Properties

Label 6400.2.a.bf.1.2
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 320)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051 q^{3} +1.26795 q^{7} -2.46410 q^{9} +O(q^{10})\) \(q+0.732051 q^{3} +1.26795 q^{7} -2.46410 q^{9} +3.46410 q^{11} -3.46410 q^{13} -3.46410 q^{17} -2.00000 q^{19} +0.928203 q^{21} +8.19615 q^{23} -4.00000 q^{27} -9.46410 q^{31} +2.53590 q^{33} -6.00000 q^{37} -2.53590 q^{39} -2.53590 q^{41} +10.1962 q^{43} +8.19615 q^{47} -5.39230 q^{49} -2.53590 q^{51} +10.3923 q^{53} -1.46410 q^{57} -6.00000 q^{59} -12.9282 q^{61} -3.12436 q^{63} -10.1962 q^{67} +6.00000 q^{69} +4.39230 q^{71} -14.3923 q^{73} +4.39230 q^{77} -12.0000 q^{79} +4.46410 q^{81} -4.73205 q^{83} -0.928203 q^{89} -4.39230 q^{91} -6.92820 q^{93} +6.39230 q^{97} -8.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 6 q^{7} + 2 q^{9} - 4 q^{19} - 12 q^{21} + 6 q^{23} - 8 q^{27} - 12 q^{31} + 12 q^{33} - 12 q^{37} - 12 q^{39} - 12 q^{41} + 10 q^{43} + 6 q^{47} + 10 q^{49} - 12 q^{51} + 4 q^{57} - 12 q^{59} - 12 q^{61} + 18 q^{63} - 10 q^{67} + 12 q^{69} - 12 q^{71} - 8 q^{73} - 12 q^{77} - 24 q^{79} + 2 q^{81} - 6 q^{83} + 12 q^{89} + 12 q^{91} - 8 q^{97} - 24 q^{99} + O(q^{100}) \)

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
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.26795 0.479240 0.239620 0.970867i \(-0.422977\pi\)
0.239620 + 0.970867i \(0.422977\pi\)
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −3.46410 −0.960769 −0.480384 0.877058i \(-0.659503\pi\)
−0.480384 + 0.877058i \(0.659503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0.928203 0.202551
\(22\) 0 0
\(23\) 8.19615 1.70902 0.854508 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −9.46410 −1.69980 −0.849901 0.526942i \(-0.823339\pi\)
−0.849901 + 0.526942i \(0.823339\pi\)
\(32\) 0 0
\(33\) 2.53590 0.441443
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −2.53590 −0.406069
\(40\) 0 0
\(41\) −2.53590 −0.396041 −0.198020 0.980198i \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(42\) 0 0
\(43\) 10.1962 1.55490 0.777449 0.628946i \(-0.216513\pi\)
0.777449 + 0.628946i \(0.216513\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.19615 1.19553 0.597766 0.801671i \(-0.296055\pi\)
0.597766 + 0.801671i \(0.296055\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 0 0
\(51\) −2.53590 −0.355097
\(52\) 0 0
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.46410 −0.193925
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −12.9282 −1.65529 −0.827643 0.561254i \(-0.810319\pi\)
−0.827643 + 0.561254i \(0.810319\pi\)
\(62\) 0 0
\(63\) −3.12436 −0.393632
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −10.1962 −1.24566 −0.622829 0.782358i \(-0.714017\pi\)
−0.622829 + 0.782358i \(0.714017\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 4.39230 0.521271 0.260635 0.965437i \(-0.416068\pi\)
0.260635 + 0.965437i \(0.416068\pi\)
\(72\) 0 0
\(73\) −14.3923 −1.68449 −0.842246 0.539093i \(-0.818767\pi\)
−0.842246 + 0.539093i \(0.818767\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.39230 0.500550
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) −4.73205 −0.519410 −0.259705 0.965688i \(-0.583625\pi\)
−0.259705 + 0.965688i \(0.583625\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) 0 0
\(91\) −4.39230 −0.460439
\(92\) 0 0
\(93\) −6.92820 −0.718421
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.39230 0.649040 0.324520 0.945879i \(-0.394797\pi\)
0.324520 + 0.945879i \(0.394797\pi\)
\(98\) 0 0
\(99\) −8.53590 −0.857890
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 8.19615 0.807591 0.403795 0.914849i \(-0.367691\pi\)
0.403795 + 0.914849i \(0.367691\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.7321 −1.61755 −0.808774 0.588119i \(-0.799869\pi\)
−0.808774 + 0.588119i \(0.799869\pi\)
