Properties

Label 770.2.a.k.1.2
Level $770$
Weight $2$
Character 770.1
Self dual yes
Analytic conductor $6.148$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.14848095564\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 770.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{12} +6.74456 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.74456 q^{17} +1.00000 q^{18} -6.74456 q^{19} +1.00000 q^{20} +2.00000 q^{21} -1.00000 q^{22} -6.74456 q^{23} +2.00000 q^{24} +1.00000 q^{25} +6.74456 q^{26} -4.00000 q^{27} +1.00000 q^{28} +8.74456 q^{29} +2.00000 q^{30} +4.74456 q^{31} +1.00000 q^{32} -2.00000 q^{33} -6.74456 q^{34} +1.00000 q^{35} +1.00000 q^{36} +0.744563 q^{37} -6.74456 q^{38} +13.4891 q^{39} +1.00000 q^{40} -4.00000 q^{41} +2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -6.74456 q^{46} +4.74456 q^{47} +2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -13.4891 q^{51} +6.74456 q^{52} -12.7446 q^{53} -4.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} -13.4891 q^{57} +8.74456 q^{58} -8.74456 q^{59} +2.00000 q^{60} +1.25544 q^{61} +4.74456 q^{62} +1.00000 q^{63} +1.00000 q^{64} +6.74456 q^{65} -2.00000 q^{66} -4.00000 q^{67} -6.74456 q^{68} -13.4891 q^{69} +1.00000 q^{70} -4.00000 q^{71} +1.00000 q^{72} +10.7446 q^{73} +0.744563 q^{74} +2.00000 q^{75} -6.74456 q^{76} -1.00000 q^{77} +13.4891 q^{78} +6.74456 q^{79} +1.00000 q^{80} -11.0000 q^{81} -4.00000 q^{82} +8.00000 q^{83} +2.00000 q^{84} -6.74456 q^{85} +4.00000 q^{86} +17.4891 q^{87} -1.00000 q^{88} +15.4891 q^{89} +1.00000 q^{90} +6.74456 q^{91} -6.74456 q^{92} +9.48913 q^{93} +4.74456 q^{94} -6.74456 q^{95} +2.00000 q^{96} -16.7446 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} - 2 q^{11} + 4 q^{12} + 2 q^{13} + 2 q^{14} + 4 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{20} + 4 q^{21} - 2 q^{22} - 2 q^{23} + 4 q^{24} + 2 q^{25} + 2 q^{26} - 8 q^{27} + 2 q^{28} + 6 q^{29} + 4 q^{30} - 2 q^{31} + 2 q^{32} - 4 q^{33} - 2 q^{34} + 2 q^{35} + 2 q^{36} - 10 q^{37} - 2 q^{38} + 4 q^{39} + 2 q^{40} - 8 q^{41} + 4 q^{42} + 8 q^{43} - 2 q^{44} + 2 q^{45} - 2 q^{46} - 2 q^{47} + 4 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} + 2 q^{52} - 14 q^{53} - 8 q^{54} - 2 q^{55} + 2 q^{56} - 4 q^{57} + 6 q^{58} - 6 q^{59} + 4 q^{60} + 14 q^{61} - 2 q^{62} + 2 q^{63} + 2 q^{64} + 2 q^{65} - 4 q^{66} - 8 q^{67} - 2 q^{68} - 4 q^{69} + 2 q^{70} - 8 q^{71} + 2 q^{72} + 10 q^{73} - 10 q^{74} + 4 q^{75} - 2 q^{76} - 2 q^{77} + 4 q^{78} + 2 q^{79} + 2 q^{80} - 22 q^{81} - 8 q^{82} + 16 q^{83} + 4 q^{84} - 2 q^{85} + 8 q^{86} + 12 q^{87} - 2 q^{88} + 8 q^{89} + 2 q^{90} + 2 q^{91} - 2 q^{92} - 4 q^{93} - 2 q^{94} - 2 q^{95} + 4 q^{96} - 22 q^{97} + 2 q^{98} - 2 q^{99} + O(q^{100}) \)

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 \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) 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 \(-0.115127\pi\)
0.935303 + 0.353849i \(0.115127\pi\)
\(14\) 1.00000 0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −6.74456 −1.63580 −0.817898 0.575363i \(-0.804861\pi\)
−0.817898 + 0.575363i \(0.804861\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.74456 −1.54731 −0.773654 0.633608i \(-0.781573\pi\)
−0.773654 + 0.633608i \(0.781573\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) −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
\(28\) 1.00000 0.188982
\(29\) 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 \(-0.359896\pi\)
0.426074 + 0.904688i \(0.359896\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) −6.74456 −1.15668
\(35\) 1.00000 0.169031
\(36\) 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 \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 2.00000 0.308607
\(43\) 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 \(-0.387529\pi\)
0.346033 + 0.938222i \(0.387529\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) 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
