Properties

Label 6864.2.a.cf.1.1
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.2172244.1
Defining polynomial: \(x^{5} - 2 x^{4} - 8 x^{3} + 16 x^{2} + 5 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.19338\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.47314 q^{5} +3.67591 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.47314 q^{5} +3.67591 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -3.47314 q^{15} +2.76229 q^{17} -6.53152 q^{19} +3.67591 q^{21} +2.05410 q^{23} +7.06267 q^{25} +1.00000 q^{27} +5.71513 q^{29} -4.73430 q^{31} +1.00000 q^{33} -12.7670 q^{35} -5.04350 q^{37} +1.00000 q^{39} +9.96544 q^{41} -0.157873 q^{43} -3.47314 q^{45} +12.5681 q^{47} +6.51235 q^{49} +2.76229 q^{51} -9.05447 q^{53} -3.47314 q^{55} -6.53152 q^{57} -0.0906653 q^{59} +10.0719 q^{61} +3.67591 q^{63} -3.47314 q^{65} +0.535436 q^{67} +2.05410 q^{69} +13.9113 q^{71} -12.6222 q^{73} +7.06267 q^{75} +3.67591 q^{77} -6.77087 q^{79} +1.00000 q^{81} -10.4063 q^{83} -9.59382 q^{85} +5.71513 q^{87} -7.25547 q^{89} +3.67591 q^{91} -4.73430 q^{93} +22.6849 q^{95} -0.269986 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + 5q^{11} + 5q^{13} + q^{15} + 4q^{19} + 5q^{21} + 5q^{23} + 4q^{25} + 5q^{27} + 11q^{29} + 8q^{31} + 5q^{33} + 5q^{35} - 8q^{37} + 5q^{39} - q^{41} - q^{43} + q^{45} + 18q^{47} + 10q^{49} + 2q^{53} + q^{55} + 4q^{57} + 13q^{59} + 9q^{61} + 5q^{63} + q^{65} - 5q^{67} + 5q^{69} + 24q^{71} - 13q^{73} + 4q^{75} + 5q^{77} + 6q^{79} + 5q^{81} + 22q^{83} - 22q^{85} + 11q^{87} - 14q^{89} + 5q^{91} + 8q^{93} + 32q^{95} - 20q^{97} + 5q^{99} + O(q^{100}) \)

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
\(4\) 0 0
\(5\) −3.47314 −1.55323 −0.776617 0.629973i \(-0.783066\pi\)
−0.776617 + 0.629973i \(0.783066\pi\)
\(6\) 0 0
\(7\) 3.67591 1.38937 0.694683 0.719316i \(-0.255545\pi\)
0.694683 + 0.719316i \(0.255545\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −3.47314 −0.896760
\(16\) 0 0
\(17\) 2.76229 0.669955 0.334977 0.942226i \(-0.391271\pi\)
0.334977 + 0.942226i \(0.391271\pi\)
\(18\) 0 0
\(19\) −6.53152 −1.49843 −0.749217 0.662325i \(-0.769570\pi\)
−0.749217 + 0.662325i \(0.769570\pi\)
\(20\) 0 0
\(21\) 3.67591 0.802150
\(22\) 0 0
\(23\) 2.05410 0.428309 0.214155 0.976800i \(-0.431300\pi\)
0.214155 + 0.976800i \(0.431300\pi\)
\(24\) 0 0
\(25\) 7.06267 1.41253
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 5.71513 1.06127 0.530636 0.847600i \(-0.321953\pi\)
0.530636 + 0.847600i \(0.321953\pi\)
\(30\) 0 0
\(31\) −4.73430 −0.850305 −0.425153 0.905122i \(-0.639780\pi\)
−0.425153 + 0.905122i \(0.639780\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −12.7670 −2.15801
\(36\) 0 0
\(37\) −5.04350 −0.829146 −0.414573 0.910016i \(-0.636069\pi\)
−0.414573 + 0.910016i \(0.636069\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 9.96544 1.55634 0.778170 0.628053i \(-0.216148\pi\)
0.778170 + 0.628053i \(0.216148\pi\)
\(42\) 0 0
\(43\) −0.157873 −0.0240753 −0.0120377 0.999928i \(-0.503832\pi\)
−0.0120377 + 0.999928i \(0.503832\pi\)
\(44\) 0 0
\(45\) −3.47314 −0.517744
\(46\) 0 0
\(47\) 12.5681 1.83324 0.916622 0.399755i \(-0.130905\pi\)
0.916622 + 0.399755i \(0.130905\pi\)
\(48\) 0 0
\(49\) 6.51235 0.930336
\(50\) 0 0
\(51\) 2.76229 0.386799
\(52\) 0 0
\(53\) −9.05447 −1.24373 −0.621863 0.783126i \(-0.713624\pi\)
−0.621863 + 0.783126i \(0.713624\pi\)
\(54\) 0 0
\(55\) −3.47314 −0.468318
\(56\) 0 0
\(57\) −6.53152 −0.865121
\(58\) 0 0
\(59\) −0.0906653 −0.0118036 −0.00590181 0.999983i \(-0.501879\pi\)
−0.00590181 + 0.999983i \(0.501879\pi\)
\(60\) 0 0
\(61\) 10.0719 1.28957 0.644785 0.764364i \(-0.276947\pi\)
0.644785 + 0.764364i \(0.276947\pi\)
\(62\) 0 0
\(63\) 3.67591 0.463122
\(64\) 0 0
\(65\) −3.47314 −0.430789
\(66\) 0 0
\(67\) 0.535436 0.0654140 0.0327070 0.999465i \(-0.489587\pi\)
0.0327070 + 0.999465i \(0.489587\pi\)
\(68\) 0 0
\(69\) 2.05410 0.247284
\(70\) 0 0
\(71\) 13.9113 1.65097 0.825486 0.564422i \(-0.190901\pi\)
0.825486 + 0.564422i \(0.190901\pi\)
\(72\) 0 0
\(73\) −12.6222 −1.47732 −0.738658 0.674081i \(-0.764540\pi\)
−0.738658 + 0.674081i \(0.764540\pi\)
