Properties

Label 6048.2.h.f.2591.5
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.5
Root \(0.965926 - 0.258819i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.f.2591.4

$q$-expansion

\(f(q)\) \(=\) \(q+0.896575i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+0.896575i q^{5} -1.00000i q^{7} +3.34607 q^{11} +4.00000 q^{13} +3.48477i q^{17} +1.00000i q^{19} +0.138701 q^{23} +4.19615 q^{25} +2.07055i q^{29} +0.267949i q^{31} +0.896575 q^{35} -8.26795 q^{37} +9.28032i q^{41} +0.535898i q^{43} +5.93426 q^{47} -1.00000 q^{49} -2.82843i q^{53} +3.00000i q^{55} -1.03528 q^{59} +8.39230 q^{61} +3.58630i q^{65} -2.92820i q^{67} +1.55291 q^{71} +6.39230 q^{73} -3.34607i q^{77} -0.535898i q^{79} -12.6264 q^{83} -3.12436 q^{85} -10.7961i q^{89} -4.00000i q^{91} -0.896575 q^{95} -10.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 32q^{13} - 8q^{25} - 80q^{37} - 8q^{49} - 16q^{61} - 32q^{73} + 72q^{85} - 32q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.896575i 0.400961i 0.979698 + 0.200480i \(0.0642503\pi\)
−0.979698 + 0.200480i \(0.935750\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.34607 1.00888 0.504438 0.863448i \(-0.331700\pi\)
0.504438 + 0.863448i \(0.331700\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.48477i 0.845180i 0.906321 + 0.422590i \(0.138879\pi\)
−0.906321 + 0.422590i \(0.861121\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i 0.993399 + 0.114708i \(0.0365932\pi\)
−0.993399 + 0.114708i \(0.963407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.138701 0.0289211 0.0144605 0.999895i \(-0.495397\pi\)
0.0144605 + 0.999895i \(0.495397\pi\)
\(24\) 0 0
\(25\) 4.19615 0.839230
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.07055i 0.384492i 0.981347 + 0.192246i \(0.0615771\pi\)
−0.981347 + 0.192246i \(0.938423\pi\)
\(30\) 0 0
\(31\) 0.267949i 0.0481251i 0.999710 + 0.0240625i \(0.00766009\pi\)
−0.999710 + 0.0240625i \(0.992340\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.896575 0.151549
\(36\) 0 0
\(37\) −8.26795 −1.35924 −0.679621 0.733563i \(-0.737856\pi\)
−0.679621 + 0.733563i \(0.737856\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.28032i 1.44934i 0.689095 + 0.724671i \(0.258008\pi\)
−0.689095 + 0.724671i \(0.741992\pi\)
\(42\) 0 0
\(43\) 0.535898i 0.0817237i 0.999165 + 0.0408619i \(0.0130104\pi\)
−0.999165 + 0.0408619i \(0.986990\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.93426 0.865600 0.432800 0.901490i \(-0.357526\pi\)
0.432800 + 0.901490i \(0.357526\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 2.82843i − 0.388514i −0.980951 0.194257i \(-0.937770\pi\)
0.980951 0.194257i \(-0.0622296\pi\)
\(54\) 0 0
\(55\) 3.00000i 0.404520i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.03528 −0.134781 −0.0673907 0.997727i \(-0.521467\pi\)
−0.0673907 + 0.997727i \(0.521467\pi\)
\(60\) 0 0
\(61\) 8.39230 1.07452 0.537262 0.843415i \(-0.319459\pi\)
0.537262 + 0.843415i \(0.319459\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.58630i 0.444826i
\(66\) 0 0
\(67\) − 2.92820i − 0.357737i −0.983873 0.178868i \(-0.942756\pi\)
0.983873 0.178868i \(-0.0572437\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.55291 0.184297 0.0921485 0.995745i \(-0.470627\pi\)
0.0921485 + 0.995745i \(0.470627\pi\)
\(72\) 0 0
\(73\) 6.39230 0.748163 0.374081 0.927396i \(-0.377958\pi\)
0.374081 + 0.927396i \(0.377958\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.34607i − 0.381320i
\(78\) 0 0
\(79\) − 0.535898i − 0.0602933i −0.999545 0.0301466i \(-0.990403\pi\)
0.999545 0.0301466i \(-0.00959743\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.6264 −1.38593 −0.692963 0.720973i \(-0.743695\pi\)
−0.692963 + 0.720973i \(0.743695\pi\)
\(84\) 0 0
\(85\) −3.12436 −0.338884
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 10.7961i − 1.14438i −0.820121 0.572191i \(-0.806094\pi\)