\(56\) 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 \(0.692756\pi\)
\(60\) 2.00000 0.258199
\(61\) 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
\(65\) 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
\(70\) 1.00000 0.119523
\(71\) −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 \(-0.283555\pi\)
0.628778 + 0.777585i \(0.283555\pi\)
\(74\) 0.744563 0.0865536
\(75\) 2.00000 0.230940
\(76\) −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 \(0.376127\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) −4.00000 −0.441726
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 2.00000 0.218218
\(85\) −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 \(-0.193460\pi\)
0.820922 + 0.571040i \(0.193460\pi\)
\(90\) 1.00000 0.105409
\(91\) 6.74456 0.707022
\(92\) −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 \(-0.823444\pi\)
−0.850076 + 0.526659i \(0.823444\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 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 \(-0.511679\pi\)
−0.0366820 + 0.999327i \(0.511679\pi\)
\(104\) 6.74456 0.661359
\(105\) 2.00000 0.195180
\(106\) −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
\(112\) 1.00000 0.0944911
\(113\) 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
\(119\) −6.74456 −0.618273
\(120\) 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
\(126\) 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
\(130\) 6.74456 0.591537
\(131\) −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
\(135\) −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 \(-0.284973\pi\)
0.625309 + 0.780377i \(0.284973\pi\)
\(140\) 1.00000 0.0845154
\(141\) 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
\(147\) 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 \(0.509709\pi\)
\(150\) 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
\(154\) −1.00000 −0.0805823
\(155\) 4.74456 0.381092
\(156\) 13.4891 1.07999
\(157\) 0.510875 0.0407722 0.0203861 0.999792i \(-0.493510\pi\)
0.0203861 + 0.999792i \(0.493510\pi\)
\(158\) 6.74456 0.536569
\(159\) −25.4891 −2.02142
\(160\) 1.00000 0.0790569
\(161\) −6.74456 −0.531546
\(162\) −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 \(0.5\pi\)
\(168\) 2.00000 0.154303
\(169\) 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 \(-0.220670\pi\)
0.769169 + 0.639045i \(0.220670\pi\)
\(174\) 17.4891 1.32585
\(175\) 1.00000 0.0755929
\(176\) −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 \(-0.757840\pi\)
−0.724308 + 0.689477i \(0.757840\pi\)
\(182\) 6.74456 0.499940
\(183\) 2.51087 0.185609
\(184\) −6.74456 −0.497216
\(185\) 0.744563 0.0547413
\(186\) 9.48913 0.695776
\(187\) 6.74456 0.493211
\(188\) 4.74456 0.346033
\(189\) −4.00000 −0.290957
\(190\) −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
\(196\) 1.00000 0.0714286
\(197\) 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 \(-0.463190\pi\)
0.115386 + 0.993321i \(0.463190\pi\)
\(200\) 1.00000 0.0707107
\(201\) −8.00000 −0.564276
\(202\) 2.74456 0.193107
\(203\) 8.74456 0.613748
\(204\) −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
\(210\) 2.00000 0.138013
\(211\) −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
\(217\) 4.74456 0.322082
\(218\) −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 \(-0.329353\pi\)
0.510790 + 0.859706i \(0.329353\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 10.0000 0.665190
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −13.4891 −0.893339
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −6.74456 −0.444723
\(231\) −2.00000 −0.131590
\(232\) 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
\(238\) −6.74456 −0.437185
\(239\) −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 \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 1.00000 0.0642824
\(243\) −10.0000 −0.641500
\(244\) 1.25544 0.0803711
\(245\) 1.00000 0.0638877
