Properties

Label 6400.2.a.cf.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: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1600)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.46410 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.46410 q^{7} -2.00000 q^{9} -3.00000 q^{11} +3.46410 q^{13} -3.00000 q^{17} +1.00000 q^{19} -3.46410 q^{21} -5.00000 q^{27} -10.3923 q^{29} +6.92820 q^{31} -3.00000 q^{33} -10.3923 q^{37} +3.46410 q^{39} +9.00000 q^{41} +4.00000 q^{43} +10.3923 q^{47} +5.00000 q^{49} -3.00000 q^{51} +1.00000 q^{57} +12.0000 q^{59} +3.46410 q^{61} +6.92820 q^{63} +11.0000 q^{67} -10.3923 q^{71} +7.00000 q^{73} +10.3923 q^{77} -10.3923 q^{79} +1.00000 q^{81} +15.0000 q^{83} -10.3923 q^{87} +3.00000 q^{89} -12.0000 q^{91} +6.92820 q^{93} -14.0000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{9} - 6q^{11} - 6q^{17} + 2q^{19} - 10q^{27} - 6q^{33} + 18q^{41} + 8q^{43} + 10q^{49} - 6q^{51} + 2q^{57} + 24q^{59} + 22q^{67} + 14q^{73} + 2q^{81} + 30q^{83} + 6q^{89} - 24q^{91} - 28q^{97} + 12q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 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
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −3.46410 −0.755929
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −10.3923 −1.92980 −0.964901 0.262613i \(-0.915416\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.3923 −1.70848 −0.854242 0.519875i \(-0.825978\pi\)
−0.854242 + 0.519875i \(0.825978\pi\)
\(38\) 0 0
\(39\) 3.46410 0.554700
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.3923 1.51587 0.757937 0.652328i \(-0.226208\pi\)
0.757937 + 0.652328i \(0.226208\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 3.46410 0.443533 0.221766 0.975100i \(-0.428818\pi\)
0.221766 + 0.975100i \(0.428818\pi\)
\(62\) 0 0
\(63\) 6.92820 0.872872
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.0000 1.34386 0.671932 0.740613i \(-0.265465\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923 1.18431
\(78\) 0 0
\(79\) −10.3923 −1.16923 −0.584613 0.811312i \(-0.698754\pi\)
−0.584613 + 0.811312i \(0.698754\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.0000 1.64646 0.823232 0.567705i \(-0.192169\pi\)
0.823232 + 0.567705i \(0.192169\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −10.3923 −1.11417
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 6.92820 0.718421
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) −10.3923 −1.02398 −0.511992 0.858990i \(-0.671092\pi\)
−0.511992 + 0.858990i \(0.671092\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) 3.46410 0.331801 0.165900 0.986143i \(-0.446947\pi\)
0.165900 + 0.986143i \(0.446947\pi\)
\(110\) 0 0
\(111\) −10.3923 −0.986394
\(112\) 0 0
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −6.92820 −0.640513
\(118\) 0 0
\(119\) 10.3923 0.952661
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 9.00000 0.811503
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −3.46410 −0.300376
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 10.3923 0.875190
\(142\) 0 0
\(143\) −10.3923 −0.869048
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.00000 0.412393
\(148\) 0 0
\(149\) 10.3923 0.851371 0.425685 0.904871i \(-0.360033\pi\)
0.425685 + 0.904871i \(0.360033\pi\)
\(150\) 0 0
\(151\) 3.46410 0.281905 0.140952 0.990016i \(-0.454984\pi\)
0.140952 + 0.990016i \(0.454984\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.92820 0.552931 0.276465 0.961024i \(-0.410837\pi\)
