Properties

Label 6160.2.a.bf.1.3
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.24914 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.05863 q^{9} +O(q^{10})\) \(q+2.24914 q^{3} +1.00000 q^{5} +1.00000 q^{7} +2.05863 q^{9} -1.00000 q^{11} +0.941367 q^{13} +2.24914 q^{15} +6.49828 q^{17} +4.36641 q^{19} +2.24914 q^{21} -6.24914 q^{23} +1.00000 q^{25} -2.11727 q^{27} +8.74742 q^{29} +9.55691 q^{31} -2.24914 q^{33} +1.00000 q^{35} +4.24914 q^{37} +2.11727 q^{39} -2.13187 q^{41} -7.67418 q^{43} +2.05863 q^{45} -11.1138 q^{47} +1.00000 q^{49} +14.6155 q^{51} -4.74742 q^{53} -1.00000 q^{55} +9.82066 q^{57} +1.88273 q^{59} +9.11383 q^{61} +2.05863 q^{63} +0.941367 q^{65} -12.9966 q^{67} -14.0552 q^{69} +14.6155 q^{71} -10.4983 q^{73} +2.24914 q^{75} -1.00000 q^{77} -8.36641 q^{79} -10.9379 q^{81} +8.49828 q^{83} +6.49828 q^{85} +19.6742 q^{87} +12.3810 q^{89} +0.941367 q^{91} +21.4948 q^{93} +4.36641 q^{95} -15.3630 q^{97} -2.05863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{3} + 3q^{5} + 3q^{7} + 7q^{9} + O(q^{10}) \) \( 3q - 2q^{3} + 3q^{5} + 3q^{7} + 7q^{9} - 3q^{11} + 2q^{13} - 2q^{15} + 2q^{17} + 6q^{19} - 2q^{21} - 10q^{23} + 3q^{25} - 8q^{27} + 12q^{31} + 2q^{33} + 3q^{35} + 4q^{37} + 8q^{39} + 4q^{41} - 8q^{43} + 7q^{45} + 3q^{49} + 28q^{51} + 12q^{53} - 3q^{55} - 8q^{57} + 4q^{59} - 6q^{61} + 7q^{63} + 2q^{65} - 4q^{67} - 8q^{69} + 28q^{71} - 14q^{73} - 2q^{75} - 3q^{77} - 18q^{79} + 3q^{81} + 8q^{83} + 2q^{85} + 44q^{87} + 18q^{89} + 2q^{91} + 12q^{93} + 6q^{95} - 4q^{97} - 7q^{99} + O(q^{100}) \)

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
\(3\) 2.24914 1.29854 0.649271 0.760557i \(-0.275074\pi\)
0.649271 + 0.760557i \(0.275074\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 2.05863 0.686211
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.941367 0.261088 0.130544 0.991443i \(-0.458328\pi\)
0.130544 + 0.991443i \(0.458328\pi\)
\(14\) 0 0
\(15\) 2.24914 0.580726
\(16\) 0 0
\(17\) 6.49828 1.57606 0.788032 0.615634i \(-0.211100\pi\)
0.788032 + 0.615634i \(0.211100\pi\)
\(18\) 0 0
\(19\) 4.36641 1.00172 0.500861 0.865528i \(-0.333017\pi\)
0.500861 + 0.865528i \(0.333017\pi\)
\(20\) 0 0
\(21\) 2.24914 0.490803
\(22\) 0 0
\(23\) −6.24914 −1.30304 −0.651518 0.758633i \(-0.725867\pi\)
−0.651518 + 0.758633i \(0.725867\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.11727 −0.407468
\(28\) 0 0
\(29\) 8.74742 1.62436 0.812178 0.583410i \(-0.198282\pi\)
0.812178 + 0.583410i \(0.198282\pi\)
\(30\) 0 0
\(31\) 9.55691 1.71647 0.858236 0.513255i \(-0.171560\pi\)
0.858236 + 0.513255i \(0.171560\pi\)
\(32\) 0 0
\(33\) −2.24914 −0.391525
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 4.24914 0.698554 0.349277 0.937019i \(-0.386427\pi\)
0.349277 + 0.937019i \(0.386427\pi\)
\(38\) 0 0
\(39\) 2.11727 0.339034
\(40\) 0 0
\(41\) −2.13187 −0.332943 −0.166471 0.986046i \(-0.553237\pi\)
−0.166471 + 0.986046i \(0.553237\pi\)
\(42\) 0 0
\(43\) −7.67418 −1.17030 −0.585151 0.810925i \(-0.698965\pi\)
−0.585151 + 0.810925i \(0.698965\pi\)
\(44\) 0 0
\(45\) 2.05863 0.306883
\(46\) 0 0
\(47\) −11.1138 −1.62112 −0.810559 0.585657i \(-0.800837\pi\)
−0.810559 + 0.585657i \(0.800837\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 14.6155 2.04659
\(52\) 0 0
\(53\) −4.74742 −0.652109 −0.326054 0.945351i \(-0.605719\pi\)
−0.326054 + 0.945351i \(0.605719\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 9.82066 1.30078
\(58\) 0 0
\(59\) 1.88273 0.245111 0.122556 0.992462i \(-0.460891\pi\)
0.122556 + 0.992462i \(0.460891\pi\)
\(60\) 0 0
\(61\) 9.11383 1.16691 0.583453 0.812147i \(-0.301701\pi\)
0.583453 + 0.812147i \(0.301701\pi\)
\(62\) 0 0
\(63\) 2.05863 0.259363
\(64\) 0 0
\(65\) 0.941367 0.116762
\(66\) 0 0
\(67\) −12.9966 −1.58778 −0.793891 0.608060i \(-0.791948\pi\)
−0.793891 + 0.608060i \(0.791948\pi\)
\(68\) 0 0
\(69\) −14.0552 −1.69205
\(70\) 0 0
\(71\) 14.6155 1.73455 0.867273 0.497833i \(-0.165871\pi\)
0.867273 + 0.497833i \(0.165871\pi\)
\(72\) 0 0
\(73\) −10.4983 −1.22873 −0.614365 0.789022i \(-0.710588\pi\)
−0.614365 + 0.789022i \(0.710588\pi\)
\(74\) 0 0
