Properties

Label 6400.2.a.ce.1.2
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.73205 q^{3} +0.732051 q^{7} +4.46410 q^{9} +O(q^{10})\) \(q+2.73205 q^{3} +0.732051 q^{7} +4.46410 q^{9} -2.00000 q^{11} -3.46410 q^{13} -3.46410 q^{17} -0.535898 q^{19} +2.00000 q^{21} -6.19615 q^{23} +4.00000 q^{27} -6.92820 q^{29} -5.46410 q^{31} -5.46410 q^{33} +2.00000 q^{37} -9.46410 q^{39} -1.46410 q^{41} +5.26795 q^{43} -3.26795 q^{47} -6.46410 q^{49} -9.46410 q^{51} -11.4641 q^{53} -1.46410 q^{57} -7.46410 q^{59} +8.92820 q^{61} +3.26795 q^{63} +10.7321 q^{67} -16.9282 q^{69} -5.46410 q^{71} +7.46410 q^{73} -1.46410 q^{77} -1.07180 q^{79} -2.46410 q^{81} +1.26795 q^{83} -18.9282 q^{87} -8.92820 q^{89} -2.53590 q^{91} -14.9282 q^{93} +14.3923 q^{97} -8.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 8 q^{19} + 4 q^{21} - 2 q^{23} + 8 q^{27} - 4 q^{31} - 4 q^{33} + 4 q^{37} - 12 q^{39} + 4 q^{41} + 14 q^{43} - 10 q^{47} - 6 q^{49} - 12 q^{51} - 16 q^{53} + 4 q^{57} - 8 q^{59} + 4 q^{61} + 10 q^{63} + 18 q^{67} - 20 q^{69} - 4 q^{71} + 8 q^{73} + 4 q^{77} - 16 q^{79} + 2 q^{81} + 6 q^{83} - 24 q^{87} - 4 q^{89} - 12 q^{91} - 16 q^{93} + 8 q^{97} - 4 q^{99} + O(q^{100}) \)

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
\(3\) 2.73205 1.57735 0.788675 0.614810i \(-0.210767\pi\)
0.788675 + 0.614810i \(0.210767\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.732051 0.276689 0.138345 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) −0.535898 −0.122944 −0.0614718 0.998109i \(-0.519579\pi\)
−0.0614718 + 0.998109i \(0.519579\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −6.19615 −1.29199 −0.645994 0.763343i \(-0.723557\pi\)
−0.645994 + 0.763343i \(0.723557\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 0 0
\(31\) −5.46410 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) 0 0
\(33\) −5.46410 −0.951178
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −9.46410 −1.51547
\(40\) 0 0
\(41\) −1.46410 −0.228654 −0.114327 0.993443i \(-0.536471\pi\)
−0.114327 + 0.993443i \(0.536471\pi\)
\(42\) 0 0
\(43\) 5.26795 0.803355 0.401677 0.915781i \(-0.368427\pi\)
0.401677 + 0.915781i \(0.368427\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.26795 −0.476679 −0.238340 0.971182i \(-0.576603\pi\)
−0.238340 + 0.971182i \(0.576603\pi\)
\(48\) 0 0
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) −9.46410 −1.32524
\(52\) 0 0
\(53\) −11.4641 −1.57472 −0.787358 0.616496i \(-0.788551\pi\)
−0.787358 + 0.616496i \(0.788551\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.46410 −0.193925
\(58\) 0 0
\(59\) −7.46410 −0.971743 −0.485872 0.874030i \(-0.661498\pi\)
−0.485872 + 0.874030i \(0.661498\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) 0 0
\(63\) 3.26795 0.411723
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.7321 1.31113 0.655564 0.755139i \(-0.272431\pi\)
0.655564 + 0.755139i \(0.272431\pi\)
\(68\) 0 0
\(69\) −16.9282 −2.03792
\(70\) 0 0
\(71\) −5.46410 −0.648470 −0.324235 0.945977i \(-0.605107\pi\)
−0.324235 + 0.945977i \(0.605107\pi\)
\(72\) 0 0
\(73\) 7.46410 0.873607 0.436804 0.899557i \(-0.356111\pi\)
0.436804 + 0.899557i \(0.356111\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.46410 −0.166850
\(78\) 0 0
\(79\) −1.07180 −0.120587 −0.0602933 0.998181i \(-0.519204\pi\)
−0.0602933 + 0.998181i \(0.519204\pi\)
\(80\) 0 0
\(81\) −2.46410 −0.273789
\(82\) 0 0
\(83\) 1.26795 0.139176 0.0695878 0.997576i \(-0.477832\pi\)
0.0695878 + 0.997576i \(0.477832\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −18.9282 −2.02932
\(88\) 0 0
\(89\) −8.92820 −0.946388 −0.473194 0.880958i \(-0.656899\pi\)
−0.473194 + 0.880958i \(0.656899\pi\)
\(90\) 0 0
\(91\) −2.53590 −0.265834
\(92\) 0 0
\(93\) −14.9282 −1.54798
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.3923 1.46132 0.730659 0.682743i \(-0.239213\pi\)
0.730659 + 0.682743i \(0.239213\pi\)
\(98\) 0 0
\(99\) −8.92820 −0.897318
\(100\) 0 0
\(101\) 2.92820 0.291367 0.145684 0.989331i \(-0.453462\pi\)
0.145684 + 0.989331i \(0.453462\pi\)
\(102\) 0 0
