Properties

Label 5070.2.a.bb.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -3.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -3.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.12311 q^{11} +1.00000 q^{12} +3.56155 q^{14} -1.00000 q^{15} +1.00000 q^{16} +5.12311 q^{17} -1.00000 q^{18} +3.56155 q^{19} -1.00000 q^{20} -3.56155 q^{21} -4.12311 q^{22} -7.68466 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -3.56155 q^{28} +6.56155 q^{29} +1.00000 q^{30} -5.68466 q^{31} -1.00000 q^{32} +4.12311 q^{33} -5.12311 q^{34} +3.56155 q^{35} +1.00000 q^{36} -4.12311 q^{37} -3.56155 q^{38} +1.00000 q^{40} +4.24621 q^{41} +3.56155 q^{42} +4.56155 q^{43} +4.12311 q^{44} -1.00000 q^{45} +7.68466 q^{46} -7.00000 q^{47} +1.00000 q^{48} +5.68466 q^{49} -1.00000 q^{50} +5.12311 q^{51} +4.43845 q^{53} -1.00000 q^{54} -4.12311 q^{55} +3.56155 q^{56} +3.56155 q^{57} -6.56155 q^{58} -10.5616 q^{59} -1.00000 q^{60} +6.00000 q^{61} +5.68466 q^{62} -3.56155 q^{63} +1.00000 q^{64} -4.12311 q^{66} -14.2462 q^{67} +5.12311 q^{68} -7.68466 q^{69} -3.56155 q^{70} +4.87689 q^{71} -1.00000 q^{72} +15.3693 q^{73} +4.12311 q^{74} +1.00000 q^{75} +3.56155 q^{76} -14.6847 q^{77} +7.43845 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.24621 q^{82} -1.12311 q^{83} -3.56155 q^{84} -5.12311 q^{85} -4.56155 q^{86} +6.56155 q^{87} -4.12311 q^{88} -1.80776 q^{89} +1.00000 q^{90} -7.68466 q^{92} -5.68466 q^{93} +7.00000 q^{94} -3.56155 q^{95} -1.00000 q^{96} -1.12311 q^{97} -5.68466 q^{98} +4.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} + 3q^{14} - 2q^{15} + 2q^{16} + 2q^{17} - 2q^{18} + 3q^{19} - 2q^{20} - 3q^{21} - 3q^{23} - 2q^{24} + 2q^{25} + 2q^{27} - 3q^{28} + 9q^{29} + 2q^{30} + q^{31} - 2q^{32} - 2q^{34} + 3q^{35} + 2q^{36} - 3q^{38} + 2q^{40} - 8q^{41} + 3q^{42} + 5q^{43} - 2q^{45} + 3q^{46} - 14q^{47} + 2q^{48} - q^{49} - 2q^{50} + 2q^{51} + 13q^{53} - 2q^{54} + 3q^{56} + 3q^{57} - 9q^{58} - 17q^{59} - 2q^{60} + 12q^{61} - q^{62} - 3q^{63} + 2q^{64} - 12q^{67} + 2q^{68} - 3q^{69} - 3q^{70} + 18q^{71} - 2q^{72} + 6q^{73} + 2q^{75} + 3q^{76} - 17q^{77} + 19q^{79} - 2q^{80} + 2q^{81} + 8q^{82} + 6q^{83} - 3q^{84} - 2q^{85} - 5q^{86} + 9q^{87} + 17q^{89} + 2q^{90} - 3q^{92} + q^{93} + 14q^{94} - 3q^{95} - 2q^{96} + 6q^{97} + q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −3.56155 −1.34614 −0.673070 0.739579i \(-0.735025\pi\)
−0.673070 + 0.739579i \(0.735025\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.12311 1.24316 0.621582 0.783349i \(-0.286490\pi\)
0.621582 + 0.783349i \(0.286490\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 3.56155 0.951865
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 5.12311 1.24254 0.621268 0.783598i \(-0.286618\pi\)
0.621268 + 0.783598i \(0.286618\pi\)
\(18\) −1.00000 −0.235702
\(19\) 3.56155 0.817076 0.408538 0.912741i \(-0.366039\pi\)
0.408538 + 0.912741i \(0.366039\pi\)
\(20\) −1.00000 −0.223607
\(21\) −3.56155 −0.777195
\(22\) −4.12311 −0.879049
\(23\) −7.68466 −1.60236 −0.801181 0.598422i \(-0.795795\pi\)
−0.801181 + 0.598422i \(0.795795\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −3.56155 −0.673070
\(29\) 6.56155 1.21845 0.609225 0.792998i \(-0.291481\pi\)
0.609225 + 0.792998i \(0.291481\pi\)
\(30\) 1.00000 0.182574
\(31\) −5.68466 −1.02099 −0.510497 0.859879i \(-0.670539\pi\)
−0.510497 + 0.859879i \(0.670539\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.12311 0.717741
\(34\) −5.12311 −0.878605
\(35\) 3.56155 0.602012
\(36\) 1.00000 0.166667
\(37\) −4.12311 −0.677834 −0.338917 0.940816i \(-0.610061\pi\)
−0.338917 + 0.940816i \(0.610061\pi\)
\(38\) −3.56155 −0.577760
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 4.24621 0.663147 0.331573 0.943429i \(-0.392421\pi\)
0.331573 + 0.943429i \(0.392421\pi\)
\(42\) 3.56155 0.549560
\(43\) 4.56155 0.695630 0.347815 0.937563i \(-0.386924\pi\)
0.347815 + 0.937563i \(0.386924\pi\)
\(44\) 4.12311 0.621582
\(45\) −1.00000 −0.149071
\(46\) 7.68466 1.13304
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 1.00000 0.144338
\(49\) 5.68466 0.812094
\(50\) −1.00000 −0.141421
\(51\) 5.12311 0.717378
\(52\) 0 0
\(53\) 4.43845 0.609668 0.304834 0.952406i \(-0.401399\pi\)
0.304834 + 0.952406i \(0.401399\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.12311 −0.555959
\(56\) 3.56155 0.475933
\(57\) 3.56155 0.471739
\(58\) −6.56155 −0.861574
\(59\) −10.5616 −1.37500 −0.687499 0.726186i \(-0.741291\pi\)
−0.687499 + 0.726186i \(0.741291\pi\)
\(60\) −1.00000 −0.129099
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 5.68466 0.721952
\(63\) −3.56155 −0.448713
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.12311 −0.507519
\(67\) −14.2462 −1.74045 −0.870226 0.492653i \(-0.836027\pi\)
−0.870226 + 0.492653i \(0.836027\pi\)
\(68\) 5.12311 0.621268
\(69\) −7.68466 −0.925124
\(70\) −3.56155 −0.425687
\(71\) 4.87689 0.578781 0.289390 0.957211i \(-0.406547\pi\)
0.289390 + 0.957211i \(0.406547\pi\)
\(72\) −1.00000 −0.117851
