Properties

Label 800.2.f.e.49.3
Level $800$
Weight $2$
Character 800.49
Analytic conductor $6.388$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.3
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 800.49
Dual form 800.2.f.e.49.4

$q$-expansion

\(f(q)\) \(=\) \(q+2.73205 q^{3} -0.732051i q^{7} +4.46410 q^{9} +O(q^{10})\) \(q+2.73205 q^{3} -0.732051i q^{7} +4.46410 q^{9} -2.00000i q^{11} +3.46410 q^{13} +3.46410i q^{17} -0.535898i q^{19} -2.00000i q^{21} -6.19615i q^{23} +4.00000 q^{27} +6.92820i q^{29} +5.46410 q^{31} -5.46410i q^{33} -2.00000 q^{37} +9.46410 q^{39} +1.46410 q^{41} -5.26795 q^{43} -3.26795i q^{47} +6.46410 q^{49} +9.46410i q^{51} -11.4641 q^{53} -1.46410i q^{57} +7.46410i q^{59} +8.92820i q^{61} -3.26795i q^{63} -10.7321 q^{67} -16.9282i q^{69} -5.46410 q^{71} -7.46410i q^{73} -1.46410 q^{77} -1.07180 q^{79} -2.46410 q^{81} +1.26795 q^{83} +18.9282i q^{87} -8.92820 q^{89} -2.53590i q^{91} +14.9282 q^{93} -14.3923i q^{97} -8.92820i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{9} + 16q^{27} + 8q^{31} - 8q^{37} + 24q^{39} - 8q^{41} - 28q^{43} + 12q^{49} - 32q^{53} - 36q^{67} - 8q^{71} + 8q^{77} - 32q^{79} + 4q^{81} + 12q^{83} - 8q^{89} + 32q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 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
\(7\) − 0.732051i − 0.276689i −0.990384 0.138345i \(-0.955822\pi\)
0.990384 0.138345i \(-0.0441781\pi\)
\(8\) 0 0
\(9\) 4.46410 1.48803
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 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
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) − 0.535898i − 0.122944i −0.998109 0.0614718i \(-0.980421\pi\)
0.998109 0.0614718i \(-0.0195794\pi\)
\(20\) 0 0
\(21\) − 2.00000i − 0.436436i
\(22\) 0 0
\(23\) − 6.19615i − 1.29199i −0.763343 0.645994i \(-0.776443\pi\)
0.763343 0.645994i \(-0.223557\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 0 0
\(33\) − 5.46410i − 0.951178i
\(34\) 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 \(-0.463529\pi\)
0.114327 + 0.993443i \(0.463529\pi\)
\(42\) 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
\(47\) − 3.26795i − 0.476679i −0.971182 0.238340i \(-0.923397\pi\)
0.971182 0.238340i \(-0.0766032\pi\)
\(48\) 0 0
\(49\) 6.46410 0.923443
\(50\) 0 0
\(51\) 9.46410i 1.32524i
\(52\) 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
\(57\) − 1.46410i − 0.193925i
\(58\) 0 0
\(59\) 7.46410i 0.971743i 0.874030 + 0.485872i \(0.161498\pi\)
−0.874030 + 0.485872i \(0.838502\pi\)
\(60\) 0 0
\(61\) 8.92820i 1.14314i 0.820554 + 0.571570i \(0.193665\pi\)
−0.820554 + 0.571570i \(0.806335\pi\)
\(62\) 0 0
\(63\) − 3.26795i − 0.411723i
\(64\) 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
\(69\) − 16.9282i − 2.03792i
\(70\) 0 0
\(71\) −5.46410 −0.648470 −0.324235 0.945977i \(-0.605107\pi\)
−0.324235 + 0.945977i \(0.605107\pi\)
\(72\) 0 0
\(73\) − 7.46410i − 0.873607i −0.899557 0.436804i \(-0.856111\pi\)
0.899557 0.436804i \(-0.143889\pi\)
\(74\) 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
\(87\) 18.9282i 2.02932i
\(88\) 0 0
\(89\) −8.92820 −0.946388 −0.473194 0.880958i \(-0.656899\pi\)
−0.473194 + 0.880958i \(0.656899\pi\)
\(90\) 0 0
\(91\) − 2.53590i − 0.265834i
\(92\) 0 0
\(93\) 14.9282 1.54798
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 14.3923i − 1.46132i −0.682743 0.730659i \(-0.739213\pi\)
