Properties

Label 3380.2.a.k.1.3
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.24698 q^{3} -1.00000 q^{5} +1.24698 q^{7} -1.44504 q^{9} +O(q^{10})\) \(q+1.24698 q^{3} -1.00000 q^{5} +1.24698 q^{7} -1.44504 q^{9} +0.198062 q^{11} -1.24698 q^{15} -4.85086 q^{17} +1.35690 q^{19} +1.55496 q^{21} -1.44504 q^{23} +1.00000 q^{25} -5.54288 q^{27} +1.13706 q^{29} -6.85086 q^{31} +0.246980 q^{33} -1.24698 q^{35} +3.00000 q^{37} -3.54288 q^{41} -7.89977 q^{43} +1.44504 q^{45} +8.87263 q^{47} -5.44504 q^{49} -6.04892 q^{51} -1.86831 q^{53} -0.198062 q^{55} +1.69202 q^{57} +0.878002 q^{59} -8.19806 q^{61} -1.80194 q^{63} -0.207751 q^{67} -1.80194 q^{69} -0.664874 q^{71} +1.72587 q^{73} +1.24698 q^{75} +0.246980 q^{77} -12.5700 q^{79} -2.57673 q^{81} -12.3230 q^{83} +4.85086 q^{85} +1.41789 q^{87} +15.8659 q^{89} -8.54288 q^{93} -1.35690 q^{95} +1.31336 q^{97} -0.286208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{3} - 3q^{5} - q^{7} - 4q^{9} + O(q^{10}) \) \( 3q - q^{3} - 3q^{5} - q^{7} - 4q^{9} + 5q^{11} + q^{15} - q^{17} + 5q^{21} - 4q^{23} + 3q^{25} + 2q^{27} - 2q^{29} - 7q^{31} - 4q^{33} + q^{35} + 9q^{37} + 8q^{41} - q^{43} + 4q^{45} + 10q^{47} - 16q^{49} - 9q^{51} - 8q^{53} - 5q^{55} - 17q^{59} - 29q^{61} - q^{63} + 17q^{67} - q^{69} - 3q^{71} + 16q^{73} - q^{75} - 4q^{77} - 13q^{79} - 5q^{81} - 17q^{83} + q^{85} + 10q^{87} + 9q^{89} - 7q^{93} + 6q^{97} - 9q^{99} + O(q^{100}) \)

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
\(3\) 1.24698 0.719944 0.359972 0.932963i \(-0.382786\pi\)
0.359972 + 0.932963i \(0.382786\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.24698 0.471314 0.235657 0.971836i \(-0.424276\pi\)
0.235657 + 0.971836i \(0.424276\pi\)
\(8\) 0 0
\(9\) −1.44504 −0.481681
\(10\) 0 0
\(11\) 0.198062 0.0597180 0.0298590 0.999554i \(-0.490494\pi\)
0.0298590 + 0.999554i \(0.490494\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −1.24698 −0.321969
\(16\) 0 0
\(17\) −4.85086 −1.17651 −0.588253 0.808677i \(-0.700184\pi\)
−0.588253 + 0.808677i \(0.700184\pi\)
\(18\) 0 0
\(19\) 1.35690 0.311293 0.155647 0.987813i \(-0.450254\pi\)
0.155647 + 0.987813i \(0.450254\pi\)
\(20\) 0 0
\(21\) 1.55496 0.339320
\(22\) 0 0
\(23\) −1.44504 −0.301312 −0.150656 0.988586i \(-0.548139\pi\)
−0.150656 + 0.988586i \(0.548139\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.54288 −1.06673
\(28\) 0 0
\(29\) 1.13706 0.211147 0.105574 0.994411i \(-0.466332\pi\)
0.105574 + 0.994411i \(0.466332\pi\)
\(30\) 0 0
\(31\) −6.85086 −1.23045 −0.615225 0.788352i \(-0.710935\pi\)
−0.615225 + 0.788352i \(0.710935\pi\)
\(32\) 0 0
\(33\) 0.246980 0.0429936
\(34\) 0 0
\(35\) −1.24698 −0.210778
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.54288 −0.553304 −0.276652 0.960970i \(-0.589225\pi\)
−0.276652 + 0.960970i \(0.589225\pi\)
\(42\) 0 0
\(43\) −7.89977 −1.20470 −0.602352 0.798231i \(-0.705770\pi\)
−0.602352 + 0.798231i \(0.705770\pi\)
\(44\) 0 0
\(45\) 1.44504 0.215414
\(46\) 0 0
\(47\) 8.87263 1.29421 0.647103 0.762403i \(-0.275980\pi\)
0.647103 + 0.762403i \(0.275980\pi\)
\(48\) 0 0
\(49\) −5.44504 −0.777863
\(50\) 0 0
\(51\) −6.04892 −0.847018
\(52\) 0 0
\(53\) −1.86831 −0.256633 −0.128316 0.991733i \(-0.540957\pi\)
−0.128316 + 0.991733i \(0.540957\pi\)
\(54\) 0 0
\(55\) −0.198062 −0.0267067
\(56\) 0 0
\(57\) 1.69202 0.224114
\(58\) 0 0
\(59\) 0.878002 0.114306 0.0571531 0.998365i \(-0.481798\pi\)
0.0571531 + 0.998365i \(0.481798\pi\)
\(60\) 0 0
\(61\) −8.19806 −1.04965 −0.524827 0.851209i \(-0.675870\pi\)
−0.524827 + 0.851209i \(0.675870\pi\)
\(62\) 0 0
\(63\) −1.80194 −0.227023
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.207751 −0.0253808 −0.0126904 0.999919i \(-0.504040\pi\)
−0.0126904 + 0.999919i \(0.504040\pi\)
\(68\) 0 0
\(69\) −1.80194 −0.216928
\(70\) 0 0
\(71\) −0.664874 −0.0789061 −0.0394530 0.999221i \(-0.512562\pi\)
−0.0394530 + 0.999221i \(0.512562\pi\)
\(72\) 0 0
\(73\) 1.72587 0.201998 0.100999 0.994887i \(-0.467796\pi\)
0.100999 + 0.994887i \(0.467796\pi\)
\(74\) 0 0
\(75\) 1.24698 0.143989
\(76\) 0 0
\(77\) 0.246980 0.0281459
\(78\) 0 0