\(103\) 15.6603 1.54305 0.771525 0.636199i \(-0.219494\pi\)
0.771525 + 0.636199i \(0.219494\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.73205 −0.264117 −0.132059 0.991242i \(-0.542159\pi\)
−0.132059 + 0.991242i \(0.542159\pi\)
\(108\) 0 0
\(109\) −16.9282 −1.62143 −0.810714 0.585443i \(-0.800921\pi\)
−0.810714 + 0.585443i \(0.800921\pi\)
\(110\) 0 0
\(111\) 5.46410 0.518630
\(112\) 0 0
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −15.4641 −1.42966
\(118\) 0 0
\(119\) −2.53590 −0.232465
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.7321 −1.48473 −0.742365 0.669996i \(-0.766296\pi\)
−0.742365 + 0.669996i \(0.766296\pi\)
\(128\) 0 0
\(129\) 14.3923 1.26717
\(130\) 0 0
\(131\) 19.8564 1.73486 0.867431 0.497557i \(-0.165770\pi\)
0.867431 + 0.497557i \(0.165770\pi\)
\(132\) 0 0
\(133\) −0.392305 −0.0340171
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.92820 0.421045 0.210522 0.977589i \(-0.432484\pi\)
0.210522 + 0.977589i \(0.432484\pi\)
\(138\) 0 0
\(139\) −0.535898 −0.0454543 −0.0227272 0.999742i \(-0.507235\pi\)
−0.0227272 + 0.999742i \(0.507235\pi\)
\(140\) 0 0
\(141\) −8.92820 −0.751890
\(142\) 0 0
\(143\) 6.92820 0.579365
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −17.6603 −1.45659
\(148\) 0 0
\(149\) −7.85641 −0.643622 −0.321811 0.946804i \(-0.604292\pi\)
−0.321811 + 0.946804i \(0.604292\pi\)
\(150\) 0 0
\(151\) 12.3923 1.00847 0.504236 0.863566i \(-0.331774\pi\)
0.504236 + 0.863566i \(0.331774\pi\)
\(152\) 0 0
\(153\) −15.4641 −1.25020
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.07180 −0.245156 −0.122578 0.992459i \(-0.539116\pi\)
−0.122578 + 0.992459i \(0.539116\pi\)
\(158\) 0 0
\(159\) −31.3205 −2.48388
\(160\) 0 0
\(161\) −4.53590 −0.357479
\(162\) 0 0
\(163\) −0.196152 −0.0153638 −0.00768192 0.999970i \(-0.502445\pi\)
−0.00768192 + 0.999970i \(0.502445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.80385 −0.758645 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −2.39230 −0.182944
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −20.3923 −1.53278
\(178\) 0 0
\(179\) 8.53590 0.638003 0.319002 0.947754i \(-0.396652\pi\)
0.319002 + 0.947754i \(0.396652\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 24.3923 1.80313
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.92820 0.506640
\(188\) 0 0
\(189\) 2.92820 0.212995
\(190\) 0 0
\(191\) 15.3205 1.10855 0.554277 0.832333i \(-0.312995\pi\)
0.554277 + 0.832333i \(0.312995\pi\)
\(192\) 0 0
\(193\) −0.535898 −0.0385748 −0.0192874 0.999814i \(-0.506140\pi\)
−0.0192874 + 0.999814i \(0.506140\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.4641 1.38676 0.693380 0.720572i \(-0.256121\pi\)
0.693380 + 0.720572i \(0.256121\pi\)
\(198\) 0 0
\(199\) 1.85641 0.131597 0.0657986 0.997833i \(-0.479041\pi\)
0.0657986 + 0.997833i \(0.479041\pi\)
\(200\) 0 0
\(201\) 29.3205 2.06811
\(202\) 0 0
\(203\) −5.07180 −0.355970
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −27.6603 −1.92252
\(208\) 0 0
\(209\) 1.07180 0.0741377
\(210\) 0 0
\(211\) −26.7846 −1.84393 −0.921964 0.387275i \(-0.873416\pi\)
−0.921964 + 0.387275i \(0.873416\pi\)
\(212\) 0 0
\(213\) −14.9282 −1.02286
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 20.3923 1.37798
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 5.80385 0.388654 0.194327 0.980937i \(-0.437748\pi\)
0.194327 + 0.980937i \(0.437748\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.0526 0.667212 0.333606 0.942713i \(-0.391735\pi\)
0.333606 + 0.942713i \(0.391735\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) −5.32051 −0.348558 −0.174279 0.984696i \(-0.555759\pi\)
−0.174279 + 0.984696i \(0.555759\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.92820 −0.190207
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) 16.3923 1.05592 0.527961 0.849269i \(-0.322957\pi\)
0.527961 + 0.849269i \(0.322957\pi\)
\(242\) 0 0
\(243\) −18.7321 −1.20166
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.85641 0.118120
\(248\) 0 0
\(249\) 3.46410 0.219529
\(250\) 0 0
\(251\) 24.9282 1.57345 0.786727 0.617301i \(-0.211774\pi\)
