Properties

Label 5390.2.a.bq.1.1
Level $5390$
Weight $2$
Character 5390.1
Self dual yes
Analytic conductor $43.039$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.0393666895\)
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 \(\chi\) \(=\) 5390.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -2.00000 q^{12} -6.74456 q^{13} +2.00000 q^{15} +1.00000 q^{16} +6.74456 q^{17} +1.00000 q^{18} +6.74456 q^{19} -1.00000 q^{20} -1.00000 q^{22} -6.74456 q^{23} -2.00000 q^{24} +1.00000 q^{25} -6.74456 q^{26} +4.00000 q^{27} +8.74456 q^{29} +2.00000 q^{30} -4.74456 q^{31} +1.00000 q^{32} +2.00000 q^{33} +6.74456 q^{34} +1.00000 q^{36} +0.744563 q^{37} +6.74456 q^{38} +13.4891 q^{39} -1.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -6.74456 q^{46} -4.74456 q^{47} -2.00000 q^{48} +1.00000 q^{50} -13.4891 q^{51} -6.74456 q^{52} -12.7446 q^{53} +4.00000 q^{54} +1.00000 q^{55} -13.4891 q^{57} +8.74456 q^{58} +8.74456 q^{59} +2.00000 q^{60} -1.25544 q^{61} -4.74456 q^{62} +1.00000 q^{64} +6.74456 q^{65} +2.00000 q^{66} -4.00000 q^{67} +6.74456 q^{68} +13.4891 q^{69} -4.00000 q^{71} +1.00000 q^{72} -10.7446 q^{73} +0.744563 q^{74} -2.00000 q^{75} +6.74456 q^{76} +13.4891 q^{78} +6.74456 q^{79} -1.00000 q^{80} -11.0000 q^{81} +4.00000 q^{82} -8.00000 q^{83} -6.74456 q^{85} +4.00000 q^{86} -17.4891 q^{87} -1.00000 q^{88} -15.4891 q^{89} -1.00000 q^{90} -6.74456 q^{92} +9.48913 q^{93} -4.74456 q^{94} -6.74456 q^{95} -2.00000 q^{96} +16.7446 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} - 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} - 2 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 2 q^{19} - 2 q^{20} - 2 q^{22} - 2 q^{23} - 4 q^{24} + 2 q^{25} - 2 q^{26} + 8 q^{27} + 6 q^{29} + 4 q^{30} + 2 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{34} + 2 q^{36} - 10 q^{37} + 2 q^{38} + 4 q^{39} - 2 q^{40} + 8 q^{41} + 8 q^{43} - 2 q^{44} - 2 q^{45} - 2 q^{46} + 2 q^{47} - 4 q^{48} + 2 q^{50} - 4 q^{51} - 2 q^{52} - 14 q^{53} + 8 q^{54} + 2 q^{55} - 4 q^{57} + 6 q^{58} + 6 q^{59} + 4 q^{60} - 14 q^{61} + 2 q^{62} + 2 q^{64} + 2 q^{65} + 4 q^{66} - 8 q^{67} + 2 q^{68} + 4 q^{69} - 8 q^{71} + 2 q^{72} - 10 q^{73} - 10 q^{74} - 4 q^{75} + 2 q^{76} + 4 q^{78} + 2 q^{79} - 2 q^{80} - 22 q^{81} + 8 q^{82} - 16 q^{83} - 2 q^{85} + 8 q^{86} - 12 q^{87} - 2 q^{88} - 8 q^{89} - 2 q^{90} - 2 q^{92} - 4 q^{93} + 2 q^{94} - 2 q^{95} - 4 q^{96} + 22 q^{97} - 2 q^{99} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −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 \(0.884873\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 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 \(-0.218427\pi\)
0.773654 + 0.633608i \(0.218427\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.74456 −1.40634 −0.703169 0.711022i \(-0.748232\pi\)
−0.703169 + 0.711022i \(0.748232\pi\)
\(24\) −2.00000 −0.408248
\(25\) 1.00000 0.200000
\(26\) −6.74456 −1.32272
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 8.74456 1.62382 0.811912 0.583779i \(-0.198427\pi\)
0.811912 + 0.583779i \(0.198427\pi\)
\(30\) 2.00000 0.365148
\(31\) −4.74456 −0.852149 −0.426074 0.904688i \(-0.640104\pi\)
−0.426074 + 0.904688i \(0.640104\pi\)
\(32\) 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 \(-0.480506\pi\)
0.0612027 + 0.998125i \(0.480506\pi\)
\(38\) 6.74456 1.09411
\(39\) 13.4891 2.15999
\(40\) −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