\(79\) −12.5700 −1.41424 −0.707119 0.707094i \(-0.750006\pi\)
−0.707119 + 0.707094i \(0.750006\pi\)
\(80\) 0 0
\(81\) −2.57673 −0.286303
\(82\) 0 0
\(83\) −12.3230 −1.35263 −0.676315 0.736613i \(-0.736424\pi\)
−0.676315 + 0.736613i \(0.736424\pi\)
\(84\) 0 0
\(85\) 4.85086 0.526149
\(86\) 0 0
\(87\) 1.41789 0.152014
\(88\) 0 0
\(89\) 15.8659 1.68178 0.840892 0.541203i \(-0.182031\pi\)
0.840892 + 0.541203i \(0.182031\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.54288 −0.885855
\(94\) 0 0
\(95\) −1.35690 −0.139215
\(96\) 0 0
\(97\) 1.31336 0.133351 0.0666755 0.997775i \(-0.478761\pi\)
0.0666755 + 0.997775i \(0.478761\pi\)
\(98\) 0 0
\(99\) −0.286208 −0.0287650
\(100\) 0 0
\(101\) 13.7506 1.36824 0.684119 0.729370i \(-0.260187\pi\)
0.684119 + 0.729370i \(0.260187\pi\)
\(102\) 0 0
\(103\) −11.9608 −1.17853 −0.589265 0.807940i \(-0.700583\pi\)
−0.589265 + 0.807940i \(0.700583\pi\)
\(104\) 0 0
\(105\) −1.55496 −0.151748
\(106\) 0 0
\(107\) −5.47219 −0.529016 −0.264508 0.964383i \(-0.585210\pi\)
−0.264508 + 0.964383i \(0.585210\pi\)
\(108\) 0 0
\(109\) −18.8116 −1.80183 −0.900914 0.433999i \(-0.857102\pi\)
−0.900914 + 0.433999i \(0.857102\pi\)
\(110\) 0 0
\(111\) 3.74094 0.355074
\(112\) 0 0
\(113\) 3.72886 0.350781 0.175391 0.984499i \(-0.443881\pi\)
0.175391 + 0.984499i \(0.443881\pi\)
\(114\) 0 0
\(115\) 1.44504 0.134751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.04892 −0.554503
\(120\) 0 0
\(121\) −10.9608 −0.996434
\(122\) 0 0
\(123\) −4.41789 −0.398348
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 7.66786 0.680412 0.340206 0.940351i \(-0.389503\pi\)
0.340206 + 0.940351i \(0.389503\pi\)
\(128\) 0 0
\(129\) −9.85086 −0.867319
\(130\) 0 0
\(131\) −9.32304 −0.814558 −0.407279 0.913304i \(-0.633522\pi\)
−0.407279 + 0.913304i \(0.633522\pi\)
\(132\) 0 0
\(133\) 1.69202 0.146717
\(134\) 0 0
\(135\) 5.54288 0.477055
\(136\) 0 0
\(137\) 4.72348 0.403554 0.201777 0.979431i \(-0.435328\pi\)
0.201777 + 0.979431i \(0.435328\pi\)
\(138\) 0 0
\(139\) 0.818331 0.0694099 0.0347050 0.999398i \(-0.488951\pi\)
0.0347050 + 0.999398i \(0.488951\pi\)
\(140\) 0 0
\(141\) 11.0640 0.931755
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.13706 −0.0944280
\(146\) 0 0
\(147\) −6.78986 −0.560018
\(148\) 0 0
\(149\) 23.6233 1.93529 0.967646 0.252311i \(-0.0811907\pi\)
0.967646 + 0.252311i \(0.0811907\pi\)
\(150\) 0 0
\(151\) 6.78448 0.552113 0.276057 0.961141i \(-0.410972\pi\)
0.276057 + 0.961141i \(0.410972\pi\)
\(152\) 0 0
\(153\) 7.00969 0.566700
\(154\) 0 0
\(155\) 6.85086 0.550274
\(156\) 0 0
\(157\) −16.4276 −1.31106 −0.655532 0.755167i \(-0.727556\pi\)
−0.655532 + 0.755167i \(0.727556\pi\)
\(158\) 0 0
\(159\) −2.32975 −0.184761
\(160\) 0 0
\(161\) −1.80194 −0.142013
\(162\) 0 0
\(163\) −12.7463 −0.998368 −0.499184 0.866496i \(-0.666367\pi\)
−0.499184 + 0.866496i \(0.666367\pi\)
\(164\) 0 0
\(165\) −0.246980 −0.0192273
\(166\) 0 0
\(167\) −15.2054 −1.17663 −0.588313 0.808633i \(-0.700208\pi\)
−0.588313 + 0.808633i \(0.700208\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −1.96077 −0.149944
\(172\) 0 0
\(173\) −13.7845 −1.04801 −0.524007 0.851714i \(-0.675564\pi\)
−0.524007 + 0.851714i \(0.675564\pi\)
\(174\) 0 0
\(175\) 1.24698 0.0942628
\(176\) 0 0
\(177\) 1.09485 0.0822940
\(178\) 0 0
\(179\) −6.51035 −0.486607 −0.243303 0.969950i \(-0.578231\pi\)
−0.243303 + 0.969950i \(0.578231\pi\)
\(180\) 0 0
\(181\) −4.45712 −0.331295 −0.165648 0.986185i \(-0.552971\pi\)
−0.165648 + 0.986185i \(0.552971\pi\)
\(182\) 0 0
\(183\) −10.2228 −0.755692
\(184\) 0 0
\(185\) −3.00000 −0.220564
\(186\) 0 0
\(187\) −0.960771 −0.0702586
\(188\) 0 0
\(189\) −6.91185 −0.502763
\(190\) 0 0
\(191\) 23.1564 1.67554 0.837771 0.546022i \(-0.183859\pi\)
0.837771 + 0.546022i \(0.183859\pi\)
\(192\) 0 0
\(193\) 0.692021 0.0498128 0.0249064 0.999690i \(-0.492071\pi\)
0.0249064 + 0.999690i \(0.492071\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.98792 −0.497869 −0.248934 0.968520i \(-0.580080\pi\)
−0.248934 + 0.968520i \(0.580080\pi\)
\(198\) 0 0
\(199\) 18.1400 1.28591 0.642957 0.765902i \(-0.277707\pi\)
0.642957 + 0.765902i \(0.277707\pi\)
