Properties

Label 430.2.a.g.1.1
Level 430
Weight 2
Character 430.1
Self dual yes
Analytic conductor 3.434
Analytic rank 0
Dimension 2
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 430.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\)
Character \(\chi\) = 430.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.41421 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.41421 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +1.00000 q^{10} +3.41421 q^{11} -1.41421 q^{12} +3.82843 q^{13} +1.00000 q^{14} -1.41421 q^{15} +1.00000 q^{16} -1.41421 q^{17} -1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} -1.41421 q^{21} +3.41421 q^{22} +9.07107 q^{23} -1.41421 q^{24} +1.00000 q^{25} +3.82843 q^{26} +5.65685 q^{27} +1.00000 q^{28} -7.24264 q^{29} -1.41421 q^{30} +2.41421 q^{31} +1.00000 q^{32} -4.82843 q^{33} -1.41421 q^{34} +1.00000 q^{35} -1.00000 q^{36} -6.24264 q^{37} -1.00000 q^{38} -5.41421 q^{39} +1.00000 q^{40} +3.82843 q^{41} -1.41421 q^{42} +1.00000 q^{43} +3.41421 q^{44} -1.00000 q^{45} +9.07107 q^{46} -7.07107 q^{47} -1.41421 q^{48} -6.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} +3.82843 q^{52} +5.65685 q^{53} +5.65685 q^{54} +3.41421 q^{55} +1.00000 q^{56} +1.41421 q^{57} -7.24264 q^{58} -6.82843 q^{59} -1.41421 q^{60} -14.0711 q^{61} +2.41421 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.82843 q^{65} -4.82843 q^{66} +7.24264 q^{67} -1.41421 q^{68} -12.8284 q^{69} +1.00000 q^{70} -7.89949 q^{71} -1.00000 q^{72} -0.757359 q^{73} -6.24264 q^{74} -1.41421 q^{75} -1.00000 q^{76} +3.41421 q^{77} -5.41421 q^{78} -5.58579 q^{79} +1.00000 q^{80} -5.00000 q^{81} +3.82843 q^{82} -7.65685 q^{83} -1.41421 q^{84} -1.41421 q^{85} +1.00000 q^{86} +10.2426 q^{87} +3.41421 q^{88} +5.07107 q^{89} -1.00000 q^{90} +3.82843 q^{91} +9.07107 q^{92} -3.41421 q^{93} -7.07107 q^{94} -1.00000 q^{95} -1.41421 q^{96} -17.5563 q^{97} -6.00000 q^{98} -3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + 2q^{8} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + 2q^{8} - 2q^{9} + 2q^{10} + 4q^{11} + 2q^{13} + 2q^{14} + 2q^{16} - 2q^{18} - 2q^{19} + 2q^{20} + 4q^{22} + 4q^{23} + 2q^{25} + 2q^{26} + 2q^{28} - 6q^{29} + 2q^{31} + 2q^{32} - 4q^{33} + 2q^{35} - 2q^{36} - 4q^{37} - 2q^{38} - 8q^{39} + 2q^{40} + 2q^{41} + 2q^{43} + 4q^{44} - 2q^{45} + 4q^{46} - 12q^{49} + 2q^{50} + 4q^{51} + 2q^{52} + 4q^{55} + 2q^{56} - 6q^{58} - 8q^{59} - 14q^{61} + 2q^{62} - 2q^{63} + 2q^{64} + 2q^{65} - 4q^{66} + 6q^{67} - 20q^{69} + 2q^{70} + 4q^{71} - 2q^{72} - 10q^{73} - 4q^{74} - 2q^{76} + 4q^{77} - 8q^{78} - 14q^{79} + 2q^{80} - 10q^{81} + 2q^{82} - 4q^{83} + 2q^{86} + 12q^{87} + 4q^{88} - 4q^{89} - 2q^{90} + 2q^{91} + 4q^{92} - 4q^{93} - 2q^{95} - 4q^{97} - 12q^{98} - 4q^{99} + O(q^{100}) \)

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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.41421 −0.577350
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.00000 −0.333333
\(10\) 1.00000 0.316228
\(11\) 3.41421 1.02942 0.514712 0.857363i \(-0.327899\pi\)
0.514712 + 0.857363i \(0.327899\pi\)
\(12\) −1.41421 −0.408248
\(13\) 3.82843 1.06181 0.530907 0.847430i \(-0.321851\pi\)
0.530907 + 0.847430i \(0.321851\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.41421 −0.365148
\(16\) 1.00000 0.250000
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.41421 −0.308607
\(22\) 3.41421 0.727913
\(23\) 9.07107 1.89145 0.945724 0.324970i \(-0.105354\pi\)
0.945724 + 0.324970i \(0.105354\pi\)
\(24\) −1.41421 −0.288675
\(25\) 1.00000 0.200000
\(26\) 3.82843 0.750816
\(27\) 5.65685 1.08866
\(28\) 1.00000 0.188982
\(29\) −7.24264 −1.34492 −0.672462 0.740131i \(-0.734763\pi\)
−0.672462 + 0.740131i \(0.734763\pi\)
\(30\) −1.41421 −0.258199
\(31\) 2.41421 0.433606 0.216803 0.976215i \(-0.430437\pi\)
0.216803 + 0.976215i \(0.430437\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.82843 −0.840521
\(34\) −1.41421 −0.242536
\(35\) 1.00000 0.169031
\(36\) −1.00000 −0.166667
\(37\) −6.24264 −1.02628 −0.513142 0.858304i \(-0.671519\pi\)
−0.513142 + 0.858304i \(0.671519\pi\)
\(38\) −1.00000 −0.162221
\(39\) −5.41421 −0.866968
\(40\) 1.00000 0.158114
\(41\) 3.82843 0.597900 0.298950 0.954269i \(-0.403364\pi\)
0.298950 + 0.954269i \(0.403364\pi\)
\(42\) −1.41421 −0.218218
\(43\) 1.00000 0.152499
\(44\) 3.41421 0.514712
\(45\) −1.00000 −0.149071
\(46\) 9.07107 1.33746
\(47\) −7.07107 −1.03142 −0.515711 0.856763i \(-0.672472\pi\)
−0.515711 + 0.856763i \(0.672472\pi\)
\(48\) −1.41421 −0.204124
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 2.00000 0.280056
\(52\) 3.82843 0.530907
\(53\) 5.65685 0.777029 0.388514 0.921443i \(-0.372988\pi\)
0.388514 + 0.921443i \(0.372988\pi\)
\(54\) 5.65685 0.769800
\(55\) 3.41421 0.460372
\(56\) 1.00000 0.133631
\(57\) 1.41421 0.187317
\(58\) −7.24264 −0.951005
\(59\) −6.82843 −0.888985 −0.444493 0.895782i \(-0.646616\pi\)
−0.444493 + 0.895782i \(0.646616\pi\)
\(60\) −1.41421 −0.182574
\(61\) −14.0711 −1.80162 −0.900808 0.434218i \(-0.857025\pi\)
−0.900808 + 0.434218i \(0.857025\pi\)
\(62\) 2.41421 0.306605
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 3.82843 0.474858
