Properties

Label 6864.2.a.cf.1.2
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 1.2
Root \(0.679950\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.05398 q^{5} -4.24902 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.05398 q^{5} -4.24902 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.05398 q^{15} -4.55494 q^{17} +0.977426 q^{19} -4.24902 q^{21} +2.82631 q^{23} -3.88912 q^{25} +1.00000 q^{27} +4.75121 q^{29} +8.28043 q^{31} +1.00000 q^{33} +4.47839 q^{35} -6.14251 q^{37} +1.00000 q^{39} -6.92368 q^{41} -7.56208 q^{43} -1.05398 q^{45} -0.967375 q^{47} +11.0542 q^{49} -4.55494 q^{51} -5.76058 q^{53} -1.05398 q^{55} +0.977426 q^{57} +5.16364 q^{59} +12.8023 q^{61} -4.24902 q^{63} -1.05398 q^{65} -8.76942 q^{67} +2.82631 q^{69} -3.74999 q^{71} +0.141064 q^{73} -3.88912 q^{75} -4.24902 q^{77} +12.2704 q^{79} +1.00000 q^{81} +8.73747 q^{83} +4.80082 q^{85} +4.75121 q^{87} -16.5309 q^{89} -4.24902 q^{91} +8.28043 q^{93} -1.03019 q^{95} -7.42271 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + O(q^{10}) \) \( 5q + 5q^{3} + q^{5} + 5q^{7} + 5q^{9} + 5q^{11} + 5q^{13} + q^{15} + 4q^{19} + 5q^{21} + 5q^{23} + 4q^{25} + 5q^{27} + 11q^{29} + 8q^{31} + 5q^{33} + 5q^{35} - 8q^{37} + 5q^{39} - q^{41} - q^{43} + q^{45} + 18q^{47} + 10q^{49} + 2q^{53} + q^{55} + 4q^{57} + 13q^{59} + 9q^{61} + 5q^{63} + q^{65} - 5q^{67} + 5q^{69} + 24q^{71} - 13q^{73} + 4q^{75} + 5q^{77} + 6q^{79} + 5q^{81} + 22q^{83} - 22q^{85} + 11q^{87} - 14q^{89} + 5q^{91} + 8q^{93} + 32q^{95} - 20q^{97} + 5q^{99} + O(q^{100}) \)

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
\(5\) −1.05398 −0.471354 −0.235677 0.971831i \(-0.575731\pi\)
−0.235677 + 0.971831i \(0.575731\pi\)
\(6\) 0 0
\(7\) −4.24902 −1.60598 −0.802990 0.595992i \(-0.796759\pi\)
−0.802990 + 0.595992i \(0.796759\pi\)
\(8\) 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
\(15\) −1.05398 −0.272137
\(16\) 0 0
\(17\) −4.55494 −1.10474 −0.552368 0.833600i \(-0.686276\pi\)
−0.552368 + 0.833600i \(0.686276\pi\)
\(18\) 0 0
\(19\) 0.977426 0.224237 0.112118 0.993695i \(-0.464236\pi\)
0.112118 + 0.993695i \(0.464236\pi\)
\(20\) 0 0
\(21\) −4.24902 −0.927213
\(22\) 0 0
\(23\) 2.82631 0.589327 0.294663 0.955601i \(-0.404793\pi\)
0.294663 + 0.955601i \(0.404793\pi\)
\(24\) 0 0
\(25\) −3.88912 −0.777825
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.75121 0.882277 0.441139 0.897439i \(-0.354575\pi\)
0.441139 + 0.897439i \(0.354575\pi\)
\(30\) 0 0
\(31\) 8.28043 1.48721 0.743605 0.668619i \(-0.233114\pi\)
0.743605 + 0.668619i \(0.233114\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 4.47839 0.756986
\(36\) 0 0
\(37\) −6.14251 −1.00982 −0.504912 0.863171i \(-0.668475\pi\)
−0.504912 + 0.863171i \(0.668475\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −6.92368 −1.08130 −0.540648 0.841249i \(-0.681821\pi\)
−0.540648 + 0.841249i \(0.681821\pi\)
\(42\) 0 0
\(43\) −7.56208 −1.15321 −0.576603 0.817024i \(-0.695622\pi\)
−0.576603 + 0.817024i \(0.695622\pi\)
\(44\) 0 0
\(45\) −1.05398 −0.157118
\(46\) 0 0
\(47\) −0.967375 −0.141106 −0.0705530 0.997508i \(-0.522476\pi\)
−0.0705530 + 0.997508i \(0.522476\pi\)
\(48\) 0 0
\(49\) 11.0542 1.57917
\(50\) 0 0
\(51\) −4.55494 −0.637820
\(52\) 0 0
\(53\) −5.76058 −0.791277 −0.395638 0.918406i \(-0.629477\pi\)
−0.395638 + 0.918406i \(0.629477\pi\)
\(54\) 0 0
\(55\) −1.05398 −0.142119
\(56\) 0 0
\(57\) 0.977426 0.129463
\(58\) 0 0
\(59\) 5.16364 0.672248 0.336124 0.941818i \(-0.390884\pi\)
0.336124 + 0.941818i \(0.390884\pi\)
\(60\) 0 0
\(61\) 12.8023 1.63916 0.819582 0.572962i \(-0.194206\pi\)
0.819582 + 0.572962i \(0.194206\pi\)
\(62\) 0 0
\(63\) −4.24902 −0.535327
\(64\) 0 0
\(65\) −1.05398 −0.130730
\(66\) 0 0
\(67\) −8.76942 −1.07135 −0.535677 0.844423i \(-0.679944\pi\)
−0.535677 + 0.844423i \(0.679944\pi\)
\(68\) 0 0
\(69\) 2.82631 0.340248
\(70\) 0 0
\(71\) −3.74999 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(72\) 0 0
\(73\) 0.141064 0.0165102 0.00825512 0.999966i \(-0.497372\pi\)
0.00825512 + 0.999966i \(0.497372\pi\)
\(74\) 0 0
\(75\) −3.88912 −0.449077
\(76\) 0 0
\(77\) −4.24902 −0.484221
