Properties

Label 6400.2.a.bv.1.2
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.44949 q^{3} +2.44949 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+2.44949 q^{3} +2.44949 q^{7} +3.00000 q^{9} +4.89898 q^{11} -4.00000 q^{17} -4.89898 q^{19} +6.00000 q^{21} +2.44949 q^{23} +8.00000 q^{29} +9.79796 q^{31} +12.0000 q^{33} -4.00000 q^{37} -8.00000 q^{41} +7.34847 q^{43} +12.2474 q^{47} -1.00000 q^{49} -9.79796 q^{51} -8.00000 q^{53} -12.0000 q^{57} +4.89898 q^{59} +6.00000 q^{61} +7.34847 q^{63} +2.44949 q^{67} +6.00000 q^{69} -9.79796 q^{71} -4.00000 q^{73} +12.0000 q^{77} +9.79796 q^{79} -9.00000 q^{81} -2.44949 q^{83} +19.5959 q^{87} +2.00000 q^{89} +24.0000 q^{93} +4.00000 q^{97} +14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{9} - 8 q^{17} + 12 q^{21} + 16 q^{29} + 24 q^{33} - 8 q^{37} - 16 q^{41} - 2 q^{49} - 16 q^{53} - 24 q^{57} + 12 q^{61} + 12 q^{69} - 8 q^{73} + 24 q^{77} - 18 q^{81} + 4 q^{89} + 48 q^{93} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.44949 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.44949 0.925820 0.462910 0.886405i \(-0.346805\pi\)
0.462910 + 0.886405i \(0.346805\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) −4.89898 −1.12390 −0.561951 0.827170i \(-0.689949\pi\)
−0.561951 + 0.827170i \(0.689949\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) 0 0
\(23\) 2.44949 0.510754 0.255377 0.966842i \(-0.417800\pi\)
0.255377 + 0.966842i \(0.417800\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 9.79796 1.75977 0.879883 0.475191i \(-0.157621\pi\)
0.879883 + 0.475191i \(0.157621\pi\)
\(32\) 0 0
\(33\) 12.0000 2.08893
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 7.34847 1.12063 0.560316 0.828279i \(-0.310680\pi\)
0.560316 + 0.828279i \(0.310680\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.2474 1.78647 0.893237 0.449586i \(-0.148429\pi\)
0.893237 + 0.449586i \(0.148429\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −9.79796 −1.37199
\(52\) 0 0
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) 0 0
\(59\) 4.89898 0.637793 0.318896 0.947790i \(-0.396688\pi\)
0.318896 + 0.947790i \(0.396688\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 7.34847 0.925820
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.44949 0.299253 0.149626 0.988743i \(-0.452193\pi\)
0.149626 + 0.988743i \(0.452193\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −9.79796 −1.16280 −0.581402 0.813617i \(-0.697496\pi\)
−0.581402 + 0.813617i \(0.697496\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) 9.79796 1.10236 0.551178 0.834388i \(-0.314178\pi\)
0.551178 + 0.834388i \(0.314178\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −2.44949 −0.268866 −0.134433 0.990923i \(-0.542921\pi\)
−0.134433 + 0.990923i \(0.542921\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 19.5959 2.10090
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 24.0000 2.48868
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) 14.6969 1.47710
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) 2.44949 0.241355 0.120678 0.992692i \(-0.461493\pi\)
0.120678 + 0.992692i \(0.461493\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.34847 0.710403 0.355202 0.934790i \(-0.384412\pi\)
0.355202 + 0.934790i \(0.384412\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −9.79796 −0.929981
