Properties

Label 6930.2.a.ce.1.2
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 1.2
Root \(0.470683\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} +0.941367 q^{13} +1.00000 q^{14} +1.00000 q^{16} -6.49828 q^{17} -4.36641 q^{19} -1.00000 q^{20} +1.00000 q^{22} -6.24914 q^{23} +1.00000 q^{25} -0.941367 q^{26} -1.00000 q^{28} -8.74742 q^{29} -9.55691 q^{31} -1.00000 q^{32} +6.49828 q^{34} +1.00000 q^{35} +4.24914 q^{37} +4.36641 q^{38} +1.00000 q^{40} +2.13187 q^{41} +7.67418 q^{43} -1.00000 q^{44} +6.24914 q^{46} -11.1138 q^{47} +1.00000 q^{49} -1.00000 q^{50} +0.941367 q^{52} +4.74742 q^{53} +1.00000 q^{55} +1.00000 q^{56} +8.74742 q^{58} +1.88273 q^{59} +9.11383 q^{61} +9.55691 q^{62} +1.00000 q^{64} -0.941367 q^{65} +12.9966 q^{67} -6.49828 q^{68} -1.00000 q^{70} +14.6155 q^{71} -10.4983 q^{73} -4.24914 q^{74} -4.36641 q^{76} +1.00000 q^{77} +8.36641 q^{79} -1.00000 q^{80} -2.13187 q^{82} +8.49828 q^{83} +6.49828 q^{85} -7.67418 q^{86} +1.00000 q^{88} -12.3810 q^{89} -0.941367 q^{91} -6.24914 q^{92} +11.1138 q^{94} +4.36641 q^{95} -15.3630 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} - 3q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} - 3q^{5} - 3q^{7} - 3q^{8} + 3q^{10} - 3q^{11} + 2q^{13} + 3q^{14} + 3q^{16} - 2q^{17} - 6q^{19} - 3q^{20} + 3q^{22} - 10q^{23} + 3q^{25} - 2q^{26} - 3q^{28} - 12q^{31} - 3q^{32} + 2q^{34} + 3q^{35} + 4q^{37} + 6q^{38} + 3q^{40} - 4q^{41} + 8q^{43} - 3q^{44} + 10q^{46} + 3q^{49} - 3q^{50} + 2q^{52} - 12q^{53} + 3q^{55} + 3q^{56} + 4q^{59} - 6q^{61} + 12q^{62} + 3q^{64} - 2q^{65} + 4q^{67} - 2q^{68} - 3q^{70} + 28q^{71} - 14q^{73} - 4q^{74} - 6q^{76} + 3q^{77} + 18q^{79} - 3q^{80} + 4q^{82} + 8q^{83} + 2q^{85} - 8q^{86} + 3q^{88} - 18q^{89} - 2q^{91} - 10q^{92} + 6q^{95} - 4q^{97} - 3q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 0.941367 0.261088 0.130544 0.991443i \(-0.458328\pi\)
0.130544 + 0.991443i \(0.458328\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.49828 −1.57606 −0.788032 0.615634i \(-0.788900\pi\)
−0.788032 + 0.615634i \(0.788900\pi\)
\(18\) 0 0
\(19\) −4.36641 −1.00172 −0.500861 0.865528i \(-0.666983\pi\)
−0.500861 + 0.865528i \(0.666983\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −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
\(26\) −0.941367 −0.184617
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −8.74742 −1.62436 −0.812178 0.583410i \(-0.801718\pi\)
−0.812178 + 0.583410i \(0.801718\pi\)
\(30\) 0 0
\(31\) −9.55691 −1.71647 −0.858236 0.513255i \(-0.828440\pi\)
−0.858236 + 0.513255i \(0.828440\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.49828 1.11445
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 4.24914 0.698554 0.349277 0.937019i \(-0.386427\pi\)
0.349277 + 0.937019i \(0.386427\pi\)
\(38\) 4.36641 0.708325
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 2.13187 0.332943 0.166471 0.986046i \(-0.446763\pi\)
0.166471 + 0.986046i \(0.446763\pi\)
\(42\) 0 0
\(43\) 7.67418 1.17030 0.585151 0.810925i \(-0.301035\pi\)
0.585151 + 0.810925i \(0.301035\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 6.24914 0.921386
\(47\) −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
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 0.941367 0.130544
\(53\) 4.74742 0.652109 0.326054 0.945351i \(-0.394281\pi\)
0.326054 + 0.945351i \(0.394281\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 8.74742 1.14859
\(59\) 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 \(0.301701\pi\)
\(62\) 9.55691 1.21373
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.941367 −0.116762
\(66\) 0 0
\(67\) 12.9966 1.58778 0.793891 0.608060i \(-0.208052\pi\)
0.793891 + 0.608060i \(0.208052\pi\)
\(68\) −6.49828 −0.788032
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 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 \(0.710588\pi\)
\(74\) −4.24914 −0.493953
\(75\) 0 0
\(76\) −4.36641 −0.500861
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 8.36641 0.941294 0.470647 0.882322i \(-0.344020\pi\)