\(66\) −4.82843 −0.594338
\(67\) 7.24264 0.884829 0.442415 0.896811i \(-0.354122\pi\)
0.442415 + 0.896811i \(0.354122\pi\)
\(68\) −1.41421 −0.171499
\(69\) −12.8284 −1.54436
\(70\) 1.00000 0.119523
\(71\) −7.89949 −0.937498 −0.468749 0.883332i \(-0.655295\pi\)
−0.468749 + 0.883332i \(0.655295\pi\)
\(72\) −1.00000 −0.117851
\(73\) −0.757359 −0.0886422 −0.0443211 0.999017i \(-0.514112\pi\)
−0.0443211 + 0.999017i \(0.514112\pi\)
\(74\) −6.24264 −0.725692
\(75\) −1.41421 −0.163299
\(76\) −1.00000 −0.114708
\(77\) 3.41421 0.389086
\(78\) −5.41421 −0.613039
\(79\) −5.58579 −0.628450 −0.314225 0.949349i \(-0.601745\pi\)
−0.314225 + 0.949349i \(0.601745\pi\)
\(80\) 1.00000 0.111803
\(81\) −5.00000 −0.555556
\(82\) 3.82843 0.422779
\(83\) −7.65685 −0.840449 −0.420224 0.907420i \(-0.638049\pi\)
−0.420224 + 0.907420i \(0.638049\pi\)
\(84\) −1.41421 −0.154303
\(85\) −1.41421 −0.153393
\(86\) 1.00000 0.107833
\(87\) 10.2426 1.09813
\(88\) 3.41421 0.363956
\(89\) 5.07107 0.537532 0.268766 0.963205i \(-0.413384\pi\)
0.268766 + 0.963205i \(0.413384\pi\)
\(90\) −1.00000 −0.105409
\(91\) 3.82843 0.401328
\(92\) 9.07107 0.945724
\(93\) −3.41421 −0.354037
\(94\) −7.07107 −0.729325
\(95\) −1.00000 −0.102598
\(96\) −1.41421 −0.144338
\(97\) −17.5563 −1.78258 −0.891289 0.453436i \(-0.850198\pi\)
−0.891289 + 0.453436i \(0.850198\pi\)
\(98\) −6.00000 −0.606092
\(99\) −3.41421 −0.343141
\(100\) 1.00000 0.100000
\(101\) −1.41421 −0.140720 −0.0703598 0.997522i \(-0.522415\pi\)
−0.0703598 + 0.997522i \(0.522415\pi\)
\(102\) 2.00000 0.198030
\(103\) −4.82843 −0.475759 −0.237880 0.971295i \(-0.576452\pi\)
−0.237880 + 0.971295i \(0.576452\pi\)
\(104\) 3.82843 0.375408
\(105\) −1.41421 −0.138013
\(106\) 5.65685 0.549442
\(107\) 16.0711 1.55365 0.776824 0.629717i \(-0.216829\pi\)
0.776824 + 0.629717i \(0.216829\pi\)
\(108\) 5.65685 0.544331
\(109\) −5.17157 −0.495347 −0.247673 0.968844i \(-0.579666\pi\)
−0.247673 + 0.968844i \(0.579666\pi\)
\(110\) 3.41421 0.325532
\(111\) 8.82843 0.837957
\(112\) 1.00000 0.0944911
\(113\) −14.8995 −1.40163 −0.700813 0.713345i \(-0.747179\pi\)
−0.700813 + 0.713345i \(0.747179\pi\)
\(114\) 1.41421 0.132453
\(115\) 9.07107 0.845881
\(116\) −7.24264 −0.672462
\(117\) −3.82843 −0.353938
\(118\) −6.82843 −0.628608
\(119\) −1.41421 −0.129641
\(120\) −1.41421 −0.129099
\(121\) 0.656854 0.0597140
\(122\) −14.0711 −1.27393
\(123\) −5.41421 −0.488183
\(124\) 2.41421 0.216803
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) 20.4853 1.81777 0.908887 0.417042i \(-0.136933\pi\)
0.908887 + 0.417042i \(0.136933\pi\)
\(128\) 1.00000 0.0883883
\(129\) −1.41421 −0.124515
\(130\) 3.82843 0.335775
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) −4.82843 −0.420261
\(133\) −1.00000 −0.0867110
\(134\) 7.24264 0.625669
\(135\) 5.65685 0.486864
\(136\) −1.41421 −0.121268
\(137\) 18.0711 1.54392 0.771958 0.635674i \(-0.219278\pi\)
0.771958 + 0.635674i \(0.219278\pi\)
\(138\) −12.8284 −1.09203
\(139\) 2.24264 0.190218 0.0951092 0.995467i \(-0.469680\pi\)
0.0951092 + 0.995467i \(0.469680\pi\)
\(140\) 1.00000 0.0845154
\(141\) 10.0000 0.842152
\(142\) −7.89949 −0.662911
\(143\) 13.0711 1.09306
\(144\) −1.00000 −0.0833333
\(145\) −7.24264 −0.601469
\(146\) −0.757359 −0.0626795
\(147\) 8.48528 0.699854
\(148\) −6.24264 −0.513142
\(149\) 20.5563 1.68404 0.842021 0.539445i \(-0.181366\pi\)
0.842021 + 0.539445i \(0.181366\pi\)
\(150\) −1.41421 −0.115470
\(151\) −6.48528 −0.527765 −0.263882 0.964555i \(-0.585003\pi\)
−0.263882 + 0.964555i \(0.585003\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 1.41421 0.114332
\(154\) 3.41421 0.275125
\(155\) 2.41421 0.193914
\(156\) −5.41421 −0.433484
\(157\) 1.75736 0.140253 0.0701263 0.997538i \(-0.477660\pi\)
0.0701263 + 0.997538i \(0.477660\pi\)
\(158\) −5.58579 −0.444381
\(159\) −8.00000 −0.634441
\(160\) 1.00000 0.0790569
\(161\) 9.07107 0.714900
\(162\) −5.00000 −0.392837
\(163\) 2.34315 0.183529 0.0917647 0.995781i \(-0.470749\pi\)
0.0917647 + 0.995781i \(0.470749\pi\)
\(164\) 3.82843 0.298950
\(165\) −4.82843 −0.375893
\(166\) −7.65685 −0.594287
\(167\) −17.5563 −1.35855 −0.679276 0.733883i \(-0.737706\pi\)
−0.679276 + 0.733883i \(0.737706\pi\)
\(168\) −1.41421 −0.109109
\(169\) 1.65685 0.127450
\(170\) −1.41421 −0.108465
\(171\) 1.00000 0.0764719
\(172\) 1.00000 0.0762493
\(173\) 17.4853 1.32938 0.664691 0.747119i \(-0.268563\pi\)
0.664691 + 0.747119i \(0.268563\pi\)
\(174\) 10.2426 0.776493
\(175\) 1.00000 0.0755929
\(176\) 3.41421 0.257356
\(177\) 9.65685 0.725854
\(178\) 5.07107 0.380093
\(179\) 0.514719 0.0384719 0.0192359 0.999815i \(-0.493877\pi\)
0.0192359 + 0.999815i \(0.493877\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −0.928932 −0.0690470 −0.0345235 0.999404i \(-0.510991\pi\)
−0.0345235 + 0.999404i \(0.510991\pi\)
\(182\) 3.82843 0.283782
\(183\) 19.8995 1.47101
\(184\) 9.07107 0.668728
\(185\) −6.24264 −0.458968
\(186\) −3.41421 −0.250342
\(187\) −4.82843 −0.353090
\(188\) −7.07107 −0.515711
\(189\) 5.65685 0.411476
\(190\) −1.00000 −0.0725476
\(191\) 0.100505 0.00727229 0.00363615 0.999993i \(-0.498843\pi\)
0.00363615 + 0.999993i \(0.498843\pi\)
\(192\) −1.41421 −0.102062