\(45\) −1.00000 −0.149071
\(46\) −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
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −13.4891 −1.88886
\(52\) −6.74456 −0.935303
\(53\) −12.7446 −1.75060 −0.875300 0.483580i \(-0.839336\pi\)
−0.875300 + 0.483580i \(0.839336\pi\)
\(54\) 4.00000 0.544331
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −13.4891 −1.78668
\(58\) 8.74456 1.14822
\(59\) 8.74456 1.13845 0.569223 0.822183i \(-0.307244\pi\)
0.569223 + 0.822183i \(0.307244\pi\)
\(60\) 2.00000 0.258199
\(61\) −1.25544 −0.160742 −0.0803711 0.996765i \(-0.525611\pi\)
−0.0803711 + 0.996765i \(0.525611\pi\)
\(62\) −4.74456 −0.602560
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 6.74456 0.836560
\(66\) 2.00000 0.246183
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.74456 0.817898
\(69\) 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
\(75\) −2.00000 −0.230940
\(76\) 6.74456 0.773654
\(77\) 0 0
\(78\) 13.4891 1.52734
\(79\) 6.74456 0.758823 0.379411 0.925228i \(-0.376127\pi\)
0.379411 + 0.925228i \(0.376127\pi\)
\(80\) −1.00000 −0.111803
\(81\) −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 \(0.644687\pi\)
\(84\) 0 0
\(85\) −6.74456 −0.731551
\(86\) 4.00000 0.431331
\(87\) −17.4891 −1.87503
\(88\) −1.00000 −0.106600
\(89\) −15.4891 −1.64184 −0.820922 0.571040i \(-0.806540\pi\)
−0.820922 + 0.571040i \(0.806540\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −6.74456 −0.703169
\(93\) 9.48913 0.983976
\(94\) −4.74456 −0.489364
\(95\) −6.74456 −0.691978
\(96\) −2.00000 −0.204124
\(97\) 16.7446 1.70015 0.850076 0.526659i \(-0.176556\pi\)
0.850076 + 0.526659i \(0.176556\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) −2.74456 −0.273094 −0.136547 0.990634i \(-0.543601\pi\)
−0.136547 + 0.990634i \(0.543601\pi\)
\(102\) −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 \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 4.00000 0.384900
\(109\) −18.2337 −1.74647 −0.873235 0.487299i \(-0.837982\pi\)
−0.873235 + 0.487299i \(0.837982\pi\)
\(110\) 1.00000 0.0953463
\(111\) −1.48913 −0.141342
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) −13.4891 −1.26337
\(115\) 6.74456 0.628934
\(116\) 8.74456 0.811912
\(117\) −6.74456 −0.623535
\(118\) 8.74456 0.805002
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) −1.25544 −0.113662
\(123\) −8.00000 −0.721336
\(124\) −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
\(129\) −8.00000 −0.704361
\(130\) 6.74456 0.591537
\(131\) 2.74456 0.239794 0.119897 0.992786i \(-0.461744\pi\)
0.119897 + 0.992786i \(0.461744\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) −4.00000 −0.344265
\(136\) 6.74456 0.578341
\(137\) 3.48913 0.298096 0.149048 0.988830i \(-0.452379\pi\)
0.149048 + 0.988830i \(0.452379\pi\)
\(138\) 13.4891 1.14827
\(139\) −14.7446 −1.25062 −0.625309 0.780377i \(-0.715027\pi\)
−0.625309 + 0.780377i \(0.715027\pi\)
\(140\) 0 0
\(141\) 9.48913 0.799129
\(142\) −4.00000 −0.335673
\(143\) 6.74456 0.564009
\(144\) 1.00000 0.0833333
\(145\) −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 \(-0.509709\pi\)
−0.0304985 + 0.999535i \(0.509709\pi\)
\(150\) −2.00000 −0.163299
\(151\) −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
\(154\) 0 0
\(155\) 4.74456 0.381092
\(156\) 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
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 4.00000 0.312348