\(200\) 0 0
\(201\) −0.259061 −0.0182728
\(202\) 0 0
\(203\) 1.41789 0.0995167
\(204\) 0 0
\(205\) 3.54288 0.247445
\(206\) 0 0
\(207\) 2.08815 0.145136
\(208\) 0 0
\(209\) 0.268750 0.0185898
\(210\) 0 0
\(211\) 16.9095 1.16410 0.582048 0.813155i \(-0.302252\pi\)
0.582048 + 0.813155i \(0.302252\pi\)
\(212\) 0 0
\(213\) −0.829085 −0.0568080
\(214\) 0 0
\(215\) 7.89977 0.538760
\(216\) 0 0
\(217\) −8.54288 −0.579928
\(218\) 0 0
\(219\) 2.15213 0.145427
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −12.0532 −0.807144 −0.403572 0.914948i \(-0.632231\pi\)
−0.403572 + 0.914948i \(0.632231\pi\)
\(224\) 0 0
\(225\) −1.44504 −0.0963361
\(226\) 0 0
\(227\) 15.1642 1.00648 0.503242 0.864146i \(-0.332140\pi\)
0.503242 + 0.864146i \(0.332140\pi\)
\(228\) 0 0
\(229\) −20.5375 −1.35716 −0.678578 0.734528i \(-0.737403\pi\)
−0.678578 + 0.734528i \(0.737403\pi\)
\(230\) 0 0
\(231\) 0.307979 0.0202635
\(232\) 0 0
\(233\) −17.2198 −1.12811 −0.564054 0.825738i \(-0.690759\pi\)
−0.564054 + 0.825738i \(0.690759\pi\)
\(234\) 0 0
\(235\) −8.87263 −0.578786
\(236\) 0 0
\(237\) −15.6746 −1.01817
\(238\) 0 0
\(239\) −15.4112 −0.996867 −0.498434 0.866928i \(-0.666091\pi\)
−0.498434 + 0.866928i \(0.666091\pi\)
\(240\) 0 0
\(241\) −10.2892 −0.662785 −0.331393 0.943493i \(-0.607518\pi\)
−0.331393 + 0.943493i \(0.607518\pi\)
\(242\) 0 0
\(243\) 13.4155 0.860605
\(244\) 0 0
\(245\) 5.44504 0.347871
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −15.3666 −0.973818
\(250\) 0 0
\(251\) −8.09246 −0.510791 −0.255396 0.966837i \(-0.582206\pi\)
−0.255396 + 0.966837i \(0.582206\pi\)
\(252\) 0 0
\(253\) −0.286208 −0.0179938
\(254\) 0 0
\(255\) 6.04892 0.378798
\(256\) 0 0
\(257\) 31.4228 1.96010 0.980050 0.198750i \(-0.0636883\pi\)
0.980050 + 0.198750i \(0.0636883\pi\)
\(258\) 0 0
\(259\) 3.74094 0.232451
\(260\) 0 0
\(261\) −1.64310 −0.101706
\(262\) 0 0
\(263\) 6.84117 0.421844 0.210922 0.977503i \(-0.432353\pi\)
0.210922 + 0.977503i \(0.432353\pi\)
\(264\) 0 0
\(265\) 1.86831 0.114770
\(266\) 0 0
\(267\) 19.7845 1.21079
\(268\) 0 0
\(269\) 0.735562 0.0448480 0.0224240 0.999749i \(-0.492862\pi\)
0.0224240 + 0.999749i \(0.492862\pi\)
\(270\) 0 0
\(271\) 0.994623 0.0604191 0.0302095 0.999544i \(-0.490383\pi\)
0.0302095 + 0.999544i \(0.490383\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.198062 0.0119436
\(276\) 0 0
\(277\) −27.3110 −1.64096 −0.820478 0.571678i \(-0.806293\pi\)
−0.820478 + 0.571678i \(0.806293\pi\)
\(278\) 0 0
\(279\) 9.89977 0.592684
\(280\) 0 0
\(281\) −13.3327 −0.795364 −0.397682 0.917523i \(-0.630185\pi\)
−0.397682 + 0.917523i \(0.630185\pi\)
\(282\) 0 0
\(283\) 0.907542 0.0539478 0.0269739 0.999636i \(-0.491413\pi\)
0.0269739 + 0.999636i \(0.491413\pi\)
\(284\) 0 0
\(285\) −1.69202 −0.100227
\(286\) 0 0
\(287\) −4.41789 −0.260780
\(288\) 0 0
\(289\) 6.53079 0.384164
\(290\) 0 0
\(291\) 1.63773 0.0960053
\(292\) 0 0
\(293\) −6.68425 −0.390498 −0.195249 0.980754i \(-0.562552\pi\)
−0.195249 + 0.980754i \(0.562552\pi\)
\(294\) 0 0
\(295\) −0.878002 −0.0511193
\(296\) 0 0
\(297\) −1.09783 −0.0637028
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −9.85086 −0.567794
\(302\) 0 0
\(303\) 17.1468 0.985055
\(304\) 0 0
\(305\) 8.19806 0.469420
\(306\) 0 0
\(307\) 30.9269 1.76509 0.882546 0.470226i \(-0.155827\pi\)
0.882546 + 0.470226i \(0.155827\pi\)
\(308\) 0 0
\(309\) −14.9148 −0.848475
\(310\) 0 0
\(311\) −19.9095 −1.12896 −0.564481 0.825446i \(-0.690924\pi\)
−0.564481 + 0.825446i \(0.690924\pi\)
\(312\) 0 0
\(313\) 26.9681 1.52433 0.762163 0.647386i \(-0.224138\pi\)
0.762163 + 0.647386i \(0.224138\pi\)
\(314\) 0 0
\(315\) 1.80194 0.101528
\(316\) 0 0
\(317\) 33.6829 1.89182 0.945911 0.324427i \(-0.105171\pi\)
0.945911 + 0.324427i \(0.105171\pi\)
\(318\) 0 0
\(319\) 0.225209 0.0126093
\(320\) 0 0
\(321\) −6.82371 −0.380862
\(322\) 0 0
\(323\) −6.58211 −0.366238
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −23.4577 −1.29721
\(328\) 0 0
\(329\) 11.0640 0.609977
\(330\) 0 0
\(331\) 0.115293 0.00633708 0.00316854 0.999995i \(-0.498991\pi\)
0.00316854 + 0.999995i \(0.498991\pi\)
\(332\) 0 0
