Properties

Label 3850.2.a.bc.1.1
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7424047782\)
Analytic rank: \(1\)
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 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 3850.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} -6.74456 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.74456 q^{17} -1.00000 q^{18} -6.74456 q^{19} +2.00000 q^{21} +1.00000 q^{22} +6.74456 q^{23} +2.00000 q^{24} +6.74456 q^{26} +4.00000 q^{27} -1.00000 q^{28} +8.74456 q^{29} +4.74456 q^{31} -1.00000 q^{32} +2.00000 q^{33} -6.74456 q^{34} +1.00000 q^{36} -0.744563 q^{37} +6.74456 q^{38} +13.4891 q^{39} -4.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} -6.74456 q^{46} -4.74456 q^{47} -2.00000 q^{48} +1.00000 q^{49} -13.4891 q^{51} -6.74456 q^{52} +12.7446 q^{53} -4.00000 q^{54} +1.00000 q^{56} +13.4891 q^{57} -8.74456 q^{58} -8.74456 q^{59} +1.25544 q^{61} -4.74456 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +4.00000 q^{67} +6.74456 q^{68} -13.4891 q^{69} -4.00000 q^{71} -1.00000 q^{72} -10.7446 q^{73} +0.744563 q^{74} -6.74456 q^{76} +1.00000 q^{77} -13.4891 q^{78} +6.74456 q^{79} -11.0000 q^{81} +4.00000 q^{82} -8.00000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -17.4891 q^{87} +1.00000 q^{88} +15.4891 q^{89} +6.74456 q^{91} +6.74456 q^{92} -9.48913 q^{93} +4.74456 q^{94} +2.00000 q^{96} +16.7446 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{3} + 2q^{4} + 4q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{3} + 2q^{4} + 4q^{6} - 2q^{7} - 2q^{8} + 2q^{9} - 2q^{11} - 4q^{12} - 2q^{13} + 2q^{14} + 2q^{16} + 2q^{17} - 2q^{18} - 2q^{19} + 4q^{21} + 2q^{22} + 2q^{23} + 4q^{24} + 2q^{26} + 8q^{27} - 2q^{28} + 6q^{29} - 2q^{31} - 2q^{32} + 4q^{33} - 2q^{34} + 2q^{36} + 10q^{37} + 2q^{38} + 4q^{39} - 8q^{41} - 4q^{42} - 8q^{43} - 2q^{44} - 2q^{46} + 2q^{47} - 4q^{48} + 2q^{49} - 4q^{51} - 2q^{52} + 14q^{53} - 8q^{54} + 2q^{56} + 4q^{57} - 6q^{58} - 6q^{59} + 14q^{61} + 2q^{62} - 2q^{63} + 2q^{64} - 4q^{66} + 8q^{67} + 2q^{68} - 4q^{69} - 8q^{71} - 2q^{72} - 10q^{73} - 10q^{74} - 2q^{76} + 2q^{77} - 4q^{78} + 2q^{79} - 22q^{81} + 8q^{82} - 16q^{83} + 4q^{84} + 8q^{86} - 12q^{87} + 2q^{88} + 8q^{89} + 2q^{91} + 2q^{92} + 4q^{93} - 2q^{94} + 4q^{96} + 22q^{97} - 2q^{98} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −2.00000 −0.577350
\(13\) −6.74456 −1.87061 −0.935303 0.353849i \(-0.884873\pi\)
−0.935303 + 0.353849i \(0.884873\pi\)
\(14\) 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 \(0.195139\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.74456 −1.54731 −0.773654 0.633608i \(-0.781573\pi\)
−0.773654 + 0.633608i \(0.781573\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 1.00000 0.213201
\(23\) 6.74456 1.40634 0.703169 0.711022i \(-0.251768\pi\)
0.703169 + 0.711022i \(0.251768\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 6.74456 1.32272
\(27\) 4.00000 0.769800
\(28\) −1.00000 −0.188982
\(29\) 8.74456 1.62382 0.811912 0.583779i \(-0.198427\pi\)
0.811912 + 0.583779i \(0.198427\pi\)
\(30\) 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
\(33\) 2.00000 0.348155
\(34\) −6.74456 −1.15668
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −0.744563 −0.122405 −0.0612027 0.998125i \(-0.519494\pi\)
−0.0612027 + 0.998125i \(0.519494\pi\)
\(38\) 6.74456 1.09411
\(39\) 13.4891 2.15999
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −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 \(0.612471\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −13.4891 −1.88886
\(52\) −6.74456 −0.935303
\(53\) 12.7446 1.75060 0.875300 0.483580i \(-0.160664\pi\)
0.875300 + 0.483580i \(0.160664\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 13.4891 1.78668
\(58\) −8.74456 −1.14822
\(59\) −8.74456 −1.13845 −0.569223 0.822183i \(-0.692756\pi\)
−0.569223 + 0.822183i \(0.692756\pi\)
\(60\) 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
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 6.74456 0.817898
\(69\) −13.4891 −1.62390