\(165\) −2.00000 −0.155700
\(166\) −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
\(171\) 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 \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 19.4891 1.44862 0.724308 0.689477i \(-0.242160\pi\)
0.724308 + 0.689477i \(0.242160\pi\)
\(182\) 0 0
\(183\) 2.51087 0.185609
\(184\) −6.74456 −0.497216
\(185\) −0.744563 −0.0547413
\(186\) 9.48913 0.695776
\(187\) −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 \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −2.00000 −0.144338
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 16.7446 1.20219
\(195\) −13.4891 −0.965976
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −3.25544 −0.230772 −0.115386 0.993321i \(-0.536810\pi\)
−0.115386 + 0.993321i \(0.536810\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) −2.74456 −0.193107
\(203\) 0 0
\(204\) −13.4891 −0.944428
\(205\) −4.00000 −0.279372
\(206\) 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
\(215\) −4.00000 −0.272798
\(216\) 4.00000 0.272166
\(217\) 0 0
\(218\) −18.2337 −1.23494
\(219\) 21.4891 1.45210
\(220\) 1.00000 0.0674200
\(221\) −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 \(0.670647\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 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 \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 6.74456 0.444723
\(231\) 0 0
\(232\) 8.74456 0.574109
\(233\) −24.9783 −1.63638 −0.818190 0.574948i \(-0.805022\pi\)
−0.818190 + 0.574948i \(0.805022\pi\)
\(234\) −6.74456 −0.440906
\(235\) 4.74456 0.309501
\(236\) 8.74456 0.569223
\(237\) −13.4891 −0.876213
\(238\) 0 0
\(239\) −14.7446 −0.953746 −0.476873 0.878972i \(-0.658230\pi\)
−0.476873 + 0.878972i \(0.658230\pi\)
\(240\) 2.00000 0.129099
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 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
\(250\) −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
\(255\) 13.4891 0.844722
\(256\) 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 0
\(260\) 6.74456 0.418280
\(261\) 8.74456 0.541275
\(262\) 2.74456 0.169560
\(263\) 26.9783 1.66355 0.831775 0.555113i \(-0.187325\pi\)
0.831775 + 0.555113i \(0.187325\pi\)
\(264\) 2.00000 0.123091
\(265\) 12.7446 0.782892
\(266\) 0 0
\(267\) 30.9783 1.89584
\(268\) −4.00000 −0.244339
\(269\) −20.9783 −1.27907 −0.639533 0.768763i \(-0.720872\pi\)
−0.639533 + 0.768763i \(0.720872\pi\)
\(270\) −4.00000 −0.243432
\(271\) −14.9783 −0.909864 −0.454932 0.890526i \(-0.650336\pi\)
−0.454932 + 0.890526i \(0.650336\pi\)
\(272\) 6.74456 0.408949
\(273\) 0 0
\(274\) 3.48913 0.210786
\(275\) −1.00000 −0.0603023
\(276\) 13.4891 0.811950
\(277\) 15.4891 0.930651 0.465326 0.885140i \(-0.345937\pi\)
0.465326 + 0.885140i \(0.345937\pi\)
\(278\) −14.7446 −0.884320
\(279\) −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
\(285\) 13.4891 0.799027
\(286\) 6.74456 0.398814
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 28.4891 1.67583
\(290\) −8.74456 −0.513498
\(291\) −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 \(0.626556\pi\)
\(294\) 0 0
\(295\) −8.74456 −0.509128
\(296\) 0.744563 0.0432768
\(297\) −4.00000 −0.232104
\(298\) −0.744563 −0.0431314
\(299\) 45.4891 2.63070
\(300\) −2.00000 −0.115470
\(301\) 0 0
\(302\) −9.25544 −0.532591
\(303\) 5.48913 0.315342
\(304\) 6.74456 0.386827
\(305\) 1.25544 0.0718861
\(306\) 6.74456 0.385561