\(73\) 15.3693 1.79884 0.899421 0.437083i \(-0.143988\pi\)
0.899421 + 0.437083i \(0.143988\pi\)
\(74\) 4.12311 0.479301
\(75\) 1.00000 0.115470
\(76\) 3.56155 0.408538
\(77\) −14.6847 −1.67347
\(78\) 0 0
\(79\) 7.43845 0.836891 0.418445 0.908242i \(-0.362575\pi\)
0.418445 + 0.908242i \(0.362575\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −4.24621 −0.468916
\(83\) −1.12311 −0.123277 −0.0616384 0.998099i \(-0.519633\pi\)
−0.0616384 + 0.998099i \(0.519633\pi\)
\(84\) −3.56155 −0.388597
\(85\) −5.12311 −0.555679
\(86\) −4.56155 −0.491885
\(87\) 6.56155 0.703472
\(88\) −4.12311 −0.439525
\(89\) −1.80776 −0.191623 −0.0958113 0.995400i \(-0.530545\pi\)
−0.0958113 + 0.995400i \(0.530545\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −7.68466 −0.801181
\(93\) −5.68466 −0.589472
\(94\) 7.00000 0.721995
\(95\) −3.56155 −0.365408
\(96\) −1.00000 −0.102062
\(97\) −1.12311 −0.114034 −0.0570170 0.998373i \(-0.518159\pi\)
−0.0570170 + 0.998373i \(0.518159\pi\)
\(98\) −5.68466 −0.574237
\(99\) 4.12311 0.414388
\(100\) 1.00000 0.100000
\(101\) −17.1231 −1.70381 −0.851906 0.523694i \(-0.824553\pi\)
−0.851906 + 0.523694i \(0.824553\pi\)
\(102\) −5.12311 −0.507263
\(103\) 0.438447 0.0432015 0.0216007 0.999767i \(-0.493124\pi\)
0.0216007 + 0.999767i \(0.493124\pi\)
\(104\) 0 0
\(105\) 3.56155 0.347572
\(106\) −4.43845 −0.431100
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 1.00000 0.0962250
\(109\) 20.2462 1.93924 0.969618 0.244625i \(-0.0786650\pi\)
0.969618 + 0.244625i \(0.0786650\pi\)
\(110\) 4.12311 0.393123
\(111\) −4.12311 −0.391348
\(112\) −3.56155 −0.336535
\(113\) −3.68466 −0.346624 −0.173312 0.984867i \(-0.555447\pi\)
−0.173312 + 0.984867i \(0.555447\pi\)
\(114\) −3.56155 −0.333570
\(115\) 7.68466 0.716598
\(116\) 6.56155 0.609225
\(117\) 0 0
\(118\) 10.5616 0.972270
\(119\) −18.2462 −1.67263
\(120\) 1.00000 0.0912871
\(121\) 6.00000 0.545455
\(122\) −6.00000 −0.543214
\(123\) 4.24621 0.382868
\(124\) −5.68466 −0.510497
\(125\) −1.00000 −0.0894427
\(126\) 3.56155 0.317288
\(127\) −4.43845 −0.393849 −0.196924 0.980419i \(-0.563095\pi\)
−0.196924 + 0.980419i \(0.563095\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.56155 0.401622
\(130\) 0 0
\(131\) 6.12311 0.534978 0.267489 0.963561i \(-0.413806\pi\)
0.267489 + 0.963561i \(0.413806\pi\)
\(132\) 4.12311 0.358870
\(133\) −12.6847 −1.09990
\(134\) 14.2462 1.23069
\(135\) −1.00000 −0.0860663
\(136\) −5.12311 −0.439303
\(137\) 8.80776 0.752498 0.376249 0.926519i \(-0.377214\pi\)
0.376249 + 0.926519i \(0.377214\pi\)
\(138\) 7.68466 0.654162
\(139\) 8.43845 0.715740 0.357870 0.933771i \(-0.383503\pi\)
0.357870 + 0.933771i \(0.383503\pi\)
\(140\) 3.56155 0.301006
\(141\) −7.00000 −0.589506
\(142\) −4.87689 −0.409260
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.56155 −0.544907
\(146\) −15.3693 −1.27197
\(147\) 5.68466 0.468863
\(148\) −4.12311 −0.338917
\(149\) 22.8078 1.86848 0.934242 0.356639i \(-0.116077\pi\)
0.934242 + 0.356639i \(0.116077\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −9.36932 −0.762464 −0.381232 0.924479i \(-0.624500\pi\)
−0.381232 + 0.924479i \(0.624500\pi\)
\(152\) −3.56155 −0.288880
\(153\) 5.12311 0.414179
\(154\) 14.6847 1.18332
\(155\) 5.68466 0.456603
\(156\) 0 0
\(157\) 22.1231 1.76562 0.882808 0.469734i \(-0.155650\pi\)
0.882808 + 0.469734i \(0.155650\pi\)
\(158\) −7.43845 −0.591771
\(159\) 4.43845 0.351992
\(160\) 1.00000 0.0790569
\(161\) 27.3693 2.15700
\(162\) −1.00000 −0.0785674
\(163\) 18.5616 1.45385 0.726927 0.686715i \(-0.240948\pi\)
0.726927 + 0.686715i \(0.240948\pi\)
\(164\) 4.24621 0.331573
\(165\) −4.12311 −0.320983
\(166\) 1.12311 0.0871699
\(167\) 20.3693 1.57623 0.788113 0.615531i \(-0.211058\pi\)
0.788113 + 0.615531i \(0.211058\pi\)
\(168\) 3.56155 0.274780
\(169\) 0 0
\(170\) 5.12311 0.392924
\(171\) 3.56155 0.272359
\(172\) 4.56155 0.347815
\(173\) −25.1771 −1.91418 −0.957089 0.289794i \(-0.906413\pi\)
−0.957089 + 0.289794i \(0.906413\pi\)
\(174\) −6.56155 −0.497430
\(175\) −3.56155 −0.269228
\(176\) 4.12311 0.310791
\(177\) −10.5616 −0.793855
\(178\) 1.80776 0.135498
\(179\) −0.315342 −0.0235697 −0.0117849 0.999931i \(-0.503751\pi\)
−0.0117849 + 0.999931i \(0.503751\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 11.1231 0.826774 0.413387 0.910555i \(-0.364346\pi\)
0.413387 + 0.910555i \(0.364346\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 7.68466 0.566521
\(185\) 4.12311 0.303137
\(186\) 5.68466 0.416819
\(187\) 21.1231 1.54467
\(188\) −7.00000 −0.510527
\(189\) −3.56155 −0.259065
\(190\) 3.56155 0.258382
\(191\) 19.1231 1.38370 0.691850 0.722042i \(-0.256796\pi\)
0.691850 + 0.722042i \(0.256796\pi\)
\(192\) 1.00000 0.0721688
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 1.12311 0.0806343
\(195\) 0 0
\(196\) 5.68466 0.406047
\(197\) 7.80776 0.556280 0.278140 0.960541i \(-0.410282\pi\)
0.278140 + 0.960541i \(0.410282\pi\)
\(198\) −4.12311 −0.293016
\(199\) 11.1231 0.788496 0.394248 0.919004i \(-0.371005\pi\)