0.820121 0.572191i \(-0.193906\pi\)
\(90\) 0 0
\(91\) − 4.00000i − 0.419314i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.896575 −0.0919867
\(96\) 0 0
\(97\) −10.9282 −1.10959 −0.554795 0.831987i \(-0.687203\pi\)
−0.554795 + 0.831987i \(0.687203\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.14162i 0.909625i 0.890587 + 0.454813i \(0.150294\pi\)
−0.890587 + 0.454813i \(0.849706\pi\)
\(102\) 0 0
\(103\) − 6.12436i − 0.603451i −0.953395 0.301725i \(-0.902437\pi\)
0.953395 0.301725i \(-0.0975626\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.1769 0.983838 0.491919 0.870641i \(-0.336296\pi\)
0.491919 + 0.870641i \(0.336296\pi\)
\(108\) 0 0
\(109\) −10.4641 −1.00228 −0.501140 0.865366i \(-0.667086\pi\)
−0.501140 + 0.865366i \(0.667086\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.62158i 0.434761i 0.976087 + 0.217381i \(0.0697513\pi\)
−0.976087 + 0.217381i \(0.930249\pi\)
\(114\) 0 0
\(115\) 0.124356i 0.0115962i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.48477 0.319448
\(120\) 0 0
\(121\) 0.196152 0.0178320
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.24504i 0.737459i
\(126\) 0 0
\(127\) − 4.39230i − 0.389754i −0.980828 0.194877i \(-0.937569\pi\)
0.980828 0.194877i \(-0.0624308\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.0764 1.75408 0.877041 0.480415i \(-0.159514\pi\)
0.877041 + 0.480415i \(0.159514\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.2832i 1.56204i 0.624504 + 0.781021i \(0.285301\pi\)
−0.624504 + 0.781021i \(0.714699\pi\)
\(138\) 0 0
\(139\) 7.46410i 0.633097i 0.948576 + 0.316548i \(0.102524\pi\)
−0.948576 + 0.316548i \(0.897476\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 13.3843 1.11925
\(144\) 0 0
\(145\) −1.85641 −0.154166
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 1.79315i − 0.146901i −0.997299 0.0734503i \(-0.976599\pi\)
0.997299 0.0734503i \(-0.0234010\pi\)
\(150\) 0 0
\(151\) − 17.3205i − 1.40952i −0.709444 0.704761i \(-0.751054\pi\)
0.709444 0.704761i \(-0.248946\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.240237 −0.0192963
\(156\) 0 0
\(157\) 5.46410 0.436083 0.218041 0.975940i \(-0.430033\pi\)
0.218041 + 0.975940i \(0.430033\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 0.138701i − 0.0109311i
\(162\) 0 0
\(163\) 15.3205i 1.19999i 0.800002 + 0.599997i \(0.204832\pi\)
−0.800002 + 0.599997i \(0.795168\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.6617 1.05717 0.528586 0.848880i \(-0.322723\pi\)
0.528586 + 0.848880i \(0.322723\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.0735i 0.841900i 0.907084 + 0.420950i \(0.138303\pi\)
−0.907084 + 0.420950i \(0.861697\pi\)
\(174\) 0 0
\(175\) − 4.19615i − 0.317199i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.20736 −0.239730 −0.119865 0.992790i \(-0.538246\pi\)
−0.119865 + 0.992790i \(0.538246\pi\)
\(180\) 0 0
\(181\) −3.46410 −0.257485 −0.128742 0.991678i \(-0.541094\pi\)
−0.128742 + 0.991678i \(0.541094\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 7.41284i − 0.545003i
\(186\) 0 0
\(187\) 11.6603i 0.852682i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.8709 1.14838 0.574190 0.818722i \(-0.305317\pi\)
0.574190 + 0.818722i \(0.305317\pi\)
\(192\) 0 0
\(193\) 1.07180 0.0771496 0.0385748 0.999256i \(-0.487718\pi\)
0.0385748 + 0.999256i \(0.487718\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.93426i 0.422798i 0.977400 + 0.211399i \(0.0678020\pi\)
−0.977400 + 0.211399i \(0.932198\pi\)
\(198\) 0 0
\(199\) − 19.5359i − 1.38486i −0.721484 0.692432i \(-0.756539\pi\)
0.721484 0.692432i \(-0.243461\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.07055 0.145324
\(204\) 0 0
\(205\) −8.32051 −0.581129
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.34607i 0.231452i
\(210\) 0 0
\(211\) 21.3205i 1.46776i 0.679277 + 0.733882i \(0.262294\pi\)