\(108\) 0 0
\(109\) −0.928203 −0.0889057 −0.0444529 0.999011i \(-0.514154\pi\)
−0.0444529 + 0.999011i \(0.514154\pi\)
\(110\) 0 0
\(111\) −4.39230 −0.416899
\(112\) 0 0
\(113\) −0.928203 −0.0873180 −0.0436590 0.999046i \(-0.513902\pi\)
−0.0436590 + 0.999046i \(0.513902\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 8.53590 0.789144
\(118\) 0 0
\(119\) −4.39230 −0.402642
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −1.85641 −0.167387
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.80385 0.337537 0.168768 0.985656i \(-0.446021\pi\)
0.168768 + 0.985656i \(0.446021\pi\)
\(128\) 0 0
\(129\) 7.46410 0.657178
\(130\) 0 0
\(131\) −10.3923 −0.907980 −0.453990 0.891007i \(-0.650000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(132\) 0 0
\(133\) −2.53590 −0.219890
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.9282 1.10453 0.552265 0.833668i \(-0.313763\pi\)
0.552265 + 0.833668i \(0.313763\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.94744 −0.325579
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −2.53590 −0.206368 −0.103184 0.994662i \(-0.532903\pi\)
−0.103184 + 0.994662i \(0.532903\pi\)
\(152\) 0 0
\(153\) 8.53590 0.690086
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.928203 −0.0740787 −0.0370393 0.999314i \(-0.511793\pi\)
−0.0370393 + 0.999314i \(0.511793\pi\)
\(158\) 0 0
\(159\) 7.60770 0.603329
\(160\) 0 0
\(161\) 10.3923 0.819028
\(162\) 0 0
\(163\) 5.80385 0.454592 0.227296 0.973826i \(-0.427011\pi\)
0.227296 + 0.973826i \(0.427011\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.19615 −0.634237 −0.317119 0.948386i \(-0.602715\pi\)
−0.317119 + 0.948386i \(0.602715\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 4.92820 0.376869
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.39230 −0.330146
\(178\) 0 0
\(179\) −19.8564 −1.48414 −0.742069 0.670324i \(-0.766155\pi\)
−0.742069 + 0.670324i \(0.766155\pi\)
\(180\) 0 0
\(181\) −6.92820 −0.514969 −0.257485 0.966282i \(-0.582894\pi\)
−0.257485 + 0.966282i \(0.582894\pi\)
\(182\) 0 0
\(183\) −9.46410 −0.699607
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) −5.07180 −0.368919
\(190\) 0 0
\(191\) 16.3923 1.18611 0.593053 0.805164i \(-0.297923\pi\)
0.593053 + 0.805164i \(0.297923\pi\)
\(192\) 0 0
\(193\) −6.39230 −0.460128 −0.230064 0.973175i \(-0.573894\pi\)
−0.230064 + 0.973175i \(0.573894\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.3923 −0.740421 −0.370211 0.928948i \(-0.620714\pi\)
−0.370211 + 0.928948i \(0.620714\pi\)
\(198\) 0 0
\(199\) 6.92820 0.491127 0.245564 0.969380i \(-0.421027\pi\)
0.245564 + 0.969380i \(0.421027\pi\)
\(200\) 0 0
\(201\) −7.46410 −0.526477
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −20.1962 −1.40373
\(208\) 0 0
\(209\) −6.92820 −0.479234
\(210\) 0 0
\(211\) 6.39230 0.440064 0.220032 0.975493i \(-0.429384\pi\)
0.220032 + 0.975493i \(0.429384\pi\)
\(212\) 0 0
\(213\) 3.21539 0.220315
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) −10.5359 −0.711950
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 15.1244 1.01280 0.506401 0.862298i \(-0.330976\pi\)
0.506401 + 0.862298i \(0.330976\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −18.5885 −1.23376 −0.616880 0.787058i \(-0.711603\pi\)
−0.616880 + 0.787058i \(0.711603\pi\)
\(228\) 0 0
\(229\) 18.9282 1.25081 0.625405 0.780300i \(-0.284934\pi\)
0.625405 + 0.780300i \(0.284934\pi\)
\(230\) 0 0
\(231\) 3.21539 0.211557
\(232\) 0 0
\(233\) −1.60770 −0.105324 −0.0526618 0.998612i \(-0.516771\pi\)
−0.0526618 + 0.998612i \(0.516771\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −8.78461 −0.570622
\(238\) 0 0
\(239\) 20.7846 1.34444 0.672222 0.740349i \(-0.265340\pi\)
0.672222 + 0.740349i \(0.265340\pi\)
\(240\) 0 0
\(241\) −20.3923 −1.31358 −0.656792 0.754072i \(-0.728087\pi\)
