Properties

Label 39.2.a.b.1.2
Level $39$
Weight $2$
Character 39.1
Self dual yes
Analytic conductor $0.311$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.311416567883\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 39.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -2.82843 q^{5} +0.414214 q^{6} +2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} +1.00000 q^{3} -1.82843 q^{4} -2.82843 q^{5} +0.414214 q^{6} +2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -1.17157 q^{10} -2.00000 q^{11} -1.82843 q^{12} -1.00000 q^{13} +1.17157 q^{14} -2.82843 q^{15} +3.00000 q^{16} +7.65685 q^{17} +0.414214 q^{18} -2.82843 q^{19} +5.17157 q^{20} +2.82843 q^{21} -0.828427 q^{22} -4.00000 q^{23} -1.58579 q^{24} +3.00000 q^{25} -0.414214 q^{26} +1.00000 q^{27} -5.17157 q^{28} +2.00000 q^{29} -1.17157 q^{30} -1.17157 q^{31} +4.41421 q^{32} -2.00000 q^{33} +3.17157 q^{34} -8.00000 q^{35} -1.82843 q^{36} -7.65685 q^{37} -1.17157 q^{38} -1.00000 q^{39} +4.48528 q^{40} +5.17157 q^{41} +1.17157 q^{42} -1.65685 q^{43} +3.65685 q^{44} -2.82843 q^{45} -1.65685 q^{46} -11.6569 q^{47} +3.00000 q^{48} +1.00000 q^{49} +1.24264 q^{50} +7.65685 q^{51} +1.82843 q^{52} -2.00000 q^{53} +0.414214 q^{54} +5.65685 q^{55} -4.48528 q^{56} -2.82843 q^{57} +0.828427 q^{58} +7.65685 q^{59} +5.17157 q^{60} +13.3137 q^{61} -0.485281 q^{62} +2.82843 q^{63} -4.17157 q^{64} +2.82843 q^{65} -0.828427 q^{66} +6.82843 q^{67} -14.0000 q^{68} -4.00000 q^{69} -3.31371 q^{70} +2.00000 q^{71} -1.58579 q^{72} +0.343146 q^{73} -3.17157 q^{74} +3.00000 q^{75} +5.17157 q^{76} -5.65685 q^{77} -0.414214 q^{78} -11.3137 q^{79} -8.48528 q^{80} +1.00000 q^{81} +2.14214 q^{82} +3.65685 q^{83} -5.17157 q^{84} -21.6569 q^{85} -0.686292 q^{86} +2.00000 q^{87} +3.17157 q^{88} +14.8284 q^{89} -1.17157 q^{90} -2.82843 q^{91} +7.31371 q^{92} -1.17157 q^{93} -4.82843 q^{94} +8.00000 q^{95} +4.41421 q^{96} +3.65685 q^{97} +0.414214 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 6q^{8} + 2q^{9} - 8q^{10} - 4q^{11} + 2q^{12} - 2q^{13} + 8q^{14} + 6q^{16} + 4q^{17} - 2q^{18} + 16q^{20} + 4q^{22} - 8q^{23} - 6q^{24} + 6q^{25} + 2q^{26} + 2q^{27} - 16q^{28} + 4q^{29} - 8q^{30} - 8q^{31} + 6q^{32} - 4q^{33} + 12q^{34} - 16q^{35} + 2q^{36} - 4q^{37} - 8q^{38} - 2q^{39} - 8q^{40} + 16q^{41} + 8q^{42} + 8q^{43} - 4q^{44} + 8q^{46} - 12q^{47} + 6q^{48} + 2q^{49} - 6q^{50} + 4q^{51} - 2q^{52} - 4q^{53} - 2q^{54} + 8q^{56} - 4q^{58} + 4q^{59} + 16q^{60} + 4q^{61} + 16q^{62} - 14q^{64} + 4q^{66} + 8q^{67} - 28q^{68} - 8q^{69} + 16q^{70} + 4q^{71} - 6q^{72} + 12q^{73} - 12q^{74} + 6q^{75} + 16q^{76} + 2q^{78} + 2q^{81} - 24q^{82} - 4q^{83} - 16q^{84} - 32q^{85} - 24q^{86} + 4q^{87} + 12q^{88} + 24q^{89} - 8q^{90} - 8q^{92} - 8q^{93} - 4q^{94} + 16q^{95} + 6q^{96} - 4q^{97} - 2q^{98} - 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.82843 −0.914214
\(5\) −2.82843 −1.26491 −0.632456 0.774597i \(-0.717953\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0.414214 0.169102
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) −1.17157 −0.370484
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.82843 −0.527821
\(13\) −1.00000 −0.277350
\(14\) 1.17157 0.313116
\(15\) −2.82843 −0.730297
\(16\) 3.00000 0.750000
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 0.414214 0.0976311
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 5.17157 1.15640
\(21\) 2.82843 0.617213
\(22\) −0.828427 −0.176621
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.58579 −0.323697
\(25\) 3.00000 0.600000
\(26\) −0.414214 −0.0812340
\(27\) 1.00000 0.192450
\(28\) −5.17157 −0.977335
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.17157 −0.213899
\(31\) −1.17157 −0.210421 −0.105210 0.994450i \(-0.533552\pi\)
−0.105210 + 0.994450i \(0.533552\pi\)
\(32\) 4.41421 0.780330
\(33\) −2.00000 −0.348155
\(34\) 3.17157 0.543920
\(35\) −8.00000 −1.35225
\(36\) −1.82843 −0.304738
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) −1.17157 −0.190054
\(39\) −1.00000 −0.160128
\(40\) 4.48528 0.709185
\(41\) 5.17157 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(42\) 1.17157 0.180778
\(43\) −1.65685 −0.252668 −0.126334 0.991988i \(-0.540321\pi\)
−0.126334 + 0.991988i \(0.540321\pi\)
\(44\) 3.65685 0.551292
\(45\) −2.82843 −0.421637
\(46\) −1.65685 −0.244290
\(47\) −11.6569 −1.70033 −0.850163 0.526519i \(-0.823497\pi\)
−0.850163 + 0.526519i \(0.823497\pi\)
\(48\) 3.00000 0.433013
\(49\) 1.00000 0.142857
\(50\) 1.24264 0.175736
\(51\) 7.65685 1.07217
\(52\) 1.82843 0.253557
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0.414214 0.0563673
\(55\) 5.65685 0.762770
\(56\) −4.48528 −0.599371
\(57\) −2.82843 −0.374634
\(58\) 0.828427 0.108778
\(59\) 7.65685 0.996838 0.498419 0.866936i \(-0.333914\pi\)
0.498419 + 0.866936i \(0.333914\pi\)
\(60\) 5.17157 0.667647
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) −0.485281 −0.0616308
\(63\) 2.82843 0.356348
\(64\) −4.17157 −0.521447
\(65\) 2.82843 0.350823
\(66\) −0.828427 −0.101972
\(67\) 6.82843 0.834225 0.417113 0.908855i \(-0.363042\pi\)
0.417113 + 0.908855i \(0.363042\pi\)
\(68\) −14.0000 −1.69775
\(69\) −4.00000 −0.481543
\(70\) −3.31371 −0.396064