\(74\) 0 0
\(75\) 7.06267 0.815527
\(76\) 0 0
\(77\) 3.67591 0.418909
\(78\) 0 0
\(79\) −6.77087 −0.761782 −0.380891 0.924620i \(-0.624383\pi\)
−0.380891 + 0.924620i \(0.624383\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.4063 −1.14224 −0.571120 0.820867i \(-0.693491\pi\)
−0.571120 + 0.820867i \(0.693491\pi\)
\(84\) 0 0
\(85\) −9.59382 −1.04060
\(86\) 0 0
\(87\) 5.71513 0.612726
\(88\) 0 0
\(89\) −7.25547 −0.769078 −0.384539 0.923109i \(-0.625640\pi\)
−0.384539 + 0.923109i \(0.625640\pi\)
\(90\) 0 0
\(91\) 3.67591 0.385341
\(92\) 0 0
\(93\) −4.73430 −0.490924
\(94\) 0 0
\(95\) 22.6849 2.32742
\(96\) 0 0
\(97\) −0.269986 −0.0274130 −0.0137065 0.999906i \(-0.504363\pi\)
−0.0137065 + 0.999906i \(0.504363\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 1.34590 0.133922 0.0669612 0.997756i \(-0.478670\pi\)
0.0669612 + 0.997756i \(0.478670\pi\)
\(102\) 0 0
\(103\) 6.85561 0.675503 0.337751 0.941235i \(-0.390334\pi\)
0.337751 + 0.941235i \(0.390334\pi\)
\(104\) 0 0
\(105\) −12.7670 −1.24593
\(106\) 0 0
\(107\) 4.84703 0.468580 0.234290 0.972167i \(-0.424723\pi\)
0.234290 + 0.972167i \(0.424723\pi\)
\(108\) 0 0
\(109\) 7.83643 0.750594 0.375297 0.926905i \(-0.377541\pi\)
0.375297 + 0.926905i \(0.377541\pi\)
\(110\) 0 0
\(111\) −5.04350 −0.478708
\(112\) 0 0
\(113\) 15.6377 1.47107 0.735535 0.677487i \(-0.236931\pi\)
0.735535 + 0.677487i \(0.236931\pi\)
\(114\) 0 0
\(115\) −7.13416 −0.665264
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 10.1540 0.930812
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 9.96544 0.898554
\(124\) 0 0
\(125\) −7.16394 −0.640762
\(126\) 0 0
\(127\) −0.562162 −0.0498838 −0.0249419 0.999689i \(-0.507940\pi\)
−0.0249419 + 0.999689i \(0.507940\pi\)
\(128\) 0 0
\(129\) −0.157873 −0.0138999
\(130\) 0 0
\(131\) 6.74741 0.589524 0.294762 0.955571i \(-0.404760\pi\)
0.294762 + 0.955571i \(0.404760\pi\)
\(132\) 0 0
\(133\) −24.0093 −2.08187
\(134\) 0 0
\(135\) −3.47314 −0.298920
\(136\) 0 0
\(137\) −16.4643 −1.40664 −0.703321 0.710873i \(-0.748300\pi\)
−0.703321 + 0.710873i \(0.748300\pi\)
\(138\) 0 0
\(139\) 10.3894 0.881218 0.440609 0.897699i \(-0.354763\pi\)
0.440609 + 0.897699i \(0.354763\pi\)
\(140\) 0 0
\(141\) 12.5681 1.05842
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −19.8494 −1.64840
\(146\) 0 0
\(147\) 6.51235 0.537129
\(148\) 0 0
\(149\) −3.75801 −0.307868 −0.153934 0.988081i \(-0.549194\pi\)
−0.153934 + 0.988081i \(0.549194\pi\)
\(150\) 0 0
\(151\) 9.10984 0.741348 0.370674 0.928763i \(-0.379127\pi\)
0.370674 + 0.928763i \(0.379127\pi\)
\(152\) 0 0
\(153\) 2.76229 0.223318
\(154\) 0 0
\(155\) 16.4429 1.32072
\(156\) 0 0
\(157\) −14.2318 −1.13582 −0.567909 0.823091i \(-0.692248\pi\)
−0.567909 + 0.823091i \(0.692248\pi\)
\(158\) 0 0
\(159\) −9.05447 −0.718066
\(160\) 0 0
\(161\) 7.55069 0.595078
\(162\) 0 0
\(163\) −16.5636 −1.29736 −0.648679 0.761062i \(-0.724678\pi\)
−0.648679 + 0.761062i \(0.724678\pi\)
\(164\) 0 0
\(165\) −3.47314 −0.270383
\(166\) 0 0
\(167\) 2.56608 0.198569 0.0992845 0.995059i \(-0.468345\pi\)
0.0992845 + 0.995059i \(0.468345\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −6.53152 −0.499478
\(172\) 0 0
\(173\) 6.52583 0.496149 0.248075 0.968741i \(-0.420202\pi\)
0.248075 + 0.968741i \(0.420202\pi\)
\(174\) 0 0
\(175\) 25.9618 1.96253
\(176\) 0 0
\(177\) −0.0906653 −0.00681482
\(178\) 0 0
\(179\) 18.9183 1.41402 0.707009 0.707204i \(-0.250044\pi\)
0.707009 + 0.707204i \(0.250044\pi\)
\(180\) 0 0
\(181\) 21.2809 1.58180 0.790900 0.611946i \(-0.209613\pi\)
0.790900 + 0.611946i \(0.209613\pi\)
\(182\) 0 0
\(183\) 10.0719 0.744534
\(184\) 0 0
\(185\) 17.5168 1.28786
\(186\) 0 0
\(187\) 2.76229 0.201999
\(188\) 0 0
\(189\) 3.67591 0.267383
\(190\) 0 0
\(191\) 0.0735212 0.00531981 0.00265990 0.999996i \(-0.499153\pi\)
0.00265990 + 0.999996i \(0.499153\pi\)
\(192\) 0 0
\(193\) 15.4778 1.11412 0.557058 0.830474i \(-0.311930\pi\)
0.557058 + 0.830474i \(0.311930\pi\)
\(194\) 0 0
\(195\) −3.47314 −0.248716
\(196\) 0 0
\(197\) 10.5229 0.749729 0.374865 0.927080i \(-0.377689\pi\)
0.374865 + 0.927080i \(0.377689\pi\)
\(198\) 0 0