−0.679277 + 0.733882i \(0.737706\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.480473 −0.0327680
\(216\) 0 0
\(217\) 0.267949 0.0181896
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.9391i 0.937643i
\(222\) 0 0
\(223\) − 14.4641i − 0.968588i −0.874905 0.484294i \(-0.839077\pi\)
0.874905 0.484294i \(-0.160923\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.37945 −0.357047 −0.178523 0.983936i \(-0.557132\pi\)
−0.178523 + 0.983936i \(0.557132\pi\)
\(228\) 0 0
\(229\) 23.3205 1.54106 0.770531 0.637402i \(-0.219991\pi\)
0.770531 + 0.637402i \(0.219991\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 4.62158i − 0.302770i −0.988475 0.151385i \(-0.951627\pi\)
0.988475 0.151385i \(-0.0483733\pi\)
\(234\) 0 0
\(235\) 5.32051i 0.347072i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.3495 1.12225 0.561123 0.827732i \(-0.310369\pi\)
0.561123 + 0.827732i \(0.310369\pi\)
\(240\) 0 0
\(241\) 0.928203 0.0597908 0.0298954 0.999553i \(-0.490483\pi\)
0.0298954 + 0.999553i \(0.490483\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 0.896575i − 0.0572801i
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.3538 1.28472 0.642360 0.766403i \(-0.277955\pi\)
0.642360 + 0.766403i \(0.277955\pi\)
\(252\) 0 0
\(253\) 0.464102 0.0291778
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.51815i 0.344213i 0.985078 + 0.172106i \(0.0550573\pi\)
−0.985078 + 0.172106i \(0.944943\pi\)
\(258\) 0 0
\(259\) 8.26795i 0.513745i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0072 −0.740396 −0.370198 0.928953i \(-0.620710\pi\)
−0.370198 + 0.928953i \(0.620710\pi\)
\(264\) 0 0
\(265\) 2.53590 0.155779
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.9382i 1.52051i 0.649625 + 0.760255i \(0.274926\pi\)
−0.649625 + 0.760255i \(0.725074\pi\)
\(270\) 0 0
\(271\) − 0.535898i − 0.0325535i −0.999868 0.0162768i \(-0.994819\pi\)
0.999868 0.0162768i \(-0.00518128\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.0406 0.846680
\(276\) 0 0
\(277\) −11.1962 −0.672712 −0.336356 0.941735i \(-0.609194\pi\)
−0.336356 + 0.941735i \(0.609194\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 19.3185i 1.15245i 0.817292 + 0.576223i \(0.195474\pi\)
−0.817292 + 0.576223i \(0.804526\pi\)
\(282\) 0 0
\(283\) 9.85641i 0.585903i 0.956127 + 0.292951i \(0.0946374\pi\)
−0.956127 + 0.292951i \(0.905363\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.28032 0.547800
\(288\) 0 0
\(289\) 4.85641 0.285671
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 11.9700i − 0.699298i −0.936881 0.349649i \(-0.886301\pi\)
0.936881 0.349649i \(-0.113699\pi\)
\(294\) 0 0
\(295\) − 0.928203i − 0.0540421i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.554803 0.0320851
\(300\) 0 0
\(301\) 0.535898 0.0308887
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.52433i 0.430842i
\(306\) 0 0
\(307\) 5.00000i 0.285365i 0.989769 + 0.142683i \(0.0455728\pi\)
−0.989769 + 0.142683i \(0.954427\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.14110 0.234821 0.117410 0.993083i \(-0.462541\pi\)
0.117410 + 0.993083i \(0.462541\pi\)
\(312\) 0 0
\(313\) −8.39230 −0.474361 −0.237181 0.971466i \(-0.576223\pi\)
−0.237181 + 0.971466i \(0.576223\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.6312i 1.15876i 0.815056 + 0.579382i \(0.196706\pi\)
−0.815056 + 0.579382i \(0.803294\pi\)
\(318\) 0 0
\(319\) 6.92820i 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.48477 −0.193898
\(324\) 0 0
\(325\) 16.7846 0.931043
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 5.93426i − 0.327166i
\(330\) 0 0
\(331\) 26.2487i 1.44276i 0.692540 + 0.721380i \(0.256492\pi\)
−0.692540 + 0.721380i \(0.743508\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.62536 0.143438
\(336\) 0 0
\(337\) −32.7128 −1.78198 −0.890990 0.454022i \(-0.849989\pi\)