0.470647 + 0.882322i \(0.344020\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −2.13187 −0.235426
\(83\) 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
\(86\) −7.67418 −0.827528
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −12.3810 −1.31238 −0.656192 0.754594i \(-0.727834\pi\)
−0.656192 + 0.754594i \(0.727834\pi\)
\(90\) 0 0
\(91\) −0.941367 −0.0986821
\(92\) −6.24914 −0.651518
\(93\) 0 0
\(94\) 11.1138 1.14630
\(95\) 4.36641 0.447984
\(96\) 0 0
\(97\) −15.3630 −1.55987 −0.779937 0.625859i \(-0.784749\pi\)
−0.779937 + 0.625859i \(0.784749\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 16.8793 1.67955 0.839776 0.542932i \(-0.182686\pi\)
0.839776 + 0.542932i \(0.182686\pi\)
\(102\) 0 0
\(103\) −6.61555 −0.651849 −0.325925 0.945396i \(-0.605676\pi\)
−0.325925 + 0.945396i \(0.605676\pi\)
\(104\) −0.941367 −0.0923086
\(105\) 0 0
\(106\) −4.74742 −0.461110
\(107\) 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 \(0.503798\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −10.9966 −1.03447 −0.517235 0.855844i \(-0.673039\pi\)
−0.517235 + 0.855844i \(0.673039\pi\)
\(114\) 0 0
\(115\) 6.24914 0.582735
\(116\) −8.74742 −0.812178
\(117\) 0 0
\(118\) −1.88273 −0.173320
\(119\) 6.49828 0.595696
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −9.11383 −0.825127
\(123\) 0 0
\(124\) −9.55691 −0.858236
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −5.88273 −0.522008 −0.261004 0.965338i \(-0.584054\pi\)
−0.261004 + 0.965338i \(0.584054\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0.941367 0.0825633
\(131\) 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
\(134\) −12.9966 −1.12273
\(135\) 0 0
\(136\) 6.49828 0.557223
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 10.9820 0.931477 0.465739 0.884922i \(-0.345789\pi\)
0.465739 + 0.884922i \(0.345789\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −14.6155 −1.22651
\(143\) −0.941367 −0.0787210
\(144\) 0 0
\(145\) 8.74742 0.726434
\(146\) 10.4983 0.868844
\(147\) 0 0
\(148\) 4.24914 0.349277
\(149\) −0.0146079 −0.00119673 −0.000598363 1.00000i \(-0.500190\pi\)
−0.000598363 1.00000i \(0.500190\pi\)
\(150\) 0 0
\(151\) −15.2457 −1.24068 −0.620339 0.784334i \(-0.713005\pi\)
−0.620339 + 0.784334i \(0.713005\pi\)
\(152\) 4.36641 0.354162
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 9.55691 0.767630
\(156\) 0 0
\(157\) −5.50172 −0.439085 −0.219542 0.975603i \(-0.570456\pi\)
−0.219542 + 0.975603i \(0.570456\pi\)
\(158\) −8.36641 −0.665596
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 6.24914 0.492501
\(162\) 0 0
\(163\) −18.2277 −1.42770 −0.713850 0.700298i \(-0.753050\pi\)
−0.713850 + 0.700298i \(0.753050\pi\)
\(164\) 2.13187 0.166471
\(165\) 0 0
\(166\) −8.49828 −0.659595
\(167\) 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
\(170\) −6.49828 −0.498395
\(171\) 0 0
\(172\) 7.67418 0.585151
\(173\) −0.117266 −0.00891559 −0.00445780 0.999990i \(-0.501419\pi\)
−0.00445780 + 0.999990i \(0.501419\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 12.3810 0.927996
\(179\) 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 \(0.217259\pi\)
\(182\) 0.941367 0.0697788
\(183\) 0 0
\(184\) 6.24914 0.460693
\(185\) −4.24914 −0.312403
\(186\) 0 0
\(187\) 6.49828 0.475201
\(188\) −11.1138 −0.810559
\(189\) 0 0
\(190\) −4.36641 −0.316772
\(191\) −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 \(0.571308\pi\)
\(194\) 15.3630 1.10300
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.73281 −0.479693 −0.239847 0.970811i \(-0.577097\pi\)
−0.239847 + 0.970811i \(0.577097\pi\)
\(198\) 0 0
\(199\) −13.2311 −0.937927 −0.468964 0.883217i \(-0.655373\pi\)
−0.468964 + 0.883217i \(0.655373\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −16.8793 −1.18762
\(203\) 8.74742 0.613949
\(204\) 0 0
\(205\) −2.13187 −0.148897
\(206\) 6.61555 0.460927
\(207\) 0 0
\(208\) 0.941367 0.0652720
\(209\) 4.36641 0.302031
\(210\) 0 0
