Properties

Label 6864.2.a.cf.1.3
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 1.3
Root \(-2.71661\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.169303 q^{5} +1.46187 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.169303 q^{5} +1.46187 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.169303 q^{15} +6.72578 q^{17} +6.80398 q^{19} +1.46187 q^{21} -5.29802 q^{23} -4.97134 q^{25} +1.00000 q^{27} -4.23175 q^{29} +7.17280 q^{31} +1.00000 q^{33} +0.247499 q^{35} +5.03028 q^{37} +1.00000 q^{39} +0.720338 q^{41} -12.1322 q^{43} +0.169303 q^{45} +10.4213 q^{47} -4.86292 q^{49} +6.72578 q^{51} +12.9346 q^{53} +0.169303 q^{55} +6.80398 q^{57} -3.92725 q^{59} +1.23168 q^{61} +1.46187 q^{63} +0.169303 q^{65} -0.504017 q^{67} -5.29802 q^{69} +12.0184 q^{71} -3.12327 q^{73} -4.97134 q^{75} +1.46187 q^{77} -6.05246 q^{79} +1.00000 q^{81} +16.0109 q^{83} +1.13869 q^{85} -4.23175 q^{87} -1.80392 q^{89} +1.46187 q^{91} +7.17280 q^{93} +1.15193 q^{95} -9.83614 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + 5q^{11} + 5q^{13} + q^{15} + 4q^{19} + 5q^{21} + 5q^{23} + 4q^{25} + 5q^{27} + 11q^{29} + 8q^{31} + 5q^{33} + 5q^{35} - 8q^{37} + 5q^{39} - q^{41} - q^{43} + q^{45} + 18q^{47} + 10q^{49} + 2q^{53} + q^{55} + 4q^{57} + 13q^{59} + 9q^{61} + 5q^{63} + q^{65} - 5q^{67} + 5q^{69} + 24q^{71} - 13q^{73} + 4q^{75} + 5q^{77} + 6q^{79} + 5q^{81} + 22q^{83} - 22q^{85} + 11q^{87} - 14q^{89} + 5q^{91} + 8q^{93} + 32q^{95} - 20q^{97} + 5q^{99} + O(q^{100}) \)

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
\(5\) 0.169303 0.0757145 0.0378572 0.999283i \(-0.487947\pi\)
0.0378572 + 0.999283i \(0.487947\pi\)
\(6\) 0 0
\(7\) 1.46187 0.552537 0.276268 0.961081i \(-0.410902\pi\)
0.276268 + 0.961081i \(0.410902\pi\)
\(8\) 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
\(15\) 0.169303 0.0437138
\(16\) 0 0
\(17\) 6.72578 1.63124 0.815621 0.578587i \(-0.196396\pi\)
0.815621 + 0.578587i \(0.196396\pi\)
\(18\) 0 0
\(19\) 6.80398 1.56094 0.780470 0.625193i \(-0.214980\pi\)
0.780470 + 0.625193i \(0.214980\pi\)
\(20\) 0 0
\(21\) 1.46187 0.319007
\(22\) 0 0
\(23\) −5.29802 −1.10471 −0.552356 0.833608i \(-0.686271\pi\)
−0.552356 + 0.833608i \(0.686271\pi\)
\(24\) 0 0
\(25\) −4.97134 −0.994267
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.23175 −0.785815 −0.392908 0.919578i \(-0.628531\pi\)
−0.392908 + 0.919578i \(0.628531\pi\)
\(30\) 0 0
\(31\) 7.17280 1.28827 0.644137 0.764910i \(-0.277217\pi\)
0.644137 + 0.764910i \(0.277217\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 0.247499 0.0418350
\(36\) 0 0
\(37\) 5.03028 0.826973 0.413486 0.910510i \(-0.364311\pi\)
0.413486 + 0.910510i \(0.364311\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0.720338 0.112498 0.0562489 0.998417i \(-0.482086\pi\)
0.0562489 + 0.998417i \(0.482086\pi\)
\(42\) 0 0
\(43\) −12.1322 −1.85015 −0.925073 0.379790i \(-0.875996\pi\)
−0.925073 + 0.379790i \(0.875996\pi\)
\(44\) 0 0
\(45\) 0.169303 0.0252382
\(46\) 0 0
\(47\) 10.4213 1.52010 0.760050 0.649864i \(-0.225174\pi\)
0.760050 + 0.649864i \(0.225174\pi\)
\(48\) 0 0
\(49\) −4.86292 −0.694703
\(50\) 0 0
\(51\) 6.72578 0.941798
\(52\) 0 0
\(53\) 12.9346 1.77671 0.888355 0.459158i \(-0.151849\pi\)
0.888355 + 0.459158i \(0.151849\pi\)
\(54\) 0 0
\(55\) 0.169303 0.0228288
\(56\) 0 0
\(57\) 6.80398 0.901209
\(58\) 0 0
\(59\) −3.92725 −0.511284 −0.255642 0.966771i \(-0.582287\pi\)
−0.255642 + 0.966771i \(0.582287\pi\)
\(60\) 0 0
\(61\) 1.23168 0.157701 0.0788504 0.996886i \(-0.474875\pi\)
0.0788504 + 0.996886i \(0.474875\pi\)
\(62\) 0 0
\(63\) 1.46187 0.184179
\(64\) 0 0
\(65\) 0.169303 0.0209994
\(66\) 0 0
\(67\) −0.504017 −0.0615754 −0.0307877 0.999526i \(-0.509802\pi\)
−0.0307877 + 0.999526i \(0.509802\pi\)
\(68\) 0 0
\(69\) −5.29802 −0.637806
\(70\) 0 0
\(71\) 12.0184 1.42632 0.713158 0.701003i \(-0.247264\pi\)
0.713158 + 0.701003i \(0.247264\pi\)
\(72\) 0 0
\(73\) −3.12327 −0.365551 −0.182775 0.983155i \(-0.558508\pi\)
−0.182775 + 0.983155i \(0.558508\pi\)
\(74\) 0 0
\(75\) −4.97134 −0.574041
\(76\) 0 0
\(77\) 1.46187 0.166596
\(78\) 0 0