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −1.58579 −0.186887
\(73\) 0.343146 0.0401622 0.0200811 0.999798i \(-0.493608\pi\)
0.0200811 + 0.999798i \(0.493608\pi\)
\(74\) −3.17157 −0.368688
\(75\) 3.00000 0.346410
\(76\) 5.17157 0.593220
\(77\) −5.65685 −0.644658
\(78\) −0.414214 −0.0469005
\(79\) −11.3137 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(80\) −8.48528 −0.948683
\(81\) 1.00000 0.111111
\(82\) 2.14214 0.236559
\(83\) 3.65685 0.401392 0.200696 0.979654i \(-0.435680\pi\)
0.200696 + 0.979654i \(0.435680\pi\)
\(84\) −5.17157 −0.564265
\(85\) −21.6569 −2.34902
\(86\) −0.686292 −0.0740047
\(87\) 2.00000 0.214423
\(88\) 3.17157 0.338091
\(89\) 14.8284 1.57181 0.785905 0.618347i \(-0.212197\pi\)
0.785905 + 0.618347i \(0.212197\pi\)
\(90\) −1.17157 −0.123495
\(91\) −2.82843 −0.296500
\(92\) 7.31371 0.762507
\(93\) −1.17157 −0.121486
\(94\) −4.82843 −0.498014
\(95\) 8.00000 0.820783
\(96\) 4.41421 0.450524
\(97\) 3.65685 0.371297 0.185649 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) 0.414214 0.0418419
\(99\) −2.00000 −0.201008
\(100\) −5.48528 −0.548528
\(101\) 7.65685 0.761885 0.380943 0.924599i \(-0.375599\pi\)
0.380943 + 0.924599i \(0.375599\pi\)
\(102\) 3.17157 0.314033
\(103\) 2.34315 0.230877 0.115439 0.993315i \(-0.463173\pi\)
0.115439 + 0.993315i \(0.463173\pi\)
\(104\) 1.58579 0.155499
\(105\) −8.00000 −0.780720
\(106\) −0.828427 −0.0804640
\(107\) −11.3137 −1.09374 −0.546869 0.837218i \(-0.684180\pi\)
−0.546869 + 0.837218i \(0.684180\pi\)
\(108\) −1.82843 −0.175940
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) 2.34315 0.223410
\(111\) −7.65685 −0.726756
\(112\) 8.48528 0.801784
\(113\) −5.31371 −0.499872 −0.249936 0.968262i \(-0.580410\pi\)
−0.249936 + 0.968262i \(0.580410\pi\)
\(114\) −1.17157 −0.109728
\(115\) 11.3137 1.05501
\(116\) −3.65685 −0.339530
\(117\) −1.00000 −0.0924500
\(118\) 3.17157 0.291967
\(119\) 21.6569 1.98528
\(120\) 4.48528 0.409448
\(121\) −7.00000 −0.636364
\(122\) 5.51472 0.499279
\(123\) 5.17157 0.466305
\(124\) 2.14214 0.192369
\(125\) 5.65685 0.505964
\(126\) 1.17157 0.104372
\(127\) 5.65685 0.501965 0.250982 0.967992i \(-0.419246\pi\)
0.250982 + 0.967992i \(0.419246\pi\)
\(128\) −10.5563 −0.933058
\(129\) −1.65685 −0.145878
\(130\) 1.17157 0.102754
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 3.65685 0.318288
\(133\) −8.00000 −0.693688
\(134\) 2.82843 0.244339
\(135\) −2.82843 −0.243432
\(136\) −12.1421 −1.04118
\(137\) −10.8284 −0.925135 −0.462567 0.886584i \(-0.653072\pi\)
−0.462567 + 0.886584i \(0.653072\pi\)
\(138\) −1.65685 −0.141041
\(139\) −7.31371 −0.620341 −0.310170 0.950681i \(-0.600386\pi\)
−0.310170 + 0.950681i \(0.600386\pi\)
\(140\) 14.6274 1.23624
\(141\) −11.6569 −0.981684
\(142\) 0.828427 0.0695201
\(143\) 2.00000 0.167248
\(144\) 3.00000 0.250000
\(145\) −5.65685 −0.469776
\(146\) 0.142136 0.0117632
\(147\) 1.00000 0.0824786
\(148\) 14.0000 1.15079
\(149\) −9.17157 −0.751365 −0.375682 0.926749i \(-0.622592\pi\)
−0.375682 + 0.926749i \(0.622592\pi\)
\(150\) 1.24264 0.101461
\(151\) −3.51472 −0.286024 −0.143012 0.989721i \(-0.545679\pi\)
−0.143012 + 0.989721i \(0.545679\pi\)
\(152\) 4.48528 0.363804
\(153\) 7.65685 0.619020
\(154\) −2.34315 −0.188816
\(155\) 3.31371 0.266163
\(156\) 1.82843 0.146391
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −4.68629 −0.372821
\(159\) −2.00000 −0.158610
\(160\) −12.4853 −0.987048
\(161\) −11.3137 −0.891645
\(162\) 0.414214 0.0325437
\(163\) 18.8284 1.47476 0.737378 0.675480i \(-0.236064\pi\)
0.737378 + 0.675480i \(0.236064\pi\)
\(164\) −9.45584 −0.738377
\(165\) 5.65685 0.440386
\(166\) 1.51472 0.117565
\(167\) −3.65685 −0.282976 −0.141488 0.989940i \(-0.545189\pi\)
−0.141488 + 0.989940i \(0.545189\pi\)
\(168\) −4.48528 −0.346047
\(169\) 1.00000 0.0769231
\(170\) −8.97056 −0.688011
\(171\) −2.82843 −0.216295
\(172\) 3.02944 0.230992
\(173\) −11.6569 −0.886254 −0.443127 0.896459i \(-0.646131\pi\)
−0.443127 + 0.896459i \(0.646131\pi\)
\(174\) 0.828427 0.0628029
\(175\) 8.48528 0.641427
\(176\) −6.00000 −0.452267
\(177\) 7.65685 0.575524
\(178\) 6.14214 0.460373
\(179\) −23.3137 −1.74255 −0.871274 0.490797i \(-0.836706\pi\)
−0.871274 + 0.490797i \(0.836706\pi\)
\(180\) 5.17157 0.385466
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −1.17157 −0.0868428
\(183\) 13.3137 0.984178
\(184\) 6.34315 0.467623
\(185\) 21.6569 1.59224
\(186\) −0.485281 −0.0355826
\(187\) −15.3137 −1.11985
\(188\) 21.3137 1.55446
\(189\) 2.82843 0.205738
\(190\) 3.31371 0.240402
\(191\) 3.31371 0.239772 0.119886 0.992788i \(-0.461747\pi\)
0.119886 + 0.992788i \(0.461747\pi\)
\(192\) −4.17157 −0.301057
\(193\) 5.31371 0.382489 0.191245 0.981542i \(-0.438748\pi\)
0.191245 + 0.981542i \(0.438748\pi\)
\(194\) 1.51472 0.108750
\(195\) 2.82843 0.202548
\(196\) −1.82843 −0.130602
\(197\) 0.485281 0.0345749 0.0172874 0.999851i \(-0.494497\pi\)
0.0172874 + 0.999851i \(0.494497\pi\)
\(198\) −0.828427 −0.0588738
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) −4.75736 −0.336396
\(201\) 6.82843 0.481640
\(202\) 3.17157 0.223151
\(203\) 5.65685 0.397033
\(204\) −14.0000 −0.980196
\(205\) −14.6274 −1.02162
\(206\) 0.970563 0.0676223
\(207\) −4.00000 −0.278019