\(307\) 1.48913 0.0849889 0.0424944 0.999097i \(-0.486470\pi\)
0.0424944 + 0.999097i \(0.486470\pi\)
\(308\) 0 0
\(309\) −1.48913 −0.0847134
\(310\) 4.74456 0.269473
\(311\) 20.7446 1.17632 0.588158 0.808746i \(-0.299853\pi\)
0.588158 + 0.808746i \(0.299853\pi\)
\(312\) 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 \(-0.480020\pi\)
0.0627282 + 0.998031i \(0.480020\pi\)
\(318\) 25.4891 1.42936
\(319\) −8.74456 −0.489602
\(320\) −1.00000 −0.0559017
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 45.4891 2.53108
\(324\) −11.0000 −0.611111
\(325\) −6.74456 −0.374121
\(326\) −4.00000 −0.221540
\(327\) 36.4674 2.01665
\(328\) 4.00000 0.220863
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) 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
\(335\) 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
\(339\) −20.0000 −1.08625
\(340\) −6.74456 −0.365775
\(341\) 4.74456 0.256932
\(342\) 6.74456 0.364704
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) −13.4891 −0.726230
\(346\) −20.2337 −1.08777
\(347\) −22.9783 −1.23354 −0.616769 0.787145i \(-0.711559\pi\)
−0.616769 + 0.787145i \(0.711559\pi\)
\(348\) −17.4891 −0.937516
\(349\) −30.7446 −1.64572 −0.822859 0.568245i \(-0.807623\pi\)
−0.822859 + 0.568245i \(0.807623\pi\)
\(350\) 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
\(355\) 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 \(-0.510547\pi\)
−0.0331297 + 0.999451i \(0.510547\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 26.4891 1.39416
\(362\) 19.4891 1.02433
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) 10.7446 0.562396
\(366\) 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
\(370\) −0.744563 −0.0387080
\(371\) 0 0
\(372\) 9.48913 0.491988
\(373\) −16.9783 −0.879100 −0.439550 0.898218i \(-0.644862\pi\)
−0.439550 + 0.898218i \(0.644862\pi\)
\(374\) −6.74456 −0.348753
\(375\) 2.00000 0.103280
\(376\) −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 \(0.668100\pi\)
\(384\) −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 \(0.560802\pi\)
\(390\) −13.4891 −0.683048
\(391\) −45.4891 −2.30048
\(392\) 0 0
\(393\) −5.48913 −0.276890
\(394\) 10.0000 0.503793
\(395\) −6.74456 −0.339356
\(396\) −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
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 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
\(405\) 11.0000 0.546594
\(406\) 0 0
\(407\) −0.744563 −0.0369066
\(408\) −13.4891 −0.667811
\(409\) 21.4891 1.06257 0.531284 0.847194i \(-0.321710\pi\)
0.531284 + 0.847194i \(0.321710\pi\)
\(410\) −4.00000 −0.197546
\(411\) −6.97825 −0.344212
\(412\) 0.744563 0.0366820
\(413\) 0 0
\(414\) −6.74456 −0.331477
\(415\) 8.00000 0.392705
\(416\) −6.74456 −0.330679
\(417\) 29.4891 1.44409
\(418\) −6.74456 −0.329887
\(419\) 0.744563 0.0363743 0.0181871 0.999835i \(-0.494211\pi\)
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
\(423\) −4.74456 −0.230689
\(424\) −12.7446 −0.618931
\(425\) 6.74456 0.327159
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) −13.4891 −0.651261
\(430\) −4.00000 −0.192897
\(431\) −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 \(0.844423\pi\)
\(434\) 0 0
\(435\) 17.4891 0.838539
\(436\) −18.2337 −0.873235
\(437\) −45.4891 −2.17604
\(438\) 21.4891 1.02679
\(439\) −14.9783 −0.714873 −0.357436 0.933937i \(-0.616349\pi\)
−0.357436 + 0.933937i \(0.616349\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −45.4891 −2.16370
