Properties

Label 6400.2.a.bt.1.1
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: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3200)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607 q^{3} +2.00000 q^{9} +O(q^{10})\) \(q-2.23607 q^{3} +2.00000 q^{9} +2.23607 q^{11} +4.00000 q^{13} +3.00000 q^{17} +2.23607 q^{19} +8.94427 q^{23} +2.23607 q^{27} +4.00000 q^{29} -8.94427 q^{31} -5.00000 q^{33} +8.00000 q^{37} -8.94427 q^{39} +5.00000 q^{41} +8.94427 q^{43} -8.94427 q^{47} -7.00000 q^{49} -6.70820 q^{51} +4.00000 q^{53} -5.00000 q^{57} -8.94427 q^{59} +8.00000 q^{61} -6.70820 q^{67} -20.0000 q^{69} -8.94427 q^{71} +9.00000 q^{73} -11.0000 q^{81} -6.70820 q^{83} -8.94427 q^{87} +15.0000 q^{89} +20.0000 q^{93} +2.00000 q^{97} +4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{9} + 8q^{13} + 6q^{17} + 8q^{29} - 10q^{33} + 16q^{37} + 10q^{41} - 14q^{49} + 8q^{53} - 10q^{57} + 16q^{61} - 40q^{69} + 18q^{73} - 22q^{81} + 30q^{89} + 40q^{93} + 4q^{97} + O(q^{100}) \)

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
\(3\) −2.23607 −1.29099 −0.645497 0.763763i \(-0.723350\pi\)
−0.645497 + 0.763763i \(0.723350\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) 2.23607 0.674200 0.337100 0.941469i \(-0.390554\pi\)
0.337100 + 0.941469i \(0.390554\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 2.23607 0.512989 0.256495 0.966546i \(-0.417432\pi\)
0.256495 + 0.966546i \(0.417432\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.94427 1.86501 0.932505 0.361158i \(-0.117618\pi\)
0.932505 + 0.361158i \(0.117618\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.23607 0.430331
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −8.94427 −1.60644 −0.803219 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) −8.94427 −1.43223
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 8.94427 1.36399 0.681994 0.731357i \(-0.261113\pi\)
0.681994 + 0.731357i \(0.261113\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.94427 −1.30466 −0.652328 0.757937i \(-0.726208\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −6.70820 −0.939336
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) 0 0
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.70820 −0.819538 −0.409769 0.912189i \(-0.634391\pi\)
−0.409769 + 0.912189i \(0.634391\pi\)
\(68\) 0 0
\(69\) −20.0000 −2.40772
\(70\) 0 0
\(71\) −8.94427 −1.06149 −0.530745 0.847532i \(-0.678088\pi\)
−0.530745 + 0.847532i \(0.678088\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −6.70820 −0.736321 −0.368161 0.929762i \(-0.620012\pi\)
−0.368161 + 0.929762i \(0.620012\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.94427 −0.958927
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 20.0000 2.07390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 4.47214 0.449467
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 8.94427 0.881305 0.440653 0.897678i \(-0.354747\pi\)
0.440653 + 0.897678i \(0.354747\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.23607 −0.216169 −0.108084 0.994142i \(-0.534472\pi\)
−0.108084 + 0.994142i \(0.534472\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −17.8885 −1.69791
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.00000 0.739600
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) 0 0
\(123\) −11.1803 −1.00810
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.94427 −0.793676 −0.396838 0.917889i \(-0.629892\pi\)
−0.396838 + 0.917889i \(0.629892\pi\)
\(128\) 0 0
\(129\) −20.0000 −1.76090
\(130\) 0 0