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.79796 −0.898177
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) −19.5959 −1.76690
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.34847 0.652071 0.326036 0.945357i \(-0.394287\pi\)
0.326036 + 0.945357i \(0.394287\pi\)
\(128\) 0 0
\(129\) 18.0000 1.58481
\(130\) 0 0
\(131\) 4.89898 0.428026 0.214013 0.976831i \(-0.431347\pi\)
0.214013 + 0.976831i \(0.431347\pi\)
\(132\) 0 0
\(133\) −12.0000 −1.04053
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) −4.89898 −0.415526 −0.207763 0.978179i \(-0.566618\pi\)
−0.207763 + 0.978179i \(0.566618\pi\)
\(140\) 0 0
\(141\) 30.0000 2.52646
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.44949 −0.202031
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −20.0000 −1.59617 −0.798087 0.602542i \(-0.794154\pi\)
−0.798087 + 0.602542i \(0.794154\pi\)
\(158\) 0 0
\(159\) −19.5959 −1.55406
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 2.44949 0.191859 0.0959294 0.995388i \(-0.469418\pi\)
0.0959294 + 0.995388i \(0.469418\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.44949 −0.189547 −0.0947736 0.995499i \(-0.530213\pi\)
−0.0947736 + 0.995499i \(0.530213\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −14.6969 −1.12390
\(172\) 0 0
\(173\) −4.00000 −0.304114 −0.152057 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −14.6969 −1.09850 −0.549250 0.835658i \(-0.685087\pi\)
−0.549250 + 0.835658i \(0.685087\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 14.6969 1.08643
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −19.5959 −1.43300
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.5959 −1.41791 −0.708955 0.705253i \(-0.750833\pi\)
−0.708955 + 0.705253i \(0.750833\pi\)
\(192\) 0 0
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) 9.79796 0.694559 0.347279 0.937762i \(-0.387106\pi\)
0.347279 + 0.937762i \(0.387106\pi\)
\(200\) 0 0
\(201\) 6.00000 0.423207
\(202\) 0 0
\(203\) 19.5959 1.37536
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.34847 0.510754
\(208\) 0 0
\(209\) −24.0000 −1.66011
\(210\) 0 0
\(211\) 24.4949 1.68630 0.843149 0.537680i \(-0.180699\pi\)
0.843149 + 0.537680i \(0.180699\pi\)
\(212\) 0 0
\(213\) −24.0000 −1.64445
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 0 0
\(219\) −9.79796 −0.662085
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −7.34847 −0.492090 −0.246045 0.969258i \(-0.579131\pi\)
−0.246045 + 0.969258i \(0.579131\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.44949 −0.162578 −0.0812892 0.996691i \(-0.525904\pi\)
−0.0812892 + 0.996691i \(0.525904\pi\)
\(228\) 0 0
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 29.3939 1.93398
\(232\) 0 0
\(233\) −20.0000 −1.31024 −0.655122 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 24.0000 1.55897
\(238\) 0 0
\(239\) −9.79796 −0.633777 −0.316889 0.948463i \(-0.602638\pi\)
−0.316889 + 0.948463i \(0.602638\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) −22.0454 −1.41421
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 4.89898 0.309221 0.154610 0.987976i \(-0.450588\pi\)
0.154610 + 0.987976i \(0.450588\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 0 0
\(259\) −9.79796 −0.608816
\(260\) 0 0
\(261\) 24.0000 1.48556
\(262\) 0 0
\(263\) −22.0454 −1.35938 −0.679689 0.733500i \(-0.737885\pi\)
