Properties

Label 6930.2.a.bo.1.2
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +6.74456 q^{13} -1.00000 q^{14} +1.00000 q^{16} +6.74456 q^{17} -6.74456 q^{19} -1.00000 q^{20} -1.00000 q^{22} +6.74456 q^{23} +1.00000 q^{25} -6.74456 q^{26} +1.00000 q^{28} -8.74456 q^{29} +4.74456 q^{31} -1.00000 q^{32} -6.74456 q^{34} -1.00000 q^{35} +0.744563 q^{37} +6.74456 q^{38} +1.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -6.74456 q^{46} -4.74456 q^{47} +1.00000 q^{49} -1.00000 q^{50} +6.74456 q^{52} +12.7446 q^{53} -1.00000 q^{55} -1.00000 q^{56} +8.74456 q^{58} +8.74456 q^{59} +1.25544 q^{61} -4.74456 q^{62} +1.00000 q^{64} -6.74456 q^{65} -4.00000 q^{67} +6.74456 q^{68} +1.00000 q^{70} +4.00000 q^{71} +10.7446 q^{73} -0.744563 q^{74} -6.74456 q^{76} +1.00000 q^{77} +6.74456 q^{79} -1.00000 q^{80} -4.00000 q^{82} -8.00000 q^{83} -6.74456 q^{85} -4.00000 q^{86} -1.00000 q^{88} -15.4891 q^{89} +6.74456 q^{91} +6.74456 q^{92} +4.74456 q^{94} +6.74456 q^{95} -16.7446 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + 2q^{10} + 2q^{11} + 2q^{13} - 2q^{14} + 2q^{16} + 2q^{17} - 2q^{19} - 2q^{20} - 2q^{22} + 2q^{23} + 2q^{25} - 2q^{26} + 2q^{28} - 6q^{29} - 2q^{31} - 2q^{32} - 2q^{34} - 2q^{35} - 10q^{37} + 2q^{38} + 2q^{40} + 8q^{41} + 8q^{43} + 2q^{44} - 2q^{46} + 2q^{47} + 2q^{49} - 2q^{50} + 2q^{52} + 14q^{53} - 2q^{55} - 2q^{56} + 6q^{58} + 6q^{59} + 14q^{61} + 2q^{62} + 2q^{64} - 2q^{65} - 8q^{67} + 2q^{68} + 2q^{70} + 8q^{71} + 10q^{73} + 10q^{74} - 2q^{76} + 2q^{77} + 2q^{79} - 2q^{80} - 8q^{82} - 16q^{83} - 2q^{85} - 8q^{86} - 2q^{88} - 8q^{89} + 2q^{91} + 2q^{92} - 2q^{94} + 2q^{95} - 22q^{97} - 2q^{98} + O(q^{100}) \)

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
\(13\) 6.74456 1.87061 0.935303 0.353849i \(-0.115127\pi\)
0.935303 + 0.353849i \(0.115127\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.74456 1.63580 0.817898 0.575363i \(-0.195139\pi\)
0.817898 + 0.575363i \(0.195139\pi\)
\(18\) 0 0
\(19\) −6.74456 −1.54731 −0.773654 0.633608i \(-0.781573\pi\)
−0.773654 + 0.633608i \(0.781573\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 6.74456 1.40634 0.703169 0.711022i \(-0.251768\pi\)
0.703169 + 0.711022i \(0.251768\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −6.74456 −1.32272
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −8.74456 −1.62382 −0.811912 0.583779i \(-0.801573\pi\)
−0.811912 + 0.583779i \(0.801573\pi\)
\(30\) 0 0
\(31\) 4.74456 0.852149 0.426074 0.904688i \(-0.359896\pi\)
0.426074 + 0.904688i \(0.359896\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −6.74456 −1.15668
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 0.744563 0.122405 0.0612027 0.998125i \(-0.480506\pi\)
0.0612027 + 0.998125i \(0.480506\pi\)
\(38\) 6.74456 1.09411
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −6.74456 −0.994432
\(47\) −4.74456 −0.692066 −0.346033 0.938222i \(-0.612471\pi\)
−0.346033 + 0.938222i \(0.612471\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 6.74456 0.935303
\(53\) 12.7446 1.75060 0.875300 0.483580i \(-0.160664\pi\)
0.875300 + 0.483580i \(0.160664\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 8.74456 1.14822
\(59\) 8.74456 1.13845 0.569223 0.822183i \(-0.307244\pi\)
0.569223 + 0.822183i \(0.307244\pi\)
\(60\) 0 0
\(61\) 1.25544 0.160742 0.0803711 0.996765i \(-0.474389\pi\)
0.0803711 + 0.996765i \(0.474389\pi\)
\(62\) −4.74456 −0.602560
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.74456 −0.836560
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.74456 0.817898
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 10.7446 1.25756 0.628778 0.777585i \(-0.283555\pi\)
0.628778 + 0.777585i \(0.283555\pi\)
\(74\) −0.744563 −0.0865536
\(75\) 0 0
\(76\) −6.74456 −0.773654
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 6.74456 0.758823 0.379411 0.925228i \(-0.376127\pi\)
0.379411 + 0.925228i \(0.376127\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −4.00000 −0.441726
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) −6.74456 −0.731551