0.394248 + 0.919004i \(0.371005\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −14.2462 −1.00485
\(202\) 17.1231 1.20478
\(203\) −23.3693 −1.64020
\(204\) 5.12311 0.358689
\(205\) −4.24621 −0.296568
\(206\) −0.438447 −0.0305481
\(207\) −7.68466 −0.534121
\(208\) 0 0
\(209\) 14.6847 1.01576
\(210\) −3.56155 −0.245770
\(211\) 6.93087 0.477141 0.238570 0.971125i \(-0.423321\pi\)
0.238570 + 0.971125i \(0.423321\pi\)
\(212\) 4.43845 0.304834
\(213\) 4.87689 0.334159
\(214\) −2.00000 −0.136717
\(215\) −4.56155 −0.311095
\(216\) −1.00000 −0.0680414
\(217\) 20.2462 1.37440
\(218\) −20.2462 −1.37125
\(219\) 15.3693 1.03856
\(220\) −4.12311 −0.277980
\(221\) 0 0
\(222\) 4.12311 0.276725
\(223\) 26.3002 1.76119 0.880595 0.473869i \(-0.157143\pi\)
0.880595 + 0.473869i \(0.157143\pi\)
\(224\) 3.56155 0.237966
\(225\) 1.00000 0.0666667
\(226\) 3.68466 0.245100
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 3.56155 0.235870
\(229\) 7.75379 0.512385 0.256192 0.966626i \(-0.417532\pi\)
0.256192 + 0.966626i \(0.417532\pi\)
\(230\) −7.68466 −0.506711
\(231\) −14.6847 −0.966180
\(232\) −6.56155 −0.430787
\(233\) −17.6847 −1.15856 −0.579280 0.815128i \(-0.696666\pi\)
−0.579280 + 0.815128i \(0.696666\pi\)
\(234\) 0 0
\(235\) 7.00000 0.456630
\(236\) −10.5616 −0.687499
\(237\) 7.43845 0.483179
\(238\) 18.2462 1.18273
\(239\) −13.3693 −0.864789 −0.432395 0.901684i \(-0.642331\pi\)
−0.432395 + 0.901684i \(0.642331\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 19.8769 1.28038 0.640192 0.768215i \(-0.278855\pi\)
0.640192 + 0.768215i \(0.278855\pi\)
\(242\) −6.00000 −0.385695
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) −5.68466 −0.363180
\(246\) −4.24621 −0.270729
\(247\) 0 0
\(248\) 5.68466 0.360976
\(249\) −1.12311 −0.0711739
\(250\) 1.00000 0.0632456
\(251\) 0.123106 0.00777036 0.00388518 0.999992i \(-0.498763\pi\)
0.00388518 + 0.999992i \(0.498763\pi\)
\(252\) −3.56155 −0.224357
\(253\) −31.6847 −1.99200
\(254\) 4.43845 0.278493
\(255\) −5.12311 −0.320821
\(256\) 1.00000 0.0625000
\(257\) −1.43845 −0.0897279 −0.0448639 0.998993i \(-0.514285\pi\)
−0.0448639 + 0.998993i \(0.514285\pi\)
\(258\) −4.56155 −0.283990
\(259\) 14.6847 0.912460
\(260\) 0 0
\(261\) 6.56155 0.406150
\(262\) −6.12311 −0.378287
\(263\) −13.0000 −0.801614 −0.400807 0.916162i \(-0.631270\pi\)
−0.400807 + 0.916162i \(0.631270\pi\)
\(264\) −4.12311 −0.253760
\(265\) −4.43845 −0.272652
\(266\) 12.6847 0.777746
\(267\) −1.80776 −0.110633
\(268\) −14.2462 −0.870226
\(269\) −3.36932 −0.205431 −0.102715 0.994711i \(-0.532753\pi\)
−0.102715 + 0.994711i \(0.532753\pi\)
\(270\) 1.00000 0.0608581
\(271\) −32.1771 −1.95462 −0.977309 0.211818i \(-0.932062\pi\)
−0.977309 + 0.211818i \(0.932062\pi\)
\(272\) 5.12311 0.310634
\(273\) 0 0
\(274\) −8.80776 −0.532096
\(275\) 4.12311 0.248633
\(276\) −7.68466 −0.462562
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) −8.43845 −0.506104
\(279\) −5.68466 −0.340332
\(280\) −3.56155 −0.212843
\(281\) 0.246211 0.0146877 0.00734387 0.999973i \(-0.497662\pi\)
0.00734387 + 0.999973i \(0.497662\pi\)
\(282\) 7.00000 0.416844
\(283\) 11.4384 0.679945 0.339973 0.940435i \(-0.389582\pi\)
0.339973 + 0.940435i \(0.389582\pi\)
\(284\) 4.87689 0.289390
\(285\) −3.56155 −0.210968
\(286\) 0 0
\(287\) −15.1231 −0.892689
\(288\) −1.00000 −0.0589256
\(289\) 9.24621 0.543895
\(290\) 6.56155 0.385308
\(291\) −1.12311 −0.0658376
\(292\) 15.3693 0.899421
\(293\) 24.9309 1.45648 0.728238 0.685324i \(-0.240339\pi\)
0.728238 + 0.685324i \(0.240339\pi\)
\(294\) −5.68466 −0.331536
\(295\) 10.5616 0.614917
\(296\) 4.12311 0.239651
\(297\) 4.12311 0.239247
\(298\) −22.8078 −1.32122
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −16.2462 −0.936416
\(302\) 9.36932 0.539144
\(303\) −17.1231 −0.983697
\(304\) 3.56155 0.204269
\(305\) −6.00000 −0.343559
\(306\) −5.12311 −0.292868
\(307\) 15.6155 0.891225 0.445613 0.895226i \(-0.352986\pi\)
0.445613 + 0.895226i \(0.352986\pi\)
\(308\) −14.6847 −0.836736
\(309\) 0.438447 0.0249424
\(310\) −5.68466 −0.322867
\(311\) −18.7386 −1.06257 −0.531285 0.847193i \(-0.678291\pi\)
−0.531285 + 0.847193i \(0.678291\pi\)
\(312\) 0 0
\(313\) 6.63068 0.374788 0.187394 0.982285i \(-0.439996\pi\)
0.187394 + 0.982285i \(0.439996\pi\)
\(314\) −22.1231 −1.24848
\(315\) 3.56155 0.200671
\(316\) 7.43845 0.418445
\(317\) 4.19224 0.235459 0.117730 0.993046i \(-0.462438\pi\)
0.117730 + 0.993046i \(0.462438\pi\)
\(318\) −4.43845 −0.248896
\(319\) 27.0540 1.51473
\(320\) −1.00000 −0.0559017
\(321\) 2.00000 0.111629
\(322\) −27.3693 −1.52523
\(323\) 18.2462 1.01525
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −18.5616 −1.02803
\(327\) 20.2462 1.11962
\(328\) −4.24621 −0.234458
\(329\) 24.9309 1.37448
\(330\) 4.12311 0.226969
\(331\) 18.7386 1.02997 0.514984 0.857200i \(-0.327798\pi\)
0.514984 + 0.857200i \(0.327798\pi\)
\(332\) −1.12311 −0.0616384
\(333\) −4.12311 −0.225945
\(334\) −20.3693 −1.11456
\(335\) 14.2462 0.778354
\(336\) −3.56155 −0.194299
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) −3.68466 −0.200123
