.0838198
\(428\) 0 0
\(429\) 41.5692i 2.00698i
\(430\) 0 0
\(431\) 27.7128 1.33488 0.667440 0.744664i \(-0.267390\pi\)
0.667440 + 0.744664i \(0.267390\pi\)
\(432\) 0 0
\(433\) 10.3923i 0.499422i −0.968320 0.249711i \(-0.919664\pi\)
0.968320 0.249711i \(-0.0803357\pi\)
\(434\) 0 0
\(435\) 20.7846 0.996546
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −10.3923 −0.495998 −0.247999 0.968760i \(-0.579773\pi\)
−0.247999 + 0.968760i \(0.579773\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 1.73205 0.0822922 0.0411461 0.999153i \(-0.486899\pi\)
0.0411461 + 0.999153i \(0.486899\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) −38.1051 −1.80231
\(448\) 0 0
\(449\) 6.92820i 0.326962i −0.986546 0.163481i \(-0.947728\pi\)
0.986546 0.163481i \(-0.0522723\pi\)
\(450\) 0 0
\(451\) 18.0000i 0.847587i
\(452\) 0 0
\(453\) 27.7128i 1.30206i
\(454\) 0 0
\(455\) 6.92820 0.324799
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 0 0
\(459\) 12.0000i 0.560112i
\(460\) 0 0
\(461\) 22.5167i 1.04871i −0.851501 0.524353i \(-0.824307\pi\)
0.851501 0.524353i \(-0.175693\pi\)
\(462\) 0 0
\(463\) 25.0000i 1.16185i −0.813958 0.580924i \(-0.802691\pi\)
0.813958 0.580924i \(-0.197309\pi\)
\(464\) 0 0
\(465\) 36.0000 1.66946
\(466\) 0 0
\(467\) 15.5885 0.721348 0.360674 0.932692i \(-0.382547\pi\)
0.360674 + 0.932692i \(0.382547\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −27.0000 −1.24146
\(474\) 0 0
\(475\) −3.46410 8.00000i −0.158944 0.367065i
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 24.0000i 1.09659i −0.836286 0.548294i \(-0.815277\pi\)
0.836286 0.548294i \(-0.184723\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 24.0000 1.08978
\(486\) 0 0
\(487\) −27.7128 −1.25579 −0.627894 0.778299i \(-0.716083\pi\)
−0.627894 + 0.778299i \(0.716083\pi\)
\(488\) 0 0
\(489\) 34.6410i 1.56652i
\(490\) 0 0
\(491\) −24.2487 −1.09433 −0.547165 0.837025i \(-0.684293\pi\)
−0.547165 + 0.837025i \(0.684293\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) 9.00000i 0.404520i
\(496\) 0 0
\(497\) 6.92820i 0.310772i
\(498\) 0 0
\(499\) −36.3731 −1.62828 −0.814141 0.580667i \(-0.802792\pi\)
−0.814141 + 0.580667i \(0.802792\pi\)
\(500\) 0 0
\(501\) 34.6410i 1.54765i
\(502\) 0 0
\(503\) 24.0000i 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 0 0
\(507\) 6.00000i 0.266469i
\(508\) 0 0
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 11.0000i 0.486611i
\(512\) 0 0
\(513\) 16.0000 6.92820i 0.706417 0.305888i
\(514\) 0 0
\(515\) 12.0000i 0.528783i
\(516\) 0 0
\(517\) 46.7654i 2.05674i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.3205i 0.758825i −0.925228 0.379413i \(-0.876126\pi\)
0.925228 0.379413i \(-0.123874\pi\)
\(522\) 0 0
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) −31.1769 −1.35809
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 13.8564i 0.600188i
\(534\) 0 0
\(535\) −20.7846 −0.898597
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) −31.1769 −1.34288
\(540\) 0 0
\(541\) 8.66025i 0.372333i 0.982518 + 0.186167i \(0.0596065\pi\)
−0.982518 + 0.186167i \(0.940394\pi\)
\(542\) 0 0
\(543\) 40.0000i 1.71656i
\(544\) 0 0
\(545\) 24.2487i 1.03870i
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 0 0
\(549\) 1.73205i 0.0739221i
\(550\) 0 0
\(551\) 10.3923 + 24.0000i 0.442727 + 1.02243i
\(552\) 0 0
\(553\) 3.46410i 0.147309i
\(554\) 0 0
\(555\) −6.92820 −0.294086
\(556\) 0 0