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.7446 −1.25756 −0.628778 0.777585i \(-0.716445\pi\)
−0.628778 + 0.777585i \(0.716445\pi\)
\(74\) 0.744563 0.0865536
\(75\) 0 0
\(76\) −6.74456 −0.773654
\(77\) 1.00000 0.113961
\(78\) −13.4891 −1.52734
\(79\) 6.74456 0.758823 0.379411 0.925228i \(-0.376127\pi\)
0.379411 + 0.925228i \(0.376127\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 4.00000 0.441726
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −17.4891 −1.87503
\(88\) 1.00000 0.106600
\(89\) 15.4891 1.64184 0.820922 0.571040i \(-0.193460\pi\)
0.820922 + 0.571040i \(0.193460\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) 6.74456 0.703169
\(93\) −9.48913 −0.983976
\(94\) 4.74456 0.489364
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 16.7446 1.70015 0.850076 0.526659i \(-0.176556\pi\)
0.850076 + 0.526659i \(0.176556\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 2.74456 0.273094 0.136547 0.990634i \(-0.456399\pi\)
0.136547 + 0.990634i \(0.456399\pi\)
\(102\) 13.4891 1.33562
\(103\) 0.744563 0.0733639 0.0366820 0.999327i \(-0.488321\pi\)
0.0366820 + 0.999327i \(0.488321\pi\)
\(104\) 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 \(0.303036\pi\)
\(108\) 4.00000 0.384900
\(109\) −18.2337 −1.74647 −0.873235 0.487299i \(-0.837982\pi\)
−0.873235 + 0.487299i \(0.837982\pi\)
\(110\) 0 0
\(111\) 1.48913 0.141342
\(112\) −1.00000 −0.0944911
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) −13.4891 −1.26337
\(115\) 0 0
\(116\) 8.74456 0.811912
\(117\) −6.74456 −0.623535
\(118\) 8.74456 0.805002
\(119\) −6.74456 −0.618273
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.25544 −0.113662
\(123\) 8.00000 0.721336
\(124\) 4.74456 0.426074
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −2.74456 −0.239794 −0.119897 0.992786i \(-0.538256\pi\)
−0.119897 + 0.992786i \(0.538256\pi\)
\(132\) 2.00000 0.174078
\(133\) 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 \(0.547621\pi\)
\(138\) 13.4891 1.14827
\(139\) 14.7446 1.25062 0.625309 0.780377i \(-0.284973\pi\)
0.625309 + 0.780377i \(0.284973\pi\)
\(140\) 0 0
\(141\) 9.48913 0.799129
\(142\) 4.00000 0.335673
\(143\) 6.74456 0.564009
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 10.7446 0.889226
\(147\) −2.00000 −0.164957
\(148\) −0.744563 −0.0612027
\(149\) −0.744563 −0.0609969 −0.0304985 0.999535i \(-0.509709\pi\)
−0.0304985 + 0.999535i \(0.509709\pi\)
\(150\) 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
\(153\) 6.74456 0.545266
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 13.4891 1.07999
\(157\) −0.510875 −0.0407722 −0.0203861 0.999792i \(-0.506490\pi\)
−0.0203861 + 0.999792i \(0.506490\pi\)
\(158\) −6.74456 −0.536569
\(159\) −25.4891 −2.02142
\(160\) 0 0
\(161\) −6.74456 −0.531546
\(162\) 11.0000 0.864242
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) 32.4891 2.49916
\(170\) 0 0
\(171\) −6.74456 −0.515770
\(172\) −4.00000 −0.304997
\(173\) −20.2337 −1.53834 −0.769169 0.639045i \(-0.779330\pi\)
−0.769169 + 0.639045i \(0.779330\pi\)
\(174\) 17.4891 1.32585
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 17.4891 1.31456
\(178\) −15.4891 −1.16096
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 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
\(183\) −2.51087 −0.185609
\(184\) −6.74456 −0.497216
\(185\) 0 0
\(186\) 9.48913 0.695776
\(187\) −6.74456 −0.493211
\(188\) −4.74456 −0.346033
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −2.00000 −0.144338
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −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 \(0.615940\pi\)
\(198\) 1.00000 0.0710669
\(199\) 3.25544 0.230772 0.115386 0.993321i \(-0.463190\pi\)
0.115386 + 0.993321i \(0.463190\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) −2.74456 −0.193107
\(203\) −8.74456 −0.613748
\(204\) −13.4891 −0.944428
\(205\) 0 0
\(206\) −0.744563 −0.0518761
\(207\) 6.74456 0.468780
\(208\) −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
\(213\) 8.00000 0.548151