\(193\) 1.17157 0.0843317 0.0421658 0.999111i \(-0.486574\pi\)
0.0421658 + 0.999111i \(0.486574\pi\)
\(194\) −17.5563 −1.26047
\(195\) −5.41421 −0.387720
\(196\) −6.00000 −0.428571
\(197\) −3.14214 −0.223868 −0.111934 0.993716i \(-0.535705\pi\)
−0.111934 + 0.993716i \(0.535705\pi\)
\(198\) −3.41421 −0.242638
\(199\) −19.8995 −1.41064 −0.705319 0.708890i \(-0.749196\pi\)
−0.705319 + 0.708890i \(0.749196\pi\)
\(200\) 1.00000 0.0707107
\(201\) −10.2426 −0.722460
\(202\) −1.41421 −0.0995037
\(203\) −7.24264 −0.508334
\(204\) 2.00000 0.140028
\(205\) 3.82843 0.267389
\(206\) −4.82843 −0.336412
\(207\) −9.07107 −0.630483
\(208\) 3.82843 0.265454
\(209\) −3.41421 −0.236166
\(210\) −1.41421 −0.0975900
\(211\) 9.65685 0.664805 0.332403 0.943138i \(-0.392141\pi\)
0.332403 + 0.943138i \(0.392141\pi\)
\(212\) 5.65685 0.388514
\(213\) 11.1716 0.765464
\(214\) 16.0711 1.09860
\(215\) 1.00000 0.0681994
\(216\) 5.65685 0.384900
\(217\) 2.41421 0.163887
\(218\) −5.17157 −0.350263
\(219\) 1.07107 0.0723761
\(220\) 3.41421 0.230186
\(221\) −5.41421 −0.364199
\(222\) 8.82843 0.592525
\(223\) −12.1421 −0.813098 −0.406549 0.913629i \(-0.633268\pi\)
−0.406549 + 0.913629i \(0.633268\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) −14.8995 −0.991100
\(227\) 14.8284 0.984197 0.492099 0.870539i \(-0.336230\pi\)
0.492099 + 0.870539i \(0.336230\pi\)
\(228\) 1.41421 0.0936586
\(229\) −4.82843 −0.319071 −0.159536 0.987192i \(-0.551000\pi\)
−0.159536 + 0.987192i \(0.551000\pi\)
\(230\) 9.07107 0.598128
\(231\) −4.82843 −0.317687
\(232\) −7.24264 −0.475503
\(233\) −8.48528 −0.555889 −0.277945 0.960597i \(-0.589653\pi\)
−0.277945 + 0.960597i \(0.589653\pi\)
\(234\) −3.82843 −0.250272
\(235\) −7.07107 −0.461266
\(236\) −6.82843 −0.444493
\(237\) 7.89949 0.513127
\(238\) −1.41421 −0.0916698
\(239\) 14.4142 0.932378 0.466189 0.884685i \(-0.345627\pi\)
0.466189 + 0.884685i \(0.345627\pi\)
\(240\) −1.41421 −0.0912871
\(241\) −18.5858 −1.19722 −0.598608 0.801042i \(-0.704279\pi\)
−0.598608 + 0.801042i \(0.704279\pi\)
\(242\) 0.656854 0.0422242
\(243\) −9.89949 −0.635053
\(244\) −14.0711 −0.900808
\(245\) −6.00000 −0.383326
\(246\) −5.41421 −0.345198
\(247\) −3.82843 −0.243597
\(248\) 2.41421 0.153303
\(249\) 10.8284 0.686224
\(250\) 1.00000 0.0632456
\(251\) 22.9706 1.44989 0.724945 0.688807i \(-0.241865\pi\)
0.724945 + 0.688807i \(0.241865\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 30.9706 1.94710
\(254\) 20.4853 1.28536
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −10.8995 −0.679892 −0.339946 0.940445i \(-0.610409\pi\)
−0.339946 + 0.940445i \(0.610409\pi\)
\(258\) −1.41421 −0.0880451
\(259\) −6.24264 −0.387899
\(260\) 3.82843 0.237429
\(261\) 7.24264 0.448308
\(262\) 6.00000 0.370681
\(263\) 0.514719 0.0317389 0.0158695 0.999874i \(-0.494948\pi\)
0.0158695 + 0.999874i \(0.494948\pi\)
\(264\) −4.82843 −0.297169
\(265\) 5.65685 0.347498
\(266\) −1.00000 −0.0613139
\(267\) −7.17157 −0.438893
\(268\) 7.24264 0.442415
\(269\) 14.8284 0.904105 0.452053 0.891991i \(-0.350692\pi\)
0.452053 + 0.891991i \(0.350692\pi\)
\(270\) 5.65685 0.344265
\(271\) −10.8995 −0.662097 −0.331049 0.943614i \(-0.607402\pi\)
−0.331049 + 0.943614i \(0.607402\pi\)
\(272\) −1.41421 −0.0857493
\(273\) −5.41421 −0.327683
\(274\) 18.0711 1.09171
\(275\) 3.41421 0.205885
\(276\) −12.8284 −0.772181
\(277\) 16.4853 0.990505 0.495252 0.868749i \(-0.335076\pi\)
0.495252 + 0.868749i \(0.335076\pi\)
\(278\) 2.24264 0.134505
\(279\) −2.41421 −0.144535
\(280\) 1.00000 0.0597614
\(281\) −12.6569 −0.755045 −0.377522 0.926000i \(-0.623224\pi\)
−0.377522 + 0.926000i \(0.623224\pi\)
\(282\) 10.0000 0.595491
\(283\) 27.2426 1.61941 0.809703 0.586839i \(-0.199628\pi\)
0.809703 + 0.586839i \(0.199628\pi\)
\(284\) −7.89949 −0.468749
\(285\) 1.41421 0.0837708
\(286\) 13.0711 0.772908
\(287\) 3.82843 0.225985
\(288\) −1.00000 −0.0589256
\(289\) −15.0000 −0.882353
\(290\) −7.24264 −0.425303
\(291\) 24.8284 1.45547
\(292\) −0.757359 −0.0443211
\(293\) −22.9706 −1.34195 −0.670977 0.741478i \(-0.734125\pi\)
−0.670977 + 0.741478i \(0.734125\pi\)
\(294\) 8.48528 0.494872
\(295\) −6.82843 −0.397566
\(296\) −6.24264 −0.362846
\(297\) 19.3137 1.12070
\(298\) 20.5563 1.19080
\(299\) 34.7279 2.00837
\(300\) −1.41421 −0.0816497
\(301\) 1.00000 0.0576390
\(302\) −6.48528 −0.373186
\(303\) 2.00000 0.114897
\(304\) −1.00000 −0.0573539
\(305\) −14.0711 −0.805707
\(306\) 1.41421 0.0808452
\(307\) −20.0711 −1.14552 −0.572758 0.819724i \(-0.694127\pi\)
−0.572758 + 0.819724i \(0.694127\pi\)
\(308\) 3.41421 0.194543
\(309\) 6.82843 0.388456
\(310\) 2.41421 0.137118
\(311\) 10.0711 0.571078 0.285539 0.958367i \(-0.407827\pi\)
0.285539 + 0.958367i \(0.407827\pi\)
\(312\) −5.41421 −0.306519
\(313\) 21.3137 1.20472 0.602361 0.798224i \(-0.294227\pi\)
0.602361 + 0.798224i \(0.294227\pi\)
\(314\) 1.75736 0.0991735
\(315\) −1.00000 −0.0563436
\(316\) −5.58579 −0.314225
\(317\) 29.1421 1.63679 0.818393 0.574659i \(-0.194865\pi\)
0.818393 + 0.574659i \(0.194865\pi\)
\(318\) −8.00000 −0.448618
\(319\) −24.7279 −1.38450
\(320\) 1.00000 0.0559017
\(321\) −22.7279 −1.26855
\(322\) 9.07107 0.505511