−0.679689 + 0.733500i \(0.737885\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.89898 0.299813
\(268\) 0 0
\(269\) 26.0000 1.58525 0.792624 0.609711i \(-0.208714\pi\)
0.792624 + 0.609711i \(0.208714\pi\)
\(270\) 0 0
\(271\) 9.79796 0.595184 0.297592 0.954693i \(-0.403817\pi\)
0.297592 + 0.954693i \(0.403817\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 0 0
\(279\) 29.3939 1.75977
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) −26.9444 −1.60168 −0.800839 0.598880i \(-0.795613\pi\)
−0.800839 + 0.598880i \(0.795613\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −19.5959 −1.15671
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 9.79796 0.574367
\(292\) 0 0
\(293\) 20.0000 1.16841 0.584206 0.811605i \(-0.301406\pi\)
0.584206 + 0.811605i \(0.301406\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 18.0000 1.03750
\(302\) 0 0
\(303\) 19.5959 1.12576
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −17.1464 −0.978598 −0.489299 0.872116i \(-0.662747\pi\)
−0.489299 + 0.872116i \(0.662747\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 19.5959 1.11118 0.555591 0.831456i \(-0.312492\pi\)
0.555591 + 0.831456i \(0.312492\pi\)
\(312\) 0 0
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.00000 0.449325 0.224662 0.974437i \(-0.427872\pi\)
0.224662 + 0.974437i \(0.427872\pi\)
\(318\) 0 0
\(319\) 39.1918 2.19432
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) 19.5959 1.09035
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −24.4949 −1.35457
\(328\) 0 0
\(329\) 30.0000 1.65395
\(330\) 0 0
\(331\) −4.89898 −0.269272 −0.134636 0.990895i \(-0.542987\pi\)
−0.134636 + 0.990895i \(0.542987\pi\)
\(332\) 0 0
\(333\) −12.0000 −0.657596
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000 0.871576 0.435788 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 0 0
\(339\) 39.1918 2.12861
\(340\) 0 0
\(341\) 48.0000 2.59935
\(342\) 0 0
\(343\) −19.5959 −1.05808
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.2474 0.657477 0.328739 0.944421i \(-0.393376\pi\)
0.328739 + 0.944421i \(0.393376\pi\)
\(348\) 0 0
\(349\) 24.0000 1.28469 0.642345 0.766415i \(-0.277962\pi\)
0.642345 + 0.766415i \(0.277962\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.00000 0.425797 0.212899 0.977074i \(-0.431710\pi\)
0.212899 + 0.977074i \(0.431710\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −24.0000 −1.27021
\(358\) 0 0
\(359\) −19.5959 −1.03423 −0.517116 0.855915i \(-0.672995\pi\)
−0.517116 + 0.855915i \(0.672995\pi\)
\(360\) 0 0
\(361\) 5.00000 0.263158
\(362\) 0 0
\(363\) 31.8434 1.67134
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −31.8434 −1.66221 −0.831105 0.556115i \(-0.812291\pi\)
−0.831105 + 0.556115i \(0.812291\pi\)
\(368\) 0 0
\(369\) −24.0000 −1.24939
\(370\) 0 0
\(371\) −19.5959 −1.01737
\(372\) 0 0
\(373\) 20.0000 1.03556 0.517780 0.855514i \(-0.326758\pi\)
0.517780 + 0.855514i \(0.326758\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 14.6969 0.754931 0.377466 0.926024i \(-0.376796\pi\)
0.377466 + 0.926024i \(0.376796\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 0 0
\(383\) −26.9444 −1.37679 −0.688397 0.725334i \(-0.741685\pi\)
−0.688397 + 0.725334i \(0.741685\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 22.0454 1.12063
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −9.79796 −0.495504