\(199\) −11.1267 −0.788754 −0.394377 0.918949i \(-0.629040\pi\)
−0.394377 + 0.918949i \(0.629040\pi\)
\(200\) 0 0
\(201\) 0.535436 0.0377668
\(202\) 0 0
\(203\) 21.0083 1.47450
\(204\) 0 0
\(205\) −34.6113 −2.41736
\(206\) 0 0
\(207\) 2.05410 0.142770
\(208\) 0 0
\(209\) −6.53152 −0.451795
\(210\) 0 0
\(211\) −26.3391 −1.81326 −0.906629 0.421929i \(-0.861353\pi\)
−0.906629 + 0.421929i \(0.861353\pi\)
\(212\) 0 0
\(213\) 13.9113 0.953190
\(214\) 0 0
\(215\) 0.548313 0.0373946
\(216\) 0 0
\(217\) −17.4029 −1.18138
\(218\) 0 0
\(219\) −12.6222 −0.852928
\(220\) 0 0
\(221\) 2.76229 0.185812
\(222\) 0 0
\(223\) 19.4380 1.30166 0.650831 0.759223i \(-0.274421\pi\)
0.650831 + 0.759223i \(0.274421\pi\)
\(224\) 0 0
\(225\) 7.06267 0.470845
\(226\) 0 0
\(227\) 25.3994 1.68582 0.842908 0.538058i \(-0.180842\pi\)
0.842908 + 0.538058i \(0.180842\pi\)
\(228\) 0 0
\(229\) 15.9460 1.05374 0.526871 0.849945i \(-0.323365\pi\)
0.526871 + 0.849945i \(0.323365\pi\)
\(230\) 0 0
\(231\) 3.67591 0.241857
\(232\) 0 0
\(233\) −13.9924 −0.916674 −0.458337 0.888779i \(-0.651555\pi\)
−0.458337 + 0.888779i \(0.651555\pi\)
\(234\) 0 0
\(235\) −43.6507 −2.84746
\(236\) 0 0
\(237\) −6.77087 −0.439815
\(238\) 0 0
\(239\) −12.8121 −0.828745 −0.414373 0.910107i \(-0.635999\pi\)
−0.414373 + 0.910107i \(0.635999\pi\)
\(240\) 0 0
\(241\) −5.64919 −0.363896 −0.181948 0.983308i \(-0.558240\pi\)
−0.181948 + 0.983308i \(0.558240\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −22.6183 −1.44503
\(246\) 0 0
\(247\) −6.53152 −0.415591
\(248\) 0 0
\(249\) −10.4063 −0.659472
\(250\) 0 0
\(251\) 26.7690 1.68964 0.844821 0.535049i \(-0.179707\pi\)
0.844821 + 0.535049i \(0.179707\pi\)
\(252\) 0 0
\(253\) 2.05410 0.129140
\(254\) 0 0
\(255\) −9.59382 −0.600788
\(256\) 0 0
\(257\) 25.3436 1.58089 0.790445 0.612532i \(-0.209849\pi\)
0.790445 + 0.612532i \(0.209849\pi\)
\(258\) 0 0
\(259\) −18.5395 −1.15199
\(260\) 0 0
\(261\) 5.71513 0.353758
\(262\) 0 0
\(263\) 7.11046 0.438450 0.219225 0.975674i \(-0.429647\pi\)
0.219225 + 0.975674i \(0.429647\pi\)
\(264\) 0 0
\(265\) 31.4474 1.93180
\(266\) 0 0
\(267\) −7.25547 −0.444028
\(268\) 0 0
\(269\) −3.90037 −0.237810 −0.118905 0.992906i \(-0.537938\pi\)
−0.118905 + 0.992906i \(0.537938\pi\)
\(270\) 0 0
\(271\) 23.1368 1.40546 0.702730 0.711457i \(-0.251964\pi\)
0.702730 + 0.711457i \(0.251964\pi\)
\(272\) 0 0
\(273\) 3.67591 0.222476
\(274\) 0 0
\(275\) 7.06267 0.425895
\(276\) 0 0
\(277\) 14.2773 0.857839 0.428920 0.903343i \(-0.358894\pi\)
0.428920 + 0.903343i \(0.358894\pi\)
\(278\) 0 0
\(279\) −4.73430 −0.283435
\(280\) 0 0
\(281\) −30.6045 −1.82571 −0.912857 0.408280i \(-0.866129\pi\)
−0.912857 + 0.408280i \(0.866129\pi\)
\(282\) 0 0
\(283\) −5.13681 −0.305352 −0.152676 0.988276i \(-0.548789\pi\)
−0.152676 + 0.988276i \(0.548789\pi\)
\(284\) 0 0
\(285\) 22.6849 1.34373
\(286\) 0 0
\(287\) 36.6321 2.16233
\(288\) 0 0
\(289\) −9.36973 −0.551161
\(290\) 0 0
\(291\) −0.269986 −0.0158269
\(292\) 0 0
\(293\) −13.6584 −0.797931 −0.398966 0.916966i \(-0.630631\pi\)
−0.398966 + 0.916966i \(0.630631\pi\)
\(294\) 0 0
\(295\) 0.314893 0.0183338
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 2.05410 0.118792
\(300\) 0 0
\(301\) −0.580326 −0.0334494
\(302\) 0 0
\(303\) 1.34590 0.0773202
\(304\) 0 0
\(305\) −34.9809 −2.00300
\(306\) 0 0
\(307\) −9.02002 −0.514800 −0.257400 0.966305i \(-0.582866\pi\)
−0.257400 + 0.966305i \(0.582866\pi\)
\(308\) 0 0
\(309\) 6.85561 0.390002
\(310\) 0 0
\(311\) 30.5299 1.73119 0.865595 0.500745i \(-0.166941\pi\)
0.865595 + 0.500745i \(0.166941\pi\)
\(312\) 0 0
\(313\) −6.37842 −0.360529 −0.180265 0.983618i \(-0.557695\pi\)
−0.180265 + 0.983618i \(0.557695\pi\)
\(314\) 0 0
\(315\) −12.7670 −0.719336
\(316\) 0 0
\(317\) 8.71917 0.489717 0.244859 0.969559i \(-0.421258\pi\)
0.244859 + 0.969559i \(0.421258\pi\)
\(318\) 0 0
\(319\) 5.71513 0.319986
\(320\) 0 0
\(321\) 4.84703 0.270535
\(322\) 0 0
\(323\) −18.0420 −1.00388
\(324\) 0 0
\(325\) 7.06267 0.391767
\(326\) 0 0
\(327\) 7.83643 0.433356
\(328\) 0 0
\(329\) 46.1992 2.54705
\(330\) 0 0
\(331\) 10.7678 0.591853 0.295926 0.955211i \(-0.404372\pi\)