\(131\) 8.94427 0.781465 0.390732 0.920504i \(-0.372222\pi\)
0.390732 + 0.920504i \(0.372222\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.0000 −1.11066 −0.555332 0.831628i \(-0.687409\pi\)
−0.555332 + 0.831628i \(0.687409\pi\)
\(138\) 0 0
\(139\) 20.1246 1.70695 0.853474 0.521136i \(-0.174492\pi\)
0.853474 + 0.521136i \(0.174492\pi\)
\(140\) 0 0
\(141\) 20.0000 1.68430
\(142\) 0 0
\(143\) 8.94427 0.747958
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 15.6525 1.29099
\(148\) 0 0
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) −17.8885 −1.45575 −0.727875 0.685710i \(-0.759492\pi\)
−0.727875 + 0.685710i \(0.759492\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.0000 −0.957704 −0.478852 0.877896i \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 0 0
\(159\) −8.94427 −0.709327
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 11.1803 0.875712 0.437856 0.899045i \(-0.355738\pi\)
0.437856 + 0.899045i \(0.355738\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.94427 −0.692129 −0.346064 0.938211i \(-0.612482\pi\)
−0.346064 + 0.938211i \(0.612482\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 4.47214 0.341993
\(172\) 0 0
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 20.0000 1.50329
\(178\) 0 0
\(179\) −15.6525 −1.16992 −0.584960 0.811062i \(-0.698890\pi\)
−0.584960 + 0.811062i \(0.698890\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) −17.8885 −1.32236
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.70820 0.490552
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.94427 0.647185 0.323592 0.946197i \(-0.395109\pi\)
0.323592 + 0.946197i \(0.395109\pi\)
\(192\) 0 0
\(193\) −21.0000 −1.51161 −0.755807 0.654795i \(-0.772755\pi\)
−0.755807 + 0.654795i \(0.772755\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 17.8885 1.26809 0.634043 0.773298i \(-0.281394\pi\)
0.634043 + 0.773298i \(0.281394\pi\)
\(200\) 0 0
\(201\) 15.0000 1.05802
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 17.8885 1.24334
\(208\) 0 0
\(209\) 5.00000 0.345857
\(210\) 0 0
\(211\) 6.70820 0.461812 0.230906 0.972976i \(-0.425831\pi\)
0.230906 + 0.972976i \(0.425831\pi\)
\(212\) 0 0
\(213\) 20.0000 1.37038
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −20.1246 −1.35990
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 17.8885 1.19791 0.598953 0.800784i \(-0.295584\pi\)
0.598953 + 0.800784i \(0.295584\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.94427 −0.593652 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.94427 −0.578557 −0.289278 0.957245i \(-0.593415\pi\)
−0.289278 + 0.957245i \(0.593415\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) 17.8885 1.14755
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.94427 0.569110
\(248\) 0 0
\(249\) 15.0000 0.950586
\(250\) 0 0
\(251\) −29.0689 −1.83481 −0.917406 0.397953i \(-0.869721\pi\)
−0.917406 + 0.397953i \(0.869721\pi\)
\(252\) 0 0
\(253\) 20.0000 1.25739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 0 0
\(263\) −8.94427 −0.551527 −0.275764 0.961225i \(-0.588931\pi\)
−0.275764 + 0.961225i \(0.588931\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −33.5410 −2.05268
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 26.8328 1.62998 0.814989 0.579477i \(-0.196743\pi\)
0.814989 + 0.579477i \(0.196743\pi\)
\(272\) 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
\(279\) −17.8885 −1.07096
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −24.5967 −1.46212 −0.731062 0.682311i \(-0.760975\pi\)
−0.731062 + 0.682311i \(0.760975\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −4.47214 −0.262161