\(443\) −29.4891 −1.40107 −0.700535 0.713618i \(-0.747055\pi\)
−0.700535 + 0.713618i \(0.747055\pi\)
\(444\) −1.48913 −0.0706708
\(445\) 15.4891 0.734255
\(446\) −15.2554 −0.722366
\(447\) 1.48913 0.0704332
\(448\) 0 0
\(449\) −4.97825 −0.234938 −0.117469 0.993077i \(-0.537478\pi\)
−0.117469 + 0.993077i \(0.537478\pi\)
\(450\) 1.00000 0.0471405
\(451\) −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 \(-0.473995\pi\)
0.0816072 + 0.996665i \(0.473995\pi\)
\(458\) 6.00000 0.280362
\(459\) 26.9783 1.25924
\(460\) 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
\(465\) −9.48913 −0.440048
\(466\) −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
\(469\) 0 0
\(470\) 4.74456 0.218850
\(471\) 1.02175 0.0470797
\(472\) 8.74456 0.402501
\(473\) −4.00000 −0.183920
\(474\) −13.4891 −0.619576
\(475\) 6.74456 0.309462
\(476\) 0 0
\(477\) −12.7446 −0.583533
\(478\) −14.7446 −0.674401
\(479\) 41.4891 1.89569 0.947843 0.318737i \(-0.103259\pi\)
0.947843 + 0.318737i \(0.103259\pi\)
\(480\) 2.00000 0.0912871
\(481\) −5.02175 −0.228972
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −16.7446 −0.760331
\(486\) 10.0000 0.453609
\(487\) 9.25544 0.419404 0.209702 0.977765i \(-0.432751\pi\)
0.209702 + 0.977765i \(0.432751\pi\)
\(488\) −1.25544 −0.0568310
\(489\) 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
\(495\) 1.00000 0.0449467
\(496\) −4.74456 −0.213037
\(497\) 0 0
\(498\) 16.0000 0.716977
\(499\) −13.4891 −0.603856 −0.301928 0.953331i \(-0.597630\pi\)
−0.301928 + 0.953331i \(0.597630\pi\)
\(500\) −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
\(507\) −64.9783 −2.88579
\(508\) 0 0
\(509\) 35.4891 1.57303 0.786514 0.617573i \(-0.211884\pi\)
0.786514 + 0.617573i \(0.211884\pi\)
\(510\) 13.4891 0.597309
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 26.9783 1.19112
\(514\) 10.2337 0.451389
\(515\) −0.744563 −0.0328094
\(516\) −8.00000 −0.352180
\(517\) 4.74456 0.208666
\(518\) 0 0
\(519\) 40.4674 1.77632
\(520\) 6.74456 0.295769
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 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
\(530\) 12.7446 0.553588
\(531\) 8.74456 0.379482
\(532\) 0 0
\(533\) −26.9783 −1.16856
\(534\) 30.9783 1.34056
\(535\) 12.0000 0.518805
\(536\) −4.00000 −0.172774
\(537\) 8.00000 0.345225
\(538\) −20.9783 −0.904437
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) 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
\(545\) 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
\(549\) −1.25544 −0.0535808
\(550\) −1.00000 −0.0426401
\(551\) 58.9783 2.51256
\(552\) 13.4891 0.574135
\(553\) 0 0
\(554\) 15.4891 0.658070
\(555\) 1.48913 0.0632098
\(556\) −14.7446 −0.625309
\(557\) −0.978251 −0.0414498 −0.0207249 0.999785i \(-0.506597\pi\)
−0.0207249 + 0.999785i \(0.506597\pi\)
\(558\) −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
\(565\) −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 \(-0.387511\pi\)
0.346086 + 0.938203i \(0.387511\pi\)
\(570\) 13.4891 0.564997
\(571\) −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
\(574\) 0 0
\(575\) −6.74456 −0.281268
\(576\) 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
\(580\) −8.74456 −0.363098
\(581\) 0 0
\(582\) −33.4891 −1.38817
\(583\) 12.7446 0.527826
\(584\) −10.7446 −0.444613
\(585\) 6.74456 0.278853
\(586\) −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