−0.656792 + 0.754072i \(0.728087\pi\)
\(242\) 0 0
\(243\) 15.2679 0.979439
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.92820 0.440831
\(248\) 0 0
\(249\) −3.46410 −0.219529
\(250\) 0 0
\(251\) −15.4641 −0.976085 −0.488043 0.872820i \(-0.662289\pi\)
−0.488043 + 0.872820i \(0.662289\pi\)
\(252\) 0 0
\(253\) 28.3923 1.78501
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −7.60770 −0.472719
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.1962 −1.24535 −0.622674 0.782481i \(-0.713954\pi\)
−0.622674 + 0.782481i \(0.713954\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.679492 −0.0415842
\(268\) 0 0
\(269\) 14.7846 0.901434 0.450717 0.892667i \(-0.351168\pi\)
0.450717 + 0.892667i \(0.351168\pi\)
\(270\) 0 0
\(271\) 4.39230 0.266814 0.133407 0.991061i \(-0.457408\pi\)
0.133407 + 0.991061i \(0.457408\pi\)
\(272\) 0 0
\(273\) −3.21539 −0.194604
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 19.8564 1.19306 0.596528 0.802592i \(-0.296546\pi\)
0.596528 + 0.802592i \(0.296546\pi\)
\(278\) 0 0
\(279\) 23.3205 1.39616
\(280\) 0 0
\(281\) 28.3923 1.69374 0.846871 0.531798i \(-0.178483\pi\)
0.846871 + 0.531798i \(0.178483\pi\)
\(282\) 0 0
\(283\) 10.5885 0.629418 0.314709 0.949188i \(-0.398093\pi\)
0.314709 + 0.949188i \(0.398093\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.21539 −0.189798
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 4.67949 0.274317
\(292\) 0 0
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −13.8564 −0.804030
\(298\) 0 0
\(299\) −28.3923 −1.64197
\(300\) 0 0
\(301\) 12.9282 0.745169
\(302\) 0 0
\(303\) 8.78461 0.504663
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −34.1962 −1.95168 −0.975839 0.218492i \(-0.929886\pi\)
−0.975839 + 0.218492i \(0.929886\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 7.60770 0.431393 0.215696 0.976460i \(-0.430798\pi\)
0.215696 + 0.976460i \(0.430798\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.3923 1.25768 0.628839 0.777536i \(-0.283531\pi\)
0.628839 + 0.777536i \(0.283531\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.2487 −0.683656
\(322\) 0 0
\(323\) 6.92820 0.385496
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −0.679492 −0.0375760
\(328\) 0 0
\(329\) 10.3923 0.572946
\(330\) 0 0
\(331\) −5.60770 −0.308227 −0.154113 0.988053i \(-0.549252\pi\)
−0.154113 + 0.988053i \(0.549252\pi\)
\(332\) 0 0
\(333\) 14.7846 0.810192
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −0.679492 −0.0369049
\(340\) 0 0
\(341\) −32.7846 −1.77539
\(342\) 0 0
\(343\) −15.7128 −0.848412
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.0526 −1.50594 −0.752970 0.658055i \(-0.771380\pi\)
−0.752970 + 0.658055i \(0.771380\pi\)
\(348\) 0 0
\(349\) −8.78461 −0.470229 −0.235115 0.971968i \(-0.575547\pi\)
−0.235115 + 0.971968i \(0.575547\pi\)
\(350\) 0 0
\(351\) 13.8564 0.739600
\(352\) 0 0
\(353\) 14.7846 0.786905 0.393453 0.919345i \(-0.371281\pi\)
0.393453 + 0.919345i \(0.371281\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −3.21539 −0.170177
\(358\) 0 0
\(359\) −8.78461 −0.463634 −0.231817 0.972759i \(-0.574467\pi\)
−0.231817 + 0.972759i \(0.574467\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0.732051 0.0384227
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −10.0526 −0.524739 −0.262370 0.964967i \(-0.584504\pi\)
−0.262370 + 0.964967i \(0.584504\pi\)
\(368\) 0 0
\(369\) 6.24871 0.325295
\(370\) 0 0
\(371\) 13.1769 0.684111
\(372\) 0 0
\(373\) −7.85641 −0.406789 −0.203395 0.979097i \(-0.565197\pi\)
−0.203395 + 0.979097i \(0.565197\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) 2.78461 0.142660
\(382\) 0 0
\(383\) −3.80385 −0.194368 −0.0971838 0.995266i \(-0.530983\pi\)
−0.0971838 + 0.995266i \(0.530983\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −25.1244 −1.27714