\(292\) 0 0
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 35.7771 2.06904
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.23607 −0.127619 −0.0638096 0.997962i \(-0.520325\pi\)
−0.0638096 + 0.997962i \(0.520325\pi\)
\(308\) 0 0
\(309\) −20.0000 −1.13776
\(310\) 0 0
\(311\) 17.8885 1.01437 0.507183 0.861838i \(-0.330687\pi\)
0.507183 + 0.861838i \(0.330687\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 32.0000 1.79730 0.898650 0.438667i \(-0.144549\pi\)
0.898650 + 0.438667i \(0.144549\pi\)
\(318\) 0 0
\(319\) 8.94427 0.500783
\(320\) 0 0
\(321\) 5.00000 0.279073
\(322\) 0 0
\(323\) 6.70820 0.373254
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.23607 0.122905 0.0614527 0.998110i \(-0.480427\pi\)
0.0614527 + 0.998110i \(0.480427\pi\)
\(332\) 0 0
\(333\) 16.0000 0.876795
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 33.0000 1.79762 0.898812 0.438334i \(-0.144431\pi\)
0.898812 + 0.438334i \(0.144431\pi\)
\(338\) 0 0
\(339\) −2.23607 −0.121447
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 33.5410 1.80058 0.900288 0.435294i \(-0.143356\pi\)
0.900288 + 0.435294i \(0.143356\pi\)
\(348\) 0 0
\(349\) −24.0000 −1.28469 −0.642345 0.766415i \(-0.722038\pi\)
−0.642345 + 0.766415i \(0.722038\pi\)
\(350\) 0 0
\(351\) 8.94427 0.477410
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 17.8885 0.944121 0.472061 0.881566i \(-0.343510\pi\)
0.472061 + 0.881566i \(0.343510\pi\)
\(360\) 0 0
\(361\) −14.0000 −0.736842
\(362\) 0 0
\(363\) 13.4164 0.704179
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 16.0000 0.824042
\(378\) 0 0
\(379\) −29.0689 −1.49317 −0.746584 0.665291i \(-0.768307\pi\)
−0.746584 + 0.665291i \(0.768307\pi\)
\(380\) 0 0
\(381\) 20.0000 1.02463
\(382\) 0 0
\(383\) −17.8885 −0.914062 −0.457031 0.889451i \(-0.651087\pi\)
−0.457031 + 0.889451i \(0.651087\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 17.8885 0.909326
\(388\) 0 0
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) 26.8328 1.35699
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) −35.7771 −1.78218
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 17.8885 0.886702
\(408\) 0 0
\(409\) −19.0000 −0.939490 −0.469745 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 0 0
\(411\) 29.0689 1.43386
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −45.0000 −2.20366
\(418\) 0 0
\(419\) 20.1246 0.983152 0.491576 0.870835i \(-0.336421\pi\)
0.491576 + 0.870835i \(0.336421\pi\)
\(420\) 0 0
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) 0 0
\(423\) −17.8885 −0.869771
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −20.0000 −0.965609
\(430\) 0 0
\(431\) 17.8885 0.861661 0.430830 0.902433i \(-0.358221\pi\)
0.430830 + 0.902433i \(0.358221\pi\)
\(432\) 0 0
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.0000 0.956730
\(438\) 0 0
\(439\) 26.8328 1.28066 0.640330 0.768100i \(-0.278798\pi\)
0.640330 + 0.768100i \(0.278798\pi\)
\(440\) 0 0
\(441\) −14.0000 −0.666667
\(442\) 0 0
\(443\) 11.1803 0.531194 0.265597 0.964084i \(-0.414431\pi\)
0.265597 + 0.964084i \(0.414431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 44.7214 2.11525
\(448\) 0 0
\(449\) −11.0000 −0.519122 −0.259561 0.965727i \(-0.583578\pi\)
−0.259561 + 0.965727i \(0.583578\pi\)
\(450\) 0 0
\(451\) 11.1803 0.526462
\(452\) 0 0
\(453\) 40.0000 1.87936
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −37.0000 −1.73079 −0.865393 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 0 0
\(459\) 6.70820 0.313112