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 0 0
\(399\) −29.3939 −1.47153
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −19.5959 −0.971334
\(408\) 0 0
\(409\) −24.0000 −1.18672 −0.593362 0.804936i \(-0.702200\pi\)
−0.593362 + 0.804936i \(0.702200\pi\)
\(410\) 0 0
\(411\) −19.5959 −0.966595
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −14.6969 −0.717992 −0.358996 0.933339i \(-0.616881\pi\)
−0.358996 + 0.933339i \(0.616881\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 36.7423 1.78647
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 14.6969 0.711235
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 29.3939 1.41585 0.707927 0.706286i \(-0.249631\pi\)
0.707927 + 0.706286i \(0.249631\pi\)
\(432\) 0 0
\(433\) −36.0000 −1.73005 −0.865025 0.501729i \(-0.832697\pi\)
−0.865025 + 0.501729i \(0.832697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) −19.5959 −0.935262 −0.467631 0.883924i \(-0.654892\pi\)
−0.467631 + 0.883924i \(0.654892\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −26.9444 −1.28017 −0.640083 0.768306i \(-0.721100\pi\)
−0.640083 + 0.768306i \(0.721100\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4.89898 0.231714
\(448\) 0 0
\(449\) 32.0000 1.51017 0.755087 0.655625i \(-0.227595\pi\)
0.755087 + 0.655625i \(0.227595\pi\)
\(450\) 0 0
\(451\) −39.1918 −1.84547
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) 0 0
\(463\) −31.8434 −1.47989 −0.739943 0.672669i \(-0.765148\pi\)
−0.739943 + 0.672669i \(0.765148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.7423 1.70023 0.850117 0.526595i \(-0.176531\pi\)
0.850117 + 0.526595i \(0.176531\pi\)
\(468\) 0 0
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) −48.9898 −2.25733
\(472\) 0 0
\(473\) 36.0000 1.65528
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −24.0000 −1.09888
\(478\) 0 0
\(479\) 19.5959 0.895360 0.447680 0.894194i \(-0.352250\pi\)
0.447680 + 0.894194i \(0.352250\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 14.6969 0.668734
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −17.1464 −0.776979 −0.388489 0.921453i \(-0.627003\pi\)
−0.388489 + 0.921453i \(0.627003\pi\)
\(488\) 0 0
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) 14.6969 0.663264 0.331632 0.943409i \(-0.392401\pi\)
0.331632 + 0.943409i \(0.392401\pi\)
\(492\) 0 0
\(493\) −32.0000 −1.44121
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) 4.89898 0.219308 0.109654 0.993970i \(-0.465026\pi\)
0.109654 + 0.993970i \(0.465026\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) −17.1464 −0.764521 −0.382261 0.924055i \(-0.624854\pi\)
−0.382261 + 0.924055i \(0.624854\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −31.8434 −1.41421
\(508\) 0 0
\(509\) 8.00000 0.354594 0.177297 0.984157i \(-0.443265\pi\)
0.177297 + 0.984157i \(0.443265\pi\)
\(510\) 0 0
\(511\) −9.79796 −0.433436
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 60.0000 2.63880
\(518\) 0 0
\(519\) −9.79796 −0.430083
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 31.8434 1.39241 0.696207 0.717841i \(-0.254870\pi\)
0.696207 + 0.717841i \(0.254870\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −39.1918 −1.70722
\(528\) 0 0
\(529\) −17.0000 −0.739130
\(530\) 0 0
\(531\) 14.6969 0.637793
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −36.0000 −1.55351
\(538\) 0 0
\(539\) −4.89898 −0.211014
