Properties

Label 6400.2.a.z.1.1
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 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

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

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 \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 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 \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 3.46410 0.960769 0.480384 0.877058i \(-0.340497\pi\)
0.480384 + 0.877058i \(0.340497\pi\)
\(14\) 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 \(-0.480421\pi\)
0.0614718 + 0.998109i \(0.480421\pi\)
\(20\) 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 \(-0.277578\pi\)
0.643268 + 0.765641i \(0.277578\pi\)
\(30\) 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 \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 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 \(-0.631573\pi\)
−0.401677 + 0.915781i \(0.631573\pi\)
\(44\) 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 \(-0.211449\pi\)
0.787358 + 0.616496i \(0.211449\pi\)
\(54\) 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 \(-0.338502\pi\)
0.485872 + 0.874030i \(0.338502\pi\)
\(60\) 0 0
\(61\) −8.92820 −1.14314 −0.571570 0.820554i \(-0.693665\pi\)
−0.571570 + 0.820554i \(0.693665\pi\)
\(62\) 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 \(-0.727569\pi\)
−0.655564 + 0.755139i \(0.727569\pi\)
\(68\) 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 \(-0.522168\pi\)
−0.0695878 + 0.997576i \(0.522168\pi\)
\(84\) 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 \(-0.546538\pi\)
−0.145684 + 0.989331i \(0.546538\pi\)
\(102\) 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 \(-0.457841\pi\)
0.132059 + 0.991242i \(0.457841\pi\)
\(108\) 0 0
\(109\) 16.9282 1.62143 0.810714 0.585443i \(-0.199079\pi\)
0.810714 + 0.585443i \(0.199079\pi\)
\(110\) 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 \(-0.834230\pi\)
−0.867431 + 0.497557i \(0.834230\pi\)
\(132\) 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 \(-0.492765\pi\)
0.0227272 + 0.999742i \(0.492765\pi\)
\(140\) 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 \(-0.395708\pi\)
0.321811 + 0.946804i \(0.395708\pi\)
\(150\) 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 \(-0.460884\pi\)
0.122578 + 0.992459i \(0.460884\pi\)
\(158\) 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 \(-0.497555\pi\)
0.00768192 + 0.999970i \(0.497555\pi\)
\(164\) 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 \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 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 \(-0.603348\pi\)
−0.319002 + 0.947754i \(0.603348\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 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 \(-0.743879\pi\)
−0.693380 + 0.720572i \(0.743879\pi\)
\(198\) 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 \(-0.126584\pi\)
0.921964 + 0.387275i \(0.126584\pi\)
\(212\) 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 \(-0.608265\pi\)
−0.333606 + 0.942713i \(0.608265\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 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 \(-0.788226\pi\)
−0.786727 + 0.617301i \(0.788226\pi\)
\(252\) 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 \(-0.587745\pi\)
−0.272181 + 0.962246i \(0.587745\pi\)
\(270\) 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 \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 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 \(-0.592698\pi\)
−0.287121 + 0.957894i \(0.592698\pi\)
\(284\) 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 \(-0.653288\pi\)
−0.463171 + 0.886269i \(0.653288\pi\)
\(294\) 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 \(-0.247398\pi\)
0.712864 + 0.701303i \(0.247398\pi\)
\(308\) 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 \(-0.577053\pi\)
−0.239712 + 0.970844i \(0.577053\pi\)
\(318\) 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 \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) 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 \(-0.514190\pi\)