\(208\) −3.00000 −0.208013
\(209\) 5.65685 0.391293
\(210\) −3.31371 −0.228668
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 3.65685 0.251154
\(213\) 2.00000 0.137038
\(214\) −4.68629 −0.320348
\(215\) 4.68629 0.319602
\(216\) −1.58579 −0.107899
\(217\) −3.31371 −0.224949
\(218\) 2.20101 0.149071
\(219\) 0.343146 0.0231876
\(220\) −10.3431 −0.697335
\(221\) −7.65685 −0.515056
\(222\) −3.17157 −0.212862
\(223\) −12.4853 −0.836076 −0.418038 0.908429i \(-0.637282\pi\)
−0.418038 + 0.908429i \(0.637282\pi\)
\(224\) 12.4853 0.834208
\(225\) 3.00000 0.200000
\(226\) −2.20101 −0.146409
\(227\) −17.3137 −1.14915 −0.574576 0.818452i \(-0.694833\pi\)
−0.574576 + 0.818452i \(0.694833\pi\)
\(228\) 5.17157 0.342496
\(229\) −1.31371 −0.0868123 −0.0434062 0.999058i \(-0.513821\pi\)
−0.0434062 + 0.999058i \(0.513821\pi\)
\(230\) 4.68629 0.309005
\(231\) −5.65685 −0.372194
\(232\) −3.17157 −0.208224
\(233\) 6.97056 0.456657 0.228328 0.973584i \(-0.426674\pi\)
0.228328 + 0.973584i \(0.426674\pi\)
\(234\) −0.414214 −0.0270780
\(235\) 32.9706 2.15076
\(236\) −14.0000 −0.911322
\(237\) −11.3137 −0.734904
\(238\) 8.97056 0.581475
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) −8.48528 −0.547723
\(241\) 0.343146 0.0221040 0.0110520 0.999939i \(-0.496482\pi\)
0.0110520 + 0.999939i \(0.496482\pi\)
\(242\) −2.89949 −0.186387
\(243\) 1.00000 0.0641500
\(244\) −24.3431 −1.55841
\(245\) −2.82843 −0.180702
\(246\) 2.14214 0.136578
\(247\) 2.82843 0.179969
\(248\) 1.85786 0.117975
\(249\) 3.65685 0.231744
\(250\) 2.34315 0.148194
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −5.17157 −0.325778
\(253\) 8.00000 0.502956
\(254\) 2.34315 0.147022
\(255\) −21.6569 −1.35620
\(256\) 3.97056 0.248160
\(257\) −4.34315 −0.270918 −0.135459 0.990783i \(-0.543251\pi\)
−0.135459 + 0.990783i \(0.543251\pi\)
\(258\) −0.686292 −0.0427266
\(259\) −21.6569 −1.34569
\(260\) −5.17157 −0.320727
\(261\) 2.00000 0.123797
\(262\) −3.31371 −0.204722
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 3.17157 0.195197
\(265\) 5.65685 0.347498
\(266\) −3.31371 −0.203177
\(267\) 14.8284 0.907485
\(268\) −12.4853 −0.762660
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −1.17157 −0.0712997
\(271\) −27.7990 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(272\) 22.9706 1.39279
\(273\) −2.82843 −0.171184
\(274\) −4.48528 −0.270966
\(275\) −6.00000 −0.361814
\(276\) 7.31371 0.440234
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −3.02944 −0.181694
\(279\) −1.17157 −0.0701402
\(280\) 12.6863 0.758151
\(281\) 21.1716 1.26299 0.631495 0.775380i \(-0.282442\pi\)
0.631495 + 0.775380i \(0.282442\pi\)
\(282\) −4.82843 −0.287529
\(283\) 28.9706 1.72212 0.861061 0.508502i \(-0.169801\pi\)
0.861061 + 0.508502i \(0.169801\pi\)
\(284\) −3.65685 −0.216994
\(285\) 8.00000 0.473879
\(286\) 0.828427 0.0489859
\(287\) 14.6274 0.863429
\(288\) 4.41421 0.260110
\(289\) 41.6274 2.44867
\(290\) −2.34315 −0.137594
\(291\) 3.65685 0.214369
\(292\) −0.627417 −0.0367168
\(293\) 2.14214 0.125145 0.0625724 0.998040i \(-0.480070\pi\)
0.0625724 + 0.998040i \(0.480070\pi\)
\(294\) 0.414214 0.0241574
\(295\) −21.6569 −1.26091
\(296\) 12.1421 0.705747
\(297\) −2.00000 −0.116052
\(298\) −3.79899 −0.220070
\(299\) 4.00000 0.231326
\(300\) −5.48528 −0.316693
\(301\) −4.68629 −0.270113
\(302\) −1.45584 −0.0837744
\(303\) 7.65685 0.439875
\(304\) −8.48528 −0.486664
\(305\) −37.6569 −2.15623
\(306\) 3.17157 0.181307
\(307\) −22.8284 −1.30289 −0.651444 0.758697i \(-0.725836\pi\)
−0.651444 + 0.758697i \(0.725836\pi\)
\(308\) 10.3431 0.589355
\(309\) 2.34315 0.133297
\(310\) 1.37258 0.0779575
\(311\) −10.6274 −0.602626 −0.301313 0.953525i \(-0.597425\pi\)
−0.301313 + 0.953525i \(0.597425\pi\)
\(312\) 1.58579 0.0897775
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −4.14214 −0.233754
\(315\) −8.00000 −0.450749
\(316\) 20.6863 1.16369
\(317\) 8.48528 0.476581 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(318\) −0.828427 −0.0464559
\(319\) −4.00000 −0.223957
\(320\) 11.7990 0.659584
\(321\) −11.3137 −0.631470
\(322\) −4.68629 −0.261157
\(323\) −21.6569 −1.20502
\(324\) −1.82843 −0.101579
\(325\) −3.00000 −0.166410
\(326\) 7.79899 0.431946
\(327\) 5.31371 0.293849
\(328\) −8.20101 −0.452825
\(329\) −32.9706 −1.81773
\(330\) 2.34315 0.128986
\(331\) 26.1421 1.43690 0.718451 0.695578i \(-0.244852\pi\)
0.718451 + 0.695578i \(0.244852\pi\)
\(332\) −6.68629 −0.366958
\(333\) −7.65685 −0.419593
\(334\) −1.51472 −0.0828817
\(335\) −19.3137 −1.05522
\(336\) 8.48528 0.462910
\(337\) 9.31371 0.507350 0.253675 0.967290i \(-0.418361\pi\)
0.253675 + 0.967290i \(0.418361\pi\)
\(338\) 0.414214 0.0225302
\(339\) −5.31371 −0.288601
\(340\) 39.5980 2.14750
\(341\) 2.34315 0.126888
\(342\) −1.17157 −0.0633514
\(343\) −16.9706 −0.916324
\(344\) 2.62742 0.141661
\(345\) 11.3137 0.609110
\(346\) −4.82843 −0.259578
\(347\) −8.68629 −0.466305 −0.233152 0.972440i \(-0.574904\pi\)
−0.233152 + 0.972440i \(0.574904\pi\)
\(348\) −3.65685 −0.196028
\(349\) 3.65685 0.195747 0.0978735 0.995199i \(-0.468796\pi\)
0.0978735 + 0.995199i \(0.468796\pi\)
\(350\) 3.51472 0.187870
\(351\) −1.00000 −0.0533761
\(352\) −8.82843 −0.470557