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) −4.74456 −0.322082
\(218\) 18.2337 1.23494
\(219\) 21.4891 1.45210
\(220\) 0 0
\(221\) −45.4891 −3.05993
\(222\) −1.48913 −0.0999435
\(223\) −15.2554 −1.02158 −0.510790 0.859706i \(-0.670647\pi\)
−0.510790 + 0.859706i \(0.670647\pi\)
\(224\) 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 \(0.731025\pi\)
\(228\) 13.4891 0.893339
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) −8.74456 −0.574109
\(233\) 24.9783 1.63638 0.818190 0.574948i \(-0.194978\pi\)
0.818190 + 0.574948i \(0.194978\pi\)
\(234\) 6.74456 0.440906
\(235\) 0 0
\(236\) −8.74456 −0.569223
\(237\) −13.4891 −0.876213
\(238\) 6.74456 0.437185
\(239\) −14.7446 −0.953746 −0.476873 0.878972i \(-0.658230\pi\)
−0.476873 + 0.878972i \(0.658230\pi\)
\(240\) 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
\(243\) 10.0000 0.641500
\(244\) 1.25544 0.0803711
\(245\) 0 0
\(246\) −8.00000 −0.510061
\(247\) 45.4891 2.89440
\(248\) −4.74456 −0.301280
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) −26.2337 −1.65586 −0.827928 0.560835i \(-0.810480\pi\)
−0.827928 + 0.560835i \(0.810480\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −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 \(0.396593\pi\)
\(258\) −8.00000 −0.498058
\(259\) 0.744563 0.0462649
\(260\) 0 0
\(261\) 8.74456 0.541275
\(262\) 2.74456 0.169560
\(263\) −26.9783 −1.66355 −0.831775 0.555113i \(-0.812675\pi\)
−0.831775 + 0.555113i \(0.812675\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) −6.74456 −0.413536
\(267\) −30.9783 −1.89584
\(268\) 4.00000 0.244339
\(269\) 20.9783 1.27907 0.639533 0.768763i \(-0.279128\pi\)
0.639533 + 0.768763i \(0.279128\pi\)
\(270\) 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
\(273\) −13.4891 −0.816399
\(274\) 3.48913 0.210786
\(275\) 0 0
\(276\) −13.4891 −0.811950
\(277\) −15.4891 −0.930651 −0.465326 0.885140i \(-0.654063\pi\)
−0.465326 + 0.885140i \(0.654063\pi\)
\(278\) −14.7446 −0.884320
\(279\) 4.74456 0.284050
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) −9.48913 −0.565069
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) −6.74456 −0.398814
\(287\) 4.00000 0.236113
\(288\) −1.00000 −0.0589256
\(289\) 28.4891 1.67583
\(290\) 0 0
\(291\) −33.4891 −1.96317
\(292\) −10.7446 −0.628778
\(293\) −13.2554 −0.774391 −0.387195 0.921998i \(-0.626556\pi\)
−0.387195 + 0.921998i \(0.626556\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 0.744563 0.0432768
\(297\) −4.00000 −0.232104
\(298\) 0.744563 0.0431314
\(299\) −45.4891 −2.63070
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 9.25544 0.532591
\(303\) −5.48913 −0.315342
\(304\) −6.74456 −0.386827
\(305\) 0 0
\(306\) −6.74456 −0.385561
\(307\) 1.48913 0.0849889 0.0424944 0.999097i \(-0.486470\pi\)
0.0424944 + 0.999097i \(0.486470\pi\)
\(308\) 1.00000 0.0569803
\(309\) −1.48913 −0.0847134
\(310\) 0 0
\(311\) −20.7446 −1.17632 −0.588158 0.808746i \(-0.700147\pi\)
−0.588158 + 0.808746i \(0.700147\pi\)
\(312\) −13.4891 −0.763671
\(313\) 2.23369 0.126256 0.0631278 0.998005i \(-0.479892\pi\)
0.0631278 + 0.998005i \(0.479892\pi\)
\(314\) 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 \(0.519980\pi\)
\(318\) 25.4891 1.42936
\(319\) −8.74456 −0.489602
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 6.74456 0.375860
\(323\) −45.4891 −2.53108
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 36.4674 2.01665
\(328\) 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
\(333\) −0.744563 −0.0408018
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) −32.4891 −1.76718
\(339\) 20.0000 1.08625
\(340\) 0 0
\(341\) −4.74456 −0.256932
\(342\) 6.74456 0.364704
\(343\) −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 \(0.288441\pi\)
\(348\) −17.4891 −0.937516
\(349\) 30.7446 1.64572 0.822859 0.568245i \(-0.192377\pi\)
0.822859 + 0.568245i \(0.192377\pi\)
\(350\) 0 0
\(351\) −26.9783 −1.43999
\(352\) 1.00000 0.0533002
\(353\) −8.74456 −0.465426 −0.232713 0.972545i \(-0.574760\pi\)
−0.232713 + 0.972545i \(0.574760\pi\)