0.682743 0.730659i \(-0.260787\pi\)
\(98\) 0 0
\(99\) − 8.92820i − 0.897318i
\(100\) 0 0
\(101\) − 2.92820i − 0.291367i −0.989331 0.145684i \(-0.953462\pi\)
0.989331 0.145684i \(-0.0465381\pi\)
\(102\) 0 0
\(103\) 15.6603i 1.54305i 0.636199 + 0.771525i \(0.280506\pi\)
−0.636199 + 0.771525i \(0.719494\pi\)
\(104\) 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
\(109\) 16.9282i 1.62143i 0.585443 + 0.810714i \(0.300921\pi\)
−0.585443 + 0.810714i \(0.699079\pi\)
\(110\) 0 0
\(111\) −5.46410 −0.518630
\(112\) 0 0
\(113\) 12.9282i 1.21618i 0.793867 + 0.608092i \(0.208065\pi\)
−0.793867 + 0.608092i \(0.791935\pi\)
\(114\) 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
\(127\) − 16.7321i − 1.48473i −0.669996 0.742365i \(-0.733704\pi\)
0.669996 0.742365i \(-0.266296\pi\)
\(128\) 0 0
\(129\) −14.3923 −1.26717
\(130\) 0 0
\(131\) − 19.8564i − 1.73486i −0.497557 0.867431i \(-0.665770\pi\)
0.497557 0.867431i \(-0.334230\pi\)
\(132\) 0 0
\(133\) −0.392305 −0.0340171
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.92820i 0.421045i 0.977589 + 0.210522i \(0.0675165\pi\)
−0.977589 + 0.210522i \(0.932484\pi\)
\(138\) 0 0
\(139\) 0.535898i 0.0454543i 0.999742 + 0.0227272i \(0.00723490\pi\)
−0.999742 + 0.0227272i \(0.992765\pi\)
\(140\) 0 0
\(141\) − 8.92820i − 0.751890i
\(142\) 0 0
\(143\) − 6.92820i − 0.579365i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 17.6603 1.45659
\(148\) 0 0
\(149\) − 7.85641i − 0.643622i −0.946804 0.321811i \(-0.895708\pi\)
0.946804 0.321811i \(-0.104292\pi\)
\(150\) 0 0
\(151\) 12.3923 1.00847 0.504236 0.863566i \(-0.331774\pi\)
0.504236 + 0.863566i \(0.331774\pi\)
\(152\) 0 0
\(153\) 15.4641i 1.25020i
\(154\) 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
\(167\) 9.80385i 0.758645i 0.925265 + 0.379322i \(0.123843\pi\)
−0.925265 + 0.379322i \(0.876157\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) − 2.39230i − 0.182944i
\(172\) 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
\(177\) 20.3923i 1.53278i
\(178\) 0 0
\(179\) 8.53590i 0.638003i 0.947754 + 0.319002i \(0.103348\pi\)
−0.947754 + 0.319002i \(0.896652\pi\)
\(180\) 0 0
\(181\) − 16.0000i − 1.18927i −0.803996 0.594635i \(-0.797296\pi\)
0.803996 0.594635i \(-0.202704\pi\)
\(182\) 0 0
\(183\) 24.3923i 1.80313i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.92820 0.506640
\(188\) 0 0
\(189\) − 2.92820i − 0.212995i
\(190\) 0 0
\(191\) −15.3205 −1.10855 −0.554277 0.832333i \(-0.687005\pi\)
−0.554277 + 0.832333i \(0.687005\pi\)
\(192\) 0 0
\(193\) − 0.535898i − 0.0385748i −0.999814 0.0192874i \(-0.993860\pi\)
0.999814 0.0192874i \(-0.00613975\pi\)
\(194\) 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 \(-0.520959\pi\)
−0.0657986 + 0.997833i \(0.520959\pi\)
\(200\) 0 0
\(201\) −29.3205 −2.06811
\(202\) 0 0
\(203\) 5.07180 0.355970
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 27.6603i − 1.92252i
\(208\) 0 0
\(209\) −1.07180 −0.0741377
\(210\) 0 0
\(211\) 26.7846i 1.84393i 0.387275 + 0.921964i \(0.373416\pi\)
−0.387275 + 0.921964i \(0.626584\pi\)
\(212\) 0 0
\(213\) −14.9282 −1.02286
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) 0 0
\(219\) − 20.3923i − 1.37798i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) − 5.80385i − 0.388654i −0.980937 0.194327i \(-0.937748\pi\)
0.980937 0.194327i \(-0.0622523\pi\)
\(224\) 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