−0.890990 + 0.454022i \(0.849989\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.896575i 0.0485523i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 33.3220 1.78882 0.894408 0.447252i \(-0.147597\pi\)
0.894408 + 0.447252i \(0.147597\pi\)
\(348\) 0 0
\(349\) −21.3205 −1.14126 −0.570630 0.821207i \(-0.693301\pi\)
−0.570630 + 0.821207i \(0.693301\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.5235i 0.985905i 0.870056 + 0.492953i \(0.164082\pi\)
−0.870056 + 0.492953i \(0.835918\pi\)
\(354\) 0 0
\(355\) 1.39230i 0.0738959i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.96524 0.209277 0.104639 0.994510i \(-0.466631\pi\)
0.104639 + 0.994510i \(0.466631\pi\)
\(360\) 0 0
\(361\) 18.0000 0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.73118i 0.299984i
\(366\) 0 0
\(367\) 11.5359i 0.602169i 0.953597 + 0.301084i \(0.0973486\pi\)
−0.953597 + 0.301084i \(0.902651\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.82843 −0.146845
\(372\) 0 0
\(373\) −5.39230 −0.279203 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.28221i 0.426555i
\(378\) 0 0
\(379\) − 17.3205i − 0.889695i −0.895606 0.444847i \(-0.853258\pi\)
0.895606 0.444847i \(-0.146742\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10.2784 0.525203 0.262602 0.964904i \(-0.415419\pi\)
0.262602 + 0.964904i \(0.415419\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.6579i 0.793886i 0.917843 + 0.396943i \(0.129929\pi\)
−0.917843 + 0.396943i \(0.870071\pi\)
\(390\) 0 0
\(391\) 0.483340i 0.0244435i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.480473 0.0241752
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 31.6675i − 1.58140i −0.612204 0.790700i \(-0.709717\pi\)
0.612204 0.790700i \(-0.290283\pi\)
\(402\) 0 0
\(403\) 1.07180i 0.0533900i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −27.6651 −1.37131
\(408\) 0 0
\(409\) −5.60770 −0.277283 −0.138641 0.990343i \(-0.544274\pi\)
−0.138641 + 0.990343i \(0.544274\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.03528i 0.0509426i
\(414\) 0 0
\(415\) − 11.3205i − 0.555702i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.3490 0.603287 0.301644 0.953421i \(-0.402465\pi\)
0.301644 + 0.953421i \(0.402465\pi\)
\(420\) 0 0
\(421\) 13.5885 0.662261 0.331130 0.943585i \(-0.392570\pi\)
0.331130 + 0.943585i \(0.392570\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 14.6226i 0.709301i
\(426\) 0 0
\(427\) − 8.39230i − 0.406132i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.2819 −1.16962 −0.584808 0.811172i \(-0.698830\pi\)
−0.584808 + 0.811172i \(0.698830\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.138701i 0.00663495i
\(438\) 0 0
\(439\) − 13.0718i − 0.623883i −0.950101 0.311941i \(-0.899021\pi\)
0.950101 0.311941i \(-0.100979\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.31079 −0.109789 −0.0548945 0.998492i \(-0.517482\pi\)
−0.0548945 + 0.998492i \(0.517482\pi\)
\(444\) 0 0
\(445\) 9.67949 0.458852
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.76268i 0.413537i 0.978390 + 0.206768i \(0.0662946\pi\)
−0.978390 + 0.206768i \(0.933705\pi\)
\(450\) 0 0
\(451\) 31.0526i 1.46221i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.58630 0.168128
\(456\) 0 0
\(457\) 37.7846 1.76749 0.883745 0.467969i \(-0.155014\pi\)
0.883745 + 0.467969i \(0.155014\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 13.6988i − 0.638018i −0.947752 0.319009i \(-0.896650\pi\)
0.947752 0.319009i \(-0.103350\pi\)
\(462\) 0 0
\(463\) − 39.8564i − 1.85228i −0.377175 0.926142i \(-0.623104\pi\)
0.377175 0.926142i \(-0.376896\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.2480 0.798141 0.399070 0.916920i \(-0.369333\pi\)
0.399070 + 0.916920i \(0.369333\pi\)
\(468\) 0 0
\(469\) −2.92820 −0.135212
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.79315i 0.0824492i
\(474\) 0 0
\(475\) 4.19615i 0.192533i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −36.5665 −1.67077 −0.835383 0.549669i \(-0.814754\pi\)