\(75\) 2.24914 0.259708
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −8.36641 −0.941294 −0.470647 0.882322i \(-0.655980\pi\)
−0.470647 + 0.882322i \(0.655980\pi\)
\(80\) 0 0
\(81\) −10.9379 −1.21533
\(82\) 0 0
\(83\) 8.49828 0.932808 0.466404 0.884572i \(-0.345549\pi\)
0.466404 + 0.884572i \(0.345549\pi\)
\(84\) 0 0
\(85\) 6.49828 0.704838
\(86\) 0 0
\(87\) 19.6742 2.10929
\(88\) 0 0
\(89\) 12.3810 1.31238 0.656192 0.754594i \(-0.272166\pi\)
0.656192 + 0.754594i \(0.272166\pi\)
\(90\) 0 0
\(91\) 0.941367 0.0986821
\(92\) 0 0
\(93\) 21.4948 2.22891
\(94\) 0 0
\(95\) 4.36641 0.447984
\(96\) 0 0
\(97\) −15.3630 −1.55987 −0.779937 0.625859i \(-0.784749\pi\)
−0.779937 + 0.625859i \(0.784749\pi\)
\(98\) 0 0
\(99\) −2.05863 −0.206900
\(100\) 0 0
\(101\) −16.8793 −1.67955 −0.839776 0.542932i \(-0.817314\pi\)
−0.839776 + 0.542932i \(0.817314\pi\)
\(102\) 0 0
\(103\) 6.61555 0.651849 0.325925 0.945396i \(-0.394324\pi\)
0.325925 + 0.945396i \(0.394324\pi\)
\(104\) 0 0
\(105\) 2.24914 0.219494
\(106\) 0 0
\(107\) 14.5535 1.40694 0.703469 0.710726i \(-0.251633\pi\)
0.703469 + 0.710726i \(0.251633\pi\)
\(108\) 0 0
\(109\) −0.249141 −0.0238633 −0.0119317 0.999929i \(-0.503798\pi\)
−0.0119317 + 0.999929i \(0.503798\pi\)
\(110\) 0 0
\(111\) 9.55691 0.907102
\(112\) 0 0
\(113\) 10.9966 1.03447 0.517235 0.855844i \(-0.326961\pi\)
0.517235 + 0.855844i \(0.326961\pi\)
\(114\) 0 0
\(115\) −6.24914 −0.582735
\(116\) 0 0
\(117\) 1.93793 0.179162
\(118\) 0 0
\(119\) 6.49828 0.595696
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −4.79488 −0.432340
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 5.88273 0.522008 0.261004 0.965338i \(-0.415946\pi\)
0.261004 + 0.965338i \(0.415946\pi\)
\(128\) 0 0
\(129\) −17.2603 −1.51969
\(130\) 0 0
\(131\) 1.25258 0.109438 0.0547191 0.998502i \(-0.482574\pi\)
0.0547191 + 0.998502i \(0.482574\pi\)
\(132\) 0 0
\(133\) 4.36641 0.378615
\(134\) 0 0
\(135\) −2.11727 −0.182225
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −10.9820 −0.931477 −0.465739 0.884922i \(-0.654211\pi\)
−0.465739 + 0.884922i \(0.654211\pi\)
\(140\) 0 0
\(141\) −24.9966 −2.10509
\(142\) 0 0
\(143\) −0.941367 −0.0787210
\(144\) 0 0
\(145\) 8.74742 0.726434
\(146\) 0 0
\(147\) 2.24914 0.185506
\(148\) 0 0
\(149\) 0.0146079 0.00119673 0.000598363 1.00000i \(-0.499810\pi\)
0.000598363 1.00000i \(0.499810\pi\)
\(150\) 0 0
\(151\) 15.2457 1.24068 0.620339 0.784334i \(-0.286995\pi\)
0.620339 + 0.784334i \(0.286995\pi\)
\(152\) 0 0
\(153\) 13.3776 1.08151
\(154\) 0 0
\(155\) 9.55691 0.767630
\(156\) 0 0
\(157\) −5.50172 −0.439085 −0.219542 0.975603i \(-0.570456\pi\)
−0.219542 + 0.975603i \(0.570456\pi\)
\(158\) 0 0
\(159\) −10.6776 −0.846790
\(160\) 0 0
\(161\) −6.24914 −0.492501
\(162\) 0 0
\(163\) 18.2277 1.42770 0.713850 0.700298i \(-0.246950\pi\)
0.713850 + 0.700298i \(0.246950\pi\)
\(164\) 0 0
\(165\) −2.24914 −0.175095
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −12.1138 −0.931833
\(170\) 0 0
\(171\) 8.98883 0.687393
\(172\) 0 0
\(173\) 0.117266 0.00891559 0.00445780 0.999990i \(-0.498581\pi\)
0.00445780 + 0.999990i \(0.498581\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 4.23453 0.318287
\(178\) 0 0
\(179\) 22.5535 1.68573 0.842863 0.538128i \(-0.180868\pi\)
0.842863 + 0.538128i \(0.180868\pi\)
\(180\) 0 0
\(181\) 20.8793 1.55195 0.775973 0.630766i \(-0.217259\pi\)
0.775973 + 0.630766i \(0.217259\pi\)
\(182\) 0 0
\(183\) 20.4983 1.51528
\(184\) 0 0
\(185\) 4.24914 0.312403
\(186\) 0 0
\(187\) −6.49828 −0.475201
\(188\) 0 0
\(189\) −2.11727 −0.154008
\(190\) 0 0
\(191\) −3.11383 −0.225309 −0.112654 0.993634i \(-0.535935\pi\)
−0.112654 + 0.993634i \(0.535935\pi\)
\(192\) 0 0
\(193\) −6.17246 −0.444304 −0.222152 0.975012i \(-0.571308\pi\)
−0.222152 + 0.975012i \(0.571308\pi\)
\(194\) 0 0
\(195\) 2.11727 0.151621
\(196\) 0 0
\(197\) 6.73281 0.479693 0.239847 0.970811i \(-0.422903\pi\)
0.239847 + 0.970811i \(0.422903\pi\)
\(198\) 0 0
\(199\) 13.2311 0.937927 0.468964 0.883217i \(-0.344627\pi\)
0.468964 + 0.883217i \(0.344627\pi\)
\(200\) 0 0
\(201\) −29.2311 −2.06180
\(202\) 0 0
\(203\) 8.74742 0.613949