\(211\) 23.1138 1.59122 0.795611 0.605808i \(-0.207150\pi\)
0.795611 + 0.605808i \(0.207150\pi\)
\(212\) 4.74742 0.326054
\(213\) 0 0
\(214\) −14.5535 −0.994855
\(215\) −7.67418 −0.523375
\(216\) 0 0
\(217\) 9.55691 0.648766
\(218\) 0.249141 0.0168739
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −6.11727 −0.411492
\(222\) 0 0
\(223\) 10.3810 0.695164 0.347582 0.937650i \(-0.387003\pi\)
0.347582 + 0.937650i \(0.387003\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 10.9966 0.731480
\(227\) −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 \(0.203473\pi\)
\(230\) −6.24914 −0.412056
\(231\) 0 0
\(232\) 8.74742 0.574296
\(233\) −6.88617 −0.451128 −0.225564 0.974228i \(-0.572422\pi\)
−0.225564 + 0.974228i \(0.572422\pi\)
\(234\) 0 0
\(235\) 11.1138 0.724986
\(236\) 1.88273 0.122556
\(237\) 0 0
\(238\) −6.49828 −0.421221
\(239\) 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 \(0.475716\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 9.11383 0.583453
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −4.11039 −0.261538
\(248\) 9.55691 0.606865
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −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
\(254\) 5.88273 0.369116
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 2.86469 0.178694 0.0893472 0.996001i \(-0.471522\pi\)
0.0893472 + 0.996001i \(0.471522\pi\)
\(258\) 0 0
\(259\) −4.24914 −0.264029
\(260\) −0.941367 −0.0583811
\(261\) 0 0
\(262\) −1.25258 −0.0773846
\(263\) −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
\(266\) −4.36641 −0.267722
\(267\) 0 0
\(268\) 12.9966 0.793891
\(269\) −8.94137 −0.545165 −0.272582 0.962132i \(-0.587878\pi\)
−0.272582 + 0.962132i \(0.587878\pi\)
\(270\) 0 0
\(271\) 21.4948 1.30572 0.652859 0.757479i \(-0.273569\pi\)
0.652859 + 0.757479i \(0.273569\pi\)
\(272\) −6.49828 −0.394016
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 7.64820 0.459536 0.229768 0.973245i \(-0.426203\pi\)
0.229768 + 0.973245i \(0.426203\pi\)
\(278\) −10.9820 −0.658654
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 28.6155 1.70706 0.853530 0.521043i \(-0.174457\pi\)
0.853530 + 0.521043i \(0.174457\pi\)
\(282\) 0 0
\(283\) −2.87930 −0.171156 −0.0855782 0.996331i \(-0.527274\pi\)
−0.0855782 + 0.996331i \(0.527274\pi\)
\(284\) 14.6155 0.867273
\(285\) 0 0
\(286\) 0.941367 0.0556642
\(287\) −2.13187 −0.125841
\(288\) 0 0
\(289\) 25.2277 1.48398
\(290\) −8.74742 −0.513666
\(291\) 0 0
\(292\) −10.4983 −0.614365
\(293\) 12.9414 0.756043 0.378021 0.925797i \(-0.376605\pi\)
0.378021 + 0.925797i \(0.376605\pi\)
\(294\) 0 0
\(295\) −1.88273 −0.109617
\(296\) −4.24914 −0.246976
\(297\) 0 0
\(298\) 0.0146079 0.000846213 0
\(299\) −5.88273 −0.340207
\(300\) 0 0
\(301\) −7.67418 −0.442332
\(302\) 15.2457 0.877292
\(303\) 0 0
\(304\) −4.36641 −0.250431
\(305\) −9.11383 −0.521856
\(306\) 0 0
\(307\) 0.498281 0.0284384 0.0142192 0.999899i \(-0.495474\pi\)
0.0142192 + 0.999899i \(0.495474\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −9.55691 −0.542796
\(311\) 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 \(0.414692\pi\)
\(314\) 5.50172 0.310480
\(315\) 0 0
\(316\) 8.36641 0.470647
\(317\) 9.97852 0.560449 0.280225 0.959934i \(-0.409591\pi\)
0.280225 + 0.959934i \(0.409591\pi\)
\(318\) 0 0
\(319\) 8.74742 0.489762
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −6.24914 −0.348251
\(323\) 28.3741 1.57878
\(324\) 0 0
\(325\) 0.941367 0.0522176
\(326\) 18.2277 1.00954
\(327\) 0 0
\(328\) −2.13187 −0.117713
\(329\) 11.1138 0.612725
\(330\) 0 0
\(331\) −20.4362 −1.12328 −0.561638 0.827383i \(-0.689829\pi\)
−0.561638 + 0.827383i \(0.689829\pi\)
\(332\) 8.49828 0.466404
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) −12.9966 −0.710078
\(336\) 0 0
\(337\) 7.88273 0.429400 0.214700 0.976680i \(-0.431123\pi\)
0.214700 + 0.976680i \(0.431123\pi\)
\(338\) 12.1138 0.658905
\(339\) 0 0
\(340\) 6.49828 0.352419
\(341\) 9.55691 0.517536
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −7.67418 −0.413764