\(354\) −17.4891 −0.929537
\(355\) 0 0
\(356\) 15.4891 0.820922
\(357\) 13.4891 0.713920
\(358\) 4.00000 0.211407
\(359\) −1.25544 −0.0662594 −0.0331297 0.999451i \(-0.510547\pi\)
−0.0331297 + 0.999451i \(0.510547\pi\)
\(360\) 0 0
\(361\) 26.4891 1.39416
\(362\) 19.4891 1.02433
\(363\) −2.00000 −0.104973
\(364\) 6.74456 0.353511
\(365\) 0 0
\(366\) 2.51087 0.131246
\(367\) 8.74456 0.456462 0.228231 0.973607i \(-0.426706\pi\)
0.228231 + 0.973607i \(0.426706\pi\)
\(368\) 6.74456 0.351585
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) −12.7446 −0.661665
\(372\) −9.48913 −0.491988
\(373\) 16.9783 0.879100 0.439550 0.898218i \(-0.355138\pi\)
0.439550 + 0.898218i \(0.355138\pi\)
\(374\) 6.74456 0.348753
\(375\) 0 0
\(376\) 4.74456 0.244682
\(377\) −58.9783 −3.03753
\(378\) 4.00000 0.205738
\(379\) −37.4891 −1.92569 −0.962844 0.270060i \(-0.912956\pi\)
−0.962844 + 0.270060i \(0.912956\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) −19.7228 −1.00779 −0.503894 0.863765i \(-0.668100\pi\)
−0.503894 + 0.863765i \(0.668100\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) 16.7446 0.850076
\(389\) −7.48913 −0.379714 −0.189857 0.981812i \(-0.560802\pi\)
−0.189857 + 0.981812i \(0.560802\pi\)
\(390\) 0 0
\(391\) 45.4891 2.30048
\(392\) −1.00000 −0.0505076
\(393\) 5.48913 0.276890
\(394\) 10.0000 0.503793
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −35.4891 −1.78115 −0.890574 0.454838i \(-0.849697\pi\)
−0.890574 + 0.454838i \(0.849697\pi\)
\(398\) −3.25544 −0.163180
\(399\) −13.4891 −0.675301
\(400\) 0 0
\(401\) 23.4891 1.17299 0.586495 0.809953i \(-0.300507\pi\)
0.586495 + 0.809953i \(0.300507\pi\)
\(402\) 8.00000 0.399004
\(403\) −32.0000 −1.59403
\(404\) 2.74456 0.136547
\(405\) 0 0
\(406\) 8.74456 0.433985
\(407\) 0.744563 0.0369066
\(408\) 13.4891 0.667811
\(409\) −21.4891 −1.06257 −0.531284 0.847194i \(-0.678290\pi\)
−0.531284 + 0.847194i \(0.678290\pi\)
\(410\) 0 0
\(411\) 6.97825 0.344212
\(412\) 0.744563 0.0366820
\(413\) 8.74456 0.430292
\(414\) −6.74456 −0.331477
\(415\) 0 0
\(416\) 6.74456 0.330679
\(417\) −29.4891 −1.44409
\(418\) −6.74456 −0.329887
\(419\) −0.744563 −0.0363743 −0.0181871 0.999835i \(-0.505789\pi\)
−0.0181871 + 0.999835i \(0.505789\pi\)
\(420\) 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
\(423\) −4.74456 −0.230689
\(424\) −12.7446 −0.618931
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) −1.25544 −0.0607549
\(428\) 12.0000 0.580042
\(429\) −13.4891 −0.651261
\(430\) 0 0
\(431\) −17.2554 −0.831165 −0.415583 0.909555i \(-0.636422\pi\)
−0.415583 + 0.909555i \(0.636422\pi\)
\(432\) 4.00000 0.192450
\(433\) −36.7446 −1.76583 −0.882915 0.469532i \(-0.844423\pi\)
−0.882915 + 0.469532i \(0.844423\pi\)
\(434\) 4.74456 0.227746
\(435\) 0 0
\(436\) −18.2337 −0.873235
\(437\) −45.4891 −2.17604
\(438\) −21.4891 −1.02679
\(439\) 14.9783 0.714873 0.357436 0.933937i \(-0.383651\pi\)
0.357436 + 0.933937i \(0.383651\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 45.4891 2.16370
\(443\) 29.4891 1.40107 0.700535 0.713618i \(-0.252945\pi\)
0.700535 + 0.713618i \(0.252945\pi\)
\(444\) 1.48913 0.0706708
\(445\) 0 0
\(446\) 15.2554 0.722366
\(447\) 1.48913 0.0704332
\(448\) −1.00000 −0.0472456
\(449\) −4.97825 −0.234938 −0.117469 0.993077i \(-0.537478\pi\)
−0.117469 + 0.993077i \(0.537478\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) −10.0000 −0.470360
\(453\) 18.5109 0.869717
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −13.4891 −0.631686
\(457\) −3.48913 −0.163214 −0.0816072 0.996665i \(-0.526005\pi\)
−0.0816072 + 0.996665i \(0.526005\pi\)
\(458\) 6.00000 0.280362
\(459\) 26.9783 1.25924
\(460\) 0 0
\(461\) 33.7228 1.57063 0.785314 0.619098i \(-0.212502\pi\)
0.785314 + 0.619098i \(0.212502\pi\)
\(462\) 2.00000 0.0930484
\(463\) −1.25544 −0.0583451 −0.0291726 0.999574i \(-0.509287\pi\)
−0.0291726 + 0.999574i \(0.509287\pi\)
\(464\) 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 \(0.628504\pi\)
\(468\) −6.74456 −0.311768
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 1.02175 0.0470797