\(246\) −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 \(-0.810480\pi\)
−0.827928 + 0.560835i \(0.810480\pi\)
\(252\) 1.00000 0.0629941
\(253\) 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 \(-0.603407\pi\)
−0.319180 + 0.947694i \(0.603407\pi\)
\(258\) 8.00000 0.498058
\(259\) 0.744563 0.0462649
\(260\) 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
\(266\) −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 \(0.279128\pi\)
\(270\) −4.00000 −0.243432
\(271\) 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
\(275\) −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
\(280\) 1.00000 0.0597614
\(281\) 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 \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −4.00000 −0.237356
\(285\) −13.4891 −0.799027
\(286\) −6.74456 −0.398814
\(287\) −4.00000 −0.236113
\(288\) 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 \(-0.373444\pi\)
0.387195 + 0.921998i \(0.373444\pi\)
\(294\) 2.00000 0.116642
\(295\) −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
\(301\) 4.00000 0.230556
\(302\) −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 \(-0.513530\pi\)
−0.0424944 + 0.999097i \(0.513530\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −1.48913 −0.0847134
\(310\) 4.74456 0.269473
\(311\) −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 \(-0.520108\pi\)
−0.0631278 + 0.998005i \(0.520108\pi\)
\(314\) 0.510875 0.0288303
\(315\) 1.00000 0.0563436
\(316\) 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
\(322\) −6.74456 −0.375860
\(323\) 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
\(329\) 4.74456 0.261576
\(330\) −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
\(336\) 2.00000 0.109109
\(337\) 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
\(343\) 1.00000 0.0539949
\(344\) 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 \(-0.192377\pi\)
0.822859 + 0.568245i \(0.192377\pi\)
\(350\) 1.00000 0.0534522
\(351\) −26.9783 −1.43999
\(352\) −1.00000 −0.0533002
\(353\) 8.74456 0.465426 0.232713 0.972545i \(-0.425240\pi\)
0.232713 + 0.972545i \(0.425240\pi\)
\(354\) −17.4891 −0.929537
\(355\) −4.00000 −0.212298
\(356\) 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 \(0.510547\pi\)
\(360\) 1.00000 0.0527046
\(361\) 26.4891 1.39416
\(362\) −19.4891 −1.02433
\(363\) 2.00000 0.104973
\(364\) 6.74456 0.353511
\(365\) 10.7446 0.562396
\(366\) 2.51087 0.131246
\(367\) −8.74456 −0.456462 −0.228231 0.973607i \(-0.573294\pi\)
−0.228231 + 0.973607i \(0.573294\pi\)
\(368\) −6.74456 −0.351585
\(369\) −4.00000 −0.208232
\(370\) 0.744563 0.0387080
\(371\) −12.7446 −0.661665
\(372\) 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
\(378\) −4.00000 −0.205738
\(379\) −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 \(-0.331900\pi\)
0.503894 + 0.863765i \(0.331900\pi\)
\(384\) 2.00000 0.102062
\(385\) −1.00000 −0.0509647
\(386\) −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
\(392\) 1.00000 0.0505076
\(393\) −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 \(-0.150303\pi\)
0.890574 + 0.454838i \(0.150303\pi\)
\(398\) 3.25544 0.163180
\(399\) −13.4891 −0.675301
\(400\) 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
\(406\) 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 \(0.678290\pi\)
\(410\) −4.00000 −0.197546
\(411\) 6.97825 0.344212
\(412\) −0.744563 −0.0366820
\(413\) −8.74456 −0.430292
\(414\) −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 \(-0.505789\pi\)
−0.0181871 + 0.999835i \(0.505789\pi\)
\(420\) 2.00000 0.0975900
\(421\) 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
\(427\) 1.25544 0.0607549
\(428\) −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 \(-0.155577\pi\)
0.882915 + 0.469532i \(0.155577\pi\)
\(434\) 4.74456 0.227746
\(435\) 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 \(-0.383651\pi\)