\(340\) −5.12311 −0.277839
\(341\) −23.4384 −1.26926
\(342\) −3.56155 −0.192587
\(343\) 4.68466 0.252948
\(344\) −4.56155 −0.245942
\(345\) 7.68466 0.413728
\(346\) 25.1771 1.35353
\(347\) 9.12311 0.489754 0.244877 0.969554i \(-0.421252\pi\)
0.244877 + 0.969554i \(0.421252\pi\)
\(348\) 6.56155 0.351736
\(349\) 24.4924 1.31105 0.655525 0.755174i \(-0.272448\pi\)
0.655525 + 0.755174i \(0.272448\pi\)
\(350\) 3.56155 0.190373
\(351\) 0 0
\(352\) −4.12311 −0.219762
\(353\) −31.8617 −1.69583 −0.847915 0.530133i \(-0.822142\pi\)
−0.847915 + 0.530133i \(0.822142\pi\)
\(354\) 10.5616 0.561340
\(355\) −4.87689 −0.258839
\(356\) −1.80776 −0.0958113
\(357\) −18.2462 −0.965692
\(358\) 0.315342 0.0166663
\(359\) −4.87689 −0.257393 −0.128696 0.991684i \(-0.541079\pi\)
−0.128696 + 0.991684i \(0.541079\pi\)
\(360\) 1.00000 0.0527046
\(361\) −6.31534 −0.332386
\(362\) −11.1231 −0.584617
\(363\) 6.00000 0.314918
\(364\) 0 0
\(365\) −15.3693 −0.804467
\(366\) −6.00000 −0.313625
\(367\) −20.4924 −1.06970 −0.534848 0.844948i \(-0.679631\pi\)
−0.534848 + 0.844948i \(0.679631\pi\)
\(368\) −7.68466 −0.400591
\(369\) 4.24621 0.221049
\(370\) −4.12311 −0.214350
\(371\) −15.8078 −0.820698
\(372\) −5.68466 −0.294736
\(373\) −5.19224 −0.268844 −0.134422 0.990924i \(-0.542918\pi\)
−0.134422 + 0.990924i \(0.542918\pi\)
\(374\) −21.1231 −1.09225
\(375\) −1.00000 −0.0516398
\(376\) 7.00000 0.360997
\(377\) 0 0
\(378\) 3.56155 0.183187
\(379\) 5.31534 0.273031 0.136515 0.990638i \(-0.456410\pi\)
0.136515 + 0.990638i \(0.456410\pi\)
\(380\) −3.56155 −0.182704
\(381\) −4.43845 −0.227389
\(382\) −19.1231 −0.978423
\(383\) −20.8078 −1.06323 −0.531614 0.846987i \(-0.678414\pi\)
−0.531614 + 0.846987i \(0.678414\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 14.6847 0.748399
\(386\) 0 0
\(387\) 4.56155 0.231877
\(388\) −1.12311 −0.0570170
\(389\) 19.0540 0.966075 0.483037 0.875600i \(-0.339533\pi\)
0.483037 + 0.875600i \(0.339533\pi\)
\(390\) 0 0
\(391\) −39.3693 −1.99099
\(392\) −5.68466 −0.287119
\(393\) 6.12311 0.308870
\(394\) −7.80776 −0.393349
\(395\) −7.43845 −0.374269
\(396\) 4.12311 0.207194
\(397\) 11.8769 0.596084 0.298042 0.954553i \(-0.403666\pi\)
0.298042 + 0.954553i \(0.403666\pi\)
\(398\) −11.1231 −0.557551
\(399\) −12.6847 −0.635027
\(400\) 1.00000 0.0500000
\(401\) −12.6847 −0.633442 −0.316721 0.948519i \(-0.602582\pi\)
−0.316721 + 0.948519i \(0.602582\pi\)
\(402\) 14.2462 0.710536
\(403\) 0 0
\(404\) −17.1231 −0.851906
\(405\) −1.00000 −0.0496904
\(406\) 23.3693 1.15980
\(407\) −17.0000 −0.842659
\(408\) −5.12311 −0.253632
\(409\) −1.80776 −0.0893882 −0.0446941 0.999001i \(-0.514231\pi\)
−0.0446941 + 0.999001i \(0.514231\pi\)
\(410\) 4.24621 0.209705
\(411\) 8.80776 0.434455
\(412\) 0.438447 0.0216007
\(413\) 37.6155 1.85094
\(414\) 7.68466 0.377680
\(415\) 1.12311 0.0551311
\(416\) 0 0
\(417\) 8.43845 0.413233
\(418\) −14.6847 −0.718250
\(419\) −0.492423 −0.0240564 −0.0120282 0.999928i \(-0.503829\pi\)
−0.0120282 + 0.999928i \(0.503829\pi\)
\(420\) 3.56155 0.173786
\(421\) −0.492423 −0.0239992 −0.0119996 0.999928i \(-0.503820\pi\)
−0.0119996 + 0.999928i \(0.503820\pi\)
\(422\) −6.93087 −0.337389
\(423\) −7.00000 −0.340352
\(424\) −4.43845 −0.215550
\(425\) 5.12311 0.248507
\(426\) −4.87689 −0.236286
\(427\) −21.3693 −1.03413
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) 4.56155 0.219978
\(431\) −26.7386 −1.28795 −0.643977 0.765045i \(-0.722717\pi\)
−0.643977 + 0.765045i \(0.722717\pi\)
\(432\) 1.00000 0.0481125
\(433\) 30.7386 1.47720 0.738602 0.674141i \(-0.235486\pi\)
0.738602 + 0.674141i \(0.235486\pi\)
\(434\) −20.2462 −0.971849
\(435\) −6.56155 −0.314602
\(436\) 20.2462 0.969618
\(437\) −27.3693 −1.30925
\(438\) −15.3693 −0.734374
\(439\) 16.8769 0.805490 0.402745 0.915312i \(-0.368056\pi\)
0.402745 + 0.915312i \(0.368056\pi\)
\(440\) 4.12311 0.196561
\(441\) 5.68466 0.270698
\(442\) 0 0
\(443\) −4.87689 −0.231708 −0.115854 0.993266i \(-0.536961\pi\)
−0.115854 + 0.993266i \(0.536961\pi\)
\(444\) −4.12311 −0.195674
\(445\) 1.80776 0.0856962
\(446\) −26.3002 −1.24535
\(447\) 22.8078 1.07877
\(448\) −3.56155 −0.168268
\(449\) −25.1771 −1.18818 −0.594090 0.804399i \(-0.702488\pi\)
−0.594090 + 0.804399i \(0.702488\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 17.5076 0.824400
\(452\) −3.68466 −0.173312
\(453\) −9.36932 −0.440209
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) −3.56155 −0.166785
\(457\) −3.75379 −0.175595 −0.0877974 0.996138i \(-0.527983\pi\)
−0.0877974 + 0.996138i \(0.527983\pi\)
\(458\) −7.75379 −0.362311
\(459\) 5.12311 0.239126
\(460\) 7.68466 0.358299
\(461\) −7.05398 −0.328536 −0.164268 0.986416i \(-0.552526\pi\)
−0.164268 + 0.986416i \(0.552526\pi\)
\(462\) 14.6847 0.683192
\(463\) −33.6155 −1.56225 −0.781123 0.624377i \(-0.785353\pi\)
−0.781123 + 0.624377i \(0.785353\pi\)
\(464\) 6.56155 0.304612
\(465\) 5.68466 0.263620
\(466\) 17.6847 0.819226
\(467\) 39.8617 1.84458 0.922291 0.386497i \(-0.126315\pi\)