\(472\) 8.74456 0.402501
\(473\) 4.00000 0.183920
\(474\) 13.4891 0.619576
\(475\) 0 0
\(476\) −6.74456 −0.309137
\(477\) 12.7446 0.583533
\(478\) 14.7446 0.674401
\(479\) −41.4891 −1.89569 −0.947843 0.318737i \(-0.896741\pi\)
−0.947843 + 0.318737i \(0.896741\pi\)
\(480\) 0 0
\(481\) 5.02175 0.228972
\(482\) −20.0000 −0.910975
\(483\) 13.4891 0.613776
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) −9.25544 −0.419404 −0.209702 0.977765i \(-0.567249\pi\)
−0.209702 + 0.977765i \(0.567249\pi\)
\(488\) −1.25544 −0.0568310
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) 30.9783 1.39803 0.699014 0.715108i \(-0.253622\pi\)
0.699014 + 0.715108i \(0.253622\pi\)
\(492\) 8.00000 0.360668
\(493\) 58.9783 2.65625
\(494\) −45.4891 −2.04665
\(495\) 0 0
\(496\) 4.74456 0.213037
\(497\) 4.00000 0.179425
\(498\) −16.0000 −0.716977
\(499\) −13.4891 −0.603856 −0.301928 0.953331i \(-0.597630\pi\)
−0.301928 + 0.953331i \(0.597630\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 26.2337 1.17087
\(503\) −6.51087 −0.290306 −0.145153 0.989409i \(-0.546367\pi\)
−0.145153 + 0.989409i \(0.546367\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 6.74456 0.299832
\(507\) −64.9783 −2.88579
\(508\) 0 0
\(509\) −35.4891 −1.57303 −0.786514 0.617573i \(-0.788116\pi\)
−0.786514 + 0.617573i \(0.788116\pi\)
\(510\) 0 0
\(511\) 10.7446 0.475311
\(512\) −1.00000 −0.0441942
\(513\) −26.9783 −1.19112
\(514\) −10.2337 −0.451389
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 4.74456 0.208666
\(518\) −0.744563 −0.0327142
\(519\) 40.4674 1.77632
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) −8.74456 −0.382739
\(523\) 17.4891 0.764746 0.382373 0.924008i \(-0.375107\pi\)
0.382373 + 0.924008i \(0.375107\pi\)
\(524\) −2.74456 −0.119897
\(525\) 0 0
\(526\) 26.9783 1.17631
\(527\) 32.0000 1.39394
\(528\) 2.00000 0.0870388
\(529\) 22.4891 0.977788
\(530\) 0 0
\(531\) −8.74456 −0.379482
\(532\) 6.74456 0.292414
\(533\) 26.9783 1.16856
\(534\) 30.9783 1.34056
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 8.00000 0.345225
\(538\) −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
\(543\) 38.9783 1.67272
\(544\) −6.74456 −0.289171
\(545\) 0 0
\(546\) 13.4891 0.577281
\(547\) −14.9783 −0.640424 −0.320212 0.947346i \(-0.603754\pi\)
−0.320212 + 0.947346i \(0.603754\pi\)
\(548\) −3.48913 −0.149048
\(549\) 1.25544 0.0535808
\(550\) 0 0
\(551\) −58.9783 −2.51256
\(552\) 13.4891 0.574135
\(553\) −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 \(0.493403\pi\)
\(558\) −4.74456 −0.200853
\(559\) 26.9783 1.14106
\(560\) 0 0
\(561\) 13.4891 0.569511
\(562\) −14.0000 −0.590554
\(563\) −5.48913 −0.231339 −0.115670 0.993288i \(-0.536901\pi\)
−0.115670 + 0.993288i \(0.536901\pi\)
\(564\) 9.48913 0.399564
\(565\) 0 0
\(566\) 28.0000 1.17693
\(567\) 11.0000 0.461957
\(568\) 4.00000 0.167836
\(569\) 16.5109 0.692172 0.346086 0.938203i \(-0.387511\pi\)
0.346086 + 0.938203i \(0.387511\pi\)
\(570\) 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
\(573\) 32.0000 1.33682
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 8.74456 0.364041 0.182020 0.983295i \(-0.441736\pi\)
0.182020 + 0.983295i \(0.441736\pi\)
\(578\) −28.4891 −1.18499
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 33.4891 1.38817
\(583\) −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 \(0.467240\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 20.0000 0.822690
\(592\) −0.744563 −0.0306013
\(593\) 40.2337 1.65220 0.826100 0.563524i \(-0.190555\pi\)
0.826100 + 0.563524i \(0.190555\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −0.744563 −0.0304985
\(597\) −6.51087 −0.266472
\(598\) 45.4891 1.86019
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 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
\(603\) 4.00000 0.162893
\(604\) −9.25544 −0.376598
\(605\) 0 0
\(606\) 5.48913 0.222980
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 6.74456 0.273528
\(609\) 17.4891 0.708695
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 6.74456 0.272633