0.276465 + 0.961024i \(0.410837\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 19.0000 1.48819 0.744097 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.3923 0.804181 0.402090 0.915600i \(-0.368284\pi\)
0.402090 + 0.915600i \(0.368284\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −20.7846 −1.58022 −0.790112 0.612962i \(-0.789978\pi\)
−0.790112 + 0.612962i \(0.789978\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) 13.8564 1.02994 0.514969 0.857209i \(-0.327803\pi\)
0.514969 + 0.857209i \(0.327803\pi\)
\(182\) 0 0
\(183\) 3.46410 0.256074
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) 17.3205 1.25988
\(190\) 0 0
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 0 0
\(193\) −7.00000 −0.503871 −0.251936 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.7846 1.48084 0.740421 0.672143i \(-0.234626\pi\)
0.740421 + 0.672143i \(0.234626\pi\)
\(198\) 0 0
\(199\) −13.8564 −0.982255 −0.491127 0.871088i \(-0.663415\pi\)
−0.491127 + 0.871088i \(0.663415\pi\)
\(200\) 0 0
\(201\) 11.0000 0.775880
\(202\) 0 0
\(203\) 36.0000 2.52670
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −25.0000 −1.72107 −0.860535 0.509390i \(-0.829871\pi\)
−0.860535 + 0.509390i \(0.829871\pi\)
\(212\) 0 0
\(213\) −10.3923 −0.712069
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −24.0000 −1.62923
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) −10.3923 −0.699062
\(222\) 0 0
\(223\) 13.8564 0.927894 0.463947 0.885863i \(-0.346433\pi\)
0.463947 + 0.885863i \(0.346433\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 6.92820 0.457829 0.228914 0.973447i \(-0.426482\pi\)
0.228914 + 0.973447i \(0.426482\pi\)
\(230\) 0 0
\(231\) 10.3923 0.683763
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −10.3923 −0.675053
\(238\) 0 0
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.46410 0.220416
\(248\) 0 0
\(249\) 15.0000 0.950586
\(250\) 0 0
\(251\) −9.00000 −0.568075 −0.284037 0.958813i \(-0.591674\pi\)
−0.284037 + 0.958813i \(0.591674\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 36.0000 2.23693
\(260\) 0 0
\(261\) 20.7846 1.28654
\(262\) 0 0
\(263\) 20.7846 1.28163 0.640817 0.767694i \(-0.278596\pi\)
0.640817 + 0.767694i \(0.278596\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3.00000 0.183597
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −20.7846 −1.26258 −0.631288 0.775549i \(-0.717473\pi\)
−0.631288 + 0.775549i \(0.717473\pi\)
\(272\) 0 0
\(273\) −12.0000 −0.726273
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.3205 1.04069 0.520344 0.853957i \(-0.325804\pi\)
0.520344 + 0.853957i \(0.325804\pi\)
\(278\) 0 0
\(279\) −13.8564 −0.829561
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 11.0000 0.653882 0.326941 0.945045i \(-0.393982\pi\)
0.326941 + 0.945045i \(0.393982\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −31.1769 −1.84032
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 10.3923 0.607125 0.303562 0.952812i \(-0.401824\pi\)
0.303562 + 0.952812i \(0.401824\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 15.0000 0.870388
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −13.8564 −0.798670
\(302\) 0 0
\(303\) −10.3923 −0.597022
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −19.0000 −1.08439 −0.542194 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) 0 0
\(309\) −10.3923 −0.591198
\(310\) 0 0
\(311\) 10.3923 0.589294 0.294647 0.955606i \(-0.404798\pi\)