\(79\) −6.05246 −0.680955 −0.340478 0.940253i \(-0.610589\pi\)
−0.340478 + 0.940253i \(0.610589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.0109 1.75742 0.878712 0.477353i \(-0.158404\pi\)
0.878712 + 0.477353i \(0.158404\pi\)
\(84\) 0 0
\(85\) 1.13869 0.123509
\(86\) 0 0
\(87\) −4.23175 −0.453691
\(88\) 0 0
\(89\) −1.80392 −0.191215 −0.0956074 0.995419i \(-0.530479\pi\)
−0.0956074 + 0.995419i \(0.530479\pi\)
\(90\) 0 0
\(91\) 1.46187 0.153246
\(92\) 0 0
\(93\) 7.17280 0.743785
\(94\) 0 0
\(95\) 1.15193 0.118186
\(96\) 0 0
\(97\) −9.83614 −0.998709 −0.499354 0.866398i \(-0.666429\pi\)
−0.499354 + 0.866398i \(0.666429\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −17.3218 −1.72359 −0.861793 0.507261i \(-0.830658\pi\)
−0.861793 + 0.507261i \(0.830658\pi\)
\(102\) 0 0
\(103\) −4.26585 −0.420327 −0.210164 0.977666i \(-0.567400\pi\)
−0.210164 + 0.977666i \(0.567400\pi\)
\(104\) 0 0
\(105\) 0.247499 0.0241535
\(106\) 0 0
\(107\) −1.59253 −0.153956 −0.0769781 0.997033i \(-0.524527\pi\)
−0.0769781 + 0.997033i \(0.524527\pi\)
\(108\) 0 0
\(109\) −1.32480 −0.126893 −0.0634463 0.997985i \(-0.520209\pi\)
−0.0634463 + 0.997985i \(0.520209\pi\)
\(110\) 0 0
\(111\) 5.03028 0.477453
\(112\) 0 0
\(113\) −19.8056 −1.86315 −0.931577 0.363544i \(-0.881566\pi\)
−0.931577 + 0.363544i \(0.881566\pi\)
\(114\) 0 0
\(115\) −0.896969 −0.0836428
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 9.83225 0.901321
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0.720338 0.0649507
\(124\) 0 0
\(125\) −1.68817 −0.150995
\(126\) 0 0
\(127\) 15.8243 1.40418 0.702089 0.712089i \(-0.252251\pi\)
0.702089 + 0.712089i \(0.252251\pi\)
\(128\) 0 0
\(129\) −12.1322 −1.06818
\(130\) 0 0
\(131\) 10.3302 0.902552 0.451276 0.892384i \(-0.350969\pi\)
0.451276 + 0.892384i \(0.350969\pi\)
\(132\) 0 0
\(133\) 9.94656 0.862477
\(134\) 0 0
\(135\) 0.169303 0.0145713
\(136\) 0 0
\(137\) 5.00894 0.427943 0.213972 0.976840i \(-0.431360\pi\)
0.213972 + 0.976840i \(0.431360\pi\)
\(138\) 0 0
\(139\) −18.3521 −1.55660 −0.778302 0.627890i \(-0.783919\pi\)
−0.778302 + 0.627890i \(0.783919\pi\)
\(140\) 0 0
\(141\) 10.4213 0.877631
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −0.716446 −0.0594976
\(146\) 0 0
\(147\) −4.86292 −0.401087
\(148\) 0 0
\(149\) −10.0624 −0.824347 −0.412174 0.911105i \(-0.635230\pi\)
−0.412174 + 0.911105i \(0.635230\pi\)
\(150\) 0 0
\(151\) 10.9862 0.894044 0.447022 0.894523i \(-0.352485\pi\)
0.447022 + 0.894523i \(0.352485\pi\)
\(152\) 0 0
\(153\) 6.72578 0.543747
\(154\) 0 0
\(155\) 1.21438 0.0975410
\(156\) 0 0
\(157\) 9.43133 0.752702 0.376351 0.926477i \(-0.377179\pi\)
0.376351 + 0.926477i \(0.377179\pi\)
\(158\) 0 0
\(159\) 12.9346 1.02578
\(160\) 0 0
\(161\) −7.74504 −0.610394
\(162\) 0 0
\(163\) 5.75502 0.450768 0.225384 0.974270i \(-0.427636\pi\)
0.225384 + 0.974270i \(0.427636\pi\)
\(164\) 0 0
\(165\) 0.169303 0.0131802
\(166\) 0 0
\(167\) −1.52432 −0.117955 −0.0589776 0.998259i \(-0.518784\pi\)
−0.0589776 + 0.998259i \(0.518784\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 6.80398 0.520313
\(172\) 0 0
\(173\) −3.99656 −0.303853 −0.151927 0.988392i \(-0.548548\pi\)
−0.151927 + 0.988392i \(0.548548\pi\)
\(174\) 0 0
\(175\) −7.26747 −0.549369
\(176\) 0 0
\(177\) −3.92725 −0.295190
\(178\) 0 0
\(179\) −4.23719 −0.316703 −0.158351 0.987383i \(-0.550618\pi\)
−0.158351 + 0.987383i \(0.550618\pi\)
\(180\) 0 0
\(181\) 13.4718 1.00135 0.500676 0.865635i \(-0.333085\pi\)
0.500676 + 0.865635i \(0.333085\pi\)
\(182\) 0 0
\(183\) 1.23168 0.0910486
\(184\) 0 0
\(185\) 0.851640 0.0626138
\(186\) 0 0
\(187\) 6.72578 0.491838
\(188\) 0 0
\(189\) 1.46187 0.106336
\(190\) 0 0
\(191\) 13.2739 0.960465 0.480232 0.877141i \(-0.340552\pi\)
0.480232 + 0.877141i \(0.340552\pi\)
\(192\) 0 0
\(193\) −5.14259 −0.370171 −0.185086 0.982722i \(-0.559256\pi\)
−0.185086 + 0.982722i \(0.559256\pi\)
\(194\) 0 0
\(195\) 0.169303 0.0121240
\(196\) 0 0
\(197\) 1.86934 0.133185 0.0665925 0.997780i \(-0.478787\pi\)
0.0665925 + 0.997780i \(0.478787\pi\)
\(198\) 0 0
\(199\) 13.4134 0.950853 0.475426 0.879755i \(-0.342294\pi\)
0.475426 + 0.879755i \(0.342294\pi\)