\(78\) 0 0
\(79\) 12.2704 1.38053 0.690263 0.723559i \(-0.257495\pi\)
0.690263 + 0.723559i \(0.257495\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.73747 0.959062 0.479531 0.877525i \(-0.340807\pi\)
0.479531 + 0.877525i \(0.340807\pi\)
\(84\) 0 0
\(85\) 4.80082 0.520722
\(86\) 0 0
\(87\) 4.75121 0.509383
\(88\) 0 0
\(89\) −16.5309 −1.75227 −0.876136 0.482063i \(-0.839887\pi\)
−0.876136 + 0.482063i \(0.839887\pi\)
\(90\) 0 0
\(91\) −4.24902 −0.445419
\(92\) 0 0
\(93\) 8.28043 0.858641
\(94\) 0 0
\(95\) −1.03019 −0.105695
\(96\) 0 0
\(97\) −7.42271 −0.753662 −0.376831 0.926282i \(-0.622986\pi\)
−0.376831 + 0.926282i \(0.622986\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 10.2076 1.01569 0.507845 0.861448i \(-0.330442\pi\)
0.507845 + 0.861448i \(0.330442\pi\)
\(102\) 0 0
\(103\) 7.27160 0.716492 0.358246 0.933627i \(-0.383375\pi\)
0.358246 + 0.933627i \(0.383375\pi\)
\(104\) 0 0
\(105\) 4.47839 0.437046
\(106\) 0 0
\(107\) 16.9870 1.64220 0.821099 0.570785i \(-0.193361\pi\)
0.821099 + 0.570785i \(0.193361\pi\)
\(108\) 0 0
\(109\) 20.3032 1.94470 0.972349 0.233534i \(-0.0750289\pi\)
0.972349 + 0.233534i \(0.0750289\pi\)
\(110\) 0 0
\(111\) −6.14251 −0.583022
\(112\) 0 0
\(113\) −1.72404 −0.162184 −0.0810919 0.996707i \(-0.525841\pi\)
−0.0810919 + 0.996707i \(0.525841\pi\)
\(114\) 0 0
\(115\) −2.97888 −0.277782
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 19.3541 1.77418
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.92368 −0.624287
\(124\) 0 0
\(125\) 9.36897 0.837986
\(126\) 0 0
\(127\) −11.7382 −1.04160 −0.520800 0.853679i \(-0.674366\pi\)
−0.520800 + 0.853679i \(0.674366\pi\)
\(128\) 0 0
\(129\) −7.56208 −0.665804
\(130\) 0 0
\(131\) 5.61898 0.490932 0.245466 0.969405i \(-0.421059\pi\)
0.245466 + 0.969405i \(0.421059\pi\)
\(132\) 0 0
\(133\) −4.15311 −0.360120
\(134\) 0 0
\(135\) −1.05398 −0.0907122
\(136\) 0 0
\(137\) 3.70315 0.316381 0.158191 0.987409i \(-0.449434\pi\)
0.158191 + 0.987409i \(0.449434\pi\)
\(138\) 0 0
\(139\) 20.3501 1.72607 0.863036 0.505143i \(-0.168560\pi\)
0.863036 + 0.505143i \(0.168560\pi\)
\(140\) 0 0
\(141\) −0.967375 −0.0814676
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −5.00768 −0.415865
\(146\) 0 0
\(147\) 11.0542 0.911736
\(148\) 0 0
\(149\) −2.30277 −0.188651 −0.0943253 0.995541i \(-0.530069\pi\)
−0.0943253 + 0.995541i \(0.530069\pi\)
\(150\) 0 0
\(151\) −8.19528 −0.666922 −0.333461 0.942764i \(-0.608217\pi\)
−0.333461 + 0.942764i \(0.608217\pi\)
\(152\) 0 0
\(153\) −4.55494 −0.368245
\(154\) 0 0
\(155\) −8.72741 −0.701003
\(156\) 0 0
\(157\) −11.9477 −0.953530 −0.476765 0.879031i \(-0.658191\pi\)
−0.476765 + 0.879031i \(0.658191\pi\)
\(158\) 0 0
\(159\) −5.76058 −0.456844
\(160\) 0 0
\(161\) −12.0091 −0.946447
\(162\) 0 0
\(163\) 20.5822 1.61212 0.806062 0.591831i \(-0.201595\pi\)
0.806062 + 0.591831i \(0.201595\pi\)
\(164\) 0 0
\(165\) −1.05398 −0.0820523
\(166\) 0 0
\(167\) 11.9463 0.924429 0.462214 0.886768i \(-0.347055\pi\)
0.462214 + 0.886768i \(0.347055\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0.977426 0.0747457
\(172\) 0 0
\(173\) 18.8879 1.43602 0.718010 0.696033i \(-0.245053\pi\)
0.718010 + 0.696033i \(0.245053\pi\)
\(174\) 0 0
\(175\) 16.5250 1.24917
\(176\) 0 0
\(177\) 5.16364 0.388123
\(178\) 0 0
\(179\) 8.38247 0.626536 0.313268 0.949665i \(-0.398576\pi\)
0.313268 + 0.949665i \(0.398576\pi\)
\(180\) 0 0
\(181\) −10.2721 −0.763517 −0.381759 0.924262i \(-0.624681\pi\)
−0.381759 + 0.924262i \(0.624681\pi\)
\(182\) 0 0
\(183\) 12.8023 0.946371
\(184\) 0 0
\(185\) 6.47409 0.475985
\(186\) 0 0
\(187\) −4.55494 −0.333091
\(188\) 0 0
\(189\) −4.24902 −0.309071
\(190\) 0 0
\(191\) 18.2672 1.32177 0.660885 0.750487i \(-0.270181\pi\)
0.660885 + 0.750487i \(0.270181\pi\)
\(192\) 0 0
\(193\) 3.13054 0.225341 0.112670 0.993632i \(-0.464060\pi\)
0.112670 + 0.993632i \(0.464060\pi\)
\(194\) 0 0
\(195\) −1.05398 −0.0754771
\(196\) 0 0
\(197\) 14.7380 1.05004 0.525020 0.851090i \(-0.324058\pi\)
0.525020 + 0.851090i \(0.324058\pi\)
\(198\) 0 0
\(199\) 24.7859 1.75702 0.878511 0.477722i \(-0.158537\pi\)