\(204\) 0 0
\(205\) −2.13187 −0.148897
\(206\) 0 0
\(207\) −12.8647 −0.894158
\(208\) 0 0
\(209\) −4.36641 −0.302031
\(210\) 0 0
\(211\) −23.1138 −1.59122 −0.795611 0.605808i \(-0.792850\pi\)
−0.795611 + 0.605808i \(0.792850\pi\)
\(212\) 0 0
\(213\) 32.8724 2.25238
\(214\) 0 0
\(215\) −7.67418 −0.523375
\(216\) 0 0
\(217\) 9.55691 0.648766
\(218\) 0 0
\(219\) −23.6121 −1.59556
\(220\) 0 0
\(221\) 6.11727 0.411492
\(222\) 0 0
\(223\) −10.3810 −0.695164 −0.347582 0.937650i \(-0.612997\pi\)
−0.347582 + 0.937650i \(0.612997\pi\)
\(224\) 0 0
\(225\) 2.05863 0.137242
\(226\) 0 0
\(227\) −16.1104 −1.06928 −0.534642 0.845079i \(-0.679554\pi\)
−0.534642 + 0.845079i \(0.679554\pi\)
\(228\) 0 0
\(229\) 24.2897 1.60511 0.802555 0.596578i \(-0.203473\pi\)
0.802555 + 0.596578i \(0.203473\pi\)
\(230\) 0 0
\(231\) −2.24914 −0.147983
\(232\) 0 0
\(233\) 6.88617 0.451128 0.225564 0.974228i \(-0.427578\pi\)
0.225564 + 0.974228i \(0.427578\pi\)
\(234\) 0 0
\(235\) −11.1138 −0.724986
\(236\) 0 0
\(237\) −18.8172 −1.22231
\(238\) 0 0
\(239\) 11.1353 0.720283 0.360142 0.932898i \(-0.382728\pi\)
0.360142 + 0.932898i \(0.382728\pi\)
\(240\) 0 0
\(241\) 2.36641 0.152434 0.0762168 0.997091i \(-0.475716\pi\)
0.0762168 + 0.997091i \(0.475716\pi\)
\(242\) 0 0
\(243\) −18.2491 −1.17068
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 4.11039 0.261538
\(248\) 0 0
\(249\) 19.1138 1.21129
\(250\) 0 0
\(251\) −25.1070 −1.58474 −0.792368 0.610043i \(-0.791152\pi\)
−0.792368 + 0.610043i \(0.791152\pi\)
\(252\) 0 0
\(253\) 6.24914 0.392880
\(254\) 0 0
\(255\) 14.6155 0.915261
\(256\) 0 0
\(257\) −2.86469 −0.178694 −0.0893472 0.996001i \(-0.528478\pi\)
−0.0893472 + 0.996001i \(0.528478\pi\)
\(258\) 0 0
\(259\) 4.24914 0.264029
\(260\) 0 0
\(261\) 18.0077 1.11465
\(262\) 0 0
\(263\) −3.76547 −0.232189 −0.116094 0.993238i \(-0.537037\pi\)
−0.116094 + 0.993238i \(0.537037\pi\)
\(264\) 0 0
\(265\) −4.74742 −0.291632
\(266\) 0 0
\(267\) 27.8466 1.70419
\(268\) 0 0
\(269\) 8.94137 0.545165 0.272582 0.962132i \(-0.412122\pi\)
0.272582 + 0.962132i \(0.412122\pi\)
\(270\) 0 0
\(271\) −21.4948 −1.30572 −0.652859 0.757479i \(-0.726431\pi\)
−0.652859 + 0.757479i \(0.726431\pi\)
\(272\) 0 0
\(273\) 2.11727 0.128143
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 7.64820 0.459536 0.229768 0.973245i \(-0.426203\pi\)
0.229768 + 0.973245i \(0.426203\pi\)
\(278\) 0 0
\(279\) 19.6742 1.17786
\(280\) 0 0
\(281\) −28.6155 −1.70706 −0.853530 0.521043i \(-0.825543\pi\)
−0.853530 + 0.521043i \(0.825543\pi\)
\(282\) 0 0
\(283\) 2.87930 0.171156 0.0855782 0.996331i \(-0.472726\pi\)
0.0855782 + 0.996331i \(0.472726\pi\)
\(284\) 0 0
\(285\) 9.82066 0.581726
\(286\) 0 0
\(287\) −2.13187 −0.125841
\(288\) 0 0
\(289\) 25.2277 1.48398
\(290\) 0 0
\(291\) −34.5535 −2.02556
\(292\) 0 0
\(293\) −12.9414 −0.756043 −0.378021 0.925797i \(-0.623395\pi\)
−0.378021 + 0.925797i \(0.623395\pi\)
\(294\) 0 0
\(295\) 1.88273 0.109617
\(296\) 0 0
\(297\) 2.11727 0.122856
\(298\) 0 0
\(299\) −5.88273 −0.340207
\(300\) 0 0
\(301\) −7.67418 −0.442332
\(302\) 0 0
\(303\) −37.9639 −2.18097
\(304\) 0 0
\(305\) 9.11383 0.521856
\(306\) 0 0
\(307\) −0.498281 −0.0284384 −0.0142192 0.999899i \(-0.504526\pi\)
−0.0142192 + 0.999899i \(0.504526\pi\)
\(308\) 0 0
\(309\) 14.8793 0.846454
\(310\) 0 0
\(311\) 23.4396 1.32914 0.664570 0.747226i \(-0.268615\pi\)
0.664570 + 0.747226i \(0.268615\pi\)
\(312\) 0 0
\(313\) 9.36984 0.529615 0.264807 0.964301i \(-0.414692\pi\)
0.264807 + 0.964301i \(0.414692\pi\)
\(314\) 0 0
\(315\) 2.05863 0.115991
\(316\) 0 0
\(317\) −9.97852 −0.560449 −0.280225 0.959934i \(-0.590409\pi\)
−0.280225 + 0.959934i \(0.590409\pi\)
\(318\) 0 0
\(319\) −8.74742 −0.489762
\(320\) 0 0
\(321\) 32.7328 1.82697
\(322\) 0 0
\(323\) 28.3741 1.57878
\(324\) 0 0
\(325\) 0.941367 0.0522176
\(326\) 0 0
\(327\) −0.560352 −0.0309875
\(328\) 0 0
\(329\) −11.1138 −0.612725
\(330\) 0 0
\(331\) 20.4362 1.12328 0.561638 0.827383i \(-0.310171\pi\)
0.561638 + 0.827383i \(0.310171\pi\)
\(332\) 0 0
\(333\) 8.74742 0.479356
\(334\) 0 0
\(335\) −12.9966 −0.710078
\(336\) 0 0