\(345\) 0 0
\(346\) 0.117266 0.00630428
\(347\) −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 \(0.592743\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 20.8578 1.11015 0.555075 0.831801i \(-0.312690\pi\)
0.555075 + 0.831801i \(0.312690\pi\)
\(354\) 0 0
\(355\) −14.6155 −0.775713
\(356\) −12.3810 −0.656192
\(357\) 0 0
\(358\) −22.5535 −1.19199
\(359\) −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
\(362\) −20.8793 −1.09739
\(363\) 0 0
\(364\) −0.941367 −0.0493410
\(365\) 10.4983 0.549505
\(366\) 0 0
\(367\) 20.8432 1.08801 0.544003 0.839083i \(-0.316908\pi\)
0.544003 + 0.839083i \(0.316908\pi\)
\(368\) −6.24914 −0.325759
\(369\) 0 0
\(370\) 4.24914 0.220902
\(371\) −4.74742 −0.246474
\(372\) 0 0
\(373\) −5.37758 −0.278440 −0.139220 0.990261i \(-0.544460\pi\)
−0.139220 + 0.990261i \(0.544460\pi\)
\(374\) −6.49828 −0.336018
\(375\) 0 0
\(376\) 11.1138 0.573152
\(377\) −8.23453 −0.424100
\(378\) 0 0
\(379\) 3.53093 0.181372 0.0906860 0.995880i \(-0.471094\pi\)
0.0906860 + 0.995880i \(0.471094\pi\)
\(380\) 4.36641 0.223992
\(381\) 0 0
\(382\) 3.11383 0.159317
\(383\) −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
\(386\) 6.17246 0.314170
\(387\) 0 0
\(388\) −15.3630 −0.779937
\(389\) −15.7294 −0.797511 −0.398756 0.917057i \(-0.630558\pi\)
−0.398756 + 0.917057i \(0.630558\pi\)
\(390\) 0 0
\(391\) 40.6087 2.05367
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 6.73281 0.339194
\(395\) −8.36641 −0.420960
\(396\) 0 0
\(397\) −17.6121 −0.883926 −0.441963 0.897033i \(-0.645718\pi\)
−0.441963 + 0.897033i \(0.645718\pi\)
\(398\) 13.2311 0.663215
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 32.8172 1.63881 0.819407 0.573212i \(-0.194303\pi\)
0.819407 + 0.573212i \(0.194303\pi\)
\(402\) 0 0
\(403\) −8.99656 −0.448151
\(404\) 16.8793 0.839776
\(405\) 0 0
\(406\) −8.74742 −0.434127
\(407\) −4.24914 −0.210622
\(408\) 0 0
\(409\) −33.3561 −1.64935 −0.824676 0.565605i \(-0.808643\pi\)
−0.824676 + 0.565605i \(0.808643\pi\)
\(410\) 2.13187 0.105286
\(411\) 0 0
\(412\) −6.61555 −0.325925
\(413\) −1.88273 −0.0926433
\(414\) 0 0
\(415\) −8.49828 −0.417164
\(416\) −0.941367 −0.0461543
\(417\) 0 0
\(418\) −4.36641 −0.213568
\(419\) −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 \(0.569423\pi\)
\(422\) −23.1138 −1.12516
\(423\) 0 0
\(424\) −4.74742 −0.230555
\(425\) −6.49828 −0.315213
\(426\) 0 0
\(427\) −9.11383 −0.441049
\(428\) 14.5535 0.703469
\(429\) 0 0
\(430\) 7.67418 0.370082
\(431\) −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 \(0.447254\pi\)
\(434\) −9.55691 −0.458747
\(435\) 0 0
\(436\) −0.249141 −0.0119317
\(437\) 27.2863 1.30528
\(438\) 0 0
\(439\) −11.8466 −0.565409 −0.282705 0.959207i \(-0.591232\pi\)
−0.282705 + 0.959207i \(0.591232\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 6.11727 0.290969
\(443\) −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
\(446\) −10.3810 −0.491555
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 16.8172 0.793654 0.396827 0.917893i \(-0.370111\pi\)
0.396827 + 0.917893i \(0.370111\pi\)
\(450\) 0 0
\(451\) −2.13187 −0.100386
\(452\) −10.9966 −0.517235
\(453\) 0 0
\(454\) 16.1104 0.756098
\(455\) 0.941367 0.0441320
\(456\) 0 0
\(457\) −12.2277 −0.571986 −0.285993 0.958232i \(-0.592323\pi\)
−0.285993 + 0.958232i \(0.592323\pi\)
\(458\) −24.2897 −1.13498
\(459\) 0 0
\(460\) 6.24914 0.291368
\(461\) 16.6155 0.773863 0.386932 0.922108i \(-0.373535\pi\)
0.386932 + 0.922108i \(0.373535\pi\)
\(462\) 0 0
\(463\) 18.4837 0.859009 0.429505 0.903065i \(-0.358688\pi\)
0.429505 + 0.903065i \(0.358688\pi\)
\(464\) −8.74742 −0.406089
\(465\) 0 0
\(466\) 6.88617 0.318996
\(467\) −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
\(470\) −11.1138 −0.512643
\(471\) 0 0
\(472\) −1.88273 −0.0866598
\(473\) −7.67418 −0.352859
\(474\) 0 0
\(475\) −4.36641 −0.200344
\(476\) 6.49828 0.297848
\(477\) 0 0
\(478\) −11.1353 −0.509317
\(479\) 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