\(591\) −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 \(0.578838\pi\)
\(600\) −2.00000 −0.0816497
\(601\) 33.4891 1.36605 0.683025 0.730395i \(-0.260664\pi\)
0.683025 + 0.730395i \(0.260664\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −9.25544 −0.376598
\(605\) −1.00000 −0.0406558
\(606\) 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
\(609\) 0 0
\(610\) 1.25544 0.0508312
\(611\) 32.0000 1.29458
\(612\) 6.74456 0.272633
\(613\) 11.4891 0.464041 0.232021 0.972711i \(-0.425466\pi\)
0.232021 + 0.972711i \(0.425466\pi\)
\(614\) 1.48913 0.0600962
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) −1.48913 −0.0599014
\(619\) −0.744563 −0.0299265 −0.0149632 0.999888i \(-0.504763\pi\)
−0.0149632 + 0.999888i \(0.504763\pi\)
\(620\) 4.74456 0.190546
\(621\) −26.9783 −1.08260
\(622\) 20.7446 0.831781
\(623\) 0 0
\(624\) 13.4891 0.539997
\(625\) 1.00000 0.0400000
\(626\) 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
\(637\) 0 0
\(638\) −8.74456 −0.346201
\(639\) −4.00000 −0.158238
\(640\) −1.00000 −0.0395285
\(641\) 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 \(0.868797\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 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
\(650\) −6.74456 −0.264544
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 15.7228 0.615281 0.307641 0.951503i \(-0.400461\pi\)
0.307641 + 0.951503i \(0.400461\pi\)
\(654\) 36.4674 1.42599
\(655\) −2.74456 −0.107239
\(656\) 4.00000 0.156174
\(657\) −10.7446 −0.419185
\(658\) 0 0
\(659\) −41.4891 −1.61619 −0.808093 0.589054i \(-0.799500\pi\)
−0.808093 + 0.589054i \(0.799500\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 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
\(670\) 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
\(675\) 4.00000 0.153960
\(676\) 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
\(679\) 0 0
\(680\) −6.74456 −0.258642
\(681\) 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
\(685\) −3.48913 −0.133313
\(686\) 0 0
\(687\) −12.0000 −0.457829
\(688\) 4.00000 0.152499
\(689\) 85.9565 3.27468
\(690\) −13.4891 −0.513522
\(691\) −1.76631 −0.0671937 −0.0335968 0.999435i \(-0.510696\pi\)
−0.0335968 + 0.999435i \(0.510696\pi\)
\(692\) −20.2337 −0.769169
\(693\) 0 0
\(694\) −22.9783 −0.872242
\(695\) 14.7446 0.559293
\(696\) −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
\(705\) −9.48913 −0.357381
\(706\) −8.74456 −0.329106
\(707\) 0 0
\(708\) −17.4891 −0.657282
\(709\) 0.510875 0.0191863 0.00959315 0.999954i \(-0.496946\pi\)
0.00959315 + 0.999954i \(0.496946\pi\)
\(710\) 4.00000 0.150117
\(711\) 6.74456 0.252941
\(712\) −15.4891 −0.580480
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) −6.74456 −0.252232
\(716\) −4.00000 −0.149487
\(717\) 29.4891 1.10129
\(718\) −1.25544 −0.0468525
\(719\) −7.72281 −0.288012 −0.144006 0.989577i \(-0.545999\pi\)
−0.144006 + 0.989577i \(0.545999\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 26.4891 0.985823
\(723\) 40.0000 1.48762
\(724\) 19.4891 0.724308
\(725\) 8.74456 0.324765
\(726\) −2.00000 −0.0742270
\(727\) −14.2337 −0.527898 −0.263949 0.964537i \(-0.585025\pi\)
−0.263949 + 0.964537i \(0.585025\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 10.7446 0.397674
\(731\) 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 \(0.692970\pi\)
\(740\) −0.744563 −0.0273707
\(741\) 90.9783 3.34217
\(742\) 0 0