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −35.7771 −1.66270 −0.831351 0.555748i \(-0.812432\pi\)
−0.831351 + 0.555748i \(0.812432\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.94427 −0.413892 −0.206946 0.978352i \(-0.566352\pi\)
−0.206946 + 0.978352i \(0.566352\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 26.8328 1.23639
\(472\) 0 0
\(473\) 20.0000 0.919601
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.00000 0.366295
\(478\) 0 0
\(479\) 26.8328 1.22602 0.613011 0.790074i \(-0.289958\pi\)
0.613011 + 0.790074i \(0.289958\pi\)
\(480\) 0 0
\(481\) 32.0000 1.45907
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 26.8328 1.21591 0.607955 0.793971i \(-0.291990\pi\)
0.607955 + 0.793971i \(0.291990\pi\)
\(488\) 0 0
\(489\) −25.0000 −1.13054
\(490\) 0 0
\(491\) 26.8328 1.21095 0.605474 0.795865i \(-0.292984\pi\)
0.605474 + 0.795865i \(0.292984\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −26.8328 −1.20120 −0.600601 0.799549i \(-0.705072\pi\)
−0.600601 + 0.799549i \(0.705072\pi\)
\(500\) 0 0
\(501\) 20.0000 0.893534
\(502\) 0 0
\(503\) 17.8885 0.797611 0.398805 0.917036i \(-0.369425\pi\)
0.398805 + 0.917036i \(0.369425\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.70820 −0.297922
\(508\) 0 0
\(509\) −4.00000 −0.177297 −0.0886484 0.996063i \(-0.528255\pi\)
−0.0886484 + 0.996063i \(0.528255\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −20.0000 −0.879599
\(518\) 0 0
\(519\) −53.6656 −2.35566
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 33.5410 1.46665 0.733323 0.679880i \(-0.237968\pi\)
0.733323 + 0.679880i \(0.237968\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −26.8328 −1.16886
\(528\) 0 0
\(529\) 57.0000 2.47826
\(530\) 0 0
\(531\) −17.8885 −0.776297
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 35.0000 1.51036
\(538\) 0 0
\(539\) −15.6525 −0.674200
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 0 0
\(543\) −44.7214 −1.91918
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −20.1246 −0.860466 −0.430233 0.902718i \(-0.641569\pi\)
−0.430233 + 0.902718i \(0.641569\pi\)
\(548\) 0 0
\(549\) 16.0000 0.682863
\(550\) 0 0
\(551\) 8.94427 0.381039
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) 35.7771 1.51321
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) −8.94427 −0.376956 −0.188478 0.982077i \(-0.560355\pi\)
−0.188478 + 0.982077i \(0.560355\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11.0000 −0.461144 −0.230572 0.973055i \(-0.574060\pi\)
−0.230572 + 0.973055i \(0.574060\pi\)
\(570\) 0 0
\(571\) −44.7214 −1.87153 −0.935765 0.352623i \(-0.885290\pi\)
−0.935765 + 0.352623i \(0.885290\pi\)
\(572\) 0 0
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −23.0000 −0.957503 −0.478751 0.877951i \(-0.658910\pi\)
−0.478751 + 0.877951i \(0.658910\pi\)
\(578\) 0 0
\(579\) 46.9574 1.95148
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 8.94427 0.370434
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.70820 −0.276877 −0.138439 0.990371i \(-0.544208\pi\)
−0.138439 + 0.990371i \(0.544208\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −26.8328 −1.10375
\(592\) 0 0
\(593\) 9.00000 0.369586 0.184793 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −40.0000 −1.63709
\(598\) 0 0
\(599\) 35.7771 1.46181 0.730906 0.682478i \(-0.239098\pi\)
0.730906 + 0.682478i \(0.239098\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) 0 0
\(603\) −13.4164 −0.546358
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −17.8885 −0.726074 −0.363037 0.931775i \(-0.618260\pi\)