\(200\) 0 0
\(201\) −0.504017 −0.0355506
\(202\) 0 0
\(203\) −6.18628 −0.434192
\(204\) 0 0
\(205\) 0.121955 0.00851772
\(206\) 0 0
\(207\) −5.29802 −0.368238
\(208\) 0 0
\(209\) 6.80398 0.470641
\(210\) 0 0
\(211\) 8.21586 0.565603 0.282801 0.959178i \(-0.408736\pi\)
0.282801 + 0.959178i \(0.408736\pi\)
\(212\) 0 0
\(213\) 12.0184 0.823484
\(214\) 0 0
\(215\) −2.05402 −0.140083
\(216\) 0 0
\(217\) 10.4857 0.711818
\(218\) 0 0
\(219\) −3.12327 −0.211051
\(220\) 0 0
\(221\) 6.72578 0.452425
\(222\) 0 0
\(223\) −1.32530 −0.0887489 −0.0443745 0.999015i \(-0.514129\pi\)
−0.0443745 + 0.999015i \(0.514129\pi\)
\(224\) 0 0
\(225\) −4.97134 −0.331422
\(226\) 0 0
\(227\) 20.2447 1.34368 0.671842 0.740694i \(-0.265503\pi\)
0.671842 + 0.740694i \(0.265503\pi\)
\(228\) 0 0
\(229\) −13.8516 −0.915337 −0.457669 0.889123i \(-0.651315\pi\)
−0.457669 + 0.889123i \(0.651315\pi\)
\(230\) 0 0
\(231\) 1.46187 0.0961843
\(232\) 0 0
\(233\) 22.2878 1.46012 0.730060 0.683383i \(-0.239492\pi\)
0.730060 + 0.683383i \(0.239492\pi\)
\(234\) 0 0
\(235\) 1.76435 0.115094
\(236\) 0 0
\(237\) −6.05246 −0.393150
\(238\) 0 0
\(239\) −6.30445 −0.407801 −0.203900 0.978992i \(-0.565362\pi\)
−0.203900 + 0.978992i \(0.565362\pi\)
\(240\) 0 0
\(241\) −18.7821 −1.20986 −0.604932 0.796277i \(-0.706800\pi\)
−0.604932 + 0.796277i \(0.706800\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.823306 −0.0525991
\(246\) 0 0
\(247\) 6.80398 0.432927
\(248\) 0 0
\(249\) 16.0109 1.01465
\(250\) 0 0
\(251\) 15.6981 0.990856 0.495428 0.868649i \(-0.335011\pi\)
0.495428 + 0.868649i \(0.335011\pi\)
\(252\) 0 0
\(253\) −5.29802 −0.333083
\(254\) 0 0
\(255\) 1.13869 0.0713078
\(256\) 0 0
\(257\) 5.50047 0.343110 0.171555 0.985175i \(-0.445121\pi\)
0.171555 + 0.985175i \(0.445121\pi\)
\(258\) 0 0
\(259\) 7.35364 0.456933
\(260\) 0 0
\(261\) −4.23175 −0.261938
\(262\) 0 0
\(263\) 16.8625 1.03979 0.519894 0.854231i \(-0.325971\pi\)
0.519894 + 0.854231i \(0.325971\pi\)
\(264\) 0 0
\(265\) 2.18987 0.134523
\(266\) 0 0
\(267\) −1.80392 −0.110398
\(268\) 0 0
\(269\) −13.9227 −0.848883 −0.424441 0.905455i \(-0.639530\pi\)
−0.424441 + 0.905455i \(0.639530\pi\)
\(270\) 0 0
\(271\) 26.7189 1.62306 0.811529 0.584312i \(-0.198636\pi\)
0.811529 + 0.584312i \(0.198636\pi\)
\(272\) 0 0
\(273\) 1.46187 0.0884767
\(274\) 0 0
\(275\) −4.97134 −0.299783
\(276\) 0 0
\(277\) −12.0560 −0.724376 −0.362188 0.932105i \(-0.617970\pi\)
−0.362188 + 0.932105i \(0.617970\pi\)
\(278\) 0 0
\(279\) 7.17280 0.429424
\(280\) 0 0
\(281\) 14.5560 0.868339 0.434170 0.900831i \(-0.357042\pi\)
0.434170 + 0.900831i \(0.357042\pi\)
\(282\) 0 0
\(283\) 20.0219 1.19018 0.595090 0.803659i \(-0.297117\pi\)
0.595090 + 0.803659i \(0.297117\pi\)
\(284\) 0 0
\(285\) 1.15193 0.0682346
\(286\) 0 0
\(287\) 1.05304 0.0621592
\(288\) 0 0
\(289\) 28.2362 1.66095
\(290\) 0 0
\(291\) −9.83614 −0.576605
\(292\) 0 0
\(293\) −29.9852 −1.75175 −0.875876 0.482537i \(-0.839716\pi\)
−0.875876 + 0.482537i \(0.839716\pi\)
\(294\) 0 0
\(295\) −0.664894 −0.0387116
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) −5.29802 −0.306392
\(300\) 0 0
\(301\) −17.7358 −1.02227
\(302\) 0 0
\(303\) −17.3218 −0.995112
\(304\) 0 0
\(305\) 0.208527 0.0119402
\(306\) 0 0
\(307\) −31.9883 −1.82567 −0.912833 0.408332i \(-0.866110\pi\)
−0.912833 + 0.408332i \(0.866110\pi\)
\(308\) 0 0
\(309\) −4.26585 −0.242676
\(310\) 0 0
\(311\) −4.84618 −0.274802 −0.137401 0.990516i \(-0.543875\pi\)
−0.137401 + 0.990516i \(0.543875\pi\)
\(312\) 0 0
\(313\) −18.2931 −1.03399 −0.516993 0.855989i \(-0.672949\pi\)
−0.516993 + 0.855989i \(0.672949\pi\)
\(314\) 0 0
\(315\) 0.247499 0.0139450
\(316\) 0 0
\(317\) 8.11944 0.456033 0.228017 0.973657i \(-0.426776\pi\)
0.228017 + 0.973657i \(0.426776\pi\)
\(318\) 0 0
\(319\) −4.23175 −0.236932
\(320\) 0 0
\(321\) −1.59253 −0.0888866
\(322\) 0 0
\(323\) 45.7621 2.54627
\(324\) 0 0
\(325\) −4.97134 −0.275760
\(326\) 0 0
\(327\) −1.32480 −0.0732615
\(328\) 0 0
\(329\) 15.2346 0.839911
\(330\) 0 0
\(331\) −6.05901 −0.333033 −0.166517 0.986039i \(-0.553252\pi\)
−0.166517 + 0.986039i \(0.553252\pi\)
\(332\) 0 0
