Properties

Label 1216.2.c.f.609.4
Level $1216$
Weight $2$
Character 1216.609
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 9\)
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 609.4
Root \(1.65831 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1216.609
Dual form 1216.2.c.f.609.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} +3.31662i q^{5} +3.31662 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} +3.31662i q^{5} +3.31662 q^{7} -1.00000 q^{9} +5.00000i q^{11} -6.63325 q^{15} +5.00000 q^{17} -1.00000i q^{19} +6.63325i q^{21} +6.63325 q^{23} -6.00000 q^{25} +4.00000i q^{27} -6.63325i q^{29} -10.0000 q^{33} +11.0000i q^{35} -6.63325i q^{37} -6.00000 q^{41} -1.00000i q^{43} -3.31662i q^{45} +9.94987 q^{47} +4.00000 q^{49} +10.0000i q^{51} -13.2665i q^{53} -16.5831 q^{55} +2.00000 q^{57} -6.00000i q^{59} -9.94987i q^{61} -3.31662 q^{63} +8.00000i q^{67} +13.2665i q^{69} -6.63325 q^{71} -9.00000 q^{73} -12.0000i q^{75} +16.5831i q^{77} -13.2665 q^{79} -11.0000 q^{81} -4.00000i q^{83} +16.5831i q^{85} +13.2665 q^{87} -4.00000 q^{89} +3.31662 q^{95} -12.0000 q^{97} -5.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 20q^{17} - 24q^{25} - 40q^{33} - 24q^{41} + 16q^{49} + 8q^{57} - 36q^{73} - 44q^{81} - 16q^{89} - 48q^{97} + O(q^{100}) \)

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
\(5\) 3.31662i 1.48324i 0.670820 + 0.741620i \(0.265942\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 3.31662 1.25357 0.626783 0.779194i \(-0.284371\pi\)
0.626783 + 0.779194i \(0.284371\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000i 1.50756i 0.657129 + 0.753778i \(0.271771\pi\)
−0.657129 + 0.753778i \(0.728229\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −6.63325 −1.71270
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i
\(20\) 0 0
\(21\) 6.63325i 1.44749i
\(22\) 0 0
\(23\) 6.63325 1.38313 0.691564 0.722315i \(-0.256922\pi\)
0.691564 + 0.722315i \(0.256922\pi\)
\(24\) 0 0
\(25\) −6.00000 −1.20000
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) − 6.63325i − 1.23176i −0.787839 0.615882i \(-0.788800\pi\)
0.787839 0.615882i \(-0.211200\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −10.0000 −1.74078
\(34\) 0 0
\(35\) 11.0000i 1.85934i
\(36\) 0 0
\(37\) − 6.63325i − 1.09050i −0.838274 0.545250i \(-0.816435\pi\)
0.838274 0.545250i \(-0.183565\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 1.00000i − 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 0 0
\(45\) − 3.31662i − 0.494413i
\(46\) 0 0
\(47\) 9.94987 1.45134 0.725669 0.688044i \(-0.241530\pi\)
0.725669 + 0.688044i \(0.241530\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) 10.0000i 1.40028i
\(52\) 0 0
\(53\) − 13.2665i − 1.82229i −0.412082 0.911147i \(-0.635198\pi\)
0.412082 0.911147i \(-0.364802\pi\)
\(54\) 0 0
\(55\) −16.5831 −2.23607
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) − 6.00000i − 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) − 9.94987i − 1.27395i −0.770884 0.636975i \(-0.780185\pi\)
0.770884 0.636975i \(-0.219815\pi\)
\(62\) 0 0
\(63\) −3.31662 −0.417855
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 13.2665i 1.59710i
\(70\) 0 0
\(71\) −6.63325 −0.787222 −0.393611 0.919277i \(-0.628774\pi\)
−0.393611 + 0.919277i \(0.628774\pi\)
\(72\) 0 0
\(73\) −9.00000 −1.05337 −0.526685 0.850060i \(-0.676565\pi\)
−0.526685 + 0.850060i \(0.676565\pi\)
\(74\) 0 0
\(75\) − 12.0000i − 1.38564i
\(76\) 0 0
\(77\) 16.5831i 1.88982i
\(78\) 0 0
\(79\) −13.2665 −1.49260 −0.746299 0.665611i \(-0.768171\pi\)
−0.746299 + 0.665611i \(0.768171\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 16.5831i 1.79869i
\(86\) 0 0
\(87\) 13.2665 1.42232