−0.835383 + 0.549669i \(0.814754\pi\)
\(480\) 0 0
\(481\) −33.0718 −1.50794
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 9.79796i − 0.444902i
\(486\) 0 0
\(487\) − 25.7128i − 1.16516i −0.812774 0.582579i \(-0.802044\pi\)
0.812774 0.582579i \(-0.197956\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −18.4219 −0.831371 −0.415685 0.909509i \(-0.636458\pi\)
−0.415685 + 0.909509i \(0.636458\pi\)
\(492\) 0 0
\(493\) −7.21539 −0.324965
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1.55291i − 0.0696577i
\(498\) 0 0
\(499\) − 18.0000i − 0.805791i −0.915246 0.402895i \(-0.868004\pi\)
0.915246 0.402895i \(-0.131996\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 29.0421 1.29493 0.647463 0.762097i \(-0.275830\pi\)
0.647463 + 0.762097i \(0.275830\pi\)
\(504\) 0 0
\(505\) −8.19615 −0.364724
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 41.0122i − 1.81783i −0.416978 0.908917i \(-0.636911\pi\)
0.416978 0.908917i \(-0.363089\pi\)
\(510\) 0 0
\(511\) − 6.39230i − 0.282779i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.49095 0.241960
\(516\) 0 0
\(517\) 19.8564 0.873284
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.8709i 0.695317i 0.937621 + 0.347659i \(0.113023\pi\)
−0.937621 + 0.347659i \(0.886977\pi\)
\(522\) 0 0
\(523\) − 11.3397i − 0.495852i −0.968779 0.247926i \(-0.920251\pi\)
0.968779 0.247926i \(-0.0797491\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.933740 −0.0406744
\(528\) 0 0
\(529\) −22.9808 −0.999164
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 37.1213i 1.60790i
\(534\) 0 0
\(535\) 9.12436i 0.394480i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.34607 −0.144125
\(540\) 0 0
\(541\) −34.6603 −1.49016 −0.745080 0.666975i \(-0.767589\pi\)
−0.745080 + 0.666975i \(0.767589\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 9.38186i − 0.401875i
\(546\) 0 0
\(547\) − 2.14359i − 0.0916534i −0.998949 0.0458267i \(-0.985408\pi\)
0.998949 0.0458267i \(-0.0145922\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.07055 −0.0882085
\(552\) 0 0
\(553\) −0.535898 −0.0227887
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 36.2891i − 1.53762i −0.639479 0.768809i \(-0.720850\pi\)
0.639479 0.768809i \(-0.279150\pi\)
\(558\) 0 0
\(559\) 2.14359i 0.0906643i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −17.7284 −0.747165 −0.373582 0.927597i \(-0.621871\pi\)
−0.373582 + 0.927597i \(0.621871\pi\)
\(564\) 0 0
\(565\) −4.14359 −0.174322
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.8343i 0.873418i 0.899603 + 0.436709i \(0.143856\pi\)
−0.899603 + 0.436709i \(0.856144\pi\)
\(570\) 0 0
\(571\) 26.3923i 1.10448i 0.833684 + 0.552242i \(0.186227\pi\)
−0.833684 + 0.552242i \(0.813773\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.582009 0.0242715
\(576\) 0 0
\(577\) 2.67949 0.111549 0.0557744 0.998443i \(-0.482237\pi\)
0.0557744 + 0.998443i \(0.482237\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.6264i 0.523831i
\(582\) 0 0
\(583\) − 9.46410i − 0.391963i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.23835 −0.0511121 −0.0255560 0.999673i \(-0.508136\pi\)
−0.0255560 + 0.999673i \(0.508136\pi\)
\(588\) 0 0
\(589\) −0.267949 −0.0110407
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 11.7298i − 0.481686i −0.970564 0.240843i \(-0.922576\pi\)
0.970564 0.240843i \(-0.0774238\pi\)
\(594\) 0 0
\(595\) 3.12436i 0.128086i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.8367 1.01480 0.507399 0.861711i \(-0.330607\pi\)
0.507399 + 0.861711i \(0.330607\pi\)
\(600\) 0 0
\(601\) 24.7846 1.01099 0.505493 0.862831i \(-0.331311\pi\)
0.505493 + 0.862831i \(0.331311\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.175865i 0.00714995i
\(606\) 0 0
\(607\) − 20.7846i − 0.843621i −0.906684 0.421811i \(-0.861395\pi\)
