Properties

Label 1216.2.c.h.609.1
Level $1216$
Weight $2$
Character 1216.609
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.303595776.1
Defining polynomial: \(x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 609.1
Root \(1.26217 - 1.18614i\) of defining polynomial
Character \(\chi\) \(=\) 1216.609
Dual form 1216.2.c.h.609.8

$q$-expansion

\(f(q)\) \(=\) \(q-3.37228i q^{3} -2.52434i q^{5} -3.31662 q^{7} -8.37228 q^{9} +O(q^{10})\) \(q-3.37228i q^{3} -2.52434i q^{5} -3.31662 q^{7} -8.37228 q^{9} +2.37228i q^{11} -5.84096i q^{13} -8.51278 q^{15} +5.00000 q^{17} +1.00000i q^{19} +11.1846i q^{21} -0.792287 q^{23} -1.37228 q^{25} +18.1168i q^{27} +2.67181i q^{29} -3.46410 q^{31} +8.00000 q^{33} +8.37228i q^{35} -10.0974i q^{37} -19.6974 q^{39} -3.62772i q^{43} +21.1345i q^{45} +0.644810 q^{47} +4.00000 q^{49} -16.8614i q^{51} -6.13592i q^{53} +5.98844 q^{55} +3.37228 q^{57} +1.37228i q^{59} +14.5012i q^{61} +27.7677 q^{63} -14.7446 q^{65} +2.62772i q^{67} +2.67181i q^{69} +6.63325 q^{71} +8.48913 q^{73} +4.62772i q^{75} -7.86797i q^{77} +0.294954 q^{79} +35.9783 q^{81} -8.00000i q^{83} -12.6217i q^{85} +9.01011 q^{87} -15.4891 q^{89} +19.3723i q^{91} +11.6819i q^{93} +2.52434 q^{95} -2.74456 q^{97} -19.8614i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 44q^{9} + O(q^{10}) \) \( 8q - 44q^{9} + 40q^{17} + 12q^{25} + 64q^{33} + 32q^{49} + 4q^{57} - 72q^{65} - 24q^{73} + 104q^{81} - 32q^{89} + 24q^{97} + O(q^{100}) \)

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
\(3\) − 3.37228i − 1.94699i −0.228714 0.973494i \(-0.573452\pi\)
0.228714 0.973494i \(-0.426548\pi\)
\(4\) 0 0
\(5\) − 2.52434i − 1.12892i −0.825461 0.564459i \(-0.809085\pi\)
0.825461 0.564459i \(-0.190915\pi\)
\(6\) 0 0
\(7\) −3.31662 −1.25357 −0.626783 0.779194i \(-0.715629\pi\)
−0.626783 + 0.779194i \(0.715629\pi\)
\(8\) 0 0
\(9\) −8.37228 −2.79076
\(10\) 0 0
\(11\) 2.37228i 0.715270i 0.933862 + 0.357635i \(0.116417\pi\)
−0.933862 + 0.357635i \(0.883583\pi\)
\(12\) 0 0
\(13\) − 5.84096i − 1.61999i −0.586436 0.809996i \(-0.699469\pi\)
0.586436 0.809996i \(-0.300531\pi\)
\(14\) 0 0
\(15\) −8.51278 −2.19799
\(16\) 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
\(21\) 11.1846i 2.44068i
\(22\) 0 0
\(23\) −0.792287 −0.165203 −0.0826016 0.996583i \(-0.526323\pi\)
−0.0826016 + 0.996583i \(0.526323\pi\)
\(24\) 0 0
\(25\) −1.37228 −0.274456
\(26\) 0 0
\(27\) 18.1168i 3.48659i
\(28\) 0 0
\(29\) 2.67181i 0.496144i 0.968742 + 0.248072i \(0.0797969\pi\)
−0.968742 + 0.248072i \(0.920203\pi\)
\(30\) 0 0
\(31\) −3.46410 −0.622171 −0.311086 0.950382i \(-0.600693\pi\)
−0.311086 + 0.950382i \(0.600693\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 8.37228i 1.41517i
\(36\) 0 0
\(37\) − 10.0974i − 1.65999i −0.557768 0.829997i \(-0.688342\pi\)
0.557768 0.829997i \(-0.311658\pi\)
\(38\) 0 0
\(39\) −19.6974 −3.15410
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 3.62772i − 0.553222i −0.960982 0.276611i \(-0.910789\pi\)
0.960982 0.276611i \(-0.0892113\pi\)
\(44\) 0 0
\(45\) 21.1345i 3.15054i
\(46\) 0 0
\(47\) 0.644810 0.0940552 0.0470276 0.998894i \(-0.485025\pi\)
0.0470276 + 0.998894i \(0.485025\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) − 16.8614i − 2.36107i
\(52\) 0 0
\(53\) − 6.13592i − 0.842833i −0.906867 0.421416i \(-0.861533\pi\)
0.906867 0.421416i \(-0.138467\pi\)
\(54\) 0 0
\(55\) 5.98844 0.807481
\(56\) 0 0
\(57\) 3.37228 0.446670
\(58\) 0 0
\(59\) 1.37228i 0.178656i 0.996002 + 0.0893279i \(0.0284719\pi\)
−0.996002 + 0.0893279i \(0.971528\pi\)
\(60\) 0 0
\(61\) 14.5012i 1.85669i 0.371719 + 0.928345i \(0.378768\pi\)
−0.371719 + 0.928345i \(0.621232\pi\)
\(62\) 0 0
\(63\) 27.7677 3.49840
\(64\) 0 0
\(65\) −14.7446 −1.82884
\(66\) 0 0
\(67\) 2.62772i 0.321027i 0.987034 + 0.160513i \(0.0513150\pi\)
−0.987034 + 0.160513i \(0.948685\pi\)
\(68\) 0 0
\(69\) 2.67181i 0.321649i
\(70\) 0 0
\(71\) 6.63325 0.787222 0.393611 0.919277i \(-0.371226\pi\)
0.393611 + 0.919277i \(0.371226\pi\)
\(72\) 0 0
\(73\) 8.48913 0.993577 0.496788 0.867872i \(-0.334512\pi\)
0.496788 + 0.867872i \(0.334512\pi\)
\(74\) 0 0
\(75\) 4.62772i 0.534363i
\(76\) 0 0
\(77\) − 7.86797i − 0.896638i
\(78\) 0 0