\(743\) 26.9783 0.989736 0.494868 0.868968i \(-0.335216\pi\)
0.494868 + 0.868968i \(0.335216\pi\)
\(744\) 9.48913 0.347888
\(745\) 0.744563 0.0272787
\(746\) −16.9783 −0.621618
\(747\) −8.00000 −0.292705
\(748\) −6.74456 −0.246606
\(749\) 0 0
\(750\) 2.00000 0.0730297
\(751\) 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
\(755\) 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
\(759\) −13.4891 −0.489624
\(760\) −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
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) −6.74456 −0.243850
\(766\) −19.7228 −0.712614
\(767\) −58.9783 −2.12958
\(768\) −2.00000 −0.0721688
\(769\) 8.00000 0.288487 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) 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
\(775\) −4.74456 −0.170430
\(776\) 16.7446 0.601095
\(777\) 0 0
\(778\) −7.48913 −0.268498
\(779\) 26.9783 0.966596
\(780\) −13.4891 −0.482988
\(781\) 4.00000 0.143131
\(782\) −45.4891 −1.62669
\(783\) 34.9783 1.25002
\(784\) 0 0
\(785\) 0.510875 0.0182339
\(786\) −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
\(790\) −6.74456 −0.239961
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 8.46738 0.300685
\(794\) −35.4891 −1.25946
\(795\) −25.4891 −0.904006
\(796\) −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
\(801\) −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 \(0.292811\pi\)
\(810\) 11.0000 0.386501
\(811\) 18.7446 0.658211 0.329105 0.944293i \(-0.393253\pi\)
0.329105 + 0.944293i \(0.393253\pi\)
\(812\) 0 0
\(813\) 29.9565 1.05062
\(814\) −0.744563 −0.0260969
\(815\) 4.00000 0.140114
\(816\) −13.4891 −0.472214
\(817\) 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 \(-0.456972\pi\)
0.134764 + 0.990878i \(0.456972\pi\)
\(822\) −6.97825 −0.243394
\(823\) 7.76631 0.270717 0.135358 0.990797i \(-0.456781\pi\)
0.135358 + 0.990797i \(0.456781\pi\)
\(824\) 0.744563 0.0259381
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) −6.74456 −0.234390
\(829\) −14.4674 −0.502473 −0.251236 0.967926i \(-0.580837\pi\)
−0.251236 + 0.967926i \(0.580837\pi\)
\(830\) 8.00000 0.277684
\(831\) −30.9783 −1.07462
\(832\) −6.74456 −0.233826
\(833\) 0 0
\(834\) 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 \(-0.482103\pi\)
0.0561951 + 0.998420i \(0.482103\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) 4.51087 0.155455
\(843\) −28.0000 −0.964371
\(844\) −12.0000 −0.413057
\(845\) −32.4891 −1.11766
\(846\) −4.74456 −0.163121
\(847\) 0 0
\(848\) −12.7446 −0.437650
\(849\) 56.0000 1.92192
\(850\) 6.74456 0.231337
\(851\) −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 \(0.372689\pi\)
\(854\) 0 0
\(855\) −6.74456 −0.230659
\(856\) −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 \(-0.333940\pi\)
0.498349 + 0.866976i \(0.333940\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −17.2554 −0.587723
\(863\) −40.2337 −1.36957 −0.684785 0.728745i \(-0.740104\pi\)
−0.684785 + 0.728745i \(0.740104\pi\)
\(864\) 4.00000 0.136083
\(865\) 20.2337 0.687966
\(866\) −36.7446 −1.24863
\(867\) −56.9783 −1.93508
\(868\) 0 0
\(869\) −6.74456 −0.228794
\(870\) 17.4891 0.592937
\(871\) 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 \(-0.647461\pi\)
−0.446870 + 0.894599i \(0.647461\pi\)
\(878\) −14.9783 −0.505491
\(879\) 26.5109 0.894190
\(880\) 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
\(885\) 17.4891 0.587891
\(886\) −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