0.786727 + 0.617301i \(0.211774\pi\)
\(252\) 0 0
\(253\) 12.3923 0.779098
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 1.46410 0.0909748
\(260\) 0 0
\(261\) −30.9282 −1.91441
\(262\) 0 0
\(263\) −11.6603 −0.719002 −0.359501 0.933145i \(-0.617053\pi\)
−0.359501 + 0.933145i \(0.617053\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −24.3923 −1.49278
\(268\) 0 0
\(269\) 8.92820 0.544362 0.272181 0.962246i \(-0.412255\pi\)
0.272181 + 0.962246i \(0.412255\pi\)
\(270\) 0 0
\(271\) −19.3205 −1.17364 −0.586819 0.809718i \(-0.699620\pi\)
−0.586819 + 0.809718i \(0.699620\pi\)
\(272\) 0 0
\(273\) −6.92820 −0.419314
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) −24.3923 −1.46033
\(280\) 0 0
\(281\) −10.5359 −0.628519 −0.314260 0.949337i \(-0.601756\pi\)
−0.314260 + 0.949337i \(0.601756\pi\)
\(282\) 0 0
\(283\) 9.66025 0.574242 0.287121 0.957894i \(-0.407302\pi\)
0.287121 + 0.957894i \(0.407302\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.07180 −0.0632662
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 39.3205 2.30501
\(292\) 0 0
\(293\) 15.8564 0.926341 0.463171 0.886269i \(-0.346712\pi\)
0.463171 + 0.886269i \(0.346712\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −8.00000 −0.464207
\(298\) 0 0
\(299\) 21.4641 1.24130
\(300\) 0 0
\(301\) 3.85641 0.222280
\(302\) 0 0
\(303\) 8.00000 0.459588
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −24.9808 −1.42573 −0.712864 0.701303i \(-0.752602\pi\)
−0.712864 + 0.701303i \(0.752602\pi\)
\(308\) 0 0
\(309\) 42.7846 2.43393
\(310\) 0 0
\(311\) −31.3205 −1.77602 −0.888012 0.459821i \(-0.847914\pi\)
−0.888012 + 0.459821i \(0.847914\pi\)
\(312\) 0 0
\(313\) −4.14359 −0.234210 −0.117105 0.993120i \(-0.537361\pi\)
−0.117105 + 0.993120i \(0.537361\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.53590 0.479424 0.239712 0.970844i \(-0.422947\pi\)
0.239712 + 0.970844i \(0.422947\pi\)
\(318\) 0 0
\(319\) 13.8564 0.775810
\(320\) 0 0
\(321\) −7.46410 −0.416606
\(322\) 0 0
\(323\) 1.85641 0.103293
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −46.2487 −2.55756
\(328\) 0 0
\(329\) −2.39230 −0.131892
\(330\) 0 0
\(331\) −14.0000 −0.769510 −0.384755 0.923019i \(-0.625714\pi\)
−0.384755 + 0.923019i \(0.625714\pi\)
\(332\) 0 0
\(333\) 8.92820 0.489263
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 19.8564 1.08165 0.540824 0.841136i \(-0.318113\pi\)
0.540824 + 0.841136i \(0.318113\pi\)
\(338\) 0 0
\(339\) 35.3205 1.91835
\(340\) 0 0
\(341\) 10.9282 0.591795
\(342\) 0 0
\(343\) −9.85641 −0.532196
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.66025 0.0891271 0.0445636 0.999007i \(-0.485810\pi\)
0.0445636 + 0.999007i \(0.485810\pi\)
\(348\) 0 0
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) −13.8564 −0.739600
\(352\) 0 0
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.92820 −0.366679
\(358\) 0 0
\(359\) −18.9282 −0.998992 −0.499496 0.866316i \(-0.666482\pi\)
−0.499496 + 0.866316i \(0.666482\pi\)
\(360\) 0 0
\(361\) −18.7128 −0.984885
\(362\) 0 0
\(363\) −19.1244 −1.00377
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.87564 −0.150107 −0.0750537 0.997179i \(-0.523913\pi\)
−0.0750537 + 0.997179i \(0.523913\pi\)
\(368\) 0 0
\(369\) −6.53590 −0.340245
\(370\) 0 0
\(371\) −8.39230 −0.435707
\(372\) 0 0
\(373\) 25.7128 1.33136 0.665679 0.746238i \(-0.268142\pi\)
0.665679 + 0.746238i \(0.268142\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) −36.2487 −1.86197 −0.930986 0.365056i \(-0.881050\pi\)
−0.930986 + 0.365056i \(0.881050\pi\)
\(380\) 0 0
\(381\) −45.7128 −2.34194
\(382\) 0 0
\(383\) −21.1244 −1.07940 −0.539702 0.841856i \(-0.681463\pi\)
−0.539702 + 0.841856i \(0.681463\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 23.5167 1.19542
\(388\) 0 0
\(389\) 6.78461 0.343993 0.171997 0.985098i \(-0.444978\pi\)
0.171997 + 0.985098i \(0.444978\pi\)
\(390\) 0 0
\(391\) 21.4641 1.08549
\(392\) 0 0
\(393\) 54.2487 2.73649
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 32.2487 1.61852 0.809258 0.587453i \(-0.199869\pi\)