\(333\) 5.03028 0.275658
\(334\) 0 0
\(335\) −0.0853314 −0.00466215
\(336\) 0 0
\(337\) −3.19803 −0.174208 −0.0871039 0.996199i \(-0.527761\pi\)
−0.0871039 + 0.996199i \(0.527761\pi\)
\(338\) 0 0
\(339\) −19.8056 −1.07569
\(340\) 0 0
\(341\) 7.17280 0.388429
\(342\) 0 0
\(343\) −17.3421 −0.936386
\(344\) 0 0
\(345\) −0.896969 −0.0482912
\(346\) 0 0
\(347\) 3.58261 0.192324 0.0961622 0.995366i \(-0.469343\pi\)
0.0961622 + 0.995366i \(0.469343\pi\)
\(348\) 0 0
\(349\) 4.75243 0.254392 0.127196 0.991878i \(-0.459402\pi\)
0.127196 + 0.991878i \(0.459402\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 5.89049 0.313519 0.156760 0.987637i \(-0.449895\pi\)
0.156760 + 0.987637i \(0.449895\pi\)
\(354\) 0 0
\(355\) 2.03474 0.107993
\(356\) 0 0
\(357\) 9.83225 0.520378
\(358\) 0 0
\(359\) 17.1342 0.904306 0.452153 0.891940i \(-0.350656\pi\)
0.452153 + 0.891940i \(0.350656\pi\)
\(360\) 0 0
\(361\) 27.2941 1.43653
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −0.528778 −0.0276775
\(366\) 0 0
\(367\) −23.2841 −1.21542 −0.607711 0.794158i \(-0.707912\pi\)
−0.607711 + 0.794158i \(0.707912\pi\)
\(368\) 0 0
\(369\) 0.720338 0.0374993
\(370\) 0 0
\(371\) 18.9088 0.981697
\(372\) 0 0
\(373\) −10.8175 −0.560110 −0.280055 0.959984i \(-0.590353\pi\)
−0.280055 + 0.959984i \(0.590353\pi\)
\(374\) 0 0
\(375\) −1.68817 −0.0871770
\(376\) 0 0
\(377\) −4.23175 −0.217946
\(378\) 0 0
\(379\) −25.4050 −1.30497 −0.652484 0.757803i \(-0.726273\pi\)
−0.652484 + 0.757803i \(0.726273\pi\)
\(380\) 0 0
\(381\) 15.8243 0.810702
\(382\) 0 0
\(383\) 22.4029 1.14474 0.572368 0.819997i \(-0.306025\pi\)
0.572368 + 0.819997i \(0.306025\pi\)
\(384\) 0 0
\(385\) 0.247499 0.0126137
\(386\) 0 0
\(387\) −12.1322 −0.616715
\(388\) 0 0
\(389\) −8.84646 −0.448534 −0.224267 0.974528i \(-0.571999\pi\)
−0.224267 + 0.974528i \(0.571999\pi\)
\(390\) 0 0
\(391\) −35.6333 −1.80205
\(392\) 0 0
\(393\) 10.3302 0.521089
\(394\) 0 0
\(395\) −1.02470 −0.0515582
\(396\) 0 0
\(397\) −5.44164 −0.273108 −0.136554 0.990633i \(-0.543603\pi\)
−0.136554 + 0.990633i \(0.543603\pi\)
\(398\) 0 0
\(399\) 9.94656 0.497951
\(400\) 0 0
\(401\) 4.80646 0.240023 0.120011 0.992773i \(-0.461707\pi\)
0.120011 + 0.992773i \(0.461707\pi\)
\(402\) 0 0
\(403\) 7.17280 0.357303
\(404\) 0 0
\(405\) 0.169303 0.00841272
\(406\) 0 0
\(407\) 5.03028 0.249342
\(408\) 0 0
\(409\) −10.3006 −0.509330 −0.254665 0.967029i \(-0.581965\pi\)
−0.254665 + 0.967029i \(0.581965\pi\)
\(410\) 0 0
\(411\) 5.00894 0.247073
\(412\) 0 0
\(413\) −5.74114 −0.282503
\(414\) 0 0
\(415\) 2.71069 0.133062
\(416\) 0 0
\(417\) −18.3521 −0.898706
\(418\) 0 0
\(419\) −6.71741 −0.328167 −0.164083 0.986446i \(-0.552467\pi\)
−0.164083 + 0.986446i \(0.552467\pi\)
\(420\) 0 0
\(421\) 26.2699 1.28032 0.640158 0.768244i \(-0.278869\pi\)
0.640158 + 0.768244i \(0.278869\pi\)
\(422\) 0 0
\(423\) 10.4213 0.506700
\(424\) 0 0
\(425\) −33.4361 −1.62189
\(426\) 0 0
\(427\) 1.80057 0.0871355
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 5.97238 0.287680 0.143840 0.989601i \(-0.454055\pi\)
0.143840 + 0.989601i \(0.454055\pi\)
\(432\) 0 0
\(433\) 19.1175 0.918728 0.459364 0.888248i \(-0.348077\pi\)
0.459364 + 0.888248i \(0.348077\pi\)
\(434\) 0 0
\(435\) −0.716446 −0.0343510
\(436\) 0 0
\(437\) −36.0476 −1.72439
\(438\) 0 0
\(439\) 25.0129 1.19380 0.596901 0.802315i \(-0.296399\pi\)
0.596901 + 0.802315i \(0.296399\pi\)
\(440\) 0 0
\(441\) −4.86292 −0.231568
\(442\) 0 0
\(443\) 37.8615 1.79885 0.899427 0.437071i \(-0.143984\pi\)
0.899427 + 0.437071i \(0.143984\pi\)
\(444\) 0 0
\(445\) −0.305408 −0.0144777
\(446\) 0 0
\(447\) −10.0624 −0.475937
\(448\) 0 0
\(449\) −36.1295 −1.70506 −0.852528 0.522681i \(-0.824932\pi\)
−0.852528 + 0.522681i \(0.824932\pi\)
\(450\) 0 0
\(451\) 0.720338 0.0339194
\(452\) 0 0
\(453\) 10.9862 0.516176
\(454\) 0 0
\(455\) 0.247499 0.0116030
\(456\) 0 0
\(457\) −9.63104 −0.450521 −0.225261 0.974299i \(-0.572323\pi\)
−0.225261 + 0.974299i \(0.572323\pi\)
\(458\) 0 0
\(459\) 6.72578 0.313933
\(460\) 0 0
\(461\) 39.0377 1.81817 0.909083 0.416615i \(-0.136784\pi\)
0.909083 + 0.416615i \(0.136784\pi\)