\(79\) 0.294954 0.0331849 0.0165924 0.999862i \(-0.494718\pi\)
0.0165924 + 0.999862i \(0.494718\pi\)
\(80\) 0 0
\(81\) 35.9783 3.99758
\(82\) 0 0
\(83\) − 8.00000i − 0.878114i −0.898459 0.439057i \(-0.855313\pi\)
0.898459 0.439057i \(-0.144687\pi\)
\(84\) 0 0
\(85\) − 12.6217i − 1.36901i
\(86\) 0 0
\(87\) 9.01011 0.965985
\(88\) 0 0
\(89\) −15.4891 −1.64184 −0.820922 0.571040i \(-0.806540\pi\)
−0.820922 + 0.571040i \(0.806540\pi\)
\(90\) 0 0
\(91\) 19.3723i 2.03077i
\(92\) 0 0
\(93\) 11.6819i 1.21136i
\(94\) 0 0
\(95\) 2.52434 0.258992
\(96\) 0 0
\(97\) −2.74456 −0.278668 −0.139334 0.990245i \(-0.544496\pi\)
−0.139334 + 0.990245i \(0.544496\pi\)
\(98\) 0 0
\(99\) − 19.8614i − 1.99615i
\(100\) 0 0
\(101\) 13.8564i 1.37876i 0.724398 + 0.689382i \(0.242118\pi\)
−0.724398 + 0.689382i \(0.757882\pi\)
\(102\) 0 0
\(103\) −11.9769 −1.18012 −0.590058 0.807361i \(-0.700895\pi\)
−0.590058 + 0.807361i \(0.700895\pi\)
\(104\) 0 0
\(105\) 28.2337 2.75533
\(106\) 0 0
\(107\) − 13.3723i − 1.29275i −0.763021 0.646374i \(-0.776285\pi\)
0.763021 0.646374i \(-0.223715\pi\)
\(108\) 0 0
\(109\) − 3.96143i − 0.379437i −0.981839 0.189718i \(-0.939243\pi\)
0.981839 0.189718i \(-0.0607575\pi\)
\(110\) 0 0
\(111\) −34.0511 −3.23199
\(112\) 0 0
\(113\) −14.7446 −1.38705 −0.693526 0.720432i \(-0.743944\pi\)
−0.693526 + 0.720432i \(0.743944\pi\)
\(114\) 0 0
\(115\) 2.00000i 0.186501i
\(116\) 0 0
\(117\) 48.9022i 4.52101i
\(118\) 0 0
\(119\) −16.5831 −1.52017
\(120\) 0 0
\(121\) 5.37228 0.488389
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 9.15759i − 0.819080i
\(126\) 0 0
\(127\) −12.2718 −1.08895 −0.544475 0.838777i \(-0.683271\pi\)
−0.544475 + 0.838777i \(0.683271\pi\)
\(128\) 0 0
\(129\) −12.2337 −1.07712
\(130\) 0 0
\(131\) − 4.37228i − 0.382008i −0.981589 0.191004i \(-0.938826\pi\)
0.981589 0.191004i \(-0.0611743\pi\)
\(132\) 0 0
\(133\) − 3.31662i − 0.287588i
\(134\) 0 0
\(135\) 45.7330 3.93607
\(136\) 0 0
\(137\) 5.00000 0.427179 0.213589 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 0 0
\(139\) − 3.62772i − 0.307699i −0.988094 0.153850i \(-0.950833\pi\)
0.988094 0.153850i \(-0.0491671\pi\)
\(140\) 0 0
\(141\) − 2.17448i − 0.183124i
\(142\) 0 0
\(143\) 13.8564 1.15873
\(144\) 0 0
\(145\) 6.74456 0.560105
\(146\) 0 0
\(147\) − 13.4891i − 1.11256i
\(148\) 0 0
\(149\) − 4.69882i − 0.384942i −0.981303 0.192471i \(-0.938350\pi\)
0.981303 0.192471i \(-0.0616502\pi\)
\(150\) 0 0
\(151\) −15.7359 −1.28057 −0.640286 0.768137i \(-0.721184\pi\)
−0.640286 + 0.768137i \(0.721184\pi\)
\(152\) 0 0
\(153\) −41.8614 −3.38429
\(154\) 0 0
\(155\) 8.74456i 0.702380i
\(156\) 0 0
\(157\) 18.6101i 1.48525i 0.669708 + 0.742625i \(0.266419\pi\)
−0.669708 + 0.742625i \(0.733581\pi\)
\(158\) 0 0
\(159\) −20.6920 −1.64099
\(160\) 0 0
\(161\) 2.62772 0.207093
\(162\) 0 0
\(163\) 10.7446i 0.841579i 0.907158 + 0.420790i \(0.138247\pi\)
−0.907158 + 0.420790i \(0.861753\pi\)
\(164\) 0 0
\(165\) − 20.1947i − 1.57216i
\(166\) 0 0
\(167\) −6.33830 −0.490472 −0.245236 0.969463i \(-0.578865\pi\)
−0.245236 + 0.969463i \(0.578865\pi\)
\(168\) 0 0
\(169\) −21.1168 −1.62437
\(170\) 0 0
\(171\) − 8.37228i − 0.640244i
\(172\) 0 0
\(173\) − 15.1460i − 1.15153i −0.817615 0.575766i \(-0.804704\pi\)
0.817615 0.575766i \(-0.195296\pi\)
\(174\) 0 0
\(175\) 4.55134 0.344049
\(176\) 0 0
\(177\) 4.62772 0.347841
\(178\) 0 0
\(179\) 22.9783i 1.71748i 0.512416 + 0.858738i \(0.328751\pi\)
−0.512416 + 0.858738i \(0.671249\pi\)
\(180\) 0 0
\(181\) − 8.21782i − 0.610826i −0.952220 0.305413i \(-0.901205\pi\)
0.952220 0.305413i \(-0.0987945\pi\)
\(182\) 0 0
\(183\) 48.9022 3.61495
\(184\) 0 0
\(185\) −25.4891 −1.87400
\(186\) 0 0
\(187\) 11.8614i 0.867392i
\(188\) 0 0
\(189\) − 60.0868i − 4.37067i
\(190\) 0 0
\(191\) 7.07568 0.511978 0.255989 0.966680i \(-0.417599\pi\)
0.255989 + 0.966680i \(0.417599\pi\)
\(192\) 0 0
\(193\) 3.25544 0.234332 0.117166 0.993112i \(-0.462619\pi\)
0.117166 + 0.993112i \(0.462619\pi\)
\(194\) 0 0
\(195\) 49.7228i 3.56072i
\(196\) 0 0
\(197\) 6.33830i 0.451585i 0.974175 + 0.225792i \(0.0724971\pi\)
−0.974175 + 0.225792i \(0.927503\pi\)
\(198\) 0 0
\(199\) 0.147477 0.0104544 0.00522718 0.999986i \(-0.498336\pi\)
0.00522718 + 0.999986i \(0.498336\pi\)