\(333\) −4.33513 −0.237563
\(334\) 0 0
\(335\) 0.207751 0.0113506
\(336\) 0 0
\(337\) 25.0006 1.36187 0.680934 0.732344i \(-0.261574\pi\)
0.680934 + 0.732344i \(0.261574\pi\)
\(338\) 0 0
\(339\) 4.64981 0.252543
\(340\) 0 0
\(341\) −1.35690 −0.0734800
\(342\) 0 0
\(343\) −15.5187 −0.837932
\(344\) 0 0
\(345\) 1.80194 0.0970131
\(346\) 0 0
\(347\) 5.92931 0.318302 0.159151 0.987254i \(-0.449124\pi\)
0.159151 + 0.987254i \(0.449124\pi\)
\(348\) 0 0
\(349\) 28.0441 1.50117 0.750584 0.660775i \(-0.229772\pi\)
0.750584 + 0.660775i \(0.229772\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.1957 1.18136 0.590678 0.806907i \(-0.298860\pi\)
0.590678 + 0.806907i \(0.298860\pi\)
\(354\) 0 0
\(355\) 0.664874 0.0352879
\(356\) 0 0
\(357\) −7.54288 −0.399211
\(358\) 0 0
\(359\) −5.31037 −0.280271 −0.140135 0.990132i \(-0.544754\pi\)
−0.140135 + 0.990132i \(0.544754\pi\)
\(360\) 0 0
\(361\) −17.1588 −0.903097
\(362\) 0 0
\(363\) −13.6679 −0.717377
\(364\) 0 0
\(365\) −1.72587 −0.0903363
\(366\) 0 0
\(367\) −3.29696 −0.172100 −0.0860500 0.996291i \(-0.527424\pi\)
−0.0860500 + 0.996291i \(0.527424\pi\)
\(368\) 0 0
\(369\) 5.11960 0.266516
\(370\) 0 0
\(371\) −2.32975 −0.120955
\(372\) 0 0
\(373\) −8.70410 −0.450681 −0.225341 0.974280i \(-0.572350\pi\)
−0.225341 + 0.974280i \(0.572350\pi\)
\(374\) 0 0
\(375\) −1.24698 −0.0643937
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 15.9269 0.818111 0.409055 0.912510i \(-0.365858\pi\)
0.409055 + 0.912510i \(0.365858\pi\)
\(380\) 0 0
\(381\) 9.56166 0.489859
\(382\) 0 0
\(383\) −4.13169 −0.211119 −0.105560 0.994413i \(-0.533663\pi\)
−0.105560 + 0.994413i \(0.533663\pi\)
\(384\) 0 0
\(385\) −0.246980 −0.0125872
\(386\) 0 0
\(387\) 11.4155 0.580283
\(388\) 0 0
\(389\) −15.1347 −0.767358 −0.383679 0.923466i \(-0.625343\pi\)
−0.383679 + 0.923466i \(0.625343\pi\)
\(390\) 0 0
\(391\) 7.00969 0.354495
\(392\) 0 0
\(393\) −11.6256 −0.586436
\(394\) 0 0
\(395\) 12.5700 0.632467
\(396\) 0 0
\(397\) 19.5080 0.979076 0.489538 0.871982i \(-0.337165\pi\)
0.489538 + 0.871982i \(0.337165\pi\)
\(398\) 0 0
\(399\) 2.10992 0.105628
\(400\) 0 0
\(401\) −27.8974 −1.39313 −0.696564 0.717494i \(-0.745289\pi\)
−0.696564 + 0.717494i \(0.745289\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.57673 0.128039
\(406\) 0 0
\(407\) 0.594187 0.0294527
\(408\) 0 0
\(409\) −6.46788 −0.319816 −0.159908 0.987132i \(-0.551120\pi\)
−0.159908 + 0.987132i \(0.551120\pi\)
\(410\) 0 0
\(411\) 5.89008 0.290536
\(412\) 0 0
\(413\) 1.09485 0.0538741
\(414\) 0 0
\(415\) 12.3230 0.604914
\(416\) 0 0
\(417\) 1.02044 0.0499713
\(418\) 0 0
\(419\) 20.9541 1.02367 0.511837 0.859083i \(-0.328965\pi\)
0.511837 + 0.859083i \(0.328965\pi\)
\(420\) 0 0
\(421\) −11.5211 −0.561504 −0.280752 0.959780i \(-0.590584\pi\)
−0.280752 + 0.959780i \(0.590584\pi\)
\(422\) 0 0
\(423\) −12.8213 −0.623394
\(424\) 0 0
\(425\) −4.85086 −0.235301
\(426\) 0 0
\(427\) −10.2228 −0.494717
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.33273 0.305037 0.152519 0.988301i \(-0.451262\pi\)
0.152519 + 0.988301i \(0.451262\pi\)
\(432\) 0 0
\(433\) −11.5332 −0.554250 −0.277125 0.960834i \(-0.589382\pi\)
−0.277125 + 0.960834i \(0.589382\pi\)
\(434\) 0 0
\(435\) −1.41789 −0.0679829
\(436\) 0 0
\(437\) −1.96077 −0.0937964
\(438\) 0 0
\(439\) −27.7036 −1.32222 −0.661111 0.750288i \(-0.729915\pi\)
−0.661111 + 0.750288i \(0.729915\pi\)
\(440\) 0 0
\(441\) 7.86831 0.374682
\(442\) 0 0
\(443\) −12.2814 −0.583508 −0.291754 0.956493i \(-0.594239\pi\)
−0.291754 + 0.956493i \(0.594239\pi\)
\(444\) 0 0
\(445\) −15.8659 −0.752117
\(446\) 0 0
\(447\) 29.4577 1.39330
\(448\) 0 0
\(449\) −25.2556 −1.19189 −0.595943 0.803027i \(-0.703222\pi\)
−0.595943 + 0.803027i \(0.703222\pi\)
\(450\) 0 0
\(451\) −0.701710 −0.0330422
\(452\) 0 0
\(453\) 8.46011 0.397491
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 31.1608 1.45764 0.728819 0.684706i \(-0.240069\pi\)
0.728819 + 0.684706i \(0.240069\pi\)
\(458\) 0 0
\(459\) 26.8877 1.25501
\(460\) 0 0
\(461\) 0.605203 0.0281871 0.0140936 0.999901i \(-0.495514\pi\)
0.0140936 + 0.999901i \(0.495514\pi\)
\(462\) 0 0