\(229\) − 4.00000i − 0.264327i −0.991228 0.132164i \(-0.957808\pi\)
0.991228 0.132164i \(-0.0421925\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 5.32051i 0.348558i 0.984696 + 0.174279i \(0.0557595\pi\)
−0.984696 + 0.174279i \(0.944241\pi\)
\(234\) 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
\(247\) − 1.85641i − 0.118120i
\(248\) 0 0
\(249\) 3.46410 0.219529
\(250\) 0 0
\(251\) 24.9282i 1.57345i 0.617301 + 0.786727i \(0.288226\pi\)
−0.617301 + 0.786727i \(0.711774\pi\)
\(252\) 0 0
\(253\) −12.3923 −0.779098
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 0 0
\(259\) 1.46410i 0.0909748i
\(260\) 0 0
\(261\) 30.9282i 1.91441i
\(262\) 0 0
\(263\) − 11.6603i − 0.719002i −0.933145 0.359501i \(-0.882947\pi\)
0.933145 0.359501i \(-0.117053\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −24.3923 −1.49278
\(268\) 0 0
\(269\) − 8.92820i − 0.544362i −0.962246 0.272181i \(-0.912255\pi\)
0.962246 0.272181i \(-0.0877450\pi\)
\(270\) 0 0
\(271\) 19.3205 1.17364 0.586819 0.809718i \(-0.300380\pi\)
0.586819 + 0.809718i \(0.300380\pi\)
\(272\) 0 0
\(273\) − 6.92820i − 0.419314i
\(274\) 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 \(-0.398244\pi\)
0.314260 + 0.949337i \(0.398244\pi\)
\(282\) 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
\(287\) − 1.07180i − 0.0632662i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) − 39.3205i − 2.30501i
\(292\) 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
\(297\) − 8.00000i − 0.464207i
\(298\) 0 0
\(299\) − 21.4641i − 1.24130i
\(300\) 0 0
\(301\) 3.85641i 0.222280i
\(302\) 0 0
\(303\) − 8.00000i − 0.459588i
\(304\) 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
\(309\) 42.7846i 2.43393i
\(310\) 0 0
\(311\) −31.3205 −1.77602 −0.888012 0.459821i \(-0.847914\pi\)
−0.888012 + 0.459821i \(0.847914\pi\)
\(312\) 0 0
\(313\) 4.14359i 0.234210i 0.993120 + 0.117105i \(0.0373614\pi\)
−0.993120 + 0.117105i \(0.962639\pi\)
\(314\) 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
\(327\) 46.2487i 2.55756i
\(328\) 0 0
\(329\) −2.39230 −0.131892
\(330\) 0 0
\(331\) − 14.0000i − 0.769510i −0.923019 0.384755i \(-0.874286\pi\)
0.923019 0.384755i \(-0.125714\pi\)
\(332\) 0 0
\(333\) −8.92820 −0.489263
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 19.8564i − 1.08165i −0.841136 0.540824i \(-0.818113\pi\)
0.841136 0.540824i \(-0.181887\pi\)
\(338\) 0 0
\(339\) 35.3205i 1.91835i
\(340\) 0 0
\(341\) − 10.9282i − 0.591795i
\(342\) 0 0
\(343\) − 9.85641i − 0.532196i
\(344\) 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
\(349\) − 28.0000i − 1.49881i −0.662114 0.749403i \(-0.730341\pi\)
0.662114 0.749403i \(-0.269659\pi\)
\(350\) 0 0
\(351\) 13.8564 0.739600
\(352\) 0 0
\(353\) − 12.9282i − 0.688099i −0.938952 0.344049i \(-0.888201\pi\)
0.938952 0.344049i \(-0.111799\pi\)
\(354\) 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 \(-0.333518\pi\)
0.499496 + 0.866316i \(0.333518\pi\)
\(360\) 0 0
\(361\) 18.7128 0.984885
\(362\) 0 0
\(363\) 19.1244 1.00377
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 2.87564i − 0.150107i −0.997179 0.0750537i \(-0.976087\pi\)
0.997179 0.0750537i \(-0.0239128\pi\)
\(368\) 0 0
\(369\) 6.53590 0.340245
\(370\) 0 0
\(371\) 8.39230i 0.435707i
\(372\) 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
\(377\) 24.0000i 1.23606i
\(378\) 0 0