\(353\) −33.4558 −1.78067 −0.890337 0.455301i \(-0.849532\pi\)
−0.890337 + 0.455301i \(0.849532\pi\)
\(354\) 3.17157 0.168567
\(355\) −5.65685 −0.300235
\(356\) −27.1127 −1.43697
\(357\) 21.6569 1.14620
\(358\) −9.65685 −0.510381
\(359\) 34.9706 1.84568 0.922838 0.385189i \(-0.125864\pi\)
0.922838 + 0.385189i \(0.125864\pi\)
\(360\) 4.48528 0.236395
\(361\) −11.0000 −0.578947
\(362\) 5.79899 0.304788
\(363\) −7.00000 −0.367405
\(364\) 5.17157 0.271064
\(365\) −0.970563 −0.0508016
\(366\) 5.51472 0.288259
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) −12.0000 −0.625543
\(369\) 5.17157 0.269221
\(370\) 8.97056 0.466357
\(371\) −5.65685 −0.293689
\(372\) 2.14214 0.111065
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −6.34315 −0.327996
\(375\) 5.65685 0.292119
\(376\) 18.4853 0.953306
\(377\) −2.00000 −0.103005
\(378\) 1.17157 0.0602592
\(379\) 0.485281 0.0249272 0.0124636 0.999922i \(-0.496033\pi\)
0.0124636 + 0.999922i \(0.496033\pi\)
\(380\) −14.6274 −0.750371
\(381\) 5.65685 0.289809
\(382\) 1.37258 0.0702275
\(383\) 30.9706 1.58252 0.791261 0.611479i \(-0.209425\pi\)
0.791261 + 0.611479i \(0.209425\pi\)
\(384\) −10.5563 −0.538701
\(385\) 16.0000 0.815436
\(386\) 2.20101 0.112028
\(387\) −1.65685 −0.0842226
\(388\) −6.68629 −0.339445
\(389\) −26.9706 −1.36746 −0.683731 0.729734i \(-0.739644\pi\)
−0.683731 + 0.729734i \(0.739644\pi\)
\(390\) 1.17157 0.0593249
\(391\) −30.6274 −1.54890
\(392\) −1.58579 −0.0800943
\(393\) −8.00000 −0.403547
\(394\) 0.201010 0.0101267
\(395\) 32.0000 1.61009
\(396\) 3.65685 0.183764
\(397\) 30.9706 1.55437 0.777184 0.629273i \(-0.216647\pi\)
0.777184 + 0.629273i \(0.216647\pi\)
\(398\) 8.97056 0.449654
\(399\) −8.00000 −0.400501
\(400\) 9.00000 0.450000
\(401\) 26.1421 1.30548 0.652738 0.757584i \(-0.273620\pi\)
0.652738 + 0.757584i \(0.273620\pi\)
\(402\) 2.82843 0.141069
\(403\) 1.17157 0.0583602
\(404\) −14.0000 −0.696526
\(405\) −2.82843 −0.140546
\(406\) 2.34315 0.116288
\(407\) 15.3137 0.759072
\(408\) −12.1421 −0.601125
\(409\) −34.9706 −1.72918 −0.864592 0.502475i \(-0.832423\pi\)
−0.864592 + 0.502475i \(0.832423\pi\)
\(410\) −6.05887 −0.299226
\(411\) −10.8284 −0.534127
\(412\) −4.28427 −0.211071
\(413\) 21.6569 1.06566
\(414\) −1.65685 −0.0814299
\(415\) −10.3431 −0.507725
\(416\) −4.41421 −0.216425
\(417\) −7.31371 −0.358154
\(418\) 2.34315 0.114607
\(419\) 14.6274 0.714596 0.357298 0.933990i \(-0.383698\pi\)
0.357298 + 0.933990i \(0.383698\pi\)
\(420\) 14.6274 0.713745
\(421\) 37.3137 1.81856 0.909279 0.416186i \(-0.136634\pi\)
0.909279 + 0.416186i \(0.136634\pi\)
\(422\) −4.97056 −0.241963
\(423\) −11.6569 −0.566776
\(424\) 3.17157 0.154025
\(425\) 22.9706 1.11424
\(426\) 0.828427 0.0401374
\(427\) 37.6569 1.82234
\(428\) 20.6863 0.999910
\(429\) 2.00000 0.0965609
\(430\) 1.94113 0.0936094
\(431\) 8.34315 0.401875 0.200938 0.979604i \(-0.435601\pi\)
0.200938 + 0.979604i \(0.435601\pi\)
\(432\) 3.00000 0.144338
\(433\) −21.3137 −1.02427 −0.512136 0.858905i \(-0.671146\pi\)
−0.512136 + 0.858905i \(0.671146\pi\)
\(434\) −1.37258 −0.0658861
\(435\) −5.65685 −0.271225
\(436\) −9.71573 −0.465299
\(437\) 11.3137 0.541208
\(438\) 0.142136 0.00679150
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) −8.97056 −0.427655
\(441\) 1.00000 0.0476190
\(442\) −3.17157 −0.150856
\(443\) −25.9411 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(444\) 14.0000 0.664411
\(445\) −41.9411 −1.98820
\(446\) −5.17157 −0.244881
\(447\) −9.17157 −0.433801
\(448\) −11.7990 −0.557450
\(449\) −31.7990 −1.50069 −0.750344 0.661048i \(-0.770112\pi\)
−0.750344 + 0.661048i \(0.770112\pi\)
\(450\) 1.24264 0.0585786
\(451\) −10.3431 −0.487040
\(452\) 9.71573 0.456989
\(453\) −3.51472 −0.165136
\(454\) −7.17157 −0.336579
\(455\) 8.00000 0.375046
\(456\) 4.48528 0.210043
\(457\) −7.65685 −0.358173 −0.179086 0.983833i \(-0.557314\pi\)
−0.179086 + 0.983833i \(0.557314\pi\)
\(458\) −0.544156 −0.0254267
\(459\) 7.65685 0.357391
\(460\) −20.6863 −0.964503
\(461\) 5.17157 0.240864 0.120432 0.992722i \(-0.461572\pi\)
0.120432 + 0.992722i \(0.461572\pi\)
\(462\) −2.34315 −0.109013
\(463\) −24.4853 −1.13793 −0.568964 0.822363i \(-0.692656\pi\)
−0.568964 + 0.822363i \(0.692656\pi\)
\(464\) 6.00000 0.278543
\(465\) 3.31371 0.153670
\(466\) 2.88730 0.133752
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 1.82843 0.0845191
\(469\) 19.3137 0.891824
\(470\) 13.6569 0.629944
\(471\) −10.0000 −0.460776
\(472\) −12.1421 −0.558887
\(473\) 3.31371 0.152364
\(474\) −4.68629 −0.215248
\(475\) −8.48528 −0.389331
\(476\) −39.5980 −1.81497
\(477\) −2.00000 −0.0915737
\(478\) 0.828427 0.0378914
\(479\) 25.3137 1.15661 0.578306 0.815820i \(-0.303714\pi\)
0.578306 + 0.815820i \(0.303714\pi\)
\(480\) −12.4853 −0.569873
\(481\) 7.65685 0.349123
\(482\) 0.142136 0.00647410
\(483\) −11.3137 −0.514792
\(484\) 12.7990 0.581772
\(485\) −10.3431 −0.469658
\(486\) 0.414214 0.0187891
\(487\) −7.79899 −0.353406 −0.176703 0.984264i \(-0.556543\pi\)
−0.176703 + 0.984264i \(0.556543\pi\)
\(488\) −21.1127 −0.955727
\(489\) 18.8284 0.851451
\(490\) −1.17157 −0.0529263
\(491\) 30.6274 1.38220 0.691098 0.722761i \(-0.257127\pi\)
0.691098 + 0.722761i \(0.257127\pi\)