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 0 0
\(543\) 19.5959 0.840941
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −22.0454 −0.942594 −0.471297 0.881975i \(-0.656214\pi\)
−0.471297 + 0.881975i \(0.656214\pi\)
\(548\) 0 0
\(549\) 18.0000 0.768221
\(550\) 0 0
\(551\) −39.1918 −1.66963
\(552\) 0 0
\(553\) 24.0000 1.02058
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.00000 −0.169485 −0.0847427 0.996403i \(-0.527007\pi\)
−0.0847427 + 0.996403i \(0.527007\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −48.0000 −2.02656
\(562\) 0 0
\(563\) 2.44949 0.103234 0.0516168 0.998667i \(-0.483563\pi\)
0.0516168 + 0.998667i \(0.483563\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −22.0454 −0.925820
\(568\) 0 0
\(569\) −8.00000 −0.335377 −0.167689 0.985840i \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) 0 0
\(571\) 44.0908 1.84514 0.922572 0.385826i \(-0.126083\pi\)
0.922572 + 0.385826i \(0.126083\pi\)
\(572\) 0 0
\(573\) −48.0000 −2.00523
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −24.0000 −0.999133 −0.499567 0.866276i \(-0.666507\pi\)
−0.499567 + 0.866276i \(0.666507\pi\)
\(578\) 0 0
\(579\) −29.3939 −1.22157
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) −39.1918 −1.62316
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.2474 0.505506 0.252753 0.967531i \(-0.418664\pi\)
0.252753 + 0.967531i \(0.418664\pi\)
\(588\) 0 0
\(589\) −48.0000 −1.97781
\(590\) 0 0
\(591\) −19.5959 −0.806068
\(592\) 0 0
\(593\) 8.00000 0.328521 0.164260 0.986417i \(-0.447476\pi\)
0.164260 + 0.986417i \(0.447476\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) −9.79796 −0.400334 −0.200167 0.979762i \(-0.564148\pi\)
−0.200167 + 0.979762i \(0.564148\pi\)
\(600\) 0 0
\(601\) −16.0000 −0.652654 −0.326327 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(602\) 0 0
\(603\) 7.34847 0.299253
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −7.34847 −0.298265 −0.149133 0.988817i \(-0.547648\pi\)
−0.149133 + 0.988817i \(0.547648\pi\)
\(608\) 0 0
\(609\) 48.0000 1.94506
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28.0000 −1.12724 −0.563619 0.826035i \(-0.690591\pi\)
−0.563619 + 0.826035i \(0.690591\pi\)
\(618\) 0 0
\(619\) 24.4949 0.984533 0.492267 0.870445i \(-0.336169\pi\)
0.492267 + 0.870445i \(0.336169\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.89898 0.196273
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −58.7878 −2.34776
\(628\) 0 0
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 60.0000 2.38479
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −29.3939 −1.16280
\(640\) 0 0
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) 0 0
\(643\) −2.44949 −0.0965984 −0.0482992 0.998833i \(-0.515380\pi\)
−0.0482992 + 0.998833i \(0.515380\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.0454 0.866694 0.433347 0.901227i \(-0.357332\pi\)
0.433347 + 0.901227i \(0.357332\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 58.7878 2.30407
\(652\) 0 0
\(653\) 16.0000 0.626128 0.313064 0.949732i \(-0.398644\pi\)
0.313064 + 0.949732i \(0.398644\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) −14.6969 −0.572511 −0.286256 0.958153i \(-0.592411\pi\)
−0.286256 + 0.958153i \(0.592411\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 19.5959 0.758757
\(668\) 0 0
\(669\) −18.0000 −0.695920
\(670\) 0 0