0.294647 + 0.955606i \(0.404798\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.7846 −1.16738 −0.583690 0.811977i \(-0.698392\pi\)
−0.583690 + 0.811977i \(0.698392\pi\)
\(318\) 0 0
\(319\) 31.1769 1.74557
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.46410 0.191565
\(328\) 0 0
\(329\) −36.0000 −1.98474
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 0 0
\(333\) 20.7846 1.13899
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.00000 0.381314 0.190657 0.981657i \(-0.438938\pi\)
0.190657 + 0.981657i \(0.438938\pi\)
\(338\) 0 0
\(339\) 15.0000 0.814688
\(340\) 0 0
\(341\) −20.7846 −1.12555
\(342\) 0 0
\(343\) 6.92820 0.374088
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 33.0000 1.77153 0.885766 0.464131i \(-0.153633\pi\)
0.885766 + 0.464131i \(0.153633\pi\)
\(348\) 0 0
\(349\) 10.3923 0.556287 0.278144 0.960539i \(-0.410281\pi\)
0.278144 + 0.960539i \(0.410281\pi\)
\(350\) 0 0
\(351\) −17.3205 −0.924500
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 10.3923 0.550019
\(358\) 0 0
\(359\) 10.3923 0.548485 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6.92820 −0.361649 −0.180825 0.983515i \(-0.557877\pi\)
−0.180825 + 0.983515i \(0.557877\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 13.8564 0.717458 0.358729 0.933442i \(-0.383210\pi\)
0.358729 + 0.933442i \(0.383210\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 19.0000 0.975964 0.487982 0.872854i \(-0.337733\pi\)
0.487982 + 0.872854i \(0.337733\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −31.1769 −1.59307 −0.796533 0.604595i \(-0.793335\pi\)
−0.796533 + 0.604595i \(0.793335\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −20.7846 −1.05382 −0.526911 0.849921i \(-0.676650\pi\)
−0.526911 + 0.849921i \(0.676650\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −27.7128 −1.39087 −0.695433 0.718591i \(-0.744787\pi\)
−0.695433 + 0.718591i \(0.744787\pi\)
\(398\) 0 0
\(399\) −3.46410 −0.173422
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 24.0000 1.19553
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 31.1769 1.54538
\(408\) 0 0
\(409\) 25.0000 1.23617 0.618085 0.786111i \(-0.287909\pi\)
0.618085 + 0.786111i \(0.287909\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) 0 0
\(413\) −41.5692 −2.04549
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.00000 0.244851
\(418\) 0 0
\(419\) 21.0000 1.02592 0.512959 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) 0 0
\(421\) 20.7846 1.01298 0.506490 0.862246i \(-0.330943\pi\)
0.506490 + 0.862246i \(0.330943\pi\)
\(422\) 0 0
\(423\) −20.7846 −1.01058
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) 0 0
\(429\) −10.3923 −0.501745
\(430\) 0 0
\(431\) 10.3923 0.500580 0.250290 0.968171i \(-0.419474\pi\)
0.250290 + 0.968171i \(0.419474\pi\)
\(432\) 0 0
\(433\) 1.00000 0.0480569 0.0240285 0.999711i \(-0.492351\pi\)
0.0240285 + 0.999711i \(0.492351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −17.3205 −0.826663 −0.413331 0.910581i \(-0.635635\pi\)
−0.413331 + 0.910581i \(0.635635\pi\)
\(440\) 0 0
\(441\) −10.0000 −0.476190
\(442\) 0 0
\(443\) 39.0000 1.85295 0.926473 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 10.3923 0.491539
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) −27.0000 −1.27138
\(452\) 0 0
\(453\) 3.46410 0.162758
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.0000 1.73079 0.865393 0.501093i \(-0.167069\pi\)
0.865393 + 0.501093i \(0.167069\pi\)
\(458\) 0 0
\(459\) 15.0000 0.700140
\(460\) 0 0