\(379\) 36.2487i 1.86197i 0.365056 + 0.930986i \(0.381050\pi\)
−0.365056 + 0.930986i \(0.618950\pi\)
\(380\) 0 0
\(381\) − 45.7128i − 2.34194i
\(382\) 0 0
\(383\) 21.1244i 1.07940i 0.841856 + 0.539702i \(0.181463\pi\)
−0.841856 + 0.539702i \(0.818537\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −23.5167 −1.19542
\(388\) 0 0
\(389\) 6.78461i 0.343993i 0.985098 + 0.171997i \(0.0550218\pi\)
−0.985098 + 0.171997i \(0.944978\pi\)
\(390\) 0 0
\(391\) 21.4641 1.08549
\(392\) 0 0
\(393\) − 54.2487i − 2.73649i
\(394\) 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
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) 11.3205 0.559763 0.279882 0.960035i \(-0.409705\pi\)
0.279882 + 0.960035i \(0.409705\pi\)
\(410\) 0 0
\(411\) 13.4641i 0.664135i
\(412\) 0 0
\(413\) 5.46410 0.268871
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.46410i 0.0716974i
\(418\) 0 0
\(419\) − 18.3923i − 0.898523i −0.893400 0.449261i \(-0.851687\pi\)
0.893400 0.449261i \(-0.148313\pi\)
\(420\) 0 0
\(421\) − 0.143594i − 0.00699832i −0.999994 0.00349916i \(-0.998886\pi\)
0.999994 0.00349916i \(-0.00111382\pi\)
\(422\) 0 0
\(423\) − 14.5885i − 0.709315i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.53590 0.316294
\(428\) 0 0
\(429\) − 18.9282i − 0.913862i
\(430\) 0 0
\(431\) 21.4641 1.03389 0.516945 0.856019i \(-0.327069\pi\)
0.516945 + 0.856019i \(0.327069\pi\)
\(432\) 0 0
\(433\) 19.4641i 0.935385i 0.883891 + 0.467693i \(0.154915\pi\)
−0.883891 + 0.467693i \(0.845085\pi\)
\(434\) 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 \(-0.0737605\pi\)
0.973272 + 0.229657i \(0.0737605\pi\)
\(440\) 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
\(447\) − 21.4641i − 1.01522i
\(448\) 0 0
\(449\) 23.3205 1.10056 0.550281 0.834979i \(-0.314520\pi\)
0.550281 + 0.834979i \(0.314520\pi\)
\(450\) 0 0
\(451\) − 2.92820i − 0.137884i
\(452\) 0 0
\(453\) 33.8564 1.59071
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 26.7846i − 1.25293i −0.779449 0.626466i \(-0.784501\pi\)
0.779449 0.626466i \(-0.215499\pi\)
\(458\) 0 0
\(459\) 13.8564i 0.646762i
\(460\) 0 0
\(461\) 10.9282i 0.508977i 0.967076 + 0.254489i \(0.0819071\pi\)
−0.967076 + 0.254489i \(0.918093\pi\)
\(462\) 0 0
\(463\) − 11.2679i − 0.523666i −0.965113 0.261833i \(-0.915673\pi\)
0.965113 0.261833i \(-0.0843270\pi\)
\(464\) 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
\(469\) 7.85641i 0.362775i
\(470\) 0 0
\(471\) −8.39230 −0.386697
\(472\) 0 0
\(473\) 10.5359i 0.484441i
\(474\) 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
\(487\) − 6.58846i − 0.298551i −0.988796 0.149276i \(-0.952306\pi\)
0.988796 0.149276i \(-0.0476942\pi\)
\(488\) 0 0
\(489\) −0.535898 −0.0242342
\(490\) 0 0
\(491\) − 16.9282i − 0.763959i −0.924171 0.381980i \(-0.875242\pi\)
0.924171 0.381980i \(-0.124758\pi\)
\(492\) 0 0
\(493\) −24.0000 −1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.00000i 0.179425i
\(498\) 0 0
\(499\) − 31.4641i − 1.40853i −0.709939 0.704263i \(-0.751277\pi\)
0.709939 0.704263i \(-0.248723\pi\)
\(500\) 0 0
\(501\) 26.7846i 1.19665i
\(502\) 0 0
\(503\) − 0.339746i − 0.0151485i −0.999971 0.00757426i \(-0.997589\pi\)
0.999971 0.00757426i \(-0.00241099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −2.73205 −0.121335
\(508\) 0 0
\(509\) − 1.85641i − 0.0822838i −0.999153 0.0411419i \(-0.986900\pi\)