0.295926 + 0.955211i \(0.404372\pi\)
\(332\) 0 0
\(333\) −5.04350 −0.276382
\(334\) 0 0
\(335\) −1.85964 −0.101603
\(336\) 0 0
\(337\) 7.19746 0.392070 0.196035 0.980597i \(-0.437193\pi\)
0.196035 + 0.980597i \(0.437193\pi\)
\(338\) 0 0
\(339\) 15.6377 0.849323
\(340\) 0 0
\(341\) −4.73430 −0.256377
\(342\) 0 0
\(343\) −1.79256 −0.0967894
\(344\) 0 0
\(345\) −7.13416 −0.384090
\(346\) 0 0
\(347\) −8.45206 −0.453730 −0.226865 0.973926i \(-0.572848\pi\)
−0.226865 + 0.973926i \(0.572848\pi\)
\(348\) 0 0
\(349\) −28.6958 −1.53605 −0.768026 0.640418i \(-0.778761\pi\)
−0.768026 + 0.640418i \(0.778761\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 10.8859 0.579396 0.289698 0.957118i \(-0.406445\pi\)
0.289698 + 0.957118i \(0.406445\pi\)
\(354\) 0 0
\(355\) −48.3160 −2.56435
\(356\) 0 0
\(357\) 10.1540 0.537404
\(358\) 0 0
\(359\) 0.215887 0.0113941 0.00569705 0.999984i \(-0.498187\pi\)
0.00569705 + 0.999984i \(0.498187\pi\)
\(360\) 0 0
\(361\) 23.6608 1.24530
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 43.8386 2.29462
\(366\) 0 0
\(367\) 32.4070 1.69163 0.845817 0.533473i \(-0.179114\pi\)
0.845817 + 0.533473i \(0.179114\pi\)
\(368\) 0 0
\(369\) 9.96544 0.518780
\(370\) 0 0
\(371\) −33.2835 −1.72799
\(372\) 0 0
\(373\) 36.5925 1.89469 0.947345 0.320215i \(-0.103755\pi\)
0.947345 + 0.320215i \(0.103755\pi\)
\(374\) 0 0
\(375\) −7.16394 −0.369944
\(376\) 0 0
\(377\) 5.71513 0.294344
\(378\) 0 0
\(379\) 16.8592 0.866000 0.433000 0.901394i \(-0.357455\pi\)
0.433000 + 0.901394i \(0.357455\pi\)
\(380\) 0 0
\(381\) −0.562162 −0.0288004
\(382\) 0 0
\(383\) 22.6567 1.15771 0.578853 0.815432i \(-0.303501\pi\)
0.578853 + 0.815432i \(0.303501\pi\)
\(384\) 0 0
\(385\) −12.7670 −0.650664
\(386\) 0 0
\(387\) −0.157873 −0.00802511
\(388\) 0 0
\(389\) −3.25220 −0.164893 −0.0824466 0.996595i \(-0.526273\pi\)
−0.0824466 + 0.996595i \(0.526273\pi\)
\(390\) 0 0
\(391\) 5.67403 0.286948
\(392\) 0 0
\(393\) 6.74741 0.340362
\(394\) 0 0
\(395\) 23.5161 1.18323
\(396\) 0 0
\(397\) 8.08044 0.405545 0.202773 0.979226i \(-0.435005\pi\)
0.202773 + 0.979226i \(0.435005\pi\)
\(398\) 0 0
\(399\) −24.0093 −1.20197
\(400\) 0 0
\(401\) 34.0056 1.69816 0.849080 0.528264i \(-0.177157\pi\)
0.849080 + 0.528264i \(0.177157\pi\)
\(402\) 0 0
\(403\) −4.73430 −0.235832
\(404\) 0 0
\(405\) −3.47314 −0.172581
\(406\) 0 0
\(407\) −5.04350 −0.249997
\(408\) 0 0
\(409\) −26.6961 −1.32004 −0.660018 0.751250i \(-0.729451\pi\)
−0.660018 + 0.751250i \(0.729451\pi\)
\(410\) 0 0
\(411\) −16.4643 −0.812125
\(412\) 0 0
\(413\) −0.333278 −0.0163995
\(414\) 0 0
\(415\) 36.1425 1.77416
\(416\) 0 0
\(417\) 10.3894 0.508771
\(418\) 0 0
\(419\) 6.16191 0.301029 0.150514 0.988608i \(-0.451907\pi\)
0.150514 + 0.988608i \(0.451907\pi\)
\(420\) 0 0
\(421\) −32.3136 −1.57487 −0.787433 0.616400i \(-0.788590\pi\)
−0.787433 + 0.616400i \(0.788590\pi\)
\(422\) 0 0
\(423\) 12.5681 0.611081
\(424\) 0 0
\(425\) 19.5092 0.946334
\(426\) 0 0
\(427\) 37.0233 1.79168
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 2.21967 0.106918 0.0534590 0.998570i \(-0.482975\pi\)
0.0534590 + 0.998570i \(0.482975\pi\)
\(432\) 0 0
\(433\) 24.6612 1.18514 0.592570 0.805519i \(-0.298114\pi\)
0.592570 + 0.805519i \(0.298114\pi\)
\(434\) 0 0
\(435\) −19.8494 −0.951707
\(436\) 0 0
\(437\) −13.4164 −0.641793
\(438\) 0 0
\(439\) 33.9454 1.62012 0.810062 0.586344i \(-0.199433\pi\)
0.810062 + 0.586344i \(0.199433\pi\)
\(440\) 0 0
\(441\) 6.51235 0.310112
\(442\) 0 0
\(443\) 13.6590 0.648959 0.324479 0.945893i \(-0.394811\pi\)
0.324479 + 0.945893i \(0.394811\pi\)
\(444\) 0 0
\(445\) 25.1992 1.19456
\(446\) 0 0
\(447\) −3.75801 −0.177748
\(448\) 0 0
\(449\) −4.75021 −0.224176 −0.112088 0.993698i \(-0.535754\pi\)
−0.112088 + 0.993698i \(0.535754\pi\)
\(450\) 0 0
\(451\) 9.96544 0.469254
\(452\) 0 0
\(453\) 9.10984 0.428017
\(454\) 0 0
\(455\) −12.7670 −0.598524
\(456\) 0 0
\(457\) 19.4730 0.910909 0.455454 0.890259i \(-0.349477\pi\)
0.455454 + 0.890259i \(0.349477\pi\)
\(458\) 0 0
\(459\) 2.76229 0.128933
\(460\) 0 0
\(461\) 4.64226 0.216211 0.108106 0.994139i \(-0.465521\pi\)