\(463\) 21.3575 0.992567 0.496283 0.868161i \(-0.334698\pi\)
0.496283 + 0.868161i \(0.334698\pi\)
\(464\) 0 0
\(465\) 8.54288 0.396166
\(466\) 0 0
\(467\) 26.4886 1.22575 0.612873 0.790182i \(-0.290014\pi\)
0.612873 + 0.790182i \(0.290014\pi\)
\(468\) 0 0
\(469\) −0.259061 −0.0119623
\(470\) 0 0
\(471\) −20.4849 −0.943893
\(472\) 0 0
\(473\) −1.56465 −0.0719425
\(474\) 0 0
\(475\) 1.35690 0.0622587
\(476\) 0 0
\(477\) 2.69979 0.123615
\(478\) 0 0
\(479\) 5.08038 0.232128 0.116064 0.993242i \(-0.462972\pi\)
0.116064 + 0.993242i \(0.462972\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −2.24698 −0.102241
\(484\) 0 0
\(485\) −1.31336 −0.0596364
\(486\) 0 0
\(487\) −23.7646 −1.07688 −0.538439 0.842664i \(-0.680986\pi\)
−0.538439 + 0.842664i \(0.680986\pi\)
\(488\) 0 0
\(489\) −15.8944 −0.718769
\(490\) 0 0
\(491\) −12.9071 −0.582488 −0.291244 0.956649i \(-0.594069\pi\)
−0.291244 + 0.956649i \(0.594069\pi\)
\(492\) 0 0
\(493\) −5.51573 −0.248416
\(494\) 0 0
\(495\) 0.286208 0.0128641
\(496\) 0 0
\(497\) −0.829085 −0.0371895
\(498\) 0 0
\(499\) −40.3749 −1.80743 −0.903715 0.428134i \(-0.859171\pi\)
−0.903715 + 0.428134i \(0.859171\pi\)
\(500\) 0 0
\(501\) −18.9608 −0.847105
\(502\) 0 0
\(503\) −10.2634 −0.457621 −0.228811 0.973471i \(-0.573484\pi\)
−0.228811 + 0.973471i \(0.573484\pi\)
\(504\) 0 0
\(505\) −13.7506 −0.611895
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.40342 0.106530 0.0532649 0.998580i \(-0.483037\pi\)
0.0532649 + 0.998580i \(0.483037\pi\)
\(510\) 0 0
\(511\) 2.15213 0.0952046
\(512\) 0 0
\(513\) −7.52111 −0.332065
\(514\) 0 0
\(515\) 11.9608 0.527055
\(516\) 0 0
\(517\) 1.75733 0.0772874
\(518\) 0 0
\(519\) −17.1890 −0.754512
\(520\) 0 0
\(521\) 23.1806 1.01556 0.507780 0.861487i \(-0.330466\pi\)
0.507780 + 0.861487i \(0.330466\pi\)
\(522\) 0 0
\(523\) 40.0073 1.74940 0.874698 0.484668i \(-0.161059\pi\)
0.874698 + 0.484668i \(0.161059\pi\)
\(524\) 0 0
\(525\) 1.55496 0.0678639
\(526\) 0 0
\(527\) 33.2325 1.44763
\(528\) 0 0
\(529\) −20.9119 −0.909211
\(530\) 0 0
\(531\) −1.26875 −0.0550591
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.47219 0.236583
\(536\) 0 0
\(537\) −8.11828 −0.350330
\(538\) 0 0
\(539\) −1.07846 −0.0464524
\(540\) 0 0
\(541\) 3.13813 0.134919 0.0674593 0.997722i \(-0.478511\pi\)
0.0674593 + 0.997722i \(0.478511\pi\)
\(542\) 0 0
\(543\) −5.55794 −0.238514
\(544\) 0 0
\(545\) 18.8116 0.805802
\(546\) 0 0
\(547\) 10.9280 0.467247 0.233623 0.972327i \(-0.424942\pi\)
0.233623 + 0.972327i \(0.424942\pi\)
\(548\) 0 0
\(549\) 11.8465 0.505598
\(550\) 0 0
\(551\) 1.54288 0.0657288
\(552\) 0 0
\(553\) −15.6746 −0.666550
\(554\) 0 0
\(555\) −3.74094 −0.158794
\(556\) 0 0
\(557\) 6.19508 0.262494 0.131247 0.991350i \(-0.458102\pi\)
0.131247 + 0.991350i \(0.458102\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −1.19806 −0.0505822
\(562\) 0 0
\(563\) 33.0411 1.39252 0.696259 0.717790i \(-0.254846\pi\)
0.696259 + 0.717790i \(0.254846\pi\)
\(564\) 0 0
\(565\) −3.72886 −0.156874
\(566\) 0 0
\(567\) −3.21313 −0.134939
\(568\) 0 0
\(569\) −14.4577 −0.606099 −0.303049 0.952975i \(-0.598005\pi\)
−0.303049 + 0.952975i \(0.598005\pi\)
\(570\) 0 0
\(571\) 34.6765 1.45117 0.725583 0.688135i \(-0.241570\pi\)
0.725583 + 0.688135i \(0.241570\pi\)
\(572\) 0 0
\(573\) 28.8756 1.20630
\(574\) 0 0
\(575\) −1.44504 −0.0602624
\(576\) 0 0
\(577\) 36.9842 1.53967 0.769836 0.638242i \(-0.220338\pi\)
0.769836 + 0.638242i \(0.220338\pi\)
\(578\) 0 0
\(579\) 0.862937 0.0358624
\(580\) 0 0
\(581\) −15.3666 −0.637513
\(582\) 0 0
\(583\) −0.370042 −0.0153256
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0043 1.48606 0.743028 0.669260i \(-0.233389\pi\)
0.743028 + 0.669260i \(0.233389\pi\)
\(588\) 0 0
\(589\) −9.29590 −0.383031
\(590\) 0 0
\(591\) −8.71379 −0.358437
\(592\) 0 0
\(593\) −43.4161 −1.78289 −0.891443 0.453134i \(-0.850306\pi\)
−0.891443 + 0.453134i \(0.850306\pi\)
\(594\) 0 0
\(595\) 6.04892 0.247981
\(596\) 0 0
\(597\) 22.6203 0.925786
\(598\) 0 0
\(599\) 38.7549 1.58348 0.791742 0.610856i \(-0.209174\pi\)
0.791742 + 0.610856i \(0.209174\pi\)
\(600\) 0 0