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.31662 0.340279
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) − 5.00000i − 0.502519i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −6.63325 −0.653594 −0.326797 0.945095i \(-0.605969\pi\)
−0.326797 + 0.945095i \(0.605969\pi\)
\(104\) 0 0
\(105\) −22.0000 −2.14698
\(106\) 0 0
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) − 13.2665i − 1.27070i −0.772224 0.635350i \(-0.780856\pi\)
0.772224 0.635350i \(-0.219144\pi\)
\(110\) 0 0
\(111\) 13.2665 1.25920
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 22.0000i 2.05151i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.5831 1.52017
\(120\) 0 0
\(121\) −14.0000 −1.27273
\(122\) 0 0
\(123\) − 12.0000i − 1.08200i
\(124\) 0 0
\(125\) − 3.31662i − 0.296648i
\(126\) 0 0
\(127\) 6.63325 0.588606 0.294303 0.955712i \(-0.404913\pi\)
0.294303 + 0.955712i \(0.404913\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) − 15.0000i − 1.31056i −0.755388 0.655278i \(-0.772551\pi\)
0.755388 0.655278i \(-0.227449\pi\)
\(132\) 0 0
\(133\) − 3.31662i − 0.287588i
\(134\) 0 0
\(135\) −13.2665 −1.14180
\(136\) 0 0
\(137\) 5.00000 0.427179 0.213589 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 0 0
\(139\) 11.0000i 0.933008i 0.884519 + 0.466504i \(0.154487\pi\)
−0.884519 + 0.466504i \(0.845513\pi\)
\(140\) 0 0
\(141\) 19.8997i 1.67586i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 22.0000 1.82700
\(146\) 0 0
\(147\) 8.00000i 0.659829i
\(148\) 0 0
\(149\) 3.31662i 0.271708i 0.990729 + 0.135854i \(0.0433779\pi\)
−0.990729 + 0.135854i \(0.956622\pi\)
\(150\) 0 0
\(151\) 6.63325 0.539806 0.269903 0.962887i \(-0.413008\pi\)
0.269903 + 0.962887i \(0.413008\pi\)
\(152\) 0 0
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 26.5330 2.10420
\(160\) 0 0
\(161\) 22.0000 1.73384
\(162\) 0 0
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 0 0
\(165\) − 33.1662i − 2.58199i
\(166\) 0 0
\(167\) −6.63325 −0.513296 −0.256648 0.966505i \(-0.582618\pi\)
−0.256648 + 0.966505i \(0.582618\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 1.00000i 0.0764719i
\(172\) 0 0
\(173\) 19.8997i 1.51295i 0.654023 + 0.756475i \(0.273080\pi\)
−0.654023 + 0.756475i \(0.726920\pi\)
\(174\) 0 0
\(175\) −19.8997 −1.50428
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 18.0000i 1.34538i 0.739923 + 0.672692i \(0.234862\pi\)
−0.739923 + 0.672692i \(0.765138\pi\)
\(180\) 0 0
\(181\) 19.8997i 1.47914i 0.673081 + 0.739568i \(0.264970\pi\)
−0.673081 + 0.739568i \(0.735030\pi\)
\(182\) 0 0
\(183\) 19.8997 1.47103
\(184\) 0 0
\(185\) 22.0000 1.61747
\(186\) 0 0
\(187\) 25.0000i 1.82818i
\(188\) 0 0
\(189\) 13.2665i 0.964996i
\(190\) 0 0
\(191\) −16.5831 −1.19991 −0.599956 0.800033i \(-0.704815\pi\)
−0.599956 + 0.800033i \(0.704815\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 26.5330i − 1.89040i −0.326495 0.945199i \(-0.605868\pi\)
0.326495 0.945199i \(-0.394132\pi\)
\(198\) 0 0
\(199\) −16.5831 −1.17555 −0.587773 0.809026i \(-0.699995\pi\)
−0.587773 + 0.809026i \(0.699995\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) − 22.0000i − 1.54410i
\(204\) 0 0
\(205\) − 19.8997i − 1.38986i
\(206\) 0 0
\(207\) −6.63325 −0.461043
\(208\) 0 0
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) − 14.0000i − 0.963800i −0.876226 0.481900i \(-0.839947\pi\)
0.876226 0.481900i \(-0.160053\pi\)
\(212\) 0 0
\(213\) − 13.2665i − 0.909006i
\(214\) 0 0
\(215\) 3.31662 0.226192
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 18.0000i − 1.21633i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.8997 1.33259 0.666293 0.745690i \(-0.267880\pi\)