\(492\) −9.45584 −0.426302
\(493\) 15.3137 0.689695
\(494\) 1.17157 0.0527116
\(495\) 5.65685 0.254257
\(496\) −3.51472 −0.157816
\(497\) 5.65685 0.253745
\(498\) 1.51472 0.0678762
\(499\) 26.1421 1.17028 0.585141 0.810931i \(-0.301039\pi\)
0.585141 + 0.810931i \(0.301039\pi\)
\(500\) −10.3431 −0.462560
\(501\) −3.65685 −0.163376
\(502\) 0 0
\(503\) −7.31371 −0.326102 −0.163051 0.986618i \(-0.552134\pi\)
−0.163051 + 0.986618i \(0.552134\pi\)
\(504\) −4.48528 −0.199790
\(505\) −21.6569 −0.963717
\(506\) 3.31371 0.147312
\(507\) 1.00000 0.0444116
\(508\) −10.3431 −0.458903
\(509\) 11.7990 0.522981 0.261491 0.965206i \(-0.415786\pi\)
0.261491 + 0.965206i \(0.415786\pi\)
\(510\) −8.97056 −0.397223
\(511\) 0.970563 0.0429352
\(512\) 22.7574 1.00574
\(513\) −2.82843 −0.124878
\(514\) −1.79899 −0.0793500
\(515\) −6.62742 −0.292039
\(516\) 3.02944 0.133364
\(517\) 23.3137 1.02534
\(518\) −8.97056 −0.394144
\(519\) −11.6569 −0.511679
\(520\) −4.48528 −0.196693
\(521\) 25.3137 1.10901 0.554507 0.832179i \(-0.312907\pi\)
0.554507 + 0.832179i \(0.312907\pi\)
\(522\) 0.828427 0.0362593
\(523\) −15.3137 −0.669622 −0.334811 0.942285i \(-0.608672\pi\)
−0.334811 + 0.942285i \(0.608672\pi\)
\(524\) 14.6274 0.639002
\(525\) 8.48528 0.370328
\(526\) 4.97056 0.216727
\(527\) −8.97056 −0.390764
\(528\) −6.00000 −0.261116
\(529\) −7.00000 −0.304348
\(530\) 2.34315 0.101780
\(531\) 7.65685 0.332279
\(532\) 14.6274 0.634179
\(533\) −5.17157 −0.224006
\(534\) 6.14214 0.265796
\(535\) 32.0000 1.38348
\(536\) −10.8284 −0.467717
\(537\) −23.3137 −1.00606
\(538\) 7.45584 0.321444
\(539\) −2.00000 −0.0861461
\(540\) 5.17157 0.222549
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −11.5147 −0.494600
\(543\) 14.0000 0.600798
\(544\) 33.7990 1.44912
\(545\) −15.0294 −0.643790
\(546\) −1.17157 −0.0501387
\(547\) 23.3137 0.996822 0.498411 0.866941i \(-0.333917\pi\)
0.498411 + 0.866941i \(0.333917\pi\)
\(548\) 19.7990 0.845771
\(549\) 13.3137 0.568215
\(550\) −2.48528 −0.105973
\(551\) −5.65685 −0.240990
\(552\) 6.34315 0.269982
\(553\) −32.0000 −1.36078
\(554\) −0.828427 −0.0351965
\(555\) 21.6569 0.919282
\(556\) 13.3726 0.567124
\(557\) −7.79899 −0.330454 −0.165227 0.986256i \(-0.552836\pi\)
−0.165227 + 0.986256i \(0.552836\pi\)
\(558\) −0.485281 −0.0205436
\(559\) 1.65685 0.0700775
\(560\) −24.0000 −1.01419
\(561\) −15.3137 −0.646545
\(562\) 8.76955 0.369921
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 21.3137 0.897469
\(565\) 15.0294 0.632293
\(566\) 12.0000 0.504398
\(567\) 2.82843 0.118783
\(568\) −3.17157 −0.133076
\(569\) −42.9706 −1.80142 −0.900710 0.434421i \(-0.856953\pi\)
−0.900710 + 0.434421i \(0.856953\pi\)
\(570\) 3.31371 0.138796
\(571\) −12.9706 −0.542801 −0.271401 0.962466i \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) −3.65685 −0.152901
\(573\) 3.31371 0.138432
\(574\) 6.05887 0.252893
\(575\) −12.0000 −0.500435
\(576\) −4.17157 −0.173816
\(577\) −31.9411 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(578\) 17.2426 0.717199
\(579\) 5.31371 0.220830
\(580\) 10.3431 0.429476
\(581\) 10.3431 0.429106
\(582\) 1.51472 0.0627871
\(583\) 4.00000 0.165663
\(584\) −0.544156 −0.0225173
\(585\) 2.82843 0.116941
\(586\) 0.887302 0.0366541
\(587\) −10.9706 −0.452804 −0.226402 0.974034i \(-0.572696\pi\)
−0.226402 + 0.974034i \(0.572696\pi\)
\(588\) −1.82843 −0.0754031
\(589\) 3.31371 0.136539
\(590\) −8.97056 −0.369312
\(591\) 0.485281 0.0199618
\(592\) −22.9706 −0.944084
\(593\) −20.4853 −0.841230 −0.420615 0.907239i \(-0.638186\pi\)
−0.420615 + 0.907239i \(0.638186\pi\)
\(594\) −0.828427 −0.0339908
\(595\) −61.2548 −2.51120
\(596\) 16.7696 0.686908
\(597\) 21.6569 0.886356
\(598\) 1.65685 0.0677538
\(599\) −23.3137 −0.952572 −0.476286 0.879290i \(-0.658017\pi\)
−0.476286 + 0.879290i \(0.658017\pi\)
\(600\) −4.75736 −0.194218
\(601\) −0.627417 −0.0255929 −0.0127964 0.999918i \(-0.504073\pi\)
−0.0127964 + 0.999918i \(0.504073\pi\)
\(602\) −1.94113 −0.0791144
\(603\) 6.82843 0.278075
\(604\) 6.42641 0.261487
\(605\) 19.7990 0.804943
\(606\) 3.17157 0.128836
\(607\) 41.9411 1.70234 0.851169 0.524892i \(-0.175894\pi\)
0.851169 + 0.524892i \(0.175894\pi\)
\(608\) −12.4853 −0.506345
\(609\) 5.65685 0.229227
\(610\) −15.5980 −0.631544
\(611\) 11.6569 0.471586
\(612\) −14.0000 −0.565916
\(613\) −47.6569 −1.92484 −0.962421 0.271561i \(-0.912460\pi\)
−0.962421 + 0.271561i \(0.912460\pi\)
\(614\) −9.45584 −0.381607
\(615\) −14.6274 −0.589834
\(616\) 8.97056 0.361434
\(617\) −34.8284 −1.40214 −0.701070 0.713093i \(-0.747294\pi\)
−0.701070 + 0.713093i \(0.747294\pi\)
\(618\) 0.970563 0.0390418
\(619\) 23.7990 0.956562 0.478281 0.878207i \(-0.341260\pi\)
0.478281 + 0.878207i \(0.341260\pi\)
\(620\) −6.05887 −0.243330
\(621\) −4.00000 −0.160514
\(622\) −4.40202 −0.176505
\(623\) 41.9411 1.68034
\(624\) −3.00000 −0.120096
\(625\) −31.0000 −1.24000
\(626\) 2.48528 0.0993318
\(627\) 5.65685 0.225913
\(628\) 18.2843 0.729622
\(629\) −58.6274 −2.33763
\(630\) −3.31371 −0.132021
\(631\) 43.1127 1.71629 0.858145 0.513408i \(-0.171617\pi\)
0.858145 + 0.513408i \(0.171617\pi\)
\(632\) 17.9411 0.713660
\(633\) −12.0000 −0.476957
\(634\) 3.51472 0.139587
\(635\) −16.0000 −0.634941