\(200\) 0 0
\(201\) 8.86141 0.625035
\(202\) 0 0
\(203\) − 8.86141i − 0.621949i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.63325 0.461043
\(208\) 0 0
\(209\) −2.37228 −0.164094
\(210\) 0 0
\(211\) − 2.62772i − 0.180900i −0.995901 0.0904498i \(-0.971170\pi\)
0.995901 0.0904498i \(-0.0288305\pi\)
\(212\) 0 0
\(213\) − 22.3692i − 1.53271i
\(214\) 0 0
\(215\) −9.15759 −0.624542
\(216\) 0 0
\(217\) 11.4891 0.779933
\(218\) 0 0
\(219\) − 28.6277i − 1.93448i
\(220\) 0 0
\(221\) − 29.2048i − 1.96453i
\(222\) 0 0
\(223\) 6.92820 0.463947 0.231973 0.972722i \(-0.425482\pi\)
0.231973 + 0.972722i \(0.425482\pi\)
\(224\) 0 0
\(225\) 11.4891 0.765942
\(226\) 0 0
\(227\) − 14.1168i − 0.936968i −0.883472 0.468484i \(-0.844800\pi\)
0.883472 0.468484i \(-0.155200\pi\)
\(228\) 0 0
\(229\) − 12.6217i − 0.834065i −0.908892 0.417032i \(-0.863070\pi\)
0.908892 0.417032i \(-0.136930\pi\)
\(230\) 0 0
\(231\) −26.5330 −1.74574
\(232\) 0 0
\(233\) −10.8832 −0.712979 −0.356490 0.934299i \(-0.616026\pi\)
−0.356490 + 0.934299i \(0.616026\pi\)
\(234\) 0 0
\(235\) − 1.62772i − 0.106181i
\(236\) 0 0
\(237\) − 0.994667i − 0.0646105i
\(238\) 0 0
\(239\) 17.4680 1.12991 0.564955 0.825122i \(-0.308894\pi\)
0.564955 + 0.825122i \(0.308894\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) − 66.9783i − 4.29666i
\(244\) 0 0
\(245\) − 10.0974i − 0.645096i
\(246\) 0 0
\(247\) 5.84096 0.371652
\(248\) 0 0
\(249\) −26.9783 −1.70968
\(250\) 0 0
\(251\) − 17.8614i − 1.12740i −0.825979 0.563701i \(-0.809377\pi\)
0.825979 0.563701i \(-0.190623\pi\)
\(252\) 0 0
\(253\) − 1.87953i − 0.118165i
\(254\) 0 0
\(255\) −42.5639 −2.66545
\(256\) 0 0
\(257\) −16.2337 −1.01263 −0.506315 0.862349i \(-0.668993\pi\)
−0.506315 + 0.862349i \(0.668993\pi\)
\(258\) 0 0
\(259\) 33.4891i 2.08091i
\(260\) 0 0
\(261\) − 22.3692i − 1.38462i
\(262\) 0 0
\(263\) −18.9600 −1.16912 −0.584561 0.811349i \(-0.698733\pi\)
−0.584561 + 0.811349i \(0.698733\pi\)
\(264\) 0 0
\(265\) −15.4891 −0.951489
\(266\) 0 0
\(267\) 52.2337i 3.19665i
\(268\) 0 0
\(269\) − 1.87953i − 0.114597i −0.998357 0.0572984i \(-0.981751\pi\)
0.998357 0.0572984i \(-0.0182486\pi\)
\(270\) 0 0
\(271\) −2.37686 −0.144384 −0.0721920 0.997391i \(-0.522999\pi\)
−0.0721920 + 0.997391i \(0.522999\pi\)
\(272\) 0 0
\(273\) 65.3288 3.95388
\(274\) 0 0
\(275\) − 3.25544i − 0.196310i
\(276\) 0 0
\(277\) 16.3807i 0.984224i 0.870532 + 0.492112i \(0.163775\pi\)
−0.870532 + 0.492112i \(0.836225\pi\)
\(278\) 0 0
\(279\) 29.0024 1.73633
\(280\) 0 0
\(281\) 6.23369 0.371871 0.185935 0.982562i \(-0.440469\pi\)
0.185935 + 0.982562i \(0.440469\pi\)
\(282\) 0 0
\(283\) 17.6277i 1.04786i 0.851762 + 0.523930i \(0.175534\pi\)
−0.851762 + 0.523930i \(0.824466\pi\)
\(284\) 0 0
\(285\) − 8.51278i − 0.504253i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 9.25544i 0.542563i
\(292\) 0 0
\(293\) − 4.84630i − 0.283124i −0.989929 0.141562i \(-0.954788\pi\)
0.989929 0.141562i \(-0.0452124\pi\)
\(294\) 0 0
\(295\) 3.46410 0.201688
\(296\) 0 0
\(297\) −42.9783 −2.49385
\(298\) 0 0
\(299\) 4.62772i 0.267628i
\(300\) 0 0
\(301\) 12.0318i 0.693500i
\(302\) 0 0
\(303\) 46.7277 2.68444
\(304\) 0 0
\(305\) 36.6060 2.09605
\(306\) 0 0
\(307\) − 15.2554i − 0.870674i −0.900268 0.435337i \(-0.856629\pi\)
0.900268 0.435337i \(-0.143371\pi\)
\(308\) 0 0
\(309\) 40.3894i 2.29767i
\(310\) 0 0
\(311\) −23.2164 −1.31648 −0.658240 0.752808i \(-0.728699\pi\)
−0.658240 + 0.752808i \(0.728699\pi\)
\(312\) 0 0
\(313\) 3.88316 0.219489 0.109744 0.993960i \(-0.464997\pi\)
0.109744 + 0.993960i \(0.464997\pi\)
\(314\) 0 0
\(315\) − 70.0951i − 3.94941i
\(316\) 0 0
\(317\) − 21.5769i − 1.21188i −0.795511 0.605940i \(-0.792797\pi\)
0.795511 0.605940i \(-0.207203\pi\)
\(318\) 0 0
\(319\) −6.33830 −0.354876
\(320\) 0 0
\(321\) −45.0951 −2.51696
\(322\) 0 0
\(323\) 5.00000i 0.278207i
\(324\) 0 0
\(325\) 8.01544i 0.444617i
\(326\) 0 0
\(327\) −13.3591 −0.738758
\(328\) 0 0
\(329\) −2.13859 −0.117904
\(330\) 0 0
\(331\) − 10.8614i − 0.596997i −0.954410 0.298498i \(-0.903514\pi\)
0.954410 0.298498i \(-0.0964858\pi\)
\(332\) 0 0
\(333\) 84.5379i 4.63265i
\(334\) 0 0
\(335\) 6.63325 0.362413
\(336\) 0 0
\(337\) −13.2554 −0.722070 −0.361035 0.932552i \(-0.617576\pi\)