\(601\) −18.1293 −0.739509 −0.369755 0.929129i \(-0.620558\pi\)
−0.369755 + 0.929129i \(0.620558\pi\)
\(602\) 0 0
\(603\) 0.300209 0.0122254
\(604\) 0 0
\(605\) 10.9608 0.445619
\(606\) 0 0
\(607\) −30.1094 −1.22210 −0.611052 0.791590i \(-0.709253\pi\)
−0.611052 + 0.791590i \(0.709253\pi\)
\(608\) 0 0
\(609\) 1.76809 0.0716465
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 40.9197 1.65273 0.826366 0.563133i \(-0.190404\pi\)
0.826366 + 0.563133i \(0.190404\pi\)
\(614\) 0 0
\(615\) 4.41789 0.178147
\(616\) 0 0
\(617\) −28.7308 −1.15666 −0.578329 0.815804i \(-0.696295\pi\)
−0.578329 + 0.815804i \(0.696295\pi\)
\(618\) 0 0
\(619\) −32.1118 −1.29068 −0.645342 0.763894i \(-0.723285\pi\)
−0.645342 + 0.763894i \(0.723285\pi\)
\(620\) 0 0
\(621\) 8.00969 0.321418
\(622\) 0 0
\(623\) 19.7845 0.792648
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.335126 0.0133836
\(628\) 0 0
\(629\) −14.5526 −0.580249
\(630\) 0 0
\(631\) 34.2010 1.36152 0.680761 0.732506i \(-0.261649\pi\)
0.680761 + 0.732506i \(0.261649\pi\)
\(632\) 0 0
\(633\) 21.0858 0.838083
\(634\) 0 0
\(635\) −7.66786 −0.304290
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0.960771 0.0380075
\(640\) 0 0
\(641\) 4.39506 0.173594 0.0867972 0.996226i \(-0.472337\pi\)
0.0867972 + 0.996226i \(0.472337\pi\)
\(642\) 0 0
\(643\) −36.3773 −1.43458 −0.717291 0.696774i \(-0.754618\pi\)
−0.717291 + 0.696774i \(0.754618\pi\)
\(644\) 0 0
\(645\) 9.85086 0.387877
\(646\) 0 0
\(647\) 18.1129 0.712092 0.356046 0.934469i \(-0.384125\pi\)
0.356046 + 0.934469i \(0.384125\pi\)
\(648\) 0 0
\(649\) 0.173899 0.00682614
\(650\) 0 0
\(651\) −10.6528 −0.417516
\(652\) 0 0
\(653\) −16.6601 −0.651960 −0.325980 0.945377i \(-0.605694\pi\)
−0.325980 + 0.945377i \(0.605694\pi\)
\(654\) 0 0
\(655\) 9.32304 0.364281
\(656\) 0 0
\(657\) −2.49396 −0.0972986
\(658\) 0 0
\(659\) 30.8471 1.20163 0.600817 0.799387i \(-0.294842\pi\)
0.600817 + 0.799387i \(0.294842\pi\)
\(660\) 0 0
\(661\) −22.4896 −0.874746 −0.437373 0.899280i \(-0.644091\pi\)
−0.437373 + 0.899280i \(0.644091\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.69202 −0.0656138
\(666\) 0 0
\(667\) −1.64310 −0.0636212
\(668\) 0 0
\(669\) −15.0301 −0.581098
\(670\) 0 0
\(671\) −1.62373 −0.0626833
\(672\) 0 0
\(673\) −48.7405 −1.87881 −0.939403 0.342814i \(-0.888620\pi\)
−0.939403 + 0.342814i \(0.888620\pi\)
\(674\) 0 0
\(675\) −5.54288 −0.213345
\(676\) 0 0
\(677\) −2.81295 −0.108111 −0.0540553 0.998538i \(-0.517215\pi\)
−0.0540553 + 0.998538i \(0.517215\pi\)
\(678\) 0 0
\(679\) 1.63773 0.0628502
\(680\) 0 0
\(681\) 18.9095 0.724612
\(682\) 0 0
\(683\) −0.701710 −0.0268502 −0.0134251 0.999910i \(-0.504273\pi\)
−0.0134251 + 0.999910i \(0.504273\pi\)
\(684\) 0 0
\(685\) −4.72348 −0.180475
\(686\) 0 0
\(687\) −25.6098 −0.977076
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 35.1377 1.33670 0.668350 0.743847i \(-0.267001\pi\)
0.668350 + 0.743847i \(0.267001\pi\)
\(692\) 0 0
\(693\) −0.356896 −0.0135574
\(694\) 0 0
\(695\) −0.818331 −0.0310411
\(696\) 0 0
\(697\) 17.1860 0.650965
\(698\) 0 0
\(699\) −21.4728 −0.812175
\(700\) 0 0
\(701\) 8.23968 0.311209 0.155604 0.987819i \(-0.450268\pi\)
0.155604 + 0.987819i \(0.450268\pi\)
\(702\) 0 0
\(703\) 4.07069 0.153529
\(704\) 0 0
\(705\) −11.0640 −0.416694
\(706\) 0 0
\(707\) 17.1468 0.644870
\(708\) 0 0
\(709\) −35.6359 −1.33834 −0.669168 0.743111i \(-0.733349\pi\)
−0.669168 + 0.743111i \(0.733349\pi\)
\(710\) 0 0
\(711\) 18.1642 0.681211
\(712\) 0 0
\(713\) 9.89977 0.370749
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −19.2174 −0.717688
\(718\) 0 0
\(719\) 45.3497 1.69126 0.845630 0.533770i \(-0.179225\pi\)
0.845630 + 0.533770i \(0.179225\pi\)
\(720\) 0 0
\(721\) −14.9148 −0.555458
\(722\) 0 0
\(723\) −12.8304 −0.477168
\(724\) 0 0
\(725\) 1.13706 0.0422295
\(726\) 0 0
\(727\) 35.2664 1.30796 0.653978 0.756513i \(-0.273099\pi\)
0.653978 + 0.756513i \(0.273099\pi\)
\(728\) 0 0
\(729\) 24.4590 0.905890
\(730\) 0 0
\(731\) 38.3207 1.41734
\(732\) 0 0
\(733\) −15.6722 −0.578865 −0.289432 0.957198i \(-0.593467\pi\)
−0.289432 + 0.957198i \(0.593467\pi\)
\(734\) 0 0