\(462\) 0 0
\(463\) −10.4168 −0.484110 −0.242055 0.970263i \(-0.577821\pi\)
−0.242055 + 0.970263i \(0.577821\pi\)
\(464\) 0 0
\(465\) 1.21438 0.0563153
\(466\) 0 0
\(467\) −6.82610 −0.315874 −0.157937 0.987449i \(-0.550484\pi\)
−0.157937 + 0.987449i \(0.550484\pi\)
\(468\) 0 0
\(469\) −0.736809 −0.0340227
\(470\) 0 0
\(471\) 9.43133 0.434573
\(472\) 0 0
\(473\) −12.1322 −0.557840
\(474\) 0 0
\(475\) −33.8249 −1.55199
\(476\) 0 0
\(477\) 12.9346 0.592236
\(478\) 0 0
\(479\) −26.2244 −1.19822 −0.599112 0.800665i \(-0.704480\pi\)
−0.599112 + 0.800665i \(0.704480\pi\)
\(480\) 0 0
\(481\) 5.03028 0.229361
\(482\) 0 0
\(483\) −7.74504 −0.352411
\(484\) 0 0
\(485\) −1.66529 −0.0756167
\(486\) 0 0
\(487\) −22.1940 −1.00571 −0.502853 0.864372i \(-0.667716\pi\)
−0.502853 + 0.864372i \(0.667716\pi\)
\(488\) 0 0
\(489\) 5.75502 0.260251
\(490\) 0 0
\(491\) −31.9338 −1.44115 −0.720576 0.693376i \(-0.756122\pi\)
−0.720576 + 0.693376i \(0.756122\pi\)
\(492\) 0 0
\(493\) −28.4618 −1.28186
\(494\) 0 0
\(495\) 0.169303 0.00760959
\(496\) 0 0
\(497\) 17.5693 0.788092
\(498\) 0 0
\(499\) −39.3209 −1.76025 −0.880123 0.474747i \(-0.842540\pi\)
−0.880123 + 0.474747i \(0.842540\pi\)
\(500\) 0 0
\(501\) −1.52432 −0.0681015
\(502\) 0 0
\(503\) 16.9623 0.756310 0.378155 0.925742i \(-0.376559\pi\)
0.378155 + 0.925742i \(0.376559\pi\)
\(504\) 0 0
\(505\) −2.93263 −0.130500
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 30.2437 1.34053 0.670264 0.742123i \(-0.266181\pi\)
0.670264 + 0.742123i \(0.266181\pi\)
\(510\) 0 0
\(511\) −4.56583 −0.201980
\(512\) 0 0
\(513\) 6.80398 0.300403
\(514\) 0 0
\(515\) −0.722221 −0.0318249
\(516\) 0 0
\(517\) 10.4213 0.458328
\(518\) 0 0
\(519\) −3.99656 −0.175430
\(520\) 0 0
\(521\) 23.2926 1.02047 0.510233 0.860036i \(-0.329559\pi\)
0.510233 + 0.860036i \(0.329559\pi\)
\(522\) 0 0
\(523\) −10.2861 −0.449782 −0.224891 0.974384i \(-0.572203\pi\)
−0.224891 + 0.974384i \(0.572203\pi\)
\(524\) 0 0
\(525\) −7.26747 −0.317178
\(526\) 0 0
\(527\) 48.2427 2.10149
\(528\) 0 0
\(529\) 5.06899 0.220391
\(530\) 0 0
\(531\) −3.92725 −0.170428
\(532\) 0 0
\(533\) 0.720338 0.0312013
\(534\) 0 0
\(535\) −0.269621 −0.0116567
\(536\) 0 0
\(537\) −4.23719 −0.182848
\(538\) 0 0
\(539\) −4.86292 −0.209461
\(540\) 0 0
\(541\) 31.4278 1.35119 0.675594 0.737274i \(-0.263887\pi\)
0.675594 + 0.737274i \(0.263887\pi\)
\(542\) 0 0
\(543\) 13.4718 0.578130
\(544\) 0 0
\(545\) −0.224292 −0.00960761
\(546\) 0 0
\(547\) 6.41396 0.274241 0.137121 0.990554i \(-0.456215\pi\)
0.137121 + 0.990554i \(0.456215\pi\)
\(548\) 0 0
\(549\) 1.23168 0.0525669
\(550\) 0 0
\(551\) −28.7927 −1.22661
\(552\) 0 0
\(553\) −8.84794 −0.376253
\(554\) 0 0
\(555\) 0.851640 0.0361501
\(556\) 0 0
\(557\) 40.6435 1.72212 0.861061 0.508501i \(-0.169800\pi\)
0.861061 + 0.508501i \(0.169800\pi\)
\(558\) 0 0
\(559\) −12.1322 −0.513138
\(560\) 0 0
\(561\) 6.72578 0.283963
\(562\) 0 0
\(563\) −35.7972 −1.50867 −0.754335 0.656489i \(-0.772041\pi\)
−0.754335 + 0.656489i \(0.772041\pi\)
\(564\) 0 0
\(565\) −3.35314 −0.141068
\(566\) 0 0
\(567\) 1.46187 0.0613930
\(568\) 0 0
\(569\) −0.718784 −0.0301330 −0.0150665 0.999886i \(-0.504796\pi\)
−0.0150665 + 0.999886i \(0.504796\pi\)
\(570\) 0 0
\(571\) 41.9442 1.75531 0.877654 0.479294i \(-0.159107\pi\)
0.877654 + 0.479294i \(0.159107\pi\)
\(572\) 0 0
\(573\) 13.2739 0.554525
\(574\) 0 0
\(575\) 26.3382 1.09838
\(576\) 0 0
\(577\) 27.2971 1.13639 0.568196 0.822893i \(-0.307641\pi\)
0.568196 + 0.822893i \(0.307641\pi\)
\(578\) 0 0
\(579\) −5.14259 −0.213719
\(580\) 0 0
\(581\) 23.4059 0.971041
\(582\) 0 0
\(583\) 12.9346 0.535698
\(584\) 0 0
\(585\) 0.169303 0.00699981
\(586\) 0 0
\(587\) 1.53119 0.0631990 0.0315995 0.999501i \(-0.489940\pi\)
0.0315995 + 0.999501i \(0.489940\pi\)
\(588\) 0 0
\(589\) 48.8036 2.01092
\(590\) 0 0
\(591\) 1.86934 0.0768944
\(592\) 0 0
\(593\) 15.2913 0.627939 0.313970 0.949433i \(-0.398341\pi\)
0.313970 + 0.949433i \(0.398341\pi\)
\(594\) 0 0
\(595\) 1.66463 0.0682431
\(596\) 0 0
\(597\) 13.4134 0.548975
\(598\) 0 0
\(599\) 32.3443 1.32155 0.660776 0.750583i \(-0.270227\pi\)