−0.361035 + 0.932552i \(0.617576\pi\)
\(338\) 0 0
\(339\) 49.7228i 2.70057i
\(340\) 0 0
\(341\) − 8.21782i − 0.445020i
\(342\) 0 0
\(343\) 9.94987 0.537243
\(344\) 0 0
\(345\) 6.74456 0.363115
\(346\) 0 0
\(347\) − 37.1168i − 1.99254i −0.0863104 0.996268i \(-0.527508\pi\)
0.0863104 0.996268i \(-0.472492\pi\)
\(348\) 0 0
\(349\) − 13.2116i − 0.707201i −0.935397 0.353600i \(-0.884957\pi\)
0.935397 0.353600i \(-0.115043\pi\)
\(350\) 0 0
\(351\) 105.820 5.64824
\(352\) 0 0
\(353\) 3.88316 0.206680 0.103340 0.994646i \(-0.467047\pi\)
0.103340 + 0.994646i \(0.467047\pi\)
\(354\) 0 0
\(355\) − 16.7446i − 0.888709i
\(356\) 0 0
\(357\) 55.9230i 2.95976i
\(358\) 0 0
\(359\) 16.5831 0.875224 0.437612 0.899164i \(-0.355824\pi\)
0.437612 + 0.899164i \(0.355824\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) − 18.1168i − 0.950888i
\(364\) 0 0
\(365\) − 21.4294i − 1.12167i
\(366\) 0 0
\(367\) −4.05401 −0.211618 −0.105809 0.994386i \(-0.533743\pi\)
−0.105809 + 0.994386i \(0.533743\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 20.3505i 1.05655i
\(372\) 0 0
\(373\) − 14.9436i − 0.773753i −0.922132 0.386876i \(-0.873554\pi\)
0.922132 0.386876i \(-0.126446\pi\)
\(374\) 0 0
\(375\) −30.8820 −1.59474
\(376\) 0 0
\(377\) 15.6060 0.803748
\(378\) 0 0
\(379\) − 36.3505i − 1.86720i −0.358315 0.933601i \(-0.616649\pi\)
0.358315 0.933601i \(-0.383351\pi\)
\(380\) 0 0
\(381\) 41.3841i 2.12017i
\(382\) 0 0
\(383\) 25.2434 1.28988 0.644938 0.764235i \(-0.276883\pi\)
0.644938 + 0.764235i \(0.276883\pi\)
\(384\) 0 0
\(385\) −19.8614 −1.01223
\(386\) 0 0
\(387\) 30.3723i 1.54391i
\(388\) 0 0
\(389\) − 24.3036i − 1.23224i −0.787651 0.616121i \(-0.788703\pi\)
0.787651 0.616121i \(-0.211297\pi\)
\(390\) 0 0
\(391\) −3.96143 −0.200338
\(392\) 0 0
\(393\) −14.7446 −0.743765
\(394\) 0 0
\(395\) − 0.744563i − 0.0374630i
\(396\) 0 0
\(397\) − 0.644810i − 0.0323621i −0.999869 0.0161810i \(-0.994849\pi\)
0.999869 0.0161810i \(-0.00515081\pi\)
\(398\) 0 0
\(399\) −11.1846 −0.559930
\(400\) 0 0
\(401\) 35.7228 1.78391 0.891956 0.452122i \(-0.149333\pi\)
0.891956 + 0.452122i \(0.149333\pi\)
\(402\) 0 0
\(403\) 20.2337i 1.00791i
\(404\) 0 0
\(405\) − 90.8213i − 4.51294i
\(406\) 0 0
\(407\) 23.9538 1.18734
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) − 16.8614i − 0.831712i
\(412\) 0 0
\(413\) − 4.55134i − 0.223957i
\(414\) 0 0
\(415\) −20.1947 −0.991319
\(416\) 0 0
\(417\) −12.2337 −0.599086
\(418\) 0 0
\(419\) 0.233688i 0.0114164i 0.999984 + 0.00570820i \(0.00181699\pi\)
−0.999984 + 0.00570820i \(0.998183\pi\)
\(420\) 0 0
\(421\) 20.3971i 0.994093i 0.867724 + 0.497046i \(0.165582\pi\)
−0.867724 + 0.497046i \(0.834418\pi\)
\(422\) 0 0
\(423\) −5.39853 −0.262486
\(424\) 0 0
\(425\) −6.86141 −0.332827
\(426\) 0 0
\(427\) − 48.0951i − 2.32748i
\(428\) 0 0
\(429\) − 46.7277i − 2.25603i
\(430\) 0 0
\(431\) 21.7793 1.04907 0.524535 0.851389i \(-0.324239\pi\)
0.524535 + 0.851389i \(0.324239\pi\)
\(432\) 0 0
\(433\) −1.48913 −0.0715628 −0.0357814 0.999360i \(-0.511392\pi\)
−0.0357814 + 0.999360i \(0.511392\pi\)
\(434\) 0 0
\(435\) − 22.7446i − 1.09052i
\(436\) 0 0
\(437\) − 0.792287i − 0.0379002i
\(438\) 0 0
\(439\) 6.92820 0.330665 0.165333 0.986238i \(-0.447130\pi\)
0.165333 + 0.986238i \(0.447130\pi\)
\(440\) 0 0
\(441\) −33.4891 −1.59472
\(442\) 0 0
\(443\) − 41.3505i − 1.96462i −0.187254 0.982312i \(-0.559959\pi\)
0.187254 0.982312i \(-0.440041\pi\)
\(444\) 0 0
\(445\) 39.0998i 1.85351i
\(446\) 0 0
\(447\) −15.8457 −0.749478
\(448\) 0 0
\(449\) 37.4891 1.76922 0.884611 0.466330i \(-0.154424\pi\)
0.884611 + 0.466330i \(0.154424\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 53.0660i 2.49326i
\(454\) 0 0
\(455\) 48.9022 2.29257
\(456\) 0 0
\(457\) 31.0000 1.45012 0.725059 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 0 0
\(459\) 90.5842i 4.22811i
\(460\) 0 0
\(461\) − 7.86797i − 0.366448i −0.983071 0.183224i \(-0.941347\pi\)
0.983071 0.183224i \(-0.0586533\pi\)
\(462\) 0 0
\(463\) −31.5268 −1.46517 −0.732587 0.680674i \(-0.761687\pi\)
−0.732587 + 0.680674i \(0.761687\pi\)
\(464\) 0 0
\(465\) 29.4891 1.36753
\(466\) 0 0
\(467\) 10.8832i 0.503612i 0.967778 + 0.251806i \(0.0810246\pi\)