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −15.4891 −1.64184 −0.820922 0.571040i \(-0.806540\pi\)
−0.820922 + 0.571040i \(0.806540\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) 6.74456 0.703169
\(93\) 0 0
\(94\) 4.74456 0.489364
\(95\) 6.74456 0.691978
\(96\) 0 0
\(97\) −16.7446 −1.70015 −0.850076 0.526659i \(-0.823444\pi\)
−0.850076 + 0.526659i \(0.823444\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −2.74456 −0.273094 −0.136547 0.990634i \(-0.543601\pi\)
−0.136547 + 0.990634i \(0.543601\pi\)
\(102\) 0 0
\(103\) −0.744563 −0.0733639 −0.0366820 0.999327i \(-0.511679\pi\)
−0.0366820 + 0.999327i \(0.511679\pi\)
\(104\) −6.74456 −0.661359
\(105\) 0 0
\(106\) −12.7446 −1.23786
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −18.2337 −1.74647 −0.873235 0.487299i \(-0.837982\pi\)
−0.873235 + 0.487299i \(0.837982\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) −6.74456 −0.628934
\(116\) −8.74456 −0.811912
\(117\) 0 0
\(118\) −8.74456 −0.805002
\(119\) 6.74456 0.618273
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.25544 −0.113662
\(123\) 0 0
\(124\) 4.74456 0.426074
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 6.74456 0.591537
\(131\) 2.74456 0.239794 0.119897 0.992786i \(-0.461744\pi\)
0.119897 + 0.992786i \(0.461744\pi\)
\(132\) 0 0
\(133\) −6.74456 −0.584828
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.74456 −0.578341
\(137\) −3.48913 −0.298096 −0.149048 0.988830i \(-0.547621\pi\)
−0.149048 + 0.988830i \(0.547621\pi\)
\(138\) 0 0
\(139\) 14.7446 1.25062 0.625309 0.780377i \(-0.284973\pi\)
0.625309 + 0.780377i \(0.284973\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 6.74456 0.564009
\(144\) 0 0
\(145\) 8.74456 0.726196
\(146\) −10.7446 −0.889226
\(147\) 0 0
\(148\) 0.744563 0.0612027
\(149\) 0.744563 0.0609969 0.0304985 0.999535i \(-0.490291\pi\)
0.0304985 + 0.999535i \(0.490291\pi\)
\(150\) 0 0
\(151\) −9.25544 −0.753197 −0.376598 0.926377i \(-0.622906\pi\)
−0.376598 + 0.926377i \(0.622906\pi\)
\(152\) 6.74456 0.547056
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −4.74456 −0.381092
\(156\) 0 0
\(157\) 0.510875 0.0407722 0.0203861 0.999792i \(-0.493510\pi\)
0.0203861 + 0.999792i \(0.493510\pi\)
\(158\) −6.74456 −0.536569
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 6.74456 0.531546
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 32.4891 2.49916
\(170\) 6.74456 0.517284
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −20.2337 −1.53834 −0.769169 0.639045i \(-0.779330\pi\)
−0.769169 + 0.639045i \(0.779330\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 15.4891 1.16096
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −19.4891 −1.44862 −0.724308 0.689477i \(-0.757840\pi\)
−0.724308 + 0.689477i \(0.757840\pi\)
\(182\) −6.74456 −0.499940
\(183\) 0 0
\(184\) −6.74456 −0.497216
\(185\) −0.744563 −0.0547413
\(186\) 0 0
\(187\) 6.74456 0.493211
\(188\) −4.74456 −0.346033
\(189\) 0 0
\(190\) −6.74456 −0.489302
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 16.7446 1.20219
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 3.25544 0.230772 0.115386 0.993321i \(-0.463190\pi\)
0.115386 + 0.993321i \(0.463190\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 2.74456 0.193107
\(203\) −8.74456 −0.613748
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0.744563 0.0518761
\(207\) 0 0
\(208\) 6.74456 0.467651
\(209\) −6.74456 −0.466531
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 12.7446 0.875300
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 4.74456 0.322082
\(218\) 18.2337 1.23494
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 45.4891 3.05993
\(222\) 0 0
\(223\) 15.2554 1.02158 0.510790 0.859706i \(-0.329353\pi\)
0.510790 + 0.859706i \(0.329353\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 6.74456 0.444723
\(231\) 0 0
\(232\) 8.74456 0.574109
\(233\) 24.9783 1.63638 0.818190 0.574948i \(-0.194978\pi\)
0.818190 + 0.574948i \(0.194978\pi\)
\(234\) 0 0
\(235\) 4.74456 0.309501