\(461\) −20.7846 −0.968036 −0.484018 0.875058i \(-0.660823\pi\)
−0.484018 + 0.875058i \(0.660823\pi\)
\(462\) 0 0
\(463\) 20.7846 0.965943 0.482971 0.875636i \(-0.339558\pi\)
0.482971 + 0.875636i \(0.339558\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −38.1051 −1.75953
\(470\) 0 0
\(471\) 6.92820 0.319235
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −41.5692 −1.89935 −0.949673 0.313243i \(-0.898585\pi\)
−0.949673 + 0.313243i \(0.898585\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.8564 0.627894 0.313947 0.949441i \(-0.398349\pi\)
0.313947 + 0.949441i \(0.398349\pi\)
\(488\) 0 0
\(489\) 19.0000 0.859210
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 31.1769 1.40414
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 36.0000 1.61482
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) 10.3923 0.464294
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −20.7846 −0.921262 −0.460631 0.887592i \(-0.652377\pi\)
−0.460631 + 0.887592i \(0.652377\pi\)
\(510\) 0 0
\(511\) −24.2487 −1.07270
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −31.1769 −1.37116
\(518\) 0 0
\(519\) −20.7846 −0.912343
\(520\) 0 0
\(521\) 21.0000 0.920027 0.460013 0.887912i \(-0.347845\pi\)
0.460013 + 0.887912i \(0.347845\pi\)
\(522\) 0 0
\(523\) 1.00000 0.0437269 0.0218635 0.999761i \(-0.493040\pi\)
0.0218635 + 0.999761i \(0.493040\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.7846 −0.905392
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) 31.1769 1.35042
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.00000 −0.129460
\(538\) 0 0
\(539\) −15.0000 −0.646096
\(540\) 0 0
\(541\) 6.92820 0.297867 0.148933 0.988847i \(-0.452416\pi\)
0.148933 + 0.988847i \(0.452416\pi\)
\(542\) 0 0
\(543\) 13.8564 0.594635
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.00000 0.0427569 0.0213785 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) 0 0
\(549\) −6.92820 −0.295689
\(550\) 0 0
\(551\) −10.3923 −0.442727
\(552\) 0 0
\(553\) 36.0000 1.53088
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.3923 0.440336 0.220168 0.975462i \(-0.429339\pi\)
0.220168 + 0.975462i \(0.429339\pi\)
\(558\) 0 0
\(559\) 13.8564 0.586064
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.46410 −0.145479
\(568\) 0 0
\(569\) −3.00000 −0.125767 −0.0628833 0.998021i \(-0.520030\pi\)
−0.0628833 + 0.998021i \(0.520030\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 20.7846 0.868290
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −17.0000 −0.707719 −0.353860 0.935299i \(-0.615131\pi\)
−0.353860 + 0.935299i \(0.615131\pi\)
\(578\) 0 0
\(579\) −7.00000 −0.290910
\(580\) 0 0
\(581\) −51.9615 −2.15573
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.0000 0.619116 0.309558 0.950881i \(-0.399819\pi\)
0.309558 + 0.950881i \(0.399819\pi\)
\(588\) 0 0
\(589\) 6.92820 0.285472
\(590\) 0 0
\(591\) 20.7846 0.854965
\(592\) 0 0
\(593\) 3.00000 0.123195 0.0615976 0.998101i \(-0.480380\pi\)
0.0615976 + 0.998101i \(0.480380\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −13.8564 −0.567105
\(598\) 0 0
\(599\) 31.1769 1.27385 0.636927 0.770924i \(-0.280205\pi\)
0.636927 + 0.770924i \(0.280205\pi\)
\(600\) 0 0
\(601\) −41.0000 −1.67242 −0.836212 0.548406i \(-0.815235\pi\)
−0.836212 + 0.548406i \(0.815235\pi\)
\(602\) 0 0
\(603\) −22.0000 −0.895909
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 20.7846 0.843621 0.421811 0.906684i \(-0.361395\pi\)
0.421811 + 0.906684i \(0.361395\pi\)