\(388\) 0 0
\(389\) −26.7846 −1.35803 −0.679017 0.734123i \(-0.737594\pi\)
−0.679017 + 0.734123i \(0.737594\pi\)
\(390\) 0 0
\(391\) −28.3923 −1.43586
\(392\) 0 0
\(393\) −7.60770 −0.383757
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −27.4641 −1.37838 −0.689192 0.724579i \(-0.742034\pi\)
−0.689192 + 0.724579i \(0.742034\pi\)
\(398\) 0 0
\(399\) −1.85641 −0.0929366
\(400\) 0 0
\(401\) −4.14359 −0.206921 −0.103461 0.994634i \(-0.532992\pi\)
−0.103461 + 0.994634i \(0.532992\pi\)
\(402\) 0 0
\(403\) 32.7846 1.63312
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.7846 −1.03025
\(408\) 0 0
\(409\) −3.60770 −0.178389 −0.0891945 0.996014i \(-0.528429\pi\)
−0.0891945 + 0.996014i \(0.528429\pi\)
\(410\) 0 0
\(411\) 9.46410 0.466830
\(412\) 0 0
\(413\) −7.60770 −0.374350
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.32051 −0.358487
\(418\) 0 0
\(419\) −0.928203 −0.0453457 −0.0226728 0.999743i \(-0.507218\pi\)
−0.0226728 + 0.999743i \(0.507218\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) −20.1962 −0.981971
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −16.3923 −0.793279
\(428\) 0 0
\(429\) −8.78461 −0.424125
\(430\) 0 0
\(431\) 28.3923 1.36761 0.683805 0.729665i \(-0.260324\pi\)
0.683805 + 0.729665i \(0.260324\pi\)
\(432\) 0 0
\(433\) −26.3923 −1.26833 −0.634167 0.773196i \(-0.718657\pi\)
−0.634167 + 0.773196i \(0.718657\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −16.3923 −0.784150
\(438\) 0 0
\(439\) −18.9282 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(440\) 0 0
\(441\) 13.2872 0.632723
\(442\) 0 0
\(443\) −0.339746 −0.0161418 −0.00807091 0.999967i \(-0.502569\pi\)
−0.00807091 + 0.999967i \(0.502569\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −13.1769 −0.623247
\(448\) 0 0
\(449\) −2.53590 −0.119676 −0.0598382 0.998208i \(-0.519058\pi\)
−0.0598382 + 0.998208i \(0.519058\pi\)
\(450\) 0 0
\(451\) −8.78461 −0.413651
\(452\) 0 0
\(453\) −1.85641 −0.0872216
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 22.7846 1.06582 0.532910 0.846172i \(-0.321099\pi\)
0.532910 + 0.846172i \(0.321099\pi\)
\(458\) 0 0
\(459\) 13.8564 0.646762
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 15.8038 0.734467 0.367234 0.930129i \(-0.380305\pi\)
0.367234 + 0.930129i \(0.380305\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 22.9808 1.06342 0.531711 0.846926i \(-0.321549\pi\)
0.531711 + 0.846926i \(0.321549\pi\)
\(468\) 0 0
\(469\) −12.9282 −0.596969
\(470\) 0 0
\(471\) −0.679492 −0.0313093
\(472\) 0 0
\(473\) 35.3205 1.62404
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −25.6077 −1.17250
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 20.7846 0.947697
\(482\) 0 0
\(483\) 7.60770 0.346162
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 39.1244 1.77289 0.886447 0.462830i \(-0.153166\pi\)
0.886447 + 0.462830i \(0.153166\pi\)
\(488\) 0 0
\(489\) 4.24871 0.192133
\(490\) 0 0
\(491\) −22.3923 −1.01055 −0.505275 0.862958i \(-0.668609\pi\)
−0.505275 + 0.862958i \(0.668609\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.56922 0.249814
\(498\) 0 0
\(499\) −39.5692 −1.77136 −0.885681 0.464295i \(-0.846308\pi\)
−0.885681 + 0.464295i \(0.846308\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) 8.19615 0.365448 0.182724 0.983164i \(-0.441508\pi\)
0.182724 + 0.983164i \(0.441508\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.732051 −0.0325115
\(508\) 0 0
\(509\) 8.78461 0.389371 0.194685 0.980866i \(-0.437631\pi\)
0.194685 + 0.980866i \(0.437631\pi\)
\(510\) 0 0
\(511\) −18.2487 −0.807275
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 28.3923 1.24869
\(518\) 0 0
\(519\) −4.39230 −0.192801
\(520\) 0 0
\(521\) −31.8564 −1.39565 −0.697827 0.716266i \(-0.745850\pi\)
−0.697827 + 0.716266i \(0.745850\pi\)
\(522\) 0 0