\(482\) −2.36641 −0.107787
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 15.3630 0.697596
\(486\) 0 0
\(487\) 10.0958 0.457484 0.228742 0.973487i \(-0.426539\pi\)
0.228742 + 0.973487i \(0.426539\pi\)
\(488\) −9.11383 −0.412564
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 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
\(494\) 4.11039 0.184935
\(495\) 0 0
\(496\) −9.55691 −0.429118
\(497\) −14.6155 −0.655597
\(498\) 0 0
\(499\) −8.79488 −0.393713 −0.196857 0.980432i \(-0.563073\pi\)
−0.196857 + 0.980432i \(0.563073\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 25.1070 1.12058
\(503\) 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
\(506\) −6.24914 −0.277808
\(507\) 0 0
\(508\) −5.88273 −0.261004
\(509\) −21.8759 −0.969630 −0.484815 0.874617i \(-0.661113\pi\)
−0.484815 + 0.874617i \(0.661113\pi\)
\(510\) 0 0
\(511\) 10.4983 0.464417
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −2.86469 −0.126356
\(515\) 6.61555 0.291516
\(516\) 0 0
\(517\) 11.1138 0.488786
\(518\) 4.24914 0.186697
\(519\) 0 0
\(520\) 0.941367 0.0412817
\(521\) −14.0292 −0.614631 −0.307316 0.951608i \(-0.599431\pi\)
−0.307316 + 0.951608i \(0.599431\pi\)
\(522\) 0 0
\(523\) 1.14992 0.0502825 0.0251412 0.999684i \(-0.491996\pi\)
0.0251412 + 0.999684i \(0.491996\pi\)
\(524\) 1.25258 0.0547191
\(525\) 0 0
\(526\) 3.76547 0.164182
\(527\) 62.1035 2.70527
\(528\) 0 0
\(529\) 16.0518 0.697902
\(530\) 4.74742 0.206215
\(531\) 0 0
\(532\) 4.36641 0.189308
\(533\) 2.00688 0.0869274
\(534\) 0 0
\(535\) −14.5535 −0.629202
\(536\) −12.9966 −0.561366
\(537\) 0 0
\(538\) 8.94137 0.385490
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −6.47680 −0.278459 −0.139230 0.990260i \(-0.544463\pi\)
−0.139230 + 0.990260i \(0.544463\pi\)
\(542\) −21.4948 −0.923283
\(543\) 0 0
\(544\) 6.49828 0.278612
\(545\) 0.249141 0.0106720
\(546\) 0 0
\(547\) 12.9966 0.555693 0.277846 0.960626i \(-0.410379\pi\)
0.277846 + 0.960626i \(0.410379\pi\)
\(548\) −10.0000 −0.427179
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 38.1948 1.62715
\(552\) 0 0
\(553\) −8.36641 −0.355776
\(554\) −7.64820 −0.324941
\(555\) 0 0
\(556\) 10.9820 0.465739
\(557\) 16.4914 0.698763 0.349382 0.936981i \(-0.386392\pi\)
0.349382 + 0.936981i \(0.386392\pi\)
\(558\) 0 0
\(559\) 7.22422 0.305552
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −28.6155 −1.20707
\(563\) 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
\(566\) 2.87930 0.121026
\(567\) 0 0
\(568\) −14.6155 −0.613255
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) 12.7328 0.532852 0.266426 0.963855i \(-0.414157\pi\)
0.266426 + 0.963855i \(0.414157\pi\)
\(572\) −0.941367 −0.0393605
\(573\) 0 0
\(574\) 2.13187 0.0889827
\(575\) −6.24914 −0.260607
\(576\) 0 0
\(577\) −11.8613 −0.493790 −0.246895 0.969042i \(-0.579410\pi\)
−0.246895 + 0.969042i \(0.579410\pi\)
\(578\) −25.2277 −1.04933
\(579\) 0 0
\(580\) 8.74742 0.363217
\(581\) −8.49828 −0.352568
\(582\) 0 0
\(583\) −4.74742 −0.196618
\(584\) 10.4983 0.434422
\(585\) 0 0
\(586\) −12.9414 −0.534603
\(587\) 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
\(590\) 1.88273 0.0775109
\(591\) 0 0
\(592\) 4.24914 0.174639
\(593\) −10.7328 −0.440744 −0.220372 0.975416i \(-0.570727\pi\)
−0.220372 + 0.975416i \(0.570727\pi\)
\(594\) 0 0
\(595\) −6.49828 −0.266404
\(596\) −0.0146079 −0.000598363 0
\(597\) 0 0
\(598\) 5.88273 0.240563
\(599\) −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 \(0.744710\pi\)
\(602\) 7.67418 0.312776
\(603\) 0 0
\(604\) −15.2457 −0.620339
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 11.6742 0.473840 0.236920 0.971529i \(-0.423862\pi\)
0.236920 + 0.971529i \(0.423862\pi\)
\(608\) 4.36641 0.177081
\(609\) 0 0
\(610\) 9.11383 0.369008
\(611\) −10.4622 −0.423255
\(612\) 0 0
\(613\) −34.2637 −1.38390 −0.691950 0.721946i \(-0.743248\pi\)
−0.691950 + 0.721946i \(0.743248\pi\)
\(614\) −0.498281 −0.0201090
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −39.3707 −1.58500 −0.792502 0.609869i \(-0.791222\pi\)