\(608\) 0 0
\(609\) 36.0000 1.45879
\(610\) 0 0
\(611\) 36.0000 1.45640
\(612\) 0 0
\(613\) −20.7846 −0.839482 −0.419741 0.907644i \(-0.637879\pi\)
−0.419741 + 0.907644i \(0.637879\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.3923 −0.416359
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) 0 0
\(629\) 31.1769 1.24310
\(630\) 0 0
\(631\) −13.8564 −0.551615 −0.275807 0.961213i \(-0.588945\pi\)
−0.275807 + 0.961213i \(0.588945\pi\)
\(632\) 0 0
\(633\) −25.0000 −0.993661
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 17.3205 0.686264
\(638\) 0 0
\(639\) 20.7846 0.822226
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −41.5692 −1.63425 −0.817127 0.576457i \(-0.804435\pi\)
−0.817127 + 0.576457i \(0.804435\pi\)
\(648\) 0 0
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) 0 0
\(653\) −31.1769 −1.22005 −0.610023 0.792383i \(-0.708840\pi\)
−0.610023 + 0.792383i \(0.708840\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −33.0000 −1.28550 −0.642749 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) 10.3923 0.404214 0.202107 0.979363i \(-0.435221\pi\)
0.202107 + 0.979363i \(0.435221\pi\)
\(662\) 0 0
\(663\) −10.3923 −0.403604
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 13.8564 0.535720
\(670\) 0 0
\(671\) −10.3923 −0.401190
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20.7846 0.798817 0.399409 0.916773i \(-0.369215\pi\)
0.399409 + 0.916773i \(0.369215\pi\)
\(678\) 0 0
\(679\) 48.4974 1.86116
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 9.00000 0.344375 0.172188 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 6.92820 0.264327
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 37.0000 1.40755 0.703773 0.710425i \(-0.251497\pi\)
0.703773 + 0.710425i \(0.251497\pi\)
\(692\) 0 0
\(693\) −20.7846 −0.789542
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −27.0000 −1.02270
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 31.1769 1.17754 0.588768 0.808302i \(-0.299613\pi\)
0.588768 + 0.808302i \(0.299613\pi\)
\(702\) 0 0
\(703\) −10.3923 −0.391953
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.0000 1.35392
\(708\) 0 0
\(709\) 34.6410 1.30097 0.650485 0.759519i \(-0.274566\pi\)
0.650485 + 0.759519i \(0.274566\pi\)
\(710\) 0 0
\(711\) 20.7846 0.779484
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 10.3923 0.388108
\(718\) 0 0
\(719\) −10.3923 −0.387568 −0.193784 0.981044i \(-0.562076\pi\)
−0.193784 + 0.981044i \(0.562076\pi\)
\(720\) 0 0
\(721\) 36.0000 1.34071
\(722\) 0 0
\(723\) 5.00000 0.185952
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −45.0333 −1.67019 −0.835097 0.550103i \(-0.814588\pi\)
−0.835097 + 0.550103i \(0.814588\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 31.1769 1.15155 0.575773 0.817610i \(-0.304701\pi\)
0.575773 + 0.817610i \(0.304701\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −33.0000 −1.21557
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 3.46410 0.127257
\(742\) 0 0
\(743\) 41.5692 1.52503 0.762513 0.646972i \(-0.223965\pi\)
0.762513 + 0.646972i \(0.223965\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −30.0000 −1.09764
\(748\) 0 0
\(749\) −31.1769 −1.13918
\(750\) 0 0
\(751\) 51.9615 1.89610 0.948051 0.318117i \(-0.103050\pi\)
0.948051 + 0.318117i \(0.103050\pi\)
\(752\) 0 0
\(753\) −9.00000 −0.327978
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −6.92820 −0.251810 −0.125905 0.992042i \(-0.540183\pi\)