0.666293 + 0.745690i \(0.267880\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 0 0
\(227\) − 8.00000i − 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 0 0
\(229\) − 23.2164i − 1.53418i −0.641539 0.767091i \(-0.721704\pi\)
0.641539 0.767091i \(-0.278296\pi\)
\(230\) 0 0
\(231\) −33.1662 −2.18218
\(232\) 0 0
\(233\) 3.00000 0.196537 0.0982683 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 0 0
\(235\) 33.0000i 2.15268i
\(236\) 0 0
\(237\) − 26.5330i − 1.72350i
\(238\) 0 0
\(239\) 3.31662 0.214535 0.107267 0.994230i \(-0.465790\pi\)
0.107267 + 0.994230i \(0.465790\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) − 10.0000i − 0.641500i
\(244\) 0 0
\(245\) 13.2665i 0.847566i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 5.00000i 0.315597i 0.987471 + 0.157799i \(0.0504397\pi\)
−0.987471 + 0.157799i \(0.949560\pi\)
\(252\) 0 0
\(253\) 33.1662i 2.08514i
\(254\) 0 0
\(255\) −33.1662 −2.07695
\(256\) 0 0
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) − 22.0000i − 1.36701i
\(260\) 0 0
\(261\) 6.63325i 0.410588i
\(262\) 0 0
\(263\) −3.31662 −0.204512 −0.102256 0.994758i \(-0.532606\pi\)
−0.102256 + 0.994758i \(0.532606\pi\)
\(264\) 0 0
\(265\) 44.0000 2.70290
\(266\) 0 0
\(267\) − 8.00000i − 0.489592i
\(268\) 0 0
\(269\) 13.2665i 0.808873i 0.914566 + 0.404436i \(0.132532\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 19.8997 1.20882 0.604412 0.796672i \(-0.293408\pi\)
0.604412 + 0.796672i \(0.293408\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 30.0000i − 1.80907i
\(276\) 0 0
\(277\) − 3.31662i − 0.199277i −0.995024 0.0996383i \(-0.968231\pi\)
0.995024 0.0996383i \(-0.0317686\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 11.0000i 0.653882i 0.945045 + 0.326941i \(0.106018\pi\)
−0.945045 + 0.326941i \(0.893982\pi\)
\(284\) 0 0
\(285\) 6.63325i 0.392920i
\(286\) 0 0
\(287\) −19.8997 −1.17465
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) − 24.0000i − 1.40690i
\(292\) 0 0
\(293\) 26.5330i 1.55007i 0.631916 + 0.775037i \(0.282269\pi\)
−0.631916 + 0.775037i \(0.717731\pi\)
\(294\) 0 0
\(295\) 19.8997 1.15861
\(296\) 0 0
\(297\) −20.0000 −1.16052
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 3.31662i − 0.191167i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 33.0000 1.88957
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) − 13.2665i − 0.754705i
\(310\) 0 0
\(311\) −16.5831 −0.940343 −0.470171 0.882575i \(-0.655808\pi\)
−0.470171 + 0.882575i \(0.655808\pi\)
\(312\) 0 0
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 0 0
\(315\) − 11.0000i − 0.619780i
\(316\) 0 0
\(317\) − 6.63325i − 0.372560i −0.982497 0.186280i \(-0.940357\pi\)
0.982497 0.186280i \(-0.0596432\pi\)
\(318\) 0 0
\(319\) 33.1662 1.85695
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) − 5.00000i − 0.278207i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 26.5330 1.46728
\(328\) 0 0
\(329\) 33.0000 1.81935
\(330\) 0 0
\(331\) 4.00000i 0.219860i 0.993939 + 0.109930i \(0.0350627\pi\)
−0.993939 + 0.109930i \(0.964937\pi\)
\(332\) 0 0
\(333\) 6.63325i 0.363500i
\(334\) 0 0
\(335\) −26.5330 −1.44965
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 36.0000i 1.95525i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −9.94987 −0.537243
\(344\) 0 0
\(345\) −44.0000 −2.36888
\(346\) 0 0
\(347\) 27.0000i 1.44944i 0.689046 + 0.724718i \(0.258030\pi\)
−0.689046 + 0.724718i \(0.741970\pi\)
\(348\) 0 0
\(349\) − 9.94987i − 0.532605i −0.963890 0.266302i \(-0.914198\pi\)
0.963890 0.266302i \(-0.0858019\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) − 22.0000i − 1.16764i
\(356\) 0 0
\(357\) 33.1662i 1.75534i
\(358\) 0 0
\(359\) 23.2164 1.22531 0.612657 0.790349i \(-0.290101\pi\)