\(337\) 7.88273 0.429400 0.214700 0.976680i \(-0.431123\pi\)
0.214700 + 0.976680i \(0.431123\pi\)
\(338\) 0 0
\(339\) 24.7328 1.34330
\(340\) 0 0
\(341\) −9.55691 −0.517536
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −14.0552 −0.756706
\(346\) 0 0
\(347\) −13.5569 −0.727773 −0.363887 0.931443i \(-0.618550\pi\)
−0.363887 + 0.931443i \(0.618550\pi\)
\(348\) 0 0
\(349\) −10.7328 −0.574514 −0.287257 0.957853i \(-0.592743\pi\)
−0.287257 + 0.957853i \(0.592743\pi\)
\(350\) 0 0
\(351\) −1.99312 −0.106385
\(352\) 0 0
\(353\) −20.8578 −1.11015 −0.555075 0.831801i \(-0.687310\pi\)
−0.555075 + 0.831801i \(0.687310\pi\)
\(354\) 0 0
\(355\) 14.6155 0.775713
\(356\) 0 0
\(357\) 14.6155 0.773537
\(358\) 0 0
\(359\) −0.366407 −0.0193382 −0.00966911 0.999953i \(-0.503078\pi\)
−0.00966911 + 0.999953i \(0.503078\pi\)
\(360\) 0 0
\(361\) 0.0655089 0.00344783
\(362\) 0 0
\(363\) 2.24914 0.118049
\(364\) 0 0
\(365\) −10.4983 −0.549505
\(366\) 0 0
\(367\) −20.8432 −1.08801 −0.544003 0.839083i \(-0.683092\pi\)
−0.544003 + 0.839083i \(0.683092\pi\)
\(368\) 0 0
\(369\) −4.38875 −0.228469
\(370\) 0 0
\(371\) −4.74742 −0.246474
\(372\) 0 0
\(373\) −5.37758 −0.278440 −0.139220 0.990261i \(-0.544460\pi\)
−0.139220 + 0.990261i \(0.544460\pi\)
\(374\) 0 0
\(375\) 2.24914 0.116145
\(376\) 0 0
\(377\) 8.23453 0.424100
\(378\) 0 0
\(379\) −3.53093 −0.181372 −0.0906860 0.995880i \(-0.528906\pi\)
−0.0906860 + 0.995880i \(0.528906\pi\)
\(380\) 0 0
\(381\) 13.2311 0.677850
\(382\) 0 0
\(383\) −1.38445 −0.0707422 −0.0353711 0.999374i \(-0.511261\pi\)
−0.0353711 + 0.999374i \(0.511261\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) −15.7983 −0.803074
\(388\) 0 0
\(389\) 15.7294 0.797511 0.398756 0.917057i \(-0.369442\pi\)
0.398756 + 0.917057i \(0.369442\pi\)
\(390\) 0 0
\(391\) −40.6087 −2.05367
\(392\) 0 0
\(393\) 2.81722 0.142110
\(394\) 0 0
\(395\) −8.36641 −0.420960
\(396\) 0 0
\(397\) −17.6121 −0.883926 −0.441963 0.897033i \(-0.645718\pi\)
−0.441963 + 0.897033i \(0.645718\pi\)
\(398\) 0 0
\(399\) 9.82066 0.491648
\(400\) 0 0
\(401\) −32.8172 −1.63881 −0.819407 0.573212i \(-0.805697\pi\)
−0.819407 + 0.573212i \(0.805697\pi\)
\(402\) 0 0
\(403\) 8.99656 0.448151
\(404\) 0 0
\(405\) −10.9379 −0.543510
\(406\) 0 0
\(407\) −4.24914 −0.210622
\(408\) 0 0
\(409\) −33.3561 −1.64935 −0.824676 0.565605i \(-0.808643\pi\)
−0.824676 + 0.565605i \(0.808643\pi\)
\(410\) 0 0
\(411\) 22.4914 1.10942
\(412\) 0 0
\(413\) 1.88273 0.0926433
\(414\) 0 0
\(415\) 8.49828 0.417164
\(416\) 0 0
\(417\) −24.7000 −1.20956
\(418\) 0 0
\(419\) −12.3449 −0.603089 −0.301544 0.953452i \(-0.597502\pi\)
−0.301544 + 0.953452i \(0.597502\pi\)
\(420\) 0 0
\(421\) −8.87930 −0.432750 −0.216375 0.976310i \(-0.569423\pi\)
−0.216375 + 0.976310i \(0.569423\pi\)
\(422\) 0 0
\(423\) −22.8793 −1.11243
\(424\) 0 0
\(425\) 6.49828 0.315213
\(426\) 0 0
\(427\) 9.11383 0.441049
\(428\) 0 0
\(429\) −2.11727 −0.102223
\(430\) 0 0
\(431\) −18.2784 −0.880437 −0.440219 0.897891i \(-0.645099\pi\)
−0.440219 + 0.897891i \(0.645099\pi\)
\(432\) 0 0
\(433\) 6.86469 0.329896 0.164948 0.986302i \(-0.447254\pi\)
0.164948 + 0.986302i \(0.447254\pi\)
\(434\) 0 0
\(435\) 19.6742 0.943305
\(436\) 0 0
\(437\) −27.2863 −1.30528
\(438\) 0 0
\(439\) 11.8466 0.565409 0.282705 0.959207i \(-0.408768\pi\)
0.282705 + 0.959207i \(0.408768\pi\)
\(440\) 0 0
\(441\) 2.05863 0.0980302
\(442\) 0 0
\(443\) −17.8827 −0.849634 −0.424817 0.905279i \(-0.639662\pi\)
−0.424817 + 0.905279i \(0.639662\pi\)
\(444\) 0 0
\(445\) 12.3810 0.586916
\(446\) 0 0
\(447\) 0.0328552 0.00155400
\(448\) 0 0
\(449\) −16.8172 −0.793654 −0.396827 0.917893i \(-0.629889\pi\)
−0.396827 + 0.917893i \(0.629889\pi\)
\(450\) 0 0
\(451\) 2.13187 0.100386
\(452\) 0 0
\(453\) 34.2897 1.61107
\(454\) 0 0
\(455\) 0.941367 0.0441320
\(456\) 0 0
\(457\) −12.2277 −0.571986 −0.285993 0.958232i \(-0.592323\pi\)
−0.285993 + 0.958232i \(0.592323\pi\)
\(458\) 0 0
\(459\) −13.7586 −0.642196
\(460\) 0 0
\(461\) −16.6155 −0.773863 −0.386932 0.922108i \(-0.626465\pi\)
−0.386932 + 0.922108i \(0.626465\pi\)
\(462\) 0 0
\(463\) −18.4837 −0.859009 −0.429505 0.903065i \(-0.641312\pi\)