0.999153 0.0411419i \(-0.0130996\pi\)
\(510\) 0 0
\(511\) −5.46410 −0.241718
\(512\) 0 0
\(513\) − 2.14359i − 0.0946420i
\(514\) 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 \(-0.910456\pi\)
−0.960692 + 0.277616i \(0.910456\pi\)
\(522\) 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
\(527\) 18.9282i 0.824525i
\(528\) 0 0
\(529\) −15.3923 −0.669231
\(530\) 0 0
\(531\) 33.3205i 1.44599i
\(532\) 0 0
\(533\) 5.07180 0.219684
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 23.3205i 1.00635i
\(538\) 0 0
\(539\) − 12.9282i − 0.556857i
\(540\) 0 0
\(541\) 26.9282i 1.15773i 0.815422 + 0.578867i \(0.196505\pi\)
−0.815422 + 0.578867i \(0.803495\pi\)
\(542\) 0 0
\(543\) − 43.7128i − 1.87590i
\(544\) 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
\(549\) 39.8564i 1.70103i
\(550\) 0 0
\(551\) 3.71281 0.158171
\(552\) 0 0
\(553\) 0.784610i 0.0333650i
\(554\) 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
\(567\) 1.80385i 0.0757545i
\(568\) 0 0
\(569\) 13.4641 0.564445 0.282222 0.959349i \(-0.408928\pi\)
0.282222 + 0.959349i \(0.408928\pi\)
\(570\) 0 0
\(571\) 6.78461i 0.283927i 0.989872 + 0.141964i \(0.0453416\pi\)
−0.989872 + 0.141964i \(0.954658\pi\)
\(572\) 0 0
\(573\) −41.8564 −1.74858
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 39.5692i 1.64729i 0.567107 + 0.823644i \(0.308063\pi\)
−0.567107 + 0.823644i \(0.691937\pi\)
\(578\) 0 0
\(579\) − 1.46410i − 0.0608460i
\(580\) 0 0
\(581\) − 0.928203i − 0.0385084i
\(582\) 0 0
\(583\) 22.9282i 0.949589i
\(584\) 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
\(589\) − 2.92820i − 0.120655i
\(590\) 0 0
\(591\) −53.1769 −2.18741
\(592\) 0 0
\(593\) − 32.6410i − 1.34041i −0.742178 0.670203i \(-0.766207\pi\)
0.742178 0.670203i \(-0.233793\pi\)
\(594\) 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 \(-0.750266\pi\)
−0.707697 + 0.706516i \(0.750266\pi\)
\(600\) 0 0
\(601\) 18.5359 0.756095 0.378048 0.925786i \(-0.376596\pi\)
0.378048 + 0.925786i \(0.376596\pi\)
\(602\) 0 0
\(603\) −47.9090 −1.95100
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 30.9808i − 1.25747i −0.777619 0.628735i \(-0.783573\pi\)
0.777619 0.628735i \(-0.216427\pi\)
\(608\) 0 0
\(609\) 13.8564 0.561490
\(610\) 0 0
\(611\) − 11.3205i − 0.457979i
\(612\) 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
\(617\) − 20.5359i − 0.826744i −0.910562 0.413372i \(-0.864351\pi\)
0.910562 0.413372i \(-0.135649\pi\)
\(618\) 0 0
\(619\) − 1.32051i − 0.0530757i −0.999648 0.0265379i \(-0.991552\pi\)
0.999648 0.0265379i \(-0.00844825\pi\)
\(620\) 0 0
\(621\) − 24.7846i − 0.994572i
\(622\) 0 0
\(623\) 6.53590i 0.261855i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −2.92820 −0.116941
\(628\) 0 0
\(629\) − 6.92820i − 0.276246i
\(630\) 0 0
\(631\) 23.3205 0.928375 0.464187 0.885737i \(-0.346346\pi\)
0.464187 + 0.885737i \(0.346346\pi\)
\(632\) 0 0
\(633\) 73.1769i 2.90852i
\(634\) 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
\(647\) − 16.7321i − 0.657805i −0.944364 0.328902i \(-0.893321\pi\)
0.944364 0.328902i \(-0.106679\pi\)
\(648\) 0 0
\(649\) 14.9282 0.585983
\(650\) 0 0
\(651\) − 10.9282i − 0.428310i
\(652\) 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
\(657\) − 33.3205i − 1.29996i
\(658\) 0 0
\(659\) 17.3205i 0.674711i 0.941377 + 0.337356i \(0.109532\pi\)
−0.941377 + 0.337356i \(0.890468\pi\)
\(660\) 0 0