\(523\) −14.9808 −0.655063 −0.327531 0.944840i \(-0.606217\pi\)
−0.327531 + 0.944840i \(0.606217\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 32.7846 1.42812
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) 0 0
\(531\) 14.7846 0.641597
\(532\) 0 0
\(533\) 8.78461 0.380504
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −14.5359 −0.627270
\(538\) 0 0
\(539\) −18.6795 −0.804583
\(540\) 0 0
\(541\) 15.7128 0.675547 0.337773 0.941227i \(-0.390326\pi\)
0.337773 + 0.941227i \(0.390326\pi\)
\(542\) 0 0
\(543\) −5.07180 −0.217652
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.80385 −0.0771270 −0.0385635 0.999256i \(-0.512278\pi\)
−0.0385635 + 0.999256i \(0.512278\pi\)
\(548\) 0 0
\(549\) 31.8564 1.35960
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −15.2154 −0.647024
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.7846 0.626444 0.313222 0.949680i \(-0.398592\pi\)
0.313222 + 0.949680i \(0.398592\pi\)
\(558\) 0 0
\(559\) −35.3205 −1.49390
\(560\) 0 0
\(561\) −8.78461 −0.370887
\(562\) 0 0
\(563\) −26.1962 −1.10404 −0.552018 0.833832i \(-0.686142\pi\)
−0.552018 + 0.833832i \(0.686142\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 5.66025 0.237708
\(568\) 0 0
\(569\) 42.2487 1.77116 0.885579 0.464489i \(-0.153762\pi\)
0.885579 + 0.464489i \(0.153762\pi\)
\(570\) 0 0
\(571\) 30.3923 1.27188 0.635939 0.771739i \(-0.280613\pi\)
0.635939 + 0.771739i \(0.280613\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) −4.67949 −0.194473
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.5167 0.557892 0.278946 0.960307i \(-0.410015\pi\)
0.278946 + 0.960307i \(0.410015\pi\)
\(588\) 0 0
\(589\) 18.9282 0.779923
\(590\) 0 0
\(591\) −7.60770 −0.312939
\(592\) 0 0
\(593\) 0.928203 0.0381167 0.0190584 0.999818i \(-0.493933\pi\)
0.0190584 + 0.999818i \(0.493933\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.07180 0.207575
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 20.3923 0.831819 0.415910 0.909406i \(-0.363463\pi\)
0.415910 + 0.909406i \(0.363463\pi\)
\(602\) 0 0
\(603\) 25.1244 1.02314
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.19615 0.332672 0.166336 0.986069i \(-0.446806\pi\)
0.166336 + 0.986069i \(0.446806\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −28.3923 −1.14863
\(612\) 0 0
\(613\) 13.6077 0.549610 0.274805 0.961500i \(-0.411387\pi\)
0.274805 + 0.961500i \(0.411387\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.6077 0.547825 0.273913 0.961755i \(-0.411682\pi\)
0.273913 + 0.961755i \(0.411682\pi\)
\(618\) 0 0
\(619\) −6.78461 −0.272696 −0.136348 0.990661i \(-0.543537\pi\)
−0.136348 + 0.990661i \(0.543537\pi\)
\(620\) 0 0
\(621\) −32.7846 −1.31560
\(622\) 0 0
\(623\) −1.17691 −0.0471521
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −5.07180 −0.202548
\(628\) 0 0
\(629\) 20.7846 0.828737
\(630\) 0 0
\(631\) −21.4641 −0.854472 −0.427236 0.904140i \(-0.640513\pi\)
−0.427236 + 0.904140i \(0.640513\pi\)
\(632\) 0 0
\(633\) 4.67949 0.185993
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 18.6795 0.740108
\(638\) 0 0
\(639\) −10.8231 −0.428155
\(640\) 0 0
\(641\) −4.39230 −0.173486 −0.0867428 0.996231i \(-0.527646\pi\)
−0.0867428 + 0.996231i \(0.527646\pi\)
\(642\) 0 0
\(643\) −10.5885 −0.417568 −0.208784 0.977962i \(-0.566951\pi\)
−0.208784 + 0.977962i \(0.566951\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −36.5885 −1.43844 −0.719220 0.694782i \(-0.755501\pi\)
−0.719220 + 0.694782i \(0.755501\pi\)
\(648\) 0 0
\(649\) −20.7846 −0.815867
\(650\) 0 0
\(651\) −8.78461 −0.344296
\(652\) 0 0
\(653\) −19.1769 −0.750451 −0.375225 0.926934i \(-0.622435\pi\)
−0.375225 + 0.926934i \(0.622435\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 35.4641 1.38359
\(658\) 0 0
\(659\) 40.6410 1.58315 0.791575 0.611073i \(-0.209262\pi\)