0.612657 + 0.790349i \(0.290101\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) − 28.0000i − 1.46962i
\(364\) 0 0
\(365\) − 29.8496i − 1.56240i
\(366\) 0 0
\(367\) 6.63325 0.346253 0.173126 0.984900i \(-0.444613\pi\)
0.173126 + 0.984900i \(0.444613\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) − 44.0000i − 2.28437i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 6.63325 0.342540
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.00000i 0.308199i 0.988055 + 0.154100i \(0.0492477\pi\)
−0.988055 + 0.154100i \(0.950752\pi\)
\(380\) 0 0
\(381\) 13.2665i 0.679663i
\(382\) 0 0
\(383\) 13.2665 0.677886 0.338943 0.940807i \(-0.389931\pi\)
0.338943 + 0.940807i \(0.389931\pi\)
\(384\) 0 0
\(385\) −55.0000 −2.80306
\(386\) 0 0
\(387\) 1.00000i 0.0508329i
\(388\) 0 0
\(389\) 16.5831i 0.840798i 0.907339 + 0.420399i \(0.138110\pi\)
−0.907339 + 0.420399i \(0.861890\pi\)
\(390\) 0 0
\(391\) 33.1662 1.67729
\(392\) 0 0
\(393\) 30.0000 1.51330
\(394\) 0 0
\(395\) − 44.0000i − 2.21388i
\(396\) 0 0
\(397\) 9.94987i 0.499370i 0.968327 + 0.249685i \(0.0803271\pi\)
−0.968327 + 0.249685i \(0.919673\pi\)
\(398\) 0 0
\(399\) 6.63325 0.332078
\(400\) 0 0
\(401\) 16.0000 0.799002 0.399501 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 36.4829i − 1.81285i
\(406\) 0 0
\(407\) 33.1662 1.64399
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 10.0000i 0.493264i
\(412\) 0 0
\(413\) − 19.8997i − 0.979203i
\(414\) 0 0
\(415\) 13.2665 0.651227
\(416\) 0 0
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) − 4.00000i − 0.195413i −0.995215 0.0977064i \(-0.968849\pi\)
0.995215 0.0977064i \(-0.0311506\pi\)
\(420\) 0 0
\(421\) 13.2665i 0.646570i 0.946302 + 0.323285i \(0.104787\pi\)
−0.946302 + 0.323285i \(0.895213\pi\)
\(422\) 0 0
\(423\) −9.94987 −0.483779
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) − 33.0000i − 1.59698i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −26.5330 −1.27805 −0.639025 0.769186i \(-0.720662\pi\)
−0.639025 + 0.769186i \(0.720662\pi\)
\(432\) 0 0
\(433\) −38.0000 −1.82616 −0.913082 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 0 0
\(435\) 44.0000i 2.10964i
\(436\) 0 0
\(437\) − 6.63325i − 0.317311i
\(438\) 0 0
\(439\) −19.8997 −0.949763 −0.474882 0.880050i \(-0.657509\pi\)
−0.474882 + 0.880050i \(0.657509\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) − 13.0000i − 0.617649i −0.951119 0.308824i \(-0.900064\pi\)
0.951119 0.308824i \(-0.0999355\pi\)
\(444\) 0 0
\(445\) − 13.2665i − 0.628892i
\(446\) 0 0
\(447\) −6.63325 −0.313742
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) − 30.0000i − 1.41264i
\(452\) 0 0
\(453\) 13.2665i 0.623315i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 0 0
\(459\) 20.0000i 0.933520i
\(460\) 0 0
\(461\) 16.5831i 0.772353i 0.922425 + 0.386177i \(0.126204\pi\)
−0.922425 + 0.386177i \(0.873796\pi\)
\(462\) 0 0
\(463\) −23.2164 −1.07896 −0.539478 0.842000i \(-0.681378\pi\)
−0.539478 + 0.842000i \(0.681378\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.00000i 0.138823i 0.997588 + 0.0694117i \(0.0221122\pi\)
−0.997588 + 0.0694117i \(0.977888\pi\)
\(468\) 0 0
\(469\) 26.5330i 1.22518i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.00000 0.229900
\(474\) 0 0
\(475\) 6.00000i 0.275299i
\(476\) 0 0
\(477\) 13.2665i 0.607431i
\(478\) 0 0
\(479\) 6.63325 0.303081 0.151540 0.988451i \(-0.451577\pi\)
0.151540 + 0.988451i \(0.451577\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 44.0000i 2.00207i
\(484\) 0 0
\(485\) − 39.7995i − 1.80720i
\(486\) 0 0
\(487\) −33.1662 −1.50291 −0.751453 0.659787i \(-0.770647\pi\)
−0.751453 + 0.659787i \(0.770647\pi\)
\(488\) 0 0