\(236\) 8.74456 0.569223
\(237\) 0 0
\(238\) −6.74456 −0.437185
\(239\) 14.7446 0.953746 0.476873 0.878972i \(-0.341770\pi\)
0.476873 + 0.878972i \(0.341770\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 1.25544 0.0803711
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −45.4891 −2.89440
\(248\) −4.74456 −0.301280
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 26.2337 1.65586 0.827928 0.560835i \(-0.189520\pi\)
0.827928 + 0.560835i \(0.189520\pi\)
\(252\) 0 0
\(253\) 6.74456 0.424027
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.2337 0.638360 0.319180 0.947694i \(-0.396593\pi\)
0.319180 + 0.947694i \(0.396593\pi\)
\(258\) 0 0
\(259\) 0.744563 0.0462649
\(260\) −6.74456 −0.418280
\(261\) 0 0
\(262\) −2.74456 −0.169560
\(263\) −26.9783 −1.66355 −0.831775 0.555113i \(-0.812675\pi\)
−0.831775 + 0.555113i \(0.812675\pi\)
\(264\) 0 0
\(265\) −12.7446 −0.782892
\(266\) 6.74456 0.413536
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −20.9783 −1.27907 −0.639533 0.768763i \(-0.720872\pi\)
−0.639533 + 0.768763i \(0.720872\pi\)
\(270\) 0 0
\(271\) 14.9783 0.909864 0.454932 0.890526i \(-0.349664\pi\)
0.454932 + 0.890526i \(0.349664\pi\)
\(272\) 6.74456 0.408949
\(273\) 0 0
\(274\) 3.48913 0.210786
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 15.4891 0.930651 0.465326 0.885140i \(-0.345937\pi\)
0.465326 + 0.885140i \(0.345937\pi\)
\(278\) −14.7446 −0.884320
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) −6.74456 −0.398814
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 28.4891 1.67583
\(290\) −8.74456 −0.513498
\(291\) 0 0
\(292\) 10.7446 0.628778
\(293\) −13.2554 −0.774391 −0.387195 0.921998i \(-0.626556\pi\)
−0.387195 + 0.921998i \(0.626556\pi\)
\(294\) 0 0
\(295\) −8.74456 −0.509128
\(296\) −0.744563 −0.0432768
\(297\) 0 0
\(298\) −0.744563 −0.0431314
\(299\) 45.4891 2.63070
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 9.25544 0.532591
\(303\) 0 0
\(304\) −6.74456 −0.386827
\(305\) −1.25544 −0.0718861
\(306\) 0 0
\(307\) −1.48913 −0.0849889 −0.0424944 0.999097i \(-0.513530\pi\)
−0.0424944 + 0.999097i \(0.513530\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 4.74456 0.269473
\(311\) 20.7446 1.17632 0.588158 0.808746i \(-0.299853\pi\)
0.588158 + 0.808746i \(0.299853\pi\)
\(312\) 0 0
\(313\) −2.23369 −0.126256 −0.0631278 0.998005i \(-0.520108\pi\)
−0.0631278 + 0.998005i \(0.520108\pi\)
\(314\) −0.510875 −0.0288303
\(315\) 0 0
\(316\) 6.74456 0.379411
\(317\) −2.23369 −0.125456 −0.0627282 0.998031i \(-0.519980\pi\)
−0.0627282 + 0.998031i \(0.519980\pi\)
\(318\) 0 0
\(319\) −8.74456 −0.489602
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −6.74456 −0.375860
\(323\) −45.4891 −2.53108
\(324\) 0 0
\(325\) 6.74456 0.374121
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −4.00000 −0.220863
\(329\) −4.74456 −0.261576
\(330\) 0 0
\(331\) 14.9783 0.823279 0.411640 0.911347i \(-0.364956\pi\)
0.411640 + 0.911347i \(0.364956\pi\)
\(332\) −8.00000 −0.439057
\(333\) 0 0
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −32.4891 −1.76718
\(339\) 0 0
\(340\) −6.74456 −0.365775
\(341\) 4.74456 0.256932
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 20.2337 1.08777
\(347\) 22.9783 1.23354 0.616769 0.787145i \(-0.288441\pi\)
0.616769 + 0.787145i \(0.288441\pi\)
\(348\) 0 0
\(349\) 30.7446 1.64572 0.822859 0.568245i \(-0.192377\pi\)
0.822859 + 0.568245i \(0.192377\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −8.74456 −0.465426 −0.232713 0.972545i \(-0.574760\pi\)
−0.232713 + 0.972545i \(0.574760\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) −15.4891 −0.820922
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 1.25544 0.0662594 0.0331297 0.999451i \(-0.489453\pi\)
0.0331297 + 0.999451i \(0.489453\pi\)
\(360\) 0 0
\(361\) 26.4891 1.39416
\(362\) 19.4891 1.02433
\(363\) 0 0
\(364\) 6.74456 0.353511
\(365\) −10.7446 −0.562396
\(366\) 0 0
\(367\) −8.74456 −0.456462 −0.228231 0.973607i \(-0.573294\pi\)
−0.228231 + 0.973607i \(0.573294\pi\)
\(368\) 6.74456 0.351585