−0.363037 + 0.931775i \(0.618260\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −35.7771 −1.44739
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) −8.94427 −0.359501 −0.179750 0.983712i \(-0.557529\pi\)
−0.179750 + 0.983712i \(0.557529\pi\)
\(620\) 0 0
\(621\) 20.0000 0.802572
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −11.1803 −0.446500
\(628\) 0 0
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 8.94427 0.356066 0.178033 0.984025i \(-0.443027\pi\)
0.178033 + 0.984025i \(0.443027\pi\)
\(632\) 0 0
\(633\) −15.0000 −0.596196
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −28.0000 −1.10940
\(638\) 0 0
\(639\) −17.8885 −0.707660
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −8.94427 −0.352728 −0.176364 0.984325i \(-0.556434\pi\)
−0.176364 + 0.984325i \(0.556434\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.8885 0.703271 0.351636 0.936137i \(-0.385626\pi\)
0.351636 + 0.936137i \(0.385626\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 18.0000 0.702247
\(658\) 0 0
\(659\) 6.70820 0.261315 0.130657 0.991428i \(-0.458291\pi\)
0.130657 + 0.991428i \(0.458291\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) 0 0
\(663\) −26.8328 −1.04210
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 35.7771 1.38529
\(668\) 0 0
\(669\) −40.0000 −1.54649
\(670\) 0 0
\(671\) 17.8885 0.690580
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.00000 −0.307465 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) 15.6525 0.598925 0.299463 0.954108i \(-0.403193\pi\)
0.299463 + 0.954108i \(0.403193\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 35.7771 1.36498
\(688\) 0 0
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) −15.6525 −0.595448 −0.297724 0.954652i \(-0.596228\pi\)
−0.297724 + 0.954652i \(0.596228\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 15.0000 0.568166
\(698\) 0 0
\(699\) −13.4164 −0.507455
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 17.8885 0.674679
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −80.0000 −2.99602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.0000 0.746914
\(718\) 0 0
\(719\) 17.8885 0.667130 0.333565 0.942727i \(-0.391748\pi\)
0.333565 + 0.942727i \(0.391748\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −11.1803 −0.415801
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −26.8328 −0.995174 −0.497587 0.867414i \(-0.665780\pi\)
−0.497587 + 0.867414i \(0.665780\pi\)
\(728\) 0 0
\(729\) −7.00000 −0.259259
\(730\) 0 0
\(731\) 26.8328 0.992448
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −15.0000 −0.552532
\(738\) 0 0
\(739\) −26.8328 −0.987061 −0.493531 0.869728i \(-0.664294\pi\)
−0.493531 + 0.869728i \(0.664294\pi\)
\(740\) 0 0
\(741\) −20.0000 −0.734718
\(742\) 0 0
\(743\) 8.94427 0.328134 0.164067 0.986449i \(-0.447539\pi\)
0.164067 + 0.986449i \(0.447539\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −13.4164 −0.490881
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 26.8328 0.979143 0.489572 0.871963i \(-0.337153\pi\)
0.489572 + 0.871963i \(0.337153\pi\)
\(752\) 0 0
\(753\) 65.0000 2.36873
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 0 0
\(759\) −44.7214 −1.62328
\(760\) 0 0
\(761\) 13.0000 0.471250 0.235625 0.971844i \(-0.424286\pi\)
0.235625 + 0.971844i \(0.424286\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −35.7771 −1.29184
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 40.2492 1.44954
\(772\) 0 0
\(773\) −4.00000 −0.143870 −0.0719350 0.997409i \(-0.522917\pi\)