\(661\) 8.14359i 0.316749i 0.987379 + 0.158375i \(0.0506253\pi\)
−0.987379 + 0.158375i \(0.949375\pi\)
\(662\) 0 0
\(663\) 32.7846i 1.27325i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 42.9282 1.66219
\(668\) 0 0
\(669\) − 15.8564i − 0.613044i
\(670\) 0 0
\(671\) 17.8564 0.689339
\(672\) 0 0
\(673\) − 12.5359i − 0.483223i −0.970373 0.241612i \(-0.922324\pi\)
0.970373 0.241612i \(-0.0776760\pi\)
\(674\) 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
\(687\) − 10.9282i − 0.416937i
\(688\) 0 0
\(689\) −39.7128 −1.51294
\(690\) 0 0
\(691\) 18.0000i 0.684752i 0.939563 + 0.342376i \(0.111232\pi\)
−0.939563 + 0.342376i \(0.888768\pi\)
\(692\) 0 0
\(693\) −6.53590 −0.248278
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.07180i 0.192108i
\(698\) 0 0
\(699\) 14.5359i 0.549798i
\(700\) 0 0
\(701\) − 19.0718i − 0.720332i −0.932888 0.360166i \(-0.882720\pi\)
0.932888 0.360166i \(-0.117280\pi\)
\(702\) 0 0
\(703\) 1.07180i 0.0404236i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.14359 −0.0806181
\(708\) 0 0
\(709\) − 12.7846i − 0.480136i −0.970756 0.240068i \(-0.922830\pi\)
0.970756 0.240068i \(-0.0771698\pi\)
\(710\) 0 0
\(711\) −4.78461 −0.179437
\(712\) 0 0
\(713\) − 33.8564i − 1.26793i
\(714\) 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
\(727\) 24.0526i 0.892060i 0.895018 + 0.446030i \(0.147163\pi\)
−0.895018 + 0.446030i \(0.852837\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 0 0
\(731\) − 18.2487i − 0.674953i
\(732\) 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
\(737\) 21.4641i 0.790640i
\(738\) 0 0
\(739\) − 29.3205i − 1.07857i −0.842123 0.539286i \(-0.818694\pi\)
0.842123 0.539286i \(-0.181306\pi\)
\(740\) 0 0
\(741\) − 5.07180i − 0.186317i
\(742\) 0 0
\(743\) 10.9808i 0.402845i 0.979504 + 0.201423i \(0.0645564\pi\)
−0.979504 + 0.201423i \(0.935444\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.66025 0.207098
\(748\) 0 0
\(749\) 2.00000i 0.0730784i
\(750\) 0 0
\(751\) −26.2487 −0.957829 −0.478915 0.877862i \(-0.658970\pi\)
−0.478915 + 0.877862i \(0.658970\pi\)
\(752\) 0 0
\(753\) 68.1051i 2.48189i
\(754\) 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 \(-0.533018\pi\)
−0.103545 + 0.994625i \(0.533018\pi\)
\(762\) 0 0
\(763\) 12.3923 0.448632
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25.8564i 0.933621i
\(768\) 0 0
\(769\) −12.9282 −0.466203 −0.233101 0.972452i \(-0.574887\pi\)
−0.233101 + 0.972452i \(0.574887\pi\)
\(770\) 0 0
\(771\) − 5.46410i − 0.196785i
\(772\) 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
\(777\) 4.00000i 0.143499i
\(778\) 0 0
\(779\) − 0.784610i − 0.0281116i
\(780\) 0 0
\(781\) 10.9282i 0.391042i
\(782\) 0 0
\(783\) 27.7128i 0.990375i
\(784\) 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
\(789\) − 31.8564i − 1.13412i
\(790\) 0 0
\(791\) 9.46410 0.336505
\(792\) 0 0
\(793\) 30.9282i 1.09829i
\(794\) 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
\(807\) − 24.3923i − 0.858650i
\(808\) 0 0
\(809\) −23.8564 −0.838747 −0.419373 0.907814i \(-0.637750\pi\)
−0.419373 + 0.907814i \(0.637750\pi\)
\(810\) 0 0
\(811\) − 28.9282i − 1.01581i −0.861414 0.507903i \(-0.830421\pi\)
0.861414 0.507903i \(-0.169579\pi\)
\(812\) 0 0
\(813\) 52.7846 1.85124
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.82309i 0.0987673i
\(818\) 0 0
\(819\) − 11.3205i − 0.395571i
\(820\) 0 0