−0.429505 + 0.903065i \(0.641312\pi\)
\(464\) 0 0
\(465\) 21.4948 0.996799
\(466\) 0 0
\(467\) −24.3956 −1.12889 −0.564447 0.825469i \(-0.690911\pi\)
−0.564447 + 0.825469i \(0.690911\pi\)
\(468\) 0 0
\(469\) −12.9966 −0.600125
\(470\) 0 0
\(471\) −12.3741 −0.570170
\(472\) 0 0
\(473\) 7.67418 0.352859
\(474\) 0 0
\(475\) 4.36641 0.200344
\(476\) 0 0
\(477\) −9.77320 −0.447484
\(478\) 0 0
\(479\) 27.7655 1.26864 0.634318 0.773072i \(-0.281281\pi\)
0.634318 + 0.773072i \(0.281281\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) −14.0552 −0.639534
\(484\) 0 0
\(485\) −15.3630 −0.697596
\(486\) 0 0
\(487\) −10.0958 −0.457484 −0.228742 0.973487i \(-0.573461\pi\)
−0.228742 + 0.973487i \(0.573461\pi\)
\(488\) 0 0
\(489\) 40.9966 1.85393
\(490\) 0 0
\(491\) 32.1104 1.44912 0.724561 0.689211i \(-0.242043\pi\)
0.724561 + 0.689211i \(0.242043\pi\)
\(492\) 0 0
\(493\) 56.8432 2.56009
\(494\) 0 0
\(495\) −2.05863 −0.0925287
\(496\) 0 0
\(497\) 14.6155 0.655597
\(498\) 0 0
\(499\) 8.79488 0.393713 0.196857 0.980432i \(-0.436927\pi\)
0.196857 + 0.980432i \(0.436927\pi\)
\(500\) 0 0
\(501\) 17.9931 0.803874
\(502\) 0 0
\(503\) 15.0034 0.668970 0.334485 0.942401i \(-0.391438\pi\)
0.334485 + 0.942401i \(0.391438\pi\)
\(504\) 0 0
\(505\) −16.8793 −0.751119
\(506\) 0 0
\(507\) −27.2457 −1.21002
\(508\) 0 0
\(509\) 21.8759 0.969630 0.484815 0.874617i \(-0.338887\pi\)
0.484815 + 0.874617i \(0.338887\pi\)
\(510\) 0 0
\(511\) −10.4983 −0.464417
\(512\) 0 0
\(513\) −9.24485 −0.408170
\(514\) 0 0
\(515\) 6.61555 0.291516
\(516\) 0 0
\(517\) 11.1138 0.488786
\(518\) 0 0
\(519\) 0.263748 0.0115773
\(520\) 0 0
\(521\) 14.0292 0.614631 0.307316 0.951608i \(-0.400569\pi\)
0.307316 + 0.951608i \(0.400569\pi\)
\(522\) 0 0
\(523\) −1.14992 −0.0502825 −0.0251412 0.999684i \(-0.508004\pi\)
−0.0251412 + 0.999684i \(0.508004\pi\)
\(524\) 0 0
\(525\) 2.24914 0.0981605
\(526\) 0 0
\(527\) 62.1035 2.70527
\(528\) 0 0
\(529\) 16.0518 0.697902
\(530\) 0 0
\(531\) 3.87586 0.168198
\(532\) 0 0
\(533\) −2.00688 −0.0869274
\(534\) 0 0
\(535\) 14.5535 0.629202
\(536\) 0 0
\(537\) 50.7259 2.18899
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −6.47680 −0.278459 −0.139230 0.990260i \(-0.544463\pi\)
−0.139230 + 0.990260i \(0.544463\pi\)
\(542\) 0 0
\(543\) 46.9605 2.01527
\(544\) 0 0
\(545\) −0.249141 −0.0106720
\(546\) 0 0
\(547\) −12.9966 −0.555693 −0.277846 0.960626i \(-0.589621\pi\)
−0.277846 + 0.960626i \(0.589621\pi\)
\(548\) 0 0
\(549\) 18.7620 0.800744
\(550\) 0 0
\(551\) 38.1948 1.62715
\(552\) 0 0
\(553\) −8.36641 −0.355776
\(554\) 0 0
\(555\) 9.55691 0.405668
\(556\) 0 0
\(557\) −16.4914 −0.698763 −0.349382 0.936981i \(-0.613608\pi\)
−0.349382 + 0.936981i \(0.613608\pi\)
\(558\) 0 0
\(559\) −7.22422 −0.305552
\(560\) 0 0
\(561\) −14.6155 −0.617069
\(562\) 0 0
\(563\) 16.7620 0.706435 0.353218 0.935541i \(-0.385088\pi\)
0.353218 + 0.935541i \(0.385088\pi\)
\(564\) 0 0
\(565\) 10.9966 0.462629
\(566\) 0 0
\(567\) −10.9379 −0.459350
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −12.7328 −0.532852 −0.266426 0.963855i \(-0.585843\pi\)
−0.266426 + 0.963855i \(0.585843\pi\)
\(572\) 0 0
\(573\) −7.00344 −0.292573
\(574\) 0 0
\(575\) −6.24914 −0.260607
\(576\) 0 0
\(577\) −11.8613 −0.493790 −0.246895 0.969042i \(-0.579410\pi\)
−0.246895 + 0.969042i \(0.579410\pi\)
\(578\) 0 0
\(579\) −13.8827 −0.576947
\(580\) 0 0
\(581\) 8.49828 0.352568
\(582\) 0 0
\(583\) 4.74742 0.196618
\(584\) 0 0
\(585\) 1.93793 0.0801235
\(586\) 0 0
\(587\) 8.60094 0.354999 0.177499 0.984121i \(-0.443199\pi\)
0.177499 + 0.984121i \(0.443199\pi\)
\(588\) 0 0
\(589\) 41.7294 1.71943
\(590\) 0 0
\(591\) 15.1430 0.622902
\(592\) 0 0
\(593\) 10.7328 0.440744 0.220372 0.975416i \(-0.429273\pi\)
0.220372 + 0.975416i \(0.429273\pi\)
\(594\) 0 0
\(595\) 6.49828 0.266404
\(596\) 0 0
\(597\) 29.7586 1.21794
\(598\) 0 0
\(599\) −16.0812 −0.657059 −0.328529 0.944494i \(-0.606553\pi\)
−0.328529 + 0.944494i \(0.606553\pi\)
\(600\) 0 0
\(601\) −34.0889 −1.39052 −0.695258 0.718760i \(-0.744710\pi\)
−0.695258 + 0.718760i \(0.744710\pi\)
\(602\) 0 0
\(603\) −26.7552 −1.08955