0.906684 0.421811i \(-0.138605\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 23.7370 0.960297
\(612\) 0 0
\(613\) 32.4641 1.31121 0.655606 0.755103i \(-0.272413\pi\)
0.655606 + 0.755103i \(0.272413\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 36.8439i 1.48328i 0.670799 + 0.741639i \(0.265951\pi\)
−0.670799 + 0.741639i \(0.734049\pi\)
\(618\) 0 0
\(619\) − 12.1244i − 0.487319i −0.969861 0.243659i \(-0.921652\pi\)
0.969861 0.243659i \(-0.0783479\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.7961 −0.432535
\(624\) 0 0
\(625\) 13.5885 0.543538
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 28.8119i − 1.14880i
\(630\) 0 0
\(631\) − 2.78461i − 0.110854i −0.998463 0.0554268i \(-0.982348\pi\)
0.998463 0.0554268i \(-0.0176519\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.93803 0.156276
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.1774i 0.599472i 0.954022 + 0.299736i \(0.0968986\pi\)
−0.954022 + 0.299736i \(0.903101\pi\)
\(642\) 0 0
\(643\) − 10.0718i − 0.397193i −0.980081 0.198596i \(-0.936362\pi\)
0.980081 0.198596i \(-0.0636383\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 31.3901 1.23407 0.617036 0.786935i \(-0.288333\pi\)
0.617036 + 0.786935i \(0.288333\pi\)
\(648\) 0 0
\(649\) −3.46410 −0.135978
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.2480i 0.674965i 0.941332 + 0.337482i \(0.109575\pi\)
−0.941332 + 0.337482i \(0.890425\pi\)
\(654\) 0 0
\(655\) 18.0000i 0.703318i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.4673 −1.57638 −0.788192 0.615429i \(-0.788983\pi\)
−0.788192 + 0.615429i \(0.788983\pi\)
\(660\) 0 0
\(661\) 24.7846 0.964010 0.482005 0.876169i \(-0.339909\pi\)
0.482005 + 0.876169i \(0.339909\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.896575i 0.0347677i
\(666\) 0 0
\(667\) 0.287187i 0.0111199i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 28.0812 1.08406
\(672\) 0 0
\(673\) 20.0000 0.770943 0.385472 0.922720i \(-0.374039\pi\)
0.385472 + 0.922720i \(0.374039\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.34658i 0.320785i 0.987053 + 0.160393i \(0.0512760\pi\)
−0.987053 + 0.160393i \(0.948724\pi\)
\(678\) 0 0
\(679\) 10.9282i 0.419386i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.27551 0.0488061 0.0244031 0.999702i \(-0.492231\pi\)
0.0244031 + 0.999702i \(0.492231\pi\)
\(684\) 0 0
\(685\) −16.3923 −0.626318
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 11.3137i − 0.431018i
\(690\) 0 0
\(691\) 31.4641i 1.19695i 0.801141 + 0.598475i \(0.204227\pi\)
−0.801141 + 0.598475i \(0.795773\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.69213 −0.253847
\(696\) 0 0
\(697\) −32.3397 −1.22496
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 44.0908i 1.66529i 0.553809 + 0.832644i \(0.313174\pi\)
−0.553809 + 0.832644i \(0.686826\pi\)
\(702\) 0 0
\(703\) − 8.26795i − 0.311832i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.14162 0.343806
\(708\) 0 0
\(709\) 21.3923 0.803405 0.401702 0.915770i \(-0.368419\pi\)
0.401702 + 0.915770i \(0.368419\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.0371647i 0.00139183i
\(714\) 0 0
\(715\) 12.0000i 0.448775i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 27.8038 1.03691 0.518453 0.855106i \(-0.326508\pi\)
0.518453 + 0.855106i \(0.326508\pi\)
\(720\) 0 0
\(721\) −6.12436 −0.228083
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.68835i 0.322677i
\(726\) 0 0
\(727\) − 49.3205i − 1.82920i −0.404364 0.914598i \(-0.632507\pi\)
0.404364 0.914598i \(-0.367493\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.86748 −0.0690713
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 9.79796i − 0.360912i
\(738\) 0 0
\(739\) 8.14359i 0.299567i 0.988719 + 0.149783i \(0.0478577\pi\)
−0.988719 + 0.149783i \(0.952142\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32.4625 −1.19094 −0.595468 0.803379i \(-0.703033\pi\)