\(735\) 6.78986 0.250448
\(736\) 0 0
\(737\) −0.0411476 −0.00151569
\(738\) 0 0
\(739\) −8.80433 −0.323873 −0.161936 0.986801i \(-0.551774\pi\)
−0.161936 + 0.986801i \(0.551774\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −26.8974 −0.986769 −0.493385 0.869811i \(-0.664240\pi\)
−0.493385 + 0.869811i \(0.664240\pi\)
\(744\) 0 0
\(745\) −23.6233 −0.865489
\(746\) 0 0
\(747\) 17.8073 0.651536
\(748\) 0 0
\(749\) −6.82371 −0.249333
\(750\) 0 0
\(751\) −37.0200 −1.35088 −0.675439 0.737416i \(-0.736046\pi\)
−0.675439 + 0.737416i \(0.736046\pi\)
\(752\) 0 0
\(753\) −10.0911 −0.367741
\(754\) 0 0
\(755\) −6.78448 −0.246912
\(756\) 0 0
\(757\) −20.3730 −0.740470 −0.370235 0.928938i \(-0.620723\pi\)
−0.370235 + 0.928938i \(0.620723\pi\)
\(758\) 0 0
\(759\) −0.356896 −0.0129545
\(760\) 0 0
\(761\) −39.7362 −1.44043 −0.720217 0.693749i \(-0.755958\pi\)
−0.720217 + 0.693749i \(0.755958\pi\)
\(762\) 0 0
\(763\) −23.4577 −0.849226
\(764\) 0 0
\(765\) −7.00969 −0.253436
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 41.6171 1.50075 0.750376 0.661011i \(-0.229872\pi\)
0.750376 + 0.661011i \(0.229872\pi\)
\(770\) 0 0
\(771\) 39.1836 1.41116
\(772\) 0 0
\(773\) 38.2948 1.37737 0.688685 0.725061i \(-0.258188\pi\)
0.688685 + 0.725061i \(0.258188\pi\)
\(774\) 0 0
\(775\) −6.85086 −0.246090
\(776\) 0 0
\(777\) 4.66487 0.167351
\(778\) 0 0
\(779\) −4.80731 −0.172240
\(780\) 0 0
\(781\) −0.131687 −0.00471211
\(782\) 0 0
\(783\) −6.30260 −0.225237
\(784\) 0 0
\(785\) 16.4276 0.586326
\(786\) 0 0
\(787\) 52.0847 1.85662 0.928309 0.371809i \(-0.121262\pi\)
0.928309 + 0.371809i \(0.121262\pi\)
\(788\) 0 0
\(789\) 8.53079 0.303704
\(790\) 0 0
\(791\) 4.64981 0.165328
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 2.32975 0.0826277
\(796\) 0 0
\(797\) −19.8278 −0.702335 −0.351168 0.936313i \(-0.614215\pi\)
−0.351168 + 0.936313i \(0.614215\pi\)
\(798\) 0 0
\(799\) −43.0398 −1.52264
\(800\) 0 0
\(801\) −22.9269 −0.810083
\(802\) 0 0
\(803\) 0.341830 0.0120629
\(804\) 0 0
\(805\) 1.80194 0.0635100
\(806\) 0 0
\(807\) 0.917231 0.0322881
\(808\) 0 0
\(809\) −2.59073 −0.0910852 −0.0455426 0.998962i \(-0.514502\pi\)
−0.0455426 + 0.998962i \(0.514502\pi\)
\(810\) 0 0
\(811\) −0.946297 −0.0332290 −0.0166145 0.999862i \(-0.505289\pi\)
−0.0166145 + 0.999862i \(0.505289\pi\)
\(812\) 0 0
\(813\) 1.24027 0.0434983
\(814\) 0 0
\(815\) 12.7463 0.446484
\(816\) 0 0
\(817\) −10.7192 −0.375016
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.83148 −0.203520 −0.101760 0.994809i \(-0.532447\pi\)
−0.101760 + 0.994809i \(0.532447\pi\)
\(822\) 0 0
\(823\) 25.2083 0.878708 0.439354 0.898314i \(-0.355207\pi\)
0.439354 + 0.898314i \(0.355207\pi\)
\(824\) 0 0
\(825\) 0.246980 0.00859873
\(826\) 0 0
\(827\) 27.0127 0.939323 0.469661 0.882847i \(-0.344376\pi\)
0.469661 + 0.882847i \(0.344376\pi\)
\(828\) 0 0
\(829\) −20.4064 −0.708744 −0.354372 0.935105i \(-0.615305\pi\)
−0.354372 + 0.935105i \(0.615305\pi\)
\(830\) 0 0
\(831\) −34.0562 −1.18140
\(832\) 0 0
\(833\) 26.4131 0.915160
\(834\) 0 0
\(835\) 15.2054 0.526203
\(836\) 0 0
\(837\) 37.9734 1.31255
\(838\) 0 0
\(839\) −39.1463 −1.35148 −0.675740 0.737140i \(-0.736176\pi\)
−0.675740 + 0.737140i \(0.736176\pi\)
\(840\) 0 0
\(841\) −27.7071 −0.955417
\(842\) 0 0
\(843\) −16.6256 −0.572618
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −13.6679 −0.469633
\(848\) 0 0
\(849\) 1.13169 0.0388394
\(850\) 0 0
\(851\) −4.33513 −0.148606
\(852\) 0 0
\(853\) 4.51009 0.154423 0.0772113 0.997015i \(-0.475398\pi\)
0.0772113 + 0.997015i \(0.475398\pi\)
\(854\) 0 0
\(855\) 1.96077 0.0670570
\(856\) 0 0
\(857\) 19.3948 0.662514 0.331257 0.943541i \(-0.392527\pi\)
0.331257 + 0.943541i \(0.392527\pi\)
\(858\) 0 0
\(859\) 39.7429 1.35601 0.678004 0.735058i \(-0.262845\pi\)
0.678004 + 0.735058i \(0.262845\pi\)
\(860\) 0 0
\(861\) −5.50902 −0.187747
\(862\) 0 0
\(863\) 26.7827 0.911693 0.455846 0.890058i \(-0.349337\pi\)
0.455846 + 0.890058i \(0.349337\pi\)
\(864\) 0 0
\(865\) 13.7845 0.468686
\(866\) 0 0
\(867\) 8.14377 0.276577
\(868\) 0 0
\(869\) −2.48965 −0.0844555