\(369\) 0 0
\(370\) 0.744563 0.0387080
\(371\) 12.7446 0.661665
\(372\) 0 0
\(373\) −16.9783 −0.879100 −0.439550 0.898218i \(-0.644862\pi\)
−0.439550 + 0.898218i \(0.644862\pi\)
\(374\) −6.74456 −0.348753
\(375\) 0 0
\(376\) 4.74456 0.244682
\(377\) −58.9783 −3.03753
\(378\) 0 0
\(379\) −37.4891 −1.92569 −0.962844 0.270060i \(-0.912956\pi\)
−0.962844 + 0.270060i \(0.912956\pi\)
\(380\) 6.74456 0.345989
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −19.7228 −1.00779 −0.503894 0.863765i \(-0.668100\pi\)
−0.503894 + 0.863765i \(0.668100\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) −16.7446 −0.850076
\(389\) 7.48913 0.379714 0.189857 0.981812i \(-0.439198\pi\)
0.189857 + 0.981812i \(0.439198\pi\)
\(390\) 0 0
\(391\) 45.4891 2.30048
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 10.0000 0.503793
\(395\) −6.74456 −0.339356
\(396\) 0 0
\(397\) 35.4891 1.78115 0.890574 0.454838i \(-0.150303\pi\)
0.890574 + 0.454838i \(0.150303\pi\)
\(398\) −3.25544 −0.163180
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −23.4891 −1.17299 −0.586495 0.809953i \(-0.699493\pi\)
−0.586495 + 0.809953i \(0.699493\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) −2.74456 −0.136547
\(405\) 0 0
\(406\) 8.74456 0.433985
\(407\) 0.744563 0.0369066
\(408\) 0 0
\(409\) −21.4891 −1.06257 −0.531284 0.847194i \(-0.678290\pi\)
−0.531284 + 0.847194i \(0.678290\pi\)
\(410\) 4.00000 0.197546
\(411\) 0 0
\(412\) −0.744563 −0.0366820
\(413\) 8.74456 0.430292
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) −6.74456 −0.330679
\(417\) 0 0
\(418\) 6.74456 0.329887
\(419\) 0.744563 0.0363743 0.0181871 0.999835i \(-0.494211\pi\)
0.0181871 + 0.999835i \(0.494211\pi\)
\(420\) 0 0
\(421\) 4.51087 0.219847 0.109923 0.993940i \(-0.464939\pi\)
0.109923 + 0.993940i \(0.464939\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) −12.7446 −0.618931
\(425\) 6.74456 0.327159
\(426\) 0 0
\(427\) 1.25544 0.0607549
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 17.2554 0.831165 0.415583 0.909555i \(-0.363578\pi\)
0.415583 + 0.909555i \(0.363578\pi\)
\(432\) 0 0
\(433\) 36.7446 1.76583 0.882915 0.469532i \(-0.155577\pi\)
0.882915 + 0.469532i \(0.155577\pi\)
\(434\) −4.74456 −0.227746
\(435\) 0 0
\(436\) −18.2337 −0.873235
\(437\) −45.4891 −2.17604
\(438\) 0 0
\(439\) 14.9783 0.714873 0.357436 0.933937i \(-0.383651\pi\)
0.357436 + 0.933937i \(0.383651\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −45.4891 −2.16370
\(443\) 29.4891 1.40107 0.700535 0.713618i \(-0.252945\pi\)
0.700535 + 0.713618i \(0.252945\pi\)
\(444\) 0 0
\(445\) 15.4891 0.734255
\(446\) −15.2554 −0.722366
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 4.97825 0.234938 0.117469 0.993077i \(-0.462522\pi\)
0.117469 + 0.993077i \(0.462522\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) −6.74456 −0.316190
\(456\) 0 0
\(457\) 3.48913 0.163214 0.0816072 0.996665i \(-0.473995\pi\)
0.0816072 + 0.996665i \(0.473995\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) −6.74456 −0.314467
\(461\) −33.7228 −1.57063 −0.785314 0.619098i \(-0.787498\pi\)
−0.785314 + 0.619098i \(0.787498\pi\)
\(462\) 0 0
\(463\) 1.25544 0.0583451 0.0291726 0.999574i \(-0.490713\pi\)
0.0291726 + 0.999574i \(0.490713\pi\)
\(464\) −8.74456 −0.405956
\(465\) 0 0
\(466\) −24.9783 −1.15710
\(467\) −16.9783 −0.785660 −0.392830 0.919611i \(-0.628504\pi\)
−0.392830 + 0.919611i \(0.628504\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −4.74456 −0.218850
\(471\) 0 0
\(472\) −8.74456 −0.402501
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −6.74456 −0.309462
\(476\) 6.74456 0.309137
\(477\) 0 0
\(478\) −14.7446 −0.674401
\(479\) 41.4891 1.89569 0.947843 0.318737i \(-0.103259\pi\)
0.947843 + 0.318737i \(0.103259\pi\)
\(480\) 0 0
\(481\) 5.02175 0.228972
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 16.7446 0.760331
\(486\) 0 0
\(487\) 9.25544 0.419404 0.209702 0.977765i \(-0.432751\pi\)
0.209702 + 0.977765i \(0.432751\pi\)
\(488\) −1.25544 −0.0568310