\(671\) 29.3939 1.13474
\(672\) 0 0
\(673\) −36.0000 −1.38770 −0.693849 0.720121i \(-0.744086\pi\)
−0.693849 + 0.720121i \(0.744086\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −32.0000 −1.22986 −0.614930 0.788582i \(-0.710816\pi\)
−0.614930 + 0.788582i \(0.710816\pi\)
\(678\) 0 0
\(679\) 9.79796 0.376011
\(680\) 0 0
\(681\) −6.00000 −0.229920
\(682\) 0 0
\(683\) 7.34847 0.281181 0.140591 0.990068i \(-0.455100\pi\)
0.140591 + 0.990068i \(0.455100\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 19.5959 0.747631
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −14.6969 −0.559098 −0.279549 0.960131i \(-0.590185\pi\)
−0.279549 + 0.960131i \(0.590185\pi\)
\(692\) 0 0
\(693\) 36.0000 1.36753
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 32.0000 1.21209
\(698\) 0 0
\(699\) −48.9898 −1.85296
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 19.5959 0.739074
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.5959 0.736980
\(708\) 0 0
\(709\) −40.0000 −1.50223 −0.751116 0.660171i \(-0.770484\pi\)
−0.751116 + 0.660171i \(0.770484\pi\)
\(710\) 0 0
\(711\) 29.3939 1.10236
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) 48.9898 1.82701 0.913506 0.406826i \(-0.133365\pi\)
0.913506 + 0.406826i \(0.133365\pi\)
\(720\) 0 0
\(721\) 6.00000 0.223452
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −36.7423 −1.36270 −0.681349 0.731959i \(-0.738606\pi\)
−0.681349 + 0.731959i \(0.738606\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −29.3939 −1.08717
\(732\) 0 0
\(733\) −12.0000 −0.443230 −0.221615 0.975134i \(-0.571133\pi\)
−0.221615 + 0.975134i \(0.571133\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −14.6969 −0.540636 −0.270318 0.962771i \(-0.587129\pi\)
−0.270318 + 0.962771i \(0.587129\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.1464 0.629041 0.314521 0.949251i \(-0.398156\pi\)
0.314521 + 0.949251i \(0.398156\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −7.34847 −0.268866
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −12.0000 −0.436147 −0.218074 0.975932i \(-0.569977\pi\)
−0.218074 + 0.975932i \(0.569977\pi\)
\(758\) 0 0
\(759\) 29.3939 1.06693
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) −24.4949 −0.886775
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) 19.5959 0.705730
\(772\) 0 0
\(773\) −8.00000 −0.287740 −0.143870 0.989597i \(-0.545955\pi\)
−0.143870 + 0.989597i \(0.545955\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −24.0000 −0.860995
\(778\) 0 0
\(779\) 39.1918 1.40419
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 41.6413 1.48435 0.742176 0.670205i \(-0.233794\pi\)
0.742176 + 0.670205i \(0.233794\pi\)
\(788\) 0 0
\(789\) −54.0000 −1.92245
\(790\) 0 0
\(791\) 39.1918 1.39350
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) −48.9898 −1.73313
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) −19.5959 −0.691525
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 63.6867 2.24188
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) −53.8888 −1.89229 −0.946145 0.323742i \(-0.895059\pi\)
−0.946145 + 0.323742i \(0.895059\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −14.0000 −0.488603 −0.244302 0.969699i \(-0.578559\pi\)
−0.244302 + 0.969699i \(0.578559\pi\)
\(822\) 0 0
\(823\) 22.0454 0.768455 0.384227 0.923239i \(-0.374468\pi\)