0.922291 + 0.386497i \(0.126315\pi\)
\(468\) 0 0
\(469\) 50.7386 2.34289
\(470\) −7.00000 −0.322886
\(471\) 22.1231 1.01938
\(472\) 10.5616 0.486135
\(473\) 18.8078 0.864782
\(474\) −7.43845 −0.341659
\(475\) 3.56155 0.163415
\(476\) −18.2462 −0.836314
\(477\) 4.43845 0.203223
\(478\) 13.3693 0.611498
\(479\) 21.7538 0.993956 0.496978 0.867763i \(-0.334443\pi\)
0.496978 + 0.867763i \(0.334443\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) −19.8769 −0.905368
\(483\) 27.3693 1.24535
\(484\) 6.00000 0.272727
\(485\) 1.12311 0.0509976
\(486\) −1.00000 −0.0453609
\(487\) 24.0540 1.08999 0.544995 0.838439i \(-0.316532\pi\)
0.544995 + 0.838439i \(0.316532\pi\)
\(488\) −6.00000 −0.271607
\(489\) 18.5616 0.839382
\(490\) 5.68466 0.256807
\(491\) −21.5616 −0.973059 −0.486530 0.873664i \(-0.661737\pi\)
−0.486530 + 0.873664i \(0.661737\pi\)
\(492\) 4.24621 0.191434
\(493\) 33.6155 1.51397
\(494\) 0 0
\(495\) −4.12311 −0.185320
\(496\) −5.68466 −0.255249
\(497\) −17.3693 −0.779120
\(498\) 1.12311 0.0503276
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 20.3693 0.910034
\(502\) −0.123106 −0.00549447
\(503\) −5.94602 −0.265120 −0.132560 0.991175i \(-0.542320\pi\)
−0.132560 + 0.991175i \(0.542320\pi\)
\(504\) 3.56155 0.158644
\(505\) 17.1231 0.761968
\(506\) 31.6847 1.40855
\(507\) 0 0
\(508\) −4.43845 −0.196924
\(509\) 29.5464 1.30962 0.654811 0.755793i \(-0.272748\pi\)
0.654811 + 0.755793i \(0.272748\pi\)
\(510\) 5.12311 0.226855
\(511\) −54.7386 −2.42149
\(512\) −1.00000 −0.0441942
\(513\) 3.56155 0.157246
\(514\) 1.43845 0.0634472
\(515\) −0.438447 −0.0193203
\(516\) 4.56155 0.200811
\(517\) −28.8617 −1.26934
\(518\) −14.6847 −0.645207
\(519\) −25.1771 −1.10515
\(520\) 0 0
\(521\) −18.6847 −0.818590 −0.409295 0.912402i \(-0.634225\pi\)
−0.409295 + 0.912402i \(0.634225\pi\)
\(522\) −6.56155 −0.287191
\(523\) −9.19224 −0.401948 −0.200974 0.979597i \(-0.564411\pi\)
−0.200974 + 0.979597i \(0.564411\pi\)
\(524\) 6.12311 0.267489
\(525\) −3.56155 −0.155439
\(526\) 13.0000 0.566827
\(527\) −29.1231 −1.26862
\(528\) 4.12311 0.179435
\(529\) 36.0540 1.56756
\(530\) 4.43845 0.192794
\(531\) −10.5616 −0.458332
\(532\) −12.6847 −0.549950
\(533\) 0 0
\(534\) 1.80776 0.0782296
\(535\) −2.00000 −0.0864675
\(536\) 14.2462 0.615343
\(537\) −0.315342 −0.0136080
\(538\) 3.36932 0.145262
\(539\) 23.4384 1.00957
\(540\) −1.00000 −0.0430331
\(541\) 2.63068 0.113102 0.0565510 0.998400i \(-0.481990\pi\)
0.0565510 + 0.998400i \(0.481990\pi\)
\(542\) 32.1771 1.38212
\(543\) 11.1231 0.477338
\(544\) −5.12311 −0.219651
\(545\) −20.2462 −0.867252
\(546\) 0 0
\(547\) 35.6155 1.52281 0.761405 0.648276i \(-0.224510\pi\)
0.761405 + 0.648276i \(0.224510\pi\)
\(548\) 8.80776 0.376249
\(549\) 6.00000 0.256074
\(550\) −4.12311 −0.175810
\(551\) 23.3693 0.995566
\(552\) 7.68466 0.327081
\(553\) −26.4924 −1.12657
\(554\) 1.00000 0.0424859
\(555\) 4.12311 0.175016
\(556\) 8.43845 0.357870
\(557\) 4.43845 0.188063 0.0940315 0.995569i \(-0.470025\pi\)
0.0940315 + 0.995569i \(0.470025\pi\)
\(558\) 5.68466 0.240651
\(559\) 0 0
\(560\) 3.56155 0.150503
\(561\) 21.1231 0.891818
\(562\) −0.246211 −0.0103858
\(563\) 32.9848 1.39015 0.695073 0.718939i \(-0.255372\pi\)
0.695073 + 0.718939i \(0.255372\pi\)
\(564\) −7.00000 −0.294753
\(565\) 3.68466 0.155015
\(566\) −11.4384 −0.480794
\(567\) −3.56155 −0.149571
\(568\) −4.87689 −0.204630
\(569\) −19.1771 −0.803945 −0.401973 0.915652i \(-0.631675\pi\)
−0.401973 + 0.915652i \(0.631675\pi\)
\(570\) 3.56155 0.149177
\(571\) 11.3153 0.473532 0.236766 0.971567i \(-0.423912\pi\)
0.236766 + 0.971567i \(0.423912\pi\)
\(572\) 0 0
\(573\) 19.1231 0.798879
\(574\) 15.1231 0.631226
\(575\) −7.68466 −0.320472
\(576\) 1.00000 0.0416667
\(577\) −8.73863 −0.363794 −0.181897 0.983318i \(-0.558224\pi\)
−0.181897 + 0.983318i \(0.558224\pi\)
\(578\) −9.24621 −0.384592
\(579\) 0 0
\(580\) −6.56155 −0.272454
\(581\) 4.00000 0.165948
\(582\) 1.12311 0.0465542
\(583\) 18.3002 0.757916
\(584\) −15.3693 −0.635987
\(585\) 0 0
\(586\) −24.9309 −1.02988
\(587\) 23.7538 0.980424 0.490212 0.871603i \(-0.336919\pi\)
0.490212 + 0.871603i \(0.336919\pi\)
\(588\) 5.68466 0.234431
\(589\) −20.2462 −0.834231
\(590\) −10.5616 −0.434812
\(591\) 7.80776 0.321168
\(592\) −4.12311 −0.169459
\(593\) 12.1771 0.500053 0.250026 0.968239i \(-0.419561\pi\)
0.250026 + 0.968239i \(0.419561\pi\)
\(594\) −4.12311 −0.169173
\(595\) 18.2462 0.748022
\(596\) 22.8078 0.934242
\(597\) 11.1231 0.455238
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −35.9848 −1.46785 −0.733926 0.679229i \(-0.762314\pi\)
−0.733926 + 0.679229i \(0.762314\pi\)
\(602\) 16.2462 0.662146
\(603\) −14.2462 −0.580151
\(604\) −9.36932 −0.381232
\(605\) −6.00000 −0.243935
\(606\) 17.1231 0.695579
\(607\) −29.4233 −1.19425 −0.597127 0.802146i \(-0.703691\pi\)
−0.597127 + 0.802146i \(0.703691\pi\)
\(608\) −3.56155 −0.144440
\(609\) −23.3693 −0.946973
\(610\) 6.00000 0.242933
\(611\) 0 0
\(612\) 5.12311 0.207089
\(613\) −28.1231 −1.13588 −0.567941 0.823069i \(-0.692260\pi\)