0.878511 + 0.477722i \(0.158537\pi\)
\(200\) 0 0
\(201\) −8.76942 −0.618547
\(202\) 0 0
\(203\) −20.1880 −1.41692
\(204\) 0 0
\(205\) 7.29742 0.509674
\(206\) 0 0
\(207\) 2.82631 0.196442
\(208\) 0 0
\(209\) 0.977426 0.0676100
\(210\) 0 0
\(211\) 5.46319 0.376101 0.188051 0.982159i \(-0.439783\pi\)
0.188051 + 0.982159i \(0.439783\pi\)
\(212\) 0 0
\(213\) −3.74999 −0.256945
\(214\) 0 0
\(215\) 7.97029 0.543569
\(216\) 0 0
\(217\) −35.1838 −2.38843
\(218\) 0 0
\(219\) 0.141064 0.00953220
\(220\) 0 0
\(221\) −4.55494 −0.306399
\(222\) 0 0
\(223\) −25.2765 −1.69264 −0.846321 0.532673i \(-0.821187\pi\)
−0.846321 + 0.532673i \(0.821187\pi\)
\(224\) 0 0
\(225\) −3.88912 −0.259275
\(226\) 0 0
\(227\) −0.869929 −0.0577392 −0.0288696 0.999583i \(-0.509191\pi\)
−0.0288696 + 0.999583i \(0.509191\pi\)
\(228\) 0 0
\(229\) −18.3646 −1.21357 −0.606783 0.794867i \(-0.707541\pi\)
−0.606783 + 0.794867i \(0.707541\pi\)
\(230\) 0 0
\(231\) −4.24902 −0.279565
\(232\) 0 0
\(233\) −23.2407 −1.52255 −0.761273 0.648432i \(-0.775425\pi\)
−0.761273 + 0.648432i \(0.775425\pi\)
\(234\) 0 0
\(235\) 1.01959 0.0665110
\(236\) 0 0
\(237\) 12.2704 0.797047
\(238\) 0 0
\(239\) 22.1838 1.43495 0.717475 0.696585i \(-0.245298\pi\)
0.717475 + 0.696585i \(0.245298\pi\)
\(240\) 0 0
\(241\) 22.7567 1.46589 0.732943 0.680290i \(-0.238146\pi\)
0.732943 + 0.680290i \(0.238146\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −11.6509 −0.744350
\(246\) 0 0
\(247\) 0.977426 0.0621921
\(248\) 0 0
\(249\) 8.73747 0.553714
\(250\) 0 0
\(251\) −13.3920 −0.845297 −0.422648 0.906294i \(-0.638899\pi\)
−0.422648 + 0.906294i \(0.638899\pi\)
\(252\) 0 0
\(253\) 2.82631 0.177689
\(254\) 0 0
\(255\) 4.80082 0.300639
\(256\) 0 0
\(257\) −17.1612 −1.07049 −0.535243 0.844698i \(-0.679780\pi\)
−0.535243 + 0.844698i \(0.679780\pi\)
\(258\) 0 0
\(259\) 26.0997 1.62176
\(260\) 0 0
\(261\) 4.75121 0.294092
\(262\) 0 0
\(263\) 15.2116 0.937985 0.468992 0.883202i \(-0.344617\pi\)
0.468992 + 0.883202i \(0.344617\pi\)
\(264\) 0 0
\(265\) 6.07154 0.372972
\(266\) 0 0
\(267\) −16.5309 −1.01168
\(268\) 0 0
\(269\) 9.36806 0.571181 0.285590 0.958352i \(-0.407810\pi\)
0.285590 + 0.958352i \(0.407810\pi\)
\(270\) 0 0
\(271\) 21.4721 1.30434 0.652168 0.758074i \(-0.273860\pi\)
0.652168 + 0.758074i \(0.273860\pi\)
\(272\) 0 0
\(273\) −4.24902 −0.257163
\(274\) 0 0
\(275\) −3.88912 −0.234523
\(276\) 0 0
\(277\) 24.4894 1.47143 0.735714 0.677292i \(-0.236847\pi\)
0.735714 + 0.677292i \(0.236847\pi\)
\(278\) 0 0
\(279\) 8.28043 0.495736
\(280\) 0 0
\(281\) 17.6553 1.05323 0.526614 0.850105i \(-0.323461\pi\)
0.526614 + 0.850105i \(0.323461\pi\)
\(282\) 0 0
\(283\) −13.9691 −0.830374 −0.415187 0.909736i \(-0.636284\pi\)
−0.415187 + 0.909736i \(0.636284\pi\)
\(284\) 0 0
\(285\) −1.03019 −0.0610231
\(286\) 0 0
\(287\) 29.4189 1.73654
\(288\) 0 0
\(289\) 3.74752 0.220442
\(290\) 0 0
\(291\) −7.42271 −0.435127
\(292\) 0 0
\(293\) 1.06528 0.0622346 0.0311173 0.999516i \(-0.490093\pi\)
0.0311173 + 0.999516i \(0.490093\pi\)
\(294\) 0 0
\(295\) −5.44237 −0.316867
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 2.82631 0.163450
\(300\) 0 0
\(301\) 32.1315 1.85203
\(302\) 0 0
\(303\) 10.2076 0.586409
\(304\) 0 0
\(305\) −13.4933 −0.772627
\(306\) 0 0
\(307\) −17.5349 −1.00077 −0.500385 0.865803i \(-0.666808\pi\)
−0.500385 + 0.865803i \(0.666808\pi\)
\(308\) 0 0
\(309\) 7.27160 0.413667
\(310\) 0 0
\(311\) 7.55761 0.428553 0.214276 0.976773i \(-0.431261\pi\)
0.214276 + 0.976773i \(0.431261\pi\)
\(312\) 0 0
\(313\) −10.2350 −0.578519 −0.289259 0.957251i \(-0.593409\pi\)
−0.289259 + 0.957251i \(0.593409\pi\)
\(314\) 0 0
\(315\) 4.47839 0.252329
\(316\) 0 0
\(317\) −3.76319 −0.211362 −0.105681 0.994400i \(-0.533702\pi\)
−0.105681 + 0.994400i \(0.533702\pi\)
\(318\) 0 0
\(319\) 4.75121 0.266017
\(320\) 0 0
\(321\) 16.9870 0.948124
\(322\) 0 0
\(323\) −4.45212 −0.247723
\(324\) 0 0
\(325\) −3.88912 −0.215730
\(326\) 0 0
\(327\) 20.3032 1.12277
\(328\) 0 0
\(329\) 4.11040 0.226614
\(330\) 0 0
\(331\) 24.5851 1.35132 0.675660 0.737213i \(-0.263859\pi\)
0.675660 + 0.737213i \(0.263859\pi\)