0.108106 + 0.994139i \(0.465521\pi\)
\(462\) 0 0
\(463\) 6.24009 0.290001 0.145001 0.989432i \(-0.453682\pi\)
0.145001 + 0.989432i \(0.453682\pi\)
\(464\) 0 0
\(465\) 16.4429 0.762520
\(466\) 0 0
\(467\) −23.0698 −1.06754 −0.533772 0.845628i \(-0.679226\pi\)
−0.533772 + 0.845628i \(0.679226\pi\)
\(468\) 0 0
\(469\) 1.96822 0.0908839
\(470\) 0 0
\(471\) −14.2318 −0.655765
\(472\) 0 0
\(473\) −0.157873 −0.00725899
\(474\) 0 0
\(475\) −46.1300 −2.11659
\(476\) 0 0
\(477\) −9.05447 −0.414576
\(478\) 0 0
\(479\) 28.1806 1.28760 0.643802 0.765192i \(-0.277356\pi\)
0.643802 + 0.765192i \(0.277356\pi\)
\(480\) 0 0
\(481\) −5.04350 −0.229964
\(482\) 0 0
\(483\) 7.55069 0.343568
\(484\) 0 0
\(485\) 0.937699 0.0425787
\(486\) 0 0
\(487\) 12.1584 0.550949 0.275474 0.961308i \(-0.411165\pi\)
0.275474 + 0.961308i \(0.411165\pi\)
\(488\) 0 0
\(489\) −16.5636 −0.749030
\(490\) 0 0
\(491\) 33.2716 1.50153 0.750764 0.660571i \(-0.229686\pi\)
0.750764 + 0.660571i \(0.229686\pi\)
\(492\) 0 0
\(493\) 15.7869 0.711005
\(494\) 0 0
\(495\) −3.47314 −0.156106
\(496\) 0 0
\(497\) 51.1369 2.29380
\(498\) 0 0
\(499\) 25.2022 1.12821 0.564103 0.825704i \(-0.309222\pi\)
0.564103 + 0.825704i \(0.309222\pi\)
\(500\) 0 0
\(501\) 2.56608 0.114644
\(502\) 0 0
\(503\) −1.27414 −0.0568113 −0.0284056 0.999596i \(-0.509043\pi\)
−0.0284056 + 0.999596i \(0.509043\pi\)
\(504\) 0 0
\(505\) −4.67451 −0.208013
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 2.16092 0.0957811 0.0478905 0.998853i \(-0.484750\pi\)
0.0478905 + 0.998853i \(0.484750\pi\)
\(510\) 0 0
\(511\) −46.3981 −2.05253
\(512\) 0 0
\(513\) −6.53152 −0.288374
\(514\) 0 0
\(515\) −23.8104 −1.04921
\(516\) 0 0
\(517\) 12.5681 0.552744
\(518\) 0 0
\(519\) 6.52583 0.286452
\(520\) 0 0
\(521\) −36.1007 −1.58160 −0.790801 0.612074i \(-0.790336\pi\)
−0.790801 + 0.612074i \(0.790336\pi\)
\(522\) 0 0
\(523\) −2.28286 −0.0998226 −0.0499113 0.998754i \(-0.515894\pi\)
−0.0499113 + 0.998754i \(0.515894\pi\)
\(524\) 0 0
\(525\) 25.9618 1.13306
\(526\) 0 0
\(527\) −13.0775 −0.569666
\(528\) 0 0
\(529\) −18.7807 −0.816551
\(530\) 0 0
\(531\) −0.0906653 −0.00393454
\(532\) 0 0
\(533\) 9.96544 0.431651
\(534\) 0 0
\(535\) −16.8344 −0.727815
\(536\) 0 0
\(537\) 18.9183 0.816384
\(538\) 0 0
\(539\) 6.51235 0.280507
\(540\) 0 0
\(541\) 9.86029 0.423927 0.211964 0.977278i \(-0.432014\pi\)
0.211964 + 0.977278i \(0.432014\pi\)
\(542\) 0 0
\(543\) 21.2809 0.913252
\(544\) 0 0
\(545\) −27.2170 −1.16585
\(546\) 0 0
\(547\) 7.92623 0.338901 0.169451 0.985539i \(-0.445801\pi\)
0.169451 + 0.985539i \(0.445801\pi\)
\(548\) 0 0
\(549\) 10.0719 0.429857
\(550\) 0 0
\(551\) −37.3285 −1.59025
\(552\) 0 0
\(553\) −24.8891 −1.05839
\(554\) 0 0
\(555\) 17.5168 0.743545
\(556\) 0 0
\(557\) −36.9855 −1.56712 −0.783562 0.621313i \(-0.786600\pi\)
−0.783562 + 0.621313i \(0.786600\pi\)
\(558\) 0 0
\(559\) −0.157873 −0.00667730
\(560\) 0 0
\(561\) 2.76229 0.116624
\(562\) 0 0
\(563\) −4.05599 −0.170940 −0.0854698 0.996341i \(-0.527239\pi\)
−0.0854698 + 0.996341i \(0.527239\pi\)
\(564\) 0 0
\(565\) −54.3118 −2.28492
\(566\) 0 0
\(567\) 3.67591 0.154374
\(568\) 0 0
\(569\) −13.2846 −0.556920 −0.278460 0.960448i \(-0.589824\pi\)
−0.278460 + 0.960448i \(0.589824\pi\)
\(570\) 0 0
\(571\) −38.0764 −1.59345 −0.796724 0.604343i \(-0.793436\pi\)
−0.796724 + 0.604343i \(0.793436\pi\)
\(572\) 0 0
\(573\) 0.0735212 0.00307139
\(574\) 0 0
\(575\) 14.5074 0.605001
\(576\) 0 0
\(577\) −25.9104 −1.07867 −0.539333 0.842093i \(-0.681323\pi\)
−0.539333 + 0.842093i \(0.681323\pi\)
\(578\) 0 0
\(579\) 15.4778 0.643235
\(580\) 0 0
\(581\) −38.2527 −1.58699
\(582\) 0 0
\(583\) −9.05447 −0.374998
\(584\) 0 0
\(585\) −3.47314 −0.143596
\(586\) 0 0
\(587\) 18.4856 0.762981 0.381491 0.924373i \(-0.375411\pi\)
0.381491 + 0.924373i \(0.375411\pi\)
\(588\) 0 0
\(589\) 30.9222 1.27413
\(590\) 0 0
\(591\) 10.5229 0.432856
\(592\) 0 0
\(593\) −41.8715 −1.71945 −0.859727 0.510753i \(-0.829367\pi\)
−0.859727 + 0.510753i \(0.829367\pi\)
\(594\) 0 0
\(595\) −35.2661 −1.44577
\(596\) 0 0
\(597\) −11.1267 −0.455388
\(598\) 0 0
\(599\) −41.8712 −1.71081 −0.855406 0.517958i \(-0.826692\pi\)