−0.967778 + 0.251806i \(0.918975\pi\)
\(468\) 0 0
\(469\) − 8.71516i − 0.402429i
\(470\) 0 0
\(471\) 62.7586 2.89176
\(472\) 0 0
\(473\) 8.60597 0.395703
\(474\) 0 0
\(475\) − 1.37228i − 0.0629646i
\(476\) 0 0
\(477\) 51.3716i 2.35214i
\(478\) 0 0
\(479\) −27.4179 −1.25275 −0.626377 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(480\) 0 0
\(481\) −58.9783 −2.68918
\(482\) 0 0
\(483\) − 8.86141i − 0.403208i
\(484\) 0 0
\(485\) 6.92820i 0.314594i
\(486\) 0 0
\(487\) −22.6641 −1.02701 −0.513505 0.858087i \(-0.671653\pi\)
−0.513505 + 0.858087i \(0.671653\pi\)
\(488\) 0 0
\(489\) 36.2337 1.63854
\(490\) 0 0
\(491\) − 30.9783i − 1.39803i −0.715108 0.699014i \(-0.753622\pi\)
0.715108 0.699014i \(-0.246378\pi\)
\(492\) 0 0
\(493\) 13.3591i 0.601662i
\(494\) 0 0
\(495\) −50.1369 −2.25349
\(496\) 0 0
\(497\) −22.0000 −0.986835
\(498\) 0 0
\(499\) − 4.37228i − 0.195730i −0.995200 0.0978651i \(-0.968799\pi\)
0.995200 0.0978651i \(-0.0312014\pi\)
\(500\) 0 0
\(501\) 21.3745i 0.954943i
\(502\) 0 0
\(503\) 11.4795 0.511848 0.255924 0.966697i \(-0.417620\pi\)
0.255924 + 0.966697i \(0.417620\pi\)
\(504\) 0 0
\(505\) 34.9783 1.55651
\(506\) 0 0
\(507\) 71.2119i 3.16263i
\(508\) 0 0
\(509\) 37.8102i 1.67591i 0.545742 + 0.837953i \(0.316248\pi\)
−0.545742 + 0.837953i \(0.683752\pi\)
\(510\) 0 0
\(511\) −28.1552 −1.24551
\(512\) 0 0
\(513\) −18.1168 −0.799878
\(514\) 0 0
\(515\) 30.2337i 1.33226i
\(516\) 0 0
\(517\) 1.52967i 0.0672749i
\(518\) 0 0
\(519\) −51.0767 −2.24202
\(520\) 0 0
\(521\) −5.76631 −0.252627 −0.126313 0.991990i \(-0.540314\pi\)
−0.126313 + 0.991990i \(0.540314\pi\)
\(522\) 0 0
\(523\) 25.3723i 1.10945i 0.832033 + 0.554726i \(0.187177\pi\)
−0.832033 + 0.554726i \(0.812823\pi\)
\(524\) 0 0
\(525\) − 15.3484i − 0.669859i
\(526\) 0 0
\(527\) −17.3205 −0.754493
\(528\) 0 0
\(529\) −22.3723 −0.972708
\(530\) 0 0
\(531\) − 11.4891i − 0.498586i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −33.7562 −1.45941
\(536\) 0 0
\(537\) 77.4891 3.34390
\(538\) 0 0
\(539\) 9.48913i 0.408726i
\(540\) 0 0
\(541\) 23.6039i 1.01481i 0.861707 + 0.507405i \(0.169395\pi\)
−0.861707 + 0.507405i \(0.830605\pi\)
\(542\) 0 0
\(543\) −27.7128 −1.18927
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) − 2.51087i − 0.107357i −0.998558 0.0536786i \(-0.982905\pi\)
0.998558 0.0536786i \(-0.0170947\pi\)
\(548\) 0 0
\(549\) − 121.408i − 5.18158i
\(550\) 0 0
\(551\) −2.67181 −0.113823
\(552\) 0 0
\(553\) −0.978251 −0.0415994
\(554\) 0 0
\(555\) 85.9565i 3.64865i
\(556\) 0 0
\(557\) − 25.1885i − 1.06727i −0.845715 0.533635i \(-0.820826\pi\)
0.845715 0.533635i \(-0.179174\pi\)
\(558\) 0 0
\(559\) −21.1894 −0.896215
\(560\) 0 0
\(561\) 40.0000 1.68880
\(562\) 0 0
\(563\) − 13.4891i − 0.568499i −0.958750 0.284249i \(-0.908256\pi\)
0.958750 0.284249i \(-0.0917444\pi\)
\(564\) 0 0
\(565\) 37.2203i 1.56587i
\(566\) 0 0
\(567\) −119.326 −5.01124
\(568\) 0 0
\(569\) −26.9783 −1.13099 −0.565494 0.824753i \(-0.691314\pi\)
−0.565494 + 0.824753i \(0.691314\pi\)
\(570\) 0 0
\(571\) − 42.9783i − 1.79858i −0.437349 0.899292i \(-0.644083\pi\)
0.437349 0.899292i \(-0.355917\pi\)
\(572\) 0 0
\(573\) − 23.8612i − 0.996815i
\(574\) 0 0
\(575\) 1.08724 0.0453411
\(576\) 0 0
\(577\) 33.4674 1.39327 0.696633 0.717428i \(-0.254681\pi\)
0.696633 + 0.717428i \(0.254681\pi\)
\(578\) 0 0
\(579\) − 10.9783i − 0.456241i
\(580\) 0 0
\(581\) 26.5330i 1.10077i
\(582\) 0 0
\(583\) 14.5561 0.602853
\(584\) 0 0
\(585\) 123.446 5.10385
\(586\) 0 0
\(587\) 31.8614i 1.31506i 0.753428 + 0.657530i \(0.228399\pi\)
−0.753428 + 0.657530i \(0.771601\pi\)
\(588\) 0 0
\(589\) − 3.46410i − 0.142736i
\(590\) 0 0
\(591\) 21.3745 0.879230
\(592\) 0 0
\(593\) 4.97825 0.204432 0.102216 0.994762i \(-0.467407\pi\)
0.102216 + 0.994762i \(0.467407\pi\)
\(594\) 0 0
\(595\) 41.8614i 1.71615i
\(596\) 0 0
\(597\) − 0.497333i − 0.0203545i
\(598\) 0 0
\(599\) 39.6897 1.62168 0.810838 0.585270i \(-0.199012\pi\)
0.810838 + 0.585270i \(0.199012\pi\)
\(600\) 0 0
\(601\) −22.4674 −0.916463 −0.458232 0.888833i \(-0.651517\pi\)
−0.458232 + 0.888833i \(0.651517\pi\)
\(602\) 0 0
\(603\) − 22.0000i − 0.895909i
\(604\) 0 0
\(605\) − 13.5615i − 0.551351i
\(606\) 0 0