\(332\) 0 0
\(333\) −6.14251 −0.336608
\(334\) 0 0
\(335\) 9.24280 0.504988
\(336\) 0 0
\(337\) 17.4966 0.953100 0.476550 0.879147i \(-0.341887\pi\)
0.476550 + 0.879147i \(0.341887\pi\)
\(338\) 0 0
\(339\) −1.72404 −0.0936368
\(340\) 0 0
\(341\) 8.28043 0.448411
\(342\) 0 0
\(343\) −17.2265 −0.930141
\(344\) 0 0
\(345\) −2.97888 −0.160377
\(346\) 0 0
\(347\) 3.03602 0.162982 0.0814909 0.996674i \(-0.474032\pi\)
0.0814909 + 0.996674i \(0.474032\pi\)
\(348\) 0 0
\(349\) −0.587880 −0.0314685 −0.0157343 0.999876i \(-0.505009\pi\)
−0.0157343 + 0.999876i \(0.505009\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −0.259864 −0.0138312 −0.00691558 0.999976i \(-0.502201\pi\)
−0.00691558 + 0.999976i \(0.502201\pi\)
\(354\) 0 0
\(355\) 3.95242 0.209772
\(356\) 0 0
\(357\) 19.3541 1.02433
\(358\) 0 0
\(359\) 6.59640 0.348145 0.174072 0.984733i \(-0.444307\pi\)
0.174072 + 0.984733i \(0.444307\pi\)
\(360\) 0 0
\(361\) −18.0446 −0.949718
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −0.148678 −0.00778218
\(366\) 0 0
\(367\) 5.13108 0.267840 0.133920 0.990992i \(-0.457243\pi\)
0.133920 + 0.990992i \(0.457243\pi\)
\(368\) 0 0
\(369\) −6.92368 −0.360432
\(370\) 0 0
\(371\) 24.4769 1.27078
\(372\) 0 0
\(373\) 2.66849 0.138169 0.0690846 0.997611i \(-0.477992\pi\)
0.0690846 + 0.997611i \(0.477992\pi\)
\(374\) 0 0
\(375\) 9.36897 0.483811
\(376\) 0 0
\(377\) 4.75121 0.244700
\(378\) 0 0
\(379\) 17.2900 0.888129 0.444065 0.895995i \(-0.353536\pi\)
0.444065 + 0.895995i \(0.353536\pi\)
\(380\) 0 0
\(381\) −11.7382 −0.601368
\(382\) 0 0
\(383\) 26.7826 1.36853 0.684264 0.729235i \(-0.260124\pi\)
0.684264 + 0.729235i \(0.260124\pi\)
\(384\) 0 0
\(385\) 4.47839 0.228240
\(386\) 0 0
\(387\) −7.56208 −0.384402
\(388\) 0 0
\(389\) 25.8661 1.31146 0.655732 0.754994i \(-0.272360\pi\)
0.655732 + 0.754994i \(0.272360\pi\)
\(390\) 0 0
\(391\) −12.8737 −0.651050
\(392\) 0 0
\(393\) 5.61898 0.283440
\(394\) 0 0
\(395\) −12.9327 −0.650717
\(396\) 0 0
\(397\) −0.913162 −0.0458303 −0.0229151 0.999737i \(-0.507295\pi\)
−0.0229151 + 0.999737i \(0.507295\pi\)
\(398\) 0 0
\(399\) −4.15311 −0.207915
\(400\) 0 0
\(401\) 11.1048 0.554547 0.277274 0.960791i \(-0.410569\pi\)
0.277274 + 0.960791i \(0.410569\pi\)
\(402\) 0 0
\(403\) 8.28043 0.412478
\(404\) 0 0
\(405\) −1.05398 −0.0523727
\(406\) 0 0
\(407\) −6.14251 −0.304473
\(408\) 0 0
\(409\) 6.25242 0.309162 0.154581 0.987980i \(-0.450597\pi\)
0.154581 + 0.987980i \(0.450597\pi\)
\(410\) 0 0
\(411\) 3.70315 0.182663
\(412\) 0 0
\(413\) −21.9404 −1.07962
\(414\) 0 0
\(415\) −9.20912 −0.452058
\(416\) 0 0
\(417\) 20.3501 0.996548
\(418\) 0 0
\(419\) −21.7682 −1.06345 −0.531723 0.846918i \(-0.678455\pi\)
−0.531723 + 0.846918i \(0.678455\pi\)
\(420\) 0 0
\(421\) −38.3099 −1.86711 −0.933556 0.358433i \(-0.883311\pi\)
−0.933556 + 0.358433i \(0.883311\pi\)
\(422\) 0 0
\(423\) −0.967375 −0.0470354
\(424\) 0 0
\(425\) 17.7147 0.859291
\(426\) 0 0
\(427\) −54.3972 −2.63246
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) −32.3906 −1.56020 −0.780099 0.625656i \(-0.784831\pi\)
−0.780099 + 0.625656i \(0.784831\pi\)
\(432\) 0 0
\(433\) 13.2025 0.634471 0.317236 0.948347i \(-0.397245\pi\)
0.317236 + 0.948347i \(0.397245\pi\)
\(434\) 0 0
\(435\) −5.00768 −0.240100
\(436\) 0 0
\(437\) 2.76251 0.132149
\(438\) 0 0
\(439\) −36.2684 −1.73100 −0.865498 0.500912i \(-0.832998\pi\)
−0.865498 + 0.500912i \(0.832998\pi\)
\(440\) 0 0
\(441\) 11.0542 0.526391
\(442\) 0 0
\(443\) 24.3416 1.15650 0.578251 0.815859i \(-0.303735\pi\)
0.578251 + 0.815859i \(0.303735\pi\)
\(444\) 0 0
\(445\) 17.4233 0.825942
\(446\) 0 0
\(447\) −2.30277 −0.108917
\(448\) 0 0
\(449\) 34.8932 1.64671 0.823356 0.567525i \(-0.192099\pi\)
0.823356 + 0.567525i \(0.192099\pi\)
\(450\) 0 0
\(451\) −6.92368 −0.326023
\(452\) 0 0
\(453\) −8.19528 −0.385048
\(454\) 0 0
\(455\) 4.47839 0.209950
\(456\) 0 0
\(457\) −22.1410 −1.03571 −0.517856 0.855468i \(-0.673270\pi\)
−0.517856 + 0.855468i \(0.673270\pi\)
\(458\) 0 0
\(459\) −4.55494 −0.212607
\(460\) 0 0
\(461\) −30.8891 −1.43865 −0.719325 0.694673i \(-0.755549\pi\)