0.791575 + 0.611073i \(0.209262\pi\)
\(660\) 0 0
\(661\) 35.5692 1.38348 0.691741 0.722146i \(-0.256844\pi\)
0.691741 + 0.722146i \(0.256844\pi\)
\(662\) 0 0
\(663\) 8.78461 0.341166
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 11.0718 0.428060
\(670\) 0 0
\(671\) −44.7846 −1.72889
\(672\) 0 0
\(673\) 39.1769 1.51016 0.755080 0.655633i \(-0.227598\pi\)
0.755080 + 0.655633i \(0.227598\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −43.1769 −1.65942 −0.829712 0.558192i \(-0.811495\pi\)
−0.829712 + 0.558192i \(0.811495\pi\)
\(678\) 0 0
\(679\) 8.10512 0.311046
\(680\) 0 0
\(681\) −13.6077 −0.521448
\(682\) 0 0
\(683\) −11.6603 −0.446167 −0.223084 0.974799i \(-0.571612\pi\)
−0.223084 + 0.974799i \(0.571612\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 13.8564 0.528655
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 5.60770 0.213327 0.106663 0.994295i \(-0.465983\pi\)
0.106663 + 0.994295i \(0.465983\pi\)
\(692\) 0 0
\(693\) −10.8231 −0.411135
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 8.78461 0.332741
\(698\) 0 0
\(699\) −1.17691 −0.0445150
\(700\) 0 0
\(701\) −14.7846 −0.558407 −0.279204 0.960232i \(-0.590070\pi\)
−0.279204 + 0.960232i \(0.590070\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.2154 0.572234
\(708\) 0 0
\(709\) −10.1436 −0.380951 −0.190475 0.981692i \(-0.561003\pi\)
−0.190475 + 0.981692i \(0.561003\pi\)
\(710\) 0 0
\(711\) 29.5692 1.10893
\(712\) 0 0
\(713\) −77.5692 −2.90499
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 15.2154 0.568229
\(718\) 0 0
\(719\) 44.7846 1.67018 0.835092 0.550110i \(-0.185414\pi\)
0.835092 + 0.550110i \(0.185414\pi\)
\(720\) 0 0
\(721\) 10.3923 0.387030
\(722\) 0 0
\(723\) −14.9282 −0.555186
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 34.0526 1.26294 0.631470 0.775401i \(-0.282452\pi\)
0.631470 + 0.775401i \(0.282452\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) −35.3205 −1.30638
\(732\) 0 0
\(733\) −38.7846 −1.43254 −0.716271 0.697822i \(-0.754153\pi\)
−0.716271 + 0.697822i \(0.754153\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −35.3205 −1.30105
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 0 0
\(741\) 5.07180 0.186317
\(742\) 0 0
\(743\) 36.5885 1.34230 0.671150 0.741321i \(-0.265801\pi\)
0.671150 + 0.741321i \(0.265801\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 11.6603 0.426626
\(748\) 0 0
\(749\) −21.2154 −0.775193
\(750\) 0 0
\(751\) −40.3923 −1.47394 −0.736968 0.675928i \(-0.763743\pi\)
−0.736968 + 0.675928i \(0.763743\pi\)
\(752\) 0 0
\(753\) −11.3205 −0.412542
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −23.0718 −0.838559 −0.419279 0.907857i \(-0.637717\pi\)
−0.419279 + 0.907857i \(0.637717\pi\)
\(758\) 0 0
\(759\) 20.7846 0.754434
\(760\) 0 0
\(761\) 21.7128 0.787089 0.393544 0.919306i \(-0.371249\pi\)
0.393544 + 0.919306i \(0.371249\pi\)
\(762\) 0 0
\(763\) −1.17691 −0.0426072
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.7846 0.750489
\(768\) 0 0
\(769\) −6.78461 −0.244659 −0.122330 0.992490i \(-0.539037\pi\)
−0.122330 + 0.992490i \(0.539037\pi\)
\(770\) 0 0
\(771\) −4.39230 −0.158185
\(772\) 0 0
\(773\) 46.3923 1.66862 0.834308 0.551299i \(-0.185868\pi\)
0.834308 + 0.551299i \(0.185868\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −5.56922 −0.199795
\(778\) 0 0
\(779\) 5.07180 0.181716
\(780\) 0 0
\(781\) 15.2154 0.544449
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5.80385 0.206885 0.103442 0.994635i \(-0.467014\pi\)
0.103442 + 0.994635i \(0.467014\pi\)
\(788\) 0 0
\(789\) −14.7846 −0.526346
\(790\) 0 0
\(791\) −1.17691 −0.0418463
\(792\) 0 0
\(793\) 44.7846 1.59035
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.60770 −0.0569475 −0.0284737 0.999595i \(-0.509065\pi\)
−0.0284737 + 0.999595i \(0.509065\pi\)
\(798\) 0 0