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −11.6742 −0.473840 −0.236920 0.971529i \(-0.576138\pi\)
−0.236920 + 0.971529i \(0.576138\pi\)
\(608\) 0 0
\(609\) 19.6742 0.797238
\(610\) 0 0
\(611\) −10.4622 −0.423255
\(612\) 0 0
\(613\) −34.2637 −1.38390 −0.691950 0.721946i \(-0.743248\pi\)
−0.691950 + 0.721946i \(0.743248\pi\)
\(614\) 0 0
\(615\) −4.79488 −0.193348
\(616\) 0 0
\(617\) 39.3707 1.58500 0.792502 0.609869i \(-0.208778\pi\)
0.792502 + 0.609869i \(0.208778\pi\)
\(618\) 0 0
\(619\) −16.1104 −0.647531 −0.323766 0.946137i \(-0.604949\pi\)
−0.323766 + 0.946137i \(0.604949\pi\)
\(620\) 0 0
\(621\) 13.2311 0.530946
\(622\) 0 0
\(623\) 12.3810 0.496035
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −9.82066 −0.392199
\(628\) 0 0
\(629\) 27.6121 1.10097
\(630\) 0 0
\(631\) −13.8827 −0.552663 −0.276331 0.961062i \(-0.589119\pi\)
−0.276331 + 0.961062i \(0.589119\pi\)
\(632\) 0 0
\(633\) −51.9862 −2.06627
\(634\) 0 0
\(635\) 5.88273 0.233449
\(636\) 0 0
\(637\) 0.941367 0.0372983
\(638\) 0 0
\(639\) 30.0881 1.19026
\(640\) 0 0
\(641\) −39.3415 −1.55390 −0.776948 0.629565i \(-0.783233\pi\)
−0.776948 + 0.629565i \(0.783233\pi\)
\(642\) 0 0
\(643\) −38.3595 −1.51275 −0.756376 0.654137i \(-0.773032\pi\)
−0.756376 + 0.654137i \(0.773032\pi\)
\(644\) 0 0
\(645\) −17.2603 −0.679624
\(646\) 0 0
\(647\) −25.7294 −1.01153 −0.505763 0.862672i \(-0.668789\pi\)
−0.505763 + 0.862672i \(0.668789\pi\)
\(648\) 0 0
\(649\) −1.88273 −0.0739038
\(650\) 0 0
\(651\) 21.4948 0.842449
\(652\) 0 0
\(653\) 18.4768 0.723053 0.361526 0.932362i \(-0.382256\pi\)
0.361526 + 0.932362i \(0.382256\pi\)
\(654\) 0 0
\(655\) 1.25258 0.0489423
\(656\) 0 0
\(657\) −21.6121 −0.843169
\(658\) 0 0
\(659\) 39.3776 1.53393 0.766966 0.641687i \(-0.221765\pi\)
0.766966 + 0.641687i \(0.221765\pi\)
\(660\) 0 0
\(661\) −34.2208 −1.33103 −0.665517 0.746383i \(-0.731789\pi\)
−0.665517 + 0.746383i \(0.731789\pi\)
\(662\) 0 0
\(663\) 13.7586 0.534339
\(664\) 0 0
\(665\) 4.36641 0.169322
\(666\) 0 0
\(667\) −54.6639 −2.11659
\(668\) 0 0
\(669\) −23.3484 −0.902700
\(670\) 0 0
\(671\) −9.11383 −0.351835
\(672\) 0 0
\(673\) −24.4622 −0.942948 −0.471474 0.881880i \(-0.656278\pi\)
−0.471474 + 0.881880i \(0.656278\pi\)
\(674\) 0 0
\(675\) −2.11727 −0.0814936
\(676\) 0 0
\(677\) −39.1070 −1.50300 −0.751501 0.659732i \(-0.770670\pi\)
−0.751501 + 0.659732i \(0.770670\pi\)
\(678\) 0 0
\(679\) −15.3630 −0.589577
\(680\) 0 0
\(681\) −36.2345 −1.38851
\(682\) 0 0
\(683\) −23.3776 −0.894518 −0.447259 0.894404i \(-0.647600\pi\)
−0.447259 + 0.894404i \(0.647600\pi\)
\(684\) 0 0
\(685\) 10.0000 0.382080
\(686\) 0 0
\(687\) 54.6310 2.08430
\(688\) 0 0
\(689\) −4.46907 −0.170258
\(690\) 0 0
\(691\) −8.49828 −0.323290 −0.161645 0.986849i \(-0.551680\pi\)
−0.161645 + 0.986849i \(0.551680\pi\)
\(692\) 0 0
\(693\) −2.05863 −0.0782010
\(694\) 0 0
\(695\) −10.9820 −0.416569
\(696\) 0 0
\(697\) −13.8535 −0.524739
\(698\) 0 0
\(699\) 15.4880 0.585809
\(700\) 0 0
\(701\) 14.9751 0.565601 0.282800 0.959179i \(-0.408737\pi\)
0.282800 + 0.959179i \(0.408737\pi\)
\(702\) 0 0
\(703\) 18.5535 0.699758
\(704\) 0 0
\(705\) −24.9966 −0.941425
\(706\) 0 0
\(707\) −16.8793 −0.634811
\(708\) 0 0
\(709\) 40.7259 1.52949 0.764747 0.644330i \(-0.222864\pi\)
0.764747 + 0.644330i \(0.222864\pi\)
\(710\) 0 0
\(711\) −17.2234 −0.645927
\(712\) 0 0
\(713\) −59.7225 −2.23663
\(714\) 0 0
\(715\) −0.941367 −0.0352051
\(716\) 0 0
\(717\) 25.0449 0.935318
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 6.61555 0.246376
\(722\) 0 0
\(723\) 5.32238 0.197942
\(724\) 0 0
\(725\) 8.74742 0.324871
\(726\) 0 0
\(727\) 51.6413 1.91527 0.957635 0.287984i \(-0.0929848\pi\)
0.957635 + 0.287984i \(0.0929848\pi\)
\(728\) 0 0
\(729\) −8.23109 −0.304855
\(730\) 0 0
\(731\) −49.8690 −1.84447
\(732\) 0 0
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) 0 0
\(735\) 2.24914 0.0829608
\(736\) 0 0
\(737\) 12.9966 0.478735
\(738\) 0 0
\(739\) 49.1070 1.80643 0.903214 0.429190i \(-0.141201\pi\)
0.903214 + 0.429190i \(0.141201\pi\)
\(740\) 0 0
\(741\) 9.24485 0.339618
\(742\) 0 0
\(743\) 53.2311 1.95286 0.976430 0.215836i \(-0.0692475\pi\)