\(323\) 1.41421 0.0786889
\(324\) −5.00000 −0.277778
\(325\) 3.82843 0.212363
\(326\) 2.34315 0.129775
\(327\) 7.31371 0.404449
\(328\) 3.82843 0.211390
\(329\) −7.07107 −0.389841
\(330\) −4.82843 −0.265796
\(331\) −2.68629 −0.147652 −0.0738260 0.997271i \(-0.523521\pi\)
−0.0738260 + 0.997271i \(0.523521\pi\)
\(332\) −7.65685 −0.420224
\(333\) 6.24264 0.342095
\(334\) −17.5563 −0.960641
\(335\) 7.24264 0.395708
\(336\) −1.41421 −0.0771517
\(337\) −19.6569 −1.07078 −0.535389 0.844606i \(-0.679835\pi\)
−0.535389 + 0.844606i \(0.679835\pi\)
\(338\) 1.65685 0.0901210
\(339\) 21.0711 1.14442
\(340\) −1.41421 −0.0766965
\(341\) 8.24264 0.446364
\(342\) 1.00000 0.0540738
\(343\) −13.0000 −0.701934
\(344\) 1.00000 0.0539164
\(345\) −12.8284 −0.690659
\(346\) 17.4853 0.940015
\(347\) −13.7574 −0.738534 −0.369267 0.929323i \(-0.620391\pi\)
−0.369267 + 0.929323i \(0.620391\pi\)
\(348\) 10.2426 0.549063
\(349\) −27.4558 −1.46968 −0.734839 0.678242i \(-0.762742\pi\)
−0.734839 + 0.678242i \(0.762742\pi\)
\(350\) 1.00000 0.0534522
\(351\) 21.6569 1.15596
\(352\) 3.41421 0.181978
\(353\) 4.38478 0.233378 0.116689 0.993168i \(-0.462772\pi\)
0.116689 + 0.993168i \(0.462772\pi\)
\(354\) 9.65685 0.513256
\(355\) −7.89949 −0.419262
\(356\) 5.07107 0.268766
\(357\) 2.00000 0.105851
\(358\) 0.514719 0.0272037
\(359\) −37.5269 −1.98059 −0.990297 0.138965i \(-0.955623\pi\)
−0.990297 + 0.138965i \(0.955623\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.0000 −0.947368
\(362\) −0.928932 −0.0488236
\(363\) −0.928932 −0.0487563
\(364\) 3.82843 0.200664
\(365\) −0.757359 −0.0396420
\(366\) 19.8995 1.04016
\(367\) −8.68629 −0.453421 −0.226710 0.973962i \(-0.572797\pi\)
−0.226710 + 0.973962i \(0.572797\pi\)
\(368\) 9.07107 0.472862
\(369\) −3.82843 −0.199300
\(370\) −6.24264 −0.324539
\(371\) 5.65685 0.293689
\(372\) −3.41421 −0.177019
\(373\) −10.7279 −0.555471 −0.277735 0.960658i \(-0.589584\pi\)
−0.277735 + 0.960658i \(0.589584\pi\)
\(374\) −4.82843 −0.249672
\(375\) −1.41421 −0.0730297
\(376\) −7.07107 −0.364662
\(377\) −27.7279 −1.42806
\(378\) 5.65685 0.290957
\(379\) −24.1421 −1.24010 −0.620049 0.784563i \(-0.712887\pi\)
−0.620049 + 0.784563i \(0.712887\pi\)
\(380\) −1.00000 −0.0512989
\(381\) −28.9706 −1.48421
\(382\) 0.100505 0.00514229
\(383\) 14.3137 0.731396 0.365698 0.930734i \(-0.380830\pi\)
0.365698 + 0.930734i \(0.380830\pi\)
\(384\) −1.41421 −0.0721688
\(385\) 3.41421 0.174004
\(386\) 1.17157 0.0596315
\(387\) −1.00000 −0.0508329
\(388\) −17.5563 −0.891289
\(389\) 15.1716 0.769229 0.384615 0.923077i \(-0.374334\pi\)
0.384615 + 0.923077i \(0.374334\pi\)
\(390\) −5.41421 −0.274159
\(391\) −12.8284 −0.648761
\(392\) −6.00000 −0.303046
\(393\) −8.48528 −0.428026
\(394\) −3.14214 −0.158299
\(395\) −5.58579 −0.281051
\(396\) −3.41421 −0.171571
\(397\) −3.51472 −0.176399 −0.0881993 0.996103i \(-0.528111\pi\)
−0.0881993 + 0.996103i \(0.528111\pi\)
\(398\) −19.8995 −0.997472
\(399\) 1.41421 0.0707992
\(400\) 1.00000 0.0500000
\(401\) 38.1127 1.90326 0.951629 0.307251i \(-0.0994090\pi\)
0.951629 + 0.307251i \(0.0994090\pi\)
\(402\) −10.2426 −0.510856
\(403\) 9.24264 0.460409
\(404\) −1.41421 −0.0703598
\(405\) −5.00000 −0.248452
\(406\) −7.24264 −0.359446
\(407\) −21.3137 −1.05648
\(408\) 2.00000 0.0990148
\(409\) 15.2132 0.752244 0.376122 0.926570i \(-0.377257\pi\)
0.376122 + 0.926570i \(0.377257\pi\)
\(410\) 3.82843 0.189073
\(411\) −25.5563 −1.26060
\(412\) −4.82843 −0.237880
\(413\) −6.82843 −0.336005
\(414\) −9.07107 −0.445819
\(415\) −7.65685 −0.375860
\(416\) 3.82843 0.187704
\(417\) −3.17157 −0.155313
\(418\) −3.41421 −0.166995
\(419\) 26.7990 1.30922 0.654608 0.755968i \(-0.272834\pi\)
0.654608 + 0.755968i \(0.272834\pi\)
\(420\) −1.41421 −0.0690066
\(421\) −7.24264 −0.352985 −0.176492 0.984302i \(-0.556475\pi\)
−0.176492 + 0.984302i \(0.556475\pi\)
\(422\) 9.65685 0.470088
\(423\) 7.07107 0.343807
\(424\) 5.65685 0.274721
\(425\) −1.41421 −0.0685994
\(426\) 11.1716 0.541264
\(427\) −14.0711 −0.680947
\(428\) 16.0711 0.776824
\(429\) −18.4853 −0.892478
\(430\) 1.00000 0.0482243
\(431\) 34.9706 1.68447 0.842236 0.539108i \(-0.181239\pi\)
0.842236 + 0.539108i \(0.181239\pi\)
\(432\) 5.65685 0.272166
\(433\) −21.9289 −1.05384 −0.526919 0.849916i \(-0.676653\pi\)
−0.526919 + 0.849916i \(0.676653\pi\)
\(434\) 2.41421 0.115886
\(435\) 10.2426 0.491097
\(436\) −5.17157 −0.247673
\(437\) −9.07107 −0.433928
\(438\) 1.07107 0.0511776
\(439\) 7.31371 0.349064 0.174532 0.984651i \(-0.444159\pi\)
0.174532 + 0.984651i \(0.444159\pi\)
\(440\) 3.41421 0.162766
\(441\) 6.00000 0.285714
\(442\) −5.41421 −0.257528
\(443\) −24.0711 −1.14365 −0.571825 0.820375i \(-0.693765\pi\)
−0.571825 + 0.820375i \(0.693765\pi\)
\(444\) 8.82843 0.418979
\(445\) 5.07107 0.240392
\(446\) −12.1421 −0.574947
\(447\) −29.0711 −1.37501
\(448\) 1.00000 0.0472456
\(449\) 13.7574 0.649250 0.324625 0.945843i \(-0.394762\pi\)
0.324625 + 0.945843i \(0.394762\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 13.0711 0.615493
\(452\) −14.8995 −0.700813
\(453\) 9.17157 0.430918
\(454\) 14.8284 0.695933
\(455\) 3.82843 0.179479
\(456\) 1.41421 0.0662266