−0.0445636 + 0.999007i \(0.514190\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 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 \(-0.731858\pi\)
−0.665679 + 0.746238i \(0.731858\pi\)
\(374\) 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 \(-0.118950\pi\)
0.930986 + 0.365056i \(0.118950\pi\)
\(380\) 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 \(-0.555022\pi\)
−0.171997 + 0.985098i \(0.555022\pi\)
\(390\) 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 \(-0.800131\pi\)
−0.809258 + 0.587453i \(0.800131\pi\)
\(398\) 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 \(-0.351687\pi\)
0.449261 + 0.893400i \(0.351687\pi\)
\(420\) 0 0
\(421\) −0.143594 −0.00699832 −0.00349916 0.999994i \(-0.501114\pi\)
−0.00349916 + 0.999994i \(0.501114\pi\)
\(422\) 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 \(-0.333916\pi\)
0.498413 + 0.866940i \(0.333916\pi\)
\(444\) 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 \(-0.581907\pi\)
−0.254489 + 0.967076i \(0.581907\pi\)
\(462\) 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 \(-0.297664\pi\)
0.593707 + 0.804681i \(0.297664\pi\)
\(468\) 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 \(-0.375242\pi\)
0.381980 + 0.924171i \(0.375242\pi\)
\(492\) 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 \(-0.251277\pi\)
0.704263 + 0.709939i \(0.251277\pi\)
\(500\) 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 \(-0.513100\pi\)
−0.0411419 + 0.999153i \(0.513100\pi\)
\(510\) 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 \(-0.583088\pi\)
−0.258073 + 0.966125i \(0.583088\pi\)
\(524\) 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 \(-0.696505\pi\)
−0.578867 + 0.815422i \(0.696505\pi\)
\(542\) 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 \(-0.751856\pi\)
−0.711217 + 0.702972i \(0.751856\pi\)
\(548\) 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 \(-0.601408\pi\)
−0.313222 + 0.949680i \(0.601408\pi\)
\(558\) 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 \(-0.346161\pi\)
0.464702 + 0.885467i \(0.346161\pi\)
\(564\) 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 \(-0.545342\pi\)
−0.141964 + 0.989872i \(0.545342\pi\)
\(572\) 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 \(-0.525013\pi\)
−0.0785008 + 0.996914i \(0.525013\pi\)
\(588\) 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 \(-0.321069\pi\)
0.532988 + 0.846123i \(0.321069\pi\)
\(614\) 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 \(-0.508448\pi\)
−0.0265379 + 0.999648i \(0.508448\pi\)
\(620\) 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 \(-0.780472\pi\)
−0.771457 + 0.636281i \(0.780472\pi\)
\(644\) 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 \(-0.577037\pi\)
−0.239665 + 0.970856i \(0.577037\pi\)
\(654\) 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 \(-0.609532\pi\)
−0.337356 + 0.941377i \(0.609532\pi\)
\(660\) 0 0
\(661\) 8.14359 0.316749 0.158375 0.987379i \(-0.449375\pi\)
0.158375 + 0.987379i \(0.449375\pi\)
\(662\) 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 \(-0.390128\pi\)
0.338359 + 0.941017i \(0.390128\pi\)
\(678\) 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 \(-0.605322\pi\)
−0.324875 + 0.945757i \(0.605322\pi\)
\(684\) 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 \(-0.388768\pi\)
0.342376 + 0.939563i \(0.388768\pi\)
\(692\) 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 \(-0.382720\pi\)
0.360166 + 0.932888i \(0.382720\pi\)
\(702\) 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 \(-0.422830\pi\)
0.240068 + 0.970756i \(0.422830\pi\)
\(710\) 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 \(-0.275730\pi\)
0.647703 + 0.761893i \(0.275730\pi\)
\(734\) 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 \(-0.318694\pi\)
0.539286 + 0.842123i \(0.318694\pi\)
\(740\) 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 \(-0.612660\pi\)
−0.346588 + 0.938017i \(0.612660\pi\)
\(758\) 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 \(-0.631927\pi\)
−0.402698 + 0.915333i \(0.631927\pi\)
\(774\) 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 \(-0.595539\pi\)
−0.295657 + 0.955294i \(0.595539\pi\)