0.976430 + 0.215836i \(0.0692475\pi\)
\(744\) 0 0
\(745\) 0.0146079 0.000535192 0
\(746\) 0 0
\(747\) 17.4948 0.640103
\(748\) 0 0
\(749\) 14.5535 0.531772
\(750\) 0 0
\(751\) 31.6121 1.15354 0.576771 0.816906i \(-0.304312\pi\)
0.576771 + 0.816906i \(0.304312\pi\)
\(752\) 0 0
\(753\) −56.4691 −2.05785
\(754\) 0 0
\(755\) 15.2457 0.554848
\(756\) 0 0
\(757\) −1.24570 −0.0452758 −0.0226379 0.999744i \(-0.507206\pi\)
−0.0226379 + 0.999744i \(0.507206\pi\)
\(758\) 0 0
\(759\) 14.0552 0.510171
\(760\) 0 0
\(761\) −10.6009 −0.384284 −0.192142 0.981367i \(-0.561543\pi\)
−0.192142 + 0.981367i \(0.561543\pi\)
\(762\) 0 0
\(763\) −0.249141 −0.00901949
\(764\) 0 0
\(765\) 13.3776 0.483667
\(766\) 0 0
\(767\) 1.77234 0.0639956
\(768\) 0 0
\(769\) −19.1284 −0.689789 −0.344895 0.938641i \(-0.612085\pi\)
−0.344895 + 0.938641i \(0.612085\pi\)
\(770\) 0 0
\(771\) −6.44309 −0.232042
\(772\) 0 0
\(773\) 39.9931 1.43845 0.719226 0.694776i \(-0.244496\pi\)
0.719226 + 0.694776i \(0.244496\pi\)
\(774\) 0 0
\(775\) 9.55691 0.343294
\(776\) 0 0
\(777\) 9.55691 0.342852
\(778\) 0 0
\(779\) −9.30863 −0.333516
\(780\) 0 0
\(781\) −14.6155 −0.522985
\(782\) 0 0
\(783\) −18.5206 −0.661873
\(784\) 0 0
\(785\) −5.50172 −0.196365
\(786\) 0 0
\(787\) −37.9931 −1.35431 −0.677154 0.735841i \(-0.736787\pi\)
−0.677154 + 0.735841i \(0.736787\pi\)
\(788\) 0 0
\(789\) −8.46907 −0.301507
\(790\) 0 0
\(791\) 10.9966 0.390993
\(792\) 0 0
\(793\) 8.57946 0.304665
\(794\) 0 0
\(795\) −10.6776 −0.378696
\(796\) 0 0
\(797\) −20.3810 −0.721933 −0.360966 0.932579i \(-0.617553\pi\)
−0.360966 + 0.932579i \(0.617553\pi\)
\(798\) 0 0
\(799\) −72.2208 −2.55499
\(800\) 0 0
\(801\) 25.4880 0.900573
\(802\) 0 0
\(803\) 10.4983 0.370476
\(804\) 0 0
\(805\) −6.24914 −0.220253
\(806\) 0 0
\(807\) 20.1104 0.707919
\(808\) 0 0
\(809\) −31.7294 −1.11555 −0.557773 0.829994i \(-0.688344\pi\)
−0.557773 + 0.829994i \(0.688344\pi\)
\(810\) 0 0
\(811\) −43.5095 −1.52782 −0.763912 0.645321i \(-0.776724\pi\)
−0.763912 + 0.645321i \(0.776724\pi\)
\(812\) 0 0
\(813\) −48.3449 −1.69553
\(814\) 0 0
\(815\) 18.2277 0.638487
\(816\) 0 0
\(817\) −33.5086 −1.17232
\(818\) 0 0
\(819\) 1.93793 0.0677167
\(820\) 0 0
\(821\) −24.7766 −0.864711 −0.432355 0.901703i \(-0.642317\pi\)
−0.432355 + 0.901703i \(0.642317\pi\)
\(822\) 0 0
\(823\) −28.8647 −1.00616 −0.503080 0.864240i \(-0.667800\pi\)
−0.503080 + 0.864240i \(0.667800\pi\)
\(824\) 0 0
\(825\) −2.24914 −0.0783050
\(826\) 0 0
\(827\) −34.2277 −1.19021 −0.595106 0.803647i \(-0.702890\pi\)
−0.595106 + 0.803647i \(0.702890\pi\)
\(828\) 0 0
\(829\) 23.6381 0.820985 0.410492 0.911864i \(-0.365357\pi\)
0.410492 + 0.911864i \(0.365357\pi\)
\(830\) 0 0
\(831\) 17.2019 0.596727
\(832\) 0 0
\(833\) 6.49828 0.225152
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −20.2345 −0.699408
\(838\) 0 0
\(839\) 24.9053 0.859826 0.429913 0.902870i \(-0.358544\pi\)
0.429913 + 0.902870i \(0.358544\pi\)
\(840\) 0 0
\(841\) 47.5174 1.63853
\(842\) 0 0
\(843\) −64.3604 −2.21669
\(844\) 0 0
\(845\) −12.1138 −0.416728
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 6.47594 0.222254
\(850\) 0 0
\(851\) −26.5535 −0.910241
\(852\) 0 0
\(853\) 22.9966 0.787387 0.393694 0.919242i \(-0.371197\pi\)
0.393694 + 0.919242i \(0.371197\pi\)
\(854\) 0 0
\(855\) 8.98883 0.307411
\(856\) 0 0
\(857\) −31.2603 −1.06783 −0.533916 0.845538i \(-0.679280\pi\)
−0.533916 + 0.845538i \(0.679280\pi\)
\(858\) 0 0
\(859\) −20.3449 −0.694160 −0.347080 0.937836i \(-0.612827\pi\)
−0.347080 + 0.937836i \(0.612827\pi\)
\(860\) 0 0
\(861\) −4.79488 −0.163409
\(862\) 0 0
\(863\) 58.3595 1.98658 0.993291 0.115644i \(-0.0368930\pi\)
0.993291 + 0.115644i \(0.0368930\pi\)
\(864\) 0 0
\(865\) 0.117266 0.00398717
\(866\) 0 0
\(867\) 56.7405 1.92701
\(868\) 0 0
\(869\) 8.36641 0.283811
\(870\) 0 0
\(871\) −12.2345 −0.414551
\(872\) 0 0
\(873\) −31.6267 −1.07040
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 11.9639 0.403992 0.201996 0.979386i \(-0.435257\pi\)
0.201996 + 0.979386i \(0.435257\pi\)
\(878\) 0 0
\(879\) −29.1070 −0.981753
\(880\) 0 0