−0.125905 + 0.992042i \(0.540183\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) 0 0
\(763\) −12.0000 −0.434429
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 41.5692 1.50098
\(768\) 0 0
\(769\) −7.00000 −0.252426 −0.126213 0.992003i \(-0.540282\pi\)
−0.126213 + 0.992003i \(0.540282\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) −10.3923 −0.373785 −0.186893 0.982380i \(-0.559842\pi\)
−0.186893 + 0.982380i \(0.559842\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 36.0000 1.29149
\(778\) 0 0
\(779\) 9.00000 0.322458
\(780\) 0 0
\(781\) 31.1769 1.11560
\(782\) 0 0
\(783\) 51.9615 1.85695
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 16.0000 0.570338 0.285169 0.958477i \(-0.407950\pi\)
0.285169 + 0.958477i \(0.407950\pi\)
\(788\) 0 0
\(789\) 20.7846 0.739952
\(790\) 0 0
\(791\) −51.9615 −1.84754
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.7846 0.736229 0.368114 0.929781i \(-0.380004\pi\)
0.368114 + 0.929781i \(0.380004\pi\)
\(798\) 0 0
\(799\) −31.1769 −1.10296
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −21.0000 −0.741074
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) −20.7846 −0.728948
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −34.6410 −1.20751 −0.603755 0.797170i \(-0.706329\pi\)
−0.603755 + 0.797170i \(0.706329\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 0 0
\(829\) −10.3923 −0.360940 −0.180470 0.983581i \(-0.557762\pi\)
−0.180470 + 0.983581i \(0.557762\pi\)
\(830\) 0 0
\(831\) 17.3205 0.600842
\(832\) 0 0
\(833\) −15.0000 −0.519719
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −34.6410 −1.19737
\(838\) 0 0
\(839\) −41.5692 −1.43513 −0.717564 0.696492i \(-0.754743\pi\)
−0.717564 + 0.696492i \(0.754743\pi\)
\(840\) 0 0
\(841\) 79.0000 2.72414
\(842\) 0 0
\(843\) −18.0000 −0.619953
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 6.92820 0.238056
\(848\) 0 0
\(849\) 11.0000 0.377519
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −55.4256 −1.89774 −0.948869 0.315671i \(-0.897770\pi\)
−0.948869 + 0.315671i \(0.897770\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39.0000 −1.33221 −0.666107 0.745856i \(-0.732041\pi\)
−0.666107 + 0.745856i \(0.732041\pi\)
\(858\) 0 0
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) 0 0
\(861\) −31.1769 −1.06251
\(862\) 0 0
\(863\) −10.3923 −0.353758 −0.176879 0.984233i \(-0.556600\pi\)
−0.176879 + 0.984233i \(0.556600\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 31.1769 1.05760
\(870\) 0 0
\(871\) 38.1051 1.29114
\(872\) 0 0
\(873\) 28.0000 0.947656
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 48.4974 1.63764 0.818821 0.574049i \(-0.194628\pi\)
0.818821 + 0.574049i \(0.194628\pi\)
\(878\) 0 0
\(879\) 10.3923 0.350524
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −35.0000 −1.17784 −0.588922 0.808190i \(-0.700447\pi\)
−0.588922 + 0.808190i \(0.700447\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) 10.3923 0.347765
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −72.0000 −2.40133
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −13.8564 −0.461112
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) 20.7846 0.689382
\(910\) 0 0
\(911\) 20.7846 0.688625 0.344312 0.938855i \(-0.388112\pi\)
0.344312 + 0.938855i \(0.388112\pi\)
\(912\) 0 0
\(913\) −45.0000 −1.48928
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −27.7128 −0.914161 −0.457081 0.889425i \(-0.651105\pi\)
−0.457081 + 0.889425i \(0.651105\pi\)
\(920\) 0 0
\(921\) −19.0000 −0.626071