−0.792502 + 0.609869i \(0.791222\pi\)
\(618\) 0 0
\(619\) 16.1104 0.647531 0.323766 0.946137i \(-0.395051\pi\)
0.323766 + 0.946137i \(0.395051\pi\)
\(620\) 9.55691 0.383815
\(621\) 0 0
\(622\) −23.4396 −0.939844
\(623\) 12.3810 0.496035
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −9.36984 −0.374494
\(627\) 0 0
\(628\) −5.50172 −0.219542
\(629\) −27.6121 −1.10097
\(630\) 0 0
\(631\) 13.8827 0.552663 0.276331 0.961062i \(-0.410881\pi\)
0.276331 + 0.961062i \(0.410881\pi\)
\(632\) −8.36641 −0.332798
\(633\) 0 0
\(634\) −9.97852 −0.396298
\(635\) 5.88273 0.233449
\(636\) 0 0
\(637\) 0.941367 0.0372983
\(638\) −8.74742 −0.346314
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 39.3415 1.55390 0.776948 0.629565i \(-0.216767\pi\)
0.776948 + 0.629565i \(0.216767\pi\)
\(642\) 0 0
\(643\) 38.3595 1.51275 0.756376 0.654137i \(-0.226968\pi\)
0.756376 + 0.654137i \(0.226968\pi\)
\(644\) 6.24914 0.246251
\(645\) 0 0
\(646\) −28.3741 −1.11637
\(647\) −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
\(650\) −0.941367 −0.0369234
\(651\) 0 0
\(652\) −18.2277 −0.713850
\(653\) −18.4768 −0.723053 −0.361526 0.932362i \(-0.617744\pi\)
−0.361526 + 0.932362i \(0.617744\pi\)
\(654\) 0 0
\(655\) −1.25258 −0.0489423
\(656\) 2.13187 0.0832357
\(657\) 0 0
\(658\) −11.1138 −0.433262
\(659\) 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 \(0.731789\pi\)
\(662\) 20.4362 0.794276
\(663\) 0 0
\(664\) −8.49828 −0.329797
\(665\) −4.36641 −0.169322
\(666\) 0 0
\(667\) 54.6639 2.11659
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 12.9966 0.502101
\(671\) −9.11383 −0.351835
\(672\) 0 0
\(673\) −24.4622 −0.942948 −0.471474 0.881880i \(-0.656278\pi\)
−0.471474 + 0.881880i \(0.656278\pi\)
\(674\) −7.88273 −0.303632
\(675\) 0 0
\(676\) −12.1138 −0.465916
\(677\) 39.1070 1.50300 0.751501 0.659732i \(-0.229330\pi\)
0.751501 + 0.659732i \(0.229330\pi\)
\(678\) 0 0
\(679\) 15.3630 0.589577
\(680\) −6.49828 −0.249198
\(681\) 0 0
\(682\) −9.55691 −0.365953
\(683\) −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
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 7.67418 0.292575
\(689\) 4.46907 0.170258
\(690\) 0 0
\(691\) 8.49828 0.323290 0.161645 0.986849i \(-0.448320\pi\)
0.161645 + 0.986849i \(0.448320\pi\)
\(692\) −0.117266 −0.00445780
\(693\) 0 0
\(694\) 13.5569 0.514613
\(695\) −10.9820 −0.416569
\(696\) 0 0
\(697\) −13.8535 −0.524739
\(698\) 10.7328 0.406243
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −14.9751 −0.565601 −0.282800 0.959179i \(-0.591263\pi\)
−0.282800 + 0.959179i \(0.591263\pi\)
\(702\) 0 0
\(703\) −18.5535 −0.699758
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −20.8578 −0.784994
\(707\) −16.8793 −0.634811
\(708\) 0 0
\(709\) 40.7259 1.52949 0.764747 0.644330i \(-0.222864\pi\)
0.764747 + 0.644330i \(0.222864\pi\)
\(710\) 14.6155 0.548512
\(711\) 0 0
\(712\) 12.3810 0.463998
\(713\) 59.7225 2.23663
\(714\) 0 0
\(715\) 0.941367 0.0352051
\(716\) 22.5535 0.842863
\(717\) 0 0
\(718\) 0.366407 0.0136742
\(719\) 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
\(722\) −0.0655089 −0.00243799
\(723\) 0 0
\(724\) 20.8793 0.775973
\(725\) −8.74742 −0.324871
\(726\) 0 0
\(727\) −51.6413 −1.91527 −0.957635 0.287984i \(-0.907015\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(728\) 0.941367 0.0348894
\(729\) 0 0
\(730\) −10.4983 −0.388559
\(731\) −49.8690 −1.84447
\(732\) 0 0
\(733\) −42.0000 −1.55131 −0.775653 0.631160i \(-0.782579\pi\)
−0.775653 + 0.631160i \(0.782579\pi\)
\(734\) −20.8432 −0.769337
\(735\) 0 0
\(736\) 6.24914 0.230346
\(737\) −12.9966 −0.478735
\(738\) 0 0
\(739\) −49.1070 −1.80643 −0.903214 0.429190i \(-0.858799\pi\)
−0.903214 + 0.429190i \(0.858799\pi\)
\(740\) −4.24914 −0.156202
\(741\) 0 0
\(742\) 4.74742 0.174283
\(743\) 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
\(746\) 5.37758 0.196887
\(747\) 0 0
\(748\) 6.49828 0.237601
\(749\) −14.5535 −0.531772
\(750\) 0 0