\(636\) 3.65685 0.145004
\(637\) −1.00000 −0.0396214
\(638\) −1.65685 −0.0655955
\(639\) 2.00000 0.0791188
\(640\) 29.8579 1.18024
\(641\) 30.2843 1.19616 0.598078 0.801438i \(-0.295931\pi\)
0.598078 + 0.801438i \(0.295931\pi\)
\(642\) −4.68629 −0.184953
\(643\) 22.8284 0.900265 0.450133 0.892962i \(-0.351377\pi\)
0.450133 + 0.892962i \(0.351377\pi\)
\(644\) 20.6863 0.815154
\(645\) 4.68629 0.184523
\(646\) −8.97056 −0.352942
\(647\) 11.3137 0.444788 0.222394 0.974957i \(-0.428613\pi\)
0.222394 + 0.974957i \(0.428613\pi\)
\(648\) −1.58579 −0.0622956
\(649\) −15.3137 −0.601116
\(650\) −1.24264 −0.0487404
\(651\) −3.31371 −0.129874
\(652\) −34.4264 −1.34824
\(653\) −25.3137 −0.990602 −0.495301 0.868721i \(-0.664942\pi\)
−0.495301 + 0.868721i \(0.664942\pi\)
\(654\) 2.20101 0.0860663
\(655\) 22.6274 0.884126
\(656\) 15.5147 0.605748
\(657\) 0.343146 0.0133874
\(658\) −13.6569 −0.532400
\(659\) −47.3137 −1.84308 −0.921540 0.388283i \(-0.873068\pi\)
−0.921540 + 0.388283i \(0.873068\pi\)
\(660\) −10.3431 −0.402606
\(661\) −34.9706 −1.36020 −0.680099 0.733121i \(-0.738063\pi\)
−0.680099 + 0.733121i \(0.738063\pi\)
\(662\) 10.8284 0.420859
\(663\) −7.65685 −0.297368
\(664\) −5.79899 −0.225044
\(665\) 22.6274 0.877454
\(666\) −3.17157 −0.122896
\(667\) −8.00000 −0.309761
\(668\) 6.68629 0.258700
\(669\) −12.4853 −0.482709
\(670\) −8.00000 −0.309067
\(671\) −26.6274 −1.02794
\(672\) 12.4853 0.481630
\(673\) 16.6274 0.640940 0.320470 0.947259i \(-0.396159\pi\)
0.320470 + 0.947259i \(0.396159\pi\)
\(674\) 3.85786 0.148599
\(675\) 3.00000 0.115470
\(676\) −1.82843 −0.0703241
\(677\) 26.6863 1.02564 0.512819 0.858497i \(-0.328601\pi\)
0.512819 + 0.858497i \(0.328601\pi\)
\(678\) −2.20101 −0.0845293
\(679\) 10.3431 0.396934
\(680\) 34.3431 1.31700
\(681\) −17.3137 −0.663463
\(682\) 0.970563 0.0371648
\(683\) 47.9411 1.83442 0.917208 0.398408i \(-0.130437\pi\)
0.917208 + 0.398408i \(0.130437\pi\)
\(684\) 5.17157 0.197740
\(685\) 30.6274 1.17021
\(686\) −7.02944 −0.268385
\(687\) −1.31371 −0.0501211
\(688\) −4.97056 −0.189501
\(689\) 2.00000 0.0761939
\(690\) 4.68629 0.178404
\(691\) −5.85786 −0.222844 −0.111422 0.993773i \(-0.535540\pi\)
−0.111422 + 0.993773i \(0.535540\pi\)
\(692\) 21.3137 0.810226
\(693\) −5.65685 −0.214886
\(694\) −3.59798 −0.136577
\(695\) 20.6863 0.784676
\(696\) −3.17157 −0.120218
\(697\) 39.5980 1.49988
\(698\) 1.51472 0.0573329
\(699\) 6.97056 0.263651
\(700\) −15.5147 −0.586401
\(701\) 5.02944 0.189959 0.0949796 0.995479i \(-0.469721\pi\)
0.0949796 + 0.995479i \(0.469721\pi\)
\(702\) −0.414214 −0.0156335
\(703\) 21.6569 0.816804
\(704\) 8.34315 0.314444
\(705\) 32.9706 1.24174
\(706\) −13.8579 −0.521548
\(707\) 21.6569 0.814490
\(708\) −14.0000 −0.526152
\(709\) −4.62742 −0.173786 −0.0868931 0.996218i \(-0.527694\pi\)
−0.0868931 + 0.996218i \(0.527694\pi\)
\(710\) −2.34315 −0.0879367
\(711\) −11.3137 −0.424297
\(712\) −23.5147 −0.881251
\(713\) 4.68629 0.175503
\(714\) 8.97056 0.335715
\(715\) −5.65685 −0.211554
\(716\) 42.6274 1.59306
\(717\) 2.00000 0.0746914
\(718\) 14.4853 0.540586
\(719\) −29.9411 −1.11662 −0.558308 0.829634i \(-0.688549\pi\)
−0.558308 + 0.829634i \(0.688549\pi\)
\(720\) −8.48528 −0.316228
\(721\) 6.62742 0.246818
\(722\) −4.55635 −0.169570
\(723\) 0.343146 0.0127617
\(724\) −25.5980 −0.951341
\(725\) 6.00000 0.222834
\(726\) −2.89949 −0.107610
\(727\) −10.3431 −0.383606 −0.191803 0.981433i \(-0.561433\pi\)
−0.191803 + 0.981433i \(0.561433\pi\)
\(728\) 4.48528 0.166236
\(729\) 1.00000 0.0370370
\(730\) −0.402020 −0.0148794
\(731\) −12.6863 −0.469219
\(732\) −24.3431 −0.899749
\(733\) −36.6274 −1.35286 −0.676432 0.736505i \(-0.736475\pi\)
−0.676432 + 0.736505i \(0.736475\pi\)
\(734\) −9.94113 −0.366934
\(735\) −2.82843 −0.104328
\(736\) −17.6569 −0.650840
\(737\) −13.6569 −0.503057
\(738\) 2.14214 0.0788531
\(739\) −18.1421 −0.667369 −0.333685 0.942685i \(-0.608292\pi\)
−0.333685 + 0.942685i \(0.608292\pi\)
\(740\) −39.5980 −1.45565
\(741\) 2.82843 0.103905
\(742\) −2.34315 −0.0860196
\(743\) 2.00000 0.0733729 0.0366864 0.999327i \(-0.488320\pi\)
0.0366864 + 0.999327i \(0.488320\pi\)
\(744\) 1.85786 0.0681126
\(745\) 25.9411 0.950409
\(746\) 4.14214 0.151654
\(747\) 3.65685 0.133797
\(748\) 28.0000 1.02378
\(749\) −32.0000 −1.16925
\(750\) 2.34315 0.0855596
\(751\) −0.970563 −0.0354163 −0.0177082 0.999843i \(-0.505637\pi\)
−0.0177082 + 0.999843i \(0.505637\pi\)
\(752\) −34.9706 −1.27525
\(753\) 0 0
\(754\) −0.828427 −0.0301695
\(755\) 9.94113 0.361795
\(756\) −5.17157 −0.188088
\(757\) 51.9411 1.88783 0.943916 0.330185i \(-0.107111\pi\)
0.943916 + 0.330185i \(0.107111\pi\)
\(758\) 0.201010 0.00730102
\(759\) 8.00000 0.290382
\(760\) −12.6863 −0.460180
\(761\) 32.4853 1.17759 0.588795 0.808282i \(-0.299602\pi\)
0.588795 + 0.808282i \(0.299602\pi\)
\(762\) 2.34315 0.0848832
\(763\) 15.0294 0.544102
\(764\) −6.05887 −0.219202
\(765\) −21.6569 −0.783005
\(766\) 12.8284 0.463510
\(767\) −7.65685 −0.276473
\(768\) 3.97056 0.143275
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 6.62742 0.238836
\(771\) −4.34315 −0.156415
\(772\) −9.71573 −0.349677
\(773\) 34.1421 1.22801 0.614004 0.789303i \(-0.289558\pi\)
0.614004 + 0.789303i \(0.289558\pi\)