\(489\) 40.0000 1.80886
\(490\) 0 0
\(491\) 20.0000i 0.902587i 0.892375 + 0.451294i \(0.149037\pi\)
−0.892375 + 0.451294i \(0.850963\pi\)
\(492\) 0 0
\(493\) − 33.1662i − 1.49373i
\(494\) 0 0
\(495\) 16.5831 0.745356
\(496\) 0 0
\(497\) −22.0000 −0.986835
\(498\) 0 0
\(499\) − 3.00000i − 0.134298i −0.997743 0.0671492i \(-0.978610\pi\)
0.997743 0.0671492i \(-0.0213904\pi\)
\(500\) 0 0
\(501\) − 13.2665i − 0.592703i
\(502\) 0 0
\(503\) −19.8997 −0.887286 −0.443643 0.896204i \(-0.646314\pi\)
−0.443643 + 0.896204i \(0.646314\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 26.0000i 1.15470i
\(508\) 0 0
\(509\) 26.5330i 1.17605i 0.808841 + 0.588027i \(0.200095\pi\)
−0.808841 + 0.588027i \(0.799905\pi\)
\(510\) 0 0
\(511\) −29.8496 −1.32047
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) − 22.0000i − 0.969436i
\(516\) 0 0
\(517\) 49.7494i 2.18797i
\(518\) 0 0
\(519\) −39.7995 −1.74700
\(520\) 0 0
\(521\) 16.0000 0.700973 0.350486 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) 0 0
\(523\) 6.00000i 0.262362i 0.991358 + 0.131181i \(0.0418769\pi\)
−0.991358 + 0.131181i \(0.958123\pi\)
\(524\) 0 0
\(525\) − 39.7995i − 1.73699i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 21.0000 0.913043
\(530\) 0 0
\(531\) 6.00000i 0.260378i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 19.8997 0.860341
\(536\) 0 0
\(537\) −36.0000 −1.55351
\(538\) 0 0
\(539\) 20.0000i 0.861461i
\(540\) 0 0
\(541\) − 9.94987i − 0.427779i −0.976858 0.213889i \(-0.931387\pi\)
0.976858 0.213889i \(-0.0686132\pi\)
\(542\) 0 0
\(543\) −39.7995 −1.70796
\(544\) 0 0
\(545\) 44.0000 1.88475
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) 0 0
\(549\) 9.94987i 0.424650i
\(550\) 0 0
\(551\) −6.63325 −0.282586
\(552\) 0 0
\(553\) −44.0000 −1.87107
\(554\) 0 0
\(555\) 44.0000i 1.86770i
\(556\) 0 0
\(557\) − 3.31662i − 0.140530i −0.997528 0.0702650i \(-0.977616\pi\)
0.997528 0.0702650i \(-0.0223845\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −50.0000 −2.11100
\(562\) 0 0
\(563\) 14.0000i 0.590030i 0.955493 + 0.295015i \(0.0953246\pi\)
−0.955493 + 0.295015i \(0.904675\pi\)
\(564\) 0 0
\(565\) 59.6992i 2.51157i
\(566\) 0 0
\(567\) −36.4829 −1.53214
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) 44.0000i 1.84134i 0.390339 + 0.920671i \(0.372358\pi\)
−0.390339 + 0.920671i \(0.627642\pi\)
\(572\) 0 0
\(573\) − 33.1662i − 1.38554i
\(574\) 0 0
\(575\) −39.7995 −1.65975
\(576\) 0 0
\(577\) −43.0000 −1.79011 −0.895057 0.445952i \(-0.852865\pi\)
−0.895057 + 0.445952i \(0.852865\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 13.2665i − 0.550387i
\(582\) 0 0
\(583\) 66.3325 2.74721
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.00000i 0.206372i 0.994662 + 0.103186i \(0.0329037\pi\)
−0.994662 + 0.103186i \(0.967096\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 53.0660 2.18284
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 55.0000i 2.25478i
\(596\) 0 0
\(597\) − 33.1662i − 1.35740i
\(598\) 0 0
\(599\) −13.2665 −0.542054 −0.271027 0.962572i \(-0.587363\pi\)
−0.271027 + 0.962572i \(0.587363\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 0 0
\(603\) − 8.00000i − 0.325785i
\(604\) 0 0
\(605\) − 46.4327i − 1.88776i
\(606\) 0 0
\(607\) 26.5330 1.07694 0.538471 0.842644i \(-0.319002\pi\)
0.538471 + 0.842644i \(0.319002\pi\)
\(608\) 0 0
\(609\) 44.0000 1.78297
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 3.31662i − 0.133957i −0.997754 0.0669786i \(-0.978664\pi\)
0.997754 0.0669786i \(-0.0213359\pi\)
\(614\) 0 0
\(615\) 39.7995 1.60487
\(616\) 0 0
\(617\) −41.0000 −1.65060 −0.825299 0.564696i \(-0.808993\pi\)
−0.825299 + 0.564696i \(0.808993\pi\)
\(618\) 0 0