−0.0719350 + 0.997409i \(0.522917\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 11.1803 0.400577
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) 0 0
\(783\) 8.94427 0.319642
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 26.8328 0.956487 0.478243 0.878227i \(-0.341274\pi\)
0.478243 + 0.878227i \(0.341274\pi\)
\(788\) 0 0
\(789\) 20.0000 0.712019
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 32.0000 1.13635
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 52.0000 1.84193 0.920967 0.389640i \(-0.127401\pi\)
0.920967 + 0.389640i \(0.127401\pi\)
\(798\) 0 0
\(799\) −26.8328 −0.949277
\(800\) 0 0
\(801\) 30.0000 1.06000
\(802\) 0 0
\(803\) 20.1246 0.710182
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 26.8328 0.942228 0.471114 0.882072i \(-0.343852\pi\)
0.471114 + 0.882072i \(0.343852\pi\)
\(812\) 0 0
\(813\) −60.0000 −2.10429
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 20.0000 0.699711
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −28.0000 −0.977207 −0.488603 0.872506i \(-0.662493\pi\)
−0.488603 + 0.872506i \(0.662493\pi\)
\(822\) 0 0
\(823\) 35.7771 1.24711 0.623555 0.781779i \(-0.285688\pi\)
0.623555 + 0.781779i \(0.285688\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −24.5967 −0.855313 −0.427656 0.903941i \(-0.640661\pi\)
−0.427656 + 0.903941i \(0.640661\pi\)
\(828\) 0 0
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) 0 0
\(831\) −26.8328 −0.930820
\(832\) 0 0
\(833\) −21.0000 −0.727607
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −20.0000 −0.691301
\(838\) 0 0
\(839\) 35.7771 1.23516 0.617581 0.786507i \(-0.288113\pi\)
0.617581 + 0.786507i \(0.288113\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −22.3607 −0.770143
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 55.0000 1.88760
\(850\) 0 0
\(851\) 71.5542 2.45285
\(852\) 0 0
\(853\) −4.00000 −0.136957 −0.0684787 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.0000 0.922302 0.461151 0.887322i \(-0.347437\pi\)
0.461151 + 0.887322i \(0.347437\pi\)
\(858\) 0 0
\(859\) 38.0132 1.29699 0.648496 0.761218i \(-0.275398\pi\)
0.648496 + 0.761218i \(0.275398\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 26.8328 0.913400 0.456700 0.889621i \(-0.349031\pi\)
0.456700 + 0.889621i \(0.349031\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 17.8885 0.607527
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −26.8328 −0.909195
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 0 0
\(879\) −53.6656 −1.81010
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 33.5410 1.12875 0.564373 0.825520i \(-0.309118\pi\)
0.564373 + 0.825520i \(0.309118\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −17.8885 −0.600639 −0.300319 0.953839i \(-0.597093\pi\)
−0.300319 + 0.953839i \(0.597093\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −24.5967 −0.824022
\(892\) 0 0
\(893\) −20.0000 −0.669274
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −80.0000 −2.67112
\(898\) 0 0
\(899\) −35.7771 −1.19323
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 8.94427 0.296990 0.148495 0.988913i \(-0.452557\pi\)
0.148495 + 0.988913i \(0.452557\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −35.7771 −1.18535 −0.592674 0.805443i \(-0.701928\pi\)
−0.592674 + 0.805443i \(0.701928\pi\)
\(912\) 0 0
\(913\) −15.0000 −0.496428
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −8.94427 −0.295044 −0.147522 0.989059i \(-0.547130\pi\)
−0.147522 + 0.989059i \(0.547130\pi\)
\(920\) 0 0
\(921\) 5.00000 0.164756
\(922\) 0 0
\(923\) −35.7771 −1.17762