\(613\) −11.4891 −0.464041 −0.232021 0.972711i \(-0.574534\pi\)
−0.232021 + 0.972711i \(0.574534\pi\)
\(614\) −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 \(0.435486\pi\)
\(618\) 1.48913 0.0599014
\(619\) 0.744563 0.0299265 0.0149632 0.999888i \(-0.495237\pi\)
0.0149632 + 0.999888i \(0.495237\pi\)
\(620\) 0 0
\(621\) 26.9783 1.08260
\(622\) 20.7446 0.831781
\(623\) −15.4891 −0.620559
\(624\) 13.4891 0.539997
\(625\) 0 0
\(626\) −2.23369 −0.0892761
\(627\) −13.4891 −0.538704
\(628\) −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
\(633\) 24.0000 0.953914
\(634\) 2.23369 0.0887111
\(635\) 0 0
\(636\) −25.4891 −1.01071
\(637\) −6.74456 −0.267229
\(638\) 8.74456 0.346201
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) 11.4891 0.453793 0.226897 0.973919i \(-0.427142\pi\)
0.226897 + 0.973919i \(0.427142\pi\)
\(642\) 24.0000 0.947204
\(643\) −46.4674 −1.83249 −0.916247 0.400613i \(-0.868797\pi\)
−0.916247 + 0.400613i \(0.868797\pi\)
\(644\) −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 \(0.554988\pi\)
\(648\) 11.0000 0.432121
\(649\) 8.74456 0.343254
\(650\) 0 0
\(651\) 9.48913 0.371908
\(652\) 4.00000 0.156652
\(653\) −15.7228 −0.615281 −0.307641 0.951503i \(-0.599539\pi\)
−0.307641 + 0.951503i \(0.599539\pi\)
\(654\) −36.4674 −1.42599
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) −10.7446 −0.419185
\(658\) −4.74456 −0.184962
\(659\) −41.4891 −1.61619 −0.808093 0.589054i \(-0.799500\pi\)
−0.808093 + 0.589054i \(0.799500\pi\)
\(660\) 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
\(663\) 90.9783 3.53330
\(664\) 8.00000 0.310460
\(665\) 0 0
\(666\) 0.744563 0.0288512
\(667\) 58.9783 2.28365
\(668\) 0 0
\(669\) 30.5109 1.17962
\(670\) 0 0
\(671\) −1.25544 −0.0484656
\(672\) −2.00000 −0.0771517
\(673\) −20.9783 −0.808652 −0.404326 0.914615i \(-0.632494\pi\)
−0.404326 + 0.914615i \(0.632494\pi\)
\(674\) 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 \(0.745166\pi\)
\(678\) −20.0000 −0.768095
\(679\) −16.7446 −0.642597
\(680\) 0 0
\(681\) 40.0000 1.53280
\(682\) 4.74456 0.181679
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −6.74456 −0.257885
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 12.0000 0.457829
\(688\) −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
\(693\) 1.00000 0.0379869
\(694\) −22.9783 −0.872242
\(695\) 0 0
\(696\) 17.4891 0.662924
\(697\) −26.9783 −1.02187
\(698\) −30.7446 −1.16370
\(699\) −49.9565 −1.88953
\(700\) 0 0
\(701\) 22.2337 0.839755 0.419877 0.907581i \(-0.362073\pi\)
0.419877 + 0.907581i \(0.362073\pi\)
\(702\) 26.9783 1.01823
\(703\) 5.02175 0.189399
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 8.74456 0.329106
\(707\) −2.74456 −0.103220
\(708\) 17.4891 0.657282
\(709\) 0.510875 0.0191863 0.00959315 0.999954i \(-0.496946\pi\)
0.00959315 + 0.999954i \(0.496946\pi\)
\(710\) 0 0
\(711\) 6.74456 0.252941
\(712\) −15.4891 −0.580480
\(713\) 32.0000 1.19841
\(714\) −13.4891 −0.504818
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 29.4891 1.10129
\(718\) 1.25544 0.0468525
\(719\) 7.72281 0.288012 0.144006 0.989577i \(-0.454001\pi\)
0.144006 + 0.989577i \(0.454001\pi\)
\(720\) 0 0
\(721\) −0.744563 −0.0277290
\(722\) −26.4891 −0.985823
\(723\) −40.0000 −1.48762
\(724\) −19.4891 −0.724308
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) −14.2337 −0.527898 −0.263949 0.964537i \(-0.585025\pi\)
−0.263949 + 0.964537i \(0.585025\pi\)
\(728\) −6.74456 −0.249970
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −26.9783 −0.997827
\(732\) −2.51087 −0.0928046
\(733\) 16.2337 0.599605 0.299802 0.954001i \(-0.403079\pi\)
0.299802 + 0.954001i \(0.403079\pi\)
\(734\) −8.74456 −0.322768
\(735\) 0 0
\(736\) −6.74456 −0.248608
\(737\) −4.00000 −0.147342
\(738\) 4.00000 0.147242
\(739\) −30.9783 −1.13955 −0.569777 0.821800i \(-0.692970\pi\)
−0.569777 + 0.821800i \(0.692970\pi\)
\(740\) 0 0
\(741\) −90.9783 −3.34217
\(742\) 12.7446 0.467868
\(743\) −26.9783 −0.989736 −0.494868 0.868968i \(-0.664784\pi\)