−0.567941 + 0.823069i \(0.692260\pi\)
\(614\) −15.6155 −0.630191
\(615\) −4.24621 −0.171224
\(616\) 14.6847 0.591662
\(617\) 41.6847 1.67816 0.839081 0.544007i \(-0.183094\pi\)
0.839081 + 0.544007i \(0.183094\pi\)
\(618\) −0.438447 −0.0176369
\(619\) 16.4384 0.660717 0.330358 0.943856i \(-0.392830\pi\)
0.330358 + 0.943856i \(0.392830\pi\)
\(620\) 5.68466 0.228301
\(621\) −7.68466 −0.308375
\(622\) 18.7386 0.751351
\(623\) 6.43845 0.257951
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −6.63068 −0.265015
\(627\) 14.6847 0.586449
\(628\) 22.1231 0.882808
\(629\) −21.1231 −0.842233
\(630\) −3.56155 −0.141896
\(631\) 30.2462 1.20408 0.602041 0.798465i \(-0.294354\pi\)
0.602041 + 0.798465i \(0.294354\pi\)
\(632\) −7.43845 −0.295886
\(633\) 6.93087 0.275477
\(634\) −4.19224 −0.166495
\(635\) 4.43845 0.176134
\(636\) 4.43845 0.175996
\(637\) 0 0
\(638\) −27.0540 −1.07108
\(639\) 4.87689 0.192927
\(640\) 1.00000 0.0395285
\(641\) −15.5616 −0.614644 −0.307322 0.951606i \(-0.599433\pi\)
−0.307322 + 0.951606i \(0.599433\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −38.2462 −1.50828 −0.754142 0.656712i \(-0.771947\pi\)
−0.754142 + 0.656712i \(0.771947\pi\)
\(644\) 27.3693 1.07850
\(645\) −4.56155 −0.179611
\(646\) −18.2462 −0.717888
\(647\) −2.05398 −0.0807501 −0.0403751 0.999185i \(-0.512855\pi\)
−0.0403751 + 0.999185i \(0.512855\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −43.5464 −1.70935
\(650\) 0 0
\(651\) 20.2462 0.793512
\(652\) 18.5616 0.726927
\(653\) −40.4384 −1.58248 −0.791239 0.611507i \(-0.790564\pi\)
−0.791239 + 0.611507i \(0.790564\pi\)
\(654\) −20.2462 −0.791690
\(655\) −6.12311 −0.239250
\(656\) 4.24621 0.165787
\(657\) 15.3693 0.599614
\(658\) −24.9309 −0.971906
\(659\) 31.0540 1.20969 0.604846 0.796343i \(-0.293235\pi\)
0.604846 + 0.796343i \(0.293235\pi\)
\(660\) −4.12311 −0.160492
\(661\) 11.6155 0.451792 0.225896 0.974151i \(-0.427469\pi\)
0.225896 + 0.974151i \(0.427469\pi\)
\(662\) −18.7386 −0.728298
\(663\) 0 0
\(664\) 1.12311 0.0435850
\(665\) 12.6847 0.491890
\(666\) 4.12311 0.159767
\(667\) −50.4233 −1.95240
\(668\) 20.3693 0.788113
\(669\) 26.3002 1.01682
\(670\) −14.2462 −0.550379
\(671\) 24.7386 0.955024
\(672\) 3.56155 0.137390
\(673\) 42.2462 1.62847 0.814236 0.580534i \(-0.197156\pi\)
0.814236 + 0.580534i \(0.197156\pi\)
\(674\) 6.00000 0.231111
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −28.8769 −1.10983 −0.554915 0.831907i \(-0.687249\pi\)
−0.554915 + 0.831907i \(0.687249\pi\)
\(678\) 3.68466 0.141508
\(679\) 4.00000 0.153506
\(680\) 5.12311 0.196462
\(681\) 20.0000 0.766402
\(682\) 23.4384 0.897505
\(683\) −8.87689 −0.339665 −0.169832 0.985473i \(-0.554323\pi\)
−0.169832 + 0.985473i \(0.554323\pi\)
\(684\) 3.56155 0.136179
\(685\) −8.80776 −0.336527
\(686\) −4.68466 −0.178861
\(687\) 7.75379 0.295825
\(688\) 4.56155 0.173908
\(689\) 0 0
\(690\) −7.68466 −0.292550
\(691\) 16.4384 0.625348 0.312674 0.949860i \(-0.398775\pi\)
0.312674 + 0.949860i \(0.398775\pi\)
\(692\) −25.1771 −0.957089
\(693\) −14.6847 −0.557824
\(694\) −9.12311 −0.346308
\(695\) −8.43845 −0.320089
\(696\) −6.56155 −0.248715
\(697\) 21.7538 0.823984
\(698\) −24.4924 −0.927052
\(699\) −17.6847 −0.668895
\(700\) −3.56155 −0.134614
\(701\) 17.3002 0.653419 0.326710 0.945125i \(-0.394060\pi\)
0.326710 + 0.945125i \(0.394060\pi\)
\(702\) 0 0
\(703\) −14.6847 −0.553842
\(704\) 4.12311 0.155395
\(705\) 7.00000 0.263635
\(706\) 31.8617 1.19913
\(707\) 60.9848 2.29357
\(708\) −10.5616 −0.396927
\(709\) −17.7538 −0.666758 −0.333379 0.942793i \(-0.608189\pi\)
−0.333379 + 0.942793i \(0.608189\pi\)
\(710\) 4.87689 0.183027
\(711\) 7.43845 0.278964
\(712\) 1.80776 0.0677488
\(713\) 43.6847 1.63600
\(714\) 18.2462 0.682847
\(715\) 0 0
\(716\) −0.315342 −0.0117849
\(717\) −13.3693 −0.499286
\(718\) 4.87689 0.182004
\(719\) −20.9848 −0.782603 −0.391301 0.920263i \(-0.627975\pi\)
−0.391301 + 0.920263i \(0.627975\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −1.56155 −0.0581553
\(722\) 6.31534 0.235033
\(723\) 19.8769 0.739230
\(724\) 11.1231 0.413387
\(725\) 6.56155 0.243690
\(726\) −6.00000 −0.222681
\(727\) −23.4233 −0.868722 −0.434361 0.900739i \(-0.643026\pi\)
−0.434361 + 0.900739i \(0.643026\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 15.3693 0.568844
\(731\) 23.3693 0.864345
\(732\) 6.00000 0.221766
\(733\) −48.9309 −1.80730 −0.903651 0.428269i \(-0.859124\pi\)
−0.903651 + 0.428269i \(0.859124\pi\)
\(734\) 20.4924 0.756389
\(735\) −5.68466 −0.209682
\(736\) 7.68466 0.283260
\(737\) −58.7386 −2.16367
\(738\) −4.24621 −0.156305
\(739\) −42.5464 −1.56509 −0.782547 0.622591i \(-0.786080\pi\)
−0.782547 + 0.622591i \(0.786080\pi\)
\(740\) 4.12311 0.151568
\(741\) 0 0
\(742\) 15.8078 0.580321
\(743\) 32.1771 1.18046 0.590231 0.807234i \(-0.299037\pi\)
0.590231 + 0.807234i \(0.299037\pi\)
\(744\) 5.68466 0.208410
\(745\) −22.8078 −0.835612
\(746\) 5.19224 0.190101
\(747\) −1.12311 −0.0410923
\(748\) 21.1231 0.772337
\(749\) −7.12311 −0.260273