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) −30.9783 −1.39803 −0.699014 0.715108i \(-0.746378\pi\)
−0.699014 + 0.715108i \(0.746378\pi\)
\(492\) 0 0
\(493\) −58.9783 −2.65625
\(494\) 45.4891 2.04665
\(495\) 0 0
\(496\) 4.74456 0.213037
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) −13.4891 −0.603856 −0.301928 0.953331i \(-0.597630\pi\)
−0.301928 + 0.953331i \(0.597630\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −26.2337 −1.17087
\(503\) −6.51087 −0.290306 −0.145153 0.989409i \(-0.546367\pi\)
−0.145153 + 0.989409i \(0.546367\pi\)
\(504\) 0 0
\(505\) 2.74456 0.122131
\(506\) −6.74456 −0.299832
\(507\) 0 0
\(508\) 0 0
\(509\) 35.4891 1.57303 0.786514 0.617573i \(-0.211884\pi\)
0.786514 + 0.617573i \(0.211884\pi\)
\(510\) 0 0
\(511\) 10.7446 0.475311
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −10.2337 −0.451389
\(515\) 0.744563 0.0328094
\(516\) 0 0
\(517\) −4.74456 −0.208666
\(518\) −0.744563 −0.0327142
\(519\) 0 0
\(520\) 6.74456 0.295769
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 0 0
\(523\) −17.4891 −0.764746 −0.382373 0.924008i \(-0.624893\pi\)
−0.382373 + 0.924008i \(0.624893\pi\)
\(524\) 2.74456 0.119897
\(525\) 0 0
\(526\) 26.9783 1.17631
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) 22.4891 0.977788
\(530\) 12.7446 0.553588
\(531\) 0 0
\(532\) −6.74456 −0.292414
\(533\) 26.9783 1.16856
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 20.9783 0.904437
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 1.76631 0.0759397 0.0379698 0.999279i \(-0.487911\pi\)
0.0379698 + 0.999279i \(0.487911\pi\)
\(542\) −14.9783 −0.643371
\(543\) 0 0
\(544\) −6.74456 −0.289171
\(545\) 18.2337 0.781045
\(546\) 0 0
\(547\) 14.9783 0.640424 0.320212 0.947346i \(-0.396246\pi\)
0.320212 + 0.947346i \(0.396246\pi\)
\(548\) −3.48913 −0.149048
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 58.9783 2.51256
\(552\) 0 0
\(553\) 6.74456 0.286808
\(554\) −15.4891 −0.658070
\(555\) 0 0
\(556\) 14.7446 0.625309
\(557\) 0.978251 0.0414498 0.0207249 0.999785i \(-0.493403\pi\)
0.0207249 + 0.999785i \(0.493403\pi\)
\(558\) 0 0
\(559\) 26.9783 1.14106
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 14.0000 0.590554
\(563\) −5.48913 −0.231339 −0.115670 0.993288i \(-0.536901\pi\)
−0.115670 + 0.993288i \(0.536901\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −16.5109 −0.692172 −0.346086 0.938203i \(-0.612489\pi\)
−0.346086 + 0.938203i \(0.612489\pi\)
\(570\) 0 0
\(571\) −17.4891 −0.731897 −0.365949 0.930635i \(-0.619255\pi\)
−0.365949 + 0.930635i \(0.619255\pi\)
\(572\) 6.74456 0.282004
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 6.74456 0.281268
\(576\) 0 0
\(577\) −8.74456 −0.364041 −0.182020 0.983295i \(-0.558264\pi\)
−0.182020 + 0.983295i \(0.558264\pi\)
\(578\) −28.4891 −1.18499
\(579\) 0 0
\(580\) 8.74456 0.363098
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 12.7446 0.527826
\(584\) −10.7446 −0.444613
\(585\) 0 0
\(586\) 13.2554 0.547577
\(587\) 4.97825 0.205474 0.102737 0.994709i \(-0.467240\pi\)
0.102737 + 0.994709i \(0.467240\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 8.74456 0.360008
\(591\) 0 0
\(592\) 0.744563 0.0306013
\(593\) 40.2337 1.65220 0.826100 0.563524i \(-0.190555\pi\)
0.826100 + 0.563524i \(0.190555\pi\)
\(594\) 0 0
\(595\) −6.74456 −0.276500
\(596\) 0.744563 0.0304985
\(597\) 0 0
\(598\) −45.4891 −1.86019
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −33.4891 −1.36605 −0.683025 0.730395i \(-0.739336\pi\)
−0.683025 + 0.730395i \(0.739336\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) −9.25544 −0.376598
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 6.74456 0.273528
\(609\) 0 0
\(610\) 1.25544 0.0508312
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) 11.4891 0.464041 0.232021 0.972711i \(-0.425466\pi\)
0.232021 + 0.972711i \(0.425466\pi\)
\(614\) 1.48913 0.0600962
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) 0 0
\(619\) 0.744563 0.0299265 0.0149632 0.999888i \(-0.495237\pi\)
0.0149632 + 0.999888i \(0.495237\pi\)