\(751\) −31.6121 −1.15354 −0.576771 0.816906i \(-0.695688\pi\)
−0.576771 + 0.816906i \(0.695688\pi\)
\(752\) −11.1138 −0.405280
\(753\) 0 0
\(754\) 8.23453 0.299884
\(755\) 15.2457 0.554848
\(756\) 0 0
\(757\) −1.24570 −0.0452758 −0.0226379 0.999744i \(-0.507206\pi\)
−0.0226379 + 0.999744i \(0.507206\pi\)
\(758\) −3.53093 −0.128249
\(759\) 0 0
\(760\) −4.36641 −0.158386
\(761\) 10.6009 0.384284 0.192142 0.981367i \(-0.438457\pi\)
0.192142 + 0.981367i \(0.438457\pi\)
\(762\) 0 0
\(763\) 0.249141 0.00901949
\(764\) −3.11383 −0.112654
\(765\) 0 0
\(766\) 1.38445 0.0500223
\(767\) 1.77234 0.0639956
\(768\) 0 0
\(769\) −19.1284 −0.689789 −0.344895 0.938641i \(-0.612085\pi\)
−0.344895 + 0.938641i \(0.612085\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −6.17246 −0.222152
\(773\) −39.9931 −1.43845 −0.719226 0.694776i \(-0.755504\pi\)
−0.719226 + 0.694776i \(0.755504\pi\)
\(774\) 0 0
\(775\) −9.55691 −0.343294
\(776\) 15.3630 0.551498
\(777\) 0 0
\(778\) 15.7294 0.563925
\(779\) −9.30863 −0.333516
\(780\) 0 0
\(781\) −14.6155 −0.522985
\(782\) −40.6087 −1.45216
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 5.50172 0.196365
\(786\) 0 0
\(787\) 37.9931 1.35431 0.677154 0.735841i \(-0.263213\pi\)
0.677154 + 0.735841i \(0.263213\pi\)
\(788\) −6.73281 −0.239847
\(789\) 0 0
\(790\) 8.36641 0.297663
\(791\) 10.9966 0.390993
\(792\) 0 0
\(793\) 8.57946 0.304665
\(794\) 17.6121 0.625030
\(795\) 0 0
\(796\) −13.2311 −0.468964
\(797\) 20.3810 0.721933 0.360966 0.932579i \(-0.382447\pi\)
0.360966 + 0.932579i \(0.382447\pi\)
\(798\) 0 0
\(799\) 72.2208 2.55499
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −32.8172 −1.15882
\(803\) 10.4983 0.370476
\(804\) 0 0
\(805\) −6.24914 −0.220253
\(806\) 8.99656 0.316890
\(807\) 0 0
\(808\) −16.8793 −0.593812
\(809\) 31.7294 1.11555 0.557773 0.829994i \(-0.311656\pi\)
0.557773 + 0.829994i \(0.311656\pi\)
\(810\) 0 0
\(811\) 43.5095 1.52782 0.763912 0.645321i \(-0.223276\pi\)
0.763912 + 0.645321i \(0.223276\pi\)
\(812\) 8.74742 0.306974
\(813\) 0 0
\(814\) 4.24914 0.148932
\(815\) 18.2277 0.638487
\(816\) 0 0
\(817\) −33.5086 −1.17232
\(818\) 33.3561 1.16627
\(819\) 0 0
\(820\) −2.13187 −0.0744483
\(821\) 24.7766 0.864711 0.432355 0.901703i \(-0.357683\pi\)
0.432355 + 0.901703i \(0.357683\pi\)
\(822\) 0 0
\(823\) 28.8647 1.00616 0.503080 0.864240i \(-0.332200\pi\)
0.503080 + 0.864240i \(0.332200\pi\)
\(824\) 6.61555 0.230464
\(825\) 0 0
\(826\) 1.88273 0.0655087
\(827\) −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 \(0.365357\pi\)
\(830\) 8.49828 0.294980
\(831\) 0 0
\(832\) 0.941367 0.0326360
\(833\) −6.49828 −0.225152
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 4.36641 0.151015
\(837\) 0 0
\(838\) 12.3449 0.426448
\(839\) 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
\(842\) 8.87930 0.306001
\(843\) 0 0
\(844\) 23.1138 0.795611
\(845\) 12.1138 0.416728
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 4.74742 0.163027
\(849\) 0 0
\(850\) 6.49828 0.222889
\(851\) −26.5535 −0.910241
\(852\) 0 0
\(853\) 22.9966 0.787387 0.393694 0.919242i \(-0.371197\pi\)
0.393694 + 0.919242i \(0.371197\pi\)
\(854\) 9.11383 0.311869
\(855\) 0 0
\(856\) −14.5535 −0.497428
\(857\) 31.2603 1.06783 0.533916 0.845538i \(-0.320720\pi\)
0.533916 + 0.845538i \(0.320720\pi\)
\(858\) 0 0
\(859\) 20.3449 0.694160 0.347080 0.937836i \(-0.387173\pi\)
0.347080 + 0.937836i \(0.387173\pi\)
\(860\) −7.67418 −0.261687
\(861\) 0 0
\(862\) 18.2784 0.622563
\(863\) 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
\(866\) −6.86469 −0.233272
\(867\) 0 0
\(868\) 9.55691 0.324383
\(869\) −8.36641 −0.283811
\(870\) 0 0
\(871\) 12.2345 0.414551
\(872\) 0.249141 0.00843696
\(873\) 0 0
\(874\) −27.2863 −0.922973
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 11.9639 0.403992 0.201996 0.979386i \(-0.435257\pi\)
0.201996 + 0.979386i \(0.435257\pi\)
\(878\) 11.8466 0.399805
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) −5.88961 −0.198426 −0.0992130 0.995066i \(-0.531633\pi\)