\(607\) 38.8048 1.57504 0.787520 0.616289i \(-0.211365\pi\)
0.787520 + 0.616289i \(0.211365\pi\)
\(608\) 0 0
\(609\) −29.8832 −1.21093
\(610\) 0 0
\(611\) − 3.76631i − 0.152369i
\(612\) 0 0
\(613\) 18.5552i 0.749439i 0.927138 + 0.374719i \(0.122261\pi\)
−0.927138 + 0.374719i \(0.877739\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −40.8397 −1.64414 −0.822071 0.569384i \(-0.807182\pi\)
−0.822071 + 0.569384i \(0.807182\pi\)
\(618\) 0 0
\(619\) 16.2337i 0.652487i 0.945286 + 0.326244i \(0.105783\pi\)
−0.945286 + 0.326244i \(0.894217\pi\)
\(620\) 0 0
\(621\) − 14.3537i − 0.575996i
\(622\) 0 0
\(623\) 51.3716 2.05816
\(624\) 0 0
\(625\) −29.9783 −1.19913
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) − 50.4868i − 2.01304i
\(630\) 0 0
\(631\) 37.8651 1.50738 0.753692 0.657227i \(-0.228271\pi\)
0.753692 + 0.657227i \(0.228271\pi\)
\(632\) 0 0
\(633\) −8.86141 −0.352209
\(634\) 0 0
\(635\) 30.9783i 1.22933i
\(636\) 0 0
\(637\) − 23.3639i − 0.925709i
\(638\) 0 0
\(639\) −55.5354 −2.19695
\(640\) 0 0
\(641\) −10.9783 −0.433615 −0.216807 0.976214i \(-0.569564\pi\)
−0.216807 + 0.976214i \(0.569564\pi\)
\(642\) 0 0
\(643\) 36.3723i 1.43438i 0.696876 + 0.717191i \(0.254573\pi\)
−0.696876 + 0.717191i \(0.745427\pi\)
\(644\) 0 0
\(645\) 30.8820i 1.21598i
\(646\) 0 0
\(647\) 16.5831 0.651950 0.325975 0.945378i \(-0.394307\pi\)
0.325975 + 0.945378i \(0.394307\pi\)
\(648\) 0 0
\(649\) −3.25544 −0.127787
\(650\) 0 0
\(651\) − 38.7446i − 1.51852i
\(652\) 0 0
\(653\) 35.3956i 1.38514i 0.721352 + 0.692569i \(0.243521\pi\)
−0.721352 + 0.692569i \(0.756479\pi\)
\(654\) 0 0
\(655\) −11.0371 −0.431256
\(656\) 0 0
\(657\) −71.0733 −2.77284
\(658\) 0 0
\(659\) − 46.8614i − 1.82546i −0.408562 0.912731i \(-0.633970\pi\)
0.408562 0.912731i \(-0.366030\pi\)
\(660\) 0 0
\(661\) − 10.5947i − 0.412085i −0.978543 0.206043i \(-0.933941\pi\)
0.978543 0.206043i \(-0.0660586\pi\)
\(662\) 0 0
\(663\) −98.4868 −3.82491
\(664\) 0 0
\(665\) −8.37228 −0.324663
\(666\) 0 0
\(667\) − 2.11684i − 0.0819645i
\(668\) 0 0
\(669\) − 23.3639i − 0.903299i
\(670\) 0 0
\(671\) −34.4010 −1.32803
\(672\) 0 0
\(673\) 16.7446 0.645455 0.322728 0.946492i \(-0.395400\pi\)
0.322728 + 0.946492i \(0.395400\pi\)
\(674\) 0 0
\(675\) − 24.8614i − 0.956916i
\(676\) 0 0
\(677\) 5.84096i 0.224486i 0.993681 + 0.112243i \(0.0358036\pi\)
−0.993681 + 0.112243i \(0.964196\pi\)
\(678\) 0 0
\(679\) 9.10268 0.349329
\(680\) 0 0
\(681\) −47.6060 −1.82426
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) − 12.6217i − 0.482250i
\(686\) 0 0
\(687\) −42.5639 −1.62391
\(688\) 0 0
\(689\) −35.8397 −1.36538
\(690\) 0 0
\(691\) 17.8614i 0.679480i 0.940519 + 0.339740i \(0.110339\pi\)
−0.940519 + 0.339740i \(0.889661\pi\)
\(692\) 0 0
\(693\) 65.8728i 2.50230i
\(694\) 0 0
\(695\) −9.15759 −0.347367
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 36.7011i 1.38816i
\(700\) 0 0
\(701\) − 27.7128i − 1.04670i −0.852118 0.523349i \(-0.824682\pi\)
0.852118 0.523349i \(-0.175318\pi\)
\(702\) 0 0
\(703\) 10.0974 0.380829
\(704\) 0 0
\(705\) −5.48913 −0.206732
\(706\) 0 0
\(707\) − 45.9565i − 1.72837i
\(708\) 0 0
\(709\) 45.7330i 1.71754i 0.512361 + 0.858770i \(0.328771\pi\)
−0.512361 + 0.858770i \(0.671229\pi\)
\(710\) 0 0
\(711\) −2.46943 −0.0926110
\(712\) 0 0
\(713\) 2.74456 0.102785
\(714\) 0 0
\(715\) − 34.9783i − 1.30811i
\(716\) 0 0
\(717\) − 58.9070i − 2.19992i
\(718\) 0 0
\(719\) 40.5369 1.51177 0.755885 0.654704i \(-0.227207\pi\)
0.755885 + 0.654704i \(0.227207\pi\)
\(720\) 0 0
\(721\) 39.7228 1.47935
\(722\) 0 0
\(723\) − 26.9783i − 1.00333i
\(724\) 0 0
\(725\) − 3.66648i − 0.136170i
\(726\) 0 0
\(727\) 14.0039 0.519375 0.259688 0.965693i \(-0.416380\pi\)
0.259688 + 0.965693i \(0.416380\pi\)
\(728\) 0 0
\(729\) −117.935 −4.36795
\(730\) 0 0
\(731\) − 18.1386i − 0.670880i
\(732\) 0 0
\(733\) 21.1894i 0.782647i 0.920253 + 0.391324i \(0.127983\pi\)
−0.920253 + 0.391324i \(0.872017\pi\)
\(734\) 0 0
\(735\) −34.0511 −1.25599
\(736\) 0 0
\(737\) −6.23369 −0.229621
\(738\) 0 0
\(739\) − 12.6060i − 0.463718i −0.972749 0.231859i \(-0.925519\pi\)
0.972749 0.231859i \(-0.0744808\pi\)
\(740\) 0 0
\(741\) − 19.6974i − 0.723601i
\(742\) 0 0