0.384227 + 0.923239i \(0.374468\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −46.5403 −1.61836 −0.809182 0.587557i \(-0.800090\pi\)
−0.809182 + 0.587557i \(0.800090\pi\)
\(828\) 0 0
\(829\) −22.0000 −0.764092 −0.382046 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) 0 0
\(831\) 29.3939 1.01966
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 39.1918 1.35305 0.676526 0.736419i \(-0.263485\pi\)
0.676526 + 0.736419i \(0.263485\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −39.1918 −1.34984
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 31.8434 1.09415
\(848\) 0 0
\(849\) −66.0000 −2.26511
\(850\) 0 0
\(851\) −9.79796 −0.335870
\(852\) 0 0
\(853\) 32.0000 1.09566 0.547830 0.836590i \(-0.315454\pi\)
0.547830 + 0.836590i \(0.315454\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −32.0000 −1.09310 −0.546550 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) 0 0
\(859\) 24.4949 0.835755 0.417878 0.908503i \(-0.362774\pi\)
0.417878 + 0.908503i \(0.362774\pi\)
\(860\) 0 0
\(861\) −48.0000 −1.63584
\(862\) 0 0
\(863\) −51.4393 −1.75101 −0.875507 0.483206i \(-0.839472\pi\)
−0.875507 + 0.483206i \(0.839472\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.44949 −0.0831890
\(868\) 0 0
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −28.0000 −0.945493 −0.472746 0.881199i \(-0.656737\pi\)
−0.472746 + 0.881199i \(0.656737\pi\)
\(878\) 0 0
\(879\) 48.9898 1.65238
\(880\) 0 0
\(881\) −8.00000 −0.269527 −0.134763 0.990878i \(-0.543027\pi\)
−0.134763 + 0.990878i \(0.543027\pi\)
\(882\) 0 0
\(883\) −36.7423 −1.23648 −0.618239 0.785990i \(-0.712154\pi\)
−0.618239 + 0.785990i \(0.712154\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.44949 0.0822458 0.0411229 0.999154i \(-0.486906\pi\)
0.0411229 + 0.999154i \(0.486906\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) −44.0908 −1.47710
\(892\) 0 0
\(893\) −60.0000 −2.00782
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 78.3837 2.61424
\(900\) 0 0
\(901\) 32.0000 1.06607
\(902\) 0 0
\(903\) 44.0908 1.46725
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −12.2474 −0.406670 −0.203335 0.979109i \(-0.565178\pi\)
−0.203335 + 0.979109i \(0.565178\pi\)
\(908\) 0 0
\(909\) 24.0000 0.796030
\(910\) 0 0
\(911\) −29.3939 −0.973863 −0.486931 0.873440i \(-0.661884\pi\)
−0.486931 + 0.873440i \(0.661884\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −48.9898 −1.61602 −0.808012 0.589166i \(-0.799456\pi\)
−0.808012 + 0.589166i \(0.799456\pi\)
\(920\) 0 0
\(921\) −42.0000 −1.38395
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 7.34847 0.241355
\(928\) 0 0
\(929\) 16.0000 0.524943 0.262471 0.964940i \(-0.415462\pi\)
0.262471 + 0.964940i \(0.415462\pi\)
\(930\) 0 0
\(931\) 4.89898 0.160558
\(932\) 0 0
\(933\) 48.0000 1.57145
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 60.0000 1.96011 0.980057 0.198715i \(-0.0636769\pi\)
0.980057 + 0.198715i \(0.0636769\pi\)
\(938\) 0 0
\(939\) −58.7878 −1.91847
\(940\) 0 0
\(941\) 8.00000 0.260793 0.130396 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) 0 0
\(943\) −19.5959 −0.638131
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −17.1464 −0.557184 −0.278592 0.960410i \(-0.589868\pi\)
−0.278592 + 0.960410i \(0.589868\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 19.5959 0.635441
\(952\) 0 0
\(953\) −32.0000 −1.03658 −0.518291 0.855204i \(-0.673432\pi\)