\(619\) 4.00000i 0.160774i 0.996764 + 0.0803868i \(0.0256155\pi\)
−0.996764 + 0.0803868i \(0.974384\pi\)
\(620\) 0 0
\(621\) 26.5330i 1.06473i
\(622\) 0 0
\(623\) −13.2665 −0.531511
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 10.0000i 0.399362i
\(628\) 0 0
\(629\) − 33.1662i − 1.32242i
\(630\) 0 0
\(631\) 9.94987 0.396098 0.198049 0.980192i \(-0.436539\pi\)
0.198049 + 0.980192i \(0.436539\pi\)
\(632\) 0 0
\(633\) 28.0000 1.11290
\(634\) 0 0
\(635\) 22.0000i 0.873043i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.63325 0.262407
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) − 5.00000i − 0.197181i −0.995128 0.0985904i \(-0.968567\pi\)
0.995128 0.0985904i \(-0.0314334\pi\)
\(644\) 0 0
\(645\) 6.63325i 0.261184i
\(646\) 0 0
\(647\) −16.5831 −0.651950 −0.325975 0.945378i \(-0.605693\pi\)
−0.325975 + 0.945378i \(0.605693\pi\)
\(648\) 0 0
\(649\) 30.0000 1.17760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.4829i 1.42769i 0.700306 + 0.713843i \(0.253047\pi\)
−0.700306 + 0.713843i \(0.746953\pi\)
\(654\) 0 0
\(655\) 49.7494 1.94387
\(656\) 0 0
\(657\) 9.00000 0.351123
\(658\) 0 0
\(659\) − 26.0000i − 1.01282i −0.862294 0.506408i \(-0.830973\pi\)
0.862294 0.506408i \(-0.169027\pi\)
\(660\) 0 0
\(661\) − 39.7995i − 1.54802i −0.633173 0.774011i \(-0.718248\pi\)
0.633173 0.774011i \(-0.281752\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.0000 0.426562
\(666\) 0 0
\(667\) − 44.0000i − 1.70369i
\(668\) 0 0
\(669\) 39.7995i 1.53874i
\(670\) 0 0
\(671\) 49.7494 1.92055
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) − 24.0000i − 0.923760i
\(676\) 0 0
\(677\) 19.8997i 0.764809i 0.923995 + 0.382405i \(0.124904\pi\)
−0.923995 + 0.382405i \(0.875096\pi\)
\(678\) 0 0
\(679\) −39.7995 −1.52736
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) 0 0
\(683\) − 24.0000i − 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(684\) 0 0
\(685\) 16.5831i 0.633609i
\(686\) 0 0
\(687\) 46.4327 1.77152
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 31.0000i 1.17930i 0.807661 + 0.589648i \(0.200733\pi\)
−0.807661 + 0.589648i \(0.799267\pi\)
\(692\) 0 0
\(693\) − 16.5831i − 0.629941i
\(694\) 0 0
\(695\) −36.4829 −1.38387
\(696\) 0 0
\(697\) −30.0000 −1.13633
\(698\) 0 0
\(699\) 6.00000i 0.226941i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −6.63325 −0.250178
\(704\) 0 0
\(705\) −66.0000 −2.48570
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 13.2665i 0.498234i 0.968473 + 0.249117i \(0.0801403\pi\)
−0.968473 + 0.249117i \(0.919860\pi\)
\(710\) 0 0
\(711\) 13.2665 0.497533
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.63325i 0.247723i
\(718\) 0 0
\(719\) −3.31662 −0.123689 −0.0618446 0.998086i \(-0.519698\pi\)
−0.0618446 + 0.998086i \(0.519698\pi\)
\(720\) 0 0
\(721\) −22.0000 −0.819323
\(722\) 0 0
\(723\) 52.0000i 1.93390i
\(724\) 0 0
\(725\) 39.7995i 1.47812i
\(726\) 0 0
\(727\) −16.5831 −0.615034 −0.307517 0.951543i \(-0.599498\pi\)
−0.307517 + 0.951543i \(0.599498\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) − 5.00000i − 0.184932i
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −26.5330 −0.978684
\(736\) 0 0
\(737\) −40.0000 −1.47342
\(738\) 0 0
\(739\) 45.0000i 1.65535i 0.561206 + 0.827676i \(0.310337\pi\)
−0.561206 + 0.827676i \(0.689663\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 53.0660 1.94680 0.973401 0.229107i \(-0.0735805\pi\)
0.973401 + 0.229107i \(0.0735805\pi\)
\(744\) 0 0
\(745\) −11.0000 −0.403009
\(746\) 0 0
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) − 19.8997i − 0.727121i
\(750\) 0 0
\(751\) −39.7995 −1.45230 −0.726152 0.687534i \(-0.758693\pi\)
−0.726152 + 0.687534i \(0.758693\pi\)