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1.89785 −0.0642326
\(874\) 0 0
\(875\) −1.24698 −0.0421556
\(876\) 0 0
\(877\) 12.2185 0.412590 0.206295 0.978490i \(-0.433859\pi\)
0.206295 + 0.978490i \(0.433859\pi\)
\(878\) 0 0
\(879\) −8.33513 −0.281137
\(880\) 0 0
\(881\) −1.72827 −0.0582268 −0.0291134 0.999576i \(-0.509268\pi\)
−0.0291134 + 0.999576i \(0.509268\pi\)
\(882\) 0 0
\(883\) 38.6340 1.30014 0.650069 0.759875i \(-0.274740\pi\)
0.650069 + 0.759875i \(0.274740\pi\)
\(884\) 0 0
\(885\) −1.09485 −0.0368030
\(886\) 0 0
\(887\) −36.6055 −1.22909 −0.614547 0.788880i \(-0.710661\pi\)
−0.614547 + 0.788880i \(0.710661\pi\)
\(888\) 0 0
\(889\) 9.56166 0.320688
\(890\) 0 0
\(891\) −0.510353 −0.0170975
\(892\) 0 0
\(893\) 12.0392 0.402877
\(894\) 0 0
\(895\) 6.51035 0.217617
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −7.78986 −0.259806
\(900\) 0 0
\(901\) 9.06292 0.301930
\(902\) 0 0
\(903\) −12.2838 −0.408780
\(904\) 0 0
\(905\) 4.45712 0.148160
\(906\) 0 0
\(907\) 49.5881 1.64654 0.823272 0.567646i \(-0.192146\pi\)
0.823272 + 0.567646i \(0.192146\pi\)
\(908\) 0 0
\(909\) −19.8702 −0.659054
\(910\) 0 0
\(911\) 32.7982 1.08665 0.543327 0.839521i \(-0.317164\pi\)
0.543327 + 0.839521i \(0.317164\pi\)
\(912\) 0 0
\(913\) −2.44073 −0.0807764
\(914\) 0 0
\(915\) 10.2228 0.337956
\(916\) 0 0
\(917\) −11.6256 −0.383913
\(918\) 0 0
\(919\) −14.6944 −0.484724 −0.242362 0.970186i \(-0.577922\pi\)
−0.242362 + 0.970186i \(0.577922\pi\)
\(920\) 0 0
\(921\) 38.5652 1.27077
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.00000 0.0986394
\(926\) 0 0
\(927\) 17.2838 0.567675
\(928\) 0 0
\(929\) −2.53127 −0.0830482 −0.0415241 0.999138i \(-0.513221\pi\)
−0.0415241 + 0.999138i \(0.513221\pi\)
\(930\) 0 0
\(931\) −7.38835 −0.242144
\(932\) 0 0
\(933\) −24.8267 −0.812789
\(934\) 0 0
\(935\) 0.960771 0.0314206
\(936\) 0 0
\(937\) −14.3951 −0.470266 −0.235133 0.971963i \(-0.575553\pi\)
−0.235133 + 0.971963i \(0.575553\pi\)
\(938\) 0 0
\(939\) 33.6286 1.09743
\(940\) 0 0
\(941\) 18.7885 0.612489 0.306244 0.951953i \(-0.400928\pi\)
0.306244 + 0.951953i \(0.400928\pi\)
\(942\) 0 0
\(943\) 5.11960 0.166717
\(944\) 0 0
\(945\) 6.91185 0.224843
\(946\) 0 0
\(947\) −50.4543 −1.63954 −0.819772 0.572691i \(-0.805900\pi\)
−0.819772 + 0.572691i \(0.805900\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 42.0019 1.36201
\(952\) 0 0
\(953\) 56.6708 1.83575 0.917874 0.396871i \(-0.129904\pi\)
0.917874 + 0.396871i \(0.129904\pi\)
\(954\) 0 0
\(955\) −23.1564 −0.749325
\(956\) 0 0
\(957\) 0.280831 0.00907799
\(958\) 0 0
\(959\) 5.89008 0.190201
\(960\) 0 0
\(961\) 15.9342 0.514007
\(962\) 0 0
\(963\) 7.90754 0.254817
\(964\) 0 0
\(965\) −0.692021 −0.0222770
\(966\) 0 0
\(967\) 13.4421 0.432267 0.216134 0.976364i \(-0.430655\pi\)
0.216134 + 0.976364i \(0.430655\pi\)
\(968\) 0 0
\(969\) −8.20775 −0.263671
\(970\) 0 0
\(971\) 41.6963 1.33810 0.669050 0.743218i \(-0.266701\pi\)
0.669050 + 0.743218i \(0.266701\pi\)
\(972\) 0 0
\(973\) 1.02044 0.0327139
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.2228 0.487021 0.243511 0.969898i \(-0.421701\pi\)
0.243511 + 0.969898i \(0.421701\pi\)
\(978\) 0 0
\(979\) 3.14244 0.100433
\(980\) 0 0
\(981\) 27.1836 0.867905
\(982\) 0 0
\(983\) −42.7888 −1.36475 −0.682375 0.731002i \(-0.739053\pi\)
−0.682375 + 0.731002i \(0.739053\pi\)
\(984\) 0 0
\(985\) 6.98792 0.222654
\(986\) 0 0
\(987\) 13.7966 0.439149
\(988\) 0 0
\(989\) 11.4155 0.362992
\(990\) 0 0
\(991\) 18.2857 0.580865 0.290433 0.956895i \(-0.406201\pi\)
0.290433 + 0.956895i \(0.406201\pi\)
\(992\) 0 0
\(993\) 0.143768 0.00456234
\(994\) 0 0
\(995\) −18.1400 −0.575078
\(996\) 0 0
\(997\) −29.7415 −0.941924 −0.470962 0.882154i \(-0.656093\pi\)
−0.470962 + 0.882154i \(0.656093\pi\)
\(998\) 0 0
\(999\) −16.6286 −0.526107
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 3380.2.a.k.1.3 3
13.5 odd 4 3380.2.f.g.3041.6 6
13.8 odd 4 3380.2.f.g.3041.5 6
13.12 even 2 3380.2.a.l.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.2.a.k.1.3 3 1.1 even 1 trivial
3380.2.a.l.1.3 yes 3 13.12 even 2
3380.2.f.g.3041.5 6 13.8 odd 4
3380.2.f.g.3041.6 6 13.5 odd 4