−0.595468 + 0.803379i \(0.703033\pi\)
\(744\) 0 0
\(745\) 1.60770 0.0589014
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 10.1769i − 0.371856i
\(750\) 0 0
\(751\) − 36.9282i − 1.34753i −0.738946 0.673765i \(-0.764676\pi\)
0.738946 0.673765i \(-0.235324\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15.5291 0.565163
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 22.5259i − 0.816563i −0.912856 0.408281i \(-0.866128\pi\)
0.912856 0.408281i \(-0.133872\pi\)
\(762\) 0 0
\(763\) 10.4641i 0.378826i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.14110 −0.149527
\(768\) 0 0
\(769\) 10.2487 0.369578 0.184789 0.982778i \(-0.440840\pi\)
0.184789 + 0.982778i \(0.440840\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 10.1397i − 0.364701i −0.983234 0.182350i \(-0.941629\pi\)
0.983234 0.182350i \(-0.0583705\pi\)
\(774\) 0 0
\(775\) 1.12436i 0.0403880i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −9.28032 −0.332502
\(780\) 0 0
\(781\) 5.19615 0.185933
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.89898i 0.174852i
\(786\) 0 0
\(787\) 26.3923i 0.940784i 0.882458 + 0.470392i \(0.155887\pi\)
−0.882458 + 0.470392i \(0.844113\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.62158 0.164324
\(792\) 0 0
\(793\) 33.5692 1.19208
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 28.7747i − 1.01925i −0.860396 0.509626i \(-0.829784\pi\)
0.860396 0.509626i \(-0.170216\pi\)
\(798\) 0 0
\(799\) 20.6795i 0.731588i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 21.3891 0.754804
\(804\) 0 0
\(805\) 0.124356 0.00438296
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 42.2233i − 1.48449i −0.670127 0.742247i \(-0.733760\pi\)
0.670127 0.742247i \(-0.266240\pi\)
\(810\) 0 0
\(811\) 46.5167i 1.63342i 0.577048 + 0.816710i \(0.304205\pi\)
−0.577048 + 0.816710i \(0.695795\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −13.7360 −0.481151
\(816\) 0 0
\(817\) −0.535898 −0.0187487
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 17.0449i − 0.594871i −0.954742 0.297435i \(-0.903869\pi\)
0.954742 0.297435i \(-0.0961313\pi\)
\(822\) 0 0
\(823\) − 17.7128i − 0.617430i −0.951155 0.308715i \(-0.900101\pi\)
0.951155 0.308715i \(-0.0998989\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −26.7042 −0.928594 −0.464297 0.885679i \(-0.653693\pi\)
−0.464297 + 0.885679i \(0.653693\pi\)
\(828\) 0 0
\(829\) −15.1769 −0.527116 −0.263558 0.964644i \(-0.584896\pi\)
−0.263558 + 0.964644i \(0.584896\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 3.48477i − 0.120740i
\(834\) 0 0
\(835\) 12.2487i 0.423884i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16.4158 −0.566735 −0.283367 0.959011i \(-0.591452\pi\)
−0.283367 + 0.959011i \(0.591452\pi\)
\(840\) 0 0
\(841\) 24.7128 0.852166
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.68973i 0.0925294i
\(846\) 0 0
\(847\) − 0.196152i − 0.00673988i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.14677 −0.0393108
\(852\) 0 0
\(853\) −17.8564 −0.611392 −0.305696 0.952129i \(-0.598889\pi\)
−0.305696 + 0.952129i \(0.598889\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15.8981i 0.543069i 0.962429 + 0.271535i \(0.0875312\pi\)
−0.962429 + 0.271535i \(0.912469\pi\)
\(858\) 0 0
\(859\) 12.7128i 0.433756i 0.976199 + 0.216878i \(0.0695873\pi\)
−0.976199 + 0.216878i \(0.930413\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.3896 −0.898312 −0.449156 0.893453i \(-0.648275\pi\)
−0.449156 + 0.893453i \(0.648275\pi\)
\(864\) 0 0
\(865\) −9.92820 −0.337569
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 1.79315i − 0.0608285i
\(870\) 0 0
\(871\) − 11.7128i − 0.396874i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 8.24504 0.278733
\(876\) 0 0
\(877\) 9.21539 0.311182 0.155591 0.987822i \(-0.450272\pi\)
0.155591 + 0.987822i \(0.450272\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 41.2252i − 1.38891i −0.719535 0.694457i \(-0.755645\pi\)