\(750\) 1.00000 0.0365148
\(751\) −3.19224 −0.116486 −0.0582432 0.998302i \(-0.518550\pi\)
−0.0582432 + 0.998302i \(0.518550\pi\)
\(752\) −7.00000 −0.255264
\(753\) 0.123106 0.00448622
\(754\) 0 0
\(755\) 9.36932 0.340984
\(756\) −3.56155 −0.129532
\(757\) 33.4233 1.21479 0.607395 0.794400i \(-0.292215\pi\)
0.607395 + 0.794400i \(0.292215\pi\)
\(758\) −5.31534 −0.193062
\(759\) −31.6847 −1.15008
\(760\) 3.56155 0.129191
\(761\) −10.9309 −0.396244 −0.198122 0.980177i \(-0.563484\pi\)
−0.198122 + 0.980177i \(0.563484\pi\)
\(762\) 4.43845 0.160788
\(763\) −72.1080 −2.61048
\(764\) 19.1231 0.691850
\(765\) −5.12311 −0.185226
\(766\) 20.8078 0.751815
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 45.6847 1.64743 0.823715 0.567003i \(-0.191897\pi\)
0.823715 + 0.567003i \(0.191897\pi\)
\(770\) −14.6847 −0.529198
\(771\) −1.43845 −0.0518044
\(772\) 0 0
\(773\) 21.1771 0.761687 0.380843 0.924640i \(-0.375634\pi\)
0.380843 + 0.924640i \(0.375634\pi\)
\(774\) −4.56155 −0.163962
\(775\) −5.68466 −0.204199
\(776\) 1.12311 0.0403171
\(777\) 14.6847 0.526809
\(778\) −19.0540 −0.683118
\(779\) 15.1231 0.541841
\(780\) 0 0
\(781\) 20.1080 0.719519
\(782\) 39.3693 1.40784
\(783\) 6.56155 0.234491
\(784\) 5.68466 0.203024
\(785\) −22.1231 −0.789607
\(786\) −6.12311 −0.218404
\(787\) −43.6847 −1.55719 −0.778595 0.627527i \(-0.784067\pi\)
−0.778595 + 0.627527i \(0.784067\pi\)
\(788\) 7.80776 0.278140
\(789\) −13.0000 −0.462812
\(790\) 7.43845 0.264648
\(791\) 13.1231 0.466604
\(792\) −4.12311 −0.146508
\(793\) 0 0
\(794\) −11.8769 −0.421495
\(795\) −4.43845 −0.157415
\(796\) 11.1231 0.394248
\(797\) 4.87689 0.172748 0.0863742 0.996263i \(-0.472472\pi\)
0.0863742 + 0.996263i \(0.472472\pi\)
\(798\) 12.6847 0.449032
\(799\) −35.8617 −1.26870
\(800\) −1.00000 −0.0353553
\(801\) −1.80776 −0.0638742
\(802\) 12.6847 0.447911
\(803\) 63.3693 2.23625
\(804\) −14.2462 −0.502425
\(805\) −27.3693 −0.964642
\(806\) 0 0
\(807\) −3.36932 −0.118606
\(808\) 17.1231 0.602389
\(809\) 18.4924 0.650159 0.325079 0.945687i \(-0.394609\pi\)
0.325079 + 0.945687i \(0.394609\pi\)
\(810\) 1.00000 0.0351364
\(811\) 22.5464 0.791711 0.395856 0.918313i \(-0.370448\pi\)
0.395856 + 0.918313i \(0.370448\pi\)
\(812\) −23.3693 −0.820102
\(813\) −32.1771 −1.12850
\(814\) 17.0000 0.595850
\(815\) −18.5616 −0.650183
\(816\) 5.12311 0.179345
\(817\) 16.2462 0.568383
\(818\) 1.80776 0.0632070
\(819\) 0 0
\(820\) −4.24621 −0.148284
\(821\) 12.5616 0.438401 0.219201 0.975680i \(-0.429655\pi\)
0.219201 + 0.975680i \(0.429655\pi\)
\(822\) −8.80776 −0.307206
\(823\) −32.0540 −1.11733 −0.558666 0.829393i \(-0.688687\pi\)
−0.558666 + 0.829393i \(0.688687\pi\)
\(824\) −0.438447 −0.0152740
\(825\) 4.12311 0.143548
\(826\) −37.6155 −1.30881
\(827\) 11.7538 0.408719 0.204360 0.978896i \(-0.434489\pi\)
0.204360 + 0.978896i \(0.434489\pi\)
\(828\) −7.68466 −0.267060
\(829\) 53.6155 1.86214 0.931072 0.364835i \(-0.118875\pi\)
0.931072 + 0.364835i \(0.118875\pi\)
\(830\) −1.12311 −0.0389836
\(831\) −1.00000 −0.0346896
\(832\) 0 0
\(833\) 29.1231 1.00906
\(834\) −8.43845 −0.292200
\(835\) −20.3693 −0.704909
\(836\) 14.6847 0.507880
\(837\) −5.68466 −0.196491
\(838\) 0.492423 0.0170105
\(839\) 12.7386 0.439786 0.219893 0.975524i \(-0.429429\pi\)
0.219893 + 0.975524i \(0.429429\pi\)
\(840\) −3.56155 −0.122885
\(841\) 14.0540 0.484620
\(842\) 0.492423 0.0169700
\(843\) 0.246211 0.00847997
\(844\) 6.93087 0.238570
\(845\) 0 0
\(846\) 7.00000 0.240665
\(847\) −21.3693 −0.734258
\(848\) 4.43845 0.152417
\(849\) 11.4384 0.392566
\(850\) −5.12311 −0.175721
\(851\) 31.6847 1.08614
\(852\) 4.87689 0.167080
\(853\) 51.3002 1.75648 0.878242 0.478216i \(-0.158716\pi\)
0.878242 + 0.478216i \(0.158716\pi\)
\(854\) 21.3693 0.731243
\(855\) −3.56155 −0.121803
\(856\) −2.00000 −0.0683586
\(857\) −26.8078 −0.915736 −0.457868 0.889020i \(-0.651387\pi\)
−0.457868 + 0.889020i \(0.651387\pi\)
\(858\) 0 0
\(859\) −46.7926 −1.59654 −0.798272 0.602298i \(-0.794252\pi\)
−0.798272 + 0.602298i \(0.794252\pi\)
\(860\) −4.56155 −0.155548
\(861\) −15.1231 −0.515394
\(862\) 26.7386 0.910721
\(863\) −6.56155 −0.223358 −0.111679 0.993744i \(-0.535623\pi\)
−0.111679 + 0.993744i \(0.535623\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 25.1771 0.856046
\(866\) −30.7386 −1.04454
\(867\) 9.24621 0.314018
\(868\) 20.2462 0.687201
\(869\) 30.6695 1.04039
\(870\) 6.56155 0.222457
\(871\) 0 0
\(872\) −20.2462 −0.685623
\(873\) −1.12311 −0.0380114
\(874\) 27.3693 0.925781
\(875\) 3.56155 0.120402
\(876\) 15.3693 0.519281
\(877\) −12.5616 −0.424173 −0.212087 0.977251i \(-0.568026\pi\)
−0.212087 + 0.977251i \(0.568026\pi\)
\(878\) −16.8769 −0.569568
\(879\) 24.9309 0.840897
\(880\) −4.12311 −0.138990
\(881\) −20.4384 −0.688589 −0.344294 0.938862i \(-0.611882\pi\)
−0.344294 + 0.938862i \(0.611882\pi\)
\(882\) −5.68466 −0.191412
\(883\) 26.1771 0.880929 0.440464 0.897770i \(-0.354814\pi\)
0.440464 + 0.897770i \(0.354814\pi\)
\(884\) 0 0
\(885\) 10.5616 0.355023