\(620\) −4.74456 −0.190546
\(621\) 0 0
\(622\) −20.7446 −0.831781
\(623\) −15.4891 −0.620559
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.23369 0.0892761
\(627\) 0 0
\(628\) 0.510875 0.0203861
\(629\) 5.02175 0.200230
\(630\) 0 0
\(631\) −2.97825 −0.118562 −0.0592811 0.998241i \(-0.518881\pi\)
−0.0592811 + 0.998241i \(0.518881\pi\)
\(632\) −6.74456 −0.268284
\(633\) 0 0
\(634\) 2.23369 0.0887111
\(635\) 0 0
\(636\) 0 0
\(637\) 6.74456 0.267229
\(638\) 8.74456 0.346201
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −11.4891 −0.453793 −0.226897 0.973919i \(-0.572858\pi\)
−0.226897 + 0.973919i \(0.572858\pi\)
\(642\) 0 0
\(643\) 46.4674 1.83249 0.916247 0.400613i \(-0.131203\pi\)
0.916247 + 0.400613i \(0.131203\pi\)
\(644\) 6.74456 0.265773
\(645\) 0 0
\(646\) 45.4891 1.78975
\(647\) −8.74456 −0.343784 −0.171892 0.985116i \(-0.554988\pi\)
−0.171892 + 0.985116i \(0.554988\pi\)
\(648\) 0 0
\(649\) 8.74456 0.343254
\(650\) −6.74456 −0.264544
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −15.7228 −0.615281 −0.307641 0.951503i \(-0.599539\pi\)
−0.307641 + 0.951503i \(0.599539\pi\)
\(654\) 0 0
\(655\) −2.74456 −0.107239
\(656\) 4.00000 0.156174
\(657\) 0 0
\(658\) 4.74456 0.184962
\(659\) 41.4891 1.61619 0.808093 0.589054i \(-0.200500\pi\)
0.808093 + 0.589054i \(0.200500\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −14.9783 −0.582146
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) 6.74456 0.261543
\(666\) 0 0
\(667\) −58.9783 −2.28365
\(668\) 0 0
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) 1.25544 0.0484656
\(672\) 0 0
\(673\) 20.9783 0.808652 0.404326 0.914615i \(-0.367506\pi\)
0.404326 + 0.914615i \(0.367506\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) 32.4891 1.24958
\(677\) −36.2337 −1.39257 −0.696287 0.717764i \(-0.745166\pi\)
−0.696287 + 0.717764i \(0.745166\pi\)
\(678\) 0 0
\(679\) −16.7446 −0.642597
\(680\) 6.74456 0.258642
\(681\) 0 0
\(682\) −4.74456 −0.181679
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 3.48913 0.133313
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 85.9565 3.27468
\(690\) 0 0
\(691\) 1.76631 0.0671937 0.0335968 0.999435i \(-0.489304\pi\)
0.0335968 + 0.999435i \(0.489304\pi\)
\(692\) −20.2337 −0.769169
\(693\) 0 0
\(694\) −22.9783 −0.872242
\(695\) −14.7446 −0.559293
\(696\) 0 0
\(697\) 26.9783 1.02187
\(698\) −30.7446 −1.16370
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −22.2337 −0.839755 −0.419877 0.907581i \(-0.637927\pi\)
−0.419877 + 0.907581i \(0.637927\pi\)
\(702\) 0 0
\(703\) −5.02175 −0.189399
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 8.74456 0.329106
\(707\) −2.74456 −0.103220
\(708\) 0 0
\(709\) 0.510875 0.0191863 0.00959315 0.999954i \(-0.496946\pi\)
0.00959315 + 0.999954i \(0.496946\pi\)
\(710\) 4.00000 0.150117
\(711\) 0 0
\(712\) 15.4891 0.580480
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) −6.74456 −0.252232
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) −1.25544 −0.0468525
\(719\) −7.72281 −0.288012 −0.144006 0.989577i \(-0.545999\pi\)
−0.144006 + 0.989577i \(0.545999\pi\)
\(720\) 0 0
\(721\) −0.744563 −0.0277290
\(722\) −26.4891 −0.985823
\(723\) 0 0
\(724\) −19.4891 −0.724308
\(725\) −8.74456 −0.324765
\(726\) 0 0
\(727\) 14.2337 0.527898 0.263949 0.964537i \(-0.414975\pi\)
0.263949 + 0.964537i \(0.414975\pi\)
\(728\) −6.74456 −0.249970
\(729\) 0 0
\(730\) 10.7446 0.397674
\(731\) 26.9783 0.997827
\(732\) 0 0
\(733\) −16.2337 −0.599605 −0.299802 0.954001i \(-0.596921\pi\)
−0.299802 + 0.954001i \(0.596921\pi\)
\(734\) 8.74456 0.322768
\(735\) 0 0
\(736\) −6.74456 −0.248608
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −30.9783 −1.13955 −0.569777 0.821800i \(-0.692970\pi\)
−0.569777 + 0.821800i \(0.692970\pi\)
\(740\) −0.744563 −0.0273707
\(741\) 0 0
\(742\) −12.7446 −0.467868
\(743\) −26.9783 −0.989736 −0.494868 0.868968i \(-0.664784\pi\)
−0.494868 + 0.868968i \(0.664784\pi\)
\(744\) 0 0
\(745\) −0.744563 −0.0272787
\(746\) 16.9783 0.621618
\(747\) 0 0