0.660776 + 0.750583i \(0.270227\pi\)
\(600\) 0 0
\(601\) 15.8783 0.647691 0.323846 0.946110i \(-0.395024\pi\)
0.323846 + 0.946110i \(0.395024\pi\)
\(602\) 0 0
\(603\) −0.504017 −0.0205251
\(604\) 0 0
\(605\) 0.169303 0.00688314
\(606\) 0 0
\(607\) 15.8243 0.642288 0.321144 0.947030i \(-0.395933\pi\)
0.321144 + 0.947030i \(0.395933\pi\)
\(608\) 0 0
\(609\) −6.18628 −0.250681
\(610\) 0 0
\(611\) 10.4213 0.421600
\(612\) 0 0
\(613\) −25.0853 −1.01318 −0.506592 0.862186i \(-0.669095\pi\)
−0.506592 + 0.862186i \(0.669095\pi\)
\(614\) 0 0
\(615\) 0.121955 0.00491771
\(616\) 0 0
\(617\) −42.3649 −1.70555 −0.852773 0.522282i \(-0.825081\pi\)
−0.852773 + 0.522282i \(0.825081\pi\)
\(618\) 0 0
\(619\) 8.87965 0.356903 0.178452 0.983949i \(-0.442891\pi\)
0.178452 + 0.983949i \(0.442891\pi\)
\(620\) 0 0
\(621\) −5.29802 −0.212602
\(622\) 0 0
\(623\) −2.63710 −0.105653
\(624\) 0 0
\(625\) 24.5709 0.982835
\(626\) 0 0
\(627\) 6.80398 0.271725
\(628\) 0 0
\(629\) 33.8326 1.34899
\(630\) 0 0
\(631\) −23.0862 −0.919048 −0.459524 0.888165i \(-0.651980\pi\)
−0.459524 + 0.888165i \(0.651980\pi\)
\(632\) 0 0
\(633\) 8.21586 0.326551
\(634\) 0 0
\(635\) 2.67909 0.106317
\(636\) 0 0
\(637\) −4.86292 −0.192676
\(638\) 0 0
\(639\) 12.0184 0.475439
\(640\) 0 0
\(641\) −49.0640 −1.93791 −0.968955 0.247237i \(-0.920477\pi\)
−0.968955 + 0.247237i \(0.920477\pi\)
\(642\) 0 0
\(643\) −12.7668 −0.503472 −0.251736 0.967796i \(-0.581001\pi\)
−0.251736 + 0.967796i \(0.581001\pi\)
\(644\) 0 0
\(645\) −2.05402 −0.0808768
\(646\) 0 0
\(647\) −10.3683 −0.407621 −0.203810 0.979010i \(-0.565333\pi\)
−0.203810 + 0.979010i \(0.565333\pi\)
\(648\) 0 0
\(649\) −3.92725 −0.154158
\(650\) 0 0
\(651\) 10.4857 0.410968
\(652\) 0 0
\(653\) 18.9307 0.740817 0.370409 0.928869i \(-0.379218\pi\)
0.370409 + 0.928869i \(0.379218\pi\)
\(654\) 0 0
\(655\) 1.74893 0.0683363
\(656\) 0 0
\(657\) −3.12327 −0.121850
\(658\) 0 0
\(659\) 9.33917 0.363803 0.181901 0.983317i \(-0.441775\pi\)
0.181901 + 0.983317i \(0.441775\pi\)
\(660\) 0 0
\(661\) 3.10985 0.120959 0.0604796 0.998169i \(-0.480737\pi\)
0.0604796 + 0.998169i \(0.480737\pi\)
\(662\) 0 0
\(663\) 6.72578 0.261208
\(664\) 0 0
\(665\) 1.68398 0.0653020
\(666\) 0 0
\(667\) 22.4199 0.868100
\(668\) 0 0
\(669\) −1.32530 −0.0512392
\(670\) 0 0
\(671\) 1.23168 0.0475486
\(672\) 0 0
\(673\) 15.3905 0.593262 0.296631 0.954992i \(-0.404137\pi\)
0.296631 + 0.954992i \(0.404137\pi\)
\(674\) 0 0
\(675\) −4.97134 −0.191347
\(676\) 0 0
\(677\) −26.3887 −1.01420 −0.507100 0.861887i \(-0.669282\pi\)
−0.507100 + 0.861887i \(0.669282\pi\)
\(678\) 0 0
\(679\) −14.3792 −0.551823
\(680\) 0 0
\(681\) 20.2447 0.775777
\(682\) 0 0
\(683\) −26.3448 −1.00805 −0.504027 0.863688i \(-0.668149\pi\)
−0.504027 + 0.863688i \(0.668149\pi\)
\(684\) 0 0
\(685\) 0.848028 0.0324015
\(686\) 0 0
\(687\) −13.8516 −0.528470
\(688\) 0 0
\(689\) 12.9346 0.492771
\(690\) 0 0
\(691\) −18.3557 −0.698284 −0.349142 0.937070i \(-0.613527\pi\)
−0.349142 + 0.937070i \(0.613527\pi\)
\(692\) 0 0
\(693\) 1.46187 0.0555320
\(694\) 0 0
\(695\) −3.10706 −0.117858
\(696\) 0 0
\(697\) 4.84483 0.183511
\(698\) 0 0
\(699\) 22.2878 0.843001
\(700\) 0 0
\(701\) −11.9392 −0.450939 −0.225469 0.974250i \(-0.572392\pi\)
−0.225469 + 0.974250i \(0.572392\pi\)
\(702\) 0 0
\(703\) 34.2259 1.29086
\(704\) 0 0
\(705\) 1.76435 0.0664494
\(706\) 0 0
\(707\) −25.3223 −0.952344
\(708\) 0 0
\(709\) 35.4427 1.33108 0.665539 0.746363i \(-0.268202\pi\)
0.665539 + 0.746363i \(0.268202\pi\)
\(710\) 0 0
\(711\) −6.05246 −0.226985
\(712\) 0 0
\(713\) −38.0016 −1.42317
\(714\) 0 0
\(715\) 0.169303 0.00633156
\(716\) 0 0
\(717\) −6.30445 −0.235444
\(718\) 0 0
\(719\) −21.5206 −0.802581 −0.401291 0.915951i \(-0.631438\pi\)
−0.401291 + 0.915951i \(0.631438\pi\)
\(720\) 0 0
\(721\) −6.23614 −0.232246
\(722\) 0 0
\(723\) −18.7821 −0.698515
\(724\) 0 0
\(725\) 21.0374 0.781311
\(726\) 0 0
\(727\) −19.9536 −0.740037 −0.370018 0.929024i \(-0.620649\pi\)
−0.370018 + 0.929024i \(0.620649\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −81.5986 −3.01803
\(732\) 0 0
\(733\) −9.02391 −0.333306 −0.166653 0.986016i \(-0.553296\pi\)