0.809258 + 0.587453i \(0.199869\pi\)
\(398\) 0 0
\(399\) −1.07180 −0.0536570
\(400\) 0 0
\(401\) −7.85641 −0.392330 −0.196165 0.980571i \(-0.562849\pi\)
−0.196165 + 0.980571i \(0.562849\pi\)
\(402\) 0 0
\(403\) 18.9282 0.942881
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 11.3205 0.559763 0.279882 0.960035i \(-0.409705\pi\)
0.279882 + 0.960035i \(0.409705\pi\)
\(410\) 0 0
\(411\) 13.4641 0.664135
\(412\) 0 0
\(413\) −5.46410 −0.268871
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.46410 −0.0716974
\(418\) 0 0
\(419\) −18.3923 −0.898523 −0.449261 0.893400i \(-0.648313\pi\)
−0.449261 + 0.893400i \(0.648313\pi\)
\(420\) 0 0
\(421\) 0.143594 0.00699832 0.00349916 0.999994i \(-0.498886\pi\)
0.00349916 + 0.999994i \(0.498886\pi\)
\(422\) 0 0
\(423\) −14.5885 −0.709315
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.53590 0.316294
\(428\) 0 0
\(429\) 18.9282 0.913862
\(430\) 0 0
\(431\) −21.4641 −1.03389 −0.516945 0.856019i \(-0.672931\pi\)
−0.516945 + 0.856019i \(0.672931\pi\)
\(432\) 0 0
\(433\) 19.4641 0.935385 0.467693 0.883891i \(-0.345085\pi\)
0.467693 + 0.883891i \(0.345085\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.32051 0.158841
\(438\) 0 0
\(439\) −40.7846 −1.94654 −0.973272 0.229657i \(-0.926240\pi\)
−0.973272 + 0.229657i \(0.926240\pi\)
\(440\) 0 0
\(441\) −28.8564 −1.37411
\(442\) 0 0
\(443\) −20.9808 −0.996826 −0.498413 0.866940i \(-0.666084\pi\)
−0.498413 + 0.866940i \(0.666084\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −21.4641 −1.01522
\(448\) 0 0
\(449\) −23.3205 −1.10056 −0.550281 0.834979i \(-0.685480\pi\)
−0.550281 + 0.834979i \(0.685480\pi\)
\(450\) 0 0
\(451\) 2.92820 0.137884
\(452\) 0 0
\(453\) 33.8564 1.59071
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −26.7846 −1.25293 −0.626466 0.779449i \(-0.715499\pi\)
−0.626466 + 0.779449i \(0.715499\pi\)
\(458\) 0 0
\(459\) −13.8564 −0.646762
\(460\) 0 0
\(461\) 10.9282 0.508977 0.254489 0.967076i \(-0.418093\pi\)
0.254489 + 0.967076i \(0.418093\pi\)
\(462\) 0 0
\(463\) 11.2679 0.523666 0.261833 0.965113i \(-0.415673\pi\)
0.261833 + 0.965113i \(0.415673\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −25.6603 −1.18741 −0.593707 0.804681i \(-0.702336\pi\)
−0.593707 + 0.804681i \(0.702336\pi\)
\(468\) 0 0
\(469\) 7.85641 0.362775
\(470\) 0 0
\(471\) −8.39230 −0.386697
\(472\) 0 0
\(473\) −10.5359 −0.484441
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −51.1769 −2.34323
\(478\) 0 0
\(479\) 5.85641 0.267586 0.133793 0.991009i \(-0.457284\pi\)
0.133793 + 0.991009i \(0.457284\pi\)
\(480\) 0 0
\(481\) −6.92820 −0.315899
\(482\) 0 0
\(483\) −12.3923 −0.563869
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.58846 0.298551 0.149276 0.988796i \(-0.452306\pi\)
0.149276 + 0.988796i \(0.452306\pi\)
\(488\) 0 0
\(489\) −0.535898 −0.0242342
\(490\) 0 0
\(491\) −16.9282 −0.763959 −0.381980 0.924171i \(-0.624758\pi\)
−0.381980 + 0.924171i \(0.624758\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −31.4641 −1.40853 −0.704263 0.709939i \(-0.748723\pi\)
−0.704263 + 0.709939i \(0.748723\pi\)
\(500\) 0 0
\(501\) −26.7846 −1.19665
\(502\) 0 0
\(503\) −0.339746 −0.0151485 −0.00757426 0.999971i \(-0.502411\pi\)
−0.00757426 + 0.999971i \(0.502411\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.73205 −0.121335
\(508\) 0 0
\(509\) 1.85641 0.0822838 0.0411419 0.999153i \(-0.486900\pi\)
0.0411419 + 0.999153i \(0.486900\pi\)
\(510\) 0 0
\(511\) 5.46410 0.241718
\(512\) 0 0
\(513\) −2.14359 −0.0946420
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 6.53590 0.287448
\(518\) 0 0
\(519\) 5.46410 0.239847
\(520\) 0 0
\(521\) 43.8564 1.92138 0.960692 0.277616i \(-0.0895444\pi\)
0.960692 + 0.277616i \(0.0895444\pi\)
\(522\) 0 0
\(523\) 11.8038 0.516146 0.258073 0.966125i \(-0.416912\pi\)
0.258073 + 0.966125i \(0.416912\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.9282 0.824525
\(528\) 0 0
\(529\) 15.3923 0.669231
\(530\) 0 0
\(531\) −33.3205 −1.44599
\(532\) 0 0
\(533\) 5.07180 0.219684
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 23.3205 1.00635