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 17.8885 0.587537
\(928\) 0 0
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −15.6525 −0.512989
\(932\) 0 0
\(933\) −40.0000 −1.30954
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −23.0000 −0.751377 −0.375689 0.926746i \(-0.622594\pi\)
−0.375689 + 0.926746i \(0.622594\pi\)
\(938\) 0 0
\(939\) 13.4164 0.437828
\(940\) 0 0
\(941\) −60.0000 −1.95594 −0.977972 0.208736i \(-0.933065\pi\)
−0.977972 + 0.208736i \(0.933065\pi\)
\(942\) 0 0
\(943\) 44.7214 1.45633
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.94427 −0.290650 −0.145325 0.989384i \(-0.546423\pi\)
−0.145325 + 0.989384i \(0.546423\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) −71.5542 −2.32030
\(952\) 0 0
\(953\) −21.0000 −0.680257 −0.340128 0.940379i \(-0.610471\pi\)
−0.340128 + 0.940379i \(0.610471\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −20.0000 −0.646508
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 49.0000 1.58065
\(962\) 0 0
\(963\) −4.47214 −0.144113
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −53.6656 −1.72577 −0.862885 0.505400i \(-0.831345\pi\)
−0.862885 + 0.505400i \(0.831345\pi\)
\(968\) 0 0
\(969\) −15.0000 −0.481869
\(970\) 0 0
\(971\) −51.4296 −1.65045 −0.825227 0.564802i \(-0.808953\pi\)
−0.825227 + 0.564802i \(0.808953\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27.0000 0.863807 0.431903 0.901920i \(-0.357842\pi\)
0.431903 + 0.901920i \(0.357842\pi\)
\(978\) 0 0
\(979\) 33.5410 1.07198
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.8885 −0.570556 −0.285278 0.958445i \(-0.592086\pi\)
−0.285278 + 0.958445i \(0.592086\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 80.0000 2.54385
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) −5.00000 −0.158670
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 0 0
\(999\) 17.8885 0.565968
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.bt.1.1 2
4.3 odd 2 inner 6400.2.a.bt.1.2 2
5.4 even 2 6400.2.a.bq.1.2 2
8.3 odd 2 6400.2.a.br.1.1 2
8.5 even 2 6400.2.a.br.1.2 2
16.3 odd 4 3200.2.d.p.1601.4 yes 4
16.5 even 4 3200.2.d.p.1601.3 yes 4
16.11 odd 4 3200.2.d.p.1601.1 yes 4
16.13 even 4 3200.2.d.p.1601.2 yes 4
20.19 odd 2 6400.2.a.bq.1.1 2
40.19 odd 2 6400.2.a.bs.1.2 2
40.29 even 2 6400.2.a.bs.1.1 2
80.3 even 4 3200.2.f.m.449.2 4
80.13 odd 4 3200.2.f.m.449.3 4
80.19 odd 4 3200.2.d.o.1601.1 4
80.27 even 4 3200.2.f.m.449.1 4
80.29 even 4 3200.2.d.o.1601.3 yes 4
80.37 odd 4 3200.2.f.m.449.4 4
80.43 even 4 3200.2.f.n.449.3 4
80.53 odd 4 3200.2.f.n.449.2 4
80.59 odd 4 3200.2.d.o.1601.4 yes 4
80.67 even 4 3200.2.f.n.449.4 4
80.69 even 4 3200.2.d.o.1601.2 yes 4
80.77 odd 4 3200.2.f.n.449.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3200.2.d.o.1601.1 4 80.19 odd 4
3200.2.d.o.1601.2 yes 4 80.69 even 4
3200.2.d.o.1601.3 yes 4 80.29 even 4
3200.2.d.o.1601.4 yes 4 80.59 odd 4
3200.2.d.p.1601.1 yes 4 16.11 odd 4
3200.2.d.p.1601.2 yes 4 16.13 even 4
3200.2.d.p.1601.3 yes 4 16.5 even 4
3200.2.d.p.1601.4 yes 4 16.3 odd 4
3200.2.f.m.449.1 4 80.27 even 4
3200.2.f.m.449.2 4 80.3 even 4
3200.2.f.m.449.3 4 80.13 odd 4
3200.2.f.m.449.4 4 80.37 odd 4
3200.2.f.n.449.1 4 80.77 odd 4
3200.2.f.n.449.2 4 80.53 odd 4
3200.2.f.n.449.3 4 80.43 even 4
3200.2.f.n.449.4 4 80.67 even 4
6400.2.a.bq.1.1 2 20.19 odd 2
6400.2.a.bq.1.2 2 5.4 even 2
6400.2.a.br.1.1 2 8.3 odd 2
6400.2.a.br.1.2 2 8.5 even 2
6400.2.a.bs.1.1 2 40.29 even 2
6400.2.a.bs.1.2 2 40.19 odd 2
6400.2.a.bt.1.1 2 1.1 even 1 trivial
6400.2.a.bt.1.2 2 4.3 odd 2 inner