−0.166653 + 0.986016i \(0.553296\pi\)
\(734\) 0 0
\(735\) −0.823306 −0.0303681
\(736\) 0 0
\(737\) −0.504017 −0.0185657
\(738\) 0 0
\(739\) −52.6309 −1.93606 −0.968029 0.250839i \(-0.919293\pi\)
−0.968029 + 0.250839i \(0.919293\pi\)
\(740\) 0 0
\(741\) 6.80398 0.249950
\(742\) 0 0
\(743\) 29.0711 1.06651 0.533257 0.845953i \(-0.320968\pi\)
0.533257 + 0.845953i \(0.320968\pi\)
\(744\) 0 0
\(745\) −1.70360 −0.0624150
\(746\) 0 0
\(747\) 16.0109 0.585808
\(748\) 0 0
\(749\) −2.32809 −0.0850664
\(750\) 0 0
\(751\) −6.44794 −0.235289 −0.117644 0.993056i \(-0.537534\pi\)
−0.117644 + 0.993056i \(0.537534\pi\)
\(752\) 0 0
\(753\) 15.6981 0.572071
\(754\) 0 0
\(755\) 1.85999 0.0676921
\(756\) 0 0
\(757\) −36.6510 −1.33210 −0.666052 0.745905i \(-0.732017\pi\)
−0.666052 + 0.745905i \(0.732017\pi\)
\(758\) 0 0
\(759\) −5.29802 −0.192306
\(760\) 0 0
\(761\) −31.6839 −1.14854 −0.574270 0.818666i \(-0.694714\pi\)
−0.574270 + 0.818666i \(0.694714\pi\)
\(762\) 0 0
\(763\) −1.93669 −0.0701128
\(764\) 0 0
\(765\) 1.13869 0.0411696
\(766\) 0 0
\(767\) −3.92725 −0.141805
\(768\) 0 0
\(769\) 5.40643 0.194961 0.0974804 0.995237i \(-0.468922\pi\)
0.0974804 + 0.995237i \(0.468922\pi\)
\(770\) 0 0
\(771\) 5.50047 0.198094
\(772\) 0 0
\(773\) −25.1099 −0.903139 −0.451569 0.892236i \(-0.649136\pi\)
−0.451569 + 0.892236i \(0.649136\pi\)
\(774\) 0 0
\(775\) −35.6584 −1.28089
\(776\) 0 0
\(777\) 7.35364 0.263810
\(778\) 0 0
\(779\) 4.90116 0.175602
\(780\) 0 0
\(781\) 12.0184 0.430051
\(782\) 0 0
\(783\) −4.23175 −0.151230
\(784\) 0 0
\(785\) 1.59675 0.0569904
\(786\) 0 0
\(787\) −46.6625 −1.66334 −0.831669 0.555272i \(-0.812614\pi\)
−0.831669 + 0.555272i \(0.812614\pi\)
\(788\) 0 0
\(789\) 16.8625 0.600322
\(790\) 0 0
\(791\) −28.9533 −1.02946
\(792\) 0 0
\(793\) 1.23168 0.0437383
\(794\) 0 0
\(795\) 2.18987 0.0776667
\(796\) 0 0
\(797\) 42.8734 1.51865 0.759327 0.650709i \(-0.225528\pi\)
0.759327 + 0.650709i \(0.225528\pi\)
\(798\) 0 0
\(799\) 70.0913 2.47965
\(800\) 0 0
\(801\) −1.80392 −0.0637383
\(802\) 0 0
\(803\) −3.12327 −0.110218
\(804\) 0 0
\(805\) −1.31126 −0.0462157
\(806\) 0 0
\(807\) −13.9227 −0.490103
\(808\) 0 0
\(809\) −15.5992 −0.548439 −0.274219 0.961667i \(-0.588419\pi\)
−0.274219 + 0.961667i \(0.588419\pi\)
\(810\) 0 0
\(811\) 1.72773 0.0606688 0.0303344 0.999540i \(-0.490343\pi\)
0.0303344 + 0.999540i \(0.490343\pi\)
\(812\) 0 0
\(813\) 26.7189 0.937073
\(814\) 0 0
\(815\) 0.974340 0.0341296
\(816\) 0 0
\(817\) −82.5473 −2.88797
\(818\) 0 0
\(819\) 1.46187 0.0510820
\(820\) 0 0
\(821\) 22.9823 0.802088 0.401044 0.916059i \(-0.368647\pi\)
0.401044 + 0.916059i \(0.368647\pi\)
\(822\) 0 0
\(823\) −24.7293 −0.862011 −0.431005 0.902349i \(-0.641841\pi\)
−0.431005 + 0.902349i \(0.641841\pi\)
\(824\) 0 0
\(825\) −4.97134 −0.173080
\(826\) 0 0
\(827\) −5.12470 −0.178203 −0.0891016 0.996023i \(-0.528400\pi\)
−0.0891016 + 0.996023i \(0.528400\pi\)
\(828\) 0 0
\(829\) −47.5783 −1.65246 −0.826231 0.563332i \(-0.809519\pi\)
−0.826231 + 0.563332i \(0.809519\pi\)
\(830\) 0 0
\(831\) −12.0560 −0.418219
\(832\) 0 0
\(833\) −32.7070 −1.13323
\(834\) 0 0
\(835\) −0.258071 −0.00893092
\(836\) 0 0
\(837\) 7.17280 0.247928
\(838\) 0 0
\(839\) −42.4415 −1.46524 −0.732621 0.680637i \(-0.761703\pi\)
−0.732621 + 0.680637i \(0.761703\pi\)
\(840\) 0 0
\(841\) −11.0923 −0.382494
\(842\) 0 0
\(843\) 14.5560 0.501336
\(844\) 0 0
\(845\) 0.169303 0.00582419
\(846\) 0 0
\(847\) 1.46187 0.0502306
\(848\) 0 0
\(849\) 20.0219 0.687150
\(850\) 0 0
\(851\) −26.6505 −0.913568
\(852\) 0 0
\(853\) −28.4447 −0.973927 −0.486963 0.873422i \(-0.661896\pi\)
−0.486963 + 0.873422i \(0.661896\pi\)
\(854\) 0 0
\(855\) 1.15193 0.0393953
\(856\) 0 0
\(857\) 38.5420 1.31657 0.658284 0.752769i \(-0.271282\pi\)
0.658284 + 0.752769i \(0.271282\pi\)
\(858\) 0 0
\(859\) −52.5831 −1.79411 −0.897056 0.441918i \(-0.854298\pi\)
−0.897056 + 0.441918i \(0.854298\pi\)
\(860\) 0 0
\(861\) 1.05304 0.0358876
\(862\) 0 0
\(863\) 16.1893 0.551091 0.275546 0.961288i \(-0.411141\pi\)
0.275546 + 0.961288i \(0.411141\pi\)
\(864\) 0 0
\(865\) −0.676629 −0.0230061
\(866\) 0 0