\(538\) 0 0
\(539\) 12.9282 0.556857
\(540\) 0 0
\(541\) 26.9282 1.15773 0.578867 0.815422i \(-0.303495\pi\)
0.578867 + 0.815422i \(0.303495\pi\)
\(542\) 0 0
\(543\) 43.7128 1.87590
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 33.2679 1.42243 0.711217 0.702972i \(-0.248144\pi\)
0.711217 + 0.702972i \(0.248144\pi\)
\(548\) 0 0
\(549\) 39.8564 1.70103
\(550\) 0 0
\(551\) 3.71281 0.158171
\(552\) 0 0
\(553\) −0.784610 −0.0333650
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.7846 0.626444 0.313222 0.949680i \(-0.398592\pi\)
0.313222 + 0.949680i \(0.398592\pi\)
\(558\) 0 0
\(559\) −18.2487 −0.771838
\(560\) 0 0
\(561\) 18.9282 0.799149
\(562\) 0 0
\(563\) −22.0526 −0.929405 −0.464702 0.885467i \(-0.653839\pi\)
−0.464702 + 0.885467i \(0.653839\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.80385 −0.0757545
\(568\) 0 0
\(569\) 13.4641 0.564445 0.282222 0.959349i \(-0.408928\pi\)
0.282222 + 0.959349i \(0.408928\pi\)
\(570\) 0 0
\(571\) 6.78461 0.283927 0.141964 0.989872i \(-0.454658\pi\)
0.141964 + 0.989872i \(0.454658\pi\)
\(572\) 0 0
\(573\) 41.8564 1.74858
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −39.5692 −1.64729 −0.823644 0.567107i \(-0.808063\pi\)
−0.823644 + 0.567107i \(0.808063\pi\)
\(578\) 0 0
\(579\) −1.46410 −0.0608460
\(580\) 0 0
\(581\) 0.928203 0.0385084
\(582\) 0 0
\(583\) 22.9282 0.949589
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.80385 0.157002 0.0785008 0.996914i \(-0.474987\pi\)
0.0785008 + 0.996914i \(0.474987\pi\)
\(588\) 0 0
\(589\) 2.92820 0.120655
\(590\) 0 0
\(591\) 53.1769 2.18741
\(592\) 0 0
\(593\) −32.6410 −1.34041 −0.670203 0.742178i \(-0.733793\pi\)
−0.670203 + 0.742178i \(0.733793\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.07180 0.207575
\(598\) 0 0
\(599\) 34.6410 1.41539 0.707697 0.706516i \(-0.249734\pi\)
0.707697 + 0.706516i \(0.249734\pi\)
\(600\) 0 0
\(601\) −18.5359 −0.756095 −0.378048 0.925786i \(-0.623404\pi\)
−0.378048 + 0.925786i \(0.623404\pi\)
\(602\) 0 0
\(603\) 47.9090 1.95100
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −30.9808 −1.25747 −0.628735 0.777619i \(-0.716427\pi\)
−0.628735 + 0.777619i \(0.716427\pi\)
\(608\) 0 0
\(609\) −13.8564 −0.561490
\(610\) 0 0
\(611\) 11.3205 0.457979
\(612\) 0 0
\(613\) −26.3923 −1.06598 −0.532988 0.846123i \(-0.678931\pi\)
−0.532988 + 0.846123i \(0.678931\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −20.5359 −0.826744 −0.413372 0.910562i \(-0.635649\pi\)
−0.413372 + 0.910562i \(0.635649\pi\)
\(618\) 0 0
\(619\) 1.32051 0.0530757 0.0265379 0.999648i \(-0.491552\pi\)
0.0265379 + 0.999648i \(0.491552\pi\)
\(620\) 0 0
\(621\) −24.7846 −0.994572
\(622\) 0 0
\(623\) −6.53590 −0.261855
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2.92820 0.116941
\(628\) 0 0
\(629\) −6.92820 −0.276246
\(630\) 0 0
\(631\) 23.3205 0.928375 0.464187 0.885737i \(-0.346346\pi\)
0.464187 + 0.885737i \(0.346346\pi\)
\(632\) 0 0
\(633\) −73.1769 −2.90852
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 22.3923 0.887215
\(638\) 0 0
\(639\) −24.3923 −0.964945
\(640\) 0 0
\(641\) 0.392305 0.0154951 0.00774755 0.999970i \(-0.497534\pi\)
0.00774755 + 0.999970i \(0.497534\pi\)
\(642\) 0 0
\(643\) 39.1244 1.54291 0.771457 0.636281i \(-0.219528\pi\)
0.771457 + 0.636281i \(0.219528\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.7321 0.657805 0.328902 0.944364i \(-0.393321\pi\)
0.328902 + 0.944364i \(0.393321\pi\)
\(648\) 0 0
\(649\) 14.9282 0.585983
\(650\) 0 0
\(651\) −10.9282 −0.428310
\(652\) 0 0
\(653\) 12.2487 0.479329 0.239665 0.970856i \(-0.422963\pi\)
0.239665 + 0.970856i \(0.422963\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 33.3205 1.29996
\(658\) 0 0
\(659\) 17.3205 0.674711 0.337356 0.941377i \(-0.390468\pi\)
0.337356 + 0.941377i \(0.390468\pi\)
\(660\) 0 0
\(661\) −8.14359 −0.316749 −0.158375 0.987379i \(-0.550625\pi\)
−0.158375 + 0.987379i \(0.550625\pi\)
\(662\) 0 0
\(663\) 32.7846 1.27325
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 42.9282 1.66219
\(668\) 0 0