\(748\) 6.74456 0.246606
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −4.74456 −0.173016
\(753\) 0 0
\(754\) 58.9783 2.14786
\(755\) 9.25544 0.336840
\(756\) 0 0
\(757\) −19.2554 −0.699851 −0.349925 0.936778i \(-0.613793\pi\)
−0.349925 + 0.936778i \(0.613793\pi\)
\(758\) 37.4891 1.36167
\(759\) 0 0
\(760\) −6.74456 −0.244651
\(761\) 29.4891 1.06898 0.534490 0.845175i \(-0.320504\pi\)
0.534490 + 0.845175i \(0.320504\pi\)
\(762\) 0 0
\(763\) −18.2337 −0.660104
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) 19.7228 0.712614
\(767\) 58.9783 2.12958
\(768\) 0 0
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 51.4891 1.85194 0.925968 0.377603i \(-0.123252\pi\)
0.925968 + 0.377603i \(0.123252\pi\)
\(774\) 0 0
\(775\) 4.74456 0.170430
\(776\) 16.7446 0.601095
\(777\) 0 0
\(778\) −7.48913 −0.268498
\(779\) −26.9783 −0.966596
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) −45.4891 −1.62669
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −0.510875 −0.0182339
\(786\) 0 0
\(787\) 5.48913 0.195666 0.0978331 0.995203i \(-0.468809\pi\)
0.0978331 + 0.995203i \(0.468809\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) 6.74456 0.239961
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 8.46738 0.300685
\(794\) −35.4891 −1.25946
\(795\) 0 0
\(796\) 3.25544 0.115386
\(797\) 42.4674 1.50427 0.752136 0.659008i \(-0.229024\pi\)
0.752136 + 0.659008i \(0.229024\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 23.4891 0.829430
\(803\) 10.7446 0.379167
\(804\) 0 0
\(805\) −6.74456 −0.237715
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) 2.74456 0.0965534
\(809\) −34.4674 −1.21181 −0.605904 0.795538i \(-0.707189\pi\)
−0.605904 + 0.795538i \(0.707189\pi\)
\(810\) 0 0
\(811\) −18.7446 −0.658211 −0.329105 0.944293i \(-0.606747\pi\)
−0.329105 + 0.944293i \(0.606747\pi\)
\(812\) −8.74456 −0.306874
\(813\) 0 0
\(814\) −0.744563 −0.0260969
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) −26.9783 −0.943850
\(818\) 21.4891 0.751350
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −7.72281 −0.269528 −0.134764 0.990878i \(-0.543028\pi\)
−0.134764 + 0.990878i \(0.543028\pi\)
\(822\) 0 0
\(823\) 7.76631 0.270717 0.135358 0.990797i \(-0.456781\pi\)
0.135358 + 0.990797i \(0.456781\pi\)
\(824\) 0.744563 0.0259381
\(825\) 0 0
\(826\) −8.74456 −0.304262
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) 14.4674 0.502473 0.251236 0.967926i \(-0.419163\pi\)
0.251236 + 0.967926i \(0.419163\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) 6.74456 0.233826
\(833\) 6.74456 0.233685
\(834\) 0 0
\(835\) 0 0
\(836\) −6.74456 −0.233266
\(837\) 0 0
\(838\) −0.744563 −0.0257205
\(839\) 3.25544 0.112390 0.0561951 0.998420i \(-0.482103\pi\)
0.0561951 + 0.998420i \(0.482103\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) −4.51087 −0.155455
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) −32.4891 −1.11766
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 12.7446 0.437650
\(849\) 0 0
\(850\) −6.74456 −0.231337
\(851\) 5.02175 0.172143
\(852\) 0 0
\(853\) −22.7446 −0.778759 −0.389379 0.921077i \(-0.627311\pi\)
−0.389379 + 0.921077i \(0.627311\pi\)
\(854\) −1.25544 −0.0429602
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 43.2119 1.47609 0.738046 0.674751i \(-0.235749\pi\)
0.738046 + 0.674751i \(0.235749\pi\)
\(858\) 0 0
\(859\) −29.2119 −0.996698 −0.498349 0.866976i \(-0.666060\pi\)
−0.498349 + 0.866976i \(0.666060\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −17.2554 −0.587723
\(863\) 40.2337 1.36957 0.684785 0.728745i \(-0.259896\pi\)
0.684785 + 0.728745i \(0.259896\pi\)
\(864\) 0 0
\(865\) 20.2337 0.687966
\(866\) −36.7446 −1.24863
\(867\) 0 0
\(868\) 4.74456 0.161041
\(869\) 6.74456 0.228794
\(870\) 0 0
\(871\) −26.9783 −0.914123
\(872\) 18.2337 0.617471
\(873\) 0 0
\(874\) 45.4891 1.53869
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −26.4674 −0.893740 −0.446870 0.894599i \(-0.647461\pi\)
−0.446870 + 0.894599i \(0.647461\pi\)
\(878\) −14.9783 −0.505491