−0.719325 + 0.694673i \(0.755549\pi\)
\(462\) 0 0
\(463\) −13.4244 −0.623885 −0.311943 0.950101i \(-0.600980\pi\)
−0.311943 + 0.950101i \(0.600980\pi\)
\(464\) 0 0
\(465\) −8.72741 −0.404724
\(466\) 0 0
\(467\) −14.4030 −0.666493 −0.333247 0.942840i \(-0.608144\pi\)
−0.333247 + 0.942840i \(0.608144\pi\)
\(468\) 0 0
\(469\) 37.2615 1.72057
\(470\) 0 0
\(471\) −11.9477 −0.550521
\(472\) 0 0
\(473\) −7.56208 −0.347705
\(474\) 0 0
\(475\) −3.80133 −0.174417
\(476\) 0 0
\(477\) −5.76058 −0.263759
\(478\) 0 0
\(479\) −36.4321 −1.66463 −0.832313 0.554306i \(-0.812984\pi\)
−0.832313 + 0.554306i \(0.812984\pi\)
\(480\) 0 0
\(481\) −6.14251 −0.280075
\(482\) 0 0
\(483\) −12.0091 −0.546431
\(484\) 0 0
\(485\) 7.82340 0.355242
\(486\) 0 0
\(487\) 16.2709 0.737306 0.368653 0.929567i \(-0.379819\pi\)
0.368653 + 0.929567i \(0.379819\pi\)
\(488\) 0 0
\(489\) 20.5822 0.930760
\(490\) 0 0
\(491\) −40.6615 −1.83503 −0.917513 0.397705i \(-0.869807\pi\)
−0.917513 + 0.397705i \(0.869807\pi\)
\(492\) 0 0
\(493\) −21.6415 −0.974683
\(494\) 0 0
\(495\) −1.05398 −0.0473729
\(496\) 0 0
\(497\) 15.9338 0.714728
\(498\) 0 0
\(499\) −7.96515 −0.356569 −0.178285 0.983979i \(-0.557055\pi\)
−0.178285 + 0.983979i \(0.557055\pi\)
\(500\) 0 0
\(501\) 11.9463 0.533719
\(502\) 0 0
\(503\) 36.6300 1.63325 0.816625 0.577169i \(-0.195843\pi\)
0.816625 + 0.577169i \(0.195843\pi\)
\(504\) 0 0
\(505\) −10.7586 −0.478750
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −23.2934 −1.03246 −0.516231 0.856449i \(-0.672666\pi\)
−0.516231 + 0.856449i \(0.672666\pi\)
\(510\) 0 0
\(511\) −0.599383 −0.0265151
\(512\) 0 0
\(513\) 0.977426 0.0431544
\(514\) 0 0
\(515\) −7.66412 −0.337722
\(516\) 0 0
\(517\) −0.967375 −0.0425451
\(518\) 0 0
\(519\) 18.8879 0.829087
\(520\) 0 0
\(521\) −2.85802 −0.125212 −0.0626060 0.998038i \(-0.519941\pi\)
−0.0626060 + 0.998038i \(0.519941\pi\)
\(522\) 0 0
\(523\) −26.1624 −1.14400 −0.572001 0.820253i \(-0.693833\pi\)
−0.572001 + 0.820253i \(0.693833\pi\)
\(524\) 0 0
\(525\) 16.5250 0.721210
\(526\) 0 0
\(527\) −37.7169 −1.64297
\(528\) 0 0
\(529\) −15.0120 −0.652694
\(530\) 0 0
\(531\) 5.16364 0.224083
\(532\) 0 0
\(533\) −6.92368 −0.299898
\(534\) 0 0
\(535\) −17.9040 −0.774058
\(536\) 0 0
\(537\) 8.38247 0.361730
\(538\) 0 0
\(539\) 11.0542 0.476139
\(540\) 0 0
\(541\) −41.4318 −1.78129 −0.890646 0.454697i \(-0.849747\pi\)
−0.890646 + 0.454697i \(0.849747\pi\)
\(542\) 0 0
\(543\) −10.2721 −0.440817
\(544\) 0 0
\(545\) −21.3992 −0.916642
\(546\) 0 0
\(547\) −15.9239 −0.680857 −0.340429 0.940270i \(-0.610572\pi\)
−0.340429 + 0.940270i \(0.610572\pi\)
\(548\) 0 0
\(549\) 12.8023 0.546388
\(550\) 0 0
\(551\) 4.64395 0.197839
\(552\) 0 0
\(553\) −52.1372 −2.21710
\(554\) 0 0
\(555\) 6.47409 0.274810
\(556\) 0 0
\(557\) 33.6251 1.42474 0.712370 0.701804i \(-0.247622\pi\)
0.712370 + 0.701804i \(0.247622\pi\)
\(558\) 0 0
\(559\) −7.56208 −0.319842
\(560\) 0 0
\(561\) −4.55494 −0.192310
\(562\) 0 0
\(563\) −15.4510 −0.651181 −0.325590 0.945511i \(-0.605563\pi\)
−0.325590 + 0.945511i \(0.605563\pi\)
\(564\) 0 0
\(565\) 1.81710 0.0764460
\(566\) 0 0
\(567\) −4.24902 −0.178442
\(568\) 0 0
\(569\) 15.2238 0.638214 0.319107 0.947719i \(-0.396617\pi\)
0.319107 + 0.947719i \(0.396617\pi\)
\(570\) 0 0
\(571\) −2.18083 −0.0912651 −0.0456325 0.998958i \(-0.514530\pi\)
−0.0456325 + 0.998958i \(0.514530\pi\)
\(572\) 0 0
\(573\) 18.2672 0.763125
\(574\) 0 0
\(575\) −10.9919 −0.458393
\(576\) 0 0
\(577\) 15.3464 0.638879 0.319440 0.947607i \(-0.396505\pi\)
0.319440 + 0.947607i \(0.396505\pi\)
\(578\) 0 0
\(579\) 3.13054 0.130101
\(580\) 0 0
\(581\) −37.1257 −1.54023
\(582\) 0 0
\(583\) −5.76058 −0.238579
\(584\) 0 0
\(585\) −1.05398 −0.0435767
\(586\) 0 0
\(587\) 33.8295 1.39629 0.698147 0.715954i \(-0.254008\pi\)
0.698147 + 0.715954i \(0.254008\pi\)
\(588\) 0 0
\(589\) 8.09351 0.333487
\(590\) 0 0
\(591\) 14.7380 0.606241
\(592\) 0 0
\(593\) 0.689961 0.0283333 0.0141666 0.999900i \(-0.495490\pi\)
0.0141666 + 0.999900i \(0.495490\pi\)
\(594\) 0 0
\(595\) −20.3988 −0.836270
\(596\) 0 0
\(597\) 24.7859 1.01442
\(598\) 0 0