\(752\) 0 0
\(753\) −10.0000 −0.364420
\(754\) 0 0
\(755\) 22.0000i 0.800662i
\(756\) 0 0
\(757\) 9.94987i 0.361634i 0.983517 + 0.180817i \(0.0578742\pi\)
−0.983517 + 0.180817i \(0.942126\pi\)
\(758\) 0 0
\(759\) −66.3325 −2.40772
\(760\) 0 0
\(761\) 1.00000 0.0362500 0.0181250 0.999836i \(-0.494230\pi\)
0.0181250 + 0.999836i \(0.494230\pi\)
\(762\) 0 0
\(763\) − 44.0000i − 1.59291i
\(764\) 0 0
\(765\) − 16.5831i − 0.599564i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 33.0000 1.19001 0.595005 0.803722i \(-0.297150\pi\)
0.595005 + 0.803722i \(0.297150\pi\)
\(770\) 0 0
\(771\) − 16.0000i − 0.576226i
\(772\) 0 0
\(773\) 6.63325i 0.238581i 0.992859 + 0.119291i \(0.0380620\pi\)
−0.992859 + 0.119291i \(0.961938\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 44.0000 1.57849
\(778\) 0 0
\(779\) 6.00000i 0.214972i
\(780\) 0 0
\(781\) − 33.1662i − 1.18678i
\(782\) 0 0
\(783\) 26.5330 0.948212
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) 0 0
\(789\) − 6.63325i − 0.236150i
\(790\) 0 0
\(791\) 59.6992 2.12266
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 88.0000i 3.12104i
\(796\) 0 0
\(797\) 26.5330i 0.939847i 0.882707 + 0.469924i \(0.155719\pi\)
−0.882707 + 0.469924i \(0.844281\pi\)
\(798\) 0 0
\(799\) 49.7494 1.76001
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) − 45.0000i − 1.58802i
\(804\) 0 0
\(805\) 72.9657i 2.57170i
\(806\) 0 0
\(807\) −26.5330 −0.934006
\(808\) 0 0
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) 28.0000i 0.983213i 0.870817 + 0.491606i \(0.163590\pi\)
−0.870817 + 0.491606i \(0.836410\pi\)
\(812\) 0 0
\(813\) 39.7995i 1.39583i
\(814\) 0 0
\(815\) 66.3325 2.32353
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 9.94987i − 0.347253i −0.984812 0.173627i \(-0.944451\pi\)
0.984812 0.173627i \(-0.0555486\pi\)
\(822\) 0 0
\(823\) −29.8496 −1.04049 −0.520246 0.854016i \(-0.674160\pi\)
−0.520246 + 0.854016i \(0.674160\pi\)
\(824\) 0 0
\(825\) 60.0000 2.08893
\(826\) 0 0
\(827\) − 32.0000i − 1.11275i −0.830932 0.556375i \(-0.812192\pi\)
0.830932 0.556375i \(-0.187808\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 6.63325 0.230105
\(832\) 0 0
\(833\) 20.0000 0.692959
\(834\) 0 0
\(835\) − 22.0000i − 0.761341i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 46.4327 1.60304 0.801518 0.597970i \(-0.204026\pi\)
0.801518 + 0.597970i \(0.204026\pi\)
\(840\) 0 0
\(841\) −15.0000 −0.517241
\(842\) 0 0
\(843\) − 52.0000i − 1.79098i
\(844\) 0 0
\(845\) 43.1161i 1.48324i
\(846\) 0 0
\(847\) −46.4327 −1.59545
\(848\) 0 0
\(849\) −22.0000 −0.755038
\(850\) 0 0
\(851\) − 44.0000i − 1.50830i
\(852\) 0 0
\(853\) − 53.0660i − 1.81695i −0.417945 0.908473i \(-0.637249\pi\)
0.417945 0.908473i \(-0.362751\pi\)
\(854\) 0 0
\(855\) −3.31662 −0.113426
\(856\) 0 0
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 0 0
\(859\) − 41.0000i − 1.39890i −0.714681 0.699451i \(-0.753428\pi\)
0.714681 0.699451i \(-0.246572\pi\)
\(860\) 0 0
\(861\) − 39.7995i − 1.35636i
\(862\) 0 0
\(863\) −33.1662 −1.12899 −0.564496 0.825436i \(-0.690929\pi\)
−0.564496 + 0.825436i \(0.690929\pi\)
\(864\) 0 0
\(865\) −66.0000 −2.24407
\(866\) 0 0
\(867\) 16.0000i 0.543388i
\(868\) 0 0
\(869\) − 66.3325i − 2.25018i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) − 11.0000i − 0.371868i
\(876\) 0 0
\(877\) − 6.63325i − 0.223989i −0.993709 0.111994i \(-0.964276\pi\)
0.993709 0.111994i \(-0.0357239\pi\)
\(878\) 0 0
\(879\) −53.0660 −1.78987
\(880\) 0 0
\(881\) 51.0000 1.71823 0.859117 0.511780i \(-0.171014\pi\)
0.859117 + 0.511780i \(0.171014\pi\)
\(882\) 0 0
\(883\) − 31.0000i − 1.04323i −0.853180 0.521617i \(-0.825329\pi\)
0.853180 0.521617i \(-0.174671\pi\)