\(867\) 28.2362 0.958950
\(868\) 0 0
\(869\) −6.05246 −0.205316
\(870\) 0 0
\(871\) −0.504017 −0.0170780
\(872\) 0 0
\(873\) −9.83614 −0.332903
\(874\) 0 0
\(875\) −2.46790 −0.0834302
\(876\) 0 0
\(877\) −42.1952 −1.42483 −0.712415 0.701758i \(-0.752399\pi\)
−0.712415 + 0.701758i \(0.752399\pi\)
\(878\) 0 0
\(879\) −29.9852 −1.01137
\(880\) 0 0
\(881\) −44.8413 −1.51074 −0.755370 0.655298i \(-0.772543\pi\)
−0.755370 + 0.655298i \(0.772543\pi\)
\(882\) 0 0
\(883\) 0.137720 0.00463464 0.00231732 0.999997i \(-0.499262\pi\)
0.00231732 + 0.999997i \(0.499262\pi\)
\(884\) 0 0
\(885\) −0.664894 −0.0223502
\(886\) 0 0
\(887\) −16.0809 −0.539943 −0.269972 0.962868i \(-0.587014\pi\)
−0.269972 + 0.962868i \(0.587014\pi\)
\(888\) 0 0
\(889\) 23.1331 0.775860
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 70.9062 2.37279
\(894\) 0 0
\(895\) −0.717368 −0.0239790
\(896\) 0 0
\(897\) −5.29802 −0.176896
\(898\) 0 0
\(899\) −30.3535 −1.01235
\(900\) 0 0
\(901\) 86.9956 2.89824
\(902\) 0 0
\(903\) −17.7358 −0.590210
\(904\) 0 0
\(905\) 2.28081 0.0758168
\(906\) 0 0
\(907\) −45.3518 −1.50588 −0.752942 0.658087i \(-0.771366\pi\)
−0.752942 + 0.658087i \(0.771366\pi\)
\(908\) 0 0
\(909\) −17.3218 −0.574528
\(910\) 0 0
\(911\) −1.90417 −0.0630878 −0.0315439 0.999502i \(-0.510042\pi\)
−0.0315439 + 0.999502i \(0.510042\pi\)
\(912\) 0 0
\(913\) 16.0109 0.529883
\(914\) 0 0
\(915\) 0.208527 0.00689370
\(916\) 0 0
\(917\) 15.1014 0.498693
\(918\) 0 0
\(919\) 60.0572 1.98110 0.990552 0.137137i \(-0.0437900\pi\)
0.990552 + 0.137137i \(0.0437900\pi\)
\(920\) 0 0
\(921\) −31.9883 −1.05405
\(922\) 0 0
\(923\) 12.0184 0.395589
\(924\) 0 0
\(925\) −25.0072 −0.822232
\(926\) 0 0
\(927\) −4.26585 −0.140109
\(928\) 0 0
\(929\) 15.4685 0.507506 0.253753 0.967269i \(-0.418335\pi\)
0.253753 + 0.967269i \(0.418335\pi\)
\(930\) 0 0
\(931\) −33.0872 −1.08439
\(932\) 0 0
\(933\) −4.84618 −0.158657
\(934\) 0 0
\(935\) 1.13869 0.0372393
\(936\) 0 0
\(937\) −47.9243 −1.56562 −0.782810 0.622261i \(-0.786214\pi\)
−0.782810 + 0.622261i \(0.786214\pi\)
\(938\) 0 0
\(939\) −18.2931 −0.596972
\(940\) 0 0
\(941\) −3.31714 −0.108136 −0.0540678 0.998537i \(-0.517219\pi\)
−0.0540678 + 0.998537i \(0.517219\pi\)
\(942\) 0 0
\(943\) −3.81636 −0.124278
\(944\) 0 0
\(945\) 0.247499 0.00805116
\(946\) 0 0
\(947\) 20.7581 0.674547 0.337274 0.941407i \(-0.390495\pi\)
0.337274 + 0.941407i \(0.390495\pi\)
\(948\) 0 0
\(949\) −3.12327 −0.101386
\(950\) 0 0
\(951\) 8.11944 0.263291
\(952\) 0 0
\(953\) 14.2806 0.462596 0.231298 0.972883i \(-0.425703\pi\)
0.231298 + 0.972883i \(0.425703\pi\)
\(954\) 0 0
\(955\) 2.24731 0.0727211
\(956\) 0 0
\(957\) −4.23175 −0.136793
\(958\) 0 0
\(959\) 7.32245 0.236454
\(960\) 0 0
\(961\) 20.4491 0.659648
\(962\) 0 0
\(963\) −1.59253 −0.0513187
\(964\) 0 0
\(965\) −0.870654 −0.0280273
\(966\) 0 0
\(967\) 34.9044 1.12245 0.561224 0.827664i \(-0.310330\pi\)
0.561224 + 0.827664i \(0.310330\pi\)
\(968\) 0 0
\(969\) 45.7621 1.47009
\(970\) 0 0
\(971\) 45.4184 1.45754 0.728772 0.684756i \(-0.240091\pi\)
0.728772 + 0.684756i \(0.240091\pi\)
\(972\) 0 0
\(973\) −26.8285 −0.860081
\(974\) 0 0
\(975\) −4.97134 −0.159210
\(976\) 0 0
\(977\) 57.9835 1.85506 0.927529 0.373752i \(-0.121929\pi\)
0.927529 + 0.373752i \(0.121929\pi\)
\(978\) 0 0
\(979\) −1.80392 −0.0576534
\(980\) 0 0
\(981\) −1.32480 −0.0422975
\(982\) 0 0
\(983\) −15.4019 −0.491244 −0.245622 0.969366i \(-0.578992\pi\)
−0.245622 + 0.969366i \(0.578992\pi\)
\(984\) 0 0
\(985\) 0.316484 0.0100840
\(986\) 0 0
\(987\) 15.2346 0.484923
\(988\) 0 0
\(989\) 64.2767 2.04388
\(990\) 0 0
\(991\) 10.1165 0.321360 0.160680 0.987007i \(-0.448631\pi\)
0.160680 + 0.987007i \(0.448631\pi\)
\(992\) 0 0
\(993\) −6.05901 −0.192277
\(994\) 0 0
\(995\) 2.27093 0.0719933
\(996\) 0 0
\(997\) −44.1473 −1.39816 −0.699080 0.715044i \(-0.746407\pi\)
−0.699080 + 0.715044i \(0.746407\pi\)
\(998\) 0 0
\(999\) 5.03028 0.159151
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.1.3 5
4.3 odd 2 3432.2.a.w.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.w.1.3 5 4.3 odd 2
6864.2.a.cf.1.3 5 1.1 even 1 trivial