−0.0992130 + 0.995066i \(0.531633\pi\)
\(882\) 0 0
\(883\) −19.4880 −0.655822 −0.327911 0.944709i \(-0.606345\pi\)
−0.327911 + 0.944709i \(0.606345\pi\)
\(884\) −6.11727 −0.205746
\(885\) 0 0
\(886\) 17.8827 0.600782
\(887\) −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
\(890\) −12.3810 −0.415013
\(891\) 0 0
\(892\) 10.3810 0.347582
\(893\) 48.5275 1.62391
\(894\) 0 0
\(895\) −22.5535 −0.753880
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −16.8172 −0.561198
\(899\) 83.5984 2.78816
\(900\) 0 0
\(901\) −30.8501 −1.02777
\(902\) 2.13187 0.0709836
\(903\) 0 0
\(904\) 10.9966 0.365740
\(905\) −20.8793 −0.694051
\(906\) 0 0
\(907\) −16.8432 −0.559269 −0.279635 0.960106i \(-0.590213\pi\)
−0.279635 + 0.960106i \(0.590213\pi\)
\(908\) −16.1104 −0.534642
\(909\) 0 0
\(910\) −0.941367 −0.0312060
\(911\) 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
\(914\) 12.2277 0.404455
\(915\) 0 0
\(916\) 24.2897 0.802555
\(917\) −1.25258 −0.0413638
\(918\) 0 0
\(919\) 17.2818 0.570074 0.285037 0.958517i \(-0.407994\pi\)
0.285037 + 0.958517i \(0.407994\pi\)
\(920\) −6.24914 −0.206028
\(921\) 0 0
\(922\) −16.6155 −0.547204
\(923\) 13.7586 0.452870
\(924\) 0 0
\(925\) 4.24914 0.139711
\(926\) −18.4837 −0.607411
\(927\) 0 0
\(928\) 8.74742 0.287148
\(929\) −2.91539 −0.0956508 −0.0478254 0.998856i \(-0.515229\pi\)
−0.0478254 + 0.998856i \(0.515229\pi\)
\(930\) 0 0
\(931\) −4.36641 −0.143103
\(932\) −6.88617 −0.225564
\(933\) 0 0
\(934\) 24.3956 0.798249
\(935\) −6.49828 −0.212517
\(936\) 0 0
\(937\) 22.5795 0.737639 0.368819 0.929501i \(-0.379762\pi\)
0.368819 + 0.929501i \(0.379762\pi\)
\(938\) 12.9966 0.424353
\(939\) 0 0
\(940\) 11.1138 0.362493
\(941\) 21.7655 0.709534 0.354767 0.934955i \(-0.384560\pi\)
0.354767 + 0.934955i \(0.384560\pi\)
\(942\) 0 0
\(943\) −13.3224 −0.433836
\(944\) 1.88273 0.0612778
\(945\) 0 0
\(946\) 7.67418 0.249509
\(947\) 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
\(950\) 4.36641 0.141665
\(951\) 0 0
\(952\) −6.49828 −0.210610
\(953\) 4.83098 0.156491 0.0782453 0.996934i \(-0.475068\pi\)
0.0782453 + 0.996934i \(0.475068\pi\)
\(954\) 0 0
\(955\) 3.11383 0.100761
\(956\) 11.1353 0.360142
\(957\) 0 0
\(958\) −27.7655 −0.897062
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) 60.3346 1.94628
\(962\) −4.00000 −0.128965
\(963\) 0 0
\(964\) 2.36641 0.0762168
\(965\) 6.17246 0.198699
\(966\) 0 0
\(967\) −51.9278 −1.66989 −0.834943 0.550336i \(-0.814499\pi\)
−0.834943 + 0.550336i \(0.814499\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −15.3630 −0.493275
\(971\) −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
\(974\) −10.0958 −0.323490
\(975\) 0 0
\(976\) 9.11383 0.291727
\(977\) −31.2311 −0.999171 −0.499586 0.866265i \(-0.666514\pi\)
−0.499586 + 0.866265i \(0.666514\pi\)
\(978\) 0 0
\(979\) 12.3810 0.395699
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −32.1104 −1.02468
\(983\) 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
\(986\) −56.8432 −1.81026
\(987\) 0 0
\(988\) −4.11039 −0.130769
\(989\) −47.9570 −1.52494
\(990\) 0 0
\(991\) 22.4102 0.711884 0.355942 0.934508i \(-0.384160\pi\)
0.355942 + 0.934508i \(0.384160\pi\)
\(992\) 9.55691 0.303432
\(993\) 0 0
\(994\) 14.6155 0.463577
\(995\) 13.2311 0.419454
\(996\) 0 0
\(997\) −27.3415 −0.865914 −0.432957 0.901415i \(-0.642530\pi\)
−0.432957 + 0.901415i \(0.642530\pi\)
\(998\) 8.79488 0.278397
\(999\) 0 0
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 6930.2.a.ce.1.2 3
3.2 odd 2 770.2.a.m.1.1 3
12.11 even 2 6160.2.a.bf.1.3 3
15.2 even 4 3850.2.c.ba.1849.6 6
15.8 even 4 3850.2.c.ba.1849.1 6
15.14 odd 2 3850.2.a.bt.1.3 3
21.20 even 2 5390.2.a.ca.1.3 3
33.32 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 3.2 odd 2
3850.2.a.bt.1.3 3 15.14 odd 2
3850.2.c.ba.1849.1 6 15.8 even 4
3850.2.c.ba.1849.6 6 15.2 even 4
5390.2.a.ca.1.3 3 21.20 even 2
6160.2.a.bf.1.3 3 12.11 even 2
6930.2.a.ce.1.2 3 1.1 even 1 trivial
8470.2.a.ci.1.1 3 33.32 even 2