−0.518291 + 0.855204i \(0.673432\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 96.0000 3.10324
\(958\) 0 0
\(959\) −19.5959 −0.632785
\(960\) 0 0
\(961\) 65.0000 2.09677
\(962\) 0 0
\(963\) 22.0454 0.710403
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 17.1464 0.551392 0.275696 0.961245i \(-0.411092\pi\)
0.275696 + 0.961245i \(0.411092\pi\)
\(968\) 0 0
\(969\) 48.0000 1.54198
\(970\) 0 0
\(971\) −24.4949 −0.786079 −0.393039 0.919522i \(-0.628576\pi\)
−0.393039 + 0.919522i \(0.628576\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.00000 0.127971 0.0639857 0.997951i \(-0.479619\pi\)
0.0639857 + 0.997951i \(0.479619\pi\)
\(978\) 0 0
\(979\) 9.79796 0.313144
\(980\) 0 0
\(981\) −30.0000 −0.957826
\(982\) 0 0
\(983\) −22.0454 −0.703139 −0.351570 0.936162i \(-0.614352\pi\)
−0.351570 + 0.936162i \(0.614352\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 73.4847 2.33904
\(988\) 0 0
\(989\) 18.0000 0.572367
\(990\) 0 0
\(991\) −39.1918 −1.24497 −0.622485 0.782632i \(-0.713877\pi\)
−0.622485 + 0.782632i \(0.713877\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 56.0000 1.77354 0.886769 0.462213i \(-0.152944\pi\)
0.886769 + 0.462213i \(0.152944\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6400.2.a.bv.1.2 2
4.3 odd 2 inner 6400.2.a.bv.1.1 2
5.2 odd 4 1280.2.c.m.769.1 4
5.3 odd 4 1280.2.c.m.769.4 4
5.4 even 2 6400.2.a.bx.1.1 2
8.3 odd 2 6400.2.a.bu.1.2 2
8.5 even 2 6400.2.a.bu.1.1 2
16.3 odd 4 3200.2.d.k.1601.2 4
16.5 even 4 3200.2.d.k.1601.1 4
16.11 odd 4 3200.2.d.k.1601.4 4
16.13 even 4 3200.2.d.k.1601.3 4
20.3 even 4 1280.2.c.m.769.2 4
20.7 even 4 1280.2.c.m.769.3 4
20.19 odd 2 6400.2.a.bx.1.2 2
40.3 even 4 1280.2.c.e.769.3 4
40.13 odd 4 1280.2.c.e.769.1 4
40.19 odd 2 6400.2.a.bw.1.1 2
40.27 even 4 1280.2.c.e.769.2 4
40.29 even 2 6400.2.a.bw.1.2 2
40.37 odd 4 1280.2.c.e.769.4 4
80.3 even 4 640.2.f.g.449.3 yes 4
80.13 odd 4 640.2.f.g.449.1 yes 4
80.19 odd 4 3200.2.d.l.1601.3 4
80.27 even 4 640.2.f.g.449.4 yes 4
80.29 even 4 3200.2.d.l.1601.2 4
80.37 odd 4 640.2.f.g.449.2 yes 4
80.43 even 4 640.2.f.c.449.2 yes 4
80.53 odd 4 640.2.f.c.449.4 yes 4
80.59 odd 4 3200.2.d.l.1601.1 4
80.67 even 4 640.2.f.c.449.1 4
80.69 even 4 3200.2.d.l.1601.4 4
80.77 odd 4 640.2.f.c.449.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.f.c.449.1 4 80.67 even 4
640.2.f.c.449.2 yes 4 80.43 even 4
640.2.f.c.449.3 yes 4 80.77 odd 4
640.2.f.c.449.4 yes 4 80.53 odd 4
640.2.f.g.449.1 yes 4 80.13 odd 4
640.2.f.g.449.2 yes 4 80.37 odd 4
640.2.f.g.449.3 yes 4 80.3 even 4
640.2.f.g.449.4 yes 4 80.27 even 4
1280.2.c.e.769.1 4 40.13 odd 4
1280.2.c.e.769.2 4 40.27 even 4
1280.2.c.e.769.3 4 40.3 even 4
1280.2.c.e.769.4 4 40.37 odd 4
1280.2.c.m.769.1 4 5.2 odd 4
1280.2.c.m.769.2 4 20.3 even 4
1280.2.c.m.769.3 4 20.7 even 4
1280.2.c.m.769.4 4 5.3 odd 4
3200.2.d.k.1601.1 4 16.5 even 4
3200.2.d.k.1601.2 4 16.3 odd 4
3200.2.d.k.1601.3 4 16.13 even 4
3200.2.d.k.1601.4 4 16.11 odd 4
3200.2.d.l.1601.1 4 80.59 odd 4
3200.2.d.l.1601.2 4 80.29 even 4
3200.2.d.l.1601.3 4 80.19 odd 4
3200.2.d.l.1601.4 4 80.69 even 4
6400.2.a.bu.1.1 2 8.5 even 2
6400.2.a.bu.1.2 2 8.3 odd 2
6400.2.a.bv.1.1 2 4.3 odd 2 inner
6400.2.a.bv.1.2 2 1.1 even 1 trivial
6400.2.a.bw.1.1 2 40.19 odd 2
6400.2.a.bw.1.2 2 40.29 even 2
6400.2.a.bx.1.1 2 5.4 even 2
6400.2.a.bx.1.2 2 20.19 odd 2