\(884\) 0 0
\(885\) 39.7995i 1.33785i
\(886\) 0 0
\(887\) 19.8997 0.668168 0.334084 0.942543i \(-0.391573\pi\)
0.334084 + 0.942543i \(0.391573\pi\)
\(888\) 0 0
\(889\) 22.0000 0.737856
\(890\) 0 0
\(891\) − 55.0000i − 1.84257i
\(892\) 0 0
\(893\) − 9.94987i − 0.332960i
\(894\) 0 0
\(895\) −59.6992 −1.99553
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) − 66.3325i − 2.20986i
\(902\) 0 0
\(903\) 6.63325 0.220741
\(904\) 0 0
\(905\) −66.0000 −2.19391
\(906\) 0 0
\(907\) 32.0000i 1.06254i 0.847202 + 0.531271i \(0.178286\pi\)
−0.847202 + 0.531271i \(0.821714\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −19.8997 −0.659308 −0.329654 0.944102i \(-0.606932\pi\)
−0.329654 + 0.944102i \(0.606932\pi\)
\(912\) 0 0
\(913\) 20.0000 0.661903
\(914\) 0 0
\(915\) 66.0000i 2.18189i
\(916\) 0 0
\(917\) − 49.7494i − 1.64287i
\(918\) 0 0
\(919\) −33.1662 −1.09405 −0.547027 0.837115i \(-0.684240\pi\)
−0.547027 + 0.837115i \(0.684240\pi\)
\(920\) 0 0
\(921\) −24.0000 −0.790827
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 39.7995i 1.30860i
\(926\) 0 0
\(927\) 6.63325 0.217865
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) − 4.00000i − 0.131095i
\(932\) 0 0
\(933\) − 33.1662i − 1.08581i
\(934\) 0 0
\(935\) −82.9156 −2.71163
\(936\) 0 0
\(937\) 1.00000 0.0326686 0.0163343 0.999867i \(-0.494800\pi\)
0.0163343 + 0.999867i \(0.494800\pi\)
\(938\) 0 0
\(939\) − 68.0000i − 2.21910i
\(940\) 0 0
\(941\) 46.4327i 1.51366i 0.653609 + 0.756832i \(0.273254\pi\)
−0.653609 + 0.756832i \(0.726746\pi\)
\(942\) 0 0
\(943\) −39.7995 −1.29605
\(944\) 0 0
\(945\) −44.0000 −1.43132
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 13.2665 0.430196
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 0 0
\(955\) − 55.0000i − 1.77976i
\(956\) 0 0
\(957\) 66.3325i 2.14423i
\(958\) 0 0
\(959\) 16.5831 0.535497
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 6.00000i 0.193347i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 33.1662 1.06655 0.533277 0.845940i \(-0.320960\pi\)
0.533277 + 0.845940i \(0.320960\pi\)
\(968\) 0 0
\(969\) 10.0000 0.321246
\(970\) 0 0
\(971\) − 32.0000i − 1.02693i −0.858111 0.513464i \(-0.828362\pi\)
0.858111 0.513464i \(-0.171638\pi\)
\(972\) 0 0
\(973\) 36.4829i 1.16959i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.0000 0.511885 0.255943 0.966692i \(-0.417614\pi\)
0.255943 + 0.966692i \(0.417614\pi\)
\(978\) 0 0
\(979\) − 20.0000i − 0.639203i
\(980\) 0 0
\(981\) 13.2665i 0.423567i
\(982\) 0 0
\(983\) −13.2665 −0.423136 −0.211568 0.977363i \(-0.567857\pi\)
−0.211568 + 0.977363i \(0.567857\pi\)
\(984\) 0 0
\(985\) 88.0000 2.80391
\(986\) 0 0
\(987\) 66.0000i 2.10080i
\(988\) 0 0
\(989\) − 6.63325i − 0.210925i
\(990\) 0 0
\(991\) −33.1662 −1.05356 −0.526780 0.850001i \(-0.676601\pi\)
−0.526780 + 0.850001i \(0.676601\pi\)
\(992\) 0 0
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) − 55.0000i − 1.74362i
\(996\) 0 0
\(997\) − 43.1161i − 1.36550i −0.730651 0.682751i \(-0.760784\pi\)
0.730651 0.682751i \(-0.239216\pi\)
\(998\) 0 0
\(999\) 26.5330 0.839467
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 1216.2.c.f.609.4 yes 4
4.3 odd 2 inner 1216.2.c.f.609.2 yes 4
8.3 odd 2 inner 1216.2.c.f.609.3 yes 4
8.5 even 2 inner 1216.2.c.f.609.1 4
16.3 odd 4 4864.2.a.y.1.2 2
16.5 even 4 4864.2.a.y.1.1 2
16.11 odd 4 4864.2.a.r.1.1 2
16.13 even 4 4864.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.f.609.1 4 8.5 even 2 inner
1216.2.c.f.609.2 yes 4 4.3 odd 2 inner
1216.2.c.f.609.3 yes 4 8.3 odd 2 inner
1216.2.c.f.609.4 yes 4 1.1 even 1 trivial
4864.2.a.r.1.1 2 16.11 odd 4
4864.2.a.r.1.2 2 16.13 even 4
4864.2.a.y.1.1 2 16.5 even 4
4864.2.a.y.1.2 2 16.3 odd 4