\(599\) −4.15034 −0.169578 −0.0847891 0.996399i \(-0.527022\pi\)
−0.0847891 + 0.996399i \(0.527022\pi\)
\(600\) 0 0
\(601\) 6.88767 0.280954 0.140477 0.990084i \(-0.455136\pi\)
0.140477 + 0.990084i \(0.455136\pi\)
\(602\) 0 0
\(603\) −8.76942 −0.357118
\(604\) 0 0
\(605\) −1.05398 −0.0428504
\(606\) 0 0
\(607\) −11.7382 −0.476441 −0.238220 0.971211i \(-0.576564\pi\)
−0.238220 + 0.971211i \(0.576564\pi\)
\(608\) 0 0
\(609\) −20.1880 −0.818059
\(610\) 0 0
\(611\) −0.967375 −0.0391358
\(612\) 0 0
\(613\) −14.6477 −0.591616 −0.295808 0.955247i \(-0.595589\pi\)
−0.295808 + 0.955247i \(0.595589\pi\)
\(614\) 0 0
\(615\) 7.29742 0.294260
\(616\) 0 0
\(617\) −10.9752 −0.441844 −0.220922 0.975291i \(-0.570907\pi\)
−0.220922 + 0.975291i \(0.570907\pi\)
\(618\) 0 0
\(619\) −32.2138 −1.29478 −0.647391 0.762158i \(-0.724140\pi\)
−0.647391 + 0.762158i \(0.724140\pi\)
\(620\) 0 0
\(621\) 2.82631 0.113416
\(622\) 0 0
\(623\) 70.2402 2.81412
\(624\) 0 0
\(625\) 9.57092 0.382837
\(626\) 0 0
\(627\) 0.977426 0.0390346
\(628\) 0 0
\(629\) 27.9788 1.11559
\(630\) 0 0
\(631\) −45.0712 −1.79426 −0.897128 0.441770i \(-0.854351\pi\)
−0.897128 + 0.441770i \(0.854351\pi\)
\(632\) 0 0
\(633\) 5.46319 0.217142
\(634\) 0 0
\(635\) 12.3719 0.490963
\(636\) 0 0
\(637\) 11.0542 0.437984
\(638\) 0 0
\(639\) −3.74999 −0.148347
\(640\) 0 0
\(641\) 28.1430 1.11158 0.555791 0.831322i \(-0.312415\pi\)
0.555791 + 0.831322i \(0.312415\pi\)
\(642\) 0 0
\(643\) 7.10237 0.280090 0.140045 0.990145i \(-0.455275\pi\)
0.140045 + 0.990145i \(0.455275\pi\)
\(644\) 0 0
\(645\) 7.97029 0.313830
\(646\) 0 0
\(647\) 28.2768 1.11167 0.555837 0.831291i \(-0.312398\pi\)
0.555837 + 0.831291i \(0.312398\pi\)
\(648\) 0 0
\(649\) 5.16364 0.202690
\(650\) 0 0
\(651\) −35.1838 −1.37896
\(652\) 0 0
\(653\) 12.1708 0.476279 0.238140 0.971231i \(-0.423462\pi\)
0.238140 + 0.971231i \(0.423462\pi\)
\(654\) 0 0
\(655\) −5.92229 −0.231403
\(656\) 0 0
\(657\) 0.141064 0.00550342
\(658\) 0 0
\(659\) 31.9183 1.24336 0.621681 0.783271i \(-0.286450\pi\)
0.621681 + 0.783271i \(0.286450\pi\)
\(660\) 0 0
\(661\) −33.6021 −1.30697 −0.653485 0.756939i \(-0.726694\pi\)
−0.653485 + 0.756939i \(0.726694\pi\)
\(662\) 0 0
\(663\) −4.55494 −0.176899
\(664\) 0 0
\(665\) 4.37730 0.169744
\(666\) 0 0
\(667\) 13.4284 0.519949
\(668\) 0 0
\(669\) −25.2765 −0.977247
\(670\) 0 0
\(671\) 12.8023 0.494226
\(672\) 0 0
\(673\) 28.3314 1.09210 0.546048 0.837754i \(-0.316132\pi\)
0.546048 + 0.837754i \(0.316132\pi\)
\(674\) 0 0
\(675\) −3.88912 −0.149692
\(676\) 0 0
\(677\) 2.74310 0.105426 0.0527129 0.998610i \(-0.483213\pi\)
0.0527129 + 0.998610i \(0.483213\pi\)
\(678\) 0 0
\(679\) 31.5393 1.21037
\(680\) 0 0
\(681\) −0.869929 −0.0333358
\(682\) 0 0
\(683\) 23.3066 0.891803 0.445901 0.895082i \(-0.352883\pi\)
0.445901 + 0.895082i \(0.352883\pi\)
\(684\) 0 0
\(685\) −3.90304 −0.149128
\(686\) 0 0
\(687\) −18.3646 −0.700653
\(688\) 0 0
\(689\) −5.76058 −0.219461
\(690\) 0 0
\(691\) 9.97295 0.379389 0.189694 0.981843i \(-0.439250\pi\)
0.189694 + 0.981843i \(0.439250\pi\)
\(692\) 0 0
\(693\) −4.24902 −0.161407
\(694\) 0 0
\(695\) −21.4486 −0.813591
\(696\) 0 0
\(697\) 31.5370 1.19455
\(698\) 0 0
\(699\) −23.2407 −0.879042
\(700\) 0 0
\(701\) 13.1096 0.495144 0.247572 0.968869i \(-0.420367\pi\)
0.247572 + 0.968869i \(0.420367\pi\)
\(702\) 0 0
\(703\) −6.00386 −0.226440
\(704\) 0 0
\(705\) 1.01959 0.0384001
\(706\) 0 0
\(707\) −43.3722 −1.63118
\(708\) 0 0
\(709\) 23.7733 0.892826 0.446413 0.894827i \(-0.352701\pi\)
0.446413 + 0.894827i \(0.352701\pi\)
\(710\) 0 0
\(711\) 12.2704 0.460175
\(712\) 0 0
\(713\) 23.4031 0.876452
\(714\) 0 0
\(715\) −1.05398 −0.0394166
\(716\) 0 0
\(717\) 22.1838 0.828468
\(718\) 0 0
\(719\) 21.1219 0.787712 0.393856 0.919172i \(-0.371141\pi\)
0.393856 + 0.919172i \(0.371141\pi\)
\(720\) 0 0
\(721\) −30.8972 −1.15067
\(722\) 0 0
\(723\) 22.7567 0.846330
\(724\) 0 0
\(725\) −18.4780 −0.686257
\(726\) 0 0
\(727\) −10.5157 −0.390006 −0.195003 0.980803i \(-0.562472\pi\)
−0.195003 + 0.980803i \(0.562472\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 34.4449 1.27399
\(732\) 0 0