\(881\) 5.88961 0.198426 0.0992130 0.995066i \(-0.468367\pi\)
0.0992130 + 0.995066i \(0.468367\pi\)
\(882\) 0 0
\(883\) 19.4880 0.655822 0.327911 0.944709i \(-0.393655\pi\)
0.327911 + 0.944709i \(0.393655\pi\)
\(884\) 0 0
\(885\) 4.23453 0.142342
\(886\) 0 0
\(887\) −50.4622 −1.69435 −0.847177 0.531310i \(-0.821700\pi\)
−0.847177 + 0.531310i \(0.821700\pi\)
\(888\) 0 0
\(889\) 5.88273 0.197301
\(890\) 0 0
\(891\) 10.9379 0.366434
\(892\) 0 0
\(893\) −48.5275 −1.62391
\(894\) 0 0
\(895\) 22.5535 0.753880
\(896\) 0 0
\(897\) −13.2311 −0.441773
\(898\) 0 0
\(899\) 83.5984 2.78816
\(900\) 0 0
\(901\) −30.8501 −1.02777
\(902\) 0 0
\(903\) −17.2603 −0.574387
\(904\) 0 0
\(905\) 20.8793 0.694051
\(906\) 0 0
\(907\) 16.8432 0.559269 0.279635 0.960106i \(-0.409787\pi\)
0.279635 + 0.960106i \(0.409787\pi\)
\(908\) 0 0
\(909\) −34.7483 −1.15253
\(910\) 0 0
\(911\) 38.6155 1.27939 0.639695 0.768629i \(-0.279061\pi\)
0.639695 + 0.768629i \(0.279061\pi\)
\(912\) 0 0
\(913\) −8.49828 −0.281252
\(914\) 0 0
\(915\) 20.4983 0.677652
\(916\) 0 0
\(917\) 1.25258 0.0413638
\(918\) 0 0
\(919\) −17.2818 −0.570074 −0.285037 0.958517i \(-0.592006\pi\)
−0.285037 + 0.958517i \(0.592006\pi\)
\(920\) 0 0
\(921\) −1.12070 −0.0369285
\(922\) 0 0
\(923\) 13.7586 0.452870
\(924\) 0 0
\(925\) 4.24914 0.139711
\(926\) 0 0
\(927\) 13.6190 0.447306
\(928\) 0 0
\(929\) 2.91539 0.0956508 0.0478254 0.998856i \(-0.484771\pi\)
0.0478254 + 0.998856i \(0.484771\pi\)
\(930\) 0 0
\(931\) 4.36641 0.143103
\(932\) 0 0
\(933\) 52.7191 1.72594
\(934\) 0 0
\(935\) −6.49828 −0.212517
\(936\) 0 0
\(937\) 22.5795 0.737639 0.368819 0.929501i \(-0.379762\pi\)
0.368819 + 0.929501i \(0.379762\pi\)
\(938\) 0 0
\(939\) 21.0741 0.687727
\(940\) 0 0
\(941\) −21.7655 −0.709534 −0.354767 0.934955i \(-0.615440\pi\)
−0.354767 + 0.934955i \(0.615440\pi\)
\(942\) 0 0
\(943\) 13.3224 0.433836
\(944\) 0 0
\(945\) −2.11727 −0.0688747
\(946\) 0 0
\(947\) 1.49484 0.0485759 0.0242879 0.999705i \(-0.492268\pi\)
0.0242879 + 0.999705i \(0.492268\pi\)
\(948\) 0 0
\(949\) −9.88273 −0.320807
\(950\) 0 0
\(951\) −22.4431 −0.727767
\(952\) 0 0
\(953\) −4.83098 −0.156491 −0.0782453 0.996934i \(-0.524932\pi\)
−0.0782453 + 0.996934i \(0.524932\pi\)
\(954\) 0 0
\(955\) −3.11383 −0.100761
\(956\) 0 0
\(957\) −19.6742 −0.635976
\(958\) 0 0
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) 60.3346 1.94628
\(962\) 0 0
\(963\) 29.9603 0.965456
\(964\) 0 0
\(965\) −6.17246 −0.198699
\(966\) 0 0
\(967\) 51.9278 1.66989 0.834943 0.550336i \(-0.185501\pi\)
0.834943 + 0.550336i \(0.185501\pi\)
\(968\) 0 0
\(969\) 63.8174 2.05011
\(970\) 0 0
\(971\) −59.6933 −1.91565 −0.957824 0.287354i \(-0.907224\pi\)
−0.957824 + 0.287354i \(0.907224\pi\)
\(972\) 0 0
\(973\) −10.9820 −0.352065
\(974\) 0 0
\(975\) 2.11727 0.0678068
\(976\) 0 0
\(977\) 31.2311 0.999171 0.499586 0.866265i \(-0.333486\pi\)
0.499586 + 0.866265i \(0.333486\pi\)
\(978\) 0 0
\(979\) −12.3810 −0.395699
\(980\) 0 0
\(981\) −0.512889 −0.0163753
\(982\) 0 0
\(983\) 22.6155 0.721324 0.360662 0.932697i \(-0.382551\pi\)
0.360662 + 0.932697i \(0.382551\pi\)
\(984\) 0 0
\(985\) 6.73281 0.214525
\(986\) 0 0
\(987\) −24.9966 −0.795649
\(988\) 0 0
\(989\) 47.9570 1.52494
\(990\) 0 0
\(991\) −22.4102 −0.711884 −0.355942 0.934508i \(-0.615840\pi\)
−0.355942 + 0.934508i \(0.615840\pi\)
\(992\) 0 0
\(993\) 45.9639 1.45862
\(994\) 0 0
\(995\) 13.2311 0.419454
\(996\) 0 0
\(997\) −27.3415 −0.865914 −0.432957 0.901415i \(-0.642530\pi\)
−0.432957 + 0.901415i \(0.642530\pi\)
\(998\) 0 0
\(999\) −8.99656 −0.284639
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 6160.2.a.bf.1.3 3
4.3 odd 2 770.2.a.m.1.1 3
12.11 even 2 6930.2.a.ce.1.2 3
20.3 even 4 3850.2.c.ba.1849.1 6
20.7 even 4 3850.2.c.ba.1849.6 6
20.19 odd 2 3850.2.a.bt.1.3 3
28.27 even 2 5390.2.a.ca.1.3 3
44.43 even 2 8470.2.a.ci.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.1 3 4.3 odd 2
3850.2.a.bt.1.3 3 20.19 odd 2
3850.2.c.ba.1849.1 6 20.3 even 4
3850.2.c.ba.1849.6 6 20.7 even 4
5390.2.a.ca.1.3 3 28.27 even 2
6160.2.a.bf.1.3 3 1.1 even 1 trivial
6930.2.a.ce.1.2 3 12.11 even 2
8470.2.a.ci.1.1 3 44.43 even 2