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) 55.4891 1.86948 0.934738 0.355338i \(-0.115634\pi\)
0.934738 + 0.355338i \(0.115634\pi\)
\(882\) 0 0
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 45.4891 1.52996
\(885\) 0 0
\(886\) −29.4891 −0.990707
\(887\) 41.4891 1.39307 0.696534 0.717524i \(-0.254724\pi\)
0.696534 + 0.717524i \(0.254724\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −15.4891 −0.519197
\(891\) 0 0
\(892\) 15.2554 0.510790
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −4.97825 −0.166126
\(899\) −41.4891 −1.38374
\(900\) 0 0
\(901\) 85.9565 2.86363
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) 19.4891 0.647840
\(906\) 0 0
\(907\) −14.5109 −0.481826 −0.240913 0.970547i \(-0.577447\pi\)
−0.240913 + 0.970547i \(0.577447\pi\)
\(908\) −20.0000 −0.663723
\(909\) 0 0
\(910\) 6.74456 0.223580
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) −3.48913 −0.115410
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 2.74456 0.0906334
\(918\) 0 0
\(919\) −55.2119 −1.82127 −0.910637 0.413207i \(-0.864408\pi\)
−0.910637 + 0.413207i \(0.864408\pi\)
\(920\) 6.74456 0.222362
\(921\) 0 0
\(922\) 33.7228 1.11060
\(923\) 26.9783 0.888000
\(924\) 0 0
\(925\) 0.744563 0.0244811
\(926\) −1.25544 −0.0412562
\(927\) 0 0
\(928\) 8.74456 0.287054
\(929\) 28.9783 0.950746 0.475373 0.879784i \(-0.342313\pi\)
0.475373 + 0.879784i \(0.342313\pi\)
\(930\) 0 0
\(931\) −6.74456 −0.221044
\(932\) 24.9783 0.818190
\(933\) 0 0
\(934\) 16.9783 0.555545
\(935\) −6.74456 −0.220571
\(936\) 0 0
\(937\) −18.7446 −0.612358 −0.306179 0.951974i \(-0.599051\pi\)
−0.306179 + 0.951974i \(0.599051\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) 4.74456 0.154751
\(941\) −12.2337 −0.398807 −0.199403 0.979917i \(-0.563900\pi\)
−0.199403 + 0.979917i \(0.563900\pi\)
\(942\) 0 0
\(943\) 26.9783 0.878533
\(944\) 8.74456 0.284611
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) 72.4674 2.35239
\(950\) 6.74456 0.218823
\(951\) 0 0
\(952\) −6.74456 −0.218593
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 14.7446 0.476873
\(957\) 0 0
\(958\) −41.4891 −1.34045
\(959\) −3.48913 −0.112670
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) −5.02175 −0.161908
\(963\) 0 0
\(964\) 20.0000 0.644157
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −50.9783 −1.63935 −0.819675 0.572829i \(-0.805846\pi\)
−0.819675 + 0.572829i \(0.805846\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −16.7446 −0.537636
\(971\) −19.7228 −0.632935 −0.316468 0.948603i \(-0.602497\pi\)
−0.316468 + 0.948603i \(0.602497\pi\)
\(972\) 0 0
\(973\) 14.7446 0.472689
\(974\) −9.25544 −0.296563
\(975\) 0 0
\(976\) 1.25544 0.0401856
\(977\) −12.9783 −0.415211 −0.207606 0.978213i \(-0.566567\pi\)
−0.207606 + 0.978213i \(0.566567\pi\)
\(978\) 0 0
\(979\) −15.4891 −0.495035
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 30.9783 0.988556
\(983\) −2.23369 −0.0712436 −0.0356218 0.999365i \(-0.511341\pi\)
−0.0356218 + 0.999365i \(0.511341\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 58.9783 1.87825
\(987\) 0 0
\(988\) −45.4891 −1.44720
\(989\) 26.9783 0.857858
\(990\) 0 0
\(991\) 1.48913 0.0473036 0.0236518 0.999720i \(-0.492471\pi\)
0.0236518 + 0.999720i \(0.492471\pi\)
\(992\) −4.74456 −0.150640
\(993\) 0 0
\(994\) −4.00000 −0.126872
\(995\) −3.25544 −0.103204
\(996\) 0 0
\(997\) 39.2119 1.24185 0.620927 0.783868i \(-0.286756\pi\)
0.620927 + 0.783868i \(0.286756\pi\)
\(998\) 13.4891 0.426991
\(999\) 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.bo.1.2 2
3.2 odd 2 770.2.a.k.1.2 2
12.11 even 2 6160.2.a.r.1.2 2
15.2 even 4 3850.2.c.y.1849.3 4
15.8 even 4 3850.2.c.y.1849.2 4
15.14 odd 2 3850.2.a.bc.1.1 2
21.20 even 2 5390.2.a.bq.1.1 2
33.32 even 2 8470.2.a.bu.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.k.1.2 2 3.2 odd 2
3850.2.a.bc.1.1 2 15.14 odd 2
3850.2.c.y.1849.2 4 15.8 even 4
3850.2.c.y.1849.3 4 15.2 even 4
5390.2.a.bq.1.1 2 21.20 even 2
6160.2.a.r.1.2 2 12.11 even 2
6930.2.a.bo.1.2 2 1.1 even 1 trivial
8470.2.a.bu.1.1 2 33.32 even 2