\(733\) −49.4643 −1.82701 −0.913503 0.406833i \(-0.866633\pi\)
−0.913503 + 0.406833i \(0.866633\pi\)
\(734\) 0 0
\(735\) −11.6509 −0.429751
\(736\) 0 0
\(737\) −8.76942 −0.323026
\(738\) 0 0
\(739\) 9.74040 0.358306 0.179153 0.983821i \(-0.442664\pi\)
0.179153 + 0.983821i \(0.442664\pi\)
\(740\) 0 0
\(741\) 0.977426 0.0359067
\(742\) 0 0
\(743\) 13.7182 0.503272 0.251636 0.967822i \(-0.419031\pi\)
0.251636 + 0.967822i \(0.419031\pi\)
\(744\) 0 0
\(745\) 2.42708 0.0889213
\(746\) 0 0
\(747\) 8.73747 0.319687
\(748\) 0 0
\(749\) −72.1783 −2.63734
\(750\) 0 0
\(751\) −22.6627 −0.826972 −0.413486 0.910510i \(-0.635689\pi\)
−0.413486 + 0.910510i \(0.635689\pi\)
\(752\) 0 0
\(753\) −13.3920 −0.488032
\(754\) 0 0
\(755\) 8.63766 0.314357
\(756\) 0 0
\(757\) −21.1376 −0.768259 −0.384130 0.923279i \(-0.625498\pi\)
−0.384130 + 0.923279i \(0.625498\pi\)
\(758\) 0 0
\(759\) 2.82631 0.102589
\(760\) 0 0
\(761\) 2.32152 0.0841551 0.0420775 0.999114i \(-0.486602\pi\)
0.0420775 + 0.999114i \(0.486602\pi\)
\(762\) 0 0
\(763\) −86.2690 −3.12315
\(764\) 0 0
\(765\) 4.80082 0.173574
\(766\) 0 0
\(767\) 5.16364 0.186448
\(768\) 0 0
\(769\) 12.1170 0.436951 0.218476 0.975842i \(-0.429892\pi\)
0.218476 + 0.975842i \(0.429892\pi\)
\(770\) 0 0
\(771\) −17.1612 −0.618046
\(772\) 0 0
\(773\) −40.2007 −1.44592 −0.722959 0.690890i \(-0.757219\pi\)
−0.722959 + 0.690890i \(0.757219\pi\)
\(774\) 0 0
\(775\) −32.2036 −1.15679
\(776\) 0 0
\(777\) 26.0997 0.936321
\(778\) 0 0
\(779\) −6.76738 −0.242467
\(780\) 0 0
\(781\) −3.74999 −0.134185
\(782\) 0 0
\(783\) 4.75121 0.169794
\(784\) 0 0
\(785\) 12.5926 0.449451
\(786\) 0 0
\(787\) 30.3882 1.08322 0.541611 0.840629i \(-0.317815\pi\)
0.541611 + 0.840629i \(0.317815\pi\)
\(788\) 0 0
\(789\) 15.2116 0.541546
\(790\) 0 0
\(791\) 7.32548 0.260464
\(792\) 0 0
\(793\) 12.8023 0.454622
\(794\) 0 0
\(795\) 6.07154 0.215335
\(796\) 0 0
\(797\) 33.9490 1.20254 0.601268 0.799047i \(-0.294662\pi\)
0.601268 + 0.799047i \(0.294662\pi\)
\(798\) 0 0
\(799\) 4.40634 0.155885
\(800\) 0 0
\(801\) −16.5309 −0.584091
\(802\) 0 0
\(803\) 0.141064 0.00497803
\(804\) 0 0
\(805\) 12.6573 0.446112
\(806\) 0 0
\(807\) 9.36806 0.329771
\(808\) 0 0
\(809\) 4.60592 0.161936 0.0809678 0.996717i \(-0.474199\pi\)
0.0809678 + 0.996717i \(0.474199\pi\)
\(810\) 0 0
\(811\) −15.5206 −0.545003 −0.272501 0.962155i \(-0.587851\pi\)
−0.272501 + 0.962155i \(0.587851\pi\)
\(812\) 0 0
\(813\) 21.4721 0.753059
\(814\) 0 0
\(815\) −21.6933 −0.759882
\(816\) 0 0
\(817\) −7.39138 −0.258592
\(818\) 0 0
\(819\) −4.24902 −0.148473
\(820\) 0 0
\(821\) 15.7361 0.549193 0.274597 0.961560i \(-0.411456\pi\)
0.274597 + 0.961560i \(0.411456\pi\)
\(822\) 0 0
\(823\) 4.77401 0.166412 0.0832058 0.996532i \(-0.473484\pi\)
0.0832058 + 0.996532i \(0.473484\pi\)
\(824\) 0 0
\(825\) −3.88912 −0.135402
\(826\) 0 0
\(827\) 0.536827 0.0186673 0.00933365 0.999956i \(-0.497029\pi\)
0.00933365 + 0.999956i \(0.497029\pi\)
\(828\) 0 0
\(829\) 26.1757 0.909120 0.454560 0.890716i \(-0.349796\pi\)
0.454560 + 0.890716i \(0.349796\pi\)
\(830\) 0 0
\(831\) 24.4894 0.849530
\(832\) 0 0
\(833\) −50.3513 −1.74457
\(834\) 0 0
\(835\) −12.5911 −0.435734
\(836\) 0 0
\(837\) 8.28043 0.286214
\(838\) 0 0
\(839\) 20.3887 0.703897 0.351949 0.936019i \(-0.385519\pi\)
0.351949 + 0.936019i \(0.385519\pi\)
\(840\) 0 0
\(841\) −6.42603 −0.221587
\(842\) 0 0
\(843\) 17.6553 0.608081
\(844\) 0 0
\(845\) −1.05398 −0.0362580
\(846\) 0 0
\(847\) −4.24902 −0.145998
\(848\) 0 0
\(849\) −13.9691 −0.479417
\(850\) 0 0
\(851\) −17.3607 −0.595116
\(852\) 0 0
\(853\) −39.4039 −1.34916 −0.674581 0.738201i \(-0.735676\pi\)
−0.674581 + 0.738201i \(0.735676\pi\)
\(854\) 0 0
\(855\) −1.03019 −0.0352317
\(856\) 0 0
\(857\) 3.72146 0.127123 0.0635614 0.997978i \(-0.479754\pi\)
0.0635614 + 0.997978i \(0.479754\pi\)
\(858\) 0 0
\(859\) −19.8699 −0.677952 −0.338976 0.940795i \(-0.610081\pi\)
−0.338976 + 0.940795i \(0.610081\pi\)
\(860\) 0 0
\(861\) 29.4189 1.00259
\(862\) 0 0
\(863\) −33.6108 −1.14413 −0.572063 0.820210i \(-0.693857\pi\)
−0.572063 + 0.820210i \(0.693857\pi\)
\(864\) 0 0
\(865\) −19.9075 −0.676874