\(774\) −0.686292 −0.0246682
\(775\) −3.51472 −0.126252
\(776\) −5.79899 −0.208172
\(777\) −21.6569 −0.776935
\(778\) −11.1716 −0.400520
\(779\) −14.6274 −0.524082
\(780\) −5.17157 −0.185172
\(781\) −4.00000 −0.143131
\(782\) −12.6863 −0.453661
\(783\) 2.00000 0.0714742
\(784\) 3.00000 0.107143
\(785\) 28.2843 1.00951
\(786\) −3.31371 −0.118196
\(787\) −40.7696 −1.45328 −0.726639 0.687020i \(-0.758919\pi\)
−0.726639 + 0.687020i \(0.758919\pi\)
\(788\) −0.887302 −0.0316088
\(789\) 12.0000 0.427211
\(790\) 13.2548 0.471586
\(791\) −15.0294 −0.534385
\(792\) 3.17157 0.112697
\(793\) −13.3137 −0.472784
\(794\) 12.8284 0.455264
\(795\) 5.65685 0.200628
\(796\) −39.5980 −1.40351
\(797\) 24.3431 0.862278 0.431139 0.902285i \(-0.358112\pi\)
0.431139 + 0.902285i \(0.358112\pi\)
\(798\) −3.31371 −0.117304
\(799\) −89.2548 −3.15761
\(800\) 13.2426 0.468198
\(801\) 14.8284 0.523937
\(802\) 10.8284 0.382365
\(803\) −0.686292 −0.0242187
\(804\) −12.4853 −0.440322
\(805\) 32.0000 1.12785
\(806\) 0.485281 0.0170933
\(807\) 18.0000 0.633630
\(808\) −12.1421 −0.427159
\(809\) 18.6863 0.656975 0.328488 0.944508i \(-0.393461\pi\)
0.328488 + 0.944508i \(0.393461\pi\)
\(810\) −1.17157 −0.0411649
\(811\) −30.1421 −1.05843 −0.529217 0.848487i \(-0.677514\pi\)
−0.529217 + 0.848487i \(0.677514\pi\)
\(812\) −10.3431 −0.362973
\(813\) −27.7990 −0.974953
\(814\) 6.34315 0.222327
\(815\) −53.2548 −1.86544
\(816\) 22.9706 0.804131
\(817\) 4.68629 0.163953
\(818\) −14.4853 −0.506466
\(819\) −2.82843 −0.0988332
\(820\) 26.7452 0.933982
\(821\) −23.7990 −0.830590 −0.415295 0.909687i \(-0.636322\pi\)
−0.415295 + 0.909687i \(0.636322\pi\)
\(822\) −4.48528 −0.156442
\(823\) 15.0294 0.523893 0.261947 0.965082i \(-0.415636\pi\)
0.261947 + 0.965082i \(0.415636\pi\)
\(824\) −3.71573 −0.129444
\(825\) −6.00000 −0.208893
\(826\) 8.97056 0.312126
\(827\) −26.0000 −0.904109 −0.452054 0.891990i \(-0.649309\pi\)
−0.452054 + 0.891990i \(0.649309\pi\)
\(828\) 7.31371 0.254169
\(829\) −17.3137 −0.601330 −0.300665 0.953730i \(-0.597209\pi\)
−0.300665 + 0.953730i \(0.597209\pi\)
\(830\) −4.28427 −0.148709
\(831\) −2.00000 −0.0693792
\(832\) 4.17157 0.144623
\(833\) 7.65685 0.265294
\(834\) −3.02944 −0.104901
\(835\) 10.3431 0.357939
\(836\) −10.3431 −0.357725
\(837\) −1.17157 −0.0404955
\(838\) 6.05887 0.209300
\(839\) 43.2548 1.49332 0.746661 0.665204i \(-0.231656\pi\)
0.746661 + 0.665204i \(0.231656\pi\)
\(840\) 12.6863 0.437719
\(841\) −25.0000 −0.862069
\(842\) 15.4558 0.532644
\(843\) 21.1716 0.729188
\(844\) 21.9411 0.755245
\(845\) −2.82843 −0.0973009
\(846\) −4.82843 −0.166005
\(847\) −19.7990 −0.680301
\(848\) −6.00000 −0.206041
\(849\) 28.9706 0.994267
\(850\) 9.51472 0.326352
\(851\) 30.6274 1.04989
\(852\) −3.65685 −0.125282
\(853\) 3.65685 0.125208 0.0626042 0.998038i \(-0.480059\pi\)
0.0626042 + 0.998038i \(0.480059\pi\)
\(854\) 15.5980 0.533752
\(855\) 8.00000 0.273594
\(856\) 17.9411 0.613215
\(857\) −49.5980 −1.69423 −0.847117 0.531406i \(-0.821664\pi\)
−0.847117 + 0.531406i \(0.821664\pi\)
\(858\) 0.828427 0.0282820
\(859\) −0.686292 −0.0234160 −0.0117080 0.999931i \(-0.503727\pi\)
−0.0117080 + 0.999931i \(0.503727\pi\)
\(860\) −8.56854 −0.292185
\(861\) 14.6274 0.498501
\(862\) 3.45584 0.117707
\(863\) −28.3431 −0.964812 −0.482406 0.875948i \(-0.660237\pi\)
−0.482406 + 0.875948i \(0.660237\pi\)
\(864\) 4.41421 0.150175
\(865\) 32.9706 1.12103
\(866\) −8.82843 −0.300002
\(867\) 41.6274 1.41374
\(868\) 6.05887 0.205652
\(869\) 22.6274 0.767583
\(870\) −2.34315 −0.0794401
\(871\) −6.82843 −0.231372
\(872\) −8.42641 −0.285354
\(873\) 3.65685 0.123766
\(874\) 4.68629 0.158516
\(875\) 16.0000 0.540899
\(876\) −0.627417 −0.0211985
\(877\) 42.2843 1.42784 0.713919 0.700228i \(-0.246918\pi\)
0.713919 + 0.700228i \(0.246918\pi\)
\(878\) 7.02944 0.237232
\(879\) 2.14214 0.0722524
\(880\) 16.9706 0.572078
\(881\) −25.5980 −0.862418 −0.431209 0.902252i \(-0.641913\pi\)
−0.431209 + 0.902252i \(0.641913\pi\)
\(882\) 0.414214 0.0139473
\(883\) 27.5980 0.928746 0.464373 0.885640i \(-0.346280\pi\)
0.464373 + 0.885640i \(0.346280\pi\)
\(884\) 14.0000 0.470871
\(885\) −21.6569 −0.727987
\(886\) −10.7452 −0.360991
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 12.1421 0.407463
\(889\) 16.0000 0.536623
\(890\) −17.3726 −0.582330
\(891\) −2.00000 −0.0670025
\(892\) 22.8284 0.764352
\(893\) 32.9706 1.10332
\(894\) −3.79899 −0.127057
\(895\) 65.9411 2.20417
\(896\) −29.8579 −0.997481
\(897\) 4.00000 0.133556
\(898\) −13.1716 −0.439541
\(899\) −2.34315 −0.0781483
\(900\) −5.48528 −0.182843
\(901\) −15.3137 −0.510174
\(902\) −4.28427 −0.142651
\(903\) −4.68629 −0.155950
\(904\) 8.42641 0.280258
\(905\) −39.5980 −1.31628
\(906\) −1.45584 −0.0483672
\(907\) −12.9706 −0.430680 −0.215340 0.976539i \(-0.569086\pi\)
−0.215340 + 0.976539i \(0.569086\pi\)
\(908\) 31.6569 1.05057
\(909\) 7.65685 0.253962
\(910\) 3.31371 0.109848
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −8.48528 −0.280976
\(913\) −7.31371 −0.242048
\(914\) −3.17157 −0.104906
\(915\) −37.6569 −1.24490
\(916\) 2.40202 0.0793650
\(917\) −22.6274 −0.747223
\(918\) 3.17157 0.104678
\(919\) 3.31371 0.109309 0.0546546 0.998505i \(-0.482594\pi\)