−0.855406 + 0.517958i \(0.826692\pi\)
\(600\) 0 0
\(601\) −15.7284 −0.641573 −0.320787 0.947152i \(-0.603947\pi\)
−0.320787 + 0.947152i \(0.603947\pi\)
\(602\) 0 0
\(603\) 0.535436 0.0218047
\(604\) 0 0
\(605\) −3.47314 −0.141203
\(606\) 0 0
\(607\) −0.562162 −0.0228174 −0.0114087 0.999935i \(-0.503632\pi\)
−0.0114087 + 0.999935i \(0.503632\pi\)
\(608\) 0 0
\(609\) 21.0083 0.851300
\(610\) 0 0
\(611\) 12.5681 0.508450
\(612\) 0 0
\(613\) 19.6031 0.791763 0.395882 0.918302i \(-0.370439\pi\)
0.395882 + 0.918302i \(0.370439\pi\)
\(614\) 0 0
\(615\) −34.6113 −1.39566
\(616\) 0 0
\(617\) −1.04930 −0.0422434 −0.0211217 0.999777i \(-0.506724\pi\)
−0.0211217 + 0.999777i \(0.506724\pi\)
\(618\) 0 0
\(619\) −43.7446 −1.75824 −0.879122 0.476597i \(-0.841870\pi\)
−0.879122 + 0.476597i \(0.841870\pi\)
\(620\) 0 0
\(621\) 2.05410 0.0824282
\(622\) 0 0
\(623\) −26.6705 −1.06853
\(624\) 0 0
\(625\) −10.4320 −0.417281
\(626\) 0 0
\(627\) −6.53152 −0.260844
\(628\) 0 0
\(629\) −13.9316 −0.555490
\(630\) 0 0
\(631\) −11.6353 −0.463194 −0.231597 0.972812i \(-0.574395\pi\)
−0.231597 + 0.972812i \(0.574395\pi\)
\(632\) 0 0
\(633\) −26.3391 −1.04688
\(634\) 0 0
\(635\) 1.95246 0.0774812
\(636\) 0 0
\(637\) 6.51235 0.258029
\(638\) 0 0
\(639\) 13.9113 0.550324
\(640\) 0 0
\(641\) −45.8876 −1.81245 −0.906226 0.422794i \(-0.861049\pi\)
−0.906226 + 0.422794i \(0.861049\pi\)
\(642\) 0 0
\(643\) −7.24732 −0.285806 −0.142903 0.989737i \(-0.545644\pi\)
−0.142903 + 0.989737i \(0.545644\pi\)
\(644\) 0 0
\(645\) 0.548313 0.0215898
\(646\) 0 0
\(647\) −23.3987 −0.919900 −0.459950 0.887945i \(-0.652133\pi\)
−0.459950 + 0.887945i \(0.652133\pi\)
\(648\) 0 0
\(649\) −0.0906653 −0.00355892
\(650\) 0 0
\(651\) −17.4029 −0.682073
\(652\) 0 0
\(653\) 6.82950 0.267259 0.133630 0.991031i \(-0.457337\pi\)
0.133630 + 0.991031i \(0.457337\pi\)
\(654\) 0 0
\(655\) −23.4347 −0.915668
\(656\) 0 0
\(657\) −12.6222 −0.492438
\(658\) 0 0
\(659\) −28.3348 −1.10377 −0.551883 0.833921i \(-0.686091\pi\)
−0.551883 + 0.833921i \(0.686091\pi\)
\(660\) 0 0
\(661\) 9.10922 0.354307 0.177154 0.984183i \(-0.443311\pi\)
0.177154 + 0.984183i \(0.443311\pi\)
\(662\) 0 0
\(663\) 2.76229 0.107279
\(664\) 0 0
\(665\) 83.3876 3.23363
\(666\) 0 0
\(667\) 11.7394 0.454553
\(668\) 0 0
\(669\) 19.4380 0.751515
\(670\) 0 0
\(671\) 10.0719 0.388820
\(672\) 0 0
\(673\) −17.2284 −0.664105 −0.332053 0.943261i \(-0.607741\pi\)
−0.332053 + 0.943261i \(0.607741\pi\)
\(674\) 0 0
\(675\) 7.06267 0.271842
\(676\) 0 0
\(677\) −39.0073 −1.49917 −0.749586 0.661907i \(-0.769747\pi\)
−0.749586 + 0.661907i \(0.769747\pi\)
\(678\) 0 0
\(679\) −0.992447 −0.0380866
\(680\) 0 0
\(681\) 25.3994 0.973306
\(682\) 0 0
\(683\) 3.03126 0.115988 0.0579940 0.998317i \(-0.481530\pi\)
0.0579940 + 0.998317i \(0.481530\pi\)
\(684\) 0 0
\(685\) 57.1828 2.18484
\(686\) 0 0
\(687\) 15.9460 0.608379
\(688\) 0 0
\(689\) −9.05447 −0.344948
\(690\) 0 0
\(691\) 50.0554 1.90420 0.952099 0.305789i \(-0.0989202\pi\)
0.952099 + 0.305789i \(0.0989202\pi\)
\(692\) 0 0
\(693\) 3.67591 0.139636
\(694\) 0 0
\(695\) −36.0838 −1.36874
\(696\) 0 0
\(697\) 27.5275 1.04268
\(698\) 0 0
\(699\) −13.9924 −0.529242
\(700\) 0 0
\(701\) 22.6512 0.855523 0.427761 0.903892i \(-0.359302\pi\)
0.427761 + 0.903892i \(0.359302\pi\)
\(702\) 0 0
\(703\) 32.9417 1.24242
\(704\) 0 0
\(705\) −43.6507 −1.64398
\(706\) 0 0
\(707\) 4.94743 0.186067
\(708\) 0 0
\(709\) −13.6217 −0.511574 −0.255787 0.966733i \(-0.582335\pi\)
−0.255787 + 0.966733i \(0.582335\pi\)
\(710\) 0 0
\(711\) −6.77087 −0.253927
\(712\) 0 0
\(713\) −9.72472 −0.364194
\(714\) 0 0
\(715\) −3.47314 −0.129888
\(716\) 0 0
\(717\) −12.8121 −0.478476
\(718\) 0 0
\(719\) 10.2561 0.382488 0.191244 0.981543i \(-0.438748\pi\)
0.191244 + 0.981543i \(0.438748\pi\)
\(720\) 0 0
\(721\) 25.2006 0.938520
\(722\) 0 0
\(723\) −5.64919 −0.210096
\(724\) 0 0
\(725\) 40.3641 1.49908
\(726\) 0 0
\(727\) 30.5316 1.13236 0.566178 0.824283i \(-0.308422\pi\)
0.566178 + 0.824283i \(0.308422\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −0.436090 −0.0161294
\(732\) 0 0