\(457\) −13.5147 −0.632192 −0.316096 0.948727i \(-0.602372\pi\)
−0.316096 + 0.948727i \(0.602372\pi\)
\(458\) −4.82843 −0.225618
\(459\) −8.00000 −0.373408
\(460\) 9.07107 0.422941
\(461\) −0.928932 −0.0432647 −0.0216323 0.999766i \(-0.506886\pi\)
−0.0216323 + 0.999766i \(0.506886\pi\)
\(462\) −4.82843 −0.224639
\(463\) 7.68629 0.357212 0.178606 0.983921i \(-0.442841\pi\)
0.178606 + 0.983921i \(0.442841\pi\)
\(464\) −7.24264 −0.336231
\(465\) −3.41421 −0.158330
\(466\) −8.48528 −0.393073
\(467\) −35.0711 −1.62290 −0.811448 0.584425i \(-0.801320\pi\)
−0.811448 + 0.584425i \(0.801320\pi\)
\(468\) −3.82843 −0.176969
\(469\) 7.24264 0.334434
\(470\) −7.07107 −0.326164
\(471\) −2.48528 −0.114516
\(472\) −6.82843 −0.314304
\(473\) 3.41421 0.156986
\(474\) 7.89949 0.362836
\(475\) −1.00000 −0.0458831
\(476\) −1.41421 −0.0648204
\(477\) −5.65685 −0.259010
\(478\) 14.4142 0.659291
\(479\) 14.8284 0.677528 0.338764 0.940871i \(-0.389991\pi\)
0.338764 + 0.940871i \(0.389991\pi\)
\(480\) −1.41421 −0.0645497
\(481\) −23.8995 −1.08972
\(482\) −18.5858 −0.846559
\(483\) −12.8284 −0.583714
\(484\) 0.656854 0.0298570
\(485\) −17.5563 −0.797193
\(486\) −9.89949 −0.449050
\(487\) 9.21320 0.417490 0.208745 0.977970i \(-0.433062\pi\)
0.208745 + 0.977970i \(0.433062\pi\)
\(488\) −14.0711 −0.636967
\(489\) −3.31371 −0.149851
\(490\) −6.00000 −0.271052
\(491\) −28.6274 −1.29194 −0.645969 0.763364i \(-0.723546\pi\)
−0.645969 + 0.763364i \(0.723546\pi\)
\(492\) −5.41421 −0.244092
\(493\) 10.2426 0.461305
\(494\) −3.82843 −0.172249
\(495\) −3.41421 −0.153457
\(496\) 2.41421 0.108401
\(497\) −7.89949 −0.354341
\(498\) 10.8284 0.485233
\(499\) 16.4558 0.736665 0.368332 0.929694i \(-0.379929\pi\)
0.368332 + 0.929694i \(0.379929\pi\)
\(500\) 1.00000 0.0447214
\(501\) 24.8284 1.10925
\(502\) 22.9706 1.02523
\(503\) −20.3431 −0.907056 −0.453528 0.891242i \(-0.649835\pi\)
−0.453528 + 0.891242i \(0.649835\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −1.41421 −0.0629317
\(506\) 30.9706 1.37681
\(507\) −2.34315 −0.104063
\(508\) 20.4853 0.908887
\(509\) −3.51472 −0.155787 −0.0778936 0.996962i \(-0.524819\pi\)
−0.0778936 + 0.996962i \(0.524819\pi\)
\(510\) 2.00000 0.0885615
\(511\) −0.757359 −0.0335036
\(512\) 1.00000 0.0441942
\(513\) −5.65685 −0.249756
\(514\) −10.8995 −0.480756
\(515\) −4.82843 −0.212766
\(516\) −1.41421 −0.0622573
\(517\) −24.1421 −1.06177
\(518\) −6.24264 −0.274286
\(519\) −24.7279 −1.08544
\(520\) 3.82843 0.167888
\(521\) 3.07107 0.134546 0.0672730 0.997735i \(-0.478570\pi\)
0.0672730 + 0.997735i \(0.478570\pi\)
\(522\) 7.24264 0.317002
\(523\) −14.7279 −0.644007 −0.322004 0.946738i \(-0.604356\pi\)
−0.322004 + 0.946738i \(0.604356\pi\)
\(524\) 6.00000 0.262111
\(525\) −1.41421 −0.0617213
\(526\) 0.514719 0.0224428
\(527\) −3.41421 −0.148725
\(528\) −4.82843 −0.210130
\(529\) 59.2843 2.57758
\(530\) 5.65685 0.245718
\(531\) 6.82843 0.296328
\(532\) −1.00000 −0.0433555
\(533\) 14.6569 0.634859
\(534\) −7.17157 −0.310344
\(535\) 16.0711 0.694813
\(536\) 7.24264 0.312834
\(537\) −0.727922 −0.0314122
\(538\) 14.8284 0.639299
\(539\) −20.4853 −0.882364
\(540\) 5.65685 0.243432
\(541\) 36.1421 1.55387 0.776936 0.629580i \(-0.216773\pi\)
0.776936 + 0.629580i \(0.216773\pi\)
\(542\) −10.8995 −0.468173
\(543\) 1.31371 0.0563766
\(544\) −1.41421 −0.0606339
\(545\) −5.17157 −0.221526
\(546\) −5.41421 −0.231707
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 18.0711 0.771958
\(549\) 14.0711 0.600539
\(550\) 3.41421 0.145583
\(551\) 7.24264 0.308547
\(552\) −12.8284 −0.546014
\(553\) −5.58579 −0.237532
\(554\) 16.4853 0.700392
\(555\) 8.82843 0.374746
\(556\) 2.24264 0.0951092
\(557\) 36.1716 1.53264 0.766319 0.642460i \(-0.222086\pi\)
0.766319 + 0.642460i \(0.222086\pi\)
\(558\) −2.41421 −0.102202
\(559\) 3.82843 0.161925
\(560\) 1.00000 0.0422577
\(561\) 6.82843 0.288296
\(562\) −12.6569 −0.533897
\(563\) −16.0711 −0.677315 −0.338657 0.940910i \(-0.609973\pi\)
−0.338657 + 0.940910i \(0.609973\pi\)
\(564\) 10.0000 0.421076
\(565\) −14.8995 −0.626826
\(566\) 27.2426 1.14509
\(567\) −5.00000 −0.209980
\(568\) −7.89949 −0.331455
\(569\) −16.7990 −0.704250 −0.352125 0.935953i \(-0.614541\pi\)
−0.352125 + 0.935953i \(0.614541\pi\)
\(570\) 1.41421 0.0592349
\(571\) 33.0000 1.38101 0.690504 0.723329i \(-0.257389\pi\)
0.690504 + 0.723329i \(0.257389\pi\)
\(572\) 13.0711 0.546529
\(573\) −0.142136 −0.00593780
\(574\) 3.82843 0.159795
\(575\) 9.07107 0.378290
\(576\) −1.00000 −0.0416667
\(577\) −8.41421 −0.350288 −0.175144 0.984543i \(-0.556039\pi\)
−0.175144 + 0.984543i \(0.556039\pi\)
\(578\) −15.0000 −0.623918
\(579\) −1.65685 −0.0688565
\(580\) −7.24264 −0.300734
\(581\) −7.65685 −0.317660
\(582\) 24.8284 1.02917
\(583\) 19.3137 0.799892
\(584\) −0.757359 −0.0313398
\(585\) −3.82843 −0.158286
\(586\) −22.9706 −0.948905
\(587\) 3.55635 0.146786 0.0733931 0.997303i \(-0.476617\pi\)
0.0733931 + 0.997303i \(0.476617\pi\)
\(588\) 8.48528 0.349927
\(589\) −2.41421 −0.0994759
\(590\) −6.82843 −0.281122
\(591\) 4.44365 0.182787
\(592\) −6.24264 −0.256571
\(593\) 1.24264 0.0510291 0.0255146 0.999674i \(-0.491878\pi\)
0.0255146 + 0.999674i \(0.491878\pi\)