−0.494868 + 0.868968i \(0.664784\pi\)
\(744\) 9.48913 0.347888
\(745\) 0 0
\(746\) −16.9783 −0.621618
\(747\) −8.00000 −0.292705
\(748\) −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
\(753\) 52.4674 1.91202
\(754\) 58.9783 2.14786
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 19.2554 0.699851 0.349925 0.936778i \(-0.386207\pi\)
0.349925 + 0.936778i \(0.386207\pi\)
\(758\) 37.4891 1.36167
\(759\) 13.4891 0.489624
\(760\) 0 0
\(761\) −29.4891 −1.06898 −0.534490 0.845175i \(-0.679496\pi\)
−0.534490 + 0.845175i \(0.679496\pi\)
\(762\) 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
\(768\) −2.00000 −0.0721688
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) −20.4674 −0.737115
\(772\) 2.00000 0.0719816
\(773\) 51.4891 1.85194 0.925968 0.377603i \(-0.123252\pi\)
0.925968 + 0.377603i \(0.123252\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −16.7446 −0.601095
\(777\) −1.48913 −0.0534221
\(778\) 7.48913 0.268498
\(779\) 26.9783 0.966596
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) −45.4891 −1.62669
\(783\) 34.9783 1.25002
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −5.48913 −0.195791
\(787\) −5.48913 −0.195666 −0.0978331 0.995203i \(-0.531191\pi\)
−0.0978331 + 0.995203i \(0.531191\pi\)
\(788\) −10.0000 −0.356235
\(789\) 53.9565 1.92090
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 1.00000 0.0355335
\(793\) −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 \(0.229024\pi\)
\(798\) 13.4891 0.477510
\(799\) −32.0000 −1.13208
\(800\) 0 0
\(801\) 15.4891 0.547281
\(802\) −23.4891 −0.829430
\(803\) 10.7446 0.379167
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 32.0000 1.12715
\(807\) −41.9565 −1.47694
\(808\) −2.74456 −0.0965534
\(809\) 34.4674 1.21181 0.605904 0.795538i \(-0.292811\pi\)
0.605904 + 0.795538i \(0.292811\pi\)
\(810\) 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
\(813\) −29.9565 −1.05062
\(814\) −0.744563 −0.0260969
\(815\) 0 0
\(816\) −13.4891 −0.472214
\(817\) 26.9783 0.943850
\(818\) 21.4891 0.751350
\(819\) 6.74456 0.235674
\(820\) 0 0
\(821\) 7.72281 0.269528 0.134764 0.990878i \(-0.456972\pi\)
0.134764 + 0.990878i \(0.456972\pi\)
\(822\) −6.97825 −0.243394
\(823\) −7.76631 −0.270717 −0.135358 0.990797i \(-0.543219\pi\)
−0.135358 + 0.990797i \(0.543219\pi\)
\(824\) −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 \(0.522155\pi\)
\(828\) 6.74456 0.234390
\(829\) 14.4674 0.502473 0.251236 0.967926i \(-0.419163\pi\)
0.251236 + 0.967926i \(0.419163\pi\)
\(830\) 0 0
\(831\) 30.9783 1.07462
\(832\) −6.74456 −0.233826
\(833\) 6.74456 0.233685
\(834\) 29.4891 1.02112
\(835\) 0 0
\(836\) 6.74456 0.233266
\(837\) 18.9783 0.655984
\(838\) 0.744563 0.0257205
\(839\) −3.25544 −0.112390 −0.0561951 0.998420i \(-0.517897\pi\)
−0.0561951 + 0.998420i \(0.517897\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) −4.51087 −0.155455
\(843\) −28.0000 −0.964371
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) 4.74456 0.163121
\(847\) −1.00000 −0.0343604
\(848\) 12.7446 0.437650
\(849\) 56.0000 1.92192
\(850\) 0 0
\(851\) −5.02175 −0.172143
\(852\) 8.00000 0.274075
\(853\) 22.7446 0.778759 0.389379 0.921077i \(-0.372689\pi\)
0.389379 + 0.921077i \(0.372689\pi\)
\(854\) 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 \(0.235749\pi\)
\(858\) 13.4891 0.460511
\(859\) −29.2119 −0.996698 −0.498349 0.866976i \(-0.666060\pi\)
−0.498349 + 0.866976i \(0.666060\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 17.2554 0.587723
\(863\) 40.2337 1.36957 0.684785 0.728745i \(-0.259896\pi\)
0.684785 + 0.728745i \(0.259896\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 36.7446 1.24863
\(867\) −56.9783 −1.93508
\(868\) −4.74456 −0.161041
\(869\) −6.74456 −0.228794
\(870\) 0 0
\(871\) −26.9783 −0.914123
\(872\) 18.2337 0.617471
\(873\) 16.7446 0.566718
\(874\) 45.4891 1.53869
\(875\) 0 0
\(876\) 21.4891 0.726050
\(877\) 26.4674 0.893740 0.446870 0.894599i \(-0.352539\pi\)
0.446870 + 0.894599i \(0.352539\pi\)
\(878\) −14.9783 −0.505491
\(879\) 26.5109 0.894190