\(821\) − 34.7846i − 1.21399i −0.794705 0.606996i \(-0.792375\pi\)
0.794705 0.606996i \(-0.207625\pi\)
\(822\) 0 0
\(823\) − 9.12436i − 0.318055i −0.987274 0.159028i \(-0.949164\pi\)
0.987274 0.159028i \(-0.0508359\pi\)
\(824\) 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
\(829\) 28.9282i 1.00472i 0.864659 + 0.502359i \(0.167534\pi\)
−0.864659 + 0.502359i \(0.832466\pi\)
\(830\) 0 0
\(831\) 5.46410 0.189548
\(832\) 0 0
\(833\) 22.3923i 0.775847i
\(834\) 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 \(-0.359278\pi\)
0.427830 + 0.903859i \(0.359278\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) 28.7846 0.991395
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 5.12436i − 0.176075i
\(848\) 0 0
\(849\) −26.3923 −0.905782
\(850\) 0 0
\(851\) 12.3923i 0.424803i
\(852\) 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
\(857\) − 19.8564i − 0.678282i −0.940736 0.339141i \(-0.889864\pi\)
0.940736 0.339141i \(-0.110136\pi\)
\(858\) 0 0
\(859\) − 28.2487i − 0.963834i −0.876217 0.481917i \(-0.839941\pi\)
0.876217 0.481917i \(-0.160059\pi\)
\(860\) 0 0
\(861\) − 2.92820i − 0.0997929i
\(862\) 0 0
\(863\) 47.6603i 1.62237i 0.584787 + 0.811187i \(0.301178\pi\)
−0.584787 + 0.811187i \(0.698822\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.6603 0.463927
\(868\) 0 0
\(869\) 2.14359i 0.0727164i
\(870\) 0 0
\(871\) −37.1769 −1.25969
\(872\) 0 0
\(873\) − 64.2487i − 2.17449i
\(874\) 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
\(887\) 13.9090i 0.467017i 0.972355 + 0.233509i \(0.0750207\pi\)
−0.972355 + 0.233509i \(0.924979\pi\)
\(888\) 0 0
\(889\) −12.2487 −0.410809
\(890\) 0 0
\(891\) 4.92820i 0.165101i
\(892\) 0 0
\(893\) −1.75129 −0.0586046
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 58.6410i − 1.95797i
\(898\) 0 0
\(899\) 37.8564i 1.26258i
\(900\) 0 0
\(901\) − 39.7128i − 1.32303i
\(902\) 0 0
\(903\) 10.5359i 0.350613i
\(904\) 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
\(909\) − 13.0718i − 0.433564i
\(910\) 0 0
\(911\) 49.1769 1.62930 0.814652 0.579950i \(-0.196928\pi\)
0.814652 + 0.579950i \(0.196928\pi\)
\(912\) 0 0
\(913\) − 2.53590i − 0.0839260i
\(914\) 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 \(-0.721921\pi\)
−0.642061 + 0.766653i \(0.721921\pi\)
\(920\) 0 0
\(921\) 68.2487 2.24887
\(922\) 0 0
\(923\) −18.9282 −0.623029
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 69.9090i 2.29611i
\(928\) 0 0
\(929\) 17.4641 0.572979 0.286489 0.958083i \(-0.407512\pi\)
0.286489 + 0.958083i \(0.407512\pi\)
\(930\) 0 0
\(931\) − 3.46410i − 0.113531i
\(932\) 0 0
\(933\) −85.5692 −2.80141
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.24871i 0.138799i 0.997589 + 0.0693997i \(0.0221084\pi\)
−0.997589 + 0.0693997i \(0.977892\pi\)
\(938\) 0 0
\(939\) 11.3205i 0.369431i
\(940\) 0 0
\(941\) 32.0000i 1.04317i 0.853199 + 0.521585i \(0.174659\pi\)
−0.853199 + 0.521585i \(0.825341\pi\)
\(942\) 0 0
\(943\) − 9.07180i − 0.295418i
\(944\) 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
\(949\) − 25.8564i − 0.839334i
\(950\) 0 0
\(951\) 23.3205 0.756219
\(952\) 0 0
\(953\) 17.2154i 0.557661i 0.960340 + 0.278831i \(0.0899468\pi\)
−0.960340 + 0.278831i \(0.910053\pi\)
\(954\) 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
\(967\) − 16.3397i − 0.525451i −0.964871 0.262725i \(-0.915379\pi\)
0.964871 0.262725i \(-0.0846213\pi\)