\(669\) 15.8564 0.613044
\(670\) 0 0
\(671\) −17.8564 −0.689339
\(672\) 0 0
\(673\) −12.5359 −0.483223 −0.241612 0.970373i \(-0.577676\pi\)
−0.241612 + 0.970373i \(0.577676\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17.6077 −0.676719 −0.338359 0.941017i \(-0.609872\pi\)
−0.338359 + 0.941017i \(0.609872\pi\)
\(678\) 0 0
\(679\) 10.5359 0.404331
\(680\) 0 0
\(681\) 27.4641 1.05243
\(682\) 0 0
\(683\) 16.9808 0.649751 0.324875 0.945757i \(-0.394678\pi\)
0.324875 + 0.945757i \(0.394678\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −10.9282 −0.416937
\(688\) 0 0
\(689\) 39.7128 1.51294
\(690\) 0 0
\(691\) −18.0000 −0.684752 −0.342376 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(692\) 0 0
\(693\) −6.53590 −0.248278
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.07180 0.192108
\(698\) 0 0
\(699\) −14.5359 −0.549798
\(700\) 0 0
\(701\) −19.0718 −0.720332 −0.360166 0.932888i \(-0.617280\pi\)
−0.360166 + 0.932888i \(0.617280\pi\)
\(702\) 0 0
\(703\) −1.07180 −0.0404236
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.14359 0.0806181
\(708\) 0 0
\(709\) −12.7846 −0.480136 −0.240068 0.970756i \(-0.577170\pi\)
−0.240068 + 0.970756i \(0.577170\pi\)
\(710\) 0 0
\(711\) −4.78461 −0.179437
\(712\) 0 0
\(713\) 33.8564 1.26793
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −54.6410 −2.04061
\(718\) 0 0
\(719\) −1.85641 −0.0692323 −0.0346161 0.999401i \(-0.511021\pi\)
−0.0346161 + 0.999401i \(0.511021\pi\)
\(720\) 0 0
\(721\) 11.4641 0.426945
\(722\) 0 0
\(723\) 44.7846 1.66556
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −24.0526 −0.892060 −0.446030 0.895018i \(-0.647163\pi\)
−0.446030 + 0.895018i \(0.647163\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) −18.2487 −0.674953
\(732\) 0 0
\(733\) −35.0718 −1.29541 −0.647703 0.761893i \(-0.724270\pi\)
−0.647703 + 0.761893i \(0.724270\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.4641 −0.790640
\(738\) 0 0
\(739\) −29.3205 −1.07857 −0.539286 0.842123i \(-0.681306\pi\)
−0.539286 + 0.842123i \(0.681306\pi\)
\(740\) 0 0
\(741\) 5.07180 0.186317
\(742\) 0 0
\(743\) 10.9808 0.402845 0.201423 0.979504i \(-0.435444\pi\)
0.201423 + 0.979504i \(0.435444\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.66025 0.207098
\(748\) 0 0
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) 26.2487 0.957829 0.478915 0.877862i \(-0.341030\pi\)
0.478915 + 0.877862i \(0.341030\pi\)
\(752\) 0 0
\(753\) 68.1051 2.48189
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 19.0718 0.693176 0.346588 0.938017i \(-0.387340\pi\)
0.346588 + 0.938017i \(0.387340\pi\)
\(758\) 0 0
\(759\) 33.8564 1.22891
\(760\) 0 0
\(761\) 5.71281 0.207089 0.103545 0.994625i \(-0.466982\pi\)
0.103545 + 0.994625i \(0.466982\pi\)
\(762\) 0 0
\(763\) −12.3923 −0.448632
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25.8564 0.933621
\(768\) 0 0
\(769\) 12.9282 0.466203 0.233101 0.972452i \(-0.425113\pi\)
0.233101 + 0.972452i \(0.425113\pi\)
\(770\) 0 0
\(771\) 5.46410 0.196785
\(772\) 0 0
\(773\) 22.3923 0.805395 0.402698 0.915333i \(-0.368073\pi\)
0.402698 + 0.915333i \(0.368073\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) 0 0
\(779\) 0.784610 0.0281116
\(780\) 0 0
\(781\) 10.9282 0.391042
\(782\) 0 0
\(783\) −27.7128 −0.990375
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 16.5885 0.591315 0.295657 0.955294i \(-0.404461\pi\)
0.295657 + 0.955294i \(0.404461\pi\)
\(788\) 0 0
\(789\) −31.8564 −1.13412
\(790\) 0 0
\(791\) 9.46410 0.336505
\(792\) 0 0
\(793\) −30.9282 −1.09829
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 50.1051 1.77481 0.887407 0.460986i \(-0.152504\pi\)
0.887407 + 0.460986i \(0.152504\pi\)
\(798\) 0 0
\(799\) 11.3205 0.400491
\(800\) 0 0
\(801\) −39.8564 −1.40826
\(802\) 0 0
\(803\) −14.9282 −0.526805
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.3923 0.858650
\(808\) 0 0
\(809\) −23.8564 −0.838747 −0.419373 0.907814i \(-0.637750\pi\)
−0.419373 + 0.907814i \(0.637750\pi\)
\(810\) 0 0
\(811\) −28.9282 −1.01581 −0.507903 0.861414i \(-0.669579\pi\)