0.357436 + 0.933937i \(0.383651\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 1.00000 0.0476190
\(442\) −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
\(448\) 1.00000 0.0472456
\(449\) −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
\(455\) 6.74456 0.316190
\(456\) −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 \(-0.212502\pi\)
0.785314 + 0.619098i \(0.212502\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 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 \(-0.371496\pi\)
0.392830 + 0.919611i \(0.371496\pi\)
\(468\) 6.74456 0.311768
\(469\) −4.00000 −0.184703
\(470\) 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
\(476\) −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 \(0.896741\pi\)
\(480\) 2.00000 0.0912871
\(481\) 5.02175 0.228972
\(482\) 20.0000 0.910975
\(483\) −13.4891 −0.613776
\(484\) 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
\(490\) 1.00000 0.0451754
\(491\) 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
\(497\) −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 \(0.597630\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −26.2337 −1.17087
\(503\) 6.51087 0.290306 0.145153 0.989409i \(-0.453633\pi\)
0.145153 + 0.989409i \(0.453633\pi\)
\(504\) 1.00000 0.0445435
\(505\) 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 \(-0.788116\pi\)
−0.786514 + 0.617573i \(0.788116\pi\)
\(510\) −13.4891 −0.597309
\(511\) 10.7446 0.475311
\(512\) 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
\(518\) 0.744563 0.0327142
\(519\) 40.4674 1.77632
\(520\) 6.74456 0.295769
\(521\) −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 \(-0.624893\pi\)
−0.382373 + 0.924008i \(0.624893\pi\)
\(524\) −2.74456 −0.119897
\(525\) 2.00000 0.0872872
\(526\) 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
\(532\) −6.74456 −0.292414
\(533\) −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
\(539\) −1.00000 −0.0430730
\(540\) −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
\(546\) 13.4891 0.577281
\(547\) 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
\(553\) 6.74456 0.286808
\(554\) 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
\(560\) 1.00000 0.0422577
\(561\) 13.4891 0.569511
\(562\) 14.0000 0.590554
\(563\) 5.48913 0.231339 0.115670 0.993288i \(-0.463099\pi\)
0.115670 + 0.993288i \(0.463099\pi\)
\(564\) 9.48913 0.399564
\(565\) 10.0000 0.420703
\(566\) 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 \(0.387511\pi\)
\(570\) −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
\(574\) −4.00000 −0.166957
\(575\) −6.74456 −0.281268
\(576\) 1.00000 0.0416667
\(577\) −8.74456 −0.364041 −0.182020 0.983295i \(-0.558264\pi\)
−0.182020 + 0.983295i \(0.558264\pi\)
\(578\) 28.4891 1.18499
\(579\) −4.00000 −0.166234
\(580\) 8.74456 0.363098
\(581\) 8.00000 0.331896
\(582\) −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 \(-0.532760\pi\)
−0.102737 + 0.994709i \(0.532760\pi\)
\(588\) 2.00000 0.0824786
\(589\) −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 \(-0.809445\pi\)
−0.826100 + 0.563524i \(0.809445\pi\)
\(594\) 4.00000 0.164122
\(595\) −6.74456 −0.276500
\(596\) −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 \(-0.739336\pi\)
−0.683025 + 0.730395i \(0.739336\pi\)
\(602\) 4.00000 0.163028
\(603\) −4.00000 −0.162893
\(604\) −9.25544 −0.376598
\(605\) 1.00000 0.0406558
\(606\) 5.48913 0.222980
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −6.74456 −0.273528
\(609\) 17.4891 0.708695
\(610\) 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
\(616\) −1.00000 −0.0402911
\(617\) −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 \(-0.495237\pi\)
0.0149632 + 0.999888i \(0.495237\pi\)
\(620\) 4.74456 0.190546
\(621\) 26.9783 1.08260
\(622\) −20.7446 −0.831781
\(623\) 15.4891 0.620559
\(624\) 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
\(630\) 1.00000 0.0398410
\(631\) −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