\(594\) 19.3137 0.792451
\(595\) −1.41421 −0.0579771
\(596\) 20.5563 0.842021
\(597\) 28.1421 1.15178
\(598\) 34.7279 1.42013
\(599\) 38.1421 1.55845 0.779223 0.626747i \(-0.215614\pi\)
0.779223 + 0.626747i \(0.215614\pi\)
\(600\) −1.41421 −0.0577350
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 1.00000 0.0407570
\(603\) −7.24264 −0.294943
\(604\) −6.48528 −0.263882
\(605\) 0.656854 0.0267049
\(606\) 2.00000 0.0812444
\(607\) 6.00000 0.243532 0.121766 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 10.2426 0.415053
\(610\) −14.0711 −0.569721
\(611\) −27.0711 −1.09518
\(612\) 1.41421 0.0571662
\(613\) 43.8284 1.77021 0.885107 0.465388i \(-0.154085\pi\)
0.885107 + 0.465388i \(0.154085\pi\)
\(614\) −20.0711 −0.810002
\(615\) −5.41421 −0.218322
\(616\) 3.41421 0.137563
\(617\) 40.2843 1.62178 0.810892 0.585196i \(-0.198982\pi\)
0.810892 + 0.585196i \(0.198982\pi\)
\(618\) 6.82843 0.274680
\(619\) −19.8995 −0.799828 −0.399914 0.916553i \(-0.630960\pi\)
−0.399914 + 0.916553i \(0.630960\pi\)
\(620\) 2.41421 0.0969571
\(621\) 51.3137 2.05915
\(622\) 10.0711 0.403813
\(623\) 5.07107 0.203168
\(624\) −5.41421 −0.216742
\(625\) 1.00000 0.0400000
\(626\) 21.3137 0.851867
\(627\) 4.82843 0.192829
\(628\) 1.75736 0.0701263
\(629\) 8.82843 0.352012
\(630\) −1.00000 −0.0398410
\(631\) 45.6985 1.81923 0.909614 0.415454i \(-0.136377\pi\)
0.909614 + 0.415454i \(0.136377\pi\)
\(632\) −5.58579 −0.222191
\(633\) −13.6569 −0.542811
\(634\) 29.1421 1.15738
\(635\) 20.4853 0.812934
\(636\) −8.00000 −0.317221
\(637\) −22.9706 −0.910127
\(638\) −24.7279 −0.978988
\(639\) 7.89949 0.312499
\(640\) 1.00000 0.0395285
\(641\) −30.7696 −1.21532 −0.607662 0.794196i \(-0.707893\pi\)
−0.607662 + 0.794196i \(0.707893\pi\)
\(642\) −22.7279 −0.897000
\(643\) 14.2721 0.562836 0.281418 0.959585i \(-0.409195\pi\)
0.281418 + 0.959585i \(0.409195\pi\)
\(644\) 9.07107 0.357450
\(645\) −1.41421 −0.0556846
\(646\) 1.41421 0.0556415
\(647\) 44.7990 1.76123 0.880615 0.473832i \(-0.157130\pi\)
0.880615 + 0.473832i \(0.157130\pi\)
\(648\) −5.00000 −0.196419
\(649\) −23.3137 −0.915143
\(650\) 3.82843 0.150163
\(651\) −3.41421 −0.133814
\(652\) 2.34315 0.0917647
\(653\) −9.41421 −0.368407 −0.184203 0.982888i \(-0.558970\pi\)
−0.184203 + 0.982888i \(0.558970\pi\)
\(654\) 7.31371 0.285989
\(655\) 6.00000 0.234439
\(656\) 3.82843 0.149475
\(657\) 0.757359 0.0295474
\(658\) −7.07107 −0.275659
\(659\) −12.3848 −0.482442 −0.241221 0.970470i \(-0.577548\pi\)
−0.241221 + 0.970470i \(0.577548\pi\)
\(660\) −4.82843 −0.187946
\(661\) −28.8701 −1.12292 −0.561458 0.827506i \(-0.689759\pi\)
−0.561458 + 0.827506i \(0.689759\pi\)
\(662\) −2.68629 −0.104406
\(663\) 7.65685 0.297368
\(664\) −7.65685 −0.297144
\(665\) −1.00000 −0.0387783
\(666\) 6.24264 0.241897
\(667\) −65.6985 −2.54386
\(668\) −17.5563 −0.679276
\(669\) 17.1716 0.663891
\(670\) 7.24264 0.279808
\(671\) −48.0416 −1.85463
\(672\) −1.41421 −0.0545545
\(673\) 17.5858 0.677882 0.338941 0.940808i \(-0.389931\pi\)
0.338941 + 0.940808i \(0.389931\pi\)
\(674\) −19.6569 −0.757154
\(675\) 5.65685 0.217732
\(676\) 1.65685 0.0637252
\(677\) −7.45584 −0.286551 −0.143276 0.989683i \(-0.545764\pi\)
−0.143276 + 0.989683i \(0.545764\pi\)
\(678\) 21.0711 0.809229
\(679\) −17.5563 −0.673751
\(680\) −1.41421 −0.0542326
\(681\) −20.9706 −0.803594
\(682\) 8.24264 0.315627
\(683\) −20.4853 −0.783848 −0.391924 0.919998i \(-0.628190\pi\)
−0.391924 + 0.919998i \(0.628190\pi\)
\(684\) 1.00000 0.0382360
\(685\) 18.0711 0.690460
\(686\) −13.0000 −0.496342
\(687\) 6.82843 0.260521
\(688\) 1.00000 0.0381246
\(689\) 21.6569 0.825060
\(690\) −12.8284 −0.488370
\(691\) 1.51472 0.0576226 0.0288113 0.999585i \(-0.490828\pi\)
0.0288113 + 0.999585i \(0.490828\pi\)
\(692\) 17.4853 0.664691
\(693\) −3.41421 −0.129695
\(694\) −13.7574 −0.522222
\(695\) 2.24264 0.0850682
\(696\) 10.2426 0.388246
\(697\) −5.41421 −0.205078
\(698\) −27.4558 −1.03922
\(699\) 12.0000 0.453882
\(700\) 1.00000 0.0377964
\(701\) 36.7279 1.38719 0.693597 0.720363i \(-0.256025\pi\)
0.693597 + 0.720363i \(0.256025\pi\)
\(702\) 21.6569 0.817385
\(703\) 6.24264 0.235446
\(704\) 3.41421 0.128678
\(705\) 10.0000 0.376622
\(706\) 4.38478 0.165023
\(707\) −1.41421 −0.0531870
\(708\) 9.65685 0.362927
\(709\) 35.2132 1.32246 0.661230 0.750183i \(-0.270035\pi\)
0.661230 + 0.750183i \(0.270035\pi\)
\(710\) −7.89949 −0.296463
\(711\) 5.58579 0.209483
\(712\) 5.07107 0.190046
\(713\) 21.8995 0.820143
\(714\) 2.00000 0.0748481
\(715\) 13.0711 0.488830
\(716\) 0.514719 0.0192359
\(717\) −20.3848 −0.761283
\(718\) −37.5269 −1.40049
\(719\) 8.62742 0.321748 0.160874 0.986975i \(-0.448569\pi\)
0.160874 + 0.986975i \(0.448569\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −4.82843 −0.179820
\(722\) −18.0000 −0.669891
\(723\) 26.2843 0.977523
\(724\) −0.928932 −0.0345235
\(725\) −7.24264 −0.268985
\(726\) −0.928932 −0.0344759
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 3.82843 0.141891
\(729\) 29.0000 1.07407
\(730\) −0.757359 −0.0280311
\(731\) −1.41421 −0.0523066
\(732\) 19.8995 0.735506
\(733\) 36.2426 1.33865 0.669326 0.742969i \(-0.266583\pi\)
0.669326 + 0.742969i \(0.266583\pi\)