\(799\) −28.3923 −1.00445
\(800\) 0 0
\(801\) 2.28719 0.0808138
\(802\) 0 0
\(803\) −49.8564 −1.75939
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10.8231 0.380991
\(808\) 0 0
\(809\) −35.5692 −1.25055 −0.625274 0.780406i \(-0.715013\pi\)
−0.625274 + 0.780406i \(0.715013\pi\)
\(810\) 0 0
\(811\) 38.3923 1.34814 0.674068 0.738669i \(-0.264545\pi\)
0.674068 + 0.738669i \(0.264545\pi\)
\(812\) 0 0
\(813\) 3.21539 0.112769
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −20.3923 −0.713436
\(818\) 0 0
\(819\) 10.8231 0.378189
\(820\) 0 0
\(821\) 50.7846 1.77240 0.886198 0.463308i \(-0.153337\pi\)
0.886198 + 0.463308i \(0.153337\pi\)
\(822\) 0 0
\(823\) 30.8372 1.07492 0.537458 0.843290i \(-0.319385\pi\)
0.537458 + 0.843290i \(0.319385\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.73205 0.164550 0.0822748 0.996610i \(-0.473781\pi\)
0.0822748 + 0.996610i \(0.473781\pi\)
\(828\) 0 0
\(829\) −50.7846 −1.76382 −0.881911 0.471416i \(-0.843743\pi\)
−0.881911 + 0.471416i \(0.843743\pi\)
\(830\) 0 0
\(831\) 14.5359 0.504245
\(832\) 0 0
\(833\) 18.6795 0.647206
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 37.8564 1.30851
\(838\) 0 0
\(839\) −8.78461 −0.303278 −0.151639 0.988436i \(-0.548455\pi\)
−0.151639 + 0.988436i \(0.548455\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 20.7846 0.715860
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.26795 0.0435672
\(848\) 0 0
\(849\) 7.75129 0.266024
\(850\) 0 0
\(851\) −49.1769 −1.68576
\(852\) 0 0
\(853\) 3.46410 0.118609 0.0593043 0.998240i \(-0.481112\pi\)
0.0593043 + 0.998240i \(0.481112\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −16.1436 −0.551455 −0.275727 0.961236i \(-0.588919\pi\)
−0.275727 + 0.961236i \(0.588919\pi\)
\(858\) 0 0
\(859\) 51.5692 1.75952 0.879760 0.475419i \(-0.157704\pi\)
0.879760 + 0.475419i \(0.157704\pi\)
\(860\) 0 0
\(861\) −2.35383 −0.0802183
\(862\) 0 0
\(863\) −40.9808 −1.39500 −0.697501 0.716584i \(-0.745705\pi\)
−0.697501 + 0.716584i \(0.745705\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −3.66025 −0.124309
\(868\) 0 0
\(869\) −41.5692 −1.41014
\(870\) 0 0
\(871\) 35.3205 1.19679
\(872\) 0 0
\(873\) −15.7513 −0.533100
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.85641 0.265292 0.132646 0.991163i \(-0.457653\pi\)
0.132646 + 0.991163i \(0.457653\pi\)
\(878\) 0 0
\(879\) −21.9615 −0.740744
\(880\) 0 0
\(881\) −7.60770 −0.256310 −0.128155 0.991754i \(-0.540905\pi\)
−0.128155 + 0.991754i \(0.540905\pi\)
\(882\) 0 0
\(883\) 5.80385 0.195315 0.0976575 0.995220i \(-0.468865\pi\)
0.0976575 + 0.995220i \(0.468865\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −21.3731 −0.717637 −0.358819 0.933407i \(-0.616820\pi\)
−0.358819 + 0.933407i \(0.616820\pi\)
\(888\) 0 0
\(889\) 4.82309 0.161761
\(890\) 0 0
\(891\) 15.4641 0.518067
\(892\) 0 0
\(893\) −16.3923 −0.548548
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −20.7846 −0.693978
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 9.46410 0.314946
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.41154 0.0468695 0.0234348 0.999725i \(-0.492540\pi\)
0.0234348 + 0.999725i \(0.492540\pi\)
\(908\) 0 0
\(909\) −29.5692 −0.980749
\(910\) 0 0
\(911\) −37.1769 −1.23173 −0.615863 0.787853i \(-0.711193\pi\)
−0.615863 + 0.787853i \(0.711193\pi\)
\(912\) 0 0
\(913\) −16.3923 −0.542506
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −13.1769 −0.435140
\(918\) 0 0
\(919\) −39.7128 −1.31000 −0.655002 0.755627i \(-0.727332\pi\)
−0.655002 + 0.755627i \(0.727332\pi\)
\(920\) 0 0
\(921\) −25.0333 −0.824876
\(922\) 0 0
\(923\) −15.2154 −0.500821
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −20.1962 −0.663329
\(928\) 0 0
\(929\) 7.60770 0.249600 0.124800 0.992182i \(-0.460171\pi\)
0.124800 + 0.992182i \(0.460171\pi\)
\(930\) 0 0
\(931\) 10.7846 0.353451