\(640\) 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 \(-0.131203\pi\)
0.916247 + 0.400613i \(0.131203\pi\)
\(644\) −6.74456 −0.265773
\(645\) 8.00000 0.315000
\(646\) 45.4891 1.78975
\(647\) 8.74456 0.343784 0.171892 0.985116i \(-0.445012\pi\)
0.171892 + 0.985116i \(0.445012\pi\)
\(648\) −11.0000 −0.432121
\(649\) 8.74456 0.343254
\(650\) 6.74456 0.264544
\(651\) 9.48913 0.371908
\(652\) −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
\(658\) 4.74456 0.184962
\(659\) −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 \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 14.9783 0.582146
\(663\) −90.9783 −3.53330
\(664\) 8.00000 0.310460
\(665\) −6.74456 −0.261543
\(666\) 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
\(672\) 2.00000 0.0771517
\(673\) 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 \(-0.254834\pi\)
0.696287 + 0.717764i \(0.254834\pi\)
\(678\) 20.0000 0.768095
\(679\) −16.7446 −0.642597
\(680\) −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
\(686\) 1.00000 0.0381802
\(687\) −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 \(-0.489304\pi\)
0.0335968 + 0.999435i \(0.489304\pi\)
\(692\) 20.2337 0.769169
\(693\) −1.00000 −0.0379869
\(694\) −22.9783 −0.872242
\(695\) 14.7446 0.559293
\(696\) 17.4891 0.662924
\(697\) 26.9783 1.02187
\(698\) 30.7446 1.16370
\(699\) −49.9565 −1.88953
\(700\) 1.00000 0.0377964
\(701\) 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
\(707\) 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 \(0.496946\pi\)
\(710\) −4.00000 −0.150117
\(711\) 6.74456 0.252941
\(712\) 15.4891 0.580480
\(713\) −32.0000 −1.19841
\(714\) −13.4891 −0.504818
\(715\) −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 \(-0.454001\pi\)
0.144006 + 0.989577i \(0.454001\pi\)
\(720\) 1.00000 0.0372678
\(721\) −0.744563 −0.0277290
\(722\) 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 \(-0.414975\pi\)
0.263949 + 0.964537i \(0.414975\pi\)
\(728\) 6.74456 0.249970
\(729\) 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 \(-0.596921\pi\)
−0.299802 + 0.954001i \(0.596921\pi\)
\(734\) −8.74456 −0.322768
\(735\) 2.00000 0.0737711
\(736\) −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
\(742\) −12.7446 −0.467868
\(743\) 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
\(749\) −12.0000 −0.438470
\(750\) 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
\(756\) −4.00000 −0.145479
\(757\) −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 \(-0.679496\pi\)
−0.534490 + 0.845175i \(0.679496\pi\)
\(762\) 0 0
\(763\) −18.2337 −0.660104
\(764\) −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 \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) −1.00000 −0.0360375
\(771\) −20.4674 −0.737115
\(772\) −2.00000 −0.0719816
\(773\) −51.4891 −1.85194 −0.925968 0.377603i \(-0.876748\pi\)
−0.925968 + 0.377603i \(0.876748\pi\)
\(774\) 4.00000 0.143777
\(775\) 4.74456 0.170430
\(776\) −16.7446 −0.601095
\(777\) 1.48913 0.0534221
\(778\) −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
\(784\) 1.00000 0.0357143
\(785\) 0.510875 0.0182339
\(786\) −5.48913 −0.195791
\(787\) 5.48913 0.195666 0.0978331 0.995203i \(-0.468809\pi\)
0.0978331 + 0.995203i \(0.468809\pi\)
\(788\) 10.0000 0.356235
\(789\) 53.9565 1.92090
\(790\) 6.74456 0.239961
\(791\) 10.0000 0.355559
\(792\) −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 \(-0.770976\pi\)
−0.752136 + 0.659008i \(0.770976\pi\)
\(798\) −13.4891 −0.477510
\(799\) −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
\(805\) −6.74456 −0.237715
\(806\) 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 \(-0.606747\pi\)
−0.329105 + 0.944293i \(0.606747\pi\)
\(812\) 8.74456 0.306874
\(813\) 29.9565 1.05062
\(814\) −0.744563 −0.0260969
\(815\) −4.00000 −0.140114
\(816\) −13.4891 −0.472214
\(817\) −26.9783 −0.943850
\(818\) −21.4891 −0.751350
\(819\) 6.74456 0.235674
\(820\) −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