\(743\) 6.63325 0.243350 0.121675 0.992570i \(-0.461173\pi\)
0.121675 + 0.992570i \(0.461173\pi\)
\(744\) 0 0
\(745\) −11.8614 −0.434568
\(746\) 0 0
\(747\) 66.9783i 2.45061i
\(748\) 0 0
\(749\) 44.3508i 1.62054i
\(750\) 0 0
\(751\) 9.50744 0.346932 0.173466 0.984840i \(-0.444503\pi\)
0.173466 + 0.984840i \(0.444503\pi\)
\(752\) 0 0
\(753\) −60.2337 −2.19504
\(754\) 0 0
\(755\) 39.7228i 1.44566i
\(756\) 0 0
\(757\) − 18.8502i − 0.685121i −0.939496 0.342561i \(-0.888706\pi\)
0.939496 0.342561i \(-0.111294\pi\)
\(758\) 0 0
\(759\) −6.33830 −0.230066
\(760\) 0 0
\(761\) −11.0000 −0.398750 −0.199375 0.979923i \(-0.563891\pi\)
−0.199375 + 0.979923i \(0.563891\pi\)
\(762\) 0 0
\(763\) 13.1386i 0.475649i
\(764\) 0 0
\(765\) 105.672i 3.82059i
\(766\) 0 0
\(767\) 8.01544 0.289421
\(768\) 0 0
\(769\) 37.4674 1.35111 0.675554 0.737310i \(-0.263904\pi\)
0.675554 + 0.737310i \(0.263904\pi\)
\(770\) 0 0
\(771\) 54.7446i 1.97158i
\(772\) 0 0
\(773\) 24.1561i 0.868836i 0.900711 + 0.434418i \(0.143046\pi\)
−0.900711 + 0.434418i \(0.856954\pi\)
\(774\) 0 0
\(775\) 4.75372 0.170759
\(776\) 0 0
\(777\) 112.935 4.05151
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 15.7359i 0.563076i
\(782\) 0 0
\(783\) −48.4048 −1.72985
\(784\) 0 0
\(785\) 46.9783 1.67673
\(786\) 0 0
\(787\) − 19.6060i − 0.698877i −0.936959 0.349439i \(-0.886372\pi\)
0.936959 0.349439i \(-0.113628\pi\)
\(788\) 0 0
\(789\) 63.9384i 2.27627i
\(790\) 0 0
\(791\) 48.9022 1.73876
\(792\) 0 0
\(793\) 84.7011 3.00782
\(794\) 0 0
\(795\) 52.2337i 1.85254i
\(796\) 0 0
\(797\) 14.6487i 0.518883i 0.965759 + 0.259442i \(0.0835385\pi\)
−0.965759 + 0.259442i \(0.916461\pi\)
\(798\) 0 0
\(799\) 3.22405 0.114059
\(800\) 0 0
\(801\) 129.679 4.58199
\(802\) 0 0
\(803\) 20.1386i 0.710676i
\(804\) 0 0
\(805\) − 6.63325i − 0.233791i
\(806\) 0 0
\(807\) −6.33830 −0.223119
\(808\) 0 0
\(809\) −37.9783 −1.33524 −0.667622 0.744500i \(-0.732688\pi\)
−0.667622 + 0.744500i \(0.732688\pi\)
\(810\) 0 0
\(811\) − 28.8614i − 1.01346i −0.862105 0.506731i \(-0.830854\pi\)
0.862105 0.506731i \(-0.169146\pi\)
\(812\) 0 0
\(813\) 8.01544i 0.281114i
\(814\) 0 0
\(815\) 27.1229 0.950074
\(816\) 0 0
\(817\) 3.62772 0.126918
\(818\) 0 0
\(819\) − 162.190i − 5.66738i
\(820\) 0 0
\(821\) − 27.0680i − 0.944680i −0.881416 0.472340i \(-0.843409\pi\)
0.881416 0.472340i \(-0.156591\pi\)
\(822\) 0 0
\(823\) −7.37063 −0.256924 −0.128462 0.991714i \(-0.541004\pi\)
−0.128462 + 0.991714i \(0.541004\pi\)
\(824\) 0 0
\(825\) −10.9783 −0.382214
\(826\) 0 0
\(827\) − 40.3505i − 1.40313i −0.712608 0.701563i \(-0.752486\pi\)
0.712608 0.701563i \(-0.247514\pi\)
\(828\) 0 0
\(829\) − 3.26172i − 0.113284i −0.998395 0.0566421i \(-0.981961\pi\)
0.998395 0.0566421i \(-0.0180394\pi\)
\(830\) 0 0
\(831\) 55.2405 1.91627
\(832\) 0 0
\(833\) 20.0000 0.692959
\(834\) 0 0
\(835\) 16.0000i 0.553703i
\(836\) 0 0
\(837\) − 62.7586i − 2.16925i
\(838\) 0 0
\(839\) 9.80240 0.338416 0.169208 0.985580i \(-0.445879\pi\)
0.169208 + 0.985580i \(0.445879\pi\)
\(840\) 0 0
\(841\) 21.8614 0.753842
\(842\) 0 0
\(843\) − 21.0217i − 0.724028i
\(844\) 0 0
\(845\) 53.3060i 1.83378i
\(846\) 0 0
\(847\) −17.8178 −0.612228
\(848\) 0 0
\(849\) 59.4456 2.04017
\(850\) 0 0
\(851\) 8.00000i 0.274236i
\(852\) 0 0
\(853\) 40.3894i 1.38291i 0.722421 + 0.691453i \(0.243029\pi\)
−0.722421 + 0.691453i \(0.756971\pi\)
\(854\) 0 0
\(855\) −21.1345 −0.722783
\(856\) 0 0
\(857\) 22.2337 0.759488 0.379744 0.925092i \(-0.376012\pi\)
0.379744 + 0.925092i \(0.376012\pi\)
\(858\) 0 0
\(859\) 8.60597i 0.293632i 0.989164 + 0.146816i \(0.0469025\pi\)
−0.989164 + 0.146816i \(0.953097\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.76439 −0.0941009 −0.0470504 0.998893i \(-0.514982\pi\)
−0.0470504 + 0.998893i \(0.514982\pi\)
\(864\) 0 0
\(865\) −38.2337 −1.29998
\(866\) 0 0
\(867\) − 26.9783i − 0.916229i
\(868\) 0 0
\(869\) 0.699713i 0.0237361i
\(870\) 0 0
\(871\) 15.3484 0.520061
\(872\) 0 0
\(873\) 22.9783 0.777696
\(874\) 0 0
\(875\) 30.3723i 1.02677i
\(876\) 0 0
\(877\) 1.38219i 0.0466734i 0.999728 + 0.0233367i \(0.00742897\pi\)
−0.999728 + 0.0233367i \(0.992571\pi\)
\(878\) 0 0
\(879\) −16.3431 −0.551238
\(880\) 0 0
\(881\) −4.37228 −0.147306 −0.0736530 0.997284i \(-0.523466\pi\)