\(557\) 8.66025i 0.366947i 0.983025 + 0.183473i \(0.0587341\pi\)
−0.983025 + 0.183473i \(0.941266\pi\)
\(558\) 0 0
\(559\) −20.7846 −0.879095
\(560\) 0 0
\(561\) 31.1769i 1.31629i
\(562\) 0 0
\(563\) 30.0000i 1.26435i 0.774826 + 0.632175i \(0.217837\pi\)
−0.774826 + 0.632175i \(0.782163\pi\)
\(564\) 0 0
\(565\) −30.0000 −1.26211
\(566\) 0 0
\(567\) 11.0000i 0.461957i
\(568\) 0 0
\(569\) 27.7128i 1.16178i 0.813982 + 0.580891i \(0.197296\pi\)
−0.813982 + 0.580891i \(0.802704\pi\)
\(570\) 0 0
\(571\) 3.46410 0.144968 0.0724841 0.997370i \(-0.476907\pi\)
0.0724841 + 0.997370i \(0.476907\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 29.0000 1.20729 0.603643 0.797255i \(-0.293715\pi\)
0.603643 + 0.797255i \(0.293715\pi\)
\(578\) 0 0
\(579\) −34.6410 −1.43963
\(580\) 0 0
\(581\) 3.46410i 0.143715i
\(582\) 0 0
\(583\) −31.1769 −1.29122
\(584\) 0 0
\(585\) 6.92820i 0.286446i
\(586\) 0 0
\(587\) 32.9090 1.35830 0.679149 0.734000i \(-0.262349\pi\)
0.679149 + 0.734000i \(0.262349\pi\)
\(588\) 0 0
\(589\) 18.0000 + 41.5692i 0.741677 + 1.71283i
\(590\) 0 0
\(591\) −41.5692 −1.70993
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −5.19615 −0.213021
\(596\) 0 0
\(597\) −38.0000 −1.55524
\(598\) 0 0
\(599\) 38.1051 1.55693 0.778466 0.627686i \(-0.215998\pi\)
0.778466 + 0.627686i \(0.215998\pi\)
\(600\) 0 0
\(601\) 41.5692i 1.69564i −0.530281 0.847822i \(-0.677914\pi\)
0.530281 0.847822i \(-0.322086\pi\)
\(602\) 0 0
\(603\) 2.00000i 0.0814463i
\(604\) 0 0
\(605\) 27.7128i 1.12669i
\(606\) 0 0
\(607\) −6.92820 −0.281207 −0.140604 0.990066i \(-0.544904\pi\)
−0.140604 + 0.990066i \(0.544904\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 36.0000i 1.45640i
\(612\) 0 0
\(613\) 39.8372i 1.60901i −0.593947 0.804504i \(-0.702431\pi\)
0.593947 0.804504i \(-0.297569\pi\)
\(614\) 0 0
\(615\) 12.0000i 0.483887i
\(616\) 0 0
\(617\) −3.00000 −0.120775 −0.0603877 0.998175i \(-0.519234\pi\)
−0.0603877 + 0.998175i \(0.519234\pi\)
\(618\) 0 0
\(619\) −24.2487 −0.974638 −0.487319 0.873224i \(-0.662025\pi\)
−0.487319 + 0.873224i \(0.662025\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.46410 −0.138786
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) −41.5692 + 18.0000i −1.66011 + 0.718851i
\(628\) 0 0
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 7.00000i 0.278666i 0.990246 + 0.139333i \(0.0444958\pi\)
−0.990246 + 0.139333i \(0.955504\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 18.0000i 0.714308i
\(636\) 0 0
\(637\) −24.0000 −0.950915
\(638\) 0 0
\(639\) 6.92820 0.274075
\(640\) 0 0
\(641\) 34.6410i 1.36824i 0.729370 + 0.684119i \(0.239813\pi\)
−0.729370 + 0.684119i \(0.760187\pi\)
\(642\) 0 0
\(643\) 5.19615 0.204916 0.102458 0.994737i \(-0.467329\pi\)
0.102458 + 0.994737i \(0.467329\pi\)
\(644\) 0 0
\(645\) −18.0000 −0.708749
\(646\) 0 0
\(647\) 9.00000i 0.353827i −0.984226 0.176913i \(-0.943389\pi\)
0.984226 0.176913i \(-0.0566112\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 20.7846 0.814613
\(652\) 0 0
\(653\) 8.66025i 0.338902i 0.985539 + 0.169451i \(0.0541994\pi\)
−0.985539 + 0.169451i \(0.945801\pi\)
\(654\) 0 0
\(655\) 3.00000i 0.117220i
\(656\) 0 0
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 0 0
\(663\) 24.0000i 0.932083i
\(664\) 0 0
\(665\) 3.00000 + 6.92820i 0.116335 + 0.268664i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 20.7846i 0.803579i
\(670\) 0 0
\(671\) 9.00000i 0.347441i