\(886\) 4.87689 0.163842
\(887\) 41.1080 1.38027 0.690135 0.723681i \(-0.257551\pi\)
0.690135 + 0.723681i \(0.257551\pi\)
\(888\) 4.12311 0.138362
\(889\) 15.8078 0.530175
\(890\) −1.80776 −0.0605964
\(891\) 4.12311 0.138129
\(892\) 26.3002 0.880595
\(893\) −24.9309 −0.834280
\(894\) −22.8078 −0.762806
\(895\) 0.315342 0.0105407
\(896\) 3.56155 0.118983
\(897\) 0 0
\(898\) 25.1771 0.840170
\(899\) −37.3002 −1.24403
\(900\) 1.00000 0.0333333
\(901\) 22.7386 0.757534
\(902\) −17.5076 −0.582939
\(903\) −16.2462 −0.540640
\(904\) 3.68466 0.122550
\(905\) −11.1231 −0.369745
\(906\) 9.36932 0.311275
\(907\) 40.9157 1.35858 0.679292 0.733868i \(-0.262287\pi\)
0.679292 + 0.733868i \(0.262287\pi\)
\(908\) 20.0000 0.663723
\(909\) −17.1231 −0.567938
\(910\) 0 0
\(911\) −43.3693 −1.43689 −0.718445 0.695584i \(-0.755146\pi\)
−0.718445 + 0.695584i \(0.755146\pi\)
\(912\) 3.56155 0.117935
\(913\) −4.63068 −0.153253
\(914\) 3.75379 0.124164
\(915\) −6.00000 −0.198354
\(916\) 7.75379 0.256192
\(917\) −21.8078 −0.720156
\(918\) −5.12311 −0.169088
\(919\) 41.3693 1.36465 0.682324 0.731050i \(-0.260969\pi\)
0.682324 + 0.731050i \(0.260969\pi\)
\(920\) −7.68466 −0.253356
\(921\) 15.6155 0.514549
\(922\) 7.05398 0.232310
\(923\) 0 0
\(924\) −14.6847 −0.483090
\(925\) −4.12311 −0.135567
\(926\) 33.6155 1.10467
\(927\) 0.438447 0.0144005
\(928\) −6.56155 −0.215394
\(929\) 16.8769 0.553713 0.276856 0.960911i \(-0.410707\pi\)
0.276856 + 0.960911i \(0.410707\pi\)
\(930\) −5.68466 −0.186407
\(931\) 20.2462 0.663543
\(932\) −17.6847 −0.579280
\(933\) −18.7386 −0.613475
\(934\) −39.8617 −1.30432
\(935\) −21.1231 −0.690799
\(936\) 0 0
\(937\) −29.2311 −0.954937 −0.477468 0.878649i \(-0.658446\pi\)
−0.477468 + 0.878649i \(0.658446\pi\)
\(938\) −50.7386 −1.65668
\(939\) 6.63068 0.216384
\(940\) 7.00000 0.228315
\(941\) 0.630683 0.0205597 0.0102798 0.999947i \(-0.496728\pi\)
0.0102798 + 0.999947i \(0.496728\pi\)
\(942\) −22.1231 −0.720810
\(943\) −32.6307 −1.06260
\(944\) −10.5616 −0.343749
\(945\) 3.56155 0.115857
\(946\) −18.8078 −0.611493
\(947\) −13.8617 −0.450446 −0.225223 0.974307i \(-0.572311\pi\)
−0.225223 + 0.974307i \(0.572311\pi\)
\(948\) 7.43845 0.241590
\(949\) 0 0
\(950\) −3.56155 −0.115552
\(951\) 4.19224 0.135943
\(952\) 18.2462 0.591363
\(953\) 5.82292 0.188623 0.0943114 0.995543i \(-0.469935\pi\)
0.0943114 + 0.995543i \(0.469935\pi\)
\(954\) −4.43845 −0.143700
\(955\) −19.1231 −0.618809
\(956\) −13.3693 −0.432395
\(957\) 27.0540 0.874531
\(958\) −21.7538 −0.702833
\(959\) −31.3693 −1.01297
\(960\) −1.00000 −0.0322749
\(961\) 1.31534 0.0424304
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) 19.8769 0.640192
\(965\) 0 0
\(966\) −27.3693 −0.880593
\(967\) −3.31534 −0.106614 −0.0533071 0.998578i \(-0.516976\pi\)
−0.0533071 + 0.998578i \(0.516976\pi\)
\(968\) −6.00000 −0.192847
\(969\) 18.2462 0.586153
\(970\) −1.12311 −0.0360607
\(971\) −16.6847 −0.535436 −0.267718 0.963497i \(-0.586270\pi\)
−0.267718 + 0.963497i \(0.586270\pi\)
\(972\) 1.00000 0.0320750
\(973\) −30.0540 −0.963486
\(974\) −24.0540 −0.770739
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −5.30019 −0.169568 −0.0847840 0.996399i \(-0.527020\pi\)
−0.0847840 + 0.996399i \(0.527020\pi\)
\(978\) −18.5616 −0.593533
\(979\) −7.45360 −0.238218
\(980\) −5.68466 −0.181590
\(981\) 20.2462 0.646412
\(982\) 21.5616 0.688057
\(983\) −23.8769 −0.761555 −0.380777 0.924667i \(-0.624344\pi\)
−0.380777 + 0.924667i \(0.624344\pi\)
\(984\) −4.24621 −0.135364
\(985\) −7.80776 −0.248776
\(986\) −33.6155 −1.07054
\(987\) 24.9309 0.793558
\(988\) 0 0
\(989\) −35.0540 −1.11465
\(990\) 4.12311 0.131041
\(991\) −12.4233 −0.394639 −0.197319 0.980339i \(-0.563224\pi\)
−0.197319 + 0.980339i \(0.563224\pi\)
\(992\) 5.68466 0.180488
\(993\) 18.7386 0.594653
\(994\) 17.3693 0.550921
\(995\) −11.1231 −0.352626
\(996\) −1.12311 −0.0355870
\(997\) 32.5464 1.03075 0.515377 0.856963i \(-0.327652\pi\)
0.515377 + 0.856963i \(0.327652\pi\)
\(998\) 4.00000 0.126618
\(999\) −4.12311 −0.130449
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 5070.2.a.bb.1.1 2
13.4 even 6 390.2.i.g.211.1 yes 4
13.5 odd 4 5070.2.b.r.1351.3 4
13.8 odd 4 5070.2.b.r.1351.2 4
13.10 even 6 390.2.i.g.61.1 4
13.12 even 2 5070.2.a.bi.1.2 2
39.17 odd 6 1170.2.i.o.991.1 4
39.23 odd 6 1170.2.i.o.451.1 4
65.4 even 6 1950.2.i.bi.601.2 4
65.17 odd 12 1950.2.z.n.1849.3 8
65.23 odd 12 1950.2.z.n.1699.3 8
65.43 odd 12 1950.2.z.n.1849.2 8
65.49 even 6 1950.2.i.bi.451.2 4
65.62 odd 12 1950.2.z.n.1699.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.g.61.1 4 13.10 even 6
390.2.i.g.211.1 yes 4 13.4 even 6
1170.2.i.o.451.1 4 39.23 odd 6
1170.2.i.o.991.1 4 39.17 odd 6
1950.2.i.bi.451.2 4 65.49 even 6
1950.2.i.bi.601.2 4 65.4 even 6
1950.2.z.n.1699.2 8 65.62 odd 12
1950.2.z.n.1699.3 8 65.23 odd 12
1950.2.z.n.1849.2 8 65.43 odd 12
1950.2.z.n.1849.3 8 65.17 odd 12
5070.2.a.bb.1.1 2 1.1 even 1 trivial
5070.2.a.bi.1.2 2 13.12 even 2
5070.2.b.r.1351.2 4 13.8 odd 4
5070.2.b.r.1351.3 4 13.5 odd 4