0.0546546 + 0.998505i \(0.482594\pi\)
\(920\) −17.9411 −0.591501
\(921\) −22.8284 −0.752222
\(922\) 2.14214 0.0705475
\(923\) −2.00000 −0.0658308
\(924\) 10.3431 0.340265
\(925\) −22.9706 −0.755267
\(926\) −10.1421 −0.333291
\(927\) 2.34315 0.0769590
\(928\) 8.82843 0.289807
\(929\) 11.7990 0.387112 0.193556 0.981089i \(-0.437998\pi\)
0.193556 + 0.981089i \(0.437998\pi\)
\(930\) 1.37258 0.0450088
\(931\) −2.82843 −0.0926980
\(932\) −12.7452 −0.417482
\(933\) −10.6274 −0.347926
\(934\) −3.31371 −0.108428
\(935\) 43.3137 1.41651
\(936\) 1.58579 0.0518331
\(937\) −21.3137 −0.696289 −0.348144 0.937441i \(-0.613188\pi\)
−0.348144 + 0.937441i \(0.613188\pi\)
\(938\) 8.00000 0.261209
\(939\) 6.00000 0.195803
\(940\) −60.2843 −1.96626
\(941\) 34.1421 1.11300 0.556501 0.830847i \(-0.312144\pi\)
0.556501 + 0.830847i \(0.312144\pi\)
\(942\) −4.14214 −0.134958
\(943\) −20.6863 −0.673638
\(944\) 22.9706 0.747628
\(945\) −8.00000 −0.260240
\(946\) 1.37258 0.0446265
\(947\) −21.0294 −0.683365 −0.341682 0.939815i \(-0.610997\pi\)
−0.341682 + 0.939815i \(0.610997\pi\)
\(948\) 20.6863 0.671860
\(949\) −0.343146 −0.0111390
\(950\) −3.51472 −0.114033
\(951\) 8.48528 0.275154
\(952\) −34.3431 −1.11307
\(953\) 40.3431 1.30684 0.653421 0.756994i \(-0.273333\pi\)
0.653421 + 0.756994i \(0.273333\pi\)
\(954\) −0.828427 −0.0268213
\(955\) −9.37258 −0.303290
\(956\) −3.65685 −0.118271
\(957\) −4.00000 −0.129302
\(958\) 10.4853 0.338764
\(959\) −30.6274 −0.989011
\(960\) 11.7990 0.380811
\(961\) −29.6274 −0.955723
\(962\) 3.17157 0.102256
\(963\) −11.3137 −0.364579
\(964\) −0.627417 −0.0202077
\(965\) −15.0294 −0.483815
\(966\) −4.68629 −0.150779
\(967\) 18.1421 0.583412 0.291706 0.956508i \(-0.405777\pi\)
0.291706 + 0.956508i \(0.405777\pi\)
\(968\) 11.1005 0.356784
\(969\) −21.6569 −0.695718
\(970\) −4.28427 −0.137560
\(971\) −15.3137 −0.491440 −0.245720 0.969341i \(-0.579024\pi\)
−0.245720 + 0.969341i \(0.579024\pi\)
\(972\) −1.82843 −0.0586468
\(973\) −20.6863 −0.663172
\(974\) −3.23045 −0.103510
\(975\) −3.00000 −0.0960769
\(976\) 39.9411 1.27848
\(977\) 42.1421 1.34825 0.674123 0.738619i \(-0.264522\pi\)
0.674123 + 0.738619i \(0.264522\pi\)
\(978\) 7.79899 0.249384
\(979\) −29.6569 −0.947837
\(980\) 5.17157 0.165200
\(981\) 5.31371 0.169654
\(982\) 12.6863 0.404836
\(983\) 25.3137 0.807382 0.403691 0.914895i \(-0.367727\pi\)
0.403691 + 0.914895i \(0.367727\pi\)
\(984\) −8.20101 −0.261439
\(985\) −1.37258 −0.0437341
\(986\) 6.34315 0.202007
\(987\) −32.9706 −1.04946
\(988\) −5.17157 −0.164530
\(989\) 6.62742 0.210740
\(990\) 2.34315 0.0744701
\(991\) 4.68629 0.148865 0.0744325 0.997226i \(-0.476285\pi\)
0.0744325 + 0.997226i \(0.476285\pi\)
\(992\) −5.17157 −0.164198
\(993\) 26.1421 0.829596
\(994\) 2.34315 0.0743201
\(995\) −61.2548 −1.94191
\(996\) −6.68629 −0.211863
\(997\) −39.2548 −1.24321 −0.621607 0.783330i \(-0.713520\pi\)
−0.621607 + 0.783330i \(0.713520\pi\)
\(998\) 10.8284 0.342768
\(999\) −7.65685 −0.242252
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 39.2.a.b.1.2 2
3.2 odd 2 117.2.a.c.1.1 2
4.3 odd 2 624.2.a.k.1.1 2
5.2 odd 4 975.2.c.h.274.3 4
5.3 odd 4 975.2.c.h.274.2 4
5.4 even 2 975.2.a.l.1.1 2
7.6 odd 2 1911.2.a.h.1.2 2
8.3 odd 2 2496.2.a.bi.1.2 2
8.5 even 2 2496.2.a.bf.1.2 2
9.2 odd 6 1053.2.e.e.352.2 4
9.4 even 3 1053.2.e.m.703.1 4
9.5 odd 6 1053.2.e.e.703.2 4
9.7 even 3 1053.2.e.m.352.1 4
11.10 odd 2 4719.2.a.p.1.1 2
12.11 even 2 1872.2.a.w.1.2 2
13.2 odd 12 507.2.j.f.316.3 8
13.3 even 3 507.2.e.h.22.1 4
13.4 even 6 507.2.e.d.484.2 4
13.5 odd 4 507.2.b.e.337.2 4
13.6 odd 12 507.2.j.f.361.2 8
13.7 odd 12 507.2.j.f.361.3 8
13.8 odd 4 507.2.b.e.337.3 4
13.9 even 3 507.2.e.h.484.1 4
13.10 even 6 507.2.e.d.22.2 4
13.11 odd 12 507.2.j.f.316.2 8
13.12 even 2 507.2.a.h.1.1 2
15.2 even 4 2925.2.c.u.2224.2 4
15.8 even 4 2925.2.c.u.2224.3 4
15.14 odd 2 2925.2.a.v.1.2 2
21.20 even 2 5733.2.a.u.1.1 2
24.5 odd 2 7488.2.a.cl.1.1 2
24.11 even 2 7488.2.a.co.1.1 2
39.5 even 4 1521.2.b.j.1351.3 4
39.8 even 4 1521.2.b.j.1351.2 4
39.38 odd 2 1521.2.a.f.1.2 2
52.51 odd 2 8112.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.2 2 1.1 even 1 trivial
117.2.a.c.1.1 2 3.2 odd 2
507.2.a.h.1.1 2 13.12 even 2
507.2.b.e.337.2 4 13.5 odd 4
507.2.b.e.337.3 4 13.8 odd 4
507.2.e.d.22.2 4 13.10 even 6
507.2.e.d.484.2 4 13.4 even 6
507.2.e.h.22.1 4 13.3 even 3
507.2.e.h.484.1 4 13.9 even 3
507.2.j.f.316.2 8 13.11 odd 12
507.2.j.f.316.3 8 13.2 odd 12
507.2.j.f.361.2 8 13.6 odd 12
507.2.j.f.361.3 8 13.7 odd 12
624.2.a.k.1.1 2 4.3 odd 2
975.2.a.l.1.1 2 5.4 even 2
975.2.c.h.274.2 4 5.3 odd 4
975.2.c.h.274.3 4 5.2 odd 4
1053.2.e.e.352.2 4 9.2 odd 6
1053.2.e.e.703.2 4 9.5 odd 6
1053.2.e.m.352.1 4 9.7 even 3
1053.2.e.m.703.1 4 9.4 even 3
1521.2.a.f.1.2 2 39.38 odd 2
1521.2.b.j.1351.2 4 39.8 even 4
1521.2.b.j.1351.3 4 39.5 even 4
1872.2.a.w.1.2 2 12.11 even 2
1911.2.a.h.1.2 2 7.6 odd 2
2496.2.a.bf.1.2 2 8.5 even 2
2496.2.a.bi.1.2 2 8.3 odd 2
2925.2.a.v.1.2 2 15.14 odd 2
2925.2.c.u.2224.2 4 15.2 even 4
2925.2.c.u.2224.3 4 15.8 even 4
4719.2.a.p.1.1 2 11.10 odd 2
5733.2.a.u.1.1 2 21.20 even 2
7488.2.a.cl.1.1 2 24.5 odd 2
7488.2.a.co.1.1 2 24.11 even 2
8112.2.a.bm.1.2 2 52.51 odd 2