\(880\) 0 0
\(881\) −55.4891 −1.86948 −0.934738 0.355338i \(-0.884366\pi\)
−0.934738 + 0.355338i \(0.884366\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −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 \(0.254724\pi\)
\(888\) −1.48913 −0.0499718
\(889\) 0 0
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) −15.2554 −0.510790
\(893\) 32.0000 1.07084
\(894\) −1.48913 −0.0498038
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 90.9783 3.03768
\(898\) 4.97825 0.166126
\(899\) 41.4891 1.38374
\(900\) 0 0
\(901\) 85.9565 2.86363
\(902\) −4.00000 −0.133185
\(903\) −8.00000 −0.266223
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) −18.5109 −0.614983
\(907\) 14.5109 0.481826 0.240913 0.970547i \(-0.422553\pi\)
0.240913 + 0.970547i \(0.422553\pi\)
\(908\) −20.0000 −0.663723
\(909\) 2.74456 0.0910314
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 13.4891 0.446670
\(913\) 8.00000 0.264761
\(914\) 3.48913 0.115410
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 2.74456 0.0906334
\(918\) −26.9783 −0.890415
\(919\) −55.2119 −1.82127 −0.910637 0.413207i \(-0.864408\pi\)
−0.910637 + 0.413207i \(0.864408\pi\)
\(920\) 0 0
\(921\) −2.97825 −0.0981367
\(922\) −33.7228 −1.11060
\(923\) 26.9783 0.888000
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 1.25544 0.0412562
\(927\) 0.744563 0.0244546
\(928\) −8.74456 −0.287054
\(929\) −28.9783 −0.950746 −0.475373 0.879784i \(-0.657687\pi\)
−0.475373 + 0.879784i \(0.657687\pi\)
\(930\) 0 0
\(931\) −6.74456 −0.221044
\(932\) 24.9783 0.818190
\(933\) 41.4891 1.35829
\(934\) 16.9783 0.555545
\(935\) 0 0
\(936\) 6.74456 0.220453
\(937\) 18.7446 0.612358 0.306179 0.951974i \(-0.400949\pi\)
0.306179 + 0.951974i \(0.400949\pi\)
\(938\) 4.00000 0.130605
\(939\) −4.46738 −0.145787
\(940\) 0 0
\(941\) 12.2337 0.398807 0.199403 0.979917i \(-0.436100\pi\)
0.199403 + 0.979917i \(0.436100\pi\)
\(942\) −1.02175 −0.0332904
\(943\) −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 \(0.458508\pi\)
\(948\) −13.4891 −0.438106
\(949\) 72.4674 2.35239
\(950\) 0 0
\(951\) 4.46738 0.144865
\(952\) 6.74456 0.218593
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −12.7446 −0.412620
\(955\) 0 0
\(956\) −14.7446 −0.476873
\(957\) 17.4891 0.565343
\(958\) 41.4891 1.34045
\(959\) 3.48913 0.112670
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) −5.02175 −0.161908
\(963\) 12.0000 0.386695
\(964\) 20.0000 0.644157
\(965\) 0 0
\(966\) −13.4891 −0.434005
\(967\) 50.9783 1.63935 0.819675 0.572829i \(-0.194154\pi\)
0.819675 + 0.572829i \(0.194154\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 90.9783 2.92264
\(970\) 0 0
\(971\) 19.7228 0.632935 0.316468 0.948603i \(-0.397503\pi\)
0.316468 + 0.948603i \(0.397503\pi\)
\(972\) 10.0000 0.320750
\(973\) −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 \(0.566567\pi\)
\(978\) 8.00000 0.255812
\(979\) −15.4891 −0.495035
\(980\) 0 0
\(981\) −18.2337 −0.582157
\(982\) −30.9783 −0.988556
\(983\) −2.23369 −0.0712436 −0.0356218 0.999365i \(-0.511341\pi\)
−0.0356218 + 0.999365i \(0.511341\pi\)
\(984\) −8.00000 −0.255031
\(985\) 0 0
\(986\) −58.9783 −1.87825
\(987\) −9.48913 −0.302042
\(988\) 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
\(993\) −29.9565 −0.950641
\(994\) −4.00000 −0.126872
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) −39.2119 −1.24185 −0.620927 0.783868i \(-0.713244\pi\)
−0.620927 + 0.783868i \(0.713244\pi\)
\(998\) 13.4891 0.426991
\(999\) −2.97825 −0.0942277
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 3850.2.a.bc.1.1 2
5.2 odd 4 3850.2.c.y.1849.2 4
5.3 odd 4 3850.2.c.y.1849.3 4
5.4 even 2 770.2.a.k.1.2 2
15.14 odd 2 6930.2.a.bo.1.2 2
20.19 odd 2 6160.2.a.r.1.2 2
35.34 odd 2 5390.2.a.bq.1.1 2
55.54 odd 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 5.4 even 2
3850.2.a.bc.1.1 2 1.1 even 1 trivial
3850.2.c.y.1849.2 4 5.2 odd 4
3850.2.c.y.1849.3 4 5.3 odd 4
5390.2.a.bq.1.1 2 35.34 odd 2
6160.2.a.r.1.2 2 20.19 odd 2
6930.2.a.bo.1.2 2 15.14 odd 2
8470.2.a.bu.1.1 2 55.54 odd 2