\(733\) 12.2705 0.453222 0.226611 0.973985i \(-0.427235\pi\)
0.226611 + 0.973985i \(0.427235\pi\)
\(734\) 0 0
\(735\) −22.6183 −0.834287
\(736\) 0 0
\(737\) 0.535436 0.0197231
\(738\) 0 0
\(739\) −27.8695 −1.02520 −0.512598 0.858629i \(-0.671317\pi\)
−0.512598 + 0.858629i \(0.671317\pi\)
\(740\) 0 0
\(741\) −6.53152 −0.239941
\(742\) 0 0
\(743\) −15.8705 −0.582232 −0.291116 0.956688i \(-0.594027\pi\)
−0.291116 + 0.956688i \(0.594027\pi\)
\(744\) 0 0
\(745\) 13.0521 0.478191
\(746\) 0 0
\(747\) −10.4063 −0.380747
\(748\) 0 0
\(749\) 17.8173 0.651029
\(750\) 0 0
\(751\) −44.3597 −1.61871 −0.809355 0.587320i \(-0.800183\pi\)
−0.809355 + 0.587320i \(0.800183\pi\)
\(752\) 0 0
\(753\) 26.7690 0.975515
\(754\) 0 0
\(755\) −31.6397 −1.15149
\(756\) 0 0
\(757\) 12.6678 0.460420 0.230210 0.973141i \(-0.426059\pi\)
0.230210 + 0.973141i \(0.426059\pi\)
\(758\) 0 0
\(759\) 2.05410 0.0745591
\(760\) 0 0
\(761\) −42.5016 −1.54068 −0.770341 0.637632i \(-0.779914\pi\)
−0.770341 + 0.637632i \(0.779914\pi\)
\(762\) 0 0
\(763\) 28.8061 1.04285
\(764\) 0 0
\(765\) −9.59382 −0.346865
\(766\) 0 0
\(767\) −0.0906653 −0.00327373
\(768\) 0 0
\(769\) −2.60442 −0.0939178 −0.0469589 0.998897i \(-0.514953\pi\)
−0.0469589 + 0.998897i \(0.514953\pi\)
\(770\) 0 0
\(771\) 25.3436 0.912728
\(772\) 0 0
\(773\) −52.5354 −1.88957 −0.944783 0.327697i \(-0.893728\pi\)
−0.944783 + 0.327697i \(0.893728\pi\)
\(774\) 0 0
\(775\) −33.4368 −1.20109
\(776\) 0 0
\(777\) −18.5395 −0.665100
\(778\) 0 0
\(779\) −65.0895 −2.33207
\(780\) 0 0
\(781\) 13.9113 0.497787
\(782\) 0 0
\(783\) 5.71513 0.204242
\(784\) 0 0
\(785\) 49.4288 1.76419
\(786\) 0 0
\(787\) 21.8081 0.777376 0.388688 0.921369i \(-0.372928\pi\)
0.388688 + 0.921369i \(0.372928\pi\)
\(788\) 0 0
\(789\) 7.11046 0.253139
\(790\) 0 0
\(791\) 57.4828 2.04385
\(792\) 0 0
\(793\) 10.0719 0.357662
\(794\) 0 0
\(795\) 31.4474 1.11532
\(796\) 0 0
\(797\) 6.70416 0.237474 0.118737 0.992926i \(-0.462116\pi\)
0.118737 + 0.992926i \(0.462116\pi\)
\(798\) 0 0
\(799\) 34.7168 1.22819
\(800\) 0 0
\(801\) −7.25547 −0.256359
\(802\) 0 0
\(803\) −12.6222 −0.445427
\(804\) 0 0
\(805\) −26.2246 −0.924295
\(806\) 0 0
\(807\) −3.90037 −0.137300
\(808\) 0 0
\(809\) 24.5335 0.862554 0.431277 0.902220i \(-0.358063\pi\)
0.431277 + 0.902220i \(0.358063\pi\)
\(810\) 0 0
\(811\) −7.17969 −0.252113 −0.126057 0.992023i \(-0.540232\pi\)
−0.126057 + 0.992023i \(0.540232\pi\)
\(812\) 0 0
\(813\) 23.1368 0.811443
\(814\) 0 0
\(815\) 57.5275 2.01510
\(816\) 0 0
\(817\) 1.03115 0.0360753
\(818\) 0 0
\(819\) 3.67591 0.128447
\(820\) 0 0
\(821\) 30.9938 1.08169 0.540846 0.841122i \(-0.318104\pi\)
0.540846 + 0.841122i \(0.318104\pi\)
\(822\) 0 0
\(823\) 6.28586 0.219111 0.109556 0.993981i \(-0.465057\pi\)
0.109556 + 0.993981i \(0.465057\pi\)
\(824\) 0 0
\(825\) 7.06267 0.245891
\(826\) 0 0
\(827\) −27.4508 −0.954559 −0.477280 0.878752i \(-0.658377\pi\)
−0.477280 + 0.878752i \(0.658377\pi\)
\(828\) 0 0
\(829\) 11.7463 0.407965 0.203983 0.978975i \(-0.434611\pi\)
0.203983 + 0.978975i \(0.434611\pi\)
\(830\) 0 0
\(831\) 14.2773 0.495274
\(832\) 0 0
\(833\) 17.9890 0.623283
\(834\) 0 0
\(835\) −8.91233 −0.308424
\(836\) 0 0
\(837\) −4.73430 −0.163641
\(838\) 0 0
\(839\) −7.69255 −0.265576 −0.132788 0.991144i \(-0.542393\pi\)
−0.132788 + 0.991144i \(0.542393\pi\)
\(840\) 0 0
\(841\) 3.66268 0.126299
\(842\) 0 0
\(843\) −30.6045 −1.05408
\(844\) 0 0
\(845\) −3.47314 −0.119479
\(846\) 0 0
\(847\) 3.67591 0.126306
\(848\) 0 0
\(849\) −5.13681 −0.176295
\(850\) 0 0
\(851\) −10.3598 −0.355131
\(852\) 0 0
\(853\) 38.1816 1.30731 0.653656 0.756792i \(-0.273234\pi\)
0.653656 + 0.756792i \(0.273234\pi\)
\(854\) 0 0
\(855\) 22.6849 0.775806
\(856\) 0 0
\(857\) 39.0580 1.33420 0.667098 0.744970i \(-0.267536\pi\)
0.667098 + 0.744970i \(0.267536\pi\)
\(858\) 0 0
\(859\) −35.3952 −1.20767 −0.603834 0.797110i \(-0.706361\pi\)
−0.603834 + 0.797110i \(0.706361\pi\)
\(860\) 0 0
\(861\) 36.6321 1.24842
\(862\) 0 0
\(863\) −26.4550 −0.900537 −0.450269 0.892893i \(-0.648672\pi\)
−0.450269 + 0.892893i \(0.648672\pi\)
\(864\) 0 0
\(865\) −22.6651 −0.770636
\(866\) 0 0