\(788\) 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 \(-0.847496\pi\)
−0.887407 + 0.460986i \(0.847496\pi\)
\(798\) 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 \(-0.330421\pi\)
0.507903 + 0.861414i \(0.330421\pi\)
\(812\) 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 \(-0.707625\pi\)
−0.606996 + 0.794705i \(0.707625\pi\)
\(822\) 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 \(-0.368296\pi\)
0.402056 + 0.915615i \(0.368296\pi\)
\(828\) 0 0
\(829\) 28.9282 1.00472 0.502359 0.864659i \(-0.332466\pi\)
0.502359 + 0.864659i \(0.332466\pi\)
\(830\) 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 \(-0.620614\pi\)
−0.369917 + 0.929065i \(0.620614\pi\)
\(854\) 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 \(-0.660059\pi\)
−0.481917 + 0.876217i \(0.660059\pi\)
\(860\) 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 \(-0.490794\pi\)
0.0289188 + 0.999582i \(0.490794\pi\)
\(878\) 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 \(-0.344396\pi\)
0.469606 + 0.882876i \(0.344396\pi\)
\(884\) 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 \(-0.474206\pi\)
0.0809466 + 0.996718i \(0.474206\pi\)
\(908\) 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 \(-0.674659\pi\)
−0.521585 + 0.853199i \(0.674659\pi\)
\(942\) 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 \(-0.483834\pi\)
0.0507640 + 0.998711i \(0.483834\pi\)
\(948\) 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 \(-0.701875\pi\)
−0.592541 + 0.805540i \(0.701875\pi\)
\(972\) 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 \(-0.556635\pi\)
−0.176988 + 0.984213i \(0.556635\pi\)
\(998\) 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.z.1.1 2
4.3 odd 2 6400.2.a.cj.1.2 2
5.4 even 2 1280.2.a.o.1.2 2
8.3 odd 2 6400.2.a.be.1.1 2
8.5 even 2 6400.2.a.ce.1.2 2
16.3 odd 4 800.2.d.e.401.4 4
16.5 even 4 200.2.d.f.101.3 4
16.11 odd 4 800.2.d.e.401.1 4
16.13 even 4 200.2.d.f.101.4 4
20.19 odd 2 1280.2.a.d.1.1 2
40.19 odd 2 1280.2.a.n.1.2 2
40.29 even 2 1280.2.a.a.1.1 2
48.5 odd 4 1800.2.k.j.901.2 4
48.11 even 4 7200.2.k.j.3601.4 4
48.29 odd 4 1800.2.k.j.901.1 4
48.35 even 4 7200.2.k.j.3601.3 4
80.3 even 4 800.2.f.c.49.1 4
80.13 odd 4 200.2.f.c.149.3 4
80.19 odd 4 160.2.d.a.81.1 4
80.27 even 4 800.2.f.c.49.2 4
80.29 even 4 40.2.d.a.21.1 4
80.37 odd 4 200.2.f.c.149.4 4
80.43 even 4 800.2.f.e.49.3 4
80.53 odd 4 200.2.f.e.149.1 4
80.59 odd 4 160.2.d.a.81.4 4
80.67 even 4 800.2.f.e.49.4 4
80.69 even 4 40.2.d.a.21.2 yes 4
80.77 odd 4 200.2.f.e.149.2 4
240.29 odd 4 360.2.k.e.181.4 4
240.53 even 4 1800.2.d.l.1549.4 4
240.59 even 4 1440.2.k.e.721.1 4
240.77 even 4 1800.2.d.l.1549.3 4
240.83 odd 4 7200.2.d.o.2449.2 4
240.107 odd 4 7200.2.d.o.2449.3 4
240.149 odd 4 360.2.k.e.181.3 4
240.173 even 4 1800.2.d.p.1549.2 4
240.179 even 4 1440.2.k.e.721.3 4
240.197 even 4 1800.2.d.p.1549.1 4
240.203 odd 4 7200.2.d.n.2449.2 4
240.227 odd 4 7200.2.d.n.2449.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.d.a.21.1 4 80.29 even 4
40.2.d.a.21.2 yes 4 80.69 even 4
160.2.d.a.81.1 4 80.19 odd 4
160.2.d.a.81.4 4 80.59 odd 4
200.2.d.f.101.3 4 16.5 even 4
200.2.d.f.101.4 4 16.13 even 4
200.2.f.c.149.3 4 80.13 odd 4
200.2.f.c.149.4 4 80.37 odd 4
200.2.f.e.149.1 4 80.53 odd 4
200.2.f.e.149.2 4 80.77 odd 4
360.2.k.e.181.3 4 240.149 odd 4
360.2.k.e.181.4 4 240.29 odd 4
800.2.d.e.401.1 4 16.11 odd 4
800.2.d.e.401.4 4 16.3 odd 4
800.2.f.c.49.1 4 80.3 even 4
800.2.f.c.49.2 4 80.27 even 4
800.2.f.e.49.3 4 80.43 even 4
800.2.f.e.49.4 4 80.67 even 4
1280.2.a.a.1.1 2 40.29 even 2
1280.2.a.d.1.1 2 20.19 odd 2
1280.2.a.n.1.2 2 40.19 odd 2
1280.2.a.o.1.2 2 5.4 even 2
1440.2.k.e.721.1 4 240.59 even 4
1440.2.k.e.721.3 4 240.179 even 4
1800.2.d.l.1549.3 4 240.77 even 4
1800.2.d.l.1549.4 4 240.53 even 4
1800.2.d.p.1549.1 4 240.197 even 4
1800.2.d.p.1549.2 4 240.173 even 4
1800.2.k.j.901.1 4 48.29 odd 4
1800.2.k.j.901.2 4 48.5 odd 4
6400.2.a.z.1.1 2 1.1 even 1 trivial
6400.2.a.be.1.1 2 8.3 odd 2
6400.2.a.ce.1.2 2 8.5 even 2
6400.2.a.cj.1.2 2 4.3 odd 2
7200.2.d.n.2449.2 4 240.203 odd 4
7200.2.d.n.2449.3 4 240.227 odd 4
7200.2.d.o.2449.2 4 240.83 odd 4
7200.2.d.o.2449.3 4 240.107 odd 4
7200.2.k.j.3601.3 4 48.35 even 4
7200.2.k.j.3601.4 4 48.11 even 4