\(890\) 15.4891 0.519197
\(891\) 11.0000 0.368514
\(892\) −15.2554 −0.510790
\(893\) −32.0000 −1.07084
\(894\) 1.48913 0.0498038
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) −90.9783 −3.03768
\(898\) −4.97825 −0.166126
\(899\) −41.4891 −1.38374
\(900\) 1.00000 0.0333333
\(901\) −85.9565 −2.86363
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) −19.4891 −0.647840
\(906\) 18.5109 0.614983
\(907\) −14.5109 −0.481826 −0.240913 0.970547i \(-0.577447\pi\)
−0.240913 + 0.970547i \(0.577447\pi\)
\(908\) −20.0000 −0.663723
\(909\) −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
\(915\) −2.51087 −0.0830070
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) 26.9783 0.890415
\(919\) −55.2119 −1.82127 −0.910637 0.413207i \(-0.864408\pi\)
−0.910637 + 0.413207i \(0.864408\pi\)
\(920\) 6.74456 0.222362
\(921\) −2.97825 −0.0981367
\(922\) −33.7228 −1.11060
\(923\) 26.9783 0.888000
\(924\) 0 0
\(925\) 0.744563 0.0244811
\(926\) 1.25544 0.0412562
\(927\) 0.744563 0.0244546
\(928\) 8.74456 0.287054
\(929\) 28.9783 0.950746 0.475373 0.879784i \(-0.342313\pi\)
0.475373 + 0.879784i \(0.342313\pi\)
\(930\) −9.48913 −0.311161
\(931\) 0 0
\(932\) −24.9783 −0.818190
\(933\) −41.4891 −1.35829
\(934\) −16.9783 −0.555545
\(935\) 6.74456 0.220571
\(936\) −6.74456 −0.220453
\(937\) 18.7446 0.612358 0.306179 0.951974i \(-0.400949\pi\)
0.306179 + 0.951974i \(0.400949\pi\)
\(938\) 0 0
\(939\) −4.46738 −0.145787
\(940\) 4.74456 0.154751
\(941\) −12.2337 −0.398807 −0.199403 0.979917i \(-0.563900\pi\)
−0.199403 + 0.979917i \(0.563900\pi\)
\(942\) 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 \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) −13.4891 −0.438106
\(949\) 72.4674 2.35239
\(950\) 6.74456 0.218823
\(951\) −4.46738 −0.144865
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −12.7446 −0.412620
\(955\) 16.0000 0.517748
\(956\) −14.7446 −0.476873
\(957\) 17.4891 0.565343
\(958\) 41.4891 1.34045
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) −8.48913 −0.273843
\(962\) −5.02175 −0.161908
\(963\) −12.0000 −0.386695
\(964\) −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
\(969\) −90.9783 −2.92264
\(970\) −16.7446 −0.537636
\(971\) −19.7228 −0.632935 −0.316468 0.948603i \(-0.602497\pi\)
−0.316468 + 0.948603i \(0.602497\pi\)
\(972\) 10.0000 0.320750
\(973\) 0 0
\(974\) 9.25544 0.296563
\(975\) 13.4891 0.431998
\(976\) −1.25544 −0.0401856
\(977\) 12.9783 0.415211 0.207606 0.978213i \(-0.433433\pi\)
0.207606 + 0.978213i \(0.433433\pi\)
\(978\) 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
\(985\) −10.0000 −0.318626
\(986\) 58.9783 1.87825
\(987\) 0 0
\(988\) −45.4891 −1.44720
\(989\) −26.9783 −0.857858
\(990\) 1.00000 0.0317821
\(991\) 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
\(994\) 0 0
\(995\) 3.25544 0.103204
\(996\) 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 5390.2.a.bq.1.1 2
7.6 odd 2 770.2.a.k.1.2 2
21.20 even 2 6930.2.a.bo.1.2 2
28.27 even 2 6160.2.a.r.1.2 2
35.13 even 4 3850.2.c.y.1849.2 4
35.27 even 4 3850.2.c.y.1849.3 4
35.34 odd 2 3850.2.a.bc.1.1 2
77.76 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 7.6 odd 2
3850.2.a.bc.1.1 2 35.34 odd 2
3850.2.c.y.1849.2 4 35.13 even 4
3850.2.c.y.1849.3 4 35.27 even 4
5390.2.a.bq.1.1 2 1.1 even 1 trivial
6160.2.a.r.1.2 2 28.27 even 2
6930.2.a.bo.1.2 2 21.20 even 2
8470.2.a.bu.1.1 2 77.76 even 2