−0.507903 + 0.861414i \(0.669579\pi\)
\(812\) 0 0
\(813\) −52.7846 −1.85124
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.82309 −0.0987673
\(818\) 0 0
\(819\) −11.3205 −0.395571
\(820\) 0 0
\(821\) 34.7846 1.21399 0.606996 0.794705i \(-0.292375\pi\)
0.606996 + 0.794705i \(0.292375\pi\)
\(822\) 0 0
\(823\) −9.12436 −0.318055 −0.159028 0.987274i \(-0.550836\pi\)
−0.159028 + 0.987274i \(0.550836\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −23.1244 −0.804113 −0.402056 0.915615i \(-0.631704\pi\)
−0.402056 + 0.915615i \(0.631704\pi\)
\(828\) 0 0
\(829\) −28.9282 −1.00472 −0.502359 0.864659i \(-0.667534\pi\)
−0.502359 + 0.864659i \(0.667534\pi\)
\(830\) 0 0
\(831\) −5.46410 −0.189548
\(832\) 0 0
\(833\) 22.3923 0.775847
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −21.8564 −0.755468
\(838\) 0 0
\(839\) −24.7846 −0.855660 −0.427830 0.903859i \(-0.640722\pi\)
−0.427830 + 0.903859i \(0.640722\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) 0 0
\(843\) −28.7846 −0.991395
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5.12436 −0.176075
\(848\) 0 0
\(849\) 26.3923 0.905782
\(850\) 0 0
\(851\) −12.3923 −0.424803
\(852\) 0 0
\(853\) 21.6077 0.739833 0.369917 0.929065i \(-0.379386\pi\)
0.369917 + 0.929065i \(0.379386\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −19.8564 −0.678282 −0.339141 0.940736i \(-0.610136\pi\)
−0.339141 + 0.940736i \(0.610136\pi\)
\(858\) 0 0
\(859\) 28.2487 0.963834 0.481917 0.876217i \(-0.339941\pi\)
0.481917 + 0.876217i \(0.339941\pi\)
\(860\) 0 0
\(861\) −2.92820 −0.0997929
\(862\) 0 0
\(863\) −47.6603 −1.62237 −0.811187 0.584787i \(-0.801178\pi\)
−0.811187 + 0.584787i \(0.801178\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −13.6603 −0.463927
\(868\) 0 0
\(869\) 2.14359 0.0727164
\(870\) 0 0
\(871\) −37.1769 −1.25969
\(872\) 0 0
\(873\) 64.2487 2.17449
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.71281 −0.0578376 −0.0289188 0.999582i \(-0.509206\pi\)
−0.0289188 + 0.999582i \(0.509206\pi\)
\(878\) 0 0
\(879\) 43.3205 1.46116
\(880\) 0 0
\(881\) 9.46410 0.318854 0.159427 0.987210i \(-0.449035\pi\)
0.159427 + 0.987210i \(0.449035\pi\)
\(882\) 0 0
\(883\) −27.9090 −0.939211 −0.469606 0.882876i \(-0.655604\pi\)
−0.469606 + 0.882876i \(0.655604\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −13.9090 −0.467017 −0.233509 0.972355i \(-0.575021\pi\)
−0.233509 + 0.972355i \(0.575021\pi\)
\(888\) 0 0
\(889\) −12.2487 −0.410809
\(890\) 0 0
\(891\) 4.92820 0.165101
\(892\) 0 0
\(893\) 1.75129 0.0586046
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 58.6410 1.95797
\(898\) 0 0
\(899\) 37.8564 1.26258
\(900\) 0 0
\(901\) 39.7128 1.32303
\(902\) 0 0
\(903\) 10.5359 0.350613
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.87564 −0.161893 −0.0809466 0.996718i \(-0.525794\pi\)
−0.0809466 + 0.996718i \(0.525794\pi\)
\(908\) 0 0
\(909\) 13.0718 0.433564
\(910\) 0 0
\(911\) −49.1769 −1.62930 −0.814652 0.579950i \(-0.803072\pi\)
−0.814652 + 0.579950i \(0.803072\pi\)
\(912\) 0 0
\(913\) −2.53590 −0.0839260
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 14.5359 0.480018
\(918\) 0 0
\(919\) 38.9282 1.28412 0.642061 0.766653i \(-0.278079\pi\)
0.642061 + 0.766653i \(0.278079\pi\)
\(920\) 0 0
\(921\) −68.2487 −2.24887
\(922\) 0 0
\(923\) 18.9282 0.623029
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 69.9090 2.29611
\(928\) 0 0
\(929\) −17.4641 −0.572979 −0.286489 0.958083i \(-0.592488\pi\)
−0.286489 + 0.958083i \(0.592488\pi\)
\(930\) 0 0
\(931\) 3.46410 0.113531
\(932\) 0 0
\(933\) −85.5692 −2.80141
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.24871 0.138799 0.0693997 0.997589i \(-0.477892\pi\)
0.0693997 + 0.997589i \(0.477892\pi\)
\(938\) 0 0
\(939\) −11.3205 −0.369431
\(940\) 0 0
\(941\) 32.0000 1.04317 0.521585 0.853199i \(-0.325341\pi\)
0.521585 + 0.853199i \(0.325341\pi\)
\(942\) 0 0
\(943\) 9.07180 0.295418
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.12436 −0.101528 −0.0507640 0.998711i \(-0.516166\pi\)
−0.0507640 + 0.998711i \(0.516166\pi\)