\(672\) 0 0
\(673\) 13.8564i 0.534125i −0.963679 0.267063i \(-0.913947\pi\)
0.963679 0.267063i \(-0.0860531\pi\)
\(674\) 0 0
\(675\) 8.00000i 0.307920i
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 13.8564 0.531760
\(680\) 0 0
\(681\) 48.0000 1.83936
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) 15.5885i 0.595604i
\(686\) 0 0
\(687\) −51.9615 −1.98246
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −5.19615 −0.197671 −0.0988355 0.995104i \(-0.531512\pi\)
−0.0988355 + 0.995104i \(0.531512\pi\)
\(692\) 0 0
\(693\) 5.19615i 0.197386i
\(694\) 0 0
\(695\) 3.00000i 0.113796i
\(696\) 0 0
\(697\) 10.3923i 0.393637i
\(698\) 0 0
\(699\) 18.0000i 0.680823i
\(700\) 0 0
\(701\) 27.7128i 1.04670i −0.852118 0.523349i \(-0.824682\pi\)
0.852118 0.523349i \(-0.175318\pi\)
\(702\) 0 0
\(703\) −3.46410 8.00000i −0.130651 0.301726i
\(704\) 0 0
\(705\) 31.1769i 1.17419i
\(706\) 0 0
\(707\) 13.8564 0.521124
\(708\) 0 0
\(709\) 13.8564i 0.520388i 0.965556 + 0.260194i \(0.0837866\pi\)
−0.965556 + 0.260194i \(0.916213\pi\)
\(710\) 0 0
\(711\) 3.46410 0.129914
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 36.0000i 1.34632i
\(716\) 0 0
\(717\) −18.0000 −0.672222
\(718\) 0 0
\(719\) 51.0000i 1.90198i 0.309223 + 0.950990i \(0.399931\pi\)
−0.309223 + 0.950990i \(0.600069\pi\)
\(720\) 0 0
\(721\) 6.92820i 0.258020i
\(722\) 0 0
\(723\) 6.92820 0.257663
\(724\) 0 0
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) 43.0000i 1.59478i −0.603463 0.797391i \(-0.706213\pi\)
0.603463 0.797391i \(-0.293787\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 15.5885 0.576560
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −20.7846 −0.766652
\(736\) 0 0
\(737\) 10.3923i 0.382805i
\(738\) 0 0
\(739\) 15.5885 0.573431 0.286715 0.958016i \(-0.407437\pi\)
0.286715 + 0.958016i \(0.407437\pi\)
\(740\) 0 0
\(741\) −32.0000 + 13.8564i −1.17555 + 0.509028i
\(742\) 0 0
\(743\) 24.2487 0.889599 0.444799 0.895630i \(-0.353275\pi\)
0.444799 + 0.895630i \(0.353275\pi\)
\(744\) 0 0
\(745\) −33.0000 −1.20903
\(746\) 0 0
\(747\) −3.46410 −0.126745
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 27.7128 1.01125 0.505627 0.862752i \(-0.331261\pi\)
0.505627 + 0.862752i \(0.331261\pi\)
\(752\) 0 0
\(753\) 24.2487i 0.883672i
\(754\) 0 0
\(755\) 24.0000i 0.873449i
\(756\) 0 0
\(757\) 12.1244i 0.440667i 0.975425 + 0.220334i \(0.0707146\pi\)
−0.975425 + 0.220334i \(0.929285\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 0 0
\(763\) 14.0000i 0.506834i
\(764\) 0 0
\(765\) 5.19615i 0.187867i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −23.0000 −0.829401 −0.414701 0.909958i \(-0.636114\pi\)
−0.414701 + 0.909958i \(0.636114\pi\)
\(770\) 0 0
\(771\) −55.4256 −1.99611
\(772\) 0 0
\(773\) 24.0000 0.863220 0.431610 0.902060i \(-0.357946\pi\)
0.431610 + 0.902060i \(0.357946\pi\)
\(774\) 0 0
\(775\) −20.7846 −0.746605
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) −13.8564 + 6.00000i −0.496457 + 0.214972i
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 0 0
\(783\) 24.0000i 0.857690i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −17.3205 −0.615846
\(792\) 0 0
\(793\) 6.92820i 0.246028i
\(794\) 0 0
\(795\) −20.7846 −0.737154
\(796\) 0 0
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 27.0000i 0.955191i
\(800\) 0 0
\(801\) 3.46410i 0.122398i
\(802\) 0 0
\(803\) 57.1577 2.01705
\(804\) 0 0
\(805\) 0 0 <