\(826\) −8.74456 −0.304262
\(827\) 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 \(-0.419163\pi\)
0.251236 + 0.967926i \(0.419163\pi\)
\(830\) 8.00000 0.277684
\(831\) 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 \(0.517897\pi\)
\(840\) 2.00000 0.0690066
\(841\) 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
\(847\) 1.00000 0.0343604
\(848\) −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 \(-0.627311\pi\)
−0.389379 + 0.921077i \(0.627311\pi\)
\(854\) 1.25544 0.0429602
\(855\) −6.74456 −0.230659
\(856\) −12.0000 −0.410152
\(857\) −43.2119 −1.47609 −0.738046 0.674751i \(-0.764251\pi\)
−0.738046 + 0.674751i \(0.764251\pi\)
\(858\) −13.4891 −0.460511
\(859\) −29.2119 −0.996698 −0.498349 0.866976i \(-0.666060\pi\)
−0.498349 + 0.866976i \(0.666060\pi\)
\(860\) 4.00000 0.136399
\(861\) −8.00000 −0.272639
\(862\) −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
\(868\) 4.74456 0.161041
\(869\) −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
\(875\) 1.00000 0.0338062
\(876\) 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 \(-0.884366\pi\)
−0.934738 + 0.355338i \(0.884366\pi\)
\(882\) 1.00000 0.0336718
\(883\) −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 \(-0.745276\pi\)
−0.696534 + 0.717524i \(0.745276\pi\)
\(888\) 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
\(896\) 1.00000 0.0334077
\(897\) −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
\(903\) 8.00000 0.266223
\(904\) 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
\(910\) 6.74456 0.223580
\(911\) 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
\(917\) −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 \(0.864408\pi\)
\(920\) −6.74456 −0.222362
\(921\) −2.97825 −0.0981367
\(922\) 33.7228 1.11060
\(923\) −26.9783 −0.888000
\(924\) −2.00000 −0.0657952
\(925\) 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 \(-0.657687\pi\)
−0.475373 + 0.879784i \(0.657687\pi\)
\(930\) 9.48913 0.311161
\(931\) −6.74456 −0.221044
\(932\) −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 \(-0.599051\pi\)
−0.306179 + 0.951974i \(0.599051\pi\)
\(938\) −4.00000 −0.130605
\(939\) −4.46738 −0.145787
\(940\) 4.74456 0.154751
\(941\) 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
\(945\) −4.00000 −0.130120
\(946\) −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
\(952\) −6.74456 −0.218593
\(953\) 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
\(959\) 3.48913 0.112670
\(960\) 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
\(966\) −13.4891 −0.434005
\(967\) −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 \(-0.397503\pi\)
0.316468 + 0.948603i \(0.397503\pi\)
\(972\) −10.0000 −0.320750
\(973\) 14.7446 0.472689
\(974\) 9.25544 0.296563
\(975\) 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
\(980\) 1.00000 0.0319438
\(981\) −18.2337 −0.582157
\(982\) 30.9783 0.988556
\(983\) 2.23369 0.0712436 0.0356218 0.999365i \(-0.488659\pi\)
0.0356218 + 0.999365i \(0.488659\pi\)
\(984\) −8.00000 −0.255031
\(985\) 10.0000 0.318626
\(986\) −58.9783 −1.87825
\(987\) 9.48913 0.302042
\(988\) −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
\(994\) −4.00000 −0.126872
\(995\) 3.25544 0.103204
\(996\) 16.0000 0.506979
\(997\) 39.2119 1.24185 0.620927 0.783868i \(-0.286756\pi\)
0.620927 + 0.783868i \(0.286756\pi\)
\(998\) −13.4891 −0.426991
\(999\) −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 770.2.a.k.1.2 2
3.2 odd 2 6930.2.a.bo.1.2 2
4.3 odd 2 6160.2.a.r.1.2 2
5.2 odd 4 3850.2.c.y.1849.3 4
5.3 odd 4 3850.2.c.y.1849.2 4
5.4 even 2 3850.2.a.bc.1.1 2
7.6 odd 2 5390.2.a.bq.1.1 2
11.10 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 1.1 even 1 trivial
3850.2.a.bc.1.1 2 5.4 even 2
3850.2.c.y.1849.2 4 5.3 odd 4
3850.2.c.y.1849.3 4 5.2 odd 4
5390.2.a.bq.1.1 2 7.6 odd 2
6160.2.a.r.1.2 2 4.3 odd 2
6930.2.a.bo.1.2 2 3.2 odd 2
8470.2.a.bu.1.1 2 11.10 odd 2