0.719535 0.694457i \(-0.244355\pi\)
\(882\) 0 0
\(883\) 35.3205i 1.18863i 0.804232 + 0.594315i \(0.202577\pi\)
−0.804232 + 0.594315i \(0.797423\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −13.5873 −0.456218 −0.228109 0.973636i \(-0.573254\pi\)
−0.228109 + 0.973636i \(0.573254\pi\)
\(888\) 0 0
\(889\) −4.39230 −0.147313
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.93426i 0.198582i
\(894\) 0 0
\(895\) − 2.87564i − 0.0961222i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.554803 −0.0185037
\(900\) 0 0
\(901\) 9.85641 0.328365
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 3.10583i − 0.103241i
\(906\) 0 0
\(907\) 2.78461i 0.0924614i 0.998931 + 0.0462307i \(0.0147209\pi\)
−0.998931 + 0.0462307i \(0.985279\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −50.5328 −1.67422 −0.837112 0.547031i \(-0.815758\pi\)
−0.837112 + 0.547031i \(0.815758\pi\)
\(912\) 0 0
\(913\) −42.2487 −1.39823
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 20.0764i − 0.662981i
\(918\) 0 0
\(919\) 37.0718i 1.22289i 0.791289 + 0.611443i \(0.209411\pi\)
−0.791289 + 0.611443i \(0.790589\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.21166 0.204459
\(924\) 0 0
\(925\) −34.6936 −1.14072
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 23.0807i − 0.757253i −0.925550 0.378626i \(-0.876397\pi\)
0.925550 0.378626i \(-0.123603\pi\)
\(930\) 0 0
\(931\) − 1.00000i − 0.0327737i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −10.4543 −0.341892
\(936\) 0 0
\(937\) 11.3205 0.369825 0.184912 0.982755i \(-0.440800\pi\)
0.184912 + 0.982755i \(0.440800\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 44.9131i − 1.46412i −0.681238 0.732062i \(-0.738558\pi\)
0.681238 0.732062i \(-0.261442\pi\)
\(942\) 0 0
\(943\) 1.28719i 0.0419166i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −45.1905 −1.46849 −0.734246 0.678883i \(-0.762464\pi\)
−0.734246 + 0.678883i \(0.762464\pi\)
\(948\) 0 0
\(949\) 25.5692 0.830012
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 15.6579i − 0.507209i −0.967308 0.253604i \(-0.918384\pi\)
0.967308 0.253604i \(-0.0816161\pi\)
\(954\) 0 0
\(955\) 14.2295i 0.460455i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.2832 0.590397
\(960\) 0 0
\(961\) 30.9282 0.997684
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.960947i 0.0309340i
\(966\) 0 0
\(967\) − 19.4641i − 0.625923i −0.949766 0.312962i \(-0.898679\pi\)
0.949766 0.312962i \(-0.101321\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −5.45378 −0.175020 −0.0875101 0.996164i \(-0.527891\pi\)
−0.0875101 + 0.996164i \(0.527891\pi\)
\(972\) 0 0
\(973\) 7.46410 0.239288
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 5.10205i − 0.163229i −0.996664 0.0816145i \(-0.973992\pi\)
0.996664 0.0816145i \(-0.0260076\pi\)
\(978\) 0 0
\(979\) − 36.1244i − 1.15454i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −10.4815 −0.334308 −0.167154 0.985931i \(-0.553458\pi\)
−0.167154 + 0.985931i \(0.553458\pi\)
\(984\) 0 0
\(985\) −5.32051 −0.169525
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.0743295i 0.00236354i
\(990\) 0 0
\(991\) − 3.60770i − 0.114602i −0.998357 0.0573011i \(-0.981750\pi\)
0.998357 0.0573011i \(-0.0182495\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 17.5154 0.555276
\(996\) 0 0
\(997\) 12.2487 0.387921 0.193960 0.981009i \(-0.437867\pi\)
0.193960 + 0.981009i \(0.437867\pi\)
\(998\) 0 0
\(999\) 0 0
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 6048.2.h.f.2591.5 yes 8
3.2 odd 2 inner 6048.2.h.f.2591.3 8
4.3 odd 2 inner 6048.2.h.f.2591.6 yes 8
12.11 even 2 inner 6048.2.h.f.2591.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.f.2591.3 8 3.2 odd 2 inner
6048.2.h.f.2591.4 yes 8 12.11 even 2 inner
6048.2.h.f.2591.5 yes 8 1.1 even 1 trivial
6048.2.h.f.2591.6 yes 8 4.3 odd 2 inner