\(968\) 0 0
\(969\) 5.07180 0.162930
\(970\) 0 0
\(971\) 36.9282i 1.18508i 0.805540 + 0.592541i \(0.201875\pi\)
−0.805540 + 0.592541i \(0.798125\pi\)
\(972\) 0 0
\(973\) 0.392305 0.0125767
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 24.5359i − 0.784973i −0.919758 0.392486i \(-0.871615\pi\)
0.919758 0.392486i \(-0.128385\pi\)
\(978\) 0 0
\(979\) 17.8564i 0.570693i
\(980\) 0 0
\(981\) 75.5692i 2.41274i
\(982\) 0 0
\(983\) − 48.7321i − 1.55431i −0.629309 0.777156i \(-0.716662\pi\)
0.629309 0.777156i \(-0.283338\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −6.53590 −0.208040
\(988\) 0 0
\(989\) 32.6410i 1.03792i
\(990\) 0 0
\(991\) −41.4641 −1.31715 −0.658575 0.752515i \(-0.728841\pi\)
−0.658575 + 0.752515i \(0.728841\pi\)
\(992\) 0 0
\(993\) − 38.2487i − 1.21379i
\(994\) 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 800.2.f.e.49.3 4
3.2 odd 2 7200.2.d.n.2449.2 4
4.3 odd 2 200.2.f.e.149.1 4
5.2 odd 4 800.2.d.e.401.1 4
5.3 odd 4 160.2.d.a.81.4 4
5.4 even 2 800.2.f.c.49.2 4
8.3 odd 2 200.2.f.c.149.3 4
8.5 even 2 800.2.f.c.49.1 4
12.11 even 2 1800.2.d.l.1549.4 4
15.2 even 4 7200.2.k.j.3601.4 4
15.8 even 4 1440.2.k.e.721.1 4
15.14 odd 2 7200.2.d.o.2449.3 4
20.3 even 4 40.2.d.a.21.2 yes 4
20.7 even 4 200.2.d.f.101.3 4
20.19 odd 2 200.2.f.c.149.4 4
24.5 odd 2 7200.2.d.o.2449.2 4
24.11 even 2 1800.2.d.p.1549.2 4
40.3 even 4 40.2.d.a.21.1 4
40.13 odd 4 160.2.d.a.81.1 4
40.19 odd 2 200.2.f.e.149.2 4
40.27 even 4 200.2.d.f.101.4 4
40.29 even 2 inner 800.2.f.e.49.4 4
40.37 odd 4 800.2.d.e.401.4 4
60.23 odd 4 360.2.k.e.181.3 4
60.47 odd 4 1800.2.k.j.901.2 4
60.59 even 2 1800.2.d.p.1549.1 4
80.3 even 4 1280.2.a.o.1.2 2
80.13 odd 4 1280.2.a.d.1.1 2
80.27 even 4 6400.2.a.ce.1.2 2
80.37 odd 4 6400.2.a.be.1.1 2
80.43 even 4 1280.2.a.a.1.1 2
80.53 odd 4 1280.2.a.n.1.2 2
80.67 even 4 6400.2.a.z.1.1 2
80.77 odd 4 6400.2.a.cj.1.2 2
120.29 odd 2 7200.2.d.n.2449.3 4
120.53 even 4 1440.2.k.e.721.3 4
120.59 even 2 1800.2.d.l.1549.3 4
120.77 even 4 7200.2.k.j.3601.3 4
120.83 odd 4 360.2.k.e.181.4 4
120.107 odd 4 1800.2.k.j.901.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.d.a.21.1 4 40.3 even 4
40.2.d.a.21.2 yes 4 20.3 even 4
160.2.d.a.81.1 4 40.13 odd 4
160.2.d.a.81.4 4 5.3 odd 4
200.2.d.f.101.3 4 20.7 even 4
200.2.d.f.101.4 4 40.27 even 4
200.2.f.c.149.3 4 8.3 odd 2
200.2.f.c.149.4 4 20.19 odd 2
200.2.f.e.149.1 4 4.3 odd 2
200.2.f.e.149.2 4 40.19 odd 2
360.2.k.e.181.3 4 60.23 odd 4
360.2.k.e.181.4 4 120.83 odd 4
800.2.d.e.401.1 4 5.2 odd 4
800.2.d.e.401.4 4 40.37 odd 4
800.2.f.c.49.1 4 8.5 even 2
800.2.f.c.49.2 4 5.4 even 2
800.2.f.e.49.3 4 1.1 even 1 trivial
800.2.f.e.49.4 4 40.29 even 2 inner
1280.2.a.a.1.1 2 80.43 even 4
1280.2.a.d.1.1 2 80.13 odd 4
1280.2.a.n.1.2 2 80.53 odd 4
1280.2.a.o.1.2 2 80.3 even 4
1440.2.k.e.721.1 4 15.8 even 4
1440.2.k.e.721.3 4 120.53 even 4
1800.2.d.l.1549.3 4 120.59 even 2
1800.2.d.l.1549.4 4 12.11 even 2
1800.2.d.p.1549.1 4 60.59 even 2
1800.2.d.p.1549.2 4 24.11 even 2
1800.2.k.j.901.1 4 120.107 odd 4
1800.2.k.j.901.2 4 60.47 odd 4
6400.2.a.z.1.1 2 80.67 even 4
6400.2.a.be.1.1 2 80.37 odd 4
6400.2.a.ce.1.2 2 80.27 even 4
6400.2.a.cj.1.2 2 80.77 odd 4
7200.2.d.n.2449.2 4 3.2 odd 2
7200.2.d.n.2449.3 4 120.29 odd 2
7200.2.d.o.2449.2 4 24.5 odd 2
7200.2.d.o.2449.3 4 15.14 odd 2
7200.2.k.j.3601.3 4 120.77 even 4
7200.2.k.j.3601.4 4 15.2 even 4