\(867\) −9.36973 −0.318213
\(868\) 0 0
\(869\) −6.77087 −0.229686
\(870\) 0 0
\(871\) 0.535436 0.0181426
\(872\) 0 0
\(873\) −0.269986 −0.00913765
\(874\) 0 0
\(875\) −26.3340 −0.890252
\(876\) 0 0
\(877\) −43.7998 −1.47901 −0.739506 0.673150i \(-0.764941\pi\)
−0.739506 + 0.673150i \(0.764941\pi\)
\(878\) 0 0
\(879\) −13.6584 −0.460686
\(880\) 0 0
\(881\) 1.26646 0.0426680 0.0213340 0.999772i \(-0.493209\pi\)
0.0213340 + 0.999772i \(0.493209\pi\)
\(882\) 0 0
\(883\) 27.5866 0.928363 0.464182 0.885740i \(-0.346348\pi\)
0.464182 + 0.885740i \(0.346348\pi\)
\(884\) 0 0
\(885\) 0.314893 0.0105850
\(886\) 0 0
\(887\) −51.9630 −1.74475 −0.872374 0.488840i \(-0.837420\pi\)
−0.872374 + 0.488840i \(0.837420\pi\)
\(888\) 0 0
\(889\) −2.06646 −0.0693068
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −82.0887 −2.74699
\(894\) 0 0
\(895\) −65.7057 −2.19630
\(896\) 0 0
\(897\) 2.05410 0.0685844
\(898\) 0 0
\(899\) −27.0571 −0.902406
\(900\) 0 0
\(901\) −25.0111 −0.833241
\(902\) 0 0
\(903\) −0.580326 −0.0193120
\(904\) 0 0
\(905\) −73.9116 −2.45690
\(906\) 0 0
\(907\) −45.9092 −1.52439 −0.762195 0.647348i \(-0.775878\pi\)
−0.762195 + 0.647348i \(0.775878\pi\)
\(908\) 0 0
\(909\) 1.34590 0.0446408
\(910\) 0 0
\(911\) −0.500631 −0.0165866 −0.00829332 0.999966i \(-0.502640\pi\)
−0.00829332 + 0.999966i \(0.502640\pi\)
\(912\) 0 0
\(913\) −10.4063 −0.344398
\(914\) 0 0
\(915\) −34.9809 −1.15643
\(916\) 0 0
\(917\) 24.8029 0.819064
\(918\) 0 0
\(919\) 25.8572 0.852951 0.426476 0.904499i \(-0.359755\pi\)
0.426476 + 0.904499i \(0.359755\pi\)
\(920\) 0 0
\(921\) −9.02002 −0.297220
\(922\) 0 0
\(923\) 13.9113 0.457897
\(924\) 0 0
\(925\) −35.6206 −1.17120
\(926\) 0 0
\(927\) 6.85561 0.225168
\(928\) 0 0
\(929\) −16.0106 −0.525291 −0.262645 0.964892i \(-0.584595\pi\)
−0.262645 + 0.964892i \(0.584595\pi\)
\(930\) 0 0
\(931\) −42.5355 −1.39405
\(932\) 0 0
\(933\) 30.5299 0.999503
\(934\) 0 0
\(935\) −9.59382 −0.313752
\(936\) 0 0
\(937\) −9.62509 −0.314438 −0.157219 0.987564i \(-0.550253\pi\)
−0.157219 + 0.987564i \(0.550253\pi\)
\(938\) 0 0
\(939\) −6.37842 −0.208152
\(940\) 0 0
\(941\) 23.6425 0.770723 0.385361 0.922766i \(-0.374077\pi\)
0.385361 + 0.922766i \(0.374077\pi\)
\(942\) 0 0
\(943\) 20.4700 0.666595
\(944\) 0 0
\(945\) −12.7670 −0.415309
\(946\) 0 0
\(947\) 42.0705 1.36711 0.683553 0.729900i \(-0.260434\pi\)
0.683553 + 0.729900i \(0.260434\pi\)
\(948\) 0 0
\(949\) −12.6222 −0.409733
\(950\) 0 0
\(951\) 8.71917 0.282738
\(952\) 0 0
\(953\) 32.1039 1.03995 0.519974 0.854182i \(-0.325942\pi\)
0.519974 + 0.854182i \(0.325942\pi\)
\(954\) 0 0
\(955\) −0.255349 −0.00826290
\(956\) 0 0
\(957\) 5.71513 0.184744
\(958\) 0 0
\(959\) −60.5214 −1.95434
\(960\) 0 0
\(961\) −8.58641 −0.276981
\(962\) 0 0
\(963\) 4.84703 0.156193
\(964\) 0 0
\(965\) −53.7565 −1.73048
\(966\) 0 0
\(967\) 16.8199 0.540892 0.270446 0.962735i \(-0.412829\pi\)
0.270446 + 0.962735i \(0.412829\pi\)
\(968\) 0 0
\(969\) −18.0420 −0.579592
\(970\) 0 0
\(971\) 19.2716 0.618456 0.309228 0.950988i \(-0.399929\pi\)
0.309228 + 0.950988i \(0.399929\pi\)
\(972\) 0 0
\(973\) 38.1906 1.22433
\(974\) 0 0
\(975\) 7.06267 0.226186
\(976\) 0 0
\(977\) −25.9084 −0.828884 −0.414442 0.910076i \(-0.636023\pi\)
−0.414442 + 0.910076i \(0.636023\pi\)
\(978\) 0 0
\(979\) −7.25547 −0.231886
\(980\) 0 0
\(981\) 7.83643 0.250198
\(982\) 0 0
\(983\) −1.20529 −0.0384427 −0.0192213 0.999815i \(-0.506119\pi\)
−0.0192213 + 0.999815i \(0.506119\pi\)
\(984\) 0 0
\(985\) −36.5476 −1.16450
\(986\) 0 0
\(987\) 46.1992 1.47054
\(988\) 0 0
\(989\) −0.324286 −0.0103117
\(990\) 0 0
\(991\) 1.20970 0.0384272 0.0192136 0.999815i \(-0.493884\pi\)
0.0192136 + 0.999815i \(0.493884\pi\)
\(992\) 0 0
\(993\) 10.7678 0.341706
\(994\) 0 0
\(995\) 38.6447 1.22512
\(996\) 0 0
\(997\) 5.22232 0.165393 0.0826963 0.996575i \(-0.473647\pi\)
0.0826963 + 0.996575i \(0.473647\pi\)
\(998\) 0 0
\(999\) −5.04350 −0.159569
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 6864.2.a.cf.1.1 5
4.3 odd 2 3432.2.a.w.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.w.1.1 5 4.3 odd 2
6864.2.a.cf.1.1 5 1.1 even 1 trivial