\(866\) 0 0
\(867\) 3.74752 0.127272
\(868\) 0 0
\(869\) 12.2704 0.416244
\(870\) 0 0
\(871\) −8.76942 −0.297140
\(872\) 0 0
\(873\) −7.42271 −0.251221
\(874\) 0 0
\(875\) −39.8090 −1.34579
\(876\) 0 0
\(877\) 52.9946 1.78950 0.894750 0.446567i \(-0.147354\pi\)
0.894750 + 0.446567i \(0.147354\pi\)
\(878\) 0 0
\(879\) 1.06528 0.0359311
\(880\) 0 0
\(881\) 16.6459 0.560816 0.280408 0.959881i \(-0.409530\pi\)
0.280408 + 0.959881i \(0.409530\pi\)
\(882\) 0 0
\(883\) 42.1900 1.41981 0.709903 0.704299i \(-0.248739\pi\)
0.709903 + 0.704299i \(0.248739\pi\)
\(884\) 0 0
\(885\) −5.44237 −0.182943
\(886\) 0 0
\(887\) 55.4874 1.86308 0.931542 0.363633i \(-0.118464\pi\)
0.931542 + 0.363633i \(0.118464\pi\)
\(888\) 0 0
\(889\) 49.8761 1.67279
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) −0.945537 −0.0316412
\(894\) 0 0
\(895\) −8.83497 −0.295320
\(896\) 0 0
\(897\) 2.82631 0.0943678
\(898\) 0 0
\(899\) 39.3420 1.31213
\(900\) 0 0
\(901\) 26.2391 0.874152
\(902\) 0 0
\(903\) 32.1315 1.06927
\(904\) 0 0
\(905\) 10.8266 0.359887
\(906\) 0 0
\(907\) −1.47170 −0.0488670 −0.0244335 0.999701i \(-0.507778\pi\)
−0.0244335 + 0.999701i \(0.507778\pi\)
\(908\) 0 0
\(909\) 10.2076 0.338564
\(910\) 0 0
\(911\) 31.3498 1.03866 0.519332 0.854573i \(-0.326181\pi\)
0.519332 + 0.854573i \(0.326181\pi\)
\(912\) 0 0
\(913\) 8.73747 0.289168
\(914\) 0 0
\(915\) −13.4933 −0.446076
\(916\) 0 0
\(917\) −23.8752 −0.788428
\(918\) 0 0
\(919\) −3.07327 −0.101378 −0.0506889 0.998714i \(-0.516142\pi\)
−0.0506889 + 0.998714i \(0.516142\pi\)
\(920\) 0 0
\(921\) −17.5349 −0.577795
\(922\) 0 0
\(923\) −3.74999 −0.123432
\(924\) 0 0
\(925\) 23.8890 0.785466
\(926\) 0 0
\(927\) 7.27160 0.238831
\(928\) 0 0
\(929\) 8.13209 0.266805 0.133403 0.991062i \(-0.457410\pi\)
0.133403 + 0.991062i \(0.457410\pi\)
\(930\) 0 0
\(931\) 10.8047 0.354109
\(932\) 0 0
\(933\) 7.55761 0.247425
\(934\) 0 0
\(935\) 4.80082 0.157004
\(936\) 0 0
\(937\) 36.7711 1.20126 0.600630 0.799527i \(-0.294916\pi\)
0.600630 + 0.799527i \(0.294916\pi\)
\(938\) 0 0
\(939\) −10.2350 −0.334008
\(940\) 0 0
\(941\) 35.5475 1.15881 0.579407 0.815038i \(-0.303284\pi\)
0.579407 + 0.815038i \(0.303284\pi\)
\(942\) 0 0
\(943\) −19.5685 −0.637237
\(944\) 0 0
\(945\) 4.47839 0.145682
\(946\) 0 0
\(947\) −55.7033 −1.81012 −0.905058 0.425289i \(-0.860172\pi\)
−0.905058 + 0.425289i \(0.860172\pi\)
\(948\) 0 0
\(949\) 0.141064 0.00457912
\(950\) 0 0
\(951\) −3.76319 −0.122030
\(952\) 0 0
\(953\) 5.19748 0.168363 0.0841815 0.996450i \(-0.473172\pi\)
0.0841815 + 0.996450i \(0.473172\pi\)
\(954\) 0 0
\(955\) −19.2533 −0.623023
\(956\) 0 0
\(957\) 4.75121 0.153585
\(958\) 0 0
\(959\) −15.7348 −0.508102
\(960\) 0 0
\(961\) 37.5656 1.21179
\(962\) 0 0
\(963\) 16.9870 0.547400
\(964\) 0 0
\(965\) −3.29952 −0.106215
\(966\) 0 0
\(967\) −21.7678 −0.700004 −0.350002 0.936749i \(-0.613819\pi\)
−0.350002 + 0.936749i \(0.613819\pi\)
\(968\) 0 0
\(969\) −4.45212 −0.143023
\(970\) 0 0
\(971\) 7.57482 0.243087 0.121544 0.992586i \(-0.461216\pi\)
0.121544 + 0.992586i \(0.461216\pi\)
\(972\) 0 0
\(973\) −86.4680 −2.77204
\(974\) 0 0
\(975\) −3.88912 −0.124552
\(976\) 0 0
\(977\) −18.0642 −0.577925 −0.288962 0.957340i \(-0.593310\pi\)
−0.288962 + 0.957340i \(0.593310\pi\)
\(978\) 0 0
\(979\) −16.5309 −0.528330
\(980\) 0 0
\(981\) 20.3032 0.648232
\(982\) 0 0
\(983\) −7.91261 −0.252373 −0.126186 0.992007i \(-0.540274\pi\)
−0.126186 + 0.992007i \(0.540274\pi\)
\(984\) 0 0
\(985\) −15.5336 −0.494941
\(986\) 0 0
\(987\) 4.11040 0.130835
\(988\) 0 0
\(989\) −21.3728 −0.679615
\(990\) 0 0
\(991\) −7.66751 −0.243567 −0.121783 0.992557i \(-0.538861\pi\)
−0.121783 + 0.992557i \(0.538861\pi\)
\(992\) 0 0
\(993\) 24.5851 0.780185
\(994\) 0 0
\(995\) −26.1238 −0.828180
\(996\) 0 0
\(997\) −40.2163 −1.27366 −0.636831 0.771004i \(-0.719755\pi\)
−0.636831 + 0.771004i \(0.719755\pi\)
\(998\) 0 0
\(999\) −6.14251 −0.194341
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.1.2 5
4.3 odd 2 3432.2.a.w.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.w.1.2 5 4.3 odd 2
6864.2.a.cf.1.2 5 1.1 even 1 trivial