\(922\) 0 0
\(923\) −36.0000 −1.18495
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 20.7846 0.682656
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) 0 0
\(933\) 10.3923 0.340229
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.0000 1.14340 0.571700 0.820463i \(-0.306284\pi\)
0.571700 + 0.820463i \(0.306284\pi\)
\(938\) 0 0
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) 51.9615 1.69390 0.846949 0.531675i \(-0.178437\pi\)
0.846949 + 0.531675i \(0.178437\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 24.2487 0.787146
\(950\) 0 0
\(951\) −20.7846 −0.673987
\(952\) 0 0
\(953\) −39.0000 −1.26333 −0.631667 0.775240i \(-0.717629\pi\)
−0.631667 + 0.775240i \(0.717629\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 31.1769 1.00781
\(958\) 0 0
\(959\) −31.1769 −1.00676
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −20.7846 −0.668388 −0.334194 0.942504i \(-0.608464\pi\)
−0.334194 + 0.942504i \(0.608464\pi\)
\(968\) 0 0
\(969\) −3.00000 −0.0963739
\(970\) 0 0
\(971\) 57.0000 1.82922 0.914609 0.404341i \(-0.132499\pi\)
0.914609 + 0.404341i \(0.132499\pi\)
\(972\) 0 0
\(973\) −17.3205 −0.555270
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27.0000 −0.863807 −0.431903 0.901920i \(-0.642158\pi\)
−0.431903 + 0.901920i \(0.642158\pi\)
\(978\) 0 0
\(979\) −9.00000 −0.287641
\(980\) 0 0
\(981\) −6.92820 −0.221201
\(982\) 0 0
\(983\) 10.3923 0.331463 0.165732 0.986171i \(-0.447001\pi\)
0.165732 + 0.986171i \(0.447001\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −36.0000 −1.14589
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −24.2487 −0.770286 −0.385143 0.922857i \(-0.625848\pi\)
−0.385143 + 0.922857i \(0.625848\pi\)
\(992\) 0 0
\(993\) −19.0000 −0.602947
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 0 0
\(999\) 51.9615 1.64399
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.cf.1.1 2
4.3 odd 2 6400.2.a.bb.1.2 2
5.4 even 2 6400.2.a.ba.1.2 2
8.3 odd 2 inner 6400.2.a.cf.1.2 2
8.5 even 2 6400.2.a.bb.1.1 2
16.3 odd 4 1600.2.d.e.801.1 4
16.5 even 4 1600.2.d.e.801.2 yes 4
16.11 odd 4 1600.2.d.e.801.3 yes 4
16.13 even 4 1600.2.d.e.801.4 yes 4
20.19 odd 2 6400.2.a.cg.1.1 2
40.19 odd 2 6400.2.a.ba.1.1 2
40.29 even 2 6400.2.a.cg.1.2 2
80.3 even 4 1600.2.f.g.1249.3 4
80.13 odd 4 1600.2.f.c.1249.2 4
80.19 odd 4 1600.2.d.f.801.4 yes 4
80.27 even 4 1600.2.f.g.1249.2 4
80.29 even 4 1600.2.d.f.801.1 yes 4
80.37 odd 4 1600.2.f.c.1249.3 4
80.43 even 4 1600.2.f.c.1249.4 4
80.53 odd 4 1600.2.f.g.1249.1 4
80.59 odd 4 1600.2.d.f.801.2 yes 4
80.67 even 4 1600.2.f.c.1249.1 4
80.69 even 4 1600.2.d.f.801.3 yes 4
80.77 odd 4 1600.2.f.g.1249.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.d.e.801.1 4 16.3 odd 4
1600.2.d.e.801.2 yes 4 16.5 even 4
1600.2.d.e.801.3 yes 4 16.11 odd 4
1600.2.d.e.801.4 yes 4 16.13 even 4
1600.2.d.f.801.1 yes 4 80.29 even 4
1600.2.d.f.801.2 yes 4 80.59 odd 4
1600.2.d.f.801.3 yes 4 80.69 even 4
1600.2.d.f.801.4 yes 4 80.19 odd 4
1600.2.f.c.1249.1 4 80.67 even 4
1600.2.f.c.1249.2 4 80.13 odd 4
1600.2.f.c.1249.3 4 80.37 odd 4
1600.2.f.c.1249.4 4 80.43 even 4
1600.2.f.g.1249.1 4 80.53 odd 4
1600.2.f.g.1249.2 4 80.27 even 4
1600.2.f.g.1249.3 4 80.3 even 4
1600.2.f.g.1249.4 4 80.77 odd 4
6400.2.a.ba.1.1 2 40.19 odd 2
6400.2.a.ba.1.2 2 5.4 even 2
6400.2.a.bb.1.1 2 8.5 even 2
6400.2.a.bb.1.2 2 4.3 odd 2
6400.2.a.cf.1.1 2 1.1 even 1 trivial
6400.2.a.cf.1.2 2 8.3 odd 2 inner
6400.2.a.cg.1.1 2 20.19 odd 2
6400.2.a.cg.1.2 2 40.29 even 2