\(948\) 0 0
\(949\) −25.8564 −0.839334
\(950\) 0 0
\(951\) 23.3205 0.756219
\(952\) 0 0
\(953\) −17.2154 −0.557661 −0.278831 0.960340i \(-0.589947\pi\)
−0.278831 + 0.960340i \(0.589947\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 37.8564 1.22372
\(958\) 0 0
\(959\) 3.60770 0.116499
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) −12.1962 −0.393016
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 16.3397 0.525451 0.262725 0.964871i \(-0.415379\pi\)
0.262725 + 0.964871i \(0.415379\pi\)
\(968\) 0 0
\(969\) 5.07180 0.162930
\(970\) 0 0
\(971\) 36.9282 1.18508 0.592541 0.805540i \(-0.298125\pi\)
0.592541 + 0.805540i \(0.298125\pi\)
\(972\) 0 0
\(973\) −0.392305 −0.0125767
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.5359 0.784973 0.392486 0.919758i \(-0.371615\pi\)
0.392486 + 0.919758i \(0.371615\pi\)
\(978\) 0 0
\(979\) 17.8564 0.570693
\(980\) 0 0
\(981\) −75.5692 −2.41274
\(982\) 0 0
\(983\) −48.7321 −1.55431 −0.777156 0.629309i \(-0.783338\pi\)
−0.777156 + 0.629309i \(0.783338\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −6.53590 −0.208040
\(988\) 0 0
\(989\) −32.6410 −1.03792
\(990\) 0 0
\(991\) 41.4641 1.31715 0.658575 0.752515i \(-0.271159\pi\)
0.658575 + 0.752515i \(0.271159\pi\)
\(992\) 0 0
\(993\) −38.2487 −1.21379
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 11.1769 0.353976 0.176988 0.984213i \(-0.443365\pi\)
0.176988 + 0.984213i \(0.443365\pi\)
\(998\) 0 0
\(999\) 8.00000 0.253109
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 6400.2.a.ce.1.2 2
4.3 odd 2 6400.2.a.be.1.1 2
5.4 even 2 1280.2.a.a.1.1 2
8.3 odd 2 6400.2.a.cj.1.2 2
8.5 even 2 6400.2.a.z.1.1 2
16.3 odd 4 800.2.d.e.401.1 4
16.5 even 4 200.2.d.f.101.4 4
16.11 odd 4 800.2.d.e.401.4 4
16.13 even 4 200.2.d.f.101.3 4
20.19 odd 2 1280.2.a.n.1.2 2
40.19 odd 2 1280.2.a.d.1.1 2
40.29 even 2 1280.2.a.o.1.2 2
48.5 odd 4 1800.2.k.j.901.1 4
48.11 even 4 7200.2.k.j.3601.3 4
48.29 odd 4 1800.2.k.j.901.2 4
48.35 even 4 7200.2.k.j.3601.4 4
80.3 even 4 800.2.f.e.49.3 4
80.13 odd 4 200.2.f.e.149.1 4
80.19 odd 4 160.2.d.a.81.4 4
80.27 even 4 800.2.f.e.49.4 4
80.29 even 4 40.2.d.a.21.2 yes 4
80.37 odd 4 200.2.f.e.149.2 4
80.43 even 4 800.2.f.c.49.1 4
80.53 odd 4 200.2.f.c.149.3 4
80.59 odd 4 160.2.d.a.81.1 4
80.67 even 4 800.2.f.c.49.2 4
80.69 even 4 40.2.d.a.21.1 4
80.77 odd 4 200.2.f.c.149.4 4
240.29 odd 4 360.2.k.e.181.3 4
240.53 even 4 1800.2.d.p.1549.2 4
240.59 even 4 1440.2.k.e.721.3 4
240.77 even 4 1800.2.d.p.1549.1 4
240.83 odd 4 7200.2.d.n.2449.2 4
240.107 odd 4 7200.2.d.n.2449.3 4
240.149 odd 4 360.2.k.e.181.4 4
240.173 even 4 1800.2.d.l.1549.4 4
240.179 even 4 1440.2.k.e.721.1 4
240.197 even 4 1800.2.d.l.1549.3 4
240.203 odd 4 7200.2.d.o.2449.2 4
240.227 odd 4 7200.2.d.o.2449.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.d.a.21.1 4 80.69 even 4
40.2.d.a.21.2 yes 4 80.29 even 4
160.2.d.a.81.1 4 80.59 odd 4
160.2.d.a.81.4 4 80.19 odd 4
200.2.d.f.101.3 4 16.13 even 4
200.2.d.f.101.4 4 16.5 even 4
200.2.f.c.149.3 4 80.53 odd 4
200.2.f.c.149.4 4 80.77 odd 4
200.2.f.e.149.1 4 80.13 odd 4
200.2.f.e.149.2 4 80.37 odd 4
360.2.k.e.181.3 4 240.29 odd 4
360.2.k.e.181.4 4 240.149 odd 4
800.2.d.e.401.1 4 16.3 odd 4
800.2.d.e.401.4 4 16.11 odd 4
800.2.f.c.49.1 4 80.43 even 4
800.2.f.c.49.2 4 80.67 even 4
800.2.f.e.49.3 4 80.3 even 4
800.2.f.e.49.4 4 80.27 even 4
1280.2.a.a.1.1 2 5.4 even 2
1280.2.a.d.1.1 2 40.19 odd 2
1280.2.a.n.1.2 2 20.19 odd 2
1280.2.a.o.1.2 2 40.29 even 2
1440.2.k.e.721.1 4 240.179 even 4
1440.2.k.e.721.3 4 240.59 even 4
1800.2.d.l.1549.3 4 240.197 even 4
1800.2.d.l.1549.4 4 240.173 even 4
1800.2.d.p.1549.1 4 240.77 even 4
1800.2.d.p.1549.2 4 240.53 even 4
1800.2.k.j.901.1 4 48.5 odd 4
1800.2.k.j.901.2 4 48.29 odd 4
6400.2.a.z.1.1 2 8.5 even 2
6400.2.a.be.1.1 2 4.3 odd 2
6400.2.a.ce.1.2 2 1.1 even 1 trivial
6400.2.a.cj.1.2 2 8.3 odd 2
7200.2.d.n.2449.2 4 240.83 odd 4
7200.2.d.n.2449.3 4 240.107 odd 4
7200.2.d.o.2449.2 4 240.203 odd 4
7200.2.d.o.2449.3 4 240.227 odd 4
7200.2.k.j.3601.3 4 48.11 even 4
7200.2.k.j.3601.4 4 48.35 even 4