−0.0736530 + 0.997284i \(0.523466\pi\)
\(882\) 0 0
\(883\) − 18.8832i − 0.635469i −0.948180 0.317734i \(-0.897078\pi\)
0.948180 0.317734i \(-0.102922\pi\)
\(884\) 0 0
\(885\) − 11.6819i − 0.392684i
\(886\) 0 0
\(887\) −45.0333 −1.51207 −0.756035 0.654531i \(-0.772866\pi\)
−0.756035 + 0.654531i \(0.772866\pi\)
\(888\) 0 0
\(889\) 40.7011 1.36507
\(890\) 0 0
\(891\) 85.3505i 2.85935i
\(892\) 0 0
\(893\) 0.644810i 0.0215777i
\(894\) 0 0
\(895\) 58.0049 1.93889
\(896\) 0 0
\(897\) 15.6060 0.521068
\(898\) 0 0
\(899\) − 9.25544i − 0.308686i
\(900\) 0 0
\(901\) − 30.6796i − 1.02209i
\(902\) 0 0
\(903\) 40.5746 1.35024
\(904\) 0 0
\(905\) −20.7446 −0.689573
\(906\) 0 0
\(907\) 46.8614i 1.55601i 0.628260 + 0.778004i \(0.283768\pi\)
−0.628260 + 0.778004i \(0.716232\pi\)
\(908\) 0 0
\(909\) − 116.010i − 3.84780i
\(910\) 0 0
\(911\) −47.6126 −1.57747 −0.788737 0.614730i \(-0.789265\pi\)
−0.788737 + 0.614730i \(0.789265\pi\)
\(912\) 0 0
\(913\) 18.9783 0.628088
\(914\) 0 0
\(915\) − 123.446i − 4.08099i
\(916\) 0 0
\(917\) 14.5012i 0.478872i
\(918\) 0 0
\(919\) 8.71516 0.287486 0.143743 0.989615i \(-0.454086\pi\)
0.143743 + 0.989615i \(0.454086\pi\)
\(920\) 0 0
\(921\) −51.4456 −1.69519
\(922\) 0 0
\(923\) − 38.7446i − 1.27529i
\(924\) 0 0
\(925\) 13.8564i 0.455596i
\(926\) 0 0
\(927\) 100.274 3.29342
\(928\) 0 0
\(929\) −27.8832 −0.914817 −0.457408 0.889257i \(-0.651222\pi\)
−0.457408 + 0.889257i \(0.651222\pi\)
\(930\) 0 0
\(931\) 4.00000i 0.131095i
\(932\) 0 0
\(933\) 78.2921i 2.56317i
\(934\) 0 0
\(935\) 29.9422 0.979215
\(936\) 0 0
\(937\) −21.9783 −0.717998 −0.358999 0.933338i \(-0.616882\pi\)
−0.358999 + 0.933338i \(0.616882\pi\)
\(938\) 0 0
\(939\) − 13.0951i − 0.427342i
\(940\) 0 0
\(941\) − 10.8896i − 0.354992i −0.984122 0.177496i \(-0.943200\pi\)
0.984122 0.177496i \(-0.0567997\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −151.679 −4.93413
\(946\) 0 0
\(947\) − 43.2119i − 1.40420i −0.712079 0.702100i \(-0.752246\pi\)
0.712079 0.702100i \(-0.247754\pi\)
\(948\) 0 0
\(949\) − 49.5847i − 1.60959i
\(950\) 0 0
\(951\) −72.7634 −2.35951
\(952\) 0 0
\(953\) 26.2337 0.849793 0.424896 0.905242i \(-0.360311\pi\)
0.424896 + 0.905242i \(0.360311\pi\)
\(954\) 0 0
\(955\) − 17.8614i − 0.577982i
\(956\) 0 0
\(957\) 21.3745i 0.690940i
\(958\) 0 0
\(959\) −16.5831 −0.535497
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 111.957i 3.60775i
\(964\) 0 0
\(965\) − 8.21782i − 0.264541i
\(966\) 0 0
\(967\) 55.1307 1.77288 0.886441 0.462841i \(-0.153170\pi\)
0.886441 + 0.462841i \(0.153170\pi\)
\(968\) 0 0
\(969\) 16.8614 0.541666
\(970\) 0 0
\(971\) − 4.00000i − 0.128366i −0.997938 0.0641831i \(-0.979556\pi\)
0.997938 0.0641831i \(-0.0204442\pi\)
\(972\) 0 0
\(973\) 12.0318i 0.385721i
\(974\) 0 0
\(975\) 27.0303 0.865663
\(976\) 0 0
\(977\) 27.4891 0.879455 0.439728 0.898131i \(-0.355075\pi\)
0.439728 + 0.898131i \(0.355075\pi\)
\(978\) 0 0
\(979\) − 36.7446i − 1.17436i
\(980\) 0 0
\(981\) 33.1662i 1.05892i
\(982\) 0 0
\(983\) 52.6612 1.67963 0.839816 0.542871i \(-0.182663\pi\)
0.839816 + 0.542871i \(0.182663\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 0 0
\(987\) 7.21194i 0.229559i
\(988\) 0 0
\(989\) 2.87419i 0.0913941i
\(990\) 0 0
\(991\) −7.92287 −0.251678 −0.125839 0.992051i \(-0.540162\pi\)
−0.125839 + 0.992051i \(0.540162\pi\)
\(992\) 0 0
\(993\) −36.6277 −1.16235
\(994\) 0 0
\(995\) − 0.372281i − 0.0118021i
\(996\) 0 0
\(997\) 32.8164i 1.03931i 0.854378 + 0.519653i \(0.173939\pi\)
−0.854378 + 0.519653i \(0.826061\pi\)
\(998\) 0 0
\(999\) 182.932 5.78772
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.h.609.1 8
4.3 odd 2 inner 1216.2.c.h.609.7 yes 8
8.3 odd 2 inner 1216.2.c.h.609.2 yes 8
8.5 even 2 inner 1216.2.c.h.609.8 yes 8
16.3 odd 4 4864.2.a.bi.1.1 4
16.5 even 4 4864.2.a.bi.1.2 4
16.11 odd 4 4864.2.a.bl.1.4 4
16.13 even 4 4864.2.a.bl.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.h.609.1 8 1.1 even 1 trivial
1216.2.c.h.609.2 yes 8 8.3 odd 2 inner
1216.2.c.h.609.7 yes 8 4.3 odd 2 inner
1216.2.c.h.609.8 yes 8 8.5 even 2 inner
4864.2.a.bi.1.1 4 16.3 odd 4
4864.2.a.bi.1.2 4 16.5 even 4
4864.2.a.bl.1.3 4 16.13 even 4
4864.2.a.bl.1.4 4 16.11 odd 4