\(734\) −8.68629 −0.320617
\(735\) 8.48528 0.312984
\(736\) 9.07107 0.334364
\(737\) 24.7279 0.910865
\(738\) −3.82843 −0.140926
\(739\) 41.9706 1.54391 0.771956 0.635676i \(-0.219279\pi\)
0.771956 + 0.635676i \(0.219279\pi\)
\(740\) −6.24264 −0.229484
\(741\) 5.41421 0.198896
\(742\) 5.65685 0.207670
\(743\) −20.3137 −0.745238 −0.372619 0.927984i \(-0.621540\pi\)
−0.372619 + 0.927984i \(0.621540\pi\)
\(744\) −3.41421 −0.125171
\(745\) 20.5563 0.753126
\(746\) −10.7279 −0.392777
\(747\) 7.65685 0.280150
\(748\) −4.82843 −0.176545
\(749\) 16.0711 0.587224
\(750\) −1.41421 −0.0516398
\(751\) 16.7279 0.610411 0.305205 0.952287i \(-0.401275\pi\)
0.305205 + 0.952287i \(0.401275\pi\)
\(752\) −7.07107 −0.257855
\(753\) −32.4853 −1.18383
\(754\) −27.7279 −1.00979
\(755\) −6.48528 −0.236024
\(756\) 5.65685 0.205738
\(757\) −48.4264 −1.76009 −0.880044 0.474892i \(-0.842487\pi\)
−0.880044 + 0.474892i \(0.842487\pi\)
\(758\) −24.1421 −0.876882
\(759\) −43.7990 −1.58980
\(760\) −1.00000 −0.0362738
\(761\) −39.1716 −1.41997 −0.709984 0.704218i \(-0.751298\pi\)
−0.709984 + 0.704218i \(0.751298\pi\)
\(762\) −28.9706 −1.04949
\(763\) −5.17157 −0.187224
\(764\) 0.100505 0.00363615
\(765\) 1.41421 0.0511310
\(766\) 14.3137 0.517175
\(767\) −26.1421 −0.943938
\(768\) −1.41421 −0.0510310
\(769\) 21.0000 0.757279 0.378640 0.925544i \(-0.376392\pi\)
0.378640 + 0.925544i \(0.376392\pi\)
\(770\) 3.41421 0.123040
\(771\) 15.4142 0.555129
\(772\) 1.17157 0.0421658
\(773\) −10.9289 −0.393086 −0.196543 0.980495i \(-0.562972\pi\)
−0.196543 + 0.980495i \(0.562972\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 2.41421 0.0867211
\(776\) −17.5563 −0.630236
\(777\) 8.82843 0.316718
\(778\) 15.1716 0.543927
\(779\) −3.82843 −0.137168
\(780\) −5.41421 −0.193860
\(781\) −26.9706 −0.965083
\(782\) −12.8284 −0.458744
\(783\) −40.9706 −1.46417
\(784\) −6.00000 −0.214286
\(785\) 1.75736 0.0627228
\(786\) −8.48528 −0.302660
\(787\) 22.7574 0.811212 0.405606 0.914048i \(-0.367060\pi\)
0.405606 + 0.914048i \(0.367060\pi\)
\(788\) −3.14214 −0.111934
\(789\) −0.727922 −0.0259147
\(790\) −5.58579 −0.198733
\(791\) −14.8995 −0.529765
\(792\) −3.41421 −0.121319
\(793\) −53.8701 −1.91298
\(794\) −3.51472 −0.124733
\(795\) −8.00000 −0.283731
\(796\) −19.8995 −0.705319
\(797\) 24.9411 0.883460 0.441730 0.897148i \(-0.354365\pi\)
0.441730 + 0.897148i \(0.354365\pi\)
\(798\) 1.41421 0.0500626
\(799\) 10.0000 0.353775
\(800\) 1.00000 0.0353553
\(801\) −5.07107 −0.179177
\(802\) 38.1127 1.34581
\(803\) −2.58579 −0.0912504
\(804\) −10.2426 −0.361230
\(805\) 9.07107 0.319713
\(806\) 9.24264 0.325558
\(807\) −20.9706 −0.738199
\(808\) −1.41421 −0.0497519
\(809\) 10.7990 0.379672 0.189836 0.981816i \(-0.439204\pi\)
0.189836 + 0.981816i \(0.439204\pi\)
\(810\) −5.00000 −0.175682
\(811\) 14.1127 0.495564 0.247782 0.968816i \(-0.420298\pi\)
0.247782 + 0.968816i \(0.420298\pi\)
\(812\) −7.24264 −0.254167
\(813\) 15.4142 0.540600
\(814\) −21.3137 −0.747045
\(815\) 2.34315 0.0820768
\(816\) 2.00000 0.0700140
\(817\) −1.00000 −0.0349856
\(818\) 15.2132 0.531917
\(819\) −3.82843 −0.133776
\(820\) 3.82843 0.133694
\(821\) 8.14214 0.284162 0.142081 0.989855i \(-0.454621\pi\)
0.142081 + 0.989855i \(0.454621\pi\)
\(822\) −25.5563 −0.891380
\(823\) 30.4853 1.06265 0.531325 0.847168i \(-0.321694\pi\)
0.531325 + 0.847168i \(0.321694\pi\)
\(824\) −4.82843 −0.168206
\(825\) −4.82843 −0.168104
\(826\) −6.82843 −0.237591
\(827\) 21.5269 0.748564 0.374282 0.927315i \(-0.377889\pi\)
0.374282 + 0.927315i \(0.377889\pi\)
\(828\) −9.07107 −0.315241
\(829\) −5.24264 −0.182084 −0.0910422 0.995847i \(-0.529020\pi\)
−0.0910422 + 0.995847i \(0.529020\pi\)
\(830\) −7.65685 −0.265773
\(831\) −23.3137 −0.808744
\(832\) 3.82843 0.132727
\(833\) 8.48528 0.293998
\(834\) −3.17157 −0.109823
\(835\) −17.5563 −0.607563
\(836\) −3.41421 −0.118083
\(837\) 13.6569 0.472050
\(838\) 26.7990 0.925756
\(839\) 31.0122 1.07066 0.535330 0.844643i \(-0.320187\pi\)
0.535330 + 0.844643i \(0.320187\pi\)
\(840\) −1.41421 −0.0487950
\(841\) 23.4558 0.808822
\(842\) −7.24264 −0.249598
\(843\) 17.8995 0.616491
\(844\) 9.65685 0.332403
\(845\) 1.65685 0.0569975
\(846\) 7.07107 0.243108
\(847\) 0.656854 0.0225698
\(848\) 5.65685 0.194257
\(849\) −38.5269 −1.32224
\(850\) −1.41421 −0.0485071
\(851\) −56.6274 −1.94116
\(852\) 11.1716 0.382732
\(853\) 31.9411 1.09364 0.546822 0.837249i \(-0.315838\pi\)
0.546822 + 0.837249i \(0.315838\pi\)
\(854\) −14.0711 −0.481502
\(855\) 1.00000 0.0341993
\(856\) 16.0711 0.549298
\(857\) 32.9289 1.12483 0.562415 0.826855i \(-0.309872\pi\)
0.562415 + 0.826855i \(0.309872\pi\)
\(858\) −18.4853 −0.631077
\(859\) −33.4853 −1.14250 −0.571252 0.820775i \(-0.693542\pi\)
−0.571252 + 0.820775i \(0.693542\pi\)
\(860\) 1.00000 0.0340997
\(861\) −5.41421 −0.184516
\(862\) 34.9706 1.19110
\(863\) −35.1716 −1.19725 −0.598627 0.801028i \(-0.704287\pi\)
−0.598627 + 0.801028i \(0.704287\pi\)
\(864\) 5.65685 0.192450
\(865\) 17.4853 0.594517
\(866\) −21.9289 −0.745175
\(867\) 21.2132 0.720438
\(868\) 2.41421 0.0819437
\(869\) −19.0711 −0.646942
\(870\) 10.2426 0.347258
\(871\) 27.7279 0.939525
\(872\) −5.17157 −0.175132