\(932\) 0 0
\(933\) 5.56922 0.182328
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −9.60770 −0.313870 −0.156935 0.987609i \(-0.550161\pi\)
−0.156935 + 0.987609i \(0.550161\pi\)
\(938\) 0 0
\(939\) 16.1051 0.525571
\(940\) 0 0
\(941\) 20.7846 0.677559 0.338779 0.940866i \(-0.389986\pi\)
0.338779 + 0.940866i \(0.389986\pi\)
\(942\) 0 0
\(943\) −20.7846 −0.676840
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.8038 1.48843 0.744213 0.667943i \(-0.232825\pi\)
0.744213 + 0.667943i \(0.232825\pi\)
\(948\) 0 0
\(949\) 49.8564 1.61841
\(950\) 0 0
\(951\) 16.3923 0.531557
\(952\) 0 0
\(953\) 24.9282 0.807504 0.403752 0.914869i \(-0.367706\pi\)
0.403752 + 0.914869i \(0.367706\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 16.3923 0.529335
\(960\) 0 0
\(961\) 58.5692 1.88933
\(962\) 0 0
\(963\) 41.2295 1.32860
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.19615 −0.263570 −0.131785 0.991278i \(-0.542071\pi\)
−0.131785 + 0.991278i \(0.542071\pi\)
\(968\) 0 0
\(969\) 5.07180 0.162930
\(970\) 0 0
\(971\) 33.0333 1.06009 0.530045 0.847970i \(-0.322175\pi\)
0.530045 + 0.847970i \(0.322175\pi\)
\(972\) 0 0
\(973\) −12.6795 −0.406486
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.32051 −0.170218 −0.0851091 0.996372i \(-0.527124\pi\)
−0.0851091 + 0.996372i \(0.527124\pi\)
\(978\) 0 0
\(979\) −3.21539 −0.102764
\(980\) 0 0
\(981\) 2.28719 0.0730243
\(982\) 0 0
\(983\) −11.4115 −0.363972 −0.181986 0.983301i \(-0.558253\pi\)
−0.181986 + 0.983301i \(0.558253\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 7.60770 0.242156
\(988\) 0 0
\(989\) 83.5692 2.65735
\(990\) 0 0
\(991\) 44.1051 1.40105 0.700523 0.713630i \(-0.252950\pi\)
0.700523 + 0.713630i \(0.252950\pi\)
\(992\) 0 0
\(993\) −4.10512 −0.130272
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25.6077 0.811004 0.405502 0.914094i \(-0.367097\pi\)
0.405502 + 0.914094i \(0.367097\pi\)
\(998\) 0 0
\(999\) 24.0000 0.759326
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.bf.1.2 2
4.3 odd 2 6400.2.a.cd.1.1 2
5.4 even 2 1280.2.a.m.1.1 2
8.3 odd 2 6400.2.a.y.1.2 2
8.5 even 2 6400.2.a.ck.1.1 2
16.3 odd 4 1600.2.d.h.801.2 4
16.5 even 4 1600.2.d.b.801.2 4
16.11 odd 4 1600.2.d.h.801.3 4
16.13 even 4 1600.2.d.b.801.3 4
20.19 odd 2 1280.2.a.b.1.2 2
40.19 odd 2 1280.2.a.p.1.1 2
40.29 even 2 1280.2.a.c.1.2 2
80.3 even 4 1600.2.f.e.1249.3 4
80.13 odd 4 1600.2.f.h.1249.2 4
80.19 odd 4 320.2.d.a.161.3 yes 4
80.27 even 4 1600.2.f.e.1249.4 4
80.29 even 4 320.2.d.b.161.2 yes 4
80.37 odd 4 1600.2.f.h.1249.1 4
80.43 even 4 1600.2.f.i.1249.1 4
80.53 odd 4 1600.2.f.d.1249.4 4
80.59 odd 4 320.2.d.a.161.2 4
80.67 even 4 1600.2.f.i.1249.2 4
80.69 even 4 320.2.d.b.161.3 yes 4
80.77 odd 4 1600.2.f.d.1249.3 4
240.29 odd 4 2880.2.k.l.1441.1 4
240.59 even 4 2880.2.k.e.1441.4 4
240.149 odd 4 2880.2.k.l.1441.3 4
240.179 even 4 2880.2.k.e.1441.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.2 4 80.59 odd 4
320.2.d.a.161.3 yes 4 80.19 odd 4
320.2.d.b.161.2 yes 4 80.29 even 4
320.2.d.b.161.3 yes 4 80.69 even 4
1280.2.a.b.1.2 2 20.19 odd 2
1280.2.a.c.1.2 2 40.29 even 2
1280.2.a.m.1.1 2 5.4 even 2
1280.2.a.p.1.1 2 40.19 odd 2
1600.2.d.b.801.2 4 16.5 even 4
1600.2.d.b.801.3 4 16.13 even 4
1600.2.d.h.801.2 4 16.3 odd 4
1600.2.d.h.801.3 4 16.11 odd 4
1600.2.f.d.1249.3 4 80.77 odd 4
1600.2.f.d.1249.4 4 80.53 odd 4
1600.2.f.e.1249.3 4 80.3 even 4
1600.2.f.e.1249.4 4 80.27 even 4
1600.2.f.h.1249.1 4 80.37 odd 4
1600.2.f.h.1249.2 4 80.13 odd 4
1600.2.f.i.1249.1 4 80.43 even 4
1600.2.f.i.1249.2 4 80.67 even 4
2880.2.k.e.1441.2 4 240.179 even 4
2880.2.k.e.1441.4 4 240.59 even 4
2880.2.k.l.1441.1 4 240.29 odd 4
2880.2.k.l.1441.3 4 240.149 odd 4
6400.2.a.y.1.2 2 8.3 odd 2
6400.2.a.bf.1.2 2 1.1 even 1 trivial
6400.2.a.cd.1.1 2 4.3 odd 2
6400.2.a.ck.1.1 2 8.5 even 2