\(873\) 17.5563 0.594192
\(874\) −9.07107 −0.306833
\(875\) 1.00000 0.0338062
\(876\) 1.07107 0.0361880
\(877\) 26.6274 0.899144 0.449572 0.893244i \(-0.351577\pi\)
0.449572 + 0.893244i \(0.351577\pi\)
\(878\) 7.31371 0.246826
\(879\) 32.4853 1.09570
\(880\) 3.41421 0.115093
\(881\) −40.7990 −1.37455 −0.687276 0.726396i \(-0.741194\pi\)
−0.687276 + 0.726396i \(0.741194\pi\)
\(882\) 6.00000 0.202031
\(883\) −16.2132 −0.545618 −0.272809 0.962068i \(-0.587953\pi\)
−0.272809 + 0.962068i \(0.587953\pi\)
\(884\) −5.41421 −0.182100
\(885\) 9.65685 0.324612
\(886\) −24.0711 −0.808683
\(887\) 6.11270 0.205244 0.102622 0.994720i \(-0.467277\pi\)
0.102622 + 0.994720i \(0.467277\pi\)
\(888\) 8.82843 0.296263
\(889\) 20.4853 0.687054
\(890\) 5.07107 0.169983
\(891\) −17.0711 −0.571902
\(892\) −12.1421 −0.406549
\(893\) 7.07107 0.236624
\(894\) −29.0711 −0.972282
\(895\) 0.514719 0.0172051
\(896\) 1.00000 0.0334077
\(897\) −49.1127 −1.63983
\(898\) 13.7574 0.459089
\(899\) −17.4853 −0.583167
\(900\) −1.00000 −0.0333333
\(901\) −8.00000 −0.266519
\(902\) 13.0711 0.435219
\(903\) −1.41421 −0.0470621
\(904\) −14.8995 −0.495550
\(905\) −0.928932 −0.0308788
\(906\) 9.17157 0.304705
\(907\) −4.89949 −0.162685 −0.0813425 0.996686i \(-0.525921\pi\)
−0.0813425 + 0.996686i \(0.525921\pi\)
\(908\) 14.8284 0.492099
\(909\) 1.41421 0.0469065
\(910\) 3.82843 0.126911
\(911\) 16.3848 0.542852 0.271426 0.962459i \(-0.412505\pi\)
0.271426 + 0.962459i \(0.412505\pi\)
\(912\) 1.41421 0.0468293
\(913\) −26.1421 −0.865178
\(914\) −13.5147 −0.447027
\(915\) 19.8995 0.657857
\(916\) −4.82843 −0.159536
\(917\) 6.00000 0.198137
\(918\) −8.00000 −0.264039
\(919\) −15.1005 −0.498120 −0.249060 0.968488i \(-0.580122\pi\)
−0.249060 + 0.968488i \(0.580122\pi\)
\(920\) 9.07107 0.299064
\(921\) 28.3848 0.935310
\(922\) −0.928932 −0.0305928
\(923\) −30.2426 −0.995449
\(924\) −4.82843 −0.158844
\(925\) −6.24264 −0.205257
\(926\) 7.68629 0.252587
\(927\) 4.82843 0.158586
\(928\) −7.24264 −0.237751
\(929\) 25.7990 0.846437 0.423219 0.906028i \(-0.360900\pi\)
0.423219 + 0.906028i \(0.360900\pi\)
\(930\) −3.41421 −0.111956
\(931\) 6.00000 0.196642
\(932\) −8.48528 −0.277945
\(933\) −14.2426 −0.466283
\(934\) −35.0711 −1.14756
\(935\) −4.82843 −0.157906
\(936\) −3.82843 −0.125136
\(937\) 50.4264 1.64736 0.823679 0.567056i \(-0.191918\pi\)
0.823679 + 0.567056i \(0.191918\pi\)
\(938\) 7.24264 0.236481
\(939\) −30.1421 −0.983651
\(940\) −7.07107 −0.230633
\(941\) −38.9289 −1.26905 −0.634523 0.772904i \(-0.718804\pi\)
−0.634523 + 0.772904i \(0.718804\pi\)
\(942\) −2.48528 −0.0809748
\(943\) 34.7279 1.13090
\(944\) −6.82843 −0.222246
\(945\) 5.65685 0.184017
\(946\) 3.41421 0.111006
\(947\) −17.1838 −0.558397 −0.279199 0.960233i \(-0.590069\pi\)
−0.279199 + 0.960233i \(0.590069\pi\)
\(948\) 7.89949 0.256564
\(949\) −2.89949 −0.0941216
\(950\) −1.00000 −0.0324443
\(951\) −41.2132 −1.33643
\(952\) −1.41421 −0.0458349
\(953\) −28.8995 −0.936146 −0.468073 0.883690i \(-0.655052\pi\)
−0.468073 + 0.883690i \(0.655052\pi\)
\(954\) −5.65685 −0.183147
\(955\) 0.100505 0.00325227
\(956\) 14.4142 0.466189
\(957\) 34.9706 1.13044
\(958\) 14.8284 0.479085
\(959\) 18.0711 0.583545
\(960\) −1.41421 −0.0456435
\(961\) −25.1716 −0.811986
\(962\) −23.8995 −0.770551
\(963\) −16.0711 −0.517883
\(964\) −18.5858 −0.598608
\(965\) 1.17157 0.0377143
\(966\) −12.8284 −0.412748
\(967\) 27.7574 0.892616 0.446308 0.894879i \(-0.352738\pi\)
0.446308 + 0.894879i \(0.352738\pi\)
\(968\) 0.656854 0.0211121
\(969\) −2.00000 −0.0642493
\(970\) −17.5563 −0.563700
\(971\) 6.92893 0.222360 0.111180 0.993800i \(-0.464537\pi\)
0.111180 + 0.993800i \(0.464537\pi\)
\(972\) −9.89949 −0.317526
\(973\) 2.24264 0.0718958
\(974\) 9.21320 0.295210
\(975\) −5.41421 −0.173394
\(976\) −14.0711 −0.450404
\(977\) −2.92893 −0.0937048 −0.0468524 0.998902i \(-0.514919\pi\)
−0.0468524 + 0.998902i \(0.514919\pi\)
\(978\) −3.31371 −0.105961
\(979\) 17.3137 0.553349
\(980\) −6.00000 −0.191663
\(981\) 5.17157 0.165116
\(982\) −28.6274 −0.913538
\(983\) −38.6569 −1.23296 −0.616481 0.787370i \(-0.711442\pi\)
−0.616481 + 0.787370i \(0.711442\pi\)
\(984\) −5.41421 −0.172599
\(985\) −3.14214 −0.100117
\(986\) 10.2426 0.326192
\(987\) 10.0000 0.318304
\(988\) −3.82843 −0.121798
\(989\) 9.07107 0.288443
\(990\) −3.41421 −0.108511
\(991\) −49.2548 −1.56463 −0.782316 0.622882i \(-0.785962\pi\)
−0.782316 + 0.622882i \(0.785962\pi\)
\(992\) 2.41421 0.0766514
\(993\) 3.79899 0.120557
\(994\) −7.89949 −0.250557
\(995\) −19.8995 −0.630856
\(996\) 10.8284 0.343112
\(997\) 42.8284 1.35639 0.678195 0.734882i \(-0.262762\pi\)
0.678195 + 0.734882i \(0.262762\pi\)
\(998\) 16.4558 0.520901
\(999\) −35.3137 −1.11728
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 430.2.a.g.1.1 2
3.2 odd 2 3870.2.a.bc.1.1 2
4.3 odd 2 3440.2.a.j.1.2 2
5.2 odd 4 2150.2.b.o.1549.4 4
5.3 odd 4 2150.2.b.o.1549.1 4
5.4 even 2 2150.2.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
430.2.a.g.1.1 2 1.1 even 1 trivial
2150.2.a.v.1.2 2 5.4 even 2
2150.2.b.o.1549.1 4 5.3 odd 4
2150.2.b.o.1549.4 4 5.2 odd 4
3440.2.a.j.1.2 2 4.3 odd 2
3870.2.a.bc.1.1 2 3.2 odd 2